Theoretical and numerical combustion third edition 2012

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Skip to main content. Search SpringerLink Search. Abstract The dynamic power-law wrinkling model proposed by Charlette et al. References 1. Flame 1 , — Article Google Scholar 6. Flame 11 , — Article Google Scholar 7. Flame 4 , — Article Google Scholar 8.

Flame 4 , — Article Google Scholar 9. Flame 4 , — Article Google Scholar Flame , — Article Google Scholar Flame 3 , — Article Google Scholar Flame 8 , — Article Google Scholar Flame 2 , — Article Google Scholar Flame 9 , — Article Google Scholar Flame 7 , — Article Google Scholar Tabbal, H. Numerical computation of bifurcation phenomena and pattern formation in combustion. Bayliss, B. Matkowsky, M. An explicit Runge-Kutta method for turbulent reacting flows calculations.

Knock prediction in spark ignition engines. Cox, R. Myhill, W. A local extinction of the thermo-diffusive premixed flame at low Lewis number. Computer simulation of detailed processes occurring at near extinction conditions of flame spread over solid fuels. Induction period generation of a supersonic flame. A numerical solution for reacting and non reacting flow. On the equations for reactive granular flow.

Implicit schemes for subsonic combustion problems. Page 1 Navigate to page number of 2. Because of thermal expansion, ub is expected to be larger than uu , leading to a positive turbulent flux as suggested by Eq. This counter-gradient turbulent transport i.

This phenomenon is observed both experimentally Libby and Bray , Shepherd et al. A simple physical insight into counter-gradient turbulent transport Veynante et al. The first part of the chamber referred to as region A is controlled by the vortices shed behind the flame holder. When the centerline velocity increases in region B because of expansion induced by the heat release , the upper lower coherent structures start to turn counterclockwise clockwise. A simple geometrical analysis, based on Eq.

This last term is not related to turbulent motions. Once again, an additional term, corresponding to intermittency between fresh and burnt gases and not to actual turbulent motions, occurs. Because of the intermittency term, e k measures a velocity fluctuation which does not correspond to the real turbulence levels either in the fresh or in the burnt gases. This point can be demonstrated experimentally by conditional velocity measurements Cheng and Shepherd92 , Cho et al.

Laser Doppler velocity measurements have been performed in a V-shape turbulent premixed flame stabilized behind a small rod Veynante et al. In the fresh gases region location a , a gaussian-type probability density function pdf is observed as in a classical turbulent flow field. The RMS 5. The flame can no longer be identified as a laminar flame front but is still a wrinkled flame. The question is now: are Kolmogorov scales able to enter the reaction zone?

Then, two regimes are identified: — When 1 Ka,r , both diffusion and reaction zone are affected by turbulent motions. No laminar structure could be longer identified. Vorticity field thin lines and reaction rate bold lines. Very thick line denotes the fresh gases boundary of the preheat zone. Same initial turbulent field for both simulations. The flame thickness is 5 times larger in case b than in a.

In case a the whole flame structure remains laminar-like. In case b the preheat zone is modified by small turbulent scales. Kolmogorov vortices are the most efficient in terms of induced strain rate but, because of viscous dissipation, have a short lifetime and therefore only limited effects on combustion. For example, these structures are probably unable to effectively quench the flame.

The flame response depends on how long it is submitted to the vortex stretch and how fast the vortex is dissipated by viscous effects. Following the previous comments, a more refined diagram should be derived analyzing the ability of vortices to quench a premixed flame front and the importance of unsteady phenomena.

Quenching in turbulent premixed combustion Flame quenching occurs when a flame front is submitted to external perturbations such as heat losses or sufficiently strong aerodynamic stretch to decrease the reaction rate to a negligible value, and, in some cases, to completely suppress the combustion process. For example, asymptotic studies of laminar stagnation point flame established by the counterflow of reactants and products Bush and Fendell74 , Libby and Williams , Libby et al.

