Modelling of the Curvature Term of the Flame Surface Density Transport Equation for Large Eddy Simulations

A simplified chemistry based three-dimensional Direct Numerical Simulation (DNS) database of freely propagating statistically planar turbulent premixed flames with a range of different values of turbulent Reynolds number has been used for the a priori modelling of the curvature term of the generalised Flame Surface Density (FSD) transport equation in the context of Large Eddy Simulation (LES). The curvature term has been split into the contributions arising due to the reaction and normal diffusion components of displacement speed and the term originating from the tangential diffusion component of displacement speed. Subsequently, these contributions of the curvature term have been split into the resolved and subgrid contributions. New models have been proposed for the subgrid curvature terms arising from the combined reaction and normal diffusion components and the tangential diffusion component of displacement speed. The performances of the new model and the existing models for the subgrid curvature term have been compared with the corresponding quantity extracted from the explicitly filtered DNS data. The new model for the subgrid curvature term is shown to perform satisfactorily in all cases considered in the current study, accounting for wide variations in LES filter size.

The final term on the right hand side of (1) originates due to flame curvature κ m = (∂N i /∂x i )/2 and thus this term (i.e., (S d ∇ • N) s Σ gen ) is referred to as the curvature term [4-7, 9, 11].It is evident from (1) that the curvature dependence of S d plays a key role in the statistical behaviours of (S d ∇ • N) s Σ gen and this was confirmed in previous a priori Direct Numerical Simulation (DNS) based analyses [9,11].It was previously demonstrated [9,11] that the existing models for the subgrid curvature term C sg often do not capture its correct qualitative and quantitative behaviours, particularly in the Thin Reaction Zones (TRZ) regime flames.Moreover, the model parameters for the existing C sg models were found to be strong functions of LES filter width Δ [9,11].The modelling of (S d ∇ • N) s Σ gen therefore remains one of the weakest points in the LES modelling of the Σ gen transport equation.This gap in the existing literature is addressed in this paper by explicitly LES filtering a DNS database of freely propagating statistically planar turbulent premixed flames with different values of turbulent Reynolds number Re t .In this regard, the main objectives of the present study are as follows: (1) to analyse the statistical behaviours of the subgrid FSD curvature term in the context of LES for the flames with different values of Re t ; (2) to propose models for the subgrid FSD curvature term and assess their performances in comparison to the corresponding quantities extracted from DNS data.
The necessary mathematical background and numerical details will be provided in the next section.Following this, the results will be presented and subsequently discussed.Finally the main findings will be summarised and conclusions will be drawn.

Mathematical Background and Numerical Implementation
Although three-dimensional DNS with detailed chemical mechanism is currently possible, it remains extremely computationally intensive [13] and is often not suitable for a detailed parametric analysis.Thus the chemical mechanism is simplified here using a single step Arrhenius type chemical reaction in order to carry out a parametric variation in terms of Re t .For the convenience of modelling, (S d ∇ • N) s Σ gen is often split as [4-7, 9, 11] where C mean and C sg are the resolved and subgrid curvature terms, respectively.Chakraborty and Cant [9,11] analysed the possibility of using three different expressions of C mean : where (N i ) s = −(∂c/∂x i )/Σ gen and M i = −(∂c/∂x i )/|∇c| are the ith component of surface-weighted and resolved flame normal vector, respectively.Previous a priori DNS analyses suggested that C mean = (S d ) s [∂(N i ) s /∂x i ]Σ gen is the most preferred expression for the resolved curvature term out of the three options presented in (3), as it allows for the smallest magnitude of C sg , while satisfactorily capturing the qualitative behaviour of (S d ∇ • N) s Σ gen [9,11].Moreover, C mean = (S d ) s [∂(N i ) s /∂x i ]Σ gen was used in previous LES simulations [5][6][7]12].It is useful to split /ρ|∇c| in the following manner [9-11, 14, 15] for obtaining further insight into where ẇ is the reaction rate and D is the progress variable diffusivity.The following expression for C sg can be obtained using (4) and where Equation (6i) indicates that the curvature κ m = ∇ • N/2 dependences of (S r + S n ) and |∇c| are likely to influence the statistical behavior of C sg1 .According to (6ii), C sg2 remains deterministically negative throughout the flame brush.
