Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 4, pp. 859-871, Warsaw 2013 LES-CMC AND LES-FLAMELET SIMULATION OF NON-PREMIXED METHANE FLAME (SANDIA F) Artur Tyliszczak Czestochowa University of Technology, Institute of Thermal Machinery, Częstochowa, Poland e-mail: atyl@imc.pcz.czest.pl In this paper, the Large Eddy Simulation (LES) together with the Conditional Moment Closure (CMC) andflamelet combustionmodels havebeen applied formodelling ofmethane flameSandiaF. In the case of theCMCmodel, both instantaneous and time averagedvalues predicted numerically agree well with measurements. Attention was devoted to modelling aspects of the conditional scalar dissipation rate (SDR), which is a key quantity of theCMC approach. The two methods of computing SDR are compared with emphasis on a correct prediction of localised extinctions and on their influence on the mean values. It was found that themethod ofmodelling of SDR has ratherminor impact on the instantaneous values, whereas larger differences were observed in statistics. In the case of the flamelet model, although it is not able to predict extinctions and re-ignition, the mean values were in good agreement with the experiment. Key words: localised extinction, piloted flame, large eddy simulation, conditional moment closure, flamelet 1. Introduction The Large Eddy Simulation (LES)method is becoming a standard tool in academic research in virtually all aspects of contemporary CFD (Computational Fluid Dynamics) including reactive flows with combustion. The LES approach, contrary to the classical (u)RANS ((unsteady) Rey- nolds Averaged Navier-Stokes) methods, gives a very deep insight into the unsteady turbulent flow phenomena. In the case of fundamental research, the LES method is often combined with very sophisticated combustionmodels such as theConditionalMomentClosure (CMC) (Klimen- ko and Bilger, 1999) or Eulerian PDF approach (Jones and Prasad, 2010), which are currently regarded as the most accurate. The CMC model applied in this paper allows for analysis of very complicated physical processes including lifted flames (Navarro-Martinez andKronenburg, 2011), local extinction (Garmory and Mastorakos, 2011), auto-ignition (Stankovic et al., 2011; Tyliszczak, 2013) or forced ignition (Triantafyllidis et al., 2009). A big disadvantage of the CMC model is very high computational cost both from the po- int of view of memory requirements as well as from the point of view of computational time. Hence, LES-CMC simulations even for relatively simple problems always involve a number of optimisation steps which in many cases open fields for simplifications and various modelling strategies. The paper concentrates onmodelling aspects of the conditional scalar dissipation rate (whe- rein after abbreviated as SDR) which is a key parameter of the CMC model. We focus on its influence on time averaged quantities and on correctness of capturing localised extinctions. The test case is a piloted methane flame issuing into ambient air – the so-called Sandia flame. We consider Sandia flame F in which, due to high fuel velocity, the localised extinctions occur in a large extent. Analysis of possible influence of SDR modeling was motivated by Garmory and 860 A. Tyliszczak Mastorakos (2011) where it was found that propermodelling of SandiaF flame required calibra- tion of themodel constant in the formula for SDR. In the present work, rather than tuning the model constant, we compare two different approaches to calculate conditional values of SDR. Considering the complexity of the CMC approach, on the opposite side there is a flamelet combustion model (Peters, 2000; Poinsot and Veynante, 2001) which is both simple in formula- tion and very efficient from the computational point of view. Known limitation of the flamelet model is its inability to predict the extinction caused, for instance, by a strong velocity gradient. In this work, the LES-Flamelet approach is applied for purpose of comparison with the CMC method, it is shown that lack of excellent features of the CMC method does not necessarily mean wrong results. Thepaper is organised as follows: in the next section presentation of LES andCMCmethods is limited to the fundamental formulation andappropriate papers are cited for interested readers; the main attention is paid to possible variants of modelling of the scalar dissipation rate which are then compared in computations; in Section 3 numerical schemes and algorithms used in the LES and CMC codes are briefly characterised; the obtained results are presented in Section 4, which is followed by conclusions. 