http://www.cfd-online.com/W/index.php?title=Special:Contributions/Salva&feed=atom&limit=50&target=Salva&year=&month=CFD-Wiki - User contributions [en]2016-10-01T08:31:25ZFrom CFD-WikiMediaWiki 1.16.5http://www.cfd-online.com/Wiki/Conditional_filteringConditional filtering2006-05-08T12:38:07Z<p>Salva: </p>
<hr />
<div>A conditional filtering operation of a variable <math>\Phi </math> is defined as<br />
:<math><br />
\overline{\Phi|\eta} \equiv \frac{\int_V \Phi \psi_\eta \left(<br />
\xi(\mathbf{x'},t) - \eta<br />
\right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'}{\bar{P}(\eta)}<br />
</math><br />
<br />
where <math>G </math>is a positive defined space filter with filter width <math> \Delta </math> (see [[LES filters]]),<br />
<math> \psi_\eta </math> is a fine-grained [[probability density function]],<br />
which is taken as a Dirac delta <math> \psi_\eta \equiv \delta ( \xi - \eta ) </math>.<br />
The probability density function <br />
<math>\bar{P}(\eta) </math> is a [[subgrid PDF]] and <math> \eta </math> is the sample space of the passive scalar<br />
<math> \xi </math>. <br />
In variable density flows, the conditional density-weighted (Favre) filtering is used.<br />
Using the density-weighted PDF , <math> \tilde {P}(\eta) </math>, the conditionally Favre filtered operation is<br />
<br />
:<math><br />
\bar{\rho} \widetilde{\Phi|\eta} \equiv \frac{\int_V \rho \Phi \psi_\eta \left(<br />
\xi(\mathbf{x'},t) - \eta<br />
\right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'}{\tilde {P}(\eta)}<br />
</math><br />
<br />
The relation between Favre and conventional PDF's is<br />
:<math> <br />
\bar{\rho} \tilde{P}(\eta) = \overline{\rho|\eta}\bar{P}(\eta)<br />
</math></div>Salvahttp://www.cfd-online.com/Wiki/Subgrid_PDFSubgrid PDF2006-05-08T12:30:17Z<p>Salva: </p>
<hr />
<div>A subgrid [[probability density function]] <math> \bar{P}(\eta) </math> ,<br />
also known as filtered density function (FDF),<br />
is the distribution function of scalar <math> Z </math> at subgrid scales.<br />
<br />
The probability of observing values between <math> \eta < Z < \eta + d\eta </math><br />
within the filter volume is <math> \bar{P}(\eta) d\eta </math><br />
<br />
:<math><br />
\bar{P}(\eta) <br />
\equiv \int_V \delta \left(<br />
Z(\mathbf{x'},t) - \eta<br />
\right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'<br />
</math><br />
<br />
where <math> \delta </math> is the Dirac delta function,<br />
<math> G </math> is a positive defined filter function<br />
with filter width <math> \Delta </math>.<br />
<br />
The joint subgrid PDF of <math> N </math> scalars is<br />
<br />
:<math><br />
\bar{P}(\underline{\psi}) <br />
\equiv \int_V<br />
\prod_i^N \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right)<br />
G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'<br />
</math><br />
<br />
where <math> \underline{\psi} = ( \psi_1,\psi_2,.. \psi_N)</math><br />
is the phase space for the scalar variables<br />
<math> \underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)</math>.<br />
<br />
A density weighted FDF, <math> \tilde{P}(\eta) </math>, can be obtained as<br />
<br />
:<math><br />
\bar{\rho} \tilde{P}(\eta) <br />
\equiv \int_V \rho \delta \left(<br />
Z(\mathbf{x'},t) - \eta<br />
\right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'<br />
</math><br />
<br />
and in the same manner for the joint FDF<br />
:<math><br />
\bar{\rho} \tilde{P}(\underline{\psi}) <br />
\equiv \int_V \rho<br />
\prod_i^N \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right)<br />
G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'<br />
</math></div>Salvahttp://www.cfd-online.com/Wiki/Subgrid_PDFSubgrid PDF2006-05-08T12:29:47Z<p>Salva: </p>
<hr />
<div>A subgrid [[probability density function]] <math> \bar{P}(\eta) </math> ,<br />
also known as filtered density function (FDF),<br />
is the distribution function of scalar <math> Z </math> at subgrid scales.<br />
<br />
The probability of observing values between <math> \eta < Z < \eta + d\eta </math><br />
within the filter volume is <math> \bar{P}(\eta) d\eta </math><br />
<br />
:<math><br />
\bar{P}(\eta) <br />
\equiv \int_V \delta \left(<br />
Z(\mathbf{x'},t) - \eta<br />
\right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'<br />
</math><br />
<br />
where <math> \delta </math> is the Dirac delta function,<br />
<math> G </math> is a positive defined filter function<br />
with filter width <math> \Delta </math>.<br />
<br />
The joint subgrid PDF of <math> N </math> scalars is<br />
<br />
:<math><br />
\bar{P}(\underline{\psi}) <br />
\equiv \int_V<br />
\prod_i^N \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right)<br />
G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'<br />
</math><br />
<br />
where <math> \underline{\psi} = ( \psi_1,\psi_2,.. \psi_N)</math><br />
is the phase space for the scalar variables<br />
<math> \underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)</math>.<br />
<br />
A density weighted FDF, <math> \tilde{P}(\eta) </math>, can be obtained as<br />
<br />
:<math><br />
\bar{\rho} \tilde{P}(\eta) <br />
\equiv \int_V \rho \delta \left(<br />
Z(\mathbf{x'},t) - \eta<br />
\right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'<br />
</math><br />
<br />
and in the same manner for the joint FDF<br />
:<math><br />
\bar{\rho} \tilde{P}(\underline{\psi}) <br />
\equiv \int_V<br />
\prod_i^N \rho \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right)<br />
G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'<br />
</math></div>Salvahttp://www.cfd-online.com/Wiki/Subgrid_PDFSubgrid PDF2006-05-08T12:25:36Z<p>Salva: </p>
<hr />
<div>A subgrid [[probability density function]] <math> \bar{P}(\eta) </math> ,<br />
also known as filtered density function (FDF),<br />
is the distribution function of scalar <math> Z </math> at subgrid scales.<br />
<br />
The probability of observing values between <math> \eta < Z < \eta + d\eta </math><br />
within the filter volume is <math> \bar{P}(\eta) d\eta </math><br />
<br />
:<math><br />
\bar{P}(\eta) <br />
\equiv \int_V \delta \left(<br />
Z(\mathbf{x'},t) - \eta<br />
\right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'<br />
</math><br />
<br />
where <math> \delta </math> is the Dirac delta function,<br />
<math> G </math> is a positive defined filter function<br />
with filter width <math> \Delta </math>.<br />
<br />
The joint subgrid PDF of <math> N </math> scalars is<br />
<br />
:<math><br />
\bar{P}(\underline{\psi}) <br />
\equiv \int_V<br />
\prod_i^N \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right)<br />
G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'<br />
</math><br />
<br />
where <math> \underline{\psi} = ( \psi_1,\psi_2,.. \psi_N)</math><br />
is the phase space for the scalar variables<br />
<math> \underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)</math></div>Salvahttp://www.cfd-online.com/Wiki/Kinetic_energy_subgrid-scale_modelKinetic energy subgrid-scale model2006-05-08T12:21:17Z<p>Salva: </p>
