# Local Strain Rate and Curvature Dependences of Scalar Dissipation Rate Transport in Turbulent Premixed Flames: A Direct Numerical Simulation Analysis

Authors: Y. Gao; N. Chakraborty; N. Swaminathan
Journal: Journal of Combustion
 (2014)

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## Abstract

The statistical behaviours of the instantaneous scalar dissipation rateNc of reaction progress variable c in turbulent premixed flames have been analysed based on three-dimensional direct numerical simulation data of freely propagating statistically planar flame and V-flame configurations with different turbulent Reynolds number Ret. The statistical behaviours of Nc and different terms of its transport equation for planar and V-flames are found to be qualitatively similar. The mean contribution of the density-variation term T1 is positive, whereas the molecular dissipation term (-D2) acts as a leading order sink. The mean contribution of the strain rate term T2 is predominantly negative for the cases considered here. The mean reaction rate contribution T3 is positive (negative) towards the unburned (burned) gas side of the flame, whereas the mean contribution of the diffusivity gradient term (D) assumes negative (positive) values towards the unburned (burned) gas side. The local statistical behaviours of Nc, T1, T2, T3, (-D2), and f(D) have been analysed in terms of their marginal probability density functions (pdfs) and their joint pdfs with local tangential strain rate aT and curvature km. Detailed physical explanations have been provided for the observed behaviour.


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## Body

## 1. Introduction

Scalar dissipation rate (SDR) plays a pivotal role in turbulent reacting flows [1, 2] and thus its statistical behaviour is of fundamental importance to the modelling of turbulent premixed combustion. In turbulent premixed combustion the mean/filtered reaction rate w˙¯ of a reaction progress variable c is directly related to the Favre mean/filtered value of SDR N~c=ρD∇c·∇c¯/ρ- [1–4], where ρ is the fluid density and D is the progress variable diffusivity with the overbar indicating a Reynolds averaging/large eddy simulation (LES) filtering process as applicable. It is well known that strain rate and curvature can significantly affect the local flame propagation behaviour and |∇c| statistics in turbulent premixed flames [5–16]. Thus, strain rate and curvature are expected to have appreciable influences on local statistics of SDR Nc and its transport. The transport equation of the instantaneous SDR of reaction progress variable Nc is given as [3, 17](1a)∂(ρNc)∂t+∂(ρujNc)∂xj=∂∂xj(ρD∂Nc∂xj)-2Dρ∂ρ∂xj∂c∂xj[w˙+∇·(ρD∇c)]-2ρD∂c∂xi∂ui∂xj∂c∂xj+2D∂w˙∂xj∂c∂xj-2ρD2∂2c∂xi∂xj∂2c∂xi∂xj+f(D),
where
(1b)f(D)=2D∂c∂xk∂(ρD)∂xk∂2c∂xj∂xj︸TD1+2D∂c∂xk∂2(ρD)∂xj∂xk∂c∂xj︸TD2-∂∂xj(ρNc∂D∂xj)︸TD3-2ρD∂D∂xj∂(∇c·∇c)∂xj︸TD4+ρ∇c·∇c[∂D∂t+uj∂D∂xj]︸TD5.The first two terms on the left hand side of (1a) represent the transient and advection effects, whereas the first term on the right hand side (i.e., D1=∇·(ρD∇Nc)) denotes molecular diffusion of SDR. The second term on the right hand side of (1a) (i.e., T1=-2D∇ρ·∇c[w˙+∇·(ρD∇c)]/ρ) originates due to density variation and will henceforth be referred to as the density variation term. The third term on the right hand side of (1a) (i.e., T2=-2ρD(∂c/∂xi)(∂ui/∂xj)(∂c/∂xj)) represents the effects of fluid-dynamic straining, whereas the fourth term (i.e., T3=2D(∂w˙/∂xi)(∂c/∂xi)) denotes the reaction rate contribution to the SDR transport. The penultimate term on the right hand side of (1a) (i.e., -D2=-2ρD2(∂2c/∂xi∂xj)(∂2c/∂xi∂xj)) denotes molecular dissipation of Nc, and the terms involving temporal and spatial gradients of diffusivity are collectively referred to as f(D) (see (1b)).Although the statistical behaviours of|∇c| and the terms of its transport equation were analysed earlier, the terms of Nc transport equation are fundamentally different from the terms of the |∇c| transport equation, which can be written for a givenc isosurface in the following manner [11, 13–16]:
(2)∂|∇c|∂t+∂(uj|∇c|)∂xj=(δij-ninj)∂ui∂xj|∇c|+Sd∂ni∂xi-∂(Sdni|∇c|)∂xi,
where ni=-(∂c/∂xi)/|∇c| is the ith component of flame normal vector and Sd=(w˙+∇·(ρD∇c))/ρ|∇c| is the local flame displacement speed. It is evident from (1a) and (1b) and (2) that the statistical behaviour of Nc transport is likely to be different from |∇c| transport although the quantities Nc and |∇c| are closely related to each other (i.e., Nc=D|∇c|2).It is often necessary to solve a transport equation forN~c in the context of Reynolds averaged Navier-Stokes (RANS) simulations and LES [17–30]. The transport equation for N~c can be obtained by Reynolds averaging or LES filtering of (1a) and (1b) as
(3)∂(ρ-N~c)∂t+∂(ρ-u~jN~c)∂xj=∂∂xj(ρD∂Nc∂xj)¯-∂[ρuiNc¯-ρ-u~iN~c]∂xi+T¯1+T¯2+T¯3-D¯2+f(D)¯.The termsT¯1, T¯2, T¯3,(-D¯2), and f(D)¯ are unclosed and therefore it is important to understand the statistical behaviours of Nc, T1, T2, T3, (-D2), and f(D) (since lim⁡Δ→0N~c=Nc, lim⁡Δ→0T¯1=T1, lim⁡Δ→0T¯2=T2, lim⁡Δ→0T¯3=T3, lim⁡Δ→0(-D¯2)=(-D2), and lim⁡Δ→0f(D)¯=f(D), where Δ is the LES filter width) and their local strain rate and curvature dependences in order to model these quantities in the context of LES, where the local strain rate and curvature dependences of these terms need to be adequately captured. The local strain rate and curvature dependences of Nc and the terms of its transport equation (i.e., T1, T2, T3, and (-D2)) are yet to be analysed in detail in the existing literature. This paper aims to address this gap by analysing local tangential strain rate aT=(δij-ninj)∂ui/∂xj and curvature κm=0.5(∂ni/∂xi) (for the above definition of κm, the flame elements convex towards the reactants has a positive curvature) dependences of Nc, T1, T2, T3, (-D2), and f(D) at different locations within the flame using direct numerical simulations (DNS) data of turbulent premixed freely propagating statistically planar flame and turbulent V-flame configurations. In this respect, the main objectives of this study are as follows:(1)
to analyse local statistical behaviours of instantaneous SDR (i.e.,Nc) and the terms of its transport equation T1, T2, T3, (-D2), and f(D);(2)
to explain the observed strain rate and curvature dependences ofNc, T1, T2, T3, (-D2), and f(D);(3)
to compare the statistical behaviours of instantaneous SDR and the terms of its transport equation obtained from DNS in a canonical configuration with constant thermophysical properties with DNS of a laboratory configuration (e.g., turbulent V-flame configuration) with temperature-dependent thermophysical properties.The rest of the paper will be organised as follows. The necessary mathematical modelling and the information related to the numerical implementation of DNS simulations will be presented in the next section. This will be followed by the presentation of the results and the subsequent discussion. The main findings will be summarised and conclusions will be drawn in the final section of this paper.


