http://www.cfd-online.com/W/index.php?title=Special:Contributions/DavidF&feed=atom&limit=50&target=DavidF&year=&month=CFD-Wiki - User contributions [en]2016-05-01T03:28:33ZFrom CFD-WikiMediaWiki 1.16.5http://www.cfd-online.com/Wiki/NACA0012_airfoilNACA0012 airfoil2009-11-11T09:44:39Z<p>DavidF: /* Introduction */</p>
<hr />
<div>== Introduction ==<br />
The NACA 0012 airfoil is widely used. The simple geometry and the large amount of wind tunnel data provide an excellent 2D validation case. For this case I use the [[Spalart-Allmaras model | Spalart-Allmaras]] turbulence model.<br />
<br><br />
<br />
== Mesh ==<br />
<br />
[[Image:Naca0012 mesh final.JPG]]<br />
<br><br><br />
The mesh is a 30,000 cell C-grid. The width of the first cell at the airfoil boundary is 0.02 mm. At Re = 3e6 and zero angle of attack, this results in a wall ''y+'' = 1.3 ± 0.4, which is low enough for the turbulence model to resolve the sub layer. The mesh shown is for an angle of attack of 6 degrees.<br />
<br><br />
<br />
== Drag Coefficient ==<br />
<br />
[[Image:Naca0012 cd tripwire2.JPG]]<br />
<br><br><br />
The drag coefficient at zero angle of attack depends on the Reynold's number. The experimental data is for an airfoil with a trip wire, which forces the boundary layer to be completely turbulent.[1] This corresponds to the Fluent model, which has an active turbulence model over the complete airfoil. Note that the calculated drag coefficient is somewhat higher than the experimental one. Possibly, the modeled boundary layer is turbulent from the beginning, while in reality the trip wire is not at the very leading edge of the foil.<br />
<br />
<br><br />
Modeling the NACA 0012 airfoil without a trip wire is more complicated, since Fluent itself is unable to predict the point along the chord where the transition from a laminar to a turbulent boundary layer takes place. One option is to manually set this transition point at Re = 5.5e6.<br />
<br><br />
<br />
== Lift Curve ==<br />
<br />
[[Image:Naca0012 lift curve3.JPG]]<br />
<br><br><br />
The lift coefficient depends on the angle of attack. For Re = 2e6 I compare the lift coefficient to experimental results.[2] Because the lift coefficient is less sensitive to the transition point, the experimental data is for an airfoil without trip wire.<br />
<br><br />
<br />
== Lift Curve Slope ==<br />
<br />
[[Image:Naca0012 lift curve slope.JPG]]<br />
<br><br><br />
The initial slope of the lift curve depends on the Reynold's number. Here I compare the lift curve slope to experimental results.[1] Experimental results are both with and without trip wire; again the lift coefficient is less sensetive to the transition point than the drag coefficient.<br />
<br><br />
<br />
== References ==<br />
1. W. J. McCroskey, ''A Critical Assessment of Wind Tunnel Results for the NACA 0012 Airfoil'', NASA Technical Memorandum 10001 9 (1987) <br><br />
2. L. Lazauskus, [http://www.cyberiad.net/library/airfoils/foildata/n0012cl.htm NACA 0012 Lift Data]</div>DavidFhttp://www.cfd-online.com/Wiki/NACA0012_airfoilNACA0012 airfoil2009-11-11T09:43:21Z<p>DavidF: /* Mesh */</p>
<hr />
<div>== Introduction ==<br />
The NACA 0012 airfoil is widely used. The simple geometry and the large amount of wind tunnel data provide an excellent 2D validation case. For this case I use the Spalart-Allmaras turbulence model.<br />
<br><br />
== Mesh ==<br />
<br />
[[Image:Naca0012 mesh final.JPG]]<br />
<br><br><br />
The mesh is a 30,000 cell C-grid. The width of the first cell at the airfoil boundary is 0.02 mm. At Re = 3e6 and zero angle of attack, this results in a wall ''y+'' = 1.3 ± 0.4, which is low enough for the turbulence model to resolve the sub layer. The mesh shown is for an angle of attack of 6 degrees.<br />
<br><br />
<br />
== Drag Coefficient ==<br />
<br />
[[Image:Naca0012 cd tripwire2.JPG]]<br />
<br><br><br />
The drag coefficient at zero angle of attack depends on the Reynold's number. The experimental data is for an airfoil with a trip wire, which forces the boundary layer to be completely turbulent.[1] This corresponds to the Fluent model, which has an active turbulence model over the complete airfoil. Note that the calculated drag coefficient is somewhat higher than the experimental one. Possibly, the modeled boundary layer is turbulent from the beginning, while in reality the trip wire is not at the very leading edge of the foil.<br />
<br />
<br><br />
Modeling the NACA 0012 airfoil without a trip wire is more complicated, since Fluent itself is unable to predict the point along the chord where the transition from a laminar to a turbulent boundary layer takes place. One option is to manually set this transition point at Re = 5.5e6.<br />
<br><br />
<br />
== Lift Curve ==<br />
<br />
[[Image:Naca0012 lift curve3.JPG]]<br />
<br><br><br />
The lift coefficient depends on the angle of attack. For Re = 2e6 I compare the lift coefficient to experimental results.[2] Because the lift coefficient is less sensitive to the transition point, the experimental data is for an airfoil without trip wire.<br />
<br><br />
<br />
== Lift Curve Slope ==<br />
<br />
[[Image:Naca0012 lift curve slope.JPG]]<br />
<br><br><br />
The initial slope of the lift curve depends on the Reynold's number. Here I compare the lift curve slope to experimental results.[1] Experimental results are both with and without trip wire; again the lift coefficient is less sensetive to the transition point than the drag coefficient.<br />
<br><br />
<br />
== References ==<br />
1. W. J. McCroskey, ''A Critical Assessment of Wind Tunnel Results for the NACA 0012 Airfoil'', NASA Technical Memorandum 10001 9 (1987) <br><br />
2. L. Lazauskus, [http://www.cyberiad.net/library/airfoils/foildata/n0012cl.htm NACA 0012 Lift Data]</div>DavidFhttp://www.cfd-online.com/Wiki/CombustionCombustion2009-01-28T13:23:33Z<p>DavidF: /* Main Specificities of Combustion Chemistry */</p>
<hr />
<div>''The power of Fire, or Flame, for instance, which we designate by some trivial chemical name, thereby hiding from ourselves the essential character of wonder that dwells in it as in all things, is with these old Northmen, Loke, a most swift subtle Demon of the brood of the J\"otuns... From us too no Chemistry, if it had not Stupidity to help it, would hide that Flame is a wonder. What is Flame?''<br />
<br />
'''''Carlyle on''''' Heroes '''''Odin and Scandinavian Mythology.''''', [1].<br />
<br />
<br />
== What is combustion -- physics versus modelling ==<br />
<br />
Combustion phenomena consist of many physical and chemical processes which exhibit a <br />
broad range of time and length scales. A mathematical description of combustion is not <br />
always trivial, although some analytical solutions exist for simple situations of <br />
laminar flame. Such analytical models are usually restricted to problems in zero or one-dimensional space.<br />
<br />
= Fundamental Aspects =<br />
<br />
== Main Specificities of Combustion Chemistry ==<br />
<br />
Combustion can be split into two processes interacting with each other: thermal, and chemical. <br />
<br />
The chemistry is highly exothermal (this is the reason of its use) but also highly temperature dependent, thus highly self-accelerating. In a simplified form, combustion can be represented by a single irreversible reaction involving 'a' fuel and 'an' oxidizer:<br />
:<math> \frac{\nu_F}{\bar M_F}Y_F + \frac{\nu_O}{\bar M_O} Y_O \rightarrow Product + Heat </math><br />
Although very simplified compared to real chemistry involving hundreds of species (and their individual transport properties) and elemental reactions, this rudimentary chemistry has been the cornerstone of combustion analysis and modelling.<br />
<br />
The most widely used form for the rate of the above reaction is the Arrh&eacute;nius law:<br />
:<math> \dot\omega = \rho^2 A \left ( \frac{Y_F}{\bar M_F} \right )^{n_F} \left (\frac{Y_O}{\bar M_O}\right )^{n_O} \exp^{-T_a/T} </math><br />
<math> T_a </math> is the activation temperature, high in combustion, consistently with the temperature dependence.<br />
This is where the high non-linearity in temperature is modelled. ''A'' is the pre-exponential constant. One of the interpretation of the Arrh&eacute;nius law comes from gas kinetic theory: the number of molecules whose kinetic energy is larger than the minimum value allowing a collision to be energetic enough to trig a reaction is proportional to the exponential term introduced above divided by the square root of the temperature. This interpretation allows one to think that the temperature dependence of ''A'' is very weak compared to the exponential term. ''A'' is eventually considered as constant.<br />
The reaction rate is also naturally proportional to the molecular density of each of the reactant. Nonetheless, the orders of reaction <math> n_i</math> are different from the stoichiometric coefficients as the single-step reaction is global, not governed by collision for it represents hundreds of elementary reactions.<br />
If one goes into the details, combustion chemistry is based on chain reactions, decomposed into three main steps: (i) generation (where radicals are created from the fresh mixture), (ii) branching (where products and new radicals appear from interaction of radicals with reactants), and (iii) termination (where radicals collide and turn into products). The branching step tends to accelerate the production of active radicals (autocatalytic). The impact is nevertheless small compared to the high non-linearity in temperature. This explains why single-step chemistry has been sufficient for most of the combustion modelling work up to now.<br />
<br />
The fact that a flame is a very thin reaction zone separating, and making the transition between, a frozen mixture and an equilibrium is explained by the high temperature <br />
dependence of the reaction term, modelled by a large activation temperature, and a large heat release (the ratio of the burned and fresh gas temperatures is about 7 for typical hydrocarbon flames) leading to a sharp self-acceleration in a very narrow area. To evaluate the order of magnitude of the quantities, the terms in the exponential argument are normalized:<br />
:<math> \beta=\alpha\frac{T_a}{T_s} \qquad \alpha=\frac{T_s-T_f}{T_s} </math><br />
<math>\beta</math> is named the Zeldovitch number and <math>\alpha</math> the heat release factor. <br />
Here, <math> T_s</math> has been used instead of <math> T_b</math>, the conventional notation for burned gas temperature (at final equilibrium). <math> T_s</math> is actually <math> T_b</math> <br />
for a mixture at stoichiometry and when the system is adiabatic, i.e. this is the reference highest temperature that can be<br />
obtained in the system. <math> T_f </math> is the ambient temperature of the fresh gases. That said, typical value for <math>\beta</math> and <math>\alpha</math> are 10 and 0.9, giving <br />
a good taste of the level of non-linearity of the combustion process with respect to temperature. <br />
Actually, the reaction rate is rewritten as:<br />
:<math> \dot\omega = \rho^2 A \left ( \frac{Y_F}{\bar M_F} \right )^{n_F} \left (\frac{Y_O}{\bar M_O}\right )^{n_O} <br />
\exp^{-\frac{\beta}{\alpha}}\exp^{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}} </math><br />
where the non-dimensionalized temperature is:<br />
:<math>\theta=\frac{T-T_f}{T_s-T_f}</math><br />
The non-linearity of the reaction rate is seen from the exponential term:<br />
:* <math> {\mathcal O}(\exp^{-\beta}) </math> for <math>\theta</math> far from unity (in the fresh gas)<br />
:* <math> {\mathcal O}(1) </math> for <math>\theta</math> close to unity (in the reaction zone close to the burned gas whose temperature must be close to the adiabatic one <math> T_s </math>), more exactly <math> 1-\theta \sim {\mathcal O}(\beta^{-1})</math><br />
<br />
[[Image:NonLinearite.jpg|thumb|Temperature Non-Linearity of the Source Term: the Temperature-Dependent Factor of the Reaction Term for Some Values of the Zeldovitch and Heat Release Parameters]]Note that for an infinitely high activation energy, the reaction rate is piloted by a <math>\delta(\theta)</math> function. The figure, beside, illustrates how common values of <math>\beta</math> around 10 tend to make the reaction rate singular around <math>\theta</math> of unity. Two set of values are presented: <math><br />
\beta = 10</math> and <math>\beta = 8</math>. The first magnitude is the representative value while the second one is a smoother one usually used to ease numerical simulations. In the same way, two values for the heat release <math>\alpha</math>, 0.9 and 0.75, are explored. The heat release is seen to have a minor impact on the temperature non-linearity.<br />
<br />
== Transport Equations ==<br />
<br />
Additionally to the Navier-Stokes equations, at least with variable density, the transport equations for a reacting flow are the energy and species transport equations. In usual notations, the specie ''i'' transport equation is written as:<br />
:<math>\frac{D \rho Y_i}{D t} = \nabla\cdot \rho D_i\vec\nabla Y_i - \nu_i\bar M_i\dot\omega</math><br />
and the temperature transport equation:<br />
:<math>\frac{D\rho C_p T}{Dt} = \nabla\cdot \lambda\vec\nabla T + Q\nu_F \bar M_F \dot\omega </math><br />
The diffusion is modelled thanks to Fick's law that is a (usually good) approximation to the rigorous diffusion velocity calculation. Regarding the temperature transport equation, it is derived from the energy transport equation under the assumption of a low-Mach number flow (compressibility and viscous heating neglected). The low-Mach number approximation is suitable for the deflagration regime (as it will be demonstrated [[#Premixed|below]]), which is the main focus of combustion modelling. Hence, the transport equation for temperature, as a simplified version of the energy transport equation, is usually retained for the study of combustion and its modelling.<br />
<br />
'''Note:''' The species and temperature equations are not closed as the fields of velocity and density also need to be computed. Through intense heat release in a very small area (the jump in temperature in typical hydrocarbon flames is about seven and so is the drop in density in this isobaric process, and the thickness of a flame is of the order of the millimetre), combustion influences the flow field. Nevertheless, the vast majority of combustion modelling has been developed based on the species and temperature equations, assuming simple flow fields, sometimes including <br />
hydrodynamics perturbations.<br />
<br />
<br />
=== Low-Mach Number Equations ===<br />
<br />
In compressible flows, when the motion of the fluid is not negligible compared to the speed of sound (which is the speed at which the molecules can reorganize themselves), the heap of molecules results in a local increase of pressure and temperature moving as an acoustic wave. It means that, in such a system, a proper reference velocity is the speed of sound and a proper pressure reference is the kinetic pressure. A contrario, in low-Mach number flows, the reference speed is the natural representative speed of the flow and the reference pressure is the thermodynamic pressure. Hence, the set of reference quantities to characterize a low-Mach number flow is given in the table below:<br />
<br />
Density <math>\rho_o</math> A reference density (upstream, average, etc.)<br />
<br />
Velocity <math>U_o</math> A reference velocity (inlet average, etc.)<br />
<br />
Temperature <math>T_o</math> A reference temperature (upstream, average, etc.)<br />
<br />
Pressure (static) <math>P_o=\rho_o \bar r T_o</math> From Boyle-Mariotte<br />
<br />
Length <math>L_o</math> A reference length (representative of the domain)<br />
<br />
Time <math>L_o/U_o</math><br />
<br />
Energy <math>C_p T_o</math> Internal energy at constant reference pressure. <math> C_p </math> must also be chosen in a reference thermodynamical state. <br />
<br />
The equations for fluid mechanics properly adimensionalized can be written:<br />
<br />
Mass conservation:<br />
:<math> \frac{D\rho}{Dt} =0 </math><br />
<br />
Momentum:<br />
:<math>\frac{D\rho\vec U}{Dt}=-\frac{1}{\gamma M}\vec\nabla P+\nabla\cdot\frac{1}{Re}\bar\bar\Sigma</math><br />
<br />
Total energy:<br />
:<math>\frac{D\rho e_T}{Dt}-\frac{1}{RePr}\nabla\cdot\lambda\vec\nabla T=-\frac{\gamma-1}{\gamma}\nabla\cdot P\vec U<br />
+\frac{1}{Re}M^2(\gamma-1)\nabla\cdot \vec U\bar\bar\Sigma + \frac{\rho}{C_pT_o(U_o/L_o)}Q\nu_F \bar M_F\dot\omega</math><br />
<br />
Specie:<br />
:<math>\frac{D\rho Y}{Dt}-\frac{1}{ScRe}\nabla\cdot\rho D\vec\nabla Y=-\frac{\rho}{U_o/L_o}\nu\bar M\dot\omega</math><br />
<br />
State law:<br />
:<math> P=\rho T</math><br />
<br />
The low-Mach number equations are obtained considering that <math> M^2 </math> is small. 0.1 is usually taken as the limit, which recovers the value of a Mach number of 0.3 to characterize the incompressible regime.<br />
<br />
Considering the energy equation, in addition to the terms with <math> M^2 </math> in factor in the equation, the total energy reduces to internal energy as: <math> e_T = T/\gamma+M^2(\gamma-1)\vec U^2/2 </math>. Moreover, the work of pressure is considered as negligible because the gradient of pressure is negligible (low-Mach number approximation is indeed also named ''isobaric'' approximation) and the flow is assumed close to a divergence-free state.<br />
For the same reason, volumic energy and enthalpy variations are assumed equal as they only differ through the addition of pressure. Hence, redimensionalized, the low-Mach number energy equation leads to the temperature equation as used in combustion analysis:<br />
:<math>\frac{D\rho C_p T}{Dt} = \nabla\cdot \lambda\vec\nabla T + Q\nu_F \bar M_F \dot\omega </math><br />
<br />
=== The Damk&ouml;hler Number ===<br />
<br />
A flame is a reaction zone. From this simple point of view, two aspects have to be considered: (i) the rate at which it is fed by reactants, let call <math> \tau_d </math> the characteristic time, and<br />
the strength of the chemistry to consume them, let call the characteristic chemical time <math> \tau_c </math>. In combustion, the Damk&ouml;hler number, ''Da'', compares these both time scales and, for that <br />
reason, it is one of the most integral non-dimensional groups:<br />
:<math>Da=\frac{\tau_d}{\tau_c}</math>.<br />
If ''Da'' is large, it means that the chemistry has always the time to fully consume the fresh mixture and turn it into equilibrium. Real flames are usually close to this state. The characteristic reaction time, <math> (Ae^{-T_a/T_s})^{-1} </math>, is <br />
estimated of the order of the tenth of a ms. When ''Da'' is low, the fresh mixture cannot be converted by a too weak chemistry. The flow remains frozen. This situation happens with ignition or misfire, for instance. <br />
<br />
The picture of a deflagration lends itself to a description based on the Damk&ouml;hler number. A reacting wave progresses towards the fresh mixture through preheating of the upstream closest layer. The elevation of the temperature strengthens the chemistry and reduces its characteristic time such that the mixture changes from a low-''Da'' region (far upstream, frozen) to a high-''Da'' region in the flame (intense reaction to equilibrium).<br />
<br />
== Conservation Laws ==<br />
<br />
The processus of combustion transforms the chemical enthalpy into sensible enthalpy (i.e. rise the temperature of the gases thanks to the heat released). Simple relations can be drawn between species and temperature by studying the source terms appearing in the above equations:<br />
:<math> \frac{Y_F}{\nu_F\bar M_F} - \frac{Y_O}{\nu_O\bar M_O} = \frac{Y_{F,u}}{\nu_F\bar M_F} - \frac{Y_{O,u}}{\nu_O\bar M_O} </math><br />
:<math> Y_F + \frac{Cp T}{Q} = Y_{F,u} + \frac{CpT_u}{Q} </math><br />
Those coupling functions are named Schwalb-Zeldovich variables.<br />
<br />
Hence <math> T_b = T_u + \frac{Q Y_{F,u}}{Cp} </math>, <math> Y_{O,b} = Y_{O,u} - \frac{\nu_O \bar M_O}{\nu_F \bar M_F} Y_{F,u} = Y_{O,u} -sY_{F,u} </math> and <math> Y_{F,b} = 0 </math>. Here, the example has been taken for a lean case.<br />
<br />
As mentioned in [[#Main Specificities of Combustion Chemistry|Sec. Main Specificities]], the stoichiometric state is used to non-dimensionalize the conservation equations:<br />
:<math> Y_i^* = Y_{i,u}^* - \theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} \qquad ; \qquad Y_i^*=Y_i/Y_{i,s}</math>.<br />
<br />
A comprehensive form of the reaction rate can be reconstituted to understand the difficulty of numerically resolving the reaction zone:<br />
:<math> \dot\omega = B \prod_{i=O,F}(Y_{i,u}^*-\theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_i} \exp{\left ( -\beta\frac{1-\theta}{1-\alpha(1-\theta)}\right)} </math><br />
where <math> B </math> stands for all the constant terms present in this reaction rate, plus density.<br />
<br />
[[Image:NonLineariteii.jpg|thumb|Source Term versus Temperature]]<br />
For the stoichiometric case and a global order of two, the reaction rate is graphed versus the reduced temperature for different values of the heat release and Zeldovitch parameter. A high value of <math>\beta</math> makes the reaction rate very sharp, versus temperature. It means that reaction is significant beyond a temperature level (sometimes called ignition temperature) that is close to one (the exponential term above is non-negligible for <math>1-\theta \sim \beta^{-1}</math>). The heat release has qualitatively the same impact but not so strong. Transposed to the case of a flame sheet, it effectively shows that the reaction exists only in a fraction of the thermal thickness of the flame (the region close to the flame that the latter preheats, hence, where the reduced temperature rises from 0 to 1 here) where the temperature deviates few from the maximal one (density can be assumed as constant and equal to its burned-gas value). Numerically capturing such a sharp reaction zone can be costly and the lower values of <br />
<math>\beta</math> and <math>\alpha</math> as presented here are usually preferred whenever possible.<br />
<br />
<br />
Most problems in combustion involve turbulent flows, gas and liquid <br />
fuels, and pollution transport issues (products of combustion as well as for example noise <br />
pollution). These problems require not only extensive experimental <br />
work, but also numerical modelling. All combustion models must be validated <br />
against the experiments as each one has its own drawbacks and limits. In this article, we will address the modeling fundamentals only.<br />
<br />
The combustion models are often classified on their capability to deal with the different combustion regimes. <br><br />
[[Image:SUMMARY_OF_COMBUSTION_MODELS.jpg]]<br />
<br />
= Three Combustion Regimes =<br />
<br />
Depending on how fuel and oxidizer are brought into contact in the combustion system, different combustion modes or regimes are identified. Traditionally, two regimes have been recognized: the ''premixed'' regime and the ''non-premixed'' regime. Over the last two decades, a third regime, sometime considered as a hybrid of the two former ones to a certain extend, has risen. It has been named ''partially-premixed'' regime.<br />
<br />
== The Non-Premixed Regime ==<br />
[[Image:DiffusionFlame.jpg|thumb|Sketch of a diffusion flame]]<br />
This regime is certainly the easiest to understand. Everybody has already seen a lighter, candle or gas-powered stove. Basically, the fuel issues from a nozzle or a simple duct into the atmosphere. The combustion reaction is the oxidization of the fuel. Because fuel and oxidizer are in contact only in a limited region but are separated elsewhere (especially in the feeding system) this configuration is the safest. The non-premixed flame is also named diffusion flame as diffusion of oxidant and fuel has to occur simultaneously to reaction to sustain combustion, the flame being a surface of separation of fuel and oxidant streams. The non-premixed flame has some other advantages. By controlling the flows of both reactants, it is (theoretically) possible to locate the stoichiometric interface, and thus, the location of the flame sheet. Moreover, the strength of the flame can also be controlled through the same process. Depending on the width of the transition region from the oxidizer to the fuel side, the species (fuel and oxidizer) feed the flame at different rates.<br />
This is because the diffusion of the species is directly dependent on the unbalance (gradient) of their distribution. A sharp transition from fuel to oxidizer creates intense diffusion of those species towards the flame, increasing its burning rate. As said above, this burning rate control through the diffusion process is certainly one of the reasons of the alternate name of such a flame and combustion mode: ''diffusion'' flame and ''diffusion'' regime.<br />
<br />
Because a diffusion flame is fully determined by the inter-penetration of the fuel and oxidizer streams, it is convenient to introduce a tracer of the state of the mixture. This is the role of the ''mixture fraction'', usually called ''Z'' or ''f''. Z is usually taken as unity in the fuel stream and is null in the oxidizer stream. It varies linearly between this two bounds such that at any point of a frozen flow the fuel mass fraction is given by <math> Y_F=ZY_{F,o} </math> and the oxidizer mass fraction by <math> Y_O = (1-Z)Y_{O,o} </math>. <math> Y_{F,o} </math> and <math> Y_{O,o} </math> are the fuel and oxidizer mass fractions in the fuel and oxidizer streams, respectively.<br />
The mixture fraction posses a transport equation that is expected to not have any source term as a tracer of a mixture must be a conserved scalar. First, the fuel and oxidizer mass fraction transport equations are written in usual notations:<br />
:<math>\frac{D \rho Y_F}{D t} = \nabla\cdot \rho D\vec\nabla Y_F - \nu_F\bar M_F\dot\omega</math><br />
:<math>\frac{D \rho Y_O}{D t} = \nabla\cdot \rho D\vec\nabla Y_O - \nu_O\bar M_O\dot\omega</math><br />
The two above equations are linearly combined in a single one in a manner that the source term disappears:<br />
:<math>\frac{D \rho (\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)}{D t} = \nabla\cdot \rho D\vec\nabla (\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)</math><br />
The quantity <math>(\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)</math> is thus a conserved scalar, one recognizes<br />
one of the Schwalb-Zeldovich quantities introduced in [[#Conservation Laws|Sec. Conservation Laws]]. One <br />
also remarks that this transport equations combination is made in the equidiffusional approximation, i.e. <br />
all the scalars, including temperature, diffuse at the same rate. The last step is to normalize it such that it equals unity in the pure fuel stream (<math> Y_F=Y_{F,o}</math> and <math> Y_O=0 </math>) and is null in the pure oxidizer stream <br />
(<math> Y_F=0</math> and <math> Y_O=Y_{O,o} </math>). The resulting normalized passive scalar is the mixture fraction:<br />
:<math>Z=\frac{\Phi(Y_F/Y_{F,o})-(Y_O/Y_{O,o})+1}{\Phi+1}\qquad \Phi=\frac{\nu_O\bar M_O Y_{F,o}}{\nu_F\bar M_F Y_{O,o}}</math><br />
governed by the transport equation:<br />
:<math>\frac{D \rho Z}{D t} = \nabla\cdot \rho D\vec\nabla Z </math><br />
The stoichiometric interface location (and thus the approximate location of the flame if the flow is reacting) is where <math>\Phi(Y_F/Y_{F,o})-(Y_O/Y_{O,o}) </math> vanishes (or <math>Y_F</math> and <math> Y_O </math> are both null in the reacting case). This leads to a stoichiometry definition:<br />
:<math>Z_s=\frac{1}{1+\Phi}</math><br />
<br />
As the mixture fraction qualifies the degree of inter-penetration of fuel and oxidizer, the elements originally present in these molecules are conserved and can be directly traced back to the mixture fraction. This has led to an alternate defintion of the mixture fraction, based on ''element conservation''.<br />
First, the elemental mass fraction <math> X_{j} </math> of element ''j'' is linked to the species mass fraction <math> Y_i </math>:<br />
:<math> X_j = \sum_{i=1}^{n} \frac{a_{i,j} \bar M_j}{\bar M_i} Y_i </math><br />
where <math> a_{i,j} </math> is a matrix counting the number of element ''j'' atoms in specie molecule named ''i'' and ''n'' is the number of species in the mixture.<br />
The group pictured by the summation above is a linear combination of <math> Y_i </math>. Because the transport equations of species mass fraction, few lines earlier, are also linear, a transport equation for the elemental mass fraction can be written:<br />
:<math>\frac{D \rho X_j}{D t} = \nabla\cdot \sum_{i=1}^n\frac{a_{i,j}\bar M_j}{\bar M_i}\rho D_i\vec\nabla Y_i</math><br />
For mass is conserved, the linear combination of the source terms vanishes. Furthermore, by taking the same diffusion coefficient <math> D_i </math> for all the species, the elemental mass fraction transport equation has exactly the same form as the specie transport equation (except the source term). Notice that the assumption of equal diffusion coefficient was also made in the previous definition of the mixture fraction and is justified in turbulent combustion modelling by the turbulence diffusivity flattening the diffusion process in high Reynolds number flows. Hence, the elemental mass fraction transport equation has the same structure as the mixture fraction transport equation seen above. Properly renormalized to reach unity in the fuel stream and zero in the oxidizer stream, the elemental mass fraction is a convenient way of determining the mixture fraction field in a flow. Indeed, it is widely used in practice for this purpose.<br />
<br />
==== Simplified Diffusion Flame Solution ====<br />
<br />
Because Schwalb-Zeldovitch variables rid off any source, they allow a first approximation of the flame description independent of the kinetics.<br />
From the definition of <math>Z</math>, the Schwalb-Zeldovich variables have the following expression:<br />
<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
Y_O-sY_F & = & Y_{O,o}-Z(sY_{F,o}+Y_{O,o}) \\<br />
\frac{Q Y_F}{C_p} + T & = & T_{O,o} + Z(T_{F,o}-T_{O,o}+\frac{Q Y_{F,o}}{C_p})<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
Two simplified limit cases for the diffusion flame equation can be built from those relations:<br />
* In the frozen flow, the chemical source term is null everywhere. Hence, <math> Y_F </math> and <math> Z </math> follow<br />
strictly the same linear transport equation. Then, <math>Y_F = Y_{F,o} Z </math> as anticipated above.<br />
Follow: <math> Y_O=Y_{O,o}(1-Z) </math>, <math> T=T_{O,o}+Z(T_{F,o}-T_{O,o}) </math>.<br />
* The equilibrium solution, also with zero source term, the 'historical' diffusion flame description laid out by Burke and Schumann in 1928:<br />
** Rich side: <math> Y_O = 0 </math>, <math> Y_F = -Y_{O,o}/s + Z(Y_{F,o}+Y_{O,o}/s)</math>, <math> T=T_{F,o}+(1-Z)(T_{O,o}-T_{F,o}+Y_{O,o}\frac{Q}{sC_p})</math><br />
** Lean side: <math> Y_F = 0 </math>, <math> Y_O = Y_{O,o} - Z(sY_{F,o}+Y_{O,o})</math>, <math> T=T_{O,o}+Z(T_{F,o}-T_{O,o}+Y_{F,o}\frac{Q}{C_p})</math><br />
<br />
From the value of <math>Z</math> at stoichiometry above, the adiabatic flame temperature is obtained:<br />
:<math><br />
T_s=T_{O,o}+Z_s(T_{F,o}-T_{O,o}+Y_{F,o} \frac{Q}{C_p})<br />
</math><br />
<br />
'''Remark 1:''' The Burke-Schumann solution for a diffusion flame, despite its age, is still one of the most used model, under some refreshments such as Mixed-Is-Burned, or Equilibrium combustion model where the burned value of the quantities is obtained by Gibbs function extrema instead of the full adiabatic consumption.<br />
<br />
'''Remark 2:''' In the Burke-Schuman solution, as the chemistry is infinitely fast, the flame is reduced to an interface separating the rich and lean sides. Therefore, quantities must be continuous at the flame and their gradients observe the jump conditions, from [[#Conservation Laws|Sec. Conservation Laws]]:<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
s\rho D_F||\vec\nabla Y_F|| & = & \rho D_O||\vec\nabla Y_O|| \\<br />
\frac{Q}{C_p}\rho D_F||\vec\nabla Y_F|| & = & \lambda ||\vec\nabla T||<br />
\end{array}<br />
\right .<br />
</math><br />
<br />
'''Remark 3''' From Remark 2, it is seen that the reactants fluxes must reach the flame in stoichimetric proportion (the fluxes and not the values themselves). Therefore, the relations developed in Remark 2 are true even without the equidiffusional approximation, whereas the relation based on <math> Z </math>, no. On the other hand, the heat released by chemistry is conducted away from the flame thanks to the temperature gradients on both sides.<br />
<br />
==== Dissipation Rate and Non-Equilibrium Effect====<br />
<br />
A very important quantity, derived from the mixture fraction concept, is the ''scalar dissipation rate'', usually noted: <math> \chi </math>. In the above introduction to non-premixed combustion, it has been said that a diffusion flame is fully controlled through: (i) the position of the stoichiometric line, dictating where the flame sheet lies; (ii) the gradients of fuel on one side and oxidizer on the other side, dictating the feeding rate of the reaction zone through diffusion and thus the strength of combustion. According the the mixture fraction definition, the location of the stoichiometric line is naturally tracked through the <math>Z_s</math> iso-line and it is seen here how the mixture fraction is a convenient tracer to locate the flame.<br />
In the same manner, the mixture fraction field should also be able to give information on the strength of the chemistry as the gradients of reactants are directly linked to the mixture fraction distribution. The feeding rate of the reaction zone is characterized through the inverse of a time. Because it is done through diffusion, it must be obtained through a combination of mixture fraction gradient and diffusion coefficient (dimensional analysis):<br />
:<math> \chi_s = \frac{(\rho D)_s}{\rho_s} ||\vec \nabla Z||^2_s \qquad (\rho D)_s = \left ( \frac{\lambda}{C_p} \right )_s</math><br />
where the subscript ''s'' refers to quantities taken effectively where the reacting sheet is supposed to be, close to stoichiometry. This deduction of the scalar dissipation rate, scaling the feeding rate of the flame, is obtained here through physical arguments and is to be derived from equations below in a more mathematical manner. Note that the transport coefficient for the mixture fraction is identified to the one for temperature. This notation is usually used in the literature to emphasize that the rate of temperature diffusion (that is commensurable to the rate of species diffusion) is the retained parameter (as introduced in the following approach highlighting the role of the scalar dissipation rate). Depending if the diffusion process is free (mixing layer, boundary layer) or controlled (stagnation), the dissipation rate and then the non-premixed combustion may be of unsteady or steady nature.<br />
<br />
<br />
===== Ignition / Burning / Extinction Curve =====<br />
<br />
Let consider a mixing layer between fuel and oxidizer whose strain (and then intensity of reactant inter-diffusion) is carefully controlled. Let start from a flame already existing. Let also use the Damk\"ohler number as defined above with the inverse of the scalar dissipation rate being the characteristic mechanical time of flow of reactants feeding the reaction zone. For high Da (low dissipation rate), the reactants diffuse slowly. The reaction is not very intense but the chemistry is fully achieved such that the maximum temperature is reached.<br />
Now, let decrease slowly (to avoid any unsteady effect on the chemistry) the Da. On one hand, the rate of feeding of flame through diffusion increases, and so does the reaction rate. On the other hand, the chemistry may not have the time to `eat' every reactant molecules that begin to leak through the flame: the fully sensible value is not realized and a lower temperature is achieved. With still increasing the dissipation rate, this mechanism leads to lower temperature in the flame zone down to a level that cannot be sustained by combustion (that strongly depends on temperature). The diffusion flame leaves the diffusion-controlled burning regime and extinguishes suddenly. It is said to be quenched. This is experienced in real life when blowing off a small wood fire. Slightly blowing increases the transfer between reactants and strengthens the reaction but blowing too much extinguishes the fire.<br />
<br />
[[Image:sCurve.jpg|thumb|S-Curve Diffusion Flame.]]<br />
If now the start state is a frozen mixing layer between fuel and oxidant, at low Da (high dissipation rate), the flow remains frozen. When increasing Da, there will be a point where the chemistry will self-accelerate and the flame will light up through a sudden increase in temperature. Because the starting temperature is low, the ignition Da number is higher that the extinction Da above, exhibiting an hysteresis phenomenon.<br />
<br />
When looking at the trend of the maximum temperature only versus Da, a curve with a shape in `S' appears, named as S-Curve for diffusion flames (see beside figure).<br />
<br />
===== Flamelet Equation =====<br />
<br />
In turbulent combustion modelling, the flamelet regime is identified as a regime that saves the integrity of the flame structure ([[#The Wringkled Regime|Sec. The Wrinkled Regime]]). Turbulent eddies do not enter the structure and only contord the flame at a large scale. Therefore, it is usual to call the ''flamelet equation'', the equation that describes the diffusion flame structure when not perturbed by turbulence. The model problem is usually retained as the counterflow diffusion flame. The strain imposed to the flame mimicks the one due to inhomogeneities in a real turbulent flow.<br />
<br />
[[Image:counterFlow.jpg|thumb|Sketch of the Stretched Mixing Layer Created by the Counter-Flow Configuration.]]<br />
A counter-flow diffusion flame is basically made of two ducts, front-to-front, one issuing fuel at a mass fraction <math> Y_{F,o}</math> (may be diluted by an inert) and temperature <math> T_{F,o}</math>, and the other issuing oxidizer at a mass fraction <math> Y_{O,o}</math> (the inert is usually di-nitrogen in air such that <math> Y_{O,o} =.23310 </math> is a value commonly encountered) and temperature <math> T_{O,o}</math>. A stretched mixging layer develops at the location the flows oppose. This stretch can be modulated by the flow rate from the feeding ducts. The diffusion flame sits around the iso-surface corresponding to stoichiometry. The figure beside sketches this widely used configuration, with the coordinate frame origin taken at the stagnation location in the middle of the mixing layer. <br />
<br />
The Howarth-Dorodnitzyn transform and the Chapman approximation are applied to the above mixture fraction transport equation.<br />
In the Chapman approximation, the thermal dependence of <math>(\rho D)</math> is approximated as <math>\rho^{-1}</math>.<br />
The Howarth-Dorodnitzyn transform introduces <math>\rho</math> in the space coordinate system: <math> \vec\nabla \rightarrow \rho\vec\nabla \quad ; \quad \nabla\cdot\rightarrow \rho\nabla\cdot</math>. The effect of these both mathematical operations is to `digest' the thermal variation of quantities such as density or transport coefficient. Hence, the mixture fraction equation comes in a simpler mathematical shape:<br />
:<math>\rho\vec U \cdot \vec\nabla Z = \rho_s ( \rho D )_s \Delta Z </math><br />
Here the references are taken in the stoichiometric zone (''s'' subscript).<br />
<br />
<br />
For the sake of simplicity, we shall use the potential flow result for this counter-flow problem with two identical jets although the presence of the flame (the associated density change) jeopardizes this theory, strictly speaking.<br />
The velocity solution is thus:<br />
<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
u & = & -a\int\frac{dx}{\rho} \\<br />
v & = & \frac{a}{2}\int\frac{dr}{\rho}<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math>a</math> is named the stretch of the mixing layer and is the inverse of a time.<br />
<br />
In such a flow, the transport equation for the mixture fraction may be written as a purely 1-D problem (assuming constant properties):<br />
:<math><br />
-ax\frac{dZ}{dx} = \rho_s (\rho D)_s \frac{d^2Z}{dx^2} <br />
</math><br />
<br />
from which a `length' scale emerges for the mixing layer: <math> \sqrt{\rho_s (\rho D)_s /a} </math>. Non-dimensionalized, with <math><br />
\eta = x \sqrt{a/\rho_s (\rho D)_s} </math>:<br />
:<math><br />
\eta \frac{dZ}{d\eta} + \frac{d^2 Z}{d \eta^2} = 0<br />
</math><br />
<br />
The boundary conditions may be approximated as <math> \eta \rightarrow +\infty\quad ; \quad Z\rightarrow 1 </math> and <br />
<math> \eta \rightarrow -\infty\quad ; \quad Z\rightarrow 0 </math>. Note that this approximation requires a thin layer compared to the counter-flow distance, and thus a relatively high strain rate.<br />
Then, the solution for <math> Z </math>:<br />
:<math><br />
Z=\frac{1}{2}(1+erf{\frac{\eta}{\sqrt{2}}})<br />
</math><br />
<br />
The scalar dissipation rate is deduced from this equation as:<br />
:<math><br />
\chi = \frac{1}{2\pi}\exp\left( - [erf^{-1} (2Z)]^2 \right) \frac{a\rho^2}{\rho_s^2}<br />
</math><br />
<br />
Because combustion is highly temperature-dependent, ''T'' is certainly the scalar to which attention must be paid for. The temperature equation in the Low-Mach Number regime ([[#Low-Mach Number Equations|Sec. Low-Mach Number Equations]]) is written below in steady-state:<br />
:<math>\rho\vec U \cdot \vec\nabla T = \nabla\cdot \frac{\lambda}{C_p}\vec\nabla T + \frac{Q}{C_p}\nu_F \bar M_F \dot\omega </math><br />
After the Chapman approximation and the Howarth-Dorodnitzyn transform:<br />
:<math>\frac{\rho}{\rho_s}\vec U \cdot \vec\nabla T = \left ( \frac{\lambda}{C_p}\right )_s \nabla\cdot \vec\nabla T + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
Here the references are taken in the flame, i.e. close to the stoichiometric line (''s'' subscript).<br />
<br />
In a non-premixed system, strictly speaking, <math> \vec U </math>, is not really relevant as the flame is fully controlled by the diffusion process. Notwithstanding, in practice, non-premixed flames must be stabilized by creating a strain in the direction of diffusion. This is the reason why the velocity is left in the equation. Because a diffusion flame is fully described by the mixture fraction field, a change in coordinate can be applied:<br />
:<math> \left (<br />
\begin{array}{c}<br />
r \\<br />
x<br />
\end{array}\right ) \longrightarrow <br />
\left (<br />
\begin{array}{c}<br />
r \\<br />
Z<br />
\end{array}\right )<br />
</math><br />
The Jacobian of the transform is given as:<br />
:<math> \left [<br />
\begin{array}{cc}<br />
\frac{\partial r}{\partial r} & \frac{\partial Z}{\partial r} \\<br />
\frac{\partial r}{\partial x} & \frac{\partial Z}{\partial x} <br />
\end{array}\right ] = <br />
\left [<br />
\begin{array}{cc}<br />
1 & 0 \\<br />
0 & (\rho l_d)^{-1} <br />
\end{array}\right ]</math><br />
Note that the diffusive layer of thickness <math>l_d</math> is defined as the region of transition between fuel and oxidizer and is thus given by the gradient of ''Z'' along the ''x'' direction.<br />
This transform is applied to the vectorial operators:<br />
:<math> \nabla\cdot = \nabla_r\cdot+\nabla_x\cdot=\nabla_r\cdot+\nabla_x\cdot Z\nabla_Z\cdot</math><br />
:<math> \vec\nabla = \vec\nabla_r+\vec\nabla_x=\vec\nabla_r+\vec\nabla_x Z\nabla_Z\cdot</math><br />
<br />
With this transform, the above temperature equation looks like:<br />
:<math>\frac{\rho}{\rho_s}\vec U \cdot (\vec\nabla_r T + \vec\nabla Z \nabla_Z\cdot T) = \left (\frac{\lambda}{C_p}\right )_s \left ( \nabla_r\cdot \vec\nabla_r T + \nabla_x\cdot Z \nabla_Z\cdot \vec\nabla_x Z \nabla_Z\cdot T + \nabla_x\cdot Z \nabla_Z\cdot \vec\nabla_r T + \nabla_r\cdot \vec\nabla_x Z \nabla_Z\cdot T \right ) + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
As mentioned above, the velocity and the variation along the tangential direction to the main flame structure ''r'' are not supposed to play a major role. To be convinced, a variable scaling may be done, considering that the reaction zone extends<br />
over a small fraction <math> \varepsilon </math> of the diffusion thickness <math> l_d </math> around stoichiometry. Then<br />
the convective term of the above equation is <math> {\mathcal O} ( \varepsilon^{-1} ) </math> and the diffusive <br />
term is <math> {\mathcal O} ( \varepsilon^{-2} ) </math>. By emphasizing the role of the gradient of ''Z'' along ''x'' as a key parameter defining the configuration the following equation is obtained:<br />
:<math>0 = \left (\frac{\lambda}{C_p}\right )_s ||\vec\nabla_x Z||^2 \Delta_Z T + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
This equation (sometimes named the ''flamelet equation'') serves as the basic framework to study the structure of diffusion flames. It highlights the role of the dissipation rate with respect to the strength of the source term and shows that the dissipation rate calibrates the combustion intensity.<br />
<br />
===== Ignition =====<br />
<br />
The flamelet equation may be used to explore the behaviour of the mixing-layer for not too high strain rate. As mentioned earlier, the Damk&oumlaut;hler may be high enough to allow a self-ignition. In this regime, the following assumptions may be made:<br />
* the exponential argument in the reaction term, <math> T_a/T</math>, is linearized around the frozen temperature value (indeed, as the activation temperature is large, a small change in the temperature from its frozen value is magnified in the reaction rate magnitude): <math> T_a/T = T_a/T_o-T_a/T_o^2(T-T_o) = T_a/T_o - \phi </math>. It can be seen that a small change of temperature leading to <math> T_a/T_o^2(T-T_o)\quad {\mathcal O} (1)</math> impacts the exponential term of the reaction rate.<br />
* the reactants may be approximated by their frozen flow values: <math> Y_F=Y_{F,o}Z-\frac{Cp T_o^2}{QT_a}\phi\approx Y_{F,o}Z \quad ; \quad Y_O=Y_{O,o}(1-Z)-s\frac{Cp T_o^2}{QT_a} \phi \approx Y_{O,o}(1-Z) </math>, under the hypothesis of large activation energy.<br />
<br />
The flamelet equation is recast as:<br />
<math><br />
\Delta_Z\phi = - Z^{n_F}(1-Z)^{n_O} e^{\phi}\overbrace{\frac{Q\nu_F}{\rho_s\lambda_s}\frac{A}{\nabla^2_x Z \bar M_F^{n_F-1}\bar M_O^{n_O}} \frac{T_a}{T_o^2} e^{-\frac{T_a}{T_o}}Y_{F,o}^{n_F}Y_{O,o}^{n_O}}^{\mathcal D}<br />
</math><br />
In this expression <math>\mathcal D</math> is a Damk\"ohler number.<br />
<br />
This expression must be solved numerically but, with the help of simplifications, an analytical solution may be found. It must be noticed that this analytical solution has no quantitative prediction capabilities. However, it is sufficient to illustrate the critical Damk\"ohler number for ignition for pedagogical purpose.<br />
<br />
First, the equation is integrated from <math>Z_s </math> to <math> Z</math> 'somewhere in the rich side'.<br />
:<math><br />
\int_{Z_s}^{Z} dZ \Delta_Z\phi = \left [\frac{d\phi}{dZ} \right]_{Z_s}^{Z} = - \int_{Z_s}^{Z} dZ Z^{n_F}(1-Z)^{n_O} e^{\phi}{\mathcal D}<br />
</math><br />
<br />
<math> \phi </math> may be approximated as, on the asymptote <math> \hat{\phi}(Z-1)/(Z_s-1) </math> with a maximum <math> \hat{\phi} </math> on the stoichiometric line, where ignition is supposed the strongest. Then, on the stoichiometric line, the derivative of <math> \phi </math> vanishes and, sufficiently far from the stoichiometric line it tends to <math> \hat{\phi}/(Z_s-1) </math>. To keep the solution simple, we choose the integration limit to be <br />
on the border of the region where <math> \phi </math> remains within 10% of its maximum such that it may be approximated as constant on the right hand side. The value of <math> Z </math> that fits this trade-off of being sufficiently far from <math> Z_s </math> to allow the derivative of <math>\phi</math> to reach its asymptote and <br />
sufficiently close to prevent <math>\phi</math> from decreasing is estimated to 40% of <math>Z_s-Z</math>. <br />
This estimation, quite disputable, is based on temperature profile observed in diffusion flames at low Da when the diffusive effect strongly smoothes it.<br />
<br />
[[Image:diffusionDaIgnit.jpg|thumb|Ignition Da curve in a cold diffusion layer.]]<br />
The end equation has the form:<br />
:<math><br />
\hat{\phi}e^{-\hat{\phi}}=-{\mathcal D} \overbrace{(Z_s-1)\left (\frac{Z^2-Z_s^2}{2}-\frac{Z^3-Z_s^3}{3}\right )}^{a}<br />
</math><br />
This makes <math>\mathcal D</math> a function of <math>\hat{\phi}</math>, maximum temperature in the ignition zone at <br />
stoichiometry, that has a maximum for <math>\hat{\phi}=1</math>. This gives the bottom tipping point of the S-curve presented above. For the academical case <math> Z_s = .5 </math> and <math> n_F=n_O=1 </math>, this gives <math> \mathcal D = 16.</math> which is in a satisfactory agreement with the full numerical solution. It is interesting to note that the numerical solution follows the physics process of ignition of the diffusion layer by slowly increasing the Damk\"ohler number. Therefore, it cannot catch properly the turning point and the doubled-value relation. Instead, once the maximum Damk\"ohler number corresponding to ignition is reached, the reduced temperature soars sharply. Further increasing ''D'' in the forbidden range, no solution can be found. Physically, for larger values of the Damk&ouml;hler number, the configuration jumps from this nearly frozen situation to an equilibrium situation with a well developed diffusion flame, see the S-Curve above. The simplified equation for ignition is thus unadapted and a specific solution for a developed diffusion flame is now to be presented.<br />
<br />
<br />
===== Diffusion Burning Process =====<br />
<br />
From [[#Simplified Solution of Diffusion Flame|Sec. Simplified Solution of Diffusion Flame]], the temperature jump at stoichiometry is identified as <math> Q Y_{F,o} Z_s/Cp </math>. <br />
<br />
Although chemistry in a diffusion flame is usually very fast, one expects a very residual amount of reactants in the reaction<br />
zone laying along the stoichiometric line. It thus means that equilibrium is not reached in practice and that the temperature is somewhat lower than the equilibrium temperature. Thermodynamics is moderated by kinetics effects. <br />
<br />
To a certain extend, the flamelet equation [[#Flamelet Equation|Sec. Flamelet Equation]] may be used to present a solution for the non-premixed flame when the flame is already ignited, and the chemistry strength controlled by diffusion (we have stressed the author that an interesting aspect of diffusion flames is the ability to control the reaction rate through the level of mixing in the neighbourhood of the flame). As already emphasized, due to a strong non-linearity in temperature, the structure of a burning flame can never be far from the Burke-Schumann simplified solution.<br />
<br />
Non-dimensionalized, the temperature solution in the flamelet may be written as:<br />
:<math><br />
\theta=1+(Z-Z_s)\Delta T_o + \frac{Z_s - Z}{1-Z_s} -\varepsilon\Gamma<br />
</math><br />
in the rich side of the mixing layer, and:<br />
:<math><br />
\theta=1+(Z-Z_s)\Delta T_o + \frac{Z-Z_s}{Z_s} -\varepsilon\Gamma<br />
</math><br />
in the lean side.<br />
On recognizes the structure of the Burke-Schumann solution ([[#Simplified Diffusion Flame Solution|Sec. Simplified Diffusion Flame Solution]]). <math>\Delta T_o</math> is the non-dimensionalized temperature difference between the reactants feed streams:<br />
:<math><br />
\Delta T_o = \frac{T_{F,o}-T_{O,o}}{Q Y_{F,o} Z_s/C_p}<br />
</math><br />
Those solutions are amended by the term <math>\varepsilon\Gamma</math> with <math>\Gamma\sim{\mathcal O}(1)</math> in the reaction zone and vanishing on the sides.<br />
This amendment corresponds to the departure from the equilibrium solution due to kinetics effect. The quantity <math>\varepsilon</math> is taken small for the physical reasons explained above.<br />
<br />
It is interesting to introduce <math>\gamma=2(Z_s-1)(Z_s\Delta T_o+1)+1</math>, combining the temperature difference between the reactant feeding streams and the stoichiometry of the combustion) as a synthetic parameter to inform about the geometry of the flame structure in the (''Z,T'') diagram (Burke-Schumann):<br />
:<math><br />
\theta=1-\frac{(Z-Z_s)}{2Z_s(1-Z_s)}(\gamma+1) -\varepsilon\Gamma<br />
</math><br />
in the rich side of the mixing layer, and:<br />
:<math><br />
\theta=1-\frac{(Z-Z_s)}{2Z_s(1-Z_s)}(\gamma-1) -\varepsilon\Gamma<br />
</math><br />
in the lean side.<br />
For <math>\gamma=0</math> the slope in temperatue vs ''Z'' is the same on both sides of the flame. For <math> \gamma > 0 </math> the rich side slope is the sharpest, and for <math> \gamma <0 </math> this is the lean side.<br />
An interesting case is when <math>|\gamma|>1</math>: one for the feeding stream enters the flame zone with a temperature higher than the flame temperature. In that case, whatever the dissipation rate, the flame is stretched with a hot stream<br />
that sustains its temperature such that it is never extinguished.<br />
<br />
[[Image:diffusionProfile.jpg|thumb|Structure of a diffusion flame in temperature and species mass fractions vs mixture fraction burning under the diffusion-controlled regime.]]<br />
The departure in temperature from equilibrium in the reaction zone is reflected into the species concentrations, as it corresponds to still unburned reactants. With the help of the above [[#Conservation Laws|Conservation Laws]] one finds:<br />
* Lean side: <math> Y_F=\varepsilon\Gamma Y_{F,o}Z_s</math>, <math> Y_O=Y_{O,o}(1-Z)-s(Y_{F,o}Z-\varepsilon\Gamma Y_{F,o}Z_s)</math><br />
* Rich side: <math> Y_F=Y_{F,o}Z+\frac{1}{s}(s Y_{F,o}Z_s\varepsilon\Gamma-Y_{O,o}(1-Z)) </math>, <math> Y_O=sY_{F,o}Z_s \varepsilon\Gamma </math><br />
<br />
The picture beside illustrates the structure of a diffusion flame in its `normal' regime.<br />
<br />
Recasting the [[#Flamelet Equation|Flamelet Equation]] above in terms of the reduced temperature, we find:<br />
:<math><br />
\chi_s\frac{\partial^2 \theta}{\partial Z^2} + \dot\omega\frac{\nu_F\bar M_F}{\rho\rho_s^2 Y_{F,o} Z_s} = 0<br />
</math><br />
To describe the structure of the diffusion flame, the ''reduced mixture fraction'' is set:<br />
:<math> \xi = \frac{Z-Z_s}{2Z_s(1-Z_s)\varepsilon} </math><br />
The utility of the reduced mixture fraction is to focus on the reaction zone. This reaction zone is supposed located on the stoichiometric line (this is why the reduced mixture fraction is centred on <math>Z_s</math>) and to be very thin (reason of the introduction of the magnifying factor <math>\varepsilon</math>). The equation is also written in terms of <math>\Gamma</math>, the non-equilibrium departure rescaled by <math> \varepsilon </math> and the source term as <br />
presented in [[#Main Specificities of Combustion Chemistry|Main Specificities of Combustion Chemistry]] is explicated:<br />
:<math><br />
\left \{<br />
\begin{array}{llll}<br />
\frac{\chi_s}{4Z_s^2(1-Z_s)^2\varepsilon}\frac{\partial^2 \Gamma}{\partial \xi^2} & = & <br />
A \frac{\rho\nu_F}{\rho_s^2} \left ( \frac{Y_{F,o}}{\bar M_F} \right )^{n_F-1} \left ( \frac{Y_{O,o}}{\bar M_O} \right )^{n_O} <br />
e^{-\frac{\beta}{\alpha}} Z_s^{n_F+n_O-1} \varepsilon^{n_F+n_O} \Gamma^{n_F} \Phi^{n_O} (\Gamma-2\xi)^{n_O} <br />
e^{-\beta\varepsilon (\xi(\gamma-1)+\Gamma)} & Z<Z_s \\<br />
<br />
\frac{\chi_s}{4Z_s^2(1-Z_s)^2\varepsilon}\frac{\partial^2 \Gamma}{\partial \xi^2} & = & <br />
A \frac{\rho\nu_F}{\rho_s^2} \left ( \frac{Y_{F,o}}{\bar M_F} \right )^{n_F-1} \left ( \frac{Y_{O,o}}{\bar M_O} \right )^{n_O} <br />
e^{-\frac{\beta}{\alpha}} Z_s^{n_F+n_O-1} \varepsilon^{n_F+n_O} (\Gamma+2\xi)^{n_F} \Phi^{n_O} \Gamma^{n_O} <br />
e^{-\beta\varepsilon (\xi(\gamma+1)+\Gamma)} & Z>Z_s <br />
\end{array}<br />
\right.<br />
</math><br />
It is seen that one can form a Damk&ouml;hler number as (see [[#Damk&ouml;hler Number|Sec. Damk&ouml;hler Number]] for the general meaning of the Damk&ouml;hler):<br />
:<math><br />
Da=\frac{A\rho\nu_F}{\chi_s\rho_s^2}\left(\frac{Y_{F,o}}{\bar M_F}\right)^{n_F-1}\left(\frac{Y_{O,o}}{\bar M_O}\right )^{n_O}e^{-\frac{\beta}{\alpha}}\Phi^{n_O} 4(1-Z_s)^2 Z_s^{n_F+n_O+1}<br />
</math><br />
and that this number must scale with <math>\varepsilon^{n_F+nu_O+1}</math>. So, to have a thin reaction zone, the Damk&ouml;hler number must be high. Physically, it means that when fuel and oxidizer meet close to the stoichiometric line they combine quickly such that the reaction can be completed in a very narrow region.<br />
<br />
== The Premixed Regime ==<br />
[[Image:Premixed.jpg|thumb|Sketch of a premixed flame]]<br />
In contrast to the non-premixed regime above, the reactants are here well mixed before entering the combustion chamber. Chemical reaction can occur everywhere and this flame can propagate upstream into the feeding system as a subsonic (deflagration regime) chemical wave. This presents lots of safety issues. Some situations prevent them: (i) the mixture is made too rich (lot of fuel compared to oxidizer) or too lean (too much oxidizer) such that the flame is close to its flammability limits (it cannot easily propagate); (ii) the feeding system and regions where the flame is not wanted are designed such that they impose strong heat loss to the flame in order to quench it. For a given thermodynamical state of the mixture (composition, temperature, pressure), the flame has its own dynamics (speed, heat release, etc) on which there is few control: the wave exchanges mass and energy through diffusion process in the fresh gases. On the other hand, those well defined quantities are convenient to describe the flame characteristics. The mechanism of spontaneous propagation<br />
towards fresh gas through the thermal transfer from the combustion zone to the immediate slice of fresh gas<br />
such that the ignition temperature is eventually reached for this latter was highlighted as early as by the end of the 19th century by Mallard and LeChatelier. <br />
The reason the chemical wave is contained in a narrow region of reaction propagating upstream is the consequence of the discussion on the non-linearity of the combustion with temperature in the <br />
[[#Fundamental Aspects|Sec. Fundamental Aspects]]. It is of interest to compare the orders of magnitude of the temperature dependent term <math> \exp{(-\beta(1-\theta)/(1-\alpha(1-\theta)))}</math> of the reaction source upstream in the fresh gas (<math>\theta\rightarrow 0</math>) and in the reaction zone close to equilibrium temperature (<math>\theta\rightarrow 1 </math>) for the set of representative values: <math> \beta = 10 </math> and <br />
<math>\alpha=0.9</math>. It is found that the reaction is about <math>10^{43}</math> times slower in the fresh<br />
gas than close to the burned gas. It is known that the chemical time scale is about 0.1 ms in the reaction zone of a typical flame, then the typical reaction time in the fresh gas in normal conditions is about <br />
<math> 10^{39} s </math>. To be compared with the order of magnitude of the estimated Universe age: <math> 1 0^{17} s </math>. Non-negligible chemistry is only confined in a thin reaction zone stuck to the hot burned gas at equilibrium temperature. In this zone, the [[#Damk&ouml;hler]] number is high, in contrast to in the fresh mixture. It is natural and convenient to consider that the reaction rate is strictly zero everywhere except in this small reaction zone (one recovers the Dirac-like shape of the reaction profile, <br />
provided that one can see the upstream flow as a region of increasing temperature towards the combustion zone and the downstream flow as in fully equilibrium).<br />
<br />
As the premixed flame is a reaction wave propagating from burned to fresh gases, the basic parameter is known to be the ''progress variable''. In the fresh gas, the progress variable is conventionally put to zero. In the burned gas, it equals unity. Across the flame, the intermediate values describe the progress of the reaction to turn into burned gas the fresh gas penetrating the flame sheet. A progress variable can be set with the help of any quantity like temperature, reactant mass fraction, provided it is bounded by a single value in the burned gas and another one in the fresh gas. The progress variable is usually named ''c'', in usual notations:<br />
:<math>c=\frac{T-T_f}{T_b-T_f}</math><br />
It is seen that ''c'' is a normalization of a scalar quantity. As mentioned above, the scalar transport equations are assumed linear such that the transport equation for ''c'' can be obtained directly. <br />
Actually, the transport equation for ''T'' ([[#Transport Equations|Sec. Transport Equations]]) is linear if constant heat capacity is further assumed (combustion of hydrocarbon in air implies a large excess of nitrogen whose heat capacity is only slightly varying) and the progress variable equation is directly <br />
obtained (here for a default of fuel - lean combustion):<br />
:<math>\frac{D\rho c}{Dt}=\nabla\cdot\rho D\vec\nabla c + \frac{\nu_F\bar M_F}{Y_{F,u}}\dot\omega \qquad \rho D = \frac{\lambda}{C_p} </math><br />
<br />
The fact that the default or excess of fuel has been discussed above leads to the introduction of another quantity: the ''equivalence ratio''. The equivalence ratio, usually noted <math>\Phi</math>, is the ratio of two ratios. The first one is the ratio of the mass of fuel with the mass of oxidizer in the mixture. The second one is the same ratio for a mixture at stoichiometry. Hence, when the equivalence ratio equals unity, the mixture is at stoichiometry. If it is greater than unity, the mixture is named ''rich'' as there is an excess of fuel. In contrast, when it is smaller than unity the mixture is named ''lean''. The equivalence ratio presented here for premixed flames has little connection with the equivalence ratio introduced earlier regarding the non-premixed regime. Basically, the equivalence ratio as defined for non-premixed flames gives the equivalence ratio of a premixed mixture with the same mass of fuel and oxidizer. Moreover, the equivalence ratio as defined for a premixed mixture can be obtained based on the mixture fraction (it is thus the local equivalence ratio at a point in the non-homogeneous mixture described by the mixture fraction). From the definitions given above:<br />
:<math> \Phi=\frac{Z}{1-Z}\frac{1-Z_s}{Z_s}</math><br />
<br />
==== Premixed Flame P&eacute;clet Number ====<br />
<br />
Earlier in this section, it has been said that a premixed flame posses its own dynamics, as a free propagating surface, and has thus characteristic quantities. For this reason, a P&eacute;clet number may be defined, based on these quantities. The P&eacute;clet number has the same structure as the Reynolds number but the dynamical viscosity is replaced by the ratio of the thermal conductivity and the heat capacity of the mixture. The thickness <math>\delta_L </math> of a premixed flame is essentially thermal. It means that it corresponds to the distance of the temperature rise between fresh and burned gases. This thickness is below the millimetre for conventional flames. The width of the reaction zone inside this flame is even smaller, by about one order of magnitude. This reaction zone is stuck to the hot side of the flame due to the high thermal dependency of the combustion reactions, as seen above. Hence, the flame region is essentially governed by a convection-diffusion process, the source term being negligible in most of it.<br />
<br />
It is convenient to write the progress variable transport equation in a steady-state framework. The quantities at flame temperature (<math>_f</math>) are used to non-dimensionalize the equation:<br />
:<math> \overbrace{(\rho ||\vec S_L||)}^{||\vec M||}\frac{\vec S_L}{S_L}\vec\nabla c = (\rho D)_f \nabla\cdot (\rho D)^* \vec\nabla c</math><br />
Note that the source term is neglected, consistently with what has been said above.<br />
This convection-diffusion equation makes appear a first approximation of a flame P&eacute;clet number:<br />
:<math> Pe_f = \frac{||\vec M|| \delta_L}{(\rho D)_f} \approx 1 </math><br />
<br />
From the P&eacute;clet number, it is possible to obtain an expression for the flame velocity (remembering that <math> \delta_L/S_{L,f} \approx \tau_c</math>, vid. inf. [[#Three Turbulent-Flame Interaction Regimes| Sec. Three Turbulent-Flame Interaction Regimes]]):<br />
:<math> S_{L,f}^2\approx \frac{(\rho D)_f}{\rho_f \tau_c}</math><br />
For typical hydrocarbon flames, the speed is some tens of centimetres per second and the diffusivity is some<br />
<math> 10^{-5} </math> square metres per second. The chemical time in the reaction zone of about one tenth of a millisecond is recovered.<br />
<br />
==== Details of the Premixed Unstrained Planar Flame ====<br />
<br />
A plane combustion wave propagating in a homogeneous fresh mixture is the reference case to describe the premixed regime. At constant speed, it is convenient to see the flame at rest with a flowing upstream mixture. This is actually the way propagating flames are usually stabilized. In the frame of description, the <br />
physics is 1-D, steady with a uniform (the flame is said unstrained) mass flowing across the system. Two types of equations are thus sufficient to describe the problem, the temperature transport equation and the species transport equations, as in [[#Transport Equations|Sec. Transport Equations]]. The transport coefficients will be chosen as equal: <math> \rho D_i = \lambda / C_p </math> (unity Lewis numbers). Suppose the 1-D domain is described thanks to a conventional (''Ox'') axis with a flame propagating towards negative ''x'' (this is the conventional usage), the boundary conditions are:<br />
:* in the frozen mixture: <br />
:** <math>Y_i \rightarrow Y_{i,u} \qquad ; \qquad x\rightarrow-\infty </math><br />
:** <math>T \rightarrow T_u \qquad ; \qquad x\rightarrow-\infty </math><br />
:* in the burned gas region supposed at equilibrium:<br />
:** <math>Y_i \rightarrow Y_{i,b} \qquad ; \qquad x\rightarrow+\infty </math><br />
:** <math>T \rightarrow T_b \qquad ; \qquad x\rightarrow+\infty </math><br />
<br />
:<math>Y_{i,b}</math> and <math>T_b</math> are obtained from [[#Conservation Laws|Sec. Conservation Laws]].<br />
<br />
The quantities that have been mentioned just above (scalar and temperature profiles, mass flow rate through the system) are the solution to be sought.<br />
<br />
According to the discussions above, the temperature transport equation in its full normalized form may be written as (lean /stoichiometric case):<br />
:<math> ||\vec M|| \frac{\partial \theta}{\partial x} = \frac{\partial \ }{\partial x}\frac{\lambda}{Cp} \frac{\partial \theta}{\partial x} + \frac{\nu_F \bar M_F B}{Y_{F,s}} \prod_{i=O,F}(Y_{i,u}^*-\theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_i} \exp{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}} </math><br />
This equation is further simplified by the variable change <math>d\xi=||\vec M||/(\lambda/Cp)dx</math>:<br />
:<math>\frac{\partial \theta}{\partial \xi}=\frac{\partial^2 \theta}{\partial \xi^2} + \overbrace{\frac{\lambda/Cp}{||\vec M||^2}\frac{\nu_F \bar M_F B}{Y_{F,s}}}^{\Lambda} \prod_{i=O,F}(Y_{i,u}^*-\theta)^{n_i} \exp{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}}</math><br />
<br />
Although somewhat out of scope, the existence and unicity of the solution of this type of equation are usually demonstrated with the help of the Schauder Theorem and Maximum Principle. From the point of view of physicists and engineers, the solution that is found analytically is de facto considered as the unique solution of the equation.<br />
<br />
===== Scenarii of Combustion Process in the Phase Portrait =====<br />
<br />
In the frame moving with the flame, both phase variables are the reduced temperature and its gradient. To ease the reading with usual notations, it is written: <math> X_1 = \theta \quad ; \quad X_2 = \partial \theta / \partial \xi = \dot \theta </math>. The system arises:<br />
: <math> \left\{<br />
\begin{array}{lll}<br />
\dot X_1 & = & X_2 \\<br />
\dot X_2 & = & X_2 - \varpi(X_1) <br />
\end{array}<br />
\right.<br />
</math><br />
with <math> \varpi </math> being the full source term in the above equation.<br />
<br />
In the frame moving with the flame, two singular nodes are found in the frozen flow <math> (X_1,X_2) = (0,0)</math> and the equilibrium region <math> (\theta_b,0)</math>, i.e when <math> \varpi(X_1) </math> vanishes. <br />
<math> x_1,x_2 </math> are defined as small departures from the singular nodes such that the linearized system in their neighbourhood is:<br />
: <math> \left\{<br />
\begin{array}{lll}<br />
\dot x_1 & = & l_{1,1} x_1 + l_{1,2} x_2 \\<br />
\dot x_2 & = & l_{2,1} x_1 + l_{2,2} x_2 <br />
\end{array}<br />
\right.<br />
</math><br />
provided: <math> l_{1,1} = 0 \quad l_{1,2} = 1 \quad l_{2,1} = -\varpi'_{X_1 = 0,\theta_b} \quad l_{2,2} = 1 </math>.<br />
The characteristic polynom is, in usual notations: <math> s^2 - s + \varpi' </math> such that the eigenvalues are:<br />
:<math><br />
s^{\pm}=\frac{1\pm\sqrt{1-4\varpi'}}{2}<br />
</math><br />
A priori, those eigenvalues may be (i) real distinct, (ii) real identical, or (iii) conjugated complex. In the first case, the orbits in the phase diagram are organized, in the immediate neighbourhood of the singular node, with respect to the eigenvectors directions associated to the eigenvalues. The following task is to identify the nature of those eigenvalues and of the corresponding nodes. Because <math> 0 < X_1 < \theta_b </math> is bounded, complex eigenvalues are excluded as they would lead to a spiral node. This remark is important for the node on the cold side as it imposes a bound:<br />
:<math> \varpi'_{X_1=0} \le \frac{1}{4} </math><br />
As the mass flow rate through the flame is included into <math>\varpi</math>, it imposes a minimum value on the flame speed to tackle with the cold boundary difficulty (rise of the chemical rate in the frozen flow). In this condition, it <br />
is an unstable node (improper in case of equality).<br />
On the other hand, because <math> \varpi'|_{X_1=\theta_b} </math> is not positive, the node on the hot side is found as a saddle point. The overall scenario of combustion within the flame is thus an orbit leaving the cold node to join the hot node by branching on a trajectory compatible with the negative eigenvalue of the saddle.<br />
<br />
[[Image:sketchOrbit.jpg|thumb| Sketch of orbits for a combustion process across a premixed front. Dashed lines represent forbidden orbits from the physics. The red line describes the orbit expected in an idealized combustion process.]]<br />
It must be noted that the associated eigenvectors are of the form:<br />
:<math><br />
\left (<br />
\begin{array}{l}<br />
t \\<br />
t s^{\pm} <br />
\end{array}<br />
\right ); \quad t\in \Re^*<br />
</math><br />
that is, on the cold node, a positive departure on <math>X_1</math> following any of the two eigendirections, leads<br />
to a consistent creation of positive temperature gradient, while on the hot node, only the stable direction will allow a consistent creation of a positive temperature gradient for any departure of the temperature towards region where it is inferior to <math>\theta_b </math>. Another remark is the structure of the eigendirections. The leaving directions on the cold node have a slope larger than the one of <math> \varpi </math> while the stable direction of the hot node has a slope smaller than the one of <math> \varpi </math>. It means that there is some point where the orbit must cross the profile of the chemical term versus temperature. For that temperature, the gradient equals the reaction rate through construction of the phase space. When looking back to the equation of the premixed flame, it happens in a region of inflexion for the temperature (the second order derivative must vanish). Furthermore, at this intersection, the orbit is horizontal (if the frame of reference for <math> (X_1, X_2) </math> is Cartesian) due to the shape of the premixed flame equation above that can be recast into <math> X_{2,X_1}' = (X_2-\varpi)/X_2 </math>.<br />
Close to the cold node, the orbits have a shape of parabola whose axis is the direction with the largest eigenvalue magnitude.<br />
Close to the hot node, the orbits have an hyperboloid shape with asymptots as the eigendirections. Now the ingredients are here to draw a sketchy scenario of the combustion in a premixed flame. It will be superimposed on the reaction rate graph studied in [[#Fundamentals Aspects|Sec. Fundamentals]].<br />
Some typical orbits from the above analysis are drawn in the figure on the right. The basic geometrical arguments developed are reproduced. In particular, the dashed lines represent forbidden orbits by the physics (boundedness of <math>X_1</math>, irreversibility). Orbits must be travelled from left to right, corresponding to increasing free parameter <math> \xi </math>.<br />
It is of integral importance to remember that, in combustion in conventional conditions, the source term is highly non-linear. Therefore, it is localized in a very thin sheet and, upstream, <math> \varpi </math> and <math> \varpi' \rightarrow 0 </math>. <br />
Only the most unstable eigendirection of the cold node is compatible as an orbit. The other trajectories, being paraboloidal, are tangent to the other direction that is flat at the limit. It physically means an elevation of temperature in bulk, that is contradictory to what is expected from a highly non-linear combustion term. Hence, the additional orbit in red is the one expected in idealized combustion conventional conditions.<br />
<br />
[[Image:dnsOrbit.jpg|thumb|Sketch of orbits for a combustion process across a premixed front, DNS simulations for the usual combinations of <math>\beta</math> and <math> \alpha </math>. Note that the case with <math>\alpha = 0</math> does not mean an athermal reaction but the mathematical simplification of the denominator of the exponential argument of the source term.]]<br />
The following picture is the result of actual computations of the above 1-D flame equation (stoichiometry and <math> n_i = 1 </math>) with the help of a high-order (6) code. The already presented combinations of <math> \beta </math> and <br />
<math> \alpha </math> are used and the orbits are retrieved. For most of the cases, as predicted above, the system selects a solution leaving the cold node with the most unstable direction (identifiable with its slope close to unity for vanishing<br />
<math>\varpi'_{X_1=0}</math>). There is also an additional curve, for <math> \beta = 10 </math> with the denominator of the exponential argument suppressed. This curve is remarkable as the solution selected by the system leaves the cold node in a manner fully controlled by the <math> \varpi'_{X_1=0} </math>. This is a very singular solution, not expected in combustion in conventional conditions, as explained above. The purpose of this remark is to question the well-posedness of considering simplifying the exponential argument for <math>\beta</math> "sufficiently high", as it is usually proposed for this type of modelling. As observed, the dynamical system analysis demonstrates a switch in the nature of the solution selected.<br />
At the physical level, when the orbit follows the most unstable direction with a slope close to unity, it means that <math> X_2 </math> "follows" <math> X_1 </math>, which is a signature of a diffusion process. In other words, the preheating mechanism of a premixed flame propagation as proposed for more than one century is in work. On the other hand, when <br />
<math> X_2 </math> is dependent on the evolution of <math> \varpi'</math>, it shows that "cold" chemistry drives the solution in the frozen flow and not the acknowledged mechanism of deflagration.<br />
<br />
<br />
===== Flame Solution =====<br />
<br />
As already mentioned, the flame system may be split into three zones. Upstream, the conventional mechanism of deflagration<br />
is supported by diffusion of heat. Downstream, the mixture is at equilibrium after combustion. In between, there exists the reaction layer. For large <math>\beta</math> the reaction layer is very thin such that it can be seen as a discontinuity between the fresh and burned gases. This is this difference in scales that introduces the use of the asymptotic method to resolve some of the flame characteristics, such as speed, time, heat region thickness, or reaction zone thickness.<br />
<br />
The domain is partitioned, according to this zoning defined by the scales driving the physics with an outer domain, driven by large scales and an inner domain refining the description within the discontinuity. If the discontinuity (flame reaction zone) is at <math> \xi=0 </math>, everywhere but 0, the equation is simplified as:<br />
:<math><br />
\frac{\partial^2 \theta}{\partial \xi^2} = \frac{\partial\theta}{\partial \xi}<br />
</math><br />
*For <math> \xi>0 </math>, it is expected that the mixture has reached equilibrium chemistry, such that: <math> \forall \xi > 0, \quad \theta=\theta_b \quad ; \quad \partial \theta / \partial \xi =0 </math>.<br />
*For <math> \xi < 0 </math>, this is the preheat zone and the solution is <math> \theta = \theta_b e^{\xi} </math> with <math> \theta </math> reaching <math>\theta_b</math> at the disconstinuity and vanishing, together with its gradient, far upstream in the frozen mixture.<br />
<br />
The solution for the species and the value of <math> \theta_b </math> are obtained from [[#Conservation Equations|Conservation Equations]] above. The 'big picture' is thus an exponential variation in the thermal thickness matched with a plateau in the downstream region, the line of matching being the discontinuity (flame) that has no thickness at this scale of description.<br />
<br />
To refine the analysis in the discontinuity region, a magnifying factor <math> \varepsilon </math> is used to stretch the coordinates: <math> \xi = \varepsilon \Xi </math>. The inner solution is thus a slowly-varying function of <math> \Xi </math>. Hence, in this inner region, the equation for the premixed flame becomes:<br />
:<math><br />
\frac{1}{\varepsilon}\frac{\partial \theta}{\partial \Xi}=\frac{1}{\varepsilon^2}\frac{\partial^2 \theta}{\partial \Xi^2} +\varpi<br />
</math><br />
In order to stretch and 'look inside' a discontinuity, <math>\varepsilon</math> is very small. It yields two remarks:<br />
# convection is negligible compared to diffusion. The heat losses from the reaction zone are essentially diffusion driven.<br />
# The reaction zone is governed by a diffusion-reaction budget and the reaction term <math>\varpi</math> must be strong to balance the intense heat loss due to the sharp diffusion (the zone is very thin, hence the gradients are sharp).<br />
The mechanism is thus different from the outer region that was convection-diffusion driven.<br />
<br />
Each quantity is developed in a series of <math> \varepsilon </math>.<br />
At the leading order, for <math>\theta</math>, in the lean case, the conservation relations (Sec. [[#Conservation Laws|Conservation Laws]]) yield:<br />
:<math> \theta = 1 - \varepsilon \Gamma - (1 - Y^*_{F,u})</math><br />
where <math>\Gamma</math> is the first-order development of the departure of <math>\theta</math> from the maximum value due to the incomplete combustion, and <math>1-Y_{F,u}^*=1-\theta_b</math> is the reduction of temperature for non-stoichiometric cases. Injected into the above equation:<br />
:<math> \frac{1}{\varepsilon}\frac{\partial^2 \Gamma}{\partial \Xi^2} = \Lambda (\varepsilon \Gamma)^{n_F} (Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} +\varepsilon\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\beta\frac{1-Y_{F,u}^*+\varepsilon\Gamma}{1-\alpha(1-Y_{F,u}^*+\varepsilon\Gamma)}}</math><br />
Although the full develoment is not achieved, a number of scaling may be highlighted:<br />
# because the temperature cannot be much below unity, <math>Y_{F,u}^*</math> must be close to 1 <math> {\mathcal O}(\varepsilon)</math> . For clarity, it is not expanded in an <math>\varepsilon</math> series.<br />
# The denominator of the exponential argument simplifies to unity for small <math>\varepsilon</math>. <br />
# To get a finite rate in the reaction zone, <math>\varepsilon</math> scales with <math>\beta^{-1}</math>.<br />
<br />
The burning rate eigenvalue, <math>\Lambda</math>, is naturally expanded as: <math> \Lambda = \varepsilon^{-n_O-n_F-1}(\Lambda_0 + {\mathcal O}(\varepsilon)) </math>. The low-order equation to be solved is:<br />
:<math><br />
\frac{d^2 \Gamma}{d \Xi^2} = \frac{1}{2}\frac{d(\Gamma_{\Xi})^{'2}}{d\Gamma} = \Lambda_0\exp{-\beta (1-Y_{F,u}^*)} \Gamma^{n_F} (\frac{Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} }{\varepsilon}+\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\Gamma}</math><br />
:<math> \frac{d\Gamma}{d\Xi}(-\infty)=-\theta_b, \qquad \frac{d\Gamma}{d\Xi}(\infty)=0 </math><br />
The boundary conditions are obtained from the matching of the outer solutions on the right and left sides of the flame as written above (the outer solutions are reached at infinity for a very small magnifying factor <math>\varepsilon</math>).<br />
Once integrated with respect to those boundary conditions, the burning-rate eigenvalue (from which <math> \dot M </math> is extracted) is obtained as:<br />
:<math> \Lambda_0 = \Bigg( 2\int_0^{\infty}d\Gamma\; \Gamma^{n_F} (\beta (Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )+\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\beta (1-Y_{F,u}^*)}\exp{-\Gamma} \Bigg )^{-1}<br />
</math><br />
The RHS integral is not developed for clarity but presents no peculiar difficulties.<br />
<br />
The development has been carried out at the first order in <math> \varepsilon </math>. As soon as a second order development is attempted, some expressions are no more analytically tractable. On the other hand, a second order development allows introduce the temperature-dependent trends of some terms in <math> \Lambda </math>. Physical results are retrieved such as a slight decrease of the speed for a positive sensitivity of transport parameters to temperature around equilibrium conditions.<br />
<br />
[[Image:BellSpeed.jpg|thumb|Unstrained planar premixed flame speed with respect to fuel mass fraction (lean case) for a single global irreversible Arrh\'enius term. Symbols are obtained from a high-order DNS code. Continuous line is the theory exposed here. The usual combinations of <math>\beta</math> and <math> \alpha </math> are used. <br />
Very high values of <math> \beta </math> (and thus negligible effect of <math> \alpha </math> are also presented<br />
to show the problem of slow convergence for finite value of the Zeldovitch parameter. Not that <math>\alpha = 0</math> does not mean an athermal reaction but the mathematical simplification of the denominator of the exponential argument of the source term.]]<br />
The image beside illustrates the response of the premixed flame speed with respect to fuel concentration (chosen as the limiting component here; equivalent findings are obtained for default of oxidizer) for different values of the chemical parameters <math> \beta </math> and <math>\alpha</math> (as noted above, the pre-exponential constant <math> A</math> impacts only the overall magnitude). The theoretical expression (continuous line) is tested against a 1-D high accuracy<br />
code with the given chemistry implemented. It is seen that:<br />
* the Zeldovitch parameter drives the drop for non-stoichiomeric mixtures,<br />
* the drop is relatively well modelled by the theoretical expression, and<br />
* the absolute magnitude converges slowly towards the theoretical one when increasing <math>\beta</math> in the code (effect of the finiteness of <math> \beta </math> in the real case).<br />
<br />
[[Image:PremixedProfiles.jpg|thumb|Typical profiles in 1-D premixed flame at stoichiometry. Representative value of global chemistry parameters. Specie profiles are simply the complementary to the temperature profile for simple chemistry.]]<br />
The picture exhibiting profiles for different Zeldovitch and heat release parameters shows the factual impact: upstream, the exact exponential profile is recovered and corresponds to the pre-heating region (thermal thickness). In the reaction zone (just upstream of the extremum), the departure from the exact solution is due to the kinetic effect. This kinetic effect is more pronounced when <math> \beta </math> is lower because the lower the Zeldovitch parameter, the lower the temperature reaction zone can be without leading to extinction. The flame takes this opportunity to maximize its transfer in heat and reactant with the cold zone. This is the physical understanding of an inverse dependence of the maximum flame speed with <math> \beta </math>.<br />
<br />
<br />
==== Modification of the Flame Speed with Curvature ====<br />
<br />
[[Image:premixedCurve2d.jpg|thumb|Flame seen as an interface between fresh and burned gases. Its curved profile towards the burned side increases the transfers with the fresh side.]]<br />
When the thickness of the flame <math>(\lambda/Cp)_f/||\vec M|| </math> is considered small compared to inhomogeneities existing in the flow, the flame can be reduced to an interface between fresh and burned gases. This interface may not be strictly plan in the general case. For instance, when the interface is curved towards the burned gas, it offers a larger opportunity for transfer of mass and heat with the fresh gas. As the ability of the chemistry to burn the coming matter is limited, the flame has thus to reduce its displacement speed. The figure beside provides a 2-D sketch of this situation that happens in contorded turbulent fields. Curvature effect is thus an ingredient appearing in combustion models.<br />
<br />
To mathematically give the expression showing that the curvature influences the flame speed (for small curvature), the non-dimensionalized temperature equation across a planar premixed flame above is slightly recast:<br />
:<math>||\vec M^c||||\vec\nabla_{\xi}\theta|| - \varpi= \nabla_{\xi}\cdot\vec\nabla_{\xi}\theta=-\nabla_{\xi}\cdot(||\vec\nabla_{\xi}\theta||\vec n) = - \vec\nabla_{\xi}||\vec\nabla_{\xi}\theta||\cdot\vec n - ||\vec\nabla_{\xi}\theta||\nabla_{\xi}\cdot\vec n</math><br />
<br />
In this expression, <math>\dot M^c</math> is the flame mass flow rate when perturbed by a curvature normalized by the reference one developed above. <math> \vec n</math> is the normal to the iso-temperature pointing towards the fresh gas, what is named the normal to the flame. From the expression developed above and for a slightly perturbed flame, the gradients, production and diffusion terms are very close to the unperturbed flame. Only the last term, the normal divergence, does not exist for a non-perturbed flame. Hence, the above equation may be simplified against the one for unperturbed flame presented earlier:<br />
:<math> ||\vec M^c|| = 1 - \nabla_{\xi}\cdot \vec n </math><br />
This is the divergence of the flame normal that contains the information upon geometrical perturbation (curvature) that impacts the speed.<br />
<br />
[[Image:premixedCurvature.jpg|thumb|Local geometrical approximation of a flame surface]]<br />
The key is thus to get an idea of the geometrical significance of the normal divergence. The normal divergence theorem says that this is the sum of the principal curvatures of the flame surface at the location considered. To give a good mental picture, the simplest configuration without loss of generality is to consider the flame surface approached by an osculatory revolution ellipsoid, as in the figure beside. The location of the approximation is the intersection of the Ox axis and the flame surface in red (where the ellipsoid is tangent to the flame). At this location, the normal divergence is the sum of the curvature of the basic ellipse (before its rotation around Oy) at its minor extremum, and the curvature of the circle corresponding to the rotation of this point around Oy when the 3-D shape is formed. Giving a lecture on conical coordinate systems is beyond the purpose but the interested reader may want to follow the corresponding steps to check this result: (i) write the divergence of a vector in the ellipsoid coordinate system, (ii) consider that the vector is the normal, i.e. it has only one constant component perpendicular to the local ellipsoidal iso-coordinate, to simplify the divergence expression, (iii) split the resulting terms into two parts by identifying the second derivative of the basic shape 2-D ellipse as one principal curvature, and the inverse of the radius of the circle corresponding to the rotation of the ellipse around Oy at the point where the flame surface is approached, in the figure this is simply the minor axis of the ellipse.<br />
<br />
The expression is readily written as:<br />
:<math> ||\vec M^c|| = 1 - (R_1^{-1}+R_2^{-1})</math><br />
where <math> R_1 </math> and <math> R_2</math> are the two radius of curvature (non-dimensionalized by the reference flame thickness) local to the surface.<br />
For instance, they are the small axis of the basic ellipse and its radius of curvature at its minor maximum when the <br />
surface is approached by the osculatory ellipsoid as above.<br />
<br />
<br />
==== Natural Instabilities of Premixed Flames ====<br />
<br />
Another aspect accounted for in turbulent combustion modelling of premixed flame is the creation of <br />
flame surface (corrugation) by hydrodynamics instabilities. These instabilities have been mentioned by Darrieus as early as 1938. The motivation of discussing about this is also related to the framework of description used that is widely employed for combustion models development. This framework is named the hydrodynamics limit, where the flame is isolated as <br />
a zero-thickness interface in the flow, and has been first introduced just above. In this framework, any diffusive and energetic aspects disappear and the set of equations is limited to two incompressible Euler systems. One system in the fresh gases (with a constant density of cold mixture). One system in the burned gases (with a constant density of mixture at equilibrium).<br />
<br />
To understand the basic properties of a premixed flame leading to the birth of instabilities, it is first important to realize that a premixed flame, in the hydrodynamic limit, behaves as a dioptre with a refractive index in the burned gases larger than in the fresh gases. For a flow with an angle of attack on the flame (i.e. a flame not strictly 1-D perpendicular to the flow), the tangential component of the flow speed relative to the flame surface is conserved while the normal component is accelerated by a factor corresponding to the ratio of the density in order to conserve mass flow across the interface. Hence, the streamlines are pushed towards the normal to the flame when crossing, that is similar to rays of light entering a refracting medium. <br />
<br />
[[Image:darrieusLandau.jpg|thumb|Basic mechanism explaining the unstable nature of a planar premixed flame.]]<br />
If one considers a region of a premixed flame that bumps a little bit towards the fresh gas (see figure beside -- the same approach is symmetrically true for the bump towards the burned gas), the local stream tube slightly opens on the bumpy interface before being refracted and coming back to its original section at constant mass flow rate. Hence, just in front of the bump, the gas velocity in this stream tube decreases and does not oppose the flame motion, allowing the bump to increase in magnitude. This is the fundamental mechanism of such instabilities.<br />
<br />
The equations sets are in usual notations:<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
U_x + V_y & = & 0 \\<br />
U_t + UU_x + VU_y & = & \frac{P_x}{\rho} \\<br />
V_t +UV_x+VV_y & = & \frac{P_y}{\rho} <br />
\end{array}<br />
\right.<br />
</math><br />
<math> x </math> is the coordinate used above along the flame path and <math>y</math> is in the tangential plan.<br />
<math>u </math> and <math>v</math> are the respective velocities.<br />
Those equations may be normalized by steady-state reference quantities such as flame speed, flame length, gauge pressure and density with respect to fresh gas properties and are written on both sides of the interface.<br />
<br />
These sets of equations must match at the interface (flame surface) through conservation of mass flow rate and momentum:<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
\rho_u (\vec U_u -\vec U_{f})\cdot \vec n & = & \rho_b (\vec U_b -\vec U_{f})\cdot \vec n\\<br />
(\rho_u \vec U_u (\vec U_u -\vec U_{f})+P_u)\cdot \vec n & = & (\rho_b \vec U_b (\vec U_b -\vec U_{f})+P_b)\cdot \vec n<br />
\end{array}<br />
\right.<br />
</math><br />
The subscripts <math> u </math> and <math> b </math> stand for unburned and burned sides, respectively. The subscript <math> f </math> points the interface. In these equations, the flame motion <math> U_f </math> is reintroduced because we want to track the instability movement over the mean position of the flame. This instability motion is described by the equation <math> x=F(y,t) </math> such that the motion of the interface normal to itself is obtained as:<br />
:<math> \vec U_f\cdot\vec n = - \frac{F_t}{\sqrt{1+F^2_y}} </math><br />
with a normal to the front:<br />
:<math> \vec n = \left (<br />
\begin{array}{l}<br />
-\frac{1}{\sqrt{1+F^2_y}} \\<br />
\frac{F_y}{\sqrt{1+F^2_y}} <br />
\end{array}<br />
\right )<br />
</math><br />
<br />
As the instability motion is considered at its birth, i.e. when it is still small, the quantities are linearized with <math> \varepsilon </math> as a small parameter:<br />
:<math> F=\varepsilon f</math><br />
:<math> \vec U = \vec {\mathcal U} + \varepsilon\vec u</math><br />
:<math> P = {\mathcal P} + \varepsilon p </math><br />
<br />
The Eulerian system is reduced to (at the first order in <math>\varepsilon</math>, the order zero is simplified for homogeneous flow):<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
u_x + v_y & = & 0 \\<br />
u_t + {\mathcal U}u_x & = & \frac{p_x}{\rho} \\<br />
v_t +{\mathcal U}v_x & = & \frac{p_y}{\rho} <br />
\end{array}<br />
\right.<br />
</math><br />
<br />
The one-sided equations for the jump conditions become (terms up to the first order in <math>\varepsilon</math> are kept):<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
\rho (\vec U-\vec U_f)\cdot\vec n \times \sqrt{1+F^2_y} & = & -\rho {\mathcal U}-\rho\varepsilon u+\rho\varepsilon f_t \\<br />
(\rho \vec U (\vec U -\vec U_{f})+P)\cdot \vec n \times \sqrt{1+F^2_y} & = &<br />
\left \{<br />
\begin{array}{l}<br />
-\rho {\mathcal U}^2 -2\rho{\mathcal U}\varepsilon u+\rho{\mathcal U}\varepsilon f_t - {\mathcal P} -\varepsilon p \\<br />
-\rho {\mathcal U}\varepsilon v + {\mathcal P}\varepsilon f_y<br />
\end{array}<br />
\right.<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
The following jump conditions are obtained (by recalling that unburned steady-state quantities are chosen as references):<br />
* From the first equation at the order 0, <math>\rho_b U_b = \rho_u U_u \Leftrightarrow U_b=\rho_u/\rho_b</math><br />
* From the first equation at the order 1, <math> \rho_u f_t - \rho_u u_u = \rho_b f_t -\rho_b u_b </math><br />
* From the second equation at the order 0, <math> -\rho_u {\mathcal U}_u^2 - {\mathcal P}_u = -\rho_b {\mathcal U}_b^2 - {\mathcal P}_b \Leftrightarrow {\mathcal P}_b = 1 - \rho_u/\rho_b </math><br />
* From the second equation at the order 1, <math> -2\rho_u{\mathcal U}_u u_u +\rho_u {\mathcal U}_u f_t -p_u = <br />
-2\rho_b{\mathcal U}_b u_b +\rho_b {\mathcal U}_b f_t -p_b \Leftrightarrow p_b-p_u = 2 u_u-2 u_b </math><br />
* From the third equation at order 1 (no order 0), <math> -\rho_u {\mathcal U}_u v_u +{\mathcal P}_u f_y = -\rho_b {\mathcal U}_b v_b +{\mathcal P}_b f_y \Leftrightarrow v_b-v_u = f_y(1-\rho_u/\rho_b) </math><br />
<br />
The solution for the linearized, autonomous Euler system is:<br />
:<math><br />
\left (<br />
\begin{array}{l}<br />
u \\ <br />
v \\ <br />
p<br />
\end{array}<br />
\right )<br />
= <br />
\left (<br />
\begin{array}{l}<br />
\bar u \\ <br />
\bar v \\ <br />
\bar p <br />
\end{array}<br />
\right )<br />
\exp{(\sigma x)}\exp{(\alpha t -iky)}<br />
</math><br />
where one recognizes an account for the perturbation of the field in the <math> x </math> direction (<math>\sigma</math>), the wave number of the instability following <math> y </math>, <math> k</math> and the <br />
growth rate with time <math> \alpha </math>.<br />
The eigenvalues of the system are used to determine the <math> x </math> dependence of the solution <math> \sigma = - \alpha/U,\; k,\; -k</math>, with the positive ones applying on the fresh side and the negative ones on the burned size to have a vanishing perturbation far from the flame.<br />
<br />
The eigenmodes give the flow pertubations on either side of the flame:<br />
:<math><br />
x<0:<br />
\left ( <br />
\begin{array}{l}<br />
u \\ v \\ p<br />
\end{array}<br />
\right )<br />
=<br />
a\left ( <br />
\begin{array}{l}<br />
1 \\ -i \\ -1-\frac{\alpha}{k}<br />
\end{array}<br />
\right )<br />
e^{kx+\alpha t - iky}<br />
</math><br />
<br />
:<math><br />
x>0:<br />
\left ( <br />
\begin{array}{l}<br />
u \\ v \\ p<br />
\end{array}<br />
\right )<br />
=<br />
b\left ( <br />
\begin{array}{l}<br />
1 \\ i \\ -1+\rho_b\frac{\alpha}{k}<br />
\end{array}<br />
\right )<br />
e^{-kx+\alpha t - iky}<br />
+<br />
c\left ( <br />
\begin{array}{l}<br />
1 \\ i\rho_b\frac{\alpha}{k} \\ 0<br />
\end{array}<br />
\right )<br />
e^{-\rho_b \alpha x+\alpha t - iky}<br />
</math><br />
<br />
The jump conditions above applied to this perturbation field yield the following system (<math> f </math> has also been put into the harmonic form <math> f = \bar f e^{\alpha t-iky} </math>):<br />
:<math><br />
\left \{ <br />
\begin{array}{lll}<br />
\alpha \bar f -a & = & \rho_b \alpha \bar f -\rho_b (b+c) \\<br />
a(1-\frac{\alpha}{k}) & = & b(1+\rho_b\frac{\alpha}{k}) +2c \\<br />
a + b+ c\frac{\rho_b \alpha}{k} & = & k\bar f (\frac{1}{\rho_b}-1)<br />
\end{array}<br />
\right . <br />
</math><br />
Additionaly, from the definition of the flame path <math> F </math>, we have the kinematic relation <br />
<math> u_u = a = \alpha \bar f </math>.<br />
<br />
The above set of equations forms a system of four unknowns <math> a,\; b,\; c,\; \alpha/k </math> <br />
for a given (but unknown) shape information on the flame bump <math> k\bar f</math>. <br />
Solving for the growth rate gives:<br />
:<math><br />
\left (\frac{\alpha}{k} - \frac{1}{\rho_b}\right )\left(\frac{\alpha}{k} + 2 + \rho_b\frac{\alpha}{k} -\left (\frac{1}{\rho_b}-1\right )\left (\frac{\alpha}{k}\right )^{-1} \right )=0<br />
</math><br />
This third-degree polynom has three solutions, namely the dispersion relation found by Darrieus, a stable mode, and the trivial solution (with no physical meaning), respectively:<br />
<math><br />
\frac{\alpha}{k} = \left ( \frac{1}{\rho_b + 1}\left (\sqrt{1+\frac{1-\rho_b^2}{\rho_b}}-1 \right ),\qquad \frac{1}{\rho_b + 1}\left (-\sqrt{1+\frac{1-\rho_b^2}{\rho_b}}-1 \right ),\qquad \frac{1}{\rho_b}\right )<br />
</math><br />
<br />
'''Remark 1''' By considering the shape of the eigenvectors giving the perturbation, one recovers that, upstream of the flame, the flow is potential (the rotational of the velocity vector is null and we have chosen a constant density flow, that is barotropic and divergence free), while downstream, additionally to another <br />
perturbation of the potential type, one finds a vorticity mode (the mode with the eigenvalue <math>-\rho_b\alpha </math>. The drift of vorticity at the crossing of the flame front is a known property of flames. <br />
<br />
'''Remark 2''' In more conventional literature, the strange trivial solution for the instability growth rate is swept under the rug. Given the pedagogical nature of this electronic documentation, we can dig a little bit as the appearance of this trivial, fool solution is a good example of some modelling issue. It is important to realize that a mathematical model and the physics it describes belong to different realities. Hence, the mathematical model will generate all the solutions that the mathematics can reach in its own space. Some of them are still connected to the physics. Some others, like this trivial solution, belongs only to the mathematical solution space, an indirect way of pointing out a model limitation. The model under scope here is the hydrodynamic limit. In this model, the domain is divided into two subdomains, one upstream of the flame interface, one downstream, that are put in relation with each other through a limited number of jump conditions. In reality, the physics connects these both domains much more tightly. The curious reader will have observed that the trivial solution makes the equation system for <math> a,\; b,\; c </math> (i.e. for a fixed growth rate) undetermined, and so are the jump conditions. By generating this trivial solution, the mathematics decouples both domains. The information coming from upstream to downstream (thanks to the shape of the linearized Euler system), a solution is found only for the upstream domain, the downstream being not solved and remains undetermined in lack of information. This solution cannot happen in reality because the fresh and burned gases in a real system are connected by many aspects, and not only by the jump conditions.<br />
<br />
==== Stretch / Compression of Premixed Flame ====<br />
<br />
The last of the `big' three turbulent ingredients (including aforementioned curvature and instabilities) impacting the flame at the local level is the compression or stretch of a premixed flame due to inhomogeneities in the flow, likely to happen with turbulence. <br />
<br />
[[Image:stretchCompression.jpg|thumb|Interpretation of flow inhomogeneities stresses on a 1-D premixed flame in terms of compression or stretch.]]<br />
The physics is pictured in the beside figure. In the flame area, the mass flow rate along the main direction may decrease (or increase) with distance. In a frame of reference whose origin is the core of the flame, a stretch (compression) results due to the difference in mass flow rate entering and leaving the flame zone.<br />
'''Important Remark''' Here, we are interested in the inhomogeneity in the mass flow rate along one direction only. We do NOT write that the flame volume is a region of mass source/sink.<br />
<br />
In the same manner as above for the curvature effect, we introduce a (small) inhomogeneity of the flow as <br />
<math> \vec\nabla\vec M </math> such that the following equation can be substracted from the unperturbed flame equation.<br />
:<math>||\vec M^{s/c}||||\vec\nabla_{\xi}\theta|| - \vec n\cdot \vec\nabla\vec M \cdot \vec n ||\vec\nabla_{\xi}\theta|| - \varpi= \nabla_{\xi}\cdot\vec\nabla_{\xi}\theta<br />
</math><br />
<br />
Hence, for a small perturbation, the linearized dependence of the flame on stretch/compression is:<br />
:<math>||\vec M^{s/c}|| = 1 + \vec n \cdot \vec\nabla\vec M \cdot \vec n </math><br />
<br />
== The Partially-Premixed Regime ==<br />
[[Image:ppf.jpg|thumb| Ideal sketch of a partially-premixed flame]]<br />
This regime is somewhat less academics and has been recognized two decades ago. It is acknowledged as a hybrid of the premixed and the non-premixed regimes but the degree of interaction of these two modes of combustion to accurately describe a partially-premixed flame is still not well understood. It can be simply pictured by a lifted diffusion flame. Let us consider fuel issuing from a nozzle into the air. If the exit velocity is large enough, for some fuels, the flame lifts off the rim of the nozzle. It means that below the flame base, fuel and oxidizer have room to premix. Hence, the flame base propagates into a premixed mixture. However, it cannot be reduced to a premixed flame (although it is often simplified as this): (i) the mixing is not perfect and the different parts of the flame front constituting the flame base burn in mixtures of different thermodynamical states. This provides those parts with different deflagration capabilities such that the flame base has a complex shape. Indeed, it is convex, naturally leaded by the part burning at stoichiometry, unless `exotic' feeding temperatures are used. (ii) Because the mixture is not homogeneous, transfer of species and temperature driven by diffusion occurs in a direction perpendicular to the propagation of the flame base. Because the flame front is not flat, those transfers act as a connection vehicle across the different parts of the leading front. (iii) The unburned left downstream by the sections of the leading front not burning at stoichiometry diffuse towards each other to form a diffusion flame as described above. The connection of the leading front with the trailing diffusion flame has been evidenced as complicated and the siege of transfers of species and temperature. These two last items are the state-of-the-art difficulties in understanding those flames and do not appear in the models although it has been demonstrated they have a major impact and are certainly a fundamental characteristic of partially-premixed flames.<br />
<br />
The partially-premixed flame is usually described using ''c'' and ''Z'' as introduced earlier. Because the framework is essentially non-premixed, the mixture fraction is primarily used to describe the flame. Regarding the head of the flame where partial-premixing has an impact, each part of the front is described with a local progress variable. The need of defining a local progress variable is that each section of the partially-premixed front has a different equivalence ratio leading to a different definition of ''c'':<br />
:<math> c=\frac{T-T_u}{T_b(Z)-T_u}</math><br />
<br />
= Three Turbulent-Flame Interaction Regimes =<br />
It may appear odd to try to describe here what is the reason of combustion modelling research: the interaction of the turbulence with chemistry. However, one of the first steps in building knowledge in turbulent combustion was the qualitative exploration of what might be the dynamics of a flame in a turbulent environment. This led to what is now known as ''combustion diagrams''. As explained above, the premixed regime lends itself the easiest to such an approach as it exhibits natural intrinsic quantities which are not as objectively identifiable in the other combustion regimes. Note that these quantities may depend<br />
on the geometry of the flame: for instance turbulence can bend a flame sheet, leading to a change in its <br />
dynamics compared to the flat flame propagating in a medium at rest. In this section, the turbulence-flame interaction modes will be described for a premixed flame. Only remarks will be added regarding the non-premixed and partially-premixed regimes.<br />
<br />
An integral quantity to assess the interaction between a premixed flame sheet and the turbulence<br />
is the Karlovitz number ''Ka''. It compares the characteristic time of flame displacement with the characteristic time of the smallest structures (that are also the fastest) of the turbulence.<br />
:<math> Ka= \frac{\tau_c}{\tau_k}</math><br />
<br />
<math>\tau_c</math> is the chemical time of the flame. To estimate it, it is necessary to come back to the above progress variable transport equation in a steady-state framework.<br />
:<math> \overbrace{(\rho S_L)}^{\dot M}\frac{\vec S_L}{S_L}\vec\nabla c = (\rho D)_f\nabla\cdot (\rho D)^* \vec\nabla c + \dot\omega_c</math><br />
The premixed wave propagates at a speed <math>S_L</math> because it is fed by reactants diffusing inside the combustion zone and which are preheated because temperature diffuses in the reverse direction. The speed at which the flame progresses is thus related to the rate of species diffusion into the reaction zone which are then consumed. As the premixed flame is a free propagating wave whose speed of propagation is only limited by the chemical strength, the characteristic chemical time is based only on the diffusion and the mass flow rate experienced by the flame:<br />
:<math> \tau_c = \frac{\rho (\rho D)_f}{\dot M^2}</math><br />
<br />
The smallest eddies are the ones being dissipated by the viscous forces. Their characteristic time is estimated thanks to a combination of the viscosity and the flux of turbulent energy to be dissipated (also called turbulent dissipation <math> \varepsilon=u'^3/l_t </math>):<br />
:<math>\tau_k = \sqrt{\frac{l_t\nu}{u'^3}}</math><br />
<br />
Thanks to those definitions of chemical and small structure times, it is possible to give another definition of the Karlovitz number:<br />
:<math> Ka=\left (\frac{\delta_L}{l_k} \right)^2</math> <br />
which is the square of the ratio between the premixed flame thickness and the small structure scale: ''Ka'' actually compares scales. To arrive to this latter result, the three following assumptions must be used: (i) the flame thickness is obtained thanks to the premixed flame P&eacute;clet number (''vid. sup.''); (ii) the turbulence small structure (''Kolmogorov eddies'') scale is given by: <math> l_k=(\nu^3/\varepsilon)^{1/4} </math> following the same dimensional argument as for the estimation of its time; and (iii) scalar diffusion scales with viscosity. <br />
<br />
==== Remark Regarding the Diffusion Flame ====<br />
From what has been presented above, a diffusion flame does not have characteristic scales. Setting a turbulence combustion regime classification for non-premixed flames has still not been answered by research. Some laws of behaviour will only be drawn around the scalar dissipation rate which is the parameter of integral importance for a diffusion flame.<br />
<br />
Indeed, the dynamics of a diffusion flame is determined by the strain rate imposed by the turbulence. As for the premixed flames, the shortest eddies (Kolmogorov) are the ones having the largest impact. The diffusive layer is thus given by the size of the Kolmogrov eddies: <math> l_d\approx l_k</math> and the typical diffusion time scale (feeding rate of the reaction zone) is given by the characteristic time of the Kolmogorov eddies: <math> \tau_k^{-1}\approx \chi_s</math> as the Reynolds number of the Kolmogorov structures is unity. Here, <math>\chi_s</math> is the sample-averaging of <math>\chi</math> based on (conditioned) stoichiometric conditions, where the flame is expected to be.<br />
<br />
== The Wrinkled Regime ==<br />
[[Image:wrinkled.jpg|thumb| Wrinkled flamelet regime]]<br />
This regime is also called the ''flamelet regime''. Basically, it assumes that the flame structure is not affected by turbulence. The flame sheet is convoluted and wrinkled by eddies but none of them is small enough to enter it.<br />
Locally magnifying, the laminar flame structure is maintained.<br />
<br />
This regime exists for a Karlovitz number below unity (vid. sup.), i.e. chemical time smaller than the small structure time or flame thickness smaller than small structure scale. Notwithstanding, the laminar flame dynamics can be disrupted for <math> u'>S_L</math>. In that case, although the flame structure is not altered by the small structures, it can be convected by large structures such that areas of different locations in the front interact. It shows that, even with a small Karlovitz number, the turbulence effect is not always weak.<br />
<br />
== The Corrugated Regime ==<br />
[[Image:Corrugated.jpg|thumb|Corrugated flamelet regime]]<br />
The formal definition of a flame is the region of temperature rise. However, the volume where the reaction takes place is about one order of magnitude smaller, embedded inside the temperature rise region and close to its high temperature end. Hence, there exist some levels of turbulence creating eddies able to enter the flame zone but still large enough to not affect the internal reaction sheet. In other words, the flame thermal region is thickened by turbulence but the reaction zone is still in the wrinkled regime.<br />
This situation is called the ''Corrugated Regime''.<br />
<br />
Due to the structure of the Karlovitz number, once written in terms of length scales (vid. sup.), this situation arises for an<br />
increase of the Karlovitz number by two orders of magnitude compared to the value for the wrinkled regime. Hence, in the range <br />
<math> 1 < Ka < 100 </math>, the laminar structure of the reaction zone is still preserved but not the one of the preheat zone.<br />
<br />
== The Thickened Regime ==<br />
[[Image:thickened.jpg|thumb|Thickened regime]]<br />
In this last case, turbulence is intense enough to generate eddies able to affect the structure of the reaction zone as well. In practice, it is expected that those eddies are in the tail of the energy spectrum such that their lifetime is very short. Their impact on the reaction zone is thus limited.<br />
<br />
Obviously, ''Ka > 100''. A topological description is of little relevance here and a ''well-stirred reactor model'' fits better.<br />
<br />
== Reaction mechanisms ==<br />
<br />