<hr />
<div>The subgrid-scale kinetic energy is defined as <br><br />
:<math> k_{\rm sgs} = \frac{1}{2}\left(\overline{u_k^2} - \overline{u}_k^2 \right) </math><br />
<br />
<br />
The subgrid-scale stress can then be written as <br><br />
<math> \tau_{ij} - \frac{2}{3} k_{\rm sgs} \delta_{ij} =-2 C_k k_{\rm sgs}^{1/2} \Delta \overline{S}_{ij} </math> <br><br />
this gives us the transport equation for subgrid-scale kinetic energy <br><br />
<math> \frac{\partial \overline k_{\rm sgs}}{\partial t} + \frac{\partial \overline u_{j} \overline k_{sgs}} {\partial x_{j}} = - \tau_{ij} \frac{\partial \overline u_{i}}{\partial x_{j}} - C_{\varepsilon} \frac{k_{\rm sgs}^{3/2}}{\Delta} + \frac{\partial}{\partial x_{j}} \left( \frac{\mu_t}{\sigma_k} \frac{\partial k_{\rm sgs}}{\partial x_{j}} \right) </math><br />
<br />
<br />
<br />
The subgrid-scale eddy viscosity,<math> \mu_{t} </math>, is computed using <math> k_{\rm sgs} </math> as <br />
<br />
<br />
<math> \mu_{t} = C_k k_{\rm sgs}^{1/2} \Delta </math></div>Salvahttp://www.cfd-online.com/Wiki/Smagorinsky-Lilly_modelSmagorinsky-Lilly model2006-05-08T12:20:03Z<p>Salva: </p>
<hr />
<div>The Smagorinsky model could be summerised as:<br />
:<math><br />
\tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\left( {C_s \Delta } \right)^2 \left| {\bar S} \right|S_{ij} <br />
</math> <br><br />
<br />
In the Smagorinsky-Lilly model, the eddy viscosity is modeled by <br><br />
<br />
<br />
:<math><br />
\mu _{sgs} = \rho \left( {C_s \Delta } \right)^2 \left| {\bar S} \right|<br />
</math> <br />
<br><br />
<br />
Where the filter width is usually taken to be<br />
:<math><br />
\Delta = \left( \mbox{Volume} \right)^{\frac{1}{3}} <br />
</math> <br />
<br><br />
and <br />
:<math><br />
\bar S = \sqrt {2S_{ij} S_{ij} } <br />
</math><br />
<br />
The effective viscosity is calculated from <br><br />
:<math><br />
\mu _{eff} = \mu _{mol} + \mu _{sgs} <br />
</math><br />
The Smagorinsky constant usually has the value: <br />
:<math><br />
C_s = 0.1 - 0.2<br />
</math></div>Salvahttp://www.cfd-online.com/Wiki/Dynamic_subgrid-scale_modelDynamic subgrid-scale model2006-05-04T11:06:06Z<p>Salva: /* Original model */</p>
<hr />
<div>== Introduction ==<br />
The limitations of the [[Smagorinsky-Lilly model|Smagorinsky model]] have lead to the formulation of more general subgrid-scale models. Perhaps the best known of these newer models is the dynamic subgrid-scale (DSGS) model of [[#References|Germano et al (1991)]]. The DSGS model may be viewed as a modification of the Smagorinsky model, as the dynamic model allows the Smagorinsky constant <math>C_S</math> to vary in space and time. <math>C_S</math> is calculated locally in each timestep based upon two filterings of the flow variables, which we<br />
will denote by superscript <math>r</math> and superscript <math>t</math>. These filters are called the grid filter and the test filter, respectively, and the test filter width is assumed to be larger the grid filter width.<br />
<br />
== Original model ==<br />
Filtering with the grid filter results in the normal LES equations, with <math>\tau_{ij}</math> given by <br />
<br />
<math><br />
\tau_{ij}=(u_iu_j)^r-u_i^ru_j^r.<br />
</math><br />
<br />
Filtering again with the test filter results in a similar set of equations, but<br />
with a different subgrid-scale stress term, given by<br />
<br />
<math><br />
T_{ij}= (u_iu_j)^{rt}-u_i^{rt}u_j^{rt},<br />
</math><br />
<br />
where the superscript <math>rt</math> indicates grid filtering followed by test filtering. The two subgrid-scale stress terms are related by the Germano identity:<br />
<br />
<math><br />
\mathcal{L}_{ij}=T_{ij}-\tau_{ij}^t,<br />
</math><br />
<br />
where <br />
<br />
<math><br />
\mathcal{L}_{ij}=(u_i^ru_j^r)^t-u_i^{rt}u_j^{rt}<br />
</math><br />
<br />
is the resolved turbulent stress. The Germano identity is used to calculate dynamic local values for <math>C_S</math> by applying the Smagorinsky model to both <math>T_{ij}</math> and <math>\tau_{ij}</math>. The anisotropic part of <math>\mathcal{L}_{ij}</math> is the represented as<br />
<br />
<math><br />
\mathcal{L}_{ij}-\delta_{ij}\mathcal{L}_{kk}/3 = -2C_S M_{ij},<br />
</math><br />
<br />
where <br />
<br />
<math><br />
M_{ij}=(\Delta^t)^2|S^{rt}|S^{rt}_{ij} - (\Delta^r)^2<br />
\left(|S^{r}|S^{r}_{ij}\right)^t.<br />
</math><br />
<br />
<math>C_S</math> may now be computed as<br />
<br />
<math><br />
C_S^2=-\frac{1}{2}\frac{\mathcal{L}_{kl}S^r_{kl}}{M_{mn}S^r_{mn}}.<br />
</math><br />
<br />
In practice, the DSGS model requires stabilization. Often, this has been done by averaging<br />
<math>C_S</math> in a homogeneous direction. In cases where this is not possible, local averaging has been used in place of an average in a homogenous direction.<br />
<br />
== Alternate solution (Lilly) == <br />
[[#References|Lilly (1991)]] proposed a least squares procedure that is generally preferred to the original calculation of <math>C_S</math>:<br />
<br />
<math><br />
C_S^2=-\frac{1}{2}\frac{\mathcal{L}_{kl}M_{kl}}{M_{mn}M_{mn}}.<br />
</math><br />
<br />
Stabilization must also be employed here as well.<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Germano, M., Piomelli, U., Moin, P. and Cabot, W. H.|year=1991|title=A Dynamic Subgrid-Scale Eddy Viscosity Model|rest=Physics of Fluids A, Vol. 3, No. 7, pp. 1760-1765}}<br />
<br />
* {{reference-paper|author=Lilly, D. K.|year=1991|title=A Proposed Modification of the Germano Subgrid-Scale Closure Method|rest=Physics of Fluids A, Vol. 4, No. 3, pp. 633-635}}</div>Salvahttp://www.cfd-online.com/Wiki/Scalar_dissipationScalar dissipation2006-01-31T17:09:01Z<p>Salva: </p>
<hr />
<div>Scalar dissipation is a very important quantity in non-premixed combustion modelling.<br />
It often provides the connection between the mixing field and the combustion modelling.<br />
It is specially important in flamelet and RANS models.<br />
<br />
In a laminar flow the scalar dissipation rate is defined (units are 1/s) as<br />
<br />
:<math><br />
\chi \equiv 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2<br />
</math><br />
where <math> D </math> is the diffusion coefficient of the scalar.<br />
<br />
In turbulent flows, the scalar dissipation is seen as a scalar energy dissipation} and its role is to<br />
destroy (dissipate) scalar variance (scalar energy) analogous to the dissipation of the<br />
turbulent energy <math> \epsilon </math>. This term is known as the turbulent scalar dissipation<br />
and is written as<br />
<br />
:<math><br />
\chi_t \equiv 2 D \left( \frac{\partial \widetilde{Z''}}{\partial x_j} \right) ^2<br />
</math><br />
<br />
Opposite to the kinetic energy dissipation, most of the scalar dissipation occur at the finest scales.<br />
<br />
In [[Conditional Moment Closure (CMC)]] and [[Flamelet based on conserved scalar ]] models, the quantity of interest<br />
is the " main scalar dissipation rate", <math> \widetilde{\chi} </math>. From Favre Averaging the laminar<br />
dissipation rate<br />
<br />
:<math><br />