Hawkes and Cant [6,7] modified a version of the coherent flamelet model by Candel et al. [2] as where α N = 1 − (N k ) s (N k ) s is a resolution parameter which vanishes when the flow is fully resolved and β 1 is a model parameter.Hawkes [5] discussed a possibility of modifying the RANS model proposed by Cant et al. [1] for the purpose of LES as: where is the subgrid kinetic energy and β 2 is a model parameter.Charlette et al. [4] modelled C sg as where β 3 is a model parameter.The models given by ( 7)-( 9) (henceforth will be referred to as CSGCFM, CSGCPB, and CSGCHAR, respectively) ensure that C sg vanishes when the flow is fully resolved (i.e., (N k ) s (N k ) s = 1.0 and Σ gen = |∇c|).
Modelling of C sg1 and C sg2 using a priori analysis of DNS data, and the assessment of the models given by ( 7)-( 9), will be addressed in Section 3 of this paper.In the present study a compressible DNS database of freely propagating statistically planar turbulent premixed flames under decaying turbulence has been considered.The simulation domain of size 36.6δth × 24.1δ th × 24.1δ th , was discretised using a Cartesian mesh of size 345 × 230 × 230 with uniform mesh spacing in each direction where δ th = (T ad − T 0 )/ Max |∇ T| L is the thermal flame thickness with T ad , T 0 , and T being the adiabatic flame, unburned gas, and instantaneous gas temperatures, respectively, and the subscript L refers to the unstrained planar laminar flame quantities.The domain boundaries in the direction of mean flame propagation (i.e., x 1 -direction) are taken to be partially nonreflecting, whereas the transverse boundaries are taken to be periodic.The partially nonreflecting boundaries are specified using the well-known Navier Stokes Characteristic Boundary Conditions (NSCBC) technique [16].The simulations have been carried out using a three-dimensional compressible DNS code called SENGA [17] A 10th order central difference scheme is used for spatial differentiation for internal grid points, and the order of differentiation decreases gradually to a one-sided 2nd order scheme towards nonperiodic boundaries [17].A third order Runge-Kutta scheme was used for the purpose of time advancement [17].For all cases, the reacting flow field is initialised by a steady unstrained planar laminar flame solution, and the initial turbulent velocity fluctuations are specified using an initially homogeneous isotropic velocity field.About 10 grid points are kept within the thermal flame thickness δ th for all cases considered here.The initial values for the root-mean-square turbulent velocity fluctuation normalised by unstrained planar laminar burning velocity u /S L and the integral length scale to flame thickness ratio l/δ th are presented in Table 1 along with the values of Damköhler number Da = l.S L /u δ th , Karlovitz number Ka = (u /S L ) 3/2 (l/δ th ) −1/2 , and turbulent Reynolds number Re t = ρ 0 u l/μ 0 , where ρ 0 and μ 0 are the unburned gas density and viscosity, respectively.The heat release parameter τ = (T ad −T 0 )/T 0 and Lewis number Le are taken to be 4.5 and 1.0 for all cases considered here.Standard values are taken for Prandtl number Pr, ratio of specific heats γ, and the Zel'dovich number β = T ac (T ad − T 0 )/T 2 ad (i.e., Pr = 0.7, γ = 1.4,and β = 6.0),where T ac is the activation temperature.The turbulent Reynolds number Re t scales as Re t ∼ Da 2 Ka 2 , and thus the variation of Re t in cases A-E is brought about by modifying Da and Ka independently from each other.In cases A, C and E, Da is held constant, while Ka is held constant in cases B, C, and D. For all cases the Karlovitz number remains greater than unity indicating the TRZ regime combustion according to the regime diagram by Peters [18].The range of Re t values considered in this study remains modest, although several previous studies [3,9,11,15,[19][20][21][22][23] with comparable values of Re t have made valuable contributions to the fundamental understanding and the modelling of turbulent premixed combustion.Moreover, the range of Re t considered here is comparable to that of previous laboratory-scale experiments [24].