## 2. Mathematical Background and Numerical Implementation

DNS simulations of turbulent reacting flows should address both the three-dimensionality of turbulence and detailed chemical structure of the flames. However, limitation of computer hardware until recently restricted DNS of turbulent reacting flows either to two dimensions with detailed chemistry or to three dimensions with simplified chemistry. Although it is now possible to carry out three-dimensional DNS simulations with detailed chemistry, they remain extremely expensive [31] and are often not suitable for a detailed parametric analysis especially for simulations in relatively complex configurations (e.g., V-flame). Here, three-dimensional simulations with single step Arrhenius type chemistry have been considered for an extensive parametric analysis. The parametric analysis based on freely propagating statistically planar flames in a canonical configuration has been carried out using a well-proven compressible DNS code SENGA [32]. In the context of simple chemistry, the species field is uniquely represented by a reaction progress variable c, which can be defined in terms of a suitable reactant (product) mass fraction YR(YP) as c=(YR0-YR)/(YR0-YR∞)(c=(YP-YP0)/(YP∞-YP0), where the subscripts 0 and ∞ are used to denote the values in unburned reactants and fully burned products, respectively. For the simulations of freely propagating statistically planar flames (i.e., cases P1–P5, where “P” denotes the statistically planar flames), a rectangular domain of size 36.1δth×24.1δth×24δth is considered, where δth=(Tad-T0)/Max⁡|∇T^|L is the thermal flame thickness with Tad, T0, and T^ being the adiabatic flame, unburned reactant, and instantaneous dimensional temperatures, respectively, and the subscript “L” refers to the unstrained laminar flame quantities. For the thermochemistry used in cases P1–P5, the thermal flame thickness δth is found to be 1.785D0/SL (i.e., δth=1.785D0/SL), where D0 is the mass diffusivity in the unburned gas.The simulation domain for cases P1–P5 is discretised using a uniform Cartesian grid of345×230×230. The largest side of the domain is taken to align with the mean direction of flame propagation and the boundaries in that direction are taken to be partially nonreflecting. The partially nonreflecting boundary conditions are specified using the Navier-Stokes characteristic boundary conditions (NSCBC) technique [33]. The transverse directions are taken to be periodic and thus do not need any separate boundary conditions. A 10th order central-difference scheme is used to evaluate spatial derivatives at the internal grid points but the order of differentiation gradually drops to a one-sided 4th order scheme near nonperiodic boundaries. The time-advancement is carried out using a 3rd order low storage Runge-Kutta scheme [34]. One does not obtain any spurious fluctuations due to the 10th order central difference scheme and its transition to the lower-order finite difference scheme for sufficiently small grid spacing (e.g., Δx≤η, where Δx and η are the grid spacing and the Kolmogorov length scale, respectively). Thus it was not necessary to use numerical filter to eliminate spurious oscillations. The flames in cases P1–P5 remain sufficiently away from the domain boundaries whereas the major part of the reactive region in cases V1–V3 does not interact with the nonperiodic boundaries except for flame crossing the outlet boundary. For the present analysis, the regions of flame crossing nonperiodic boundary are not considered for extracting SDR statistics in cases V1–V3. Thus, the evaluation of SDR and the terms of its transport equation at a given point of time is nominally 10th order accurate in this analysis. It is worth noting that similar numerical schemes for spatial discretisation and time advancement were used in several previous studies [4–17, 22–32].The initial values of root-mean-square turbulent velocity fluctuation normalised by unstrained laminar burning velocityu′/SL, integral length scale to flame thickness ratio l/δth, turbulent Reynolds number Ret=ρ0u′l/μ0, Damköhler number Da=lSL/u′δth and Karlovitz number Ka=(u′/SL)3/2(l/δth)-1/2, heat release parameter τ=(Tad-T0)/T0, and Zel’dovich number β=Tac(Tad-T0)/Tad2 for cases P1–P5 are provided in Table 1, where ρ0 and μ0 are the unburned gas density and viscosity, respectively, and Tac is the activation temperature. As Ret scales as Ret~Da2Ka2 [35], the change in turbulent Reynolds number in cases P1–P5 is brought about by modifying Da and Ka independently of each other (e.g., Da (Ka) is kept unaltered in cases P1, P3, and P5 (P2, P3, and P4)). In cases P1–P5, the flame-turbulence interaction takes place under decaying turbulence, which necessitates a simulation time tsim≥Max⁡(tf,tc), where tf=l/u′ is the initial eddy turn over time and tc=δth/SL is the chemical time scale. In all cases, statistics were extracted after one chemical time scale tc, which corresponds to a time equal to 2.0tf in case P4, 3.0tf in cases P1, P3, and P5, and 4.34tf for case P2. It is worth noting that the chemical time scale tc remains the same for all cases due to identical thermochemistry. The present simulation time is comparable to the simulation times used for several previous DNS studies [5–9, 12, 36–39]. 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 P1–P5, respectively. By contrast, the integral length scale increased by factors between 1.5 and 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 elsewhere [39] and thus are not repeated here. For cases P1–P5, the thermal flame thickness δth is greater than the Kolmogorov length scale η at the time of the analysis, and this suggests that combustion in these cases takes place in the thin reaction zones regime [35]. The temporal evolutions of turbulent kinetic energy evaluated over the whole domain and the global burning rate were shown in [39], which demonstrate that these quantities were not varying rapidly with time when the statistics were extracted. It was also shown in [39] that the flame propagation statistics remain unchanged halfway through the simulation.Table 1
Initial values of simulation parameters and nondimensional numbers relevant to the DNS database considered here.
Case
Domain size/δth3
Grid size
u
′
/
S
L
l
/
δ
th
τ
Re
t
Da
Ka
β
P1
36.1
×
24.1
×
24.1
345
×
230
×
230
5.0
1.67
4.5
22.0
0.33
8.67
6.0
P2
36.1
×
24.1
×
24.1
345
×
230
×
230
6.25
1.44
4.5
23.5
0.23
13.0
6.0
P3
36.1
×
24.1
×
24.1
345
×
230
×
230
7.5
2.50
4.5
48.0
0.33
13.0
6.0
P4
36.1
×
24.1
×
24.1
345
×
230
×
230
9.0
4.31
4.5
100
0.48
13.0
6.0
P5
36.1
×
24.1
×
24.1
345
×
230
×
230
11.25
3.75
4.5
110
0.33
19.5
6.0
V1
29.7
×
29.7
×
29.7
512
×
512
×
512
1.0
3.57
2.52
18
3.59
0.529
7.1
V2
29.7
×
29.7
×
29.7
512
×
512
×
512
2.0
3.62
2.52
37
1.81
1.487
7.1
V3
29.7
×
29.7
×
29.7
512
×
512
×
512
6.0
3.43
2.52
92
0.57
7.936
7.1The V-flame cases (i.e., cases V1, V2, and V3, where “V” denotes V-shape flames here) are simulated using an updated version of SENGA and SENGA2 [40, 41] with ability to handle complex chemistry. However, the V-flames were simulated using a single step chemistry to keep the comparison with statistically planar flames consistent. All the nonperiodic boundaries are specified using the NSCBC technique [33]. Nonreflecting outflows, modified to accommodate the presence of flame on the boundary, were applied to the transverse and downstream faces [40, 41]. Inlet turbulence was taken from a precomputed simulation of fully developed homogeneous isotropic turbulence, and the velocity components were interpolated onto the inlet using a high-order scheme to ensure that the structure of the turbulence was preserved. The computational domain in cases V1–V3 is taken to be cubic with sides equal to L=29.7δth, where δth=3.563D0/SL for the thermochemistry used in these cases. A Cartesian grid of 512×512×512 with uniform grid spacing is used. The numerical schemes used for spatial discretisation and time-integration in cases V1–V3 are similar to those used for cases P1–P5. The flame holder centre is located at x1=3.48δth and has an approximate radius R=1.2δth. At the flame holder, the reaction progress variable and mean velocity distributions were imposed using a Gaussian function. It is worth noting that formation of boundary layer around the flame holder and its effect on the flow and flame dynamics are not represented in the simulation due to prohibitive computational cost. However, the possible influence of these effects on the results reported in this study is minimised by carefully selecting the region for the analysis. In the selected regions, the statistical distributions of strain and curvature experienced by flame elements are similar to those for freely propagating statistically planar flames under comparable local conditions [40, 41]. The values of turbulent Reynolds number Ret,inlet=ρ0uinlet′l/μ0, Karlovitz number Ka=(uinlet′/SL)3/2(l/δth)-1/2, and Damköhler number Da=lSL/uinlet′δth based on the root-mean-square turbulent velocity fluctuation uinlet′ at the inlet are provided in Table 1 along with the values of τ and β.To ensure that initial transients had decayed and a stationary state had been reached, the simulation was carried out for a period of one flow-through timeτFT=L/U-in before data were collected for analysis, where U-in is the mean inlet velocity. In the V-flame configuration, the flame is continuously developing downstream from the flame holder, and so the present analysis is restricted to a region spanning 14.9δth≤x1≤29.1δth in the streamwise direction, thus ensuring sufficient time for the flame to develop following ignition. For the purpose of ensuring adequate convergence of the statistics, four snapshots from the simulation were used to obtain SDR statistics presented in the next section, which are taken at an interval of 0.2τFT  after the initial flow-through time. Standard values have been taken for Prandtl number (Pr=0.7) and ratio of specific heats, γ=1.4. The global Lewis number is taken to be unity for all cases considered in this analysis.The grid spacingΔx for all cases ensures 10 grid points within δth. As Karlovitz number can be scaled as Ka~δth2/η2, the grid spacing Δx can be taken to be Δx≤δth/10~ηKa/10. This indicates that η/Δx~10/Ka assumes the smallest value in case P5 amongst the cases considered here as the value Ka is the highest in case P5. For case P5, Δx remained Δx~η/2 throughout the duration of the simulation. For other cases, the Kolmogorov scale is resolved by more than two grid points due to smaller value of Ka than in case P5. The above discussion suggests that the grid size chosen for the cases considered here is sufficient to resolve turbulence structures.The thermophysical properties such as thermal conductivity (λ), dynamic viscosity (μ), and density-weighted mass diffusivity (ρD) are taken to be constant and independent of temperature in cases P1–P5, whereas these quantities in cases V1–V3 are taken to be temperature dependent and the temperature dependence approximated by 5th order polynomials following the CHEMKIN formats [40, 41]. It is worth noting that the cases P1–P5 and cases V1–V3 were originally developed independently (see [39] for cases P1–P5 and [40, 41] for cases V1–V3), but here these cases are considered together to assess if the SDR statistics obtained from DNS data with constant thermophysical properties in a canonical configuration remain qualitatively valid in a laboratory-scale configuration (e.g., V-flame configuration) with temperature-dependent thermophysical properties.