Libby et al. These results have been confirmed by numerical methods for example, Darabiha et al. These have already been presented in Section 2. All these studies have been performed for laminar flames but, assuming a thin flame front in the wrinkled flame regime, they should also be valid in turbulent flames. This regime is called a flamelet regime here and defined as Fig.

When the local stretch induced by turbulent motions on the flame front becomes sufficiently large to quench the flame at a given location, combustion stops in the vicinity of this point and fresh reactants diffuse into the products without burning.

In this situation, the description of the reacting flow becomes much more complex and standard flamelet approaches are no longer valid. Therefore, quenching in a turbulent premixed flame determines the limit between two fundamentally different behaviors i. Flame elements are usually assumed to quench in turbulent flows for conditions similar to laminar stagnation point flames Bray This assumption is justified, once again, from asymptotic theories Clavin and Williams , Clavin and Joulin which show that, under certain assumptions mainly low levels of stretch , the whole flame structure and many important flame parameters, such as displacement and consumption speeds, depend only on stretch.

For example, the displacement speed sd defined by Eq. In the present definition, the flamelet regime corresponds only to a continuous flame front without quenching. The inner flame structure can differ from laminar flame cases. The simplest example of a positively stretched flame is the planar stagnation point flame studied in Section 2. Many stretched flamelets are also found in the leading edges of turbulent flames where flame surface is produced.

Accordingly, small scales should not be included in a description which only relies on stretch. In a turbulent flame, the flame front is stretched by turbulent eddies.

Therefore, induced stretches vary because eddies move with respect to the flame. Thus, the flame is also curved when stretched. As shown by Mikolaitis , curvature effects may be very important and should be accounted for. As described in Section 4. Each scale present on the turbulence line has a different effect on the flame front. Some vortices may lead to quenching, some may induce formations of fresh gases pockets, while others may be dissipated by viscous effects before any interaction with the flame structure.

In the same turbulent reacting flow, all three types of vortices may be found at the same time: a description based on one scale cannot take all mechanisms into account. The interaction between a flame and a vortex can be characterized by three non-dimensionalized numbers Fig. A necessary condition for strong interactions is that the speed induced by the vortex be greater than the flame speed, i.

Ka r coincides with the Karlovitz number of Eq. Since the size and velocity of the vortical structure may be controlled independently, the aerodynamic perturbation of the flame front is precisely defined. At the initial time, two counter-rotating vortices are generated upstream of a laminar flame front.

The vortex-pair configuration generates large flame stretch values and interacts efficiently with the flame front because of its self-induced velocity. An important aspect linked to the choice of the equations used for the DNS deals with the introduction of heat loss models into the computation. These losses are mainly due to radiation. On the other hand, modeling heat losses in real flames is a challenge in itself and corresponding models cannot yet be included in a DNS. In this case, the difficulty is to choose a level of 5.

More sophisticated models may be found in recent DNS Patnaik and Kailasanath but the uncertainties on the level of radiation losses remain high. Simulations of Poinsot et al. Simulations also include heat release, variable density and viscosity.

The temperature ratio between fresh and burnt gases is 4 and the Lewis number is 1. The length scale r used to characterize the size of the perturbation is the sum of the vortex diameter D and of the distance between vortex centers Fig. The velocity scale u0 r is defined as the maximum velocity induced by the vortex pair. Calculations are carried out for a broad range of parameters 0.

An example of results is plotted in Figs. Figures 5. These gases are pushed rapidly into regions where the burnt gases have been cooled due to heat losses Fig. This effect, combined with the high stretch generated by the vortices, causes almost complete extinction of the pocket after it has been separated from the bulk of the fresh gases.

In this case, the flame front is locally quenched by the vortex pair and, in addition, unburnt mixture has crossed the flame. The 5. Four typical regimes are identified: 1. Two curves are plotted in Fig. Spectral diagrams may also be constructed experimentally Roberts et al.