In all cases flame-turbulence interaction takes place under decaying turbulence.The simulations were run for a time equal to one chemical time scale (i.e., t c = δ th /S L ), which is equivalent to 2.0t f in case D; 3.0t f in cases A, C, and E; 4.34t f for case B. The aforementioned simulation times remain comparable to several studies [3,9,11,15,[19][20][21][22][23], which contributed to the FSD based modelling in the past.The global turbulent kinetic energy and burning rate were not varying significantly with time when statistics were extracted (see Figure 1 of [23]) and the qualitative nature of the statistics was found to have remained unchanged since t = 1.0l/u for all cases [23].At time t = δ th /S L , the global level of u /s l had decayed from the initial values by about 45%, 55%, 40%, 25%, and 32% in cases A-E, respectively.The values of l/δ th had increased from their initial values by a factor of about 1.5-2.25 at t = δ th /S L , but there were still enough turbulent eddies on each side of the computational domain [23].Values for u /S L and l/δ th at the time when statistics were extracted were presented in Table 2 of [23] and are not repeated here.The flame thickness δ th remained greater than the Kolmogorov length scale η for all cases when the statistics were extracted (see Table 2 of [23]), confirming the TRZ regime combustion.
For the purpose of a priori DNS analysis, the relevant quantities are explicitly filtered using a Gaussian kernel G( r) = (6/πΔ 2 ) 3/2 exp(−6 r • r/Δ 2 ).The filtered value of a general quantity Q( x, t) is evaluated using the following convolution operation: The statistical behaviours of the FSD curvature term (S d ∇ • N) s Σ gen have been analysed here for Δ ranging from Δ = 4Δ m ≈ 0.4δ th to Δ = 24Δ m ≈ 2.4δ th , where Δ m is the DNS mesh size (Δ m ≈ 0.1δ th ).These filter sizes are comparable to the range of Δ used in a priori DNS analysis in several previous studies [3,4,[9][10][11] and span a useful range of length scales (i.e., from Δ comparable to 0.4δ th , where the flame is partially resolved, up to 2.4δ th , where the flame becomes fully unresolved and Δ is comparable to the integral length scale).

Results and Discussion
The isosurfaces of c ranging from 0.01 to 0.99 at time t = δ th /S L for all cases are shown in Figure 1.A comparison between Figures 1(a)-1(e) reveals that the wrinkling of the flame surface with increasing u /S L ∼ Re 1/4 t Ka 1/2 ∼ Re 1/2 t /Da 1/2 .Figures 1(a)-1(e) further demonstrate that the flame surfaces in all cases show a range of different curvatures and this range increases with increasing u /S L ∼ Re 1/4 t Ka 1/2 ∼ Re 1/2 t /Da 1/2 .This indicates that the interrelations between S d and κ m and between |∇c| and κ m may lead to nonnegligible value of (S d ∇ • N) s Σ gen even for statistically planar flames.that the magnitudes of (S d ∇ • N) s Σ gen and C mean decrease with increasing Δ, whereas the relative magnitude of C sg in comparison to C mean increases with increasing Δ.This observation is consistent with previous findings [9,11].The smearing of local information as a result of the weightedaveraging process involved in LES filtering leads to the decrease in the magnitudes of (S d ∇ • N) s Σ gen and C mean for increasing values of Δ.The flow becomes increasingly unresolved with increasing Δ and thus the flame curvature and its influence on the FSD evolution are increasingly felt at the subgrid scale, which is reflected in the relatively high magnitudes of C sg in comparison to C mean for large values of Δ.