2. Mathematical modelling 2.1. LES formulation In LES, the scales of turbulent flow are divided into large scales which are directly solved on a given numerical mesh, and the small scales (subgrid scales) which require modelling. The separation of the scales is obtained by spatial filtering (Geurts, 2003) which applied to the continuity equation, the Navier-Stokes equations and the transport equation for the mixture fraction gives ∂ρ̄ ∂t + ∂ρ̄ũj ∂xj =0 ∂ρ̄ũi ∂t + ∂ρ̄ũiũj ∂xj =− ∂p̄ ∂xi + ∂τij ∂xj + ∂τ sgs ij ∂xj ∂ρ̄ξ̃ ∂t + ∂ρ̄ũiξ̃ ∂xi = ∂ ∂xi ( ρ̄D ∂ξ̃ ∂xi ) + ∂Jsgs ∂xi (2.1) where the overbar symbol stands for the LES filtering applied to density (ρ̄) and pressure (p̄). The wide tilde symbol stands for the density weighted filtering applied to the velocity field ũi = ρui/ρ̄ and mixture fraction ξ̃ = ρξ/ρ̄. The mixture fraction is a conserved quantity re- presenting the ratio of the mass fraction of the fuel and oxidiser, and it may be regarded as a quantity reflecting the level of local mixing. Themixture fraction is a key element of the CMC model as well as all types of the flamelet based combustion models (Poinsot and Veynante, 2001). Theviscous termsare representedby the tensor τij =µ(∂ũi/∂xj+∂ũj/∂xi−2/3δij∂ũk/∂xk), andunresolved subgrid stress terms τ sgs ij and Jsgs aremodelled by the eddyviscosity typemodel defined as τ sgs ij =2µtSij − 1 3 τkkδij Jsgs = ρ̄Dt ∂ξ̃ ∂xi (2.2) where Sij is the rate of the strain tensor of the resolved field. The subgrid viscosity µt is computed according to the model proposed by Vreman (2004) and the subgrid diffusivity is defined as Dt = µt/ρSct, where Sct is the turbulent Schmidt number is assumed constant Sct =0.4 (Triantafyllidis andMastorakos, 2009). LES-CMC and LES-Flamelet simulation of ... 861 2.2. CMC formulation TheCMCmodel has been formulated in 90s independently byKlimenko andBilger and then it was summarized in their joint paper (Klimenko and Bilger, 1999). In the context of LES, the CMCmodelwas presented byNavarro-Martinez et al. (2005) approximately ten years later. The LES-CMCmodel has been derived applying the density-weighted conditional filtering operation (Colucci et al., 1998) to the transport equations for the species mass fraction (Yk) and total enthalpy. The CMC equations in the framework of LES are given as (Navarro-Martinez et al., 2005; Triantafyllidis andMastorakos, 2009) ∂Qk ∂t + ũi|η ∂Qk ∂xi = Ñ|η ∂2Qk ∂η2 + ˜̇ωk|η+eY k=1,2, . . . ,n (2.3) where n is the number of species. The operator (·|η) = (·|ξ = η) is the conditional filtering operator with conditioning being done on themixture fraction. The symbol Qk = Ỹk|η is condi- tionally filtered speciesmass fractions, ũi|η – conditionally filtered velocity, Ñ|η – conditionally filtered SDR, and eY = ∂ ∂xi (Dt|η ∂Yk ∂xi ) represents the subgrid interactions (Triantafyllidis and Mastorakos, 2009). The conditionally filtered reaction rate is evaluated with the 1-st order clo- sure (Klimenko and Bilger, 1999) where the subgrid conditional fluctuations are neglected, i.e. ˜̇ωk|η = ωk(Q1,Q2, . . . ,Qn). The conditionally filtered variables are related to the filtered va- riables by integration over the mixture fraction space, this is defined as: f̃ = ∫1 0 f̃|ηP̃(η)dη, where P̃ is the filtered probability density function assumed here as beta-function PDF (Cook and Riley, 1994). We note that the above integral formula is also used in the flamelet model in which the term f̃|η is replaced by the laminar flamelet solution (Poinsot and Veynante, 2001) obtained solving: N ∂ 2Yk ∂ξ2 + ω̇k =0with N =N0G(ξ) defined in Eq. (2.4). The CMC equations are formulated in four-dimensional space, i.e. physical co-ordinates and mixture fraction space. This means that in every time step the solution should have been computed on Nx,y,z×Nη nodes,where Nx,y,z and Nη denote thenumberof nodes in thephysical and mixture fraction spaces. This leads to a very high computational cost which absolutely prevents application of the LES-CMC approach for realistic problems, and even in simple cases the computations are hardly feasible. A common simplifying approach is to use two separate meshes: one for the solution of the flow field (CFDmesh) and another one,much coarser for the CMCequations (CMCmesh). Although the application of the coarsermesh for theCMCmodel is a must, it is additionally justified by the fact that in the physical space the conditionally filtered variables are smoother than the LES filtered variables (Navarro-Martinez et al., 2005). Hence, they do not require the numerical resolution as good as for the flow variables (velocity, mixture fraction). In the papers cited in the introduction, the ratio of the nodes of CFD/CMC meshes varies in between 20-300 depending on the flow problem. The main difficulty of the CMC model is related to the modelling of the conditional terms appearing in Eq. (2.3), i.e. the conditionally filtered SDR, velocity and diffusivity. Among these terms, the most important is the SDR which directly influences on the solution in the mixture fraction space.As itwas shown inTriantafyllidis andMastorakos (2009), Stankovic et al. (2011), Garmory and Mastorakos (2011), Tyliszczak (2013) the modelling of Ñ|η may have a crucial impact on the results, and it may qualitatively change the flow behavior. The application of two meshes causes that the conditional terms have to be first computed based on the resolved variables on the CFDmesh and then transferred to theCMCmeshwhere theCMCequations are being solved. The conditional velocity and diffusivity are usually expres- sed directly by the filtered values (Navarro-Martinez et al., 2005; Triantafyllidis andMastorakos, 2009; Stankovic et al., 2011; Garmory and Mastorakos, 2011), i.e. ũi|η ≈ ũi and Dt|η ≈ Dt, 862 A. Tyliszczak whereas the SDR is most often computed with the AMC – Amplitude Mapping Closure model (Kim andMastorakos, 2006) defined as Ñ|η=N0G(η) G(η) = exp ( −2[erf−1(2η−1)]2 ) N0 = Ñ 1∫ 0 G(η)P̃(η) dη (2.4) where erf (x) is the error function. The filtered scalar dissipation rate Ñ is computed as the sum of the resolved and subgrid part (Garmory andMastorakos, 2011; Navarro-Martinez et al., 2005; Navarro-Martinez and Kronenburg, 2011) Ñ =D [ ∂ξ̃ ∂xi ∂ξ̃ ∂xi ] ︸ ︷︷ ︸ resolved + 1 2 CN νt ∆2 ξ̃′′2 ︸ ︷︷ ︸ subgrid (2.5) Following (Triantafyllidis and Mastorakos, 2009; Stankovic et al., 2011; Navarro-Martinez and Kronenburg, 2011), the model constant is assumed CN = 2, although different values may be found in literature (Garmory andMastorakos, 2011; Tyliszczak, 2013). There are no clear pieces of advice what value of CN should be in a particular problem, and thus the value of CN is sometimes estimated based on existing experimental orDNSdata, or sometimes it is set by trial and error. This paper does not aim to calibrate CN, and we rather focus on a methodology of calculation of conditional SDR on the CMCmesh. As it was mentioned, the application of two meshes requires that the conditional terms computed on the CFDmeshmust be transferred to the CMC mesh. Various possibilities for transferring data between the CMC and CFD meshes were discussed inTriantafyllidis andMastorakos (2009). Assuming that the conditional variables (f̃|η) have been computedon theCFDmesh, their counterparts on theCMCmesh are calculated by using a PDF weighted volume integral within the CMC cells (VCMC) to which f̃|η belong. This is defined as f̃|η ∗ = ∫ VCMC ρ̄P̃(η)f̃|η dV ′ ∫ VCMC ρ̄P̃(η) dV ′ (2.6) Thus, the conditionally filtered variable f̃|η ∗ corresponding to the CMC cell is common for a group of the CFD nodes embedded in that CMC cell. Formula (2.6) is applied for the veloci- ty ũi|η ∗ , diffusivity D̃t|η ∗ and also for the scalar dissipation rate Ñ|η ∗ . However, in this case we additionally tested another option to compute Ñ|η ∗ , i.e. we applied theAMCmodel directly on the CMC resolution (Triantafyllidis andMastorakos, 2009). This approach leads to Ñ|η ∗ =N∗0G(η) N ∗ 0 = Ñ∗ 1∫ 0 G(η)P̃∗(η) dη (2.7) where Ñ∗ and P̃∗(η) are the volume integrated values Ñ∗ = ∫ VCMC ρ̄Ñ dV ′ ∫ VCMC ρ̄ dV ′ P̃∗(η) = ∫ VCMC ρ̄P̃(η) dV ′ ∫ VCMC ρ̄ dV ′ (2.8) LES-CMC and LES-Flamelet simulation of ... 863 In this work we compare the results obtained with two variants of computing Ñ|η ∗ . Themodel defined by Eq. (2.4) with volume integration according to Eq. (2.6) will be denoted asN-1, and the model defined by Eq. (2.7) with Eq. (2.8) will be denoted asN-2. Having the conditional terms computed on theCMCmesh, theCMCequationsmay be then solved. Next, using the conditionally filtered variables the LES filtered variables, on the CFD mesh are obtained from f̃(x,t)= 1∫ 0 f̃|η ∗ P̃(η) dη (2.9) with P̃(η) evaluated separately in each CFD node andwith f̃|η ∗ being the same for a group of the CFD nodes sharing particular CMC cells. 