Large differences should be expected between experiments and simple chemistry computations such as the DNS of Poinsot et al. Qualitatively, however, the trends of the experimental results are well reproduced by DNS as shown on Fig. Modified combustion diagram A turbulent premixed combustion diagram may be deduced from the spectral diagram displayed in Fig. A single vortex structure interacts at a given time with the flame front.

Any turbulent structure located in the quenching zone of the spectral diagram locally quenches the flame front and induce a distributed reaction regime. These assumptions are rather crude. For example, turbulent scales in the quenching zone do not quench the flame front if the corresponding energy density is too low.

Therefore, assumption 2 is probably too strong and not satisfied. Nevertheless, the following construction, based on DNS, is probably more precise than analysis based on simple dimensional arguments. Under the previous assumptions, the construction of a new turbulent combustion diagram based on the spectral diagram of Fig. Point B therefore corresponds to a flamelet regime. In the case of field A, even the integral scale is not sufficiently energetic to interact with the flame and the latter remains pseudo-laminar.

Turbulent field C contains scales that may locally quench the flame double-width solid line. These scales are larger and faster by orders of magnitude than the Kolmogorov scale.

Type C turbulence creates a turbulent flame which cannot satisfy flamelet assumptions. Comparing this diagram Fig. Taking into account the important value of heat losses used for this computation, the flamelet domain is expected to be even larger in most practical cases. Peters has also pointed out that the quenching limit in Fig.

For premixed flames, equations 4. The preexponential constant B is negative. T1 , T2 and Ta are respectively the fresh gases, the adiabatic flame and the activation temperatures. Y varies from 1 in the fresh gases to 0 in the burnt gases. In Eq. How these terms influence mean values is a key issue in turbulent combustion modeling.

The same demonstration holds in an unsteady turbulent flame when initial conditions also satisfy the same property. These models are discussed in Section 5. Models for scalar turbulent fluxes are generally closed using a gradient assumption see Eq. The reaction rate is then expressed as:! In most situations, this model is completely inadequate: Fig. To first order, the largest vortices, close to the integral length scale, are assumed to produce the strongest flapping motions.

The Eddy-Break-Up model is found in most commercial codes. Despite its success, its basic form 5. The preexponential constant B or the activation temperature Ta do not appear in Eq. However, the EBU model generally gives better results than the simple Arrhenius model. For example, the x A comparison shows that Eqs. According to Eq. Experimental data exhibit a large scatter and depend on various parameters chemistry characteristics, turbulence scales, flow geometry among others.

But this term slightly modifies the propagation speed of the flame front. Such a description may also be useful for very large scale combustion system simulations. The Gequation formalism does not require to resolve the flame brush thickness in the computation but only the G-field which may be quite larger Smiljanovski et al.

A more refined formalism based on G-equation has been developed by Peters Peters , Peters , briefly described page in Section 5. Combining a statistical approach using probability density functions and a physical analysis, the BML model has evidenced some special features of turbulent premixed combustion such as counter-gradient turbulent transport and flame turbulence generation, already described in Section 5.

A one-step, irreversible chemical reaction between fresh gases R and combustion products P is considered. Usual assumptions are introduced to simplify the model formulation: perfect gases, incompressible flow, constant heat capacities and unity Lewis numbers. According to Section 5. The first two formulations, based on scalar dissipation rate and on flame crossing frequency, are discussed here. The later one, derived in terms of flame surface densities, is described in Section 5.

A balance equation may be derived, and solved, for the scalar dissipation rate, as done, for example, by Mantel and Borghi Then, according to Eq. The expression 5. Flame crossing frequencies This analysis Bray et al. Points like A on the cold side of the flame brush or B on the hot side both exhibit very few passages of flame fronts and have a low mean reaction rate. Point B has not a mean reaction rate significantly higher than point A.