It is useful to examine the statistical behaviours of C sg1 and C sg2 in order to explain the differences in the behaviours of C sg for flames with different Re t .The variations of the ensemble averaged values of ((S r + m ) s Σ gen ), C sg1 and C sg2 conditional on c isosurfaces are shown in Figure 3 for cases A-E for Δ = 8Δ m ≈ 0.8δ th and Δ = 24Δ m ≈ 2.4δ th , respectively.Figure 3 demonstrates that both ((S r + S n )∇ • N) s Σ gen and C sg1 remain predominantly positive (negative) towards the unburned (burned) gas side of the flame brush for all values of Δ considered here.The contribution of −4(Dκ 2 m ) s Σ gen and C sg2 remains deterministically negative throughout the flame brush (see Figure 3).It is evident from Figure 3 that −4(Dκ 2 m ) s Σ gen remains a leading order contributor to (S d ∇ • N) s Σ gen for all the flames at all values of Δ (see Figure 3), which is consistent with the expected behaviour in the TRZ regime, where −4(Dκ 2 m ) s Σ gen is expected to play an important role [18].Figure 3 further shows that C sg1 remains close to the magnitude of ((S r + S n )∇ • N) s Σ gen for all Δ for all cases considered here, indicating that (S r + S n ) s ∂(N i ) s /∂x i Σ gen does not play a major role in capturing the behaviour of ((S r + S n )∇ • N) s Σ gen .By contrast, there is a significant difference between −4(Dκ 2 m ) s Σ gen and C sg2 for all cases for small values of Δ, and the difference between these quantities decreases with increasing Δ.As most of the contribution of −4(Dκ 2 m ) s Σ gen remains unresolved for large values of Δ, C sg2 remains the leading order contributor to plays a progressively less important role for increasing values of Δ, where the flame is fully unresolved.However, the contribution of (−(D∂N i /∂x i ) s ∂(N i ) s /∂x i Σ gen ) remains significant for small values of Δ when the flame is partially resolved.Figure 3 further shows that the order of magnitudes of both C sg1 and C sg2 remains comparable for large values of Δ (i.e., Δ > δ th ) and thus accurate modelling of C sg1 and C sg2 is necessary for accurate modelling of C sg .As the range of κ m values obtained on a flame surface increases with increasing flame wrinkling at higher values of u /S L ∼ Re 1/4 t Ka 1/2 ∼ Re 1/2 t /Da  value of Da or Ka (see Figure 3).The positive contribution of C sg1 overcomes the negative contribution of C sg2 towards the unburned gas side of the flame brush for the flames with small and moderate values of turbulent Reynolds number (i.e., cases A-C) and yields a net positive contribution of C sg towards the reactant side of the flame brush (see Figure 2).
The statistical behaviours of ((S r + S n )∇ • N) s Σ gen and C sg1 depend on the nature of the correlations between (S r + S n ) and κ m = ∇ • N/2 and between |∇c|, κ m , and the variation of (κ m ) s across the flame brush.The correlation coefficients for the (S r + S n ) − κ m and |∇c| − κ m dependences for five different c isosurfaces across the flame brush for all cases are shown in Figures 4(a [25].The physical explanations of the observed curvature dependences of (S r + S n ) and |∇c| have been discussed elsewhere [25] and will not be repeated here.
The variation of the ensemble averaged values of (κ m ) s conditional on c isosurfaces for all cases are shown in Figures 4(c) and 4(d) for Δ = 8Δ m ≈ 0.8δ th and Δ = 24Δ m ≈ 2.4δ th , respectively, which demonstrates that (κ m ) s predominantly assumes positive (negative) values towards the unburned (burned) gas side of the flame brush and the magnitude of (κ m ) s increases with increasing Δ.The quantity (κ m ) s approaches to κ m for small values of Δ (i., and E (5th column) along with the predictions of ( 10) and ( 12).
(burned) gas side of the flame brush due to positive (negative) value of (κ m ) s .The contribution of resolved curvature term (S r + S n ) s ∂(N i ) s /∂x i Σ gen remains negligible in comparison to ((S r + S n )∇ • N) s Σ gen , due to relatively small values of ∂(N i ) s /∂x i in comparison to 2(κ m ) s = (∂N i /∂x i ) s in statistically planar flames.Thus the contributions of ((S r + S n )∇ • N) s Σ gen and C sg1 remain close to each other for all values of Δ (see Figure 3).The subgrid fluctuations of the surface-weighted contributions of (S r + S n ) and ∇ • N are taken to scale with S L and (Σ gen − |∇c|), respectively, to propose the following model for C sg1 in this analysis: where β 4 , c * , a Σ , and m are the model parameters.The function 10) is used to capture the correct qualitative behaviour of C sg1 across the flame brush.In a compressible LES simulation c is readily available and c needs to be extracted from c.The methodology of extracting c from c in the context of LES was discussed elsewhere [9,11] and will not be discussed in detail in this paper.The model parameter c * ensures that the transition from positive to negative value of C sg1 takes place at the correct location within the flame brush.The quantity (Σ gen − |∇c|) vanishes when the flow is fully resolved (i.e., lim Δ → 0 (Σ gen −|∇c|) = lim Δ → 0 (|∇c|−|∇c|) = |∇c|−|∇c| = 0.0) and thus C sg1 becomes exactly equal to zero when the flow is fully resolved (i.e., Δ → 0) according to (10).It has been found that m = 1.85 enables (10) to capture the qualitative behaviour of C sg1 when the optimum values of c * and a Σ are chosen.The optimum value