Combustion is mainly a chemical process. Although we can, to some extent, <br />
describe a flame without any chemistry information, modelling of the flame <br />
propagation requires the knowledge of speeds of reactions, product concentrations, <br />
temperature, and other parameters. <br />
Therefore fundamental information about reaction kinetics is <br />
essential for any combustion model. <br />
A fuel-oxidizer mixture will generally combust if the reaction is fast enough to <br />
prevail until all of the mixture is burned into products. If the reaction <br />
is too slow, the flame will extinguish. If too fast, explosion or even <br />
detonation will occur. The reaction rate of a typical combustion reaction <br />
is influenced mainly by the concentration of the reactants, temperature, and pressure. <br />
<br />
A stoichiometric equation for an arbitrary reaction can be written as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
\sum_{j=1}^{n}\nu' (M_j) = \sum_{j=1}^{n}\nu'' (M_j),<br />
</math></td></tr></table><br />
<br />
where <math> \nu </math> denotes the stoichiometric coefficient, and <math>M_j</math> stands for an arbitrary species. A one prime holds for the reactants while a double prime holds for the products of the reaction. <br />
<br />
The reaction rate, expressing the rate of disappearance of reactant <b>i</b>, is defined as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
RR_i = k \, \prod_{j=1}^{n}(M_j)^{\nu'},<br />
</math></td></tr></table><br />
<br />
in which <b>k</b> is the specific reaction rate constant. Arrhenius found that this constant is a function of temperature only and is defined as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
k= A T^{\beta} \, exp \left( \frac{-E}{RT}\right)<br />
</math></td></table><br />
<br />
where <b>A</b> is pre-exponential factor, <b>E</b> is the activation energy, and <math>\beta</math> is a temperature exponent. The constants vary from one reaction to another and can be found in the literature.<br />
<br />
Reaction mechanisms can be deduced from experiments (for every resolved reaction), they can also be constructed numerically by the '''automatic generation method''' (see [Griffiths (1994)] for a review on reaction mechanisms).<br />
For simple hydrocarbons, tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms, global reactions <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing equations for chemically reacting flows ==<br />
<br />
Together with the usual Navier-Stokes equations for compresible flow (See [[Governing equations]]), additional equations are needed in reacting flows.<br />
<br />
The transport equation for the mass fraction <math> Y_k </math> of <i>k-th</i> species is<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Y_k \right) +<br />
\frac{\partial}{\partial x_j} \left( \rho u_j Y_k\right) = <br />
\frac{\partial}{\partial x_j} \left( \rho D_k \frac{\partial Y_k}{\partial x_j}\right)+ \dot \omega_k<br />
</math><br />
<br />
<br />
where Ficks' law is assumed for scalar diffusion with <math> D_k </math> denoting the species difussion coefficient, and <math> \dot \omega_k </math> denoting the species reaction rate.<br />
<br />
A non-reactive (passive) scalar (like the mixture fraction <math> Z </math>) is goverened by the following transport equation<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Z \right) +<br />
\frac{\partial}{\partial x_j} \left( \rho u_j Z \right) = <br />
\frac{\partial}{\partial x_j} \left( \rho D \frac{\partial Z}{\partial x_j}\right)<br />
</math><br />
<br />
where <math> D </math> is the diffusion coefficient of the passive scalar.<br />
<br />
<br />
=== RANS equations ===<br />
<br />
In turbulent flows, [[Favre averaging]] is often used to reduce the scales (see [[Reynolds averaging]]) and the<br />
mass fraction transport equation is transformed to<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } -<br />
\overline{\rho} \widetilde{u''_i Y''_k } \right)<br />
+ \overline{\dot \omega_k}<br />
</math><br />
<br />
<br />
where the turbulent fluxes <math> \widetilde{u''_i Y''_k} </math> and reaction terms<br />
<math> \overline{\dot \omega_k} </math> need to be closed.<br />
<br />
The passive scalar turbulent transport equation is<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Z} }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z} }{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D \frac{\partial Z} {\partial x_j} } -<br />
\overline{\rho} \widetilde{u''_i Z'' } \right)<br />
</math><br />
<br />
where <math> \widetilde{u''_i Z''} </math> needs modelling. A common practice is to model the turbulent fluxes using the <br />
gradient diffusion hypothesis. For example, in the equation above the flux <math> \widetilde{u''_i Z''} </math> is modelled as<br />
<br />
:<math><br />
\widetilde{u''_i Z''} = -D_t \frac{\partial \tilde Z}{\partial x_i}<br />
</math> <br />
<br />
where <math> D_t </math> is the turbulent diffusivity. Since <math> D_t >> D </math>, the first term inside the parentheses on the right hand of the mixture fraction transport equation is often neglected (Peters (2000)). This assumption is also used below.<br />
<br />
In addition to the mean passive scalar equation,<br />
an equation for the Favre variance <math> \widetilde{Z''^2}</math> is often employed<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Z''^2} }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z''^2} }{\partial x_j}=<br />
\frac{\partial}{\partial x_j}<br />
\left( \overline{\rho} \widetilde{u''_i Z''^2} \right) -<br />
2 \overline{\rho} \widetilde{u''_i Z'' }<br />
- \overline{\rho} \widetilde{\chi}<br />
</math><br />
<br />
where <math> \widetilde{\chi} </math> is the mean [[Scalar dissipation]] rate<br />
defined as <math> \widetilde{\chi} =<br />
2 D \widetilde{\left| \frac{\partial Z''}{\partial x_j} \right|^2 } </math><br />
This term and the variance diffusion fluxes needs to be modelled.<br />
<br />
=== LES equations ===<br />
The [[Large eddy simulation (LES)]] approach for reactive flows introduces equations for the filtered species mass fractions within the compressible flow field.<br />
Similar to the [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]], the filtered mass fraction transport equation is<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } -<br />
J_j \right)<br />
+ \overline{\dot \omega_k}<br />
</math><br />
<br />
where <math> J_j </math> is the transport of subgrid fluctuations of mass fraction<br />
:<math><br />
J_j = \widetilde{u_jY_k} - \widetilde{u}_j \widetilde{Y}_k<br />
</math><br />
and has to be modelled.<br />
<br />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
are assumed to be much smaller than the apparent turbulent diffusion due to transport of subgrid fluctuations.<br />
The first term on the right hand side is then<br />
:<math><br />
\frac{\partial} {\partial x_j} <br />
\left(<br />
\overline{ \rho D_k \frac{\partial Y_k} {\partial x_j} }<br />
\right)<br />
\approx<br />
\frac{\partial} {\partial x_j} <br />
\left(<br />
\overline{\rho} D_k \frac{\partial \widetilde{Y}_k} {\partial x_j} <br />
\right)<br />
</math><br />
<br />
[[Category:Equations]]<br />
<br />
== Infinitely fast chemistry ==<br />
All combustion models can be divided into two main groups according to the <br />
assumptions on the reaction kinetics.<br />
We can either assume the reactions to be infinitely fast - compared to <br />
e.g. mixing of the species, or comparable to the time scale of the mixing <br />
process. The simple approach is to assume infinitely fast chemistry. Historically, mixing of the species is the older approach, and it is still in wide use today. It is therefore simpler to solve for [[#Finite rate chemistry]] models, at the overshoot of introducing errors to the solution.<br />
<br />
=== Premixed combustion ===<br />
Premixed flames occur when the fuel and oxidiser are homogeneously mixed prior to ignition. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical examples of premixed laminar flames is bunsen burner, where <br />
the air enters the fuel stream and the mixture burns in the wake of the <br />
riser tube walls forming nice stable flame. Another example of a premixed system is the solid rocket motor where oxidizer and fuel and properly mixed in a gum-like matrix that is uniformly distributed on the periphery of the chamber.<br />
Premixed flames have many advantages in terms of control of temperature and <br />
products and pollution concentration. However, they introduce some dangers like the autoignition (in the supply system).<br />
<br />
[[Image:Premixed.jpg]]<br />
==== Turbulent flame speed model ====<br />
<br />
==== Eddy Break-Up model ====<br />
<br />
The Eddy Break-Up model is the typical example of '''mixed-is-burnt''' combustion model. <br />
It is based on the work of Magnussen, Hjertager, and Spalding and can be found in all commercial CFD packages. <br />
The model assumes that the reactions are completed at the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. Combustion is then described by a single step global chemical reaction<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
F + \nu_s O \rightarrow (1+\nu_s) P<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
in which <b>F</b> stands for fuel, <b>O</b> for oxidiser, and <b>P</b> for products of the reaction. Alternativelly we can have a multistep scheme, where each reaction has its own mean reaction rate.<br />
The mean reaction rate is given by<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
\bar{\dot\omega}_F=A_{EB} \frac{\epsilon}{k} <br />
min\left[\bar{C}_F,\frac{\bar{C}_O}{\nu},<br />
B_{EB}\frac{\bar{C}_P}{(1+\nu)}\right]<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
<br />
where <math>\bar{C}</math> denotes the mean concentrations of fuel, oxidiser, and products <br />
respectively. <b>A</b> and <b>B</b> are model constants with typical values of 0.5 <br />
and 4.0 respectively. The values of these constants are fitted according <br />
to experimental results and they are suitable for most cases of general interest. It is important to note that these constants are only based on experimental fitting and they need not be suitable for <b>all</b> the situations. Care must be taken especially in highly strained regions, where the ratio of <math>k</math> to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In these, regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually have some remedies to overcome this problem.<br />
<br />
This model largely over-predicts temperatures and concentrations of species like <i>CO</i> and other species. However, the Eddy Break-Up model enjoys a popularity for its simplicity, steady convergence, and implementation.<br />
<br />
==== Bray-Moss-Libby model ====<br />
<br />
=== Non-premixed combustion ===<br />
<br />
Non premixed combustion is a special class of combustion where fuel and oxidizer<br />
enter separately into the combustion chamber. The diffusion and mixing of the two streams<br />
must bring the reactants together for the reaction to occur.<br />
Mixing becomes the key characteristic for diffusion flames.<br />
Diffusion burners are easier and safer to operate than premixed burners.<br />
However their efficiency is reduced compared to premixed burners.<br />
One of the major theoretical tools in non-premixed combustion<br />
is the passive scalar [[mixture fraction]] <math> Z </math> which is the<br />
backbone on most of the numerical methods in non-premixed combustion.<br />
<br />
==== Conserved scalar equilibrium models ====<br />
The reactive problem is split into two parts. First, the problem of <i> mixing </i>, which consists of the location of the flame surface which is a non-reactive problem concerning the propagation of a passive scalar; And second, the <i> flame structure </i> problem, which deals with the distribution of the reactive species inside the flamelet.<br />
<br />
To obtain the distribution inside the flame front we assume it is locally one-dimensional and<br />
depends only on time and the scalar coodinate.<br />
<br />
We first make use of the following chain rules<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}<br />
</math><br />
<br><br />
:<math><br />
\frac{\partial Y_k}{\partial x_j} = \frac{\partial Z}{\partial x_j}\frac{\partial Y_k}{\partial Z}<br />
</math> <br />
and transformation <br />
:<math><br />
\frac{\partial }{\partial t}= \frac{\partial }{\partial t} + \frac{\partial Z}{\partial t} \frac{\partial }{\partial Z}<br />
</math><br />
<br />
upon substitution into the species transport equation (see [[#Governing Equations for Reacting Flows]]), we obtain<br />
<br />
:<math><br />
\rho \frac{\partial Y_k}{\partial t} + Y_k \left[<br />
\frac{\partial \rho}{\partial t} + \frac{\partial \rho u_j}{\partial x_j}<br />
\right]<br />
+ \frac{\partial Y_k}{\partial Z} \left[<br />
\rho \frac{\partial Z}{\partial t} + \rho u_j \frac{\partial Z}{\partial x_j} -<br />
\frac{\partial}{\partial x_j}\left( \rho D \frac{\partial Z}{\partial x_j} \right)<br />
\right]<br />
=<br />
\rho D \left( \frac{\partial Z}{\partial x_j} \frac{\partial Z}{\partial x_j} \right)<br />
\frac{\partial^2 Y_k}{\partial Z^2} + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third terms in the LHS cancel out due to continuity and mixture fraction transport.<br />
At the outset, the equation boils down to<br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k<br />
</math><br />
<br />
where <math> \chi = 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2 </math> is called the <b>scalar dissipation</b><br />
which controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though the field <math> Z </math> still depends on it.<br />
<br />
:<math><br />
\dot \omega_k= -\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2}<br />
</math><br />
<br />
If the reaction is assumed to be infinetly fast, the resultant flame distribution is in equilibrium.<br />
and <math> \dot \omega_k= 0</math>. When the flame is in equilibrium, the flame configuration <math> Y_k(Z) </math> is independent of strain.<br />
<br />
===== Burke-Schumann flame structure =====<br />
The Burke-Schuman solution is valid for irreversible infinitely fast chemistry.<br />
With a reaction in the form of<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
F + \nu_s O \rightarrow (1+\nu_s) P<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
If the flame is in equilibrium and therefore the reaction term is 0.<br />
Two possible solution exists, one with pure mixing (no reaction)<br />
and a linear dependence of the species mass fraction with <math> Z </math>.<br />
Fuel mass fraction<br />
:<math><br />
Y_F=Y_F^0 Z<br />
</math><br />
Oxidizer mass fraction<br />
:<math><br />
Y_O=Y_O^0(1-Z)<br />
</math><br />
Where <math> Y_F^0 </math> and <math> Y_O^0 </math> are fuel and oxidizer mass fractions<br />
in the pure fuel and oxidizer streams respectively.<br />
<br />
<br />
The other solution is given by a discontinuous slope at stoichiometric mixture fraction<br />
<math> Z_{st}</math> and two linear profiles (in the rich and lean side) at either<br />
side of the stoichiometric mixture fraction.<br />
Both concentrations must be 0 at stoichiometric, the reactants become products infinitely fast.<br />
<br />
:<math><br />
Y_F=Y_F^0 \frac{Z-Z_{st}}{1-Z_{st}}<br />
</math><br />
and oxidizer mass fraction<br />
:<math><br />
Y_O=Y_O^0 \frac{Z-Z_{st}}{Z_{st}}<br />
</math><br />
<br />
== Finite rate chemistry ==<br />
<br />
=== Premixed combustion ===<br />
<br />
==== Coherent flame model ====<br />
<br />
==== Flamelets based on G-equation ====<br />
==== Flame surface density model ====<br />
<br />
=== Non-premixed combustion ===<br />
<br />
==== Flamelets based on conserved scalar ====<br />
<br />
Peters (2000) define Flamelets as "thin diffusion layers embedded in a turbulent non-reactive flow field".<br />
If the chemistry is fast enough, the chemistry is active within a thin region<br />
where the chemistry conditions are in (or close to) stoichiometric conditions, the "flame" surface.<br />
This thin region is assumed to be smaller than Kolmogorov length scale and therefore the<br />
region is locally laminar. The flame surface is defined as an iso-surface of a certain scalar <math> Z </math>,<br />
mixture fraction in [[#Non premixed combustion]].<br />
<br />
The same equation used in [[#Conserved scalar models]] for equilibrium chemistry<br />
is used here but with chemical source term different from 0<br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} =<br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k.<br />
</math><br />
<br />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that<br />
flamelet profiles <math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in a dtaset or file which is called a "flamelet library" with all the required complex chemistry.<br />
For the generation of such libraries ready to use software is avalable such as Softpredict's Combustion Simulation Laboratory COSILAB [http://www.softpredict.com] with its relevant solver RUN1DL, which can be used for a variety of relevant geometries; see various [http://www.softpredict.com/?page=989 publications that are available for download]. Other software tools are available such as CHEMKIN [http://www.reactiondesign.com] and<br />
CANTERA [http://www.cantera.org].<br />
<br />
===== Flamelets in turbulent combustion =====<br />
<br />
In turbulent flames the interest is <math> \widetilde{Y}_k </math>.<br />
In flamelets, the flame thickness is assumed to be much smaller than Kolmogorov scale<br />
and obviously is much smaller than the grid size.<br />
It is therefore needed a distribution of the passive scalar within the cell.<br />
<math> \widetilde{Y}_k </math> cannot be obtained directly from the flamelets library<br />
<math> \widetilde{Y}_k \neq Y_F(Z,\chi) </math>, where <math> Y_F(Z,\chi) </math> corresponds<br />
to the value obtained from the flamelets libraries.<br />
A generic solution can be expressed as<br />
:<math><br />
\widetilde{Y}_k= \int Y_F( \widetilde{Z},\widetilde{\chi}) P(Z,\chi) dZ d\chi<br />
</math><br />
where <math> P(Z,\chi) </math> is the joint [[Probability density function | Probability Density Function]] (PDF) of the mixture fraction<br />
and scalar dissipation which account for the scalar distribution inside the cell and "a priori"<br />
depends on time and space.<br />
<br />
The most simple assumption is to use a constant distribution of the scalar dissipation within the cell and<br />
the above equation reduces to<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) dZ<br />
</math><br />
<br />
<math> P(Z) </math> is the PDF of the mixture fraction scalar and simple models (such as Gaussian or a [[beta PDF]])<br />
can be build depending only on two moments of the scalar<br />
mean and variance,<math> \widetilde{Z},Z''</math>.<br />
<br />
If the mixture fraction and scalar dissipation are consider independent variables,<math> P(Z,\chi) </math><br />
can be written as <math> P(Z) P(\chi)</math>. The PDF of the scalar dissipation is assumed to be log-normal with<br />
variance unity.<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) P(\chi) dZ d\chi<br />
</math><br />
In [[Large eddy simulation (LES)]] context (see [[#LES equations]] for reacting flow),<br />
the [[probability density function]] is replaced by a [[subgrid PDF]] <math> \widetilde{P}</math>.<br />
The same equation hold by replacing averaged values with filtered values.<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) \widetilde{P}(Z) \widetilde{P}(\chi) dZ d\chi<br />
</math><br />
The assumptions made regarding the shapes of the PDFs are still justified. In LES combustion the<br />
[[subgrid variance]] is smaller than RANS counterpart (part of the large-scale fluctuations are solved)<br />
and therefore the modelled PDFs are thinner.<br />
<br />
====== Unsteady flamelets ======<br />
====Intrinsic Low Dimensional Manifolds (ILDM)====<br />
<br />
Detailed mechanisms describing ignition, flame propagation and pollutant formation typically involve several hundred species and elementary reactions, prohibiting their use in practical three-dimensional engine simulations. Conventionally reduced mechanisms often fail to predict minor radicals and pollutant precursors. The ILDM-method is an automatic reduction of a detailed mechanism, which assumes local equilibrium with respect to the fastest time scales identified by a local eigenvector analysis. In the reactive flow calculation, the species compositions are constrained to these manifolds. Conservation equations are solved for only a small number of reaction progress variables, thereby retaining the accuracy of detailed chemical mechanisms. This gives an effective way of coupling the details of complex chemistry with the time variations due to turbulence.<br />
<br />
The intrinsic low-dimensional manifold (ILDM) method {Maas:1992,Maas:1993} is a method for ''in-situ'' reduction of a detailed chemical mechanism based on a local time scale analysis. This method is based on the fact that different trajectories in state space start from a high-dimensional point and quickly relax to lower-dimensional manifolds due to the fast reactions. The movement along these lower-dimensional manifolds, however, is governed by the slow reactions. It exploits the variety of time scales to systematically reduce the detailed mechanism. For a detailed chemical mechanism with N species, N different time scales govern the process. An assumption that all the time scales are relaxed results in assuming complete equilibrium, where the only variables required to describe the system are the mixture fraction, the temperature and the pressure. This results in a zero-dimensional manifold. An assumption that all but the slowest 'n' time scales are relaxed results in a 'n' dimensional manifold, which requires the additional specification of 'n' parameters (called progress variables). In the ILDM method, the fast chemical reactions do not need to be identified a priori. An eigenvalue analysis of the detailed chemical mechanism is carried out which identifies the fast processes in dynamic equilibrium with the slow processes. The computation of ILDM points can be expensive, and hence an in-situ tabulation procedure is used, which enables the calculation of only those points that are needed during the CFD calculation.<br />
<br />
--[[User:Fredgauss|Fredgauss]] 07:37, 25 August 2006 (MDT)<br />
<br />
U. Maas, S.B. Pope. Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Comb. Flame 88,<br />
239, 1992.<br />
<br />
Ulrich Maas. Automatische Reduktion von Reaktionsmechanismen zur Simulation reaktiver Str¨omungen. Habilitationsschrift, Universit<br />
¨at Stuttgart, 1993.<br />
<br />
==== Conditional Moment Closure (CMC)====<br />
In Conditional Moment Closure (CMC) methods we assume that the species mass fractions are all correlated with<br />
the mixture fraction (in non premixed combustion).<br />
<br />
From [[Probability density function]] we have<br />
:<math><br />
\overline{Y_k}= \int <Y_k|\eta> P(\eta) d\eta<br />
</math><br />
where <math> \eta </math> is the sample space for <math> Z </math>.<br />
<br />
CMC consists of providing a set of transport equations for the conditional moments which define the<br />
flame structure.<br />
<br />
Experimentally, it has been observed that temperature and chemical radicals are<br />
strong non-linear functions of mixture fraction. For a given species mass fraction we can decomposed<br />
it into a mean and a fluctuation:<br />
:<math><br />
Y_k= \overline{Y_k} + Y'_k<br />
</math><br />
The fluctuations <math> Y_k' </math> are usually very strong in time and space which makes the closure<br />
of <math> \overline{\omega_k} </math> very difficult.<br />
However, the alternative decomposition<br />
:<math><br />
Y_k= <Y_k|\eta> + y'_k<br />
</math><br />
where <math> y'_k </math> is the fluctuation around the conditional mean or the "conditional fluctuation".<br />
Experimentally, it is observed that <math> y'_k<< Y'_k </math>, which forms the basic assumption of the CMC method.<br />
Closures. Due to this property better closure methods can be used reducing the non-linearity <br />
of the mass fraction equations.<br />
<br />
The [[Derivation of the CMC equations]] produces the following CMC transport equation<br />
where <math> Q \equiv <Y_k|\eta> </math> for simplicity.<br />
:<math><br />
\frac{ \partial Q}{\partial t} + <u_j|\eta> \frac{\partial Q}{\partial x_j} =<br />
\frac{<\chi|\eta> }{2} \frac{\partial ^2 Q}{\partial \eta^2} + <br />
\frac{ < \dot \omega_k|\eta> }{ <\rho| \eta >}<br />
</math><br />
<br />
In this equation, high order terms in Reynolds number have been neglected.<br />
(See [[Derivation of the CMC equations]] for the complete series of terms).<br />
<br />
It is well known that closure of the unconditional source term<br />
<math> \overline {\dot \omega_k} </math> as a function of the<br />
mean temperature and species (<math> \overline{Y}, \overline{T}</math>) will give rise to large errors.<br />
However, in CMC the conditional averaged mass fractions contain more information and fluctuations around the mean are much smaller.<br />
The first order closure<br />
<math><br />
< \dot \omega_k|\eta> \approx \dot \omega_k \left( Q, <T|\eta> \right)<br />
</math><br />
is a good approximation in zones which are not close to extinction.<br />
<br />
===== Second order closure =====<br />
<br />
A second order closure can be obtained if conditional fluctuations are taken into account.<br />
For a chemical source term in the form <math> \dot \omega_k = k Y_A Y_B </math> with the rate constant in Arrhenius form<br />
<math> k=A_0 T^\beta exp [-Ta/T] </math><br />
the second order closure is (Klimenko and Bilger 1999)<br />
:<math><br />
< \dot \omega_k|\eta> \approx < \dot \omega_k|\eta >^{FO}<br />
<br />
\left[1+ \frac{< Y''_A Y''_B |\eta>}{Q_A Q_B}+ \left( \beta + T_a/Q_T \right)<br />
\left(<br />
\frac{< Y''_A T'' |\eta>}{Q_AQ_T} + \frac{< Y''_B T'' |\eta>}{Q_BQ_T}<br />
\right) + ...<br />
<br />
\right]<br />
<br />
</math><br />
<br />
where <math> < \dot \omega_k|\eta >^{FO} </math> is the first order CMC closure and<br />
<math> Q_T \equiv <T|\eta> </math>.<br />
When the temperature exponent <math> \beta </math> or <math> T_a/Q_T </math><br />
are large the error of taking the first order approximation increases.<br />
Improvement of small pollutant predictions can be obtained using the above reaction <br />
rate for selected species like CO and NO.<br />
<br />
<br />
===== Double conditioning =====<br />
<br />
Close to extinction and reignition. The conditional fluctuations can be very large<br />
and the primary closure of CMC of "small" fluctuations is not longer valid.<br />
A second variable <math> h </math> can be chosen to define a double conditioned mass fraction<br />
:<math><br />
Q(x,t;\eta,\psi) \equiv <Y_i(x,t) |Z=\eta,h=\psi ><br />
</math><br />
Due to the strong dependence on chemical reactions to temperature, <math> h </math><br />
is advised to be a temperature related variable (Kronenburg 2004).<br />
Scalar dissipation is not a good choice, due to its log-normal behaviour<br />
(smaller scales give highest dissipation). A must better choice is the sensible enthalpy<br />
or a progress variable.<br />
Double conditional variables have much smaller conditional fluctuations and allow<br />
the existence of points with the same chemical composition which can be fully burning<br />
(high temperature) or just mixing (low temperature). <br />
The range of applicability is greatly increased and allows non-premixed and premixed problems<br />
to be treated without ad-hoc distinctions.<br />
The main problem is the closure of the new terms involving cross scalar transport.<br />
<br />
The double conditional CMC equation is obtained in a similar manner than the conventional<br />
CMC equations<br />
<br />
===== LES modelling =====<br />
In a LES context a [[conditional filtering]] operator can be defined<br />
and <math> Q </math> therefore represents a conditionally filtered reactive scalar.<br />
<br />
=== Linear Eddy Model ===<br />
The Linear Eddy Model (LEM) was first developed by Kerstein(1988).<br />
It is an one-dimensional model for representing the flame structure in turbulent flows.<br />
<br />
In every computational cell a molecular, diffusion and chemical model is defined as<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Y_k \right) =<br />
\frac{\partial}{\partial \eta} \left( \rho D_k \frac{\partial Y_k}{\partial \eta }\right)+ \dot \omega_k<br />
</math><br />
<br />
where <math> \eta </math>is a spatial coordinate. The scalar distribution obtained can be seen as a<br />
one-dimensional reference field between Kolmogorov scale and grid scales.<br />
<br />
In a second stage a series of re-arranging stochastic event take place.<br />
These events represent the effects<br />
of a certain turbulent structure of size <math> l </math>, smaller than the grid size at a location <math> \eta_0 </math><br />
within the one-dimensional domain. This vortex distort the <math> \eta </math> field obtain by the one-dimensional equation,<br />
creating new maxima and minima in the interval <math> (\eta_0, \eta + \eta_0) </math>.<br />
The vortex size <math> l </math> is chosen randomly based on the inertial scale range while<br />
<math> \eta_0 </math> is obtained from a uniform distribution in <math> \eta </math>.<br />
The number of events is chosen to match the turbulent diffusivity of the flow.<br />
<br />
=== PDF transport models ===<br />
<br />
[[Probability Density Function]] (PDF) methods are not exclusive to combustion, although they are particularly attractive to them. They provided more information than moment closures and they are used to compute inhomegenous turbulent flows, see reviews in Dopazo (1993) and Pope (1994).<br />
<br />
<br />
PDF methods are based on the transport equation of the joint-PDF of the scalars.<br />
Denoting <math> P \equiv P(\underline{\psi}; x, t) </math> where<br />
<math> \underline{\psi} = ( \psi_1,\psi_2 ... \psi_N) </math> is the phase space for the reactive scalars<br />
<math> \underline{Y} = ( Y_1,Y_2 ... Y_N) </math>.<br />
The transport equation of the joint PDF is:<br />
<br />
:<math><br />
\frac{\partial <\rho | \underline{Y}=\underline{\psi}> P }{\partial t} + \frac{ <br />
\partial <\rho u_j | \underline{Y}=\underline{\psi}> P }{\partial x_j} =<br />
\sum^N_\alpha \frac{\partial}{\partial \psi_\alpha}\left[ \rho \dot{\omega}_\alpha P \right]<br />
- \sum^N_\alpha \sum^N_\beta \frac{\partial^2}{\partial \psi_\alpha \psi_\beta}<br />
\left[ <D \frac{\partial Y_\alpha}{\partial x_i} \frac{\partial Y_\beta}{\partial x_i} | \underline{Y}=\underline{\psi}> \right] P<br />
</math><br />
<br />
where the chemical source term is closed. Another term appeared on the right hand side which accounts for the effects of<br />
the molecular mixing on the PDF, is the so called "micro-mixing " term.<br />
Equal diffusivities are used for simplicity <math> D_k = D </math><br />
<br />
A more general approach is the velocity-composition joint-PDF<br />
with <math> P \equiv P(\underline{V},\underline{\psi}; x, t) </math>, where <br />
<math> \underline{V} </math> is the sample space of the velocity field <br />
<math> u,v,w </math>. This approach has the advantage of avoiding gradient-diffusion<br />
modelling. A similar equation to the above is obtained combining the momentum <br />
and scalar transport equation.<br />
<br />
<br />
<br />
<br />
The PDF transport equation can be solved in two ways: through a Lagrangian approach<br />
using stochastic methods or in a Eulerian ways using stochastic fields.<br />
<br />
==== Lagrangian ====<br />
<br />
The main idea of Lagrangian methods is that the flow can be represented by an <br />
ensemble of fluid particles. Central to this approach is the stochastic differential equations and in particular the [[Langevin equation]].<br />
<br />
==== Eulerian ====<br />
<br />
Instead of stochastic particles, smooth stochastic fields can be used<br />
to represent the [[probability density function ]] (PDF) of a scalar (or joint PDF) involved in transport (convection), diffusion<br />
and chemical reaction (Valino 1998). A similar formulation was proposed by Sabelnikov and Soulard 2005, which removes<br />
part of the a-priori assumption of "smoothness" of the stochastic fields.<br />
This approach is purely Eulerian and offers implementations advantages compared to<br />
Lagrangian or semi-Eulerian methods. <br />
Transport equations for scalars are often easy to programme and normal<br />
CFD algorithms can be used (see [[Discretisation of convective term]]).<br />
Although discretization errors are introduced by solving transport equations, <br />
this is partially compesated by the error introduced in Lagrangian approaches due to the numerical evaluation of means.<br />
<br />
A new set of <math> N_s </math> scalar variables<br />
(the stochastic field <math> \xi </math>) is used to represent the<br />
PDF<br />
:<math><br />
P (\underline{\psi}; x,t) = \frac{1}{N} \sum^{N_s}_{j=1} \prod^{N}_{k=1} \delta \left[\psi_k -\xi_k^j(x,t) \right]<br />
</math><br />
<br />
===Other combustion models===<br />
<br />
<br />
==== MMC ====<br />
<br />
The Multiple Mapping Conditioning (MMC) (Klimenko and Pope 2003) is an<br />
extension of the [[#Conditional Moment Closure (CMC)]] approach combined with<br />
[[probability density function]] methods. MMC<br />
looks for the minimum set of variables that describes the particular turbulent combustion<br />
system.<br />
<br />
==== Fractals ====<br />
Derived from the [[#Eddy Dissipation Concept (EDC)]].<br />
<br />
== References ==<br />
<br />
[1] {{reference-book|author=Bone, W.A., and Townend, D.T.A.|year=1927|title=Flame and Combustion in Gases|rest=Longmans, Green, and Co., Ltd}} <br />
*{{reference-paper|author=Dopazo, C.|year=1993|title=Recent development in PDF methods|rest=Turbulent Reacting Flows, ed. P. A. Libby and F. A. Williams }}<br />
*{{reference-book|author=Fox, R.O.|year=2003|title=Computational Models for Turbulent Reacting Flows|rest=ISBN 0-521-65049-6,Cambridge University Press}}<br />
*{{reference-paper|author=Kerstein, A. R.|year=1988|title=A linear eddy model of turbulent scalar transport and mixing|rest=Comb. Science and Technology, Vol. 60,pp. 391}}<br />
*{{reference-paper|author=Klimenko, A. Y., Bilger, R. W.|year=1999|title=Conditional moment closure for turbulent combustion|rest=Progress in Energy and Combustion Science, Vol. 25,pp. 595-687}}<br />
*{{reference-paper|author=Klimenko, A. Y., Pope, S. B.|year=2003|title=The modeling of turbulent reactive flows based on multiple mapping conditioning|rest=Physics of Fluids, Vol. 15, Num. 7, pp. 1907-1925}}<br />
*{{reference-paper|author=Kronenburg, A.,|year=2004|title=Double conditioning of reactive scalar transport equations in turbulent non-premixed flames|rest=Physics of Fluids, Vol. 16, Num. 7, pp. 2640-2648}}<br />
*{{reference-paper|author=Griffiths, J. F.|year=1994|title=Reduced Kinetic Models and Their Application to Practical Combustion Systems |rest=Prog. in Energy and Combustion Science,Vol. 21, pp. 25-107}}<br />
*{{reference-book|author=Peters, N.|year=2000|title=Turbulent Combustion|rest=ISBN 0-521-66082-3,Cambridge University Press}}<br />
*{{reference-book|author=Poinsot, T.,Veynante, D.|year=2001|title=Theoretical and Numerical Combustion|rest=ISBN 1-930217-05-6, R. T Edwards}}<br />
*{{reference-paper|author=Pope, S. B.|year=1994|title=Lagrangian PDF methods for turbulent flows|rest=Annu. Rev. Fluid Mech, Vol. 26, pp. 23-63}}<br />
*{{reference-paper|author=Sabel'nikov, V.,Soulard, O.|year=2005|title=Rapidly decorrelating velocity-field model as a tool for solving one-point Fokker-Planck equations for probability density functions of turbulent reactive scalars|rest=Physical Review E, Vol. 72,pp. 016301-1-22}}<br />
*{{reference-paper|author=Valino, L.,|year=1998|title=A field montecarlo formulation for calculating the probability density function of a single scalar in a turbulent flow|rest=Flow. Turb and Combustion, Vol. 60,pp. 157-172}}<br />
*{{reference-paper|author=Westbrook, Ch. K., Dryer,F. L.,|year=1984|title=Chemical Kinetic Modeling of Hydrocarbon Combustion|rest=Prog. in Energy and Combustion Science,Vol. 10, pp. 1-57}}<br />
<br />
== External links and sources ==</div>DavidFhttp://www.cfd-online.com/Wiki/Introduction_to_turbulence/Free_turbulent_shear_flowsIntroduction to turbulence/Free turbulent shear flows2009-01-06T15:29:57Z<p>DavidF: </p>
<hr />
<div>== Introduction ==<br />
<br />
Free shear flows are inhomogeneous flows with mean velocity gradients that develop in the absence of boundaries. Turbulent free shear flows are commonly found in natural and engineering environments. The jet of of air issuing from one's nostrils or mouth upon exhaling, the turbulent plume from a smoldering cigarette, and the buoyant jet issuing from an erupting volcano - all illustrate both the omnipresence of free turbulent shear flows and the range of scales of such flows in the natural environment. Examples of the multitude of engineering free shear flows are the wakes behind moving bodies and the exhausts from jet engines. Most combustion processes and many mixing processes involve turbulent free shear flows.<br />
<br />
Free shear flows in the real world are most often turbulent. Even if generated as laminar flows, they tend to become turbulent much more rapidly than the wall-bounded flows which we will discuss later. This is because the three-dimensional vorticity necessary for the transition to turbulence can develop much more rapidly in the absence of walls that inhibit the qrowth velocity components normal to them.<br />
<br />
The tendency of free shear flows to become and remain turbulent can be greatly modified by the presence of density gradients in the flow, especially if gravitational effects are also important. Why this is the case can easily be seen by examining the vorticity equation for such flows in the absence of viscosity,<br />
<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left[ \frac{ \partial \omega_{i} }{ \partial t } + \tilde{u_{j}} \frac{ \partial \tilde{\omega_{i} } }{ \partial x_{j}} \right] = \tilde{\omega_{j} } \frac{ \partial \tilde{u_{i}} }{ \partial x_{j} } + \epsilon_{ijk} \frac{ \partial \tilde{ \rho } }{ \partial x_{j} } \frac{ \partial \tilde{p} }{ \partial {x_{k} } }<br />
</math><br />
</td><td width="5%">(1)</td></tr></table><br />
<br />
The last term can act to either increase or decrease vorticity production but only in non-barotropic flows. (Recall that a barotropic flow is one in which the gradients of density and pressure are co-linear, because the density is a function of the pressure only). For example, in the vertically-oriented buoyant plume generated by exhausting a lighter fluid into heavier one, the principal density gradient is across the flow and thus perpendicular to the gravitational force which is the principal contributor to the pressure gradient. As a consequence the turbulent buoyant plume develops much more quickly than its uniform density counterpart, the jet. On the other hand, horizontal free shear flows in a stably stratified environment (fluid density decreases with height) can be quickly suppressed since the density and pressure gradients are in opposite directions.<br />
<br />
Free turbulent shear flows are distinctly different from the homogeneous shear flows. In a free turbulent shear flow, the vortical fluid is patially confined and is separated from the surrounding fluid by an interface, the turbulent-nonturbulent interface (also known as the ”Corrsin superlayer” after itself discoverer). The turbulent/non-turbulent interface has a thickness which is characterized by the Kolmogorov microscale, thus its characterization as an interface is appropriate. The actual shape of the interface is random and it is severely distorted by the energetic turbulent processes which take place below it, with the result that at any given location the turbulence can be highly ''intermittent''. This means that at a given location, it is sometimes turbulent, sometimes not.<br />
<br />
It should not be inferred from the above that the non-turbulent fluid outside the superlayer is quiescent. Quite the opposite is true since the motion of the fluid at the interface produces motions in the surrounding stream just as would the motions of a solid wall. Alternately, the flow outside the interface can be viewed as being “induced” by the vortical motions beneath it. It is easy to show that these induced motions are irrotational. Thus since these random motions of the outer flow have no vorticity, they can not be considered turbulent.<br />
<br />
Figure 7.1 shows records of the velocity versus time at a number of locations in the mixing layer of a round jet. When turbulent fluid passes the probes, the velocity signals are characterized by bursts of activity. The smooth undulations between the bursts are the irrotational fluctuations induced by the turbulent vorticity on the other side of the interface. Note that near the center of the mixing layer where the shear is a maximum, the flow is nearly always turbulent while it becomes increasingly intermittent as one proceeds away from the region of maximum<br />
production of turbulence energy. This increasing intermittency toward the<br />
outer edge is a characteristic of all free shear flows, and is an indication of the fact that the turbulent/non-turbulent interface is constantly changing its position.<br />
<br />
One of the most important features of free shear flows is that the amount of fluid which is turbulent is continuously increased by a process known as entrainment. No matter how little fluid is in the flow initially, the turbulent part of the flow will continue to capture new fluid by entrainment as it evolves. The photograph of an air jet in Figure 1.2 illustrates this phenomenon dramatically. The mass flow of the jet increases at each cross-section due to entrainment. Entrainment<br />
is not unique to turbulent flows, but is also an important haracteristic of laminar flow, even though the actual mechanism of entrainment is quite different.<br />
<br />
There are several consequences of entrainment. The first and most obvious is that free shear flows continue to spread throughout their lifetime. (That such is the case for the air jet of Figure 7.1 is obvious). A second consequence of entrainment is that the fluid in the flow is being continuously diluted by the addition of fluid from outside it. This is the basis of many mixing processes, and without such entrainment our lives would be quite different. A third consequence is that it will never be possible to neglect the turbulent transport terms in the dynamical equations, at least in the directions in which the flow is spreading. This is because the dilution process has ensured that the flow can never reach homogeneity since it will continue to entrain and spread through its lifetime (Recall that the transport terms were identically zero in homogeneous flows). Thus in dealing with free shear flows, all of the types of terms encountered in the turbulence kinetic energy equation of Chapter 4 must be dealt with — advection, dissipation, production, and turbulent transport.<br />
<br />
Turbulent free shear flows have another distinctive feature in that they very often give rise to easily recognizable large scale structures or eddies. Figure 1.2 also illustrates this phenomenon, and coherent patterns of a scale equal to the lateral extent of the flow are clearly visible. These large eddies appear to control the shape of the turbulent/non-turbulent interface and play an important role in the entrainment process. They may also be important to the processes by which<br />
the turbulence gains and distributes energy from the mean flow.<br />
<br />
A feature which free shear flows have in common with the homogeneous flows discussed in Chapter 6 is that their scales continue to grow as long as the flow remains turbulent. The dynamical equations and boundary conditions for many free shear flows can be shown to admit to similarity solutions in which the number of independent variables is reduced by one. According to the equilibrium similarity principle set forth in this chapter, such flows might be expected to asymptotically achieve such a state; and this, in fact, occurs. In the limit of infinite Reynolds number, some such flows can even be characterized by a single time and length scale, thus satisfying the conditions under which the simplest closure models might be expected to work. Care must be taken not to infer too much from the ability of a given closure model to predict such a flow, since any model which has the proper scale relations should work.<br />
<br />
Finally there is the important question of whether free shear flows become asymptotically independent of their initial conditions (or source conditions). The conventional wisdom until very recently has been that they do. If correct, this means that there is nothing that could be done to alter the far downstream flow. There is recent theoretical and experimental evidence, however, that this traditional view may well be wrong. If so, this opens up previously un-imagined possibilities for flow control at the source.<br />
<br />
In the remainder of this chapter, the averaged equations of motion will be simplified, and similarity solutions for several ideal shear flows will be derived and discussed in detail. The role of the large eddies will be discussed, and mechanisms for turbulent entrainment will be examined. The energy balance of several turbulent free shear flows will be studied in some detail. Finally, the effects of confinement and non-homogeneous external boundary conditions will be considered.<br />
<br />
==The averaged equations==<br />
===The shear layer equations===<br />
<br />
One of the most important ideas in the history of fluid mechanics is that of the ''boundary layer approximation''. These approximations to the Navier-Stokes equations were originally proposed by Prandtl in his famous theory for wall boundary layers. By introducing a different length scale for changes perpendicular to the wall than for changes in the flow direction, he was able to explain how viscous stresses could survive near the wall at high Reynolds number. These allowed the no-slip condition at a surface to be satisfied, and resolved the paradox of how<br />
there could be drag in the limit of zero viscosity.<br />
<br />
It may seem strange to be talking about Prandtl’s boundary layer idea in a section about free shear flows, but as we shall see below, the basic approximations can be applied to all “thin” (or slowly growing) shear flows with or without a surface. In this section, we shall show that free shear flows, for the most part, satisfy the conditions for these “boundary layer approximations”. Hence they belong to the general class of flows referred as “boundary layer flows”.<br />
<br />
One important difference will lie in whether momentum is being added to the flow at the source (as in jets) or taken out (by drag, as in wakes). A related influence is the presence (or absence) of a free stream in which our free shear flow is imbedded. We shall see that stationary free shear flows fall into two general classes, those with external flow and those without. One easy way to see why this makes a difference is to remember that these flows all spread by entraining mass from the surrounding fluid. You don’t have to think very hard to see that the entrained mass is carrying its own momentum into the shear flow. You should expect (and find) that even a small free stream velocity can make a significant difference, since the momentum carrried in is mixed in with that of the fluid particles which are already part of the turbulence. The longer the flow develops (or the farther downstream one looks), the more these simple differences can make a difference in how the flow spreads. In view of this, it should be no surprise that the presence or absence of an external stream plays a major role in petermining which mean convection terms which must be retained in the governing equations.<br />
<br />
We will consider only flows which are plane (or two-dimensional) in the mean (although similar considerations can be applied to flows that are axisymmetric in the mean). In effect, this is exactly the same as assuming the flow is homogeneous in the third direction. Also we shall restrict our attention to flows which are statistically stationary, so that time derivatives of averaged quantities can be neglected. And, of course, we have already agreed to confine our attention to Newtonian flows at constant density.<br />
<br />
It will be easier to abandon tensor notation for the moment, and use the symbols <math> x, U, u </math> for the streamwise direction, mean and fluctuating velicities respectively, and <math> y, V, v </math> for cross-stream. Given all this, the mean momentum equations reduce to:<br />
<br />
'''x-component:'''<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
U \frac{\partial U}{\partial x} + V \frac{\partial U}{\partial y} = - \frac{1}{\rho} \frac{\partial P}{\partial x} - \frac{\partial \left\langle u^{2}\right\rangle}{\partial x} - \frac{\partial \left\langle uv \right\rangle}{\partial y} + \nu \frac{\partial^{2} U}{\partial x^{2}} + \nu \frac{\partial^{2} U}{\partial y^{2}}<br />
</math><br />
</td><td width="5%">(2)</td></tr></table><br />
<br />
'''y-component:'''<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
U \frac{\partial V}{\partial x} + V \frac{\partial V}{\partial y} = - \frac{1}{\rho} \frac{\partial P}{\partial y} - \frac{\partial \left\langle uv\right\rangle}{\partial x} - \frac{\partial \left\langle v^{2} \right\rangle }{\partial y} + \nu \frac{\partial^{2} V}{\partial x^{2}} + \nu \frac{\partial^{2} V}{\partial y^{2}}<br />
</math><br />
</td><td width="5%">(3)</td></tr></table><br />
<br />
In addition, we have the two-dimensional mean continuity equation which redics to:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\frac{\partial U}{\partial x} + \frac{\partial V}{\partial y} =0<br />
</math><br />
</td><td width="5%">(4)</td></tr></table><br />
<br />
=== Order of magnitude estimates ===<br />
<br />
Now</div>DavidFhttp://www.cfd-online.com/Wiki/Turbulence_length_scaleTurbulence length scale2008-11-06T14:45:50Z<p>DavidF: I think this makes more sense - the quantity is the size of the eddies, which are energy-containing.</p>
<hr />
<div>The turbulence length scale, <math>l</math> , is a physical quantity describing the size of the large energy-containing eddies in a turbulent flow. <br />
<br />
The turbulent length scale is often used to estimate the turbulent properties on the inlets of a CFD simulation. Since the turbulent length scale is a quantity which is intuitively easy to relate to the physical size of the problem it is easy to guess a reasonable value of the turbulent length scale. The turbulent length scale should normally not be larger than the dimension of the problem, since that would mean that the turbulent eddies are larger than the problem size.<br />
<br />
In the [[Standard k-epsilon model|k-epsilon model]] the turbulent length scale can be computed as:<br />
<br />
:<math>l = C_\mu \, \frac{k^\frac{3}{2}}{\epsilon}</math><br />
<br />
<math>C_\mu</math> is a model constant which in the standard version of the k-epsilon model has a value of 0.09.<br />
<br />
==Estimating the turbulence length scale==<br />
<br />
It is common to set the turbulence length scale to a certain percentage of a typical dimension of the problem. For example, at the inlet to a turbine stage a typical turbulence length scale could be say 5% of the channel height. In grid-generated turbulence the turbulence length scale is often set to something close to the size of the grid bars. <br />
<br />
===Fully developed pipe flow===<br />
<br />
In pipe flows the turbulence length scale can be estimated from the [[hydraulic diameter]]. In fully developed pipe flow the turbulence length scale is 7% of the [[hydraulic diameter]] (in the case of a circular pipe the [[hydraulic diameter]] is the same as the diameter of the pipe). Hence:<br />
<br />
:<math>l = 0.07 \; d_h</math><br />
<br />
Where <math>d_h</math> is the [[hydraulic diameter]].<br />
<br />
===Wall-bounded inlet flows===<br />
<br />
When the inlet flow is bounded by walls with turbulent boundary layers, the turbulence length scale can be estimated (approximately) from the inlet boundary layer thickness. Set <math>l</math> to half the inlet boundary layer thickness.</div>DavidFhttp://www.cfd-online.com/Wiki/Newtonian_fluidNewtonian fluid2008-10-30T14:37:10Z<p>DavidF: </p>
<hr />
<div>Division of fluids in Newtonian and non-Newtonian fluids is done on the basis of relation between stress and strain.<br />
In Newtonian fluids the relation between stress and corresponding rate of strain is direct and linear.</div>DavidFhttp://www.cfd-online.com/Wiki/Main_PageMain Page2008-10-28T09:57:59Z<p>DavidF: Removed spam</p>
<hr />
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The Baldwin-Lomax model is a classical algebraic turbulence model which is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace and turbomachinery applications. The Baldwin-Lomax model is not suitable for cases with large separated regions and significant curvature/rotation effects... <br />
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''"I consider it the obligation of scientists and intellectuals to ensure that their ideas are made accessible and thus useful to society instead of being mere playthings for specialists." '' --Bjarne Stroustrup, from "Design and Evolution of C++"</div>DavidFhttp://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFDBest practice guidelines for turbomachinery CFD2008-10-09T08:49:28Z<p>DavidF: /* Transient or Stationary */</p>
<hr />
<div>This article contains a summary of the most important knowledge and experience a CFD engineer needs in order to perform CFD simulations of turbomachinery components. The guide is mainly aimed at axial turbomachinery. The goal is to give a CFD engineer, who has just started working with turbomachinery simulations, a head start and avoid some of the most difficult pit-falls. Experienced turbomachinery CFD engineers can also use the guidelines in order to learn what other experts consider best practice. The intended audience is expected to know basic CFD terminology and have a basic turbomachinery knowledge, but no detailed knowledge about CFD for turbomachinery is needed. <br />
<br />
== Deciding what type of simulation to do ==<br />
<br />
Before starting a new turbomachinery simulation it is wise to think carefully of what it is that should be predicted and what physical phenomena that affect the results. This chapter contains a brief overview of the various types of simulations and some hints of what can be predicted with them.<br />
<br />
=== 2D, Quasi-3D or 3D ===<br />
<br />
2D simulations are often used in the early design phase in order to obtain a typical 2D section of a blade. For cases with many long blades or vanes, like low-pressure turbines, a 2D simulation can also provide reasonable results. If the area of the flow-path changes significantly in the axial direction it might be necessary to instead make a quasi-3D simulation. A quasi-3D simulation is a 2D simulation in which extra source terms are used to account for the acceleration/deceleration caused by a changing channel height or growing end-wall boundary layers. Codes focused on turbomachinery applications often have the possibility to perform quasi-3D simulations, but most general purpose CFD codes can not do this type of simulations, or require user coding to implement the correct source terms in the equations. Please contact your software distributor if your code does not have the quasi-3d possibility and you require it. Many codes require special routines or hidden commands to enable this feature.<br />
<br />
Full 3D simulations are necessary if a true 3D geometry is needed to obtain correct secondary flows and/or shock locations. For low-aspect-ratio cases with only a few short blades, like for example structurally loaded turbine outlet guide vanes, the secondary flow development is important and a 3D simulation is often necessary in order to obtain reasonable results. For applications where the end-wall boundary layers grow very quickly and interact with a large part of the flow-field it is necessary to perform a full 3D simulation. This is often the case in compressors and fans, where the negative pressure gradients make the boundary layers grow much quicker than what they do in for example turbines. For cases where the shock location is very critical, like in transonic compressors, it is also often necessary to perform a 3D simulation in order to obtain reasonable shock locations.<br />
<br />
=== Inviscid or viscid ===<br />
<br />
For attached flows close to the design point and without any large separations it is often sufficient with an inviscid Euler simulation in order to obtain reasonable blade loadings and pressure distributions. Note that inviscid Euler simulations should only be used if the boundary layers are judged to not have a significant effect on the global flow-field. A viscid Navier-Stokes simulation is necessary in order to predict losses, secondary flows and separations. As soon as separations are of interest it is of course also necessary to do a viscid simulation. Note that with todays computers it is often not time and resources that make users run inviscid Euler simulations. Running viscid Navier-Stokes simulations is now so quick that it is not a time problem any more. Euler simulations are still interesting though, since with an inviscid Euler simulation you don't have to worry about wall resolutions, y+ values, turbulence modeling errors etc.<br />
<br />
=== Transient or Stationary ===<br />
<br />
Most turbomachinery simulations are performed as stationary simulations. Transient simulations are done when some kind of transient flow behaviour has a strong influence on the global flow field. Examples of transient simulations are detailed simulations of rotor-stator interaction effects, simulations of large unsteady separations etc. Sometimes when you perform a steady stationary simulation you can see tendencies of unsteady behaviour like for example periodic vortex shedding behind blunt trailing edges. This is often first seen as periodical variations of the residuals. If the unsteady tendencies are judged to not affect the overall simulation results it might be necessary to coarsen the mesh close to the vortex shedding or run a different turbulence model in order to make the simulation converge. Sometimes you are still forced to run a transient simulation and average the results if you don't obtain a converged steady solution.<br />
<br />
== Meshing ==<br />
<br />
In turbomachinery applications structured multi-block hexahedral meshes are most often used for flow-path simulations. In most solvers a structured grid requires less memory, provides superior accuracy and allows a better boundary-layer resolution than an unstructured grid. By having cells with a large aspect ratio around sharp leading and trailing edges a structured grid also provides a better resolution of these areas. Many companies have automatic meshing tools that automatically mesh blade sections with a structured mesh without much user intervention.<br />
<br />
Unstructured meshes are used for more complex and odd geometries where a structured mesh is difficult to create. Typical examples where unstructured meshes are often used are blade tip regions, areas involving leakage flows and secondary air systems, film cooling ducts etc.<br />
<br />
When meshing avoid to create large jumps in cell sizes. Typically the cell size should not change with more than a factor of 1.25 between neighbouring cells. For structured meshes also try to create fairly continous mesh lines and avoid discontinuities where the cell directions suddenly change. For multi-block structured meshes avoid placing the singular points where blocks meet in regions with strong flow gradients since most schemes have a lower accuracy in these singular points.<br />
<br />
=== Mesh size guidelines ===<br />
<br />
It is difficult to define, a priori, the mesh size. The required mesh size depends on the purpose of the simulation. <br />
<br />
If the main goal is to obtain static pressure forces a coarse mesh is often able to obtain a good solution, especially when an accurate resolution of the boundary layers is not required. For 2D inviscid simulations of one blade a mesh with say 3,000 cells is most often sufficient. For 3D inviscid blade simulations a mesh size of about 40,000 cells is usually sufficient. On inviscid Euler simulations the cells should be fairly equal in size and no boundary layer resolution should be present. Avoid having too skewed cells.<br />
<br />
For loss predictions and cases where boundary layer development and separation is important the mesh needs to have a boundary layer resolution. The boundary layer resolution can either be coarse and suitable for a [[Wall functions|wall function]] simulation or very fine and suitable for a [[Low-Re resolved boundary layers|low-Re]] simulation. For further information about selecting the near-wall turbulence model please see the [[#Turbulence modeling|turbulence modeling]] section. In 3D single-blade simulations a decent wall-function mesh typically has around 100,000 cells. This type of mesh size is suitable for quick design iterations where it is not essential to resolve all secondary flows and vortices. A good 3D wall-function mesh of a blade section intended to resolve secondary flows well should have at least 400,000 cells . A good low-Re mesh with resolved boundary layers typically has around 1,000,000 cells. <br />
<br />
In 2D blade simulations a good wall-function mesh has around 20,000 cells and a good low-Re mesh with resolved boundary layers has around 50,000 cells.<br />
<br />
Along the suction and pressure surfaces it is a good use about 100 cells in the streamwise direction. In the radial direction a good first approach is to use something like 30 cells for a wall-function mesh and 100 cells for a low-Re mesh.<br />
<br />
It is important to resolve leading and trailing edges well. Typically at least 10 cells, preferably 20 should be used around the leading and trailing edges. For very blunt and large leading edges, like those commonly found on HP turbine blades, 30 or more cells can be necessary.<br />
<br />
Cases which are difficult to converge with a steady simulation and which show tendencies of periodic vortex shedding from the trailing edge, can sometimes be "tamed" by using a coarse mesh around the trailing edge. This, of course, reduces the accuracy but can be a trick to obtain a converged solution if time and computer resources does not allow a transient simulation to be performed.<br />
<br />
=== Boundary layer mesh ===<br />
<br />
For design iteration type of simulations where a [[Wall functions|wall function]] approach is sufficient [[Dimensionless wall distance (y plus) | y+]] for the first cell should be somewhere between 20 and 200. The outer limit is dependent on the actual Re number of the simulations. For cases with fairly low Re numbers make sure to keep the maximum y+ as low as possible. For more accurate simulations with resolved boundary layers the mesh should have a y+ for the first cell which is below 1. Some new codes are now using a hybrid wall treatment that allows a smooth transition from a coarse wall-function mesh to a resolved low-Re mesh. Use some extra care when using this type of hybrid technique since it is still fairly new and unproven.<br />
<br />
Outside of the first cell at a wall a good rule of thumb is to use a growth ratio normal to the wall in the boundary layer of maximum 1.25. For a low-Re mesh this usually gives around 40 cells in the boundary layer whereas a wall-function mesh does not require more than 10 cells in the boundary layer.<br />
<br />
If you are uncertain of which wall distance to mesh with you can use a [http://www.cfd-online.com/Links/tools.html#yplus y+ estimation tool] to estitmate the distance needed to obtain the desired y+. These estimation tools are ''very handy'' if you have not done any previous similar simulations. <br />
<br />
As a rule of thumb a wall-function mesh typically requires areound 5 to 10 cells in the boundary layer whereas a resolved low-Re mesh requires about 40 cells in the boundary layer.<br />
<br />
== Boundary conditions ==<br />
<br />
Describe different types of boundary conditions and when they should be used:<br />
<br />
* Total pressure in, static pressure out<br />
* Absorbing boundary conditions<br />
* ...<br />
<br />
There are different types of boundary conditions you can use:<br />
*Mass flow inlet, static pressure outlet.<br />
To put this BC you must allocate inlet surface enough far away from the turbomachine impeller. The velocity distribution is not<br />
constant over all the surface. If you put this "lie" away from your problem it works well.<br />
<br />
<br />
=== Turbulence inlet conditions ===<br />
<br />
Prescribing realistic turbulence inlet conditions is important and usually difficult. For two-equation turbulence models two different turbulence values need to be specified on the inlet. The most common way of specifying the inlet turbulence variables is to give an inlet [[Turbulence intensity|turbulence level]] and [[Turbulence length scale|length-scale]]. For simulations where the incoming boundary layers are estimated to not have any significant effect on the secondary flows and global flow-field downstream it is usually sufficient to specify a constant incoming turbulence level and length-scale.<br />
<br />
The incoming [[Turbulence intensity|turbulence level]] is dependent on which component is analyzed. Best is of course to have measurments from a similar application in order to be able to prescribe a fairly realistic value. Without measurements it is necessary to make an educated guess. Simulations with very complex and turbulence-generating components upstream will have a very high incoming turbulence level. For example, a high-pressure turbine just downstream of a turbulence generating combustor might have incoming turbulence levels up to 20%. Fans and low-pressure compressors with not many components upstream might have as low incoming turbulence level as 1%. For components between these two extreme examples, like high-pressure compressors, low-pressure turbines etc. a turbulence level of around 5% might be realistic. If you do not have any measurements and are unsure about which incoming turbulence level to prescribe it is always a good idea to run a few different simulations with say one half and one double incoming turbulence level in order to estimate how important it is for the simulation results.<br />
<br />
The incoming [[Turbulence length scale|turbulence length-scale]] is often even more difficult to guess than the incoming turbulence level. Measurements might sometimes include the incoming turbulence level, but very seldom also include the incoming turbulent length-scale. The best way of guessing a realistic incoming length-scale is to use the geometrical properties of the upstream components. The incoming turbulence length-scale can be estimated as say the thickness of upstream blades or somewhere between 2% and 20% of the incoming channel heigth. Fortunately the incoming turbulence length-scale is usually not that important for the end results. However, some k-omega models can have an unrelisticly strong influence on the incoming turbulence length-scale. Hence, be extra carefull about the length-scale when using k-omega models. Note that the [[SST k-omega model]] does not have this unrealistic sensitivity to the inlet turbulence length-scale.<br />
<br />
Instead of specifying an incoming turbulence length-scale it is sometimes more convenient to specify an incoming [[Eddy viscosity ratio|eddy viscosity ratio]]. For low-turbulence cases with no well-specified turbulence generating components upstream it can be difficult to estimate a turbulence length-scale. Specifying an eddy viscosity ratio instead gives a more direct control over how large the effect of the incoming turbulence is. The eddy viscosity ratio describes how large the turbulence viscosity is related to the molecular viscosity. In turbomachinery the eddy viscosity ratio can vary from say 10 in low-turbulence components like fans up to say 1,000 or in rare cases even 10,000 in high-turbulence components like high-pressure turbines. In components between these extremes a typical eddy viscosity ration might be about 200. Even if specifying an inlet turbulence length scale it can still be good to compute what eddy viscosity ratio this corresponds to in order to estimate if it sounds reasonable.<br />
<br />
== Turbulence modeling ==<br />
<br />
Selecting a suitable turbulence model for turbomachinery simulations can be a challenging task. There is no single model which is suitable for all types of simulations. Which turbulence model CFD engineers use has as much to do with beliefs and traditions as with knowledge and facts. There are many different schools. However, below follows some advices that most CFD engineers in the turbomachinery field tend to agree upon.<br />
<br />
For attached flows close to the design point a simple algebraic model like the [[Baldwin-Lomax model]] can be used. Another common choice for design-iteration type of simulations is the one-equation model by [[Spalart-Allmaras model | Spalart-Allmaras]]. This model has become more popular in the last years due to the many inherent problems in more refined two-equation models. The big advantage with both the [[Baldwin-Lomax model]] and the [[Spalart-Allmaras model]] over more advanced models is that they are very robust to use and rarely produce completely unphysical results.<br />
<br />
In order to accurately predict more difficult cases, like separating flows, rotating flows, flows strongly affected by secondary flows etc. it is often necessary to use a more refined turbulence model. Common choices are two-equation models like the <math>k-\epsilon</math> model.<br />
<br />
Two-equation models are based on the [[Boussinesq eddy viscosity assumption]] and this often leads to an over-production of turbulent energy in regions with strong acceleration or deceleration, like in the leading edge region, regions around shocks and in the suction peak on the suction side of a blade. To reduce this problems it is common to use a special model variant using, for example, [[Durbin's realizability constraint]] or the [[Kato-Launder modification]]. Note that different two-equation models behave differently in these problematic stagnation and acceleration regions. Worst is probably the standard <math>k-\epsilon</math> model. <math>k-\omega</math> model are slightly better but still do not behave well. More modern variants like Menter's [[SST k-omega model]] also has problems, wheras the [[V2-f models|v2f model ]] by Durbin behaves better.<br />
<br />
=== Near-wall treatment ===<br />
<br />
For on-design simulations without any large separated regions it is often sufficient to use a [[wall-function model]] close to the wall, preferably with some form of non-equilibrium wall-function that is sensitised to streamwise pressure gradients.<br />
<br />
For off-design simulation, or simulations involving complex secondary flows and separations, it is often necessary to use a [[low-Re model]]. There exist many low-Re models that have been used with success in turbomachinery simulations. A robust and often good choice is to use a one-equation model, like for example the [[Wolfstein model]], in the inner parts of the boundary layer. There are also several Low-Re <math>k-\epsilon</math> models that work well. Just make sure they don't suffer from the problem with overproduction of turbulent energy in regions with strong acceleration or deceleration. In the last few years Menter's low-Re <math>SST k-\omega</math> model has gained increased popularity.<br />
<br />
=== Transition prediction ===<br />
<br />
Transition refers to the process when a laminar boundary layer becomes unstable and transitions to a turbulent boundary layer. There are two types of transition - natural transition, where inherent instabilities in the boundary layer cause the transition and by-pass transition, where convection and diffusion of turbulence from the free-stream into the boundary layer cause the transition. Most transitions in turbomachinery are by-pass transitions caused by free-stream turbulence and other external disturbances like wakes, vortices and surface defects.<br />
<br />
Simulating transition in a CFD code accurately is very difficult. Often a separate transition model needs to be solved in order to specify the transition location and length. Predicting natural transition in a pure CFD code is not possible. Predicting by-pass transition in a pure CFD code is almost impossible, although there are people who claim to be able to predict by-pass transition with low-Re two-equation models. However, this is usually on special test cases and with simulations that have been tuned for these special cases, see for example [Saville 2002]. In reality transition is a very complex and sensitive process where disturbances like incoming wakes and vortices from previous stages, surface roughness effects and small steps or gaps in the surfaces play a significant role. <br />
<br />
The turbomachinery codes that have transition prediction models often use old ad-hoc models like the Abu-Ghannam and Shaw model [Abu-Ghannam 1980] or the Mayle model [Mayle 1991]. These models can be quite reliable if they have been validated and tuned for a similar application. Do not trust your transition predictions without having some form of experimental validation. Menter has also recently developed a new form of transition model that might work fairly well, but it is still too new and untested.<br />
<br />
For some turbomachinery applications, like modern high-lift low-pressure turbines, transition is critical. For these applications a CFD code with a transition model that has been tuned for this type of applications should be used.<br />
<br />
== Numerical considerations ==<br />
<br />
Use at least a second order accurate scheme for the flow variables. Some codes require a first order scheme for the turbulent variables (<math>k</math> and <math>\epsilon</math>) in order to converge well. It might be sufficient with a first order scheme only on the turbulence variables, but a second order scheme is of course preferable.<br />
<br />
=== Convergence criteria ===<br />
<br />
To know when a solution is converged is not always that easy. You need some prior experience of your CFD code and your application to judge when a simulation is converged. For normal pure aero simulations without resolved walls, i.e. with wall functions or inviscid Euler simulations, convergence can most often be estimated just by looking at the residuals. Exacty what the residuals should be is not possible to say, it all depends on how your particular code computes and scales the residuals. Hence, make sure to read the manuals and plot the convergence of a few global parameters before you decide what the residuals should be for a solution to converege. Note also that many manuals for genereal purpose CFD codes list overly agressive convergence critera that often produce unconverged results. <br />
For simulations with resolved walls it is good to look at the convergence of some global properties, like total pressure losses from the inlet to the outlet. For heat transfer simulations it is even more tricky since the aerodynamic field can look almost converged although the thermal field is not converged at all. If doing heat transfer simulations make sure to plot the heat-transfer, run for some time, and plot it again to make sure that it doesn't change anymore. With very well resolved walls and heat transfer it can sometimes take 10 times longer for the thermal field to converge.<br />
<br />
=== Single or double precision ===<br />
<br />
With todays computers and cheap memory prices it often does not cost much extra to run in double precision. Before using single precision you should first investigate how your software and hardware works with double precision. If the extra time and memory needed for double precision is negligible you should of course always run in double precision. With double precision you never have to worry about round-off errors. Always using double precision is one way of avoiding one type of pit-falls in the complex world of CFD simulations. Use double precision when you have resolved boundary layers (Y+ around 1) and when you use advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions.<br />
<br />
== Multi-stage analysis ==<br />
<br />
Multi-stage analysis can be done in different ways:<br />
<br />
* Steady mixing-plane simulations<br />
* Frozen rotor simulations<br />
* Unsteady sliding-mesh stator-rotor simulations<br />
* Other '''advanced''' multi-stage methods<br />
<br />
=== Steady mixing-plane simulations ===<br />
<br />
Since the mixing-plane method was first introduced in 1979 [Denton & Singh 1979] is has become the industry standard type of rotor-stator simulations. A mixing-plane simulation is steady and only requires one rotor blade and one stator blade per stage. Between the rotating blade passage and the steady vane passage the flow properties are circumferentially averaged in a so-called mixing-plane interface. This will of course remove all transient rotor-stator interactions, but it still gives fairly representative results. In some commercial codes (CFX for example) mixing-plane interfaces are also called stage-interfaces.<br />
<br />
=== Frozen rotor simulations ===<br />
<br />
In a frozen rotor simulation the rotating and the stationary parts have a fixed relative position. A frame transformation is done to include the rotating effect on the rotating sections. This will give a steady flow and no transient effects are included. With a frozen-rotor simulation rotating wakes, secondary flows, leading edge pressure increases etc. will always stay in exactly the same positions. This makes a frozen rotor simulation very dependent on exactly how the rotors and the stators are positioned. Most often a mixing-plane simulation gives better results. Frozen rotor simulations are mainly performed to obtain a good starting flow-field before doing a transient sliding-mesh simulation.<br />
<br />
=== Unsteady sliding-mesh stator-rotor simulations ===<br />
<br />
This is the most complete type of stator-rotor simulation. In most engines the number of stators and rotors do not have a common denominator (to avoid instabilities caused by resonance between different rings). Hence, to make a full unsteady sliding-mesh computation it is necessary to have a mesh which includes the full wheel with all stators vanes and all rotor blades. This is often not possible, instead it is necessary to reduce the number of vanes and blades by finding a denominator that is '''almost''' common and then scale the geometry slightly circumferentially. Here is an example:<br />
<br />
*Real engine: 36 stator vanes, 41 rotor blades<br />
*Approximated engine: 41 stator vanes, 41 rotor blades, making it possible to simulate only 1 stator vane and 1 rotor blade<br />
*Scaling of stator: All stator vanes are scaled by 36/41 = 0.8780 circumferentially.<br />
<br />
=== Other '''advanced''' multi-stage methods ===<br />
<br />
Time-inclinded, Adamszyk stresses ...<br />
<br />
== Heat transfer predictions ==<br />
<br />
Besides listing the general heat transfer mechanisms involved <br />
(namely conduction, convection, radiation)<br />
heat transfer prediction in CFD may be seen as or split into two cases. <br />
<br />
'''Mesh consists of fluid domain(s)''': <br />
you want to know how heat inserted into the fluid domain (e.g. via the flowing medium) changes the temperature field along the flow path. Be it a steady or transient run, a fluid which enters with a given temperature will usually experience temperature variations, for instance, caused by convective boundary conditions, prescribed temperature profiles at flow obstacles etc. <br />
<br />
'''Mesh consists of fluid and solid domain(s)''': <br />
additionally to the above, you want info too wrt the (spatial, temporal or even spectral) temperature field distribution in surrounding, confining or immersed solids like channel walls or heat exchanger tubes. This is also called conjugate heattransfer CHT in the CFD context. CHT requires a good boundary layer resolution, usually the wall mesh needs to be rather refined, to obtain realistic heatflux results at the fluid/solid interface. Flow and heat transfer convergence require different time step settings, to properly capture changes in flow and heat quantities respectively. <br />
<br />
<br />
In either case, verify (strict necessity depends on CFD code used, CFX for instance checks and assists in regard) that model dimensions, boundary conditions and properties are in consistent units, hold appropriate values. Check temperature-dependence of properties and other numbers before the run.<br />
<br />
In heat transfer predictions (depending on the CFD code in use) <br />
besides the flow solver, you may have to activate the thermal solver too, <br />
as a job specification.<br />
<br />
== Acoustics and noise ==<br />
<br />
A whole separate research subject, difficult. <br />
<br />
Tone noise possible. Often run with linearzised solvers in the frequency domain.<br />
<br />
Jet noise possible. Often run with LES or DES simulations that either also resolve the sound waves or couples to a separate acoustic solver.<br />
<br />
Turbomachinery broadband noise not possible yet, or at least a great challenge.<br />
<br />
== Errors and uncertainties ==<br />
<br />
CFD is still a tool which requires that a user has a good understanding of uncertainties and errors that might spoil a CFD similation. There exists no error control in CFD and any CFD simulation must be interpreted by an expirenced user to have some credibility. Without some knowledge about possible errors and how they can be handled a CFD simulation can not be trusted. Errors can occur at different places:<br />
<br />
* Definition of the problem - What needs to be analyzed?<br />
* Selection of the solution strategy - What physical models and what numerical tools should be used?<br />
* Development of the computational model - How should the geometry and the numerical tools be set up?<br />
* Analysis and interpretation of the results - How should the model be analyzed and the results be interpreted?<br />
<br />
There exists many different definitions on errors. In this guide the errors are classified into four types of errors: problem definition errors, model errors, numerical errors and user and code errors. The chapters below describe these errors and give some guidelines on how to avoid them.<br />
<br />
=== Problem definition errors ===<br />
<br />
Problem definition errors are the most basic form of errors. In order to obtain usefull results a CFD simulation must of course analyze the correct problem, have suitable boundary contions and be based on a relevant geometry.<br />
<br />
==== Wrong type of simulation ====<br />
<br />
It is essential to have an overview of the physics involved and how the problem can best be analyzed. Running a 2D simulation in order to understand secondary flows or running a steady simulation in order to understand transient behaviour is of course no use. When assessing a CFD simulation the first thing to consider is what physical phenomena are important for the results and if the selected type of simulation is suitable to resolve this type of phenomena. For further information about selecting the most suitable type of simulation please see the previous chapter on [[#Deciding what type of simulation to do|deciding what type of simulation to perform]].<br />
<br />
==== Incorrect or uncertain boundary conditions ====<br />
<br />
A common source of errors is that incorrect boundary conditions are used. The boundary conditions must be specified in enough detail in order to resolve all the important physical features in the problem. For further information about specifying boundary conditions please see the previous chapter on [[#Boundary conditions|boundary conditions]].<br />
<br />
==== Geometrical errors ====<br />
<br />
It is almost always necessary to simplify the geometry in some form. When assessing a CFD simulation one should consider how the geometrical simplifications can affect the interesting physical phenomena. Typical geometrical errors are:<br />
<br />
*Simplifications<br />
:Small geometrical features like fillets, small steps or gaps etc. can often be disregarded. When disregarding this type of features one should consider if they might affect the important physics. For example, a very large fillet on the suction-side of a vane might affect corner separations near the end-walls. A large tip-leakage might affect the flow physics significantly in the upper part of a compressor.<br />
*Tolerances and manufacturing discrepancies<br />
:If the geometry has very large tolerances or is manufactured in a way which might produce a non-ideal shape or position it might be necessary to perform additional CFD simulations in order to cover the whole span of possible real geometries.<br />
*Surface conditions - roughness, welds, steps, gaps etc.<br />
:Often CFD simulations assume a perfectly smooth surface. A non-smooth surface which might have welds, steps or even gaps will of course produce different results. If the physical phenomena of interest might depend on the surface conditions these should of course be condidered. Typical phenomena that might be dependent on this type of errors are transition prediction, leakage flows etc.<br />
<br />
=== Model errors ===<br />
<br />
Errors related to the computational model.<br />
<br />
==== Wrong physical models ====<br />
<br />
Once the type of simulation has been selected the next step is to select what type of physical models the simulation should use. The following points should be considered:<br />
<br />
* Gas data (incompressible/compressible, perfect gas/real gas, ...)<br />
* Turbulence modeling (type of model, type of near-wall treatment, ...)<br />
* Other models (combustion, sprays, ...)<br />
<br />
When assessing model related errors it is important to know the features of the selected model and think carefully how these features and possible short comings might affect the predicted physical behaviour. Using the wrong turbulence model or combustion model can completely destroy the results of a CFD simulation.<br />
<br />
=== Numerical errors ===<br />
<br />
Errors related to the numerical solution of the developed model. Typical examples of numerical errors are discretization errors, convergence errors and round-off errors.<br />
<br />
==== Discretization errors ====<br />
<br />
Discretization errors can either be spatial errors in space or temporal errors in time.<br />
<br />
Spatial discretization errors are what people normally call discretization errors. These errors are due to the difference between the exact solution and the numerical representation of the solution in space. Describing exactly what discretization different codes use and what errors this might lead to is not possible here. Instead some general rules to avoid these errors can be summarized as:<br />
* Use at least a 2nd order accurate scheme, preferably a 3rd order accurate scheme. Some general purpose codes have a 1st order upwind scheme as default, this is a very diffusives schemes that often gives too smooth results.<br />
* For new applications always run a simulation with a finer mesh to see how grid independent your solution is.<br />
* Be aware of checker-board errors. Checker-board errors occur close to strong shocks and other large discontinuities and can be seen as a wavy pattern with a wavelength of two cells. Some schemes, especially those who behave like central differencing schemes, are more prone to checker-board errors. Upwind schemes are a bit better and schemes like TVD or chock-capturing schemes are even better.<br />
<br />
Temporal discretization errors mainly effect transient simulations. However, some codes use a time-marching method also for steady simulations and then a temporal discretization error might affect the final steady solution slightly. The discretization in time can be done with 1st or 2nd order schemes or a Runge-Kutta method, which is more accurate and saves memory. Some codes can adapt the time-step, but often it is necessary to prescribe a time-step in advance. Think of the time-step as your grid in time and make sure that the grid-resolution in time is fine enough to resolve the highest frequencies. To avoid problems with temporal discretization errors the following should be considered:<br />
* Use at least a 2nd order scheme in time.<br />
* Do a physical estimation of the typical frequencies in time of the phenomena that you are interested in and select a time-step which is fine enough to resolve these frequencies well. After the simulation also look at the frequencies captured and make sure that they are well resolved by the chosen time-step.<br />
* For new applications try a finer time-step to ensure that your solution in time is fairly grid independent in time.<br />
<br />
==== Convergence errors ====<br />
<br />
To judge when a CFD simulation is converged is not always that easy. Different codes and different applications behave very differently. For a pure aero-simulation on a fairly coarse grid convergence is easy to judge, but for more complex simulations involving resolved boundary layers, heat transfer, combustion etc. convergence can be very tricky. Aside from looking at residuals one should always also look at how global parameters like static pressure distributions, total pressure losses, skin friction, heat transfer etc. change in time.<br />
For more information about how to avoid convergence errors see the previous chapter about [[#Convergence criteria]].<br />
<br />
==== Round-off errors ====<br />
<br />
When using single precision care needs to be taken to avoid round-off errors. Inviscid Euler simulations and simulations using wall-function meshes can most often be performed in single precision. For well resolved boundary layers with Y plus close to 1 it is often necessary to use double precision. If using double precision for very fine mesh resolutions make sure that you also create the mesh in double precision and not just run the solver in double precision. Sometimes a single precision solver converges slower than a double precision solver due to numerical errors caused by round-off errors. When using advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions it is also often necessary to use double precision.<br />
<br />
=== User and code errors ===<br />
<br />
Errors related to bugs in the code used or mistakes made by the CFD engineer.<br />
<br />
=== What to trust and what not to trust ===<br />
<br />
CFD is generally quite good at predicting surface static pressure distributions. With care CFD can also be used to predict performance, total-pressure losses and blade turning. <br />
<br />
Predicting separation, stall and off-design performance can be a challenge and results with non-attached flows should be interpreted with care. <br />
<br />
Heat transfer is often very difficult to predict accurately and it is common to obtain heat-transfer coefficients that are 100% wrong or more. Validation data is critical in order to be able to trust heat transfer simulations.<br />
<br />
Transition is almost impossible to predict accurately in general. However, there exist models that have been tuned to predict transition and these tend to give acceptable results for cases close to the ones they were tuned for.<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Abu-Ghannam, B.J. and Shaw, R.|year=1980|title=Natural Transition of Boundary Layers - the Effects of Turbulence, Pressure Gradient, and Flow History|rest=Journal of Mech. Eng. Science, vol. 22, no. 5, pp. 213–228}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1985|title=Model Equation for Simulating Flows in Multistage Turbomachinery|rest=ASME Paper 85-GT-226, also NASA TM-86869}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1999|title=Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design|rest=ASME Paper 99-GT-80.}}<br />
<br />
* {{reference-paper|author=Baldwin, B. S. and Lomax, H.|year=1978|title=Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows|rest=AIAA Paper 78-257}}<br />
<br />
* {{reference-paper|author=Chen, J. P. and Barter, J. W.|year=1998|title=Comparison of Time-Accurate Calculations for the Unsteady Interaction in Turbomachinery Stage|rest=AIAA Paper 98-3292}}<br />
<br />
* {{reference-paper|author=Denton, J. and Singh, U.|year=1979|title=Time Marching Methods for Turbomachinery Flow Calculations|rest=VKI Lecture Series 1979-7, von Karman Institute}}<br />
<br />
* {{reference-paper|author=Erdos, J., Alzner, E. and McNally, W.|year=1977|title=Numerical Solution of Periodic Transonic Flow Through a Transonic Fan Stage|rest=AIAA Journal, vol. 15, no. 11, pp. 1559-1568}}<br />
<br />
* {{reference-paper|author=Gerolymos, G., Vinteler, D., Haugeard, R., Tsange, G. and Vallet, I.|year=1996|title=On the Computation of Unsteady Turbomachinery Flows – Part II – Rotor/Stator Interaction using Euler Equations|rest=AGARD Report CP-571}}<br />
<br />
* {{reference-paper|author=Giles, M. B.|year=1990|title=Stator/Rotor Interaction in a Transonic Turbine|rest=AIAA Journal of Propulsion and Power, vol. 6, no. 5}}<br />
<br />
* {{reference-paper|author=He, L, and Ning, W.|year=1998|title=Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines|rest=AIAA Journal, vol. 36, no. 11, pp. 2005-2012}}<br />
<br />
* {{reference-paper|author=Mayle, R. E.|year=1991|title=The Role of Laminar-Turbulent Transition in Gas Turbine Engines|rest=ASME Journal of Turbomachinery, vol. 113, pp. 509-537}}<br />
<br />
* {{reference-paper|author=Menter, F.R. |year=1994|title=Two-equation eddy-viscosity turbulence models for engineering applications|rest=AIAA Journal, vol. 32, pp. 269-289}}<br />
<br />
* {{reference-paper|author=Savill, M., Dick, E., Hanjalić, K. and Voke, P.|year=2002|title=Synthesis Report of the ERCOFTAC Transition Modeling – TRANSPRETURB Thematic Network Activities 1998-2002|rest=Ecoftac Bulletin 54, September 2002, pp. 5-16}}<br />
<br />
* {{reference-paper|author=Wang, X. and Chen J.|year=2004|title=A Post-Processor to Render Turbomachinery Flows Using Phase-Lag Simulations|rest=AIAA Paper 04-615}}<br />
<br />
== External links ==<br />
<br />
* [http://www.qnet-cfd.net/newsletter/8th/n8_40-46.pdf QNET-CFD Best Practise Advice for Turbomachinery Internal Flows]<br />
* [http://www.qnet-cfd.net/newsletter/7th/n7_05.pdf State of the art in Industrial CFD for Turbomachinery Flows, QNET-CFD Network Newsletter, Volume 2, No. 3 – December 2003]<br />
* [https://pronet.wsatkins.co.uk/marnet/guidelines/guide.html MARNET-CFD Best Practice Guidelines for Marine Applications of CFD]<br />
* [http://www.stanford.edu/group/ctr/ResBriefs/temp05/weide.pdf On Large Scale Turbomachinery Computations, by E. van der Weide et. al., Center for Turbulence Research, Annual Research Briefs 2005]</div>DavidFhttp://www.cfd-online.com/Wiki/Yap_correctionYap correction2008-10-02T12:41:21Z<p>DavidF: </p>
<hr />
<div>{{Turbulence modeling}}<br />
The Yap correction [[#References|[Yap. C. J. (1987)]]] consists of a modification of the epsilon equation in the form of an extra source term, <math>S_\epsilon</math>, added to the right hand side of the epsilon equation. The source term can be written as:<br />
<br />
:<math>\rho S_\epsilon \equiv 0.83 \, \rho \, \frac{\epsilon^2}{k} \, \left(\frac{k^{1.5}}{\epsilon \, l_e} - 1 \right) \, \left(\frac{k^{1.5}}{\epsilon \, l_e} \right)^2</math><br />
<br />
Where<br />
<br />
:<math>l_e \equiv c_\mu^{-0.75} \, \kappa \, y_n</math><br />
<br />
<math>y_n</math> is the normal distance to the nearest wall.<br />
<br />
This source term should be added to the epsilon equation in the following way:<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho \epsilon \right) +<br />
\frac{\partial}{\partial x_j} <br />
\left[<br />
\rho \epsilon u_j - \left( \mu + \frac{\mu_t}{\sigma_\epsilon} \right) <br />
\frac{\partial \epsilon}{\partial x_j}<br />
\right]<br />
=<br />
\left( C_{\epsilon_1} f_1 P - C_{\epsilon_2} f_2 \rho \epsilon \right)<br />
\frac{\epsilon}{k}<br />
+ \rho E<br />
+ \rho S_\epsilon<br />
</math><br />
<br />
Where the epsilon equation has been written in the same way as is in the CFD-Wiki article on [[low-Re k-epsilon models]].<br />
<br />
The Yap correction is active in nonequilibrium flows and tends to reduce the departure of the turbulence length scale from its local equilibrium level. It is an ad-hoc fix which seldom causes any problems and often improves the predictions.<br />
<br />
Yap showed strongly improved results with the k-epsilon model in separated flows when using this extra source term. The Yap correction has also been shown to improve results in a stagnation region. Launder [[#References|[Launder, B. E. (1993)]]] recommends that the Yap correction should always be used with the epsilon equation.<br />
<br />
==Implementation issues==<br />
<br />
The Yap source term contains the explicit distance to the nearest wall, <math>y_n</math>. In an unstructured 3D solver this distance is usually not available and it can be ambiguous how to compute it in more complex topologies. This makes the Yap correction most suitable for use in a structured code where the normal wall distance is readily available. There are several alternative formulations that can be used instead though ''(anyone have the references??)''.<br />
<br />
When implementing the Yap correction it is common to use it only if the source term is positive. Hence:<br />
<br />
:<math>\rho S_\epsilon^{implemented} = max(\rho S_\epsilon, 0)</math><br />
<br />
==References==<br />
<br />
{{reference-paper|author=Launder, B. E.|year=1993|title=Modelling Convective Heat Transfer in Complex Turbulent Flows|rest=Engineering Turbulence Modeling and Experiments 2, Proceedings of the Second International Symposium, Florence, Italy, 31 May - 2 June 1993, Edited by W. Rodi and F. Martelli, Elsevier, 1993, ISBN 0444898026}}<br />
<br />
{{reference-book|author=Yap, C. J.|year=1987|title=Turbulent Heat and Momentum Transfer in Recirculating and Impinging Flows|rest=PhD Thesis, Faculty of Technology, University of Manchester, United Kingdom}}</div>DavidFhttp://www.cfd-online.com/Wiki/Template:Turbulence_modelingTemplate:Turbulence modeling2008-10-02T10:26:08Z<p>DavidF: </p>
<hr />
<div>{| class="infobox bordered" style="vertical-align: top; text-align: left;"<br />
|+ style="background: #ccf; font-size: larger;" align="center" | '''[[Turbulence modeling]]'''<br />
|-<br />
| '''[[Turbulence]]'''<br />
|-<br />
| '''[[Algebraic turbulence models|Algebraic models]]'''<br />
|-<br />
|<br />
* [[Cebeci-Smith model]]<br />
* [[Baldwin-Lomax model]]<br />
* [[Johnson-King model]]<br />
|-<br />
| '''[[One equation turbulence models|One equation models]]'''<br />
|-<br />
|<br />
* [[Prandtl's one-equation model]]<br />
* [[Baldwin-Barth model]]<br />
* [[Spalart-Allmaras model]]<br />
|-<br />
| '''[[Two equation models]]'''<br />
|-<br />
|<br />
* [[k-epsilon models]]<br />
:[[Standard k-epsilon model]]<br />
:[[Realisable k-epsilon model]]<br />
:[[RNG k-epsilon model]]<br />
:[[Near-wall treatment for k-epsilon models|Near-wall treatment]]<br />
*[[k-omega models]]<br />
:[[Wilcox's k-omega model]]<br />
:[[Wilcox's modified k-omega model]]<br />
:[[SST k-omega model]]<br />
:[[Near-wall treatment for k-omega models|Near-wall treatment]]<br />
* [[Two equation turbulence model constraints and limiters|Constraints and limiters]]<br />
:[[Kato-Launder modification]]<br />
:[[Durbin's realizability constraint]]<br />
:[[Yap correction]]<br />
:[[Realisability and Schwarz' inequality]]<br />
|}</div>DavidFhttp://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFDBest practice guidelines for turbomachinery CFD2008-10-02T09:26:08Z<p>DavidF: /* Mesh size guidelines */</p>
<hr />
<div>This article contains a summary of the most important knowledge and experience a CFD engineer needs in order to perform CFD simulations of turbomachinery components. The guide is mainly aimed at axial turbomachinery. The goal is to give a CFD engineer, who has just started working with turbomachinery simulations, a head start and avoid some of the most difficult pit-falls. Experienced turbomachinery CFD engineers can also use the guidelines in order to learn what other experts consider best practice. The intended audience is expected to know basic CFD terminology and have a basic turbomachinery knowledge, but no detailed knowledge about CFD for turbomachinery is needed. <br />
<br />
== Deciding what type of simulation to do ==<br />
<br />
Before starting a new turbomachinery simulation it is wise to think carefully of what it is that should be predicted and what physical phenomena that affect the results. This chapter contains a brief overview of the various types of simulations and some hints of what can be predicted with them.<br />
<br />
=== 2D, Quasi-3D or 3D ===<br />
<br />
2D simulations are often used in the early design phase in order to obtain a typical 2D section of a blade. For cases with many long blades or vanes, like low-pressure turbines, a 2D simulation can also provide reasonable results. If the area of the flow-path changes significantly in the axial direction it might be necessary to instead make a quasi-3D simulation. A quasi-3D simulation is a 2D simulation in which extra source terms are used to account for the acceleration/deceleration caused by a changing channel height or growing end-wall boundary layers. Codes focused on turbomachinery applications often have the possibility to perform quasi-3D simulations, but most general purpose CFD codes can not do this type of simulations, or require user coding to implement the correct source terms in the equations. Please contact your software distributor if your code does not have the quasi-3d possibility and you require it. Many codes require special routines or hidden commands to enable this feature.<br />
<br />
Full 3D simulations are necessary if a true 3D geometry is needed to obtain correct secondary flows and/or shock locations. For low-aspect-ratio cases with only a few short blades, like for example structurally loaded turbine outlet guide vanes, the secondary flow development is important and a 3D simulation is often necessary in order to obtain reasonable results. For applications where the end-wall boundary layers grow very quickly and interact with a large part of the flow-field it is necessary to perform a full 3D simulation. This is often the case in compressors and fans, where the negative pressure gradients make the boundary layers grow much quicker than what they do in for example turbines. For cases where the shock location is very critical, like in transonic compressors, it is also often necessary to perform a 3D simulation in order to obtain reasonable shock locations.<br />
<br />
=== Inviscid or viscid ===<br />
<br />
For attached flows close to the design point and without any large separations it is often sufficient with an inviscid Euler simulation in order to obtain reasonable blade loadings and pressure distributions. Note that inviscid Euler simulations should only be used if the boundary layers are judged to not have a significant effect on the global flow-field. A viscid Navier-Stokes simulation is necessary in order to predict losses, secondary flows and separations. As soon as separations are of interest it is of course also necessary to do a viscid simulation. Note that with todays computers it is often not time and resources that make users run inviscid Euler simulations. Running viscid Navier-Stokes simulations is now so quick that it is not a time problem any more. Euler simulations are still interesting though, since with an inviscid Euler simulation you don't have to worry about wall resolutions, y+ values, turbulence modeling errors etc.<br />
<br />
=== Transient or Stationary ===<br />
<br />
Most turbomachinery simlation are performed as stationary simulations. Transient simulations are done when some kind of transient flow behaviour has a strong influence on the global flow field. Examples of transient simulations are detailed simulations of rotor-stator interaction effects, simulations of large unsteady separations etc. Sometimes when you perform a steady stationary simulation you can see tendencies of unsteady behaviour like for example periodic vortex shedding behind blunt trailing edges. This is often first seen as periodical variations of the residuals. If the unsteady tendencies are judged to not affect the overall simulation results if might be necessary to coarsen the mesh close to the vortex shedding or run a different turbulence model in order to make the simulation converge. Sometimes you are still forced to run a transient simulation and average the results if you don't obtain a converged steady solution.<br />
<br />
== Meshing ==<br />
<br />
In turbomachinery applications structured multi-block hexahedral meshes are most often used for flow-path simulations. In most solvers a structured grid requires less memory, provides superior accuracy and allows a better boundary-layer resolution than an unstructured grid. By having cells with a large aspect ratio around sharp leading and trailing edges a structured grid also provides a better resolution of these areas. Many companies have automatic meshing tools that automatically mesh blade sections with a structured mesh without much user intervention.<br />
<br />
Unstructured meshes are used for more complex and odd geometries where a structured mesh is difficult to create. Typical examples where unstructured meshes are often used are blade tip regions, areas involving leakage flows and secondary air systems, film cooling ducts etc.<br />
<br />
When meshing avoid to create large jumps in cell sizes. Typically the cell size should not change with more than a factor of 1.25 between neighbouring cells. For structured meshes also try to create fairly continous mesh lines and avoid discontinuities where the cell directions suddenly change. For multi-block structured meshes avoid placing the singular points where blocks meet in regions with strong flow gradients since most schemes have a lower accuracy in these singular points.<br />
<br />
=== Mesh size guidelines ===<br />
<br />
It is difficult to define, a priori, the mesh size. The required mesh size depends on the purpose of the simulation. <br />
<br />
If the main goal is to obtain static pressure forces a coarse mesh is often able to obtain a good solution, especially when an accurate resolution of the boundary layers is not required. For 2D inviscid simulations of one blade a mesh with say 3,000 cells is most often sufficient. For 3D inviscid blade simulations a mesh size of about 40,000 cells is usually sufficient. On inviscid Euler simulations the cells should be fairly equal in size and no boundary layer resolution should be present. Avoid having too skewed cells.<br />
<br />
For loss predictions and cases where boundary layer development and separation is important the mesh needs to have a boundary layer resolution. The boundary layer resolution can either be coarse and suitable for a [[Wall functions|wall function]] simulation or very fine and suitable for a [[Low-Re resolved boundary layers|low-Re]] simulation. For further information about selecting the near-wall turbulence model please see the [[#Turbulence modeling|turbulence modeling]] section. In 3D single-blade simulations a decent wall-function mesh typically has around 100,000 cells. This type of mesh size is suitable for quick design iterations where it is not essential to resolve all secondary flows and vortices. A good 3D wall-function mesh of a blade section intended to resolve secondary flows well should have at least 400,000 cells . A good low-Re mesh with resolved boundary layers typically has around 1,000,000 cells. <br />
<br />
In 2D blade simulations a good wall-function mesh has around 20,000 cells and a good low-Re mesh with resolved boundary layers has around 50,000 cells.<br />
<br />
Along the suction and pressure surfaces it is a good use about 100 cells in the streamwise direction. In the radial direction a good first approach is to use something like 30 cells for a wall-function mesh and 100 cells for a low-Re mesh.<br />
<br />
It is important to resolve leading and trailing edges well. Typically at least 10 cells, preferably 20 should be used around the leading and trailing edges. For very blunt and large leading edges, like those commonly found on HP turbine blades, 30 or more cells can be necessary.<br />
<br />
Cases which are difficult to converge with a steady simulation and which show tendencies of periodic vortex shedding from the trailing edge, can sometimes be "tamed" by using a coarse mesh around the trailing edge. This, of course, reduces the accuracy but can be a trick to obtain a converged solution if time and computer resources does not allow a transient simulation to be performed.<br />
<br />
=== Boundary layer mesh ===<br />
<br />
For design iteration type of simulations where a [[Wall functions|wall function]] approach is sufficient [[Dimensionless wall distance (y plus) | y+]] for the first cell should be somewhere between 20 and 200. The outer limit is dependent on the actual Re number of the simulations. For cases with fairly low Re numbers make sure to keep the maximum y+ as low as possible. For more accurate simulations with resolved boundary layers the mesh should have a y+ for the first cell which is below 1. Some new codes are now using a hybrid wall treatment that allows a smooth transition from a coarse wall-function mesh to a resolved low-Re mesh. Use some extra care when using this type of hybrid technique since it is still fairly new and unproven.<br />
<br />
Outside of the first cell at a wall a good rule of thumb is to use a growth ratio normal to the wall in the boundary layer of maximum 1.25. For a low-Re mesh this usually gives around 40 cells in the boundary layer whereas a wall-function mesh does not require more than 10 cells in the boundary layer.<br />
<br />
If you are uncertain of which wall distance to mesh with you can use a [http://www.cfd-online.com/Links/tools.html#yplus y+ estimation tool] to estitmate the distance needed to obtain the desired y+. These estimation tools are ''very handy'' if you have not done any previous similar simulations. <br />
<br />
As a rule of thumb a wall-function mesh typically requires areound 5 to 10 cells in the boundary layer whereas a resolved low-Re mesh requires about 40 cells in the boundary layer.<br />
<br />
== Boundary conditions ==<br />
<br />
Describe different types of boundary conditions and when they should be used:<br />
<br />
* Total pressure in, static pressure out<br />
* Absorbing boundary conditions<br />
* ...<br />
<br />
There are different types of boundary conditions you can use:<br />
*Mass flow inlet, static pressure outlet.<br />
To put this BC you must allocate inlet surface enough far away from the turbomachine impeller. The velocity distribution is not<br />
constant over all the surface. If you put this "lie" away from your problem it works well.<br />
<br />
<br />
=== Turbulence inlet conditions ===<br />
<br />
Prescribing realistic turbulence inlet conditions is important and usually difficult. For two-equation turbulence models two different turbulence values need to be specified on the inlet. The most common way of specifying the inlet turbulence variables is to give an inlet [[Turbulence intensity|turbulence level]] and [[Turbulence length scale|length-scale]]. For simulations where the incoming boundary layers are estimated to not have any significant effect on the secondary flows and global flow-field downstream it is usually sufficient to specify a constant incoming turbulence level and length-scale.<br />
<br />
The incoming [[Turbulence intensity|turbulence level]] is dependent on which component is analyzed. Best is of course to have measurments from a similar application in order to be able to prescribe a fairly realistic value. Without measurements it is necessary to make an educated guess. Simulations with very complex and turbulence-generating components upstream will have a very high incoming turbulence level. For example, a high-pressure turbine just downstream of a turbulence generating combustor might have incoming turbulence levels up to 20%. Fans and low-pressure compressors with not many components upstream might have as low incoming turbulence level as 1%. For components between these two extreme examples, like high-pressure compressors, low-pressure turbines etc. a turbulence level of around 5% might be realistic. If you do not have any measurements and are unsure about which incoming turbulence level to prescribe it is always a good idea to run a few different simulations with say one half and one double incoming turbulence level in order to estimate how important it is for the simulation results.<br />
