A Priori Model for the Effective Lewis Numbers in Premixed Turbulent Flames

Paper # 070LT-0267 Topic: Turbulent Flames 8 th US National Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 19-22, 2013. A Priori Model for the Effective Lewis Numbers in Premixed Turbulent Flames Bruno Savard 1 Guillaume Blanquart 2 1 Graduate Aerospace Laboratories, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125 2 Mechanical Engineering, California Institute of Technology A simple a priori model for the effective Lewis numbers in a premixed turbulent flame is presented. This a priori analysis is performed using data from a series of direct numerical simulations (DNS) of lean (φ = 0.4) premixed turbulent hydrogen flames, with Karlovitz number ranging from 10 to 1562 [1]. Those simulations were chosen such that the transition from the thin reaction zone to the broken reaction zone is captured. The conditional mean of various species mass fraction (< Y i T >) vs temperature profiles are evaluated from the DNS and compared to equivalent unstretched laminar premixed flame profiles. The turbulent flame structure is found to be different from the laminar flame structure. However, the turbulent flame can still be mapped onto a laminar flame with an appropriate change in Lewis numbers. Those effective Lewis numbers were obtained by minimizing the error between the DNS results and predictions from unstretched laminar premixed flames. A transition from laminar Lewis numbers to unity Lewis numbers as the Karlovitz number increases is clearly captured - equivalently, as the turbulent Reynolds number increases, given that the ratio of the integral length scale to the laminar flame thickness is fixed throughout the series of DNS. Those results suggest the importance of using effective Lewis numbers that are neither the laminar Lewis numbers nor unity in tabulated chemistry models without considering the impact of the turbulent Reynolds number or Karlovitz number. A model for those effective Lewis numbers with respect to the turbulent Reynolds number was also developed. The model is derived from a Reynolds Averaged Navier-Stokes formulation of the reactive scalar balance equations. The dependency of the effective Lewis numbers to the Karlovitz number instead of the Reynolds number was studied and is discussed in this paper. These changes in effective Lewis numbers have significant impacts. First, the laminar flame speed and laminar flame thickness vary by a factor of two through the range of obtained effective Lewis numbers. Second, the regime diagram [2] changes because a unique pair of laminar flame speed and laminar flame thickness cannot be used. A dependency on the effective Lewis numbers have to be introduced. 1 Introduction Turbulent premixed flames relevant to industrial applications often belong to the thin reaction zone or even the broken reaction zone. In the thin reaction zone, the smallest eddies are small enough to penetrate the flame preheat zone, but too big to enter the reaction zone. In the broken reaction zone, the smallest eddies are small enough to penetrate both the preheat zone and the reaction zone [2]. The relevant non-dimensional parameter is the Karlovitz number defined as the ratio of the flame and Kolmogoroz time scales Ka = t F /t η. The Karlovitz number can also be 1

