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Combustion

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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 one-dimensional space.

Most problems in combustion involve 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 on their capability to deal with the different combustion regimes.
SUMMARY OF COMBUSTION MODELS.jpg

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 fuel-oxidizer 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:

\sum_{j=1}^{n}\nu' (M_j) = \sum_{j=1}^{n}\nu'' (M_j),

where \nu denotes the stoichiometric coefficient, and M_j 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:

RR_i = k \, \prod_{j=1}^{n}(M_j)^{\nu'},

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

k= A T^{\beta} \, exp \left( \frac{-E}{RT}\right)

where A is pre-exponential factor, E is the activation energy, and \beta 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 Navier-Stokes equations for compresible flow (See Governing equations), additional equations are needed in reacting flows.

The transport equation for the mass fraction Y_k of k-th species is

\frac{\partial}{\partial t} \left( \rho Y_k \right) + \frac{\partial}{\partial x_j} \left( \rho u_j Y_k\right) = \frac{\partial}{\partial x_j} \left( \rho D_k \frac{\partial Y_k}{\partial x_j}\right)+ \dot \omega_k


where Ficks' law is assumed for scalar diffusion with D_k denoting the species difussion coefficient, and \dot \omega_k denoting the species reaction rate.

A non-reactive (passive) scalar (like the mixture fraction Z ) is goverened by the following transport equation

\frac{\partial}{\partial t} \left( \rho Z \right) + \frac{\partial}{\partial x_j} \left( \rho u_j Z \right) = \frac{\partial}{\partial x_j} \left( \rho D \frac{\partial Z}{\partial x_j}\right)

where D 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

\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} + \frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}= \frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } - \overline{\rho} \widetilde{u''_i Y''_k } \right) + \overline{\dot \omega_k}


where the turbulent fluxes \widetilde{u''_i Y''_k} and reaction terms \overline{\dot \omega_k} need to be closed.

The passive scalar turbulent transport equation is

\frac{\partial \overline{\rho} \widetilde{Z} }{\partial t} + \frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z} }{\partial x_j}= \frac{\partial} {\partial x_j} \left( \overline{\rho D \frac{\partial Z} {\partial x_j} } - \overline{\rho} \widetilde{u''_i Z'' } \right)

where \widetilde{u''_i Z''} needs modelling. A common practice is to model the turbulent fluxes using the gradient diffusion hypothesis. For example, in the equation above the flux \widetilde{u''_i Z''} is modelled as

\widetilde{u''_i Z''} = -D_t \frac{\partial \tilde Z}{\partial x_i}

where D_t is the turbulent diffusivity. Since D_t >> D , the first term inside the parentheses on the right hand of the mixture fraction transport equation is often neglected (Peters (2000)). This assumption is also used below.

In addition to the mean passive scalar equation, an equation for the Favre variance \widetilde{Z''^2} is often employed

\frac{\partial \overline{\rho} \widetilde{Z''^2} }{\partial t} + \frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z''^2} }{\partial x_j}= \frac{\partial}{\partial x_j} \left( \overline{\rho} \widetilde{u''_i Z''^2} \right) - 2 \overline{\rho} \widetilde{u''_i Z'' } - \overline{\rho} \widetilde{\chi}

where \widetilde{\chi} is the mean Scalar dissipation rate defined as \widetilde{\chi} = 2 D \widetilde{\left| \frac{\partial Z''}{\partial x_j} \right|^2 } 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

\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} + \frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}= \frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } - J_j \right) + \overline{\dot \omega_k}

where J_j is the transport of subgrid fluctuations of mass fraction

J_j = \widetilde{u_jY_k} - \widetilde{u}_j \widetilde{Y}_k

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

\frac{\partial} {\partial x_j} \left( \overline{ \rho D_k \frac{\partial Y_k} {\partial x_j} } \right) \approx \frac{\partial} {\partial x_j} \left( \overline{\rho} D_k \frac{\partial \widetilde{Y}_k} {\partial x_j} \right)

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 pre-vaporised 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 gum-like 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).

Premixed.jpg

Turbulent flame speed model

Eddy Break-Up model

The Eddy Break-Up model is the typical example of mixed-is-burnt 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

F + \nu_s O \rightarrow (1+\nu_s) P

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

\bar{\dot\omega}_F=A_{EB} \frac{\epsilon}{k} min\left[\bar{C}_F,\frac{\bar{C}_O}{\nu}, B_{EB}\frac{\bar{C}_P}{(1+\nu)}\right]


where \bar{C} 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 k to \epsilon is large (flame-holder 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 over-predicts temperatures and concentrations of species like CO and other species. However, the Eddy Break-Up model enjoys a popularity for its simplicity, steady convergence, and implementation.

