# Effects of Turbulent Reynolds Number on the Displacement Speed Statistics in the Thin Reaction Zones Regime of Turbulent Premixed Combustion

Authors: Nilanjan Chakraborty; Markus Klein; R. S. Cant
Journal: Journal of Combustion
 (2011)

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## Abstract

The effects of turbulent Reynolds number on the statistical behaviour of the displacement speed have been studied using three-dimensional Direct Numerical Simulation of statistically planar turbulent
premixed flames. The probability of finding negative values of the displacement speed is found to
increase with increasing turbulent Reynolds number when the Damköhler number is held constant. It
has been shown that the statistical behaviour of the Surface Density Function, and its strain rate and
curvature dependence, plays a key role in determining the response of the different components of
displacement speed. Increasing the turbulent Reynolds number is shown to reduce the strength of the
correlations between tangential strain rate and dilatation rate with curvature, although the qualitative
nature of the correlations remains unaffected. The dependence of displacement speed on strain rate
and curvature is found to weaken with increasing turbulent Reynolds number when either Damköhler
or Karlovitz number is held constant, but the qualitative nature of the correlation remains unaltered.
The implications of turbulent Reynolds number effects in the context of Flame Surface Density (FSD)
modelling have also been addressed, with emphasis on the influence of displacement speed on the
curvature and propagation terms in the FSD balance equation.


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## Body

## 1. Introduction

Displacement speedSd is an important quantity for the modelling of turbulent premixed flames using both level-set [1] and flame surface density (FSD) [2] approaches. The displacement speed is defined as the speed at which the flame surface moves normal to itself with respect to an initially coincident material surface. Statistics of displacement speed for turbulent premixed flames have been studied extensively in previous work based on two-dimensional direct numerical simulations (DNS) with detailed chemistry [3–13] and on three-dimensional DNS with simplified chemistry [14–22].  These studies have looked at the effects of local strain rate and curvature effect on Sd [13–24] and have included the influence of root-mean-square (rms) turbulent velocity fluctuation magnitude u′ and Lewis number Le [9, 15, 17, 22] on these effects.  Peters [1] showed using an order of magnitude analysis that the curvature dependence of displacement speed becomes the leading order contributor in the thin reaction zones regime, whereas the curvature dependence remains relatively weak in the corrugated flamelets regime.  This observation was supported using DNS data by Chakraborty [21] who also studied the individual response of the reaction, normal displacement, and curvature components of displacement speed. However, the effects of turbulent Reynolds number on displacement speed have yet to be addressed in detail.  Hence, in this paper, the influence of Ret on the statistics of Sd has been analysed based on a three-dimensional compressible DNS database of statistically planar turbulent premixed flames.  The main objectives of this study are as follows:(1)
to demonstrate the influence ofRet on the statistical distribution of  Sd(2)
to demonstrate the influence ofRet on the local curvature and strain rate dependencies of  Sd(3)
to indicate the implications of the dependence ofSd on Ret in the context of flame surface density-based turbulent combustion modelling.In order to address the above objectives, a series of DNS runs has been carried out for turbulent Reynolds numbers ranging from 20 to 100. In one group of simulations, the change inRet is obtained by modifying the Karlovitz number Ka for a given value of Damköhler number Da, whereas in the other group of simulations the change in turbulent Reynolds number is induced by modifying the Damköhler number for a given value of Karlovitz number.The rest of the paper is organised as follows. The mathematical background and the numerical details relevant to this work are discussed in the next two sections of this paper. Following this, results will be presented and subsequently discussed. In the final section, the main findings are summarised and conclusions are drawn.


## 2. Mathematical Background

Direct numerical simulations (DNS) of turbulent reacting flows ideally should be carried out in three-dimensions with detailed chemistry. However, such simulations remain extremely expensive and are not yet feasible for parametric studies.   Previous work has indicated that displacement speed statistics obtained from three-dimensional DNS with simplified chemistry [14–22] are in agreement with those obtained from two-dimensional DNS with complex chemistry [3–13].  In the present study, three-dimensional DNS with simplified chemistry is adopted in order to allow for an extensive parametric study without excessive computational cost.  The chemical reaction is represented in terms of single step Arrhenius type chemistry, and the species field is uniquely represented by a reaction progress variablec, which may be defined based on a suitable product mass fraction YP according to(1)c=YP-YP0YP∞-YP0,
where subscripts 0 and ∞ are used to refer to quantities in unburned reactants and fully burned products, respectively. The transport equation for reaction progress variable is given by(2a)ρ∂c∂t+ρuj∂c∂xj=ẇ+∂∂xj(ρD∂c∂xj),
where ẇ is the chemical reaction rate, ρ is the fluid density, and D is the diffusivity of reaction progress variable. The transport equation for reaction progress variable can be written in kinematic form for a given  isosurface defined by c=c* [6, 7]   (2b)ρ∂c∂t+ρuj∂c∂xj=ρSd|∇c|c=c*.Comparing (2a) and (2b), the expression for the displacement speed of a given c=c* isosurface can be obtained as [1, 3–7](3)Sd=ẇ+∇⋅(ρD∇c)ρ|∇c||c=c*.
It is often useful to decompose the displacement speed Sd into three separate components [3–7](4)Sd=Sr+Sn+St,
where  Sr, Sn, and St are the reaction, normal diffusion, and tangential diffusion components which are given as                                    (5)Sr=ẇρ|∇c||c=c*,Sn=N⃗⋅∇(ρDN⃗⋅∇c)ρ|∇c||c=c*,St=-2Dκm,
where κm is the arithmetic mean of the two principal curvatures, which is defined as                         (6)κm=12∇⋅N.⃗
According to (6), flame surface that is curved convex to the unburned reactants has a positive curvature and vice versa. The definition of displacement speed and its components in (4)–(6) suggests that the statistics of the surface density function (SDF=|∇c|) [15, 25–27] and curvature κm are each likely to have a significant effect on the statistical behaviour of displacement speed.  The transport equation for the SDF for an isosurface of c is given by [15, 25–27]                                                                   (7)∂|∇c|∂t+∂(uj|∇c|)∂xj=aT|∇c|+2Sdκm|∇c|-∂(SdNi|∇c|)∂xi,
where aT is the tangential strain rate given by(8)aT=[(δij-NiNj)∂ui∂xj]c=c*.It is evident from (7) and (8) that tangential strain rate  is likely to affect the statistical behaviour of |∇c|, which in turn can affect the local curvature and strain rate dependence of Sr and Sn.  It is worth noting that in some previous studies [3–12], the statistics of the density-weighted displacement speed Sd*=ρSd/ρ0 were analysed instead of the statistics of Sd itself.  For low Mach number unity Lewis number flames, Sd* and Sd are related by Sd*=Sd(1+τc*) for a given isosurface of c, and thus the statistics of Sd*  are similar to those of Sd. For this reason, and also since Sd appears explicitly in the SDF and FSD transport equations [2, 15, 25–27], the statistics of Sd are addressed in the present work. The local strain rate and curvature dependence of Sd and its components for different values of turbulent Reynolds number will be discussed in Section 4 of this paper.It is essential for the purposes of the present study to note the relationship between the turbulent Reynolds number and both the Damköhler and Karlovitz numbers.  The turbulent Reynolds number is defined asRet=ρ0u′l/μ, where ρ0 is the density and μ is the dynamic viscosity of the unburned reactants, and l is the integral length scale of the turbulence.  For unity Lewis number flames, the turbulent Reynolds number can be scaled as [1](9)Ret~Da2Ka2~(u′SL)2Da~(u′/SL)4Ka2~(lδth)4/3Ka2/3,
where Da=lSL/u′δth  is the Damköhler number and Ka=(u′/SL)3/2/(l/δth)1/2is the Karlovitz number.  Here,  δth=(Tad-T0)/Max|∇T̂|L is the thermal flame thickness with Tad, T0 and T̂ being the adiabatic flame temperature, unburned gas temperature, and instantaneous dimensional temperature, respectively, while the subscript “L” refers to the unstrained planar laminar flame quantities. Equation (9) indicates that a change of either Da or Ka will lead to a change in the turbulent Reynolds number Ret. Thus, the influence of  Ret on the statistics of  Sd is essentially induced by changes in Da and/or Ka.