written as Ka = l 2 F /η2, where l F is the laminar flame thickness and η the Kolmogoroz length scale. Considering that the reaction zone is approximately 10 times smaller than the laminar flame thickness, scaling arguments suggest that 1 < Ka < 100 defines the thin reaction zone and Ka > 100 the broken reaction zone, as long as the premixed flame is turbulent. A large body of work has been done on simulating and modeling turbulent premixed flames up to the broken reaction zone [3 11]. However, most of them consider methane as the fuel. Methane has the characteristic that its species diffusivity equals its thermal diffusivity, i.e. it has a unity Lewis number. This has the convenient effect of greatly simplifying the equations to be solved. Fuels found in industrial applications rarely have unity Lewis numbers. As an example, the Lewis number of n-dodecane, a surrogate for kerosene, is approximately 3.5 in air. Hydrogen is an interesting example of an alternative fuel for ground-based applications such as gas turbines for electricity generation. The Lewis number of hydrogen in a lean hydrogen-air mixture is approximately 0.3, also far from unity. It is obvious that there is great interest in being able to model and simulate accurately turbulent premixed flames with non-unity Lewis numbers, especially at high Karlovitz numbers. Aspden et al. [1] recently performed direct numerical simulations (DNS) of lean (φ = 0.4) premixed hydrogen flames at Karlovitz numbers (Ka) ranging from 10 to 1526. Their results clearly show that the flame structure varies significantly between the lowest and the largest Ka flames, unlike turbulent methane flames. They observed that the largest Ka flame had a structure comparable to the one of a methane flame, i.e. the flame behaved as an effective unity Lewis number flame. Highly turbulent non-unity Lewis number diffusion flames were experimentally observed to behave very similarly to unity Lewis number diffusion flames [12]. To the best of the authors knowledge, there is no premixed flame experiment equivalent to those conducted in [12] in the literature. At high turbulence levels, dissipation of species and temperature is dominated by turbulent mixing, resulting in a unity effective Lewis number. From this argument, the same results should be observed experimentally for premixed flames. Most premixed turbulent combustion models are developed for the corrugated flamelet regime (Ka < 1) and extended to higher Karlovitz regimes. In several of those models, tabulated chemistry assumes that the flame structure can be mapped into a corresponding laminar unstretched flamelet. The results from Aspden et al. seem to suggest that tabulated chemistry is inadequate for non-unity Lewis numbers premixed turbulent flames. This is one of the motivations for the objectives pursued in this work: 1) to analyze the structure of turbulent premixed lean hydrogen flames at Karlovitz numbers ranging from 10 to 1526 to determine if a mapping with a corresponding laminar unstretched flamelet can still be found allowing modifications to the Lewis numbers, 2) to derive an a priori model for the effective Lewis numbers in premixed turbulent flames. Section 2 presents the flame structure obtained from the DNS data of Aspden et al. and compares it to corresponding laminar unstretched flamelets. Section 3 derives a simple a priori model from simplified species and temperature balance equations. Section 4 compares the model against the effective Lewis numbers computed from the DNS. Section 5 presents the impacts of effective Lewis numbers on laminar flame speed and flame thickness, the effective Karlovitz number and the regime diagram. 2

2 Flame structure In this work, the flame structure from the series of DNS simulations performed by Aspden et al. [1] is compared to the structure of laminar unstretched flamelets. The complete set of parameters used for the DNS cases can be found in [1] and are summarized in Table 1. The reactants are a lean (φ = 0.4) hydrogen-air mixture. GRI-2.11 was used as the chemical mechanism (9 species, 27 reactions for hydrogen combustion). Soret and Dufour transports as well as radiation are neglected in the DNS simulations [1]. As confirmed in [1], the transition from the thin reaction zone to the broken reaction/distributed burning zone is covered by the simulation cases. The laminar flame counterparts are simulated using FlameMaster [13]. The equivalence ratio is fixed to 0.4, the GRI-2.11 chemical mechanism is used and the Soret and Dufour transports and radiation are ignored. Case A B C D Equivalence ratio (φ) 0.4 0.4 0.4 0.4 Laminar flame speed (s L ) (m/s) 0.224 0.224 0.224 0.224 Laminar flame thickness (l F ) (mm) 0.629 0.629 0.629 0.629 Length ratio (l/l F ) 0.5 0.5 0.5 0.5 Velocity ratio (u /s L ) 3.69 17.1 32.9 106.8 Turbulent Reynolds number based on viscosity of unburnt gases (Re T ) 14.2 65.8 126.1 410.7 Karlovitz number (Ka) 10 100 266 1526 Table 1: Parameters for series of turbulent premixed hydrogen flame DNS performed in [1]. The conditional mean of the mass fractions with respect to temperature < Y i T > are calculated for an instantaneous snapshot of the established (statistically steady) flame. Figure 1 shows the conditional mean of the hydrogen mass fraction as a function of temperature for cases A through D. The temperature is considered here as a progress variable. Are also shown in Fig. 1, the profiles for laminar unstretched flamelets with 1) full transport, 2) unity Lewis numbers. The full transport and unity Lewis numbers laminar flames correspond to limiting cases of purely laminar and fully turbulent premixed flames respectively. A clear trend is observed: the DNS profiles gradually move from the purely laminar towards the fully turbulent limiting cases. As mentioned in Section 1, an objective of this work is to model the transition between the purely laminar to the fully turbulent flame structure. The presence of hot spots, characteristic of thermodiffusive instabilities, are revealed by the extension of the DNS profiles to temperatures higher than the adiabatic flame temperature ( 1, 400 K). The thermodiffusive instabilities are beyond the scope of this work. This observation justifies the choice of computing the conditional mean as < Y i T > instead of < T Y i >. 3

