Combustion
From CFDWiki
(spelling, grammar, headings  still needs cleanup for the last 3 sections) 
(celanup  still incomplete) 

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prevail until all of the mixture is burned into products. If the reaction  prevail until all of the mixture is burned into products. If the reaction  
is too slow, the flame will extinguish. If too fast, explosion or even  is too slow, the flame will extinguish. If too fast, explosion or even  
  detonation will occur. The reaction rate of typical combustion reaction  +  detonation will occur. The reaction rate of a typical combustion reaction 
  is influenced mainly by concentration of reactants, temperature, and pressure.  +  is influenced mainly by the concentration of the reactants, temperature, and pressure. 
  A stoichiometric equation  +  A stoichiometric equation for an arbitrary reaction can be written as: 
<table width="100%">  <table width="100%">  
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where <b>A</b> is preexponential 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.  where <b>A</b> is preexponential 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.  
  Reaction mechanisms can be deduced from experiments (for every resolved reaction),  +  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). 
  For simple  +  For simple hydrocarbons, tens to hundreds of reactions are involved. 
By analysis and systematic reduction of reaction mechanisms, global reactions  By analysis and systematic reduction of reaction mechanisms, global reactions  
(from one to five step reactions) can be found (see [Westbrook (1984)]).  (from one to five step reactions) can be found (see [Westbrook (1984)]).  
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Together with the usual NavierStokes equations for compresible flow (See [[Governing equations]]), additional equations are needed in reacting flows.  Together with the usual NavierStokes equations for compresible flow (See [[Governing equations]]), additional equations are needed in reacting flows.  
  The mass fraction <math> Y_k </math>  +  The transport equation for the mass fraction <math> Y_k </math> of <i>kth</i> species is 
:<math>  :<math>  
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  where Ficks' law is assumed for scalar diffusion with <math> D_k </math> denoting the species difussion coefficient, and  +  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. 
A nonreactive (passive) scalar (like the mixture fraction <math> Z </math>) is goverened by the following transport equation  A nonreactive (passive) scalar (like the mixture fraction <math> Z </math>) is goverened by the following transport equation  
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=== RANS equations ===  === RANS equations ===  
  In turbulent flows, [[Favre averaging]] is often used and the  +  In turbulent flows, [[Favre averaging]] is often used to reduce the scales (see [[Reynolds averaging]]) and the 
mass fraction transport equation is transformed to  mass fraction transport equation is transformed to  
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where the turbulent fluxes <math> \widetilde{u''_i Y''_k} </math> and reaction terms  where the turbulent fluxes <math> \widetilde{u''_i Y''_k} </math> and reaction terms  
  <math> \overline{\dot \omega_k} </math>  +  <math> \overline{\dot \omega_k} </math> need to be closed. 
The passive scalar turbulent transport equation is  The passive scalar turbulent transport equation is  
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</math>  </math>  
  where  +  where <math> \widetilde{u''_i Z''} </math> needs modelling. 
In addition to the mean passive scalar equation,  In addition to the mean passive scalar equation,  
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=== LES equations ===  === LES equations ===  
  The [[Large eddy simulation (LES)]]  +  The [[Large eddy simulation (LES)]] approach for reactive flows introduces equations for the filtered species mass fractions within the compressible flow field. 
  Similar to [[#RANS equations]], but using Favre filtering instead  +  Similar to the [[#RANS equations]], but using Favre filtering instead 
  of [[Favre averaging]]  +  of [[Favre averaging]], the filtered mass fraction transport equation is 
  +  
:<math>  :<math>  
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</math>  </math>  
and has to be modelled.  and has to be modelled.  
+  
Fluctuations of diffusion coefficients are often ignored and their contributions  Fluctuations of diffusion coefficients are often ignored and their contributions  
  much smaller than apparent turbulent diffusion due to transport of subgrid fluctuations.  +  are assumed to be much smaller than the apparent turbulent diffusion due to transport of subgrid fluctuations. 
The first term on the right hand side is then  The first term on the right hand side is then  
:<math>  :<math>  
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assumptions on the reaction kinetics.  assumptions on the reaction kinetics.  
We can either assume the reactions to be infinitely fast  compared to  We can either assume the reactions to be infinitely fast  compared to  
  e.g. mixing of the species, or  +  e.g. mixing of the species, or comparable to the time scale of the mixing 
  process. The  +  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. 