This is very different from a laminar case see Fig. In the center of the flame brush, many flame crossings are found at point C and the mean reaction rate is high. Estimating this reaction rate per crossing flame is difficult in practice so that expression 5. A high flame surface density at a given location in the flow corresponds to a high turbulent reaction rate. The turbulent flame is viewed as an ensemble of small laminar flame elements flamelets supposed here to have 5.

The consumption rate per unit flame surface sc of these flamelets may be computed including complex chemistry using a simple model of a laminar planar stagnation point flame Giovangigli and Smooke This may be done using a simple algebraic expression or solving a balance equation. Ly is the mean flame length scale. Bray et al. But, on the other hand, hsc is may be easily estimated from laminar flame theories or computations.

The flame surface density may also be derived from fractal theories, leading to Gouldin et al. D is the fractal dimension of the flame surface.

Cut-off scales obtained from DNS such as expression 5. These equations may be derived from heuristic arguments Marble and Broadwell , xiv The BML model is first based on a bimodal probability density function assumption statistical analysis, Eq. Its derivation leads to a reaction rate proportional to the scalar dissipation rate mixing description, Eq.

Finally, the model is recast in terms of flame surface density geometrical description, Eq. This derivation evidences the links between the various physical approaches available for turbulent combustion modeling described in Section 4.

These links are rigourously derived in Veynante and Vervisch As qualitatively described in Section 4. The surface averaged operator i. This exact balance equation leads to the following comments: xv This last term is generally neglected in practice, assuming that the propagation velocity in the normal direction, hsd ni is , of the order of the laminar flame speed s0L , is negligible compared to the convection velocity hui is.

This assumption is not always true as shown by Veynante et al. This is true when the flame is infinitely thin. Using Eq. Without a destruction term, the flame surface density balance equation would predict an infinite growth of flame area a property which holds for non reacting material surfaces but not for flames which annihilate when they interact.

Whether this destruction term is only due to curvature effects last RHS term in Eq. A simple phenomenological closure may be derived as follows Marble and Broadwell , Darabiha et al. The hsc is term is the consumption speed computed using complex chemistry but is the only chemical parameter appearing in balance equations 5. It can be computed using one-dimensional laminar flame codes discussed in Section 2. Ret is the turbulent Reynolds number.

Chemistry and turbulence have been treated independently. Flame surface density models have been applied to turbulent premixed flames stabilized behind bluff bodies Maistret et al. They provide a fairly good prediction of chemistry effects for example, the influence of the equivalence ratio on turbulent combustion. Flame surface density models are also equivalent to a number of other flamelet models such as the Mantel and Borghi model derived from a balance equation for the scalar dissipation rate Eq.

Comparisons of various flame surface density models may be found in Duclos et al. For most models using Eq. Peters proposes sd expressions suited to flamelet and to corrugated flame regimes. Details are out of the scope of this book and the reader is referred to the recent book by Peters , devoting a large part to this level set description. This level-set formalism is also developed for corrugated flame regimes and, a priori, does not require a flamelet assumption.

The flame is only viewed as a propagating front and the challenge is to model the flame front displacement speed sd. In this situation, turbulence models are no longer required turbulence motions are directly described through the joint pdf but this approach is very expensive.

This stochastic description has many theoretical advantages. Probability density functions may be defined in any turbulent reacting flow field. They contain all the required information to describe unsteady reacting flow fields. These functions may also be extracted from experimental data or direct numerical simulations statistical analysis of one-point measurements. The difficulty is to determine the pdf which changes at every point in the flow field.

Two main paths have been proposed Pope , Pope , Dopazo : to presume the pdf shape or to solve a balance equation for the pdf. Presumed pdf approach In general, a pdf function can take any shape and exhibit multiple extrema.

It contains information not only on the mean value of the variable but also on its variance first moment and on all higher moments. For many combustion applications, however, pdf functions often present common features, suggesting that these functions can be described using a limited number of parameters.