Figure 1: Conditional mean of the hydrogen mass fraction as a function of temperature for cases DNS A through D and unstretched laminar flames with full transport and unity Lewis numbers. 3 Model proposed 3.1 Effective Lewis Numbers Neglecting the Soret and Dufour effects, the Reynolds Averaged Navier-Stokes (RANS) species and temperature balance equations, assuming equal thermal and species eddy diffusivities and constant heat capacity, can be written as: ρỹi t ( ) + ρũỹi = [ ] ρ (D i + D T ) Ỹi + ω i, (1) ρ T ( + ρũ t T ) [ = ρ (α + D T ) T ] + ω T, (2) where ρ is the Reynolds-averaged density, ũ, Ỹi and T, the Favre-averaged velocity, species mass fraction, and temperature respectively, ω i, ω T, the averaged species mass fraction and temperature source terms. α and D i are the thermal and species molecular diffusivities and D T the eddy diffusivity. Note that Eq. 1 uses a simplified Fick s law of diffusion (species molecular weights are assumed to be the same). When using Eq. 1, numerical codes use a correction velocity in the diffusion flux such that there is no net total mass diffusion flux, i.e. j i = 0. The diffusion flux becomes j i = ρ ( ) D i Y i Y i v c, (3) with v c = D j Y j. Once filtered, j i = ρ (D i + D T ) Ỹi ρy i D j Y j. (4) One can make D T appear in the second term of the flux as follows, using the fact that species mass fractions sum up to 1: j i = ρ (D i + D T ) Ỹi ρy i (D j + D T ) Y j. (5) 4 j j

It is obvious from Eq. 5 that the concept of effective Lewis number remains valid even when considering a correction velocity in the diffusion flux. An effective Lewis number can be obtained for each species: Le i,eff = α + D T = 1 + DT α. (6) D i + D 1 T Le i + D T α 3.2 A priori model testing The Lewis numbers of all 9 species are first obtained through a full transport simulation. They are then set constant to those values in a second simulation. The Lewis numbers are modified to effective Lewis numbers for subsequent simulations according to a one-parameter transformation, corresponding to Eq. 6: Le i,eff = 1 + γ 1 Le i + γ. (7) Figure 2 compares the hydrogen mass fraction versus temperature profiles obtained from DNS and from laminar unstretched flamelets varying the free parameter γ. The Lewis number being a ratio of mass diffusion to heat diffusion, Fig. 2 is perfectly adapted to assess its effects on the flame structure. Two observations can be made from Fig. 2. Firstly, full transport and constant Lewis numbers simulations show similar profiles. This justifies the assumption of constant Lewis numbers through the flame, assumption necessary to simulate the effective Lewis numbers flamelets. Secondly, effective Lewis numbers flamelet profiles agree very well with the DNS profiles. This result suggests that the turbulent flame can still be mapped onto a laminar flame with an appropriate change in Lewis numbers. Note that for the highest value of γ, all Lewis numbers are very close to unity. Figure 2: Comparison of H 2 mass fraction profiles between DNS conditional mean and flamelets with modified Lewis numbers using γ as a fitting parameter. 5