<br />
The incoming [[Turbulence length scale|turbulence length-scale]] is often even more difficult to guess than the incoming turbulence level. Measurements might sometimes include the incoming turbulence level, but very seldom also include the incoming turbulent length-scale. The best way of guessing a realistic incoming length-scale is to use the geometrical properties of the upstream components. The incoming turbulence length-scale can be estimated as say the thickness of upstream blades or somewhere between 2% and 20% of the incoming channel heigth. Fortunately the incoming turbulence length-scale is usually not that important for the end results. However, some k-omega models can have an unrelisticly strong influence on the incoming turbulence length-scale. Hence, be extra carefull about the length-scale when using k-omega models. Note that the [[SST k-omega model]] does not have this unrealistic sensitivity to the inlet turbulence length-scale.<br />
<br />
Instead of specifying an incoming turbulence length-scale it is sometimes more convenient to specify an incoming [[Eddy viscosity ratio|eddy viscosity ratio]]. For low-turbulence cases with no well-specified turbulence generating components upstream it can be difficult to estimate a turbulence length-scale. Specifying an eddy viscosity ratio instead gives a more direct control over how large the effect of the incoming turbulence is. The eddy viscosity ratio describes how large the turbulence viscosity is related to the molecular viscosity. In turbomachinery the eddy viscosity ratio can vary from say 10 in low-turbulence components like fans up to say 1,000 or in rare cases even 10,000 in high-turbulence components like high-pressure turbines. In components between these extremes a typical eddy viscosity ration might be about 200. Even if specifying an inlet turbulence length scale it can still be good to compute what eddy viscosity ratio this corresponds to in order to estimate if it sounds reasonable.<br />
<br />
== Turbulence modeling ==<br />
<br />
Selecting a suitable turbulence model for turbomachinery simulations can be a challenging task. There is no single model which is suitable for all types of simulations. Which turbulence model CFD engineers use has as much to do with beliefs and traditions as with knowledge and facts. There are many different schools. However, below follows some advices that most CFD engineers in the turbomachinery field tend to agree upon.<br />
<br />
For attached flows close to the design point a simple algebraic model like the [[Baldwin-Lomax model]] can be used. Another common choice for design-iteration type of simulations is the one-equation model by [[Spalart-Allmaras model | Spalart-Allmaras]]. This model has become more popular in the last years due to the many inherent problems in more refined two-equation models. The big advantage with both the [[Baldwin-Lomax model]] and the [[Spalart-Allmaras model]] over more advanced models is that they are very robust to use and rarely produce completely unphysical results.<br />
<br />
In order to accurately predict more difficult cases, like separating flows, rotating flows, flows strongly affected by secondary flows etc. it is often necessary to use a more refined turbulence model. Common choices are two-equation models like the <math>k-\epsilon</math> model.<br />
<br />
Two-equation models are based on the [[Boussinesq eddy viscosity assumption]] and this often leads to an over-production of turbulent energy in regions with strong acceleration or deceleration, like in the leading edge region, regions around shocks and in the suction peak on the suction side of a blade. To reduce this problems it is common to use a special model variant using, for example, [[Durbin's realizability constraint]] or the [[Kato-Launder modification]]. Note that different two-equation models behave differently in these problematic stagnation and acceleration regions. Worst is probably the standard <math>k-\epsilon</math> model. <math>k-\omega</math> model are slightly better but still do not behave well. More modern variants like Menter's [[SST k-omega model]] also has problems, wheras the [[V2-f models|v2f model ]] by Durbin behaves better.<br />
<br />
=== Near-wall treatment ===<br />
<br />
For on-design simulations without any large separated regions it is often sufficient to use a [[wall-function model]] close to the wall, preferably with some form of non-equilibrium wall-function that is sensitised to streamwise pressure gradients.<br />
<br />
For off-design simulation, or simulations involving complex secondary flows and separations, it is often necessary to use a [[low-Re model]]. There exist many low-Re models that have been used with success in turbomachinery simulations. A robust and often good choice is to use a one-equation model, like for example the [[Wolfstein model]], in the inner parts of the boundary layer. There are also several Low-Re <math>k-\epsilon</math> models that work well. Just make sure they don't suffer from the problem with overproduction of turbulent energy in regions with strong acceleration or deceleration. In the last few years Menter's low-Re <math>SST k-\omega</math> model has gained increased popularity.<br />
<br />
=== Transition prediction ===<br />
<br />
Transition refers to the process when a laminar boundary layer becomes unstable and transitions to a turbulent boundary layer. There are two types of transition - natural transition, where inherent instabilities in the boundary layer cause the transition and by-pass transition, where convection and diffusion of turbulence from the free-stream into the boundary layer cause the transition. Most transitions in turbomachinery are by-pass transitions caused by free-stream turbulence and other external disturbances like wakes, vortices and surface defects.<br />
<br />
Simulating transition in a CFD code accurately is very difficult. Often a separate transition model needs to be solved in order to specify the transition location and length. Predicting natural transition in a pure CFD code is not possible. Predicting by-pass transition in a pure CFD code is almost impossible, although there are people who claim to be able to predict by-pass transition with low-Re two-equation models. However, this is usually on special test cases and with simulations that have been tuned for these special cases, see for example [Saville 2002]. In reality transition is a very complex and sensitive process where disturbances like incoming wakes and vortices from previous stages, surface roughness effects and small steps or gaps in the surfaces play a significant role. <br />
<br />
The turbomachinery codes that have transition prediction models often use old ad-hoc models like the Abu-Ghannam and Shaw model [Abu-Ghannam 1980] or the Mayle model [Mayle 1991]. These models can be quite reliable if they have been validated and tuned for a similar application. Do not trust your transition predictions without having some form of experimental validation. Menter has also recently developed a new form of transition model that might work fairly well, but it is still too new and untested.<br />
<br />
For some turbomachinery applications, like modern high-lift low-pressure turbines, transition is critical. For these applications a CFD code with a transition model that has been tuned for this type of applications should be used.<br />
<br />
== Numerical considerations ==<br />
<br />
Use at least a second order accurate scheme for the flow variables. Some codes require a first order scheme for the turbulent variables (<math>k</math> and <math>\epsilon</math>) in order to converge well. It might be sufficient with a first order scheme only on the turbulence variables, but a second order scheme is of course preferable.<br />
<br />
=== Convergence criteria ===<br />
<br />
To know when a solution is converged is not always that easy. You need some prior experience of your CFD code and your application to judge when a simulation is converged. For normal pure aero simulations without resolved walls, i.e. with wall functions or inviscid Euler simulations, convergence can most often be estimated just by looking at the residuals. Exacty what the residuals should be is not possible to say, it all depends on how your particular code computes and scales the residuals. Hence, make sure to read the manuals and plot the convergence of a few global parameters before you decide what the residuals should be for a solution to converege. Note also that many manuals for genereal purpose CFD codes list overly agressive convergence critera that often produce unconverged results. <br />
For simulations with resolved walls it is good to look at the convergence of some global properties, like total pressure losses from the inlet to the outlet. For heat transfer simulations it is even more tricky since the aerodynamic field can look almost converged although the thermal field is not converged at all. If doing heat transfer simulations make sure to plot the heat-transfer, run for some time, and plot it again to make sure that it doesn't change anymore. With very well resolved walls and heat transfer it can sometimes take 10 times longer for the thermal field to converge.<br />
<br />
=== Single or double precision ===<br />
<br />
With todays computers and cheap memory prices it often does not cost much extra to run in double precision. Before using single precision you should first investigate how your software and hardware works with double precision. If the extra time and memory needed for double precision is negligible you should of course always run in double precision. With double precision you never have to worry about round-off errors. Always using double precision is one way of avoiding one type of pit-falls in the complex world of CFD simulations. Use double precision when you have resolved boundary layers (Y+ around 1) and when you use advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions.<br />
<br />
== Multi-stage analysis ==<br />
<br />
Multi-stage analysis can be done in different ways:<br />
<br />
* Steady mixing-plane simulations<br />
* Frozen rotor simulations<br />
* Unsteady sliding-mesh stator-rotor simulations<br />
* Other '''advanced''' multi-stage methods<br />
<br />
=== Steady mixing-plane simulations ===<br />
<br />
Since the mixing-plane method was first introduced in 1979 [Denton & Singh 1979] is has become the industry standard type of rotor-stator simulations. A mixing-plane simulation is steady and only requires one rotor blade and one stator blade per stage. Between the rotating blade passage and the steady vane passage the flow properties are circumferentially averaged in a so-called mixing-plane interface. This will of course remove all transient rotor-stator interactions, but it still gives fairly representative results. In some commercial codes (CFX for example) mixing-plane interfaces are also called stage-interfaces.<br />
<br />
=== Frozen rotor simulations ===<br />
<br />
In a frozen rotor simulation the rotating and the stationary parts have a fixed relative position. A frame transformation is done to include the rotating effect on the rotating sections. This will give a steady flow and no transient effects are included. With a frozen-rotor simulation rotating wakes, secondary flows, leading edge pressure increases etc. will always stay in exactly the same positions. This makes a frozen rotor simulation very dependent on exactly how the rotors and the stators are positioned. Most often a mixing-plane simulation gives better results. Frozen rotor simulations are mainly performed to obtain a good starting flow-field before doing a transient sliding-mesh simulation.<br />
<br />
=== Unsteady sliding-mesh stator-rotor simulations ===<br />
<br />
This is the most complete type of stator-rotor simulation. In most engines the number of stators and rotors do not have a common denominator (to avoid instabilities caused by resonance between different rings). Hence, to make a full unsteady sliding-mesh computation it is necessary to have a mesh which includes the full wheel with all stators vanes and all rotor blades. This is often not possible, instead it is necessary to reduce the number of vanes and blades by finding a denominator that is '''almost''' common and then scale the geometry slightly circumferentially. Here is an example:<br />
<br />
*Real engine: 36 stator vanes, 41 rotor blades<br />
*Approximated engine: 41 stator vanes, 41 rotor blades, making it possible to simulate only 1 stator vane and 1 rotor blade<br />
*Scaling of stator: All stator vanes are scaled by 36/41 = 0.8780 circumferentially.<br />
<br />
=== Other '''advanced''' multi-stage methods ===<br />
<br />
Time-inclinded, Adamszyk stresses ...<br />
<br />
== Heat transfer predictions ==<br />
<br />
Besides listing the general heat transfer mechanisms involved <br />
(namely conduction, convection, radiation)<br />
heat transfer prediction in CFD may be seen as or split into two cases. <br />
<br />
'''Mesh consists of fluid domain(s)''': <br />
you want to know how heat inserted into the fluid domain (e.g. via the flowing medium) changes the temperature field along the flow path. Be it a steady or transient run, a fluid which enters with a given temperature will usually experience temperature variations, for instance, caused by convective boundary conditions, prescribed temperature profiles at flow obstacles etc. <br />
<br />
'''Mesh consists of fluid and solid domain(s)''': <br />
additionally to the above, you want info too wrt the (spatial, temporal or even spectral) temperature field distribution in surrounding, confining or immersed solids like channel walls or heat exchanger tubes. This is also called conjugate heattransfer CHT in the CFD context. CHT requires a good boundary layer resolution, usually the wall mesh needs to be rather refined, to obtain realistic heatflux results at the fluid/solid interface. Flow and heat transfer convergence require different time step settings, to properly capture changes in flow and heat quantities respectively. <br />
<br />
<br />
In either case, verify (strict necessity depends on CFD code used, CFX for instance checks and assists in regard) that model dimensions, boundary conditions and properties are in consistent units, hold appropriate values. Check temperature-dependence of properties and other numbers before the run.<br />
<br />
In heat transfer predictions (depending on the CFD code in use) <br />
besides the flow solver, you may have to activate the thermal solver too, <br />
as a job specification.<br />
<br />
== Acoustics and noise ==<br />
<br />
A whole separate research subject, difficult. <br />
<br />
Tone noise possible. Often run with linearzised solvers in the frequency domain.<br />
<br />
Jet noise possible. Often run with LES or DES simulations that either also resolve the sound waves or couples to a separate acoustic solver.<br />
<br />
Turbomachinery broadband noise not possible yet, or at least a great challenge.<br />
<br />
== Errors and uncertainties ==<br />
<br />
CFD is still a tool which requires that a user has a good understanding of uncertainties and errors that might spoil a CFD similation. There exists no error control in CFD and any CFD simulation must be interpreted by an expirenced user to have some credibility. Without some knowledge about possible errors and how they can be handled a CFD simulation can not be trusted. Errors can occur at different places:<br />
<br />
* Definition of the problem - What needs to be analyzed?<br />
* Selection of the solution strategy - What physical models and what numerical tools should be used?<br />
* Development of the computational model - How should the geometry and the numerical tools be set up?<br />
* Analysis and interpretation of the results - How should the model be analyzed and the results be interpreted?<br />
<br />
There exists many different definitions on errors. In this guide the errors are classified into four types of errors: problem definition errors, model errors, numerical errors and user and code errors. The chapters below describe these errors and give some guidelines on how to avoid them.<br />
<br />
=== Problem definition errors ===<br />
<br />
Problem definition errors are the most basic form of errors. In order to obtain usefull results a CFD simulation must of course analyze the correct problem, have suitable boundary contions and be based on a relevant geometry.<br />
<br />
==== Wrong type of simulation ====<br />
<br />
It is essential to have an overview of the physics involved and how the problem can best be analyzed. Running a 2D simulation in order to understand secondary flows or running a steady simulation in order to understand transient behaviour is of course no use. When assessing a CFD simulation the first thing to consider is what physical phenomena are important for the results and if the selected type of simulation is suitable to resolve this type of phenomena. For further information about selecting the most suitable type of simulation please see the previous chapter on [[#Deciding what type of simulation to do|deciding what type of simulation to perform]].<br />
<br />
==== Incorrect or uncertain boundary conditions ====<br />
<br />
A common source of errors is that incorrect boundary conditions are used. The boundary conditions must be specified in enough detail in order to resolve all the important physical features in the problem. For further information about specifying boundary conditions please see the previous chapter on [[#Boundary conditions|boundary conditions]].<br />
<br />
==== Geometrical errors ====<br />
<br />
It is almost always necessary to simplify the geometry in some form. When assessing a CFD simulation one should consider how the geometrical simplifications can affect the interesting physical phenomena. Typical geometrical errors are:<br />
<br />
*Simplifications<br />
:Small geometrical features like fillets, small steps or gaps etc. can often be disregarded. When disregarding this type of features one should consider if they might affect the important physics. For example, a very large fillet on the suction-side of a vane might affect corner separations near the end-walls. A large tip-leakage might affect the flow physics significantly in the upper part of a compressor.<br />
*Tolerances and manufacturing discrepancies<br />
:If the geometry has very large tolerances or is manufactured in a way which might produce a non-ideal shape or position it might be necessary to perform additional CFD simulations in order to cover the whole span of possible real geometries.<br />
*Surface conditions - roughness, welds, steps, gaps etc.<br />
:Often CFD simulations assume a perfectly smooth surface. A non-smooth surface which might have welds, steps or even gaps will of course produce different results. If the physical phenomena of interest might depend on the surface conditions these should of course be condidered. Typical phenomena that might be dependent on this type of errors are transition prediction, leakage flows etc.<br />
<br />
=== Model errors ===<br />
<br />
Errors related to the computational model.<br />
<br />
==== Wrong physical models ====<br />
<br />
Once the type of simulation has been selected the next step is to select what type of physical models the simulation should use. The following points should be considered:<br />
<br />
* Gas data (incompressible/compressible, perfect gas/real gas, ...)<br />
* Turbulence modeling (type of model, type of near-wall treatment, ...)<br />
* Other models (combustion, sprays, ...)<br />
<br />
When assessing model related errors it is important to know the features of the selected model and think carefully how these features and possible short comings might affect the predicted physical behaviour. Using the wrong turbulence model or combustion model can completely destroy the results of a CFD simulation.<br />
<br />
=== Numerical errors ===<br />
<br />
Errors related to the numerical solution of the developed model. Typical examples of numerical errors are discretization errors, convergence errors and round-off errors.<br />
<br />
==== Discretization errors ====<br />
<br />
Discretization errors can either be spatial errors in space or temporal errors in time.<br />
<br />
Spatial discretization errors are what people normally call discretization errors. These errors are due to the difference between the exact solution and the numerical representation of the solution in space. Describing exactly what discretization different codes use and what errors this might lead to is not possible here. Instead some general rules to avoid these errors can be summarized as:<br />
* Use at least a 2nd order accurate scheme, preferably a 3rd order accurate scheme. Some general purpose codes have a 1st order upwind scheme as default, this is a very diffusives schemes that often gives too smooth results.<br />
* For new applications always run a simulation with a finer mesh to see how grid independent your solution is.<br />
* Be aware of checker-board errors. Checker-board errors occur close to strong shocks and other large discontinuities and can be seen as a wavy pattern with a wavelength of two cells. Some schemes, especially those who behave like central differencing schemes, are more prone to checker-board errors. Upwind schemes are a bit better and schemes like TVD or chock-capturing schemes are even better.<br />
<br />
Temporal discretization errors mainly effect transient simulations. However, some codes use a time-marching method also for steady simulations and then a temporal discretization error might affect the final steady solution slightly. The discretization in time can be done with 1st or 2nd order schemes or a Runge-Kutta method, which is more accurate and saves memory. Some codes can adapt the time-step, but often it is necessary to prescribe a time-step in advance. Think of the time-step as your grid in time and make sure that the grid-resolution in time is fine enough to resolve the highest frequencies. To avoid problems with temporal discretization errors the following should be considered:<br />
* Use at least a 2nd order scheme in time.<br />
* Do a physical estimation of the typical frequencies in time of the phenomena that you are interested in and select a time-step which is fine enough to resolve these frequencies well. After the simulation also look at the frequencies captured and make sure that they are well resolved by the chosen time-step.<br />
* For new applications try a finer time-step to ensure that your solution in time is fairly grid independent in time.<br />
<br />
==== Convergence errors ====<br />
<br />
To judge when a CFD simulation is converged is not always that easy. Different codes and different applications behave very differently. For a pure aero-simulation on a fairly coarse grid convergence is easy to judge, but for more complex simulations involving resolved boundary layers, heat transfer, combustion etc. convergence can be very tricky. Aside from looking at residuals one should always also look at how global parameters like static pressure distributions, total pressure losses, skin friction, heat transfer etc. change in time.<br />
For more information about how to avoid convergence errors see the previous chapter about [[#Convergence criteria]].<br />
<br />
==== Round-off errors ====<br />
<br />
When using single precision care needs to be taken to avoid round-off errors. Inviscid Euler simulations and simulations using wall-function meshes can most often be performed in single precision. For well resolved boundary layers with Y plus close to 1 it is often necessary to use double precision. If using double precision for very fine mesh resolutions make sure that you also create the mesh in double precision and not just run the solver in double precision. Sometimes a single precision solver converges slower than a double precision solver due to numerical errors caused by round-off errors. When using advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions it is also often necessary to use double precision.<br />
<br />
=== User and code errors ===<br />
<br />
Errors related to bugs in the code used or mistakes made by the CFD engineer.<br />
<br />
=== What to trust and what not to trust ===<br />
<br />
CFD is generally quite good at predicting surface static pressure distributions. With care CFD can also be used to predict performance, total-pressure losses and blade turning. <br />
<br />
Predicting separation, stall and off-design performance can be a challenge and results with non-attached flows should be interpreted with care. <br />
<br />
Heat transfer is often very difficult to predict accurately and it is common to obtain heat-transfer coefficients that are 100% wrong or more. Validation data is critical in order to be able to trust heat transfer simulations.<br />
<br />
Transition is almost impossible to predict accurately in general. However, there exist models that have been tuned to predict transition and these tend to give acceptable results for cases close to the ones they were tuned for.<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Abu-Ghannam, B.J. and Shaw, R.|year=1980|title=Natural Transition of Boundary Layers - the Effects of Turbulence, Pressure Gradient, and Flow History|rest=Journal of Mech. Eng. Science, vol. 22, no. 5, pp. 213–228}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1985|title=Model Equation for Simulating Flows in Multistage Turbomachinery|rest=ASME Paper 85-GT-226, also NASA TM-86869}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1999|title=Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design|rest=ASME Paper 99-GT-80.}}<br />
<br />
* {{reference-paper|author=Baldwin, B. S. and Lomax, H.|year=1978|title=Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows|rest=AIAA Paper 78-257}}<br />
<br />
* {{reference-paper|author=Chen, J. P. and Barter, J. W.|year=1998|title=Comparison of Time-Accurate Calculations for the Unsteady Interaction in Turbomachinery Stage|rest=AIAA Paper 98-3292}}<br />
<br />
* {{reference-paper|author=Denton, J. and Singh, U.|year=1979|title=Time Marching Methods for Turbomachinery Flow Calculations|rest=VKI Lecture Series 1979-7, von Karman Institute}}<br />
<br />
* {{reference-paper|author=Erdos, J., Alzner, E. and McNally, W.|year=1977|title=Numerical Solution of Periodic Transonic Flow Through a Transonic Fan Stage|rest=AIAA Journal, vol. 15, no. 11, pp. 1559-1568}}<br />
<br />
* {{reference-paper|author=Gerolymos, G., Vinteler, D., Haugeard, R., Tsange, G. and Vallet, I.|year=1996|title=On the Computation of Unsteady Turbomachinery Flows – Part II – Rotor/Stator Interaction using Euler Equations|rest=AGARD Report CP-571}}<br />
<br />
* {{reference-paper|author=Giles, M. B.|year=1990|title=Stator/Rotor Interaction in a Transonic Turbine|rest=AIAA Journal of Propulsion and Power, vol. 6, no. 5}}<br />
<br />
* {{reference-paper|author=He, L, and Ning, W.|year=1998|title=Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines|rest=AIAA Journal, vol. 36, no. 11, pp. 2005-2012}}<br />
<br />
* {{reference-paper|author=Mayle, R. E.|year=1991|title=The Role of Laminar-Turbulent Transition in Gas Turbine Engines|rest=ASME Journal of Turbomachinery, vol. 113, pp. 509-537}}<br />
<br />
* {{reference-paper|author=Menter, F.R. |year=1994|title=Two-equation eddy-viscosity turbulence models for engineering applications|rest=AIAA Journal, vol. 32, pp. 269-289}}<br />
<br />
* {{reference-paper|author=Savill, M., Dick, E., Hanjalić, K. and Voke, P.|year=2002|title=Synthesis Report of the ERCOFTAC Transition Modeling – TRANSPRETURB Thematic Network Activities 1998-2002|rest=Ecoftac Bulletin 54, September 2002, pp. 5-16}}<br />
<br />
* {{reference-paper|author=Wang, X. and Chen J.|year=2004|title=A Post-Processor to Render Turbomachinery Flows Using Phase-Lag Simulations|rest=AIAA Paper 04-615}}<br />
<br />
== External links ==<br />
<br />
* [http://www.qnet-cfd.net/newsletter/8th/n8_40-46.pdf QNET-CFD Best Practise Advice for Turbomachinery Internal Flows]<br />
* [http://www.qnet-cfd.net/newsletter/7th/n7_05.pdf State of the art in Industrial CFD for Turbomachinery Flows, QNET-CFD Network Newsletter, Volume 2, No. 3 – December 2003]<br />
* [https://pronet.wsatkins.co.uk/marnet/guidelines/guide.html MARNET-CFD Best Practice Guidelines for Marine Applications of CFD]<br />
* [http://www.stanford.edu/group/ctr/ResBriefs/temp05/weide.pdf On Large Scale Turbomachinery Computations, by E. van der Weide et. al., Center for Turbulence Research, Annual Research Briefs 2005]</div>DavidFhttp://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFDBest practice guidelines for turbomachinery CFD2008-10-02T09:25:45Z<p>DavidF: /* Meshing */</p>
<hr />
<div>This article contains a summary of the most important knowledge and experience a CFD engineer needs in order to perform CFD simulations of turbomachinery components. The guide is mainly aimed at axial turbomachinery. The goal is to give a CFD engineer, who has just started working with turbomachinery simulations, a head start and avoid some of the most difficult pit-falls. Experienced turbomachinery CFD engineers can also use the guidelines in order to learn what other experts consider best practice. The intended audience is expected to know basic CFD terminology and have a basic turbomachinery knowledge, but no detailed knowledge about CFD for turbomachinery is needed. <br />
<br />
== Deciding what type of simulation to do ==<br />
<br />
Before starting a new turbomachinery simulation it is wise to think carefully of what it is that should be predicted and what physical phenomena that affect the results. This chapter contains a brief overview of the various types of simulations and some hints of what can be predicted with them.<br />
<br />
=== 2D, Quasi-3D or 3D ===<br />
<br />
2D simulations are often used in the early design phase in order to obtain a typical 2D section of a blade. For cases with many long blades or vanes, like low-pressure turbines, a 2D simulation can also provide reasonable results. If the area of the flow-path changes significantly in the axial direction it might be necessary to instead make a quasi-3D simulation. A quasi-3D simulation is a 2D simulation in which extra source terms are used to account for the acceleration/deceleration caused by a changing channel height or growing end-wall boundary layers. Codes focused on turbomachinery applications often have the possibility to perform quasi-3D simulations, but most general purpose CFD codes can not do this type of simulations, or require user coding to implement the correct source terms in the equations. Please contact your software distributor if your code does not have the quasi-3d possibility and you require it. Many codes require special routines or hidden commands to enable this feature.<br />
<br />
Full 3D simulations are necessary if a true 3D geometry is needed to obtain correct secondary flows and/or shock locations. For low-aspect-ratio cases with only a few short blades, like for example structurally loaded turbine outlet guide vanes, the secondary flow development is important and a 3D simulation is often necessary in order to obtain reasonable results. For applications where the end-wall boundary layers grow very quickly and interact with a large part of the flow-field it is necessary to perform a full 3D simulation. This is often the case in compressors and fans, where the negative pressure gradients make the boundary layers grow much quicker than what they do in for example turbines. For cases where the shock location is very critical, like in transonic compressors, it is also often necessary to perform a 3D simulation in order to obtain reasonable shock locations.<br />
<br />
=== Inviscid or viscid ===<br />
<br />
For attached flows close to the design point and without any large separations it is often sufficient with an inviscid Euler simulation in order to obtain reasonable blade loadings and pressure distributions. Note that inviscid Euler simulations should only be used if the boundary layers are judged to not have a significant effect on the global flow-field. A viscid Navier-Stokes simulation is necessary in order to predict losses, secondary flows and separations. As soon as separations are of interest it is of course also necessary to do a viscid simulation. Note that with todays computers it is often not time and resources that make users run inviscid Euler simulations. Running viscid Navier-Stokes simulations is now so quick that it is not a time problem any more. Euler simulations are still interesting though, since with an inviscid Euler simulation you don't have to worry about wall resolutions, y+ values, turbulence modeling errors etc.<br />
<br />
=== Transient or Stationary ===<br />
<br />
Most turbomachinery simlation are performed as stationary simulations. Transient simulations are done when some kind of transient flow behaviour has a strong influence on the global flow field. Examples of transient simulations are detailed simulations of rotor-stator interaction effects, simulations of large unsteady separations etc. Sometimes when you perform a steady stationary simulation you can see tendencies of unsteady behaviour like for example periodic vortex shedding behind blunt trailing edges. This is often first seen as periodical variations of the residuals. If the unsteady tendencies are judged to not affect the overall simulation results if might be necessary to coarsen the mesh close to the vortex shedding or run a different turbulence model in order to make the simulation converge. Sometimes you are still forced to run a transient simulation and average the results if you don't obtain a converged steady solution.<br />
<br />
== Meshing ==<br />
<br />
In turbomachinery applications structured multi-block hexahedral meshes are most often used for flow-path simulations. In most solvers a structured grid requires less memory, provides superior accuracy and allows a better boundary-layer resolution than an unstructured grid. By having cells with a large aspect ratio around sharp leading and trailing edges a structured grid also provides a better resolution of these areas. Many companies have automatic meshing tools that automatically mesh blade sections with a structured mesh without much user intervention.<br />
<br />
Unstructured meshes are used for more complex and odd geometries where a structured mesh is difficult to create. Typical examples where unstructured meshes are often used are blade tip regions, areas involving leakage flows and secondary air systems, film cooling ducts etc.<br />
<br />
When meshing avoid to create large jumps in cell sizes. Typically the cell size should not change with more than a factor of 1.25 between neighbouring cells. For structured meshes also try to create fairly continous mesh lines and avoid discontinuities where the cell directions suddenly change. For multi-block structured meshes avoid placing the singular points where blocks meet in regions with strong flow gradients since most schemes have a lower accuracy in these singular points.<br />
<br />
=== Mesh size guidelines ===<br />
<br />
It is difficult to define, a priori, the mesh size. The required mesh size depends on the purpose of the simulation. <br />
<br />
If the main goal is to obtain static pressure forces a coarse mesh is often able to obtain a good solution, especially when an accurate resolution of the boundary layers is not required. For 2D inviscid simulations of one blade a mesh with say 3,000 cells is most often sufficient. For 3D inviscid blade simulations a mesh size of about 40,000 cells is usually sufficient. On inviscid Euler simulations the cells should be fairly equal in size and no boundary layer resolution should be present. Avoid having too skewed cells.<br />
<br />
For loss predictions and cases where boundary layer development and separation is important the mesh needs to have a boundary layer resolution. The boundary layer resolution can either be coarse and suitable for a [[Wall functions|wall function]] simulation or very fine and suitable for a [[Low-Re resolved boundary layers|low-Re]] simulation. For further information about selecting the near-wall turbulence model please see the [[#Turbulence modeling|turbulence modeling]] section. In 3D single-blade simulations a decent wall-function mesh typically has around 100,000 cells. This type of mesh size is suitable for quick design iterations where it is not essential to resolve all secondary flows and vortices. A good 3D wall-function mesh of a blade section intended to resolve secondary flows well should have at least 400,000 cells . A good low-Re mesh with resolved boundary layers typically has around 1,000,000 cells. <br />
<br />
In 2D blade simulations a good wall-function mesh has around 20,000 cells and a good low-Re mesh with resolved boundary layers has around 50,000 cells.<br />
<br />
Along the suction and pressure surfaces it is a good use about 100 cells in the streamwise direction. In the radial direction a good first approach is to use something like 30 cells for a wall-function mesh and 100 cells for a low-Re mesh.<br />
<br />
It is important to resolve leading and trailing edges well. Typically at least 10 cells, preferably 20 should be used around the leading and trailing edges. For very blunt and large leading edges, like those commonly found on HP turbine blades, 30 or more cells can be necessary.<br />
<br />
Cases which are difficult to converge with a steady simulation and which show tendencies of periodic vortex shedding from the trailing edge, can sometimes be "tamed" by using a coarse mesh around the trailing edge. This, of course, reduces the accuracy but can be a trick to obtain a converged solution if time and computer resouces does not allow a transient simulation to be performed.<br />
<br />
=== Boundary layer mesh ===<br />
<br />
For design iteration type of simulations where a [[Wall functions|wall function]] approach is sufficient [[Dimensionless wall distance (y plus) | y+]] for the first cell should be somewhere between 20 and 200. The outer limit is dependent on the actual Re number of the simulations. For cases with fairly low Re numbers make sure to keep the maximum y+ as low as possible. For more accurate simulations with resolved boundary layers the mesh should have a y+ for the first cell which is below 1. Some new codes are now using a hybrid wall treatment that allows a smooth transition from a coarse wall-function mesh to a resolved low-Re mesh. Use some extra care when using this type of hybrid technique since it is still fairly new and unproven.<br />
<br />
Outside of the first cell at a wall a good rule of thumb is to use a growth ratio normal to the wall in the boundary layer of maximum 1.25. For a low-Re mesh this usually gives around 40 cells in the boundary layer whereas a wall-function mesh does not require more than 10 cells in the boundary layer.<br />
<br />
If you are uncertain of which wall distance to mesh with you can use a [http://www.cfd-online.com/Links/tools.html#yplus y+ estimation tool] to estitmate the distance needed to obtain the desired y+. These estimation tools are ''very handy'' if you have not done any previous similar simulations. <br />
<br />
As a rule of thumb a wall-function mesh typically requires areound 5 to 10 cells in the boundary layer whereas a resolved low-Re mesh requires about 40 cells in the boundary layer.<br />
<br />
== Boundary conditions ==<br />
<br />
Describe different types of boundary conditions and when they should be used:<br />
<br />
* Total pressure in, static pressure out<br />
* Absorbing boundary conditions<br />
* ...<br />
<br />
There are different types of boundary conditions you can use:<br />
*Mass flow inlet, static pressure outlet.<br />
To put this BC you must allocate inlet surface enough far away from the turbomachine impeller. The velocity distribution is not<br />
constant over all the surface. If you put this "lie" away from your problem it works well.<br />
<br />
<br />
=== Turbulence inlet conditions ===<br />
<br />
Prescribing realistic turbulence inlet conditions is important and usually difficult. For two-equation turbulence models two different turbulence values need to be specified on the inlet. The most common way of specifying the inlet turbulence variables is to give an inlet [[Turbulence intensity|turbulence level]] and [[Turbulence length scale|length-scale]]. For simulations where the incoming boundary layers are estimated to not have any significant effect on the secondary flows and global flow-field downstream it is usually sufficient to specify a constant incoming turbulence level and length-scale.<br />
<br />
The incoming [[Turbulence intensity|turbulence level]] is dependent on which component is analyzed. Best is of course to have measurments from a similar application in order to be able to prescribe a fairly realistic value. Without measurements it is necessary to make an educated guess. Simulations with very complex and turbulence-generating components upstream will have a very high incoming turbulence level. For example, a high-pressure turbine just downstream of a turbulence generating combustor might have incoming turbulence levels up to 20%. Fans and low-pressure compressors with not many components upstream might have as low incoming turbulence level as 1%. For components between these two extreme examples, like high-pressure compressors, low-pressure turbines etc. a turbulence level of around 5% might be realistic. If you do not have any measurements and are unsure about which incoming turbulence level to prescribe it is always a good idea to run a few different simulations with say one half and one double incoming turbulence level in order to estimate how important it is for the simulation results.<br />
<br />
The incoming [[Turbulence length scale|turbulence length-scale]] is often even more difficult to guess than the incoming turbulence level. Measurements might sometimes include the incoming turbulence level, but very seldom also include the incoming turbulent length-scale. The best way of guessing a realistic incoming length-scale is to use the geometrical properties of the upstream components. The incoming turbulence length-scale can be estimated as say the thickness of upstream blades or somewhere between 2% and 20% of the incoming channel heigth. Fortunately the incoming turbulence length-scale is usually not that important for the end results. However, some k-omega models can have an unrelisticly strong influence on the incoming turbulence length-scale. Hence, be extra carefull about the length-scale when using k-omega models. Note that the [[SST k-omega model]] does not have this unrealistic sensitivity to the inlet turbulence length-scale.<br />
<br />
Instead of specifying an incoming turbulence length-scale it is sometimes more convenient to specify an incoming [[Eddy viscosity ratio|eddy viscosity ratio]]. For low-turbulence cases with no well-specified turbulence generating components upstream it can be difficult to estimate a turbulence length-scale. Specifying an eddy viscosity ratio instead gives a more direct control over how large the effect of the incoming turbulence is. The eddy viscosity ratio describes how large the turbulence viscosity is related to the molecular viscosity. In turbomachinery the eddy viscosity ratio can vary from say 10 in low-turbulence components like fans up to say 1,000 or in rare cases even 10,000 in high-turbulence components like high-pressure turbines. In components between these extremes a typical eddy viscosity ration might be about 200. Even if specifying an inlet turbulence length scale it can still be good to compute what eddy viscosity ratio this corresponds to in order to estimate if it sounds reasonable.<br />
<br />
== Turbulence modeling ==<br />
<br />
Selecting a suitable turbulence model for turbomachinery simulations can be a challenging task. There is no single model which is suitable for all types of simulations. Which turbulence model CFD engineers use has as much to do with beliefs and traditions as with knowledge and facts. There are many different schools. However, below follows some advices that most CFD engineers in the turbomachinery field tend to agree upon.<br />
<br />
For attached flows close to the design point a simple algebraic model like the [[Baldwin-Lomax model]] can be used. Another common choice for design-iteration type of simulations is the one-equation model by [[Spalart-Allmaras model | Spalart-Allmaras]]. This model has become more popular in the last years due to the many inherent problems in more refined two-equation models. The big advantage with both the [[Baldwin-Lomax model]] and the [[Spalart-Allmaras model]] over more advanced models is that they are very robust to use and rarely produce completely unphysical results.<br />
<br />
In order to accurately predict more difficult cases, like separating flows, rotating flows, flows strongly affected by secondary flows etc. it is often necessary to use a more refined turbulence model. Common choices are two-equation models like the <math>k-\epsilon</math> model.<br />
<br />
Two-equation models are based on the [[Boussinesq eddy viscosity assumption]] and this often leads to an over-production of turbulent energy in regions with strong acceleration or deceleration, like in the leading edge region, regions around shocks and in the suction peak on the suction side of a blade. To reduce this problems it is common to use a special model variant using, for example, [[Durbin's realizability constraint]] or the [[Kato-Launder modification]]. Note that different two-equation models behave differently in these problematic stagnation and acceleration regions. Worst is probably the standard <math>k-\epsilon</math> model. <math>k-\omega</math> model are slightly better but still do not behave well. More modern variants like Menter's [[SST k-omega model]] also has problems, wheras the [[V2-f models|v2f model ]] by Durbin behaves better.<br />
<br />
=== Near-wall treatment ===<br />
<br />
For on-design simulations without any large separated regions it is often sufficient to use a [[wall-function model]] close to the wall, preferably with some form of non-equilibrium wall-function that is sensitised to streamwise pressure gradients.<br />
<br />
For off-design simulation, or simulations involving complex secondary flows and separations, it is often necessary to use a [[low-Re model]]. There exist many low-Re models that have been used with success in turbomachinery simulations. A robust and often good choice is to use a one-equation model, like for example the [[Wolfstein model]], in the inner parts of the boundary layer. There are also several Low-Re <math>k-\epsilon</math> models that work well. Just make sure they don't suffer from the problem with overproduction of turbulent energy in regions with strong acceleration or deceleration. In the last few years Menter's low-Re <math>SST k-\omega</math> model has gained increased popularity.<br />
<br />
=== Transition prediction ===<br />
<br />
Transition refers to the process when a laminar boundary layer becomes unstable and transitions to a turbulent boundary layer. There are two types of transition - natural transition, where inherent instabilities in the boundary layer cause the transition and by-pass transition, where convection and diffusion of turbulence from the free-stream into the boundary layer cause the transition. Most transitions in turbomachinery are by-pass transitions caused by free-stream turbulence and other external disturbances like wakes, vortices and surface defects.<br />
<br />
Simulating transition in a CFD code accurately is very difficult. Often a separate transition model needs to be solved in order to specify the transition location and length. Predicting natural transition in a pure CFD code is not possible. Predicting by-pass transition in a pure CFD code is almost impossible, although there are people who claim to be able to predict by-pass transition with low-Re two-equation models. However, this is usually on special test cases and with simulations that have been tuned for these special cases, see for example [Saville 2002]. In reality transition is a very complex and sensitive process where disturbances like incoming wakes and vortices from previous stages, surface roughness effects and small steps or gaps in the surfaces play a significant role. <br />
<br />
The turbomachinery codes that have transition prediction models often use old ad-hoc models like the Abu-Ghannam and Shaw model [Abu-Ghannam 1980] or the Mayle model [Mayle 1991]. These models can be quite reliable if they have been validated and tuned for a similar application. Do not trust your transition predictions without having some form of experimental validation. Menter has also recently developed a new form of transition model that might work fairly well, but it is still too new and untested.<br />
<br />
For some turbomachinery applications, like modern high-lift low-pressure turbines, transition is critical. For these applications a CFD code with a transition model that has been tuned for this type of applications should be used.<br />
<br />
== Numerical considerations ==<br />
<br />
Use at least a second order accurate scheme for the flow variables. Some codes require a first order scheme for the turbulent variables (<math>k</math> and <math>\epsilon</math>) in order to converge well. It might be sufficient with a first order scheme only on the turbulence variables, but a second order scheme is of course preferable.<br />
<br />
=== Convergence criteria ===<br />
<br />
To know when a solution is converged is not always that easy. You need some prior experience of your CFD code and your application to judge when a simulation is converged. For normal pure aero simulations without resolved walls, i.e. with wall functions or inviscid Euler simulations, convergence can most often be estimated just by looking at the residuals. Exacty what the residuals should be is not possible to say, it all depends on how your particular code computes and scales the residuals. Hence, make sure to read the manuals and plot the convergence of a few global parameters before you decide what the residuals should be for a solution to converege. Note also that many manuals for genereal purpose CFD codes list overly agressive convergence critera that often produce unconverged results. <br />
For simulations with resolved walls it is good to look at the convergence of some global properties, like total pressure losses from the inlet to the outlet. For heat transfer simulations it is even more tricky since the aerodynamic field can look almost converged although the thermal field is not converged at all. If doing heat transfer simulations make sure to plot the heat-transfer, run for some time, and plot it again to make sure that it doesn't change anymore. With very well resolved walls and heat transfer it can sometimes take 10 times longer for the thermal field to converge.<br />
<br />
=== Single or double precision ===<br />
<br />
With todays computers and cheap memory prices it often does not cost much extra to run in double precision. Before using single precision you should first investigate how your software and hardware works with double precision. If the extra time and memory needed for double precision is negligible you should of course always run in double precision. With double precision you never have to worry about round-off errors. Always using double precision is one way of avoiding one type of pit-falls in the complex world of CFD simulations. Use double precision when you have resolved boundary layers (Y+ around 1) and when you use advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions.<br />
<br />
== Multi-stage analysis ==<br />
<br />
Multi-stage analysis can be done in different ways:<br />
<br />
* Steady mixing-plane simulations<br />
* Frozen rotor simulations<br />
* Unsteady sliding-mesh stator-rotor simulations<br />
* Other '''advanced''' multi-stage methods<br />
<br />
=== Steady mixing-plane simulations ===<br />
<br />
Since the mixing-plane method was first introduced in 1979 [Denton & Singh 1979] is has become the industry standard type of rotor-stator simulations. A mixing-plane simulation is steady and only requires one rotor blade and one stator blade per stage. Between the rotating blade passage and the steady vane passage the flow properties are circumferentially averaged in a so-called mixing-plane interface. This will of course remove all transient rotor-stator interactions, but it still gives fairly representative results. In some commercial codes (CFX for example) mixing-plane interfaces are also called stage-interfaces.<br />
<br />
=== Frozen rotor simulations ===<br />
<br />
In a frozen rotor simulation the rotating and the stationary parts have a fixed relative position. A frame transformation is done to include the rotating effect on the rotating sections. This will give a steady flow and no transient effects are included. With a frozen-rotor simulation rotating wakes, secondary flows, leading edge pressure increases etc. will always stay in exactly the same positions. This makes a frozen rotor simulation very dependent on exactly how the rotors and the stators are positioned. Most often a mixing-plane simulation gives better results. Frozen rotor simulations are mainly performed to obtain a good starting flow-field before doing a transient sliding-mesh simulation.<br />
<br />
=== Unsteady sliding-mesh stator-rotor simulations ===<br />
<br />
This is the most complete type of stator-rotor simulation. In most engines the number of stators and rotors do not have a common denominator (to avoid instabilities caused by resonance between different rings). Hence, to make a full unsteady sliding-mesh computation it is necessary to have a mesh which includes the full wheel with all stators vanes and all rotor blades. This is often not possible, instead it is necessary to reduce the number of vanes and blades by finding a denominator that is '''almost''' common and then scale the geometry slightly circumferentially. Here is an example:<br />
<br />
*Real engine: 36 stator vanes, 41 rotor blades<br />
*Approximated engine: 41 stator vanes, 41 rotor blades, making it possible to simulate only 1 stator vane and 1 rotor blade<br />
*Scaling of stator: All stator vanes are scaled by 36/41 = 0.8780 circumferentially.<br />
<br />
=== Other '''advanced''' multi-stage methods ===<br />
<br />
Time-inclinded, Adamszyk stresses ...<br />
<br />
== Heat transfer predictions ==<br />
<br />
Besides listing the general heat transfer mechanisms involved <br />
(namely conduction, convection, radiation)<br />
heat transfer prediction in CFD may be seen as or split into two cases. <br />
<br />
'''Mesh consists of fluid domain(s)''': <br />
you want to know how heat inserted into the fluid domain (e.g. via the flowing medium) changes the temperature field along the flow path. Be it a steady or transient run, a fluid which enters with a given temperature will usually experience temperature variations, for instance, caused by convective boundary conditions, prescribed temperature profiles at flow obstacles etc. <br />
<br />
'''Mesh consists of fluid and solid domain(s)''': <br />
additionally to the above, you want info too wrt the (spatial, temporal or even spectral) temperature field distribution in surrounding, confining or immersed solids like channel walls or heat exchanger tubes. This is also called conjugate heattransfer CHT in the CFD context. CHT requires a good boundary layer resolution, usually the wall mesh needs to be rather refined, to obtain realistic heatflux results at the fluid/solid interface. Flow and heat transfer convergence require different time step settings, to properly capture changes in flow and heat quantities respectively. <br />
<br />
<br />
In either case, verify (strict necessity depends on CFD code used, CFX for instance checks and assists in regard) that model dimensions, boundary conditions and properties are in consistent units, hold appropriate values. Check temperature-dependence of properties and other numbers before the run.<br />
<br />
In heat transfer predictions (depending on the CFD code in use) <br />
besides the flow solver, you may have to activate the thermal solver too, <br />
as a job specification.<br />
<br />
== Acoustics and noise ==<br />
<br />
A whole separate research subject, difficult. <br />
<br />
Tone noise possible. Often run with linearzised solvers in the frequency domain.<br />
<br />
Jet noise possible. Often run with LES or DES simulations that either also resolve the sound waves or couples to a separate acoustic solver.<br />
<br />
Turbomachinery broadband noise not possible yet, or at least a great challenge.<br />
<br />
== Errors and uncertainties ==<br />
<br />
CFD is still a tool which requires that a user has a good understanding of uncertainties and errors that might spoil a CFD similation. There exists no error control in CFD and any CFD simulation must be interpreted by an expirenced user to have some credibility. Without some knowledge about possible errors and how they can be handled a CFD simulation can not be trusted. Errors can occur at different places:<br />
<br />
* Definition of the problem - What needs to be analyzed?<br />
* Selection of the solution strategy - What physical models and what numerical tools should be used?<br />
* Development of the computational model - How should the geometry and the numerical tools be set up?<br />
* Analysis and interpretation of the results - How should the model be analyzed and the results be interpreted?<br />
<br />
There exists many different definitions on errors. In this guide the errors are classified into four types of errors: problem definition errors, model errors, numerical errors and user and code errors. The chapters below describe these errors and give some guidelines on how to avoid them.<br />
<br />
=== Problem definition errors ===<br />
<br />
Problem definition errors are the most basic form of errors. In order to obtain usefull results a CFD simulation must of course analyze the correct problem, have suitable boundary contions and be based on a relevant geometry.<br />
<br />
==== Wrong type of simulation ====<br />
<br />
It is essential to have an overview of the physics involved and how the problem can best be analyzed. Running a 2D simulation in order to understand secondary flows or running a steady simulation in order to understand transient behaviour is of course no use. When assessing a CFD simulation the first thing to consider is what physical phenomena are important for the results and if the selected type of simulation is suitable to resolve this type of phenomena. For further information about selecting the most suitable type of simulation please see the previous chapter on [[#Deciding what type of simulation to do|deciding what type of simulation to perform]].<br />
<br />
==== Incorrect or uncertain boundary conditions ====<br />
<br />
A common source of errors is that incorrect boundary conditions are used. The boundary conditions must be specified in enough detail in order to resolve all the important physical features in the problem. For further information about specifying boundary conditions please see the previous chapter on [[#Boundary conditions|boundary conditions]].<br />
<br />
==== Geometrical errors ====<br />
<br />
It is almost always necessary to simplify the geometry in some form. When assessing a CFD simulation one should consider how the geometrical simplifications can affect the interesting physical phenomena. Typical geometrical errors are:<br />
<br />
*Simplifications<br />
:Small geometrical features like fillets, small steps or gaps etc. can often be disregarded. When disregarding this type of features one should consider if they might affect the important physics. For example, a very large fillet on the suction-side of a vane might affect corner separations near the end-walls. A large tip-leakage might affect the flow physics significantly in the upper part of a compressor.<br />
*Tolerances and manufacturing discrepancies<br />
:If the geometry has very large tolerances or is manufactured in a way which might produce a non-ideal shape or position it might be necessary to perform additional CFD simulations in order to cover the whole span of possible real geometries.<br />
*Surface conditions - roughness, welds, steps, gaps etc.<br />
:Often CFD simulations assume a perfectly smooth surface. A non-smooth surface which might have welds, steps or even gaps will of course produce different results. If the physical phenomena of interest might depend on the surface conditions these should of course be condidered. Typical phenomena that might be dependent on this type of errors are transition prediction, leakage flows etc.<br />
<br />
=== Model errors ===<br />
<br />
Errors related to the computational model.<br />
<br />
==== Wrong physical models ====<br />
<br />
Once the type of simulation has been selected the next step is to select what type of physical models the simulation should use. The following points should be considered:<br />
<br />
* Gas data (incompressible/compressible, perfect gas/real gas, ...)<br />
* Turbulence modeling (type of model, type of near-wall treatment, ...)<br />
* Other models (combustion, sprays, ...)<br />
<br />
When assessing model related errors it is important to know the features of the selected model and think carefully how these features and possible short comings might affect the predicted physical behaviour. Using the wrong turbulence model or combustion model can completely destroy the results of a CFD simulation.<br />
<br />
=== Numerical errors ===<br />
<br />
Errors related to the numerical solution of the developed model. Typical examples of numerical errors are discretization errors, convergence errors and round-off errors.<br />
<br />
==== Discretization errors ====<br />
<br />
Discretization errors can either be spatial errors in space or temporal errors in time.<br />
<br />
Spatial discretization errors are what people normally call discretization errors. These errors are due to the difference between the exact solution and the numerical representation of the solution in space. Describing exactly what discretization different codes use and what errors this might lead to is not possible here. Instead some general rules to avoid these errors can be summarized as:<br />
* Use at least a 2nd order accurate scheme, preferably a 3rd order accurate scheme. Some general purpose codes have a 1st order upwind scheme as default, this is a very diffusives schemes that often gives too smooth results.<br />
* For new applications always run a simulation with a finer mesh to see how grid independent your solution is.<br />
* Be aware of checker-board errors. Checker-board errors occur close to strong shocks and other large discontinuities and can be seen as a wavy pattern with a wavelength of two cells. Some schemes, especially those who behave like central differencing schemes, are more prone to checker-board errors. Upwind schemes are a bit better and schemes like TVD or chock-capturing schemes are even better.<br />
<br />
Temporal discretization errors mainly effect transient simulations. However, some codes use a time-marching method also for steady simulations and then a temporal discretization error might affect the final steady solution slightly. The discretization in time can be done with 1st or 2nd order schemes or a Runge-Kutta method, which is more accurate and saves memory. Some codes can adapt the time-step, but often it is necessary to prescribe a time-step in advance. Think of the time-step as your grid in time and make sure that the grid-resolution in time is fine enough to resolve the highest frequencies. To avoid problems with temporal discretization errors the following should be considered:<br />
* Use at least a 2nd order scheme in time.<br />
* Do a physical estimation of the typical frequencies in time of the phenomena that you are interested in and select a time-step which is fine enough to resolve these frequencies well. After the simulation also look at the frequencies captured and make sure that they are well resolved by the chosen time-step.<br />
* For new applications try a finer time-step to ensure that your solution in time is fairly grid independent in time.<br />
<br />
==== Convergence errors ====<br />
<br />
To judge when a CFD simulation is converged is not always that easy. Different codes and different applications behave very differently. For a pure aero-simulation on a fairly coarse grid convergence is easy to judge, but for more complex simulations involving resolved boundary layers, heat transfer, combustion etc. convergence can be very tricky. Aside from looking at residuals one should always also look at how global parameters like static pressure distributions, total pressure losses, skin friction, heat transfer etc. change in time.<br />
For more information about how to avoid convergence errors see the previous chapter about [[#Convergence criteria]].<br />
<br />
==== Round-off errors ====<br />
<br />
When using single precision care needs to be taken to avoid round-off errors. Inviscid Euler simulations and simulations using wall-function meshes can most often be performed in single precision. For well resolved boundary layers with Y plus close to 1 it is often necessary to use double precision. If using double precision for very fine mesh resolutions make sure that you also create the mesh in double precision and not just run the solver in double precision. Sometimes a single precision solver converges slower than a double precision solver due to numerical errors caused by round-off errors. When using advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions it is also often necessary to use double precision.<br />
<br />
=== User and code errors ===<br />
<br />
Errors related to bugs in the code used or mistakes made by the CFD engineer.<br />
<br />
=== What to trust and what not to trust ===<br />
<br />
CFD is generally quite good at predicting surface static pressure distributions. With care CFD can also be used to predict performance, total-pressure losses and blade turning. <br />
<br />
Predicting separation, stall and off-design performance can be a challenge and results with non-attached flows should be interpreted with care. <br />
<br />
Heat transfer is often very difficult to predict accurately and it is common to obtain heat-transfer coefficients that are 100% wrong or more. Validation data is critical in order to be able to trust heat transfer simulations.<br />
<br />
Transition is almost impossible to predict accurately in general. However, there exist models that have been tuned to predict transition and these tend to give acceptable results for cases close to the ones they were tuned for.<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Abu-Ghannam, B.J. and Shaw, R.|year=1980|title=Natural Transition of Boundary Layers - the Effects of Turbulence, Pressure Gradient, and Flow History|rest=Journal of Mech. Eng. Science, vol. 22, no. 5, pp. 213–228}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1985|title=Model Equation for Simulating Flows in Multistage Turbomachinery|rest=ASME Paper 85-GT-226, also NASA TM-86869}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1999|title=Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design|rest=ASME Paper 99-GT-80.}}<br />
<br />
* {{reference-paper|author=Baldwin, B. S. and Lomax, H.|year=1978|title=Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows|rest=AIAA Paper 78-257}}<br />
<br />
* {{reference-paper|author=Chen, J. P. and Barter, J. W.|year=1998|title=Comparison of Time-Accurate Calculations for the Unsteady Interaction in Turbomachinery Stage|rest=AIAA Paper 98-3292}}<br />
<br />
* {{reference-paper|author=Denton, J. and Singh, U.|year=1979|title=Time Marching Methods for Turbomachinery Flow Calculations|rest=VKI Lecture Series 1979-7, von Karman Institute}}<br />
<br />
* {{reference-paper|author=Erdos, J., Alzner, E. and McNally, W.|year=1977|title=Numerical Solution of Periodic Transonic Flow Through a Transonic Fan Stage|rest=AIAA Journal, vol. 15, no. 11, pp. 1559-1568}}<br />
<br />
* {{reference-paper|author=Gerolymos, G., Vinteler, D., Haugeard, R., Tsange, G. and Vallet, I.|year=1996|title=On the Computation of Unsteady Turbomachinery Flows – Part II – Rotor/Stator Interaction using Euler Equations|rest=AGARD Report CP-571}}<br />
<br />
* {{reference-paper|author=Giles, M. B.|year=1990|title=Stator/Rotor Interaction in a Transonic Turbine|rest=AIAA Journal of Propulsion and Power, vol. 6, no. 5}}<br />
<br />
* {{reference-paper|author=He, L, and Ning, W.|year=1998|title=Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines|rest=AIAA Journal, vol. 36, no. 11, pp. 2005-2012}}<br />
<br />
* {{reference-paper|author=Mayle, R. E.|year=1991|title=The Role of Laminar-Turbulent Transition in Gas Turbine Engines|rest=ASME Journal of Turbomachinery, vol. 113, pp. 509-537}}<br />
<br />
* {{reference-paper|author=Menter, F.R. |year=1994|title=Two-equation eddy-viscosity turbulence models for engineering applications|rest=AIAA Journal, vol. 32, pp. 269-289}}<br />
<br />
* {{reference-paper|author=Savill, M., Dick, E., Hanjalić, K. and Voke, P.|year=2002|title=Synthesis Report of the ERCOFTAC Transition Modeling – TRANSPRETURB Thematic Network Activities 1998-2002|rest=Ecoftac Bulletin 54, September 2002, pp. 5-16}}<br />
<br />
* {{reference-paper|author=Wang, X. and Chen J.|year=2004|title=A Post-Processor to Render Turbomachinery Flows Using Phase-Lag Simulations|rest=AIAA Paper 04-615}}<br />
<br />
== External links ==<br />
<br />
* [http://www.qnet-cfd.net/newsletter/8th/n8_40-46.pdf QNET-CFD Best Practise Advice for Turbomachinery Internal Flows]<br />
* [http://www.qnet-cfd.net/newsletter/7th/n7_05.pdf State of the art in Industrial CFD for Turbomachinery Flows, QNET-CFD Network Newsletter, Volume 2, No. 3 – December 2003]<br />
* [https://pronet.wsatkins.co.uk/marnet/guidelines/guide.html MARNET-CFD Best Practice Guidelines for Marine Applications of CFD]<br />
* [http://www.stanford.edu/group/ctr/ResBriefs/temp05/weide.pdf On Large Scale Turbomachinery Computations, by E. van der Weide et. al., Center for Turbulence Research, Annual Research Briefs 2005]</div>DavidFhttp://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFDBest practice guidelines for turbomachinery CFD2008-10-02T09:24:34Z<p>DavidF: /* Transient or Stationary */</p>
<hr />
<div>This article contains a summary of the most important knowledge and experience a CFD engineer needs in order to perform CFD simulations of turbomachinery components. The guide is mainly aimed at axial turbomachinery. The goal is to give a CFD engineer, who has just started working with turbomachinery simulations, a head start and avoid some of the most difficult pit-falls. Experienced turbomachinery CFD engineers can also use the guidelines in order to learn what other experts consider best practice. The intended audience is expected to know basic CFD terminology and have a basic turbomachinery knowledge, but no detailed knowledge about CFD for turbomachinery is needed. <br />
<br />
== Deciding what type of simulation to do ==<br />
<br />
Before starting a new turbomachinery simulation it is wise to think carefully of what it is that should be predicted and what physical phenomena that affect the results. This chapter contains a brief overview of the various types of simulations and some hints of what can be predicted with them.<br />
<br />
=== 2D, Quasi-3D or 3D ===<br />
<br />
2D simulations are often used in the early design phase in order to obtain a typical 2D section of a blade. For cases with many long blades or vanes, like low-pressure turbines, a 2D simulation can also provide reasonable results. If the area of the flow-path changes significantly in the axial direction it might be necessary to instead make a quasi-3D simulation. A quasi-3D simulation is a 2D simulation in which extra source terms are used to account for the acceleration/deceleration caused by a changing channel height or growing end-wall boundary layers. Codes focused on turbomachinery applications often have the possibility to perform quasi-3D simulations, but most general purpose CFD codes can not do this type of simulations, or require user coding to implement the correct source terms in the equations. Please contact your software distributor if your code does not have the quasi-3d possibility and you require it. Many codes require special routines or hidden commands to enable this feature.<br />
<br />
Full 3D simulations are necessary if a true 3D geometry is needed to obtain correct secondary flows and/or shock locations. For low-aspect-ratio cases with only a few short blades, like for example structurally loaded turbine outlet guide vanes, the secondary flow development is important and a 3D simulation is often necessary in order to obtain reasonable results. For applications where the end-wall boundary layers grow very quickly and interact with a large part of the flow-field it is necessary to perform a full 3D simulation. This is often the case in compressors and fans, where the negative pressure gradients make the boundary layers grow much quicker than what they do in for example turbines. For cases where the shock location is very critical, like in transonic compressors, it is also often necessary to perform a 3D simulation in order to obtain reasonable shock locations.<br />
<br />
=== Inviscid or viscid ===<br />
<br />
For attached flows close to the design point and without any large separations it is often sufficient with an inviscid Euler simulation in order to obtain reasonable blade loadings and pressure distributions. Note that inviscid Euler simulations should only be used if the boundary layers are judged to not have a significant effect on the global flow-field. A viscid Navier-Stokes simulation is necessary in order to predict losses, secondary flows and separations. As soon as separations are of interest it is of course also necessary to do a viscid simulation. Note that with todays computers it is often not time and resources that make users run inviscid Euler simulations. Running viscid Navier-Stokes simulations is now so quick that it is not a time problem any more. Euler simulations are still interesting though, since with an inviscid Euler simulation you don't have to worry about wall resolutions, y+ values, turbulence modeling errors etc.<br />
<br />
=== Transient or Stationary ===<br />
<br />
Most turbomachinery simlation are performed as stationary simulations. Transient simulations are done when some kind of transient flow behaviour has a strong influence on the global flow field. Examples of transient simulations are detailed simulations of rotor-stator interaction effects, simulations of large unsteady separations etc. Sometimes when you perform a steady stationary simulation you can see tendencies of unsteady behaviour like for example periodic vortex shedding behind blunt trailing edges. This is often first seen as periodical variations of the residuals. If the unsteady tendencies are judged to not affect the overall simulation results if might be necessary to coarsen the mesh close to the vortex shedding or run a different turbulence model in order to make the simulation converge. Sometimes you are still forced to run a transient simulation and average the results if you don't obtain a converged steady solution.<br />
<br />
== Meshing ==<br />
<br />
In turbomachinery applications structured muli-block hexahedral meshes are most often used for flow-path simulations. In most solvers a structured grid requires less memory, provides superior accuracy and allows a better boundary-layer resolution than an unstructured grid. By having cells with a large aspect ratio around sharp leading and trailing edges a structured grid also provides a better resolution of these areas. Many companies have automatic meshing tools that automatically mesh blade sections with a structured mesh without much user intervention.<br />
<br />
Unstructured meshes are used for more complex and odd geometries where a structured mesh is difficult to create. Typical examples where unstructured meshes are often used are blade tip regions, areas involving leakage flows and secondary air systems, film cooling ducts etc.<br />
<br />
When meshing avoid to create large jumps in cell sizes. Typically the cell size should not change with more than a factor of 1.25 between neighbouring cells. For structured meshes also try to create fairly continous mesh lines and avoid discontinuities where the cell directions suddenly change. For multi-block structured meshes avoid placing the singular points where blocks meet in regions with strong flow gradients since most schemes have a lower accuracy in these singular points.<br />
<br />
=== Mesh size guidelines ===<br />
<br />
It is difficult to define, a priori, the mesh size. The required mesh size depends on the purpose of the simulation. <br />
<br />
If the main goal is to obtain static pressure forces a coarse mesh is often able to obtain a good solution, especially when an accurate resolution of the boundary layers is not required. For 2D inviscid simulations of one blade a mesh with say 3,000 cells is most often sufficient. For 3D inviscid blade simulations a mesh size of about 40,000 cells is usually sufficient. On inviscid Euler simulations the cells should be fairly equal in size and no boundary layer resolution should be present. Avoid having too skewed cells.<br />
<br />
For loss predictions and cases where boundary layer development and separation is important the mesh needs to have a boundary layer resolution. The boundary layer resolution can either be coarse and suitable for a [[Wall functions|wall function]] simulation or very fine and suitable for a [[Low-Re resolved boundary layers|low-Re]] simulation. For further information about selecting the near-wall turbulence model please see the [[#Turbulence modeling|turbulence modeling]] section. In 3D single-blade simulations a decent wall-function mesh typically has around 100,000 cells. This type of mesh size is suitable for quick design iterations where it is not essential to resolve all secondary flows and vortices. A good 3D wall-function mesh of a blade section intended to resolve secondary flows well should have at least 400,000 cells . A good low-Re mesh with resolved boundary layers typically has around 1,000,000 cells. <br />
<br />
In 2D blade simulations a good wall-function mesh has around 20,000 cells and a good low-Re mesh with resolved boundary layers has around 50,000 cells.<br />
<br />
Along the suction and pressure surfaces it is a good use about 100 cells in the streamwise direction. In the radial direction a good first approach is to use something like 30 cells for a wall-function mesh and 100 cells for a low-Re mesh.<br />
<br />
It is important to resolve leading and trailing edges well. Typically at least 10 cells, preferably 20 should be used around the leading and trailing edges. For very blunt and large leading edges, like those commonly found on HP turbine blades, 30 or more cells can be necessary.<br />
<br />
Cases which are difficult to converge with a steady simulation and which show tendencies of periodic vortex shedding from the trailing edge, can sometimes be "tamed" by using a coarse mesh around the trailing edge. This, of course, reduces the accuracy but can be a trick to obtain a converged solution if time and computer resouces does not allow a transient simulation to be performed.<br />
<br />
=== Boundary layer mesh ===<br />
<br />
For design iteration type of simulations where a [[Wall functions|wall function]] approach is sufficient [[Dimensionless wall distance (y plus) | y+]] for the first cell should be somewhere between 20 and 200. The outer limit is dependent on the actual Re number of the simulations. For cases with fairly low Re numbers make sure to keep the maximum y+ as low as possible. For more accurate simulations with resolved boundary layers the mesh should have a y+ for the first cell which is below 1. Some new codes are now using a hybrid wall treatment that allows a smooth transition from a coarse wall-function mesh to a resolved low-Re mesh. Use some extra care when using this type of hybrid technique since it is still fairly new and unproven.<br />
<br />
Outside of the first cell at a wall a good rule of thumb is to use a growth ratio normal to the wall in the boundary layer of maximum 1.25. For a low-Re mesh this usually gives around 40 cells in the boundary layer whereas a wall-function mesh does not require more than 10 cells in the boundary layer.<br />
<br />
If you are uncertain of which wall distance to mesh with you can use a [http://www.cfd-online.com/Links/tools.html#yplus y+ estimation tool] to estitmate the distance needed to obtain the desired y+. These estimation tools are ''very handy'' if you have not done any previous similar simulations. <br />
<br />
As a rule of thumb a wall-function mesh typically requires areound 5 to 10 cells in the boundary layer whereas a resolved low-Re mesh requires about 40 cells in the boundary layer.<br />
<br />
== Boundary conditions ==<br />
<br />
Describe different types of boundary conditions and when they should be used:<br />
<br />
* Total pressure in, static pressure out<br />
* Absorbing boundary conditions<br />
* ...<br />
<br />
There are different types of boundary conditions you can use:<br />
*Mass flow inlet, static pressure outlet.<br />
To put this BC you must allocate inlet surface enough far away from the turbomachine impeller. The velocity distribution is not<br />
constant over all the surface. If you put this "lie" away from your problem it works well.<br />
<br />
<br />
=== Turbulence inlet conditions ===<br />
<br />
Prescribing realistic turbulence inlet conditions is important and usually difficult. For two-equation turbulence models two different turbulence values need to be specified on the inlet. The most common way of specifying the inlet turbulence variables is to give an inlet [[Turbulence intensity|turbulence level]] and [[Turbulence length scale|length-scale]]. For simulations where the incoming boundary layers are estimated to not have any significant effect on the secondary flows and global flow-field downstream it is usually sufficient to specify a constant incoming turbulence level and length-scale.<br />
<br />
The incoming [[Turbulence intensity|turbulence level]] is dependent on which component is analyzed. Best is of course to have measurments from a similar application in order to be able to prescribe a fairly realistic value. Without measurements it is necessary to make an educated guess. Simulations with very complex and turbulence-generating components upstream will have a very high incoming turbulence level. For example, a high-pressure turbine just downstream of a turbulence generating combustor might have incoming turbulence levels up to 20%. Fans and low-pressure compressors with not many components upstream might have as low incoming turbulence level as 1%. For components between these two extreme examples, like high-pressure compressors, low-pressure turbines etc. a turbulence level of around 5% might be realistic. If you do not have any measurements and are unsure about which incoming turbulence level to prescribe it is always a good idea to run a few different simulations with say one half and one double incoming turbulence level in order to estimate how important it is for the simulation results.<br />
<br />
The incoming [[Turbulence length scale|turbulence length-scale]] is often even more difficult to guess than the incoming turbulence level. Measurements might sometimes include the incoming turbulence level, but very seldom also include the incoming turbulent length-scale. The best way of guessing a realistic incoming length-scale is to use the geometrical properties of the upstream components. The incoming turbulence length-scale can be estimated as say the thickness of upstream blades or somewhere between 2% and 20% of the incoming channel heigth. Fortunately the incoming turbulence length-scale is usually not that important for the end results. However, some k-omega models can have an unrelisticly strong influence on the incoming turbulence length-scale. Hence, be extra carefull about the length-scale when using k-omega models. Note that the [[SST k-omega model]] does not have this unrealistic sensitivity to the inlet turbulence length-scale.<br />
<br />
Instead of specifying an incoming turbulence length-scale it is sometimes more convenient to specify an incoming [[Eddy viscosity ratio|eddy viscosity ratio]]. For low-turbulence cases with no well-specified turbulence generating components upstream it can be difficult to estimate a turbulence length-scale. Specifying an eddy viscosity ratio instead gives a more direct control over how large the effect of the incoming turbulence is. The eddy viscosity ratio describes how large the turbulence viscosity is related to the molecular viscosity. In turbomachinery the eddy viscosity ratio can vary from say 10 in low-turbulence components like fans up to say 1,000 or in rare cases even 10,000 in high-turbulence components like high-pressure turbines. In components between these extremes a typical eddy viscosity ration might be about 200. Even if specifying an inlet turbulence length scale it can still be good to compute what eddy viscosity ratio this corresponds to in order to estimate if it sounds reasonable.<br />
<br />
== Turbulence modeling ==<br />
<br />
Selecting a suitable turbulence model for turbomachinery simulations can be a challenging task. There is no single model which is suitable for all types of simulations. Which turbulence model CFD engineers use has as much to do with beliefs and traditions as with knowledge and facts. There are many different schools. However, below follows some advices that most CFD engineers in the turbomachinery field tend to agree upon.<br />
<br />
For attached flows close to the design point a simple algebraic model like the [[Baldwin-Lomax model]] can be used. Another common choice for design-iteration type of simulations is the one-equation model by [[Spalart-Allmaras model | Spalart-Allmaras]]. This model has become more popular in the last years due to the many inherent problems in more refined two-equation models. The big advantage with both the [[Baldwin-Lomax model]] and the [[Spalart-Allmaras model]] over more advanced models is that they are very robust to use and rarely produce completely unphysical results.<br />
<br />
In order to accurately predict more difficult cases, like separating flows, rotating flows, flows strongly affected by secondary flows etc. it is often necessary to use a more refined turbulence model. Common choices are two-equation models like the <math>k-\epsilon</math> model.<br />
<br />
Two-equation models are based on the [[Boussinesq eddy viscosity assumption]] and this often leads to an over-production of turbulent energy in regions with strong acceleration or deceleration, like in the leading edge region, regions around shocks and in the suction peak on the suction side of a blade. To reduce this problems it is common to use a special model variant using, for example, [[Durbin's realizability constraint]] or the [[Kato-Launder modification]]. Note that different two-equation models behave differently in these problematic stagnation and acceleration regions. Worst is probably the standard <math>k-\epsilon</math> model. <math>k-\omega</math> model are slightly better but still do not behave well. More modern variants like Menter's [[SST k-omega model]] also has problems, wheras the [[V2-f models|v2f model ]] by Durbin behaves better.<br />
<br />
=== Near-wall treatment ===<br />
<br />
For on-design simulations without any large separated regions it is often sufficient to use a [[wall-function model]] close to the wall, preferably with some form of non-equilibrium wall-function that is sensitised to streamwise pressure gradients.<br />
<br />
For off-design simulation, or simulations involving complex secondary flows and separations, it is often necessary to use a [[low-Re model]]. There exist many low-Re models that have been used with success in turbomachinery simulations. A robust and often good choice is to use a one-equation model, like for example the [[Wolfstein model]], in the inner parts of the boundary layer. There are also several Low-Re <math>k-\epsilon</math> models that work well. Just make sure they don't suffer from the problem with overproduction of turbulent energy in regions with strong acceleration or deceleration. In the last few years Menter's low-Re <math>SST k-\omega</math> model has gained increased popularity.<br />
<br />
=== Transition prediction ===<br />
<br />
Transition refers to the process when a laminar boundary layer becomes unstable and transitions to a turbulent boundary layer. There are two types of transition - natural transition, where inherent instabilities in the boundary layer cause the transition and by-pass transition, where convection and diffusion of turbulence from the free-stream into the boundary layer cause the transition. Most transitions in turbomachinery are by-pass transitions caused by free-stream turbulence and other external disturbances like wakes, vortices and surface defects.<br />
<br />
Simulating transition in a CFD code accurately is very difficult. Often a separate transition model needs to be solved in order to specify the transition location and length. Predicting natural transition in a pure CFD code is not possible. Predicting by-pass transition in a pure CFD code is almost impossible, although there are people who claim to be able to predict by-pass transition with low-Re two-equation models. However, this is usually on special test cases and with simulations that have been tuned for these special cases, see for example [Saville 2002]. In reality transition is a very complex and sensitive process where disturbances like incoming wakes and vortices from previous stages, surface roughness effects and small steps or gaps in the surfaces play a significant role. <br />
<br />
The turbomachinery codes that have transition prediction models often use old ad-hoc models like the Abu-Ghannam and Shaw model [Abu-Ghannam 1980] or the Mayle model [Mayle 1991]. These models can be quite reliable if they have been validated and tuned for a similar application. Do not trust your transition predictions without having some form of experimental validation. Menter has also recently developed a new form of transition model that might work fairly well, but it is still too new and untested.<br />
<br />
For some turbomachinery applications, like modern high-lift low-pressure turbines, transition is critical. For these applications a CFD code with a transition model that has been tuned for this type of applications should be used.<br />
<br />
== Numerical considerations ==<br />
<br />
Use at least a second order accurate scheme for the flow variables. Some codes require a first order scheme for the turbulent variables (<math>k</math> and <math>\epsilon</math>) in order to converge well. It might be sufficient with a first order scheme only on the turbulence variables, but a second order scheme is of course preferable.<br />
<br />
=== Convergence criteria ===<br />
<br />
To know when a solution is converged is not always that easy. You need some prior experience of your CFD code and your application to judge when a simulation is converged. For normal pure aero simulations without resolved walls, i.e. with wall functions or inviscid Euler simulations, convergence can most often be estimated just by looking at the residuals. Exacty what the residuals should be is not possible to say, it all depends on how your particular code computes and scales the residuals. Hence, make sure to read the manuals and plot the convergence of a few global parameters before you decide what the residuals should be for a solution to converege. Note also that many manuals for genereal purpose CFD codes list overly agressive convergence critera that often produce unconverged results. <br />
For simulations with resolved walls it is good to look at the convergence of some global properties, like total pressure losses from the inlet to the outlet. For heat transfer simulations it is even more tricky since the aerodynamic field can look almost converged although the thermal field is not converged at all. If doing heat transfer simulations make sure to plot the heat-transfer, run for some time, and plot it again to make sure that it doesn't change anymore. With very well resolved walls and heat transfer it can sometimes take 10 times longer for the thermal field to converge.<br />
<br />
=== Single or double precision ===<br />
<br />
With todays computers and cheap memory prices it often does not cost much extra to run in double precision. Before using single precision you should first investigate how your software and hardware works with double precision. If the extra time and memory needed for double precision is negligible you should of course always run in double precision. With double precision you never have to worry about round-off errors. Always using double precision is one way of avoiding one type of pit-falls in the complex world of CFD simulations. Use double precision when you have resolved boundary layers (Y+ around 1) and when you use advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions.<br />
<br />
== Multi-stage analysis ==<br />
<br />
Multi-stage analysis can be done in different ways:<br />
<br />
* Steady mixing-plane simulations<br />
* Frozen rotor simulations<br />
* Unsteady sliding-mesh stator-rotor simulations<br />
* Other '''advanced''' multi-stage methods<br />
<br />
=== Steady mixing-plane simulations ===<br />
<br />
Since the mixing-plane method was first introduced in 1979 [Denton & Singh 1979] is has become the industry standard type of rotor-stator simulations. A mixing-plane simulation is steady and only requires one rotor blade and one stator blade per stage. Between the rotating blade passage and the steady vane passage the flow properties are circumferentially averaged in a so-called mixing-plane interface. This will of course remove all transient rotor-stator interactions, but it still gives fairly representative results. In some commercial codes (CFX for example) mixing-plane interfaces are also called stage-interfaces.<br />
<br />
=== Frozen rotor simulations ===<br />
<br />
In a frozen rotor simulation the rotating and the stationary parts have a fixed relative position. A frame transformation is done to include the rotating effect on the rotating sections. This will give a steady flow and no transient effects are included. With a frozen-rotor simulation rotating wakes, secondary flows, leading edge pressure increases etc. will always stay in exactly the same positions. This makes a frozen rotor simulation very dependent on exactly how the rotors and the stators are positioned. Most often a mixing-plane simulation gives better results. Frozen rotor simulations are mainly performed to obtain a good starting flow-field before doing a transient sliding-mesh simulation.<br />
<br />
=== Unsteady sliding-mesh stator-rotor simulations ===<br />
<br />
This is the most complete type of stator-rotor simulation. In most engines the number of stators and rotors do not have a common denominator (to avoid instabilities caused by resonance between different rings). Hence, to make a full unsteady sliding-mesh computation it is necessary to have a mesh which includes the full wheel with all stators vanes and all rotor blades. This is often not possible, instead it is necessary to reduce the number of vanes and blades by finding a denominator that is '''almost''' common and then scale the geometry slightly circumferentially. Here is an example:<br />
<br />
*Real engine: 36 stator vanes, 41 rotor blades<br />
*Approximated engine: 41 stator vanes, 41 rotor blades, making it possible to simulate only 1 stator vane and 1 rotor blade<br />
*Scaling of stator: All stator vanes are scaled by 36/41 = 0.8780 circumferentially.<br />
<br />
=== Other '''advanced''' multi-stage methods ===<br />
<br />
Time-inclinded, Adamszyk stresses ...<br />
<br />
== Heat transfer predictions ==<br />
<br />
Besides listing the general heat transfer mechanisms involved <br />
(namely conduction, convection, radiation)<br />
heat transfer prediction in CFD may be seen as or split into two cases. <br />
<br />
'''Mesh consists of fluid domain(s)''': <br />
you want to know how heat inserted into the fluid domain (e.g. via the flowing medium) changes the temperature field along the flow path. Be it a steady or transient run, a fluid which enters with a given temperature will usually experience temperature variations, for instance, caused by convective boundary conditions, prescribed temperature profiles at flow obstacles etc. <br />
<br />
'''Mesh consists of fluid and solid domain(s)''': <br />
additionally to the above, you want info too wrt the (spatial, temporal or even spectral) temperature field distribution in surrounding, confining or immersed solids like channel walls or heat exchanger tubes. This is also called conjugate heattransfer CHT in the CFD context. CHT requires a good boundary layer resolution, usually the wall mesh needs to be rather refined, to obtain realistic heatflux results at the fluid/solid interface. Flow and heat transfer convergence require different time step settings, to properly capture changes in flow and heat quantities respectively. <br />
<br />
<br />
In either case, verify (strict necessity depends on CFD code used, CFX for instance checks and assists in regard) that model dimensions, boundary conditions and properties are in consistent units, hold appropriate values. Check temperature-dependence of properties and other numbers before the run.<br />
<br />
In heat transfer predictions (depending on the CFD code in use) <br />
besides the flow solver, you may have to activate the thermal solver too, <br />
as a job specification.<br />
<br />
== Acoustics and noise ==<br />
<br />
A whole separate research subject, difficult. <br />
<br />
Tone noise possible. Often run with linearzised solvers in the frequency domain.<br />
<br />
Jet noise possible. Often run with LES or DES simulations that either also resolve the sound waves or couples to a separate acoustic solver.<br />
<br />
Turbomachinery broadband noise not possible yet, or at least a great challenge.<br />
<br />
== Errors and uncertainties ==<br />
<br />
CFD is still a tool which requires that a user has a good understanding of uncertainties and errors that might spoil a CFD similation. There exists no error control in CFD and any CFD simulation must be interpreted by an expirenced user to have some credibility. Without some knowledge about possible errors and how they can be handled a CFD simulation can not be trusted. Errors can occur at different places:<br />
<br />
* Definition of the problem - What needs to be analyzed?<br />
* Selection of the solution strategy - What physical models and what numerical tools should be used?<br />
* Development of the computational model - How should the geometry and the numerical tools be set up?<br />
* Analysis and interpretation of the results - How should the model be analyzed and the results be interpreted?<br />
<br />
There exists many different definitions on errors. In this guide the errors are classified into four types of errors: problem definition errors, model errors, numerical errors and user and code errors. The chapters below describe these errors and give some guidelines on how to avoid them.<br />
<br />
=== Problem definition errors ===<br />
<br />
Problem definition errors are the most basic form of errors. In order to obtain usefull results a CFD simulation must of course analyze the correct problem, have suitable boundary contions and be based on a relevant geometry.<br />
<br />
==== Wrong type of simulation ====<br />
<br />
It is essential to have an overview of the physics involved and how the problem can best be analyzed. Running a 2D simulation in order to understand secondary flows or running a steady simulation in order to understand transient behaviour is of course no use. When assessing a CFD simulation the first thing to consider is what physical phenomena are important for the results and if the selected type of simulation is suitable to resolve this type of phenomena. For further information about selecting the most suitable type of simulation please see the previous chapter on [[#Deciding what type of simulation to do|deciding what type of simulation to perform]].<br />
<br />
==== Incorrect or uncertain boundary conditions ====<br />
<br />
A common source of errors is that incorrect boundary conditions are used. The boundary conditions must be specified in enough detail in order to resolve all the important physical features in the problem. For further information about specifying boundary conditions please see the previous chapter on [[#Boundary conditions|boundary conditions]].<br />
<br />
==== Geometrical errors ====<br />
<br />
It is almost always necessary to simplify the geometry in some form. When assessing a CFD simulation one should consider how the geometrical simplifications can affect the interesting physical phenomena. Typical geometrical errors are:<br />
<br />
*Simplifications<br />
:Small geometrical features like fillets, small steps or gaps etc. can often be disregarded. When disregarding this type of features one should consider if they might affect the important physics. For example, a very large fillet on the suction-side of a vane might affect corner separations near the end-walls. A large tip-leakage might affect the flow physics significantly in the upper part of a compressor.<br />
*Tolerances and manufacturing discrepancies<br />
:If the geometry has very large tolerances or is manufactured in a way which might produce a non-ideal shape or position it might be necessary to perform additional CFD simulations in order to cover the whole span of possible real geometries.<br />
*Surface conditions - roughness, welds, steps, gaps etc.<br />
:Often CFD simulations assume a perfectly smooth surface. A non-smooth surface which might have welds, steps or even gaps will of course produce different results. If the physical phenomena of interest might depend on the surface conditions these should of course be condidered. Typical phenomena that might be dependent on this type of errors are transition prediction, leakage flows etc.<br />
<br />
=== Model errors ===<br />
<br />
Errors related to the computational model.<br />
<br />
==== Wrong physical models ====<br />
<br />
Once the type of simulation has been selected the next step is to select what type of physical models the simulation should use. The following points should be considered:<br />
<br />
* Gas data (incompressible/compressible, perfect gas/real gas, ...)<br />
* Turbulence modeling (type of model, type of near-wall treatment, ...)<br />
* Other models (combustion, sprays, ...)<br />
<br />
When assessing model related errors it is important to know the features of the selected model and think carefully how these features and possible short comings might affect the predicted physical behaviour. Using the wrong turbulence model or combustion model can completely destroy the results of a CFD simulation.<br />
<br />
=== Numerical errors ===<br />
<br />
Errors related to the numerical solution of the developed model. Typical examples of numerical errors are discretization errors, convergence errors and round-off errors.<br />
<br />
==== Discretization errors ====<br />
<br />
Discretization errors can either be spatial errors in space or temporal errors in time.<br />
<br />
Spatial discretization errors are what people normally call discretization errors. These errors are due to the difference between the exact solution and the numerical representation of the solution in space. Describing exactly what discretization different codes use and what errors this might lead to is not possible here. Instead some general rules to avoid these errors can be summarized as:<br />
* Use at least a 2nd order accurate scheme, preferably a 3rd order accurate scheme. Some general purpose codes have a 1st order upwind scheme as default, this is a very diffusives schemes that often gives too smooth results.<br />
* For new applications always run a simulation with a finer mesh to see how grid independent your solution is.<br />
* Be aware of checker-board errors. Checker-board errors occur close to strong shocks and other large discontinuities and can be seen as a wavy pattern with a wavelength of two cells. Some schemes, especially those who behave like central differencing schemes, are more prone to checker-board errors. Upwind schemes are a bit better and schemes like TVD or chock-capturing schemes are even better.<br />
<br />
Temporal discretization errors mainly effect transient simulations. However, some codes use a time-marching method also for steady simulations and then a temporal discretization error might affect the final steady solution slightly. The discretization in time can be done with 1st or 2nd order schemes or a Runge-Kutta method, which is more accurate and saves memory. Some codes can adapt the time-step, but often it is necessary to prescribe a time-step in advance. Think of the time-step as your grid in time and make sure that the grid-resolution in time is fine enough to resolve the highest frequencies. To avoid problems with temporal discretization errors the following should be considered:<br />
* Use at least a 2nd order scheme in time.<br />
* Do a physical estimation of the typical frequencies in time of the phenomena that you are interested in and select a time-step which is fine enough to resolve these frequencies well. After the simulation also look at the frequencies captured and make sure that they are well resolved by the chosen time-step.<br />
* For new applications try a finer time-step to ensure that your solution in time is fairly grid independent in time.<br />
<br />
==== Convergence errors ====<br />
<br />
To judge when a CFD simulation is converged is not always that easy. Different codes and different applications behave very differently. For a pure aero-simulation on a fairly coarse grid convergence is easy to judge, but for more complex simulations involving resolved boundary layers, heat transfer, combustion etc. convergence can be very tricky. Aside from looking at residuals one should always also look at how global parameters like static pressure distributions, total pressure losses, skin friction, heat transfer etc. change in time.<br />
For more information about how to avoid convergence errors see the previous chapter about [[#Convergence criteria]].<br />
<br />
==== Round-off errors ====<br />
<br />
When using single precision care needs to be taken to avoid round-off errors. Inviscid Euler simulations and simulations using wall-function meshes can most often be performed in single precision. For well resolved boundary layers with Y plus close to 1 it is often necessary to use double precision. If using double precision for very fine mesh resolutions make sure that you also create the mesh in double precision and not just run the solver in double precision. Sometimes a single precision solver converges slower than a double precision solver due to numerical errors caused by round-off errors. When using advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions it is also often necessary to use double precision.<br />
<br />
=== User and code errors ===<br />
<br />
Errors related to bugs in the code used or mistakes made by the CFD engineer.<br />
<br />
=== What to trust and what not to trust ===<br />
<br />
CFD is generally quite good at predicting surface static pressure distributions. With care CFD can also be used to predict performance, total-pressure losses and blade turning. <br />
<br />
Predicting separation, stall and off-design performance can be a challenge and results with non-attached flows should be interpreted with care. <br />
<br />
Heat transfer is often very difficult to predict accurately and it is common to obtain heat-transfer coefficients that are 100% wrong or more. Validation data is critical in order to be able to trust heat transfer simulations.<br />
<br />
Transition is almost impossible to predict accurately in general. However, there exist models that have been tuned to predict transition and these tend to give acceptable results for cases close to the ones they were tuned for.<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Abu-Ghannam, B.J. and Shaw, R.|year=1980|title=Natural Transition of Boundary Layers - the Effects of Turbulence, Pressure Gradient, and Flow History|rest=Journal of Mech. Eng. Science, vol. 22, no. 5, pp. 213–228}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1985|title=Model Equation for Simulating Flows in Multistage Turbomachinery|rest=ASME Paper 85-GT-226, also NASA TM-86869}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1999|title=Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design|rest=ASME Paper 99-GT-80.}}<br />
<br />
* {{reference-paper|author=Baldwin, B. S. and Lomax, H.|year=1978|title=Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows|rest=AIAA Paper 78-257}}<br />
<br />
* {{reference-paper|author=Chen, J. P. and Barter, J. W.|year=1998|title=Comparison of Time-Accurate Calculations for the Unsteady Interaction in Turbomachinery Stage|rest=AIAA Paper 98-3292}}<br />
<br />
* {{reference-paper|author=Denton, J. and Singh, U.|year=1979|title=Time Marching Methods for Turbomachinery Flow Calculations|rest=VKI Lecture Series 1979-7, von Karman Institute}}<br />
<br />
* {{reference-paper|author=Erdos, J., Alzner, E. and McNally, W.|year=1977|title=Numerical Solution of Periodic Transonic Flow Through a Transonic Fan Stage|rest=AIAA Journal, vol. 15, no. 11, pp. 1559-1568}}<br />
<br />
* {{reference-paper|author=Gerolymos, G., Vinteler, D., Haugeard, R., Tsange, G. and Vallet, I.|year=1996|title=On the Computation of Unsteady Turbomachinery Flows – Part II – Rotor/Stator Interaction using Euler Equations|rest=AGARD Report CP-571}}<br />
<br />
* {{reference-paper|author=Giles, M. B.|year=1990|title=Stator/Rotor Interaction in a Transonic Turbine|rest=AIAA Journal of Propulsion and Power, vol. 6, no. 5}}<br />
<br />
* {{reference-paper|author=He, L, and Ning, W.|year=1998|title=Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines|rest=AIAA Journal, vol. 36, no. 11, pp. 2005-2012}}<br />
<br />
* {{reference-paper|author=Mayle, R. E.|year=1991|title=The Role of Laminar-Turbulent Transition in Gas Turbine Engines|rest=ASME Journal of Turbomachinery, vol. 113, pp. 509-537}}<br />
<br />
* {{reference-paper|author=Menter, F.R. |year=1994|title=Two-equation eddy-viscosity turbulence models for engineering applications|rest=AIAA Journal, vol. 32, pp. 269-289}}<br />
<br />
* {{reference-paper|author=Savill, M., Dick, E., Hanjalić, K. and Voke, P.|year=2002|title=Synthesis Report of the ERCOFTAC Transition Modeling – TRANSPRETURB Thematic Network Activities 1998-2002|rest=Ecoftac Bulletin 54, September 2002, pp. 5-16}}<br />
<br />
* {{reference-paper|author=Wang, X. and Chen J.|year=2004|title=A Post-Processor to Render Turbomachinery Flows Using Phase-Lag Simulations|rest=AIAA Paper 04-615}}<br />
<br />
== External links ==<br />
<br />
* [http://www.qnet-cfd.net/newsletter/8th/n8_40-46.pdf QNET-CFD Best Practise Advice for Turbomachinery Internal Flows]<br />
* [http://www.qnet-cfd.net/newsletter/7th/n7_05.pdf State of the art in Industrial CFD for Turbomachinery Flows, QNET-CFD Network Newsletter, Volume 2, No. 3 – December 2003]<br />
* [https://pronet.wsatkins.co.uk/marnet/guidelines/guide.html MARNET-CFD Best Practice Guidelines for Marine Applications of CFD]<br />
* [http://www.stanford.edu/group/ctr/ResBriefs/temp05/weide.pdf On Large Scale Turbomachinery Computations, by E. van der Weide et. al., Center for Turbulence Research, Annual Research Briefs 2005]</div>DavidFhttp://www.cfd-online.com/Wiki/Potential_flowPotential flow2008-10-01T13:09:46Z<p>DavidF: /* External Links */ Fixed broken link</p>
<hr />
<div>A flow in which vorticity is zero is called potential flow, or irrotational flow. Since the vorticity is zero<br />
<br />
<math><br />
\omega = \nabla \times u = 0<br />
</math><br />
<br />
it implies that the velocity is the gradient of a scalar field called the velocity potential, and usually denoted as <math>\phi</math><br />
<br />
<math><br />
u_i = \frac{\partial \phi}{\partial x_i}<br />