  +  
  +  
  +  
  === Premixed  +  === Premixed combustion === 
  Premixed  +  Premixed flames occur when the fuel and oxidiser are homogeneously mixed prior to ignition. These flames are not 
  homogeneously  +  
limited only to gas fuels, but also to the prevaporised fuels.  limited only to gas fuels, but also to the prevaporised fuels.  
  Typical  +  Typical examples of premixed laminar flames is bunsen burner, where 
  the air enters the fuel stream  +  the air enters the fuel stream and the mixture burns in the wake of the 
  riser tube walls forming nice stable flame.  +  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 gumlike matrix that is uniformly distributed on the periphery of the chamber. 
  +  Premixed flames have many advantages in terms of control of temperature and  
  products and pollution concentration,  +  products and pollution concentration. However, they introduce some dangers like the autoignition (in the supply system). 
[[Image:Premixed.jpg]]  [[Image:Premixed.jpg]]  
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==== Eddy BreakUp model ====  ==== Eddy BreakUp model ====  
  The Eddy BreakUp model is the typical example of mixedisburnt combustion model.  +  The Eddy BreakUp model is the typical example of '''mixedisburnt''' combustion model. 
  It is based on the work of Magnussen  +  It is based on the work of Magnussen, Hjertager, and Spalding and can be found in all commercial CFD packages. 
  and Spalding and can be found in all CFD packages.  +  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 
  The model assumes the reactions  +  
  +  
<table width="100%">  <table width="100%">  
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</math></td><td width="5%"></td></tr></table>  </math></td><td width="5%"></td></tr></table>  
  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.  +  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. 
  The mean reaction rate is given by  +  The mean reaction rate is given by 
<table width="100%">  <table width="100%">  
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  <math>\bar{C}</math> denotes mean concentrations  +  where <math>\bar{C}</math> denotes the mean concentrations of fuel, oxidiser, and products 
  respectively  +  respectively. <b>A</b> and <b>B</b> are model constants with typical values of 0.5 
and 4.0 respectively. The values of these constants are fitted according  and 4.0 respectively. The values of these constants are fitted according  
  to  +  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 (flameholder wakes, walls ...). In these, regions a positive reaction rate occurs and an artificial flame can be observed. 
  +  CFD codes usually have some remedies to overcome this problem.  
  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 (flameholder wakes, walls ...). In  +  
  CFD codes usually  +  
  This model largely overpredicts temperatures and concentrations of species like <i>CO</i> and other species.  +  This model largely overpredicts temperatures and concentrations of species like <i>CO</i> and other species. However, the Eddy BreakUp model enjoys a popularity for its simplicity, steady convergence, and implementation. 
  ==== BrayMossLibby  +  ==== BrayMossLibby model ==== 
  === Non premixed combustion ===  +  === Nonpremixed combustion === 
Non premixed combustion is a special class of combustion where fuel and oxidizer  Non premixed combustion is a special class of combustion where fuel and oxidizer  
  enter separately  +  enter separately into the combustion chamber. The diffusion and mixing of the two streams 
must bring the reactants together for the reaction to occur.  must bring the reactants together for the reaction to occur.  
  Mixing becomes the key characteristic  +  Mixing becomes the key characteristic for diffusion flames. 
Diffusion burners are easier and safer to operate than premixed burners.  Diffusion burners are easier and safer to operate than premixed burners.  
However their efficiency is reduced compared to premixed burners.  However their efficiency is reduced compared to premixed burners.  
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==== Conserved scalar equilibrium models ====  ==== Conserved scalar equilibrium models ====  
  The reactive  +  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 nonreactive 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. 
  First, the <i> mixing </i>, which consists of the location of the flame surface  +  
  which is a nonreactive problem concerning the propagation of a passive scalar  +  
  And second, the <i> flame structure </i>, which deals with the distribution of the reactive species inside the flamelet.  +  
To obtain the distribution inside the flame front we assume it is locally onedimensional and  To obtain the distribution inside the flame front we assume it is locally onedimensional and  
depends only on time and the scalar coodinate.  depends only on time and the scalar coodinate.  