3. Numerical methods The CMC and flamelet models have been implemented in a high-order LES solver called SA- ILOR. The SAILOR code is based on the low Mach number approach (Cook and Riley, 1996). The spatial discretisation is performed by the 6th order compact method (Lele, 1992) for the Navier-Stokes and continuity equations and with 5th order WENO scheme (Shu, 2003) for the mixture fraction. The time integration is performed by the Adams-Bashforth/Adams-Multon predictor-corrector approach. The solution algorithm is well verified, the SAILOR code was used in various LES studies for gaseous flows,multi-phase flows and flames (Kuban et al., 2010, 2012; Aniszewski et al., 2012; Tyliszczak, 2013). TheCMCequationswere solvedapplyingtheoperator splittingapproachwhere the transport in a physical space, transport in a mixture fraction space and chemistry are solved separately. In the physical space, the conditional variables are smoother than the filtered ones (Navarro- Martinez et al., 2005;Triantafyllidis andMastorakos, 2009), andhence,without significant loss of accuracy, theCMCequations could bediscretisedusing the secondorderfinite differencemethod combined with 2nd order TVD (Total Variation Diminishing) scheme with van Leer limiters for the convection terms (Hirsch, 1990). The TVD schemes guarantee stable solutions without unphysical oscillations that could have appear in regions of strong gradients – for instance in the vicinity of locally extinguishing or re-igniting flame. The time integration within the operator splitting approach consists of three steps. First, the system resulting from the spatial discretisation in thephysical space is solved applying the first order explicit Eulermethod. In the mixture fraction space, the CMC equations are stiff due to the reaction rate terms. In this case, the time integration is performed applying the implicit Euler method combined with VODPK (Brown andHindmarsh, 1989) solver that is well suited for stiff systems.The reaction rates were computed using CHEMKIN interpreter. In thiswork,weused twowell knownchemicalmechanisms: theSmookemechanism(Smooke, 1991) with 16 species and 25 reactions and the GRI-2.11 chemical mechanism (Bowman et al., [3]) containing 49 species reacting through 277 elementary reactions. The Smooke mechanism was applied both for the CMC and flamelet models, whereas theGRI-2.11mechanismwas used with the flamelet model only. As the flamelet model is computationally very efficient comparing to the CMC approach, it allowed for using much more sophisticated chemistry. On the other hand, the simulations with the CMC model and with the GRI-2.11 chemistry would probably take tremendous amount of time, e.g. the computations (initial flow evolution and gathering statistics) with the Smooke mechanism took 5 days for the flamelet model, whereas for the CMC model almost 30 days were needed. In the following Section, the presentation of the results obtained by applying the flamelet model is limited to those obtained with the GRI-2.11 mechanism as it is regarded as more accurate. 864 A. Tyliszczak 4. Computational results The Sandia flames (Barlow and Frank, 1998) are commonly used test cases for non-premixed combustion models and are probably the most often computed flames over the world. Sketch of the computational configuration is shown inFig. 1 on the lefthandside.Thenozzle consists of the inner fuel pipewith adiameter D=7.2mmandtheouter pipewithadiameter D=18.2mmfor the pilot flame. The fuel jet is a mixture of methane and air withmass fractions YCH4 =0.156, YO2 =0.197, YN2 =0.647, what corresponds to the stoichiometricmixture fraction ξST =0.351. The total amount of oxygen is small and insufficient for combustion and, therefore, this flame is classified as non-premixed.Thepilot flamecorresponds to themixture fraction ξ=0.27 andwas modelled assuming the steady flamelet solution. The temperature of the fuel jet and coflowing air is equal to 300K and their velocities are Uj =99.2m/s and Uc =1.0m/s. Fig. 1. Schematic view of the nozzle for Sandia flames (a) and the isosurfaces of instantaneous temperature (b) Preliminary computations were necessary in order to set-up the computational domain and meshes ensuring a sufficient resolution. Various cuboidal or hexahedral shapes with meshes consisting of various number of nodes and stretching parameters were analysed. Finally, the computational domainhadahexahedral shapewithdimensions: length 60D, inlet plane 6D×6D and outlet plane 30D×30D. The CFDmesh consisted of 128×216×128 nodes and the CMC meshwasmuch coarserwith 20×40×20 nodes. In both cases, themesheswere slightly stretched axially and radially towards the nozzle. The inlet boundary conditionswere specified in terms of thevelocity andmixture fraction and theycorrespondedto theexperimental profiles (Barlowand Frank, 1998). The lateral boundaries were also assumed as the inlet with zero mixture fraction and axial velocity equal to Uc. The outlet boundary was assumed as a