A possible approach Williams is then to suppose that the pdf has a fixed shape, parametrized using, for example, only one or two parameters. More sophisticated pdf shapes may be used to construct other models and the literature provides multiple examples of presumed pdf shapes Borghi50 , Bray et al.

The pdf parameters a and b are determined plugging Eq. Fuel oxidation is fast and may be modeled using flamelet assumptions and flame surface density models.

On the other hand, nitric oxide N Ox formation is slower and occur in burnt gases, mainly controlled by gases temperature.

Note also that, as N Ox and most pollutants involve very small quantities ppm , they are negligible against main species and have no influence on global balances heat release rate, main species mass fractions, temperature. Accordingly, they may be estimated by postprocessing simulation results. These simple closure schemes neglect some phenomena: for example, counter-gradient turbulent transport is not taken into account by gradient assumptions in Eqs.

They also involve some rough assumptions: as discussed in Section 6. When more than one variable is needed for chemistry for example, two or more species in addition to temperature , constructing a multi-dimensional pdf becomes more difficult Lockwood and Naguib , Pope , Gutheil But this assumption does not hold in practical situations because species mass fractions and temperature are closely related in flames and, accordingly, are not statistically independent.

This derivation is out of the scope of this book see, for example, Pope , Dopazo , Vervisch et al. The first three terms in equation 5. The main interest of the balance pdf equation is that the chemical reaction term in Eq. Accordingly, the pdf balance equation approach is able to handle any complex chemical scheme. Unfortunately, the molecular diffusion term is unclosed and is difficult to model.

This finding is not surprising: as the one-point pdf describes the chemical composition at any location, one-point quantities such as the chemical reaction rate, which depend only on the local composition, are naturally closed.

But spatial gradient terms involved in molecular diffusion processes require additional length scale informations which are not incorporated in the one-point pdf formalism. Although this method is general and powerful, at least when ad hoc models are provided for molecular mixing terms, its practical application to industrial cases remains difficult and time consuming.

However, this approach is now available in some commercial codes. In this situation, turbulence model are no longer required but additional terms are found in the pdf balance equation Pope For such flows, Eq. Non-constant density reacting flows are difficult to describe and most turbulence studies are limited to constant density situations, without chemical reactions. However both experimental data Bray et al.

This is consistent with Eq. Despite these evidences, the practical importance of counter-gradient turbulent transport remains controversial. Recent DNS studies Veynante et al. For instance, in e and Fig. Mean and conditional average velocities across the turbulent flame brush are displayed in Fig. Although this result is surprising ub is expected to be larger than uu because of thermal expansion due to the heat release , it is consistent with expression 5.

Results from the Center for Turbulence Research database CTR , exhibiting a gradient turbulent transport solid line , are compared with results from direct simulations performed by C.

Rutland dashed line where counter-gradient turbulent transport is found Veynante et al. In fact, conditional averages are not intuitive quantities and should be handled with care. As shown in Fig. Velocities are normalized by the laminar flame speed s0L Veynante et al. Veynante et al. Gradient to wrinkle the flame front. This criterion matches experimental data Kalt According to the simple criterion 5.

Note also that buoyancy effects are enhanced in ducted flames, because of pressure gradients Veynante and Poinsot Counter-gradient gradient turbulent transport are promoted under favorable adverse pressure gradients, i.

Counter-gradient transport also tends to decrease the turbulent flame speed i. Results obtained with both types of simulations are similar but, because of computational costs, only two 3D DNS are available whereas a large range of parameters is explored using two-dimensional simulations.

The closure of Eq. Direct numerical simulations can be used to analyze these different terms and thereby evaluate the general feasibility of the second-order closure approach. A typical DNS evaluation of all terms appearing in Eq. The analysis reveals the dominant terms in Eq. For instance, Fig. This imbalance is due to inherent numerical errors involved in the simulations as well as in the post-processing of the data.