\widetilde{\chi} = 2D \widetilde{\left(\frac{\partial Z}{\partial x_j}\right) ^2} \approx<br />
2 D \left( \frac{\partial \tilde{Z}}{\partial x_j} \right) ^2 + \chi_t <br />
</math><br />
<br />
Under RANS assumptions gradient of the scalar fluctuations are much larger than gradients of the means,<br />
and therefore the mean scalar dissipation rate is approximately the turbulent dissipation rate<br />
<math> \widetilde{\chi} \approx \chi_t </math></div>Salvahttp://www.cfd-online.com/Wiki/Probability_density_functionProbability density function2006-01-31T16:19:01Z<p>Salva: </p>
<hr />
<div>Stochastic methods use distribution functions to decribe the fluctuacting scalars<br />
in a turbulent field.<br />
<br />
The distribution function <math> F_\phi(\Phi) </math> of a scalar <math> \phi </math> is the probability <br />
<math> p </math> of finding a value of <math> \phi < \Phi </math><br />
<br />
:<math> <br />
F_\phi(\Phi) = p(\phi < \Phi)<br />
</math><br />
<br />
The probability of finding <math> \phi </math> in a range <math> \Phi_1,\Phi_2 </math><br />
is<br />
<br />
:<math> <br />
p(\Phi_1 <\phi < \Phi_2) = F_\phi(\Phi_2)-F_\phi(\Phi_1)<br />
</math><br />
<br />
The probability density function (PDF) is<br />
<br />
:<math> <br />
P(\Phi)= \frac{d F_\phi(\Phi)} {d \Phi}<br />
</math><br />
<br />
where <math> P(\Phi) d\Phi </math> is the probability of <math> \phi </math> being in the range <math> (\Phi,\Phi+d\Phi) </math>. It follows that<br />
:<math> <br />
\int P(\Phi) d \Phi = 1<br />
</math><br />
Integrating over all the possible values of <math> \phi </math>,<br />
<math> \Phi </math> is the sample space of the scalar variable <math> \phi </math>.<br />
The PDF of any stochastic variable depends "a-priori" on space and time. <br />
:<math> P(\Phi;x,t) </math><br />
for clarity of notation, the space and time dependence is dropped. <br />
<math> P(\Phi) \equiv P(\Phi;x,t) </math><br />
<br />
<br />
From the PDF of a variable, one can define its <math> n </math>th moment as<br />
<br />
:<math><br />
\overline{\phi}^n = \int \phi^n P(\Phi) d \Phi<br />
</math><br />
<br />
the <math> n = 1 </math> case is called the "mean".<br />
<br />
:<math><br />
\overline{\phi} = \int \phi P(\Phi) d \Phi<br />
</math><br />
<br />
Similarly the mean of a function can be obtained as<br />
<br />
:<math><br />
\overline{f} = \int f(\phi) P(\Phi) d \Phi<br />
</math><br />
<br />
Where the second central moment is called the "variance"<br />
<br />
:<math><br />
\overline{u'^2} = \int (\phi-\overline{\phi}) P(\Phi) d \Phi<br />
</math><br />
<br />
For two variables (or more) a joint-PDF of <math> \phi </math> and <math> \psi </math> is defined<br />
:<math> P(\Phi,\Psi;x,t) \equiv P (\Phi,\Psi) </math><br />
<br />
where <math> \Phi \mbox{ and } \Psi </math> form the phase-space for<br />
<math> \phi \mbox{ and } \psi </math>.<br />
The marginal PDF's are obtained by integration over the sample space of one variable.<br />
:<math><br />
P(\Phi) = \int P(\Phi,\Psi) d\Psi<br />
</math><br />
<br />
For two variables the correlation is given by<br />
<br />
:<math> <br />
\overline{\phi' \psi'} = \int (\phi-\overline{\phi}) (\psi-\overline{\psi}) P(\Phi,\Psi) d \Phi d\Psi<br />
</math><br />
<br />
This term often appears in turbulent flows the averaged Navier-Stokes (with <math> u, v </math>) and is unclosed.<br />
<br />
Using Bayes' theorem a joint-pdf can be expressed as<br />
:<math><br />
P(\Phi,\Psi) = P(\Phi|\Psi) P(\Psi)<br />
</math><br />
where <math> P(\Phi|\Psi) </math> is the conditional PDF.<br />
<br />
The conditional average of a scalar can be expressed as a function of the<br />
conditional PDF<br />
:<math><br />
<\phi | \Psi > = \int \phi P(\Phi|\Psi) d \Phi<br />
</math><br />
and the mean value of a scalar can be expressed<br />
<br />
:<math><br />
\overline{\phi} = \int <\phi | \Psi > P(\Psi) d \Psi<br />
</math> <br />
only if <math> \phi </math> and <math> \psi </math> are correlated.<br />
<br />
If two variables are uncorrelated then they are statistically independent and their joint PDF can be expressed as a product of their marginal PDFs.<br />
:<math><br />
P(\Phi,\Psi)= P(\Phi) P(\Psi)<br />
</math><br />
<br />
<br />
Finally a joint PDF of <math> N </math> scalars <math> (\phi_1,\phi_2, ...,\phi_N) </math><br />
is defined as<br />
:<math><br />
P(\underline{\psi}; x,t) \equiv P(\underline{\psi})<br />
</math><br />
where <math> \underline{\psi} = (\psi_1,\psi_2, ...,\psi_N) </math> is the sample space of the array<br />
<math> \underline{\phi} </math>.</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2006-01-31T16:15:29Z<p>Salva: /* Eulerian */</p>
<hr />
<div>== 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 />
Most problems in combustion invlove 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 selon their capability to deal with the different combustion regimes. <br><br />
[[Image:SUMMARY_OF_COMBUSTION_MODELS.jpg]]<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.<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 />
<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) + \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 areaction 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 />
and two linear profiles (in the rich and lean side) at either side of<br />
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 use 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 />
can<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 include CHEMKIN [http://www.reactiondesign.com] for the creation of "flamelet libraries" from detailed chemistry, and KINetics [http://www.reactiondesign.com] for the integration of the libraries into CFD.<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 />
<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>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2006-01-31T16:07:48Z<p>Salva: /* References */</p>
<hr />
<div>== 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 />
Most problems in combustion invlove 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 selon their capability to deal with the different combustion regimes. <br><br />
[[Image:SUMMARY_OF_COMBUSTION_MODELS.jpg]]<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.<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 />
<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) + \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 areaction 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 />
and two linear profiles (in the rich and lean side) at either side of<br />
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 use 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 />
can<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 include CHEMKIN [http://www.reactiondesign.com] for the creation of "flamelet libraries" from detailed chemistry, and KINetics [http://www.reactiondesign.com] for the integration of the libraries into CFD.<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 />
<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2006-01-23T14:49:48Z<p>Salva: /* References */</p>