## 3. Results and Discussion

### 3.1. Flame-Turbulence Interaction

The contours ofc in the central x1-x2 plane for cases P1–P5 and V1–V3 are shown in Figures 1(a)–1(h), respectively. It is evident from Figures 1(a)–1(e) that the level of wrinkling increases with increasing u′/SL~Ret1/4Ka1/2~Ret1/2/Da1/2. Turbulent eddies penetrate into the preheat zone in the thin reaction zones regime combustion (Ka~δth2/η2~Ret1/2/Da>1), but the reaction zone remains unperturbed because the Kolmogorov length scale is larger than the reaction zone thickness. The isosurfaces of c representing the preheat zone (i.e., c≤0.5) show more distortion than the isosurfaces representing the reaction zones (i.e., 0.7≤c≤0.9) due to penetration of turbulent eddies within the preheat zone. However, this tendency is more prevalent for high values of u′/SL and Ka (e.g., cases P3, P4, P5, and V3) but the isosurfaces of c remain mostly parallel to each other for small values of u′/SL and Ka (e.g., cases P1, P2, V1, and V2) indicating that the internal flame structure is weakly affected by turbulence in these cases.Distributions ofc in the central x1-x2 plane for cases P1–P5 (a–e) at tc=δth/SL and for cases V1–V3 (f–h).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)


### 3.2. Statistical Behaviour of the Mean Values ofNc and the Unclosed Terms of Its Transport Equation

The variations of〈Nc〉×δth/SL with c for statistically planar flames and V-flames are shown in Figures 2(a) and 2(b), where 〈Q〉 indicates the mean value of Q, which is obtained by ensemble averaging the quantity in question on a given c isosurface in the manner previously used by Boger et al. [42], Chakraborty and Cant [10, 11], and Chakraborty and Klein [15, 16]. It is worth noting that 〈Q〉 should not be confused with either Reynolds averaging or conventional conditional averaging operation in the context of RANS simulations because 〈Q〉 is evaluated using all the samples for a given c value over the whole domain. Figures 2(a) and 2(b) show that the variations of 〈Nc〉×δth/SL for statistically planar and V-flames are qualitatively similar to each other. For both statistically planar and V-flame configurations, the location of the maximum value of 〈Nc〉×δth/SL is skewed slightly towards the burned gas side of the flame (i.e., c≈0.7). The peak magnitude of 〈Nc〉×δth/SL does not change significantly in response to u′/SL as the standard deviation for the case in the middle of the parameter range (i.e., cases P3 and V2) is found to exceed the difference in 〈Nc〉×δth/SL values for the cases considered here for both statistically planar and V-flame configurations. In order to understand the distribution of 〈Nc〉×δth/SL across the flame front, the variations of the mean values of the terms 〈T1〉,〈T2〉,〈T3〉, 〈(-D2)〉, and 〈f(D)〉 conditional on c for planar and V-flames are shown in Figure 3. The variations of the mean values of the terms in cases P2, P3, and P4 (cases V2) are qualitatively similar to those in cases P1 and P5 (case V1) and thus are not explicitly shown here. It is evident from Figure 3 that the qualitative behaviour of these terms remains similar for all cases considered here. In all cases, 〈T1〉 remains positive throughout the flame. By contrast, 〈(-D2)〉 assumes negative values throughout the flame in all cases as dictated by (1a). Expressing ρ=ρ0/(1+τc) for low Mach number, unity Lewis number flames give rise to an alternative expression for T1 [3, 17, 25, 28, 29]:
(4)T1=2ρ∂uj∂xjNc.Variation of the mean value of〈Nc〉×δth/SL conditional on c values across the flame front for (a) statistically planar cases with the bar indicating the standard deviation for case P3 and (b) V-flame cases with the bar indicating the standard deviation for case V2.
(a)
(b)Figure 3
Variation of the mean values of〈T1〉,〈T2〉,〈T3〉,〈-D2〉, and 〈f(D)〉 conditional on c values across the flame for cases P1, P5, V1, and V3. All the terms of the transport equation of Nc are normalised with respect to the respective values of ρ0SL2/δth2.As dilatation rate∂uj/∂xj is predominantly positive in premixed flames, 〈T1〉 for all values of c is positive across the flame and vanishes on both ends of the flame.The quantity〈T2〉 assumes negative values throughout the flame front for cases P1 and V1. Although 〈T2〉 remains negative for the major portion of the flame, small positive values can be discerned in cases P5 and V3. In order to understand this behaviour, the term T2 can be expressed in the following manner [3, 22–24, 28, 30]:
(5)T2=-2ρNc(eαcos⁡2α+eβcos⁡2β+eγcos⁡2γ),
where eα,eβ, and eγ are the most extensive, intermediate, and most compressive principal strain rates and α,β, and γ are the angles of these principal strain rates with ∇c. Equation (5) demonstrates that the predominant alignment of eα (eγ) with ∇c leads to a negative (positive) contribution to T2.It has been discussed in the previous analyses [23, 24, 28, 30] that the alignment of ∇c with eα and eγ is determined by relative strengths of the strain rate induced by flame normal acceleration achem and turbulent straining aturb. It has been demonstrated earlier that ∇c preferentially aligns with eα(eγ) when achem(aturb) dominates over aturb(achem). The strain rate induced by flame normal acceleration due to chemical heat release can be scaled as achem~τf(Ka)SL/δth, where f(Ka) is expected to decrease with increasing Ka [43]. Following Meneveau and Poinsot [44], aturb can be scaled as aturb~u′/l, which gives rise to achem/aturb~τf(Ka)SLl/u′δth~τf(Ka)Da~τf(Ret1/2/Da)Da. Alternatively, turbulent straining can be scaled as [45] aturb~u′/λ (where λ is the Taylor microscale), which yields achem/aturb~τf(Ka)SLλ/u′δth~τf(Ka)Da/Ret1/2~τf(Ret1/2/Da)Da/Ret1/2~τf(Ka)/Ka. The above scaling relations suggest that achem strengthens with respect to aturb with increasing Da for a given value of Ret. Previous analyses [22–24, 28, 30] demonstrated that ∇c predominantly aligns with eα for Da≫1 flames, whereas ∇c aligns with eγ in Da<1 flames for comparable values of Ret. Both achem/aturb~τf(Ka)Da and achem/aturb~τf(Ret1/2/Da)Da/Ret1/2~τf(Ka)/Ka indicate that an increase in Ka~Ret1/2/Da for a given value of Da (e.g., cases P1, P3, and P5) gives rise to weakening of achem in comparison to aturb. This increases the extent of ∇c alignment with eγ with increasing Ka when Da is held constant as in cases P1, P3, and P5. In cases P1 and P3, ∇c predominantly aligns with eα; however the extent of this alignment decreases from P1 to P3. This predominant alignment of ∇c with eα in cases P1 and P3 leads to a negative contribution of 〈T2〉 in these cases. In case P5, ∇c predominantly aligns with eγ in the unburned and fully burned gases but achem overcomes aturb in the regions of intense heat release close to the middle of the flame and as a result ∇c aligns with eα in the reaction zone. Thus the mean value of 〈T2〉 in case P5 assumes positive values towards both the unburned and burned gas sides, whereas the mean contribution of 〈T2〉 remains negative close to the middle of the flame. The relation achem/aturb~τf(Ka)Da/Ret1/2 indicates that achem weakens in comparison to aturb with decreasing τDa/Ret1/2. The quantity τDa/Ret1/2 assumes values equal to 0.96, 0.55, and 0.49 for cases P2, P3, and P4, respectively, when the statistics were extracted. This leads to larger extent of ∇c aligning with eγ in case P4 (case P3) than in case P3 (case P2). This leads to predominantly negative contribution of 〈T2〉 in cases P2 and P3, whereas 〈T2〉 assumes positive values towards the unburned and burned gas sides of the flame in case P4. However, achem overcomes aturb in the regions of intense heat release at the middle of the flame and ∇c starts to align with eα in the reaction zone giving rise to negative values of 〈T2〉 in case P4. In cases V1 and V2, the values of achem/aturb~τf(Ka)Da/Ret1/2 are larger than the corresponding value in case V3 (see the parameters in Table 1). Thus, the extent of ∇c alignment with eα(eγ) decreases (increases) from case V1 to case V3. This gives rise to positive values of 〈T2〉 towards both unburned and burned gas sides of the flame in case V3. This tendency is less prevalent in cases V1 and V2 due to smaller extent of ∇c alignment with eγ than in case V3. However, the mean contribution of 〈T2〉 is negative in the middle of the flame for cases V1–V3 due to the alignment of ∇c with eα in the heat releasing zone.The contribution of〈T3〉 remains positive (negative) towards the unburned (burned) gas side of the flame with the transition from positive to negative value taking place close to c≈0.85. In order to explain this behaviour, T3 can be rewritten as
(6)T3=-2Dni∂w˙∂xi|∇c|=-2D∂w˙∂n|∇c|,
where n is the spatial coordinate in the local flame normal direction and the flame normal vector n→ points towards the unburned gas side of the flame. For single step chemistry considered here, the maximum w˙ occurs close to c≈0.85 [10, 14]. This suggests that the probability of finding negative (positive) values of ∂w˙/∂n is significant for c<0.85 (c>0.85), which gives rise to positive (negative) value of 〈T3〉 towards the unburned (burned) gas side of the flame.Figure3 shows that 〈f(D)〉 is weakly negative towards the unburned gas side before becoming positive towards the burned gas side in all the cases. The magnitude of the mean contribution of 〈f(D)〉 remains comparable to that of 〈T1〉 in all cases indicating that 〈f(D)〉 cannot be neglected even for cases P1–P5, where ρD is considered to be constant. In cases P1–P5, ρNc[∂D/∂t+uj∂D/∂xj] can be expressed using ρ=ρ0/(1+τc) for globally adiabatic Le=1.0 flames as T1/2=ρNc∂uj/∂xj (i.e., ρNc[∂D/∂t+uj∂D/∂xj]=ρNc(∂uj/∂xj) for constant ρD) and the first two terms on the right hand side of (1b) vanish for constant values of ρD. The contributions of 〈(TD3+TD4)〉 are responsible for the change in sign of 〈f(D)〉 in cases P1–P5. These terms are also principally responsible for sign change of 〈f(D)〉 in cases V1–V3.