Figure 3 shows the same results for species O 2, H 2 O and OH. For O 2 and H 2 O the variations in profiles from case A to D are marginal. The same trend is observed with the laminar flames. The DNS profiles for OH show globally the same trend and magnitude as the laminar profiles, but less accurately than for H 2. However, it is less likely to get the radicals right and the influence of hot spots on the radicals may be far from negligible as opposed to hydrogen. As a consequence, H 2 mass fraction is the best candidate to describe the flame structure. Figure 3: Comparison of O 2, H 2 O and OH mass fraction profiles between DNS conditional mean and flamelets with modified Lewis numbers using γ as a fitting parameter. 3.3 Turbulent model Using a k ɛ model for the eddy viscosity [14], the eddy diffusivity can be written as Using the same definition for the integral length scale as in [1], and the turbulent kinetic energy D T = ν T P r T = C µk 2 P r T ɛ. (8) l = u 3 RMS ɛ (9) k = 3 2 u RMS2, (10) one obtains D T = 9 C µ u RMS l. (11) 4 P r T Using the definition of the Prandtl number, the ratio of the turbulent to the thermal diffusivities can be written as D T α = 9 C µ u RMS lp r = 9 4 νp r T 4 C P r µ Re T. (12) P r T The species effective Lewis number becomes Le i,eff = 1 + 9C 4 µ P r P r T Re T 1 Le i + 9C. (13) 4 µ P r P r T Re T 6

From the DNS parameters, Re T is a known quantity. P r is a function of temperature and mixture composition and can be obtained from a flamelet simulation. C µ = 0.09 from standard k ɛ model [14]. Finally, Le i is known for all species (can be obtained from a flamelet simulation). However, the turbulent (eddy) Prandtl number P r T is not known. Since the thermal and species eddy diffusivities are assumed equal, it is of the same order of approximation to assume that the eddy viscosity is also equal to the eddy diffusivities. P r T is therefore set to unity. 4 Validation of the model 4.1 Sensitivity analysis to species Lewis numbers The laminar unstretched flamelet profiles are not equally sensitive to all species effective Lewis number. Species like O 2 have a laminar Lewis number close to unity ( 1.1 for O 2 ) and therefore as their effective Lewis numbers progress towards unity, very small changes in the laminar flame profiles should be expected. Therefore, a sensitivity analysis is performed first. One species Lewis number at the time is modified, the other ones being fixed to the species Lewis numbers obtained from the full transport simulation. 100 flamelets are simulated for each modified species, their Lewis number varying from their laminar value towards unity. The resulting H 2 and OH mass fraction versus temperature profiles are compared with the DNS profiles presented previously in Fig. 2-3. As discussed in section 2, the profiles for O 2 and H 2 O are not suitable for this kind of analysis. Figure 4 shows the results for H 2 (top) and OH (bottom) mass fractions for modified Lewis numbers of H 2, H, and all species but H 2 and H, respectively from left to right. Modifying the Lewis number of H 2 has by far the most influence, whereas modifying the Lewis number of H has small effect on H 2 mass fraction and small but non negligible effect on OH mass fraction. Modifying the other species Lewis number has negligible influence on the flame structure. It is obvious that the Lewis numbers of H 2 and H have the most influence. Two explanations are suggested and cannot be differentiated in this study: 1) H 2 and H are the species with Lewis numbers the furthest from unity and 2) H 2 being the fuel and H the most important radical, their diffusion controls the flame structure. 4.2 Computing the effective Lewis numbers The range of variations in the flame structure due to changes in the H 2 Lewis number is sufficiently wide and encompasses all the DNS data. As a result, it is expected to be able to find a best set of Lewis numbers to compare to the DNS. Unfortunately, because of the small sensitivity of the results on the Lewis numbers of species other than H 2 (and H to a lesser extent), such a set cannot be found for these species. Therefore, the effective Lewis numbers for H 2 are computed for cases A through D. Only the Lewis number of H 2 is modified. The L2-norm of the error between the DNS and the laminar profiles is minimized to obtain the hydrogen effective Lewis number, i.e., for 7