of c * (a Σ ) tends to increase with decreasing (increasing) Δ.The κ m dependences of (S r + S n ) and |∇c| are reflected mostly in the resolved scale but these effects weaken with increasing values of Δ [9,11].As the resolved and subgrid curvature terms are closely related [9,11], the qualitative behaviour of C sg1 is also affected by the κ m dependences of (S r + S n ) and |∇c|, which leads to the variation of the optimum values of a Σ and c * .The model parameter β 4 is found to decrease with decreasing values of Σ gen for satisfactory quantitative prediction of C sg1 , which is accounted for by expressing β 4 as β 4 = 9.80Σ gen ×δ th .The prediction of (10) ensemble averaged on c isosurfaces is compared with the ensemble averaged values of C sg1 in Figure 5 for all cases for the optimum values of c * and a Σ , for Δ = 0.8δ th and Δ = 2.4δ th , when β 4 and m are taken to be β 4 = 9.80Σ gen × δ th and m = 1.85.The optimum values of c * and a Σ are estimated by calibrating the prediction of (10) with respect to the values of C sg1 obtained from DNS data and the variation of the global mean optimum values of c * and a Σ with Δ/δ th for all cases are shown in Figure 6.The optimum values of c * and a Σ are parameterised here as , where k 1 = 0.75 + 0.15 1.0 + exp(−5.0(k 4 − 4.6)) , , Figure 5 shows that (10) satisfactorily predicts C sg1 when β 4 = 9.80Σ gen × δ th and m = 1.85, and (11i), (11ii), and (11iii) are used for c * and a Σ .
Here the contribution of (Dκ , where the subgrid fluctuations of D is taken to scale with S L /Σ gen (i.e., D ∼ S L /Σ gen ).The above relations are utilised here to propose a model for C sg2 (see (6ii)) in the following manner: where Ξ Δ = Σ gen /|∇c| is the wrinkling factor [8,10,19], β 5 is a model parameter, and c(1 − c) is used to capture the correct qualitative behaviour of C sg2 .According to (12), C sg2 vanishes when the flow is fully resolved (i.e., lim It has been found that (12) satisfactorily captures the behaviour of C sg2 throughout the flame brush for n = 1.0 in all cases considered here when a suitable value of β 5 is used.The variation of the global mean optimum values of β 5 with Δ/δ th for all cases is shown in Figure 6.The optimum values of β 5 has been parameterised here in the following manner: The predictions of ( 12) ensemble averaged on c isosurfaces are compared with the ensemble averaged values of C sg2 in Figure 5 all for cases for Δ = 0.8δ th and Δ = 2.4δ th , which show that (12) satisfactorily predicts the statistical behaviour of C sg2 when n = 1.0 and (13i), (13ii), and (13iii) are used for β 5 .Equations ( 10) and ( 12) can be combined to propose a model for C sg in the following manner: The above model will henceforth be referred to CSGNEW model in this paper.Equation ( 14) allows for a positive contribution of C sg through the contribution of which is absent in the CSGCAND, CSGCANT, and CSGCHAR models.The predictions of the CSGCAND, CSG-CANT, CSGCHAR, and CSGNEW models for Δ = 0.8δ th and Δ = 2.4δ th are compared with C sg obtained from DNS in Figure 7 for optimum values of β 1 , β 2 , and β 3 , where the optimum values are estimated by calibrating the models based on the ensemble averaged value of C sg obtained from DNS data.The variations of the optimum values of β 1 , β 2 , and β 3 with Δ for cases A-E are also shown in Figures 6(a)-6(e), respectively, which demonstrate that the model constants β 1 , β 2 , and β 3 remain greater than unity for all cases.This is found to be consistent with the realisability analysis by Hawkes and Cant [26].Figures 6(a)-6(e) demonstrate that the optimum values of β 1 , β 2 , and β 3 change with respect to Δ, which is also consistent with earlier findings [9].Moreover, the optimum values of β 1 , β 2 , and β 3 for a given value of Δ vary between cases considered here (see Figures 6(a)-6(e)).The optimum values of β 1 , β 2 , and β 3 can also be parameterised in the same manner in which β 5 is parameterised in (13i), (13ii), and (13iii).However, this is not presented here as the models given by ( 7)-( 9) fail to capture the positive contribution of C sg for cases A-C.Moreover, the CSGCAND, CSGCANT, and CSGCHAR models do not capture the correct qualitative behaviour of C sg even when the optimum values of β 1 , β 2 , and β 3 are used.The CSGCHAR model tends to overpredict the negative values of C sg towards the unburned gas side and this behaviour becomes more prominent with increasing filter size.Figure 7 shows that for Δ = 24Δ m = 2.4δ th , the CSGCHAR model predicts the maximum magnitude of C sg near the middle of the flame whereas the actual maximum magnitude of C sg is attained slightly towards the burned gas side.The CSGCAND and CSGCANT models predict the correct magnitude of C sg for optimum values of β 1 and β 2 , but they do not satisfactorily capture the qualitative behaviour of C sg and underpredict (overpredict) its magnitude towards the burned gas side (middle) of the flame brush.Figure 7 demonstrates that the CSGNEW model captures the qualitative behaviour of C sg in a better manner than the CSGCAND and CSGCANT models, and the quantitative agreement between C sg and the CSGNEW model remains better than the CSGCAND, CSGCANT, and CSGCHAR models for the major part of the flame brush for all cases, when optimum values of β 4 , β 5 , a Σ , and c * are used.