### 3.3. Local Behaviour ofNc and Its Curvature and Strain Rate Dependences

The marginal probability density functions (pdfs) of normalisedNc+ (i.e., Nc×δth/SL) for different c isosurfaces across the flame are shown in Figures 4(a) and 4(b) in log-log scale for cases P3 and V2, respectively. The pdfs of Nc in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not explicitly shown here. The pdfs for c<0.5 are not shown in Figures 4(a) and 4(b), as Nc assumes small values in the preheat zone of the flame due to small magnitude of scalar gradient ∇c. It is evident from Figures 4(a) and 4(b) that the pdfs of Nc are qualitatively similar for statistically planar and V-flames and in both cases the probability of finding high values of Nc is most prevalent in the middle of the flame with slight skewness towards the burned gas side (i.e., c≈0.7) and the probability of finding high values of Nc decreases on both unburned and burned gas sides of the flame front. This is consistent with the observed behaviour of the mean values of Nc conditional on c shown in Figure 2. It can be seen in Figure 4 that a log-normal distribution captures the qualitative behaviour of the pdf of Nc although there are some disagreements in the pdf tails. This is consistent with several previous experimental [46–52] and numerical [53–55] studies investigating the scalar dissipation rate pdf of a passive scalar. An approximate log-normal distribution of SDR in turbulent premixed flames has also been reported in a previous analysis [56].The marginal pdf of normalisedNc+ (i.e., Nc×δth/SL) and the log-normal distribution in log-log scale for c=0.5, 0.7, and 0.9 across the flame for cases (a) P3 and (b) V2.
(a)
(b)The joint pdfs ofNc and tangential strain rate aT for cases P1, P5, V1, and V3 are shown in Figure 5(a) for c=0.8 isosurface, which is close to the most reactive region for the present thermochemistry. It can be seen from Figure 5(a) that Nc and aT are positively correlated on c=0.8 isosurface for cases P1, P5, V1, and V3 and similar qualitative behaviour has been observed also for other c isosurfaces in all cases considered here. This positive correlation between Nc and aT can be explained in the following manner.(i)
The dilatation rate∇·u→ can be expressed as ∇·u→=aT+an, where an=ninj∂ui/∂xj is the normal strain rate. For unity Lewis number flames, ∇·u→ can be scaled as ∇·u→~achem~τf(Ka)SL/δth, whereas aT can be taken to scale with turbulent strain rate aturb (i.e., aT~aturb~u′/l according to Meneveau and Poinsot [44] and aT~aturb~u′/λ according to Tennekes and Lumley [45]).(ii)
Above scalings indicate that∇·u→/aT scales as ∇·u→/aT~τf(Ret1/2/Da)Da and ∇·u→/aT~τf(Ret1/2/Da)Da/Ret1/2~τf(Ka)/Ka according to the scaling arguments by Meneveau and Poinsot [44] and Tennekes and Lumley [45], respectively. Both ∇·u→/aT~τf(Ret1/2/Da)Da and ∇·u→/aT~τf(Ret1/2/Da)Da/Ret1/2~τf(Ka)/Ka suggest that the magnitude of aT is likely to supersede the magnitude of ∇·u→ in most locations within the flame for small values of Da and high values of Ka.(iii)
It has been shown in several previous analyses [10, 30] that both ∇·u→ and aT assume predominantly positive values and thus a higher magnitude of aT than ∇·u→ induces a negative (i.e., compressive) normal strain rate an. Thus, an increase in aT often leads to a decrease in an=∇·u→-aT for small (high) values of Da (Ka). Thus, the isoscalar lines come close to each other under the action of decreasing an, which leads to increase in the magnitude of scalar gradient ∇c. This is reflected in the positive correlation between Nc and aT.(a) Joint pdfs betweenNc×δth/SL and normalised tangential strain rate aT×δth/SL on c=0.8 isosurface for cases P1, P5, V1, and V3. (b) Joint pdf between Nc×δth/SL and normalised curvature κm×δth on c=0.8 isosurface for cases P1, P5, V1, and V3.
(a)
(b)The joint pdfs betweenNc and curvature κm for cases P1, P5, V1, and V3 are shown in Figure 5(b) for c=0.8 isosurface. Cases P2 and P3 (case V2) are not explicitly shown here due to their similarities to cases P1 and P5 (case V1), respectively. It can be seen from Figure 5(b) that the joint pdf between Nc and κm exhibits both positive and negative correlating branches on c=0.8 isosurface for cases P5 and V3, and as a result of this, the net correlation between Nc and κm remains weak. The positive correlation branch between Nc and κm remains weak for small values of u′/SL in statistically planar flames (see Figure 5(b) for case P1) and this branch disappears completely in the V-flames with small values of u′/SL (see Figure 5(b) for case V1). Similar behaviour is observed for other c isosurfaces in all cases considered here and the correlation between Nc and κm is weak throughout the flame for high values of u′/SL (e.g., cases P3–P5 and V3). However, the disappearance of the positive correlating branch in the joint pdf of Nc and κm in Figure 5(b) indicates that Nc and κm are negatively correlated with each other throughout the flame for small values of u′/SL (e.g., cases P1, P2, V1, and V2). The observed behaviour can be explained based on the following physical mechanisms.(i)
Previous analyses (e.g., [57]) demonstrated that both aT and ∇·u→ remain negatively correlated with κm in turbulent premixed flames, and thus the behaviour of an at locations with large positive curvature is principally determined by aT since ∇·u→ is small in these zones due to defocusing of heat. Small values of aT are associated with high values of κm at these locations, which lead to small values of Nc at high values of positive κm due to positive correlation between Nc and aT. This leads to a negative correlating branch between Nc and κm at the positively curved zones.(ii)
The dilatation rate∇·u→ is large in the negatively curved locations due to strong focussing of heat and the magnitude of ∇·u→ can locally be high enough to supersede the magnitude of aT, which leads to a positive value of an. This tendency strengthens with decreasing κm, especially in the zones with large negative curvature, which gives rise to an increase in an with decreasing curvature. As the distance between the isoscalar lines increases with increasing an, the magnitude of scalar gradient ∇c decreases with decreasing κm in the negatively curved zones. This leads to the positive correlating branch in the joint pdf of Nc and κm (see Figure 5(b) for cases P5 and V3).(iii)
The relative strengths of the positive and negative correlating branches ultimately determine the net correlation betweenNc and κm in the high u′/SL cases. The probability of finding high negative curvature remains small for small values of u′/SL and as a result the probability of finding high values of ∇·u→, which locally overcomes aT, to induce a positive value of an, becomes rare (e.g., cases P1 and V1). Thus the combination of positive correlations between Nc and aT and negative correlations between aT and κm leads to a predominantly negative correlating branch between Nc and κm in the low u′/SL cases (e.g., cases P1 and V1; see Figure 5(b)).The strain rate and curvature dependences ofNc discussed above, in turn, affect the local statistical behaviours of T1, T2, T3, (-D2), and f(D) in response to aT and κm. The curvature and strain rate dependences of T1, T2, T3, (-D2), and f(D) are discussed next.


### 3.4. Local Behaviour ofT1 and Its Curvature and Strain Rate Dependences

The marginal pdfs ofT1 for different c isosurfaces across the flame are shown in Figures 6(a) and 6(b) for cases P3 and V2, respectively. The pdfs of T1 in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not explicitly shown here. It is evident from Figures 6(a) and 6(b) that the pdfs of T1 are qualitatively similar for statistically planar and V-flames and in both cases T1=2ρ(∇·u→)Nc assumes predominantly positive values throughout the flame. As dilatation rate ∇·u→ is principally positive due to thermal expansion in premixed flames [10, 30], the contribution of T1=2ρ(∇·u→)Nc is predominantly positive throughout the flame. Moreover, Figures 6(a) and 6(b) demonstrate that the probability of finding high values of T1 is most prevalent in the middle of the flame with slight skewness towards the burned gas side (i.e., c≈0.7) and the probability of finding high values of T1 decreases on both unburned and burned gas sides of the flame. This is consistent with the observed behaviour of the mean values of T1 conditional on c shown in Figure 3. The probability of finding large magnitudes of ∇·u→ is the highest at a location which is slightly skewed towards the burned gas side of the flame [30]. As the distributions of Nc and ∇·u→ are slightly skewed towards the burned gas side of the flame, the probability of finding large values of T1=2ρ(∇·u→)Nc becomes high around c≈0.7.The marginal pdfs ofT1×δth2/ρ0SL2 for c=0.1, 0.3, 0.5, 0.7, and 0.9 for cases (a) P3 and (b) V2. Joint pdf between T1×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL on c=0.8 isosurface for cases (c) P3 and (d) V2. Joint pdf between T1×δth2/ρ0SL2 and normalised curvature κm×δth on c=0.8 isosurface for cases (e) P3 and (f) V2.
(a)
(b)
(c)
(d)
(e)
(f)The joint pdfs betweenT1 and aT for cases P3 and V2 are shown in Figures 6(c) and 6(d), respectively, for c=0.8 isosurface. It can be seen from Figures 6(c) and 6(d) that T1 and aT are positively correlated on c=0.8 isosurface for cases P3 and V2 and similar qualitative behaviours have been observed for other c isosurfaces in all cases considered here. Both ∇·u→ and aT are positively correlated for all flames considered here, which along with positive correlation between Nc and aT (see Figure 5) gives rise to a positive correlation between T1=2ρ(∇·u→)Nc and aT.The joint pdfs betweenT1 and κm for cases P3 and V2 are shown in Figures 6(e) and 6(f), respectively, for c=0.8 isosurface. It can be seen from Figures 6(e) and 6(f) that the joint pdf between T1 and κm exhibits a negative correlation on c=0.8 isosurface for cases P3 and V2, and similar qualitative behaviour has been observed for other c isosurfaces in all cases considered here. In all cases, the net correlation between Nc and κm is weak (see Figure 5(b)), but ∇·u→ assumes high (small) values at negatively (positively) curved locations because of focussing (defocusing) of heat. This leads to a predominantly negative correlation between ∇·u→ and κm [57]. The negative correlation between ∇·u→ and κm is principally responsible for the negative correlation between T1=2ρ(∇·u→)Nc and κm.