## 3. Numerical Implementation

Three-dimensional compressible DNS runs for statistically planar turbulent premixed flames have been carried out using a DNS code called SENGA [28] in which the conservation equations of mass, momentum, energy, and species are solved in nondimensional form. The simulations have been carried out in a rectangular domain of size 36.6δth×24.1δth×24.1δth. The simulation domain has been discretised using a Cartesian mesh of size 345 × 230 × 230 with uniform grid spacing in each direction. The mean direction of flame propagation is assumed to align with the x1-direction and the transverse directions are assumed to be periodic. The boundaries in the  direction of mean flame propagation are taken to be partially nonreflecting and the boundary conditions are specified using the Navier-Stokes characteristic boundary  conditions (NSCBC) technique  [29]. The spatial discretisation is carried out using a 10th order central difference scheme for the internal points and the order of discretisation drops gradually to a one-sided 2nd order scheme at nonperiodic boundaries. The time advancement is carried out using a low storage 3rd order Runge-Kutta scheme [30]. The turbulent velocity fluctuations are initialised using an initially homogeneous isotropic field using a pseudospectral method proposed by Rogallo [31]. The combustion is initailised by a steady unstrained planar laminar premixed flame solution. For the present study, the thermophysical properties such as dynamic viscosity μ, thermal conductivity λ, specific heats CP and CV and the density-weighted mass diffusivity ρD are taken to be constant and independent of temperature following several previous studies [11–18, 32–34].  For the present study, standard values are taken for Prandtl number (Pr = 0.7) and the ratio of specific heats (γ=CP/CV=1.4). The flame Mach number is taken to be Ma=SL/γRT0=0.014.  The Lewis number Le is taken to be equal to unity for all the cases considered here. As Da and Ka are often used to characterise the combustion situation on combustion regime diagram, the variation of Ret in the present study is brought about by modifying Da and Ka independently of each other. In order to achieve this, the values of both u′/SL and l/δth are modified as both Da=lSL/δthu′ and Ka=(u′/SL)3/2(l/δth)-1/2 are dependent on the values of u′/SL and l/δth. The initial values of normalised rms turbulent velocity fluctuation u′/SL, integral length scale to flame thickness ratio l/δth, heat release parameter τ=(Tad-T0)/T0, Damköhler number Da, Karlovitz number Ka  and turbulent Reynolds number Ret are listed in Table 1.  From Table 1, it is evident that Ka remains greater than unity for all cases, which indicates that the combustion belongs to the thin reaction zones regime according to the regime diagram by Peters [1]. In cases B, C, and D the values of  u′/SL and l/δth were chosen in such a manner as to vary the turbulent Reynolds number  Ret by changing the Damköhler number Da while keeping the Karlovitz number Ka constant (see (9)).  On the other hand, in cases A, C, and E, u′/SL and l/δth were chosen to vary Ret by changing Ka, while Da was kept constant. The range of turbulent Reynolds number values considered in this study remains modest although several previous studies [3–22] with comparable values of Ret have made valuable contributions to the fundamental understanding of displacement speed in turbulent premixed flames. It is also worth noting that the range of Ret considered here is comparable to that of previous laboratory-scale experiments (e.g., Kobayashi et al. [35]).  Despite the limited range, a number of important Ret effects on the displacement speed statistics have been identified in the present parametric study and are discussed in detail in Section 4 of this paper.Table 1
Simulation parameters and nondimensional numbers relevant to the DNS database.
Caseu′/SLl/δthτRetDaKaA5.01.674.5220.338.67B6.251.444.523.50.2313.0C7.52.54.549.00.3313.0D9.04.314.5100.00.4813.0E11.253.754.51100.3319.5In all cases, the flame-turbulence interaction takes place under conditions of decaying turbulence, in which simulations should be carried out for a minimum timetsim=Max(tf,tc), where tf=l/u′ is the initial eddy turn over time and tc=δth/SL is the chemical time scale. All the cases considered here were run for one chemical time scale tc, which corresponds to a time equal to 2.0tf in case D, 3.0tf in cases A, C, and E, and 4.34tf for case B. The present simulation time is comparable to the simulation times used for previous DNS studies which focused on the analysis of displacement speed statistics [3–21, 23–27]. By the time statistics were extracted, the global turbulent kinetic energy and volume-averaged burning rate were no longer changing rapidly with time. The temporal variations of these quantities were presented in [36], and thus are not repeated here for the sake of conciseness. Analysis of the volume-averaged burning rate indicates that the effects of unsteadiness have become weak by the time when statistics were extracted. At this time, the global level of turbulent velocity fluctuation had decayed by 52.66%, 61.11%, 45%, 24%, and 34% in comparison to the initial values for cases A to E, respectively. By contrast, the integral length scale had increased by factors of between 1.5 to 2.25, ensuring that sufficient numbers of turbulent eddies were retained in each direction to obtain useful statistics. The values for  u′/SL, l/δth,  and  δth/η at the time when statistics were extracted have been presented in  Table 2 of [36]. For all cases, the thermal flame thickness δth remains greater than the Kolmogorov length scale η at that time, which indicates the combustion in all cases still belongs to the thin reaction zones regime. Moreover, the turbulent Reynolds number Ret for all cases remains greater than unity (i.e., Ret>1) for the values of u′/SL and l/δth  at the time when statistics were extracted, which suggests that all the flames remain turbulent  according to the regime diagram by Peters [1].


## 4. Results and Discussion

### 4.1. Flame-Turbulence Interaction

The contours of the reaction progress variablec  in the x1-x2 mid-plane of the domain at time tsim=1.0δth/SL are shown in Figures 1(a)–1(e) for cases A to E, respectively. It is evident that the extent of the wrinkling of the c isosurfaces increases with increasing  u′, that is, in going from case A through to case E (see Table 1). Moreover, the contours of c representing the preheat zone (i.e., c<0.5) are much more distorted than those representing the reaction zone (i.e., 0.7<c<0.9). This tendency is more prevalent for the case with the highest Karlovitz number (case E), since the scale separation between δth and η increases with increasing Ka, allowing more energetic eddies to enter into the preheat zone and causing greater distortion of the flame.Contours of reaction progress variablec in the x1-x2  midplane of the domain after t=1.0δth/SL for: (a) case A; (b) case B; (c) case C; (d) case D; (e) case E.
(a)(b)(c)(d)(e)