Figure 4: Comparison of H 2, and OH mass fraction profiles between DNS conditional mean and flamelets with modified H 2 (left), H (center), and all species but H 2 and H (right) Lewis numbers. each DNS case (A through D): 1 Le H2,eff = arg min Le N H 2 N i=1 ( Y lam H 2 (Le H 2, T i ) YH DNS 2 (T i ) ) 2 (14) subject to Le H 2 [Le H2, 1], where YH DNS 2 (T i ) corresponds to the interpolated value of < Y H2 T > (from the DNS) at T = T i, and YH lam 2 (Le H 2, T i ) corresponds to the value of Y H2 obtained from the laminar unstretched flamelet simulation with Le H2 = Le H 2, interpolated at T = T i. The temperature is discretized uniformly from the minimum temperature in the domain T u to the adiabatic flame temperature T ad such that T i = T u + i (T N+1 ad T u ) (the hot spots are avoided). Table 2 presents the effective Lewis numbers obtained and Fig. 5 compares them against the model presented in Section 3. Note that the Prandtl number is fixed to the unburnt value and a fitting coefficient of 0.5 is used in front of C µ in Eq. 13. This fitting coefficient is of the order of the relative uncertainties on the coefficient in front of Re T in Eq. 13. In particular, the fitting coefficient does not change the slope in the semilog plot of Fig. 5. The results are in very good agreement with the model. Since the model does not make a difference between a species or another, it is expected that each species should follow the model presented. Therefore, each Lewis number is modified according to Eq. 7 in a subsequent series of flamelet simulations. Defining Le eff as the vector containing all species effective Lewis numbers, 8

this vector is obtained, equivalently to Eq. 14, as follows: 1 Le eff = arg min Le N N i=1 ( Y lam H 2 (Le, T i ) YH DNS 2 (T i ) ) 2 Le j = 1+γ, 1 +γ Le j subject to γ [0, + ). (15) The hydrogen effective Lewis numbers obtained are presented in Table 2. They are also compared against the model in Fig. 5. The agreement with the model remains relatively good, but the slope shown by the model seems to be slightly off, which suggests that the power of Re T in Eq. 13 may be incorrect. Case A B C D Le H2,eff obtained through Eq. 14 (only Le H2 modified) 0.43 0.66 0.77 0.88 Le H2,eff obtained through Eq. 15 (all Le i modified) 0.44 0.73 0.85 0.99 Table 2: Parameters for series of turbulent premixed hydrogen flame DNS performed in [1]. Figure 5: Effective Lewis numbers versus turbulent Reynolds number obtained by L2-norm minimization (DNS cases A through D) for only H 2 and all Lewis numbers modified in flamelet simulations. 4.3 Reynolds versus Karlovitz number The series of simulations performed in [1] have a fixed l/l F ratio. Consequently, the effects of Ka cannot be differentiated from those of Re T, as described by the following relation: Re T Ka 2/3 eff, (16) 9

using the effective Karlovitz number Ka eff based on the effective laminar flame thickness. Ka eff is discussed in greater details in Section 5. Based on the observation made in subsection 4.2, Re T is replaced by Ka eff in Eq. 13 to form a non-physical model. This model is compared against the Lewis numbers presented in Table 2 (obtained from Eq. 15), as depicted in Fig. 6. Here, no fitting coefficient is used. The agreement with the DNS values is better than for the Re T -based model. This better agreement with the non-physical Ka eff -based model raises interesting questions. It seems reasonable that the effective Lewis numbers would depend on laminar flame characteristics, which is not the case with the Re T -based model. The assumption that all thermophysical properties are constant through the flame hides the flame characteristics. A more complete model is necessary to verify if a Ka dependency is physical. However, in the DNS analyzed the largest relevant turbulent scale is smaller than the flame thickness and its corresponding turnover velocity is u. Supposing that D T u T l T it is not clear that a Ka dependence should have been observed with the series of DNS studied. Figure 6: Effective Lewis numbers versus Karlovitz number obtained by L2-norm minimization (DNS cases A through D) for all species Lewis numbers modified in flamelet simulations. 5 Effective Ka and regime diagram 5.1 Impact on s L and l F As the effective Lewis number of the fuel changes with turbulent Reynolds number, the laminar flame speed and flame thickness are expected to change significantly. Since the turbulent flame can be mapped onto a laminar unstretched flamelet with appropriate effective Lewis numbers, the reference laminar flame speed and flame thickness for the turbulent flame must be recalculated consequently. From a one-step reaction, matched asymptotic expansion with only the fuel Lewis number different than unity, the following ratios are obtained [15]: ( ) 0.5 ( ) 0.5 s L,2 LeF,2 l s L,1 = Le F,1, F,2 LeF,1 l F,1 = Le F,2, (17) where s L,i and l F,i are the laminar flame speed and flame thickness of a flame with corresponding fuel Lewis number Le F,i. Note that the only different parameter between flames 1 and 2 is the fuel Lewis number. Using the Re T -based model, the laminar flame speed and flame thickness obtained 10