### 3.5. Local Behaviour ofT2 and Its Curvature and Strain Rate Dependences

The marginal pdfs ofT2 for different c isosurfaces across the flame are shown in Figures 7(a) and 7(b) for cases P3 and V2, respectively. The pdfs of T2 in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not explicitly shown here. Figures 7(a) and 7(b) show that the probability of finding negative values of T2 supersedes the probability of finding positive values. The probability of finding negative values of T2 increases as the heat releasing zone (see the pdfs for c=0.7 isosurface) is approached. It has been discussed earlier that the effects of achem overcome the effects of aturb in the heat releasing zone to give rise to a preferential alignment of ∇c with eα even for small values of Da. This preferential alignment of ∇c with eα in these zones gives rise to negative values of T2 according to (5). The extent of ∇c alignment with eα(eγ) decreases (increases) towards both unburned and burned gas sides of the flame due to diminishing effects of achem.The marginal pdfs ofT2×δth2/ρ0SL2 for c=0.1, 0.3, 0.5, 0.7, and 0.9 for cases (a) P3 and (b) V2. Joint pdf between T2×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL on c=0.8 isosurface for cases (c) P3 and (d) V2. Joint pdf between T2×δth2/ρ0SL2 and normalised curvature κm×δth on c=0.8 isosurface for cases (e) P3 and (f) V2.
(a)
(b)
(c)
(d)
(e)
(f)The contours of joint pdfs betweenT2 and aT for c=0.8 are shown in Figures 7(c) and 7(d) for cases P3 and V2 and the correlation coefficients between T2 and aT for different c isosurfaces across the flame for all cases are shown in Table 2. It is evident from Figures 7(c) and 7(d) and Table 2 that T2 and aT are positively correlated for high u′/SL cases (e.g., cases P5 and V3) although the strength of the correlation changes through the flame. However, T2 and aT are weakly correlated with each other within the flame, where the effects of heat release are significant for cases with small and moderate values of u′/SL (see Table 2). In order to explain this behaviour, it is useful to rewrite T2 in the following manner:
(7)T2=-2ρanNc.Table 2
Correlation coefficients betweenT2 and aT and between T2 and κm on c= 0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces.
Case
T
2
-
a
T
T
2
-
κ
m
c
=
0.1
c
=
0.3
c
=
0.5
c
=
0.7
c
=
0.9
c
=
0.1
c
=
0.3
c
=
0.5
c
=
0.7
c
=
0.9
P1
0.642
−0.098
−0.217
−0.092
0.685
0.141
0.509
0.720
0.614
−0.250
P2
0.673
−0.090
−0.208
−0.084
0.676
0.116
0.506
0.719
0.616
−0.227
P3
0.751
0.376
0.263
0.263
0.648
0.544
0.252
0.412
0.423
−0.065
P4
0.802
0.593
0.616
0.689
0.827
0.052
0.196
0.235
0.198
−0.027
P5
0.783
0.662
0.614
0.616
0.787
0.028
0.137
0.223
0.242
−0.014
V1
0.245
0.367
0.749
0.816
0.971
0.612
0.623
0.354
0.135
−0.043
V2
0.591
0.335
0.495
0.699
0.920
0.345
0.565
0.497
0.328
−0.067
V3
0.767
0.721
0.738
0.768
0.888
−0.009
0.057
0.077
0.028
−0.155Based on (7) the strain rate dependences of T2 can be explained in the following manner.(i)
It has already been demonstrated thatNc and aT are positively correlated with each other (see Figure 5(a)). The quantity (-an)=aT-∇·u→ tends to increase with increasing aT in the regions where the effects of ∇·u→ are weak. This along with positive correlation between Nc and aT leads to a positive correlation between T2 and aT for both unburned and burned gas sides of the flame for all cases.(ii)
The magnitudes of∇·u→ and aT increase with decreasing κm, and thus (-an)=aT-∇·u→ might not increase (even decrease) with increasing aT in the heat releasing zone of the flame where the effects of ∇·u→ are strong. The aT dependences of (-an) and Nc ultimately determine the nature of the correlation between T2 and aT. The strain rate and curvature dependences of ∇·u→ weaken with increasing u′/SL [58], so (-an)=aT-∇·u→ increases with increasing aT, which leads to a positive correlation between T2 and aT for the major portion of the flame for cases with high values of u′/SL (see Table 2).The joint pdfs betweenT2 and κm for cases P3 and V2 are shown in Figures 7(e) and 7(f), respectively, for c=0.8 isosurface and the correlation coefficients between T2 and κm for different c isosurfaces across the flame are shown in Table 2 for all cases considered here. It is evident from Figures 7(e) and 7(f) and Table 2 that T2 and κm remain weakly positively correlated except the burned gas side of the flame. The observed curvature dependence of T2 could be explained based on the following physical mechanisms.(i)
The effects of dilatation rate∇·u→ and thermal expansion are particularly strong in the negatively curved regions due to focussing of heat. By the same token, the effects of heat release are weak in the positively curved zones due to defocusing of heat. Thus, the effects of achem are more likely to dominate over the effects of aturb in the negatively curved zones, which increase the extent of ∇c alignment with eα as demonstrated earlier by Hartung et al. [58]. Weakening of the heat release effects at positively curved zones due to defocusing of heat leads to a greater (lesser) extent of ∇c alignment with eγ(eα) in the positively curved zones. The extent of ∇c alignment with eα increases in the negatively curved zones, which in turn makes T2 increasingly negative (see (5)) and the magnitude of the negative contribution of T2 decreases for positive curvature locations. This gives rise to a positive correlation between T2 and κm, as observed from Figures 7(e) and 7(f) and Table 2.(ii)
However, the effects ofaturb are more likely to dominate over the effects of achem towards the burned gas side and thus the extent of ∇c alignment with eγ is determined by local turbulent flow conditions. The effects of flame-generated turbulence become stronger at the negatively curved zones due to stronger thermal expansion effects resulting from focussing of heat especially in the heat releasing zone. The straining induced by flame-generated turbulence may overcome relatively weak effects of ∇·u→ towards the burned gas side, which can give rise to an increasing extent of ∇c alignment with eγ increases in the negative curved zones. This in turn gives rise to an increase in T2 (see (5)) with decreasing κm towards the burned gas side and leads to a negative correlation between T2 and κm (see Table 2).