### 4.2. Probability Density Functions of Displacement SpeedSd

The probability density functions (pdfs) of normalised displacement speedSd/SL on the c=0.1,0.3,0.5,0.7, and 0.9 isosurfaces are shown in Figures 2(a)–2(e), respectively. It is evident in all cases that there is a small but finite probability of finding negative displacement speed although the mean value of Sd/SL remains strongly positive throughout the flame brush. Negative values of displacement speed suggest that the progress variable isosurface in question is moving locally in a direction opposite to that of mean flame propagation. This situation arises when the negative contribution of the molecular diffusion rate dominates over the positive semi-definite chemical reaction rate, as discussed in detail by Gran et al. [4]. Moreover, Figure 2(e) indicates that the probability of finding negative values of Sd/SL is greatest for the highest Karlovitz number (case E). This behaviour can be explained in terms of the scaling analysis of Peters [1] for unity Lewis number flames, which suggested that the different components of displacement speed scale as(10)Sr+Snvη~SLvη~O(1Ka),Stvη~-2Dκmvη~O(1),
where vη is the Kolmogorov velocity scale.  Here, in accordance with Peters [1], (Sr+Sn) is taken to scale with  SL and the curvature κm  is taken to scale with the Kolmogorov length scale 1/η.  These scalings of (Sr+Sn) and κm were confirmed by DNS data presented in [21].  Equation (10) clearly suggests that in the thin reaction zones regime (i.e., Ka>1) the effects of (Sr+Sn) are likely to weaken progressively in comparison to the contribution of  St  with increasing Karlovitz number. This suggests that for large values of Karlovitz number the negative contribution of  St  more readily overcomes the predominantly positive contribution of (Sr+Sn) to yield a negative value of  Sd. This eventually gives rise to an increased probability of finding negative values of  Sd  for flames with increasing value of Karlovitz number, as suggested by Figure 2.  In order to demonstrate that the observed negative values of  Sd  originates principally due to  St, the pdfs of (Sr+Sn)/SL are shown in Figure 3, respectively, for all cases. Comparing Figures 2 and 3 reveals that the probability of finding negative values of (Sr+Sn)  is significantly smaller than that of finding negative values of  Sd.Pdfs of normalised displacement speedSd/SL on the c =    (a) 0.1, (b) 0.3, (c) 0.5, (d) 0.7,  and (e) 0.9  isosurfaces.
(a)(b)(c)(d)(e)Pdfs of normalised combined reaction and normal diffusion component of displacement speed(Sr+Sn)/SL on the c  =      (a) 0.1, (b) 0.3, (c) 0.5, (d) 0.7, and (e) 0.9 isosurfaces.
(a)(b)(c)(d)(e)Comparing Figures2(a)–2(e) and Figures 3(a)–3(e), it can be seen that the most probable values of  Sd/SL and (Sr+Sn)/SL  remain of the order of  ρ0/ρ and that the widths of the displacement speed pdf for a given c isosurface increases with increasing u′  (i.e., going from case A to case E).   In order to explain this behavior, it is useful to examine the pdfs of the normalised tangential component of displacement speed St/SL which are depicted in Figure 4 and show that these pdfs remain almost symmetrical with a peak close to a value of zero. This behaviour is consistent with the statistically planar nature of these flames.  However, the greater extent of flame wrinkling observed in Figure 1 for larger values of u′ results in a greater spread of flame curvature, which in turn increases the width of the St/SL pdfs according to (5).  The negligible mean value of St essentially leads to the mean values of Sd and (Sr+Sn)  becoming almost equal to each other, but the statistical variation of Sd induced by St leads to broader pdfs of Sd  than those of (Sr+Sn).  These results clearly suggest that the distributions of Sd and its components Sr,Sn and St generally broaden with  u′/SL, which essentially leads to broadening of the distributions with increasing Ret~(u′/SL)2Da~(u′/SL)4/Ka2~(l/δth)4/3Ka2/3  when either Da or Ka is held constant.Pdfs of normalised tangential diffusion component of displacement speedSt/SL on the c= (a) 0.1, (b) 0.3, (c) 0.5, (d) 0.7, and (e) 0.9 isosurfaces.
(a)(b)(c)(d)(e)


### 4.3. Strain Rate and Curvature Dependences of|∇c|

For low Mach number unity Lewis number flames, both densityρ and nondimensional temperature T remain uniform on a given isosurface of c and thus the statistical behaviours of Sr and Sn (see (5)) are mainly affected by the dependence of the SDF |∇c| on strain rate and curvature. The values of |∇c|×δth conditional on normalised tangential strain rate aT×δth/SL are shown in Figure 5(a) for the isosurface at c=0.8 (close to the location of maximum reaction rate), which suggests clearly that |∇c| and aT are positively correlated for all cases.  The correlation coefficients between |∇c| and aT are shown in Figure 5(b) for values of c  between 0.1 and 0.9, and these indicate that the positive correlation is maintained throughout the flame brush.  This correlation can be explained by noting that the dilatation rate ∇·u⃗ can be expressed as (11a)∇⋅u⃗=aT+an.
For unity Lewis number flames, the dilatation rate can be scaled as ∇·u⃗~τSL/δth  [21], whereas aT can be scaled as aT~u′/l  [37], which suggests that(11b)∇⋅u⃗aT~τSLlδthu′~τDa.(a) Variation of mean values of|∇c|×δth conditional on normalised tangential strain rate aT×δth/SL for the isosurface at c=0.8.  (b) Variation of the correlation coefficients between |∇c| and aT on the isosurfaces at c = 0.1 to 0.9 for all cases.
(a)(b)Equation (11b) essentially shows that for low values of the Damköhler number, aT remains much greater than ∇·u⃗. This essentially suggests that an increase in tangential strain rate is associated with a decrease in the normal strain rate, since the normal strain rate scales as an~-aT when dilatation rate effects are much weaker than those of turbulent straining.  Under decreasing normal straining, the isoscalar lines are brought closer to each other and hence give rise to an increase in |∇c|. The decreasing trend of normal strain rate with increasing tangential strain rate is reflected in the positive correlation between |∇c| and aT throughout the flame brush.  It can be seen from Figure 5 that the correlation between  |∇c| and aT does not show any monotonic trend with either Karlovitz number (Damköhler number) or turbulent Reynolds number variations when Damköhler number (Karlovitz number) is held constant.The variation of|∇c|×δth conditional on normalised curvature κm×δth  is shown in Figure 6(a) for the isosurface at c=0.8. Both positive and negative correlation trends can be observed, which is consistent with several previous studies [14–22, 26, 27]. As a result of these opposing trends, the net correlation between |∇c|×δth and κm×δth turns out be weak throughout the flame brush, as evident from the correlation coefficients between |∇c| and κm presented for different values of c across the flame brush in Figure 6(b). Both the positively and negatively correlating branches can be explained using (11a). In turbulent premixed flames the dilatation rate ∇·u⃗ is negatively correlated with curvature κm, because of focusing (defocusing) of heat in the negatively (positively) curved regions of the flame surface [18, 21]. The tangential strain rate aT is also found to be negatively correlated with curvature [18, 21]. The correlation coefficients for the ∇·u⃗-κm and aT-κm correlations for different c values across the flame brush are shown in Figures 7(a) and 7(b), respectively.  Comparing the correlation coefficients of the ∇·u⃗-κm and aT-κm correlations for different cases reveals that the strength of both of these negative correlations decreases with increasing u′/SL and Ret~(l/δth)4/3Ka2/3 when either Da or Ka is held constant.(a) Variation of mean values of|∇c|×δth conditional on normalised curvature κm×δth for the isosurface at c=0.8.  (b) Variation of the correlation coefficients between |∇c| and κm on the isosurfaces at c=0.1 to 0.9 for all cases.
(a)(b)Variation of the correlation coefficients between (a) tangential strain rateaT and curvature κm and between (b) dilatation rate ∇·u⃗ and curvature κm on the isosurfaces at c = 0.1 to 0.9 for all cases.
(a)(b)The combination of the positive correlation between|∇c|  and aT and the negative correlation between aT and κm leads to small values of |∇c| in regions of high positive curvature. This gives rise to the negative correlation trend between |∇c| and κm, as observed in Figure 6.  However, the dilatation rate ∇·u⃗ may locally attain large values in the highly negatively curved regions of the flame due to strong focussing of heat.  Hence, ∇·u⃗ may locally exceed aT and induce an increase in an according to (11a). As isoscalar lines move apart from each other under increasing normal strain rate, the increase in an with increasing negative curvature leads to a positive correlation between |∇c| and κm in negatively curved regions, as also observed in Figure 6.