from Eq. 17 are plotted as a function of the turbulent Reynolds number in Fig. 7. The analytic solutions are compared to the flamelet calculation results using the effective Lewis numbers according to the Re T -based model. The agreement between the curves is relatively good. The discrepancies can be due to the numerous simplifying assumptions made in the matched asymptotic expansion. However, it is clear that both the laminar flame speed and flame thickness vary from almost a factor of 2 between the laminar Lewis numbers flamelet and the unity Lewis numbers flamelet (hydrogen premixed flame with φ = 0.4). Figure 7: Effective laminar flame speed and flame thickness versus turbulent Reynolds number. 5.2 Impact on regime diagram The result from subsection 5.1 suggests that the Karlovitz number should take into account the effective flame thickness or flame speed. Defining an effective Karlovitz number as it can be related to the traditional Karlovitz number as follows: Ka eff = l2 F,eff η 2, (18) Ka eff = Le F Le F,eff Ka. (19) Recall the relation derived from Peters in [2] from which the iso-ka are obtained in the premixed regime diagram: ( ) u 1/3 l = Ka 2/3. (20) s L l F Following the results shown in this paper, the relevant quantity to differentiate whether the smallest eddies will penetrate the preheat zone or the reaction zone is the effective Karlovitz number. Therefore, Eq. 20 can be rewritten as ( ) u 2/3 ( ) 1/3 = Ka 2/3 LeF,eff l eff. (21) s L Le F l F 11

Using the Re-based model presented in this paper, the ratio of effective to laminar fuel Lewis numbers can be expressed as Le F,eff = 1 + β u l s L l F, (22) Le F 1 + Le F β u l l F with β = 9C 4 µ P r s L l F P r T a constant. Hence, fixing the effective Karlovitz number to relevant values ν as 1 or 100 [2], one obtains an implicit equation (combining Eq. 21 and Eq. 22) which can be solved numerically to obtain a modified regime diagram as shown in Fig. 8. The shifts observed for the delimiting lines are non-negligible. DNS simulations A through D are positioned in Fig. 8. Aspden et al. found in [1] that the two lowest Ka (10 and 100) DNS simulations showed a thin reaction zone behavior, whereas the highest Ka (1526) simulation was clearly in the broken reaction or distributed burning zone and the last simulation (Ka = 266) was a transition case, whereas scaling arguments [2] suggest that the transition should be around Ka = 100. This is not inconsistent with the scaling. However, predicting more accurately where transition occurs would find important applications. These observations from Aspden et al. agree very well with Fig. 8, even though the authors do not claim to be able to explain the transition Ka found in [1]. Aspden et al. also found an influence of the equivalence ratio on the transition Ka (from the thin reaction zone to the broken reaction or distributed burning zone). The effective Lewis number however has a non-negligible impact on the transition Ka for the l/l F ratio studied (0.5). s L Figure 8: Regime diagram taking into account the effective Karlovitz number. 6 Conclusions The average structure of turbulent premixed lean hydrogen flames (φ = 0.4), obtained through DNS simulations [1], is found to vary considerably with Karlovitz number ranging from 10 to 1562. These flames cover the transition from the thin reaction zone to the broken reaction/distributed burning zone. The turbulent flames are shown to have the same structure in average as laminar unstretched flamelets with appropriate effective Lewis numbers. The fact that flames in those regimes can be mapped onto laminar unstretched premixed flamelets with an appropriate change in Lewis 12