</math><br />
<br />
At high Reynolds numbers, flow past slender bodies is attached (no boundary layer separation) and the boundary layers are thin. In such situations vorticity is confined to the thin boundary layers and the rest of the flow is irrotational.<br />
<br />
== Governing equations ==<br />
<br />
<br />
<br />
== External Links ==<br />
* [http://simscience.org/fluid/red/superpos.html Applet Simulating 2D Potential Flow]</div>DavidFhttp://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFDBest practice guidelines for turbomachinery CFD2008-09-29T10:34:51Z<p>DavidF: /* 2D, Quasi-3D or 3D */</p>
<hr />
<div>This article contains a summary of the most important knowledge and experience a CFD engineer needs in order to perform CFD simulations of turbomachinery components. The guide is mainly aimed at axial turbomachinery. The goal is to give a CFD engineer, who has just started working with turbomachinery simulations, a head start and avoid some of the most difficult pit-falls. Experienced turbomachinery CFD engineers can also use the guidelines in order to learn what other experts consider best practice. The intended audience is expected to know basic CFD terminology and have a basic turbomachinery knowledge, but no detailed knowledge about CFD for turbomachinery is needed. <br />
<br />
== Deciding what type of simulation to do ==<br />
<br />
Before starting a new turbomachinery simulation it is wise to think carefully of what it is that should be predicted and what physical phenomena that affect the results. This chapter contains a brief overview of the various types of simulations and some hints of what can be predicted with them.<br />
<br />
=== 2D, Quasi-3D or 3D ===<br />
<br />
2D simulations are often used in the early design phase in order to obtain a typical 2D section of a blade. For cases with many long blades or vanes, like low-pressure turbines, a 2D simulation can also provide reasonable results. If the area of the flow-path changes significantly in the axial direction it might be necessary to instead make a quasi-3D simulation. A quasi-3D simulation is a 2D simulation in which extra source terms are used to account for the acceleration/deceleration caused by a changing channel height or growing end-wall boundary layers. Codes focused on turbomachinery applications often have the possibility to perform quasi-3D simulations, but most general purpose CFD codes can not do this type of simulations, or require user coding to implement the correct source terms in the equations. Please contact your software distributor if your code does not have the quasi-3d possibility and you require it. Many codes require special routines or hidden commands to enable this feature.<br />
<br />
Full 3D simulations are necessary if a true 3D geometry is needed to obtain correct secondary flows and/or shock locations. For low-aspect-ratio cases with only a few short blades, like for example structurally loaded turbine outlet guide vanes, the secondary flow development is important and a 3D simulation is often necessary in order to obtain reasonable results. For applications where the end-wall boundary layers grow very quickly and interact with a large part of the flow-field it is necessary to perform a full 3D simulation. This is often the case in compressors and fans, where the negative pressure gradients make the boundary layers grow much quicker than what they do in for example turbines. For cases where the shock location is very critical, like in transonic compressors, it is also often necessary to perform a 3D simulation in order to obtain reasonable shock locations.<br />
<br />
=== Inviscid or viscid ===<br />
<br />
For attached flows close to the design point and without any large separations it is often sufficient with an inviscid Euler simulation in order to obtain reasonable blade loadings and pressure distributions. Note that inviscid Euler simulations should only be used if the boundary layers are judged to not have a significant effect on the global flow-field. A viscid Navier-Stokes simulation is necessary in order to predict losses, secondary flows and separations. As soon as separations are of interest it is of course also necessary to do a viscid simulation. Note that with todays computers it is often not time and resources that make users run inviscid Euler simulations. Running viscid Navier-Stokes simulations is now so quick that it is not a time problem any more. Euler simulations are still interesting though, since with an inviscid Euler simulation you don't have to worry about wall resolutions, y+ values, turbulence modeling errors etc.<br />
<br />
=== Transient or Stationary ===<br />
<br />
Most turbomachinery simlation are performed as stationary simulations. Transient simulations are done when some kind of transient flow behaviour has a strong influence on the global flow field. Examples of transient simulations are detailed simulations of rotor-stator interaction effects, simulations of large unsteady separations etc. Sometimes when you perform a steady stationary simulation you can see tendencies of unsteady behvaiour like for example periodic vortex shedding behind blunt trailing edges. This is often first seen as periodical variations of the residuals. If the unsteady tendencies are judged to not affect the overall simulation results if might be necessary to coarsen the mesh close to the vortex shedding or run a different turbulence model in order to make the simulation converge. Sometimes you are still forced to run a transient simulation and average the results if you don't obtain a converged steady solution.<br />
<br />
== Meshing ==<br />
<br />
In turbomachinery applications structured muli-block hexahedral meshes are most often used for flow-path simulations. In most solvers a structured grid requires less memory, provides superior accuracy and allows a better boundary-layer resolution than an unstructured grid. By having cells with a large aspect ratio around sharp leading and trailing edges a structured grid also provides a better resolution of these areas. Many companies have automatic meshing tools that automatically mesh blade sections with a structured mesh without much user intervention.<br />
<br />
Unstructured meshes are used for more complex and odd geometries where a structured mesh is difficult to create. Typical examples where unstructured meshes are often used are blade tip regions, areas involving leakage flows and secondary air systems, film cooling ducts etc.<br />
<br />
When meshing avoid to create large jumps in cell sizes. Typically the cell size should not change with more than a factor of 1.25 between neighbouring cells. For structured meshes also try to create fairly continous mesh lines and avoid discontinuities where the cell directions suddenly change. For multi-block structured meshes avoid placing the singular points where blocks meet in regions with strong flow gradients since most schemes have a lower accuracy in these singular points.<br />
<br />
=== Mesh size guidelines ===<br />
<br />
It is difficult to define, a priori, the mesh size. The required mesh size depends on the purpose of the simulation. <br />
<br />
If the main goal is to obtain static pressure forces a coarse mesh is often able to obtain a good solution, especially when an accurate resolution of the boundary layers is not required. For 2D inviscid simulations of one blade a mesh with say 3,000 cells is most often sufficient. For 3D inviscid blade simulations a mesh size of about 40,000 cells is usually sufficient. On inviscid Euler simulations the cells should be fairly equal in size and no boundary layer resolution should be present. Avoid having too skewed cells.<br />
<br />
For loss predictions and cases where boundary layer development and separation is important the mesh needs to have a boundary layer resolution. The boundary layer resolution can either be coarse and suitable for a [[Wall functions|wall function]] simulation or very fine and suitable for a [[Low-Re resolved boundary layers|low-Re]] simulation. For further information about selecting the near-wall turbulence model please see the [[#Turbulence modeling|turbulence modeling]] section. In 3D single-blade simulations a decent wall-function mesh typically has around 100,000 cells. This type of mesh size is suitable for quick design iterations where it is not essential to resolve all secondary flows and vortices. A good 3D wall-function mesh of a blade section intended to resolve secondary flows well should have at least 400,000 cells . A good low-Re mesh with resolved boundary layers typically has around 1,000,000 cells. <br />
<br />
In 2D blade simulations a good wall-function mesh has around 20,000 cells and a good low-Re mesh with resolved boundary layers has around 50,000 cells.<br />
<br />
Along the suction and pressure surfaces it is a good use about 100 cells in the streamwise direction. In the radial direction a good first approach is to use something like 30 cells for a wall-function mesh and 100 cells for a low-Re mesh.<br />
<br />
It is important to resolve leading and trailing edges well. Typically at least 10 cells, preferably 20 should be used around the leading and trailing edges. For very blunt and large leading edges, like those commonly found on HP turbine blades, 30 or more cells can be necessary.<br />
<br />
Cases which are difficult to converge with a steady simulation and which show tendencies of periodic vortex shedding from the trailing edge, can sometimes be "tamed" by using a coarse mesh around the trailing edge. This, of course, reduces the accuracy but can be a trick to obtain a converged solution if time and computer resouces does not allow a transient simulation to be performed.<br />
<br />
=== Boundary layer mesh ===<br />
<br />
For design iteration type of simulations where a [[Wall functions|wall function]] approach is sufficient [[Dimensionless wall distance (y plus) | y+]] for the first cell should be somewhere between 20 and 200. The outer limit is dependent on the actual Re number of the simulations. For cases with fairly low Re numbers make sure to keep the maximum y+ as low as possible. For more accurate simulations with resolved boundary layers the mesh should have a y+ for the first cell which is below 1. Some new codes are now using a hybrid wall treatment that allows a smooth transition from a coarse wall-function mesh to a resolved low-Re mesh. Use some extra care when using this type of hybrid technique since it is still fairly new and unproven.<br />
<br />
Outside of the first cell at a wall a good rule of thumb is to use a growth ratio normal to the wall in the boundary layer of maximum 1.25. For a low-Re mesh this usually gives around 40 cells in the boundary layer whereas a wall-function mesh does not require more than 10 cells in the boundary layer.<br />
<br />
If you are uncertain of which wall distance to mesh with you can use a [http://www.cfd-online.com/Links/tools.html#yplus y+ estimation tool] to estitmate the distance needed to obtain the desired y+. These estimation tools are ''very handy'' if you have not done any previous similar simulations. <br />
<br />
As a rule of thumb a wall-function mesh typically requires areound 5 to 10 cells in the boundary layer whereas a resolved low-Re mesh requires about 40 cells in the boundary layer.<br />
<br />
== Boundary conditions ==<br />
<br />
Describe different types of boundary conditions and when they should be used:<br />
<br />
* Total pressure in, static pressure out<br />
* Absorbing boundary conditions<br />
* ...<br />
<br />
There are different types of boundary conditions you can use:<br />
*Mass flow inlet, static pressure outlet.<br />
To put this BC you must allocate inlet surface enough far away from the turbomachine impeller. The velocity distribution is not<br />
constant over all the surface. If you put this "lie" away from your problem it works well.<br />
<br />
<br />
=== Turbulence inlet conditions ===<br />
<br />
Prescribing realistic turbulence inlet conditions is important and usually difficult. For two-equation turbulence models two different turbulence values need to be specified on the inlet. The most common way of specifying the inlet turbulence variables is to give an inlet [[Turbulence intensity|turbulence level]] and [[Turbulence length scale|length-scale]]. For simulations where the incoming boundary layers are estimated to not have any significant effect on the secondary flows and global flow-field downstream it is usually sufficient to specify a constant incoming turbulence level and length-scale.<br />
<br />
The incoming [[Turbulence intensity|turbulence level]] is dependent on which component is analyzed. Best is of course to have measurments from a similar application in order to be able to prescribe a fairly realistic value. Without measurements it is necessary to make an educated guess. Simulations with very complex and turbulence-generating components upstream will have a very high incoming turbulence level. For example, a high-pressure turbine just downstream of a turbulence generating combustor might have incoming turbulence levels up to 20%. Fans and low-pressure compressors with not many components upstream might have as low incoming turbulence level as 1%. For components between these two extreme examples, like high-pressure compressors, low-pressure turbines etc. a turbulence level of around 5% might be realistic. If you do not have any measurements and are unsure about which incoming turbulence level to prescribe it is always a good idea to run a few different simulations with say one half and one double incoming turbulence level in order to estimate how important it is for the simulation results.<br />
<br />
The incoming [[Turbulence length scale|turbulence length-scale]] is often even more difficult to guess than the incoming turbulence level. Measurements might sometimes include the incoming turbulence level, but very seldom also include the incoming turbulent length-scale. The best way of guessing a realistic incoming length-scale is to use the geometrical properties of the upstream components. The incoming turbulence length-scale can be estimated as say the thickness of upstream blades or somewhere between 2% and 20% of the incoming channel heigth. Fortunately the incoming turbulence length-scale is usually not that important for the end results. However, some k-omega models can have an unrelisticly strong influence on the incoming turbulence length-scale. Hence, be extra carefull about the length-scale when using k-omega models. Note that the [[SST k-omega model]] does not have this unrealistic sensitivity to the inlet turbulence length-scale.<br />
<br />
Instead of specifying an incoming turbulence length-scale it is sometimes more convenient to specify an incoming [[Eddy viscosity ratio|eddy viscosity ratio]]. For low-turbulence cases with no well-specified turbulence generating components upstream it can be difficult to estimate a turbulence length-scale. Specifying an eddy viscosity ratio instead gives a more direct control over how large the effect of the incoming turbulence is. The eddy viscosity ratio describes how large the turbulence viscosity is related to the molecular viscosity. In turbomachinery the eddy viscosity ratio can vary from say 10 in low-turbulence components like fans up to say 1,000 or in rare cases even 10,000 in high-turbulence components like high-pressure turbines. In components between these extremes a typical eddy viscosity ration might be about 200. Even if specifying an inlet turbulence length scale it can still be good to compute what eddy viscosity ratio this corresponds to in order to estimate if it sounds reasonable.<br />
<br />
== Turbulence modeling ==<br />
<br />
Selecting a suitable turbulence model for turbomachinery simulations can be a challenging task. There is no single model which is suitable for all types of simulations. Which turbulence model CFD engineers use has as much to do with beliefs and traditions as with knowledge and facts. There are many different schools. However, below follows some advices that most CFD engineers in the turbomachinery field tend to agree upon.<br />
<br />
For attached flows close to the design point a simple algebraic model like the [[Baldwin-Lomax model]] can be used. Another common choice for design-iteration type of simulations is the one-equation model by [[Spalart-Allmaras model | Spalart-Allmaras]]. This model has become more popular in the last years due to the many inherent problems in more refined two-equation models. The big advantage with both the [[Baldwin-Lomax model]] and the [[Spalart-Allmaras model]] over more advanced models is that they are very robust to use and rarely produce completely unphysical results.<br />
<br />
In order to accurately predict more difficult cases, like separating flows, rotating flows, flows strongly affected by secondary flows etc. it is often necessary to use a more refined turbulence model. Common choices are two-equation models like the <math>k-\epsilon</math> model.<br />
<br />
Two-equation models are based on the [[Boussinesq eddy viscosity assumption]] and this often leads to an over-production of turbulent energy in regions with strong acceleration or deceleration, like in the leading edge region, regions around shocks and in the suction peak on the suction side of a blade. To reduce this problems it is common to use a special model variant using, for example, [[Durbin's realizability constraint]] or the [[Kato-Launder modification]]. Note that different two-equation models behave differently in these problematic stagnation and acceleration regions. Worst is probably the standard <math>k-\epsilon</math> model. <math>k-\omega</math> model are slightly better but still do not behave well. More modern variants like Menter's [[SST k-omega model]] also has problems, wheras the [[V2-f models|v2f model ]] by Durbin behaves better.<br />
<br />
=== Near-wall treatment ===<br />
<br />
For on-design simulations without any large separated regions it is often sufficient to use a [[wall-function model]] close to the wall, preferably with some form of non-equilibrium wall-function that is sensitised to streamwise pressure gradients.<br />
<br />
For off-design simulation, or simulations involving complex secondary flows and separations, it is often necessary to use a [[low-Re model]]. There exist many low-Re models that have been used with success in turbomachinery simulations. A robust and often good choice is to use a one-equation model, like for example the [[Wolfstein model]], in the inner parts of the boundary layer. There are also several Low-Re <math>k-\epsilon</math> models that work well. Just make sure they don't suffer from the problem with overproduction of turbulent energy in regions with strong acceleration or deceleration. In the last few years Menter's low-Re <math>SST k-\omega</math> model has gained increased popularity.<br />
<br />
=== Transition prediction ===<br />
<br />
Transition refers to the process when a laminar boundary layer becomes unstable and transitions to a turbulent boundary layer. There are two types of transition - natural transition, where inherent instabilities in the boundary layer cause the transition and by-pass transition, where convection and diffusion of turbulence from the free-stream into the boundary layer cause the transition. Most transitions in turbomachinery are by-pass transitions caused by free-stream turbulence and other external disturbances like wakes, vortices and surface defects.<br />
<br />
Simulating transition in a CFD code accurately is very difficult. Often a separate transition model needs to be solved in order to specify the transition location and length. Predicting natural transition in a pure CFD code is not possible. Predicting by-pass transition in a pure CFD code is almost impossible, although there are people who claim to be able to predict by-pass transition with low-Re two-equation models. However, this is usually on special test cases and with simulations that have been tuned for these special cases, see for example [Saville 2002]. In reality transition is a very complex and sensitive process where disturbances like incoming wakes and vortices from previous stages, surface roughness effects and small steps or gaps in the surfaces play a significant role. <br />
<br />
The turbomachinery codes that have transition prediction models often use old ad-hoc models like the Abu-Ghannam and Shaw model [Abu-Ghannam 1980] or the Mayle model [Mayle 1991]. These models can be quite reliable if they have been validated and tuned for a similar application. Do not trust your transition predictions without having some form of experimental validation. Menter has also recently developed a new form of transition model that might work fairly well, but it is still too new and untested.<br />
<br />
For some turbomachinery applications, like modern high-lift low-pressure turbines, transition is critical. For these applications a CFD code with a transition model that has been tuned for this type of applications should be used.<br />
<br />
== Numerical considerations ==<br />
<br />
Use at least a second order accurate scheme for the flow variables. Some codes require a first order scheme for the turbulent variables (<math>k</math> and <math>\epsilon</math>) in order to converge well. It might be sufficient with a first order scheme only on the turbulence variables, but a second order scheme is of course preferable.<br />
<br />
=== Convergence criteria ===<br />
<br />
To know when a solution is converged is not always that easy. You need some prior experience of your CFD code and your application to judge when a simulation is converged. For normal pure aero simulations without resolved walls, i.e. with wall functions or inviscid Euler simulations, convergence can most often be estimated just by looking at the residuals. Exacty what the residuals should be is not possible to say, it all depends on how your particular code computes and scales the residuals. Hence, make sure to read the manuals and plot the convergence of a few global parameters before you decide what the residuals should be for a solution to converege. Note also that many manuals for genereal purpose CFD codes list overly agressive convergence critera that often produce unconverged results. <br />
For simulations with resolved walls it is good to look at the convergence of some global properties, like total pressure losses from the inlet to the outlet. For heat transfer simulations it is even more tricky since the aerodynamic field can look almost converged although the thermal field is not converged at all. If doing heat transfer simulations make sure to plot the heat-transfer, run for some time, and plot it again to make sure that it doesn't change anymore. With very well resolved walls and heat transfer it can sometimes take 10 times longer for the thermal field to converge.<br />
<br />
=== Single or double precision ===<br />
<br />
With todays computers and cheap memory prices it often does not cost much extra to run in double precision. Before using single precision you should first investigate how your software and hardware works with double precision. If the extra time and memory needed for double precision is negligible you should of course always run in double precision. With double precision you never have to worry about round-off errors. Always using double precision is one way of avoiding one type of pit-falls in the complex world of CFD simulations. Use double precision when you have resolved boundary layers (Y+ around 1) and when you use advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions.<br />
<br />
== Multi-stage analysis ==<br />
<br />
Multi-stage analysis can be done in different ways:<br />
<br />
* Steady mixing-plane simulations<br />
* Frozen rotor simulations<br />
* Unsteady sliding-mesh stator-rotor simulations<br />
* Other '''advanced''' multi-stage methods<br />
<br />
=== Steady mixing-plane simulations ===<br />
<br />
Since the mixing-plane method was first introduced in 1979 [Denton & Singh 1979] is has become the industry standard type of rotor-stator simulations. A mixing-plane simulation is steady and only requires one rotor blade and one stator blade per stage. Between the rotating blade passage and the steady vane passage the flow properties are circumferentially averaged in a so-called mixing-plane interface. This will of course remove all transient rotor-stator interactions, but it still gives fairly representative results. In some commercial codes (CFX for example) mixing-plane interfaces are also called stage-interfaces.<br />
<br />
=== Frozen rotor simulations ===<br />
<br />
In a frozen rotor simulation the rotating and the stationary parts have a fixed relative position. A frame transformation is done to include the rotating effect on the rotating sections. This will give a steady flow and no transient effects are included. With a frozen-rotor simulation rotating wakes, secondary flows, leading edge pressure increases etc. will always stay in exactly the same positions. This makes a frozen rotor simulation very dependent on exactly how the rotors and the stators are positioned. Most often a mixing-plane simulation gives better results. Frozen rotor simulations are mainly performed to obtain a good starting flow-field before doing a transient sliding-mesh simulation.<br />
<br />
=== Unsteady sliding-mesh stator-rotor simulations ===<br />
<br />
This is the most complete type of stator-rotor simulation. In most engines the number of stators and rotors do not have a common denominator (to avoid instabilities caused by resonance between different rings). Hence, to make a full unsteady sliding-mesh computation it is necessary to have a mesh which includes the full wheel with all stators vanes and all rotor blades. This is often not possible, instead it is necessary to reduce the number of vanes and blades by finding a denominator that is '''almost''' common and then scale the geometry slightly circumferentially. Here is an example:<br />
<br />
*Real engine: 36 stator vanes, 41 rotor blades<br />
*Approximated engine: 41 stator vanes, 41 rotor blades, making it possible to simulate only 1 stator vane and 1 rotor blade<br />
*Scaling of stator: All stator vanes are scaled by 36/41 = 0.8780 circumferentially.<br />
<br />
=== Other '''advanced''' multi-stage methods ===<br />
<br />
Time-inclinded, Adamszyk stresses ...<br />
<br />
== Heat transfer predictions ==<br />
<br />
Besides listing the general heat transfer mechanisms involved <br />
(namely conduction, convection, radiation)<br />
heat transfer prediction in CFD may be seen as or split into two cases. <br />
<br />
'''Mesh consists of fluid domain(s)''': <br />
you want to know how heat inserted into the fluid domain (e.g. via the flowing medium) changes the temperature field along the flow path. Be it a steady or transient run, a fluid which enters with a given temperature will usually experience temperature variations, for instance, caused by convective boundary conditions, prescribed temperature profiles at flow obstacles etc. <br />
<br />
'''Mesh consists of fluid and solid domain(s)''': <br />
additionally to the above, you want info too wrt the (spatial, temporal or even spectral) temperature field distribution in surrounding, confining or immersed solids like channel walls or heat exchanger tubes. This is also called conjugate heattransfer CHT in the CFD context. CHT requires a good boundary layer resolution, usually the wall mesh needs to be rather refined, to obtain realistic heatflux results at the fluid/solid interface. Flow and heat transfer convergence require different time step settings, to properly capture changes in flow and heat quantities respectively. <br />
<br />
<br />
In either case, verify (strict necessity depends on CFD code used, CFX for instance checks and assists in regard) that model dimensions, boundary conditions and properties are in consistent units, hold appropriate values. Check temperature-dependence of properties and other numbers before the run.<br />
<br />
In heat transfer predictions (depending on the CFD code in use) <br />
besides the flow solver, you may have to activate the thermal solver too, <br />
as a job specification.<br />
<br />
== Acoustics and noise ==<br />
<br />
A whole separate research subject, difficult. <br />
<br />
Tone noise possible. Often run with linearzised solvers in the frequency domain.<br />
<br />
Jet noise possible. Often run with LES or DES simulations that either also resolve the sound waves or couples to a separate acoustic solver.<br />
<br />
Turbomachinery broadband noise not possible yet, or at least a great challenge.<br />
<br />
== Errors and uncertainties ==<br />
<br />
CFD is still a tool which requires that a user has a good understanding of uncertainties and errors that might spoil a CFD similation. There exists no error control in CFD and any CFD simulation must be interpreted by an expirenced user to have some credibility. Without some knowledge about possible errors and how they can be handled a CFD simulation can not be trusted. Errors can occur at different places:<br />
<br />
* Definition of the problem - What needs to be analyzed?<br />
* Selection of the solution strategy - What physical models and what numerical tools should be used?<br />
* Development of the computational model - How should the geometry and the numerical tools be set up?<br />
* Analysis and interpretation of the results - How should the model be analyzed and the results be interpreted?<br />
<br />
There exists many different definitions on errors. In this guide the errors are classified into four types of errors: problem definition errors, model errors, numerical errors and user and code errors. The chapters below describe these errors and give some guidelines on how to avoid them.<br />
<br />
=== Problem definition errors ===<br />
<br />
Problem definition errors are the most basic form of errors. In order to obtain usefull results a CFD simulation must of course analyze the correct problem, have suitable boundary contions and be based on a relevant geometry.<br />
<br />
==== Wrong type of simulation ====<br />
<br />
It is essential to have an overview of the physics involved and how the problem can best be analyzed. Running a 2D simulation in order to understand secondary flows or running a steady simulation in order to understand transient behaviour is of course no use. When assessing a CFD simulation the first thing to consider is what physical phenomena are important for the results and if the selected type of simulation is suitable to resolve this type of phenomena. For further information about selecting the most suitable type of simulation please see the previous chapter on [[#Deciding what type of simulation to do|deciding what type of simulation to perform]].<br />
<br />
==== Incorrect or uncertain boundary conditions ====<br />
<br />
A common source of errors is that incorrect boundary conditions are used. The boundary conditions must be specified in enough detail in order to resolve all the important physical features in the problem. For further information about specifying boundary conditions please see the previous chapter on [[#Boundary conditions|boundary conditions]].<br />
<br />
==== Geometrical errors ====<br />
<br />
It is almost always necessary to simplify the geometry in some form. When assessing a CFD simulation one should consider how the geometrical simplifications can affect the interesting physical phenomena. Typical geometrical errors are:<br />
<br />
*Simplifications<br />
:Small geometrical features like fillets, small steps or gaps etc. can often be disregarded. When disregarding this type of features one should consider if they might affect the important physics. For example, a very large fillet on the suction-side of a vane might affect corner separations near the end-walls. A large tip-leakage might affect the flow physics significantly in the upper part of a compressor.<br />
*Tolerances and manufacturing discrepancies<br />
:If the geometry has very large tolerances or is manufactured in a way which might produce a non-ideal shape or position it might be necessary to perform additional CFD simulations in order to cover the whole span of possible real geometries.<br />
*Surface conditions - roughness, welds, steps, gaps etc.<br />
:Often CFD simulations assume a perfectly smooth surface. A non-smooth surface which might have welds, steps or even gaps will of course produce different results. If the physical phenomena of interest might depend on the surface conditions these should of course be condidered. Typical phenomena that might be dependent on this type of errors are transition prediction, leakage flows etc.<br />
<br />
=== Model errors ===<br />
<br />
Errors related to the computational model.<br />
<br />
==== Wrong physical models ====<br />
<br />
Once the type of simulation has been selected the next step is to select what type of physical models the simulation should use. The following points should be considered:<br />
<br />
* Gas data (incompressible/compressible, perfect gas/real gas, ...)<br />
* Turbulence modeling (type of model, type of near-wall treatment, ...)<br />
* Other models (combustion, sprays, ...)<br />
<br />
When assessing model related errors it is important to know the features of the selected model and think carefully how these features and possible short comings might affect the predicted physical behaviour. Using the wrong turbulence model or combustion model can completely destroy the results of a CFD simulation.<br />
<br />
=== Numerical errors ===<br />
<br />
Errors related to the numerical solution of the developed model. Typical examples of numerical errors are discretization errors, convergence errors and round-off errors.<br />
<br />
==== Discretization errors ====<br />
<br />
Discretization errors can either be spatial errors in space or temporal errors in time.<br />
<br />
Spatial discretization errors are what people normally call discretization errors. These errors are due to the difference between the exact solution and the numerical representation of the solution in space. Describing exactly what discretization different codes use and what errors this might lead to is not possible here. Instead some general rules to avoid these errors can be summarized as:<br />
* Use at least a 2nd order accurate scheme, preferably a 3rd order accurate scheme. Some general purpose codes have a 1st order upwind scheme as default, this is a very diffusives schemes that often gives too smooth results.<br />
* For new applications always run a simulation with a finer mesh to see how grid independent your solution is.<br />
* Be aware of checker-board errors. Checker-board errors occur close to strong shocks and other large discontinuities and can be seen as a wavy pattern with a wavelength of two cells. Some schemes, especially those who behave like central differencing schemes, are more prone to checker-board errors. Upwind schemes are a bit better and schemes like TVD or chock-capturing schemes are even better.<br />
<br />
Temporal discretization errors mainly effect transient simulations. However, some codes use a time-marching method also for steady simulations and then a temporal discretization error might affect the final steady solution slightly. The discretization in time can be done with 1st or 2nd order schemes or a Runge-Kutta method, which is more accurate and saves memory. Some codes can adapt the time-step, but often it is necessary to prescribe a time-step in advance. Think of the time-step as your grid in time and make sure that the grid-resolution in time is fine enough to resolve the highest frequencies. To avoid problems with temporal discretization errors the following should be considered:<br />
* Use at least a 2nd order scheme in time.<br />
* Do a physical estimation of the typical frequencies in time of the phenomena that you are interested in and select a time-step which is fine enough to resolve these frequencies well. After the simulation also look at the frequencies captured and make sure that they are well resolved by the chosen time-step.<br />
* For new applications try a finer time-step to ensure that your solution in time is fairly grid independent in time.<br />
<br />
==== Convergence errors ====<br />
<br />
To judge when a CFD simulation is converged is not always that easy. Different codes and different applications behave very differently. For a pure aero-simulation on a fairly coarse grid convergence is easy to judge, but for more complex simulations involving resolved boundary layers, heat transfer, combustion etc. convergence can be very tricky. Aside from looking at residuals one should always also look at how global parameters like static pressure distributions, total pressure losses, skin friction, heat transfer etc. change in time.<br />
For more information about how to avoid convergence errors see the previous chapter about [[#Convergence criteria]].<br />
<br />
==== Round-off errors ====<br />
<br />
When using single precision care needs to be taken to avoid round-off errors. Inviscid Euler simulations and simulations using wall-function meshes can most often be performed in single precision. For well resolved boundary layers with Y plus close to 1 it is often necessary to use double precision. If using double precision for very fine mesh resolutions make sure that you also create the mesh in double precision and not just run the solver in double precision. Sometimes a single precision solver converges slower than a double precision solver due to numerical errors caused by round-off errors. When using advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions it is also often necessary to use double precision.<br />
<br />
=== User and code errors ===<br />
<br />
Errors related to bugs in the code used or mistakes made by the CFD engineer.<br />
<br />
=== What to trust and what not to trust ===<br />
<br />
CFD is generally quite good at predicting surface static pressure distributions. With care CFD can also be used to predict performance, total-pressure losses and blade turning. <br />
<br />
Predicting separation, stall and off-design performance can be a challenge and results with non-attached flows should be interpreted with care. <br />
<br />
Heat transfer is often very difficult to predict accurately and it is common to obtain heat-transfer coefficients that are 100% wrong or more. Validation data is critical in order to be able to trust heat transfer simulations.<br />
<br />
Transition is almost impossible to predict accurately in general. However, there exist models that have been tuned to predict transition and these tend to give acceptable results for cases close to the ones they were tuned for.<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Abu-Ghannam, B.J. and Shaw, R.|year=1980|title=Natural Transition of Boundary Layers - the Effects of Turbulence, Pressure Gradient, and Flow History|rest=Journal of Mech. Eng. Science, vol. 22, no. 5, pp. 213–228}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1985|title=Model Equation for Simulating Flows in Multistage Turbomachinery|rest=ASME Paper 85-GT-226, also NASA TM-86869}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1999|title=Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design|rest=ASME Paper 99-GT-80.}}<br />
<br />
* {{reference-paper|author=Baldwin, B. S. and Lomax, H.|year=1978|title=Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows|rest=AIAA Paper 78-257}}<br />
<br />
* {{reference-paper|author=Chen, J. P. and Barter, J. W.|year=1998|title=Comparison of Time-Accurate Calculations for the Unsteady Interaction in Turbomachinery Stage|rest=AIAA Paper 98-3292}}<br />
<br />
* {{reference-paper|author=Denton, J. and Singh, U.|year=1979|title=Time Marching Methods for Turbomachinery Flow Calculations|rest=VKI Lecture Series 1979-7, von Karman Institute}}<br />
<br />
* {{reference-paper|author=Erdos, J., Alzner, E. and McNally, W.|year=1977|title=Numerical Solution of Periodic Transonic Flow Through a Transonic Fan Stage|rest=AIAA Journal, vol. 15, no. 11, pp. 1559-1568}}<br />
<br />
* {{reference-paper|author=Gerolymos, G., Vinteler, D., Haugeard, R., Tsange, G. and Vallet, I.|year=1996|title=On the Computation of Unsteady Turbomachinery Flows – Part II – Rotor/Stator Interaction using Euler Equations|rest=AGARD Report CP-571}}<br />
<br />
* {{reference-paper|author=Giles, M. B.|year=1990|title=Stator/Rotor Interaction in a Transonic Turbine|rest=AIAA Journal of Propulsion and Power, vol. 6, no. 5}}<br />
<br />
* {{reference-paper|author=He, L, and Ning, W.|year=1998|title=Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines|rest=AIAA Journal, vol. 36, no. 11, pp. 2005-2012}}<br />
<br />
* {{reference-paper|author=Mayle, R. E.|year=1991|title=The Role of Laminar-Turbulent Transition in Gas Turbine Engines|rest=ASME Journal of Turbomachinery, vol. 113, pp. 509-537}}<br />
<br />
* {{reference-paper|author=Menter, F.R. |year=1994|title=Two-equation eddy-viscosity turbulence models for engineering applications|rest=AIAA Journal, vol. 32, pp. 269-289}}<br />
<br />
* {{reference-paper|author=Savill, M., Dick, E., Hanjalić, K. and Voke, P.|year=2002|title=Synthesis Report of the ERCOFTAC Transition Modeling – TRANSPRETURB Thematic Network Activities 1998-2002|rest=Ecoftac Bulletin 54, September 2002, pp. 5-16}}<br />
<br />
* {{reference-paper|author=Wang, X. and Chen J.|year=2004|title=A Post-Processor to Render Turbomachinery Flows Using Phase-Lag Simulations|rest=AIAA Paper 04-615}}<br />
<br />
== External links ==<br />
<br />
* [http://www.qnet-cfd.net/newsletter/8th/n8_40-46.pdf QNET-CFD Best Practise Advice for Turbomachinery Internal Flows]<br />
* [http://www.qnet-cfd.net/newsletter/7th/n7_05.pdf State of the art in Industrial CFD for Turbomachinery Flows, QNET-CFD Network Newsletter, Volume 2, No. 3 – December 2003]<br />
* [https://pronet.wsatkins.co.uk/marnet/guidelines/guide.html MARNET-CFD Best Practice Guidelines for Marine Applications of CFD]<br />
* [http://www.stanford.edu/group/ctr/ResBriefs/temp05/weide.pdf On Large Scale Turbomachinery Computations, by E. van der Weide et. al., Center for Turbulence Research, Annual Research Briefs 2005]</div>DavidFhttp://www.cfd-online.com/Wiki/CombustionCombustion2008-09-17T15:21:12Z<p>DavidF: /* Conservation Laws */</p>
<hr />
<div>''The power of Fire, or Flame, for instance, which we designate by some trivial chemical name, thereby hiding from ourselves the essential character of wonder that dwells in it as in all things, is with these old Northmen, Loke, a most swift subtle Demon of the brood of the J\"otuns... From us too no Chemistry, if it had not Stupidity to help it, would hide that Flame is a wonder. What is Flame?''<br />
<br />
'''''Carlyle on''''' Heroes '''''Odin and Scandinavian Mythology.'''''<br />
<br />
<br />
== What is combustion -- physics versus modelling ==<br />
<br />
Combustion phenomena consist of many physical and chemical processes which exhibit a <br />
broad range of time and length scales. A mathematical description of combustion is not <br />
always trivial, although some analytical solutions exist for simple situations of <br />
laminar flame. Such analytical models are usually restricted to problems in zero or one-dimensional space.<br />
<br />
= Fundamental Aspects =<br />
<br />
== Main Specificities of Combustion Chemistry ==<br />
<br />
Combustion can be split into two processes interacting with each other: thermal, and chemical. <br />
<br />
The chemistry is highly exothermal (this is the reason of its use) but also highly temperature dependent, thus highly self-accelerating. In a simplified form, combustion can be represented by a single irreversible reaction involving 'a' fuel and 'an' oxidizer:<br />
:<math> \frac{\nu_F}{\bar M_F}Y_F + \frac{\nu_O}{\bar M_O} Y_O \rightarrow Product + Heat </math><br />
Althgough very simplified compared to real chemistry involving hundreds of species (and their individual transport properties) and elemental reactions, this rudimentary chemistry has been the cornerstone of combustion analysis and modelling.<br />
<br />
The most widely used form for the rate of the above reaction is the Arrh&eacute;nius law:<br />
:<math> \dot\omega = \rho A \left ( \frac{Y_F}{\bar M_F} \right )^{n_F} \left (\frac{Y_O}{\bar M_O}\right )^{n_O} \exp^{-T_a/T} </math><br />
<math> T_a </math> is the activation temperature, high in combustion, consistently with the temperature dependence.<br />
This is where the high non-linearity in temperature is modelled. ''A'' is the pre-exponential constant. One of the interpretation of the Arrh&eacute;nius law comes from gas kinetic theory: the number of molecules whose kinetic energy is larger than the minimum value allowing a collision to be energetic enough to trig a reaction is proportional to the exponential term introduced above divided by the square root of the temperature. This interpretation allows one to think that the temperature dependence of ''A'' is very weak compared to the exponential term. ''A'' is eventually considered as constant.<br />
The reaction rate is also naturally proportional to the molecular density of each of the reactant. Nonetheless, the orders of reaction <math> n_i</math> are different from the stoichiometric coefficients as the single-step reaction is global, not governed by collision for it represents hundreds of elementary reactions.<br />
If one goes into the details, combustion chemistry is based on chain reactions, decomposed into three main steps: (i) generation (where radicals are created from the fresh mixture), (ii) branching (where products and new radicals appear from interaction of radicals with reactants), and (iii) termination (where radicals collide and turn into products). The branching step tends to accelerate the production of active radicals (autocatalytic). The impact is nevertheless small compared to the high non-linearity in temperature. This explains why single-step chemistry has been sufficient for most of the combustion modelling work up to now.<br />
<br />
The fact that a flame is a very thin reaction zone separating, and making the transition between, a frozen mixture and an equilibrium is explained by the high temperature <br />
dependence of the reaction term, modelled by a large activation temperature, and a large heat release (the ratio of the burned and fresh gas temperatures is about 7 for typical hydrocarbon flames) leading to a sharp self-acceleration in a very narrow area. To evaluate the order of magnitude of the quantities, the terms in the exponential argument are normalized:<br />
:<math> \beta=\alpha\frac{T_a}{T_s} \qquad \alpha=\frac{T_s-T_f}{T_s} </math><br />
<math>\beta</math> is named the Zeldovitch number and <math>\alpha</math> the heat release factor. <br />
Here, <math> T_s</math> has been used instead of <math> T_b</math>, the conventional notation for burned gas temperature (at final equilibrium). <math> T_s</math> is actually <math> T_b</math> <br />
for a mixture at stoichiometry and when the flame is adiabatic, i.e. this is the reference highest temperature that can be<br />
obtained in the system. That said, typical value for <math>\beta</math> and <math>\alpha</math> are 10 and 0.9, giving <br />
a good taste of the level of non-linearity of the combustion process with respect to temperature. <br />
Actually, the reaction rate is rewritten as:<br />
:<math> \dot\omega = \rho A \left ( \frac{Y_F}{\bar M_F} \right )^{n_F} \left (\frac{Y_O}{\bar M_O}\right )^{n_O} <br />
\exp^{-\frac{\beta}{\alpha}}\exp^{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}} </math><br />
where the non-dimensionalized temperature is:<br />
:<math>\theta=\frac{T-T_f}{T_s-T_f}</math><br />
The non-linearity of the reaction rate is seen from the exponential term:<br />
:* <math> {\mathcal O}(\exp^{-\beta}) </math> for <math>\theta</math> far from unity (in the fresh gas)<br />
:* <math> {\mathcal O}(1) </math> for <math>\theta</math> close to unity (in the reaction zone close to the burned gas whose temperature must be close to the adiabatic one <math> T_s </math>), more exactly <math> 1-\theta \sim {\mathcal O}(\beta^{-1})</math><br />
<br />
[[Image:NonLinearite.jpg|thumb|Temperature Non-Linearity of the Source Term: the Temperature-Dependent Factor of the Reaction Term for Some Values of the Zeldovtich and Heat Release Parameters]]Note that for an infinitely high activation energy, the reaction rate is piloted by a <math>\delta(\theta)</math> function. The figure, beside, illustrates how common values of <math>\beta</math> around 10 tend to make the reaction rate singular around <math>\theta</math> of unity. Two set of values are presented: <math><br />
\beta = 10</math> and <math>\beta = 8</math>. The first magnitude is the representative value while the second one is a smoother one usually used to ease numerical simulations. In the same way, two values for the heat release <math>\alpha</math> 0.9 and 0.75 are explored. The heat release is seen to have a minor impact on the temperature non-linearity.<br />
<br />
== Transport Equations ==<br />
<br />
Additionally to the Navier-Stokes equations, at least with variable density, the transport equations for a reacting flow are the energy and species transport equations. In usual notations, the specie ''i'' transport equation is written as:<br />
:<math>\frac{D \rho Y_i}{D t} = \nabla\cdot \rho D_i\vec\nabla Y_i - \nu_i\bar M_i\dot\omega</math><br />
and the temperature transport equation:<br />
:<math>\frac{D\rho C_p T}{Dt} = \nabla\cdot \lambda\vec\nabla T + Q\nu_F \bar M_F \dot\omega </math><br />
The diffusion is modelled thanks to Fick's law that is a (usually good) approximation to the rigorous diffusion velocity calculation. Regarding the temperature transport equation, it is derived from the energy transport equation under the assumption of a low-Mach number flow (compressibility and viscous heating neglected). The low-Mach number approximation is suitable for the deflagration regime (as it will be demonstrated [[#Premixed|below]]), which is the main focus of combustion modelling. Hence, the transport equation for temperature, as a simplified version of the energy transport equation, is usually retained for the study of combustion and its modelling.<br />
<br />
=== Low-Mach Number Equations ===<br />
In compressible flows, when the motion of the fluid is not negligible compared to the speed of sound (which is the speed at which the molecules can reorganize themselves), the heap of molecules results in a local increase of pressure and temperature moving as an acoustic wave. It means that, in such a system, a proper reference velocity is the speed of sound and a proper pressure reference is the kinetic pressure. A contrario, in low-Mach number flows, the reference speed is the natural representative speed of the flow and the reference pressure is the thermodynamic pressure. Hence, the set of reference quantities to characterize a low-Mach number flow is given in the table below:<br />
<br />
Density <math>\rho_o</math> A reference density (upstream, average, etc.)<br />
<br />
Velocity <math>U_o</math> A reference velocity (inlet average, etc.)<br />
<br />
Temperature <math>T_o</math> A reference temperature (upstream, average, etc.)<br />
<br />
Pressure (static) <math>P_o=\rho_o \bar r T_o</math> From Boyle-Mariotte<br />
<br />
Length <math>L_o</math> A reference length (representative of the domain)<br />
<br />
Time <math>L_o/U_o</math><br />
<br />
Energy <math>C_p T_o</math> Internal energy at constant reference pressure <br />
<br />
The equations for fluid mechanics properly adimensionalized can be written:<br />
<br />
Mass conservation:<br />
:<math> \frac{D\rho}{Dt} =0 </math><br />
<br />
Momentum:<br />
:<math>\frac{D\rho\vec U}{Dt}=-\frac{1}{\gamma M}\vec\nabla P+\nabla\cdot\frac{1}{Re}\bar\bar\Sigma</math><br />
<br />
Total energy:<br />
:<math>\frac{D\rho e_T}{Dt}-\frac{1}{RePr}\nabla\cdot\lambda\vec\nabla T=-\frac{\gamma-1}{\gamma}\nabla\cdot P\vec U<br />
+\frac{1}{Re}M^2(\gamma-1)\nabla\cdot \vec U\bar\bar\Sigma + \frac{\rho}{C_pT_o(U_o/L_o)}Q\nu_F \bar M_F\dot\omega</math><br />
<br />
Specie:<br />
:<math>\frac{D\rho Y}{Dt}-\frac{1}{ScRe}\nabla\cdot\rho D\vec\nabla Y=-\frac{\rho}{U_o/L_o}\nu\bar M\dot\omega</math><br />
<br />
State law:<br />
:<math> P=\rho T</math><br />
<br />
The low-Mach number equations are obtained considering that <math> M^2 </math> is small. 0.1 is usually taken as the limit, which recovers the value of a Mach number of 0.3 to characterize the incompressible regime.<br />
<br />
Considering the energy equation, in addition to the terms with <math> M^2 </math> in factor in the equation, the total energy reduces to internal energy as: <math> e_T = T/\gamma+M^2(\gamma-1)\vec U^2/2 </math>. Moreover, the work of pressure is considered as negligible because the gradient of pressure is negligible (low-Mach number approximation is indeed also named ''isobaric'' approximation) and the flow is assumed close to a divergence-free state.<br />
For the same reason, volumic energy and enthalpy variations are assumed equal as they only differ through the addition of pressure. Hence, redimensionalized, the low-Mach number energy equation leads to the temperature equation as used in combustion analysis:<br />
:<math>\frac{D\rho C_p T}{Dt} = \nabla\cdot \lambda\vec\nabla T + Q\nu_F \bar M_F \dot\omega </math><br />
<br />
______________________________<br />
<br />
'''Note:''' The species and temperature equations are not closed as the fields of velocity and density also need to be computed. Through intense heat release in a very small area (the jump in temperature in typical hydrocarbon flames is about seven and so is the drop in density in this isobaric process, and the thickness of a flame is of the order of the millimetre), combustion influences the flow field. Nevertheless, the vast majority of combustion modelling has been developed based on the species and temperature equations, assuming simple flow fields.<br />
<br />
=== The Damk&ouml;hler Number ===<br />
<br />
A flame is a reaction zone. From this simple point of view, two aspects have to be considered: (i) the rate at which it is fed by reactants, let call <math> \tau_d </math> the characteristic time, and<br />
the strength of the chemistry to consume them, let call the characteristic chemical time <math> \tau_c </math>. In combustion, the Damk&ouml;hler number, ''Da'', compares these both time scales and, for that <br />
reason, it is one of the most integral non-dimensional groups:<br />
:<math>Da=\frac{\tau_d}{\tau_c}</math>.<br />
If ''Da'' is large, it means that the chemistry has always the time to fully consume the fresh mixture and turn it into equilibrium. Real flames are usually close to this state. The characteristic reaction time, <math> (Ae^{-T_a/T_s})^{-1} </math>, is <br />
estimated of the order of the tenth of a ms. When ''Da'' is low, the fresh mixture cannot be converted by a too weak chemistry. The flow remains frozen. This situation happens with ignition or misfire, for instance. <br />
<br />
The picture of a deflagration lends itself to a description based on the Damk&ouml;hler number. A reacting wave progresses towards the fresh mixture through preheating of the upstream closest layer. The elevation of the temperature strengthens the chemistry and reduces its characteristic time such that the mixture changes from a low-''Da'' region (far upstream, frozen) to a high-''Da'' region in the flame (intense reaction to equilibrium).<br />
<br />
== Conservation Laws ==<br />
<br />
The processus of combustion transforms the chemical enthalpy into sensible enthalpy (i.e. rise the temperature of the gases thanks to the heat released). Simple relations can be drawn between species and temperature by studying the source terms appearing in the above equations:<br />
:<math> \frac{Y_F}{\nu_F\bar M_F} - \frac{Y_O}{\nu_O\bar M_O} = \frac{Y_{F,u}}{\nu_F\bar M_F} - \frac{Y_{O,u}}{\nu_O\bar M_O} </math><br />
:<math> Y_F + \frac{Cp T}{Q} = Y_{F,u} + \frac{CpT_u}{Q} </math><br />
Hence <math> T_b = T_u + \frac{Q Y_{F,u}}{Cp} </math>, <math> Y_{O,b} = Y_{O,u} - \frac{\nu_O \bar M_O}{\nu_F \bar M_F} Y_{F,u} = Y_{O,u} -sY_{F,u} </math> and <math> Y_{F,b} = 0 </math>. Here, the example has been taken for a lean case.<br />
<br />
As mentioned in [[#Main Specificities of Combustion Chemistry|Sec. Main Specificities]], the stoichiometric state is used to non-dimensionalize the conservation equations:<br />
:<math> Y_i^* = Y_{i,u}^* - \theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} \qquad ; \qquad Y_i^*=Y_i/Y_{i,s}</math>.<br />
<br />
A comprehensive form of the reaction rate can be reconstituted to understand the difficulty of numerically resolving the reaction zone:<br />
:<math> \dot\omega = B \prod_{i=O,F}(Y_{i,u}^*-\theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_i} \exp{\left ( -\beta\frac{1-\theta}{1-\alpha(1-\theta)}\right)} </math><br />
where <math> B </math> stands for all the constant terms present in this reaction rate, plus density.<br />
<br />
[[Image:NonLineariteii.jpg|thumb|Source Term versus Temperature]]<br />
For the stoichiometric case and a global order of two, the reaction rate is graphed versus the reduced temperature for different values of the heat release and Zeldovitch parameter. A high value of <math>\beta</math> makes the reaction rate very sharp, versus temperature. It means that reaction is significant beyond a temperature level (sometimes called ignition temperature) that is close to one (the exponential term above is non-negligible for <math>1-\theta \sim \beta^{-1}</math>). The heat release has qualitatively the same impact but not so strong. Transposed to the case of a flame sheet, it effectively shows that the reaction exists only in a fraction of the thermal thickness of the flame (the region close to the flame that the latter preheats, hence, where the reduced temperature rises from 0 to 1 here) where the temperature deviates few from the maximal one (density can be assumed as constant and equal to its burned-gas value). Numerically capturing such a sharp reaction zone can be costly and the lower values of <br />
<math>\beta</math> and <math>\alpha</math> as presented here are usually preferred whenever possible.<br />
<br />
<br />
Most problems in combustion involve turbulent flows, gas and liquid <br />
fuels, and pollution transport issues (products of combustion as well as for example noise <br />
pollution). These problems require not only extensive experimental <br />
work, but also numerical modelling. All combustion models must be validated <br />
against the experiments as each one has its own drawbacks and limits. In this article, we will address the modeling fundamentals only.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new dimensionless parameters are introduced, the most important of which are the [[Karlovitz number]] and the [[Damkohler number|Damk&ouml;hler number]] which represent ratios of chemical and flow time scales, and <br />
the [[Lewis number]] which compares the diffusion speeds of species.<br />
The combustion models are often classified on their capability to deal with the different combustion regimes. <br><br />
[[Image:SUMMARY_OF_COMBUSTION_MODELS.jpg]]<br />
<br />
= Three Combustion Regimes =<br />
<br />
Depending on how fuel and oxidizer are brought into contact in the combustion system, different combustion modes or regimes are identified. Traditionally, two regimes have been recognized: the ''premixed'' regime and the ''non-premixed'' regime. Over the last two decades, a third regime, sometime considered as a hybrid of the two former ones to a certain extend, has risen. It has been named ''partially-premixed'' regime.<br />
<br />
== The Non-Premixed Regime ==<br />
[[Image:DiffusionFlame.jpg|thumb|Sketch of a diffusion flame]]<br />
This regime is certainly the easiest to understand. Everybody has already seen a lighter, candle or gas-powered stove. Basically, the fuel issues from a nozzle or a simple duct into the atmosphere. The combustion reaction is the oxidization of the fuel. Because fuel and oxidizer are in contact only in a limited region but are separated elsewhere (especially in the feeding system) this configuration is the safest. The non-premixed flame has some other advantages. By controlling the flows of both reactants, it is (theoretically) possible to locate the stoichiometric interface, and thus, the location of the flame sheet. Moreover, the strength of the flame can also be controlled through the same process. Depending on the width of the transition region from the oxidizer to the fuel side, the species (fuel and oxidizer) feed the flame at different rates.<br />
This is because the diffusion of the species is directly dependent on the unbalance (gradient) of their distribution. A sharp transition from fuel to oxidizer creates intense diffusion of those species towards the flame, increasing its burning rate. This burning rate control through the diffusion process is certainly one of the reasons of the alternate name of such a flame and combustion mode: ''diffusion'' flame and ''diffusion'' regime.<br />
<br />
Because a diffusion flame is fully determined by the inter-penetration of the fuel and oxidizer streams, it has been convenient to introduce a tracer of the state of the mixture. This is the role of the ''mixture fraction'', usually called ''Z'' or ''f''. Z is usually taken as unity in the fuel stream and is null in the oxidizer stream. It varies linearly between this two bounds such that at any point of a frozen flow the fuel mass fraction is given by <math> Y_F=ZY_{F,o} </math> and the oxidizer mass fraction by <math> Y_O = (1-Z)Y_{O,o} </math>. <math> Y_{F,o} </math> and <math> Y_{O,o} </math> are the fuel and oxidizer mass fractions in the fuel and oxidizer streams, respectively.<br />
The mixture fraction posses a transport equation that is expected to not have any source term as a tracer of a mixture must be a conserved scalar. First, the fuel and oxidizer mass fraction transport equations are written in usual notations:<br />
:<math>\frac{D \rho Y_F}{D t} = \nabla\cdot \rho D\vec\nabla Y_F - \nu_F\bar M_F\dot\omega</math><br />
:<math>\frac{D \rho Y_O}{D t} = \nabla\cdot \rho D\vec\nabla Y_O - \nu_O\bar M_O\dot\omega</math><br />
The two above equations are linearly combined in a single one in a manner that the source term disappears:<br />
:<math>\frac{D \rho (\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)}{D t} = \nabla\cdot \rho D\vec\nabla (\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)</math><br />
The quantity <math>(\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)</math> is thus a conserved scalar. The last step is to normalize it such that it equals unity in the pure fuel stream (<math> Y_F=Y_{F,o}</math> and <math> Y_O=0 </math>) and is null in the pure oxidizer stream <br />
(<math> Y_F=0</math> and <math> Y_O=Y_{O,o} </math>). The resulting normalized passive scalar is the mixture fraction:<br />
:<math>Z=\frac{\Phi(Y_F/Y_{F,o})-(Y_O/Y_{O,o})+1}{\Phi+1}\qquad \Phi=\frac{\nu_O\bar M_O Y_{F,o}}{\nu_F\bar M_F Y_{O,o}}</math><br />
governed by the transport equation<br />
:<math>\frac{D \rho Z}{D t} = \nabla\cdot \rho D\vec\nabla Z </math><br />
The stoichiometric interface location (and thus the approximate location of the flame if the flow is reacting) is where <math>\Phi(Y_F/Y_{F,o})-(Y_O/Y_{O,o}) </math> vanishes (or <math>Y_F</math> and <math> Y_O </math> are both null in the reacting case). This leads to a stoichiometry definition:<br />
:<math>Z_s=\frac{1}{1+\Phi}</math><br />
<br />
As the mixture fraction qualifies the degree of inter-penetration of fuel and oxidizer, the elements originally present in these molecules are conserved and can be directly traced back to the mixture fraction. This has led to an alternate defintion of the mixture fraction, based on ''element conservation''.<br />
First, the elemental mass fraction <math> X_{j} </math> of element ''j'' is linked to the species mass fraction <math> Y_i </math>:<br />
:<math> X_j = \sum_{i=1}^{n} \frac{a_{i,j} \bar M_j}{\bar M_i} Y_i </math><br />
where <math> a_{i,j} </math> is a matrix counting the number of element ''j'' atoms in specie molecule named ''i'' and ''n'' is the number of species in the mixture.<br />
The group pictured by the summation above is a linear combination of <math> Y_i </math>. Because the transport equations of species mass fraction, few lines earlier, are also linear, a transport equation for the elemental mass fraction can be written:<br />
:<math>\frac{D \rho X_j}{D t} = \nabla\cdot \sum_{i=1}^n\frac{a_{i,j}\bar M_j}{\bar M_i}\rho D_i\vec\nabla Y_i</math><br />
For mass is conserved, the linear combination of the source terms vanishes. Furthermore, by taking the same diffusion coefficient <math> D_i </math> for all the species, the elemental mass fraction transport equation has exactly the same form as the specie transport equation (except the source term). Notice that the assumption of equal diffusion coefficient was also made in the previous definition of the mixture fraction and is justified in turbulent combustion modelling by the turbulence diffusivity flattening the diffusion process in high Reynolds number flows. Hence, the elemental mass fraction transport equation has the same structure as the mixture fraction transport equation seen above. Properly renormalized to reach unity in the fuel stream and zero in the oxidizer stream, the elemental mass fraction is a convenient way of determining the mixture fraction field in a flow. Indeed, it is widely used in practice for this purpose.<br />
<br />
==== Dissipation Rate ====<br />
A very important quantity, derived from the mixture fraction concept, is the ''scalar dissipation rate'', usually noted: <math> \chi </math>. In the above introduction to non-premixed combustion, it has been said that a diffusion flame is fully controlled through: (i) the position of the stoichiometric line, dictating where the flame sheet lies; (ii) the gradients of fuel on one side and oxidizer on the other side, dictating the feeding rate of the reaction zone through diffusion and thus the strength of combustion. According the the mixture fraction definition, the location of the stoichiometric line is naturally tracked through the <math>Z_s</math> iso-line and it is seen here how the mixture fraction is a convenient tracer to locate the flame.<br />
In the same manner, the mixture fraction field should also be able to give information on the strength of the chemistry as the gradients of reactants are directly linked to the mixture fraction distribution. The feeding rate of the reaction zone is characterized through the inverse of a time. Because it is done through diffusion, it must be obtained through a combination of mixture fraction gradient and diffusion coefficient (dimensional analysis):<br />
:<math> \chi_s = \frac{(\rho D)_s}{\rho_s} ||\vec \nabla Z||^2_s \qquad (\rho D)_s = \left ( \frac{\lambda}{C_p} \right )_s</math><br />
where the subscript ''s'' refers to quantities taken effectively where the reacting sheet is supposed to be, close to stoichiometry. This deduction of the scalar dissipation rate, scaling the feeding rate of the flame, is obtained here through physical arguments and is to be derived from equations below in a more mathematical manner. Note that the transport coefficient for the mixture fraction is identified to the one for temperature. This notation is usually used in the literature to emphasize that the rate of temperature diffusion (that is commensurable to the rate of species diffusion) is the retained parameter (as introduced in the following approach highlighting the role of the scalar dissipation rate).<br />
<br />
Because combustion is highly temperature-dependent, ''T'' is certainly the scalar to which attention must be paid for. The temperature equation in the Low-Mach Number regime ([[#Low-Mach Number Equations|Sec. Low-Mach Number Equations]]) is written below in steady-state:<br />
:<math>\rho\vec U \cdot \vec\nabla T = \nabla\cdot \frac{\lambda}{C_p}\vec\nabla T + \frac{Q}{C_p}\nu_F \bar M_F \dot\omega </math><br />
In order to make this equation easily tractable, the Howarth-Dorodnitzyn transform and the Chapman approximation are applied.<br />
In the Chapman approximation, the thermal dependence of <math>\lambda/C_p</math> is approximated as <math>\rho^{-1}</math>.<br />
The Howarth-Dorodnitzyn transform introduces <math>\rho</math> in the space coordinate system: <math> \vec\nabla \rightarrow \rho\vec\nabla \quad ; \quad \nabla\cdot\rightarrow \rho\nabla\cdot</math>. The effect of these both mathematical operations is to `digest' the thermal variation of quantities such as density or transport coefficient. Hence, the temperature equation comes in a simpler mathematical shape:<br />
:<math>\frac{\rho}{\rho_s}\vec U \cdot \vec\nabla T = \left ( \frac{\lambda}{C_p}\right )_s \nabla\cdot \vec\nabla T + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
Here the references are taken in the flame, i.e. close to the stoichiometric line (''s'' subscript).<br />
<br />
<br />
<br />
In a non-premixed system, strictly speaking, <math> \vec U </math>, is not really relevant as the flame is fully controlled by the diffusion process. Notwithstanding, in practice, non-premixed flames must be stabilized by creating a strain in the direction of diffusion. This is the reason why the velocity is left in the equation. Because a diffusion flame is fully described by the mixture fraction field, a change in coordinate can be applied:<br />
:<math> \left (<br />
\begin{array}{c}<br />
x \\<br />
y<br />
\end{array}\right ) \longrightarrow <br />
\left (<br />
\begin{array}{c}<br />
x \\<br />
Z<br />
\end{array}\right )<br />
</math><br />
where ''x'' is the coordinate tangential to the iso-<math>Z_s</math> (hence to the flame, in a first approximation) and ''y'' is perpendicular.<br />
The Jacobian of the transform is given as:<br />
:<math> \left [<br />
\begin{array}{cc}<br />
\frac{\partial x}{\partial x} & \frac{\partial Z}{\partial x} \\<br />
\frac{\partial x}{\partial y} & \frac{\partial Z}{\partial y} <br />
\end{array}\right ] = <br />
\left [<br />
\begin{array}{cc}<br />
1 & 0 \\<br />
0 & l_d^{-1} <br />
\end{array}\right ]</math><br />
Note that the diffusive layer of thickness <math>l_d</math> is defined as the region of transition between fuel and oxidizer and is thus given by the gradient of ''Z'' along the ''y'' direction.<br />
This transform is applied to the vectorial operators:<br />
:<math> \nabla\cdot = \nabla_x\cdot+\nabla_y\cdot=\nabla_x\cdot+\nabla_y\cdot Z\nabla_Z\cdot</math><br />
:<math> \vec\nabla = \vec\nabla_x+\vec\nabla_y=\vec\nabla_x+\vec\nabla_y Z\nabla_Z\cdot</math><br />
<br />
With this transform, the above temperature equation looks like:<br />
:<math>\frac{\rho}{\rho_s}\vec U \cdot (\vec\nabla_x T + \vec\nabla Z \nabla_Z\cdot T) = \left (\frac{\lambda}{C_p}\right )_s \left ( \nabla_x\cdot \vec\nabla_x T + \nabla_y\cdot Z \nabla_Z\cdot \vec\nabla_y Z \nabla_Z\cdot T + \nabla_y\cdot Z \nabla_Z\cdot \vec\nabla_x T + \nabla_x\cdot \vec\nabla_y Z \nabla_Z\cdot T \right ) + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
As mentioned above, the velocity and the variation along the tangential direction to the main flame structure ''x'' are not supposed to play a major role. By emphasizing the role of the gradient of ''Z'' along ''y'' as a key parameter defining the configuration the following equation is obtained:<br />
:<math>0 = \left (\frac{\lambda}{C_p}\right )_s ||\vec\nabla_y Z||^2 \Delta_Z T + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
This equation (sometimes named the ''flamelet equation'') serves as the basic framework to study the structure of diffusion flames. It highlights the role of the dissipation rate with respect to the strength of the source term and shows that the dissipation rate calibrates the combustion intensity.<br />
<br />
To describe the structure of the diffusion flame, the ''reduced mixture fraction'' is set:<br />
:<math> \xi = \frac{Z-Z_s}{Z_s(1-Z_s)\varepsilon} </math><br />
The utility of the reduced mixture fraction is to focus on the reaction zone. This reaction zone is supposed located on the stoichiometric line (this is why the reduced mixture fraction is centred on <math>Z_s</math>) and to be very thin (reason of the introduction of the magnifying factor <math>\varepsilon</math>).<br />
<br />
== The Premixed Regime ==<br />
[[Image:Premixed.jpg|thumb|Sketch of a premixed flame]]<br />
In contrast to the non-premixed regime above, the reactants are here well mixed before entering the combustion chamber. Chemical reaction can occur everywhere and this flame can propagate upstream into the feeding system as a subsonic (deflagration regime) chemical wave. This presents lots of safety issues. Some situations prevent them: (i) the mixture is made too rich (lot of fuel compared to oxidizer) or too lean (too much oxidizer) such that the flame is close to its flammability limits (it cannot easily propagate); (ii) the feeding system and regions where the flame is not wanted are designed such that they impose strong heat loss to the flame in order to quench it. For a given thermodynamical state of the mixture (composition, temperature, pressure), the flame has its own dynamics (speed, heat release, etc) on which there is few control: the wave exchanges mass and energy through diffusion process in the fresh gases. On the other hand, those well defined quantities are convenient to describe the flame characteristics. The mechanism of spontaneous propagation<br />
towards fresh gas through the thermal transfer from the combustion zone to the immediate slice of fresh gas<br />
such that the ignition temperature is eventually reached for this latter was highlighted as early as by the end of the 19th century by Mallard and LeChatelier. <br />
The reason the chemical wave is contained in a narrow region of reaction propagating upstream is the consequence of the discussion on the non-linearity of the combustion with temperature in the <br />
[[#Fundamental Aspects|Sec. Fundamental Aspects]]. It is of interest to compare the orders of magnitude of the temperature dependent term <math> \exp{(-\beta(1-\theta)/(1-\alpha(1-\theta)))}</math> of the reaction source upstream in the fresh gas (<math>\theta\rightarrow 0</math>) and in the reaction zone close to equilibrium temperature (<math>\theta\rightarrow 1 </math>) for the set of representative values: <math> \beta = 10 </math> and <br />
<math>\alpha=0.9</math>. It is found that the reaction is about <math>10^{43}</math> times slower in the fresh<br />
gas than close to the burned gas. It is known that the chemical time scale is about 0.1 ms in the reaction zone of a typical flame, then the typical reaction time in the fresh gas in normal conditions is about <br />
<math> 10^{39} s </math>. To be compared with the order of magnitude of the estimated Universe age: <math> 1 0^{17} s </math>. Non-negligible chemistry is only confined in a thin reaction zone stuck to the hot burned gas at equilibrium temperature. In this zone, the [[#Damk&ouml;hler]] number is high, in contrast to in the fresh mixture. It is natural and convenient to consider that the reaction rate is strictly zero everywhere except in this small reaction zone (one recovers the Dirac-like shape of the reaction profile, <br />
provided that one can see the upstream flow as a region of increasing temperature towards the combustion zone and the downstream flow as in fully equilibrium).<br />
<br />
As the premixed flame is a reaction wave propagating from burned to fresh gases, the basic parameter is known to be the ''progress variable''. In the fresh gas, the progress variable is conventionally put to zero. In the burned gas, it equals unity. Across the flame, the intermediate values describe the progress of the reaction to turn into burned gas the fresh gas penetrating the flame sheet. A progress variable can be set with the help of any quantity like temperature, reactant mass fraction, provided it is bounded by a single value in the burned gas and another one in the fresh gas. The progress variable is usually named ''c'', in usual notations:<br />
:<math>c=\frac{T-T_f}{T_b-T_f}</math><br />
It is seen that ''c'' is a normalization of a scalar quantity. As mentioned above, the scalar transport equations are assumed linear such that the transport equation for ''c'' can be obtained directly. <br />
Actually, the transport equation for ''T'' ([[#Transport Equations|Sec. Transport Equations]]) is linear if constant heat capacity is further assumed (combustion of hydrocarbon in air implies a large excess of nitrogen whose heat capacity is only slightly varying) and the progress variable equation is directly <br />
obtained (here for a default of fuel - lean combustion):<br />
:<math>\frac{D\rho c}{Dt}=\nabla\cdot\rho D\vec\nabla c + \frac{\nu_F\bar M_F}{Y_{F,u}}\dot\omega \qquad \rho D = \frac{\lambda}{C_p} </math><br />
<br />
The fact that the default or excess of fuel has been discussed above leads to the introduction of another quantity: the ''equivalence ratio''. The equivalence ratio, usually noted <math>\Phi</math>, is the ratio of two ratios. The first one is the ratio of the mass of fuel with the mass of oxidizer in the mixture. The second one is the same ratio for a mixture at stoichiometry. Hence, when the equivalence ratio equals unity, the mixture is at stoichiometry. If it is greater than unity, the mixture is named ''rich'' as there is an excess of fuel. In contrast, when it is smaller than unity the mixture is named ''lean''. The equivalence ratio presented here for premixed flames has little connection with the equivalence ratio introduced earlier regarding the non-premixed regime. Basically, the equivalence ratio as defined for non-premixed flames gives the equivalence ratio of a premixed mixture with the same mass of fuel and oxidizer. Moreover, the equivalence ratio as defined for a premixed mixture can be obtained based on the mixture fraction (it is thus the local equivalence ratio at a point in the non-homogeneous mixture described by the mixture fraction). From the definitions given above:<br />
:<math> \Phi=\frac{Z}{1-Z}\frac{1-Z_s}{Z_s}</math><br />
<br />
==== Premixed Flame P&eacute;clet Number ====<br />
<br />
Earlier in this section, it has been said that a premixed flame posses its own dynamics, as a free propagating surface, and has thus characteristic quantities. For this reason, a P&eacute;clet number may be defined, based on these quantities. The P&eacute;clet number has the same structure as the Reynolds number but the dynamical viscosity is replaced by the ratio of the thermal conductivity and the heat capacity of the mixture. The thickness <math>\delta_L </math> of a premixed flame is essentially thermal. It means that it corresponds to the distance of the temperature rise between fresh and burned gases. This thickness is below the millimetre for conventional flames. The width of the reaction zone inside this flame is even smaller, by about one order of magnitude. This reaction zone is stuck to the hot side of the flame due to the high thermal dependency of the combustion reactions, as seen above. Hence, the flame region is essentially governed by a convection-diffusion process, the source term being negligible in most of it.<br />
<br />
It is convenient to write the progress variable transport equation in a steady-state framework. The quantities at flame temperature (<math>_f</math>) are used to non-dimensionalize the equation:<br />
:<math> \overbrace{(\rho S_L)}^{\dot M}\frac{\vec S_L}{S_L}\vec\nabla c = (\rho D)_f \nabla\cdot (\rho D)^* \vec\nabla c</math><br />
Note that the source term is neglected, consistently with what has been said above.<br />
This convection-diffusion equation makes appear a first approximation of a flame P&eacute;clet number:<br />
:<math> Pe_f = \frac{\dot M \delta_L}{(\rho D)_f} \approx 1 </math><br />
<br />
From the P&eacute;clet number, it is possible to obtain an expression for the flame velocity (remembering that <math> \delta_L/S_{L,f} \approx \tau_c</math>, vid. inf. [[#Three Turbulent-Flame Interaction Regimes| Sec. Three Turbulent-Flame Interaction Regimes]]):<br />
:<math> S_{L,f}^2\approx \frac{(\rho D)_f}{\rho_f \tau_c}</math><br />
For typical hydrocarbon flames, the speed is some tens of centimetres per second and the diffusivity is some<br />
<math> 10^{-5} </math> square metres per second. The chemical time in the reaction zone of about one tenth of a millisecond is recovered.<br />
<br />
==== Details of the Premixed Unstrained Planar Flame ====<br />
<br />
A plane combustion wave propagating in a homogeneous fresh mixture is the reference case to describe the premixed regime. At constant speed, it is convenient to see the flame at rest with a flowing upstream mixture. This is actually the way propagating flames are usually stabilized. In the frame of description, the <br />
physics is 1-D, steady with a uniform (the flame is said unstrained) mass flowing across the system. Two types of equations are thus sufficient to describe the problem, the temperature transport equation and the species transport equations, as in [[#Transport Equations|Sec. Transport Equations]]. The transport coefficients will be chosen as equal: <math> \rho D_i = \lambda / C_p </math> (unity Lewis numbers). Suppose the 1-D domain is described thanks to a conventional (''Ox'') axis with a flame propagating towards negative ''x'' (this is the conventional usage), the boundary conditions are:<br />
:* in the frozen mixture: <br />
:** <math>Y_i \rightarrow Y_{i,u} \qquad ; \qquad x\rightarrow-\infty </math><br />
:** <math>T \rightarrow T_u \qquad ; \qquad x\rightarrow-\infty </math><br />
:* in the burned gas region supposed at equilibrium:<br />
:** <math>Y_i \rightarrow Y_{i,b} \qquad ; \qquad x\rightarrow+\infty </math><br />
:** <math>T \rightarrow T_b \qquad ; \qquad x\rightarrow+\infty </math><br />
<br />
:<math>Y_{i,b}</math> and <math>T_b</math> are obtained from [[#Conservation Laws|Sec. Conservation Laws]].<br />
<br />
The quantities that have been mentioned just above (scalar and temperature profiles, mass flow rate through the system) are the solution to be sought.<br />
<br />
According to the discussions above, the temperature transport equation in its full normalized form may be written as (lean /stoichiometric case):<br />
:<math> \dot M \frac{\partial \theta}{\partial x} = \frac{\partial \ }{\partial x}\frac{\lambda}{Cp} \frac{\partial \theta}{\partial x} + \frac{\nu_F \bar M_F B}{Y_{F,s}} \prod_{i=O,F}(Y_{i,u}^*-\theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_i} \exp{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}} </math><br />
This equation is further simplified by the variable change <math>d\xi=\dot M/(\lambda/Cp)dx</math>:<br />
:<math>\frac{\partial \theta}{\partial \xi}=\frac{\partial^2 \theta}{\partial \xi^2} + \overbrace{\frac{\lambda/Cp}{\dot M^2}\frac{\nu_F \bar M_F B}{Y_{F,s}}}^{\Lambda} \prod_{i=O,F}(Y_{i,u}^*-\theta)^{n_i} \exp{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}}</math><br />
<br />
Although somewhat out of scope, the existence and unicity of the solution of this type of equation are usually demonstrated with the help of the Schauder Theorem and Maximum Principle. From the point of view of physicists and engineers, the solution that is found analytically is de facto considered as the unique solution of the equation.<br />
<br />
===== Scenarii of Combustion Process in the Phase Portrait =====<br />
<br />
In the frame moving with the flame, both phase variables are the reduced temperature and its gradient. To ease the reading with usual notations, it is written: <math> X_1 = \theta \quad ; \quad X_2 = \partial \theta / \partial \xi = \dot \theta </math>. The system arises:<br />
: <math> \left\{<br />
\begin{array}{lll}<br />
\dot X_1 & = & X_2 \\<br />
\dot X_2 & = & X_2 - \varpi(X_1) <br />
\end{array}<br />
\right.<br />
</math><br />
with <math> \varpi </math> being the full source term in the above equation.<br />
<br />
In the frame moving with the flame, two singular nodes are found in the frozen flow <math> (X_1,X_2) = (0,0)</math> and the equilibrium region <math> (\theta_b,0)</math>, i.e when <math> \varpi(X_1) </math> vanishes. <br />
<math> x_1,x_2 </math> are defined as small departures from the singular nodes such that the linearized system in their neighbourhood is:<br />
: <math> \left\{<br />
\begin{array}{lll}<br />
\dot x_1 & = & l_{1,1} x_1 + l_{1,2} x_2 \\<br />
\dot x_2 & = & l_{2,1} x_1 + l_{2,2} x_2 <br />
\end{array}<br />
\right.<br />
</math><br />
provided: <math> l_{1,1} = 0 \quad l_{1,2} = 1 \quad l_{2,1} = -\varpi'_{X_1 = 0,\theta_b} \quad l_{2,2} = 1 </math>.<br />
The characteristic polynom is, in usual notations: <math> s^2 - s + \varpi' </math> such that the eigenvalues are:<br />
:<math><br />
s^{\pm}=\frac{1\pm\sqrt{1-4\varpi'}}{2}<br />
</math><br />
A priori, those eigenvalues may be (i) real distinct, (ii) real identical, or (iii) conjugated complex. In the first case, the orbits in the phase diagram are organized, in the immediate neighbourhood of the singular node, with respect to the eigenvectors directions associated to the eigenvalues. The following task is to identify the nature of those eigenvalues and of the corresponding nodes. Because <math> 0 < X_1 < \theta_b </math> is bounded, complex eigenvalues are excluded as they would lead to a spiral node. This remark is important for the node on the cold side as it imposes a bound:<br />
:<math> \varpi'_{X_1=0} \le \frac{1}{4} </math><br />
As the mass flow rate through the flame is included into <math>\varpi</math>, it imposes a minimum value on the flame speed to tackle with the cold boundary difficulty (rise of the chemical rate in the frozen flow). In this condition, it <br />
is an unstable node (improper in case of equality).<br />
On the other hand, because <math> \varpi'|_{X_1=\theta_b} </math> is not positive, the node on the hot side is found as a saddle point. The overall scenario of combustion within the flame is thus an orbit leaving the cold node to join the hot node by branching on a trajectory compatible with the negative eigenvalue of the saddle.<br />
<br />
[[Image:sketchOrbit.jpg|thumb| Sketch of orbits for a combustion process across a premixed front. Dashed lines represent forbidden orbits from the physics. The red line describes the orbit expected in an idealized combustion process.]]<br />
It must be noted that the associated eigenvectors are of the form:<br />
:<math><br />
\left (<br />
\begin{array}{l}<br />
t \\<br />
t s^{\pm} <br />
\end{array}<br />
\right ); \quad t\in \Re^*<br />
</math><br />
that is, on the cold node, a positive departure on <math>X_1</math> following any of the two eigendirections, leads<br />
to a consistent creation of positive temperature gradient, while on the hot node, only the stable direction will allow a consistent creation of a positive temperature gradient for any departure of the temperature towards region where it is inferior to <math>\theta_b </math>. Another remark is the structure of the eigendirections. The leaving directions on the cold node have a slope larger than the one of <math> \varpi </math> while the stable direction of the hot node has a slope smaller than the one of <math> \varpi </math>. It means that there is some point where the orbit must cross the profile of the chemical term versus temperature. For that temperature, the gradient equals the reaction rate through construction of the phase space. When looking back to the equation of the premixed flame, it happens in a region of inflexion for the temperature (the second order derivative must vanish). Furthermore, at this intersection, the orbit is horizontal (if the frame of reference for <math> (X_1, X_2) </math> is Cartesian) due to the shape of the premixed flame equation above that can be recast into <math> X_{2,X_1}' = (X_2-\varpi)/X_2 </math>.<br />
Close to the cold node, the orbits have a shape of parabola whose axis is the direction with the largest eigenvalue magnitude.<br />
Close to the hot node, the orbits have an hyperboloid shape with asymptots as the eigendirections. Now the ingredients are here to draw a sketchy scenario of the combustion in a premixed flame. It will be superimposed on the reaction rate graph studied in [[#Fundamentals Aspects|Sec. Fundamentals]].<br />
Some typical orbits from the above analysis are drawn in the figure on the right. The basic geometrical arguments developed are reproduced. In particular, the dashed lines represent forbidden orbits by the physics (boundedness of <math>X_1</math>, irreversibility). Orbits must be travelled from left to right, corresponding to increasing free parameter <math> \xi </math>.<br />
It is of integral importance to remember that, in combustion in conventional conditions, the source term is highly non-linear. Therefore, it is localized in a very thin sheet and, upstream, <math> \varpi </math> and <math> \varpi' \rightarrow 0 </math>. <br />
Only the most unstable eigendirection of the cold node is compatible as an orbit. The other trajectories, being paraboloidal, are tangent to the other direction that is flat at the limit. It physically means an elevation of temperature in bulk, that is contradictory to what is expected from a highly non-linear combustion term. Hence, the additional orbit in red is the one expected in idealized combustion conventional conditions.<br />
<br />
[[Image:dnsOrbit.jpg|thumb|Sketch of orbits for a combustion process across a premixed front, DNS simulations for the usual combinations of <math>\beta</math> and <math> \alpha </math>. Note that the case with <math>\alpha = 0</math> does not mean an athermal reaction but the mathematical simplification of the denominator of the exponential argument of the source term.]]<br />
The following picture is the result of actual computations of the above 1-D flame equation (stoichiometry and <math> n_i = 1 </math>) with the help of a high-order (6) code. The already presented combinations of <math> \beta </math> and <br />
<math> \alpha </math> are used and the orbits are retrieved. For most of the cases, as predicted above, the system selects a solution leaving the cold node with the most unstable direction (identifiable with its slope close to unity for vanishing<br />
<math>\varpi'_{X_1=0}</math>). There is also an additional curve, for <math> \beta = 10 </math> with the denominator of the exponential argument suppressed. This curve is remarkable as the solution selected by the system leaves the cold node in a manner fully controlled by the <math> \varpi'_{X_1=0} </math>. This is a very singular solution, not expected in combustion in conventional conditions, as explained above. The purpose of this remark is to question the well-posedness of considering simplifying the exponential argument for <math>\beta</math> "sufficiently high", as it is usually proposed for this type of modelling. As observed, the dynamical system analysis demonstrates a switch in the nature of the solution selected.<br />
At the physical level, when the orbit follows the most unstable direction with a slope close to unity, it means that <math> X_2 </math> "follows" <math> X_1 </math>, which is a signature of a diffusion process. In other words, the preheating mechanism of a premixed flame propagation as proposed for more than one century is in work. On the other hand, when <br />
<math> X_2 </math> is dependent on the evolution of <math> \varpi'</math>, it shows that "cold" chemistry drives the solution in the frozen flow and not the acknowledged mechanism of deflagration.<br />
<br />
<br />
===== Flame Solution =====<br />
<br />
As already mentioned, the flame system may be split into three zones. Upstream, the conventional mechanism of deflagration<br />
is supported by diffusion of heat. Downstream, the mixture is at equilibrium after combustion. In between, there exists the reaction layer. For large <math>\beta</math> the reaction layer is very thin such that it can be seen as a discontinuity between the fresh and burned gases. This is this difference in scales that introduces the use of the asymptotic method to resolve some of the flame characteristics, such as speed, time, heat region thickness, or reaction zone thickness.<br />
<br />
The domain is partitioned, according to this zoning defined by the scales driving the physics with an outer domain, driven by large scales and an inner domain refining the description within the discontinuity. If the discontinuity (flame reaction zone) is at <math> \xi=0 </math>, everywhere but 0, the equation is simplified as:<br />
:<math><br />
\frac{\partial^2 \theta}{\partial \xi^2} = \frac{\partial\theta}{\partial \xi}<br />
</math><br />
*For <math> \xi>0 </math>, it is expected that the mixture has reached equilibrium chemistry, such that: <math> \forall \xi > 0, \quad \theta=\theta_b \quad ; \quad \partial \theta / \partial \xi =0 </math>.<br />
*For <math> \xi < 0 </math>, this is the preheat zone and the solution is <math> \theta = \theta_b e^{\xi} </math> with <math> \theta </math> reaching <math>\theta_b</math> at the disconstinuity and vanishing, together with its gradient, far upstream in the frozen mixture.<br />
<br />
The solution for the species and the value of <math> \theta_b </math> are obtained from [[#Conservation Equations|Conservation Equations]] above. The 'big picture' is thus an exponential variation in the thermal thickness matched with a plateau in the downstream region, the line of matching being the discontinuity (flame) that has no thickness at this scale of description.<br />
<br />
To refine the analysis in the discontinuity region, a magnifying factor <math> \varepsilon </math> is used to stretch the coordinates: <math> \xi = \varepsilon \Xi </math>. The inner solution is thus a slowly-varying function of <math> \Xi </math>. Hence, in this inner region, the equation for the premixed flame becomes:<br />
:<math><br />
\frac{1}{\varepsilon}\frac{\partial \theta}{\partial \Xi}=\frac{1}{\varepsilon^2}\frac{\partial^2 \theta}{\partial \Xi^2} +\varpi<br />
</math><br />
In order to stretch and 'look inside' a discontinuity, <math>\varepsilon</math> is very small. It yields two remarks:<br />
# convection is negligible compared to diffusion. The heat losses from the reaction zone are essentially diffusion driven.<br />
# The reaction zone is governed by a diffusion-reaction budget and the reaction term <math>\varpi</math> must be strong to balance the intense heat loss due to the sharp diffusion (the zone is very thin, hence the gradients are sharp).<br />
The mechanism is thus different from the outer region that was convection-diffusion driven.<br />
<br />
Each quantity is developed in a series of <math> \varepsilon </math>.<br />
At the leading order, for <math>\theta</math>, in the lean case, the conservation relations (Sec. [[#Conservation Laws|Conservation Laws]]) yield:<br />
:<math> \theta = 1 - \varepsilon \Gamma - (1 - Y^*_{F,u})</math><br />
where <math>\Gamma</math> is the first-order development of the departure of <math>\theta</math> from the maximum value due to the incomplete combustion, and <math>1-Y_{F,u}^*=1-\theta_b</math> is the reduction of temperature for non-stoichiometric cases. Injected into the above equation:<br />
:<math> \frac{1}{\varepsilon}\frac{\partial^2 \Gamma}{\partial \Xi^2} = \Lambda (\varepsilon \Gamma)^{n_F} (Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} +\varepsilon\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\beta\frac{1-Y_{F,u}^*+\varepsilon\Gamma}{1-\alpha(1-Y_{F,u}^*+\varepsilon\Gamma)}}</math><br />
Although the full develoment is not achieved, a number of scaling may be highlighted:<br />
# because the temperature cannot be much below unity, <math>Y_{F,u}^*</math> must be close to 1 <math> {\mathcal O}(\varepsilon)</math> . For clarity, it is not expanded in an <math>\varepsilon</math> series.<br />
# The denominator of the exponential argument simplifies to unity for small <math>\varepsilon</math>. <br />
# To get a finite rate in the reaction zone, <math>\varepsilon</math> scales with <math>\beta^{-1}</math>.<br />
<br />
The burning rate eigenvalue, <math>\Lambda</math>, is naturally expanded as: <math> \Lambda = \varepsilon^{-n_O-n_F-1}(\Lambda_0 + {\mathcal O}(\varepsilon)) </math>. The low-order equation to be solved is:<br />
:<math><br />
\frac{d^2 \Gamma}{d \Xi^2} = \frac{1}{2}\frac{d(\Gamma_{\Xi})^{'2}}{d\Gamma} = \Lambda_0\exp{-\beta (1-Y_{F,u}^*)} \Gamma^{n_F} (\frac{Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} }{\varepsilon}+\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\Gamma}</math><br />
:<math> \frac{d\Gamma}{d\Xi}(-\infty)=-\theta_b, \qquad \frac{d\Gamma}{d\Xi}(\infty)=0 </math><br />
The boundary conditions are obtained from the matching of the outer solutions on the right and left sides of the flame as written above (the outer solutions are reached at infinity for a very small magnifying factor <math>\varepsilon</math>).<br />
Once integrated with respect to those boundary conditions, the burning-rate eigenvalue (from which <math> \dot M </math> is extracted) is obtained as:<br />
:<math> \Lambda_0 = \Bigg( 2\int_0^{\infty}d\Gamma\; \Gamma^{n_F} (\beta (Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )+\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\beta (1-Y_{F,u}^*)}\exp{-\Gamma} \Bigg )^{-1}<br />
</math><br />
The RHS integral is not developed for clarity but presents no peculiar difficulties.<br />
<br />
The development has been carried out at the first order in <math> \varepsilon </math>. As soon as a second order development is attempted, some expressions are no more analytically tractable. On the other hand, a second order development allows introduce the temperature-dependent trends of some terms in <math> \Lambda </math>. Physical results are retrieved such as a slight decrease of the speed for a positive sensitivity of transport parameters to temperature around equilibrium conditions.<br />
<br />
[[Image:BellSpeed.jpg|thumb|Unstrained planar premixed flame speed with respect to fuel mass fraction (lean case) for a single global irreversible Arrh\'enius term. Symbols are obtained from a high-order DNS code. Continuous line is the theory exposed here. The usual combinations of <math>\beta</math> and <math> \alpha </math> are used. <br />
Very high values of <math> \beta </math> (and thus negligible effect of <math> \alpha </math> are also presented<br />
to show the problem of slow convergence for finite value of the Zeldovitch parameter. Not that <math>\alpha = 0</math> does not mean an athermal reaction but the mathematical simplification of the denominator of the exponential argument of the source term.]]<br />
The image beside illustrates the response of the premixed flame speed with respect to fuel concentration (chosen as the limiting component here; equivalent findings are obtained for default of oxidizer) for different values of the chemical parameters <math> \beta </math> and <math>\alpha</math> (as noted above, the pre-exponential constant <math> A</math> impacts only the overall magnitude). The theoretical expression (continuous line) is tested against a 1-D high accuracy<br />
code with the given chemistry implemented. It is seen that:<br />
* the Zeldovitch parameter drives the drop for non-stoichiomeric mixtures,<br />
* the drop is relatively well modelled by the theoretical expression, and<br />
* the absolute magnitude converges slowly towards the theoretical one when increasing <math>\beta</math> in the code (effect of the finiteness of <math> \beta </math> in the real case).<br />
<br />
[[Image:PremixedProfiles.jpg|thumb|Typical profiles in 1-D premixed flame at stoichiometry. Representative value of global chemistry parameters. Specie profiles are simply the complementary to the temperature profile for simple chemistry.]]<br />
The picture exhibiting profiles for different Zeldovitch and heat release parameters shows the factual impact: upstream, the exact exponential profile is recovered and corresponds to the pre-heating region (thermal thickness). In the reaction zone (just upstream of the extremum), the departure from the exact solution is due to the kinetic effect. This kinetic effect is more pronounced when <math> \beta </math> is lower because the lower the Zeldovitch parameter, the lower the temperature reaction zone can be without leading to extinction. The flame takes this opportunity to maximize its transfer in heat and reactant with the cold zone. This is the physical understanding of an inverse dependence of the maximum flame speed with <math> \beta </math>.<br />
<br />
<br />
==== Modification of the Flame Speed with Curvature ====<br />
<br />
[[Image:premixedCurve2d.jpg|thumb|Flame seen as an interface between fresh and burned gases. Its curved profile towards the burned side increases the transfers with the fresh side.]]<br />
When the thickness of the flame <math>(\lambda/Cp)_f/\dot M </math> is considered small compared to inhomogeneities existing in the flow, the flame can be reduced to an interface between fresh and burned gases. This interface may not be strictly plan in the general case. For instance, when the interface is curved towards the burned gas, it offers a larger opportunity for transfer of mass and heat with the fresh gas. As the ability of the chemistry to burn the coming matter is limited, the flame has thus to reduce its displacement speed. The figure beside provides a 2-D sketch of this situation that happens in contorded turbulent fields. Curvature effect is thus an ingredient appearing in combustion models.<br />
<br />
To mathematically give the expression showing that the curvature influences the flame speed (for small curvature), the non-dimensionalized temperature equation across a planar premixed flame above is slightly recast:<br />
:<math>||\dot M^c||||\vec\nabla_{\xi}\theta|| - \varpi= \nabla_{\xi}\cdot\vec\nabla_{\xi}\theta=-\nabla_{\xi}\cdot(||\vec\nabla_{\xi}\theta||\vec n) = - \vec\nabla_{\xi}||\vec\nabla_{\xi}\theta||\cdot\vec n - ||\vec\nabla_{\xi}\theta||\nabla_{\xi}\cdot\vec n</math><br />
<br />
In this expression, <math>\dot M^c</math> is the flame mass flow rate when perturbed by a curvature normalized by the reference one developed above. <math> \vec n</math> is the normal to the iso-temperature pointing towards the fresh gas, what is named the normal to the flame. From the expression developed above and for a slightly perturbed flame, the gradients, production and diffusion terms are very close to the unperturbed flame. Only the last term, the normal divergence, does not exist for a non-perturbed flame. Hence, the above equation may be simplified against the one for unperturbed flame presented earlier:<br />
:<math> \dot M^c = 1 - \nabla_{\xi}\cdot \vec n </math><br />
This is the divergence of the flame normal that contains the information upon geometrical perturbation (curvature) that impacts the speed.<br />
<br />
[[Image:premixedCurvature.jpg|thumb|Local geometrical approximation of a flame surface]]<br />
The key is thus to get an idea of the geometrical significance of the normal divergence. The normal divergence theorem says that this is the sum of the principal curvatures of the flame surface at the location considered. To give a good mental picture, the simplest configuration without loss of generality is to consider the flame surface approached by an osculatory revolution ellipsoid, as in the figure beside. The location of the approximation is the intersection of the Ox axis and the flame surface in red (where the ellipsoid is tangent to the flame). At this location, the normal divergence is the sum of the curvature of the basic ellipse (before its rotation around Oy) at its minor extremum, and the curvature of the circle corresponding to the rotation of this point around Oy when the 3-D shape is formed. Giving a lecture on conical coordinate systems is beyond the purpose but the interested reader may want to follow the corresponding steps to check this result: (i) write the divergence of a vector in the ellipsoid coordinate system, (ii) consider that the vector is the normal, i.e. it has only one constant component perpendicular to the local ellipsoidal iso-coordinate, to simplify the divergence expression, (iii) split the resulting terms into two parts by identifying the second derivative of the basic shape 2-D ellipse as one principal curvature, and the inverse of the radius of the circle corresponding to the rotation of the ellipse around Oy at the point where the flame surface is approached, in the figure this is simply the minor axis of the ellipse.<br />
<br />
The expression is readily written as:<br />
:<math> \dot M^c = 1 - (R_1^{-1}+R_2^{-1})</math><br />
where <math> R_1 </math> and <math> R_2</math> are the two radius of curvature (non-dimensionalized by the reference flame thickness) local to the surface.<br />
For instance, they are the small axis of the basic ellipse and its radius of curvature at its minor maximum when the <br />
surface is approached by the osculatory ellipsoid as above.<br />
<br />
<br />
==== Natural Instabilities of Premixed Flames ====<br />
<br />
Another aspect accounted for in turbulent combustion modelling of premixed flame is the creation of <br />
flame surface (corrugation) by hydrodynamics instabilities. These instabilities have been mentioned by Darrieus as early as 1938. The motivation of discussing about this is also related to the framework of description used that is widely employed for combustion models development. This framework is named the hydrodynamics limit, where the flame is isolated as <br />
a zero-thickness interface in the flow, and has been first introduced just above. In this framework, any diffusive and energetic aspects disappear and the set of equations is limited to two incompressible Euler systems. One system in the fresh gases (with a constant density of cold mixture). One system in the burned gases (with a constant density of mixture at equilibrium).<br />
<br />
To understand the basic properties of a premixed flame leading to the birth of instabilities, it is first important to realize that a premixed flame, in the hydrodynamic limit, behaves as a dioptre with a refractive index in the burned gases larger than in the fresh gases. For a flow with an angle of attack on the flame (i.e. a flame not strictly 1-D perpendicular to the flow), the tangential component of the flow speed relative to the flame surface is conserved while the normal component is accelerated by a factor corresponding to the ratio of the density in order to conserve mass flow across the interface. Hence, the streamlines are pushed towards the normal to the flame when crossing, that is similar to rays of light entering a refracting medium. <br />
<br />
If one considers a region of a premixed flame that bumps a little bit towards the fresh gas (the same approach is symmetrically true for the bump towards the burned gas), the local stream tube slightly opens on the bumpy interface before being refracted and coming back to its original section at constant mass flow rate. Hence, just in front of the bump, the gas velocity in this stream tube decreases and does not oppose the flame motion, allowing the bump to increase in magnitude. This is the fundamental mechanism of such instabilities.<br />
<br />
The equations sets are in usual notations:<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
U_x + V_y & = & 0 \\<br />
U_t + UU_x + VU_y & = & \frac{P_x}{\rho} \\<br />
V_t +UV_x+VV_y & = & \frac{P_y}{\rho} <br />
\end{array}<br />
\right.<br />
</math><br />
<math> x </math> is the coordinate used above along the flame path and <math>y</math> is in the tangential plan.<br />
<math>u </math> and <math>v</math> are the respective velocities.<br />
Those equations may be normalized by steady-state reference quantities such as flame speed, flame length, gauge pressure and density with respect to fresh gas properties and are written on both sides of the interface.<br />
<br />
These sets of equations must match at the interface (flame surface) through conservation of mass flow rate and momentum:<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
\rho_u (\vec U_u -\vec U_{f})\cdot \vec n & = & \rho_b (\vec U_b -\vec U_{f})\cdot \vec n\\<br />
(\rho_u \vec U_u (\vec U_u -\vec U_{f})+P_u)\cdot \vec n & = & (\rho_b \vec U_b (\vec U_b -\vec U_{f})+P_b)\cdot \vec n<br />
\end{array}<br />
\right.<br />
</math><br />
The subscripts <math> u </math> and <math> b </math> stand for unburned and burned sides, respectively. The subscript <math> f </math> points the interface. In these equations, the flame motion <math> U_f </math> is reintroduced because we want to track the instability movement over the mean position of the flame. This instability motion is described by the equation <math> x=F(y,t) </math> such that the motion of the interface normal to itself is obtained as:<br />
:<math> \vec U_f\cdot\vec n = - \frac{F_t}{\sqrt{1+F^2_y}} </math><br />
with a normal to the front:<br />
:<math> \vec n = \left (<br />
\begin{array}{l}<br />
-\frac{1}{\sqrt{1+F^2_y}} \\<br />
\frac{F_y}{\sqrt{1+F^2_y}} <br />
\end{array}<br />
\right )<br />
</math><br />
<br />
As the instability motion is considered at its birth, i.e. when it is still small, the quantities are linearized with <math> \varepsilon </math> as a small parameter:<br />
:<math> F=\varepsilon f</math><br />
:<math> \vec U = \vec {\mathcal U} + \varepsilon\vec u</math><br />
:<math> P = {\mathcal P} + \varepsilon p </math><br />
<br />
The Eulerian system is reduced to (at the first order in <math>\varepsilon</math>, the order zero is simplified for homogeneous flow):<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
u_x + v_y & = & 0 \\<br />
u_t + {\mathcal U}u_x & = & \frac{p_x}{\rho} \\<br />
v_t +{\mathcal U}v_x & = & \frac{p_y}{\rho} <br />
\end{array}<br />
\right.<br />
</math><br />
<br />
The one-sided equations for the jump conditions become (terms up to the first order in <math>\varepsilon</math> are kept):<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
\rho (\vec U-\vec U_f)\cdot\vec n \times \sqrt{1+F^2_y} & = & -\rho {\mathcal U}-\rho\varepsilon u+\rho\varepsilon f_t \\<br />
(\rho \vec U (\vec U -\vec U_{f})+P)\cdot \vec n \times \sqrt{1+F^2_y} & = &<br />
\left \{<br />
\begin{array}{l}<br />
-\rho {\mathcal U}^2 -2\rho{\mathcal U}\varepsilon u+\rho{\mathcal U}\varepsilon f_t - {\mathcal P} -\varepsilon p \\<br />
-\rho {\mathcal U}\varepsilon v + {\mathcal P}\varepsilon f_y<br />
\end{array}<br />
\right.<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
The following jump conditions are obtained (by recalling that unburned steady-state quantities are chosen as references):<br />
* From the first equation at the order 0, <math>\rho_b U_b = \rho_u U_u \Leftrightarrow U_b=\rho_u/\rho_b</math><br />
* From the first equation at the order 1, <math> \rho_u f_t - \rho_u u_u = \rho_b f_t -\rho_b u_b </math><br />
* From the second equation at the order 0, <math> -\rho_u {\mathcal U}_u^2 - {\mathcal P}_u = -\rho_b {\mathcal U}_b^2 - {\mathcal P}_b \Leftrightarrow {\mathcal P}_b = 1 - \rho_u/\rho_b </math><br />
* From the second equation at the order 1, <math> -2\rho_u{\mathcal U}_u u_u +\rho_u {\mathcal U}_u f_t -p_u = <br />
-2\rho_b{\mathcal U}_b u_b +\rho_b {\mathcal U}_b f_t -p_b \Leftrightarrow p_b-p_u = 2 u_u-2 u_b </math><br />
* From the third equation at order 1 (no order 0), <math> -\rho_u {\mathcal U}_u v_u +{\mathcal P}_u f_y = -\rho_b {\mathcal U}_b v_b +{\mathcal P}_b f_y \Leftrightarrow v_b-v_u = f_y(1-\rho_u/\rho_b) </math><br />
<br />
The solution for the linearized, autonomous Euler system is:<br />
:<math><br />
\left (<br />
\begin{array}{l}<br />
u \\ <br />
v \\ <br />
p<br />
\end{array}<br />
\right )<br />
= <br />
\left (<br />
\begin{array}{l}<br />
\bar u \\ <br />
\bar v \\ <br />
\bar p <br />
\end{array}<br />
\right )<br />
\exp{(\sigma x)}\exp{(\alpha t -iky)}<br />
</math><br />
where one recognizes an account for the perturbation of the field in the <math> x </math> direction (<math>\sigma</math>), the wave number of the instability following <math> y </math>, <math> k</math> and the <br />
growth rate with time <math> \alpha </math>.<br />
The eigenvalues of the system are used to determine the <math> x </math> dependence of the solution <math> \sigma = - \alpha/U,\; k,\; -k</math>, with the positive ones applying on the fresh side and the negative ones on the burned size to have a vanishing perturbation far from the flame.<br />
<br />
The eigenmodes give the flow pertubations on either side of the flame:<br />
:<math><br />
x<0:<br />
\left ( <br />
\begin{array}{l}<br />
u \\ v \\ p<br />
\end{array}<br />
\right )<br />
=<br />
a\left ( <br />
\begin{array}{l}<br />
1 \\ -i \\ -1-\frac{\alpha}{k}<br />
\end{array}<br />
\right )<br />
e^{kx+\alpha t - iky}<br />
</math><br />
<br />
:<math><br />
x>0:<br />
\left ( <br />
\begin{array}{l}<br />
u \\ v \\ p<br />
\end{array}<br />
\right )<br />
=<br />
b\left ( <br />
\begin{array}{l}<br />
1 \\ i \\ -1+\rho_b\frac{\alpha}{k}<br />
\end{array}<br />
\right )<br />