### 4.4. Curvature Dependence ofSd

The mean normalised displacement speedSd/SL conditional on normalised curvature κm×δth is shown in Figure 8(a) for c=0.8, which indicates that Sd/SL is negatively correlated with curvature and that the correlation is nonlinear in nature. These observations are consistent with several previous DNS studies with both detailed and simplified chemistry [3–11, 13–18, 20–22].  The variation of the correlation coefficient between Sd and κm for different c isosurfaces across the flame brush is shown in Figure 8(b),  which reveals that strength of the negative correlation decreases with increasing Ret~(u′/SL)2Da~(u′/SL)4/Ka2~(l/δth)4/3Ka2/3  when either Da (see cases A, C, and E) or Ka (see cases B, C, and D) is held constant (see Table 1).  The tangential diffusion component of displacement speed St is deterministically negatively correlated with curvature with correlation coefficient equal to unity for unity Lewis number flames (see (5)). Thus, the nonlinearity of the correlation between Sd and κm originates  due to the curvature dependence of Sr and Sn.(a) Variation of mean values ofSd/SL conditional on normalised curvature κm×δth for the isosurface at c=0.8. (b) Variation of the correlation coefficients between Sd and κm on the isosurfaces at c = 0.1 to 0.9 for all cases.
(a)(b)Mean values of the reaction componentSr/SL conditional on normalised curvature κm×δth are shown in Figure 9(a) for the isosurface at  c=0.8.  The plot shows both positive and negative correlation trends.  For low Mach number unity Lewis number flames, the curvature dependence of  Sr is principally determined by the curvature dependence of |∇c|. The negative (positive) correlation between |∇c| and κm ultimately gives rise to the positive (negative) correlation between Sr and κm as observed in Figure 9.  As a result of the weak net correlation between |∇c| and κm, the net correlation between  Sr and κm also turns out to be weak, as indicated by the correlation coefficients plotted in Figure 9(b)  Note that Figure 9(b) includes isosurfaces of c from 0.5 to 0.9 only, since the reaction rate ẇ is negligible in the preheat zone (see also (5)).(a) Variation of mean values ofSr/SL conditional on normalised curvature κm×δth for the isosurface at c=0.8. (b) Variation of the correlation coefficients between Sr and κm on the isosurfaces at c = 0.5 to 0.9 for all cases.
(a)(b)Corresponding mean values of the normal diffusion componentSn/SL conditional on normalised curvature κm×δth are shown in Figure 10(a) for  c=0.8, while the correlation coefficients between  Sn and κm are shown in Figure 10(b) for values of cfrom 0.1 to 0.9. The magnitude of the normal molecular diffusion rate in the thin reaction zones regime can be scaled as |N⃗·∇(ρDN⃗·∇c)|~ρD/δ2 where δ is the characteristic flame thickness which in turn can be scaled as |∇c|~1/δ. This suggests that Sn scales as D/δ towards the reactant side of the flame brush and as -D/δ towards the product side. As a result of this, the positively (negatively) correlating branch of the |∇c|−κm correlation gives rise to a positive (negative) correlation between Sn and κm towards the reactant side, where Sn assumes predominantly positive values. By contrast, the positive (negative) correlation between |∇c|  and κm results in a negative (positive) correlation between Sn and κm towards the product side where Sn assumes predominantly negative values. Both positive and negative correlation trends can be discerned for all the c isosurfaces shown in Figure 10(b).  As a result of this, the net correlation between Sn and κm turns out to be weak throughout the flame brush.  The positive and negative correlating trends with κm observed for both  Sr and Sn with κm give rise to positive and negative correlating branches in the variation of (Sr+Sn)conditional on κm×δth as shown in Figure 10(c) for c=0.8. This nonlinear correlation between (Sr+Sn) and κm ultimately induces a nonlinear curvature dependence of displacement speed Sd (see Figure 8(a)), noting that the correlation coefficient arising from the deterministic relation between St and κm remains close to  −1.0  throughout the flame brush for all cases. The correlation coefficients between (Sr+Sn) and κm are shown in Figure 10(d) for values of c from 0.1 to 0.9, and demonstrate that the (Sr+Sn)-κm correlation is much weaker than the St-κm correlation in all cases.  Hence, the net correlation between Sd and κm turns out to be negative throughout the flame brush for all the cases considered here (see Figure 8(a)).(a) Variation of mean values ofSn/SL conditional on normalised curvature κm×δth for the isosurface at c=0.8. (b) Variation of the correlation coefficients between Sn and κm on the isosurfaces at c = 0.1 to  0.9 for all cases. (c) Variation of mean values of (Sr+Sn)/SL conditional on normalised curvature κm×δth for the isosurface at c=0.8. (d) Variation of the correlation coefficients between (Sr+Sn) and κm on the isosurfaces at c = 0.1 to 0.9 for all cases.
(a)(b)(c)(d)


### 4.5. Tangential Strain Rate Dependence ofSd

The mean values of normalised displacement speedSd/SL conditional on normalised tangential strain rate aT×δth/SL are shown in Figure 11(a) for c=0.8 which suggests that Sd and aT are positively correlated for all cases. The observed strain rate dependence of Sd is found to be qualitatively consistent with previous DNS studies for flames with global Lewis number close to unity [3, 5, 12, 14, 15, 17–19, 21, 22].  The variations of the correlation coefficients between Sd and aT for different c isosurfaces are shown in Figure 11(b), which reveals that the strength of the correlation between Sd and aT decreases with increasing u′/SL when either Da or Ka is held constant. This is consistent with the analytical treatment by Joulin [38] who indicated that the strain rate dependence of  Sd  weakens with increasing frequency of straining.  In order to explain the observed behaviour it is useful to look into the dependence of Sr, Sn, and St on aT. For low Mach number unity, Lewis number flames the strain rate dependence of Sr is essentially determined by the strain rate dependence of |∇c|.  It has been shown already that |∇c| and aT are positively correlated (see Figure 5), which gives rise to a negative correlation between Sr and aT according to (5).  This is substantiated by the correlation coefficients between Sr and aT as shown in Figure 11(c) for the isosurfaces from c= 0.5 to 0.9. The correlation between  Sr and aT does not show any monotonic trend with either Karlovitz number (Damköhler number) or turbulent Reynolds number variations when the Damköhler number (Karlovitz number) is held constant, due to the lack of any such trend in the |∇c|-aT correlation.(a) Variation of mean values ofSd/SL conditional on normalised tangential strain rate aT×δth/SL for the isosurface at c=0.8. Variation of the correlation coefficients for all cases on the different c isosurfaces among  (b) Sd and aT, (c) Sr and aT, (d)Sn and aT, (e) (Sr+Sn) and aT and (f) St and aT.
(a)(b)(c)(d)(e)(f)It has been discussed already thatSn scales as Sn~D/δ on the reactant side of the flame brush, and as  Sn~-D/δ on the product side.  Since there is a positive correlation between |∇c|~1/δ and aT, this gives rise to a positive (negative) correlation between Sn and aT on the reactant (product) side.  This behaviour can be seen in the correlation coefficients plotted in Figure 11(d), which also indicates that the correlation coefficient between Sn and aT does not show any monotonic trend with changes in either Da or Ka. However, it has been found that the correlation between Sn and aT in general remains stronger for smaller values of u′ when either Da or Ka is held constant.SinceSr  remains negligible towards the reactant side of the flame brush and Sn is positively correlated with aT, the correlation coefficient between (Sr+Sn) and aT remains positive towards the reactant side (see Figure 11(e)). By contrast, both Sr and Sn are negatively correlated with  aT towards the product side, which leads to a locally negative correlation between (Sr+Sn) and aT.SinceaT and κm are negatively correlated, the tangential diffusion component St is positively correlated with aT because St is proportional to the negative of curvature (i.e., St=-2Dκm). This positive correlation is evident from Figure 11(f), in which the correlation coefficient between St and aT is shown for isosurfaces of  c across the flame brush.  The negative correlation between aT and κm (see Figure 7(a)) arises principally due to heat release, and thus, the strength of this correlation weakens with increasing Karlovitz number, since the effects of turbulent velocity fluctuations are likely to mask the heat release effects as the combustion tends more towards the broken reaction zones regime [1]. Moreover, with decreasing Damköhler number the effects of heat release weaken in comparison to those of turbulent straining, as suggested by (11b).  As a result of this, the strength of the negative correlation between aT  and κm weakens with decreasing (increasing) Damköhler number (Karlovitz number) when Ka (Da) is held constant (see Figure 7(a)), and this gives rise to a corresponding change in the correlation between St and aT (see Figure 11(f)).  This essentially suggests that the strength of the positive correlation between St and aT decreases with increasing turbulent Reynolds number Ret when either Ka or Da is held constant. The effects of the positive St-aT and negative aT-κm correlations remain strongest in the reaction zone and the strengths of these correlations decrease towards both the reactant and product sides of the flame brush (see Figures 11(f) and 7(b), resp.).It is evident from Figures11(e) and 11(f) that the positive correlations between (Sr+Sn) and aT, and between St and aT, aid each other towards the reactant side of the flame brush. By contrast, towards the product side the positive correlation between St and aT overcomes the negative correlation between (Sr+Sn) and aT to result in a net positive correlation between Sd and aT in all the cases considered here (see Figure 11(a)). The weakening of the correlation between St and aT with increasing Ret ultimately leads to decrease in the strength of the correlation between Sd and  aT when either Da or Ka is held constant (see Figures 11(a) and 11(f)).