Its magnitude remains small, which suggests that DNS can indeed be used to analyze the variations of second-order moments as Domingo and Bray did to derive closures for the pressure fluctuation term. Such high-order models are almost 5. Results obtained in DNS of three-dimensional isotropic turbulence with variable density, single-step Arrhenius kinetics chemistry. In this case, counter-gradient turbulent transport is observed Veynante et al.

However, the limitations imposed by gradient formulations for the turbulent transport terms remain strong and high-order formulations will probably develop in the future. Large eddy simulation LES also appears as a very promising approach to describe scalar turbulent transport: as shown in Section 5.

The issue is often critical in practical simulations. A wide range of scales, from Kolmogorov to integral scales, is likely to be involved.

A very thick flame is not wrinkled by turbulence motions like a very thin flame and this should be taken into account. The basic idea is to estimate the strain rate induced by a given pair of counter-rotating vortices size r and velocity u0. As shown in Section 5. This balance equation can also be recast in terms of flame surface density equation MB model in Table 5. Expression 5. This analysis is conducted under some restrictive assumptions such as frozen turbulence turbulent flow field is not affected by combustion and can only be derived for some combustion models.

It is, however, an efficient way to study basic model trends. More details may be found in Hakberg and Gosman , Fichot et al.

A gradient description of turbulent fluxes is assumed. The basic idea of the KPP analysis is to look for an exponential solution of Eq. For example, Table 5. Relations derived in Section 4. Using the Kolmogorov time scale as the turbulent flame time leads to a turbulent flame speed which is higher than with the integral time scale.

Frozen turbulence. The turbulent flame speed itself is not a very useful quantity as experimental results display a large scatter. Nevertheless, the KPP analysis provides a simple way to predict model trends. It can be extended to two equations models, as done by Duclos et al. The objective of this section is to briefly describe these mechanisms and to evidence the corresponding difficulties for turbulent flame numerical simulations. Additional discussion on the effects of walls on turbulent flames is given in Chapter 7.

Creating a low speed region in the flow speed allows the flame stabilization Beer and Chigier In this situation, the turbulent flame speed is able to sustain the incoming flow velocity leading to flame stabilization. This objective is generally achieved using a so-called flame-holder, inducing a large recirculation zone see Fig.

In this case, the incoming reactants are continuously ignited through a heat source such as a pilot flame small secondary flame , hot wire, hot gas stream or an electrically sustained plasma, as displayed in Fig. The flame is stabilized in a location where the turbulent flame speed sT is able to sustain the flow velocity u e. Flame Fresh gases heat fluxes Hot burnt gases Figure 5. Longitudinal velocities are larger than the turbulent flame speed.

Experiment by Moreau None of these mechanisms is explicitly modeled in most turbulent premixed combustion models. However, RANS models, through sT , generally predict flame stabilization in low speed regions such as recirculation zones. But the exact flame stabilization location is not correctly predicted because these models do not account for actual ignition mechanisms which involve complex chemistry features and control initial regions of flames in Fig. An experimental evidence of modeling difficulties is displayed in Fig.

Flame surface surface density fields are obtained from laser tomography and mean reaction rates are estimated from CH radical emission. In this region, fresh and burnt gases are separated by an interface the probability to find this interface is high so that flame surface densities are large where combustion is started but not yet fully established. Flame surface density data half top , extracted from two different data sets, are not available from 30 to 70 mm downstream the rod Veynante et al.

Most models have the same drawbacks. Nevertheless, they completely miss the flame stabilization mechanism and transient chemical effects. Left: adiabatic step wall. Right: cold wall, the wall temperature is set equal to the fresh gases temperature DNS by Veynante and Poinsot The description of flame stabilization mechanisms remains a difficult challenge for turbulent combustion modeling.