e^{-kx+\alpha t - iky}<br />
+<br />
c\left ( <br />
\begin{array}{l}<br />
1 \\ i\rho_b\frac{\alpha}{k} \\ 0<br />
\end{array}<br />
\right )<br />
e^{-\rho_b \alpha x+\alpha t - iky}<br />
</math><br />
<br />
The jump conditions above applied to this perturbation field yield the following system (<math> f </math> has also been put into the harmonic form <math> f = \bar f e^{\alpha t-iky} </math>):<br />
:<math><br />
\left \{ <br />
\begin{array}{lll}<br />
\alpha \bar f -a & = & \rho_b \alpha \bar f -\rho_b (b+c) \\<br />
a(1-\frac{\alpha}{k}) & = & b(1+\rho_b\frac{\alpha}{k}) +2c \\<br />
a + b+ c\frac{\rho_b \alpha}{k} & = & k\bar f (\frac{1}{\rho_b}-1)<br />
\end{array}<br />
\right . <br />
</math><br />
Additionaly, from the definition of the flame path <math> F </math>, we have the kinematic relation <br />
<math> u_u = a = \alpha \bar f </math>.<br />
<br />
The above set of equations forms a system of four unknowns <math> a,\; b,\; c,\; \alpha/k </math> <br />
for a given (but unknown) shape information on the flame bump <math> k\bar f</math>. <br />
Solving for the growth rate gives:<br />
:<math><br />
\left (\frac{\alpha}{k} - \frac{1}{\rho_b}\right )\left(\frac{\alpha}{k} + 2 + \rho_b\frac{\alpha}{k} -\left (\frac{1}{\rho_b}-1\right )\left (\frac{\alpha}{k}\right )^{-1} \right )=0<br />
</math><br />
This third-degree polynom has three solutions, namely the dispersion relation found by Darrieus, a stable mode, and the trivial solution (with no physical meaning), respectively:<br />
<math><br />
\frac{\alpha}{k} = \left ( \frac{1}{\rho_b + 1}\left (\sqrt{1+\frac{1-\rho_b^2}{\rho_b}}-1 \right ),\qquad \frac{1}{\rho_b + 1}\left (-\sqrt{1+\frac{1-\rho_b^2}{\rho_b}}-1 \right ),\qquad \frac{1}{\rho_b}\right )<br />
</math><br />
<br />
'''Remark 1''' By considering the shape of the eigenvectors giving the perturbation, one recovers that, upstream of the flame, the flow is potential (the rotational of the velocity vector is null and we have chosen a constant density flow, that is barotropic and divergence free), while downstream, additionally to another <br />
perturbation of the potential type, one finds a vorticity mode (the mode with the eigenvalue <math>-\rho_b\alpha </math>. The drift of vorticity at the crossing of the flame front is a known property of flames. <br />
<br />
'''Remark 2''' In more conventional literature, the strange trivial solution for the instability growth rate is swept under the rug. Given the pedagogical nature of this electronic documentation, we can dig a little bit as the appearance of this trivial, fool solution is a good example of some modelling issue. It is important to realize that a mathematical model and the physics it describes belong to different realities. Hence, the mathematical model will generate all the solutions that the mathematics can reach in its own space. Some of them are still connected to the physics. Some others, like this trivial solution, belongs only to the mathematical solution space, an indirect way of pointing out a model limitation. The model under scope here is the hydrodynamic limit. In this model, the domain is divided into two subdomains, one upstream of the flame interface, one downstream, that are put in relation with each other through a limited number of jump conditions. In reality, the physics connects these both domains much more tightly. The curious reader will have observed that the trivial solution makes the equation system for <math> a,\; b,\; c </math> (i.e. for a fixed growth rate) undetermined, and so are the jump conditions. By generating this trivial solution, the mathematics decouples both domains. The information coming from upstream to downstream (thanks to the shape of the linearized Euler system), a solution is found only for the upstream domain, the downstream being not solved and remains undetermined in lack of information. This solution cannot happen in reality because the fresh and burned gases in a real system are connected by many aspects, and not only by the jump conditions.<br />
<br />
<br />
==== Stretch / Compression of Premixed Flame ====<br />
<br />
The last of the `big' three turbulent ingredients (including aforementioned curvature and instabilities) impacting the flame at the local level is the compression or stretch of a premixed flame due to inhomogeneities in the flow, likely to happen with turbulence. <br />
<br />
[[Image:stretchCompression.jpg|thumb|Interpretation of flow inhomogeneities stresses on a 1-D premixed flame in terms of compression or stretch.]]<br />
The physics is pictured in the beside figure. In the flame area, the mass flow rate along the main direction may decrease (or increase) with distance. In a frame of reference whose origin is the core of the flame, a stretch (compression) results due to the difference in mass flow rate entering and leaving the flame zone.<br />
'''Important Remark''' Here, we are interested in the inhomogeneity in the mass flow rate along one direction only. We do NOT write that the flame volume is a region of mass source/sink.<br />
<br />
In the same manner as above for the curvature effect, we introduce a (small) inhomogeneity of the flow as <br />
<math> \vec\nabla\vec M </math> such that the following equation can be substracted from the unperturbed flame equation.<br />
:<math>||\vec M^{s/c}||||\vec\nabla_{\xi}\theta|| - \vec n\cdot \vec\nabla\vec M \cdot \vec n ||\vec\nabla_{\xi}\theta|| - \varpi= \nabla_{\xi}\cdot\vec\nabla_{\xi}\theta<br />
</math><br />
<br />
Hence, for a small perturbation, the linearized dependence of the flame on stretch/compression is:<br />
:<math>||\vec M^{s/c}|| = ||\vec M^o|| + \vec n \cdot \vec\nabla\vec M \cdot \vec n </math><br />
<br />
== The Partially-Premixed Regime ==<br />
[[Image:ppf.jpg|thumb| Ideal sketch of a partially-premixed flame]]<br />
This regime is somewhat less academics and has been recognized two decades ago. It is acknowledged as a hybrid of the premixed and the non-premixed regimes but the degree of interaction of these two modes of combustion to accurately describe a partially-premixed flame is still not well understood. It can be simply pictured by a lifted diffusion flame. Let us consider fuel issuing from a nozzle into the air. If the exit velocity is large enough, for some fuels, the flame lifts off the rim of the nozzle. It means that below the flame base, fuel and oxidizer have room to premix. Hence, the flame base propagates into a premixed mixture. However, it cannot be reduced to a premixed flame (although it is often simplified as this): (i) the mixing is not perfect and the different parts of the flame front constituting the flame base burn in mixtures of different thermodynamical states. This provides those parts with different deflagration capabilities such that the flame base has a complex shape. Indeed, it is convex, naturally leaded by the part burning at stoichiometry, unless `exotic' feeding temperatures are used. (ii) Because the mixture is not homogeneous, transfer of species and temperature driven by diffusion occurs in a direction perpendicular to the propagation of the flame base. Because the flame front is not flat, those transfers act as a connection vehicle across the different parts of the leading front. (iii) The unburned left downstream by the sections of the leading front not burning at stoichiometry diffuse towards each other to form a diffusion flame as described above. The connection of the leading front with the trailing diffusion flame has been evidenced as complicated and the siege of transfers of species and temperature. These two last items are the state-of-the-art difficulties in understanding those flames and do not appear in the models although it has been demonstrated they have a major impact and are certainly a fundamental characteristic of partially-premixed flames.<br />
<br />
The partially-premixed flame is usually described using ''c'' and ''Z'' as introduced earlier. Because the framework is essentially non-premixed, the mixture fraction is primarily used to describe the flame. Regarding the head of the flame where partial-premixing has an impact, each part of the front is described with a local progress variable. The need of defining a local progress variable is that each section of the partially-premixed front has a different equivalence ratio leading to a different definition of ''c'':<br />
:<math> c=\frac{T-T_u}{T_b(Z)-T_u}</math><br />
<br />
= Three Turbulent-Flame Interaction Regimes =<br />
It may appear odd to try to describe here what is the reason of combustion modelling research: the interaction of the turbulence with chemistry. However, one of the first steps in building knowledge in turbulent combustion was the qualitative exploration of what might be the dynamics of a flame in a turbulent environment. This led to what is now known as ''combustion diagrams''. As explained above, the premixed regime lends itself the easiest to such an approach as it exhibits natural intrinsic quantities which are not as objectively identifiable in the other combustion regimes. Note that these quantities may depend<br />
on the geometry of the flame: for instance turbulence can bend a flame sheet, leading to a change in its <br />
dynamics compared to the flat flame propagating in a medium at rest. In this section, the turbulence-flame interaction modes will be described for a premixed flame. Only remarks will be added regarding the non-premixed and partially-premixed regimes.<br />
<br />
An integral quantity to assess the interaction between a premixed flame sheet and the turbulence<br />
is the Karlovitz number ''Ka''. It compares the characteristic time of flame displacement with the characteristic time of the smallest structures (that are also the fastest) of the turbulence.<br />
:<math> Ka= \frac{\tau_c}{\tau_k}</math><br />
<br />
<math>\tau_c</math> is the chemical time of the flame. To estimate it, it is necessary to come back to the above progress variable transport equation in a steady-state framework.<br />
:<math> \overbrace{(\rho S_L)}^{\dot M}\frac{\vec S_L}{S_L}\vec\nabla c = (\rho D)_f\nabla\cdot (\rho D)^* \vec\nabla c + \dot\omega_c</math><br />
The premixed wave propagates at a speed <math>S_L</math> because it is fed by reactants diffusing inside the combustion zone and which are preheated because temperature diffuses in the reverse direction. The speed at which the flame progresses is thus related to the rate of species diffusion into the reaction zone which are then consumed. As the premixed flame is a free propagating wave whose speed of propagation is only limited by the chemical strength, the characteristic chemical time is based only on the diffusion and the mass flow rate experienced by the flame:<br />
:<math> \tau_c = \frac{\rho (\rho D)_f}{\dot M^2}</math><br />
<br />
The smallest eddies are the ones being dissipated by the viscous forces. Their characteristic time is estimated thanks to a combination of the viscosity and the flux of turbulent energy to be dissipated (also called turbulent dissipation <math> \varepsilon=u'^3/l_t </math>):<br />
:<math>\tau_k = \sqrt{\frac{l_t\nu}{u'^3}}</math><br />
<br />
Thanks to those definitions of chemical and small structure times, it is possible to give another definition of the Karlovitz number:<br />
:<math> Ka=\left (\frac{\delta_L}{l_k} \right)^2</math> <br />
which is the square of the ratio between the premixed flame thickness and the small structure scale: ''Ka'' actually compares scales. To arrive to this latter result, the three following assumptions must be used: (i) the flame thickness is obtained thanks to the premixed flame P&eacute;clet number (''vid. sup.''); (ii) the turbulence small structure (''Kolmogorov eddies'') scale is given by: <math> l_k=(\nu^3/\varepsilon)^{1/4} </math> following the same dimensional argument as for the estimation of its time; and (iii) scalar diffusion scales with viscosity. <br />
<br />
==== Remark Regarding the Diffusion Flame ====<br />
From what has been presented above, a diffusion flame does not have characteristic scales. Setting a turbulence combustion regime classification for non-premixed flames has still not been answered by research. Some laws of behaviour will only be drawn around the scalar dissipation rate which is the parameter of integral importance for a diffusion flame.<br />
<br />
Indeed, the dynamics of a diffusion flame is determined by the strain rate imposed by the turbulence. As for the premixed flames, the shortest eddies (Kolmogorov) are the ones having the largest impact. The diffusive layer is thus given by the size of the Kolmogrov eddies: <math> l_d\approx l_k</math> and the typical diffusion time scale (feeding rate of the reaction zone) is given by the characteristic time of the Kolmogorov eddies: <math> \tau_k^{-1}\approx \chi_s</math> as the Reynolds number of the Kolmogorov structures is unity. Here, <math>\chi_s</math> is the sample-averaging of <math>\chi</math> based on (conditioned) stoichiometric conditions, where the flame is expected to be.<br />
<br />
== The Wrinkled Regime ==<br />
[[Image:wrinkled.jpg|thumb| Wrinkled flamelet regime]]<br />
This regime is also called the ''flamelet regime''. Basically, it assumes that the flame structure is not affected by turbulence. The flame sheet is convoluted and wrinkled by eddies but none of them is small enough to enter it.<br />
Locally magnifying, the laminar flame structure is maintained.<br />
<br />
This regime exists for a Karlovitz number below unity (vid. sup.), i.e. chemical time smaller than the small structure time or flame thickness smaller than small structure scale. Notwithstanding, the laminar flame dynamics can be disrupted for <math> u'>S_L</math>. In that case, although the flame structure is not altered by the small structures, it can be convected by large structures such that areas of different locations in the front interact. It shows that, even with a small Karlovitz number, the turbulence effect is not always weak.<br />
<br />
== The Corrugated Regime ==<br />
[[Image:Corrugated.jpg|thumb|Corrugated flamelet regime]]<br />
The formal definition of a flame is the region of temperature rise. However, the volume where the reaction takes place is about one order of magnitude smaller, embedded inside the temperature rise region and close to its high temperature end. Hence, there exist some levels of turbulence creating eddies able to enter the flame zone but still large enough to not affect the internal reaction sheet. In other words, the flame thermal region is thickened by turbulence but the reaction zone is still in the wrinkled regime.<br />
This situation is called the ''Corrugated Regime''.<br />
<br />
Due to the structure of the Karlovitz number, once written in terms of length scales (vid. sup.), this situation arises for an<br />
increase of the Karlovitz number by two orders of magnitude compared to the value for the wrinkled regime. Hence, in the range <br />
<math> 1 < Ka < 100 </math>, the laminar structure of the reaction zone is still preserved but not the one of the preheat zone.<br />
<br />
== The Thickened Regime ==<br />
[[Image:thickened.jpg|thumb|Thickened regime]]<br />
In this last case, turbulence is intense enough to generate eddies able to affect the structure of the reaction zone as well. In practice, it is expected that those eddies are in the tail of the energy spectrum such that their lifetime is very short. Their impact on the reaction zone is thus limited.<br />
<br />
Obviously, ''Ka > 100''. A topological description is of little relevance here and a ''well-stirred reactor model'' fits better.<br />
<br />
== Reaction mechanisms ==<br />
<br />
Combustion is mainly a chemical process. Although we can, to some extent, <br />
describe a flame without any chemistry information, modelling of the flame <br />
propagation requires the knowledge of speeds of reactions, product concentrations, <br />
temperature, and other parameters. <br />
Therefore fundamental information about reaction kinetics is <br />
essential for any combustion model. <br />
A fuel-oxidizer mixture will generally combust if the reaction is fast enough to <br />
prevail until all of the mixture is burned into products. If the reaction <br />
is too slow, the flame will extinguish. If too fast, explosion or even <br />
detonation will occur. The reaction rate of a typical combustion reaction <br />
is influenced mainly by the concentration of the reactants, temperature, and pressure. <br />
<br />
A stoichiometric equation for an arbitrary reaction can be written as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
\sum_{j=1}^{n}\nu' (M_j) = \sum_{j=1}^{n}\nu'' (M_j),<br />
</math></td></tr></table><br />
<br />
where <math> \nu </math> denotes the stoichiometric coefficient, and <math>M_j</math> stands for an arbitrary species. A one prime holds for the reactants while a double prime holds for the products of the reaction. <br />
<br />
The reaction rate, expressing the rate of disappearance of reactant <b>i</b>, is defined as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
RR_i = k \, \prod_{j=1}^{n}(M_j)^{\nu'},<br />
</math></td></tr></table><br />
<br />
in which <b>k</b> is the specific reaction rate constant. Arrhenius found that this constant is a function of temperature only and is defined as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
k= A T^{\beta} \, exp \left( \frac{-E}{RT}\right)<br />
</math></td></table><br />
<br />
where <b>A</b> is pre-exponential factor, <b>E</b> is the activation energy, and <math>\beta</math> is a temperature exponent. The constants vary from one reaction to another and can be found in the literature.<br />
<br />
Reaction mechanisms can be deduced from experiments (for every resolved reaction), they can also be constructed numerically by the '''automatic generation method''' (see [Griffiths (1994)] for a review on reaction mechanisms).<br />
For simple hydrocarbons, tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms, global reactions <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing equations for chemically reacting flows ==<br />
<br />
Together with the usual Navier-Stokes equations for compresible flow (See [[Governing equations]]), additional equations are needed in reacting flows.<br />
<br />
The transport equation for the mass fraction <math> Y_k </math> of <i>k-th</i> species is<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Y_k \right) +<br />
\frac{\partial}{\partial x_j} \left( \rho u_j Y_k\right) = <br />
\frac{\partial}{\partial x_j} \left( \rho D_k \frac{\partial Y_k}{\partial x_j}\right)+ \dot \omega_k<br />
</math><br />
<br />
<br />
where Ficks' law is assumed for scalar diffusion with <math> D_k </math> denoting the species difussion coefficient, and <math> \dot \omega_k </math> denoting the species reaction rate.<br />
<br />
A non-reactive (passive) scalar (like the mixture fraction <math> Z </math>) is goverened by the following transport equation<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Z \right) +<br />
\frac{\partial}{\partial x_j} \left( \rho u_j Z \right) = <br />
\frac{\partial}{\partial x_j} \left( \rho D \frac{\partial Z}{\partial x_j}\right)<br />
</math><br />
<br />
where <math> D </math> is the diffusion coefficient of the passive scalar.<br />
<br />
<br />
=== RANS equations ===<br />
<br />
In turbulent flows, [[Favre averaging]] is often used to reduce the scales (see [[Reynolds averaging]]) and the<br />
mass fraction transport equation is transformed to<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } -<br />
\overline{\rho} \widetilde{u''_i Y''_k } \right)<br />
+ \overline{\dot \omega_k}<br />
</math><br />
<br />
<br />
where the turbulent fluxes <math> \widetilde{u''_i Y''_k} </math> and reaction terms<br />
<math> \overline{\dot \omega_k} </math> need to be closed.<br />
<br />
The passive scalar turbulent transport equation is<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Z} }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z} }{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D \frac{\partial Z} {\partial x_j} } -<br />
\overline{\rho} \widetilde{u''_i Z'' } \right)<br />
</math><br />
<br />
where <math> \widetilde{u''_i Z''} </math> needs modelling. A common practice is to model the turbulent fluxes using the <br />
gradient diffusion hypothesis. For example, in the equation above the flux <math> \widetilde{u''_i Z''} </math> is modelled as<br />
<br />
:<math><br />
\widetilde{u''_i Z''} = -D_t \frac{\partial \tilde Z}{\partial x_i}<br />
</math> <br />
<br />
where <math> D_t </math> is the turbulent diffusivity. Since <math> D_t >> D </math>, the first term inside the parentheses on the right hand of the mixture fraction transport equation is often neglected (Peters (2000)). This assumption is also used below.<br />
<br />
In addition to the mean passive scalar equation,<br />
an equation for the Favre variance <math> \widetilde{Z''^2}</math> is often employed<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Z''^2} }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z''^2} }{\partial x_j}=<br />
\frac{\partial}{\partial x_j}<br />
\left( \overline{\rho} \widetilde{u''_i Z''^2} \right) -<br />
2 \overline{\rho} \widetilde{u''_i Z'' }<br />
- \overline{\rho} \widetilde{\chi}<br />
</math><br />
<br />
where <math> \widetilde{\chi} </math> is the mean [[Scalar dissipation]] rate<br />
defined as <math> \widetilde{\chi} =<br />
2 D \widetilde{\left| \frac{\partial Z''}{\partial x_j} \right|^2 } </math><br />
This term and the variance diffusion fluxes needs to be modelled.<br />
<br />
=== LES equations ===<br />
The [[Large eddy simulation (LES)]] approach for reactive flows introduces equations for the filtered species mass fractions within the compressible flow field.<br />
Similar to the [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]], the filtered mass fraction transport equation is<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } -<br />
J_j \right)<br />
+ \overline{\dot \omega_k}<br />
</math><br />
<br />
where <math> J_j </math> is the transport of subgrid fluctuations of mass fraction<br />
:<math><br />
J_j = \widetilde{u_jY_k} - \widetilde{u}_j \widetilde{Y}_k<br />
</math><br />
and has to be modelled.<br />
<br />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
are assumed to be much smaller than the apparent turbulent diffusion due to transport of subgrid fluctuations.<br />
The first term on the right hand side is then<br />
:<math><br />
\frac{\partial} {\partial x_j} <br />
\left(<br />
\overline{ \rho D_k \frac{\partial Y_k} {\partial x_j} }<br />
\right)<br />
\approx<br />
\frac{\partial} {\partial x_j} <br />
\left(<br />
\overline{\rho} D_k \frac{\partial \widetilde{Y}_k} {\partial x_j} <br />
\right)<br />
</math><br />
<br />
[[Category:Equations]]<br />
<br />
== Infinitely fast chemistry ==<br />
All combustion models can be divided into two main groups according to the <br />
assumptions on the reaction kinetics.<br />
We can either assume the reactions to be infinitely fast - compared to <br />
e.g. mixing of the species, or comparable to the time scale of the mixing <br />
process. The simple approach is to assume infinitely fast chemistry. Historically, mixing of the species is the older approach, and it is still in wide use today. It is therefore simpler to solve for [[#Finite rate chemistry]] models, at the overshoot of introducing errors to the solution.<br />
<br />
=== Premixed combustion ===<br />
Premixed flames occur when the fuel and oxidiser are homogeneously mixed prior to ignition. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical examples of premixed laminar flames is bunsen burner, where <br />
the air enters the fuel stream and the mixture burns in the wake of the <br />
riser tube walls forming nice stable flame. Another example of a premixed system is the solid rocket motor where oxidizer and fuel and properly mixed in a gum-like matrix that is uniformly distributed on the periphery of the chamber.<br />
Premixed flames have many advantages in terms of control of temperature and <br />
products and pollution concentration. However, they introduce some dangers like the autoignition (in the supply system).<br />
<br />
[[Image:Premixed.jpg]]<br />
==== Turbulent flame speed model ====<br />
<br />
==== Eddy Break-Up model ====<br />
<br />
The Eddy Break-Up model is the typical example of '''mixed-is-burnt''' combustion model. <br />
It is based on the work of Magnussen, Hjertager, and Spalding and can be found in all commercial CFD packages. <br />
The model assumes that the reactions are completed at the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. Combustion is then described by a single step global chemical reaction<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
F + \nu_s O \rightarrow (1+\nu_s) P<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
in which <b>F</b> stands for fuel, <b>O</b> for oxidiser, and <b>P</b> for products of the reaction. Alternativelly we can have a multistep scheme, where each reaction has its own mean reaction rate.<br />
The mean reaction rate is given by<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
\bar{\dot\omega}_F=A_{EB} \frac{\epsilon}{k} <br />
min\left[\bar{C}_F,\frac{\bar{C}_O}{\nu},<br />
B_{EB}\frac{\bar{C}_P}{(1+\nu)}\right]<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
<br />
where <math>\bar{C}</math> denotes the mean concentrations of fuel, oxidiser, and products <br />
respectively. <b>A</b> and <b>B</b> are model constants with typical values of 0.5 <br />
and 4.0 respectively. The values of these constants are fitted according <br />
to experimental results and they are suitable for most cases of general interest. It is important to note that these constants are only based on experimental fitting and they need not be suitable for <b>all</b> the situations. Care must be taken especially in highly strained regions, where the ratio of <math>k</math> to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In these, regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually have some remedies to overcome this problem.<br />
<br />
This model largely over-predicts temperatures and concentrations of species like <i>CO</i> and other species. However, the Eddy Break-Up model enjoys a popularity for its simplicity, steady convergence, and implementation.<br />
<br />
==== Bray-Moss-Libby model ====<br />
<br />
=== Non-premixed combustion ===<br />
<br />
Non premixed combustion is a special class of combustion where fuel and oxidizer<br />
enter separately into the combustion chamber. The diffusion and mixing of the two streams<br />
must bring the reactants together for the reaction to occur.<br />
Mixing becomes the key characteristic for diffusion flames.<br />
Diffusion burners are easier and safer to operate than premixed burners.<br />
However their efficiency is reduced compared to premixed burners.<br />
One of the major theoretical tools in non-premixed combustion<br />
is the passive scalar [[mixture fraction]] <math> Z </math> which is the<br />
backbone on most of the numerical methods in non-premixed combustion.<br />
<br />
==== Conserved scalar equilibrium models ====<br />
The reactive problem is split into two parts. First, the problem of <i> mixing </i>, which consists of the location of the flame surface which is a non-reactive problem concerning the propagation of a passive scalar; And second, the <i> flame structure </i> problem, which deals with the distribution of the reactive species inside the flamelet.<br />
<br />
To obtain the distribution inside the flame front we assume it is locally one-dimensional and<br />
depends only on time and the scalar coodinate.<br />
<br />
We first make use of the following chain rules<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}<br />
</math><br />
<br><br />
:<math><br />
\frac{\partial Y_k}{\partial x_j} = \frac{\partial Z}{\partial x_j}\frac{\partial Y_k}{\partial Z}<br />
</math> <br />
and transformation <br />
:<math><br />
\frac{\partial }{\partial t}= \frac{\partial }{\partial t} + \frac{\partial Z}{\partial t} \frac{\partial }{\partial Z}<br />
</math><br />
<br />
upon substitution into the species transport equation (see [[#Governing Equations for Reacting Flows]]), we obtain<br />
<br />
:<math><br />
\rho \frac{\partial Y_k}{\partial t} + Y_k \left[<br />
\frac{\partial \rho}{\partial t} + \frac{\partial \rho u_j}{\partial x_j}<br />
\right]<br />
+ \frac{\partial Y_k}{\partial Z} \left[<br />
\rho \frac{\partial Z}{\partial t} + \rho u_j \frac{\partial Z}{\partial x_j} -<br />
\frac{\partial}{\partial x_j}\left( \rho D \frac{\partial Z}{\partial x_j} \right)<br />
\right]<br />
=<br />
\rho D \left( \frac{\partial Z}{\partial x_j} \frac{\partial Z}{\partial x_j} \right)<br />
\frac{\partial^2 Y_k}{\partial Z^2} + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third terms in the LHS cancel out due to continuity and mixture fraction transport.<br />
At the outset, the equation boils down to<br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k<br />
</math><br />
<br />
where <math> \chi = 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2 </math> is called the <b>scalar dissipation</b><br />
which controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though the field <math> Z </math> still depends on it.<br />
<br />
:<math><br />
\dot \omega_k= -\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2}<br />
</math><br />
<br />
If the reaction is assumed to be infinetly fast, the resultant flame distribution is in equilibrium.<br />
and <math> \dot \omega_k= 0</math>. When the flame is in equilibrium, the flame configuration <math> Y_k(Z) </math> is independent of strain.<br />
<br />
===== Burke-Schumann flame structure =====<br />
The Burke-Schuman solution is valid for irreversible infinitely fast chemistry.<br />
With a reaction in the form of<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
F + \nu_s O \rightarrow (1+\nu_s) P<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
If the flame is in equilibrium and therefore the reaction term is 0.<br />
Two possible solution exists, one with pure mixing (no reaction)<br />
and a linear dependence of the species mass fraction with <math> Z </math>.<br />
Fuel mass fraction<br />
:<math><br />
Y_F=Y_F^0 Z<br />
</math><br />
Oxidizer mass fraction<br />
:<math><br />
Y_O=Y_O^0(1-Z)<br />
</math><br />
Where <math> Y_F^0 </math> and <math> Y_O^0 </math> are fuel and oxidizer mass fractions<br />
in the pure fuel and oxidizer streams respectively.<br />
<br />
<br />
The other solution is given by a discontinuous slope at stoichiometric mixture fraction<br />
<math> Z_{st}</math> and two linear profiles (in the rich and lean side) at either<br />
side of the stoichiometric mixture fraction.<br />
Both concentrations must be 0 at stoichiometric, the reactants become products infinitely fast.<br />
<br />
:<math><br />
Y_F=Y_F^0 \frac{Z-Z_{st}}{1-Z_{st}}<br />
</math><br />
and oxidizer mass fraction<br />
:<math><br />
Y_O=Y_O^0 \frac{Z-Z_{st}}{Z_{st}}<br />
</math><br />
<br />
== Finite rate chemistry ==<br />
<br />
=== Premixed combustion ===<br />
<br />
==== Coherent flame model ====<br />
<br />
==== Flamelets based on G-equation ====<br />
==== Flame surface density model ====<br />
<br />
=== Non-premixed combustion ===<br />
<br />
==== Flamelets based on conserved scalar ====<br />
<br />
Peters (2000) define Flamelets as "thin diffusion layers embedded in a turbulent non-reactive flow field".<br />
If the chemistry is fast enough, the chemistry is active within a thin region<br />
where the chemistry conditions are in (or close to) stoichiometric conditions, the "flame" surface.<br />
This thin region is assumed to be smaller than Kolmogorov length scale and therefore the<br />
region is locally laminar. The flame surface is defined as an iso-surface of a certain scalar <math> Z </math>,<br />
mixture fraction in [[#Non premixed combustion]].<br />
<br />
The same equation used in [[#Conserved scalar models]] for equilibrium chemistry<br />
is used here but with chemical source term different from 0<br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} =<br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k.<br />
</math><br />
<br />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that<br />
flamelet profiles <math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in a dtaset or file which is called a "flamelet library" with all the required complex chemistry.<br />
For the generation of such libraries ready to use software is avalable such as Softpredict's Combustion Simulation Laboratory COSILAB [http://www.softpredict.com] with its relevant solver RUN1DL, which can be used for a variety of relevant geometries; see various [http://www.softpredict.com/?page=989 publications that are available for download]. Other software tools are available such as CHEMKIN [http://www.reactiondesign.com] and<br />
CANTERA [http://www.cantera.org].<br />
<br />
===== Flamelets in turbulent combustion =====<br />
<br />
In turbulent flames the interest is <math> \widetilde{Y}_k </math>.<br />
In flamelets, the flame thickness is assumed to be much smaller than Kolmogorov scale<br />
and obviously is much smaller than the grid size.<br />
It is therefore needed a distribution of the passive scalar within the cell.<br />
<math> \widetilde{Y}_k </math> cannot be obtained directly from the flamelets library<br />
<math> \widetilde{Y}_k \neq Y_F(Z,\chi) </math>, where <math> Y_F(Z,\chi) </math> corresponds<br />
to the value obtained from the flamelets libraries.<br />
A generic solution can be expressed as<br />
:<math><br />
\widetilde{Y}_k= \int Y_F( \widetilde{Z},\widetilde{\chi}) P(Z,\chi) dZ d\chi<br />
</math><br />
where <math> P(Z,\chi) </math> is the joint [[Probability density function | Probability Density Function]] (PDF) of the mixture fraction<br />
and scalar dissipation which account for the scalar distribution inside the cell and "a priori"<br />
depends on time and space.<br />
<br />
The most simple assumption is to use a constant distribution of the scalar dissipation within the cell and<br />
the above equation reduces to<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) dZ<br />
</math><br />
<br />
<math> P(Z) </math> is the PDF of the mixture fraction scalar and simple models (such as Gaussian or a [[beta PDF]])<br />
can be build depending only on two moments of the scalar<br />
mean and variance,<math> \widetilde{Z},Z''</math>.<br />
<br />
If the mixture fraction and scalar dissipation are consider independent variables,<math> P(Z,\chi) </math><br />
can be written as <math> P(Z) P(\chi)</math>. The PDF of the scalar dissipation is assumed to be log-normal with<br />
variance unity.<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) P(\chi) dZ d\chi<br />
</math><br />
In [[Large eddy simulation (LES)]] context (see [[#LES equations]] for reacting flow),<br />
the [[probability density function]] is replaced by a [[subgrid PDF]] <math> \widetilde{P}</math>.<br />
The same equation hold by replacing averaged values with filtered values.<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) \widetilde{P}(Z) \widetilde{P}(\chi) dZ d\chi<br />
</math><br />
The assumptions made regarding the shapes of the PDFs are still justified. In LES combustion the<br />
[[subgrid variance]] is smaller than RANS counterpart (part of the large-scale fluctuations are solved)<br />
and therefore the modelled PDFs are thinner.<br />
<br />
====== Unsteady flamelets ======<br />
====Intrinsic Low Dimensional Manifolds (ILDM)====<br />
<br />
Detailed mechanisms describing ignition, flame propagation and pollutant formation typically involve several hundred species and elementary reactions, prohibiting their use in practical three-dimensional engine simulations. Conventionally reduced mechanisms often fail to predict minor radicals and pollutant precursors. The ILDM-method is an automatic reduction of a detailed mechanism, which assumes local equilibrium with respect to the fastest time scales identified by a local eigenvector analysis. In the reactive flow calculation, the species compositions are constrained to these manifolds. Conservation equations are solved for only a small number of reaction progress variables, thereby retaining the accuracy of detailed chemical mechanisms. This gives an effective way of coupling the details of complex chemistry with the time variations due to turbulence.<br />
<br />
The intrinsic low-dimensional manifold (ILDM) method {Maas:1992,Maas:1993} is a method for ''in-situ'' reduction of a detailed chemical mechanism based on a local time scale analysis. This method is based on the fact that different trajectories in state space start from a high-dimensional point and quickly relax to lower-dimensional manifolds due to the fast reactions. The movement along these lower-dimensional manifolds, however, is governed by the slow reactions. It exploits the variety of time scales to systematically reduce the detailed mechanism. For a detailed chemical mechanism with N species, N different time scales govern the process. An assumption that all the time scales are relaxed results in assuming complete equilibrium, where the only variables required to describe the system are the mixture fraction, the temperature and the pressure. This results in a zero-dimensional manifold. An assumption that all but the slowest 'n' time scales are relaxed results in a 'n' dimensional manifold, which requires the additional specification of 'n' parameters (called progress variables). In the ILDM method, the fast chemical reactions do not need to be identified a priori. An eigenvalue analysis of the detailed chemical mechanism is carried out which identifies the fast processes in dynamic equilibrium with the slow processes. The computation of ILDM points can be expensive, and hence an in-situ tabulation procedure is used, which enables the calculation of only those points that are needed during the CFD calculation.<br />
<br />
--[[User:Fredgauss|Fredgauss]] 07:37, 25 August 2006 (MDT)<br />
<br />
U. Maas, S.B. Pope. Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Comb. Flame 88,<br />
239, 1992.<br />
<br />
Ulrich Maas. Automatische Reduktion von Reaktionsmechanismen zur Simulation reaktiver Str¨omungen. Habilitationsschrift, Universit<br />
¨at Stuttgart, 1993.<br />
<br />
==== Conditional Moment Closure (CMC)====<br />
In Conditional Moment Closure (CMC) methods we assume that the species mass fractions are all correlated with<br />
the mixture fraction (in non premixed combustion).<br />
<br />
From [[Probability density function]] we have<br />
:<math><br />
\overline{Y_k}= \int <Y_k|\eta> P(\eta) d\eta<br />
</math><br />
where <math> \eta </math> is the sample space for <math> Z </math>.<br />
<br />
CMC consists of providing a set of transport equations for the conditional moments which define the<br />
flame structure.<br />
<br />
Experimentally, it has been observed that temperature and chemical radicals are<br />
strong non-linear functions of mixture fraction. For a given species mass fraction we can decomposed<br />
it into a mean and a fluctuation:<br />
:<math><br />
Y_k= \overline{Y_k} + Y'_k<br />
</math><br />
The fluctuations <math> Y_k' </math> are usually very strong in time and space which makes the closure<br />
of <math> \overline{\omega_k} </math> very difficult.<br />
However, the alternative decomposition<br />
:<math><br />
Y_k= <Y_k|\eta> + y'_k<br />
</math><br />
where <math> y'_k </math> is the fluctuation around the conditional mean or the "conditional fluctuation".<br />
Experimentally, it is observed that <math> y'_k<< Y'_k </math>, which forms the basic assumption of the CMC method.<br />
Closures. Due to this property better closure methods can be used reducing the non-linearity <br />
of the mass fraction equations.<br />
<br />
The [[Derivation of the CMC equations]] produces the following CMC transport equation<br />
where <math> Q \equiv <Y_k|\eta> </math> for simplicity.<br />
:<math><br />
\frac{ \partial Q}{\partial t} + <u_j|\eta> \frac{\partial Q}{\partial x_j} =<br />
\frac{<\chi|\eta> }{2} \frac{\partial ^2 Q}{\partial \eta^2} + <br />
\frac{ < \dot \omega_k|\eta> }{ <\rho| \eta >}<br />
</math><br />
<br />
In this equation, high order terms in Reynolds number have been neglected.<br />
(See [[Derivation of the CMC equations]] for the complete series of terms).<br />
<br />
It is well known that closure of the unconditional source term<br />
<math> \overline {\dot \omega_k} </math> as a function of the<br />
mean temperature and species (<math> \overline{Y}, \overline{T}</math>) will give rise to large errors.<br />
However, in CMC the conditional averaged mass fractions contain more information and fluctuations around the mean are much smaller.<br />
The first order closure<br />
<math><br />
< \dot \omega_k|\eta> \approx \dot \omega_k \left( Q, <T|\eta> \right)<br />
</math><br />
is a good approximation in zones which are not close to extinction.<br />
<br />
===== Second order closure =====<br />
<br />
A second order closure can be obtained if conditional fluctuations are taken into account.<br />
For a chemical source term in the form <math> \dot \omega_k = k Y_A Y_B </math> with the rate constant in Arrhenius form<br />
<math> k=A_0 T^\beta exp [-Ta/T] </math><br />
the second order closure is (Klimenko and Bilger 1999)<br />
:<math><br />
< \dot \omega_k|\eta> \approx < \dot \omega_k|\eta >^{FO}<br />
<br />
\left[1+ \frac{< Y''_A Y''_B |\eta>}{Q_A Q_B}+ \left( \beta + T_a/Q_T \right)<br />
\left(<br />
\frac{< Y''_A T'' |\eta>}{Q_AQ_T} + \frac{< Y''_B T'' |\eta>}{Q_BQ_T}<br />
\right) + ...<br />
<br />
\right]<br />
<br />
</math><br />
<br />
where <math> < \dot \omega_k|\eta >^{FO} </math> is the first order CMC closure and<br />
<math> Q_T \equiv <T|\eta> </math>.<br />
When the temperature exponent <math> \beta </math> or <math> T_a/Q_T </math><br />
are large the error of taking the first order approximation increases.<br />
Improvement of small pollutant predictions can be obtained using the above reaction <br />
rate for selected species like CO and NO.<br />
<br />
<br />
===== Double conditioning =====<br />
<br />
Close to extinction and reignition. The conditional fluctuations can be very large<br />
and the primary closure of CMC of "small" fluctuations is not longer valid.<br />
A second variable <math> h </math> can be chosen to define a double conditioned mass fraction<br />
:<math><br />
Q(x,t;\eta,\psi) \equiv <Y_i(x,t) |Z=\eta,h=\psi ><br />
</math><br />
Due to the strong dependence on chemical reactions to temperature, <math> h </math><br />
is advised to be a temperature related variable (Kronenburg 2004).<br />
Scalar dissipation is not a good choice, due to its log-normal behaviour<br />
(smaller scales give highest dissipation). A must better choice is the sensible enthalpy<br />
or a progress variable.<br />
Double conditional variables have much smaller conditional fluctuations and allow<br />
the existence of points with the same chemical composition which can be fully burning<br />
(high temperature) or just mixing (low temperature). <br />
The range of applicability is greatly increased and allows non-premixed and premixed problems<br />
to be treated without ad-hoc distinctions.<br />
The main problem is the closure of the new terms involving cross scalar transport.<br />
<br />
The double conditional CMC equation is obtained in a similar manner than the conventional<br />
CMC equations<br />
<br />
===== LES modelling =====<br />
In a LES context a [[conditional filtering]] operator can be defined<br />
and <math> Q </math> therefore represents a conditionally filtered reactive scalar.<br />
<br />
=== Linear Eddy Model ===<br />
The Linear Eddy Model (LEM) was first developed by Kerstein(1988).<br />
It is an one-dimensional model for representing the flame structure in turbulent flows.<br />
<br />
In every computational cell a molecular, diffusion and chemical model is defined as<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Y_k \right) =<br />
\frac{\partial}{\partial \eta} \left( \rho D_k \frac{\partial Y_k}{\partial \eta }\right)+ \dot \omega_k<br />
</math><br />
<br />
where <math> \eta </math>is a spatial coordinate. The scalar distribution obtained can be seen as a<br />
one-dimensional reference field between Kolmogorov scale and grid scales.<br />
<br />
In a second stage a series of re-arranging stochastic event take place.<br />
These events represent the effects<br />
of a certain turbulent structure of size <math> l </math>, smaller than the grid size at a location <math> \eta_0 </math><br />
within the one-dimensional domain. This vortex distort the <math> \eta </math> field obtain by the one-dimensional equation,<br />
creating new maxima and minima in the interval <math> (\eta_0, \eta + \eta_0) </math>.<br />
The vortex size <math> l </math> is chosen randomly based on the inertial scale range while<br />
<math> \eta_0 </math> is obtained from a uniform distribution in <math> \eta </math>.<br />
The number of events is chosen to match the turbulent diffusivity of the flow.<br />
<br />
=== PDF transport models ===<br />
<br />
[[Probability Density Function]] (PDF) methods are not exclusive to combustion, although they are particularly attractive to them. They provided more information than moment closures and they are used to compute inhomegenous turbulent flows, see reviews in Dopazo (1993) and Pope (1994).<br />
<br />
<br />
PDF methods are based on the transport equation of the joint-PDF of the scalars.<br />
Denoting <math> P \equiv P(\underline{\psi}; x, t) </math> where<br />
<math> \underline{\psi} = ( \psi_1,\psi_2 ... \psi_N) </math> is the phase space for the reactive scalars<br />
<math> \underline{Y} = ( Y_1,Y_2 ... Y_N) </math>.<br />
The transport equation of the joint PDF is:<br />
<br />
:<math><br />
\frac{\partial <\rho | \underline{Y}=\underline{\psi}> P }{\partial t} + \frac{ <br />
\partial <\rho u_j | \underline{Y}=\underline{\psi}> P }{\partial x_j} =<br />
\sum^N_\alpha \frac{\partial}{\partial \psi_\alpha}\left[ \rho \dot{\omega}_\alpha P \right]<br />
- \sum^N_\alpha \sum^N_\beta \frac{\partial^2}{\partial \psi_\alpha \psi_\beta}<br />
\left[ <D \frac{\partial Y_\alpha}{\partial x_i} \frac{\partial Y_\beta}{\partial x_i} | \underline{Y}=\underline{\psi}> \right] P<br />
</math><br />
<br />
where the chemical source term is closed. Another term appeared on the right hand side which accounts for the effects of<br />
the molecular mixing on the PDF, is the so called "micro-mixing " term.<br />
Equal diffusivities are used for simplicity <math> D_k = D </math><br />
<br />
A more general approach is the velocity-composition joint-PDF<br />
with <math> P \equiv P(\underline{V},\underline{\psi}; x, t) </math>, where <br />
<math> \underline{V} </math> is the sample space of the velocity field <br />
<math> u,v,w </math>. This approach has the advantage of avoiding gradient-diffusion<br />
modelling. A similar equation to the above is obtained combining the momentum <br />
and scalar transport equation.<br />
<br />
<br />
<br />
<br />
The PDF transport equation can be solved in two ways: through a Lagrangian approach<br />
using stochastic methods or in a Eulerian ways using stochastic fields.<br />
<br />
==== Lagrangian ====<br />
<br />
The main idea of Lagrangian methods is that the flow can be represented by an <br />
ensemble of fluid particles. Central to this approach is the stochastic differential equations and in particular the [[Langevin equation]].<br />
<br />
==== Eulerian ====<br />
<br />
Instead of stochastic particles, smooth stochastic fields can be used<br />
to represent the [[probability density function ]] (PDF) of a scalar (or joint PDF) involved in transport (convection), diffusion<br />
and chemical reaction (Valino 1998). A similar formulation was proposed by Sabelnikov and Soulard 2005, which removes<br />
part of the a-priori assumption of "smoothness" of the stochastic fields.<br />
This approach is purely Eulerian and offers implementations advantages compared to<br />
Lagrangian or semi-Eulerian methods. <br />
Transport equations for scalars are often easy to programme and normal<br />
CFD algorithms can be used (see [[Discretisation of convective term]]).<br />
Although discretization errors are introduced by solving transport equations, <br />
this is partially compesated by the error introduced in Lagrangian approaches due to the numerical evaluation of means.<br />
<br />
A new set of <math> N_s </math> scalar variables<br />
(the stochastic field <math> \xi </math>) is used to represent the<br />
PDF<br />
:<math><br />
P (\underline{\psi}; x,t) = \frac{1}{N} \sum^{N_s}_{j=1} \prod^{N}_{k=1} \delta \left[\psi_k -\xi_k^j(x,t) \right]<br />
</math><br />
<br />
===Other combustion models===<br />
<br />
<br />
==== MMC ====<br />
<br />
The Multiple Mapping Conditioning (MMC) (Klimenko and Pope 2003) is an<br />
extension of the [[#Conditional Moment Closure (CMC)]] approach combined with<br />
[[probability density function]] methods. MMC<br />
looks for the minimum set of variables that describes the particular turbulent combustion<br />
system.<br />
<br />
==== Fractals ====<br />
Derived from the [[#Eddy Dissipation Concept (EDC)]].<br />
<br />
== References ==<br />
<br />
*{{reference-paper|author=Dopazo, C.|year=1993|title=Recent development in PDF methods|rest=Turbulent Reacting Flows, ed. P. A. Libby and F. A. Williams }}<br />
*{{reference-book|author=Fox, R.O.|year=2003|title=Computational Models for Turbulent Reacting Flows|rest=ISBN 0-521-65049-6,Cambridge University Press}}<br />
*{{reference-paper|author=Kerstein, A. R.|year=1988|title=A linear eddy model of turbulent scalar transport and mixing|rest=Comb. Science and Technology, Vol. 60,pp. 391}}<br />
*{{reference-paper|author=Klimenko, A. Y., Bilger, R. W.|year=1999|title=Conditional moment closure for turbulent combustion|rest=Progress in Energy and Combustion Science, Vol. 25,pp. 595-687}}<br />
*{{reference-paper|author=Klimenko, A. Y., Pope, S. B.|year=2003|title=The modeling of turbulent reactive flows based on multiple mapping conditioning|rest=Physics of Fluids, Vol. 15, Num. 7, pp. 1907-1925}}<br />
*{{reference-paper|author=Kronenburg, A.,|year=2004|title=Double conditioning of reactive scalar transport equations in turbulent non-premixed flames|rest=Physics of Fluids, Vol. 16, Num. 7, pp. 2640-2648}}<br />
*{{reference-paper|author=Griffiths, J. F.|year=1994|title=Reduced Kinetic Models and Their Application to Practical Combustion Systems |rest=Prog. in Energy and Combustion Science,Vol. 21, pp. 25-107}}<br />
*{{reference-book|author=Peters, N.|year=2000|title=Turbulent Combustion|rest=ISBN 0-521-66082-3,Cambridge University Press}}<br />
*{{reference-book|author=Poinsot, T.,Veynante, D.|year=2001|title=Theoretical and Numerical Combustion|rest=ISBN 1-930217-05-6, R. T Edwards}}<br />
*{{reference-paper|author=Pope, S. B.|year=1994|title=Lagrangian PDF methods for turbulent flows|rest=Annu. Rev. Fluid Mech, Vol. 26, pp. 23-63}}<br />
*{{reference-paper|author=Sabel'nikov, V.,Soulard, O.|year=2005|title=Rapidly decorrelating velocity-field model as a tool for solving one-point Fokker-Planck equations for probability density functions of turbulent reactive scalars|rest=Physical Review E, Vol. 72,pp. 016301-1-22}}<br />
*{{reference-paper|author=Valino, L.,|year=1998|title=A field montecarlo formulation for calculating the probability density function of a single scalar in a turbulent flow|rest=Flow. Turb and Combustion, Vol. 60,pp. 157-172}}<br />
*{{reference-paper|author=Westbrook, Ch. K., Dryer,F. L.,|year=1984|title=Chemical Kinetic Modeling of Hydrocarbon Combustion|rest=Prog. in Energy and Combustion Science,Vol. 10, pp. 1-57}}<br />
<br />
== External links and sources ==</div>DavidFhttp://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFDBest practice guidelines for turbomachinery CFD2008-09-16T15:30:25Z<p>DavidF: /* Wrong type of simulation */</p>
<hr />
<div>This article contains a summary of the most important knowledge and experience a CFD engineer needs in order to perform CFD simulations of turbomachinery components. The guide is mainly aimed at axial turbomachinery. The goal is to give a CFD engineer, who has just started working with turbomachinery simulations, a head start and avoid some of the most difficult pit-falls. Experienced turbomachinery CFD engineers can also use the guidelines in order to learn what other experts consider best practice. The intended audience is expected to know basic CFD terminology and have a basic turbomachinery knowledge, but no detailed knowledge about CFD for turbomachinery is needed. <br />
<br />
== Deciding what type of simulation to do ==<br />
<br />
Before starting a new turbomachinery simulation it is wise to think carefully of what it is that should be predicted and what physical phenomena that affect the results. This chapter contains a brief overview of the various types of simulations and some hints of what can be predicted with them.<br />
<br />
=== 2D, Quasi-3D or 3D ===<br />
<br />
2D simulations are often used in the early design phase in order to obtain a typical 2D section of a blade. For cases with many long blades or vanes, like low-pressure turbines, a 2D simulation can also provide reasonable results. If the area of the flow-path changes significantly in the axial direction it might be necessary to instead make a quasi-3D simulation. A quasi-3D simulation is a 2D simulation in which extra source terms are used to account for the acceleration/decelleration caused by a changing channel height or growing end-wall boundary layers. Codes focused on turbomachinery applications often have the possibility to perform quasi-3D simulations, but most general purpose CFD codes can not do this type of simulations, or require user coding to implement the correct source terms in the equations. Please contact your software distributor if your code does not have the quasi-3d possibility and you require it. Many codes require special routines or hidden commands to enable this feature.<br />
<br />
Full 3D simulations are necessary if a true 3D geometry is needed to obtain correct secondary flows and/or shock locations. For low-aspect-ratio cases with only a few short blades, like for example structurally loaded turbine outlet guide vanes, the secondary flow development is important and a 3D simulation is often necessary in order to obtain reasonable results. For applications where the end-wall boundary layers grow very quickly and interact with a large part of the flow-field it is necessary to perform a full 3D simulation. This is often the case in compressors and fans, where the negative pressure gradients make the boundary layers grow much quicker than what they do in for example turbines. For cases where the shock location is very critical, like in transonic compressors, it is also often necessary to perform a 3D simulation in order to obtain reasonable shock locations.<br />
<br />
=== Inviscid or viscid ===<br />
<br />
For attached flows close to the design point and without any large separations it is often sufficient with an inviscid Euler simulation in order to obtain reasonable blade loadings and pressure distributions. Note that inviscid Euler simulations should only be used if the boundary layers are judged to not have a significant effect on the global flow-field. A viscid Navier-Stokes simulation is necessary in order to predict losses, secondary flows and separations. As soon as separations are of interest it is of course also necessary to do a viscid simulation. Note that with todays computers it is often not time and resources that make users run inviscid Euler simulations. Running viscid Navier-Stokes simulations is now so quick that it is not a time problem any more. Euler simulations are still interesting though, since with an inviscid Euler simulation you don't have to worry about wall resolutions, y+ values, turbulence modeling errors etc.<br />
<br />
=== Transient or Stationary ===<br />
<br />
Most turbomachinery simlation are performed as stationary simulations. Transient simulations are done when some kind of transient flow behaviour has a strong influence on the global flow field. Examples of transient simulations are detailed simulations of rotor-stator interaction effects, simulations of large unsteady separations etc. Sometimes when you perform a steady stationary simulation you can see tendencies of unsteady behvaiour like for example periodic vortex shedding behind blunt trailing edges. This is often first seen as periodical variations of the residuals. If the unsteady tendencies are judged to not affect the overall simulation results if might be necessary to coarsen the mesh close to the vortex shedding or run a different turbulence model in order to make the simulation converge. Sometimes you are still forced to run a transient simulation and average the results if you don't obtain a converged steady solution.<br />
<br />
== Meshing ==<br />
<br />
In turbomachinery applications structured muli-block hexahedral meshes are most often used for flow-path simulations. In most solvers a structured grid requires less memory, provides superior accuracy and allows a better boundary-layer resolution than an unstructured grid. By having cells with a large aspect ratio around sharp leading and trailing edges a structured grid also provides a better resolution of these areas. Many companies have automatic meshing tools that automatically mesh blade sections with a structured mesh without much user intervention.<br />
<br />
Unstructured meshes are used for more complex and odd geometries where a structured mesh is difficult to create. Typical examples where unstructured meshes are often used are blade tip regions, areas involving leakage flows and secondary air systems, film cooling ducts etc.<br />
<br />
When meshing avoid to create large jumps in cell sizes. Typically the cell size should not change with more than a factor of 1.25 between neighbouring cells. For structured meshes also try to create fairly continous mesh lines and avoid discontinuities where the cell directions suddenly change. For multi-block structured meshes avoid placing the singular points where blocks meet in regions with strong flow gradients since most schemes have a lower accuracy in these singular points.<br />
<br />
=== Mesh size guidelines ===<br />
<br />
It is difficult to define, a priori, the mesh size. The required mesh size depends on the purpose of the simulation. <br />
<br />
If the main goal is to obtain static pressure forces a coarse mesh is often able to obtain a good solution, especially when an accurate resolution of the boundary layers is not required. For 2D inviscid simulations of one blade a mesh with say 3,000 cells is most often sufficient. For 3D inviscid blade simulations a mesh size of about 40,000 cells is usually sufficient. On inviscid Euler simulations the cells should be fairly equal in size and no boundary layer resolution should be present. Avoid having too skewed cells.<br />
<br />
For loss predictions and cases where boundary layer development and separation is important the mesh needs to have a boundary layer resolution. The boundary layer resolution can either be coarse and suitable for a [[Wall functions|wall function]] simulation or very fine and suitable for a [[Low-Re resolved boundary layers|low-Re]] simulation. For further information about selecting the near-wall turbulence model please see the [[#Turbulence modeling|turbulence modeling]] section. In 3D single-blade simulations a decent wall-function mesh typically has around 100,000 cells. This type of mesh size is suitable for quick design iterations where it is not essential to resolve all secondary flows and vortices. A good 3D wall-function mesh of a blade section intended to resolve secondary flows well should have at least 400,000 cells . A good low-Re mesh with resolved boundary layers typically has around 1,000,000 cells. <br />
<br />
In 2D blade simulations a good wall-function mesh has around 20,000 cells and a good low-Re mesh with resolved boundary layers has around 50,000 cells.<br />
<br />
Along the suction and pressure surfaces it is a good use about 100 cells in the streamwise direction. In the radial direction a good first approach is to use something like 30 cells for a wall-function mesh and 100 cells for a low-Re mesh.<br />
<br />
It is important to resolve leading and trailing edges well. Typically at least 10 cells, preferably 20 should be used around the leading and trailing edges. For very blunt and large leading edges, like those commonly found on HP turbine blades, 30 or more cells can be necessary.<br />
<br />
Cases which are difficult to converge with a steady simulation and which show tendencies of periodic vortex shedding from the trailing edge, can sometimes be "tamed" by using a coarse mesh around the trailing edge. This, of course, reduces the accuracy but can be a trick to obtain a converged solution if time and computer resouces does not allow a transient simulation to be performed.<br />
<br />
=== Boundary layer mesh ===<br />
<br />
For design iteration type of simulations where a [[Wall functions|wall function]] approach is sufficient [[Dimensionless wall distance (y plus) | y+]] for the first cell should be somewhere between 20 and 200. The outer limit is dependent on the actual Re number of the simulations. For cases with fairly low Re numbers make sure to keep the maximum y+ as low as possible. For more accurate simulations with resolved boundary layers the mesh should have a y+ for the first cell which is below 1. Some new codes are now using a hybrid wall treatment that allows a smooth transition from a coarse wall-function mesh to a resolved low-Re mesh. Use some extra care when using this type of hybrid technique since it is still fairly new and unproven.<br />
<br />
Outside of the first cell at a wall a good rule of thumb is to use a growth ratio normal to the wall in the boundary layer of maximum 1.25. For a low-Re mesh this usually gives around 40 cells in the boundary layer whereas a wall-function mesh does not require more than 10 cells in the boundary layer.<br />
<br />
If you are uncertain of which wall distance to mesh with you can use a [http://www.cfd-online.com/Links/tools.html#yplus y+ estimation tool] to estitmate the distance needed to obtain the desired y+. These estimation tools are ''very handy'' if you have not done any previous similar simulations. <br />
<br />
As a rule of thumb a wall-function mesh typically requires areound 5 to 10 cells in the boundary layer whereas a resolved low-Re mesh requires about 40 cells in the boundary layer.<br />
<br />
== Boundary conditions ==<br />
<br />
Describe different types of boundary conditions and when they should be used:<br />
<br />
* Total pressure in, static pressure out<br />
* Absorbing boundary conditions<br />
* ...<br />
<br />
There are different types of boundary conditions you can use:<br />
*Mass flow inlet, static pressure outlet.<br />
To put this BC you must allocate inlet surface enough far away from the turbomachine impeller. The velocity distribution is not<br />
constant over all the surface. If you put this "lie" away from your problem it works well.<br />
<br />
<br />
=== Turbulence inlet conditions ===<br />
<br />
Prescribing realistic turbulence inlet conditions is important and usually difficult. For two-equation turbulence models two different turbulence values need to be specified on the inlet. The most common way of specifying the inlet turbulence variables is to give an inlet [[Turbulence intensity|turbulence level]] and [[Turbulence length scale|length-scale]]. For simulations where the incoming boundary layers are estimated to not have any significant effect on the secondary flows and global flow-field downstream it is usually sufficient to specify a constant incoming turbulence level and length-scale.<br />
<br />
The incoming [[Turbulence intensity|turbulence level]] is dependent on which component is analyzed. Best is of course to have measurments from a similar application in order to be able to prescribe a fairly realistic value. Without measurements it is necessary to make an educated guess. Simulations with very complex and turbulence-generating components upstream will have a very high incoming turbulence level. For example, a high-pressure turbine just downstream of a turbulence generating combustor might have incoming turbulence levels up to 20%. Fans and low-pressure compressors with not many components upstream might have as low incoming turbulence level as 1%. For components between these two extreme examples, like high-pressure compressors, low-pressure turbines etc. a turbulence level of around 5% might be realistic. If you do not have any measurements and are unsure about which incoming turbulence level to prescribe it is always a good idea to run a few different simulations with say one half and one double incoming turbulence level in order to estimate how important it is for the simulation results.<br />
<br />
The incoming [[Turbulence length scale|turbulence length-scale]] is often even more difficult to guess than the incoming turbulence level. Measurements might sometimes include the incoming turbulence level, but very seldom also include the incoming turbulent length-scale. The best way of guessing a realistic incoming length-scale is to use the geometrical properties of the upstream components. The incoming turbulence length-scale can be estimated as say the thickness of upstream blades or somewhere between 2% and 20% of the incoming channel heigth. Fortunately the incoming turbulence length-scale is usually not that important for the end results. However, some k-omega models can have an unrelisticly strong influence on the incoming turbulence length-scale. Hence, be extra carefull about the length-scale when using k-omega models. Note that the [[SST k-omega model]] does not have this unrealistic sensitivity to the inlet turbulence length-scale.<br />
<br />
Instead of specifying an incoming turbulence length-scale it is sometimes more convenient to specify an incoming [[Eddy viscosity ratio|eddy viscosity ratio]]. For low-turbulence cases with no well-specified turbulence generating components upstream it can be difficult to estimate a turbulence length-scale. Specifying an eddy viscosity ratio instead gives a more direct control over how large the effect of the incoming turbulence is. The eddy viscosity ratio describes how large the turbulence viscosity is related to the molecular viscosity. In turbomachinery the eddy viscosity ratio can vary from say 10 in low-turbulence components like fans up to say 1,000 or in rare cases even 10,000 in high-turbulence components like high-pressure turbines. In components between these extremes a typical eddy viscosity ration might be about 200. Even if specifying an inlet turbulence length scale it can still be good to compute what eddy viscosity ratio this corresponds to in order to estimate if it sounds reasonable.<br />
<br />
== Turbulence modeling ==<br />
<br />
Selecting a suitable turbulence model for turbomachinery simulations can be a challenging task. There is no single model which is suitable for all types of simulations. Which turbulence model CFD engineers use has as much to do with beliefs and traditions as with knowledge and facts. There are many different schools. However, below follows some advices that most CFD engineers in the turbomachinery field tend to agree upon.<br />
<br />
For attached flows close to the design point a simple algebraic model like the [[Baldwin-Lomax model]] can be used. Another common choice for design-iteration type of simulations is the one-equation model by [[Spalart-Allmaras model | Spalart-Allmaras]]. This model has become more popular in the last years due to the many inherent problems in more refined two-equation models. The big advantage with both the [[Baldwin-Lomax model]] and the [[Spalart-Allmaras model]] over more advanced models is that they are very robust to use and rarely produce completely unphysical results.<br />
<br />
In order to accurately predict more difficult cases, like separating flows, rotating flows, flows strongly affected by secondary flows etc. it is often necessary to use a more refined turbulence model. Common choices are two-equation models like the <math>k-\epsilon</math> model.<br />
<br />
Two-equation models are based on the [[Boussinesq eddy viscosity assumption]] and this often leads to an over-production of turbulent energy in regions with strong acceleration or deceleration, like in the leading edge region, regions around shocks and in the suction peak on the suction side of a blade. To reduce this problems it is common to use a special model variant using, for example, [[Durbin's realizability constraint]] or the [[Kato-Launder modification]]. Note that different two-equation models behave differently in these problematic stagnation and acceleration regions. Worst is probably the standard <math>k-\epsilon</math> model. <math>k-\omega</math> model are slightly better but still do not behave well. More modern variants like Menter's [[SST k-omega model]] also has problems, wheras the [[V2-f models|v2f model ]] by Durbin behaves better.<br />
<br />
=== Near-wall treatment ===<br />
<br />
For on-design simulations without any large separated regions it is often sufficient to use a [[wall-function model]] close to the wall, preferably with some form of non-equilibrium wall-function that is sensitised to streamwise pressure gradients.<br />
<br />
For off-design simulation, or simulations involving complex secondary flows and separations, it is often necessary to use a [[low-Re model]]. There exist many low-Re models that have been used with success in turbomachinery simulations. A robust and often good choice is to use a one-equation model, like for example the [[Wolfstein model]], in the inner parts of the boundary layer. There are also several Low-Re <math>k-\epsilon</math> models that work well. Just make sure they don't suffer from the problem with overproduction of turbulent energy in regions with strong acceleration or deceleration. In the last few years Menter's low-Re <math>SST k-\omega</math> model has gained increased popularity.<br />
<br />
=== Transition prediction ===<br />
<br />
Transition refers to the process when a laminar boundary layer becomes unstable and transitions to a turbulent boundary layer. There are two types of transition - natural transition, where inherent instabilities in the boundary layer cause the transition and by-pass transition, where convection and diffusion of turbulence from the free-stream into the boundary layer cause the transition. Most transitions in turbomachinery are by-pass transitions caused by free-stream turbulence and other external disturbances like wakes, vortices and surface defects.<br />
<br />
Simulating transition in a CFD code accurately is very difficult. Often a separate transition model needs to be solved in order to specify the transition location and length. Predicting natural transition in a pure CFD code is not possible. Predicting by-pass transition in a pure CFD code is almost impossible, although there are people who claim to be able to predict by-pass transition with low-Re two-equation models. However, this is usually on special test cases and with simulations that have been tuned for these special cases, see for example [Saville 2002]. In reality transition is a very complex and sensitive process where disturbances like incoming wakes and vortices from previous stages, surface roughness effects and small steps or gaps in the surfaces play a significant role. <br />
<br />
The turbomachinery codes that have transition prediction models often use old ad-hoc models like the Abu-Ghannam and Shaw model [Abu-Ghannam 1980] or the Mayle model [Mayle 1991]. These models can be quite reliable if they have been validated and tuned for a similar application. Do not trust your transition predictions without having some form of experimental validation. Menter has also recently developed a new form of transition model that might work fairly well, but it is still too new and untested.<br />
<br />
For some turbomachinery applications, like modern high-lift low-pressure turbines, transition is critical. For these applications a CFD code with a transition model that has been tuned for this type of applications should be used.<br />
<br />
== Numerical considerations ==<br />
<br />
Use at least a second order accurate scheme for the flow variables. Some codes require a first order scheme for the turbulent variables (<math>k</math> and <math>\epsilon</math>) in order to converge well. It might be sufficient with a first order scheme only on the turbulence variables, but a second order scheme is of course preferable.<br />
<br />
=== Convergence criteria ===<br />
<br />
To know when a solution is converged is not always that easy. You need some prior experience of your CFD code and your application to judge when a simulation is converged. For normal pure aero simulations without resolved walls, i.e. with wall functions or inviscid Euler simulations, convergence can most often be estimated just by looking at the residuals. Exacty what the residuals should be is not possible to say, it all depends on how your particular code computes and scales the residuals. Hence, make sure to read the manuals and plot the convergence of a few global parameters before you decide what the residuals should be for a solution to converege. Note also that many manuals for genereal purpose CFD codes list overly agressive convergence critera that often produce unconverged results. <br />
For simulations with resolved walls it is good to look at the convergence of some global properties, like total pressure losses from the inlet to the outlet. For heat transfer simulations it is even more tricky since the aerodynamic field can look almost converged although the thermal field is not converged at all. If doing heat transfer simulations make sure to plot the heat-transfer, run for some time, and plot it again to make sure that it doesn't change anymore. With very well resolved walls and heat transfer it can sometimes take 10 times longer for the thermal field to converge.<br />
<br />
=== Single or double precision ===<br />
<br />
With todays computers and cheap memory prices it often does not cost much extra to run in double precision. Before using single precision you should first investigate how your software and hardware works with double precision. If the extra time and memory needed for double precision is negligible you should of course always run in double precision. With double precision you never have to worry about round-off errors. Always using double precision is one way of avoiding one type of pit-falls in the complex world of CFD simulations. Use double precision when you have resolved boundary layers (Y+ around 1) and when you use advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions.<br />
<br />
== Multi-stage analysis ==<br />
<br />
Multi-stage analysis can be done in different ways:<br />
<br />
* Steady mixing-plane simulations<br />
* Frozen rotor simulations<br />
* Unsteady sliding-mesh stator-rotor simulations<br />
* Other '''advanced''' multi-stage methods<br />
<br />
=== Steady mixing-plane simulations ===<br />
<br />
Since the mixing-plane method was first introduced in 1979 [Denton & Singh 1979] is has become the industry standard type of rotor-stator simulations. A mixing-plane simulation is steady and only requires one rotor blade and one stator blade per stage. Between the rotating blade passage and the steady vane passage the flow properties are circumferentially averaged in a so-called mixing-plane interface. This will of course remove all transient rotor-stator interactions, but it still gives fairly representative results. In some commercial codes (CFX for example) mixing-plane interfaces are also called stage-interfaces.<br />
<br />
=== Frozen rotor simulations ===<br />
<br />
In a frozen rotor simulation the rotating and the stationary parts have a fixed relative position. A frame transformation is done to include the rotating effect on the rotating sections. This will give a steady flow and no transient effects are included. With a frozen-rotor simulation rotating wakes, secondary flows, leading edge pressure increases etc. will always stay in exactly the same positions. This makes a frozen rotor simulation very dependent on exactly how the rotors and the stators are positioned. Most often a mixing-plane simulation gives better results. Frozen rotor simulations are mainly performed to obtain a good starting flow-field before doing a transient sliding-mesh simulation.<br />
<br />
=== Unsteady sliding-mesh stator-rotor simulations ===<br />
<br />
This is the most complete type of stator-rotor simulation. In most engines the number of stators and rotors do not have a common denominator (to avoid instabilities caused by resonance between different rings). Hence, to make a full unsteady sliding-mesh computation it is necessary to have a mesh which includes the full wheel with all stators vanes and all rotor blades. This is often not possible, instead it is necessary to reduce the number of vanes and blades by finding a denominator that is '''almost''' common and then scale the geometry slightly circumferentially. Here is an example:<br />
<br />
*Real engine: 36 stator vanes, 41 rotor blades<br />
*Approximated engine: 41 stator vanes, 41 rotor blades, making it possible to simulate only 1 stator vane and 1 rotor blade<br />
*Scaling of stator: All stator vanes are scaled by 36/41 = 0.8780 circumferentially.<br />
<br />
=== Other '''advanced''' multi-stage methods ===<br />
<br />
Time-inclinded, Adamszyk stresses ...<br />
<br />
== Heat transfer predictions ==<br />
<br />
Besides listing the general heat transfer mechanisms involved <br />
(namely conduction, convection, radiation)<br />
heat transfer prediction in CFD may be seen as or split into two cases. <br />
<br />
'''Mesh consists of fluid domain(s)''': <br />
you want to know how heat inserted into the fluid domain (e.g. via the flowing medium) changes the temperature field along the flow path. Be it a steady or transient run, a fluid which enters with a given temperature will usually experience temperature variations, for instance, caused by convective boundary conditions, prescribed temperature profiles at flow obstacles etc. <br />
<br />
'''Mesh consists of fluid and solid domain(s)''': <br />
additionally to the above, you want info too wrt the (spatial, temporal or even spectral) temperature field distribution in surrounding, confining or immersed solids like channel walls or heat exchanger tubes. This is also called conjugate heattransfer CHT in the CFD context. CHT requires a good boundary layer resolution, usually the wall mesh needs to be rather refined, to obtain realistic heatflux results at the fluid/solid interface. Flow and heat transfer convergence require different time step settings, to properly capture changes in flow and heat quantities respectively. <br />
<br />
<br />
In either case, verify (strict necessity depends on CFD code used, CFX for instance checks and assists in regard) that model dimensions, boundary conditions and properties are in consistent units, hold appropriate values. Check temperature-dependence of properties and other numbers before the run.<br />
<br />
In heat transfer predictions (depending on the CFD code in use) <br />
besides the flow solver, you may have to activate the thermal solver too, <br />
as a job specification.<br />
<br />
== Acoustics and noise ==<br />
<br />
A whole separate research subject, difficult. <br />
<br />
Tone noise possible. Often run with linearzised solvers in the frequency domain.<br />
<br />
Jet noise possible. Often run with LES or DES simulations that either also resolve the sound waves or couples to a separate acoustic solver.<br />
<br />
Turbomachinery broadband noise not possible yet, or at least a great challenge.<br />
<br />
== Errors and uncertainties ==<br />
<br />
CFD is still a tool which requires that a user has a good understanding of uncertainties and errors that might spoil a CFD similation. There exists no error control in CFD and any CFD simulation must be interpreted by an expirenced user to have some credibility. Without some knowledge about possible errors and how they can be handled a CFD simulation can not be trusted. Errors can occur at different places:<br />
<br />
* Definition of the problem - What needs to be analyzed?<br />
* Selection of the solution strategy - What physical models and what numerical tools should be used?<br />
* Development of the computational model - How should the geometry and the numerical tools be set up?<br />
* Analysis and interpretation of the results - How should the model be analyzed and the results be interpreted?<br />
<br />
There exists many different definitions on errors. In this guide the errors are classified into four types of errors: problem definition errors, model errors, numerical errors and user and code errors. The chapters below describe these errors and give some guidelines on how to avoid them.<br />
<br />
=== Problem definition errors ===<br />
<br />
Problem definition errors are the most basic form of errors. In order to obtain usefull results a CFD simulation must of course analyze the correct problem, have suitable boundary contions and be based on a relevant geometry.<br />
<br />
==== Wrong type of simulation ====<br />
<br />
It is essential to have an overview of the physics involved and how the problem can best be analyzed. Running a 2D simulation in order to understand secondary flows or running a steady simulation in order to understand transient behaviour is of course no use. When assessing a CFD simulation the first thing to consider is what physical phenomena are important for the results and if the selected type of simulation is suitable to resolve this type of phenomena. For further information about selecting the most suitable type of simulation please see the previous chapter on [[#Deciding what type of simulation to do|deciding what type of simulation to perform]].<br />
<br />
==== Incorrect or uncertain boundary conditions ====<br />
<br />
A common source of errors is that incorrect boundary conditions are used. The boundary conditions must be specified in enough detail in order to resolve all the important physical features in the problem. For further information about specifying boundary conditions please see the previous chapter on [[#Boundary conditions|boundary conditions]].<br />
<br />
==== Geometrical errors ====<br />
<br />
It is almost always necessary to simplify the geometry in some form. When assessing a CFD simulation one should consider how the geometrical simplifications can affect the interesting physical phenomena. Typical geometrical errors are:<br />
<br />
*Simplifications<br />
:Small geometrical features like fillets, small steps or gaps etc. can often be disregarded. When disregarding this type of features one should consider if they might affect the important physics. For example, a very large fillet on the suction-side of a vane might affect corner separations near the end-walls. A large tip-leakage might affect the flow physics significantly in the upper part of a compressor.<br />
*Tolerances and manufacturing discrepancies<br />
:If the geometry has very large tolerances or is manufactured in a way which might produce a non-ideal shape or position it might be necessary to perform additional CFD simulations in order to cover the whole span of possible real geometries.<br />
*Surface conditions - roughness, welds, steps, gaps etc.<br />
:Often CFD simulations assume a perfectly smooth surface. A non-smooth surface which might have welds, steps or even gaps will of course produce different results. If the physical phenomena of interest might depend on the surface conditions these should of course be condidered. Typical phenomena that might be dependent on this type of errors are transition prediction, leakage flows etc.<br />
<br />
=== Model errors ===<br />
<br />
Errors related to the computational model.<br />
<br />
==== Wrong physical models ====<br />
<br />
Once the type of simulation has been selected the next step is to select what type of physical models the simulation should use. The following points should be considered:<br />
<br />
* Gas data (incompressible/compressible, perfect gas/real gas, ...)<br />
* Turbulence modeling (type of model, type of near-wall treatment, ...)<br />
* Other models (combustion, sprays, ...)<br />
<br />
When assessing model related errors it is important to know the features of the selected model and think carefully how these features and possible short comings might affect the predicted physical behaviour. Using the wrong turbulence model or combustion model can completely destroy the results of a CFD simulation.<br />
<br />
=== Numerical errors ===<br />
<br />
Errors related to the numerical solution of the developed model. Typical examples of numerical errors are discretization errors, convergence errors and round-off errors.<br />
<br />
==== Discretization errors ====<br />
<br />
Discretization errors can either be spatial errors in space or temporal errors in time.<br />
<br />
Spatial discretization errors are what people normally call discretization errors. These errors are due to the difference between the exact solution and the numerical representation of the solution in space. Describing exactly what discretization different codes use and what errors this might lead to is not possible here. Instead some general rules to avoid these errors can be summarized as:<br />
* Use at least a 2nd order accurate scheme, preferably a 3rd order accurate scheme. Some general purpose codes have a 1st order upwind scheme as default, this is a very diffusives schemes that often gives too smooth results.<br />
* For new applications always run a simulation with a finer mesh to see how grid independent your solution is.<br />
* Be aware of checker-board errors. Checker-board errors occur close to strong shocks and other large discontinuities and can be seen as a wavy pattern with a wavelength of two cells. Some schemes, especially those who behave like central differencing schemes, are more prone to checker-board errors. Upwind schemes are a bit better and schemes like TVD or chock-capturing schemes are even better.<br />
<br />
Temporal discretization errors mainly effect transient simulations. However, some codes use a time-marching method also for steady simulations and then a temporal discretization error might affect the final steady solution slightly. The discretization in time can be done with 1st or 2nd order schemes or a Runge-Kutta method, which is more accurate and saves memory. Some codes can adapt the time-step, but often it is necessary to prescribe a time-step in advance. Think of the time-step as your grid in time and make sure that the grid-resolution in time is fine enough to resolve the highest frequencies. To avoid problems with temporal discretization errors the following should be considered:<br />
* Use at least a 2nd order scheme in time.<br />
* Do a physical estimation of the typical frequencies in time of the phenomena that you are interested in and select a time-step which is fine enough to resolve these frequencies well. After the simulation also look at the frequencies captured and make sure that they are well resolved by the chosen time-step.<br />
* For new applications try a finer time-step to ensure that your solution in time is fairly grid independent in time.<br />
<br />
==== Convergence errors ====<br />
<br />
To judge when a CFD simulation is converged is not always that easy. Different codes and different applications behave very differently. For a pure aero-simulation on a fairly coarse grid convergence is easy to judge, but for more complex simulations involving resolved boundary layers, heat transfer, combustion etc. convergence can be very tricky. Aside from looking at residuals one should always also look at how global parameters like static pressure distributions, total pressure losses, skin friction, heat transfer etc. change in time.<br />
For more information about how to avoid convergence errors see the previous chapter about [[#Convergence criteria]].<br />
<br />
==== Round-off errors ====<br />
<br />
When using single precision care needs to be taken to avoid round-off errors. Inviscid Euler simulations and simulations using wall-function meshes can most often be performed in single precision. For well resolved boundary layers with Y plus close to 1 it is often necessary to use double precision. If using double precision for very fine mesh resolutions make sure that you also create the mesh in double precision and not just run the solver in double precision. Sometimes a single precision solver converges slower than a double precision solver due to numerical errors caused by round-off errors. When using advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions it is also often necessary to use double precision.<br />
<br />
=== User and code errors ===<br />
<br />
Errors related to bugs in the code used or mistakes made by the CFD engineer.<br />
<br />
=== What to trust and what not to trust ===<br />
<br />
CFD is generally quite good at predicting surface static pressure distributions. With care CFD can also be used to predict performance, total-pressure losses and blade turning. <br />
<br />
Predicting separation, stall and off-design performance can be a challenge and results with non-attached flows should be interpreted with care. <br />
<br />
Heat transfer is often very difficult to predict accurately and it is common to obtain heat-transfer coefficients that are 100% wrong or more. Validation data is critical in order to be able to trust heat transfer simulations.<br />
<br />
Transition is almost impossible to predict accurately in general. However, there exist models that have been tuned to predict transition and these tend to give acceptable results for cases close to the ones they were tuned for.<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Abu-Ghannam, B.J. and Shaw, R.|year=1980|title=Natural Transition of Boundary Layers - the Effects of Turbulence, Pressure Gradient, and Flow History|rest=Journal of Mech. Eng. Science, vol. 22, no. 5, pp. 213–228}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1985|title=Model Equation for Simulating Flows in Multistage Turbomachinery|rest=ASME Paper 85-GT-226, also NASA TM-86869}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1999|title=Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design|rest=ASME Paper 99-GT-80.}}<br />
<br />
* {{reference-paper|author=Baldwin, B. S. and Lomax, H.|year=1978|title=Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows|rest=AIAA Paper 78-257}}<br />
<br />
* {{reference-paper|author=Chen, J. P. and Barter, J. W.|year=1998|title=Comparison of Time-Accurate Calculations for the Unsteady Interaction in Turbomachinery Stage|rest=AIAA Paper 98-3292}}<br />
<br />
* {{reference-paper|author=Denton, J. and Singh, U.|year=1979|title=Time Marching Methods for Turbomachinery Flow Calculations|rest=VKI Lecture Series 1979-7, von Karman Institute}}<br />
<br />
* {{reference-paper|author=Erdos, J., Alzner, E. and McNally, W.|year=1977|title=Numerical Solution of Periodic Transonic Flow Through a Transonic Fan Stage|rest=AIAA Journal, vol. 15, no. 11, pp. 1559-1568}}<br />
<br />
* {{reference-paper|author=Gerolymos, G., Vinteler, D., Haugeard, R., Tsange, G. and Vallet, I.|year=1996|title=On the Computation of Unsteady Turbomachinery Flows – Part II – Rotor/Stator Interaction using Euler Equations|rest=AGARD Report CP-571}}<br />
<br />
* {{reference-paper|author=Giles, M. B.|year=1990|title=Stator/Rotor Interaction in a Transonic Turbine|rest=AIAA Journal of Propulsion and Power, vol. 6, no. 5}}<br />
<br />
* {{reference-paper|author=He, L, and Ning, W.|year=1998|title=Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines|rest=AIAA Journal, vol. 36, no. 11, pp. 2005-2012}}<br />
<br />
* {{reference-paper|author=Mayle, R. E.|year=1991|title=The Role of Laminar-Turbulent Transition in Gas Turbine Engines|rest=ASME Journal of Turbomachinery, vol. 113, pp. 509-537}}<br />
<br />
* {{reference-paper|author=Menter, F.R. |year=1994|title=Two-equation eddy-viscosity turbulence models for engineering applications|rest=AIAA Journal, vol. 32, pp. 269-289}}<br />
<br />
* {{reference-paper|author=Savill, M., Dick, E., Hanjalić, K. and Voke, P.|year=2002|title=Synthesis Report of the ERCOFTAC Transition Modeling – TRANSPRETURB Thematic Network Activities 1998-2002|rest=Ecoftac Bulletin 54, September 2002, pp. 5-16}}<br />
<br />
* {{reference-paper|author=Wang, X. and Chen J.|year=2004|title=A Post-Processor to Render Turbomachinery Flows Using Phase-Lag Simulations|rest=AIAA Paper 04-615}}<br />
<br />
== External links ==<br />
<br />
* [http://www.qnet-cfd.net/newsletter/8th/n8_40-46.pdf QNET-CFD Best Practise Advice for Turbomachinery Internal Flows]<br />
* [http://www.qnet-cfd.net/newsletter/7th/n7_05.pdf State of the art in Industrial CFD for Turbomachinery Flows, QNET-CFD Network Newsletter, Volume 2, No. 3 – December 2003]<br />
* [https://pronet.wsatkins.co.uk/marnet/guidelines/guide.html MARNET-CFD Best Practice Guidelines for Marine Applications of CFD]<br />
* [http://www.stanford.edu/group/ctr/ResBriefs/temp05/weide.pdf On Large Scale Turbomachinery Computations, by E. van der Weide et. al., Center for Turbulence Research, Annual Research Briefs 2005]</div>DavidFhttp://www.cfd-online.com/Wiki/CombustionCombustion2008-09-15T12:44:26Z<p>DavidF: /* Conservation Laws */</p>
<hr />
<div>''The power of Fire, or Flame, for instance, which we designate by some trivial chemical name, thereby hiding from ourselves the essential character of wonder that dwells in it as in all things, is with these old Northmen, Loke, a most swift subtle Demon of the brood of the J\"otuns... From us too no Chemistry, if it had not Stupidity to help it, would hide that Flame is a wonder. What is Flame?''<br />
<br />
'''''Carlyle on''''' Heroes '''''Odin and Scandinavian Mythology.'''''<br />
<br />
<br />
== What is combustion -- physics versus modelling ==<br />
<br />
Combustion phenomena consist of many physical and chemical processes which exhibit a <br />
broad range of time and length scales. A mathematical description of combustion is not <br />
always trivial, although some analytical solutions exist for simple situations of <br />
laminar flame. Such analytical models are usually restricted to problems in zero or one-dimensional space.<br />
<br />
= Fundamental Aspects =<br />
<br />
== Main Specificities of Combustion Chemistry ==<br />
<br />
Combustion can be split into two processes interacting with each other: thermal, and chemical. <br />
<br />
The chemistry is highly exothermal (this is the reason of its use) but also highly temperature dependent, thus highly self-accelerating. In a simplified form, combustion can be represented by a single irreversible reaction involving 'a' fuel and 'an' oxidizer:<br />
:<math> \frac{\nu_F}{\bar M_F}Y_F + \frac{\nu_O}{\bar M_O} Y_O \rightarrow Product + Heat </math><br />
Althgough very simplified compared to real chemistry involving hundreds of species (and their individual transport properties) and elemental reactions, this rudimentary chemistry has been the cornerstone of combustion analysis and modelling.<br />
<br />
The most widely used form for the rate of the above reaction is the Arrh&eacute;nius law:<br />
:<math> \dot\omega = \rho A \left ( \frac{Y_F}{\bar M_F} \right )^{n_F} \left (\frac{Y_O}{\bar M_O}\right )^{n_O} \exp^{-T_a/T} </math><br />
<math> T_a </math> is the activation temperature, high in combustion, consistently with the temperature dependence.<br />
This is where the high non-linearity in temperature is modelled. ''A'' is the pre-exponential constant. One of the interpretation of the Arrh&eacute;nius law comes from gas kinetic theory: the number of molecules whose kinetic energy is larger than the minimum value allowing a collision to be energetic enough to trig a reaction is proportional to the exponential term introduced above divided by the square root of the temperature. This interpretation allows one to think that the temperature dependence of ''A'' is very weak compared to the exponential term. ''A'' is eventually considered as constant.<br />
The reaction rate is also naturally proportional to the molecular density of each of the reactant. Nonetheless, the orders of reaction <math> n_i</math> are different from the stoichiometric coefficients as the single-step reaction is global, not governed by collision for it represents hundreds of elementary reactions.<br />
If one goes into the details, combustion chemistry is based on chain reactions, decomposed into three main steps: (i) generation (where radicals are created from the fresh mixture), (ii) branching (where products and new radicals appear from interaction of radicals with reactants), and (iii) termination (where radicals collide and turn into products). The branching step tends to accelerate the production of active radicals (autocatalytic). The impact is nevertheless small compared to the high non-linearity in temperature. This explains why single-step chemistry has been sufficient for most of the combustion modelling work up to now.<br />
<br />
The fact that a flame is a very thin reaction zone separating, and making the transition between, a frozen mixture and an equilibrium is explained by the high temperature <br />
dependence of the reaction term, modelled by a large activation temperature, and a large heat release (the ratio of the burned and fresh gas temperatures is about 7 for typical hydrocarbon flames) leading to a sharp self-acceleration in a very narrow area. To evaluate the order of magnitude of the quantities, the terms in the exponential argument are normalized:<br />
:<math> \beta=\alpha\frac{T_a}{T_s} \qquad \alpha=\frac{T_s-T_f}{T_s} </math><br />
<math>\beta</math> is named the Zeldovitch number and <math>\alpha</math> the heat release factor. <br />
Here, <math> T_s</math> has been used instead of <math> T_b</math>, the conventional notation for burned gas temperature (at final equilibrium). <math> T_s</math> is actually <math> T_b</math> <br />
for a mixture at stoichiometry and when the flame is adiabatic, i.e. this is the reference highest temperature that can be<br />
obtained in the system. That said, typical value for <math>\beta</math> and <math>\alpha</math> are 10 and 0.9, giving <br />
a good taste of the level of non-linearity of the combustion process with respect to temperature. <br />
Actually, the reaction rate is rewritten as:<br />
:<math> \dot\omega = \rho A \left ( \frac{Y_F}{\bar M_F} \right )^{n_F} \left (\frac{Y_O}{\bar M_O}\right )^{n_O} <br />
\exp^{-\frac{\beta}{\alpha}}\exp^{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}} </math><br />
where the non-dimensionalized temperature is:<br />
:<math>\theta=\frac{T-T_f}{T_s-T_f}</math><br />
The non-linearity of the reaction rate is seen from the exponential term:<br />
:* <math> {\mathcal O}(\exp^{-\beta}) </math> for <math>\theta</math> far from unity (in the fresh gas)<br />
:* <math> {\mathcal O}(1) </math> for <math>\theta</math> close to unity (in the reaction zone close to the burned gas whose temperature must be close to the adiabatic one <math> T_s </math>), more exactly <math> 1-\theta \sim {\mathcal O}(\beta^{-1})</math><br />
<br />
[[Image:NonLinearite.jpg|thumb|Temperature Non-Linearity of the Source Term: the Temperature-Dependent Factor of the Reaction Term for Some Values of the Zeldovtich and Heat Release Parameters]]Note that for an infinitely high activation energy, the reaction rate is piloted by a <math>\delta(\theta)</math> function. The figure, beside, illustrates how common values of <math>\beta</math> around 10 tend to make the reaction rate singular around <math>\theta</math> of unity. Two set of values are presented: <math><br />
\beta = 10</math> and <math>\beta = 8</math>. The first magnitude is the representative value while the second one is a smoother one usually used to ease numerical simulations. In the same way, two values for the heat release <math>\alpha</math> 0.9 and 0.75 are explored. The heat release is seen to have a minor impact on the temperature non-linearity.<br />
<br />
== Transport Equations ==<br />
<br />
Additionally to the Navier-Stokes equations, at least with variable density, the transport equations for a reacting flow are the energy and species transport equations. In usual notations, the specie ''i'' transport equation is written as:<br />
:<math>\frac{D \rho Y_i}{D t} = \nabla\cdot \rho D_i\vec\nabla Y_i - \nu_i\bar M_i\dot\omega</math><br />
and the temperature transport equation:<br />
:<math>\frac{D\rho C_p T}{Dt} = \nabla\cdot \lambda\vec\nabla T + Q\nu_F \bar M_F \dot\omega </math><br />
The diffusion is modelled thanks to Fick's law that is a (usually good) approximation to the rigorous diffusion velocity calculation. Regarding the temperature transport equation, it is derived from the energy transport equation under the assumption of a low-Mach number flow (compressibility and viscous heating neglected). The low-Mach number approximation is suitable for the deflagration regime (as it will be demonstrated [[#Premixed|below]]), which is the main focus of combustion modelling. Hence, the transport equation for temperature, as a simplified version of the energy transport equation, is usually retained for the study of combustion and its modelling.<br />
<br />
=== Low-Mach Number Equations ===<br />
In compressible flows, when the motion of the fluid is not negligible compared to the speed of sound (which is the speed at which the molecules can reorganize themselves), the heap of molecules results in a local increase of pressure and temperature moving as an acoustic wave. It means that, in such a system, a proper reference velocity is the speed of sound and a proper pressure reference is the kinetic pressure. A contrario, in low-Mach number flows, the reference speed is the natural representative speed of the flow and the reference pressure is the thermodynamic pressure. Hence, the set of reference quantities to characterize a low-Mach number flow is given in the table below:<br />
<br />
Density <math>\rho_o</math> A reference density (upstream, average, etc.)<br />
<br />
Velocity <math>U_o</math> A reference velocity (inlet average, etc.)<br />
<br />
Temperature <math>T_o</math> A reference temperature (upstream, average, etc.)<br />
<br />
Pressure (static) <math>P_o=\rho_o \bar r T_o</math> From Boyle-Mariotte<br />
<br />
Length <math>L_o</math> A reference length (representative of the domain)<br />
<br />
Time <math>L_o/U_o</math><br />
<br />
Energy <math>C_p T_o</math> Internal energy at constant reference pressure <br />
<br />
The equations for fluid mechanics properly adimensionalized can be written:<br />
<br />
Mass conservation:<br />
:<math> \frac{D\rho}{Dt} =0 </math><br />
<br />
Momentum:<br />
:<math>\frac{D\rho\vec U}{Dt}=-\frac{1}{\gamma M}\vec\nabla P+\nabla\cdot\frac{1}{Re}\bar\bar\Sigma</math><br />
<br />
Total energy:<br />
:<math>\frac{D\rho e_T}{Dt}-\frac{1}{RePr}\nabla\cdot\lambda\vec\nabla T=-\frac{\gamma-1}{\gamma}\nabla\cdot P\vec U<br />
+\frac{1}{Re}M^2(\gamma-1)\nabla\cdot \vec U\bar\bar\Sigma + \frac{\rho}{C_pT_o(U_o/L_o)}Q\nu_F \bar M_F\dot\omega</math><br />
<br />
Specie:<br />
:<math>\frac{D\rho Y}{Dt}-\frac{1}{ScRe}\nabla\cdot\rho D\vec\nabla Y=-\frac{\rho}{U_o/L_o}\nu\bar M\dot\omega</math><br />
<br />
State law:<br />
:<math> P=\rho T</math><br />
<br />
The low-Mach number equations are obtained considering that <math> M^2 </math> is small. 0.1 is usually taken as the limit, which recovers the value of a Mach number of 0.3 to characterize the incompressible regime.<br />
<br />
Considering the energy equation, in addition to the terms with <math> M^2 </math> in factor in the equation, the total energy reduces to internal energy as: <math> e_T = T/\gamma+M^2(\gamma-1)\vec U^2/2 </math>. Moreover, the work of pressure is considered as negligible because the gradient of pressure is negligible (low-Mach number approximation is indeed also named ''isobaric'' approximation) and the flow is assumed close to a divergence-free state.<br />
For the same reason, volumic energy and enthalpy variations are assumed equal as they only differ through the addition of pressure. Hence, redimensionalized, the low-Mach number energy equation leads to the temperature equation as used in combustion analysis:<br />
:<math>\frac{D\rho C_p T}{Dt} = \nabla\cdot \lambda\vec\nabla T + Q\nu_F \bar M_F \dot\omega </math><br />
<br />
______________________________<br />
<br />
'''Note:''' The species and temperature equations are not closed as the fields of velocity and density also need to be computed. Through intense heat release in a very small area (the jump in temperature in typical hydrocarbon flames is about seven and so is the drop in density in this isobaric process, and the thickness of a flame is of the order of the millimetre), combustion influences the flow field. Nevertheless, the vast majority of combustion modelling has been developed based on the species and temperature equations, assuming simple flow fields.<br />
<br />
=== The Damk&ouml;hler Number ===<br />
<br />
A flame is a reaction zone. From this simple point of view, two aspects have to be considered: (i) the rate at which it is fed by reactants, let call <math> \tau_d </math> the characteristic time, and<br />
the strength of the chemistry to consume them, let call the characteristic chemical time <math> \tau_c </math>. In combustion, the Damk&ouml;hler number, ''Da'', compares these both time scales and, for that <br />
reason, it is one of the most integral non-dimensional groups:<br />
:<math>Da=\frac{\tau_d}{\tau_c}</math>.<br />
If ''Da'' is large, it means that the chemistry has always the time to fully consume the fresh mixture and turn it into equilibrium. Real flames are usually close to this state. The characteristic reaction time, <math> (Ae^{-T_a/T_s})^{-1} </math>, is <br />
estimated of the order of the tenth of a ms. When ''Da'' is low, the fresh mixture cannot be converted by a too weak chemistry. The flow remains frozen. This situation happens with ignition or misfire, for instance. <br />
<br />
The picture of a deflagration lends itself to a description based on the Damk&ouml;hler number. A reacting wave progresses towards the fresh mixture through preheating of the upstream closest layer. The elevation of the temperature strengthens the chemistry and reduces its characteristic time such that the mixture changes from a low-''Da'' region (far upstream, frozen) to a high-''Da'' region in the flame (intense reaction to equilibrium).<br />
<br />
== Conservation Laws ==<br />
<br />
The processus of combustion transforms the chemical enthalpy into sensible enthalpy (i.e. rise the temperature of the gases thanks to the heat released). Simple relations can be drawn between species and temperature by studying the source terms appearing in the above equations:<br />
:<math> \frac{Y_F}{\nu_F\bar M_F} - \frac{Y_O}{\nu_O\bar M_O} = \frac{Y_{F,u}}{\nu_F\bar M_F} - \frac{Y_{O,u}}{\nu_O\bar M_O} </math><br />
:<math> Y_F + \frac{Cp T}{Q} = Y_{F,u} + \frac{CpT_u}{Q} </math><br />
Hence <math> T_b = T_u + \frac{Q Y_{F,u}}{Cp} </math>, <math> Y_{O,b} = Y_{O,u} - \frac{\nu_O \bar M_O}{\nu_F \bar M_F} Y_{F,u} = Y_{O,u} -sY_{F,u} </math> and <math> Y_{F,b} = 0 </math>. Here, the example has been taken for a lean case.<br />
<br />
As mentioned in [[#Main Specificities of Combustion Chemistry|Sec. Main Specificities]], the stoichiometric state is used to non-dimensionalize the conservation equations:<br />
:<math> Y_i^* = Y_{i,u}^* - \theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} \qquad ; \qquad Y_i^*=Y_i/Y_{i,s}</math>.<br />
<br />
A comprehensive form of the reaction rate can be reconstituted to understand the difficulty of numerically resolving the reaction zone:<br />
:<math> \dot\omega = B \prod_{i=O,F}(Y_{i,u}^*-\theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_i} \exp{\left ( -\beta\frac{1-\theta}{1-\alpha(1-\theta)}\right)} </math><br />
where <math> B </math> stands for all the constant terms present in this reaction rate, plus density.<br />
<br />
[[Image:NonLineariteii.jpg|thumb|Source Term versus Temperature]]<br />
For the stoichiometric case and a global order of two, the reaction rate is graphed versus the reduced temperature for different values of the heat release and Zeldovitch parameter. A high value of <math>\beta</math> makes the reaction rate very sharp, versus temperature. It means that reaction is significant beyond a temperature level (sometimes called ignition temperature) that is close to one (the exponential term above is non-negligible for <math>1-\theta \sim \beta^{-1}</math>). The heat release has qualitatively the same impact but not so strong. Transposed to the case of a flame sheet, it effectively shows that the reaction exists only in a fraction of the thermal thickness of the flame (the region close to the flame that the latter preheats, hence, where the reduced temperature rises from 0 to 1 here) where the temperature deviates few from the maximal one (density can be assumed as constant and equal to its burned-gas value). Numerically capturing such a sharp reaction zone can be costly and the lower values of <br />
<math>\beta</math> and <math>\alpha</math> as presented here are usually preferred whenever possible.<br />
<br />
<br />
Most problems in combustion involve turbulent flows, gas and liquid <br />
fuels, and pollution transport issues (products of combustion as well as for example noise <br />
pollution). These problems require not only extensive experimental <br />
work, but also numerical modelling. All combustion models must be validated <br />
against the experiments as each one has its own drawbacks and limits. In this article, we will address the modeling fundamentals only.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new dimensionless parameters are introduced, the most important of which are the [[Karlovitz number]] and the [[Damkohler number]] which represent ratios of chemical and flow time scales, and <br />
the [[Lewis number]] which compares the diffusion speeds of species.<br />
The combustion models are often classified on their capability to deal with the different combustion regimes. <br><br />
[[Image:SUMMARY_OF_COMBUSTION_MODELS.jpg]]<br />
<br />
= Three Combustion Regimes =<br />
<br />
Depending on how fuel and oxidizer are brought into contact in the combustion system, different combustion modes or regimes are identified. Traditionally, two regimes have been recognized: the ''premixed'' regime and the ''non-premixed'' regime. Over the last two decades, a third regime, sometime considered as a hybrid of the two former ones to a certain extend, has risen. It has been named ''partially-premixed'' regime.<br />
<br />
== The Non-Premixed Regime ==<br />
[[Image:DiffusionFlame.jpg|thumb|Sketch of a diffusion flame]]<br />
This regime is certainly the easiest to understand. Everybody has already seen a lighter, candle or gas-powered stove. Basically, the fuel issues from a nozzle or a simple duct into the atmosphere. The combustion reaction is the oxidization of the fuel. Because fuel and oxidizer are in contact only in a limited region but are separated elsewhere (especially in the feeding system) this configuration is the safest. The non-premixed flame has some other advantages. By controlling the flows of both reactants, it is (theoretically) possible to locate the stoichiometric interface, and thus, the location of the flame sheet. Moreover, the strength of the flame can also be controlled through the same process. Depending on the width of the transition region from the oxidizer to the fuel side, the species (fuel and oxidizer) feed the flame at different rates.<br />
This is because the diffusion of the species is directly dependent on the unbalance (gradient) of their distribution. A sharp transition from fuel to oxidizer creates intense diffusion of those species towards the flame, increasing its burning rate. This burning rate control through the diffusion process is certainly one of the reasons of the alternate name of such a flame and combustion mode: ''diffusion'' flame and ''diffusion'' regime.<br />
<br />
Because a diffusion flame is fully determined by the inter-penetration of the fuel and oxidizer streams, it has been convenient to introduce a tracer of the state of the mixture. This is the role of the ''mixture fraction'', usually called ''Z'' or ''f''. Z is usually taken as unity in the fuel stream and is null in the oxidizer stream. It varies linearly between this two bounds such that at any point of a frozen flow the fuel mass fraction is given by <math> Y_F=ZY_{F,o} </math> and the oxidizer mass fraction by <math> Y_O = (1-Z)Y_{O,o} </math>. <math> Y_{F,o} </math> and <math> Y_{O,o} </math> are the fuel and oxidizer mass fractions in the fuel and oxidizer streams, respectively.<br />