  +  We first make use of the following chain rules  
:<math>  :<math>  
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}  \frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}  
</math>  </math>  
  +  <br>  
  +  
  +  
:<math>  :<math>  
\frac{\partial Y_k}{\partial x_j} = \frac{\partial Z}{\partial x_j}\frac{\partial Y_k}{\partial Z}  \frac{\partial Y_k}{\partial x_j} = \frac{\partial Z}{\partial x_j}\frac{\partial Y_k}{\partial Z}  
</math>  </math>  
  +  upon substitution into the species transport equation (see [[#Governing Equations for Reacting Flows]]), we obtain  
:<math>  :<math>  
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</math>  </math>  
  The second and third  +  The second and third terms in the LHS cancel out due to continuity and mixture fraction transport. 
  the equation  +  At the outset, the equation boils down to 
:<math>  :<math>  
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</math>  </math>  
  where <math> \chi = 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2 </math> is called the scalar dissipation  +  where <math> \chi = 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2 </math> is called the <b>scalar dissipation</b> 
  +  which controls the mixing, providing the interaction between the flow and the chemistry.  
If the flame dependence on time is dropped, even though the field <math> Z </math> still depends on it.  If the flame dependence on time is dropped, even though the field <math> Z </math> still depends on it.  
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== Finite rate chemistry ==  == Finite rate chemistry ==  
  === Premixed  +  === Premixed combustion === 
  ==== Coherent  +  ==== Coherent flame model ==== 
  ==== Flamelets based on G equation ====  +  ==== Flamelets based on Gequation ==== 
==== Flame surface density model ====  ==== Flame surface density model ====  
  === Nonpremixed  +  === Nonpremixed combustion === 
==== Flamelets based on conserved scalar ====  ==== Flamelets based on conserved scalar ==== 
Revision as of 17:36, 19 December 2005
What is combustion  physics versus modelling
Combustion phenomena consist of many physical and chemical processes which exhibit a broad range of time and length scales. A mathematical description of combustion is not always trivial, although some analytical solutions exist for simple situations of laminar flame. Such analytical models are usually restricted to problems in zero or onedimensional space.
Most problems in combustion invlove turbulent flows, gas and liquid fuels, and pollution transport issues (products of combustion as well as for example noise pollution). These problems require not only extensive experimental work, but also numerical modelling. All combustion models must be validated against the experiments as each one has its own drawbacks and limits. In this article, we will address the modeling fundamentals only.
In addition to the flow parameters used in fluid mechanics,
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
the Lewis number which compares the diffusion speeds of species.
The combustion models are often classified selon their capability to deal with the different combustion regimes.
Reaction mechanisms
Combustion is mainly a chemical process. Although we can, to some extent, describe a flame without any chemistry information, modelling of the flame propagation requires the knowledge of speeds of reactions, product concentrations, temperature, and other parameters. Therefore fundamental information about reaction kinetics is essential for any combustion model. A fueloxidizer mixture will generally combust if the reaction is fast enough to prevail until all of the mixture is burned into products. If the reaction is too slow, the flame will extinguish. If too fast, explosion or even detonation will occur. The reaction rate of a typical combustion reaction is influenced mainly by the concentration of the reactants, temperature, and pressure.
A stoichiometric equation for an arbitrary reaction can be written as:

where denotes the stoichiometric coefficient, and stands for an arbitrary species. A one prime holds for the reactants while a double prime holds for the products of the reaction.
The reaction rate, expressing the rate of disappearance of reactant i, is defined as:

in which k is the specific reaction rate constant. Arrhenius found that this constant is a function of temperature only and is defined as:

where A is preexponential factor, E is the activation energy, and is a temperature exponent. The constants vary from one reaction to another and can be found in the literature.
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). For simple hydrocarbons, tens to hundreds of reactions are involved. By analysis and systematic reduction of reaction mechanisms, global reactions (from one to five step reactions) can be found (see [Westbrook (1984)]).
Governing equations for chemically reacting flows
Together with the usual NavierStokes equations for compresible flow (See Governing equations), additional equations are needed in reacting flows.
The transport equation for the mass fraction of kth species is
where Ficks' law is assumed for scalar diffusion with denoting the species difussion coefficient, and denoting the species reaction rate.
A nonreactive (passive) scalar (like the mixture fraction ) is goverened by the following transport equation
where is the diffusion coefficient of the passive scalar.
RANS equations
In turbulent flows, Favre averaging is often used to reduce the scales (see Reynolds averaging) and the mass fraction transport equation is transformed to
where the turbulent fluxes and reaction terms
need to be closed.
The passive scalar turbulent transport equation is
where needs modelling.