Bray-Moss-Libby model

Non-premixed 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 non-premixed combustion is the passive scalar mixture fraction Z which is the backbone on most of the numerical methods in non-premixed 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 non-reactive 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 one-dimensional and depends only on time and the scalar coodinate.

We first make use of the following chain rules

\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\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}

and transformation

\frac{\partial }{\partial t}= \frac{\partial }{\partial t} + \frac{\partial Z}{\partial t} \frac{\partial }{\partial Z}

upon substitution into the species transport equation (see #Governing Equations for Reacting Flows), we obtain

\rho \frac{\partial Y_k}{\partial t} + Y_k \left[ \frac{\partial \rho}{\partial t} + \frac{\partial \rho u_j}{\partial x_j} \right] + \frac{\partial Y_k}{\partial Z} \left[ \rho \frac{\partial Z}{\partial t} + \rho u_j \frac{\partial Z}{\partial x_j} - \frac{\partial}{\partial x_j}\left( \rho D \frac{\partial Z}{\partial x_j} \right) \right] = \rho D \left( \frac{\partial Z}{\partial x_j} \frac{\partial Z}{\partial x_j} \right) \frac{\partial^2 Y_k}{\partial Z^2} + \dot \omega_k

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

\frac{\partial Y_k}{\partial t} = \frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k

where \chi = 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2 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 Z still depends on it.

\dot \omega_k= -\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2}

If the reaction is assumed to be infinetly fast, the resultant flame distribution is in equilibrium. and \dot \omega_k= 0. When the flame is in equilibrium, the flame configuration Y_k(Z) is independent of strain.

Burke-Schumann flame structure

The Burke-Schuman solution is valid for irreversible infinitely fast chemistry. With a reaction in the form of

F + \nu_s O \rightarrow (1+\nu_s) P

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 Z . Fuel mass fraction

Y_F=Y_F^0 Z

Oxidizer mass fraction

Y_O=Y_O^0(1-Z)

Where Y_F^0 and Y_O^0 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 Z_{st} 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.

Y_F=Y_F^0 \frac{Z-Z_{st}}{1-Z_{st}}

and oxidizer mass fraction

Y_O=Y_O^0 \frac{Z-Z_{st}}{Z_{st}}

Finite rate chemistry

Premixed combustion

Coherent flame model

Flamelets based on G-equation

Flame surface density model

Non-premixed combustion

Flamelets based on conserved scalar

Peters (2000) define Flamelets as "thin diffusion layers embedded in a turbulent non-reactive 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 iso-surface of a certain scalar Z , mixture fraction in #Non premixed combustion.

The same equation used in #Conserved scalar models for equilibrium chemistry is used here but with chemical source term different from 0

\frac{\partial Y_k}{\partial t} = \frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k.

This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that flamelet profiles Y_k=f(Z,\chi) can be pre-computed 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 are available such as CHEMKIN [2] and CANTERA [3].

Flamelets in turbulent combustion

In turbulent flames the interest is \widetilde{Y}_k . 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. \widetilde{Y}_k cannot be obtained directly from the flamelets library \widetilde{Y}_k \neq Y_F(Z,\chi) , where Y_F(Z,\chi) corresponds to the value obtained from the flamelets libraries. A generic solution can be expressed as

\widetilde{Y}_k= \int Y_F( \widetilde{Z},\widetilde{\chi}) P(Z,\chi) dZ d\chi

where P(Z,\chi) 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

\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) dZ

P(Z) 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, \widetilde{Z},Z''.

If the mixture fraction and scalar dissipation are consider independent variables, P(Z,\chi) can be written as P(Z) P(\chi). The PDF of the scalar dissipation is assumed to be log-normal with variance unity.

\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) P(\chi) dZ d\chi

In Large eddy simulation (LES) context (see #LES equations for reacting flow), the probability density function is replaced by a subgrid PDF \widetilde{P}. The same equation hold by replacing averaged values with filtered values.

\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) \widetilde{P}(Z) \widetilde{P}(\chi) dZ d\chi

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 large-scale fluctuations are solved) and therefore the modelled PDFs are thinner.