### 3.6. Local Behaviour ofT3 and Its Curvature and Strain Rate Dependences

The marginal pdfs of normalisedT3 for different c isosurfaces across the flame are shown in Figure 8 for cases P3 and V2, respectively. The pdfs of T3 in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not explicitly shown here. The pdfs for c<0.5 are not shown in Figure 8 because T3 assumes negligible value in the preheat zone of the flame due to negligible magnitude of w˙. It is evident that T3 assumes positive values for the major portion of the flame for both statistically planar and V-flames and the probability of finding high positive values increases towards the most reactive zone (e.g., c=0.7 in Figure 8) of the flame front. However, T3 assumes negative values only towards the burned gas side (e.g., c=0.9) of the flame front for both planar and V-flames. This is consistent with the behaviour of 〈T3〉 shown in Figure 3. The physical mechanism behind the transition from positive to negative values of the mean contribution of T3 (see (6)) is also responsible for obtaining negative (positive) values of T3 towards the burned (unburned) gas side of the flame.Figure 8
The marginal pdfs ofT3×δth2/ρ0SL2 for c=0.5, 0.7, and 0.9 for cases P3 and V2.The contours of joint pdfs betweenT3 and aT for c=0.5, 0.7, and 0.9 isosurfaces are shown in Figures 9(a)–9(f) for cases P3 and V2 and similar qualitative behaviour has been observed for other cases considered here. It is evident from Figures 9(a)–9(f) that T3 and aT remain positively correlated for the part of the flame where finding positive values of T3 is prevalent. On the other hand, T3 and aT are negatively correlated with each other towards the burned gas side of the flame where T3 is predominantly negative. The observed aT dependence of T3 can be explained in the following manner.(i)
It has been demonstrated earlier thatNc and aT are positively correlated with each other which suggests that |∇c|=|∂c/∂n| increases with increasing aT. For low Mach number, unity Lewis number flames w˙ depend only on c and thus high values of |∂w˙/∂n| are associated with high values of |∇c|=|∂c/∂n| and Nc.(ii)
AsNc and aT are positively correlated with each other, the magnitude of reaction rate contribution |T3|=|2D(∂w˙/∂n)|∇c|| is positively correlated with tangential strain rate aT. Thus, T3 is positively (negative) correlated with aT, where T3 assumes positive (negative) values.Joint pdfs betweenT3×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL on (a) c=0.5, (b) 0.7, and (c) 0.9 isosurfaces for case P3. Joint pdfs between T3×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL on (d) c=0.5, (e) 0.7, and (f) 0.9 isosurfaces for case V2.
(a)
(b)
(c)
(d)
(e)
(f)The joint pdfs betweenT3 and κm for cases P3 and V2 are shown in Figure 10 for c=0.5, 0.7, and 0.9 isosurfaces and similar qualitative behaviour has been observed for other cases considered here. It is evident from Figure 10 that the joint pdf of T3 and κm exhibits both positive and negative correlating branches and the net correlation is weak throughout the flame. The physical explanations for the observed κm dependence of T3 can be summarised in the following manner.(i)
The term|T3|=|2D(∂w˙/∂n)|∇c|| is expected to be positively (negatively) correlated with curvature κm at negatively (positively) curved locations for high values of u′/SL, as in the case of Nc (see cases P5 and V3 in Figure 5(b)), because high values of |∂w˙/∂n| are associated with high values of Nc and |∇c|=|∂c/∂n|.(ii)
As a result of the aforementioned physical mechanisms, the termT3 and κm remain positively (negatively) correlated with curvature κm at negatively (positively) curved locations in the planar flames where T3 assumes positive values. By contrast, the joint pdfs of T3 and κm exhibit negative (positive) correlation with curvature κm at negatively (positively) curved locations within the flame where T3 assumes negative values for the planar flames considered here (see Figure 10(c)). However, Nc remains predominantly negatively correlated with κm for V-flame cases (see Figure 5(b)) and thus T3 shows positive (negative) correlation with curvature where T3 assumes negative (positive) values (see Figures 10(d)–10(f)).Joint pdfs betweenT3×δth2/ρ0SL2 and normalised curvature κm×δth on (a) c=0.5, (b) 0.7, and (c) 0.9 isosurfaces for case P3. Joint pdfs between T3×δth2/ρ0SL2 and normalised curvature κm×δth on (d) c=0.5, (e) 0.7, and (f) 0.9 isosurfaces for case V2.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.7. Local Behaviour of(-D2) and Its Curvature and Strain Rate Dependences

The marginal pdfs of(-D2) for c isosurfaces representative of leading edge, reaction zone, and trailing edge of the flame (e.g., c=0.3, 0.7 and 0.9 isosurfaces) are shown in Figure 11(a) for cases P3 and V2. The pdfs of (-D2) in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to that in case P3 (case V2) and thus are not explicitly shown here. Figure 11(a) shows that (-D2) assumes negative values throughout the flame and the probability of finding high magnitude of (-D2) increases from unburned gas side towards a region of the flame which is severely skewed towards the burned gas side (e.g.,c = 0.9 isosurface). This behaviour is found to be consistent with the mean behaviour of (-D2) shown in Figure 3. It can further be seen from Figure 11(a) that the pdfs of (-D2) for statistically planar and V-flames are qualitatively similar to each other.The marginal pdfs of(-D2)×δth2/ρ0SL2 for c=0.3, 0.7, and 0.9 for cases (a) P3 and V2. Joint pdfs between (-D2)×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL on c=0.8 isosurface for cases (b) P3 and (c) V2.
(a)
(b)
(c)The contours of joint pdfs between(-D2) and aT for c=0.8 isosurface are shown in Figures 11(b) and 11(c) for cases P3 and V2 and the correlation coefficients between (-D2) and aT for different c isosurfaces across the flame for all cases considered here are shown in Table 3. Figures 11(b) and 11(c) and Table 3 show that (-D2) and aT are predominantly negatively correlated throughout the flame but the strength of this negative correlation weakens with increasing u′/SL and the correlation becomes weakly positive at the middle of the flame for high values of u′/SL (e.g., cases P4 and P5). This behaviour can be explained in the following manner.(i)
The instantaneous SDRNc and the molecular dissipation term (-D2) can be taken to scale as Nc~D/δ2 and (-D2)~(-ρD2/δ4)~(-ρNc2) (where δ is the typical local flame thickness) because in premixed flames. the gradient of progress variable is only existent within the flame thickness. Alternatively, (-D2) can be considered to be governed by small-scale eddies and thus the characteristic length scale can be taken to be the Kolmogorov length scale η. Thus one obtains (-D2)~(-ρD2/η4)~(-ρNc2)Ka2 when dissipation processes are taken to be governed by η but δ/η~Ka1/2 remains of the order of unity for all cases considered here (see Table 1) and both scalings of (-D2) (i.e., (-D2)~(-ρD2/δ4)~(-ρNc2) and (-D2)~(-ρD2/η4)~(-ρNc2)Ka2) suggest that high magnitudes of the dissipation term |-D2| are associated with high values of aT due to positive correlation between Nc and aT (see Figure 5(a)).(ii)
As(-D2) assumes negative values, the quantities (-D2) and aT are predominantly negatively correlated throughout the flame due to positive correlation between |-D2| and aT. However, the negative correlation between (-D2) and aT weakens with increasing u′/SL due to weakening of positive correlation between Nc and aT. Thus the correlation between (-D2) and aT becomes weakly positive at the middle of the flame for high values of u′/SL (e.g., cases P4 and P5).Table 3
Correlation coefficients between(-D2) and aT and between (-D2) and κm on c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces.
Case
(
-
D
2
)
-
a
T
(
-
D
2
)
-
κ
m
c
=
0.1
c
=
0.3
c
=
0.5
c
=
0.7
c
=
0.9
c
=
0.1
c
=
0.3
c
=
0.5
c
=
0.7
c
=
0.9
P1
−0.598
−0.809
−0.368
−0.183
−0.522
0.280
0.577
0.582
−0.020
0.261
P2
−0.581
−0.806
−0.389
−0.218
−0.488
0.272
0.554
0.600
0.023
0.225
P3
−0.546
−0.699
−0.028
0.050
−0.296
0.283
0.422
0.364
−0.234
−0.001
P4
−0.513
−0.652
0.221
−0.078
−0.483
0.231
0.338
−0.074
−0.412
−0.100
P5
−0.472
−0.581
0.111
−0.031
−0.378
0.205
0.288
0.174
−0.317
−0.107
V1
−0.712
−0.695
−0.069
−0.616
−0.679
0.700
0.825
−0.176
0.619
0.766
V2
−0.628
−0.525
0.089
−0.445
−0.703
0.475
0.696
0.033
0.159
0.633
V3
−0.452
−0.355
0.130
−0.312
−0.697
0.289
0.481
0.049
−0.104
0.260The joint pdfs of(-D2) and curvature κm for cases P3 and V2 are shown in Figure 12 for c=0.3, 0.7, and 0.9 isosurfaces and the correlation coefficients between (-D2) and κm for different c isosurfaces across the flame are shown in Table 3 for all cases considered here. The joint pdfs of (-D2) and κm in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not shown here. It can be seen from Figure 12 that the quantities (-D2) and κm are nonlinearly related to one another. The physical explanations behind the observed behaviour are provided below.(i)
The molecular dissipation term(-D2) can alternatively be expressed as
(8)(-D2)=-2ρD2{[∂|∇c|∂n]2+4κm|∇c|∂|∇c|∂n+4κm2|∇c|2+2[∂2c∂x1∂x2∂2c∂x1∂x2-∂2c∂x1∂x1∂2c∂x2∂x2]+2[∂2c∂x1∂x3∂2c∂x1∂x3-∂2c∂x1∂x1∂2c∂x3∂x3]+2[∂2c∂x2∂x3∂2c∂x2∂x3-∂2c∂x2∂x2∂2c∂x3∂x3]}.Joint pdfs between(-D2)×δth2/ρ0SL2 and normalised curvature κm×δth on (a) c=0.3, (b) 0.7. and (c) 0.9 isosurfaces for case P3. Joint pdfs between (-D2)×δth2/ρ0SL2 and normalised curvature κm×δth on (d) c=0.3, (e) 0.7, and (f) 0.9 isosurfaces for case V2.
(a)
(b)
(c)
(d)
(e)
(f)The above expression clearly indicates that the third term on the right hand side of (8) (i.e., -8ρD2κm2|∇c|2) induces nonlinear curvature dependence of the molecular dissipation term (-D2).(ii)
The quantity∂|∇c|/∂n remains negative (positive) towards the unburned (burned) gas side of the flame [10, 59, 60]; thus the second term on the right hand side is positively (negatively) correlated with κm towards the unburned (burned) gas side of the flame. The first term on the right hand side of (8) can be taken to scale with (-ρNc2) (i.e., -2ρD2(∂|∇c|/∂n)2~-2ρNc2). It has already been shown that the joint pdfs of Nc and κm exhibit both positive and negative correlating branches for high values of u′/SL (see cases P5 and V3 in Figure 5(b)) and thus the joint pdf of -2ρD2(∂|∇c|/∂n)2 and κm is also expected to show branches with both positive and negative correlations in these cases. The weak negative correlation between Nc and κm for small values of u′/SL (see cases P1 and V1 in Figure 5(b)) leads to weak positive correlation between -2ρD2(∂|∇c|/∂n)2~-2ρNc2 and κm. The last three terms on the right hand side vanish in the limit of small scale isotropy and for the present cases they remain weakly correlated with curvature.The relative strengths of the above mechanisms determine the net curvature dependence of(-D2). Thus, both positive and negative correlations between (-D2) and κm have been observed within the flame front in all cases considered here.