Conclusions
The LES modelling of the curvature term (S d ∇ • N) s Σ gen of the Σ gen transport equation has been analysed using a simplified chemistry based DNS database of freely propagating statistically planar turbulent premixed flames with a range of different turbulent Reynolds numbers Re t .The variation of Re t is brought about by modifying Da and Ka independently from each other.The statistical behaviours of the subgrid curvature term C sg for a range of different values of Δ have been analysed in terms of its contributions C sg1 and C sg2 , which arise from (S r + S n ) and S t = −2Dκ m , respectively.Detailed physical explanations have been provided for the observed filter size dependences of the different components of (S d ∇ • N) s Σ gen .Models have been identified for individual components of the subgrid curvature term (i.e., C sg1 and C sg2 ) and the performances of these models have been compared to the corresponding quantities extracted from DNS data.It has been found that the new models for C sg1 and C sg2 satisfactorily capture the statistical behaviours of the corresponding terms extracted from DNS data.The performance of the new model for sg has been found to be either better than or comparable to the performances of the existing models.It is worth noting that the present has been carried out using a DNS database with moderate values of Re t in the absence of the effects of detailed chemistry and differential diffusion.Thus, three-dimensional DNS data with detailed chemistry and experimental data at higher values of Re t will be necessary for more comprehensive modelling of (S d ∇ • N) s Σ gen and C sg in the context of LES.

Figure 1 :
Figure 1: Instantaneous isosurfaces of c ranging from 0.01 to 0.99 at t = 1.0δ th /S L for cases (a)-(e) A-E.

Figure 2 :
Figure 2: Variations of (S d ∇ • N) s Σ gen , C mean = (S d ) s ∂(N i ) s /∂x i Σ gen and C sg conditionally averaged in bins of c across the flame brush for Δ = 8Δ m ≈ 0.8δ th (top row) and Δ = 24Δ m ≈ 2.4δ th (bottom row) for cases A (1st column), B (2nd column), C (3rd column), D (4th column), and E (5th column).All the curvature terms in this and subsequent figures are normalised by S L /δ 2 th .
) and 4(b), respectively.For unity Lewis number flames S t = −2Dκ m is deterministically negatively correlated with κ m with a correlation coefficient equal to−1.0.

Figure 4 (
a) suggests that (S r + S n ) − κ m correlation is much weaker than the S t − κ m correlation in all cases.Moreover, Figure 4(b) demonstrates that |∇c| and κ m remain weakly correlated throughout the flame brush for all cases.Figures 4(a) and 4(b) demonstrate that the curvature dependences of (S r +S n ) and |∇c| remain qualitatively similar for all the flames e., lim Δ → 0 (κ m ) s = lim Δ → 0 κ m |∇c|/|∇c| = κ m |∇c|/|∇c| = κ m ) and the mean value of κ m = ∇ • N/2 remains negligible for all the c isosurfaces due to the statistical planar nature of the flames.However, subgrid level curvature increases with increasing Δ and thus the magnitude of (κ m ) s increases with increasing values of Δ. Relatively weak curvature dependences of (S r + S n ) and |∇c| lead to positive (negative) values of ((S r + S n )∇ • N) s Σ gen and C sg1 towards the unburned

Table 1 :
List of initial simulation parameters and non-dimensional numbers.