<hr />
<div>== 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 />
Most problems in combustion invlove 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 selon their capability to deal with the different combustion regimes. <br><br />
[[Image:SUMMARY_OF_COMBUSTION_MODELS.jpg]]<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.<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 />
<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) + \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 areaction 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 />
and two linear profiles (in the rich and lean side) at either side of<br />
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 use 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 />
can<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 include CHEMKIN [http://www.reactiondesign.com] for the creation of "flamelet libraries" from detailed chemistry, and KINetics [http://www.reactiondesign.com] for the integration of the libraries into CFD.<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 />
<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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=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>Salvahttp://www.cfd-online.com/Wiki/Subgrid_varianceSubgrid variance2006-01-11T11:19:32Z<p>Salva: </p>
<hr />
<div>The subgrid variance of a passive scalar is defined as<br />
:<math><br />
\widetilde{Z_{sgs}''^2} = \widetilde{Z^2}- \widetilde{Z}^2<br />
</math><br />
The scalar subgrid variance is also known as the subgrid scalar energy in analogy<br />
to the kinetic subgrid energy.<br />
An equation for the subgrid variance is<br />
<br />
:<math><br />
\frac{\partial \overline{\rho} \widetilde{Z_{sgs}''^2} }{\partial t} +<br />
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z_{sgs}''^2} }{\partial x_j}=<br />
\frac{\partial}{\partial x_j}\left(D \frac{\partial \widetilde{Z_{sgs}''^2} }{\partial x_j} \right)<br />
-2 D \widetilde{\frac{\partial Z}{\partial x_j}\frac{\partial Z }{\partial x_j}}<br />
+ 2 D \frac{\partial \tilde{Z}}{\partial x_j}\frac{\partial \tilde{Z}}{\partial x_j} -<br />
\frac{\partial J_j}{\partial x_j} +<br />
2 \tilde{Z} \frac{\partial}{\partial x_j}<br />
\left( \widetilde{\rho u_j Z}- \overline{\rho}\tilde{u}_j \tilde{Z} \right)<br />
</math><br />
<br />
where <math> J_j = \widetilde{\rho u_j {Z_{sgs}''^2}}- \overline{\rho} \tilde{u_j} \widetilde{Z_{sgs}''^2} </math> is a subgrid variance flux and is often modeled using a gradient approach with turbulent diffusivity.<br />
<br />
Instead of solving the above equation, algebraic models are often used.<br />
For dimensional analysis<br />
:<math><br />
\widetilde{Z_{sgs}''^2} =<br />
C_Z \Delta^2 \frac{\partial \widetilde{Z} }{\partial x_i} \frac{\partial \widetilde{Z} }{\partial x_i}<br />
</math><br />
where <math> C_Z </math> can be obtained from the scalar spectra and its value is 0.1-0.2.</div>Salvahttp://www.cfd-online.com/Wiki/File:Combustion_NO.jpgFile:Combustion NO.jpg2005-12-14T15:59:07Z<p>Salva: </p>
<hr />
<div></div>Salvahttp://www.cfd-online.com/Wiki/File:Combustion_pic.jpgFile:Combustion pic.jpg2005-12-14T13:46:11Z<p>Salva: </p>
<hr />
<div></div>Salvahttp://www.cfd-online.com/Wiki/Scalar_dissipationScalar dissipation2005-12-12T11:31:09Z<p>Salva: </p>
<hr />
<div>Scalar dissipation is a very important quantity in non-premixed combustion modelling.<br />
It often provides the connection between the mixing field and the combustion modelling.<br />
It is specially important in flamelet and RANS models.<br />
<br />
In a laminar flow the scalar dissipation rate is defined (units are 1/s) as<br />
<br />
:<math><br />
\chi \equiv 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2<br />
</math><br />
where <math> D </math> is the diffusion coefficient of the scalar.<br />
<br />
In turbulent flows, the scalar dissipation is seen as a scalar energy dissipation} and its role is to<br />
destroy (dissipate) scalar variance (scalar energy) analogous to the dissipation of the<br />
turbulent energy <math> \epsilon </math>. This term is known as the turbulent scalar dissipation<br />
and is written as<br />
<br />
:<math><br />
\chi_t \equiv 2 D \left( \frac{\partial \widetilde{Z''}}{\partial x_j} \right) ^2<br />
</math><br />
<br />
Opposite to the kinetic energy dissipation, most of the scalar dissipation occur at the finest scales.<br />
<br />
In [[Conditional Moment Closure (CMC)]] and [[Flamelet based on conserved scalar ]] models, the quantity of interest<br />
is the " main scalar dissipation rate", <math> \widetilde{\chi} </math>. From Favre Averaging the laminar<br />
dissipation rate<br />
<br />
:<math><br />
\widetilde{\chi} = 2D \widetilde{\left(\frac{\partial Z}{\partial x_j}\right) ^2} \approx<br />
2 D \left( \frac{\partial \tilde{Z}}{\partial x_j} \right) ^2 + \chi_t <br />
</math><br />
<br />
Under RANS assumptions gradient of the scalar fluctuations are much larger than gradients of the means,<br />
and therefore the mean scalar dissipation rate is approximately the turbulent dissipation rate<br />
<math> \widetilde{\chi} \approx \chi_t </math></div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-12-12T11:00:24Z<p>Salva: /* RANS equations */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 problems is split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}<br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and 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 areaction 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 />
and two linear profiles (in the rich and lean side) at either side of<br />
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 use 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 />
can<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<br />
[http://www.softpredict.com/?page=989 publications that are available for download].<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 />
<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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-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=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>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T16:20:27Z<p>Salva: /* Burke-Schumann flame structure */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 problems is split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}<br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and 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 areaction 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 />
and two linear profiles (in the rich and lean side) at either side of<br />
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 use 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 />
can<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<br />
[http://www.softpredict.com/?page=989 publications that are available for download].<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 />