The mixture fraction posses a transport equation that is expected to not have any source term as a tracer of a mixture must be a conserved scalar. First, the fuel and oxidizer mass fraction transport equations are written in usual notations:<br />
:<math>\frac{D \rho Y_F}{D t} = \nabla\cdot \rho D\vec\nabla Y_F - \nu_F\bar M_F\dot\omega</math><br />
:<math>\frac{D \rho Y_O}{D t} = \nabla\cdot \rho D\vec\nabla Y_O - \nu_O\bar M_O\dot\omega</math><br />
The two above equations are linearly combined in a single one in a manner that the source term disappears:<br />
:<math>\frac{D \rho (\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)}{D t} = \nabla\cdot \rho D\vec\nabla (\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)</math><br />
The quantity <math>(\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)</math> is thus a conserved scalar. The last step is to normalize it such that it equals unity in the pure fuel stream (<math> Y_F=Y_{F,o}</math> and <math> Y_O=0 </math>) and is null in the pure oxidizer stream <br />
(<math> Y_F=0</math> and <math> Y_O=Y_{O,o} </math>). The resulting normalized passive scalar is the mixture fraction:<br />
:<math>Z=\frac{\Phi(Y_F/Y_{F,o})-(Y_O/Y_{O,o})+1}{\Phi+1}\qquad \Phi=\frac{\nu_O\bar M_O Y_{F,o}}{\nu_F\bar M_F Y_{O,o}}</math><br />
governed by the transport equation<br />
:<math>\frac{D \rho Z}{D t} = \nabla\cdot \rho D\vec\nabla Z </math><br />
The stoichiometric interface location (and thus the approximate location of the flame if the flow is reacting) is where <math>\Phi(Y_F/Y_{F,o})-(Y_O/Y_{O,o}) </math> vanishes (or <math>Y_F</math> and <math> Y_O </math> are both null in the reacting case). This leads to a stoichiometry definition:<br />
:<math>Z_s=\frac{1}{1+\Phi}</math><br />
<br />
As the mixture fraction qualifies the degree of inter-penetration of fuel and oxidizer, the elements originally present in these molecules are conserved and can be directly traced back to the mixture fraction. This has led to an alternate defintion of the mixture fraction, based on ''element conservation''.<br />
First, the elemental mass fraction <math> X_{j} </math> of element ''j'' is linked to the species mass fraction <math> Y_i </math>:<br />
:<math> X_j = \sum_{i=1}^{n} \frac{a_{i,j} \bar M_j}{\bar M_i} Y_i </math><br />
where <math> a_{i,j} </math> is a matrix counting the number of element ''j'' atoms in specie molecule named ''i'' and ''n'' is the number of species in the mixture.<br />
The group pictured by the summation above is a linear combination of <math> Y_i </math>. Because the transport equations of species mass fraction, few lines earlier, are also linear, a transport equation for the elemental mass fraction can be written:<br />
:<math>\frac{D \rho X_j}{D t} = \nabla\cdot \sum_{i=1}^n\frac{a_{i,j}\bar M_j}{\bar M_i}\rho D_i\vec\nabla Y_i</math><br />
For mass is conserved, the linear combination of the source terms vanishes. Furthermore, by taking the same diffusion coefficient <math> D_i </math> for all the species, the elemental mass fraction transport equation has exactly the same form as the specie transport equation (except the source term). Notice that the assumption of equal diffusion coefficient was also made in the previous definition of the mixture fraction and is justified in turbulent combustion modelling by the turbulence diffusivity flattening the diffusion process in high Reynolds number flows. Hence, the elemental mass fraction transport equation has the same structure as the mixture fraction transport equation seen above. Properly renormalized to reach unity in the fuel stream and zero in the oxidizer stream, the elemental mass fraction is a convenient way of determining the mixture fraction field in a flow. Indeed, it is widely used in practice for this purpose.<br />
<br />
==== Dissipation Rate ====<br />
A very important quantity, derived from the mixture fraction concept, is the ''scalar dissipation rate'', usually noted: <math> \chi </math>. In the above introduction to non-premixed combustion, it has been said that a diffusion flame is fully controlled through: (i) the position of the stoichiometric line, dictating where the flame sheet lies; (ii) the gradients of fuel on one side and oxidizer on the other side, dictating the feeding rate of the reaction zone through diffusion and thus the strength of combustion. According the the mixture fraction definition, the location of the stoichiometric line is naturally tracked through the <math>Z_s</math> iso-line and it is seen here how the mixture fraction is a convenient tracer to locate the flame.<br />
In the same manner, the mixture fraction field should also be able to give information on the strength of the chemistry as the gradients of reactants are directly linked to the mixture fraction distribution. The feeding rate of the reaction zone is characterized through the inverse of a time. Because it is done through diffusion, it must be obtained through a combination of mixture fraction gradient and diffusion coefficient (dimensional analysis):<br />
:<math> \chi_s = \frac{(\rho D)_s}{\rho_s} ||\vec \nabla Z||^2_s \qquad (\rho D)_s = \left ( \frac{\lambda}{C_p} \right )_s</math><br />
where the subscript ''s'' refers to quantities taken effectively where the reacting sheet is supposed to be, close to stoichiometry. This deduction of the scalar dissipation rate, scaling the feeding rate of the flame, is obtained here through physical arguments and is to be derived from equations below in a more mathematical manner. Note that the transport coefficient for the mixture fraction is identified to the one for temperature. This notation is usually used in the literature to emphasize that the rate of temperature diffusion (that is commensurable to the rate of species diffusion) is the retained parameter (as introduced in the following approach highlighting the role of the scalar dissipation rate).<br />
<br />
Because combustion is highly temperature-dependent, ''T'' is certainly the scalar to which attention must be paid for. The temperature equation in the Low-Mach Number regime ([[#Low-Mach Number Equations|Sec. Low-Mach Number Equations]]) is written below in steady-state:<br />
:<math>\rho\vec U \cdot \vec\nabla T = \nabla\cdot \frac{\lambda}{C_p}\vec\nabla T + \frac{Q}{C_p}\nu_F \bar M_F \dot\omega </math><br />
In order to make this equation easily tractable, the Howarth-Dorodnitzyn transform and the Chapman approximation are applied.<br />
In the Chapman approximation, the thermal dependence of <math>\lambda/C_p</math> is approximated as <math>\rho^{-1}</math>.<br />
The Howarth-Dorodnitzyn transform introduces <math>\rho</math> in the space coordinate system: <math> \vec\nabla \rightarrow \rho\vec\nabla \quad ; \quad \nabla\cdot\rightarrow \rho\nabla\cdot</math>. The effect of these both mathematical operations is to `digest' the thermal variation of quantities such as density or transport coefficient. Hence, the temperature equation comes in a simpler mathematical shape:<br />
:<math>\frac{\rho}{\rho_s}\vec U \cdot \vec\nabla T = \left ( \frac{\lambda}{C_p}\right )_s \nabla\cdot \vec\nabla T + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
Here the references are taken in the flame, i.e. close to the stoichiometric line (''s'' subscript).<br />
<br />
<br />
<br />
In a non-premixed system, strictly speaking, <math> \vec U </math>, is not really relevant as the flame is fully controlled by the diffusion process. Notwithstanding, in practice, non-premixed flames must be stabilized by creating a strain in the direction of diffusion. This is the reason why the velocity is left in the equation. Because a diffusion flame is fully described by the mixture fraction field, a change in coordinate can be applied:<br />
:<math> \left (<br />
\begin{array}{c}<br />
x \\<br />
y<br />
\end{array}\right ) \longrightarrow <br />
\left (<br />
\begin{array}{c}<br />
x \\<br />
Z<br />
\end{array}\right )<br />
</math><br />
where ''x'' is the coordinate tangential to the iso-<math>Z_s</math> (hence to the flame, in a first approximation) and ''y'' is perpendicular.<br />
The Jacobian of the transform is given as:<br />
:<math> \left [<br />
\begin{array}{cc}<br />
\frac{\partial x}{\partial x} & \frac{\partial Z}{\partial x} \\<br />
\frac{\partial x}{\partial y} & \frac{\partial Z}{\partial y} <br />
\end{array}\right ] = <br />
\left [<br />
\begin{array}{cc}<br />
1 & 0 \\<br />
0 & l_d^{-1} <br />
\end{array}\right ]</math><br />
Note that the diffusive layer of thickness <math>l_d</math> is defined as the region of transition between fuel and oxidizer and is thus given by the gradient of ''Z'' along the ''y'' direction.<br />
This transform is applied to the vectorial operators:<br />
:<math> \nabla\cdot = \nabla_x\cdot+\nabla_y\cdot=\nabla_x\cdot+\nabla_y\cdot Z\nabla_Z\cdot</math><br />
:<math> \vec\nabla = \vec\nabla_x+\vec\nabla_y=\vec\nabla_x+\vec\nabla_y Z\nabla_Z\cdot</math><br />
<br />
With this transform, the above temperature equation looks like:<br />
:<math>\frac{\rho}{\rho_s}\vec U \cdot (\vec\nabla_x T + \vec\nabla Z \nabla_Z\cdot T) = \left (\frac{\lambda}{C_p}\right )_s \left ( \nabla_x\cdot \vec\nabla_x T + \nabla_y\cdot Z \nabla_Z\cdot \vec\nabla_y Z \nabla_Z\cdot T + \nabla_y\cdot Z \nabla_Z\cdot \vec\nabla_x T + \nabla_x\cdot \vec\nabla_y Z \nabla_Z\cdot T \right ) + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
As mentioned above, the velocity and the variation along the tangential direction to the main flame structure ''x'' are not supposed to play a major role. By emphasizing the role of the gradient of ''Z'' along ''y'' as a key parameter defining the configuration the following equation is obtained:<br />
:<math>0 = \left (\frac{\lambda}{C_p}\right )_s ||\vec\nabla_y Z||^2 \Delta_Z T + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
This equation (sometimes named the ''flamelet equation'') serves as the basic framework to study the structure of diffusion flames. It highlights the role of the dissipation rate with respect to the strength of the source term and shows that the dissipation rate calibrates the combustion intensity.<br />
<br />
To describe the structure of the diffusion flame, the ''reduced mixture fraction'' is set:<br />
:<math> \xi = \frac{Z-Z_s}{Z_s(1-Z_s)\varepsilon} </math><br />
The utility of the reduced mixture fraction is to focus on the reaction zone. This reaction zone is supposed located on the stoichiometric line (this is why the reduced mixture fraction is centred on <math>Z_s</math>) and to be very thin (reason of the introduction of the magnifying factor <math>\varepsilon</math>).<br />
<br />
== The Premixed Regime ==<br />
[[Image:Premixed.jpg|thumb|Sketch of a premixed flame]]<br />
In contrast to the non-premixed regime above, the reactants are here well mixed before entering the combustion chamber. Chemical reaction can occur everywhere and this flame can propagate upstream into the feeding system as a subsonic (deflagration regime) chemical wave. This presents lots of safety issues. Some situations prevent them: (i) the mixture is made too rich (lot of fuel compared to oxidizer) or too lean (too much oxidizer) such that the flame is close to its flammability limits (it cannot easily propagate); (ii) the feeding system and regions where the flame is not wanted are designed such that they impose strong heat loss to the flame in order to quench it. For a given thermodynamical state of the mixture (composition, temperature, pressure), the flame has its own dynamics (speed, heat release, etc) on which there is few control: the wave exchanges mass and energy through diffusion process in the fresh gases. On the other hand, those well defined quantities are convenient to describe the flame characteristics. The mechanism of spontaneous propagation<br />
towards fresh gas through the thermal transfer from the combustion zone to the immediate slice of fresh gas<br />
such that the ignition temperature is eventually reached for this latter was highlighted as early as by the end of the 19th century by Mallard and LeChatelier. <br />
The reason the chemical wave is contained in a narrow region of reaction propagating upstream is the consequence of the discussion on the non-linearity of the combustion with temperature in the <br />
[[#Fundamental Aspects|Sec. Fundamental Aspects]]. It is of interest to compare the orders of magnitude of the temperature dependent term <math> \exp{(-\beta(1-\theta)/(1-\alpha(1-\theta)))}</math> of the reaction source upstream in the fresh gas (<math>\theta\rightarrow 0</math>) and in the reaction zone close to equilibrium temperature (<math>\theta\rightarrow 1 </math>) for the set of representative values: <math> \beta = 10 </math> and <br />
<math>\alpha=0.9</math>. It is found that the reaction is about <math>10^{43}</math> times slower in the fresh<br />
gas than close to the burned gas. It is known that the chemical time scale is about 0.1 ms in the reaction zone of a typical flame, then the typical reaction time in the fresh gas in normal conditions is about <br />
<math> 10^{39} s </math>. To be compared with the order of magnitude of the estimated Universe age: <math> 1 0^{17} s </math>. Non-negligible chemistry is only confined in a thin reaction zone stuck to the hot burned gas at equilibrium temperature. In this zone, the [[#Damk&ouml;hler]] number is high, in contrast to in the fresh mixture. It is natural and convenient to consider that the reaction rate is strictly zero everywhere except in this small reaction zone (one recovers the Dirac-like shape of the reaction profile, <br />
provided that one can see the upstream flow as a region of increasing temperature towards the combustion zone and the downstream flow as in fully equilibrium).<br />
<br />
As the premixed flame is a reaction wave propagating from burned to fresh gases, the basic parameter is known to be the ''progress variable''. In the fresh gas, the progress variable is conventionally put to zero. In the burned gas, it equals unity. Across the flame, the intermediate values describe the progress of the reaction to turn into burned gas the fresh gas penetrating the flame sheet. A progress variable can be set with the help of any quantity like temperature, reactant mass fraction, provided it is bounded by a single value in the burned gas and another one in the fresh gas. The progress variable is usually named ''c'', in usual notations:<br />
:<math>c=\frac{T-T_f}{T_b-T_f}</math><br />
It is seen that ''c'' is a normalization of a scalar quantity. As mentioned above, the scalar transport equations are assumed linear such that the transport equation for ''c'' can be obtained directly. <br />
Actually, the transport equation for ''T'' ([[#Transport Equations|Sec. Transport Equations]]) is linear if constant heat capacity is further assumed (combustion of hydrocarbon in air implies a large excess of nitrogen whose heat capacity is only slightly varying) and the progress variable equation is directly <br />
obtained (here for a default of fuel - lean combustion):<br />
:<math>\frac{D\rho c}{Dt}=\nabla\cdot\rho D\vec\nabla c + \frac{\nu_F\bar M_F}{Y_{F,u}}\dot\omega \qquad \rho D = \frac{\lambda}{C_p} </math><br />
<br />
The fact that the default or excess of fuel has been discussed above leads to the introduction of another quantity: the ''equivalence ratio''. The equivalence ratio, usually noted <math>\Phi</math>, is the ratio of two ratios. The first one is the ratio of the mass of fuel with the mass of oxidizer in the mixture. The second one is the same ratio for a mixture at stoichiometry. Hence, when the equivalence ratio equals unity, the mixture is at stoichiometry. If it is greater than unity, the mixture is named ''rich'' as there is an excess of fuel. In contrast, when it is smaller than unity the mixture is named ''lean''. The equivalence ratio presented here for premixed flames has little connection with the equivalence ratio introduced earlier regarding the non-premixed regime. Basically, the equivalence ratio as defined for non-premixed flames gives the equivalence ratio of a premixed mixture with the same mass of fuel and oxidizer. Moreover, the equivalence ratio as defined for a premixed mixture can be obtained based on the mixture fraction (it is thus the local equivalence ratio at a point in the non-homogeneous mixture described by the mixture fraction). From the definitions given above:<br />
:<math> \Phi=\frac{Z}{1-Z}\frac{1-Z_s}{Z_s}</math><br />
<br />
==== Premixed Flame P&eacute;clet Number ====<br />
<br />
Earlier in this section, it has been said that a premixed flame posses its own dynamics, as a free propagating surface, and has thus characteristic quantities. For this reason, a P&eacute;clet number may be defined, based on these quantities. The P&eacute;clet number has the same structure as the Reynolds number but the dynamical viscosity is replaced by the ratio of the thermal conductivity and the heat capacity of the mixture. The thickness <math>\delta_L </math> of a premixed flame is essentially thermal. It means that it corresponds to the distance of the temperature rise between fresh and burned gases. This thickness is below the millimetre for conventional flames. The width of the reaction zone inside this flame is even smaller, by about one order of magnitude. This reaction zone is stuck to the hot side of the flame due to the high thermal dependency of the combustion reactions, as seen above. Hence, the flame region is essentially governed by a convection-diffusion process, the source term being negligible in most of it.<br />
<br />
It is convenient to write the progress variable transport equation in a steady-state framework. The quantities at flame temperature (<math>_f</math>) are used to non-dimensionalize the equation:<br />
:<math> \overbrace{(\rho S_L)}^{\dot M}\frac{\vec S_L}{S_L}\vec\nabla c = (\rho D)_f \nabla\cdot (\rho D)^* \vec\nabla c</math><br />
Note that the source term is neglected, consistently with what has been said above.<br />
This convection-diffusion equation makes appear a first approximation of a flame P&eacute;clet number:<br />
:<math> Pe_f = \frac{\dot M \delta_L}{(\rho D)_f} \approx 1 </math><br />
<br />
From the P&eacute;clet number, it is possible to obtain an expression for the flame velocity (remembering that <math> \delta_L/S_{L,f} \approx \tau_c</math>, vid. inf. [[#Three Turbulent-Flame Interaction Regimes| Sec. Three Turbulent-Flame Interaction Regimes]]):<br />
:<math> S_{L,f}^2\approx \frac{(\rho D)_f}{\rho_f \tau_c}</math><br />
For typical hydrocarbon flames, the speed is some tens of centimetres per second and the diffusivity is some<br />
<math> 10^{-5} </math> square metres per second. The chemical time in the reaction zone of about one tenth of a millisecond is recovered.<br />
<br />
==== Details of the Premixed Unstrained Planar Flame ====<br />
<br />
A plane combustion wave propagating in a homogeneous fresh mixture is the reference case to describe the premixed regime. At constant speed, it is convenient to see the flame at rest with a flowing upstream mixture. This is actually the way propagating flames are usually stabilized. In the frame of description, the <br />
physics is 1-D, steady with a uniform (the flame is said unstrained) mass flowing across the system. Two types of equations are thus sufficient to describe the problem, the temperature transport equation and the species transport equations, as in [[#Transport Equations|Sec. Transport Equations]]. The transport coefficients will be chosen as equal: <math> \rho D_i = \lambda / C_p </math> (unity Lewis numbers). Suppose the 1-D domain is described thanks to a conventional (''Ox'') axis with a flame propagating towards negative ''x'' (this is the conventional usage), the boundary conditions are:<br />
:* in the frozen mixture: <br />
:** <math>Y_i \rightarrow Y_{i,u} \qquad ; \qquad x\rightarrow-\infty </math><br />
:** <math>T \rightarrow T_u \qquad ; \qquad x\rightarrow-\infty </math><br />
:* in the burned gas region supposed at equilibrium:<br />
:** <math>Y_i \rightarrow Y_{i,b} \qquad ; \qquad x\rightarrow+\infty </math><br />
:** <math>T \rightarrow T_b \qquad ; \qquad x\rightarrow+\infty </math><br />
<br />
:<math>Y_{i,b}</math> and <math>T_b</math> are obtained from [[#Conservation Laws|Sec. Conservation Laws]].<br />
<br />
The quantities that have been mentioned just above (scalar and temperature profiles, mass flow rate through the system) are the solution to be sought.<br />
<br />
According to the discussions above, the temperature transport equation in its full normalized form may be written as (lean /stoichiometric case):<br />
:<math> \dot M \frac{\partial \theta}{\partial x} = \frac{\partial \ }{\partial x}\frac{\lambda}{Cp} \frac{\partial \theta}{\partial x} + \frac{\nu_F \bar M_F B}{Y_{F,s}} \prod_{i=O,F}(Y_{i,u}^*-\theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_i} \exp{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}} </math><br />
This equation is further simplified by the variable change <math>d\xi=\dot M/(\lambda/Cp)dx</math>:<br />
:<math>\frac{\partial \theta}{\partial \xi}=\frac{\partial^2 \theta}{\partial \xi^2} + \overbrace{\frac{\lambda/Cp}{\dot M^2}\frac{\nu_F \bar M_F B}{Y_{F,s}}}^{\Lambda} \prod_{i=O,F}(Y_{i,u}^*-\theta)^{n_i} \exp{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}}</math><br />
<br />
Although somewhat out of scope, the existence and unicity of the solution of this type of equation are usually demonstrated with the help of the Schauder Theorem and Maximum Principle. From the point of view of physicists and engineers, the solution that is found analytically is de facto considered as the unique solution of the equation.<br />
<br />
===== Scenarii of Combustion Process in the Phase Portrait =====<br />
<br />
In the frame moving with the flame, both phase variables are the reduced temperature and its gradient. To ease the reading with usual notations, it is written: <math> X_1 = \theta \quad ; \quad X_2 = \partial \theta / \partial \xi = \dot \theta </math>. The system arises:<br />
: <math> \left\{<br />
\begin{array}{lll}<br />
\dot X_1 & = & X_2 \\<br />
\dot X_2 & = & X_2 - \varpi(X_1) <br />
\end{array}<br />
\right.<br />
</math><br />
with <math> \varpi </math> being the full source term in the above equation.<br />
<br />
In the frame moving with the flame, two singular nodes are found in the frozen flow <math> (X_1,X_2) = (0,0)</math> and the equilibrium region <math> (\theta_b,0)</math>, i.e when <math> \varpi(X_1) </math> vanishes. <br />
<math> x_1,x_2 </math> are defined as small departures from the singular nodes such that the linearized system in their neighbourhood is:<br />
: <math> \left\{<br />
\begin{array}{lll}<br />
\dot x_1 & = & l_{1,1} x_1 + l_{1,2} x_2 \\<br />
\dot x_2 & = & l_{2,1} x_1 + l_{2,2} x_2 <br />
\end{array}<br />
\right.<br />
</math><br />
provided: <math> l_{1,1} = 0 \quad l_{1,2} = 1 \quad l_{2,1} = -\varpi'_{X_1 = 0,\theta_b} \quad l_{2,2} = 1 </math>.<br />
The characteristic polynom is, in usual notations: <math> s^2 - s + \varpi' </math> such that the eigenvalues are:<br />
:<math><br />
s^{\pm}=\frac{1\pm\sqrt{1-4\varpi'}}{2}<br />
</math><br />
A priori, those eigenvalues may be (i) real distinct, (ii) real identical, or (iii) conjugated complex. In the first case, the orbits in the phase diagram are organized, in the immediate neighbourhood of the singular node, with respect to the eigenvectors directions associated to the eigenvalues. The following task is to identify the nature of those eigenvalues and of the corresponding nodes. Because <math> 0 < X_1 < \theta_b </math> is bounded, complex eigenvalues are excluded as they would lead to a spiral node. This remark is important for the node on the cold side as it imposes a bound:<br />
:<math> \varpi'_{X_1=0} \le \frac{1}{4} </math><br />
As the mass flow rate through the flame is included into <math>\varpi</math>, it imposes a minimum value on the flame speed to tackle with the cold boundary difficulty (rise of the chemical rate in the frozen flow). In this condition, it <br />
is an unstable node (improper in case of equality).<br />
On the other hand, because <math> \varpi'|_{X_1=\theta_b} </math> is not positive, the node on the hot side is found as a saddle point. The overall scenario of combustion within the flame is thus an orbit leaving the cold node to join the hot node by branching on a trajectory compatible with the negative eigenvalue of the saddle.<br />
<br />
[[Image:sketchOrbit.jpg|thumb| Sketch of orbits for a combustion process across a premixed front. Dashed lines represent forbidden orbits from the physics. The red line describes the orbit expected in an idealized combustion process.]]<br />
It must be noted that the associated eigenvectors are of the form:<br />
:<math><br />
\left (<br />
\begin{array}{l}<br />
t \\<br />
t s^{\pm} <br />
\end{array}<br />
\right ); \quad t\in \Re^*<br />
</math><br />
that is, on the cold node, a positive departure on <math>X_1</math> following any of the two eigendirections, leads<br />
to a consistent creation of positive temperature gradient, while on the hot node, only the stable direction will allow a consistent creation of a positive temperature gradient for any departure of the temperature towards region where it is inferior to <math>\theta_b </math>. Another remark is the structure of the eigendirections. The leaving directions on the cold node have a slope larger than the one of <math> \varpi </math> while the stable direction of the hot node has a slope smaller than the one of <math> \varpi </math>. It means that there is some point where the orbit must cross the profile of the chemical term versus temperature. For that temperature, the gradient equals the reaction rate through construction of the phase space. When looking back to the equation of the premixed flame, it happens in a region of inflexion for the temperature (the second order derivative must vanish). Furthermore, at this intersection, the orbit is horizontal (if the frame of reference for <math> (X_1, X_2) </math> is Cartesian) due to the shape of the premixed flame equation above that can be recast into <math> X_{2,X_1}' = (X_2-\varpi)/X_2 </math>.<br />
Close to the cold node, the orbits have a shape of parabola whose axis is the direction with the largest eigenvalue magnitude.<br />
Close to the hot node, the orbits have an hyperboloid shape with asymptots as the eigendirections. Now the ingredients are here to draw a sketchy scenario of the combustion in a premixed flame. It will be superimposed on the reaction rate graph studied in [[#Fundamentals Aspects|Sec. Fundamentals]].<br />
Some typical orbits from the above analysis are drawn in the figure on the right. The basic geometrical arguments developed are reproduced. In particular, the dashed lines represent forbidden orbits by the physics (boundedness of <math>X_1</math>, irreversibility). Orbits must be travelled from left to right, corresponding to increasing free parameter <math> \xi </math>.<br />
It is of integral importance to remember that, in combustion in conventional conditions, the source term is highly non-linear. Therefore, it is localized in a very thin sheet and, upstream, <math> \varpi </math> and <math> \varpi' \rightarrow 0 </math>. <br />
Only the most unstable eigendirection of the cold node is compatible as an orbit. The other trajectories, being paraboloidal, are tangent to the other direction that is flat at the limit. It physically means an elevation of temperature in bulk, that is contradictory to what is expected from a highly non-linear combustion term. Hence, the additional orbit in red is the one expected in idealized combustion conventional conditions.<br />
<br />
[[Image:dnsOrbit.jpg|thumb|Sketch of orbits for a combustion process across a premixed front, DNS simulations for the usual combinations of <math>\beta</math> and <math> \alpha </math>. Note that the case with <math>\alpha = 0</math> does not mean an athermal reaction but the mathematical simplification of the denominator of the exponential argument of the source term.]]<br />
The following picture is the result of actual computations of the above 1-D flame equation (stoichiometry and <math> n_i = 1 </math>) with the help of a high-order (6) code. The already presented combinations of <math> \beta </math> and <br />
<math> \alpha </math> are used and the orbits are retrieved. For most of the cases, as predicted above, the system selects a solution leaving the cold node with the most unstable direction (identifiable with its slope close to unity for vanishing<br />
<math>\varpi'_{X_1=0}</math>). There is also an additional curve, for <math> \beta = 10 </math> with the denominator of the exponential argument suppressed. This curve is remarkable as the solution selected by the system leaves the cold node in a manner fully controlled by the <math> \varpi'_{X_1=0} </math>. This is a very singular solution, not expected in combustion in conventional conditions, as explained above. The purpose of this remark is to question the well-posedness of considering simplifying the exponential argument for <math>\beta</math> "sufficiently high", as it is usually proposed for this type of modelling. As observed, the dynamical system analysis demonstrates a switch in the nature of the solution selected.<br />
At the physical level, when the orbit follows the most unstable direction with a slope close to unity, it means that <math> X_2 </math> "follows" <math> X_1 </math>, which is a signature of a diffusion process. In other words, the preheating mechanism of a premixed flame propagation as proposed for more than one century is in work. On the other hand, when <br />
<math> X_2 </math> is dependent on the evolution of <math> \varpi'</math>, it shows that "cold" chemistry drives the solution in the frozen flow and not the acknowledged mechanism of deflagration.<br />
<br />
<br />
===== Flame Solution =====<br />
<br />
As already mentioned, the flame system may be split into three zones. Upstream, the conventional mechanism of deflagration<br />
is supported by diffusion of heat. Downstream, the mixture is at equilibrium after combustion. In between, there exists the reaction layer. For large <math>\beta</math> the reaction layer is very thin such that it can be seen as a discontinuity between the fresh and burned gases. This is this difference in scales that introduces the use of the asymptotic method to resolve some of the flame characteristics, such as speed, time, heat region thickness, or reaction zone thickness.<br />
<br />
The domain is partitioned, according to this zoning defined by the scales driving the physics with an outer domain, driven by large scales and an inner domain refining the description within the discontinuity. If the discontinuity (flame reaction zone) is at <math> \xi=0 </math>, everywhere but 0, the equation is simplified as:<br />
:<math><br />
\frac{\partial^2 \theta}{\partial \xi^2} = \frac{\partial\theta}{\partial \xi}<br />
</math><br />
*For <math> \xi>0 </math>, it is expected that the mixture has reached equilibrium chemistry, such that: <math> \forall \xi > 0, \quad \theta=\theta_b \quad ; \quad \partial \theta / \partial \xi =0 </math>.<br />
*For <math> \xi < 0 </math>, this is the preheat zone and the solution is <math> \theta = \theta_b e^{\xi} </math> with <math> \theta </math> reaching <math>\theta_b</math> at the disconstinuity and vanishing, together with its gradient, far upstream in the frozen mixture.<br />
<br />
The solution for the species and the value of <math> \theta_b </math> are obtained from [[#Conservation Equations|Conservation Equations]] above. The 'big picture' is thus an exponential variation in the thermal thickness matched with a plateau in the downstream region, the line of matching being the discontinuity (flame) that has no thickness at this scale of description.<br />
<br />
To refine the analysis in the discontinuity region, a magnifying factor <math> \varepsilon </math> is used to stretch the coordinates: <math> \xi = \varepsilon \Xi </math>. The inner solution is thus a slowly-varying function of <math> \Xi </math>. Hence, in this inner region, the equation for the premixed flame becomes:<br />
:<math><br />
\frac{1}{\varepsilon}\frac{\partial \theta}{\partial \Xi}=\frac{1}{\varepsilon^2}\frac{\partial^2 \theta}{\partial \Xi^2} +\varpi<br />
</math><br />
In order to stretch and 'look inside' a discontinuity, <math>\varepsilon</math> is very small. It yields two remarks:<br />
# convection is negligible compared to diffusion. The heat losses from the reaction zone are essentially diffusion driven.<br />
# The reaction zone is governed by a diffusion-reaction budget and the reaction term <math>\varpi</math> must be strong to balance the intense heat loss due to the sharp diffusion (the zone is very thin, hence the gradients are sharp).<br />
The mechanism is thus different from the outer region that was convection-diffusion driven.<br />
<br />
Each quantity is developed in a series of <math> \varepsilon </math>.<br />
At the leading order, for <math>\theta</math>, in the lean case, the conservation relations (Sec. [[#Conservation Laws|Conservation Laws]]) yield:<br />
:<math> \theta = 1 - \varepsilon \Gamma - (1 - Y^*_{F,u})</math><br />
where <math>\Gamma</math> is the first-order development of the departure of <math>\theta</math> from the maximum value due to the incomplete combustion, and <math>1-Y_{F,u}^*=1-\theta_b</math> is the reduction of temperature for non-stoichiometric cases. Injected into the above equation:<br />
:<math> \frac{1}{\varepsilon}\frac{\partial^2 \Gamma}{\partial \Xi^2} = \Lambda (\varepsilon \Gamma)^{n_F} (Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} +\varepsilon\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\beta\frac{1-Y_{F,u}^*+\varepsilon\Gamma}{1-\alpha(1-Y_{F,u}^*+\varepsilon\Gamma)}}</math><br />
Although the full develoment is not achieved, a number of scaling may be highlighted:<br />
# because the temperature cannot be much below unity, <math>Y_{F,u}^*</math> must be close to 1 <math> {\mathcal O}(\varepsilon)</math> . For clarity, it is not expanded in an <math>\varepsilon</math> series.<br />
# The denominator of the exponential argument simplifies to unity for small <math>\varepsilon</math>. <br />
# To get a finite rate in the reaction zone, <math>\varepsilon</math> scales with <math>\beta^{-1}</math>.<br />
<br />
The burning rate eigenvalue, <math>\Lambda</math>, is naturally expanded as: <math> \Lambda = \varepsilon^{-n_O-n_F-1}(\Lambda_0 + {\mathcal O}(\varepsilon)) </math>. The low-order equation to be solved is:<br />
:<math><br />
\frac{d^2 \Gamma}{d \Xi^2} = \frac{1}{2}\frac{d(\Gamma_{\Xi})^{'2}}{d\Gamma} = \Lambda_0\exp{-\beta (1-Y_{F,u}^*)} \Gamma^{n_F} (\frac{Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} }{\varepsilon}+\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\Gamma}</math><br />
:<math> \frac{d\Gamma}{d\Xi}(-\infty)=-\theta_b, \qquad \frac{d\Gamma}{d\Xi}(\infty)=0 </math><br />
The boundary conditions are obtained from the matching of the outer solutions on the right and left sides of the flame as written above (the outer solutions are reached at infinity for a very small magnifying factor <math>\varepsilon</math>).<br />
Once integrated with respect to those boundary conditions, the burning-rate eigenvalue (from which <math> \dot M </math> is extracted) is obtained as:<br />
:<math> \Lambda_0 = \Bigg( 2\int_0^{\infty}d\Gamma\; \Gamma^{n_F} (\beta (Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )+\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\beta (1-Y_{F,u}^*)}\exp{-\Gamma} \Bigg )^{-1}<br />
</math><br />
The RHS integral is not developed for clarity but presents no peculiar difficulties.<br />
<br />
The development has been carried out at the first order in <math> \varepsilon </math>. As soon as a second order development is attempted, some expressions are no more analytically tractable. On the other hand, a second order development allows introduce the temperature-dependent trends of some terms in <math> \Lambda </math>. Physical results are retrieved such as a slight decrease of the speed for a positive sensitivity of transport parameters to temperature around equilibrium conditions.<br />
<br />
[[Image:BellSpeed.jpg|thumb|Unstrained planar premixed flame speed with respect to fuel mass fraction (lean case) for a single global irreversible Arrh\'enius term. Symbols are obtained from a high-order DNS code. Continuous line is the theory exposed here. The usual combinations of <math>\beta</math> and <math> \alpha </math> are used. <br />
Very high values of <math> \beta </math> (and thus negligible effect of <math> \alpha </math> are also presented<br />
to show the problem of slow convergence for finite value of the Zeldovitch parameter. Not that <math>\alpha = 0</math> does not mean an athermal reaction but the mathematical simplification of the denominator of the exponential argument of the source term.]]<br />
The image beside illustrates the response of the premixed flame speed with respect to fuel concentration (chosen as the limiting component here; equivalent findings are obtained for default of oxidizer) for different values of the chemical parameters <math> \beta </math> and <math>\alpha</math> (as noted above, the pre-exponential constant <math> A</math> impacts only the overall magnitude). The theoretical expression (continuous line) is tested against a 1-D high accuracy<br />
code with the given chemistry implemented. It is seen that:<br />
* the Zeldovitch parameter drives the drop for non-stoichiomeric mixtures,<br />
* the drop is relatively well modelled by the theoretical expression, and<br />
* the absolute magnitude converges slowly towards the theoretical one when increasing <math>\beta</math> in the code (effect of the finiteness of <math> \beta </math> in the real case).<br />
<br />
[[Image:PremixedProfiles.jpg|thumb|Typical profiles in 1-D premixed flame at stoichiometry. Representative value of global chemistry parameters. Specie profiles are simply the complementary to the temperature profile for simple chemistry.]]<br />
The picture exhibiting profiles for different Zeldovitch and heat release parameters shows the factual impact: upstream, the exact exponential profile is recovered and corresponds to the pre-heating region (thermal thickness). In the reaction zone (just upstream of the extremum), the departure from the exact solution is due to the kinetic effect. This kinetic effect is more pronounced when <math> \beta </math> is lower because the lower the Zeldovitch parameter, the lower the temperature reaction zone can be without leading to extinction. The flame takes this opportunity to maximize its transfer in heat and reactant with the cold zone. This is the physical understanding of an inverse dependence of the maximum flame speed with <math> \beta </math>.<br />
<br />
<br />
==== Modification of the Flame Speed with Curvature ====<br />
<br />
[[Image:premixedCurve2d.jpg|thumb|Flame seen as an interface between fresh and burned gases. Its curved profile towards the burned side increases the transfers with the fresh side.]]<br />
When the thickness of the flame <math>(\lambda/Cp)_f/\dot M </math> is considered small compared to inhomogeneities existing in the flow, the flame can be reduced to an interface between fresh and burned gases. This interface may not be strictly plan in the general case. For instance, when the interface is curved towards the burned gas, it offers a larger opportunity for transfer of mass and heat with the fresh gas. As the ability of the chemistry to burn the coming matter is limited, the flame has thus to reduce its displacement speed. The figure beside provides a 2-D sketch of the situation.<br />
<br />
To mathematically give the expression showing that the curvature influences the flame speed (for small curvature), the non-dimensionalized temperature equation across a planar premixed flame above is slightly recast:<br />
:<math>||\dot M^c||||\vec\nabla_{\xi}\theta|| - \varpi= \nabla_{\xi}\cdot\vec\nabla_{\xi}\theta=-\nabla_{\xi}\cdot(||\vec\nabla_{\xi}\theta||\vec n) = - \vec\nabla_{\xi}||\vec\nabla_{\xi}\theta||\cdot\vec n - ||\vec\nabla_{\xi}\theta||\nabla_{\xi}\cdot\vec n</math><br />
<br />
In this expression, <math>\dot M^c</math> is the flame mass flow rate when perturbed by a curvature normalized by the reference one developed above. <math> \vec n</math> is the normal to the iso-temperature pointing towards the fresh gas, what is named the normal to the flame. From the expression developed above and for a slightly perturbed flame, the gradients, production and diffusion terms are very close to the unperturbed flame. Only the last term, the normal divergence, does not exist for a non-perturbed flame. Hence, the above equation may be simplified against the one for unperturbed flame presented earlier:<br />
:<math> \dot M^c = 1 - \nabla_{\xi}\cdot \vec n </math><br />
This is the divergence of the flame normal that contains the information upon geometrical perturbation (curvature) that impacts the speed.<br />
<br />
[[Image:premixedCurvature.jpg|thumb|Local geometrical approximation of a flame surface]]<br />
The key is thus to get an idea of the geometrical significance of the normal divergence. The normal divergence theorem says that this is the sum of the principal curvatures of the flame surface at the location considered. To give a good mental picture, the simplest configuration without loss of generality is to consider the flame surface approached by an osculatory revolution ellipsoid, as in the figure beside. The location of the approximation is the intersection of the Ox axis and the flame surface in red (where the ellipsoid is tangent to the flame). At this location, the normal divergence is the sum of the curvature of the basic ellipse (before its rotation around Oy) at its minor extremum, and the curvature of the circle corresponding to the rotation of this point around Oy when the 3-D shape is formed. Giving a lecture on conical coordinate systems is beyond the purpose but the interested reader may want to follow the corresponding steps to check this result: (i) write the divergence of a vector in the ellipsoid coordinate system, (ii) consider that the vector is the normal, i.e. it has only one constant component perpendicular to the local ellipsoidal iso-coordinate, to simplify the divergence expression, (iii) split the resulting terms into two parts by identifying the second derivative of the basic shape 2-D ellipse as one principal curvature, and the inverse of the radius of the circle corresponding to the rotation of the ellipse around Oy at the point where the flame surface is approached, in the figure this is simply the minor axis of the ellipse.<br />
<br />
The expression is readily written as:<br />
:<math> \dot M^c = 1 - (R_1^{-1}+R_2^{-1})</math><br />
where <math> R_1 </math> and <math> R_2</math> are the two radius of curvature (non-dimensionalized by the reference flame thickness) local to the surface.<br />
For instance, they are the small axis of the basic ellipse and its radius of curvature at its minor maximum when the <br />
surface is approached by the osculatory ellipsoid as above.<br />
<br />
<br />
===== Natural Instabilities of Premixed Flames =====<br />
<br />
It is not the goal of a document about combustion modelling to list all the cases where a premixed flame exhibits a particular geometry but there is a typical phenomenon that cannot not be easily dissociated from premixed combustion and<br />
impacts a lot the dynamics of the flame. These instabilities have been mentioned by Darrieus as early as 1938. The motivation of discussing about this is also related to the framework of description used that is widely employed for combustion models development. This framework is named the hydrodynamics limit, where the flame is isolated as <br />
a zero-thickness interface in the flow, and has been first introduced just above. In this framework, any diffusive and energetic aspects disappear and the set of equations is limited to two incompressible Euler systems. One system in the fresh gases (with a constant density of cold mixture). One system in the burned gases (with a constant density of mixture at equilibrium).<br />
<br />
To understand the basic properties of a premixed flame leading to the birth of instabilities, it is first important to realize that a premixed flame, in the hydrodynamic limit, behaves as a dioptre with a refractive index in the burned gases larger than in the fresh gases. For a flow with an angle of attack on the flame (i.e. a flame not strictly 1-D perpendicular to the flow), the tangential component of the flow speed relative to the flame surface is conserved while the normal component is accelerated by a factor corresponding to the ratio of the density in order to conserve mass flow across the interface. Hence, the streamlines are pushed towards the normal to the flame when crossing, that is similar to rays of light entering a refracting medium. <br />
<br />
If one considers a region of a premixed flame that bumps a little bit towards the fresh gas (the same approach is symmetrically true for the bump towards the burned gas), the local stream tube slightly opens on the bumpy interface before being refracted and coming back to its original section at constant mass flow rate. Hence, just in front of the bump, the gas velocity in this stream tube decreases and does not oppose the flame motion, allowing the bump to increase in magnitude. This is the fundamental mechanism of such instabilities.<br />
<br />
The equations sets are in usual notations:<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
U_x + V_y & = & 0 \\<br />
U_t + UU_x + VU_y & = & \frac{P_x}{\rho} \\<br />
V_t +UV_x+VV_y & = & \frac{P_y}{\rho} <br />
\end{array}<br />
\right.<br />
</math><br />
<math> x </math> is the coordinate used above along the flame path and <math>y</math> is in the tangential plan.<br />
<math>u </math> and <math>v</math> are the respective velocities.<br />
Those equations may be normalized by steady-state reference quantities such as flame speed, flame length, gauge pressure and density with respect to fresh gas properties and are written on both sides of the interface.<br />
<br />
These sets of equations must match at the interface (flame surface) through conservation of mass flow rate and momentum:<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
\rho_u (\vec U_u -\vec U_{f})\cdot \vec n & = & \rho_b (\vec U_b -\vec U_{f})\cdot \vec n\\<br />
(\rho_u \vec U_u (\vec U_u -\vec U_{f})+P_u)\cdot \vec n & = & (\rho_b \vec U_b (\vec U_b -\vec U_{f})+P_b)\cdot \vec n<br />
\end{array}<br />
\right.<br />
</math><br />
The subscripts <math> u </math> and <math> b </math> stand for unburned and burned sides, respectively. The subscript <math> f </math> points the interface. In these equations, the flame motion <math> U_f </math> is reintroduced because we want to track the instability movement over the mean position of the flame. This instability motion is described by the equation <math> x=F(y,t) </math> such that the motion of the interface normal to itself is obtained as:<br />
:<math> \vec U_f\cdot\vec n = - \frac{F_t}{\sqrt{1+F^2_y}} </math><br />
with a normal to the front:<br />
:<math> \vec n = \left (<br />
\begin{array}{l}<br />
-\frac{1}{\sqrt{1+F^2_y}} \\<br />
\frac{F_y}{\sqrt{1+F^2_y}} <br />
\end{array}<br />
\right )<br />
</math><br />
<br />
As the instability motion is considered at its birth, i.e. when it is still small, the quantities are linearized with <math> \varepsilon </math> as a small parameter:<br />
:<math> F=\varepsilon f</math><br />
:<math> \vec U = \vec {\mathcal U} + \varepsilon\vec u</math><br />
:<math> P = {\mathcal P} + \varepsilon p </math><br />
<br />
The Eulerian system is reduced to (at the first order in <math>\varepsilon</math>, the order zero is simplified for homogeneous flow):<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
u_x + v_y & = & 0 \\<br />
u_t + {\mathcal U}u_x & = & \frac{p_x}{\rho} \\<br />
v_t +{\mathcal U}v_x & = & \frac{p_y}{\rho} <br />
\end{array}<br />
\right.<br />
</math><br />
<br />
The one-sided equations for the jump conditions become (terms up to the first order in <math>\varepsilon</math> are kept):<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
\rho (\vec U-\vec U_f)\cdot\vec n \times \sqrt{1+F^2_y} & = & -\rho {\mathcal U}-\rho\varepsilon u+\rho\varepsilon f_t \\<br />
(\rho \vec U (\vec U -\vec U_{f})+P)\cdot \vec n \times \sqrt{1+F^2_y} & = &<br />
\left \{<br />
\begin{array}{l}<br />
-\rho {\mathcal U}^2 -2\rho{\mathcal U}\varepsilon u+\rho{\mathcal U}\varepsilon f_t - {\mathcal P} -\varepsilon p \\<br />
-\rho {\mathcal U}\varepsilon v + {\mathcal P}\varepsilon f_y<br />
\end{array}<br />
\right.<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
The following jump conditions are obtained (by recalling that unburned steady-state quantities are chosen as references):<br />
* From the first equation at the order 0, <math>\rho_b U_b = \rho_u U_u \Leftrightarrow U_b=\rho_u/\rho_b</math><br />
* From the first equation at the order 1, <math> \rho_u f_t - \rho_u u_u = \rho_b f_t -\rho_b u_b </math><br />
* From the second equation at the order 0, <math> -\rho_u {\mathcal U}_u^2 - {\mathcal P}_u = -\rho_b {\mathcal U}_b^2 - {\mathcal P}_b \Leftrightarrow {\mathcal P}_b = 1 - \rho_u/\rho_b </math><br />
* From the second equation at the order 1, <math> -2\rho_u{\mathcal U}_u u_u +\rho_u {\mathcal U}_u f_t -p_u = <br />
-2\rho_b{\mathcal U}_b u_b +\rho_b {\mathcal U}_b f_t -p_b \Leftrightarrow p_b-p_u = 2 u_u-2 u_b </math><br />
* From the third equation at order 1 (no order 0), <math> -\rho_u {\mathcal U}_u v_u +{\mathcal P}_u f_y = -\rho_b {\mathcal U}_b v_b +{\mathcal P}_b f_y \Leftrightarrow v_b-v_u = f_y(1-\rho_u/\rho_b) </math><br />
<br />
The solution for the linearized, autonomous Euler system is:<br />
:<math><br />
\left (<br />
\begin{array}{l}<br />
u \\ <br />
v \\ <br />
p<br />
\end{array}<br />
\right )<br />
= <br />
\left (<br />
\begin{array}{l}<br />
\bar u \\ <br />
\bar v \\ <br />
\bar p <br />
\end{array}<br />
\right )<br />
\exp{(\sigma x)}\exp{(\alpha t -iky)}<br />
</math><br />
where one recognizes an account for the perturbation of the field in the <math> x </math> direction (<math>\sigma</math>), the wave number of the instability following <math> y </math>, <math> k</math> and the <br />
growth rate with time <math> \alpha </math>.<br />
The eigenvalues of the system are used to determine the <math> x </math> dependence of the solution <math> \sigma = - \alpha/U,\; k,\; -k</math>, with the positive ones applying on the fresh side and the negative ones on the burned size to have a vanishing perturbation far from the flame.<br />
<br />
The eigenmodes give the flow pertubations on either side of the flame:<br />
:<math><br />
x<0:<br />
\left ( <br />
\begin{array}{l}<br />
u \\ v \\ p<br />
\end{array}<br />
\right )<br />
=<br />
a\left ( <br />
\begin{array}{l}<br />
1 \\ -i \\ -1-\frac{\alpha}{k}<br />
\end{array}<br />
\right )<br />
e^{kx+\alpha t - iky}<br />
</math><br />
<br />
:<math><br />
x>0:<br />
\left ( <br />
\begin{array}{l}<br />
u \\ v \\ p<br />
\end{array}<br />
\right )<br />
=<br />
b\left ( <br />
\begin{array}{l}<br />
1 \\ i \\ -1+\rho_b\frac{\alpha}{k}<br />
\end{array}<br />
\right )<br />
e^{-kx+\alpha t - iky}<br />
+<br />
c\left ( <br />
\begin{array}{l}<br />
1 \\ i\rho_b\frac{\alpha}{k} \\ 0<br />
\end{array}<br />
\right )<br />
e^{-\rho_b \alpha x+\alpha t - iky}<br />
</math><br />
<br />
The jump conditions above applied to this perturbation field yield the following system (<math> f </math> has also been put into the harmonic form <math> f = \bar f e^{\alpha t-iky} </math>):<br />
:<math><br />
\left \{ <br />
\begin{array}{lll}<br />
\alpha \bar f -a & = & \rho_b \alpha \bar f -\rho_b (b+c) \\<br />
a(1-\frac{\alpha}{k}) & = & b(1+\rho_b\frac{\alpha}{k}) +2c \\<br />
a + b+ c\frac{\rho_b \alpha}{k} & = & k\bar f (\frac{1}{\rho_b}-1)<br />
\end{array}<br />
\right . <br />
</math><br />
Additionaly, from the definition of the flame path <math> F </math>, we have the kinematic relation <br />
<math> u_u = a = \alpha \bar f </math>.<br />
<br />
The above set of equations forms a system of four unknowns <math> a,\; b,\; c,\; \alpha/k </math> <br />
for a given (but unknown) shape information on the flame bump <math> k\bar f</math>. <br />
Solving for the growth rate gives:<br />
:<math><br />
\left (\frac{\alpha}{k} - \frac{1}{\rho_b}\right )\left(\frac{\alpha}{k} + 2 + \rho_b\frac{\alpha}{k} -\left (\frac{1}{\rho_b}-1\right )\left (\frac{\alpha}{k}\right )^{-1} \right )=0<br />
</math><br />
This third-degree polynom has three solutions, namely the dispersion relation found by Darrieus, a stable mode, and the trivial solution (with no physical meaning), respectively:<br />
<math><br />
\frac{\alpha}{k} = \left ( \frac{1}{\rho_b + 1}\left (\sqrt{1+\frac{1-\rho_b^2}{\rho_b}}-1 \right ),\qquad \frac{1}{\rho_b + 1}\left (-\sqrt{1+\frac{1-\rho_b^2}{\rho_b}}-1 \right ),\qquad \frac{1}{\rho_b}\right )<br />
</math><br />
<br />
'''Remark 1''' By considering the shape of the eigenvectors giving the perturbation, one recovers that, upstream of the flame, the flow is potential (the rotational of the velocity vector is null and we have chosen a constant density flow, that is barotropic and divergence free), while downstream, additionally to another <br />
perturbation of the potential type, one finds a vorticity mode (the mode with the eigenvalue <math>-\rho_b\alpha </math>. The drift of vorticity at the crossing of the flame front is a known property of flames. <br />
<br />
'''Remark 2''' In more conventional literature, the strange trivial solution for the instability growth rate is swept under the rug. Given the pedagogical nature of this electronic documentation, we can dig a little bit as the appearance of this trivial, fool solution is a good example of some modelling issue. It is important to realize that a mathematical model and the physics it describes belong to different realities. Hence, the mathematical model will generate all the solutions that the mathematics can reach in its own space. Some of them are still connected to the physics. Some others, like this trivial solution, belongs only to the mathematical solution space, an indirect way of pointing out a model limitation. The model under scope here is the hydrodynamic limit. In this model, the domain is divided into two subdomains, one upstream of the flame interface, one downstream, that are put in relation with each other through a limited number of jump conditions. In reality, the physics connects these both domains much more tightly. The curious reader will have observed that the trivial solution makes the equation system for <math> a,\; b,\; c </math> (i.e. for a fixed growth rate) undetermined, and so are the jump conditions. By generating this trivial solution, the mathematics decouples both domains. The information coming from upstream to downstream (thanks to the shape of the linearized Euler system), a solution is found only for the upstream domain, the downstream being not solved and remains undetermined in lack of information. This solution cannot happen in reality because the fresh and burned gases in a real system are connected by many aspects, and not only by the jump conditions.<br />
<br />
== The Partially-Premixed Regime ==<br />
[[Image:ppf.jpg|thumb| Ideal sketch of a partially-premixed flame]]<br />
This regime is somewhat less academics and has been recognized two decades ago. It is acknowledged as a hybrid of the premixed and the non-premixed regimes but the degree of interaction of these two modes of combustion to accurately describe a partially-premixed flame is still not well understood. It can be simply pictured by a lifted diffusion flame. Let us consider fuel issuing from a nozzle into the air. If the exit velocity is large enough, for some fuels, the flame lifts off the rim of the nozzle. It means that below the flame base, fuel and oxidizer have room to premix. Hence, the flame base propagates into a premixed mixture. However, it cannot be reduced to a premixed flame (although it is often simplified as this): (i) the mixing is not perfect and the different parts of the flame front constituting the flame base burn in mixtures of different thermodynamical states. This provides those parts with different deflagration capabilities such that the flame base has a complex shape. Indeed, it is convex, naturally leaded by the part burning at stoichiometry, unless `exotic' feeding temperatures are used. (ii) Because the mixture is not homogeneous, transfer of species and temperature driven by diffusion occurs in a direction perpendicular to the propagation of the flame base. Because the flame front is not flat, those transfers act as a connection vehicle across the different parts of the leading front. (iii) The unburned left downstream by the sections of the leading front not burning at stoichiometry diffuse towards each other to form a diffusion flame as described above. The connection of the leading front with the trailing diffusion flame has been evidenced as complicated and the siege of transfers of species and temperature. These two last items are the state-of-the-art difficulties in understanding those flames and do not appear in the models although it has been demonstrated they have a major impact and are certainly a fundamental characteristic of partially-premixed flames.<br />
<br />
The partially-premixed flame is usually described using ''c'' and ''Z'' as introduced earlier. Because the framework is essentially non-premixed, the mixture fraction is primarily used to describe the flame. Regarding the head of the flame where partial-premixing has an impact, each part of the front is described with a local progress variable. The need of defining a local progress variable is that each section of the partially-premixed front has a different equivalence ratio leading to a different definition of ''c'':<br />
:<math> c=\frac{T-T_u}{T_b(Z)-T_u}</math><br />
<br />
= Three Turbulent-Flame Interaction Regimes =<br />
It may appear odd to try to describe here what is the reason of combustion modelling research: the interaction of the turbulence with chemistry. However, one of the first steps in building knowledge in turbulent combustion was the qualitative exploration of what might be the dynamics of a flame in a turbulent environment. This led to what is now known as ''combustion diagrams''. As explained above, the premixed regime lends itself the easiest to such an approach as it exhibits natural intrinsic quantities which are not as objectively identifiable in the other combustion regimes. Note that these quantities may depend<br />
on the geometry of the flame: for instance turbulence can bend a flame sheet, leading to a change in its <br />
dynamics compared to the flat flame propagating in a medium at rest. In this section, the turbulence-flame interaction modes will be described for a premixed flame. Only remarks will be added regarding the non-premixed and partially-premixed regimes.<br />
<br />
An integral quantity to assess the interaction between a premixed flame sheet and the turbulence<br />
is the Karlovitz number ''Ka''. It compares the characteristic time of flame displacement with the characteristic time of the smallest structures (that are also the fastest) of the turbulence.<br />
:<math> Ka= \frac{\tau_c}{\tau_k}</math><br />
<br />
<math>\tau_c</math> is the chemical time of the flame. To estimate it, it is necessary to come back to the above progress variable transport equation in a steady-state framework.<br />
:<math> \overbrace{(\rho S_L)}^{\dot M}\frac{\vec S_L}{S_L}\vec\nabla c = (\rho D)_f\nabla\cdot (\rho D)^* \vec\nabla c + \dot\omega_c</math><br />
The premixed wave propagates at a speed <math>S_L</math> because it is fed by reactants diffusing inside the combustion zone and which are preheated because temperature diffuses in the reverse direction. The speed at which the flame progresses is thus related to the rate of species diffusion into the reaction zone which are then consumed. As the premixed flame is a free propagating wave whose speed of propagation is only limited by the chemical strength, the characteristic chemical time is based only on the diffusion and the mass flow rate experienced by the flame:<br />
:<math> \tau_c = \frac{\rho (\rho D)_f}{\dot M^2}</math><br />
<br />
The smallest eddies are the ones being dissipated by the viscous forces. Their characteristic time is estimated thanks to a combination of the viscosity and the flux of turbulent energy to be dissipated (also called turbulent dissipation <math> \varepsilon=u'^3/l_t </math>):<br />
:<math>\tau_k = \sqrt{\frac{l_t\nu}{u'^3}}</math><br />
<br />
Thanks to those definitions of chemical and small structure times, it is possible to give another definition of the Karlovitz number:<br />
:<math> Ka=\left (\frac{\delta_L}{l_k} \right)^2</math> <br />
which is the square of the ratio between the premixed flame thickness and the small structure scale: ''Ka'' actually compares scales. To arrive to this latter result, the three following assumptions must be used: (i) the flame thickness is obtained thanks to the premixed flame P&eacute;clet number (''vid. sup.''); (ii) the turbulence small structure (''Kolmogorov eddies'') scale is given by: <math> l_k=(\nu^3/\varepsilon)^{1/4} </math> following the same dimensional argument as for the estimation of its time; and (iii) scalar diffusion scales with viscosity. <br />
<br />
==== Remark Regarding the Diffusion Flame ====<br />
From what has been presented above, a diffusion flame does not have characteristic scales. Setting a turbulence combustion regime classification for non-premixed flames has still not been answered by research. Some laws of behaviour will only be drawn around the scalar dissipation rate which is the parameter of integral importance for a diffusion flame.<br />
<br />
Indeed, the dynamics of a diffusion flame is determined by the strain rate imposed by the turbulence. As for the premixed flames, the shortest eddies (Kolmogorov) are the ones having the largest impact. The diffusive layer is thus given by the size of the Kolmogrov eddies: <math> l_d\approx l_k</math> and the typical diffusion time scale (feeding rate of the reaction zone) is given by the characteristic time of the Kolmogorov eddies: <math> \tau_k^{-1}\approx \chi_s</math> as the Reynolds number of the Kolmogorov structures is unity. Here, <math>\chi_s</math> is the sample-averaging of <math>\chi</math> based on (conditioned) stoichiometric conditions, where the flame is expected to be.<br />
<br />
== The Wrinkled Regime ==<br />
[[Image:wrinkled.jpg|thumb| Wrinkled flamelet regime]]<br />
This regime is also called the ''flamelet regime''. Basically, it assumes that the flame structure is not affected by turbulence. The flame sheet is convoluted and wrinkled by eddies but none of them is small enough to enter it.<br />
Locally magnifying, the laminar flame structure is maintained.<br />
<br />
This regime exists for a Karlovitz number below unity (vid. sup.), i.e. chemical time smaller than the small structure time or flame thickness smaller than small structure scale. Notwithstanding, the laminar flame dynamics can be disrupted for <math> u'>S_L</math>. In that case, although the flame structure is not altered by the small structures, it can be convected by large structures such that areas of different locations in the front interact. It shows that, even with a small Karlovitz number, the turbulence effect is not always weak.<br />
<br />
== The Corrugated Regime ==<br />
[[Image:Corrugated.jpg|thumb|Corrugated flamelet regime]]<br />
The formal definition of a flame is the region of temperature rise. However, the volume where the reaction takes place is about one order of magnitude smaller, embedded inside the temperature rise region and close to its high temperature end. Hence, there exist some levels of turbulence creating eddies able to enter the flame zone but still large enough to not affect the internal reaction sheet. In other words, the flame thermal region is thickened by turbulence but the reaction zone is still in the wrinkled regime.<br />
This situation is called the ''Corrugated Regime''.<br />
<br />
Due to the structure of the Karlovitz number, once written in terms of length scales (vid. sup.), this situation arises for an<br />
increase of the Karlovitz number by two orders of magnitude compared to the value for the wrinkled regime. Hence, in the range <br />
<math> 1 < Ka < 100 </math>, the laminar structure of the reaction zone is still preserved but not the one of the preheat zone.<br />
<br />
== The Thickened Regime ==<br />
[[Image:thickened.jpg|thumb|Thickened regime]]<br />
In this last case, turbulence is intense enough to generate eddies able to affect the structure of the reaction zone as well. In practice, it is expected that those eddies are in the tail of the energy spectrum such that their lifetime is very short. Their impact on the reaction zone is thus limited.<br />
<br />
Obviously, ''Ka > 100''. A topological description is of little relevance here and a ''well-stirred reactor model'' fits better.<br />
<br />
== Reaction mechanisms ==<br />
<br />
Combustion is mainly a chemical process. Although we can, to some extent, <br />
describe a flame without any chemistry information, modelling of the flame <br />
propagation requires the knowledge of speeds of reactions, product concentrations, <br />
temperature, and other parameters. <br />
Therefore fundamental information about reaction kinetics is <br />
essential for any combustion model. <br />
A fuel-oxidizer mixture will generally combust if the reaction is fast enough to <br />
prevail until all of the mixture is burned into products. If the reaction <br />
is too slow, the flame will extinguish. If too fast, explosion or even <br />
detonation will occur. The reaction rate of a typical combustion reaction <br />
is influenced mainly by the concentration of the reactants, temperature, and pressure. <br />
<br />
A stoichiometric equation for an arbitrary reaction can be written as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
\sum_{j=1}^{n}\nu' (M_j) = \sum_{j=1}^{n}\nu'' (M_j),<br />
</math></td></tr></table><br />
<br />
where <math> \nu </math> denotes the stoichiometric coefficient, and <math>M_j</math> stands for an arbitrary species. A one prime holds for the reactants while a double prime holds for the products of the reaction. <br />
<br />
The reaction rate, expressing the rate of disappearance of reactant <b>i</b>, is defined as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
RR_i = k \, \prod_{j=1}^{n}(M_j)^{\nu'},<br />
</math></td></tr></table><br />
<br />
in which <b>k</b> is the specific reaction rate constant. Arrhenius found that this constant is a function of temperature only and is defined as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
k= A T^{\beta} \, exp \left( \frac{-E}{RT}\right)<br />
</math></td></table><br />
<br />
where <b>A</b> is pre-exponential factor, <b>E</b> is the activation energy, and <math>\beta</math> is a temperature exponent. The constants vary from one reaction to another and can be found in the literature.<br />
<br />
Reaction mechanisms can be deduced from experiments (for every resolved reaction), they can also be constructed numerically by the '''automatic generation method''' (see [Griffiths (1994)] for a review on reaction mechanisms).<br />
For simple hydrocarbons, tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms, global reactions <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing equations for chemically reacting flows ==<br />
<br />
Together with the usual Navier-Stokes equations for compresible flow (See [[Governing equations]]), additional equations are needed in reacting flows.<br />
<br />
The transport equation for the mass fraction <math> Y_k </math> of <i>k-th</i> species is<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Y_k \right) +<br />
\frac{\partial}{\partial x_j} \left( \rho u_j Y_k\right) = <br />
\frac{\partial}{\partial x_j} \left( \rho D_k \frac{\partial Y_k}{\partial x_j}\right)+ \dot \omega_k<br />
</math><br />
<br />
<br />
where Ficks' law is assumed for scalar diffusion with <math> D_k </math> denoting the species difussion coefficient, and <math> \dot \omega_k </math> denoting the species reaction rate.<br />
<br />
A non-reactive (passive) scalar (like the mixture fraction <math> Z </math>) is goverened by the following transport equation<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Z \right) +<br />
\frac{\partial}{\partial x_j} \left( \rho u_j Z \right) = <br />
\frac{\partial}{\partial x_j} \left( \rho D \frac{\partial Z}{\partial x_j}\right)<br />
</math><br />
<br />
where <math> D </math> is the diffusion coefficient of the passive scalar.<br />
<br />
<br />
=== RANS equations ===<br />
<br />
In turbulent flows, [[Favre averaging]] is often used to reduce the scales (see [[Reynolds averaging]]) and the<br />
mass fraction transport equation is transformed to<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } -<br />
\overline{\rho} \widetilde{u''_i Y''_k } \right)<br />
+ \overline{\dot \omega_k}<br />
</math><br />
<br />
<br />
where the turbulent fluxes <math> \widetilde{u''_i Y''_k} </math> and reaction terms<br />
<math> \overline{\dot \omega_k} </math> need to be closed.<br />
<br />
The passive scalar turbulent transport equation is<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Z} }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z} }{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D \frac{\partial Z} {\partial x_j} } -<br />
\overline{\rho} \widetilde{u''_i Z'' } \right)<br />
</math><br />
<br />
where <math> \widetilde{u''_i Z''} </math> needs modelling. A common practice is to model the turbulent fluxes using the <br />
gradient diffusion hypothesis. For example, in the equation above the flux <math> \widetilde{u''_i Z''} </math> is modelled as<br />
<br />
:<math><br />
\widetilde{u''_i Z''} = -D_t \frac{\partial \tilde Z}{\partial x_i}<br />
</math> <br />
<br />
where <math> D_t </math> is the turbulent diffusivity. Since <math> D_t >> D </math>, the first term inside the parentheses on the right hand of the mixture fraction transport equation is often neglected (Peters (2000)). This assumption is also used below.<br />
<br />
In addition to the mean passive scalar equation,<br />
an equation for the Favre variance <math> \widetilde{Z''^2}</math> is often employed<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Z''^2} }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z''^2} }{\partial x_j}=<br />
\frac{\partial}{\partial x_j}<br />
\left( \overline{\rho} \widetilde{u''_i Z''^2} \right) -<br />
2 \overline{\rho} \widetilde{u''_i Z'' }<br />
- \overline{\rho} \widetilde{\chi}<br />
</math><br />
<br />
where <math> \widetilde{\chi} </math> is the mean [[Scalar dissipation]] rate<br />
defined as <math> \widetilde{\chi} =<br />
2 D \widetilde{\left| \frac{\partial Z''}{\partial x_j} \right|^2 } </math><br />
This term and the variance diffusion fluxes needs to be modelled.<br />
<br />
=== LES equations ===<br />
The [[Large eddy simulation (LES)]] approach for reactive flows introduces equations for the filtered species mass fractions within the compressible flow field.<br />
Similar to the [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]], the filtered mass fraction transport equation is<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } -<br />
J_j \right)<br />
+ \overline{\dot \omega_k}<br />
</math><br />
<br />
where <math> J_j </math> is the transport of subgrid fluctuations of mass fraction<br />
:<math><br />
J_j = \widetilde{u_jY_k} - \widetilde{u}_j \widetilde{Y}_k<br />
</math><br />
and has to be modelled.<br />
<br />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
are assumed to be much smaller than the apparent turbulent diffusion due to transport of subgrid fluctuations.<br />
The first term on the right hand side is then<br />
:<math><br />
\frac{\partial} {\partial x_j} <br />
\left(<br />
\overline{ \rho D_k \frac{\partial Y_k} {\partial x_j} }<br />
\right)<br />
\approx<br />
\frac{\partial} {\partial x_j} <br />
\left(<br />
\overline{\rho} D_k \frac{\partial \widetilde{Y}_k} {\partial x_j} <br />
\right)<br />
</math><br />
<br />
[[Category:Equations]]<br />
<br />
== Infinitely fast chemistry ==<br />
All combustion models can be divided into two main groups according to the <br />
assumptions on the reaction kinetics.<br />
We can either assume the reactions to be infinitely fast - compared to <br />
e.g. mixing of the species, or comparable to the time scale of the mixing <br />
process. The simple approach is to assume infinitely fast chemistry. Historically, mixing of the species is the older approach, and it is still in wide use today. It is therefore simpler to solve for [[#Finite rate chemistry]] models, at the overshoot of introducing errors to the solution.<br />
<br />
=== Premixed combustion ===<br />
Premixed flames occur when the fuel and oxidiser are homogeneously mixed prior to ignition. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical examples of premixed laminar flames is bunsen burner, where <br />
the air enters the fuel stream and the mixture burns in the wake of the <br />
riser tube walls forming nice stable flame. Another example of a premixed system is the solid rocket motor where oxidizer and fuel and properly mixed in a gum-like matrix that is uniformly distributed on the periphery of the chamber.<br />
Premixed flames have many advantages in terms of control of temperature and <br />
products and pollution concentration. However, they introduce some dangers like the autoignition (in the supply system).<br />
<br />
[[Image:Premixed.jpg]]<br />
==== Turbulent flame speed model ====<br />
<br />
==== Eddy Break-Up model ====<br />
<br />
The Eddy Break-Up model is the typical example of '''mixed-is-burnt''' combustion model. <br />
It is based on the work of Magnussen, Hjertager, and Spalding and can be found in all commercial CFD packages. <br />
The model assumes that the reactions are completed at the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. Combustion is then described by a single step global chemical reaction<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
F + \nu_s O \rightarrow (1+\nu_s) P<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
in which <b>F</b> stands for fuel, <b>O</b> for oxidiser, and <b>P</b> for products of the reaction. Alternativelly we can have a multistep scheme, where each reaction has its own mean reaction rate.<br />
The mean reaction rate is given by<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
\bar{\dot\omega}_F=A_{EB} \frac{\epsilon}{k} <br />
min\left[\bar{C}_F,\frac{\bar{C}_O}{\nu},<br />
B_{EB}\frac{\bar{C}_P}{(1+\nu)}\right]<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
<br />
where <math>\bar{C}</math> denotes the mean concentrations of fuel, oxidiser, and products <br />
respectively. <b>A</b> and <b>B</b> are model constants with typical values of 0.5 <br />
and 4.0 respectively. The values of these constants are fitted according <br />
to experimental results and they are suitable for most cases of general interest. It is important to note that these constants are only based on experimental fitting and they need not be suitable for <b>all</b> the situations. Care must be taken especially in highly strained regions, where the ratio of <math>k</math> to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In these, regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually have some remedies to overcome this problem.<br />
<br />
This model largely over-predicts temperatures and concentrations of species like <i>CO</i> and other species. However, the Eddy Break-Up model enjoys a popularity for its simplicity, steady convergence, and implementation.<br />
<br />
==== Bray-Moss-Libby model ====<br />
<br />
=== Non-premixed combustion ===<br />
<br />
Non premixed combustion is a special class of combustion where fuel and oxidizer<br />
enter separately into the combustion chamber. The diffusion and mixing of the two streams<br />
must bring the reactants together for the reaction to occur.<br />
Mixing becomes the key characteristic for diffusion flames.<br />
Diffusion burners are easier and safer to operate than premixed burners.<br />
However their efficiency is reduced compared to premixed burners.<br />
One of the major theoretical tools in non-premixed combustion<br />
is the passive scalar [[mixture fraction]] <math> Z </math> which is the<br />
backbone on most of the numerical methods in non-premixed combustion.<br />
<br />
==== Conserved scalar equilibrium models ====<br />
The reactive problem is split into two parts. First, the problem of <i> mixing </i>, which consists of the location of the flame surface which is a non-reactive problem concerning the propagation of a passive scalar; And second, the <i> flame structure </i> problem, which deals with the distribution of the reactive species inside the flamelet.<br />
<br />
To obtain the distribution inside the flame front we assume it is locally one-dimensional and<br />
depends only on time and the scalar coodinate.<br />
<br />
We first make use of the following chain rules<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}<br />
</math><br />
<br><br />
:<math><br />
\frac{\partial Y_k}{\partial x_j} = \frac{\partial Z}{\partial x_j}\frac{\partial Y_k}{\partial Z}<br />
</math> <br />
and transformation <br />
:<math><br />
\frac{\partial }{\partial t}= \frac{\partial }{\partial t} + \frac{\partial Z}{\partial t} \frac{\partial }{\partial Z}<br />
</math><br />
<br />
upon substitution into the species transport equation (see [[#Governing Equations for Reacting Flows]]), we obtain<br />
<br />
:<math><br />
\rho \frac{\partial Y_k}{\partial t} + Y_k \left[<br />
\frac{\partial \rho}{\partial t} + \frac{\partial \rho u_j}{\partial x_j}<br />
\right]<br />
+ \frac{\partial Y_k}{\partial Z} \left[<br />
\rho \frac{\partial Z}{\partial t} + \rho u_j \frac{\partial Z}{\partial x_j} -<br />
\frac{\partial}{\partial x_j}\left( \rho D \frac{\partial Z}{\partial x_j} \right)<br />
\right]<br />
=<br />
\rho D \left( \frac{\partial Z}{\partial x_j} \frac{\partial Z}{\partial x_j} \right)<br />
\frac{\partial^2 Y_k}{\partial Z^2} + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third terms in the LHS cancel out due to continuity and mixture fraction transport.<br />
At the outset, the equation boils down to<br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k<br />
</math><br />
<br />
where <math> \chi = 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2 </math> is called the <b>scalar dissipation</b><br />
which controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though the field <math> Z </math> still depends on it.<br />
<br />
:<math><br />
\dot \omega_k= -\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2}<br />
</math><br />
<br />
If the reaction is assumed to be infinetly fast, the resultant flame distribution is in equilibrium.<br />
and <math> \dot \omega_k= 0</math>. When the flame is in equilibrium, the flame configuration <math> Y_k(Z) </math> is independent of strain.<br />
<br />
===== Burke-Schumann flame structure =====<br />
The Burke-Schuman solution is valid for irreversible infinitely fast chemistry.<br />
With a reaction in the form of<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
F + \nu_s O \rightarrow (1+\nu_s) P<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
If the flame is in equilibrium and therefore the reaction term is 0.<br />
Two possible solution exists, one with pure mixing (no reaction)<br />
and a linear dependence of the species mass fraction with <math> Z </math>.<br />
Fuel mass fraction<br />
:<math><br />
Y_F=Y_F^0 Z<br />
</math><br />
Oxidizer mass fraction<br />
:<math><br />
Y_O=Y_O^0(1-Z)<br />
</math><br />
Where <math> Y_F^0 </math> and <math> Y_O^0 </math> are fuel and oxidizer mass fractions<br />
in the pure fuel and oxidizer streams respectively.<br />
<br />
<br />
The other solution is given by a discontinuous slope at stoichiometric mixture fraction<br />
<math> Z_{st}</math> and two linear profiles (in the rich and lean side) at either<br />
side of the stoichiometric mixture fraction.<br />
Both concentrations must be 0 at stoichiometric, the reactants become products infinitely fast.<br />
<br />
:<math><br />
Y_F=Y_F^0 \frac{Z-Z_{st}}{1-Z_{st}}<br />
</math><br />
and oxidizer mass fraction<br />
:<math><br />
Y_O=Y_O^0 \frac{Z-Z_{st}}{Z_{st}}<br />
</math><br />
<br />
== Finite rate chemistry ==<br />
<br />
=== Premixed combustion ===<br />
<br />
==== Coherent flame model ====<br />
<br />
==== Flamelets based on G-equation ====<br />
==== Flame surface density model ====<br />
<br />
=== Non-premixed combustion ===<br />
<br />
==== Flamelets based on conserved scalar ====<br />
<br />
Peters (2000) define Flamelets as "thin diffusion layers embedded in a turbulent non-reactive flow field".<br />
If the chemistry is fast enough, the chemistry is active within a thin region<br />
where the chemistry conditions are in (or close to) stoichiometric conditions, the "flame" surface.<br />
This thin region is assumed to be smaller than Kolmogorov length scale and therefore the<br />
region is locally laminar. The flame surface is defined as an iso-surface of a certain scalar <math> Z </math>,<br />
mixture fraction in [[#Non premixed combustion]].<br />
<br />
The same equation used in [[#Conserved scalar models]] for equilibrium chemistry<br />
is used here but with chemical source term different from 0<br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} =<br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k.<br />
</math><br />
<br />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that<br />
flamelet profiles <math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in a dtaset or file which is called a "flamelet library" with all the required complex chemistry.<br />
For the generation of such libraries ready to use software is avalable such as Softpredict's Combustion Simulation Laboratory COSILAB [http://www.softpredict.com] with its relevant solver RUN1DL, which can be used for a variety of relevant geometries; see various [http://www.softpredict.com/?page=989 publications that are available for download]. Other software tools are available such as CHEMKIN [http://www.reactiondesign.com] and<br />
CANTERA [http://www.cantera.org].<br />
<br />
===== Flamelets in turbulent combustion =====<br />
<br />
In turbulent flames the interest is <math> \widetilde{Y}_k </math>.<br />
In flamelets, the flame thickness is assumed to be much smaller than Kolmogorov scale<br />
and obviously is much smaller than the grid size.<br />
It is therefore needed a distribution of the passive scalar within the cell.<br />
<math> \widetilde{Y}_k </math> cannot be obtained directly from the flamelets library<br />
<math> \widetilde{Y}_k \neq Y_F(Z,\chi) </math>, where <math> Y_F(Z,\chi) </math> corresponds<br />
to the value obtained from the flamelets libraries.<br />
A generic solution can be expressed as<br />
:<math><br />
\widetilde{Y}_k= \int Y_F( \widetilde{Z},\widetilde{\chi}) P(Z,\chi) dZ d\chi<br />
</math><br />
where <math> P(Z,\chi) </math> is the joint [[Probability density function | Probability Density Function]] (PDF) of the mixture fraction<br />
and scalar dissipation which account for the scalar distribution inside the cell and "a priori"<br />
depends on time and space.<br />
<br />
The most simple assumption is to use a constant distribution of the scalar dissipation within the cell and<br />
the above equation reduces to<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) dZ<br />
</math><br />
<br />
<math> P(Z) </math> is the PDF of the mixture fraction scalar and simple models (such as Gaussian or a [[beta PDF]])<br />
can be build depending only on two moments of the scalar<br />
mean and variance,<math> \widetilde{Z},Z''</math>.<br />
<br />
If the mixture fraction and scalar dissipation are consider independent variables,<math> P(Z,\chi) </math><br />
can be written as <math> P(Z) P(\chi)</math>. The PDF of the scalar dissipation is assumed to be log-normal with<br />
variance unity.<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) P(\chi) dZ d\chi<br />
</math><br />
In [[Large eddy simulation (LES)]] context (see [[#LES equations]] for reacting flow),<br />
the [[probability density function]] is replaced by a [[subgrid PDF]] <math> \widetilde{P}</math>.<br />
The same equation hold by replacing averaged values with filtered values.<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) \widetilde{P}(Z) \widetilde{P}(\chi) dZ d\chi<br />
</math><br />
The assumptions made regarding the shapes of the PDFs are still justified. In LES combustion the<br />
[[subgrid variance]] is smaller than RANS counterpart (part of the large-scale fluctuations are solved)<br />
and therefore the modelled PDFs are thinner.<br />
<br />
====== Unsteady flamelets ======<br />
====Intrinsic Low Dimensional Manifolds (ILDM)====<br />
<br />
Detailed mechanisms describing ignition, flame propagation and pollutant formation typically involve several hundred species and elementary reactions, prohibiting their use in practical three-dimensional engine simulations. Conventionally reduced mechanisms often fail to predict minor radicals and pollutant precursors. The ILDM-method is an automatic reduction of a detailed mechanism, which assumes local equilibrium with respect to the fastest time scales identified by a local eigenvector analysis. In the reactive flow calculation, the species compositions are constrained to these manifolds. Conservation equations are solved for only a small number of reaction progress variables, thereby retaining the accuracy of detailed chemical mechanisms. This gives an effective way of coupling the details of complex chemistry with the time variations due to turbulence.<br />
<br />
The intrinsic low-dimensional manifold (ILDM) method {Maas:1992,Maas:1993} is a method for ''in-situ'' reduction of a detailed chemical mechanism based on a local time scale analysis. This method is based on the fact that different trajectories in state space start from a high-dimensional point and quickly relax to lower-dimensional manifolds due to the fast reactions. The movement along these lower-dimensional manifolds, however, is governed by the slow reactions. It exploits the variety of time scales to systematically reduce the detailed mechanism. For a detailed chemical mechanism with N species, N different time scales govern the process. An assumption that all the time scales are relaxed results in assuming complete equilibrium, where the only variables required to describe the system are the mixture fraction, the temperature and the pressure. This results in a zero-dimensional manifold. An assumption that all but the slowest 'n' time scales are relaxed results in a 'n' dimensional manifold, which requires the additional specification of 'n' parameters (called progress variables). In the ILDM method, the fast chemical reactions do not need to be identified a priori. An eigenvalue analysis of the detailed chemical mechanism is carried out which identifies the fast processes in dynamic equilibrium with the slow processes. The computation of ILDM points can be expensive, and hence an in-situ tabulation procedure is used, which enables the calculation of only those points that are needed during the CFD calculation.<br />
<br />
--[[User:Fredgauss|Fredgauss]] 07:37, 25 August 2006 (MDT)<br />
<br />
U. Maas, S.B. Pope. Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Comb. Flame 88,<br />
239, 1992.<br />
<br />
Ulrich Maas. Automatische Reduktion von Reaktionsmechanismen zur Simulation reaktiver Str¨omungen. Habilitationsschrift, Universit<br />
¨at Stuttgart, 1993.<br />
<br />
==== Conditional Moment Closure (CMC)====<br />
In Conditional Moment Closure (CMC) methods we assume that the species mass fractions are all correlated with<br />
the mixture fraction (in non premixed combustion).<br />
<br />
From [[Probability density function]] we have<br />
:<math><br />
\overline{Y_k}= \int <Y_k|\eta> P(\eta) d\eta<br />
</math><br />
where <math> \eta </math> is the sample space for <math> Z </math>.<br />
<br />
CMC consists of providing a set of transport equations for the conditional moments which define the<br />
flame structure.<br />
<br />
Experimentally, it has been observed that temperature and chemical radicals are<br />
strong non-linear functions of mixture fraction. For a given species mass fraction we can decomposed<br />
it into a mean and a fluctuation:<br />
:<math><br />
Y_k= \overline{Y_k} + Y'_k<br />
</math><br />
The fluctuations <math> Y_k' </math> are usually very strong in time and space which makes the closure<br />
of <math> \overline{\omega_k} </math> very difficult.<br />
However, the alternative decomposition<br />
:<math><br />
Y_k= <Y_k|\eta> + y'_k<br />
</math><br />
where <math> y'_k </math> is the fluctuation around the conditional mean or the "conditional fluctuation".<br />
Experimentally, it is observed that <math> y'_k<< Y'_k </math>, which forms the basic assumption of the CMC method.<br />
Closures. Due to this property better closure methods can be used reducing the non-linearity <br />
of the mass fraction equations.<br />
<br />
The [[Derivation of the CMC equations]] produces the following CMC transport equation<br />
where <math> Q \equiv <Y_k|\eta> </math> for simplicity.<br />
:<math><br />
\frac{ \partial Q}{\partial t} + <u_j|\eta> \frac{\partial Q}{\partial x_j} =<br />
\frac{<\chi|\eta> }{2} \frac{\partial ^2 Q}{\partial \eta^2} + <br />
\frac{ < \dot \omega_k|\eta> }{ <\rho| \eta >}<br />
</math><br />
<br />
In this equation, high order terms in Reynolds number have been neglected.<br />
(See [[Derivation of the CMC equations]] for the complete series of terms).<br />
<br />
It is well known that closure of the unconditional source term<br />
<math> \overline {\dot \omega_k} </math> as a function of the<br />
mean temperature and species (<math> \overline{Y}, \overline{T}</math>) will give rise to large errors.<br />
However, in CMC the conditional averaged mass fractions contain more information and fluctuations around the mean are much smaller.<br />
The first order closure<br />
<math><br />
< \dot \omega_k|\eta> \approx \dot \omega_k \left( Q, <T|\eta> \right)<br />
</math><br />
is a good approximation in zones which are not close to extinction.<br />
<br />
===== Second order closure =====<br />
<br />
A second order closure can be obtained if conditional fluctuations are taken into account.<br />
For a chemical source term in the form <math> \dot \omega_k = k Y_A Y_B </math> with the rate constant in Arrhenius form<br />
<math> k=A_0 T^\beta exp [-Ta/T] </math><br />
the second order closure is (Klimenko and Bilger 1999)<br />
:<math><br />
< \dot \omega_k|\eta> \approx < \dot \omega_k|\eta >^{FO}<br />
<br />
\left[1+ \frac{< Y''_A Y''_B |\eta>}{Q_A Q_B}+ \left( \beta + T_a/Q_T \right)<br />
\left(<br />
\frac{< Y''_A T'' |\eta>}{Q_AQ_T} + \frac{< Y''_B T'' |\eta>}{Q_BQ_T}<br />
\right) + ...<br />
<br />
\right]<br />
<br />
</math><br />
<br />
where <math> < \dot \omega_k|\eta >^{FO} </math> is the first order CMC closure and<br />
<math> Q_T \equiv <T|\eta> </math>.<br />
When the temperature exponent <math> \beta </math> or <math> T_a/Q_T </math><br />
are large the error of taking the first order approximation increases.<br />
Improvement of small pollutant predictions can be obtained using the above reaction <br />
rate for selected species like CO and NO.<br />
<br />
<br />
===== Double conditioning =====<br />
<br />
Close to extinction and reignition. The conditional fluctuations can be very large<br />
and the primary closure of CMC of "small" fluctuations is not longer valid.<br />
A second variable <math> h </math> can be chosen to define a double conditioned mass fraction<br />
:<math><br />
Q(x,t;\eta,\psi) \equiv <Y_i(x,t) |Z=\eta,h=\psi ><br />
</math><br />
Due to the strong dependence on chemical reactions to temperature, <math> h </math><br />
is advised to be a temperature related variable (Kronenburg 2004).<br />
Scalar dissipation is not a good choice, due to its log-normal behaviour<br />
(smaller scales give highest dissipation). A must better choice is the sensible enthalpy<br />
or a progress variable.<br />
Double conditional variables have much smaller conditional fluctuations and allow<br />
the existence of points with the same chemical composition which can be fully burning<br />
(high temperature) or just mixing (low temperature). <br />
The range of applicability is greatly increased and allows non-premixed and premixed problems<br />
to be treated without ad-hoc distinctions.<br />
The main problem is the closure of the new terms involving cross scalar transport.<br />
<br />
The double conditional CMC equation is obtained in a similar manner than the conventional<br />
CMC equations<br />
<br />
===== LES modelling =====<br />
In a LES context a [[conditional filtering]] operator can be defined<br />
and <math> Q </math> therefore represents a conditionally filtered reactive scalar.<br />
<br />
=== Linear Eddy Model ===<br />
The Linear Eddy Model (LEM) was first developed by Kerstein(1988).<br />
It is an one-dimensional model for representing the flame structure in turbulent flows.<br />
<br />
In every computational cell a molecular, diffusion and chemical model is defined as<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Y_k \right) =<br />
\frac{\partial}{\partial \eta} \left( \rho D_k \frac{\partial Y_k}{\partial \eta }\right)+ \dot \omega_k<br />
</math><br />
<br />
where <math> \eta </math>is a spatial coordinate. The scalar distribution obtained can be seen as a<br />
one-dimensional reference field between Kolmogorov scale and grid scales.<br />
<br />
In a second stage a series of re-arranging stochastic event take place.<br />
These events represent the effects<br />
of a certain turbulent structure of size <math> l </math>, smaller than the grid size at a location <math> \eta_0 </math><br />
within the one-dimensional domain. This vortex distort the <math> \eta </math> field obtain by the one-dimensional equation,<br />
creating new maxima and minima in the interval <math> (\eta_0, \eta + \eta_0) </math>.<br />
The vortex size <math> l </math> is chosen randomly based on the inertial scale range while<br />
<math> \eta_0 </math> is obtained from a uniform distribution in <math> \eta </math>.<br />
The number of events is chosen to match the turbulent diffusivity of the flow.<br />
<br />
=== PDF transport models ===<br />
<br />
[[Probability Density Function]] (PDF) methods are not exclusive to combustion, although they are particularly attractive to them. They provided more information than moment closures and they are used to compute inhomegenous turbulent flows, see reviews in Dopazo (1993) and Pope (1994).<br />
<br />
<br />
PDF methods are based on the transport equation of the joint-PDF of the scalars.<br />
Denoting <math> P \equiv P(\underline{\psi}; x, t) </math> where<br />
<math> \underline{\psi} = ( \psi_1,\psi_2 ... \psi_N) </math> is the phase space for the reactive scalars<br />
<math> \underline{Y} = ( Y_1,Y_2 ... Y_N) </math>.<br />
The transport equation of the joint PDF is:<br />
<br />
:<math><br />
\frac{\partial <\rho | \underline{Y}=\underline{\psi}> P }{\partial t} + \frac{ <br />
\partial <\rho u_j | \underline{Y}=\underline{\psi}> P }{\partial x_j} =<br />
\sum^N_\alpha \frac{\partial}{\partial \psi_\alpha}\left[ \rho \dot{\omega}_\alpha P \right]<br />
- \sum^N_\alpha \sum^N_\beta \frac{\partial^2}{\partial \psi_\alpha \psi_\beta}<br />
\left[ <D \frac{\partial Y_\alpha}{\partial x_i} \frac{\partial Y_\beta}{\partial x_i} | \underline{Y}=\underline{\psi}> \right] P<br />
</math><br />
<br />
where the chemical source term is closed. Another term appeared on the right hand side which accounts for the effects of<br />
the molecular mixing on the PDF, is the so called "micro-mixing " term.<br />
Equal diffusivities are used for simplicity <math> D_k = D </math><br />
<br />
A more general approach is the velocity-composition joint-PDF<br />
with <math> P \equiv P(\underline{V},\underline{\psi}; x, t) </math>, where <br />
<math> \underline{V} </math> is the sample space of the velocity field <br />
<math> u,v,w </math>. This approach has the advantage of avoiding gradient-diffusion<br />
modelling. A similar equation to the above is obtained combining the momentum <br />
and scalar transport equation.<br />
<br />
<br />
<br />
<br />
The PDF transport equation can be solved in two ways: through a Lagrangian approach<br />
using stochastic methods or in a Eulerian ways using stochastic fields.<br />
<br />
==== Lagrangian ====<br />
<br />
The main idea of Lagrangian methods is that the flow can be represented by an <br />
ensemble of fluid particles. Central to this approach is the stochastic differential equations and in particular the [[Langevin equation]].<br />
<br />
==== Eulerian ====<br />
<br />
Instead of stochastic particles, smooth stochastic fields can be used<br />
to represent the [[probability density function ]] (PDF) of a scalar (or joint PDF) involved in transport (convection), diffusion<br />
and chemical reaction (Valino 1998). A similar formulation was proposed by Sabelnikov and Soulard 2005, which removes<br />
part of the a-priori assumption of "smoothness" of the stochastic fields.<br />
This approach is purely Eulerian and offers implementations advantages compared to<br />
Lagrangian or semi-Eulerian methods. <br />
Transport equations for scalars are often easy to programme and normal<br />
CFD algorithms can be used (see [[Discretisation of convective term]]).<br />
Although discretization errors are introduced by solving transport equations, <br />
this is partially compesated by the error introduced in Lagrangian approaches due to the numerical evaluation of means.<br />
<br />
A new set of <math> N_s </math> scalar variables<br />
(the stochastic field <math> \xi </math>) is used to represent the<br />
PDF<br />
:<math><br />
P (\underline{\psi}; x,t) = \frac{1}{N} \sum^{N_s}_{j=1} \prod^{N}_{k=1} \delta \left[\psi_k -\xi_k^j(x,t) \right]<br />
</math><br />
<br />
===Other combustion models===<br />
<br />
<br />
==== MMC ====<br />
<br />
The Multiple Mapping Conditioning (MMC) (Klimenko and Pope 2003) is an<br />
extension of the [[#Conditional Moment Closure (CMC)]] approach combined with<br />
[[probability density function]] methods. MMC<br />
looks for the minimum set of variables that describes the particular turbulent combustion<br />
system.<br />
<br />
==== Fractals ====<br />
Derived from the [[#Eddy Dissipation Concept (EDC)]].<br />
<br />
== References ==<br />
<br />
*{{reference-paper|author=Dopazo, C.|year=1993|title=Recent development in PDF methods|rest=Turbulent Reacting Flows, ed. P. A. Libby and F. A. Williams }}<br />
*{{reference-book|author=Fox, R.O.|year=2003|title=Computational Models for Turbulent Reacting Flows|rest=ISBN 0-521-65049-6,Cambridge University Press}}<br />
*{{reference-paper|author=Kerstein, A. R.|year=1988|title=A linear eddy model of turbulent scalar transport and mixing|rest=Comb. Science and Technology, Vol. 60,pp. 391}}<br />
*{{reference-paper|author=Klimenko, A. Y., Bilger, R. W.|year=1999|title=Conditional moment closure for turbulent combustion|rest=Progress in Energy and Combustion Science, Vol. 25,pp. 595-687}}<br />
*{{reference-paper|author=Klimenko, A. Y., Pope, S. B.|year=2003|title=The modeling of turbulent reactive flows based on multiple mapping conditioning|rest=Physics of Fluids, Vol. 15, Num. 7, pp. 1907-1925}}<br />
*{{reference-paper|author=Kronenburg, A.,|year=2004|title=Double conditioning of reactive scalar transport equations in turbulent non-premixed flames|rest=Physics of Fluids, Vol. 16, Num. 7, pp. 2640-2648}}<br />
*{{reference-paper|author=Griffiths, J. F.|year=1994|title=Reduced Kinetic Models and Their Application to Practical Combustion Systems |rest=Prog. in Energy and Combustion Science,Vol. 21, pp. 25-107}}<br />
*{{reference-book|author=Peters, N.|year=2000|title=Turbulent Combustion|rest=ISBN 0-521-66082-3,Cambridge University Press}}<br />
*{{reference-book|author=Poinsot, T.,Veynante, D.|year=2001|title=Theoretical and Numerical Combustion|rest=ISBN 1-930217-05-6, R. T Edwards}}<br />
*{{reference-paper|author=Pope, S. B.|year=1994|title=Lagrangian PDF methods for turbulent flows|rest=Annu. Rev. Fluid Mech, Vol. 26, pp. 23-63}}<br />
*{{reference-paper|author=Sabel'nikov, V.,Soulard, O.|year=2005|title=Rapidly decorrelating velocity-field model as a tool for solving one-point Fokker-Planck equations for probability density functions of turbulent reactive scalars|rest=Physical Review E, Vol. 72,pp. 016301-1-22}}<br />
*{{reference-paper|author=Valino, L.,|year=1998|title=A field montecarlo formulation for calculating the probability density function of a single scalar in a turbulent flow|rest=Flow. Turb and Combustion, Vol. 60,pp. 157-172}}<br />
*{{reference-paper|author=Westbrook, Ch. K., Dryer,F. L.,|year=1984|title=Chemical Kinetic Modeling of Hydrocarbon Combustion|rest=Prog. in Energy and Combustion Science,Vol. 10, pp. 1-57}}<br />
<br />
== External links and sources ==</div>DavidFhttp://www.cfd-online.com/Wiki/Damkohler_numberDamkohler number2008-09-15T12:43:27Z<p>DavidF: Damkholer number moved to Damkohler number: Typo in title</p>
<hr />
<div>The Damkohler number is used in turbulent combustion<br />
and corresponds to the ratio of chemical time scale <math> \tau_c </math><br />
and turbulent time scale <math> \tau_t </math>.<br />
This turbulent scale is usually taken as the integral scale.<br />
<br />
:<math><br />
Da \equiv \frac{\tau_t}{\tau_c}<br />
</math><br />
<br />
Damkohler number measures<br />
how important is the interaction between chemistry and turbulence. Most combustion models<br />
are placed in the extremes of Damkohler.<br />
<br />
If <math> Da << 1 </math> the turbulence is much faster than the chemistry.<br />
This regime is the "well-stirred reactor", where products and reactants are rapidly mixed.<br />
The [[Karlovitz number]] is linked to the Damkholer number.<br />
<br />
<br />
The Damkohler number is also defined in a chemical non-equilibrium process at very high velocities of the fluid or vehicles (with reference to fluid). There are two ratios of Damkohler number that can be defined and are as follows:<br />
<br />
1. First Damkohler Number or DAM 1<br />
<br />
<math><br />
DAM 1 = \frac{t_{res}}{\tau}<br />
</math><br />
<br />
Where <math> t_{res} </math> is the residence time of the flow, where <math> \frac{L_{ref}}{U_{ref}} </math> where <math> L_{ref} </math> is the reference length of the vehicle and <math> U_{ref} </math> is the reference velocity of fluid or vehicle with respect to fluid<br />
<br />
<br />
2. Second Damkohler Number or DAM 2<br />
<br />
<math><br />
DAM 2 = \frac{q_{ne}}{h_o}<br />
</math><br />
<br />
Where <math> q_{ne} </math> is the energy involved in a non-equilibrium process and <math> h_o </math> is the total enthalpy of the flow<br />
<br />
<br />
[[Category:Dimensionless parameters]]</div>DavidFhttp://www.cfd-online.com/Wiki/Damkholer_numberDamkholer number2008-09-15T12:43:27Z<p>DavidF: Damkholer number moved to Damkohler number: Typo in title</p>
<hr />
<div>#REDIRECT [[Damkohler number]]</div>DavidFhttp://www.cfd-online.com/Wiki/CombustionCombustion2008-09-15T12:42:35Z<p>DavidF: </p>
<hr />
<div>''The power of Fire, or Flame, for instance, which we designate by some trivial chemical name, thereby hiding from ourselves the essential character of wonder that dwells in it as in all things, is with these old Northmen, Loke, a most swift subtle Demon of the brood of the J\"otuns... From us too no Chemistry, if it had not Stupidity to help it, would hide that Flame is a wonder. What is Flame?''<br />
<br />
'''''Carlyle on''''' Heroes '''''Odin and Scandinavian Mythology.'''''<br />
<br />
<br />
== What is combustion -- physics versus modelling ==<br />
<br />
Combustion phenomena consist of many physical and chemical processes which exhibit a <br />
broad range of time and length scales. A mathematical description of combustion is not <br />
always trivial, although some analytical solutions exist for simple situations of <br />
laminar flame. Such analytical models are usually restricted to problems in zero or one-dimensional space.<br />
<br />
= Fundamental Aspects =<br />
<br />
== Main Specificities of Combustion Chemistry ==<br />
<br />
Combustion can be split into two processes interacting with each other: thermal, and chemical. <br />
<br />
The chemistry is highly exothermal (this is the reason of its use) but also highly temperature dependent, thus highly self-accelerating. In a simplified form, combustion can be represented by a single irreversible reaction involving 'a' fuel and 'an' oxidizer:<br />
:<math> \frac{\nu_F}{\bar M_F}Y_F + \frac{\nu_O}{\bar M_O} Y_O \rightarrow Product + Heat </math><br />
Althgough very simplified compared to real chemistry involving hundreds of species (and their individual transport properties) and elemental reactions, this rudimentary chemistry has been the cornerstone of combustion analysis and modelling.<br />
<br />
The most widely used form for the rate of the above reaction is the Arrh&eacute;nius law:<br />
:<math> \dot\omega = \rho A \left ( \frac{Y_F}{\bar M_F} \right )^{n_F} \left (\frac{Y_O}{\bar M_O}\right )^{n_O} \exp^{-T_a/T} </math><br />
<math> T_a </math> is the activation temperature, high in combustion, consistently with the temperature dependence.<br />
This is where the high non-linearity in temperature is modelled. ''A'' is the pre-exponential constant. One of the interpretation of the Arrh&eacute;nius law comes from gas kinetic theory: the number of molecules whose kinetic energy is larger than the minimum value allowing a collision to be energetic enough to trig a reaction is proportional to the exponential term introduced above divided by the square root of the temperature. This interpretation allows one to think that the temperature dependence of ''A'' is very weak compared to the exponential term. ''A'' is eventually considered as constant.<br />
The reaction rate is also naturally proportional to the molecular density of each of the reactant. Nonetheless, the orders of reaction <math> n_i</math> are different from the stoichiometric coefficients as the single-step reaction is global, not governed by collision for it represents hundreds of elementary reactions.<br />
If one goes into the details, combustion chemistry is based on chain reactions, decomposed into three main steps: (i) generation (where radicals are created from the fresh mixture), (ii) branching (where products and new radicals appear from interaction of radicals with reactants), and (iii) termination (where radicals collide and turn into products). The branching step tends to accelerate the production of active radicals (autocatalytic). The impact is nevertheless small compared to the high non-linearity in temperature. This explains why single-step chemistry has been sufficient for most of the combustion modelling work up to now.<br />
<br />
The fact that a flame is a very thin reaction zone separating, and making the transition between, a frozen mixture and an equilibrium is explained by the high temperature <br />
dependence of the reaction term, modelled by a large activation temperature, and a large heat release (the ratio of the burned and fresh gas temperatures is about 7 for typical hydrocarbon flames) leading to a sharp self-acceleration in a very narrow area. To evaluate the order of magnitude of the quantities, the terms in the exponential argument are normalized:<br />
:<math> \beta=\alpha\frac{T_a}{T_s} \qquad \alpha=\frac{T_s-T_f}{T_s} </math><br />
<math>\beta</math> is named the Zeldovitch number and <math>\alpha</math> the heat release factor. <br />
Here, <math> T_s</math> has been used instead of <math> T_b</math>, the conventional notation for burned gas temperature (at final equilibrium). <math> T_s</math> is actually <math> T_b</math> <br />
for a mixture at stoichiometry and when the flame is adiabatic, i.e. this is the reference highest temperature that can be<br />
obtained in the system. That said, typical value for <math>\beta</math> and <math>\alpha</math> are 10 and 0.9, giving <br />
a good taste of the level of non-linearity of the combustion process with respect to temperature. <br />
Actually, the reaction rate is rewritten as:<br />
:<math> \dot\omega = \rho A \left ( \frac{Y_F}{\bar M_F} \right )^{n_F} \left (\frac{Y_O}{\bar M_O}\right )^{n_O} <br />
\exp^{-\frac{\beta}{\alpha}}\exp^{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}} </math><br />
where the non-dimensionalized temperature is:<br />
:<math>\theta=\frac{T-T_f}{T_s-T_f}</math><br />
The non-linearity of the reaction rate is seen from the exponential term:<br />
:* <math> {\mathcal O}(\exp^{-\beta}) </math> for <math>\theta</math> far from unity (in the fresh gas)<br />
:* <math> {\mathcal O}(1) </math> for <math>\theta</math> close to unity (in the reaction zone close to the burned gas whose temperature must be close to the adiabatic one <math> T_s </math>), more exactly <math> 1-\theta \sim {\mathcal O}(\beta^{-1})</math><br />
<br />
[[Image:NonLinearite.jpg|thumb|Temperature Non-Linearity of the Source Term: the Temperature-Dependent Factor of the Reaction Term for Some Values of the Zeldovtich and Heat Release Parameters]]Note that for an infinitely high activation energy, the reaction rate is piloted by a <math>\delta(\theta)</math> function. The figure, beside, illustrates how common values of <math>\beta</math> around 10 tend to make the reaction rate singular around <math>\theta</math> of unity. Two set of values are presented: <math><br />
\beta = 10</math> and <math>\beta = 8</math>. The first magnitude is the representative value while the second one is a smoother one usually used to ease numerical simulations. In the same way, two values for the heat release <math>\alpha</math> 0.9 and 0.75 are explored. The heat release is seen to have a minor impact on the temperature non-linearity.<br />
<br />
== Transport Equations ==<br />
<br />
Additionally to the Navier-Stokes equations, at least with variable density, the transport equations for a reacting flow are the energy and species transport equations. In usual notations, the specie ''i'' transport equation is written as:<br />
:<math>\frac{D \rho Y_i}{D t} = \nabla\cdot \rho D_i\vec\nabla Y_i - \nu_i\bar M_i\dot\omega</math><br />
and the temperature transport equation:<br />
:<math>\frac{D\rho C_p T}{Dt} = \nabla\cdot \lambda\vec\nabla T + Q\nu_F \bar M_F \dot\omega </math><br />
The diffusion is modelled thanks to Fick's law that is a (usually good) approximation to the rigorous diffusion velocity calculation. Regarding the temperature transport equation, it is derived from the energy transport equation under the assumption of a low-Mach number flow (compressibility and viscous heating neglected). The low-Mach number approximation is suitable for the deflagration regime (as it will be demonstrated [[#Premixed|below]]), which is the main focus of combustion modelling. Hence, the transport equation for temperature, as a simplified version of the energy transport equation, is usually retained for the study of combustion and its modelling.<br />
<br />
=== Low-Mach Number Equations ===<br />
In compressible flows, when the motion of the fluid is not negligible compared to the speed of sound (which is the speed at which the molecules can reorganize themselves), the heap of molecules results in a local increase of pressure and temperature moving as an acoustic wave. It means that, in such a system, a proper reference velocity is the speed of sound and a proper pressure reference is the kinetic pressure. A contrario, in low-Mach number flows, the reference speed is the natural representative speed of the flow and the reference pressure is the thermodynamic pressure. Hence, the set of reference quantities to characterize a low-Mach number flow is given in the table below:<br />
<br />
Density <math>\rho_o</math> A reference density (upstream, average, etc.)<br />
<br />
Velocity <math>U_o</math> A reference velocity (inlet average, etc.)<br />
<br />
Temperature <math>T_o</math> A reference temperature (upstream, average, etc.)<br />
<br />
Pressure (static) <math>P_o=\rho_o \bar r T_o</math> From Boyle-Mariotte<br />
<br />
Length <math>L_o</math> A reference length (representative of the domain)<br />
<br />
Time <math>L_o/U_o</math><br />
<br />
Energy <math>C_p T_o</math> Internal energy at constant reference pressure <br />
<br />
The equations for fluid mechanics properly adimensionalized can be written:<br />
<br />
Mass conservation:<br />
:<math> \frac{D\rho}{Dt} =0 </math><br />
<br />
Momentum:<br />
:<math>\frac{D\rho\vec U}{Dt}=-\frac{1}{\gamma M}\vec\nabla P+\nabla\cdot\frac{1}{Re}\bar\bar\Sigma</math><br />