numbers has important consequences. First, this means that the tabulated chemistry approach remains valid in the thin reaction and the broken reaction/distributed burning zones as long as the tabulated flamelets are simulated with appropriate Lewis numbers corresponding to the parameters of the turbulent flame (Ka or Re T or both). Second, effective laminar flame speed and flame thickness have to be considered. Indeed those quantities vary by almost a factor 2 between the laminar Lewis numbers and the unity Lewis numbers for premixed hydrogen flamelet (φ = 0.4) simulations. In light of these results, the effective Karlovitz number Ka eff also has to be considered. The regime diagram proposed by Peters [2] should therefore be adapted to the fuel considered whenever the fuel has a Lewis number far from unity. An a priori model for the effective Lewis numbers involving Re T is derived from RANS equations. Considerable simplifications of the physics are implied. Nevertheless, very good agreement with the effective Lewis numbers obtained from the series of DNS is found. Quantification of uncertainties has yet to be performed and the model should be seen as a first attempt to describe the transition from laminar Lewis numbers to unity effective Lewis numbers. Also, more data points are needed to fully validate the model. Flames with fuel Lewis number greater than unity have yet to be compared to the model. Acknowledgments The authors gratefully acknowledge funding from the Air Force Office of Scientific Research (Award FA9550-12-1-0144) under the supervision of Dr. Chiping Li, and from the Natural Sciences and Engineering Research Council of Canada (NSERC Postgraduate Scholarship D). This work was also made possible by a collaboration with Dr. Andy Aspden, from the University of Portsmouth, who kindly shared with the authors the DNS data presented in [1]. References [1] A.J. Aspden, M.S. Day, and J.B. Bell. J. Fluid Mech., 680 (2011) 287 320. [2] N. Peters. Turbulent Combustion. Cambridge University Press, Cambridge, 2000. [3] O. Colin, F. Ducros, D. Veynante, and T. Poinsot. Physics of Fluids, 12 (2000) 1843 1863. [4] J.P. Legier, T. Poinsot, and D. Veynante. Proc. CTR Summer Program, (2000) 157 168. [5] M.S. Anand and S.B. Pope. Comb. Flame, 67 (1987) 127 142. [6] R.P. Lindstedt and E.M. Vaos. Comb. Flame., 145 (2006) 495 511. [7] O. Gicquel, N. Darabiha, and D. Thévenin. Proc. Combust. Inst., 28 (2000) 1901 1908. [8] J.A. van Oijen, F.A. Lammers, and L.P.H. de Goey. Comb. Flame, 127 (2001) 2124 2134. [9] H. Pitsch and L. Duchamp de Lageneste. Proc. Comb. Inst., 29 (2002) 2001 2008. [10] E. Knudsen and H. Pitsch. Comb. Flame, 156 (2009) 678 696. [11] A.Y. Poludnenko and E.S. Oran. Comb. Flame, 157 (2010) 995 1011. [12] J.C. Ferreira. Flamelet modelling of stabilization in turbulent non-premixed combustion. Dissertation, ETH Zürich, 1996. [13] H. Pitsch. A c++ computer program for 0d combustion and 1d laminar flame calculations. Technical report, University of Technology (RWTH), Aachen, 1998. 13

[14] D.C. Wilcox. Turbulence Modeling for CFD. DCW Industries, Anaheim, 2000. [15] N. Peters. Fifteen lectures on laminar and turbulent combustion. Ercoftac Summer School, September 1992. 14