<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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-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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T16:05:12Z<p>Salva: /* Flamelets based on conserved scalar */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 problems is split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}<br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and 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 />
<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 use 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 />
can<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<br />
[http://www.softpredict.com/?page=989 publications that are available for download].<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 />
<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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-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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T15:55:29Z<p>Salva: /* Flamelets based on conserved scalar */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 problems is split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}<br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and 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 />
<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 />
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 use in [[#Conserved scalar models]] for equilibrium chemistry<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \omega w_k,<br />
</math><br />
can <br />
<br />
<br />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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-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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T15:52:36Z<p>Salva: /* Conserved scalar equilibrium models */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 problems is split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}<br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and 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 />
<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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-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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T15:38:22Z<p>Salva: /* PDF transport models */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 />
<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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-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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T15:37:36Z<p>Salva: /* References */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 />
<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 review in 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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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-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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T15:05:24Z<p>Salva: /* References */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 />
<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 review in 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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
*{{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=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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T15:04:55Z<p>Salva: /* PDF transport models */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 />
<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 review in 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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
*{{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=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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/Mixture_fractionMixture fraction2005-11-18T12:14:05Z<p>Salva: </p>
<hr />
<div>The mixture fraction measures the fuel/oxidizer ratio.<br />
It is normalized so <math> Z=0 </math> in the oxidizer stream and <math> Z=1 </math> in the fuel stream<br />
<br />
:<math><br />
Z = \frac{s Y_F -Y_O +Y_O^0}{s Y_F +Y_O^0}<br />
</math></div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T12:10:09Z<p>Salva: /* Flamelets based on conserved scalar */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 />
<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
*{{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=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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T12:09:22Z<p>Salva: /* Flamelets based on conserved scalar */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 />
<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
*{{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=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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T12:08:12Z<p>Salva: /* Non premixed combustion */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 />
<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
*{{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=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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T11:50:37Z<p>Salva: /* References */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy convergence and implementation.<br />
<br />
==== Bray-Moss-Libby Model ====<br />
<br />
=== Non premixed combustion ===<br />
<br />
==== Conserved scalar equilibrium models ====<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
*{{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=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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-18T11:37:49Z<p>Salva: </p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy convergence and implementation.<br />
<br />
==== Bray-Moss-Libby Model ====<br />
<br />
=== Non premixed combustion ===<br />
<br />
==== Conserved scalar equilibrium models ====<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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=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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/Karlovitz_numberKarlovitz number2005-11-17T17:36:29Z<p>Salva: </p>
<hr />
<div>The Karlovitz number is used in turbulent combustion<br />
and corresponds to the ratio of chemical time scale <math> \tau_c </math><br />
and smallest turbulent time scale <math> \tau_k </math><br />
(Kolmogorov)<br />
:<math><br />
Ka \equiv \frac{\tau_c}{\tau_k}<br />
</math><br />
<br />
If <math> Ka << 1 </math> the chemical reactions occur<br />
much faster than all turbulent scales. Turbulence do not alter the flame structure<br />
and the chemical region is in laminar conditions.<br />
<br />
The Karlovitz number is linked to the [[Damkholer number]] (<math> Da </math>).<br />
<br />
:<math><br />
Ka = \frac{1}{Da (\eta)}<br />
</math><br />
<br />