<br />
Total energy:<br />
:<math>\frac{D\rho e_T}{Dt}-\frac{1}{RePr}\nabla\cdot\lambda\vec\nabla T=-\frac{\gamma-1}{\gamma}\nabla\cdot P\vec U<br />
+\frac{1}{Re}M^2(\gamma-1)\nabla\cdot \vec U\bar\bar\Sigma + \frac{\rho}{C_pT_o(U_o/L_o)}Q\nu_F \bar M_F\dot\omega</math><br />
<br />
Specie:<br />
:<math>\frac{D\rho Y}{Dt}-\frac{1}{ScRe}\nabla\cdot\rho D\vec\nabla Y=-\frac{\rho}{U_o/L_o}\nu\bar M\dot\omega</math><br />
<br />
State law:<br />
:<math> P=\rho T</math><br />
<br />
The low-Mach number equations are obtained considering that <math> M^2 </math> is small. 0.1 is usually taken as the limit, which recovers the value of a Mach number of 0.3 to characterize the incompressible regime.<br />
<br />
Considering the energy equation, in addition to the terms with <math> M^2 </math> in factor in the equation, the total energy reduces to internal energy as: <math> e_T = T/\gamma+M^2(\gamma-1)\vec U^2/2 </math>. Moreover, the work of pressure is considered as negligible because the gradient of pressure is negligible (low-Mach number approximation is indeed also named ''isobaric'' approximation) and the flow is assumed close to a divergence-free state.<br />
For the same reason, volumic energy and enthalpy variations are assumed equal as they only differ through the addition of pressure. Hence, redimensionalized, the low-Mach number energy equation leads to the temperature equation as used in combustion analysis:<br />
:<math>\frac{D\rho C_p T}{Dt} = \nabla\cdot \lambda\vec\nabla T + Q\nu_F \bar M_F \dot\omega </math><br />
<br />
______________________________<br />
<br />
'''Note:''' The species and temperature equations are not closed as the fields of velocity and density also need to be computed. Through intense heat release in a very small area (the jump in temperature in typical hydrocarbon flames is about seven and so is the drop in density in this isobaric process, and the thickness of a flame is of the order of the millimetre), combustion influences the flow field. Nevertheless, the vast majority of combustion modelling has been developed based on the species and temperature equations, assuming simple flow fields.<br />
<br />
=== The Damk&ouml;hler Number ===<br />
<br />
A flame is a reaction zone. From this simple point of view, two aspects have to be considered: (i) the rate at which it is fed by reactants, let call <math> \tau_d </math> the characteristic time, and<br />
the strength of the chemistry to consume them, let call the characteristic chemical time <math> \tau_c </math>. In combustion, the Damk&ouml;hler number, ''Da'', compares these both time scales and, for that <br />
reason, it is one of the most integral non-dimensional groups:<br />
:<math>Da=\frac{\tau_d}{\tau_c}</math>.<br />
If ''Da'' is large, it means that the chemistry has always the time to fully consume the fresh mixture and turn it into equilibrium. Real flames are usually close to this state. The characteristic reaction time, <math> (Ae^{-T_a/T_s})^{-1} </math>, is <br />
estimated of the order of the tenth of a ms. When ''Da'' is low, the fresh mixture cannot be converted by a too weak chemistry. The flow remains frozen. This situation happens with ignition or misfire, for instance. <br />
<br />
The picture of a deflagration lends itself to a description based on the Damk&ouml;hler number. A reacting wave progresses towards the fresh mixture through preheating of the upstream closest layer. The elevation of the temperature strengthens the chemistry and reduces its characteristic time such that the mixture changes from a low-''Da'' region (far upstream, frozen) to a high-''Da'' region in the flame (intense reaction to equilibrium).<br />
<br />
== Conservation Laws ==<br />
<br />
The processus of combustion transforms the chemical enthalpy into sensible enthalpy (i.e. rise the temperature of the gases thanks to the heat released). Simple relations can be drawn between species and temperature by studying the source terms appearing in the above equations:<br />
:<math> \frac{Y_F}{\nu_F\bar M_F} - \frac{Y_O}{\nu_O\bar M_O} = \frac{Y_{F,u}}{\nu_F\bar M_F} - \frac{Y_{O,u}}{\nu_O\bar M_O} </math><br />
:<math> Y_F + \frac{Cp T}{Q} = Y_{F,u} + \frac{CpT_u}{Q} </math><br />
Hence <math> T_b = T_u + \frac{Q Y_{F,u}}{Cp} </math>, <math> Y_{O,b} = Y_{O,u} - \frac{\nu_O \bar M_O}{\nu_F \bar M_F} Y_{F,u} = Y_{O,u} -sY_{F,u} </math> and <math> Y_{F,b} = 0 </math>. Here, the example has been taken for a lean case.<br />
<br />
As mentioned in [[#Main Specificities of Combustion Chemistry|Sec. Main Specificities]], the stoichiometric state is used to non-dimensionalize the conservation equations:<br />
:<math> Y_i^* = Y_{i,u}^* - \theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} \qquad ; \qquad Y_i^*=Y_i/Y_{i,s}</math>.<br />
<br />
A comprehensive form of the reaction rate can be reconstituted to understand the difficulty of numerically resolving the reaction zone:<br />
:<math> \dot\omega = B \prod_{i=O,F}(Y_{i,u}^*-\theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_i} \exp{\left ( -\beta\frac{1-\theta}{1-\alpha(1-\theta)}\right)} </math><br />
where <math> B </math> stands for all the constant terms present in this reaction rate, plus density.<br />
<br />
[[Image:NonLineariteii.jpg|thumb|Source Term versus Temperature]]<br />
For the stoichiometric case and a global order of two, the reaction rate is graphed versus the reduced temperature for different values of the heat release and Zeldovitch parameter. A high value of <math>\beta</math> makes the reaction rate very sharp, versus temperature. It means that reaction is significant beyond a temperature level (sometimes called ignition temperature) that is close to one (the exponential term above is non-negligible for <math>1-\theta \sim \beta^{-1}</math>). The heat release has qualitatively the same impact but not so strong. Transposed to the case of a flame sheet, it effectively shows that the reaction exists only in a fraction of the thermal thickness of the flame (the region close to the flame that the latter preheats, hence, where the reduced temperature rises from 0 to 1 here) where the temperature deviates few from the maximal one (density can be assumed as constant and equal to its burned-gas value). Numerically capturing such a sharp reaction zone can be costly and the lower values of <br />
<math>\beta</math> and <math>\alpha</math> as presented here are usually preferred whenever possible.<br />
<br />
<br />
Most problems in combustion involve turbulent flows, gas and liquid <br />
fuels, and pollution transport issues (products of combustion as well as for example noise <br />
pollution). These problems require not only extensive experimental <br />
work, but also numerical modelling. All combustion models must be validated <br />
against the experiments as each one has its own drawbacks and limits. In this article, we will address the modeling fundamentals only.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new dimensionless parameters are introduced, the most important of which are the [[Karlovitz number]] and the [[Damkholer number]] which represent ratios of chemical and flow time scales, and <br />
the [[Lewis number]] which compares the diffusion speeds of species.<br />
The combustion models are often classified on their capability to deal with the different combustion regimes. <br><br />
[[Image:SUMMARY_OF_COMBUSTION_MODELS.jpg]]<br />
<br />
= Three Combustion Regimes =<br />
<br />
Depending on how fuel and oxidizer are brought into contact in the combustion system, different combustion modes or regimes are identified. Traditionally, two regimes have been recognized: the ''premixed'' regime and the ''non-premixed'' regime. Over the last two decades, a third regime, sometime considered as a hybrid of the two former ones to a certain extend, has risen. It has been named ''partially-premixed'' regime.<br />
<br />
== The Non-Premixed Regime ==<br />
[[Image:DiffusionFlame.jpg|thumb|Sketch of a diffusion flame]]<br />
This regime is certainly the easiest to understand. Everybody has already seen a lighter, candle or gas-powered stove. Basically, the fuel issues from a nozzle or a simple duct into the atmosphere. The combustion reaction is the oxidization of the fuel. Because fuel and oxidizer are in contact only in a limited region but are separated elsewhere (especially in the feeding system) this configuration is the safest. The non-premixed flame has some other advantages. By controlling the flows of both reactants, it is (theoretically) possible to locate the stoichiometric interface, and thus, the location of the flame sheet. Moreover, the strength of the flame can also be controlled through the same process. Depending on the width of the transition region from the oxidizer to the fuel side, the species (fuel and oxidizer) feed the flame at different rates.<br />
This is because the diffusion of the species is directly dependent on the unbalance (gradient) of their distribution. A sharp transition from fuel to oxidizer creates intense diffusion of those species towards the flame, increasing its burning rate. This burning rate control through the diffusion process is certainly one of the reasons of the alternate name of such a flame and combustion mode: ''diffusion'' flame and ''diffusion'' regime.<br />
<br />
Because a diffusion flame is fully determined by the inter-penetration of the fuel and oxidizer streams, it has been convenient to introduce a tracer of the state of the mixture. This is the role of the ''mixture fraction'', usually called ''Z'' or ''f''. Z is usually taken as unity in the fuel stream and is null in the oxidizer stream. It varies linearly between this two bounds such that at any point of a frozen flow the fuel mass fraction is given by <math> Y_F=ZY_{F,o} </math> and the oxidizer mass fraction by <math> Y_O = (1-Z)Y_{O,o} </math>. <math> Y_{F,o} </math> and <math> Y_{O,o} </math> are the fuel and oxidizer mass fractions in the fuel and oxidizer streams, respectively.<br />
The mixture fraction posses a transport equation that is expected to not have any source term as a tracer of a mixture must be a conserved scalar. First, the fuel and oxidizer mass fraction transport equations are written in usual notations:<br />
:<math>\frac{D \rho Y_F}{D t} = \nabla\cdot \rho D\vec\nabla Y_F - \nu_F\bar M_F\dot\omega</math><br />
:<math>\frac{D \rho Y_O}{D t} = \nabla\cdot \rho D\vec\nabla Y_O - \nu_O\bar M_O\dot\omega</math><br />
The two above equations are linearly combined in a single one in a manner that the source term disappears:<br />
:<math>\frac{D \rho (\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)}{D t} = \nabla\cdot \rho D\vec\nabla (\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)</math><br />
The quantity <math>(\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)</math> is thus a conserved scalar. The last step is to normalize it such that it equals unity in the pure fuel stream (<math> Y_F=Y_{F,o}</math> and <math> Y_O=0 </math>) and is null in the pure oxidizer stream <br />
(<math> Y_F=0</math> and <math> Y_O=Y_{O,o} </math>). The resulting normalized passive scalar is the mixture fraction:<br />
:<math>Z=\frac{\Phi(Y_F/Y_{F,o})-(Y_O/Y_{O,o})+1}{\Phi+1}\qquad \Phi=\frac{\nu_O\bar M_O Y_{F,o}}{\nu_F\bar M_F Y_{O,o}}</math><br />
governed by the transport equation<br />
:<math>\frac{D \rho Z}{D t} = \nabla\cdot \rho D\vec\nabla Z </math><br />
The stoichiometric interface location (and thus the approximate location of the flame if the flow is reacting) is where <math>\Phi(Y_F/Y_{F,o})-(Y_O/Y_{O,o}) </math> vanishes (or <math>Y_F</math> and <math> Y_O </math> are both null in the reacting case). This leads to a stoichiometry definition:<br />
:<math>Z_s=\frac{1}{1+\Phi}</math><br />
<br />
As the mixture fraction qualifies the degree of inter-penetration of fuel and oxidizer, the elements originally present in these molecules are conserved and can be directly traced back to the mixture fraction. This has led to an alternate defintion of the mixture fraction, based on ''element conservation''.<br />
First, the elemental mass fraction <math> X_{j} </math> of element ''j'' is linked to the species mass fraction <math> Y_i </math>:<br />
:<math> X_j = \sum_{i=1}^{n} \frac{a_{i,j} \bar M_j}{\bar M_i} Y_i </math><br />
where <math> a_{i,j} </math> is a matrix counting the number of element ''j'' atoms in specie molecule named ''i'' and ''n'' is the number of species in the mixture.<br />
The group pictured by the summation above is a linear combination of <math> Y_i </math>. Because the transport equations of species mass fraction, few lines earlier, are also linear, a transport equation for the elemental mass fraction can be written:<br />
:<math>\frac{D \rho X_j}{D t} = \nabla\cdot \sum_{i=1}^n\frac{a_{i,j}\bar M_j}{\bar M_i}\rho D_i\vec\nabla Y_i</math><br />
For mass is conserved, the linear combination of the source terms vanishes. Furthermore, by taking the same diffusion coefficient <math> D_i </math> for all the species, the elemental mass fraction transport equation has exactly the same form as the specie transport equation (except the source term). Notice that the assumption of equal diffusion coefficient was also made in the previous definition of the mixture fraction and is justified in turbulent combustion modelling by the turbulence diffusivity flattening the diffusion process in high Reynolds number flows. Hence, the elemental mass fraction transport equation has the same structure as the mixture fraction transport equation seen above. Properly renormalized to reach unity in the fuel stream and zero in the oxidizer stream, the elemental mass fraction is a convenient way of determining the mixture fraction field in a flow. Indeed, it is widely used in practice for this purpose.<br />
<br />
==== Dissipation Rate ====<br />
A very important quantity, derived from the mixture fraction concept, is the ''scalar dissipation rate'', usually noted: <math> \chi </math>. In the above introduction to non-premixed combustion, it has been said that a diffusion flame is fully controlled through: (i) the position of the stoichiometric line, dictating where the flame sheet lies; (ii) the gradients of fuel on one side and oxidizer on the other side, dictating the feeding rate of the reaction zone through diffusion and thus the strength of combustion. According the the mixture fraction definition, the location of the stoichiometric line is naturally tracked through the <math>Z_s</math> iso-line and it is seen here how the mixture fraction is a convenient tracer to locate the flame.<br />
In the same manner, the mixture fraction field should also be able to give information on the strength of the chemistry as the gradients of reactants are directly linked to the mixture fraction distribution. The feeding rate of the reaction zone is characterized through the inverse of a time. Because it is done through diffusion, it must be obtained through a combination of mixture fraction gradient and diffusion coefficient (dimensional analysis):<br />
:<math> \chi_s = \frac{(\rho D)_s}{\rho_s} ||\vec \nabla Z||^2_s \qquad (\rho D)_s = \left ( \frac{\lambda}{C_p} \right )_s</math><br />
where the subscript ''s'' refers to quantities taken effectively where the reacting sheet is supposed to be, close to stoichiometry. This deduction of the scalar dissipation rate, scaling the feeding rate of the flame, is obtained here through physical arguments and is to be derived from equations below in a more mathematical manner. Note that the transport coefficient for the mixture fraction is identified to the one for temperature. This notation is usually used in the literature to emphasize that the rate of temperature diffusion (that is commensurable to the rate of species diffusion) is the retained parameter (as introduced in the following approach highlighting the role of the scalar dissipation rate).<br />
<br />
Because combustion is highly temperature-dependent, ''T'' is certainly the scalar to which attention must be paid for. The temperature equation in the Low-Mach Number regime ([[#Low-Mach Number Equations|Sec. Low-Mach Number Equations]]) is written below in steady-state:<br />
:<math>\rho\vec U \cdot \vec\nabla T = \nabla\cdot \frac{\lambda}{C_p}\vec\nabla T + \frac{Q}{C_p}\nu_F \bar M_F \dot\omega </math><br />
In order to make this equation easily tractable, the Howarth-Dorodnitzyn transform and the Chapman approximation are applied.<br />
In the Chapman approximation, the thermal dependence of <math>\lambda/C_p</math> is approximated as <math>\rho^{-1}</math>.<br />
The Howarth-Dorodnitzyn transform introduces <math>\rho</math> in the space coordinate system: <math> \vec\nabla \rightarrow \rho\vec\nabla \quad ; \quad \nabla\cdot\rightarrow \rho\nabla\cdot</math>. The effect of these both mathematical operations is to `digest' the thermal variation of quantities such as density or transport coefficient. Hence, the temperature equation comes in a simpler mathematical shape:<br />
:<math>\frac{\rho}{\rho_s}\vec U \cdot \vec\nabla T = \left ( \frac{\lambda}{C_p}\right )_s \nabla\cdot \vec\nabla T + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
Here the references are taken in the flame, i.e. close to the stoichiometric line (''s'' subscript).<br />
<br />
<br />
<br />
In a non-premixed system, strictly speaking, <math> \vec U </math>, is not really relevant as the flame is fully controlled by the diffusion process. Notwithstanding, in practice, non-premixed flames must be stabilized by creating a strain in the direction of diffusion. This is the reason why the velocity is left in the equation. Because a diffusion flame is fully described by the mixture fraction field, a change in coordinate can be applied:<br />
:<math> \left (<br />
\begin{array}{c}<br />
x \\<br />
y<br />
\end{array}\right ) \longrightarrow <br />
\left (<br />
\begin{array}{c}<br />
x \\<br />
Z<br />
\end{array}\right )<br />
</math><br />
where ''x'' is the coordinate tangential to the iso-<math>Z_s</math> (hence to the flame, in a first approximation) and ''y'' is perpendicular.<br />
The Jacobian of the transform is given as:<br />
:<math> \left [<br />
\begin{array}{cc}<br />
\frac{\partial x}{\partial x} & \frac{\partial Z}{\partial x} \\<br />
\frac{\partial x}{\partial y} & \frac{\partial Z}{\partial y} <br />
\end{array}\right ] = <br />
\left [<br />
\begin{array}{cc}<br />
1 & 0 \\<br />
0 & l_d^{-1} <br />
\end{array}\right ]</math><br />
Note that the diffusive layer of thickness <math>l_d</math> is defined as the region of transition between fuel and oxidizer and is thus given by the gradient of ''Z'' along the ''y'' direction.<br />
This transform is applied to the vectorial operators:<br />
:<math> \nabla\cdot = \nabla_x\cdot+\nabla_y\cdot=\nabla_x\cdot+\nabla_y\cdot Z\nabla_Z\cdot</math><br />
:<math> \vec\nabla = \vec\nabla_x+\vec\nabla_y=\vec\nabla_x+\vec\nabla_y Z\nabla_Z\cdot</math><br />
<br />
With this transform, the above temperature equation looks like:<br />
:<math>\frac{\rho}{\rho_s}\vec U \cdot (\vec\nabla_x T + \vec\nabla Z \nabla_Z\cdot T) = \left (\frac{\lambda}{C_p}\right )_s \left ( \nabla_x\cdot \vec\nabla_x T + \nabla_y\cdot Z \nabla_Z\cdot \vec\nabla_y Z \nabla_Z\cdot T + \nabla_y\cdot Z \nabla_Z\cdot \vec\nabla_x T + \nabla_x\cdot \vec\nabla_y Z \nabla_Z\cdot T \right ) + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
As mentioned above, the velocity and the variation along the tangential direction to the main flame structure ''x'' are not supposed to play a major role. By emphasizing the role of the gradient of ''Z'' along ''y'' as a key parameter defining the configuration the following equation is obtained:<br />
:<math>0 = \left (\frac{\lambda}{C_p}\right )_s ||\vec\nabla_y Z||^2 \Delta_Z T + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math><br />
This equation (sometimes named the ''flamelet equation'') serves as the basic framework to study the structure of diffusion flames. It highlights the role of the dissipation rate with respect to the strength of the source term and shows that the dissipation rate calibrates the combustion intensity.<br />
<br />
To describe the structure of the diffusion flame, the ''reduced mixture fraction'' is set:<br />
:<math> \xi = \frac{Z-Z_s}{Z_s(1-Z_s)\varepsilon} </math><br />
The utility of the reduced mixture fraction is to focus on the reaction zone. This reaction zone is supposed located on the stoichiometric line (this is why the reduced mixture fraction is centred on <math>Z_s</math>) and to be very thin (reason of the introduction of the magnifying factor <math>\varepsilon</math>).<br />
<br />
== The Premixed Regime ==<br />
[[Image:Premixed.jpg|thumb|Sketch of a premixed flame]]<br />
In contrast to the non-premixed regime above, the reactants are here well mixed before entering the combustion chamber. Chemical reaction can occur everywhere and this flame can propagate upstream into the feeding system as a subsonic (deflagration regime) chemical wave. This presents lots of safety issues. Some situations prevent them: (i) the mixture is made too rich (lot of fuel compared to oxidizer) or too lean (too much oxidizer) such that the flame is close to its flammability limits (it cannot easily propagate); (ii) the feeding system and regions where the flame is not wanted are designed such that they impose strong heat loss to the flame in order to quench it. For a given thermodynamical state of the mixture (composition, temperature, pressure), the flame has its own dynamics (speed, heat release, etc) on which there is few control: the wave exchanges mass and energy through diffusion process in the fresh gases. On the other hand, those well defined quantities are convenient to describe the flame characteristics. The mechanism of spontaneous propagation<br />
towards fresh gas through the thermal transfer from the combustion zone to the immediate slice of fresh gas<br />
such that the ignition temperature is eventually reached for this latter was highlighted as early as by the end of the 19th century by Mallard and LeChatelier. <br />
The reason the chemical wave is contained in a narrow region of reaction propagating upstream is the consequence of the discussion on the non-linearity of the combustion with temperature in the <br />
[[#Fundamental Aspects|Sec. Fundamental Aspects]]. It is of interest to compare the orders of magnitude of the temperature dependent term <math> \exp{(-\beta(1-\theta)/(1-\alpha(1-\theta)))}</math> of the reaction source upstream in the fresh gas (<math>\theta\rightarrow 0</math>) and in the reaction zone close to equilibrium temperature (<math>\theta\rightarrow 1 </math>) for the set of representative values: <math> \beta = 10 </math> and <br />
<math>\alpha=0.9</math>. It is found that the reaction is about <math>10^{43}</math> times slower in the fresh<br />
gas than close to the burned gas. It is known that the chemical time scale is about 0.1 ms in the reaction zone of a typical flame, then the typical reaction time in the fresh gas in normal conditions is about <br />
<math> 10^{39} s </math>. To be compared with the order of magnitude of the estimated Universe age: <math> 1 0^{17} s </math>. Non-negligible chemistry is only confined in a thin reaction zone stuck to the hot burned gas at equilibrium temperature. In this zone, the [[#Damk&ouml;hler]] number is high, in contrast to in the fresh mixture. It is natural and convenient to consider that the reaction rate is strictly zero everywhere except in this small reaction zone (one recovers the Dirac-like shape of the reaction profile, <br />
provided that one can see the upstream flow as a region of increasing temperature towards the combustion zone and the downstream flow as in fully equilibrium).<br />
<br />
As the premixed flame is a reaction wave propagating from burned to fresh gases, the basic parameter is known to be the ''progress variable''. In the fresh gas, the progress variable is conventionally put to zero. In the burned gas, it equals unity. Across the flame, the intermediate values describe the progress of the reaction to turn into burned gas the fresh gas penetrating the flame sheet. A progress variable can be set with the help of any quantity like temperature, reactant mass fraction, provided it is bounded by a single value in the burned gas and another one in the fresh gas. The progress variable is usually named ''c'', in usual notations:<br />
:<math>c=\frac{T-T_f}{T_b-T_f}</math><br />
It is seen that ''c'' is a normalization of a scalar quantity. As mentioned above, the scalar transport equations are assumed linear such that the transport equation for ''c'' can be obtained directly. <br />
Actually, the transport equation for ''T'' ([[#Transport Equations|Sec. Transport Equations]]) is linear if constant heat capacity is further assumed (combustion of hydrocarbon in air implies a large excess of nitrogen whose heat capacity is only slightly varying) and the progress variable equation is directly <br />
obtained (here for a default of fuel - lean combustion):<br />
:<math>\frac{D\rho c}{Dt}=\nabla\cdot\rho D\vec\nabla c + \frac{\nu_F\bar M_F}{Y_{F,u}}\dot\omega \qquad \rho D = \frac{\lambda}{C_p} </math><br />
<br />
The fact that the default or excess of fuel has been discussed above leads to the introduction of another quantity: the ''equivalence ratio''. The equivalence ratio, usually noted <math>\Phi</math>, is the ratio of two ratios. The first one is the ratio of the mass of fuel with the mass of oxidizer in the mixture. The second one is the same ratio for a mixture at stoichiometry. Hence, when the equivalence ratio equals unity, the mixture is at stoichiometry. If it is greater than unity, the mixture is named ''rich'' as there is an excess of fuel. In contrast, when it is smaller than unity the mixture is named ''lean''. The equivalence ratio presented here for premixed flames has little connection with the equivalence ratio introduced earlier regarding the non-premixed regime. Basically, the equivalence ratio as defined for non-premixed flames gives the equivalence ratio of a premixed mixture with the same mass of fuel and oxidizer. Moreover, the equivalence ratio as defined for a premixed mixture can be obtained based on the mixture fraction (it is thus the local equivalence ratio at a point in the non-homogeneous mixture described by the mixture fraction). From the definitions given above:<br />
:<math> \Phi=\frac{Z}{1-Z}\frac{1-Z_s}{Z_s}</math><br />
<br />
==== Premixed Flame P&eacute;clet Number ====<br />
<br />
Earlier in this section, it has been said that a premixed flame posses its own dynamics, as a free propagating surface, and has thus characteristic quantities. For this reason, a P&eacute;clet number may be defined, based on these quantities. The P&eacute;clet number has the same structure as the Reynolds number but the dynamical viscosity is replaced by the ratio of the thermal conductivity and the heat capacity of the mixture. The thickness <math>\delta_L </math> of a premixed flame is essentially thermal. It means that it corresponds to the distance of the temperature rise between fresh and burned gases. This thickness is below the millimetre for conventional flames. The width of the reaction zone inside this flame is even smaller, by about one order of magnitude. This reaction zone is stuck to the hot side of the flame due to the high thermal dependency of the combustion reactions, as seen above. Hence, the flame region is essentially governed by a convection-diffusion process, the source term being negligible in most of it.<br />
<br />
It is convenient to write the progress variable transport equation in a steady-state framework. The quantities at flame temperature (<math>_f</math>) are used to non-dimensionalize the equation:<br />
:<math> \overbrace{(\rho S_L)}^{\dot M}\frac{\vec S_L}{S_L}\vec\nabla c = (\rho D)_f \nabla\cdot (\rho D)^* \vec\nabla c</math><br />
Note that the source term is neglected, consistently with what has been said above.<br />
This convection-diffusion equation makes appear a first approximation of a flame P&eacute;clet number:<br />
:<math> Pe_f = \frac{\dot M \delta_L}{(\rho D)_f} \approx 1 </math><br />
<br />
From the P&eacute;clet number, it is possible to obtain an expression for the flame velocity (remembering that <math> \delta_L/S_{L,f} \approx \tau_c</math>, vid. inf. [[#Three Turbulent-Flame Interaction Regimes| Sec. Three Turbulent-Flame Interaction Regimes]]):<br />
:<math> S_{L,f}^2\approx \frac{(\rho D)_f}{\rho_f \tau_c}</math><br />
For typical hydrocarbon flames, the speed is some tens of centimetres per second and the diffusivity is some<br />
<math> 10^{-5} </math> square metres per second. The chemical time in the reaction zone of about one tenth of a millisecond is recovered.<br />
<br />
==== Details of the Premixed Unstrained Planar Flame ====<br />
<br />
A plane combustion wave propagating in a homogeneous fresh mixture is the reference case to describe the premixed regime. At constant speed, it is convenient to see the flame at rest with a flowing upstream mixture. This is actually the way propagating flames are usually stabilized. In the frame of description, the <br />
physics is 1-D, steady with a uniform (the flame is said unstrained) mass flowing across the system. Two types of equations are thus sufficient to describe the problem, the temperature transport equation and the species transport equations, as in [[#Transport Equations|Sec. Transport Equations]]. The transport coefficients will be chosen as equal: <math> \rho D_i = \lambda / C_p </math> (unity Lewis numbers). Suppose the 1-D domain is described thanks to a conventional (''Ox'') axis with a flame propagating towards negative ''x'' (this is the conventional usage), the boundary conditions are:<br />
:* in the frozen mixture: <br />
:** <math>Y_i \rightarrow Y_{i,u} \qquad ; \qquad x\rightarrow-\infty </math><br />
:** <math>T \rightarrow T_u \qquad ; \qquad x\rightarrow-\infty </math><br />
:* in the burned gas region supposed at equilibrium:<br />
:** <math>Y_i \rightarrow Y_{i,b} \qquad ; \qquad x\rightarrow+\infty </math><br />
:** <math>T \rightarrow T_b \qquad ; \qquad x\rightarrow+\infty </math><br />
<br />
:<math>Y_{i,b}</math> and <math>T_b</math> are obtained from [[#Conservation Laws|Sec. Conservation Laws]].<br />
<br />
The quantities that have been mentioned just above (scalar and temperature profiles, mass flow rate through the system) are the solution to be sought.<br />
<br />
According to the discussions above, the temperature transport equation in its full normalized form may be written as (lean /stoichiometric case):<br />
:<math> \dot M \frac{\partial \theta}{\partial x} = \frac{\partial \ }{\partial x}\frac{\lambda}{Cp} \frac{\partial \theta}{\partial x} + \frac{\nu_F \bar M_F B}{Y_{F,s}} \prod_{i=O,F}(Y_{i,u}^*-\theta\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_i} \exp{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}} </math><br />
This equation is further simplified by the variable change <math>d\xi=\dot M/(\lambda/Cp)dx</math>:<br />
:<math>\frac{\partial \theta}{\partial \xi}=\frac{\partial^2 \theta}{\partial \xi^2} + \overbrace{\frac{\lambda/Cp}{\dot M^2}\frac{\nu_F \bar M_F B}{Y_{F,s}}}^{\Lambda} \prod_{i=O,F}(Y_{i,u}^*-\theta)^{n_i} \exp{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}}</math><br />
<br />
Although somewhat out of scope, the existence and unicity of the solution of this type of equation are usually demonstrated with the help of the Schauder Theorem and Maximum Principle. From the point of view of physicists and engineers, the solution that is found analytically is de facto considered as the unique solution of the equation.<br />
<br />
===== Scenarii of Combustion Process in the Phase Portrait =====<br />
<br />
In the frame moving with the flame, both phase variables are the reduced temperature and its gradient. To ease the reading with usual notations, it is written: <math> X_1 = \theta \quad ; \quad X_2 = \partial \theta / \partial \xi = \dot \theta </math>. The system arises:<br />
: <math> \left\{<br />
\begin{array}{lll}<br />
\dot X_1 & = & X_2 \\<br />
\dot X_2 & = & X_2 - \varpi(X_1) <br />
\end{array}<br />
\right.<br />
</math><br />
with <math> \varpi </math> being the full source term in the above equation.<br />
<br />
In the frame moving with the flame, two singular nodes are found in the frozen flow <math> (X_1,X_2) = (0,0)</math> and the equilibrium region <math> (\theta_b,0)</math>, i.e when <math> \varpi(X_1) </math> vanishes. <br />
<math> x_1,x_2 </math> are defined as small departures from the singular nodes such that the linearized system in their neighbourhood is:<br />
: <math> \left\{<br />
\begin{array}{lll}<br />
\dot x_1 & = & l_{1,1} x_1 + l_{1,2} x_2 \\<br />
\dot x_2 & = & l_{2,1} x_1 + l_{2,2} x_2 <br />
\end{array}<br />
\right.<br />
</math><br />
provided: <math> l_{1,1} = 0 \quad l_{1,2} = 1 \quad l_{2,1} = -\varpi'_{X_1 = 0,\theta_b} \quad l_{2,2} = 1 </math>.<br />
The characteristic polynom is, in usual notations: <math> s^2 - s + \varpi' </math> such that the eigenvalues are:<br />
:<math><br />
s^{\pm}=\frac{1\pm\sqrt{1-4\varpi'}}{2}<br />
</math><br />
A priori, those eigenvalues may be (i) real distinct, (ii) real identical, or (iii) conjugated complex. In the first case, the orbits in the phase diagram are organized, in the immediate neighbourhood of the singular node, with respect to the eigenvectors directions associated to the eigenvalues. The following task is to identify the nature of those eigenvalues and of the corresponding nodes. Because <math> 0 < X_1 < \theta_b </math> is bounded, complex eigenvalues are excluded as they would lead to a spiral node. This remark is important for the node on the cold side as it imposes a bound:<br />
:<math> \varpi'_{X_1=0} \le \frac{1}{4} </math><br />
As the mass flow rate through the flame is included into <math>\varpi</math>, it imposes a minimum value on the flame speed to tackle with the cold boundary difficulty (rise of the chemical rate in the frozen flow). In this condition, it <br />
is an unstable node (improper in case of equality).<br />
On the other hand, because <math> \varpi'|_{X_1=\theta_b} </math> is not positive, the node on the hot side is found as a saddle point. The overall scenario of combustion within the flame is thus an orbit leaving the cold node to join the hot node by branching on a trajectory compatible with the negative eigenvalue of the saddle.<br />
<br />
[[Image:sketchOrbit.jpg|thumb| Sketch of orbits for a combustion process across a premixed front. Dashed lines represent forbidden orbits from the physics. The red line describes the orbit expected in an idealized combustion process.]]<br />
It must be noted that the associated eigenvectors are of the form:<br />
:<math><br />
\left (<br />
\begin{array}{l}<br />
t \\<br />
t s^{\pm} <br />
\end{array}<br />
\right ); \quad t\in \Re^*<br />
</math><br />
that is, on the cold node, a positive departure on <math>X_1</math> following any of the two eigendirections, leads<br />
to a consistent creation of positive temperature gradient, while on the hot node, only the stable direction will allow a consistent creation of a positive temperature gradient for any departure of the temperature towards region where it is inferior to <math>\theta_b </math>. Another remark is the structure of the eigendirections. The leaving directions on the cold node have a slope larger than the one of <math> \varpi </math> while the stable direction of the hot node has a slope smaller than the one of <math> \varpi </math>. It means that there is some point where the orbit must cross the profile of the chemical term versus temperature. For that temperature, the gradient equals the reaction rate through construction of the phase space. When looking back to the equation of the premixed flame, it happens in a region of inflexion for the temperature (the second order derivative must vanish). Furthermore, at this intersection, the orbit is horizontal (if the frame of reference for <math> (X_1, X_2) </math> is Cartesian) due to the shape of the premixed flame equation above that can be recast into <math> X_{2,X_1}' = (X_2-\varpi)/X_2 </math>.<br />
Close to the cold node, the orbits have a shape of parabola whose axis is the direction with the largest eigenvalue magnitude.<br />
Close to the hot node, the orbits have an hyperboloid shape with asymptots as the eigendirections. Now the ingredients are here to draw a sketchy scenario of the combustion in a premixed flame. It will be superimposed on the reaction rate graph studied in [[#Fundamentals Aspects|Sec. Fundamentals]].<br />
Some typical orbits from the above analysis are drawn in the figure on the right. The basic geometrical arguments developed are reproduced. In particular, the dashed lines represent forbidden orbits by the physics (boundedness of <math>X_1</math>, irreversibility). Orbits must be travelled from left to right, corresponding to increasing free parameter <math> \xi </math>.<br />
It is of integral importance to remember that, in combustion in conventional conditions, the source term is highly non-linear. Therefore, it is localized in a very thin sheet and, upstream, <math> \varpi </math> and <math> \varpi' \rightarrow 0 </math>. <br />
Only the most unstable eigendirection of the cold node is compatible as an orbit. The other trajectories, being paraboloidal, are tangent to the other direction that is flat at the limit. It physically means an elevation of temperature in bulk, that is contradictory to what is expected from a highly non-linear combustion term. Hence, the additional orbit in red is the one expected in idealized combustion conventional conditions.<br />
<br />
[[Image:dnsOrbit.jpg|thumb|Sketch of orbits for a combustion process across a premixed front, DNS simulations for the usual combinations of <math>\beta</math> and <math> \alpha </math>. Note that the case with <math>\alpha = 0</math> does not mean an athermal reaction but the mathematical simplification of the denominator of the exponential argument of the source term.]]<br />
The following picture is the result of actual computations of the above 1-D flame equation (stoichiometry and <math> n_i = 1 </math>) with the help of a high-order (6) code. The already presented combinations of <math> \beta </math> and <br />
<math> \alpha </math> are used and the orbits are retrieved. For most of the cases, as predicted above, the system selects a solution leaving the cold node with the most unstable direction (identifiable with its slope close to unity for vanishing<br />
<math>\varpi'_{X_1=0}</math>). There is also an additional curve, for <math> \beta = 10 </math> with the denominator of the exponential argument suppressed. This curve is remarkable as the solution selected by the system leaves the cold node in a manner fully controlled by the <math> \varpi'_{X_1=0} </math>. This is a very singular solution, not expected in combustion in conventional conditions, as explained above. The purpose of this remark is to question the well-posedness of considering simplifying the exponential argument for <math>\beta</math> "sufficiently high", as it is usually proposed for this type of modelling. As observed, the dynamical system analysis demonstrates a switch in the nature of the solution selected.<br />
At the physical level, when the orbit follows the most unstable direction with a slope close to unity, it means that <math> X_2 </math> "follows" <math> X_1 </math>, which is a signature of a diffusion process. In other words, the preheating mechanism of a premixed flame propagation as proposed for more than one century is in work. On the other hand, when <br />
<math> X_2 </math> is dependent on the evolution of <math> \varpi'</math>, it shows that "cold" chemistry drives the solution in the frozen flow and not the acknowledged mechanism of deflagration.<br />
<br />
<br />
===== Flame Solution =====<br />
<br />
As already mentioned, the flame system may be split into three zones. Upstream, the conventional mechanism of deflagration<br />
is supported by diffusion of heat. Downstream, the mixture is at equilibrium after combustion. In between, there exists the reaction layer. For large <math>\beta</math> the reaction layer is very thin such that it can be seen as a discontinuity between the fresh and burned gases. This is this difference in scales that introduces the use of the asymptotic method to resolve some of the flame characteristics, such as speed, time, heat region thickness, or reaction zone thickness.<br />
<br />
The domain is partitioned, according to this zoning defined by the scales driving the physics with an outer domain, driven by large scales and an inner domain refining the description within the discontinuity. If the discontinuity (flame reaction zone) is at <math> \xi=0 </math>, everywhere but 0, the equation is simplified as:<br />
:<math><br />
\frac{\partial^2 \theta}{\partial \xi^2} = \frac{\partial\theta}{\partial \xi}<br />
</math><br />
*For <math> \xi>0 </math>, it is expected that the mixture has reached equilibrium chemistry, such that: <math> \forall \xi > 0, \quad \theta=\theta_b \quad ; \quad \partial \theta / \partial \xi =0 </math>.<br />
*For <math> \xi < 0 </math>, this is the preheat zone and the solution is <math> \theta = \theta_b e^{\xi} </math> with <math> \theta </math> reaching <math>\theta_b</math> at the disconstinuity and vanishing, together with its gradient, far upstream in the frozen mixture.<br />
<br />
The solution for the species and the value of <math> \theta_b </math> are obtained from [[#Conservation Equations|Conservation Equations]] above. The 'big picture' is thus an exponential variation in the thermal thickness matched with a plateau in the downstream region, the line of matching being the discontinuity (flame) that has no thickness at this scale of description.<br />
<br />
To refine the analysis in the discontinuity region, a magnifying factor <math> \varepsilon </math> is used to stretch the coordinates: <math> \xi = \varepsilon \Xi </math>. The inner solution is thus a slowly-varying function of <math> \Xi </math>. Hence, in this inner region, the equation for the premixed flame becomes:<br />
:<math><br />
\frac{1}{\varepsilon}\frac{\partial \theta}{\partial \Xi}=\frac{1}{\varepsilon^2}\frac{\partial^2 \theta}{\partial \Xi^2} +\varpi<br />
</math><br />
In order to stretch and 'look inside' a discontinuity, <math>\varepsilon</math> is very small. It yields two remarks:<br />
# convection is negligible compared to diffusion. The heat losses from the reaction zone are essentially diffusion driven.<br />
# The reaction zone is governed by a diffusion-reaction budget and the reaction term <math>\varpi</math> must be strong to balance the intense heat loss due to the sharp diffusion (the zone is very thin, hence the gradients are sharp).<br />
The mechanism is thus different from the outer region that was convection-diffusion driven.<br />
<br />
Each quantity is developed in a series of <math> \varepsilon </math>.<br />
At the leading order, for <math>\theta</math>, in the lean case, the conservation relations (Sec. [[#Conservation Laws|Conservation Laws]]) yield:<br />
:<math> \theta = 1 - \varepsilon \Gamma - (1 - Y^*_{F,u})</math><br />
where <math>\Gamma</math> is the first-order development of the departure of <math>\theta</math> from the maximum value due to the incomplete combustion, and <math>1-Y_{F,u}^*=1-\theta_b</math> is the reduction of temperature for non-stoichiometric cases. Injected into the above equation:<br />
:<math> \frac{1}{\varepsilon}\frac{\partial^2 \Gamma}{\partial \Xi^2} = \Lambda (\varepsilon \Gamma)^{n_F} (Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} +\varepsilon\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\beta\frac{1-Y_{F,u}^*+\varepsilon\Gamma}{1-\alpha(1-Y_{F,u}^*+\varepsilon\Gamma)}}</math><br />
Although the full develoment is not achieved, a number of scaling may be highlighted:<br />
# because the temperature cannot be much below unity, <math>Y_{F,u}^*</math> must be close to 1 <math> {\mathcal O}(\varepsilon)</math> . For clarity, it is not expanded in an <math>\varepsilon</math> series.<br />
# The denominator of the exponential argument simplifies to unity for small <math>\varepsilon</math>. <br />
# To get a finite rate in the reaction zone, <math>\varepsilon</math> scales with <math>\beta^{-1}</math>.<br />
<br />
The burning rate eigenvalue, <math>\Lambda</math>, is naturally expanded as: <math> \Lambda = \varepsilon^{-n_O-n_F-1}(\Lambda_0 + {\mathcal O}(\varepsilon)) </math>. The low-order equation to be solved is:<br />
:<math><br />
\frac{d^2 \Gamma}{d \Xi^2} = \frac{1}{2}\frac{d(\Gamma_{\Xi})^{'2}}{d\Gamma} = \Lambda_0\exp{-\beta (1-Y_{F,u}^*)} \Gamma^{n_F} (\frac{Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} }{\varepsilon}+\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\Gamma}</math><br />
:<math> \frac{d\Gamma}{d\Xi}(-\infty)=-\theta_b, \qquad \frac{d\Gamma}{d\Xi}(\infty)=0 </math><br />
The boundary conditions are obtained from the matching of the outer solutions on the right and left sides of the flame as written above (the outer solutions are reached at infinity for a very small magnifying factor <math>\varepsilon</math>).<br />
Once integrated with respect to those boundary conditions, the burning-rate eigenvalue (from which <math> \dot M </math> is extracted) is obtained as:<br />
:<math> \Lambda_0 = \Bigg( 2\int_0^{\infty}d\Gamma\; \Gamma^{n_F} (\beta (Y_{O,u}^*-Y_{F,u}^*\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )+\Gamma\frac{Y_{F,s}}{Y_{i,s}}s^{\delta_{i,O}} )^{n_O}\exp{-\beta (1-Y_{F,u}^*)}\exp{-\Gamma} \Bigg )^{-1}<br />
</math><br />
The RHS integral is not developed for clarity but presents no peculiar difficulties.<br />
<br />
The development has been carried out at the first order in <math> \varepsilon </math>. As soon as a second order development is attempted, some expressions are no more analytically tractable. On the other hand, a second order development allows introduce the temperature-dependent trends of some terms in <math> \Lambda </math>. Physical results are retrieved such as a slight decrease of the speed for a positive sensitivity of transport parameters to temperature around equilibrium conditions.<br />
<br />
[[Image:BellSpeed.jpg|thumb|Unstrained planar premixed flame speed with respect to fuel mass fraction (lean case) for a single global irreversible Arrh\'enius term. Symbols are obtained from a high-order DNS code. Continuous line is the theory exposed here. The usual combinations of <math>\beta</math> and <math> \alpha </math> are used. <br />
Very high values of <math> \beta </math> (and thus negligible effect of <math> \alpha </math> are also presented<br />
to show the problem of slow convergence for finite value of the Zeldovitch parameter. Not that <math>\alpha = 0</math> does not mean an athermal reaction but the mathematical simplification of the denominator of the exponential argument of the source term.]]<br />
The image beside illustrates the response of the premixed flame speed with respect to fuel concentration (chosen as the limiting component here; equivalent findings are obtained for default of oxidizer) for different values of the chemical parameters <math> \beta </math> and <math>\alpha</math> (as noted above, the pre-exponential constant <math> A</math> impacts only the overall magnitude). The theoretical expression (continuous line) is tested against a 1-D high accuracy<br />
code with the given chemistry implemented. It is seen that:<br />
* the Zeldovitch parameter drives the drop for non-stoichiomeric mixtures,<br />
* the drop is relatively well modelled by the theoretical expression, and<br />
* the absolute magnitude converges slowly towards the theoretical one when increasing <math>\beta</math> in the code (effect of the finiteness of <math> \beta </math> in the real case).<br />
<br />
[[Image:PremixedProfiles.jpg|thumb|Typical profiles in 1-D premixed flame at stoichiometry. Representative value of global chemistry parameters. Specie profiles are simply the complementary to the temperature profile for simple chemistry.]]<br />
The picture exhibiting profiles for different Zeldovitch and heat release parameters shows the factual impact: upstream, the exact exponential profile is recovered and corresponds to the pre-heating region (thermal thickness). In the reaction zone (just upstream of the extremum), the departure from the exact solution is due to the kinetic effect. This kinetic effect is more pronounced when <math> \beta </math> is lower because the lower the Zeldovitch parameter, the lower the temperature reaction zone can be without leading to extinction. The flame takes this opportunity to maximize its transfer in heat and reactant with the cold zone. This is the physical understanding of an inverse dependence of the maximum flame speed with <math> \beta </math>.<br />
<br />
<br />
==== Modification of the Flame Speed with Curvature ====<br />
<br />
[[Image:premixedCurve2d.jpg|thumb|Flame seen as an interface between fresh and burned gases. Its curved profile towards the burned side increases the transfers with the fresh side.]]<br />
When the thickness of the flame <math>(\lambda/Cp)_f/\dot M </math> is considered small compared to inhomogeneities existing in the flow, the flame can be reduced to an interface between fresh and burned gases. This interface may not be strictly plan in the general case. For instance, when the interface is curved towards the burned gas, it offers a larger opportunity for transfer of mass and heat with the fresh gas. As the ability of the chemistry to burn the coming matter is limited, the flame has thus to reduce its displacement speed. The figure beside provides a 2-D sketch of the situation.<br />
<br />
To mathematically give the expression showing that the curvature influences the flame speed (for small curvature), the non-dimensionalized temperature equation across a planar premixed flame above is slightly recast:<br />
:<math>||\dot M^c||||\vec\nabla_{\xi}\theta|| - \varpi= \nabla_{\xi}\cdot\vec\nabla_{\xi}\theta=-\nabla_{\xi}\cdot(||\vec\nabla_{\xi}\theta||\vec n) = - \vec\nabla_{\xi}||\vec\nabla_{\xi}\theta||\cdot\vec n - ||\vec\nabla_{\xi}\theta||\nabla_{\xi}\cdot\vec n</math><br />
<br />
In this expression, <math>\dot M^c</math> is the flame mass flow rate when perturbed by a curvature normalized by the reference one developed above. <math> \vec n</math> is the normal to the iso-temperature pointing towards the fresh gas, what is named the normal to the flame. From the expression developed above and for a slightly perturbed flame, the gradients, production and diffusion terms are very close to the unperturbed flame. Only the last term, the normal divergence, does not exist for a non-perturbed flame. Hence, the above equation may be simplified against the one for unperturbed flame presented earlier:<br />
:<math> \dot M^c = 1 - \nabla_{\xi}\cdot \vec n </math><br />
This is the divergence of the flame normal that contains the information upon geometrical perturbation (curvature) that impacts the speed.<br />
<br />
[[Image:premixedCurvature.jpg|thumb|Local geometrical approximation of a flame surface]]<br />
The key is thus to get an idea of the geometrical significance of the normal divergence. The normal divergence theorem says that this is the sum of the principal curvatures of the flame surface at the location considered. To give a good mental picture, the simplest configuration without loss of generality is to consider the flame surface approached by an osculatory revolution ellipsoid, as in the figure beside. The location of the approximation is the intersection of the Ox axis and the flame surface in red (where the ellipsoid is tangent to the flame). At this location, the normal divergence is the sum of the curvature of the basic ellipse (before its rotation around Oy) at its minor extremum, and the curvature of the circle corresponding to the rotation of this point around Oy when the 3-D shape is formed. Giving a lecture on conical coordinate systems is beyond the purpose but the interested reader may want to follow the corresponding steps to check this result: (i) write the divergence of a vector in the ellipsoid coordinate system, (ii) consider that the vector is the normal, i.e. it has only one constant component perpendicular to the local ellipsoidal iso-coordinate, to simplify the divergence expression, (iii) split the resulting terms into two parts by identifying the second derivative of the basic shape 2-D ellipse as one principal curvature, and the inverse of the radius of the circle corresponding to the rotation of the ellipse around Oy at the point where the flame surface is approached, in the figure this is simply the minor axis of the ellipse.<br />
<br />
The expression is readily written as:<br />
:<math> \dot M^c = 1 - (R_1^{-1}+R_2^{-1})</math><br />
where <math> R_1 </math> and <math> R_2</math> are the two radius of curvature (non-dimensionalized by the reference flame thickness) local to the surface.<br />
For instance, they are the small axis of the basic ellipse and its radius of curvature at its minor maximum when the <br />
surface is approached by the osculatory ellipsoid as above.<br />
<br />
<br />
===== Natural Instabilities of Premixed Flames =====<br />
<br />
It is not the goal of a document about combustion modelling to list all the cases where a premixed flame exhibits a particular geometry but there is a typical phenomenon that cannot not be easily dissociated from premixed combustion and<br />
impacts a lot the dynamics of the flame. These instabilities have been mentioned by Darrieus as early as 1938. The motivation of discussing about this is also related to the framework of description used that is widely employed for combustion models development. This framework is named the hydrodynamics limit, where the flame is isolated as <br />
a zero-thickness interface in the flow, and has been first introduced just above. In this framework, any diffusive and energetic aspects disappear and the set of equations is limited to two incompressible Euler systems. One system in the fresh gases (with a constant density of cold mixture). One system in the burned gases (with a constant density of mixture at equilibrium).<br />
<br />
To understand the basic properties of a premixed flame leading to the birth of instabilities, it is first important to realize that a premixed flame, in the hydrodynamic limit, behaves as a dioptre with a refractive index in the burned gases larger than in the fresh gases. For a flow with an angle of attack on the flame (i.e. a flame not strictly 1-D perpendicular to the flow), the tangential component of the flow speed relative to the flame surface is conserved while the normal component is accelerated by a factor corresponding to the ratio of the density in order to conserve mass flow across the interface. Hence, the streamlines are pushed towards the normal to the flame when crossing, that is similar to rays of light entering a refracting medium. <br />
<br />
If one considers a region of a premixed flame that bumps a little bit towards the fresh gas (the same approach is symmetrically true for the bump towards the burned gas), the local stream tube slightly opens on the bumpy interface before being refracted and coming back to its original section at constant mass flow rate. Hence, just in front of the bump, the gas velocity in this stream tube decreases and does not oppose the flame motion, allowing the bump to increase in magnitude. This is the fundamental mechanism of such instabilities.<br />
<br />
The equations sets are in usual notations:<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
U_x + V_y & = & 0 \\<br />
U_t + UU_x + VU_y & = & \frac{P_x}{\rho} \\<br />
V_t +UV_x+VV_y & = & \frac{P_y}{\rho} <br />
\end{array}<br />
\right.<br />
</math><br />
<math> x </math> is the coordinate used above along the flame path and <math>y</math> is in the tangential plan.<br />
<math>u </math> and <math>v</math> are the respective velocities.<br />
Those equations may be normalized by steady-state reference quantities such as flame speed, flame length, gauge pressure and density with respect to fresh gas properties and are written on both sides of the interface.<br />
<br />
These sets of equations must match at the interface (flame surface) through conservation of mass flow rate and momentum:<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
\rho_u (\vec U_u -\vec U_{f})\cdot \vec n & = & \rho_b (\vec U_b -\vec U_{f})\cdot \vec n\\<br />
(\rho_u \vec U_u (\vec U_u -\vec U_{f})+P_u)\cdot \vec n & = & (\rho_b \vec U_b (\vec U_b -\vec U_{f})+P_b)\cdot \vec n<br />
\end{array}<br />
\right.<br />
</math><br />
The subscripts <math> u </math> and <math> b </math> stand for unburned and burned sides, respectively. The subscript <math> f </math> points the interface. In these equations, the flame motion <math> U_f </math> is reintroduced because we want to track the instability movement over the mean position of the flame. This instability motion is described by the equation <math> x=F(y,t) </math> such that the motion of the interface normal to itself is obtained as:<br />
:<math> \vec U_f\cdot\vec n = - \frac{F_t}{\sqrt{1+F^2_y}} </math><br />
with a normal to the front:<br />
:<math> \vec n = \left (<br />
\begin{array}{l}<br />
-\frac{1}{\sqrt{1+F^2_y}} \\<br />
\frac{F_y}{\sqrt{1+F^2_y}} <br />
\end{array}<br />
\right )<br />
</math><br />
<br />
As the instability motion is considered at its birth, i.e. when it is still small, the quantities are linearized with <math> \varepsilon </math> as a small parameter:<br />
:<math> F=\varepsilon f</math><br />
:<math> \vec U = \vec {\mathcal U} + \varepsilon\vec u</math><br />
:<math> P = {\mathcal P} + \varepsilon p </math><br />
<br />
The Eulerian system is reduced to (at the first order in <math>\varepsilon</math>, the order zero is simplified for homogeneous flow):<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
u_x + v_y & = & 0 \\<br />
u_t + {\mathcal U}u_x & = & \frac{p_x}{\rho} \\<br />
v_t +{\mathcal U}v_x & = & \frac{p_y}{\rho} <br />
\end{array}<br />
\right.<br />
</math><br />
<br />
The one-sided equations for the jump conditions become (terms up to the first order in <math>\varepsilon</math> are kept):<br />
:<math><br />
\left \{<br />
\begin{array}{lll}<br />
\rho (\vec U-\vec U_f)\cdot\vec n \times \sqrt{1+F^2_y} & = & -\rho {\mathcal U}-\rho\varepsilon u+\rho\varepsilon f_t \\<br />
(\rho \vec U (\vec U -\vec U_{f})+P)\cdot \vec n \times \sqrt{1+F^2_y} & = &<br />
\left \{<br />
\begin{array}{l}<br />
-\rho {\mathcal U}^2 -2\rho{\mathcal U}\varepsilon u+\rho{\mathcal U}\varepsilon f_t - {\mathcal P} -\varepsilon p \\<br />
-\rho {\mathcal U}\varepsilon v + {\mathcal P}\varepsilon f_y<br />
\end{array}<br />
\right.<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
The following jump conditions are obtained (by recalling that unburned steady-state quantities are chosen as references):<br />
* From the first equation at the order 0, <math>\rho_b U_b = \rho_u U_u \Leftrightarrow U_b=\rho_u/\rho_b</math><br />
* From the first equation at the order 1, <math> \rho_u f_t - \rho_u u_u = \rho_b f_t -\rho_b u_b </math><br />
* From the second equation at the order 0, <math> -\rho_u {\mathcal U}_u^2 - {\mathcal P}_u = -\rho_b {\mathcal U}_b^2 - {\mathcal P}_b \Leftrightarrow {\mathcal P}_b = 1 - \rho_u/\rho_b </math><br />
* From the second equation at the order 1, <math> -2\rho_u{\mathcal U}_u u_u +\rho_u {\mathcal U}_u f_t -p_u = <br />
-2\rho_b{\mathcal U}_b u_b +\rho_b {\mathcal U}_b f_t -p_b \Leftrightarrow p_b-p_u = 2 u_u-2 u_b </math><br />
* From the third equation at order 1 (no order 0), <math> -\rho_u {\mathcal U}_u v_u +{\mathcal P}_u f_y = -\rho_b {\mathcal U}_b v_b +{\mathcal P}_b f_y \Leftrightarrow v_b-v_u = f_y(1-\rho_u/\rho_b) </math><br />
<br />
The solution for the linearized, autonomous Euler system is:<br />
:<math><br />
\left (<br />
\begin{array}{l}<br />
u \\ <br />
v \\ <br />
p<br />
\end{array}<br />
\right )<br />
= <br />
\left (<br />
\begin{array}{l}<br />
\bar u \\ <br />
\bar v \\ <br />
\bar p <br />
\end{array}<br />
\right )<br />
\exp{(\sigma x)}\exp{(\alpha t -iky)}<br />
</math><br />
where one recognizes an account for the perturbation of the field in the <math> x </math> direction (<math>\sigma</math>), the wave number of the instability following <math> y </math>, <math> k</math> and the <br />
growth rate with time <math> \alpha </math>.<br />
The eigenvalues of the system are used to determine the <math> x </math> dependence of the solution <math> \sigma = - \alpha/U,\; k,\; -k</math>, with the positive ones applying on the fresh side and the negative ones on the burned size to have a vanishing perturbation far from the flame.<br />
<br />
The eigenmodes give the flow pertubations on either side of the flame:<br />
:<math><br />
x<0:<br />
\left ( <br />
\begin{array}{l}<br />
u \\ v \\ p<br />
\end{array}<br />
\right )<br />
=<br />
a\left ( <br />
\begin{array}{l}<br />
1 \\ -i \\ -1-\frac{\alpha}{k}<br />
\end{array}<br />
\right )<br />
e^{kx+\alpha t - iky}<br />
</math><br />
<br />
:<math><br />
x>0:<br />
\left ( <br />
\begin{array}{l}<br />
u \\ v \\ p<br />
\end{array}<br />
\right )<br />
=<br />
b\left ( <br />
\begin{array}{l}<br />
1 \\ i \\ -1+\rho_b\frac{\alpha}{k}<br />
\end{array}<br />
\right )<br />
e^{-kx+\alpha t - iky}<br />
+<br />
c\left ( <br />
\begin{array}{l}<br />
1 \\ i\rho_b\frac{\alpha}{k} \\ 0<br />
\end{array}<br />
\right )<br />
e^{-\rho_b \alpha x+\alpha t - iky}<br />
</math><br />
<br />
The jump conditions above applied to this perturbation field yield the following system (<math> f </math> has also been put into the harmonic form <math> f = \bar f e^{\alpha t-iky} </math>):<br />
:<math><br />
\left \{ <br />
\begin{array}{lll}<br />
\alpha \bar f -a & = & \rho_b \alpha \bar f -\rho_b (b+c) \\<br />
a(1-\frac{\alpha}{k}) & = & b(1+\rho_b\frac{\alpha}{k}) +2c \\<br />
a + b+ c\frac{\rho_b \alpha}{k} & = & k\bar f (\frac{1}{\rho_b}-1)<br />
\end{array}<br />
\right . <br />
</math><br />
Additionaly, from the definition of the flame path <math> F </math>, we have the kinematic relation <br />
<math> u_u = a = \alpha \bar f </math>.<br />
<br />
The above set of equations forms a system of four unknowns <math> a,\; b,\; c,\; \alpha/k </math> <br />
for a given (but unknown) shape information on the flame bump <math> k\bar f</math>. <br />
Solving for the growth rate gives:<br />
:<math><br />
\left (\frac{\alpha}{k} - \frac{1}{\rho_b}\right )\left(\frac{\alpha}{k} + 2 + \rho_b\frac{\alpha}{k} -\left (\frac{1}{\rho_b}-1\right )\left (\frac{\alpha}{k}\right )^{-1} \right )=0<br />
</math><br />
This third-degree polynom has three solutions, namely the dispersion relation found by Darrieus, a stable mode, and the trivial solution (with no physical meaning), respectively:<br />
<math><br />
\frac{\alpha}{k} = \left ( \frac{1}{\rho_b + 1}\left (\sqrt{1+\frac{1-\rho_b^2}{\rho_b}}-1 \right ),\qquad \frac{1}{\rho_b + 1}\left (-\sqrt{1+\frac{1-\rho_b^2}{\rho_b}}-1 \right ),\qquad \frac{1}{\rho_b}\right )<br />
</math><br />
<br />
'''Remark 1''' By considering the shape of the eigenvectors giving the perturbation, one recovers that, upstream of the flame, the flow is potential (the rotational of the velocity vector is null and we have chosen a constant density flow, that is barotropic and divergence free), while downstream, additionally to another <br />
perturbation of the potential type, one finds a vorticity mode (the mode with the eigenvalue <math>-\rho_b\alpha </math>. The drift of vorticity at the crossing of the flame front is a known property of flames. <br />
<br />
'''Remark 2''' In more conventional literature, the strange trivial solution for the instability growth rate is swept under the rug. Given the pedagogical nature of this electronic documentation, we can dig a little bit as the appearance of this trivial, fool solution is a good example of some modelling issue. It is important to realize that a mathematical model and the physics it describes belong to different realities. Hence, the mathematical model will generate all the solutions that the mathematics can reach in its own space. Some of them are still connected to the physics. Some others, like this trivial solution, belongs only to the mathematical solution space, an indirect way of pointing out a model limitation. The model under scope here is the hydrodynamic limit. In this model, the domain is divided into two subdomains, one upstream of the flame interface, one downstream, that are put in relation with each other through a limited number of jump conditions. In reality, the physics connects these both domains much more tightly. The curious reader will have observed that the trivial solution makes the equation system for <math> a,\; b,\; c </math> (i.e. for a fixed growth rate) undetermined, and so are the jump conditions. By generating this trivial solution, the mathematics decouples both domains. The information coming from upstream to downstream (thanks to the shape of the linearized Euler system), a solution is found only for the upstream domain, the downstream being not solved and remains undetermined in lack of information. This solution cannot happen in reality because the fresh and burned gases in a real system are connected by many aspects, and not only by the jump conditions.<br />
<br />
== The Partially-Premixed Regime ==<br />
[[Image:ppf.jpg|thumb| Ideal sketch of a partially-premixed flame]]<br />
This regime is somewhat less academics and has been recognized two decades ago. It is acknowledged as a hybrid of the premixed and the non-premixed regimes but the degree of interaction of these two modes of combustion to accurately describe a partially-premixed flame is still not well understood. It can be simply pictured by a lifted diffusion flame. Let us consider fuel issuing from a nozzle into the air. If the exit velocity is large enough, for some fuels, the flame lifts off the rim of the nozzle. It means that below the flame base, fuel and oxidizer have room to premix. Hence, the flame base propagates into a premixed mixture. However, it cannot be reduced to a premixed flame (although it is often simplified as this): (i) the mixing is not perfect and the different parts of the flame front constituting the flame base burn in mixtures of different thermodynamical states. This provides those parts with different deflagration capabilities such that the flame base has a complex shape. Indeed, it is convex, naturally leaded by the part burning at stoichiometry, unless `exotic' feeding temperatures are used. (ii) Because the mixture is not homogeneous, transfer of species and temperature driven by diffusion occurs in a direction perpendicular to the propagation of the flame base. Because the flame front is not flat, those transfers act as a connection vehicle across the different parts of the leading front. (iii) The unburned left downstream by the sections of the leading front not burning at stoichiometry diffuse towards each other to form a diffusion flame as described above. The connection of the leading front with the trailing diffusion flame has been evidenced as complicated and the siege of transfers of species and temperature. These two last items are the state-of-the-art difficulties in understanding those flames and do not appear in the models although it has been demonstrated they have a major impact and are certainly a fundamental characteristic of partially-premixed flames.<br />
<br />
The partially-premixed flame is usually described using ''c'' and ''Z'' as introduced earlier. Because the framework is essentially non-premixed, the mixture fraction is primarily used to describe the flame. Regarding the head of the flame where partial-premixing has an impact, each part of the front is described with a local progress variable. The need of defining a local progress variable is that each section of the partially-premixed front has a different equivalence ratio leading to a different definition of ''c'':<br />
:<math> c=\frac{T-T_u}{T_b(Z)-T_u}</math><br />
<br />
= Three Turbulent-Flame Interaction Regimes =<br />
It may appear odd to try to describe here what is the reason of combustion modelling research: the interaction of the turbulence with chemistry. However, one of the first steps in building knowledge in turbulent combustion was the qualitative exploration of what might be the dynamics of a flame in a turbulent environment. This led to what is now known as ''combustion diagrams''. As explained above, the premixed regime lends itself the easiest to such an approach as it exhibits natural intrinsic quantities which are not as objectively identifiable in the other combustion regimes. Note that these quantities may depend<br />
on the geometry of the flame: for instance turbulence can bend a flame sheet, leading to a change in its <br />
dynamics compared to the flat flame propagating in a medium at rest. In this section, the turbulence-flame interaction modes will be described for a premixed flame. Only remarks will be added regarding the non-premixed and partially-premixed regimes.<br />
<br />
An integral quantity to assess the interaction between a premixed flame sheet and the turbulence<br />
is the Karlovitz number ''Ka''. It compares the characteristic time of flame displacement with the characteristic time of the smallest structures (that are also the fastest) of the turbulence.<br />
:<math> Ka= \frac{\tau_c}{\tau_k}</math><br />
<br />
<math>\tau_c</math> is the chemical time of the flame. To estimate it, it is necessary to come back to the above progress variable transport equation in a steady-state framework.<br />
:<math> \overbrace{(\rho S_L)}^{\dot M}\frac{\vec S_L}{S_L}\vec\nabla c = (\rho D)_f\nabla\cdot (\rho D)^* \vec\nabla c + \dot\omega_c</math><br />
The premixed wave propagates at a speed <math>S_L</math> because it is fed by reactants diffusing inside the combustion zone and which are preheated because temperature diffuses in the reverse direction. The speed at which the flame progresses is thus related to the rate of species diffusion into the reaction zone which are then consumed. As the premixed flame is a free propagating wave whose speed of propagation is only limited by the chemical strength, the characteristic chemical time is based only on the diffusion and the mass flow rate experienced by the flame:<br />
:<math> \tau_c = \frac{\rho (\rho D)_f}{\dot M^2}</math><br />
<br />
The smallest eddies are the ones being dissipated by the viscous forces. Their characteristic time is estimated thanks to a combination of the viscosity and the flux of turbulent energy to be dissipated (also called turbulent dissipation <math> \varepsilon=u'^3/l_t </math>):<br />
:<math>\tau_k = \sqrt{\frac{l_t\nu}{u'^3}}</math><br />
<br />
Thanks to those definitions of chemical and small structure times, it is possible to give another definition of the Karlovitz number:<br />
:<math> Ka=\left (\frac{\delta_L}{l_k} \right)^2</math> <br />
which is the square of the ratio between the premixed flame thickness and the small structure scale: ''Ka'' actually compares scales. To arrive to this latter result, the three following assumptions must be used: (i) the flame thickness is obtained thanks to the premixed flame P&eacute;clet number (''vid. sup.''); (ii) the turbulence small structure (''Kolmogorov eddies'') scale is given by: <math> l_k=(\nu^3/\varepsilon)^{1/4} </math> following the same dimensional argument as for the estimation of its time; and (iii) scalar diffusion scales with viscosity. <br />
<br />
==== Remark Regarding the Diffusion Flame ====<br />
From what has been presented above, a diffusion flame does not have characteristic scales. Setting a turbulence combustion regime classification for non-premixed flames has still not been answered by research. Some laws of behaviour will only be drawn around the scalar dissipation rate which is the parameter of integral importance for a diffusion flame.<br />
<br />
Indeed, the dynamics of a diffusion flame is determined by the strain rate imposed by the turbulence. As for the premixed flames, the shortest eddies (Kolmogorov) are the ones having the largest impact. The diffusive layer is thus given by the size of the Kolmogrov eddies: <math> l_d\approx l_k</math> and the typical diffusion time scale (feeding rate of the reaction zone) is given by the characteristic time of the Kolmogorov eddies: <math> \tau_k^{-1}\approx \chi_s</math> as the Reynolds number of the Kolmogorov structures is unity. Here, <math>\chi_s</math> is the sample-averaging of <math>\chi</math> based on (conditioned) stoichiometric conditions, where the flame is expected to be.<br />
<br />
== The Wrinkled Regime ==<br />
[[Image:wrinkled.jpg|thumb| Wrinkled flamelet regime]]<br />
This regime is also called the ''flamelet regime''. Basically, it assumes that the flame structure is not affected by turbulence. The flame sheet is convoluted and wrinkled by eddies but none of them is small enough to enter it.<br />
Locally magnifying, the laminar flame structure is maintained.<br />
<br />
This regime exists for a Karlovitz number below unity (vid. sup.), i.e. chemical time smaller than the small structure time or flame thickness smaller than small structure scale. Notwithstanding, the laminar flame dynamics can be disrupted for <math> u'>S_L</math>. In that case, although the flame structure is not altered by the small structures, it can be convected by large structures such that areas of different locations in the front interact. It shows that, even with a small Karlovitz number, the turbulence effect is not always weak.<br />
<br />
== The Corrugated Regime ==<br />
[[Image:Corrugated.jpg|thumb|Corrugated flamelet regime]]<br />
The formal definition of a flame is the region of temperature rise. However, the volume where the reaction takes place is about one order of magnitude smaller, embedded inside the temperature rise region and close to its high temperature end. Hence, there exist some levels of turbulence creating eddies able to enter the flame zone but still large enough to not affect the internal reaction sheet. In other words, the flame thermal region is thickened by turbulence but the reaction zone is still in the wrinkled regime.<br />
This situation is called the ''Corrugated Regime''.<br />
<br />
Due to the structure of the Karlovitz number, once written in terms of length scales (vid. sup.), this situation arises for an<br />
increase of the Karlovitz number by two orders of magnitude compared to the value for the wrinkled regime. Hence, in the range <br />
<math> 1 < Ka < 100 </math>, the laminar structure of the reaction zone is still preserved but not the one of the preheat zone.<br />
<br />
== The Thickened Regime ==<br />
[[Image:thickened.jpg|thumb|Thickened regime]]<br />
In this last case, turbulence is intense enough to generate eddies able to affect the structure of the reaction zone as well. In practice, it is expected that those eddies are in the tail of the energy spectrum such that their lifetime is very short. Their impact on the reaction zone is thus limited.<br />
<br />
Obviously, ''Ka > 100''. A topological description is of little relevance here and a ''well-stirred reactor model'' fits better.<br />
<br />
== Reaction mechanisms ==<br />
<br />
Combustion is mainly a chemical process. Although we can, to some extent, <br />
describe a flame without any chemistry information, modelling of the flame <br />
propagation requires the knowledge of speeds of reactions, product concentrations, <br />
temperature, and other parameters. <br />
Therefore fundamental information about reaction kinetics is <br />
essential for any combustion model. <br />
A fuel-oxidizer mixture will generally combust if the reaction is fast enough to <br />
prevail until all of the mixture is burned into products. If the reaction <br />
is too slow, the flame will extinguish. If too fast, explosion or even <br />
detonation will occur. The reaction rate of a typical combustion reaction <br />
is influenced mainly by the concentration of the reactants, temperature, and pressure. <br />
<br />
A stoichiometric equation for an arbitrary reaction can be written as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
\sum_{j=1}^{n}\nu' (M_j) = \sum_{j=1}^{n}\nu'' (M_j),<br />
</math></td></tr></table><br />
<br />
where <math> \nu </math> denotes the stoichiometric coefficient, and <math>M_j</math> stands for an arbitrary species. A one prime holds for the reactants while a double prime holds for the products of the reaction. <br />
<br />
The reaction rate, expressing the rate of disappearance of reactant <b>i</b>, is defined as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
RR_i = k \, \prod_{j=1}^{n}(M_j)^{\nu'},<br />
</math></td></tr></table><br />
<br />
in which <b>k</b> is the specific reaction rate constant. Arrhenius found that this constant is a function of temperature only and is defined as:<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
k= A T^{\beta} \, exp \left( \frac{-E}{RT}\right)<br />
</math></td></table><br />
<br />
where <b>A</b> is pre-exponential factor, <b>E</b> is the activation energy, and <math>\beta</math> is a temperature exponent. The constants vary from one reaction to another and can be found in the literature.<br />
<br />
Reaction mechanisms can be deduced from experiments (for every resolved reaction), they can also be constructed numerically by the '''automatic generation method''' (see [Griffiths (1994)] for a review on reaction mechanisms).<br />
For simple hydrocarbons, tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms, global reactions <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing equations for chemically reacting flows ==<br />
<br />
Together with the usual Navier-Stokes equations for compresible flow (See [[Governing equations]]), additional equations are needed in reacting flows.<br />
<br />
The transport equation for the mass fraction <math> Y_k </math> of <i>k-th</i> species is<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Y_k \right) +<br />
\frac{\partial}{\partial x_j} \left( \rho u_j Y_k\right) = <br />
\frac{\partial}{\partial x_j} \left( \rho D_k \frac{\partial Y_k}{\partial x_j}\right)+ \dot \omega_k<br />
</math><br />
<br />
<br />
where Ficks' law is assumed for scalar diffusion with <math> D_k </math> denoting the species difussion coefficient, and <math> \dot \omega_k </math> denoting the species reaction rate.<br />
<br />
A non-reactive (passive) scalar (like the mixture fraction <math> Z </math>) is goverened by the following transport equation<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Z \right) +<br />
\frac{\partial}{\partial x_j} \left( \rho u_j Z \right) = <br />
\frac{\partial}{\partial x_j} \left( \rho D \frac{\partial Z}{\partial x_j}\right)<br />
</math><br />
<br />
where <math> D </math> is the diffusion coefficient of the passive scalar.<br />
<br />
<br />
=== RANS equations ===<br />
<br />
In turbulent flows, [[Favre averaging]] is often used to reduce the scales (see [[Reynolds averaging]]) and the<br />
mass fraction transport equation is transformed to<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } -<br />
\overline{\rho} \widetilde{u''_i Y''_k } \right)<br />
+ \overline{\dot \omega_k}<br />
</math><br />
<br />
<br />
where the turbulent fluxes <math> \widetilde{u''_i Y''_k} </math> and reaction terms<br />
<math> \overline{\dot \omega_k} </math> need to be closed.<br />
<br />
The passive scalar turbulent transport equation is<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Z} }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z} }{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D \frac{\partial Z} {\partial x_j} } -<br />
\overline{\rho} \widetilde{u''_i Z'' } \right)<br />
</math><br />
<br />
where <math> \widetilde{u''_i Z''} </math> needs modelling. A common practice is to model the turbulent fluxes using the <br />
gradient diffusion hypothesis. For example, in the equation above the flux <math> \widetilde{u''_i Z''} </math> is modelled as<br />
<br />
:<math><br />
\widetilde{u''_i Z''} = -D_t \frac{\partial \tilde Z}{\partial x_i}<br />
</math> <br />
<br />
where <math> D_t </math> is the turbulent diffusivity. Since <math> D_t >> D </math>, the first term inside the parentheses on the right hand of the mixture fraction transport equation is often neglected (Peters (2000)). This assumption is also used below.<br />
<br />
In addition to the mean passive scalar equation,<br />
an equation for the Favre variance <math> \widetilde{Z''^2}</math> is often employed<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Z''^2} }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z''^2} }{\partial x_j}=<br />
\frac{\partial}{\partial x_j}<br />
\left( \overline{\rho} \widetilde{u''_i Z''^2} \right) -<br />
2 \overline{\rho} \widetilde{u''_i Z'' }<br />
- \overline{\rho} \widetilde{\chi}<br />
</math><br />
<br />
where <math> \widetilde{\chi} </math> is the mean [[Scalar dissipation]] rate<br />
defined as <math> \widetilde{\chi} =<br />
2 D \widetilde{\left| \frac{\partial Z''}{\partial x_j} \right|^2 } </math><br />
This term and the variance diffusion fluxes needs to be modelled.<br />
<br />
=== LES equations ===<br />
The [[Large eddy simulation (LES)]] approach for reactive flows introduces equations for the filtered species mass fractions within the compressible flow field.<br />
Similar to the [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]], the filtered mass fraction transport equation is<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}=<br />
\frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } -<br />
J_j \right)<br />
+ \overline{\dot \omega_k}<br />
</math><br />
<br />
where <math> J_j </math> is the transport of subgrid fluctuations of mass fraction<br />
:<math><br />
J_j = \widetilde{u_jY_k} - \widetilde{u}_j \widetilde{Y}_k<br />
</math><br />
and has to be modelled.<br />
<br />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
are assumed to be much smaller than the apparent turbulent diffusion due to transport of subgrid fluctuations.<br />
The first term on the right hand side is then<br />
:<math><br />
\frac{\partial} {\partial x_j} <br />
\left(<br />
\overline{ \rho D_k \frac{\partial Y_k} {\partial x_j} }<br />
\right)<br />
\approx<br />
\frac{\partial} {\partial x_j} <br />
\left(<br />
\overline{\rho} D_k \frac{\partial \widetilde{Y}_k} {\partial x_j} <br />
\right)<br />
</math><br />
<br />
[[Category:Equations]]<br />
<br />
== Infinitely fast chemistry ==<br />
All combustion models can be divided into two main groups according to the <br />
assumptions on the reaction kinetics.<br />
We can either assume the reactions to be infinitely fast - compared to <br />
e.g. mixing of the species, or comparable to the time scale of the mixing <br />
process. The simple approach is to assume infinitely fast chemistry. Historically, mixing of the species is the older approach, and it is still in wide use today. It is therefore simpler to solve for [[#Finite rate chemistry]] models, at the overshoot of introducing errors to the solution.<br />
<br />
=== Premixed combustion ===<br />
Premixed flames occur when the fuel and oxidiser are homogeneously mixed prior to ignition. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical examples of premixed laminar flames is bunsen burner, where <br />
the air enters the fuel stream and the mixture burns in the wake of the <br />
riser tube walls forming nice stable flame. Another example of a premixed system is the solid rocket motor where oxidizer and fuel and properly mixed in a gum-like matrix that is uniformly distributed on the periphery of the chamber.<br />
Premixed flames have many advantages in terms of control of temperature and <br />
products and pollution concentration. However, they introduce some dangers like the autoignition (in the supply system).<br />
<br />
[[Image:Premixed.jpg]]<br />
==== Turbulent flame speed model ====<br />
<br />
==== Eddy Break-Up model ====<br />
<br />
The Eddy Break-Up model is the typical example of '''mixed-is-burnt''' combustion model. <br />
It is based on the work of Magnussen, Hjertager, and Spalding and can be found in all commercial CFD packages. <br />
The model assumes that the reactions are completed at the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. Combustion is then described by a single step global chemical reaction<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
F + \nu_s O \rightarrow (1+\nu_s) P<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
in which <b>F</b> stands for fuel, <b>O</b> for oxidiser, and <b>P</b> for products of the reaction. Alternativelly we can have a multistep scheme, where each reaction has its own mean reaction rate.<br />
The mean reaction rate is given by<br />
<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
\bar{\dot\omega}_F=A_{EB} \frac{\epsilon}{k} <br />
min\left[\bar{C}_F,\frac{\bar{C}_O}{\nu},<br />
B_{EB}\frac{\bar{C}_P}{(1+\nu)}\right]<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
<br />
where <math>\bar{C}</math> denotes the mean concentrations of fuel, oxidiser, and products <br />
respectively. <b>A</b> and <b>B</b> are model constants with typical values of 0.5 <br />
and 4.0 respectively. The values of these constants are fitted according <br />
to experimental results and they are suitable for most cases of general interest. It is important to note that these constants are only based on experimental fitting and they need not be suitable for <b>all</b> the situations. Care must be taken especially in highly strained regions, where the ratio of <math>k</math> to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In these, regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually have some remedies to overcome this problem.<br />
<br />
This model largely over-predicts temperatures and concentrations of species like <i>CO</i> and other species. However, the Eddy Break-Up model enjoys a popularity for its simplicity, steady convergence, and implementation.<br />
<br />
==== Bray-Moss-Libby model ====<br />
<br />
=== Non-premixed combustion ===<br />
<br />
Non premixed combustion is a special class of combustion where fuel and oxidizer<br />
enter separately into the combustion chamber. The diffusion and mixing of the two streams<br />
must bring the reactants together for the reaction to occur.<br />
Mixing becomes the key characteristic for diffusion flames.<br />
Diffusion burners are easier and safer to operate than premixed burners.<br />
However their efficiency is reduced compared to premixed burners.<br />
One of the major theoretical tools in non-premixed combustion<br />
is the passive scalar [[mixture fraction]] <math> Z </math> which is the<br />
backbone on most of the numerical methods in non-premixed combustion.<br />
<br />
==== Conserved scalar equilibrium models ====<br />
The reactive problem is split into two parts. First, the problem of <i> mixing </i>, which consists of the location of the flame surface which is a non-reactive problem concerning the propagation of a passive scalar; And second, the <i> flame structure </i> problem, which deals with the distribution of the reactive species inside the flamelet.<br />
<br />
To obtain the distribution inside the flame front we assume it is locally one-dimensional and<br />
depends only on time and the scalar coodinate.<br />
<br />
We first make use of the following chain rules<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}<br />
</math><br />
<br><br />
:<math><br />
\frac{\partial Y_k}{\partial x_j} = \frac{\partial Z}{\partial x_j}\frac{\partial Y_k}{\partial Z}<br />
</math> <br />
and transformation <br />
:<math><br />
\frac{\partial }{\partial t}= \frac{\partial }{\partial t} + \frac{\partial Z}{\partial t} \frac{\partial }{\partial Z}<br />
</math><br />
<br />
upon substitution into the species transport equation (see [[#Governing Equations for Reacting Flows]]), we obtain<br />
<br />
:<math><br />
\rho \frac{\partial Y_k}{\partial t} + Y_k \left[<br />
\frac{\partial \rho}{\partial t} + \frac{\partial \rho u_j}{\partial x_j}<br />
\right]<br />
+ \frac{\partial Y_k}{\partial Z} \left[<br />
\rho \frac{\partial Z}{\partial t} + \rho u_j \frac{\partial Z}{\partial x_j} -<br />
\frac{\partial}{\partial x_j}\left( \rho D \frac{\partial Z}{\partial x_j} \right)<br />
\right]<br />
=<br />
\rho D \left( \frac{\partial Z}{\partial x_j} \frac{\partial Z}{\partial x_j} \right)<br />
\frac{\partial^2 Y_k}{\partial Z^2} + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third terms in the LHS cancel out due to continuity and mixture fraction transport.<br />
At the outset, the equation boils down to<br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k<br />
</math><br />
<br />
where <math> \chi = 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2 </math> is called the <b>scalar dissipation</b><br />
which controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though the field <math> Z </math> still depends on it.<br />
<br />
:<math><br />
\dot \omega_k= -\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2}<br />
</math><br />
<br />
If the reaction is assumed to be infinetly fast, the resultant flame distribution is in equilibrium.<br />
and <math> \dot \omega_k= 0</math>. When the flame is in equilibrium, the flame configuration <math> Y_k(Z) </math> is independent of strain.<br />
<br />
===== Burke-Schumann flame structure =====<br />
The Burke-Schuman solution is valid for irreversible infinitely fast chemistry.<br />
With a reaction in the form of<br />
<table width="100%"><br />
<tr><td><br />
:<math><br />
F + \nu_s O \rightarrow (1+\nu_s) P<br />
</math></td><td width="5%"></td></tr></table><br />
<br />
If the flame is in equilibrium and therefore the reaction term is 0.<br />
Two possible solution exists, one with pure mixing (no reaction)<br />
and a linear dependence of the species mass fraction with <math> Z </math>.<br />
Fuel mass fraction<br />
:<math><br />
Y_F=Y_F^0 Z<br />
</math><br />
Oxidizer mass fraction<br />
:<math><br />
Y_O=Y_O^0(1-Z)<br />
</math><br />
Where <math> Y_F^0 </math> and <math> Y_O^0 </math> are fuel and oxidizer mass fractions<br />
in the pure fuel and oxidizer streams respectively.<br />
<br />
<br />
The other solution is given by a discontinuous slope at stoichiometric mixture fraction<br />
<math> Z_{st}</math> and two linear profiles (in the rich and lean side) at either<br />
side of the stoichiometric mixture fraction.<br />
Both concentrations must be 0 at stoichiometric, the reactants become products infinitely fast.<br />
<br />
:<math><br />
Y_F=Y_F^0 \frac{Z-Z_{st}}{1-Z_{st}}<br />
</math><br />
and oxidizer mass fraction<br />
:<math><br />
Y_O=Y_O^0 \frac{Z-Z_{st}}{Z_{st}}<br />
</math><br />
<br />
== Finite rate chemistry ==<br />
<br />
=== Premixed combustion ===<br />
<br />
==== Coherent flame model ====<br />
<br />
==== Flamelets based on G-equation ====<br />
==== Flame surface density model ====<br />
<br />
=== Non-premixed combustion ===<br />
<br />
==== Flamelets based on conserved scalar ====<br />
<br />
Peters (2000) define Flamelets as "thin diffusion layers embedded in a turbulent non-reactive flow field".<br />
If the chemistry is fast enough, the chemistry is active within a thin region<br />
where the chemistry conditions are in (or close to) stoichiometric conditions, the "flame" surface.<br />
This thin region is assumed to be smaller than Kolmogorov length scale and therefore the<br />
region is locally laminar. The flame surface is defined as an iso-surface of a certain scalar <math> Z </math>,<br />
mixture fraction in [[#Non premixed combustion]].<br />
<br />
The same equation used in [[#Conserved scalar models]] for equilibrium chemistry<br />
is used here but with chemical source term different from 0<br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} =<br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k.<br />
</math><br />
<br />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that<br />
flamelet profiles <math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in a dtaset or file which is called a "flamelet library" with all the required complex chemistry.<br />
For the generation of such libraries ready to use software is avalable such as Softpredict's Combustion Simulation Laboratory COSILAB [http://www.softpredict.com] with its relevant solver RUN1DL, which can be used for a variety of relevant geometries; see various [http://www.softpredict.com/?page=989 publications that are available for download]. Other software tools are available such as CHEMKIN [http://www.reactiondesign.com] and<br />
CANTERA [http://www.cantera.org].<br />
<br />
===== Flamelets in turbulent combustion =====<br />
<br />
In turbulent flames the interest is <math> \widetilde{Y}_k </math>.<br />
In flamelets, the flame thickness is assumed to be much smaller than Kolmogorov scale<br />
and obviously is much smaller than the grid size.<br />
It is therefore needed a distribution of the passive scalar within the cell.<br />
<math> \widetilde{Y}_k </math> cannot be obtained directly from the flamelets library<br />
<math> \widetilde{Y}_k \neq Y_F(Z,\chi) </math>, where <math> Y_F(Z,\chi) </math> corresponds<br />
to the value obtained from the flamelets libraries.<br />
A generic solution can be expressed as<br />
:<math><br />
\widetilde{Y}_k= \int Y_F( \widetilde{Z},\widetilde{\chi}) P(Z,\chi) dZ d\chi<br />
</math><br />
where <math> P(Z,\chi) </math> is the joint [[Probability density function | Probability Density Function]] (PDF) of the mixture fraction<br />
and scalar dissipation which account for the scalar distribution inside the cell and "a priori"<br />
depends on time and space.<br />
<br />
The most simple assumption is to use a constant distribution of the scalar dissipation within the cell and<br />
the above equation reduces to<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) dZ<br />
</math><br />
<br />
<math> P(Z) </math> is the PDF of the mixture fraction scalar and simple models (such as Gaussian or a [[beta PDF]])<br />
can be build depending only on two moments of the scalar<br />
mean and variance,<math> \widetilde{Z},Z''</math>.<br />
<br />
If the mixture fraction and scalar dissipation are consider independent variables,<math> P(Z,\chi) </math><br />
can be written as <math> P(Z) P(\chi)</math>. The PDF of the scalar dissipation is assumed to be log-normal with<br />
variance unity.<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) P(\chi) dZ d\chi<br />
</math><br />
In [[Large eddy simulation (LES)]] context (see [[#LES equations]] for reacting flow),<br />
the [[probability density function]] is replaced by a [[subgrid PDF]] <math> \widetilde{P}</math>.<br />
The same equation hold by replacing averaged values with filtered values.<br />
:<math><br />
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) \widetilde{P}(Z) \widetilde{P}(\chi) dZ d\chi<br />
</math><br />
The assumptions made regarding the shapes of the PDFs are still justified. In LES combustion the<br />
[[subgrid variance]] is smaller than RANS counterpart (part of the large-scale fluctuations are solved)<br />
and therefore the modelled PDFs are thinner.<br />
<br />
====== Unsteady flamelets ======<br />
====Intrinsic Low Dimensional Manifolds (ILDM)====<br />
<br />
Detailed mechanisms describing ignition, flame propagation and pollutant formation typically involve several hundred species and elementary reactions, prohibiting their use in practical three-dimensional engine simulations. Conventionally reduced mechanisms often fail to predict minor radicals and pollutant precursors. The ILDM-method is an automatic reduction of a detailed mechanism, which assumes local equilibrium with respect to the fastest time scales identified by a local eigenvector analysis. In the reactive flow calculation, the species compositions are constrained to these manifolds. Conservation equations are solved for only a small number of reaction progress variables, thereby retaining the accuracy of detailed chemical mechanisms. This gives an effective way of coupling the details of complex chemistry with the time variations due to turbulence.<br />
<br />
The intrinsic low-dimensional manifold (ILDM) method {Maas:1992,Maas:1993} is a method for ''in-situ'' reduction of a detailed chemical mechanism based on a local time scale analysis. This method is based on the fact that different trajectories in state space start from a high-dimensional point and quickly relax to lower-dimensional manifolds due to the fast reactions. The movement along these lower-dimensional manifolds, however, is governed by the slow reactions. It exploits the variety of time scales to systematically reduce the detailed mechanism. For a detailed chemical mechanism with N species, N different time scales govern the process. An assumption that all the time scales are relaxed results in assuming complete equilibrium, where the only variables required to describe the system are the mixture fraction, the temperature and the pressure. This results in a zero-dimensional manifold. An assumption that all but the slowest 'n' time scales are relaxed results in a 'n' dimensional manifold, which requires the additional specification of 'n' parameters (called progress variables). In the ILDM method, the fast chemical reactions do not need to be identified a priori. An eigenvalue analysis of the detailed chemical mechanism is carried out which identifies the fast processes in dynamic equilibrium with the slow processes. The computation of ILDM points can be expensive, and hence an in-situ tabulation procedure is used, which enables the calculation of only those points that are needed during the CFD calculation.<br />
<br />
--[[User:Fredgauss|Fredgauss]] 07:37, 25 August 2006 (MDT)<br />
<br />
U. Maas, S.B. Pope. Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Comb. Flame 88,<br />
239, 1992.<br />
<br />
Ulrich Maas. Automatische Reduktion von Reaktionsmechanismen zur Simulation reaktiver Str¨omungen. Habilitationsschrift, Universit<br />
¨at Stuttgart, 1993.<br />
<br />
==== Conditional Moment Closure (CMC)====<br />
In Conditional Moment Closure (CMC) methods we assume that the species mass fractions are all correlated with<br />
the mixture fraction (in non premixed combustion).<br />
<br />
From [[Probability density function]] we have<br />
:<math><br />
\overline{Y_k}= \int <Y_k|\eta> P(\eta) d\eta<br />
</math><br />
where <math> \eta </math> is the sample space for <math> Z </math>.<br />
<br />
CMC consists of providing a set of transport equations for the conditional moments which define the<br />
flame structure.<br />
<br />
Experimentally, it has been observed that temperature and chemical radicals are<br />
strong non-linear functions of mixture fraction. For a given species mass fraction we can decomposed<br />
it into a mean and a fluctuation:<br />
:<math><br />
Y_k= \overline{Y_k} + Y'_k<br />
</math><br />
The fluctuations <math> Y_k' </math> are usually very strong in time and space which makes the closure<br />
of <math> \overline{\omega_k} </math> very difficult.<br />
However, the alternative decomposition<br />
:<math><br />
Y_k= <Y_k|\eta> + y'_k<br />
</math><br />
where <math> y'_k </math> is the fluctuation around the conditional mean or the "conditional fluctuation".<br />
Experimentally, it is observed that <math> y'_k<< Y'_k </math>, which forms the basic assumption of the CMC method.<br />
Closures. Due to this property better closure methods can be used reducing the non-linearity <br />
of the mass fraction equations.<br />
<br />
The [[Derivation of the CMC equations]] produces the following CMC transport equation<br />
where <math> Q \equiv <Y_k|\eta> </math> for simplicity.<br />
:<math><br />
\frac{ \partial Q}{\partial t} + <u_j|\eta> \frac{\partial Q}{\partial x_j} =<br />
\frac{<\chi|\eta> }{2} \frac{\partial ^2 Q}{\partial \eta^2} + <br />
\frac{ < \dot \omega_k|\eta> }{ <\rho| \eta >}<br />
</math><br />
<br />
In this equation, high order terms in Reynolds number have been neglected.<br />
(See [[Derivation of the CMC equations]] for the complete series of terms).<br />
<br />
It is well known that closure of the unconditional source term<br />
<math> \overline {\dot \omega_k} </math> as a function of the<br />
mean temperature and species (<math> \overline{Y}, \overline{T}</math>) will give rise to large errors.<br />
However, in CMC the conditional averaged mass fractions contain more information and fluctuations around the mean are much smaller.<br />
The first order closure<br />
<math><br />
< \dot \omega_k|\eta> \approx \dot \omega_k \left( Q, <T|\eta> \right)<br />
</math><br />
is a good approximation in zones which are not close to extinction.<br />
<br />
===== Second order closure =====<br />
<br />
A second order closure can be obtained if conditional fluctuations are taken into account.<br />
For a chemical source term in the form <math> \dot \omega_k = k Y_A Y_B </math> with the rate constant in Arrhenius form<br />
<math> k=A_0 T^\beta exp [-Ta/T] </math><br />
the second order closure is (Klimenko and Bilger 1999)<br />
:<math><br />
< \dot \omega_k|\eta> \approx < \dot \omega_k|\eta >^{FO}<br />
<br />
\left[1+ \frac{< Y''_A Y''_B |\eta>}{Q_A Q_B}+ \left( \beta + T_a/Q_T \right)<br />
\left(<br />
\frac{< Y''_A T'' |\eta>}{Q_AQ_T} + \frac{< Y''_B T'' |\eta>}{Q_BQ_T}<br />
\right) + ...<br />
<br />
\right]<br />
<br />
</math><br />
<br />
where <math> < \dot \omega_k|\eta >^{FO} </math> is the first order CMC closure and<br />
<math> Q_T \equiv <T|\eta> </math>.<br />
When the temperature exponent <math> \beta </math> or <math> T_a/Q_T </math><br />
are large the error of taking the first order approximation increases.<br />
Improvement of small pollutant predictions can be obtained using the above reaction <br />
rate for selected species like CO and NO.<br />
<br />
<br />
===== Double conditioning =====<br />
<br />
Close to extinction and reignition. The conditional fluctuations can be very large<br />
and the primary closure of CMC of "small" fluctuations is not longer valid.<br />
A second variable <math> h </math> can be chosen to define a double conditioned mass fraction<br />
:<math><br />
Q(x,t;\eta,\psi) \equiv <Y_i(x,t) |Z=\eta,h=\psi ><br />
</math><br />
Due to the strong dependence on chemical reactions to temperature, <math> h </math><br />
is advised to be a temperature related variable (Kronenburg 2004).<br />
Scalar dissipation is not a good choice, due to its log-normal behaviour<br />
(smaller scales give highest dissipation). A must better choice is the sensible enthalpy<br />
or a progress variable.<br />
Double conditional variables have much smaller conditional fluctuations and allow<br />
the existence of points with the same chemical composition which can be fully burning<br />
(high temperature) or just mixing (low temperature). <br />
The range of applicability is greatly increased and allows non-premixed and premixed problems<br />
to be treated without ad-hoc distinctions.<br />
The main problem is the closure of the new terms involving cross scalar transport.<br />
<br />
The double conditional CMC equation is obtained in a similar manner than the conventional<br />
CMC equations<br />
<br />
===== LES modelling =====<br />
In a LES context a [[conditional filtering]] operator can be defined<br />
and <math> Q </math> therefore represents a conditionally filtered reactive scalar.<br />
<br />
=== Linear Eddy Model ===<br />
The Linear Eddy Model (LEM) was first developed by Kerstein(1988).<br />
It is an one-dimensional model for representing the flame structure in turbulent flows.<br />
<br />
In every computational cell a molecular, diffusion and chemical model is defined as<br />
<br />
:<math><br />
\frac{\partial}{\partial t} \left( \rho Y_k \right) =<br />
\frac{\partial}{\partial \eta} \left( \rho D_k \frac{\partial Y_k}{\partial \eta }\right)+ \dot \omega_k<br />
</math><br />
<br />
where <math> \eta </math>is a spatial coordinate. The scalar distribution obtained can be seen as a<br />
one-dimensional reference field between Kolmogorov scale and grid scales.<br />
<br />
In a second stage a series of re-arranging stochastic event take place.<br />
These events represent the effects<br />
of a certain turbulent structure of size <math> l </math>, smaller than the grid size at a location <math> \eta_0 </math><br />
within the one-dimensional domain. This vortex distort the <math> \eta </math> field obtain by the one-dimensional equation,<br />
creating new maxima and minima in the interval <math> (\eta_0, \eta + \eta_0) </math>.<br />
The vortex size <math> l </math> is chosen randomly based on the inertial scale range while<br />
<math> \eta_0 </math> is obtained from a uniform distribution in <math> \eta </math>.<br />
The number of events is chosen to match the turbulent diffusivity of the flow.<br />
<br />
=== PDF transport models ===<br />
<br />
[[Probability Density Function]] (PDF) methods are not exclusive to combustion, although they are particularly attractive to them. They provided more information than moment closures and they are used to compute inhomegenous turbulent flows, see reviews in Dopazo (1993) and Pope (1994).<br />
<br />
<br />
PDF methods are based on the transport equation of the joint-PDF of the scalars.<br />
Denoting <math> P \equiv P(\underline{\psi}; x, t) </math> where<br />
<math> \underline{\psi} = ( \psi_1,\psi_2 ... \psi_N) </math> is the phase space for the reactive scalars<br />
<math> \underline{Y} = ( Y_1,Y_2 ... Y_N) </math>.<br />
The transport equation of the joint PDF is:<br />
<br />
:<math><br />
\frac{\partial <\rho | \underline{Y}=\underline{\psi}> P }{\partial t} + \frac{ <br />
\partial <\rho u_j | \underline{Y}=\underline{\psi}> P }{\partial x_j} =<br />
\sum^N_\alpha \frac{\partial}{\partial \psi_\alpha}\left[ \rho \dot{\omega}_\alpha P \right]<br />
- \sum^N_\alpha \sum^N_\beta \frac{\partial^2}{\partial \psi_\alpha \psi_\beta}<br />
\left[ <D \frac{\partial Y_\alpha}{\partial x_i} \frac{\partial Y_\beta}{\partial x_i} | \underline{Y}=\underline{\psi}> \right] P<br />
</math><br />
<br />
where the chemical source term is closed. Another term appeared on the right hand side which accounts for the effects of<br />
the molecular mixing on the PDF, is the so called "micro-mixing " term.<br />
Equal diffusivities are used for simplicity <math> D_k = D </math><br />
<br />
A more general approach is the velocity-composition joint-PDF<br />
with <math> P \equiv P(\underline{V},\underline{\psi}; x, t) </math>, where <br />
<math> \underline{V} </math> is the sample space of the velocity field <br />
<math> u,v,w </math>. This approach has the advantage of avoiding gradient-diffusion<br />
modelling. A similar equation to the above is obtained combining the momentum <br />
and scalar transport equation.<br />
<br />
<br />
<br />
<br />
The PDF transport equation can be solved in two ways: through a Lagrangian approach<br />
using stochastic methods or in a Eulerian ways using stochastic fields.<br />
<br />
==== Lagrangian ====<br />
<br />
The main idea of Lagrangian methods is that the flow can be represented by an <br />
ensemble of fluid particles. Central to this approach is the stochastic differential equations and in particular the [[Langevin equation]].<br />
<br />
==== Eulerian ====<br />
<br />
Instead of stochastic particles, smooth stochastic fields can be used<br />
to represent the [[probability density function ]] (PDF) of a scalar (or joint PDF) involved in transport (convection), diffusion<br />
and chemical reaction (Valino 1998). A similar formulation was proposed by Sabelnikov and Soulard 2005, which removes<br />
part of the a-priori assumption of "smoothness" of the stochastic fields.<br />
This approach is purely Eulerian and offers implementations advantages compared to<br />
Lagrangian or semi-Eulerian methods. <br />
Transport equations for scalars are often easy to programme and normal<br />
CFD algorithms can be used (see [[Discretisation of convective term]]).<br />
Although discretization errors are introduced by solving transport equations, <br />
this is partially compesated by the error introduced in Lagrangian approaches due to the numerical evaluation of means.<br />
<br />
A new set of <math> N_s </math> scalar variables<br />
(the stochastic field <math> \xi </math>) is used to represent the<br />
PDF<br />
:<math><br />
P (\underline{\psi}; x,t) = \frac{1}{N} \sum^{N_s}_{j=1} \prod^{N}_{k=1} \delta \left[\psi_k -\xi_k^j(x,t) \right]<br />
</math><br />
<br />
===Other combustion models===<br />
<br />
<br />
==== MMC ====<br />
<br />
The Multiple Mapping Conditioning (MMC) (Klimenko and Pope 2003) is an<br />
extension of the [[#Conditional Moment Closure (CMC)]] approach combined with<br />
[[probability density function]] methods. MMC<br />
looks for the minimum set of variables that describes the particular turbulent combustion<br />
system.<br />
<br />
==== Fractals ====<br />
Derived from the [[#Eddy Dissipation Concept (EDC)]].<br />
<br />
== References ==<br />
<br />
*{{reference-paper|author=Dopazo, C.|year=1993|title=Recent development in PDF methods|rest=Turbulent Reacting Flows, ed. P. A. Libby and F. A. Williams }}<br />
*{{reference-book|author=Fox, R.O.|year=2003|title=Computational Models for Turbulent Reacting Flows|rest=ISBN 0-521-65049-6,Cambridge University Press}}<br />
*{{reference-paper|author=Kerstein, A. R.|year=1988|title=A linear eddy model of turbulent scalar transport and mixing|rest=Comb. Science and Technology, Vol. 60,pp. 391}}<br />
*{{reference-paper|author=Klimenko, A. Y., Bilger, R. W.|year=1999|title=Conditional moment closure for turbulent combustion|rest=Progress in Energy and Combustion Science, Vol. 25,pp. 595-687}}<br />
*{{reference-paper|author=Klimenko, A. Y., Pope, S. B.|year=2003|title=The modeling of turbulent reactive flows based on multiple mapping conditioning|rest=Physics of Fluids, Vol. 15, Num. 7, pp. 1907-1925}}<br />
*{{reference-paper|author=Kronenburg, A.,|year=2004|title=Double conditioning of reactive scalar transport equations in turbulent non-premixed flames|rest=Physics of Fluids, Vol. 16, Num. 7, pp. 2640-2648}}<br />
*{{reference-paper|author=Griffiths, J. F.|year=1994|title=Reduced Kinetic Models and Their Application to Practical Combustion Systems |rest=Prog. in Energy and Combustion Science,Vol. 21, pp. 25-107}}<br />
*{{reference-book|author=Peters, N.|year=2000|title=Turbulent Combustion|rest=ISBN 0-521-66082-3,Cambridge University Press}}<br />
*{{reference-book|author=Poinsot, T.,Veynante, D.|year=2001|title=Theoretical and Numerical Combustion|rest=ISBN 1-930217-05-6, R. T Edwards}}<br />
*{{reference-paper|author=Pope, S. B.|year=1994|title=Lagrangian PDF methods for turbulent flows|rest=Annu. Rev. Fluid Mech, Vol. 26, pp. 23-63}}<br />
*{{reference-paper|author=Sabel'nikov, V.,Soulard, O.|year=2005|title=Rapidly decorrelating velocity-field model as a tool for solving one-point Fokker-Planck equations for probability density functions of turbulent reactive scalars|rest=Physical Review E, Vol. 72,pp. 016301-1-22}}<br />
*{{reference-paper|author=Valino, L.,|year=1998|title=A field montecarlo formulation for calculating the probability density function of a single scalar in a turbulent flow|rest=Flow. Turb and Combustion, Vol. 60,pp. 157-172}}<br />
*{{reference-paper|author=Westbrook, Ch. K., Dryer,F. L.,|year=1984|title=Chemical Kinetic Modeling of Hydrocarbon Combustion|rest=Prog. in Energy and Combustion Science,Vol. 10, pp. 1-57}}<br />
<br />
== External links and sources ==</div>DavidFhttp://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFDBest practice guidelines for turbomachinery CFD2008-09-12T09:43:30Z<p>DavidF: /* Heat transfer predictions */</p>
<hr />
<div>This article contains a summary of the most important knowledge and experience a CFD engineer needs in order to perform CFD simulations of turbomachinery components. The guide is mainly aimed at axial turbomachinery. The goal is to give a CFD engineer, who has just started working with turbomachinery simulations, a head start and avoid some of the most difficult pit-falls. Experienced turbomachinery CFD engineers can also use the guidelines in order to learn what other experts consider best practice. The intended audience is expected to know basic CFD terminology and have a basic turbomachinery knowledge, but no detailed knowledge about CFD for turbomachinery is needed. <br />
<br />
== Deciding what type of simulation to do ==<br />
<br />
Before starting a new turbomachinery simulation it is wise to think carefully of what it is that should be predicted and what physical phenomena that affect the results. This chapter contains a brief overview of the various types of simulations and some hints of what can be predicted with them.<br />
<br />
=== 2D, Quasi-3D or 3D ===<br />
<br />