convective outflow with constant pressure. The boundary conditions in the physical space for the conditional variables were specified in terms of solutions in the mixture fraction space, i.e. in every node at the boundary the solution for Qk is given the entire range 0¬ ξ ¬ 1; (i) at the inlet plane in the regions of the fuel jet and coflow the inert solution was assigned, i.e. it was assumed that the species and enthalpy vary linearly for 0 < ξ < 1; in the region of the pilot flame the steady flamelet corresponding to burning state was defined (shown schematically in Fig. 1); (ii) all remaining boundaries are assumed as the Neumann boundary conditions. LES-CMC and LES-Flamelet simulation of ... 865 4.1. Instantaneous results The Sandia flame F corresponds to a fully developed turbulent flame, the complexity of the flow structure represented by isosurfaces of the temperature is shown in Fig. 1 on the right hand side. Stronglyunsteadyflamebehaviour ismanifested by the occurrence of local extictionswhich may be well seen in instantaneous experimental data or in numerical results provided that the combustionmodel predicts the extinction phenomena properly. Figure 2 presents instantaneous contours of the temperature and isoline of the stoichiometric mixture fraction ξST =0.351. The CMC results in Fig. 2 are presented together with the solutions obtained applying the steady flamelet model. Fig. 2. Contours of instantaneous temperature obtained with the flamelet model (a) and the CMCmodel (b). The arrows point local instantaneous extinctions The contours of temperature in Fig. 2 are plotted for values above 900K. Analysing the results obtained with the flamelet, we can see that for the stoichiometric conditions (black line) the temperature is always high. On the other hand, one may notice that for the CMC results in some points (shown by the arrows) the temperature is low (i.e. < 900K) even if the mixture is at stoichiometry - such behaviour is interpreted as local extinction. A very good method of quantifying the amount of extinctions is to represent the time variation of a given variable (temperature or species) as the function of themixture fraction in scatter plots. Such results for the temperature at z/D=15 for the CMC solution with conditional SDR computed according to the method N-1 and for the flamelet solutions with maximum SDR equal to N0 = 2 and N0 =20 (seeEq. (2.4)) are shown inFig. 3. TheCMCresults obtained applying themethodN-2 are qualitatively not distinguishable from those obtained applying themethodN-1, and in both cases they remind closely the experimental data.On the other hand, the results fromtheflamelet model differ from each other and also from the experimental results. As one may observe, the instantaneous values lie close to some a priori determined lines, and indeed they correspond to laminar flamelets computed for N0 =2 and N0 =20. The small dispersions from these laminar solutions are caused only by themixture fraction variance.We note that the computations with the Smooke mechanism led to very similar qualitative behaviour, i.e. the dispersion form the laminar flamelets was very small. Hence, as expected, the obtained results clearly illustrate that the flamelet model is unable to predict extinction and, on the other hand, the CMC approach does that verywell. This is further confirmed in Fig. 4 showing a comparison of the scatter plots of CO species obtained from the measurements and from simulations applying the CMC with themethodsN-1 andN-2. The agreementwith the experiment is very goodwith respect to both the absolute values and distribution and, as one may see, the methods N-1 and N-2 give very similar solutions. Finally, one should mention that further downstream or closer to the nozzle the scatter plots from the CMC results also agree reasonable well with the experiment. 866 A. Tyliszczak Fig. 3. Scatter plots of temperature at z/D=15. Experimental data and numerical solutions obtained with the CMCmodel (Ñ|η fromN-1) and with the flamelet approach for N0 =2 and N0 =20 Fig. 4. Scatter plots of the COmass fraction at z/D=15. Experimental data and numerical soulutions obtained with the CMC solutions model (Ñ|η fromN-1 andN-2) LES-CMC and LES-Flamelet simulation of ... 