[[Category:Dimensionless parameters]]</div>Salvahttp://www.cfd-online.com/Wiki/Damkohler_numberDamkohler number2005-11-17T17:35:01Z<p>Salva: </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 />
[[Category:Dimensionless parameters]]</div>Salvahttp://www.cfd-online.com/Wiki/Karlovitz_numberKarlovitz number2005-11-17T17:02:48Z<p>Salva: </p>
<hr />
<div>The Karlovitz number is used in turbulent combustion<br />
and corresponds to the ratio of chemical time scale <math> \tau_c </math><br />
and smallest turbulent time scale <math> \tau_k </math><br />
(Kolmogorov)<br />
:<math><br />
Ka \equiv \frac{\tau_c}{\tau_k}<br />
</math><br />
<br />
If <math> Ka << 1 </math> the chemical reactions occur<br />
much faster than all turbulent scales. Turbulence do not alter the flame structure<br />
and the chemical region is in laminar conditions.<br />
<br />
The Karlovitz number is linked to the [[Damkholer number]] (<math> Da </math>).<br />
The product of both numbers is always greater than 1.<br />
:<math><br />
Da Ka > 1<br />
</math><br />
<br />
[[Category:Dimensionless parameters]]</div>Salvahttp://www.cfd-online.com/Wiki/Lewis_numberLewis number2005-11-17T16:52:12Z<p>Salva: </p>
<hr />
<div>The Lewis number for a given species <math> k </math> is<br />
:<math><br />
Le_k \equiv \frac{\lambda}{\rho C_p D_k}<br />
</math><br />
Denoting <math> D_{th}= \lambda / \rho C_p </math> the heat diffusivity coefficient the Lewis number<br />
can be expressed as<br />
:<math><br />
Le_k \equiv \frac{D_{th}}{ D_k}<br />
</math><br />
which is the ratio of the heat diffusion speed to the diffusion speed of species <math> k </math>.<br />
<br />
In many combustion models, all species are assumed to diffuse at the same speed and therefore<br />
<math> Le = 1 </math><br />
<br />
[[Category:Dimensionless parameters]]</div>Salvahttp://www.cfd-online.com/Wiki/Lewis_numberLewis number2005-11-17T16:50:59Z<p>Salva: </p>
<hr />
<div>The Lewis number for a given species <math> k </math> is<br />
:<math><br />
Le_k \equiv \frac{\lambda}{\rho C_p D_k}<br />
</math><br />
Denoting <math> D_{th}= \lambda / \rho C_p </math> the heat diffusivity coefficient the Lewis number<br />
can be expressed as<br />
:<math><br />
Le_k = \frac{D_{th}}{ D_k}<br />
</math><br />
which is the ratio of the heat diffusion speed to the diffusion speed of species <math> k </math>.<br />
<br />
In many combustion models, all species are assumed to diffuse at the same speed and therefore<br />
<math> Le \equiv 1 </math><br />
<br />
[[Category:Dimensionless parameters]]</div>Salvahttp://www.cfd-online.com/Wiki/Lewis_numberLewis number2005-11-17T16:50:00Z<p>Salva: </p>
<hr />
<div>The Lewis number for a given species <math> k </math> is<br />
:<math><br />
Le_k = \frac{\lambda}{\rho C_p D_k}<br />
</math><br />
Denoting <math> D_{th}= \lambda / \rho C_p </math> the heat diffusivity coefficient the Lewis number<br />
can be expressed as<br />
:<math><br />
Le_k = \frac{D_{th}}{ D_k}<br />
</math><br />
which is the ratio of the heat diffusion speed to the diffusion speed of species <math> k </math>.<br />
<br />
In many combustion models, all species are assumed to diffuse at the same speed and therefore<br />
<math> Le=1 </math><br />
<br />
[[Category:Dimensionless parameters]]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-17T16:40:23Z<p>Salva: /* What is combustion -- Physics versus modelling */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
In addition to the flow parameters used in fluid mechanics,<br />
new non-dimensional parameters are introduced, specially improtant are: the [[Karlovitz number]]<br />
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 selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy convergence and implementation.<br />
<br />
==== Bray-Moss-Libby Model ====<br />
<br />
=== Non premixed combustion ===<br />
<br />
==== Conserved scalar equilibrium models ====<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 />
===== Multiple Mapping Closure (MMC) =====<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
== References ==<br />
<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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-17T16:32:43Z<p>Salva: </p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
The combustion models are often classified selon their capability to deal with the different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also 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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy convergence and implementation.<br />
<br />
==== Bray-Moss-Libby Model ====<br />
<br />
=== Non premixed combustion ===<br />
<br />
==== Conserved scalar equilibrium models ====<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 />
===== Multiple Mapping Closure (MMC) =====<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
== References ==<br />
<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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-15T16:09:27Z<p>Salva: /* Premixed combustion */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
=== Premixed combustion ===<br />
<br />
[[Image:Premixed.jpg]]<br />
<br />
=== Non-premixed combustion ===<br />
The models are often classified selon their capability to deal with different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also some dangers like the autoignition (in the supply system).<br />
<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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy convergence and implementation.<br />
<br />
==== Bray-Moss-Libby Model ====<br />
<br />
=== Non premixed combustion ===<br />
<br />
==== Conserved scalar equilibrium models ====<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 />
===== Multiple Mapping Closure (MMC) =====<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
== References ==<br />
<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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/File:Premixed.jpgFile:Premixed.jpg2005-11-15T16:08:20Z<p>Salva: </p>
<hr />
<div></div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-15T13:18:04Z<p>Salva: /* Reaction mechanisms */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
=== Premixed combustion ===<br />
<br />
=== Non-premixed combustion ===<br />
The models are often classified selon their capability to deal with different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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> is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also some dangers like the autoignition (in the supply system).<br />
<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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy convergence and implementation.<br />
<br />
==== Bray-Moss-Libby Model ====<br />
<br />
=== Non premixed combustion ===<br />
<br />
==== Conserved scalar equilibrium models ====<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 />
===== Multiple Mapping Closure (MMC) =====<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
== References ==<br />