Unsteady flamelets

Intrinsic Low Dimensional Manifolds (ILDM)

Detailed mechanisms describing ignition, flame propagation and pollutant formation typically involve several hundred species and elementary reactions, prohibiting their use in practical three-dimensional engine simulations. Conventionally reduced mechanisms often fail to predict minor radicals and pollutant precursors. The ILDM-method is an automatic reduction of a detailed mechanism, which assumes local equilibrium with respect to the fastest time scales identified by a local eigenvector analysis. In the reactive flow calculation, the species compositions are constrained to these manifolds. Conservation equations are solved for only a small number of reaction progress variables, thereby retaining the accuracy of detailed chemical mechanisms. This gives an effective way of coupling the details of complex chemistry with the time variations due to turbulence.

The intrinsic low-dimensional manifold (ILDM) method {Maas:1992,Maas:1993} is a method for in-situ reduction of a detailed chemical mechanism based on a local time scale analysis. This method is based on the fact that different trajectories in state space start from a high-dimensional point and quickly relax to lower-dimensional manifolds due to the fast reactions. The movement along these lower-dimensional manifolds, however, is governed by the slow reactions. It exploits the variety of time scales to systematically reduce the detailed mechanism. For a detailed chemical mechanism with N species, N different time scales govern the process. An assumption that all the time scales are relaxed results in assuming complete equilibrium, where the only variables required to describe the system are the mixture fraction, the temperature and the pressure. This results in a zero-dimensional manifold. An assumption that all but the slowest 'n' time scales are relaxed results in a 'n' dimensional manifold, which requires the additional specification of 'n' parameters (called progress variables). In the ILDM method, the fast chemical reactions do not need to be identified a priori. An eigenvalue analysis of the detailed chemical mechanism is carried out which identifies the fast processes in dynamic equilibrium with the slow processes. The computation of ILDM points can be expensive, and hence an in-situ tabulation procedure is used, which enables the calculation of only those points that are needed during the CFD calculation.

--Fredgauss 07:37, 25 August 2006 (MDT)

U. Maas, S.B. Pope. Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Comb. Flame 88, 239, 1992.

Ulrich Maas. Automatische Reduktion von Reaktionsmechanismen zur Simulation reaktiver Str¨omungen. Habilitationsschrift, Universit ¨at Stuttgart, 1993.

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

\overline{Y_k}= \int <Y_k|\eta> P(\eta) d\eta

where \eta is the sample space for Z .

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 non-linear functions of mixture fraction. For a given species mass fraction we can decomposed it into a mean and a fluctuation:

Y_k= \overline{Y_k} + Y'_k

The fluctuations Y_k' are usually very strong in time and space which makes the closure of \overline{\omega_k} very difficult. However, the alternative decomposition

Y_k= <Y_k|\eta> + y'_k

where y'_k is the fluctuation around the conditional mean or the "conditional fluctuation". Experimentally, it is observed that y'_k<< Y'_k , which forms the basic assumption of the CMC method. Closures. Due to this property better closure methods can be used reducing the non-linearity of the mass fraction equations.

The Derivation of the CMC equations produces the following CMC transport equation where Q \equiv <Y_k|\eta> for simplicity.

\frac{ \partial Q}{\partial t} + <u_j|\eta> \frac{\partial Q}{\partial x_j} = \frac{<\chi|\eta> }{2} \frac{\partial ^2 Q}{\partial \eta^2} + \frac{ < \dot \omega_k|\eta> }{ <\rho| \eta >}

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 \overline {\dot \omega_k} as a function of the mean temperature and species ( \overline{Y}, \overline{T}) 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 < \dot \omega_k|\eta> \approx \dot \omega_k \left( Q, <T|\eta> \right) 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 \dot \omega_k = k Y_A Y_B with the rate constant in Arrhenius form k=A_0 T^\beta exp [-Ta/T] the second order closure is (Klimenko and Bilger 1999)

< \dot \omega_k|\eta> \approx < \dot \omega_k|\eta >^{FO} \left[1+ \frac{< Y''_A Y''_B |\eta>}{Q_A Q_B}+ \left( \beta + T_a/Q_T \right) \left( \frac{< Y''_A T'' |\eta>}{Q_AQ_T} + \frac{< Y''_B T'' |\eta>}{Q_BQ_T} \right) + ... \right]

where < \dot \omega_k|\eta >^{FO} is the first order CMC closure and Q_T \equiv <T|\eta> . When the temperature exponent \beta or T_a/Q_T 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 h can be chosen to define a double conditioned mass fraction

Q(x,t;\eta,\psi) \equiv <Y_i(x,t) |Z=\eta,h=\psi >

Due to the strong dependence on chemical reactions to temperature, h is advised to be a temperature related variable (Kronenburg 2004). Scalar dissipation is not a good choice, due to its log-normal 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 non-premixed and premixed problems to be treated without ad-hoc 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 Q therefore represents a conditionally filtered reactive scalar.