For example, in spark-ignited engines, the initial flame is laminar, grows and later becomes turbulent. Boudier et al. Taking into account ignition time, spark energy and flow conditions, this submodel determines when the flame becomes turbulent and the initial flame surface density to be imposed in the turbulent code. Wall effects such as heat transfer and catalytic effects have to be taken into account and may strongly affect the stabilization and the evolution of the flame see Chapt.

This point is illustrated on Fig. Models for large eddy simulations LES are now briefly reviewed. On the other hand, large eddy simulations started during the eighties for non-reacting, constant density flow fields and only a few years ago for combustion. Accordingly, LES is still at an early stage for reacting flows and only the basic principles of the main proposed approaches are summarized here.

A difficult problem is encountered in large eddy simulations of premixed flames: the thick0 of a premixed flame is about 0. In fact, the most important contribution to the reaction rate probably occurs at the subgrid scale level suggesting that LES could be impossible for reacting flows Pope To overcome this difficulty, three main approaches have been proposed: simulation of an artificially thickened flame, use of a flame front tracking technique G-equation , or filtering with a gaussian filter larger than the mesh size.

In certain LES approaches, this theoretical problem is sometimes avoided by developing subgrid scale models for filtered reaction rates 5. The subgrid turbulent kinetic energy k SGS may be given from an algebraic expression or a balance equation. In this formalism, the flame thickness problem is not taken into account: this LES-EBU model is used together with a gradient model for e field in numerical simulations. As discussed in Section 4. For sufficiently large F values, the thickened flame front is resolved on the LES computational mesh see Fig.

The laminar flame is artificially thickened but its flame speed is conserved. This route has not been yet extensively tested in large eddy simulations but seems to be promising, at least when the flow scales are much larger than the laminar flame thickness, as in combustion instabilities see Chapt.

As already discussed in Section 5. This ratio is decreased by a factor F when the flame is thickened. This point has been investigated using DNS by Angelberger et al. An efficiency function E, corresponding to a subgrid scale wrinkling factor, has been derived to account for this effect. Reaction rate and vorticity fields are superimposed. This effect can be parametrized using a subgrid scale model. The G-equation is written as Kerstein et al.

This closure is generally based on Eq. Im et al. In many cases, u0 - sT correlations obtained in experiments or used in RANS models are directly used in LES without further justification, replacing the turbulent rms velocity by the subgrid scale turbulent velocity.

Equation 5. Despite these drawbacks, the G-equation is a popular technique for large eddy simulations of turbulent premixed combustion. Nevertheless, following e may be resolved using a physical space Gaussian Boger et al.

The flame front displacement term in Eq. Proposed approaches are formally identical to the ones developed in RANS context. Algebraic expressions Boger et al. Since counter-gradient transport may be explained by differential buoyancy effects between cold fresh and hot burnt gases, all characteristic length scales are involved. Thereafter, a portion of the counter-gradient phenomena is expected to be directly described in large eddy simulations through resolved motions, even when a subgrid scale gradient-type closure is used.

This point is evidenced in Fig. Three transverse profiles of the LES resolved contribution to the downstream turbulent transport are displayed on Fig. Close to the flame-holder, this turbulent transport is of gradient type but becomes counter-gradient further downstream, according to the analysis conducted in Fig. This shows that countergradient turbulent transport, at least at the resolved level, is easily recovered in LES even 5.

Thin lines correspond to RANS turbulent transport. Data analysis from Boger et al. Such a prediction would be very difficult in RANS simulations, requiring a second-order modeling Bailly et al. Another important point has to be considered when analyzing unresolved scalar fluxes in reacting, and more generally variable density flows.

Such a 5. Because of thermal expansion due to heat release, resolved velocity gradients and resolved shear stresses contain both information on turbulence and on thermal expansion and are non-zero for laminar flames. Accordingly, these quantities are not well suited to extract actual turbulence information to model, for example, the subgrid scale flame front wrinkling due to turbulence motions.