<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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/Subgrid_varianceSubgrid variance2005-11-15T13:06:07Z<p>Salva: </p>
<hr />
<div>The subgrid variance of a passive scalar is defined as<br />
:<math><br />
\widetilde{Z_{sgs}''^2} = \widetilde{Z^2}- \widetilde{Z}^2<br />
</math><br />
The scalar subgrid variance is also known as the subgrid scalar energy in analogy<br />
to the kinetic subgrid energy.<br />
An equation for the subgrid variance is<br />
<br />
<br />
Instead of solving the above equation, algebraic models are often used.<br />
For dimensional analysis<br />
:<math><br />
\widetilde{Z_{sgs}''^2} =<br />
C_Z \Delta^2 \frac{\partial \widetilde{Z} }{\partial x_i} \frac{\partial \widetilde{Z} }{\partial x_i}<br />
</math><br />
where <math> C_Z </math> can be obtained from the scalar spectra and its value is 0.1-0.2.</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-15T11:53:18Z<p>Salva: /* What is combustion -- Physics versus modelling */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
=== Premixed combustion ===<br />
<br />
=== Non-premixed combustion ===<br />
The models are often classified selon their capability to deal with different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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 $\nu$ is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing Equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also some dangers like the autoignition (in the supply system).<br />
<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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy convergence and implementation.<br />
<br />
==== Bray-Moss-Libby Model ====<br />
<br />
=== Non premixed combustion ===<br />
<br />
==== Conserved scalar equilibrium models ====<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 />
===== Multiple Mapping Closure (MMC) =====<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
== References ==<br />
<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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-15T11:50:35Z<p>Salva: /* Flamelets based on G equation */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
<br />
The models are often classified selon their capability to deal with different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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 $\nu$ is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing Equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also some dangers like the autoignition (in the supply system).<br />
<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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy convergence and implementation.<br />
<br />
==== Bray-Moss-Libby Model ====<br />
<br />
=== Non premixed combustion ===<br />
<br />
==== Conserved scalar equilibrium models ====<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 />
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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 />
===== Multiple Mapping Closure (MMC) =====<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
== References ==<br />
<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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-15T11:48:20Z<p>Salva: /* What is combustion -- Physics versus modelling */</p>
<hr />
<div>== What is combustion -- Physics versus modelling ==<br />
<br />
Combustion phenomena consists of many physical and chemical processes with <br />
broad range of time scales. Mathematical description of combustion is not <br />
always trivial. Analytical solutions exists only for basic situations of <br />
laminar flame and <br />
because of its assumptions it is often restricted to few problems solved <br />
usually in zero or one-dimensional space. <br />
<br />
Problems solved today concern mainly turbulent flows, gas as well as liquid <br />
fuels, pollution 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. However here <br />
the modelling part will be mainly addressed.<br />
<br />
<br />
The models are often classified selon their capability to deal with different combustion regimes.<br />
[[Image:schemes.jpg]]<br />
<br />
== Reaction mechanisms ==<br />
<br />
The combustion is mainly chemical process and although we can, to some extend, <br />
describe flame without any chemistry informations, for modelling of flame <br />
propagation we need to know the speed of reactions, product concentrations, <br />
temperature and other parameters. <br />
Therefore more or less detailed information about reaction kinetics is <br />
essential for any combustion model. <br />
Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to <br />
maintain 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 typical combustion reaction <br />
is influenced mainly by concentration of reactants, temperature and pressure. <br />
<br />
A stoichiometric equation of an arbitrary equation 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 $\nu$ is the stoichiometric coefficient, <math>M_j</math> is arbitrary species. One <br />
prime specifies the reactants and double prime products of the reaction. <br />
<br />
Reaction rate, expressing the rate of disappearance of reactant <b>i</b><br />
of such a reaction, 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 <br />
constant is a function only of temperature and this function 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 activation energy and <math>\beta</math> is <br />
temperature exponent. <br />
These constants for given reactions can be found in literature. <br />
The reaction mechanism can be given from experiments for every reaction <br />
resolved, it could be also constructed numerically by automatic generation <br />
method (see [Griffiths (1994)] for review on reaction mechanisms).<br />
For simple hydrocarbon tens to hundreds of reactions are involved. <br />
By analysis and systematic reduction of reaction mechanisms global reaction <br />
(from one to five step reactions) can be found (see [Westbrook (1984)]).<br />
<br />
== Governing Equations for Reacting Flows ==<br />
<br />
Together with the usual Navier-Stokes for compresible flows (See [[Governing Equations]]), additional equations are <br />
needed in reacting flows.<br />
The mass fraction transport equation for <i>k-th</i> species <math> Y_k </math> 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>, the species difussion coefficient and <math> \dot \omega_k </math> is the species reaction rate.<br />
A non-reactive scalar (like the mixture fraction <math> Z </math>) had 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 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> needs 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 similar to the mass fraction equation, <math> \widetilde{u''_i Z''} </math> needs modelling.<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)]] equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field.<br />
Similar to [[#RANS equations]], but using Favre filtering instead<br />
of [[Favre averaging]].<br />