### 3.8. Local Behaviour off(D) and Its Curvature and Strain Rate Dependences

The marginal pdfs off(D) for c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces across the flame front are shown in Figure 13(a) for cases P3 and V2. The pdfs of f(D) in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not explicitly shown here. It is evident from Figure 13(a) that f(D) predominantly assumes negative (positive) values towards the unburned (burned) gas side of the flame (see Figure 3). The density-weighted diffusivity ρD is considered to be constant in cases P1–P5 and thus TD1 and TD2 are identically zero in these cases. The marginal pdfs of TD3 and TD4 for case P3 are shown in Figures 13(b) and 13(c), which show that both TD3 and TD4 predominantly assume positive (negative) values towards burned (unburned) gas side of the flame. As TD5=T1/2 in cases P1–P5, the pdfs of TD5 are qualitatively similar to those of T1 and thus are not shown here. This indicates that TD5 shows predominant probability of finding positive values throughout the flame (see Figure 6). The pdfs of TD1,TD2,TD3,TD4, and TD5 for case V2 are shown in Figures 13(d)–13(h), respectively. It is evident from Figures 13(f) and 13(g) that both TD3 and TD4 assume positive (negative) values towards burned (unburned) gas side of the flame, whereas TD5 assumes positive values throughout the flame, which is qualitatively similar to the behaviour of the corresponding term in case P3, where ρD is assumed to be constant. Figures 13(d) and 13(e) show that both TD1 and TD2 assume predominantly positive (negative) values towards the unburned (burned) gas side of the flame.The marginal pdfs off(D)×δth2/ρ0SL2 for c=0.1, 0.3, 0.5, 0.7, and 0.9 for cases (a) P3 and V2. The marginal pdfs of (b) TD3×δth2/ρ0SL2 and (c) TD4×δth2/ρ0SL2 for c=0.1, 0.3, 0.5, 0.7, and 0.9 across the flame for case P3. The marginal pdfs of (d) TD1×δth2/ρ0SL2, (e) TD2×δth2/ρ0SL2, (f) TD3×δth2/ρ0SL2, (g) TD4×δth2/ρ0SL2, and (h) TD5×δth2/ρ0SL2 for c= 0.1, 0.3, 0.5, 0.7, and 0.9 across the flame for case V2.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)The contours of joint pdfs betweenf(D) and aT (κm) for c=0.1, 0.5, and 0.7 isosurfaces are shown in Figure 14 (Figure 15) for cases P3 and V2, and the correlation coefficients between f(D) and aT (κm) for different c isosurfaces across the flame for cases P3 and V2 are shown in Figures 16(a) and 16(b) (Figures 16(c) and 16(d)), respectively. Both Figures 14 and 16 indicate that f(D) and aT are negatively (positively) correlated with each other towards the unburned (burned) gas side of the statistically planar flame (i.e., case P3), whereas f(D) and aT are positively correlated with each other from the middle to the burned gas side in the V-flame case (i.e., case V2), but this correlation remains weak towards the unburned gas side. It is evident from Figure 15 that f(D) and κm remain weakly correlated for both statistically planar and V-flames, which is consistent with the correlation coefficient between f(D) and κm shown in Figures 16(c) and 16(d). However, there are qualitative differences in the joints pdfs between f(D) and κm for cases P3 and V2.Joint pdfs betweenf(D)×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL for case P3 on (a) c=0.1, (b) 0.5, and (c) 0.7 isosurfaces. Joint pdfs between f(D)×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL for case V2 on (d) c=0.1, (e) 0.5, and (f) 0.7 isosurfaces.
(a)
(b)
(c)
(d)
(e)
(f)Joint pdfs betweenf(D)×δth2/ρ0SL2 and normalised curvature κm×δth on (a)c = 0.1, (b) 0.5, and (c) 0.7 isosurfaces for case P3. Joint pdfs between f(D)×δth2/ρ0SL2 and normalised curvature κm×δth on (d)c= 0.1, (e) 0.5, and (f) 0.7 isosurfaces for case V2.
(a)
(b)
(c)
(d)
(e)
(f)(a) Correlation coefficients for theTD3-aT, TD4-aT, TD5-aT, and f(D)-aT correlations on c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces for case P3. (b) Correlation coefficients for the TD1-aT, TD2-aT, TD3-aT, TD4-aT, TD5-aT, and f(D)-aT correlations on c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces for case V2. (c) Correlation coefficients for the TD3-κm, TD4-κm, TD5-κm, and f(D)-κm correlations on c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces for case P3. (d) Correlation coefficients for the TD1-κm, TD2-κm, TD3-κm, TD4-κm, TD5-κm, and f(D)-κm correlations on c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces for case V2.
(a)
(b)
(c)
(d)In order to explain the observed strain rate dependence off(D), the correlation coefficients between TD3, TD4, and TD5 (TD1,TD2,TD3,TD4, and TD5) with aT, for c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces, are also shown in Figure 16(a) (Figure 16(b)) for case P3 (case V2). It is evident from Figures 16(a) and 16(b) that both TD3 and TD4 remain negatively (positively) correlated with aT towards the unburned (burned) gas side of the flame.The strain rate dependences ofTD1,TD2,TD3,TD4, and TD5 can be explained in the following manner.(i)
The magnitudes ofTD3 and TD4 can be taken to scale as |TD3|~ρDNc/δ2~ρNc2 and |TD4|~ρDNc/δ2~ρNc2, which indicates that |TD3|~ρNc2 and |TD4|~ρNc2 remain positively correlated with aT due to positive correlation between Nc and aT (see Figure 5(a)). This suggests that the negative (positive) values of TD3 and TD4 (see Figures 13, 16(a), and 16(b)) lead to negative (positive) correlations of these terms with aT due to positive correlations between Nc and aT (also due to positive correlation between |TD3|~ρNc2 (|TD4|~ρNc2) and aT).(ii)
The termTD5 remains positively correlated with aT throughout the flame, which is consistent with the positive correlation between T1 and aT shown in Figure 6(c), as TD5=0.5T1 in statistically planar cases P1–P5 considered here (see Figure 16(a)). Even though ρD increases with increasing temperature in cases V1–V3, TD5 can still be taken to scale with T1 (i.e., TD5~T1~ρτNcSL/δ) and the positive correlation between T1 and aT leads to a positive correlation between TD5 and aT in case V2 and also in cases V1 and V3 (see Figure 16(b)).(iii)
The magnitudes of the termsTD1 and TD2 can be scaled as |TD1|~ρD2/δ4~ρNc2 and |TD2|~ρD2/δ4~ρNc2, respectively, which suggests that |TD1| and |TD2| are expected to be positively correlated with aT due to positive correlation between Nc and aT (see Figure 5(a)). As TD1 and TD2 assume predominantly positive (negative) values towards the unburned (burned) gas side of the flame, these terms scale with TD1~TD2~ρNc2 (TD1~TD2~-ρNc2) towards the reactant (product) side of the flame. Thus the positive correlation between Nc and aT leads to positive (negative) TD1-aT and TD2-aT correlations towards the unburned (burned) gas side of the flame (see Figure 16(b)).(iv)
In cases P1–P5, the termsTD3,TD4, and TD5 remain positively correlated with aT towards the burned gas side of the flame (see Figure 16(a)) and these positive correlations result in a net positive correlation between f(D) and aT towards the burned gas side of the flame. On the other hand, TD3 and TD4 remain negatively correlated with aT towards the unburned gas side of the flame (see Figure 16(a)) and these correlations dominate over the positive correlation between TD5 and aT to result in a net negative correlation between f(D) and aT towards the unburned gas side of the flame.(v)
In cases V1–V3, the termsTD3,TD4, and TD5 remain positively correlated with aT towards the burned gas side of the flame (see Figure 16(b)) and these positive correlations overcome the negative TD1-aT and TD2-aT correlations to give rise to a net positive correlation between f(D) and aT towards the burned gas side of the flame. By contrast, TD3 and TD4 remain negatively correlated with aT towards the unburned gas side of the flame (see Figure 16(b)) and these correlations oppose the positive TD1-aT,TD2-aT, and TD5-aT correlations to result in a weak correlation between f(D) and aT towards the unburned gas side of the flame.The correlation coefficients betweenTD3, TD4, and TD5 (TD1,TD2,TD3,TD4, and TD5) with κm for c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces are also shown in Figure 16(c) (Figure 16(d)) for case P3 (case V2). It is evident from Figures 16(c) and 16(d) that f(D) and κm are weakly correlated throughout the flame for both planar and V-flame cases. The curvature κm dependences of TD1,TD2,TD3,TD4,TD5, and f(D) can be explained in the following manner.(i)
BothTD3 and TD4 remain negatively (positively) correlated with aT towards the unburned (burned) gas side of the flame for both planar and V-flame cases (see Figures 16(a) and 16(b)), whereas aT and κm are negatively correlated throughout the flame. Thus, high (low) values of TD3 and TD4 are associated with high positive values of κm towards the unburned (burned) gas side of the flame, which gives rise to positive (negative) correlations of TD3 and TD4 with κm towards the unburned (burned) gas side.(ii)
AsTD5=0.5T1 for statistically planar flame cases (i.e., case P1–P5), a strong negative correlation between TD5 and κm has been observed nearc= 0.7 isosurface, which is consistent with the negative correlation between T1 and κm shown in Figure 6(e). Even though ρD increases with increasing temperature in V-flame cases (i.e., cases V1–V3), TD5 can still be taken to scale with T1 (i.e., TD5~T1~ρτNcSL/δ) and the negative correlation between T1 and κm (see Figure 6(f)) leads to a weak negative correlation between TD5 and κm in case V2 and also in cases V1 and V3.(iii)
BothTD1 and TD2 remain positively (negatively) correlated with aT towards the unburned (burned) gas side of the flame in case V2 (see Figure 16(b)), whereas aT and κm are negatively correlated throughout the flame. Thus, small (high) values of TD1 and TD2 are associated with high positive (negative) values of κm towards the unburned gas side of the flame, which gives rise to negative correlations of TD1 and TD2 with κm towards the unburned gas side in case V2. The combination of negative correlations of TD1 and TD2 with aT towards the burned gas side, as well as the strengthening of negative correlation between aT and κm towards the burned gas side of the flame, leads to the weakening of negative correlations of TD1 and TD2 with κm, as the burned gas side is approached and the correlation between TD2 with κm eventually becomes positive towards the burned gas side of the flame (see Figure 16(d)).(iv)
The termsTD3,TD4, and TD5 remain negatively correlated with κm towards the burned gas side of the flame and these negative correlations dominate over weak positive TD1-κm and TD2-κm correlations to result in a net negative correlation between f(D) and κm towards the burned gas side of the flame in case V2. The terms TD3 and TD4 remain positively correlated with κm towards the unburned gas side of the flame (see Figure 16(d)) and these correlations are opposed by the negative TD1-κm, TD2-κm, and TD5-κm correlations to result in a weak correlation between f(D) and κm towards the unburned gas side of the flame front in case V2 and other V-flame cases.