Linear Eddy Model

The Linear Eddy Model (LEM) was first developed by Kerstein(1988). It is an one-dimensional model for representing the flame structure in turbulent flows.

In every computational cell a molecular, diffusion and chemical model is defined as

\frac{\partial}{\partial t} \left( \rho Y_k \right) = \frac{\partial}{\partial \eta} \left( \rho D_k \frac{\partial Y_k}{\partial \eta }\right)+ \dot \omega_k

where \eta is a spatial coordinate. The scalar distribution obtained can be seen as a one-dimensional reference field between Kolmogorov scale and grid scales.

In a second stage a series of re-arranging stochastic event take place. These events represent the effects of a certain turbulent structure of size l , smaller than the grid size at a location \eta_0 within the one-dimensional domain. This vortex distort the \eta field obtain by the one-dimensional equation, creating new maxima and minima in the interval (\eta_0, \eta + \eta_0) . The vortex size l is chosen randomly based on the inertial scale range while \eta_0 is obtained from a uniform distribution in \eta . 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 joint-PDF of the scalars. Denoting P \equiv P(\underline{\psi}; x, t) where \underline{\psi} = ( \psi_1,\psi_2 ... \psi_N) is the phase space for the reactive scalars \underline{Y} = ( Y_1,Y_2 ... Y_N) . The transport equation of the joint PDF is:

\frac{\partial <\rho | \underline{Y}=\underline{\psi}> P }{\partial t} + \frac{ \partial <\rho u_j | \underline{Y}=\underline{\psi}> P }{\partial x_j} = \sum^N_\alpha \frac{\partial}{\partial \psi_\alpha}\left[ \rho \dot{\omega}_\alpha P \right] - \sum^N_\alpha \sum^N_\beta \frac{\partial^2}{\partial \psi_\alpha \psi_\beta} \left[ <D \frac{\partial Y_\alpha}{\partial x_i} \frac{\partial Y_\beta}{\partial x_i} | \underline{Y}=\underline{\psi}> \right] P

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 "micro-mixing " term. Equal diffusivities are used for simplicity D_k = D

A more general approach is the velocity-composition joint-PDF with P \equiv P(\underline{V},\underline{\psi}; x, t) , where \underline{V} is the sample space of the velocity field u,v,w . This approach has the advantage of avoiding gradient-diffusion 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). A similar formulation was proposed by Sabelnikov and Soulard 2005, which removes part of the a-priori assumption of "smoothness" of the stochastic fields. This approach is purely Eulerian and offers implementations advantages compared to Lagrangian or semi-Eulerian methods. Transport equations for scalars are often easy to programme and normal CFD algorithms can be used (see Discretisation of convective term). Although discretization errors are introduced by solving transport equations, this is partially compesated by the error introduced in Lagrangian approaches due to the numerical evaluation of means.

A new set of N_s scalar variables (the stochastic field \xi ) is used to represent the PDF

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]

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.
  • Fox, R.O. (2003), Computational Models for Turbulent Reacting Flows, ISBN 0-521-65049-6,Cambridge University Press.
  • 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. 595-687.
  • 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. 1907-1925.
  • Kronenburg, A., (2004), "Double conditioning of reactive scalar transport equations in turbulent non-premixed flames", Physics of Fluids, Vol. 16, Num. 7, pp. 2640-2648.
  • Griffiths, J. F. (1994), "Reduced Kinetic Models and Their Application to Practical Combustion Systems", Prog. in Energy and Combustion Science,Vol. 21, pp. 25-107.
  • Peters, N. (2000), Turbulent Combustion, ISBN 0-521-66082-3,Cambridge University Press.
  • Poinsot, T.,Veynante, D. (2001), Theoretical and Numerical Combustion, ISBN 1-930217-05-6, R. T Edwards.
  • Pope, S. B. (1994), "Lagrangian PDF methods for turbulent flows", Annu. Rev. Fluid Mech, Vol. 26, pp. 23-63.
  • Sabel'nikov, V.,Soulard, O. (2005), "Rapidly decorrelating velocity-field model as a tool for solving one-point Fokker-Planck equations for probability density functions of turbulent reactive scalars", Physical Review E, Vol. 72,pp. 016301-1-22.
  • Valino, L., (1998), "A field montecarlo formulation for calculating the probability density function of a single scalar in a turbulent flow", Flow. Turb and Combustion, Vol. 60,pp. 157-172.
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