In addition to the mean passive scalar equation, an equation for the Favre variance is often employed
where is the mean Scalar dissipation rate defined as This term and the variance diffusion fluxes needs to be modelled.
LES equations
The Large eddy simulation (LES) approach for reactive flows introduces equations for the filtered species mass fractions within the compressible flow field. Similar to the #RANS equations, but using Favre filtering instead of Favre averaging, the filtered mass fraction transport equation is
where is the transport of subgrid fluctuations of mass fraction
and has to be modelled.
Fluctuations of diffusion coefficients are often ignored and their contributions are assumed to be much smaller than the apparent turbulent diffusion due to transport of subgrid fluctuations. The first term on the right hand side is then
Infinitely fast chemistry
All combustion models can be divided into two main groups according to the assumptions on the reaction kinetics. We can either assume the reactions to be infinitely fast  compared to e.g. mixing of the species, or comparable to the time scale of the mixing 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.
Premixed combustion
Premixed flames occur when the fuel and oxidiser are homogeneously mixed prior to ignition. These flames are not limited only to gas fuels, but also to the prevaporised fuels. Typical examples of premixed laminar flames is bunsen burner, where the air enters the fuel stream and the mixture burns in the wake of the 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 gumlike matrix that is uniformly distributed on the periphery of the chamber. Premixed flames have many advantages in terms of control of temperature and products and pollution concentration. However, they introduce some dangers like the autoignition (in the supply system).
Turbulent flame speed model
Eddy BreakUp model
The Eddy BreakUp model is the typical example of mixedisburnt combustion model. It is based on the work of Magnussen, Hjertager, and Spalding and can be found in all commercial CFD packages. 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

in which F stands for fuel, O for oxidiser, and P for products of the reaction. Alternativelly we can have a multistep scheme, where each reaction has its own mean reaction rate. The mean reaction rate is given by

where denotes the mean concentrations of fuel, oxidiser, and products
respectively. A and B are model constants with typical values of 0.5
and 4.0 respectively. The values of these constants are fitted according
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 all the situations. Care must be taken especially in highly strained regions, where the ratio of to is large (flameholder wakes, walls ...). In these, regions a positive reaction rate occurs and an artificial flame can be observed.
CFD codes usually have some remedies to overcome this problem.
This model largely overpredicts temperatures and concentrations of species like CO and other species. However, the Eddy BreakUp model enjoys a popularity for its simplicity, steady convergence, and implementation.
BrayMossLibby model
Nonpremixed combustion
Non premixed combustion is a special class of combustion where fuel and oxidizer enter separately into the combustion chamber. The diffusion and mixing of the two streams must bring the reactants together for the reaction to occur. Mixing becomes the key characteristic for diffusion flames. Diffusion burners are easier and safer to operate than premixed burners. However their efficiency is reduced compared to premixed burners. One of the major theoretical tools in nonpremixed combustion is the passive scalar mixture fraction which is the backbone on most of the numerical methods in nonpremixed combustion.
Conserved scalar equilibrium models
The reactive problem is split into two parts. First, the problem of mixing , which consists of the location of the flame surface which is a nonreactive problem concerning the propagation of a passive scalar; And second, the flame structure problem, which deals with the distribution of the reactive species inside the flamelet.
To obtain the distribution inside the flame front we assume it is locally onedimensional and depends only on time and the scalar coodinate.
We first make use of the following chain rules
upon substitution into the species transport equation (see #Governing Equations for Reacting Flows), we obtain
The second and third terms in the LHS cancel out due to continuity and mixture fraction transport. At the outset, the equation boils down to
where is called the scalar dissipation which controls the mixing, providing the interaction between the flow and the chemistry.
If the flame dependence on time is dropped, even though the field still depends on it.
If the reaction is assumed to be infinetly fast, the resultant flame distribution is in equilibrium. and . When the flame is in equilibrium, the flame configuration is independent of strain.
BurkeSchumann flame structure
The BurkeSchuman solution is valid for irreversible infinitely fast chemistry. With areaction in the form of

If the flame is in equilibrium and therefore the reaction term is 0. Two possible solution exists, one with pure mixing (no reaction) and a linear dependence of the species mass fraction with . Fuel mass fraction
Oxidizer mass fraction
Where and are fuel and oxidizer mass fractions in the pure fuel and oxidizer streams respectively.