### 4.6. Implications for FSD Modelling

In the context of Reynolds Averaged Navier Stokes (RANS) modelling, the generalised FSDΣgen is defined as [39]                                                 (12a)Σgen=〈|∇c|〉,
where 〈Q〉 indicates the Reynolds averaged value of a general quantity Q. It has been demonstrated in [15, 23, 26, 27] that the statistical behaviour of |∇c| and the terms of its transport (7) in the thin reaction zones regime depend significantly on the choice of progress variable isosurface c=c*. Hence, it is more appropriate to use the generalised FSD (see(12a)) which does not depend on the choice of isosurface.  Reynolds averaging of (7) yields the transport equation for the generalised FSD [2, 23, 24](12b)∂Σgen∂t+∂〈uj〉sΣgen∂xj=〈(δij-NiNj)∂ui∂xj〉sΣgen+2〈Sdκm〉sΣgen-∂〈SdNi〉sΣgen∂xi,
where 〈Q〉s=〈Q|∇c|〉/〈|∇c|〉 indicates a surface averaging operation for a general quantity Q [39]. The first term on the left hand side is the transient term while the second term is the advection term. The three terms on the right hand side represent the effects of strain rate, curvature, and propagation, respectively. It is evident from (12b) that the statistical behaviour of the curvature and propagation terms depends on the statistics of displacement speed. These terms can be rewritten in terms of the displacement speed components as(12c)2〈Sdκm〉sΣgen=2〈(Sr+Sn)κm〉sΣgen-4〈Dκm2〉sΣgen,(12d)-∇⋅〈SdN⃗〉sΣgen=-∇⋅〈(Sr+Sn)N⃗〉sΣgen+2∇⋅〈DκmN⃗〉sΣgen.The variations of all of the terms appearing in these equations with Favre averaged reaction progress variablec̃=〈ρc〉/〈ρ〉 are shown in Figure 12. The Reynolds averages are evaluated by ensemble averaging the relevant quantities in the directions normal to the direction of mean flame propagation.Variation across the flame brush with Favre averaged reaction progress variablec̃=〈ρc〉/〈ρ〉 of the FSD curvature term 2〈Sdκm〉sΣgen and its contributions (see (12c)) 2〈(Sr+Sn)κm〉sΣgen and -4〈Dκm2〉sΣgen; the FSD propagation term -∇·〈SdN⃗〉sΣgen and its contributions (see (12d)) -∇·〈(Sr+Sn)N⃗〉sΣgen and  2∇·〈DκmN⃗〉sΣgen; together with the modelled curvature term 2〈SLκm〉sΣgen and propagation term -∇·〈SLN⃗〉sΣgen for: (a) case A; (b) case B; (c) case C; (d) case D; (e) case E.  All the terms are normalised with respect to SL/δth2 of the respective cases.It can be seen from Figure12 that the FSD curvature term 〈Sd∇·N⃗〉sΣgen (line with tick symbols) assumes mostly negative value throughout the flame brush for all cases although small positive contributions are found towards the reactant side of the flame brush for small Reynolds numbers (i.e., cases A and B). The magnitude of the term 2〈(Sr+Sn)κm〉sΣgen (line with filled triangles) remains comparable to the contribution of the term -4〈Dκm2〉sΣgen (line with open squares) for small values of  Ret (i.e., cases A and B). However, the contribution of 2〈(Sr+Sn)κm〉sΣgen remains smaller than that of -4〈Dκm2〉sΣgen for larger values of Ret and the statistical behaviour of the FSD curvature term is then principally governed by the latter contribution. It is also evident from Figure 12 that the magnitude of (-4〈Dκm2〉sΣgen) becomes increasingly more important than the magnitude of 2〈(Sr+Sn)κm〉sΣgen with increasing Karlovitz number (Damköhler number) when Da (Ka) is held constant. The statistical behaviour of 2〈(Sr+Sn)κm〉sΣgen is determined by the correlation between |∇c| and κm and between (Sr+Sn) and κm and the strength of both of these correlations tends to weaken with increasing (increasing) Karlovitz number (Damköhler number) when Da (Ka) is held constant (see Figures 6(b) and 10(d)), causing a corresponding reduction in the relative magnitude of the term 2〈(Sr+Sn)κm〉sΣgen.The FSD propagation term-∇·〈SdN⃗〉sΣgen (line with crosses) remains positive towards the reactant side, but it eventually becomes negative towards the product side.  It can be seen that the term -∇·〈(Sr+Sn)N⃗〉sΣgen (line with filled squares) remains almost equal to the FSD propagation term itself, because the contribution of 2∇·〈DκmN⃗〉sΣgen (line with open circles) remains negligible throughout the flame brush due to the statistically planar nature of the flames considered here.  The contribution of  2〈(Sr+Sn)κm〉sΣgen remains positive towards the reactant side and assumes negative values towards the product side.In some previous studies, the FSD curvature and propagation terms have been modelled by replacingSd with SL to yield the modelled expressions 2〈SLκm〉sΣgen and -∇·〈SLN⃗〉sΣgen, respectively; [40, 41].  Figure 12 further indicates that there are significant differences between the original terms and these modelled expressions, which are shown, respectively, by a line with filled circles and a line with open triangles.  This essentially indicates that Sd cannot be approximated simply by SL in the modelling of the FSD curvature and propagation terms. Moreover, Figure 12 suggests that modelling of 〈κm2〉s is necessary for the closure of the FSD curvature term especially in the thin reaction zones regime since the term -4〈Dκm2〉sΣgen  remains close to 2〈Sdκm〉sΣgen throughout the flame brush for high values of the turbulent Reynolds number. The effects of turbulent Reynolds number on the modelling of the FSD curvature and propagation terms are beyond of the scope of this study and will be addressed in future work.


## 4.1. Flame-Turbulence Interaction

The contours of the reaction progress variablec  in the x1-x2 mid-plane of the domain at time tsim=1.0δth/SL are shown in Figures 1(a)–1(e) for cases A to E, respectively. It is evident that the extent of the wrinkling of the c isosurfaces increases with increasing  u′, that is, in going from case A through to case E (see Table 1). Moreover, the contours of c representing the preheat zone (i.e., c<0.5) are much more distorted than those representing the reaction zone (i.e., 0.7<c<0.9). This tendency is more prevalent for the case with the highest Karlovitz number (case E), since the scale separation between δth and η increases with increasing Ka, allowing more energetic eddies to enter into the preheat zone and causing greater distortion of the flame.Contours of reaction progress variablec in the x1-x2  midplane of the domain after t=1.0δth/SL for: (a) case A; (b) case B; (c) case C; (d) case D; (e) case E.
(a)(b)(c)(d)(e)