867 4.2. Averaged results The instantaneous resultshave shownthat theCMCmodelpredicts local extinctionsproperly whereas the flamelet model fails in this type of analysis. Below we analyse how this deficiency influences time averaged results. Figures 5-8 show mean profiles of the mixture fraction and temperature together with their fluctuations along the radial direction in four locations from the nozzle z/D=7.5, 15, 30, 45. In these figures, the results obtainedwith theCMCmodel are presented for two variants of computing Ñ|η. They are compared with solutions obtained using the flamelet model with N0 =20 and with the experimental data. The results for the flamelet model with N0 =2 are not much different from those for N0 =20. Depending on the location and compared quantity, the results for N0 =2 are slightly better or slightly worse with respect to the experiment. Fig. 5. Radial profiles of the meanmixture fraction at selected locations from the nozzle The CMC results obtained applying the method N-1 or N-2 differ substantially from each other. From all presented results those corresponding toN-1 are in the best agreement with the experiment, although the remaining solutions are also accurate. Hence, there are two interesting findings: (i) firstly, it seems that inability to predict local extinction by the flamelet model has no crucial influence on the time averaged results; this would mean that the local extinctions in the analysed flame are very short lasting phenomenawhoose overall duration is small compared to timewhen themixture burns; (ii) the instantaneousCMCresults applying themethodN-1 or N-2werevery similarbut itwasnot thecase for the timeaveragedvalues; thismeans that theway how the conditional SDR is computed has larger impact on statistics than instantaneous values. To some extent, this is in contradiction with the results presented byGarmory andMastorakos (2011), where they stressed the importance of Ñ|η with respect to modeling of instantaneous values. They used the method N-2, and in order to predict local extinctions correctly, they calibrated the level of Ñ|η using experimental data for Sandia D flame, eventually they used the model constant CN = 42 in Eq. (2.5). Unfortunately, they did not show how the level of 868 A. Tyliszczak Fig. 6. Radial profiles of the mixture fraction RMS at selected locations from the nozzle Fig. 7. Radial profiles of the mean temperature at selected locations from the nozzle LES-CMC and LES-Flamelet simulation of ... 869 Fig. 8. Radial profiles of the temperature RMS at selected locations from the nozzle Ñ|η influences themean values andwhat are instantaneous values of the temperature or species for different CN = 42. The present solutions show that from the point of view of the time averaged results, themethodN-1 gives better results than themethodN-2, and hence, onemay suppose that calibration of the model constant for Ñ|η in Eq. (2.4) would produce even better predictions. That analysis is planned for future research. 5. Conclusions The LES-CMC and LES-Flamelet approaches have been successfully applied to the modelling of non-premixed methane flame (Sandia F). The experimental data show that due to high fuel velocity the flame locally extinguishes and re-ignites. Numerical modelling of such phenomena is a challenging task for all combustion models and only few are able to model them correctly. As shown in the cited papers and in the present work, the CMC approach allows for analysis of the extinction and re-ignition with good accuracy. The formulation of the CMC model is very complex both from the theoretical point of view as well as from the computational side where the main issue is related to very high computational cost forcing the application of two different meshes: dense mesh for the LES solver and coarse mesh for the CMC. This, in turn requires a transfer of variables between CFD and CMC mesh. In this paper, it was shown that the method of transferring the conditional scalar dissipation rate may have significant influence on time averaged values and relativelyminor on instantaneous values.These results are surprising and certainly need further analysis. Another important and interesting observation is that the flamelet model, which completely fails in the simulation of local extinction, gives averaged values in the acceptable agreementwith the experimental data. So, taking into account the computational cost of theCMCmodel, one shouldalways consider application of theflamelet approach, particularly when a very deep insight in flow physics is not the main task and when 870 A. Tyliszczak the instantaneous phenomena do not determine the flow behaviour. On the other hand, if the oposite is the case, the CMC approach is indispensable. Acknowledgements The financial support for this work was provided by the Polish Ministry of Science under grant NN501098938. 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