The other solution is given by a discontinuous slope at stoichiometric mixture fraction
and two linear profiles (in the rich and lean side) at either side of
the stoichiometric mixture fraction.
Both concentrations must be 0 at stoichiometric, the reactants become products infinitely fast.
and oxidizer mass fraction
Finite rate chemistry
Premixed combustion
Coherent flame model
Flamelets based on Gequation
Flame surface density model
Nonpremixed combustion
Flamelets based on conserved scalar
Peters (2000) define Flamelets as "thin diffusion layers embedded in a turbulent nonreactive flow field". If the chemistry is fast enough, the chemistry is active within a thin region where the chemistry conditions are in (or close to) stoichiometric conditions, the "flame" surface. This thin region is assumed to be smaller than Kolmogorov length scale and therefore the region is locally laminar. The flame surface is defined as an isosurface of a certain scalar , mixture fraction in #Non premixed combustion.
The same equation use in #Conserved scalar models for equilibrium chemistry is used here but with chemical source term different from 0.
can
This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that flamelet profiles can be precomputed and stored in a dtaset or file which is called a "flamelet library" with all the required complex chemistry. For the generation of such libraries ready to use software is avalable such as Softpredict's Combustion Simulation Laboratory COSILAB [1] with its relevant solver RUN1DL, which can be used for a variety of relevant geometries; see various publications that are available for download. Other software tools include CHEMKIN [2] for the creation of "flamelet libraries" from detailed chemistry, and KINetics [3] for the integration of the libraries into CFD.
Flamelets in turbulent combustion
In turbulent flames the interest is . In flamelets, the flame thickness is assumed to be much smaller than Kolmogorov scale and obviously is much smaller than the grid size. It is therefore needed a distribution of the passive scalar within the cell. cannot be obtained directly from the flamelets library , where corresponds to the value obtained from the flamelets libraries. A generic solution can be expressed as
where is the joint Probability Density Function (PDF) of the mixture fraction and scalar dissipation which account for the scalar distribution inside the cell and "a priori" depends on time and space.
The most simple assumption is to use a constant distribution of the scalar dissipation within the cell and the above equation reduces to
is the PDF of the mixture fraction scalar and simple models (such as Gaussian or a beta PDF) can be build depending only on two moments of the scalar mean and variance,.
If the mixture fraction and scalar dissipation are consider independent variables, can be written as . The PDF of the scalar dissipation is assumed to be lognormal with variance unity.
In Large eddy simulation (LES) context (see #LES equations for reacting flow), the probability density function is replaced by a subgrid PDF . The same equation hold by replacing averaged values with filtered values.
The assumptions made regarding the shapes of the PDFs are still justified. In LES combustion the subgrid variance is smaller than RANS counterpart (part of the largescale fluctuations are solved) and therefore the modelled PDFs are thinner.
Unsteady flamelets
Conditional Moment Closure (CMC)
In Conditional Moment Closure (CMC) methods we assume that the species mass fractions are all correlated with the mixture fraction (in non premixed combustion).
From Probability density function we have
where is the sample space for .
CMC consists of providing a set of transport equations for the conditional moments which define the flame structure.
Experimentally, it has been observed that temperature and chemical radicals are strong nonlinear functions of mixture fraction. For a given species mass fraction we can decomposed it into a mean and a fluctuation:
The fluctuations are usually very strong in time and space which makes the closure of very difficult. However, the alternative decomposition
where is the fluctuation around the conditional mean or the "conditional fluctuation". Experimentally, it is observed that , which forms the basic assumption of the CMC method. Closures. Due to this property better closure methods can be used reducing the nonlinearity of the mass fraction equations.
The Derivation of the CMC equations produces the following CMC transport equation where for simplicity.
In this equation, high order terms in Reynolds number have been neglected. (See Derivation of the CMC equations for the complete series of terms).
It is well known that closure of the unconditional source term as a function of the mean temperature and species () will give rise to large errors. However, in CMC the conditional averaged mass fractions contain more information and fluctuations around the mean are much smaller. The first order closure is a good approximation in zones which are not close to extinction.
Second order closure
A second order closure can be obtained if conditional fluctuations are taken into account. For a chemical source term in the form with the rate constant in Arrhenius form the second order closure is (Klimenko and Bilger 1999)
where is the first order CMC closure and . When the temperature exponent or are large the error of taking the first order approximation increases. Improvement of small pollutant predictions can be obtained using the above reaction rate for selected species like CO and NO.