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 />
Fluctuations of diffusion coefficients are often ignored and their contributions<br />
much smaller than 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 of the comparable time scale of the mixing <br />
process. The simpler approach assuming chemistry fast enough, that the limiting <br />
process is mixing of the species is historically older approach and even today can<br />
be appropriate approach. It is simpler to solve then [[#Finite rate chemistry]] models, <br />
but introduces errors to the solution which may or may not be important. <br />
<br />
=== Premixed Combustion ===<br />
Premixed flame occurs in mixtures of fuel and oxidiser,<br />
homogeneously premixed prior to the flame. These flames are not <br />
limited only to gas fuels, but also to the pre-vaporised fuels.<br />
Typical example of premixed laminar flame is bunsen burner, where <br />
the air enters the fuel stream. The mixture burns in the wake of the <br />
riser tube walls forming nice stable flame.<br />
The premixed flames has many advantages in terms of control of temperature and <br />
products and pollution concentration, but introduce also some dangers like the autoignition (in the supply system).<br />
<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 and Hjertager, <br />
and Spalding and can be found in all CFD packages. <br />
The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. <br />
The combustion is 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 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 />
<math>\bar{C}</math> denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. <br />
Still they are just constants based on experimental fitting and they need not <br />
be suitable for <b>all</b> the situations. <br />
Care must be taken especially in highly strained regions, where the ratio of <math>k</math> <br />
to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In those regions a positive reaction rate occurs and an artificial flame can be observed.<br />
CFD codes usually has 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. Still this model is quite popular for its simplicity and relatively easy convergence and implementation.<br />
<br />
==== Bray-Moss-Libby Model ====<br />
<br />
=== Non premixed combustion ===<br />
<br />
==== Conserved scalar equilibrium models ====<br />
<br />
== Finite rate chemistry ==<br />
<br />
=== Premixed Combustion ===<br />
<br />
==== Coherent Flame Model ====<br />
<br />
==== Flamelets based on G equation ====<br />
<br />
=== Non-premixed Combustion ===<br />
<br />
==== Flamelets based on conserved scalar ====<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 reactive problems is therefore split into two parts:<br />
First, the <i> mixing </i>, which consists of the location of the flame surface<br />
which is a non-reactive problem concerning the propagation of a passive scalar.<br />
And second, the <i> flame structure </i>, 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 />
Using the following chain rules for the time <br />
:<math><br />
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z} <br />
</math><br />
<br />
and spatial coordinate<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 />
<br />
to the species transport equation (see [[#Governing Equations for Reacting Flows]]) and re-arranging, 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) + \dot \omega_k<br />
<br />
</math><br />
<br />
The second and third term in the LHS cancel due to continuity and mixture fraction transport,<br />
the equation therefore 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 scalar dissipation<br />
and controls the mixing, providing the interaction between the flow and the chemistry.<br />
<br />
If the flame dependence on time is dropped, even though he 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 />
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that libraries of<br />
<math> Y_k=f(Z,\chi)</math><br />
can be pre-computed and stored in look-up tables with all the required complex chemistry.<br />
<br />
===== Flamelet Computation and Flamelet Libraries =====<br />
<br />
The computation of non-premixed turbulent flames based on laminar-flamelet models<br />
is generally based on two-dimensional or three-dimensional CFD codes that employ standard<br />
models for fluid-mechanical closure of the govening equations. In many cases, for that<br />
purpose standard models such as the k-epsilon model are used, but occasionally more sophisticated models such as Reynolds-Stress models are also employed.<br />
<br />
Chemical-source-term closure is a different matter. To this end, the CFD codes carry out suitable averaging procedures, such as pdf-avaraging on the basis of a beta function or a clipped Gaussian distribution. The quantities to be averaged are laminar-flamelet profiles, i.e., results from laminar-flamelet computations. Generally, these flamelet computations<br />
are carried oout a-priori, i.e, they are performed separately and prior to the turbulent-combustion simulation with the CFD code. Depending on the specific laminar-flamelet<br />
model used for the turbulent-combustion simulation, one or several parameters are varied in the laminat computatations. For instance, if the computations are based on <br />
<br />
:<math><br />
\frac{\partial Y_k}{\partial t} = <br />
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot w_k,<br />
</math><br />
<br />
then the variable parameter is the scalar dissipation rate <math> \chi </math>. The flamelet profiles for the various parameter values are stored in a dataset or file which is called<br />
a "flamelet library". 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 <br />
[http://www.softpredict.com/?page=989 publications that are available for download]. <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 />
<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 />
===== Multiple Mapping Closure (MMC) =====<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 <br />
moment closures and they are used to compute inhomegenous turbulent flows.<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).<br />
This method 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<br />
(see [[Discretisation of convective term]])<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 />
== References ==<br />
<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=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=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 ==<br />
<br />
*[http://www.softpredict.com SoftPredict's COSILAB homepage]<br />
<br />
*[http://www.softpredict.com/?page=989 SoftPredict's flamelet paper download page]</div>Salvahttp://www.cfd-online.com/Wiki/File:Schemes.jpgFile:Schemes.jpg2005-11-15T11:41:16Z<p>Salva: </p>
<hr />
<div></div>Salvahttp://www.cfd-online.com/Wiki/CombustionCombustion2005-11-15T11:17:42Z<p>Salva: /* Flamelets in turbulent combustion */</p>
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