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where <math> \widetilde{\chi} </math> is the mean scalar dissipation rate
where <math> \widetilde{\chi} </math> is the mean [[Scalar dissipation]] rate
defined as <math> \widetilde{\chi} =
defined as <math> \widetilde{\chi} =
2 D \widetilde{\left| \frac{\partial Z''}{\partial x_j} \right|^2 } </math>

Revision as of 11:00, 12 December 2005


What is combustion -- Physics versus modelling

Combustion phenomena consists of many physical and chemical processes with broad range of time scales. Mathematical description of combustion is not always trivial. Analytical solutions exists only for basic situations of laminar flame and because of its assumptions it is often restricted to few problems solved usually in zero or one-dimensional space.

Problems solved today concern mainly turbulent flows, gas as well as liquid fuels, pollution 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. However here the modelling part will be mainly addressed.

In addition to the flow parameters used in fluid mechanics, new non-dimensional parameters are introduced, specially improtant 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. Schemes.jpg

Reaction mechanisms

The combustion is mainly chemical process and although we can, to some extend, describe flame without any chemistry informations, for modelling of flame propagation we need to know the speed of reactions, product concentrations, temperature and other parameters. Therefore more or less detailed information about reaction kinetics is essential for any combustion model. Mixture will generally combust, if the reaction of fuel and oxidiser is fast enough to maintain 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 typical combustion reaction is influenced mainly by concentration of reactants, temperature and pressure.

A stoichiometric equation of an arbitrary equation can be written as:


where  \nu is the stoichiometric coefficient, M_j is arbitrary species. One prime specifies the reactants and double prime products of the reaction.

Reaction rate, expressing the rate of disappearance of reactant i of such a reaction, is defined as:


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


where A is pre--exponential factor, E is activation energy and \beta is temperature exponent. These constants for given reactions can be found in literature. The reaction mechanism can be given from experiments for every reaction resolved, it could be also constructed numerically by automatic generation method (see [Griffiths (1994)] for review on reaction mechanisms). For simple hydrocarbon tens to hundreds of reactions are involved. By analysis and systematic reduction of reaction mechanisms global reaction (from one to five step reactions) can be found (see [Westbrook (1984)]).

Governing Equations for Reacting Flows

Together with the usual Navier-Stokes for compresible flows (See Governing equations), additional equations are needed in reacting flows. The mass fraction transport equation for k-th species  Y_k is:


where Ficks law is assumed for scalar diffusion with  D_k , the species difussion coefficient and  \dot \omega_k is the species reaction rate. A non-reactive scalar (like the mixture fraction  Z ) had the following transport equation:


where  D is the diffusion coefficient of the passive scalar.

RANS equations

In turbulent flows, Favre averaging is often used and the mass fraction transport equation is transformed to

+ \overline{\dot \omega_k}

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

The passive scalar turbulent transport equation is


where similar to the mass fraction equation,  \widetilde{u''_i Z''} needs modelling.


\frac{\partial}{\partial x_j}


LES equations

The Large eddy simulation (LES) equation for reactive flows introduces equations for the filtered species mass fractions to the compressible flow field. Similar to #RANS equations, but using Favre filtering instead of Favre averaging. The filtered mass fraction transport equation is

J_j \right)
+ \overline{\dot \omega_k}

where  J_j is the transport of subgrid fluctuations of mass fraction


and has to be modelled. Fluctuations of diffusion coefficients are often ignored and their contributions much smaller than apparent turbulent diffusion due to transport of subgrid fluctuations. The first term on the right hand side is then

\frac{\partial} {\partial x_j}  
\frac{\partial} {\partial x_j}  

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 of the comparable time scale of the mixing process. The simpler approach assuming chemistry fast enough, that the limiting process is mixing of the species is historically older approach and even today can be appropriate approach. It is simpler to solve then #Finite rate chemistry models, but introduces errors to the solution which may or may not be important.

Premixed Combustion

Premixed flame occurs in mixtures of fuel and oxidiser, homogeneously premixed prior to the flame. These flames are not limited only to gas fuels, but also to the pre-vaporised fuels. Typical example of premixed laminar flame is bunsen burner, where the air enters the fuel stream. The mixture burns in the wake of the riser tube walls forming nice stable flame. The premixed flames has many advantages in terms of control of temperature and products and pollution concentration, but introduce also some dangers like the autoignition (in the supply system).


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 and Hjertager, and Spalding and can be found in all CFD packages. The model assumes the reactions to be completed in the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. The combustion is 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 multistep scheme, where each reaction has its own mean reaction rate. The mean reaction rate is given by:


\bar{C} denotes mean concentrations for 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 the experimental results and they are suitable for most of the general cases. Still they are just constants 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 those regions a positive reaction rate occurs and an artificial flame can be observed. CFD codes usually has some remedies to overcome this problem.

This model largely over-predicts temperatures and concentrations of species like CO and other species. Still this model is quite popular for its simplicity and relatively easy 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 in 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 of 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 problems is split into two parts: First, the 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 , which deals with the distribution of the reactive species inside the flamelet.


Using the following chain rules for the time


and spatial coordinate


to the species transport equation (see #Governing Equations for Reacting Flows) and re-arranging, we obtain


The second and third term in the LHS cancel due to continuity and mixture fraction transport, the equation therefore boils down to


where  \chi = 2 D  \left( \frac{\partial Z}{\partial x_j} \right)^2 is called the scalar dissipation and controls the mixing, providing the interaction between the flow and the chemistry.




Burke-Schumann flame structure

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



Y_F=Y_F^0 Z

Oxidizer mass fraction



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 G equation

Flame surface density model

Non-premixed Combustion

Flamelets based on conserved scalar


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




Flamelets in turbulent combustion












Unsteady flamelets

Conditional Moment Closure (CMC)


From Probability density function we have


where  \eta is the sample space for  Z .



Y_k= \overline{Y_k} + Y'_k


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






Second order closure



\right) + ...



Double conditioning





LES modelling


Linear Eddy Model






PDF transport models













Other combustion models




Derived from the #Eddy Dissipation Concept (EDC).


  • Dopazo, C, (1993), "Recent development in PDF methods", Turbulent Reacting Flows, ed. P. A. Libby and F. A. Williams.

External links and sources

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