Double conditioning
Close to extinction and reignition. The conditional fluctuations can be very large and the primary closure of CMC of "small" fluctuations is not longer valid. A second variable can be chosen to define a double conditioned mass fraction
Due to the strong dependence on chemical reactions to temperature, is advised to be a temperature related variable (Kronenburg 2004). Scalar dissipation is not a good choice, due to its lognormal behaviour (smaller scales give highest dissipation). A must better choice is the sensible enthalpy or a progress variable. Double conditional variables have much smaller conditional fluctuations and allow the existence of points with the same chemical composition which can be fully burning (high temperature) or just mixing (low temperature). The range of applicability is greatly increased and allows nonpremixed and premixed problems to be treated without adhoc distinctions. The main problem is the closure of the new terms involving cross scalar transport.
The double conditional CMC equation is obtained in a similar manner than the conventional CMC equations
LES modelling
In a LES context a conditional filtering operator can be defined and therefore represents a conditionally filtered reactive scalar.
Linear Eddy Model
The Linear Eddy Model (LEM) was first developed by Kerstein(1988). It is an onedimensional model for representing the flame structure in turbulent flows.
In every computational cell a molecular, diffusion and chemical model is defined as
where is a spatial coordinate. The scalar distribution obtained can be seen as a onedimensional reference field between Kolmogorov scale and grid scales.
In a second stage a series of rearranging stochastic event take place. These events represent the effects of a certain turbulent structure of size , smaller than the grid size at a location within the onedimensional domain. This vortex distort the field obtain by the onedimensional equation, creating new maxima and minima in the interval . The vortex size is chosen randomly based on the inertial scale range while is obtained from a uniform distribution in . The number of events is chosen to match the turbulent diffusivity of the flow.
PDF transport models
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).
PDF methods are based on the transport equation of the jointPDF of the scalars.
Denoting where
is the phase space for the reactive scalars
.
The transport equation of the joint PDF is:
where the chemical source term is closed. Another term appeared on the right hand side which accounts for the effects of the molecular mixing on the PDF, is the so called "micromixing " term. Equal diffusivities are used for simplicity
A more general approach is the velocitycomposition jointPDF with , where is the sample space of the velocity field . This approach has the advantage of avoiding gradientdiffusion modelling. A similar equation to the above is obtained combining the momentum and scalar transport equation.
The PDF transport equation can be solved in two ways: through a Lagrangian approach
using stochastic methods or in a Eulerian ways using stochastic fields.
Lagrangian
The main idea of Lagrangian methods is that the flow can be represented by an ensemble of fluid particles. Central to this approach is the stochastic differential equations and in particular the Langevin equation.
Eulerian
Instead of stochastic particles, smooth stochastic fields can be used to represent the probability density function (PDF) of a scalar (or joint PDF) involved in transport (convection), diffusion and chemical reaction (Valino 1998). This method is purely Eulerian and offers implementations advantages compared to Lagrangian or semiEulerian methods. Transport equations for scalars are often easy to programme and normal CFD algorithms can be used (see Discretisation of convective term)
A new set of scalar variables (the stochastic field ) is used to represent the PDF
Other combustion models
MMC
The Multiple Mapping Conditioning (MMC) (Klimenko and Pope 2003) is an extension of the #Conditional Moment Closure (CMC) approach combined with probability density function methods. MMC looks for the minimum set of variables that describes the particular turbulent combustion system.
Fractals
Derived from the #Eddy Dissipation Concept (EDC).
References
 Dopazo, C, (1993), "Recent development in PDF methods", Turbulent Reacting Flows, ed. P. A. Libby and F. A. Williams.
 Kerstein, A. R. (1988), "A linear eddy model of turbulent scalar transport and mixing", Comb. Science and Technology, Vol. 60,pp. 391.
 Klimenko, A. Y., Bilger, R. W. (1999), "Conditional moment closure for turbulent combustion", Progress in Energy and Combustion Science, Vol. 25,pp. 595687.
 Klimenko, A. Y., Pope, S. B. (2003), "The modeling of turbulent reactive flows based on multiple mapping conditioning", Physics of Fluids, Vol. 15, Num. 7, pp. 19071925.
 Kronenburg, A., (2004), "Double conditioning of reactive scalar transport equations in turbulent nonpremixed flames", Physics of Fluids, Vol. 16, Num. 7, pp. 26402648.
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