## 4.2. Probability Density Functions of Displacement SpeedSd

The probability density functions (pdfs) of normalised displacement speedSd/SL on the c=0.1,0.3,0.5,0.7, and 0.9 isosurfaces are shown in Figures 2(a)–2(e), respectively. It is evident in all cases that there is a small but finite probability of finding negative displacement speed although the mean value of Sd/SL remains strongly positive throughout the flame brush. Negative values of displacement speed suggest that the progress variable isosurface in question is moving locally in a direction opposite to that of mean flame propagation. This situation arises when the negative contribution of the molecular diffusion rate dominates over the positive semi-definite chemical reaction rate, as discussed in detail by Gran et al. [4]. Moreover, Figure 2(e) indicates that the probability of finding negative values of Sd/SL is greatest for the highest Karlovitz number (case E). This behaviour can be explained in terms of the scaling analysis of Peters [1] for unity Lewis number flames, which suggested that the different components of displacement speed scale as(10)Sr+Snvη~SLvη~O(1Ka),Stvη~-2Dκmvη~O(1),
where vη is the Kolmogorov velocity scale.  Here, in accordance with Peters [1], (Sr+Sn) is taken to scale with  SL and the curvature κm  is taken to scale with the Kolmogorov length scale 1/η.  These scalings of (Sr+Sn) and κm were confirmed by DNS data presented in [21].  Equation (10) clearly suggests that in the thin reaction zones regime (i.e., Ka>1) the effects of (Sr+Sn) are likely to weaken progressively in comparison to the contribution of  St  with increasing Karlovitz number. This suggests that for large values of Karlovitz number the negative contribution of  St  more readily overcomes the predominantly positive contribution of (Sr+Sn) to yield a negative value of  Sd. This eventually gives rise to an increased probability of finding negative values of  Sd  for flames with increasing value of Karlovitz number, as suggested by Figure 2.  In order to demonstrate that the observed negative values of  Sd  originates principally due to  St, the pdfs of (Sr+Sn)/SL are shown in Figure 3, respectively, for all cases. Comparing Figures 2 and 3 reveals that the probability of finding negative values of (Sr+Sn)  is significantly smaller than that of finding negative values of  Sd.Pdfs of normalised displacement speedSd/SL on the c =    (a) 0.1, (b) 0.3, (c) 0.5, (d) 0.7,  and (e) 0.9  isosurfaces.
(a)(b)(c)(d)(e)Pdfs of normalised combined reaction and normal diffusion component of displacement speed(Sr+Sn)/SL on the c  =      (a) 0.1, (b) 0.3, (c) 0.5, (d) 0.7, and (e) 0.9 isosurfaces.
(a)(b)(c)(d)(e)Comparing Figures2(a)–2(e) and Figures 3(a)–3(e), it can be seen that the most probable values of  Sd/SL and (Sr+Sn)/SL  remain of the order of  ρ0/ρ and that the widths of the displacement speed pdf for a given c isosurface increases with increasing u′  (i.e., going from case A to case E).   In order to explain this behavior, it is useful to examine the pdfs of the normalised tangential component of displacement speed St/SL which are depicted in Figure 4 and show that these pdfs remain almost symmetrical with a peak close to a value of zero. This behaviour is consistent with the statistically planar nature of these flames.  However, the greater extent of flame wrinkling observed in Figure 1 for larger values of u′ results in a greater spread of flame curvature, which in turn increases the width of the St/SL pdfs according to (5).  The negligible mean value of St essentially leads to the mean values of Sd and (Sr+Sn)  becoming almost equal to each other, but the statistical variation of Sd induced by St leads to broader pdfs of Sd  than those of (Sr+Sn).  These results clearly suggest that the distributions of Sd and its components Sr,Sn and St generally broaden with  u′/SL, which essentially leads to broadening of the distributions with increasing Ret~(u′/SL)2Da~(u′/SL)4/Ka2~(l/δth)4/3Ka2/3  when either Da or Ka is held constant.Pdfs of normalised tangential diffusion component of displacement speedSt/SL on the c= (a) 0.1, (b) 0.3, (c) 0.5, (d) 0.7, and (e) 0.9 isosurfaces.
(a)(b)(c)(d)(e)


## 4.3. Strain Rate and Curvature Dependences of|∇c|

For low Mach number unity Lewis number flames, both densityρ and nondimensional temperature T remain uniform on a given isosurface of c and thus the statistical behaviours of Sr and Sn (see (5)) are mainly affected by the dependence of the SDF |∇c| on strain rate and curvature. The values of |∇c|×δth conditional on normalised tangential strain rate aT×δth/SL are shown in Figure 5(a) for the isosurface at c=0.8 (close to the location of maximum reaction rate), which suggests clearly that |∇c| and aT are positively correlated for all cases.  The correlation coefficients between |∇c| and aT are shown in Figure 5(b) for values of c  between 0.1 and 0.9, and these indicate that the positive correlation is maintained throughout the flame brush.  This correlation can be explained by noting that the dilatation rate ∇·u⃗ can be expressed as (11a)∇⋅u⃗=aT+an.
For unity Lewis number flames, the dilatation rate can be scaled as ∇·u⃗~τSL/δth  [21], whereas aT can be scaled as aT~u′/l  [37], which suggests that(11b)∇⋅u⃗aT~τSLlδthu′~τDa.(a) Variation of mean values of|∇c|×δth conditional on normalised tangential strain rate aT×δth/SL for the isosurface at c=0.8.  (b) Variation of the correlation coefficients between |∇c| and aT on the isosurfaces at c = 0.1 to 0.9 for all cases.
(a)(b)Equation (11b) essentially shows that for low values of the Damköhler number, aT remains much greater than ∇·u⃗. This essentially suggests that an increase in tangential strain rate is associated with a decrease in the normal strain rate, since the normal strain rate scales as an~-aT when dilatation rate effects are much weaker than those of turbulent straining.  Under decreasing normal straining, the isoscalar lines are brought closer to each other and hence give rise to an increase in |∇c|. The decreasing trend of normal strain rate with increasing tangential strain rate is reflected in the positive correlation between |∇c| and aT throughout the flame brush.  It can be seen from Figure 5 that the correlation between  |∇c| and aT does not show any monotonic trend with either Karlovitz number (Damköhler number) or turbulent Reynolds number variations when Damköhler number (Karlovitz number) is held constant.The variation of|∇c|×δth conditional on normalised curvature κm×δth  is shown in Figure 6(a) for the isosurface at c=0.8. Both positive and negative correlation trends can be observed, which is consistent with several previous studies [14–22, 26, 27]. As a result of these opposing trends, the net correlation between |∇c|×δth and κm×δth turns out be weak throughout the flame brush, as evident from the correlation coefficients between |∇c| and κm presented for different values of c across the flame brush in Figure 6(b). Both the positively and negatively correlating branches can be explained using (11a). In turbulent premixed flames the dilatation rate ∇·u⃗ is negatively correlated with curvature κm, because of focusing (defocusing) of heat in the negatively (positively) curved regions of the flame surface [18, 21]. The tangential strain rate aT is also found to be negatively correlated with curvature [18, 21]. The correlation coefficients for the ∇·u⃗-κm and aT-κm correlations for different c values across the flame brush are shown in Figures 7(a) and 7(b), respectively.  Comparing the correlation coefficients of the ∇·u⃗-κm and aT-κm correlations for different cases reveals that the strength of both of these negative correlations decreases with increasing u′/SL and Ret~(l/δth)4/3Ka2/3 when either Da or Ka is held constant.(a) Variation of mean values of|∇c|×δth conditional on normalised curvature κm×δth for the isosurface at c=0.8.  (b) Variation of the correlation coefficients between |∇c| and κm on the isosurfaces at c=0.1 to 0.9 for all cases.
(a)(b)Variation of the correlation coefficients between (a) tangential strain rateaT and curvature κm and between (b) dilatation rate ∇·u⃗ and curvature κm on the isosurfaces at c = 0.1 to 0.9 for all cases.
(a)(b)The combination of the positive correlation between|∇c|  and aT and the negative correlation between aT and κm leads to small values of |∇c| in regions of high positive curvature. This gives rise to the negative correlation trend between |∇c| and κm, as observed in Figure 6.  However, the dilatation rate ∇·u⃗ may locally attain large values in the highly negatively curved regions of the flame due to strong focussing of heat.  Hence, ∇·u⃗ may locally exceed aT and induce an increase in an according to (11a). As isoscalar lines move apart from each other under increasing normal strain rate, the increase in an with increasing negative curvature leads to a positive correlation between |∇c| and κm in negatively curved regions, as also observed in Figure 6.