2D simulations are often used in the early design phase in order to obtain a typical 2D section of a blade. For cases with many long blades or vanes, like low-pressure turbines, a 2D simulation can also provide reasonable results. If the area of the flow-path changes significantly in the axial direction it might be necessary to instead make a quasi-3D simulation. A quasi-3D simulation is a 2D simulation in which extra source terms are used to account for the acceleration/decelleration caused by a changing channel height or growing end-wall boundary layers. Codes focused on turbomachinery applications often have the possibility to perform quasi-3D simulations, but most general purpose CFD codes can not do this type of simulations, or require user coding to implement the correct source terms in the equations. Please contact your software distributor if your code does not have the quasi-3d possibility and you require it. Many codes require special routines or hidden commands to enable this feature.<br />
<br />
Full 3D simulations are necessary if a true 3D geometry is needed to obtain correct secondary flows and/or shock locations. For low-aspect-ratio cases with only a few short blades, like for example structurally loaded turbine outlet guide vanes, the secondary flow development is important and a 3D simulation is often necessary in order to obtain reasonable results. For applications where the end-wall boundary layers grow very quickly and interact with a large part of the flow-field it is necessary to perform a full 3D simulation. This is often the case in compressors and fans, where the negative pressure gradients make the boundary layers grow much quicker than what they do in for example turbines. For cases where the shock location is very critical, like in transonic compressors, it is also often necessary to perform a 3D simulation in order to obtain reasonable shock locations.<br />
<br />
=== Inviscid or viscid ===<br />
<br />
For attached flows close to the design point and without any large separations it is often sufficient with an inviscid Euler simulation in order to obtain reasonable blade loadings and pressure distributions. Note that inviscid Euler simulations should only be used if the boundary layers are judged to not have a significant effect on the global flow-field. A viscid Navier-Stokes simulation is necessary in order to predict losses, secondary flows and separations. As soon as separations are of interest it is of course also necessary to do a viscid simulation. Note that with todays computers it is often not time and resources that make users run inviscid Euler simulations. Running viscid Navier-Stokes simulations is now so quick that it is not a time problem any more. Euler simulations are still interesting though, since with an inviscid Euler simulation you don't have to worry about wall resolutions, y+ values, turbulence modeling errors etc.<br />
<br />
=== Transient or Stationary ===<br />
<br />
Most turbomachinery simlation are performed as stationary simulations. Transient simulations are done when some kind of transient flow behaviour has a strong influence on the global flow field. Examples of transient simulations are detailed simulations of rotor-stator interaction effects, simulations of large unsteady separations etc. Sometimes when you perform a steady stationary simulation you can see tendencies of unsteady behvaiour like for example periodic vortex shedding behind blunt trailing edges. This is often first seen as periodical variations of the residuals. If the unsteady tendencies are judged to not affect the overall simulation results if might be necessary to coarsen the mesh close to the vortex shedding or run a different turbulence model in order to make the simulation converge. Sometimes you are still forced to run a transient simulation and average the results if you don't obtain a converged steady solution.<br />
<br />
== Meshing ==<br />
<br />
In turbomachinery applications structured muli-block hexahedral meshes are most often used for flow-path simulations. In most solvers a structured grid requires less memory, provides superior accuracy and allows a better boundary-layer resolution than an unstructured grid. By having cells with a large aspect ratio around sharp leading and trailing edges a structured grid also provides a better resolution of these areas. Many companies have automatic meshing tools that automatically mesh blade sections with a structured mesh without much user intervention.<br />
<br />
Unstructured meshes are used for more complex and odd geometries where a structured mesh is difficult to create. Typical examples where unstructured meshes are often used are blade tip regions, areas involving leakage flows and secondary air systems, film cooling ducts etc.<br />
<br />
When meshing avoid to create large jumps in cell sizes. Typically the cell size should not change with more than a factor of 1.25 between neighbouring cells. For structured meshes also try to create fairly continous mesh lines and avoid discontinuities where the cell directions suddenly change. For multi-block structured meshes avoid placing the singular points where blocks meet in regions with strong flow gradients since most schemes have a lower accuracy in these singular points.<br />
<br />
=== Mesh size guidelines ===<br />
<br />
It is difficult to define, a priori, the mesh size. The required mesh size depends on the purpose of the simulation. <br />
<br />
If the main goal is to obtain static pressure forces a coarse mesh is often able to obtain a good solution, especially when an accurate resolution of the boundary layers is not required. For 2D inviscid simulations of one blade a mesh with say 3,000 cells is most often sufficient. For 3D inviscid blade simulations a mesh size of about 40,000 cells is usually sufficient. On inviscid Euler simulations the cells should be fairly equal in size and no boundary layer resolution should be present. Avoid having too skewed cells.<br />
<br />
For loss predictions and cases where boundary layer development and separation is important the mesh needs to have a boundary layer resolution. The boundary layer resolution can either be coarse and suitable for a [[Wall functions|wall function]] simulation or very fine and suitable for a [[Low-Re resolved boundary layers|low-Re]] simulation. For further information about selecting the near-wall turbulence model please see the [[#Turbulence modeling|turbulence modeling]] section. In 3D single-blade simulations a decent wall-function mesh typically has around 100,000 cells. This type of mesh size is suitable for quick design iterations where it is not essential to resolve all secondary flows and vortices. A good 3D wall-function mesh of a blade section intended to resolve secondary flows well should have at least 400,000 cells . A good low-Re mesh with resolved boundary layers typically has around 1,000,000 cells. <br />
<br />
In 2D blade simulations a good wall-function mesh has around 20,000 cells and a good low-Re mesh with resolved boundary layers has around 50,000 cells.<br />
<br />
Along the suction and pressure surfaces it is a good use about 100 cells in the streamwise direction. In the radial direction a good first approach is to use something like 30 cells for a wall-function mesh and 100 cells for a low-Re mesh.<br />
<br />
It is important to resolve leading and trailing edges well. Typically at least 10 cells, preferably 20 should be used around the leading and trailing edges. For very blunt and large leading edges, like those commonly found on HP turbine blades, 30 or more cells can be necessary.<br />
<br />
Cases which are difficult to converge with a steady simulation and which show tendencies of periodic vortex shedding from the trailing edge, can sometimes be "tamed" by using a coarse mesh around the trailing edge. This, of course, reduces the accuracy but can be a trick to obtain a converged solution if time and computer resouces does not allow a transient simulation to be performed.<br />
<br />
=== Boundary layer mesh ===<br />
<br />
For design iteration type of simulations where a [[Wall functions|wall function]] approach is sufficient [[Dimensionless wall distance (y plus) | y+]] for the first cell should be somewhere between 20 and 200. The outer limit is dependent on the actual Re number of the simulations. For cases with fairly low Re numbers make sure to keep the maximum y+ as low as possible. For more accurate simulations with resolved boundary layers the mesh should have a y+ for the first cell which is below 1. Some new codes are now using a hybrid wall treatment that allows a smooth transition from a coarse wall-function mesh to a resolved low-Re mesh. Use some extra care when using this type of hybrid technique since it is still fairly new and unproven.<br />
<br />
Outside of the first cell at a wall a good rule of thumb is to use a growth ratio normal to the wall in the boundary layer of maximum 1.25. For a low-Re mesh this usually gives around 40 cells in the boundary layer whereas a wall-function mesh does not require more than 10 cells in the boundary layer.<br />
<br />
If you are uncertain of which wall distance to mesh with you can use a [http://www.cfd-online.com/Links/tools.html#yplus y+ estimation tool] to estitmate the distance needed to obtain the desired y+. These estimation tools are ''very handy'' if you have not done any previous similar simulations. <br />
<br />
As a rule of thumb a wall-function mesh typically requires areound 5 to 10 cells in the boundary layer whereas a resolved low-Re mesh requires about 40 cells in the boundary layer.<br />
<br />
== Boundary conditions ==<br />
<br />
Describe different types of boundary conditions and when they should be used:<br />
<br />
* Total pressure in, static pressure out<br />
* Absorbing boundary conditions<br />
* ...<br />
<br />
There are different types of boundary conditions you can use:<br />
*Mass flow inlet, static pressure outlet.<br />
To put this BC you must allocate inlet surface enough far away from the turbomachine impeller. The velocity distribution is not<br />
constant over all the surface. If you put this "lie" away from your problem it works well.<br />
<br />
<br />
=== Turbulence inlet conditions ===<br />
<br />
Prescribing realistic turbulence inlet conditions is important and usually difficult. For two-equation turbulence models two different turbulence values need to be specified on the inlet. The most common way of specifying the inlet turbulence variables is to give an inlet [[Turbulence intensity|turbulence level]] and [[Turbulence length scale|length-scale]]. For simulations where the incoming boundary layers are estimated to not have any significant effect on the secondary flows and global flow-field downstream it is usually sufficient to specify a constant incoming turbulence level and length-scale.<br />
<br />
The incoming [[Turbulence intensity|turbulence level]] is dependent on which component is analyzed. Best is of course to have measurments from a similar application in order to be able to prescribe a fairly realistic value. Without measurements it is necessary to make an educated guess. Simulations with very complex and turbulence-generating components upstream will have a very high incoming turbulence level. For example, a high-pressure turbine just downstream of a turbulence generating combustor might have incoming turbulence levels up to 20%. Fans and low-pressure compressors with not many components upstream might have as low incoming turbulence level as 1%. For components between these two extreme examples, like high-pressure compressors, low-pressure turbines etc. a turbulence level of around 5% might be realistic. If you do not have any measurements and are unsure about which incoming turbulence level to prescribe it is always a good idea to run a few different simulations with say one half and one double incoming turbulence level in order to estimate how important it is for the simulation results.<br />
<br />
The incoming [[Turbulence length scale|turbulence length-scale]] is often even more difficult to guess than the incoming turbulence level. Measurements might sometimes include the incoming turbulence level, but very seldom also include the incoming turbulent length-scale. The best way of guessing a realistic incoming length-scale is to use the geometrical properties of the upstream components. The incoming turbulence length-scale can be estimated as say the thickness of upstream blades or somewhere between 2% and 20% of the incoming channel heigth. Fortunately the incoming turbulence length-scale is usually not that important for the end results. However, some k-omega models can have an unrelisticly strong influence on the incoming turbulence length-scale. Hence, be extra carefull about the length-scale when using k-omega models. Note that the [[SST k-omega model]] does not have this unrealistic sensitivity to the inlet turbulence length-scale.<br />
<br />
Instead of specifying an incoming turbulence length-scale it is sometimes more convenient to specify an incoming [[Eddy viscosity ratio|eddy viscosity ratio]]. For low-turbulence cases with no well-specified turbulence generating components upstream it can be difficult to estimate a turbulence length-scale. Specifying an eddy viscosity ratio instead gives a more direct control over how large the effect of the incoming turbulence is. The eddy viscosity ratio describes how large the turbulence viscosity is related to the molecular viscosity. In turbomachinery the eddy viscosity ratio can vary from say 10 in low-turbulence components like fans up to say 1,000 or in rare cases even 10,000 in high-turbulence components like high-pressure turbines. In components between these extremes a typical eddy viscosity ration might be about 200. Even if specifying an inlet turbulence length scale it can still be good to compute what eddy viscosity ratio this corresponds to in order to estimate if it sounds reasonable.<br />
<br />
== Turbulence modeling ==<br />
<br />
Selecting a suitable turbulence model for turbomachinery simulations can be a challenging task. There is no single model which is suitable for all types of simulations. Which turbulence model CFD engineers use has as much to do with beliefs and traditions as with knowledge and facts. There are many different schools. However, below follows some advices that most CFD engineers in the turbomachinery field tend to agree upon.<br />
<br />
For attached flows close to the design point a simple algebraic model like the [[Baldwin-Lomax model]] can be used. Another common choice for design-iteration type of simulations is the one-equation model by [[Spalart-Allmaras model | Spalart-Allmaras]]. This model has become more popular in the last years due to the many inherent problems in more refined two-equation models. The big advantage with both the [[Baldwin-Lomax model]] and the [[Spalart-Allmaras model]] over more advanced models is that they are very robust to use and rarely produce completely unphysical results.<br />
<br />
In order to accurately predict more difficult cases, like separating flows, rotating flows, flows strongly affected by secondary flows etc. it is often necessary to use a more refined turbulence model. Common choices are two-equation models like the <math>k-\epsilon</math> model.<br />
<br />
Two-equation models are based on the [[Boussinesq eddy viscosity assumption]] and this often leads to an over-production of turbulent energy in regions with strong acceleration or deceleration, like in the leading edge region, regions around shocks and in the suction peak on the suction side of a blade. To reduce this problems it is common to use a special model variant using, for example, [[Durbin's realizability constraint]] or the [[Kato-Launder modification]]. Note that different two-equation models behave differently in these problematic stagnation and acceleration regions. Worst is probably the standard <math>k-\epsilon</math> model. <math>k-\omega</math> model are slightly better but still do not behave well. More modern variants like Menter's [[SST k-omega model]] also has problems, wheras the [[V2-f models|v2f model ]] by Durbin behaves better.<br />
<br />
=== Near-wall treatment ===<br />
<br />
For on-design simulations without any large separated regions it is often sufficient to use a [[wall-function model]] close to the wall, preferably with some form of non-equilibrium wall-function that is sensitised to streamwise pressure gradients.<br />
<br />
For off-design simulation, or simulations involving complex secondary flows and separations, it is often necessary to use a [[low-Re model]]. There exist many low-Re models that have been used with success in turbomachinery simulations. A robust and often good choice is to use a one-equation model, like for example the [[Wolfstein model]], in the inner parts of the boundary layer. There are also several Low-Re <math>k-\epsilon</math> models that work well. Just make sure they don't suffer from the problem with overproduction of turbulent energy in regions with strong acceleration or deceleration. In the last few years Menter's low-Re <math>SST k-\omega</math> model has gained increased popularity.<br />
<br />
=== Transition prediction ===<br />
<br />
Transition refers to the process when a laminar boundary layer becomes unstable and transitions to a turbulent boundary layer. There are two types of transition - natural transition, where inherent instabilities in the boundary layer cause the transition and by-pass transition, where convection and diffusion of turbulence from the free-stream into the boundary layer cause the transition. Most transitions in turbomachinery are by-pass transitions caused by free-stream turbulence and other external disturbances like wakes, vortices and surface defects.<br />
<br />
Simulating transition in a CFD code accurately is very difficult. Often a separate transition model needs to be solved in order to specify the transition location and length. Predicting natural transition in a pure CFD code is not possible. Predicting by-pass transition in a pure CFD code is almost impossible, although there are people who claim to be able to predict by-pass transition with low-Re two-equation models. However, this is usually on special test cases and with simulations that have been tuned for these special cases, see for example [Saville 2002]. In reality transition is a very complex and sensitive process where disturbances like incoming wakes and vortices from previous stages, surface roughness effects and small steps or gaps in the surfaces play a significant role. <br />
<br />
The turbomachinery codes that have transition prediction models often use old ad-hoc models like the Abu-Ghannam and Shaw model [Abu-Ghannam 1980] or the Mayle model [Mayle 1991]. These models can be quite reliable if they have been validated and tuned for a similar application. Do not trust your transition predictions without having some form of experimental validation. Menter has also recently developed a new form of transition model that might work fairly well, but it is still too new and untested.<br />
<br />
For some turbomachinery applications, like modern high-lift low-pressure turbines, transition is critical. For these applications a CFD code with a transition model that has been tuned for this type of applications should be used.<br />
<br />
== Numerical considerations ==<br />
<br />
Use at least a second order accurate scheme for the flow variables. Some codes require a first order scheme for the turbulent variables (<math>k</math> and <math>\epsilon</math>) in order to converge well. It might be sufficient with a first order scheme only on the turbulence variables, but a second order scheme is of course preferable.<br />
<br />
=== Convergence criteria ===<br />
<br />
To know when a solution is converged is not always that easy. You need some prior experience of your CFD code and your application to judge when a simulation is converged. For normal pure aero simulations without resolved walls, i.e. with wall functions or inviscid Euler simulations, convergence can most often be estimated just by looking at the residuals. Exacty what the residuals should be is not possible to say, it all depends on how your particular code computes and scales the residuals. Hence, make sure to read the manuals and plot the convergence of a few global parameters before you decide what the residuals should be for a solution to converege. Note also that many manuals for genereal purpose CFD codes list overly agressive convergence critera that often produce unconverged results. <br />
For simulations with resolved walls it is good to look at the convergence of some global properties, like total pressure losses from the inlet to the outlet. For heat transfer simulations it is even more tricky since the aerodynamic field can look almost converged although the thermal field is not converged at all. If doing heat transfer simulations make sure to plot the heat-transfer, run for some time, and plot it again to make sure that it doesn't change anymore. With very well resolved walls and heat transfer it can sometimes take 10 times longer for the thermal field to converge.<br />
<br />
=== Single or double precision ===<br />
<br />
With todays computers and cheap memory prices it often does not cost much extra to run in double precision. Before using single precision you should first investigate how your software and hardware works with double precision. If the extra time and memory needed for double precision is negligible you should of course always run in double precision. With double precision you never have to worry about round-off errors. Always using double precision is one way of avoiding one type of pit-falls in the complex world of CFD simulations. Use double precision when you have resolved boundary layers (Y+ around 1) and when you use advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions.<br />
<br />
== Multi-stage analysis ==<br />
<br />
Multi-stage analysis can be done in different ways:<br />
<br />
* Steady mixing-plane simulations<br />
* Frozen rotor simulations<br />
* Unsteady sliding-mesh stator-rotor simulations<br />
* Other '''advanced''' multi-stage methods<br />
<br />
=== Steady mixing-plane simulations ===<br />
<br />
Since the mixing-plane method was first introduced in 1979 [Denton & Singh 1979] is has become the industry standard type of rotor-stator simulations. A mixing-plane simulation is steady and only requires one rotor blade and one stator blade per stage. Between the rotating blade passage and the steady vane passage the flow properties are circumferentially averaged in a so-called mixing-plane interface. This will of course remove all transient rotor-stator interactions, but it still gives fairly representative results. In some commercial codes (CFX for example) mixing-plane interfaces are also called stage-interfaces.<br />
<br />
=== Frozen rotor simulations ===<br />
<br />
In a frozen rotor simulation the rotating and the stationary parts have a fixed relative position. A frame transformation is done to include the rotating effect on the rotating sections. This will give a steady flow and no transient effects are included. With a frozen-rotor simulation rotating wakes, secondary flows, leading edge pressure increases etc. will always stay in exactly the same positions. This makes a frozen rotor simulation very dependent on exactly how the rotors and the stators are positioned. Most often a mixing-plane simulation gives better results. Frozen rotor simulations are mainly performed to obtain a good starting flow-field before doing a transient sliding-mesh simulation.<br />
<br />
=== Unsteady sliding-mesh stator-rotor simulations ===<br />
<br />
This is the most complete type of stator-rotor simulation. In most engines the number of stators and rotors do not have a common denominator (to avoid instabilities caused by resonance between different rings). Hence, to make a full unsteady sliding-mesh computation it is necessary to have a mesh which includes the full wheel with all stators vanes and all rotor blades. This is often not possible, instead it is necessary to reduce the number of vanes and blades by finding a denominator that is '''almost''' common and then scale the geometry slightly circumferentially. Here is an example:<br />
<br />
*Real engine: 36 stator vanes, 41 rotor blades<br />
*Approximated engine: 41 stator vanes, 41 rotor blades, making it possible to simulate only 1 stator vane and 1 rotor blade<br />
*Scaling of stator: All stator vanes are scaled by 36/41 = 0.8780 circumferentially.<br />
<br />
=== Other '''advanced''' multi-stage methods ===<br />
<br />
Time-inclinded, Adamszyk stresses ...<br />
<br />
== Heat transfer predictions ==<br />
<br />
Besides listing the general heat transfer mechanisms involved <br />
(namely conduction, convection, radiation)<br />
heat transfer prediction in CFD may be seen as or split into two cases. <br />
<br />
'''Mesh consists of fluid domain(s)''': <br />
you want to know how heat inserted into the fluid domain (e.g. via the flowing medium) changes the temperature field along the flow path. Be it a steady or transient run, a fluid which enters with a given temperature will usually experience temperature variations, for instance, caused by convective boundary conditions, prescribed temperature profiles at flow obstacles etc. <br />
<br />
'''Mesh consists of fluid and solid domain(s)''': <br />
additionally to the above, you want info too wrt the (spatial, temporal or even spectral) temperature field distribution in surrounding, confining or immersed solids like channel walls or heat exchanger tubes. This is also called conjugate heattransfer CHT in the CFD context. CHT requires a good boundary layer resolution, usually the wall mesh needs to be rather refined, to obtain realistic heatflux results at the fluid/solid interface. Flow and heat transfer convergence require different time step settings, to properly capture changes in flow and heat quantities respectively. <br />
<br />
<br />
In either case, verify (strict necessity depends on CFD code used, CFX for instance checks and assists in regard) that model dimensions, boundary conditions and properties are in consistent units, hold appropriate values. Check temperature-dependence of properties and other numbers before the run.<br />
<br />
In heat transfer predictions (depending on the CFD code in use) <br />
besides the flow solver, you may have to activate the thermal solver too, <br />
as a job specification.<br />
<br />
== Acoustics and noise ==<br />
<br />
A whole separate research subject, difficult. <br />
<br />
Tone noise possible. Often run with linearzised solvers in the frequency domain.<br />
<br />
Jet noise possible. Often run with LES or DES simulations that either also resolve the sound waves or couples to a separate acoustic solver.<br />
<br />
Turbomachinery broadband noise not possible yet, or at least a great challenge.<br />
<br />
== Errors and uncertainties ==<br />
<br />
CFD is still a tool which requires that a user has a good understanding of uncertainties and errors that might spoil a CFD similation. There exists no error control in CFD and any CFD simulation must be interpreted by an expirenced user to have some credibility. Without some knowledge about possible errors and how they can be handled a CFD simulation can not be trusted. Errors can occur at different places:<br />
<br />
* Definition of the problem - What needs to be analyzed?<br />
* Selection of the solution strategy - What physical models and what numerical tools should be used?<br />
* Development of the computational model - How should the geometry and the numerical tools be set up?<br />
* Analysis and interpretation of the results - How should the model be analyzed and the results be interpreted?<br />
<br />
There exists many different definitions on errors. In this guide the errors are classified into four types of errors: problem definition errors, model errors, numerical errors and user and code errors. The chapters below describe these errors and give some guidelines on how to avoid them.<br />
<br />
=== Problem definition errors ===<br />
<br />
Problem definition errors are the most basic form of errors. In order to obtain usefull results a CFD simulation must of course analyze the correct problem, have suitable boundary contions and be based on a relevant geometry.<br />
<br />
==== Wrong type of simulation ====<br />
<br />
It is essential to have an overview of the physics involved and how the problem can best be analyzed. Running a 2D simulation in order to understand secondary flows or running a steady simulation in order to understand transient behaviour is of course no use. When assessing a CFD simulation the first thing to consider is what physical phenomena are important for the results and if the selected type of simulation is suitable to resolve this type of phenomena. For further information about selecting the most suitable type of simulation please see the previous chapeter on [[#Deciding what type of simulation to do|deciding what type of simulation to perform]].<br />
<br />
==== Incorrect or uncertain boundary conditions ====<br />
<br />
A common source of errors is that incorrect boundary conditions are used. The boundary conditions must be specified in enough detail in order to resolve all the important physical features in the problem. For further information about specifying boundary conditions please see the previous chapter on [[#Boundary conditions|boundary conditions]].<br />
<br />
==== Geometrical errors ====<br />
<br />
It is almost always necessary to simplify the geometry in some form. When assessing a CFD simulation one should consider how the geometrical simplifications can affect the interesting physical phenomena. Typical geometrical errors are:<br />
<br />
*Simplifications<br />
:Small geometrical features like fillets, small steps or gaps etc. can often be disregarded. When disregarding this type of features one should consider if they might affect the important physics. For example, a very large fillet on the suction-side of a vane might affect corner separations near the end-walls. A large tip-leakage might affect the flow physics significantly in the upper part of a compressor.<br />
*Tolerances and manufacturing discrepancies<br />
:If the geometry has very large tolerances or is manufactured in a way which might produce a non-ideal shape or position it might be necessary to perform additional CFD simulations in order to cover the whole span of possible real geometries.<br />
*Surface conditions - roughness, welds, steps, gaps etc.<br />
:Often CFD simulations assume a perfectly smooth surface. A non-smooth surface which might have welds, steps or even gaps will of course produce different results. If the physical phenomena of interest might depend on the surface conditions these should of course be condidered. Typical phenomena that might be dependent on this type of errors are transition prediction, leakage flows etc.<br />
<br />
=== Model errors ===<br />
<br />
Errors related to the computational model.<br />
<br />
==== Wrong physical models ====<br />
<br />
Once the type of simulation has been selected the next step is to select what type of physical models the simulation should use. The following points should be considered:<br />
<br />
* Gas data (incompressible/compressible, perfect gas/real gas, ...)<br />
* Turbulence modeling (type of model, type of near-wall treatment, ...)<br />
* Other models (combustion, sprays, ...)<br />
<br />
When assessing model related errors it is important to know the features of the selected model and think carefully how these features and possible short comings might affect the predicted physical behaviour. Using the wrong turbulence model or combustion model can completely destroy the results of a CFD simulation.<br />
<br />
=== Numerical errors ===<br />
<br />
Errors related to the numerical solution of the developed model. Typical examples of numerical errors are discretization errors, convergence errors and round-off errors.<br />
<br />
==== Discretization errors ====<br />
<br />
Discretization errors can either be spatial errors in space or temporal errors in time.<br />
<br />
Spatial discretization errors are what people normally call discretization errors. These errors are due to the difference between the exact solution and the numerical representation of the solution in space. Describing exactly what discretization different codes use and what errors this might lead to is not possible here. Instead some general rules to avoid these errors can be summarized as:<br />
* Use at least a 2nd order accurate scheme, preferably a 3rd order accurate scheme. Some general purpose codes have a 1st order upwind scheme as default, this is a very diffusives schemes that often gives too smooth results.<br />
* For new applications always run a simulation with a finer mesh to see how grid independent your solution is.<br />
* Be aware of checker-board errors. Checker-board errors occur close to strong shocks and other large discontinuities and can be seen as a wavy pattern with a wavelength of two cells. Some schemes, especially those who behave like central differencing schemes, are more prone to checker-board errors. Upwind schemes are a bit better and schemes like TVD or chock-capturing schemes are even better.<br />
<br />
Temporal discretization errors mainly effect transient simulations. However, some codes use a time-marching method also for steady simulations and then a temporal discretization error might affect the final steady solution slightly. The discretization in time can be done with 1st or 2nd order schemes or a Runge-Kutta method, which is more accurate and saves memory. Some codes can adapt the time-step, but often it is necessary to prescribe a time-step in advance. Think of the time-step as your grid in time and make sure that the grid-resolution in time is fine enough to resolve the highest frequencies. To avoid problems with temporal discretization errors the following should be considered:<br />
* Use at least a 2nd order scheme in time.<br />
* Do a physical estimation of the typical frequencies in time of the phenomena that you are interested in and select a time-step which is fine enough to resolve these frequencies well. After the simulation also look at the frequencies captured and make sure that they are well resolved by the chosen time-step.<br />
* For new applications try a finer time-step to ensure that your solution in time is fairly grid independent in time.<br />
<br />
==== Convergence errors ====<br />
<br />
To judge when a CFD simulation is converged is not always that easy. Different codes and different applications behave very differently. For a pure aero-simulation on a fairly coarse grid convergence is easy to judge, but for more complex simulations involving resolved boundary layers, heat transfer, combustion etc. convergence can be very tricky. Aside from looking at residuals one should always also look at how global parameters like static pressure distributions, total pressure losses, skin friction, heat transfer etc. change in time.<br />
For more information about how to avoid convergence errors see the previous chapter about [[#Convergence criteria]].<br />
<br />
==== Round-off errors ====<br />
<br />
When using single precision care needs to be taken to avoid round-off errors. Inviscid Euler simulations and simulations using wall-function meshes can most often be performed in single precision. For well resolved boundary layers with Y plus close to 1 it is often necessary to use double precision. If using double precision for very fine mesh resolutions make sure that you also create the mesh in double precision and not just run the solver in double precision. Sometimes a single precision solver converges slower than a double precision solver due to numerical errors caused by round-off errors. When using advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions it is also often necessary to use double precision.<br />
<br />
=== User and code errors ===<br />
<br />
Errors related to bugs in the code used or mistakes made by the CFD engineer.<br />
<br />
=== What to trust and what not to trust ===<br />
<br />
CFD is generally quite good at predicting surface static pressure distributions. With care CFD can also be used to predict performance, total-pressure losses and blade turning. <br />
<br />
Predicting separation, stall and off-design performance can be a challenge and results with non-attached flows should be interpreted with care. <br />
<br />
Heat transfer is often very difficult to predict accurately and it is common to obtain heat-transfer coefficients that are 100% wrong or more. Validation data is critical in order to be able to trust heat transfer simulations.<br />
<br />
Transition is almost impossible to predict accurately in general. However, there exist models that have been tuned to predict transition and these tend to give acceptable results for cases close to the ones they were tuned for.<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Abu-Ghannam, B.J. and Shaw, R.|year=1980|title=Natural Transition of Boundary Layers - the Effects of Turbulence, Pressure Gradient, and Flow History|rest=Journal of Mech. Eng. Science, vol. 22, no. 5, pp. 213–228}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1985|title=Model Equation for Simulating Flows in Multistage Turbomachinery|rest=ASME Paper 85-GT-226, also NASA TM-86869}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1999|title=Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design|rest=ASME Paper 99-GT-80.}}<br />
<br />
* {{reference-paper|author=Baldwin, B. S. and Lomax, H.|year=1978|title=Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows|rest=AIAA Paper 78-257}}<br />
<br />
* {{reference-paper|author=Chen, J. P. and Barter, J. W.|year=1998|title=Comparison of Time-Accurate Calculations for the Unsteady Interaction in Turbomachinery Stage|rest=AIAA Paper 98-3292}}<br />
<br />
* {{reference-paper|author=Denton, J. and Singh, U.|year=1979|title=Time Marching Methods for Turbomachinery Flow Calculations|rest=VKI Lecture Series 1979-7, von Karman Institute}}<br />
<br />
* {{reference-paper|author=Erdos, J., Alzner, E. and McNally, W.|year=1977|title=Numerical Solution of Periodic Transonic Flow Through a Transonic Fan Stage|rest=AIAA Journal, vol. 15, no. 11, pp. 1559-1568}}<br />
<br />
* {{reference-paper|author=Gerolymos, G., Vinteler, D., Haugeard, R., Tsange, G. and Vallet, I.|year=1996|title=On the Computation of Unsteady Turbomachinery Flows – Part II – Rotor/Stator Interaction using Euler Equations|rest=AGARD Report CP-571}}<br />
<br />
* {{reference-paper|author=Giles, M. B.|year=1990|title=Stator/Rotor Interaction in a Transonic Turbine|rest=AIAA Journal of Propulsion and Power, vol. 6, no. 5}}<br />
<br />
* {{reference-paper|author=He, L, and Ning, W.|year=1998|title=Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines|rest=AIAA Journal, vol. 36, no. 11, pp. 2005-2012}}<br />
<br />
* {{reference-paper|author=Mayle, R. E.|year=1991|title=The Role of Laminar-Turbulent Transition in Gas Turbine Engines|rest=ASME Journal of Turbomachinery, vol. 113, pp. 509-537}}<br />
<br />
* {{reference-paper|author=Menter, F.R. |year=1994|title=Two-equation eddy-viscosity turbulence models for engineering applications|rest=AIAA Journal, vol. 32, pp. 269-289}}<br />
<br />
* {{reference-paper|author=Savill, M., Dick, E., Hanjalić, K. and Voke, P.|year=2002|title=Synthesis Report of the ERCOFTAC Transition Modeling – TRANSPRETURB Thematic Network Activities 1998-2002|rest=Ecoftac Bulletin 54, September 2002, pp. 5-16}}<br />
<br />
* {{reference-paper|author=Wang, X. and Chen J.|year=2004|title=A Post-Processor to Render Turbomachinery Flows Using Phase-Lag Simulations|rest=AIAA Paper 04-615}}<br />
<br />
== External links ==<br />
<br />
* [http://www.qnet-cfd.net/newsletter/8th/n8_40-46.pdf QNET-CFD Best Practise Advice for Turbomachinery Internal Flows]<br />
* [http://www.qnet-cfd.net/newsletter/7th/n7_05.pdf State of the art in Industrial CFD for Turbomachinery Flows, QNET-CFD Network Newsletter, Volume 2, No. 3 – December 2003]<br />
* [https://pronet.wsatkins.co.uk/marnet/guidelines/guide.html MARNET-CFD Best Practice Guidelines for Marine Applications of CFD]<br />
* [http://www.stanford.edu/group/ctr/ResBriefs/temp05/weide.pdf On Large Scale Turbomachinery Computations, by E. van der Weide et. al., Center for Turbulence Research, Annual Research Briefs 2005]</div>DavidFhttp://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFDBest practice guidelines for turbomachinery CFD2008-09-12T09:41:16Z<p>DavidF: /* Near-wall treatment */</p>
<hr />
<div>This article contains a summary of the most important knowledge and experience a CFD engineer needs in order to perform CFD simulations of turbomachinery components. The guide is mainly aimed at axial turbomachinery. The goal is to give a CFD engineer, who has just started working with turbomachinery simulations, a head start and avoid some of the most difficult pit-falls. Experienced turbomachinery CFD engineers can also use the guidelines in order to learn what other experts consider best practice. The intended audience is expected to know basic CFD terminology and have a basic turbomachinery knowledge, but no detailed knowledge about CFD for turbomachinery is needed. <br />
<br />
== Deciding what type of simulation to do ==<br />
<br />
Before starting a new turbomachinery simulation it is wise to think carefully of what it is that should be predicted and what physical phenomena that affect the results. This chapter contains a brief overview of the various types of simulations and some hints of what can be predicted with them.<br />
<br />
=== 2D, Quasi-3D or 3D ===<br />
<br />
2D simulations are often used in the early design phase in order to obtain a typical 2D section of a blade. For cases with many long blades or vanes, like low-pressure turbines, a 2D simulation can also provide reasonable results. If the area of the flow-path changes significantly in the axial direction it might be necessary to instead make a quasi-3D simulation. A quasi-3D simulation is a 2D simulation in which extra source terms are used to account for the acceleration/decelleration caused by a changing channel height or growing end-wall boundary layers. Codes focused on turbomachinery applications often have the possibility to perform quasi-3D simulations, but most general purpose CFD codes can not do this type of simulations, or require user coding to implement the correct source terms in the equations. Please contact your software distributor if your code does not have the quasi-3d possibility and you require it. Many codes require special routines or hidden commands to enable this feature.<br />
<br />
Full 3D simulations are necessary if a true 3D geometry is needed to obtain correct secondary flows and/or shock locations. For low-aspect-ratio cases with only a few short blades, like for example structurally loaded turbine outlet guide vanes, the secondary flow development is important and a 3D simulation is often necessary in order to obtain reasonable results. For applications where the end-wall boundary layers grow very quickly and interact with a large part of the flow-field it is necessary to perform a full 3D simulation. This is often the case in compressors and fans, where the negative pressure gradients make the boundary layers grow much quicker than what they do in for example turbines. For cases where the shock location is very critical, like in transonic compressors, it is also often necessary to perform a 3D simulation in order to obtain reasonable shock locations.<br />
<br />
=== Inviscid or viscid ===<br />
<br />
For attached flows close to the design point and without any large separations it is often sufficient with an inviscid Euler simulation in order to obtain reasonable blade loadings and pressure distributions. Note that inviscid Euler simulations should only be used if the boundary layers are judged to not have a significant effect on the global flow-field. A viscid Navier-Stokes simulation is necessary in order to predict losses, secondary flows and separations. As soon as separations are of interest it is of course also necessary to do a viscid simulation. Note that with todays computers it is often not time and resources that make users run inviscid Euler simulations. Running viscid Navier-Stokes simulations is now so quick that it is not a time problem any more. Euler simulations are still interesting though, since with an inviscid Euler simulation you don't have to worry about wall resolutions, y+ values, turbulence modeling errors etc.<br />
<br />
=== Transient or Stationary ===<br />
<br />
Most turbomachinery simlation are performed as stationary simulations. Transient simulations are done when some kind of transient flow behaviour has a strong influence on the global flow field. Examples of transient simulations are detailed simulations of rotor-stator interaction effects, simulations of large unsteady separations etc. Sometimes when you perform a steady stationary simulation you can see tendencies of unsteady behvaiour like for example periodic vortex shedding behind blunt trailing edges. This is often first seen as periodical variations of the residuals. If the unsteady tendencies are judged to not affect the overall simulation results if might be necessary to coarsen the mesh close to the vortex shedding or run a different turbulence model in order to make the simulation converge. Sometimes you are still forced to run a transient simulation and average the results if you don't obtain a converged steady solution.<br />
<br />
== Meshing ==<br />
<br />
In turbomachinery applications structured muli-block hexahedral meshes are most often used for flow-path simulations. In most solvers a structured grid requires less memory, provides superior accuracy and allows a better boundary-layer resolution than an unstructured grid. By having cells with a large aspect ratio around sharp leading and trailing edges a structured grid also provides a better resolution of these areas. Many companies have automatic meshing tools that automatically mesh blade sections with a structured mesh without much user intervention.<br />
<br />
Unstructured meshes are used for more complex and odd geometries where a structured mesh is difficult to create. Typical examples where unstructured meshes are often used are blade tip regions, areas involving leakage flows and secondary air systems, film cooling ducts etc.<br />
<br />
When meshing avoid to create large jumps in cell sizes. Typically the cell size should not change with more than a factor of 1.25 between neighbouring cells. For structured meshes also try to create fairly continous mesh lines and avoid discontinuities where the cell directions suddenly change. For multi-block structured meshes avoid placing the singular points where blocks meet in regions with strong flow gradients since most schemes have a lower accuracy in these singular points.<br />
<br />
=== Mesh size guidelines ===<br />
<br />
It is difficult to define, a priori, the mesh size. The required mesh size depends on the purpose of the simulation. <br />
<br />
If the main goal is to obtain static pressure forces a coarse mesh is often able to obtain a good solution, especially when an accurate resolution of the boundary layers is not required. For 2D inviscid simulations of one blade a mesh with say 3,000 cells is most often sufficient. For 3D inviscid blade simulations a mesh size of about 40,000 cells is usually sufficient. On inviscid Euler simulations the cells should be fairly equal in size and no boundary layer resolution should be present. Avoid having too skewed cells.<br />
<br />
For loss predictions and cases where boundary layer development and separation is important the mesh needs to have a boundary layer resolution. The boundary layer resolution can either be coarse and suitable for a [[Wall functions|wall function]] simulation or very fine and suitable for a [[Low-Re resolved boundary layers|low-Re]] simulation. For further information about selecting the near-wall turbulence model please see the [[#Turbulence modeling|turbulence modeling]] section. In 3D single-blade simulations a decent wall-function mesh typically has around 100,000 cells. This type of mesh size is suitable for quick design iterations where it is not essential to resolve all secondary flows and vortices. A good 3D wall-function mesh of a blade section intended to resolve secondary flows well should have at least 400,000 cells . A good low-Re mesh with resolved boundary layers typically has around 1,000,000 cells. <br />
<br />
In 2D blade simulations a good wall-function mesh has around 20,000 cells and a good low-Re mesh with resolved boundary layers has around 50,000 cells.<br />
<br />
Along the suction and pressure surfaces it is a good use about 100 cells in the streamwise direction. In the radial direction a good first approach is to use something like 30 cells for a wall-function mesh and 100 cells for a low-Re mesh.<br />
<br />
It is important to resolve leading and trailing edges well. Typically at least 10 cells, preferably 20 should be used around the leading and trailing edges. For very blunt and large leading edges, like those commonly found on HP turbine blades, 30 or more cells can be necessary.<br />
<br />
Cases which are difficult to converge with a steady simulation and which show tendencies of periodic vortex shedding from the trailing edge, can sometimes be "tamed" by using a coarse mesh around the trailing edge. This, of course, reduces the accuracy but can be a trick to obtain a converged solution if time and computer resouces does not allow a transient simulation to be performed.<br />
<br />
=== Boundary layer mesh ===<br />
<br />
For design iteration type of simulations where a [[Wall functions|wall function]] approach is sufficient [[Dimensionless wall distance (y plus) | y+]] for the first cell should be somewhere between 20 and 200. The outer limit is dependent on the actual Re number of the simulations. For cases with fairly low Re numbers make sure to keep the maximum y+ as low as possible. For more accurate simulations with resolved boundary layers the mesh should have a y+ for the first cell which is below 1. Some new codes are now using a hybrid wall treatment that allows a smooth transition from a coarse wall-function mesh to a resolved low-Re mesh. Use some extra care when using this type of hybrid technique since it is still fairly new and unproven.<br />
<br />
Outside of the first cell at a wall a good rule of thumb is to use a growth ratio normal to the wall in the boundary layer of maximum 1.25. For a low-Re mesh this usually gives around 40 cells in the boundary layer whereas a wall-function mesh does not require more than 10 cells in the boundary layer.<br />
<br />
If you are uncertain of which wall distance to mesh with you can use a [http://www.cfd-online.com/Links/tools.html#yplus y+ estimation tool] to estitmate the distance needed to obtain the desired y+. These estimation tools are ''very handy'' if you have not done any previous similar simulations. <br />
<br />
As a rule of thumb a wall-function mesh typically requires areound 5 to 10 cells in the boundary layer whereas a resolved low-Re mesh requires about 40 cells in the boundary layer.<br />
<br />
== Boundary conditions ==<br />
<br />
Describe different types of boundary conditions and when they should be used:<br />
<br />
* Total pressure in, static pressure out<br />
* Absorbing boundary conditions<br />
* ...<br />
<br />
There are different types of boundary conditions you can use:<br />
*Mass flow inlet, static pressure outlet.<br />
To put this BC you must allocate inlet surface enough far away from the turbomachine impeller. The velocity distribution is not<br />
constant over all the surface. If you put this "lie" away from your problem it works well.<br />
<br />
<br />
=== Turbulence inlet conditions ===<br />
<br />
Prescribing realistic turbulence inlet conditions is important and usually difficult. For two-equation turbulence models two different turbulence values need to be specified on the inlet. The most common way of specifying the inlet turbulence variables is to give an inlet [[Turbulence intensity|turbulence level]] and [[Turbulence length scale|length-scale]]. For simulations where the incoming boundary layers are estimated to not have any significant effect on the secondary flows and global flow-field downstream it is usually sufficient to specify a constant incoming turbulence level and length-scale.<br />
<br />
The incoming [[Turbulence intensity|turbulence level]] is dependent on which component is analyzed. Best is of course to have measurments from a similar application in order to be able to prescribe a fairly realistic value. Without measurements it is necessary to make an educated guess. Simulations with very complex and turbulence-generating components upstream will have a very high incoming turbulence level. For example, a high-pressure turbine just downstream of a turbulence generating combustor might have incoming turbulence levels up to 20%. Fans and low-pressure compressors with not many components upstream might have as low incoming turbulence level as 1%. For components between these two extreme examples, like high-pressure compressors, low-pressure turbines etc. a turbulence level of around 5% might be realistic. If you do not have any measurements and are unsure about which incoming turbulence level to prescribe it is always a good idea to run a few different simulations with say one half and one double incoming turbulence level in order to estimate how important it is for the simulation results.<br />
<br />
The incoming [[Turbulence length scale|turbulence length-scale]] is often even more difficult to guess than the incoming turbulence level. Measurements might sometimes include the incoming turbulence level, but very seldom also include the incoming turbulent length-scale. The best way of guessing a realistic incoming length-scale is to use the geometrical properties of the upstream components. The incoming turbulence length-scale can be estimated as say the thickness of upstream blades or somewhere between 2% and 20% of the incoming channel heigth. Fortunately the incoming turbulence length-scale is usually not that important for the end results. However, some k-omega models can have an unrelisticly strong influence on the incoming turbulence length-scale. Hence, be extra carefull about the length-scale when using k-omega models. Note that the [[SST k-omega model]] does not have this unrealistic sensitivity to the inlet turbulence length-scale.<br />
<br />
Instead of specifying an incoming turbulence length-scale it is sometimes more convenient to specify an incoming [[Eddy viscosity ratio|eddy viscosity ratio]]. For low-turbulence cases with no well-specified turbulence generating components upstream it can be difficult to estimate a turbulence length-scale. Specifying an eddy viscosity ratio instead gives a more direct control over how large the effect of the incoming turbulence is. The eddy viscosity ratio describes how large the turbulence viscosity is related to the molecular viscosity. In turbomachinery the eddy viscosity ratio can vary from say 10 in low-turbulence components like fans up to say 1,000 or in rare cases even 10,000 in high-turbulence components like high-pressure turbines. In components between these extremes a typical eddy viscosity ration might be about 200. Even if specifying an inlet turbulence length scale it can still be good to compute what eddy viscosity ratio this corresponds to in order to estimate if it sounds reasonable.<br />
<br />
== Turbulence modeling ==<br />
<br />
Selecting a suitable turbulence model for turbomachinery simulations can be a challenging task. There is no single model which is suitable for all types of simulations. Which turbulence model CFD engineers use has as much to do with beliefs and traditions as with knowledge and facts. There are many different schools. However, below follows some advices that most CFD engineers in the turbomachinery field tend to agree upon.<br />
<br />
For attached flows close to the design point a simple algebraic model like the [[Baldwin-Lomax model]] can be used. Another common choice for design-iteration type of simulations is the one-equation model by [[Spalart-Allmaras model | Spalart-Allmaras]]. This model has become more popular in the last years due to the many inherent problems in more refined two-equation models. The big advantage with both the [[Baldwin-Lomax model]] and the [[Spalart-Allmaras model]] over more advanced models is that they are very robust to use and rarely produce completely unphysical results.<br />
<br />
In order to accurately predict more difficult cases, like separating flows, rotating flows, flows strongly affected by secondary flows etc. it is often necessary to use a more refined turbulence model. Common choices are two-equation models like the <math>k-\epsilon</math> model.<br />
<br />
Two-equation models are based on the [[Boussinesq eddy viscosity assumption]] and this often leads to an over-production of turbulent energy in regions with strong acceleration or deceleration, like in the leading edge region, regions around shocks and in the suction peak on the suction side of a blade. To reduce this problems it is common to use a special model variant using, for example, [[Durbin's realizability constraint]] or the [[Kato-Launder modification]]. Note that different two-equation models behave differently in these problematic stagnation and acceleration regions. Worst is probably the standard <math>k-\epsilon</math> model. <math>k-\omega</math> model are slightly better but still do not behave well. More modern variants like Menter's [[SST k-omega model]] also has problems, wheras the [[V2-f models|v2f model ]] by Durbin behaves better.<br />
<br />
=== Near-wall treatment ===<br />
<br />
For on-design simulations without any large separated regions it is often sufficient to use a [[wall-function model]] close to the wall, preferably with some form of non-equilibrium wall-function that is sensitised to streamwise pressure gradients.<br />
<br />
For off-design simulation, or simulations involving complex secondary flows and separations, it is often necessary to use a [[low-Re model]]. There exist many low-Re models that have been used with success in turbomachinery simulations. A robust and often good choice is to use a one-equation model, like for example the [[Wolfstein model]], in the inner parts of the boundary layer. There are also several Low-Re <math>k-\epsilon</math> models that work well. Just make sure they don't suffer from the problem with overproduction of turbulent energy in regions with strong acceleration or deceleration. In the last few years Menter's low-Re <math>SST k-\omega</math> model has gained increased popularity.<br />
<br />
=== Transition prediction ===<br />
<br />
Transition refers to the process when a laminar boundary layer becomes unstable and transitions to a turbulent boundary layer. There are two types of transition - natural transition, where inherent instabilities in the boundary layer cause the transition and by-pass transition, where convection and diffusion of turbulence from the free-stream into the boundary layer cause the transition. Most transitions in turbomachinery are by-pass transitions caused by free-stream turbulence and other external disturbances like wakes, vortices and surface defects.<br />
<br />
Simulating transition in a CFD code accurately is very difficult. Often a separate transition model needs to be solved in order to specify the transition location and length. Predicting natural transition in a pure CFD code is not possible. Predicting by-pass transition in a pure CFD code is almost impossible, although there are people who claim to be able to predict by-pass transition with low-Re two-equation models. However, this is usually on special test cases and with simulations that have been tuned for these special cases, see for example [Saville 2002]. In reality transition is a very complex and sensitive process where disturbances like incoming wakes and vortices from previous stages, surface roughness effects and small steps or gaps in the surfaces play a significant role. <br />
<br />
The turbomachinery codes that have transition prediction models often use old ad-hoc models like the Abu-Ghannam and Shaw model [Abu-Ghannam 1980] or the Mayle model [Mayle 1991]. These models can be quite reliable if they have been validated and tuned for a similar application. Do not trust your transition predictions without having some form of experimental validation. Menter has also recently developed a new form of transition model that might work fairly well, but it is still too new and untested.<br />
<br />
For some turbomachinery applications, like modern high-lift low-pressure turbines, transition is critical. For these applications a CFD code with a transition model that has been tuned for this type of applications should be used.<br />
<br />
== Numerical considerations ==<br />
<br />
Use at least a second order accurate scheme for the flow variables. Some codes require a first order scheme for the turbulent variables (<math>k</math> and <math>\epsilon</math>) in order to converge well. It might be sufficient with a first order scheme only on the turbulence variables, but a second order scheme is of course preferable.<br />
<br />
=== Convergence criteria ===<br />
<br />
To know when a solution is converged is not always that easy. You need some prior experience of your CFD code and your application to judge when a simulation is converged. For normal pure aero simulations without resolved walls, i.e. with wall functions or inviscid Euler simulations, convergence can most often be estimated just by looking at the residuals. Exacty what the residuals should be is not possible to say, it all depends on how your particular code computes and scales the residuals. Hence, make sure to read the manuals and plot the convergence of a few global parameters before you decide what the residuals should be for a solution to converege. Note also that many manuals for genereal purpose CFD codes list overly agressive convergence critera that often produce unconverged results. <br />
For simulations with resolved walls it is good to look at the convergence of some global properties, like total pressure losses from the inlet to the outlet. For heat transfer simulations it is even more tricky since the aerodynamic field can look almost converged although the thermal field is not converged at all. If doing heat transfer simulations make sure to plot the heat-transfer, run for some time, and plot it again to make sure that it doesn't change anymore. With very well resolved walls and heat transfer it can sometimes take 10 times longer for the thermal field to converge.<br />
<br />
=== Single or double precision ===<br />
<br />
With todays computers and cheap memory prices it often does not cost much extra to run in double precision. Before using single precision you should first investigate how your software and hardware works with double precision. If the extra time and memory needed for double precision is negligible you should of course always run in double precision. With double precision you never have to worry about round-off errors. Always using double precision is one way of avoiding one type of pit-falls in the complex world of CFD simulations. Use double precision when you have resolved boundary layers (Y+ around 1) and when you use advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions.<br />
<br />
== Multi-stage analysis ==<br />
<br />
Multi-stage analysis can be done in different ways:<br />
<br />
* Steady mixing-plane simulations<br />
* Frozen rotor simulations<br />
* Unsteady sliding-mesh stator-rotor simulations<br />
* Other '''advanced''' multi-stage methods<br />
<br />
=== Steady mixing-plane simulations ===<br />
<br />
Since the mixing-plane method was first introduced in 1979 [Denton & Singh 1979] is has become the industry standard type of rotor-stator simulations. A mixing-plane simulation is steady and only requires one rotor blade and one stator blade per stage. Between the rotating blade passage and the steady vane passage the flow properties are circumferentially averaged in a so-called mixing-plane interface. This will of course remove all transient rotor-stator interactions, but it still gives fairly representative results. In some commercial codes (CFX for example) mixing-plane interfaces are also called stage-interfaces.<br />
<br />
=== Frozen rotor simulations ===<br />
<br />
In a frozen rotor simulation the rotating and the stationary parts have a fixed relative position. A frame transformation is done to include the rotating effect on the rotating sections. This will give a steady flow and no transient effects are included. With a frozen-rotor simulation rotating wakes, secondary flows, leading edge pressure increases etc. will always stay in exactly the same positions. This makes a frozen rotor simulation very dependent on exactly how the rotors and the stators are positioned. Most often a mixing-plane simulation gives better results. Frozen rotor simulations are mainly performed to obtain a good starting flow-field before doing a transient sliding-mesh simulation.<br />
<br />
=== Unsteady sliding-mesh stator-rotor simulations ===<br />
<br />
This is the most complete type of stator-rotor simulation. In most engines the number of stators and rotors do not have a common denominator (to avoid instabilities caused by resonance between different rings). Hence, to make a full unsteady sliding-mesh computation it is necessary to have a mesh which includes the full wheel with all stators vanes and all rotor blades. This is often not possible, instead it is necessary to reduce the number of vanes and blades by finding a denominator that is '''almost''' common and then scale the geometry slightly circumferentially. Here is an example:<br />
<br />
*Real engine: 36 stator vanes, 41 rotor blades<br />
*Approximated engine: 41 stator vanes, 41 rotor blades, making it possible to simulate only 1 stator vane and 1 rotor blade<br />
*Scaling of stator: All stator vanes are scaled by 36/41 = 0.8780 circumferentially.<br />
<br />
=== Other '''advanced''' multi-stage methods ===<br />
<br />
Time-inclinded, Adamszyk stresses ...<br />
<br />
== Heat transfer predictions ==<br />
<br />
Besides listing the general heat transfer mechanisms involved <br />
(namely conduction, convection, radiation)<br />
heat transfer prediction in CFD may be seen as or split into two cases. <br />
<br />
'''Mesh consists of fluid domain(s)''': <br />
you want to know how heat inserted into the fluid domain (e.g. via the flowing medium) changes the temperature field along the flow path. Be it a steady or transient run, a fluid which enters with a given temperature will usually experience temperature variations, for instance, caused by convective boundary conditions, prescribed temperature profiles at flow obstacles etc. <br />
<br />
'''Mesh consists of fluid and solid domain(s)''': <br />
additionally to the above, you want info too wrt the (spatial, temporal or even spectral) temperature field distribution in surrounding, confining or immersed solids like channel walls or heat exchanger tubes. This is also called conjugate heattransfer CHT in the CFD context. CHT requires a good boundary layer resolution, usually the wall mesh needs to be rather refined, to obtain realistic heatflux results at the fluid/solid interface. Flow and heat transfer convergence require different time step settings, to properly capture changes in flow and heat quantities respectively. <br />
<br />
<br />
In eighter case, verify (strict necessity depends on CFD code used, CFX for instance checks and assists in regard) that model dimensions, boundary conditions and properties are in consistent units, hold appropriate values. Check temperature-dependence of properties and other numbers before the run.<br />
<br />
In heat transfer predictions (depending on the CFD code in use) <br />
besides the flow solver, you may have to activate the thermal solver too, <br />
as a job specification.<br />
<br />
== Acoustics and noise ==<br />
<br />
A whole separate research subject, difficult. <br />
<br />
Tone noise possible. Often run with linearzised solvers in the frequency domain.<br />
<br />
Jet noise possible. Often run with LES or DES simulations that either also resolve the sound waves or couples to a separate acoustic solver.<br />
<br />
Turbomachinery broadband noise not possible yet, or at least a great challenge.<br />
<br />
== Errors and uncertainties ==<br />
<br />
CFD is still a tool which requires that a user has a good understanding of uncertainties and errors that might spoil a CFD similation. There exists no error control in CFD and any CFD simulation must be interpreted by an expirenced user to have some credibility. Without some knowledge about possible errors and how they can be handled a CFD simulation can not be trusted. Errors can occur at different places:<br />
<br />
* Definition of the problem - What needs to be analyzed?<br />
* Selection of the solution strategy - What physical models and what numerical tools should be used?<br />
* Development of the computational model - How should the geometry and the numerical tools be set up?<br />
* Analysis and interpretation of the results - How should the model be analyzed and the results be interpreted?<br />
<br />
There exists many different definitions on errors. In this guide the errors are classified into four types of errors: problem definition errors, model errors, numerical errors and user and code errors. The chapters below describe these errors and give some guidelines on how to avoid them.<br />
<br />
=== Problem definition errors ===<br />
<br />
Problem definition errors are the most basic form of errors. In order to obtain usefull results a CFD simulation must of course analyze the correct problem, have suitable boundary contions and be based on a relevant geometry.<br />
<br />
==== Wrong type of simulation ====<br />
<br />
It is essential to have an overview of the physics involved and how the problem can best be analyzed. Running a 2D simulation in order to understand secondary flows or running a steady simulation in order to understand transient behaviour is of course no use. When assessing a CFD simulation the first thing to consider is what physical phenomena are important for the results and if the selected type of simulation is suitable to resolve this type of phenomena. For further information about selecting the most suitable type of simulation please see the previous chapeter on [[#Deciding what type of simulation to do|deciding what type of simulation to perform]].<br />
<br />
==== Incorrect or uncertain boundary conditions ====<br />
<br />
A common source of errors is that incorrect boundary conditions are used. The boundary conditions must be specified in enough detail in order to resolve all the important physical features in the problem. For further information about specifying boundary conditions please see the previous chapter on [[#Boundary conditions|boundary conditions]].<br />
<br />
==== Geometrical errors ====<br />
<br />
It is almost always necessary to simplify the geometry in some form. When assessing a CFD simulation one should consider how the geometrical simplifications can affect the interesting physical phenomena. Typical geometrical errors are:<br />
<br />
*Simplifications<br />
:Small geometrical features like fillets, small steps or gaps etc. can often be disregarded. When disregarding this type of features one should consider if they might affect the important physics. For example, a very large fillet on the suction-side of a vane might affect corner separations near the end-walls. A large tip-leakage might affect the flow physics significantly in the upper part of a compressor.<br />
*Tolerances and manufacturing discrepancies<br />
:If the geometry has very large tolerances or is manufactured in a way which might produce a non-ideal shape or position it might be necessary to perform additional CFD simulations in order to cover the whole span of possible real geometries.<br />
*Surface conditions - roughness, welds, steps, gaps etc.<br />
:Often CFD simulations assume a perfectly smooth surface. A non-smooth surface which might have welds, steps or even gaps will of course produce different results. If the physical phenomena of interest might depend on the surface conditions these should of course be condidered. Typical phenomena that might be dependent on this type of errors are transition prediction, leakage flows etc.<br />
<br />
=== Model errors ===<br />
<br />
Errors related to the computational model.<br />
<br />
==== Wrong physical models ====<br />
<br />
Once the type of simulation has been selected the next step is to select what type of physical models the simulation should use. The following points should be considered:<br />
<br />
* Gas data (incompressible/compressible, perfect gas/real gas, ...)<br />
* Turbulence modeling (type of model, type of near-wall treatment, ...)<br />
* Other models (combustion, sprays, ...)<br />
<br />
When assessing model related errors it is important to know the features of the selected model and think carefully how these features and possible short comings might affect the predicted physical behaviour. Using the wrong turbulence model or combustion model can completely destroy the results of a CFD simulation.<br />
<br />
=== Numerical errors ===<br />
<br />
Errors related to the numerical solution of the developed model. Typical examples of numerical errors are discretization errors, convergence errors and round-off errors.<br />
<br />
==== Discretization errors ====<br />
<br />
Discretization errors can either be spatial errors in space or temporal errors in time.<br />
<br />
Spatial discretization errors are what people normally call discretization errors. These errors are due to the difference between the exact solution and the numerical representation of the solution in space. Describing exactly what discretization different codes use and what errors this might lead to is not possible here. Instead some general rules to avoid these errors can be summarized as:<br />
* Use at least a 2nd order accurate scheme, preferably a 3rd order accurate scheme. Some general purpose codes have a 1st order upwind scheme as default, this is a very diffusives schemes that often gives too smooth results.<br />
* For new applications always run a simulation with a finer mesh to see how grid independent your solution is.<br />
* Be aware of checker-board errors. Checker-board errors occur close to strong shocks and other large discontinuities and can be seen as a wavy pattern with a wavelength of two cells. Some schemes, especially those who behave like central differencing schemes, are more prone to checker-board errors. Upwind schemes are a bit better and schemes like TVD or chock-capturing schemes are even better.<br />
<br />
Temporal discretization errors mainly effect transient simulations. However, some codes use a time-marching method also for steady simulations and then a temporal discretization error might affect the final steady solution slightly. The discretization in time can be done with 1st or 2nd order schemes or a Runge-Kutta method, which is more accurate and saves memory. Some codes can adapt the time-step, but often it is necessary to prescribe a time-step in advance. Think of the time-step as your grid in time and make sure that the grid-resolution in time is fine enough to resolve the highest frequencies. To avoid problems with temporal discretization errors the following should be considered:<br />
* Use at least a 2nd order scheme in time.<br />
* Do a physical estimation of the typical frequencies in time of the phenomena that you are interested in and select a time-step which is fine enough to resolve these frequencies well. After the simulation also look at the frequencies captured and make sure that they are well resolved by the chosen time-step.<br />
* For new applications try a finer time-step to ensure that your solution in time is fairly grid independent in time.<br />
<br />
==== Convergence errors ====<br />
<br />
To judge when a CFD simulation is converged is not always that easy. Different codes and different applications behave very differently. For a pure aero-simulation on a fairly coarse grid convergence is easy to judge, but for more complex simulations involving resolved boundary layers, heat transfer, combustion etc. convergence can be very tricky. Aside from looking at residuals one should always also look at how global parameters like static pressure distributions, total pressure losses, skin friction, heat transfer etc. change in time.<br />
For more information about how to avoid convergence errors see the previous chapter about [[#Convergence criteria]].<br />
<br />
==== Round-off errors ====<br />
<br />
When using single precision care needs to be taken to avoid round-off errors. Inviscid Euler simulations and simulations using wall-function meshes can most often be performed in single precision. For well resolved boundary layers with Y plus close to 1 it is often necessary to use double precision. If using double precision for very fine mesh resolutions make sure that you also create the mesh in double precision and not just run the solver in double precision. Sometimes a single precision solver converges slower than a double precision solver due to numerical errors caused by round-off errors. When using advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions it is also often necessary to use double precision.<br />
<br />
=== User and code errors ===<br />
<br />
Errors related to bugs in the code used or mistakes made by the CFD engineer.<br />
<br />
=== What to trust and what not to trust ===<br />
<br />
CFD is generally quite good at predicting surface static pressure distributions. With care CFD can also be used to predict performance, total-pressure losses and blade turning. <br />
<br />
Predicting separation, stall and off-design performance can be a challenge and results with non-attached flows should be interpreted with care. <br />
<br />
Heat transfer is often very difficult to predict accurately and it is common to obtain heat-transfer coefficients that are 100% wrong or more. Validation data is critical in order to be able to trust heat transfer simulations.<br />
<br />
Transition is almost impossible to predict accurately in general. However, there exist models that have been tuned to predict transition and these tend to give acceptable results for cases close to the ones they were tuned for.<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Abu-Ghannam, B.J. and Shaw, R.|year=1980|title=Natural Transition of Boundary Layers - the Effects of Turbulence, Pressure Gradient, and Flow History|rest=Journal of Mech. Eng. Science, vol. 22, no. 5, pp. 213–228}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1985|title=Model Equation for Simulating Flows in Multistage Turbomachinery|rest=ASME Paper 85-GT-226, also NASA TM-86869}}<br />
<br />
* {{reference-paper|author=Adamczyk, J. J.|year=1999|title=Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design|rest=ASME Paper 99-GT-80.}}<br />
<br />
* {{reference-paper|author=Baldwin, B. S. and Lomax, H.|year=1978|title=Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows|rest=AIAA Paper 78-257}}<br />
<br />
* {{reference-paper|author=Chen, J. P. and Barter, J. W.|year=1998|title=Comparison of Time-Accurate Calculations for the Unsteady Interaction in Turbomachinery Stage|rest=AIAA Paper 98-3292}}<br />
<br />
* {{reference-paper|author=Denton, J. and Singh, U.|year=1979|title=Time Marching Methods for Turbomachinery Flow Calculations|rest=VKI Lecture Series 1979-7, von Karman Institute}}<br />
<br />
* {{reference-paper|author=Erdos, J., Alzner, E. and McNally, W.|year=1977|title=Numerical Solution of Periodic Transonic Flow Through a Transonic Fan Stage|rest=AIAA Journal, vol. 15, no. 11, pp. 1559-1568}}<br />
<br />
* {{reference-paper|author=Gerolymos, G., Vinteler, D., Haugeard, R., Tsange, G. and Vallet, I.|year=1996|title=On the Computation of Unsteady Turbomachinery Flows – Part II – Rotor/Stator Interaction using Euler Equations|rest=AGARD Report CP-571}}<br />
<br />
* {{reference-paper|author=Giles, M. B.|year=1990|title=Stator/Rotor Interaction in a Transonic Turbine|rest=AIAA Journal of Propulsion and Power, vol. 6, no. 5}}<br />
<br />
* {{reference-paper|author=He, L, and Ning, W.|year=1998|title=Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines|rest=AIAA Journal, vol. 36, no. 11, pp. 2005-2012}}<br />
<br />
* {{reference-paper|author=Mayle, R. E.|year=1991|title=The Role of Laminar-Turbulent Transition in Gas Turbine Engines|rest=ASME Journal of Turbomachinery, vol. 113, pp. 509-537}}<br />
<br />
* {{reference-paper|author=Menter, F.R. |year=1994|title=Two-equation eddy-viscosity turbulence models for engineering applications|rest=AIAA Journal, vol. 32, pp. 269-289}}<br />
<br />
* {{reference-paper|author=Savill, M., Dick, E., Hanjalić, K. and Voke, P.|year=2002|title=Synthesis Report of the ERCOFTAC Transition Modeling – TRANSPRETURB Thematic Network Activities 1998-2002|rest=Ecoftac Bulletin 54, September 2002, pp. 5-16}}<br />
<br />
* {{reference-paper|author=Wang, X. and Chen J.|year=2004|title=A Post-Processor to Render Turbomachinery Flows Using Phase-Lag Simulations|rest=AIAA Paper 04-615}}<br />
<br />
== External links ==<br />
<br />
* [http://www.qnet-cfd.net/newsletter/8th/n8_40-46.pdf QNET-CFD Best Practise Advice for Turbomachinery Internal Flows]<br />
* [http://www.qnet-cfd.net/newsletter/7th/n7_05.pdf State of the art in Industrial CFD for Turbomachinery Flows, QNET-CFD Network Newsletter, Volume 2, No. 3 – December 2003]<br />
* [https://pronet.wsatkins.co.uk/marnet/guidelines/guide.html MARNET-CFD Best Practice Guidelines for Marine Applications of CFD]<br />
* [http://www.stanford.edu/group/ctr/ResBriefs/temp05/weide.pdf On Large Scale Turbomachinery Computations, by E. van der Weide et. al., Center for Turbulence Research, Annual Research Briefs 2005]</div>DavidFhttp://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFDBest practice guidelines for turbomachinery CFD2008-09-12T09:40:11Z<p>DavidF: /* Turbulence inlet conditions */</p>
<hr />
<div>This article contains a summary of the most important knowledge and experience a CFD engineer needs in order to perform CFD simulations of turbomachinery components. The guide is mainly aimed at axial turbomachinery. The goal is to give a CFD engineer, who has just started working with turbomachinery simulations, a head start and avoid some of the most difficult pit-falls. Experienced turbomachinery CFD engineers can also use the guidelines in order to learn what other experts consider best practice. The intended audience is expected to know basic CFD terminology and have a basic turbomachinery knowledge, but no detailed knowledge about CFD for turbomachinery is needed. <br />
<br />
== Deciding what type of simulation to do ==<br />
<br />
Before starting a new turbomachinery simulation it is wise to think carefully of what it is that should be predicted and what physical phenomena that affect the results. This chapter contains a brief overview of the various types of simulations and some hints of what can be predicted with them.<br />
<br />
=== 2D, Quasi-3D or 3D ===<br />
<br />
2D simulations are often used in the early design phase in order to obtain a typical 2D section of a blade. For cases with many long blades or vanes, like low-pressure turbines, a 2D simulation can also provide reasonable results. If the area of the flow-path changes significantly in the axial direction it might be necessary to instead make a quasi-3D simulation. A quasi-3D simulation is a 2D simulation in which extra source terms are used to account for the acceleration/decelleration caused by a changing channel height or growing end-wall boundary layers. Codes focused on turbomachinery applications often have the possibility to perform quasi-3D simulations, but most general purpose CFD codes can not do this type of simulations, or require user coding to implement the correct source terms in the equations. Please contact your software distributor if your code does not have the quasi-3d possibility and you require it. Many codes require special routines or hidden commands to enable this feature.<br />
<br />
Full 3D simulations are necessary if a true 3D geometry is needed to obtain correct secondary flows and/or shock locations. For low-aspect-ratio cases with only a few short blades, like for example structurally loaded turbine outlet guide vanes, the secondary flow development is important and a 3D simulation is often necessary in order to obtain reasonable results. For applications where the end-wall boundary layers grow very quickly and interact with a large part of the flow-field it is necessary to perform a full 3D simulation. This is often the case in compressors and fans, where the negative pressure gradients make the boundary layers grow much quicker than what they do in for example turbines. For cases where the shock location is very critical, like in transonic compressors, it is also often necessary to perform a 3D simulation in order to obtain reasonable shock locations.<br />
<br />
=== Inviscid or viscid ===<br />
<br />
For attached flows close to the design point and without any large separations it is often sufficient with an inviscid Euler simulation in order to obtain reasonable blade loadings and pressure distributions. Note that inviscid Euler simulations should only be used if the boundary layers are judged to not have a significant effect on the global flow-field. A viscid Navier-Stokes simulation is necessary in order to predict losses, secondary flows and separations. As soon as separations are of interest it is of course also necessary to do a viscid simulation. Note that with todays computers it is often not time and resources that make users run inviscid Euler simulations. Running viscid Navier-Stokes simulations is now so quick that it is not a time problem any more. Euler simulations are still interesting though, since with an inviscid Euler simulation you don't have to worry about wall resolutions, y+ values, turbulence modeling errors etc.<br />
<br />
=== Transient or Stationary ===<br />
<br />
Most turbomachinery simlation are performed as stationary simulations. Transient simulations are done when some kind of transient flow behaviour has a strong influence on the global flow field. Examples of transient simulations are detailed simulations of rotor-stator interaction effects, simulations of large unsteady separations etc. Sometimes when you perform a steady stationary simulation you can see tendencies of unsteady behvaiour like for example periodic vortex shedding behind blunt trailing edges. This is often first seen as periodical variations of the residuals. If the unsteady tendencies are judged to not affect the overall simulation results if might be necessary to coarsen the mesh close to the vortex shedding or run a different turbulence model in order to make the simulation converge. Sometimes you are still forced to run a transient simulation and average the results if you don't obtain a converged steady solution.<br />
<br />
== Meshing ==<br />
<br />
In turbomachinery applications structured muli-block hexahedral meshes are most often used for flow-path simulations. In most solvers a structured grid requires less memory, provides superior accuracy and allows a better boundary-layer resolution than an unstructured grid. By having cells with a large aspect ratio around sharp leading and trailing edges a structured grid also provides a better resolution of these areas. Many companies have automatic meshing tools that automatically mesh blade sections with a structured mesh without much user intervention.<br />
<br />
Unstructured meshes are used for more complex and odd geometries where a structured mesh is difficult to create. Typical examples where unstructured meshes are often used are blade tip regions, areas involving leakage flows and secondary air systems, film cooling ducts etc.<br />
<br />
When meshing avoid to create large jumps in cell sizes. Typically the cell size should not change with more than a factor of 1.25 between neighbouring cells. For structured meshes also try to create fairly continous mesh lines and avoid discontinuities where the cell directions suddenly change. For multi-block structured meshes avoid placing the singular points where blocks meet in regions with strong flow gradients since most schemes have a lower accuracy in these singular points.<br />
<br />
=== Mesh size guidelines ===<br />
<br />
It is difficult to define, a priori, the mesh size. The required mesh size depends on the purpose of the simulation. <br />
<br />
If the main goal is to obtain static pressure forces a coarse mesh is often able to obtain a good solution, especially when an accurate resolution of the boundary layers is not required. For 2D inviscid simulations of one blade a mesh with say 3,000 cells is most often sufficient. For 3D inviscid blade simulations a mesh size of about 40,000 cells is usually sufficient. On inviscid Euler simulations the cells should be fairly equal in size and no boundary layer resolution should be present. Avoid having too skewed cells.<br />
<br />
For loss predictions and cases where boundary layer development and separation is important the mesh needs to have a boundary layer resolution. The boundary layer resolution can either be coarse and suitable for a [[Wall functions|wall function]] simulation or very fine and suitable for a [[Low-Re resolved boundary layers|low-Re]] simulation. For further information about selecting the near-wall turbulence model please see the [[#Turbulence modeling|turbulence modeling]] section. In 3D single-blade simulations a decent wall-function mesh typically has around 100,000 cells. This type of mesh size is suitable for quick design iterations where it is not essential to resolve all secondary flows and vortices. A good 3D wall-function mesh of a blade section intended to resolve secondary flows well should have at least 400,000 cells . A good low-Re mesh with resolved boundary layers typically has around 1,000,000 cells. <br />
<br />
In 2D blade simulations a good wall-function mesh has around 20,000 cells and a good low-Re mesh with resolved boundary layers has around 50,000 cells.<br />
<br />
Along the suction and pressure surfaces it is a good use about 100 cells in the streamwise direction. In the radial direction a good first approach is to use something like 30 cells for a wall-function mesh and 100 cells for a low-Re mesh.<br />
<br />
It is important to resolve leading and trailing edges well. Typically at least 10 cells, preferably 20 should be used around the leading and trailing edges. For very blunt and large leading edges, like those commonly found on HP turbine blades, 30 or more cells can be necessary.<br />
<br />
Cases which are difficult to converge with a steady simulation and which show tendencies of periodic vortex shedding from the trailing edge, can sometimes be "tamed" by using a coarse mesh around the trailing edge. This, of course, reduces the accuracy but can be a trick to obtain a converged solution if time and computer resouces does not allow a transient simulation to be performed.<br />
<br />
=== Boundary layer mesh ===<br />
<br />
For design iteration type of simulations where a [[Wall functions|wall function]] approach is sufficient [[Dimensionless wall distance (y plus) | y+]] for the first cell should be somewhere between 20 and 200. The outer limit is dependent on the actual Re number of the simulations. For cases with fairly low Re numbers make sure to keep the maximum y+ as low as possible. For more accurate simulations with resolved boundary layers the mesh should have a y+ for the first cell which is below 1. Some new codes are now using a hybrid wall treatment that allows a smooth transition from a coarse wall-function mesh to a resolved low-Re mesh. Use some extra care when using this type of hybrid technique since it is still fairly new and unproven.<br />
<br />
Outside of the first cell at a wall a good rule of thumb is to use a growth ratio normal to the wall in the boundary layer of maximum 1.25. For a low-Re mesh this usually gives around 40 cells in the boundary layer whereas a wall-function mesh does not require more than 10 cells in the boundary layer.<br />
<br />
If you are uncertain of which wall distance to mesh with you can use a [http://www.cfd-online.com/Links/tools.html#yplus y+ estimation tool] to estitmate the distance needed to obtain the desired y+. These estimation tools are ''very handy'' if you have not done any previous similar simulations. <br />
<br />
As a rule of thumb a wall-function mesh typically requires areound 5 to 10 cells in the boundary layer whereas a resolved low-Re mesh requires about 40 cells in the boundary layer.<br />
<br />
== Boundary conditions ==<br />
<br />
Describe different types of boundary conditions and when they should be used:<br />
<br />
* Total pressure in, static pressure out<br />
* Absorbing boundary conditions<br />
* ...<br />
<br />
There are different types of boundary conditions you can use:<br />
*Mass flow inlet, static pressure outlet.<br />
To put this BC you must allocate inlet surface enough far away from the turbomachine impeller. The velocity distribution is not<br />
constant over all the surface. If you put this "lie" away from your problem it works well.<br />
<br />
<br />
=== Turbulence inlet conditions ===<br />
<br />
Prescribing realistic turbulence inlet conditions is important and usually difficult. For two-equation turbulence models two different turbulence values need to be specified on the inlet. The most common way of specifying the inlet turbulence variables is to give an inlet [[Turbulence intensity|turbulence level]] and [[Turbulence length scale|length-scale]]. For simulations where the incoming boundary layers are estimated to not have any significant effect on the secondary flows and global flow-field downstream it is usually sufficient to specify a constant incoming turbulence level and length-scale.<br />
<br />
The incoming [[Turbulence intensity|turbulence level]] is dependent on which component is analyzed. Best is of course to have measurments from a similar application in order to be able to prescribe a fairly realistic value. Without measurements it is necessary to make an educated guess. Simulations with very complex and turbulence-generating components upstream will have a very high incoming turbulence level. For example, a high-pressure turbine just downstream of a turbulence generating combustor might have incoming turbulence levels up to 20%. Fans and low-pressure compressors with not many components upstream might have as low incoming turbulence level as 1%. For components between these two extreme examples, like high-pressure compressors, low-pressure turbines etc. a turbulence level of around 5% might be realistic. If you do not have any measurements and are unsure about which incoming turbulence level to prescribe it is always a good idea to run a few different simulations with say one half and one double incoming turbulence level in order to estimate how important it is for the simulation results.<br />
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The incoming [[Turbulence length scale|turbulence length-scale]] is often even more difficult to guess than the incoming turbulence level. Measurements might sometimes include the incoming turbulence level, but very seldom also include the incoming turbulent length-scale. The best way of guessing a realistic incoming length-scale is to use the geometrical properties of the upstream components. The incoming turbulence length-scale can be estimated as say the thickness of upstream blades or somewhere between 2% and 20% of the incoming channel heigth. Fortunately the incoming turbulence length-scale is usually not that important for the end results. However, some k-omega models can have an unrelisticly strong influence on the incoming turbulence length-scale. Hence, be extra carefull about the length-scale when using k-omega models. Note that the [[SST k-omega model]] does not have this unrealistic sensitivity to the inlet turbulence length-scale.<br />
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Instead of specifying an incoming turbulence length-scale it is sometimes more convenient to specify an incoming [[Eddy viscosity ratio|eddy viscosity ratio]]. For low-turbulence cases with no well-specified turbulence generating components upstream it can be difficult to estimate a turbulence length-scale. Specifying an eddy viscosity ratio instead gives a more direct control over how large the effect of the incoming turbulence is. The eddy viscosity ratio describes how large the turbulence viscosity is related to the molecular viscosity. In turbomachinery the eddy viscosity ratio can vary from say 10 in low-turbulence components like fans up to say 1,000 or in rare cases even 10,000 in high-turbulence components like high-pressure turbines. In components between these extremes a typical eddy viscosity ration might be about 200. Even if specifying an inlet turbulence length scale it can still be good to compute what eddy viscosity ratio this corresponds to in order to estimate if it sounds reasonable.<br />
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== Turbulence modeling ==<br />
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Selecting a suitable turbulence model for turbomachinery simulations can be a challenging task. There is no single model which is suitable for all types of simulations. Which turbulence model CFD engineers use has as much to do with beliefs and traditions as with knowledge and facts. There are many different schools. However, below follows some advices that most CFD engineers in the turbomachinery field tend to agree upon.<br />
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For attached flows close to the design point a simple algebraic model like the [[Baldwin-Lomax model]] can be used. Another common choice for design-iteration type of simulations is the one-equation model by [[Spalart-Allmaras model | Spalart-Allmaras]]. This model has become more popular in the last years due to the many inherent problems in more refined two-equation models. The big advantage with both the [[Baldwin-Lomax model]] and the [[Spalart-Allmaras model]] over more advanced models is that they are very robust to use and rarely produce completely unphysical results.<br />
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In order to accurately predict more difficult cases, like separating flows, rotating flows, flows strongly affected by secondary flows etc. it is often necessary to use a more refined turbulence model. Common choices are two-equation models like the <math>k-\epsilon</math> model.<br />
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Two-equation models are based on the [[Boussinesq eddy viscosity assumption]] and this often leads to an over-production of turbulent energy in regions with strong acceleration or deceleration, like in the leading edge region, regions around shocks and in the suction peak on the suction side of a blade. To reduce this problems it is common to use a special model variant using, for example, [[Durbin's realizability constraint]] or the [[Kato-Launder modification]]. Note that different two-equation models behave differently in these problematic stagnation and acceleration regions. Worst is probably the standard <math>k-\epsilon</math> model. <math>k-\omega</math> model are slightly better but still do not behave well. More modern variants like Menter's [[SST k-omega model]] also has problems, wheras the [[V2-f models|v2f model ]] by Durbin behaves better.<br />
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=== Near-wall treatment ===<br />
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For on-design simlations without any large separated regions it is often sufficient to use a [[wall-function model]] close to the wall, preferably with some form of non-equilibrium wall-function that is sensitised to streamwise pressure gradients.<br />
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For off-design simulation, or simulations involving complex secondary flows and separations, it is often necessary to use a [[low-Re model]]. There exists many low-Re models that have been used with success in turbomachinery simulations. A robust and often good choice is to use a one-equation model, like for example the [[Wolfstein model]], in the inner parts of the boundary layer. There are also several Low-Re <math>k-\epsilon</math> models that work well. Just make sure they don't suffer from the problem with overproduction of turbulent energy in regions with strong acceleration or deceleration. In the last few years Menter's low-Re <math>SST k-\omega</math> model has gained increased popularity.<br />
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=== Transition prediction ===<br />
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Transition refers to the process when a laminar boundary layer becomes unstable and transitions to a turbulent boundary layer. There are two types of transition - natural transition, where inherent instabilities in the boundary layer cause the transition and by-pass transition, where convection and diffusion of turbulence from the free-stream into the boundary layer cause the transition. Most transitions in turbomachinery are by-pass transitions caused by free-stream turbulence and other external disturbances like wakes, vortices and surface defects.<br />
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Simulating transition in a CFD code accurately is very difficult. Often a separate transition model needs to be solved in order to specify the transition location and length. Predicting natural transition in a pure CFD code is not possible. Predicting by-pass transition in a pure CFD code is almost impossible, although there are people who claim to be able to predict by-pass transition with low-Re two-equation models. However, this is usually on special test cases and with simulations that have been tuned for these special cases, see for example [Saville 2002]. In reality transition is a very complex and sensitive process where disturbances like incoming wakes and vortices from previous stages, surface roughness effects and small steps or gaps in the surfaces play a significant role. <br />
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The turbomachinery codes that have transition prediction models often use old ad-hoc models like the Abu-Ghannam and Shaw model [Abu-Ghannam 1980] or the Mayle model [Mayle 1991]. These models can be quite reliable if they have been validated and tuned for a similar application. Do not trust your transition predictions without having some form of experimental validation. Menter has also recently developed a new form of transition model that might work fairly well, but it is still too new and untested.<br />
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For some turbomachinery applications, like modern high-lift low-pressure turbines, transition is critical. For these applications a CFD code with a transition model that has been tuned for this type of applications should be used.<br />
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== Numerical considerations ==<br />
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Use at least a second order accurate scheme for the flow variables. Some codes require a first order scheme for the turbulent variables (<math>k</math> and <math>\epsilon</math>) in order to converge well. It might be sufficient with a first order scheme only on the turbulence variables, but a second order scheme is of course preferable.<br />
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=== Convergence criteria ===<br />
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To know when a solution is converged is not always that easy. You need some prior experience of your CFD code and your application to judge when a simulation is converged. For normal pure aero simulations without resolved walls, i.e. with wall functions or inviscid Euler simulations, convergence can most often be estimated just by looking at the residuals. Exacty what the residuals should be is not possible to say, it all depends on how your particular code computes and scales the residuals. Hence, make sure to read the manuals and plot the convergence of a few global parameters before you decide what the residuals should be for a solution to converege. Note also that many manuals for genereal purpose CFD codes list overly agressive convergence critera that often produce unconverged results. <br />
For simulations with resolved walls it is good to look at the convergence of some global properties, like total pressure losses from the inlet to the outlet. For heat transfer simulations it is even more tricky since the aerodynamic field can look almost converged although the thermal field is not converged at all. If doing heat transfer simulations make sure to plot the heat-transfer, run for some time, and plot it again to make sure that it doesn't change anymore. With very well resolved walls and heat transfer it can sometimes take 10 times longer for the thermal field to converge.<br />
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=== Single or double precision ===<br />
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With todays computers and cheap memory prices it often does not cost much extra to run in double precision. Before using single precision you should first investigate how your software and hardware works with double precision. If the extra time and memory needed for double precision is negligible you should of course always run in double precision. With double precision you never have to worry about round-off errors. Always using double precision is one way of avoiding one type of pit-falls in the complex world of CFD simulations. Use double precision when you have resolved boundary layers (Y+ around 1) and when you use advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions.<br />
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== Multi-stage analysis ==<br />
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Multi-stage analysis can be done in different ways:<br />
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* Steady mixing-plane simulations<br />
* Frozen rotor simulations<br />
* Unsteady sliding-mesh stator-rotor simulations<br />
* Other '''advanced''' multi-stage methods<br />
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=== Steady mixing-plane simulations ===<br />
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Since the mixing-plane method was first introduced in 1979 [Denton & Singh 1979] is has become the industry standard type of rotor-stator simulations. A mixing-plane simulation is steady and only requires one rotor blade and one stator blade per stage. Between the rotating blade passage and the steady vane passage the flow properties are circumferentially averaged in a so-called mixing-plane interface. This will of course remove all transient rotor-stator interactions, but it still gives fairly representative results. In some commercial codes (CFX for example) mixing-plane interfaces are also called stage-interfaces.<br />
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=== Frozen rotor simulations ===<br />
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In a frozen rotor simulation the rotating and the stationary parts have a fixed relative position. A frame transformation is done to include the rotating effect on the rotating sections. This will give a steady flow and no transient effects are included. With a frozen-rotor simulation rotating wakes, secondary flows, leading edge pressure increases etc. will always stay in exactly the same positions. This makes a frozen rotor simulation very dependent on exactly how the rotors and the stators are positioned. Most often a mixing-plane simulation gives better results. Frozen rotor simulations are mainly performed to obtain a good starting flow-field before doing a transient sliding-mesh simulation.<br />
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=== Unsteady sliding-mesh stator-rotor simulations ===<br />
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This is the most complete type of stator-rotor simulation. In most engines the number of stators and rotors do not have a common denominator (to avoid instabilities caused by resonance between different rings). Hence, to make a full unsteady sliding-mesh computation it is necessary to have a mesh which includes the full wheel with all stators vanes and all rotor blades. This is often not possible, instead it is necessary to reduce the number of vanes and blades by finding a denominator that is '''almost''' common and then scale the geometry slightly circumferentially. Here is an example:<br />
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*Real engine: 36 stator vanes, 41 rotor blades<br />
*Approximated engine: 41 stator vanes, 41 rotor blades, making it possible to simulate only 1 stator vane and 1 rotor blade<br />
*Scaling of stator: All stator vanes are scaled by 36/41 = 0.8780 circumferentially.<br />
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=== Other '''advanced''' multi-stage methods ===<br />
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Time-inclinded, Adamszyk stresses ...<br />
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== Heat transfer predictions ==<br />
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Besides listing the general heat transfer mechanisms involved <br />
(namely conduction, convection, radiation)<br />
heat transfer prediction in CFD may be seen as or split into two cases. <br />
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'''Mesh consists of fluid domain(s)''': <br />
you want to know how heat inserted into the fluid domain (e.g. via the flowing medium) changes the temperature field along the flow path. Be it a steady or transient run, a fluid which enters with a given temperature will usually experience temperature variations, for instance, caused by convective boundary conditions, prescribed temperature profiles at flow obstacles etc. <br />
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'''Mesh consists of fluid and solid domain(s)''': <br />
additionally to the above, you want info too wrt the (spatial, temporal or even spectral) temperature field distribution in surrounding, confining or immersed solids like channel walls or heat exchanger tubes. This is also called conjugate heattransfer CHT in the CFD context. CHT requires a good boundary layer resolution, usually the wall mesh needs to be rather refined, to obtain realistic heatflux results at the fluid/solid interface. Flow and heat transfer convergence require different time step settings, to properly capture changes in flow and heat quantities respectively. <br />
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In eighter case, verify (strict necessity depends on CFD code used, CFX for instance checks and assists in regard) that model dimensions, boundary conditions and properties are in consistent units, hold appropriate values. Check temperature-dependence of properties and other numbers before the run.<br />
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In heat transfer predictions (depending on the CFD code in use) <br />
besides the flow solver, you may have to activate the thermal solver too, <br />
as a job specification.<br />
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== Acoustics and noise ==<br />
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A whole separate research subject, difficult. <br />
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Tone noise possible. Often run with linearzised solvers in the frequency domain.<br />
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Jet noise possible. Often run with LES or DES simulations that either also resolve the sound waves or couples to a separate acoustic solver.<br />
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Turbomachinery broadband noise not possible yet, or at least a great challenge.<br />
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== Errors and uncertainties ==<br />
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CFD is still a tool which requires that a user has a good understanding of uncertainties and errors that might spoil a CFD similation. There exists no error control in CFD and any CFD simulation must be interpreted by an expirenced user to have some credibility. Without some knowledge about possible errors and how they can be handled a CFD simulation can not be trusted. Errors can occur at different places:<br />
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* Definition of the problem - What needs to be analyzed?<br />
* Selection of the solution strategy - What physical models and what numerical tools should be used?<br />
* Development of