## 4.4. Curvature Dependence ofSd

The mean normalised displacement speedSd/SL conditional on normalised curvature κm×δth is shown in Figure 8(a) for c=0.8, which indicates that Sd/SL is negatively correlated with curvature and that the correlation is nonlinear in nature. These observations are consistent with several previous DNS studies with both detailed and simplified chemistry [3–11, 13–18, 20–22].  The variation of the correlation coefficient between Sd and κm for different c isosurfaces across the flame brush is shown in Figure 8(b),  which reveals that strength of the negative correlation decreases with increasing Ret~(u′/SL)2Da~(u′/SL)4/Ka2~(l/δth)4/3Ka2/3  when either Da (see cases A, C, and E) or Ka (see cases B, C, and D) is held constant (see Table 1).  The tangential diffusion component of displacement speed St is deterministically negatively correlated with curvature with correlation coefficient equal to unity for unity Lewis number flames (see (5)). Thus, the nonlinearity of the correlation between Sd and κm originates  due to the curvature dependence of Sr and Sn.(a) Variation of mean values ofSd/SL conditional on normalised curvature κm×δth for the isosurface at c=0.8. (b) Variation of the correlation coefficients between Sd and κm on the isosurfaces at c = 0.1 to 0.9 for all cases.
(a)(b)Mean values of the reaction componentSr/SL conditional on normalised curvature κm×δth are shown in Figure 9(a) for the isosurface at  c=0.8.  The plot shows both positive and negative correlation trends.  For low Mach number unity Lewis number flames, the curvature dependence of  Sr is principally determined by the curvature dependence of |∇c|. The negative (positive) correlation between |∇c| and κm ultimately gives rise to the positive (negative) correlation between Sr and κm as observed in Figure 9.  As a result of the weak net correlation between |∇c| and κm, the net correlation between  Sr and κm also turns out to be weak, as indicated by the correlation coefficients plotted in Figure 9(b)  Note that Figure 9(b) includes isosurfaces of c from 0.5 to 0.9 only, since the reaction rate ẇ is negligible in the preheat zone (see also (5)).(a) Variation of mean values ofSr/SL conditional on normalised curvature κm×δth for the isosurface at c=0.8. (b) Variation of the correlation coefficients between Sr and κm on the isosurfaces at c = 0.5 to 0.9 for all cases.
(a)(b)Corresponding mean values of the normal diffusion componentSn/SL conditional on normalised curvature κm×δth are shown in Figure 10(a) for  c=0.8, while the correlation coefficients between  Sn and κm are shown in Figure 10(b) for values of cfrom 0.1 to 0.9. The magnitude of the normal molecular diffusion rate in the thin reaction zones regime can be scaled as |N⃗·∇(ρDN⃗·∇c)|~ρD/δ2 where δ is the characteristic flame thickness which in turn can be scaled as |∇c|~1/δ. This suggests that Sn scales as D/δ towards the reactant side of the flame brush and as -D/δ towards the product side. As a result of this, the positively (negatively) correlating branch of the |∇c|−κm correlation gives rise to a positive (negative) correlation between Sn and κm towards the reactant side, where Sn assumes predominantly positive values. By contrast, the positive (negative) correlation between |∇c|  and κm results in a negative (positive) correlation between Sn and κm towards the product side where Sn assumes predominantly negative values. Both positive and negative correlation trends can be discerned for all the c isosurfaces shown in Figure 10(b).  As a result of this, the net correlation between Sn and κm turns out to be weak throughout the flame brush.  The positive and negative correlating trends with κm observed for both  Sr and Sn with κm give rise to positive and negative correlating branches in the variation of (Sr+Sn)conditional on κm×δth as shown in Figure 10(c) for c=0.8. This nonlinear correlation between (Sr+Sn) and κm ultimately induces a nonlinear curvature dependence of displacement speed Sd (see Figure 8(a)), noting that the correlation coefficient arising from the deterministic relation between St and κm remains close to  −1.0  throughout the flame brush for all cases. The correlation coefficients between (Sr+Sn) and κm are shown in Figure 10(d) for values of c from 0.1 to 0.9, and demonstrate that the (Sr+Sn)-κm correlation is much weaker than the St-κm correlation in all cases.  Hence, the net correlation between Sd and κm turns out to be negative throughout the flame brush for all the cases considered here (see Figure 8(a)).(a) Variation of mean values ofSn/SL conditional on normalised curvature κm×δth for the isosurface at c=0.8. (b) Variation of the correlation coefficients between Sn and κm on the isosurfaces at c = 0.1 to  0.9 for all cases. (c) Variation of mean values of (Sr+Sn)/SL conditional on normalised curvature κm×δth for the isosurface at c=0.8. (d) Variation of the correlation coefficients between (Sr+Sn) and κm on the isosurfaces at c = 0.1 to 0.9 for all cases.
(a)(b)(c)(d)


## 4.5. Tangential Strain Rate Dependence ofSd

The mean values of normalised displacement speedSd/SL conditional on normalised tangential strain rate aT×δth/SL are shown in Figure 11(a) for c=0.8 which suggests that Sd and aT are positively correlated for all cases. The observed strain rate dependence of Sd is found to be qualitatively consistent with previous DNS studies for flames with global Lewis number close to unity [3, 5, 12, 14, 15, 17–19, 21, 22].  The variations of the correlation coefficients between Sd and aT for different c isosurfaces are shown in Figure 11(b), which reveals that the strength of the correlation between Sd and aT decreases with increasing u′/SL when either Da or Ka is held constant. This is consistent with the analytical treatment by Joulin [38] who indicated that the strain rate dependence of  Sd  weakens with increasing frequency of straining.  In order to explain the observed behaviour it is useful to look into the dependence of Sr, Sn, and St on aT. For low Mach number unity, Lewis number flames the strain rate dependence of Sr is essentially determined by the strain rate dependence of |∇c|.  It has been shown already that |∇c| and aT are positively correlated (see Figure 5), which gives rise to a negative correlation between Sr and aT according to (5).  This is substantiated by the correlation coefficients between Sr and aT as shown in Figure 11(c) for the isosurfaces from c= 0.5 to 0.9. The correlation between  Sr and aT does not show any monotonic trend with either Karlovitz number (Damköhler number) or turbulent Reynolds number variations when the Damköhler number (Karlovitz number) is held constant, due to the lack of any such trend in the |∇c|-aT correlation.(a) Variation of mean values ofSd/SL conditional on normalised tangential strain rate aT×δth/SL for the isosurface at c=0.8. Variation of the correlation coefficients for all cases on the different c isosurfaces among  (b) Sd and aT, (c) Sr and aT, (d)Sn and aT, (e) (Sr+Sn) and aT and (f) St and aT.
(a)(b)(c)(d)(e)(f)It has been discussed already thatSn scales as Sn~D/δ on the reactant side of the flame brush, and as  Sn~-D/δ on the product side.  Since there is a positive correlation between |∇c|~1/δ and aT, this gives rise to a positive (negative) correlation between Sn and aT on the reactant (product) side.  This behaviour can be seen in the correlation coefficients plotted in Figure 11(d), which also indicates that the correlation coefficient between Sn and aT does not show any monotonic trend with changes in either Da or Ka. However, it has been found that the correlation between Sn and aT in general remains stronger for smaller values of u′ when either Da or Ka is held constant.SinceSr  remains negligible towards the reactant side of the flame brush and Sn is positively correlated with aT, the correlation coefficient between (Sr+Sn) and aT remains positive towards the reactant side (see Figure 11(e)). By contrast, both Sr and Sn are negatively correlated with  aT towards the product side, which leads to a locally negative correlation between (Sr+Sn) and aT.SinceaT and κm are negatively correlated, the tangential diffusion component St is positively correlated with aT because St is proportional to the negative of curvature (i.e., St=-2Dκm). This positive correlation is evident from Figure 11(f), in which the correlation coefficient between St and aT is shown for isosurfaces of  c across the flame brush.  The negative correlation between aT and κm (see Figure 7(a)) arises principally due to heat release, and thus, the strength of this correlation weakens with increasing Karlovitz number, since the effects of turbulent velocity fluctuations are likely to mask the heat release effects as the combustion tends more towards the broken reaction zones regime [1]. Moreover, with decreasing Damköhler number the effects of heat release weaken in comparison to those of turbulent straining, as suggested by (11b).  As a result of this, the strength of the negative correlation between aT  and κm weakens with decreasing (increasing) Damköhler number (Karlovitz number) when Ka (Da) is held constant (see Figure 7(a)), and this gives rise to a corresponding change in the correlation between St and aT (see Figure 11(f)).  This essentially suggests that the strength of the positive correlation between St and aT decreases with increasing turbulent Reynolds number Ret when either Ka or Da is held constant. The effects of the positive St-aT and negative aT-κm correlations remain strongest in the reaction zone and the strengths of these correlations decrease towards both the reactant and product sides of the flame brush (see Figures 11(f) and 7(b), resp.).It is evident from Figures11(e) and 11(f) that the positive correlations between (Sr+Sn) and aT, and between St and aT, aid each other towards the reactant side of the flame brush. By contrast, towards the product side the positive correlation between St and aT overcomes the negative correlation between (Sr+Sn) and aT to result in a net positive correlation between Sd and aT in all the cases considered here (see Figure 11(a)). The weakening of the correlation between St and aT with increasing Ret ultimately leads to decrease in the strength of the correlation between Sd and  aT when either Da or Ka is held constant (see Figures 11(a) and 11(f)).


## 4.6. Implications for FSD Modelling

In the context of Reynolds Averaged Navier Stokes (RANS) modelling, the generalised FSDΣgen is defined as [39]                                                 (12a)Σgen=〈|∇c|〉,
where 〈Q〉 indicates the Reynolds averaged value of a general quantity Q. It has been demonstrated in [15, 23, 26, 27] that the statistical behaviour of |∇c| and the terms of its transport (7) in the thin reaction zones regime depend significantly on the choice of progress variable isosurface c=c*. Hence, it is more appropriate to use the generalised FSD (see(12a)) which does not depend on the choice of isosurface.  Reynolds averaging of (7) yields the transport equation for the generalised FSD [2, 23, 24](12b)∂Σgen∂t+∂〈uj〉sΣgen∂xj=〈(δij-NiNj)∂ui∂xj〉sΣgen+2〈Sdκm〉sΣgen-∂〈SdNi〉sΣgen∂xi,
where 〈Q〉s=〈Q|∇c|〉/〈|∇c|〉 indicates a surface averaging operation for a general quantity Q [39]. The first term on the left hand side is the transient term while the second term is the advection term. The three terms on the right hand side represent the effects of strain rate, curvature, and propagation, respectively. It is evident from (12b) that the statistical behaviour of the curvature and propagation terms depends on the statistics of displacement speed. These terms can be rewritten in terms of the displacement speed components as(12c)2〈Sdκm〉sΣgen=2〈(Sr+Sn)κm〉sΣgen-4〈Dκm2〉sΣgen,(12d)-∇⋅〈SdN⃗〉sΣgen=-∇⋅〈(Sr+Sn)N⃗〉sΣgen+2∇⋅〈DκmN⃗〉sΣgen.The variations of all of the terms appearing in these equations with Favre averaged reaction progress variablec̃=〈ρc〉/〈ρ〉 are shown in Figure 12. The Reynolds averages are evaluated by ensemble averaging the relevant quantities in the directions normal to the direction of mean flame propagation.Variation across the flame brush with Favre averaged reaction progress variablec̃=〈ρc〉/〈ρ〉 of the FSD curvature term 2〈Sdκm〉sΣgen and its contributions (see (12c)) 2〈(Sr+Sn)κm〉sΣgen and -4〈Dκm2〉sΣgen; the FSD propagation term -∇·〈SdN⃗〉sΣgen and its contributions (see (12d)) -∇·〈(Sr+Sn)N⃗〉sΣgen and  2∇·〈DκmN⃗〉sΣgen; together with the modelled curvature term 2〈SLκm〉sΣgen and propagation term -∇·〈SLN⃗〉sΣgen for: (a) case A; (b) case B; (c) case C; (d) case D; (e) case E.  All the terms are normalised with respect to SL/δth2 of the respective cases.It can be seen from Figure12 that the FSD curvature term 〈Sd∇·N⃗〉sΣgen (line with tick symbols) assumes mostly negative value throughout the flame brush for all cases although small positive contributions are found towards the reactant side of the flame brush for small Reynolds numbers (i.e., cases A and B). The magnitude of the term 2〈(Sr+Sn)κm〉sΣgen (line with filled triangles) remains comparable to the contribution of the term -4〈Dκm2〉sΣgen (line with open squares) for small values of  Ret (i.e., cases A and B). However, the contribution of 2〈(Sr+Sn)κm〉sΣgen remains smaller than that of -4〈Dκm2〉sΣgen for larger values of Ret and the statistical behaviour of the FSD curvature term is then principally governed by the latter contribution. It is also evident from Figure 12 that the magnitude of (-4〈Dκm2〉sΣgen) becomes increasingly more important than the magnitude of 2〈(Sr+Sn)κm〉sΣgen with increasing Karlovitz number (Damköhler number) when Da (Ka) is held constant. The statistical behaviour of 2〈(Sr+Sn)κm〉sΣgen is determined by the correlation between |∇c| and κm and between (Sr+Sn) and κm and the strength of both of these correlations tends to weaken with increasing (increasing) Karlovitz number (Damköhler number) when Da (Ka) is held constant (see Figures 6(b) and 10(d)), causing a corresponding reduction in the relative magnitude of the term 2〈(Sr+Sn)κm〉sΣgen.The FSD propagation term-∇·〈SdN⃗〉sΣgen (line with crosses) remains positive towards the reactant side, but it eventually becomes negative towards the product side.  It can be seen that the term -∇·〈(Sr+Sn)N⃗〉sΣgen (line with filled squares) remains almost equal to the FSD propagation term itself, because the contribution of 2∇·〈DκmN⃗〉sΣgen (line with open circles) remains negligible throughout the flame brush due to the statistically planar nature of the flames considered here.  The contribution of  2〈(Sr+Sn)κm〉sΣgen remains positive towards the reactant side and assumes negative values towards the product side.In some previous studies, the FSD curvature and propagation terms have been modelled by replacingSd with SL to yield the modelled expressions 2〈SLκm〉sΣgen and -∇·〈SLN⃗〉sΣgen, respectively; [40, 41].  Figure 12 further indicates that there are significant differences between the original terms and these modelled expressions, which are shown, respectively, by a line with filled circles and a line with open triangles.  This essentially indicates that Sd cannot be approximated simply by SL in the modelling of the FSD curvature and propagation terms. Moreover, Figure 12 suggests that modelling of 〈κm2〉s is necessary for the closure of the FSD curvature term especially in the thin reaction zones regime since the term -4〈Dκm2〉sΣgen  remains close to 2〈Sdκm〉sΣgen throughout the flame brush for high values of the turbulent Reynolds number. The effects of turbulent Reynolds number on the modelling of the FSD curvature and propagation terms are beyond of the scope of this study and will be addressed in future work.


## 5. Conclusions

The turbulent Reynolds number dependence of the statistics of displacement speed in turbulent premixed flames has been studied using a DNS database of statistically planar flames, in which the variation of turbulent Reynolds numberRet from 20 to 100 has been brought about by modifying either Damköhler number Da or Karlovitz number Ka independently of each other. As both Da and Ka are functions of u′/SL and l/δth, the variation of Ret for the present database has been achieved by modifying the initial values of u′/SL and l/δth simultaneously.The mean value of displacement speed remains positive throughout the flame brush, but there is nonzero probability of finding negative values of displacement speed, in accordance with the findings of previous work.  It has been shown here that the probability of finding negative displacement speed increases with increasing turbulent Reynolds number when the Damköhler number is held constant.The dependences of tangential strain rate and dilatation rate on flame curvature are shown to have a significant influence on the strain rate and curvature dependences of the SDF|∇c|, which in turn affects the statistical behaviour of displacement speed in response to strain rate and curvature. It has been found that the variation of turbulent Reynolds number does not alter the qualitative nature of the correlations between tangential strain rate and dilatation rate with curvature, but the strength of these correlations is found to weaken with increasing turbulent Reynolds number when either Damköhler or Karlovitz number is held constant. Similarly, the strain rate and curvature dependences of displacement speed are shown to weaken with increasing turbulent Reynolds number when either Damköhler or Karlovitz number is kept unaltered.Detailed physical explanations are provided to explain the influences of turbulent Reynolds number on the strain rate and curvature dependences of displacement speed in terms of the individual response of the reaction, normal diffusion and tangential diffusion components of  displacement speed.The implications of the turbulent Reynolds number dependence of displacement speed for the modelling of the FSD transport equation has been explored. The statistical behaviour of the FSD curvature term contributions arising due to the combined reaction and normal diffusion components of displacement speed and to the tangential diffusion component of displacement speed have been analysed in detail. At low turbulence Reynolds numbers, the magnitudes of these contributions remain comparable.  As the turbulence Reynolds number increases, the curvature dependence of the tangential component of displacement speed ensures that this contribution becomes increasingly important and is the predominant influence on the FSD curvature term for larger values of turbulent Reynolds number.  Hence, the curvature dependence of the tangential diffusion component of displacement speed cannot be ignored for accurate modelling of the FSD curvature term for flames within the thin reaction zones regime.It is worth noting that the effects of differential diffusion of heat and mass are not taken into account in the present study. However, some previous DNS studies with detailed chemistry [3–7, 9] suggested that intermediate species may play an important role even for the flames with global Lewis number close to unity. Moreover, only a moderate range of turbulent Reynolds number has been considered in this study, so three-dimensional DNS studies with detailed chemistry and at higher values of turbulent Reynolds number will be needed for deeper understanding and for the purpose of quantitative predictions.



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*Source: 473679-2011-06-09.xml*