### 3.9. Modelling Significances

A modelled transport equation ofN~c may need to be solved alongside other modelled conservation equations in RANS/LES simulations, when the rate of generation of scalar gradients does not remain in equilibrium with its destruction rate. The local strain rate and curvature dependences of SDR are expected to play important roles in LES simulations, as the necessity of capturing local behaviour of N~c is particularly important in the context of LES. As N~c approaches Nc with decreasing filter width Δ (i.e., limΔ→0N~c=Nc) in the context of LES, the local resolved-scale strain rate and curvature dependences of N~c and the terms of its transport equation (i.e., T¯1,T¯2,T¯3,(-D¯2), and f(D)¯) are likely to be qualitatively similar to the local strain rate and curvature dependences of Nc and the terms of its transport equation (i.e., T1,T2,T3,(-D2), and f(D)), respectively. The above discussion suggests that the models for N~c,T¯1,T¯2,T¯3,(-D¯2), and f(D)¯ in LES should be developed in such a manner so that they are capable of capturing the resolved strain rate and curvature dependences of N~c,T¯1,T¯2,T¯3,(-D¯2), and f(D)¯ for a range of different filter widths and approach local strain rate and curvature dependences of Nc,T1,T2,T3,(-D2), and f(D) for small filter widths.It can further be observed from Figures5–16 that the local strain rate and curvature dependence of Nc and the terms of its transport equation for V-flames are found to be qualitatively similar to the behaviour observed for the statistically planar flame cases. As the SDR statistics are principally governed by the small-scale molecular processes, the local statistics of Nc and the terms of its transport equation are largely independent of the flow configuration. Thus, the models for N~c transport developed based on data extracted from a canonical configuration might broadly be applicable to different geometries.Moreover, it is worth noting thatρD is assumed to be constant in cases P1–P5 whereas ρD is taken to be temperature dependent in cases V1–V3. However, the statistical behaviours of Nc and the terms of its transport equation are found to be broadly similar qualitatively in all cases indicating that the models, which have been developed based on DNS databases with constant ρD, should at least be able to capture the qualitative trends of N~c transport.


## 3.1. Flame-Turbulence Interaction

The contours ofc in the central x1-x2 plane for cases P1–P5 and V1–V3 are shown in Figures 1(a)–1(h), respectively. It is evident from Figures 1(a)–1(e) that the level of wrinkling increases with increasing u′/SL~Ret1/4Ka1/2~Ret1/2/Da1/2. Turbulent eddies penetrate into the preheat zone in the thin reaction zones regime combustion (Ka~δth2/η2~Ret1/2/Da>1), but the reaction zone remains unperturbed because the Kolmogorov length scale is larger than the reaction zone thickness. The isosurfaces of c representing the preheat zone (i.e., c≤0.5) show more distortion than the isosurfaces representing the reaction zones (i.e., 0.7≤c≤0.9) due to penetration of turbulent eddies within the preheat zone. However, this tendency is more prevalent for high values of u′/SL and Ka (e.g., cases P3, P4, P5, and V3) but the isosurfaces of c remain mostly parallel to each other for small values of u′/SL and Ka (e.g., cases P1, P2, V1, and V2) indicating that the internal flame structure is weakly affected by turbulence in these cases.Distributions ofc in the central x1-x2 plane for cases P1–P5 (a–e) at tc=δth/SL and for cases V1–V3 (f–h).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)


## 3.2. Statistical Behaviour of the Mean Values ofNc and the Unclosed Terms of Its Transport Equation

The variations of〈Nc〉×δth/SL with c for statistically planar flames and V-flames are shown in Figures 2(a) and 2(b), where 〈Q〉 indicates the mean value of Q, which is obtained by ensemble averaging the quantity in question on a given c isosurface in the manner previously used by Boger et al. [42], Chakraborty and Cant [10, 11], and Chakraborty and Klein [15, 16]. It is worth noting that 〈Q〉 should not be confused with either Reynolds averaging or conventional conditional averaging operation in the context of RANS simulations because 〈Q〉 is evaluated using all the samples for a given c value over the whole domain. Figures 2(a) and 2(b) show that the variations of 〈Nc〉×δth/SL for statistically planar and V-flames are qualitatively similar to each other. For both statistically planar and V-flame configurations, the location of the maximum value of 〈Nc〉×δth/SL is skewed slightly towards the burned gas side of the flame (i.e., c≈0.7). The peak magnitude of 〈Nc〉×δth/SL does not change significantly in response to u′/SL as the standard deviation for the case in the middle of the parameter range (i.e., cases P3 and V2) is found to exceed the difference in 〈Nc〉×δth/SL values for the cases considered here for both statistically planar and V-flame configurations. In order to understand the distribution of 〈Nc〉×δth/SL across the flame front, the variations of the mean values of the terms 〈T1〉,〈T2〉,〈T3〉, 〈(-D2)〉, and 〈f(D)〉 conditional on c for planar and V-flames are shown in Figure 3. The variations of the mean values of the terms in cases P2, P3, and P4 (cases V2) are qualitatively similar to those in cases P1 and P5 (case V1) and thus are not explicitly shown here. It is evident from Figure 3 that the qualitative behaviour of these terms remains similar for all cases considered here. In all cases, 〈T1〉 remains positive throughout the flame. By contrast, 〈(-D2)〉 assumes negative values throughout the flame in all cases as dictated by (1a). Expressing ρ=ρ0/(1+τc) for low Mach number, unity Lewis number flames give rise to an alternative expression for T1 [3, 17, 25, 28, 29]:
(4)T1=2ρ∂uj∂xjNc.Variation of the mean value of〈Nc〉×δth/SL conditional on c values across the flame front for (a) statistically planar cases with the bar indicating the standard deviation for case P3 and (b) V-flame cases with the bar indicating the standard deviation for case V2.
(a)
(b)Figure 3
Variation of the mean values of〈T1〉,〈T2〉,〈T3〉,〈-D2〉, and 〈f(D)〉 conditional on c values across the flame for cases P1, P5, V1, and V3. All the terms of the transport equation of Nc are normalised with respect to the respective values of ρ0SL2/δth2.As dilatation rate∂uj/∂xj is predominantly positive in premixed flames, 〈T1〉 for all values of c is positive across the flame and vanishes on both ends of the flame.The quantity〈T2〉 assumes negative values throughout the flame front for cases P1 and V1. Although 〈T2〉 remains negative for the major portion of the flame, small positive values can be discerned in cases P5 and V3. In order to understand this behaviour, the term T2 can be expressed in the following manner [3, 22–24, 28, 30]:
(5)T2=-2ρNc(eαcos⁡2α+eβcos⁡2β+eγcos⁡2γ),
where eα,eβ, and eγ are the most extensive, intermediate, and most compressive principal strain rates and α,β, and γ are the angles of these principal strain rates with ∇c. Equation (5) demonstrates that the predominant alignment of eα (eγ) with ∇c leads to a negative (positive) contribution to T2.It has been discussed in the previous analyses [23, 24, 28, 30] that the alignment of ∇c with eα and eγ is determined by relative strengths of the strain rate induced by flame normal acceleration achem and turbulent straining aturb. It has been demonstrated earlier that ∇c preferentially aligns with eα(eγ) when achem(aturb) dominates over aturb(achem). The strain rate induced by flame normal acceleration due to chemical heat release can be scaled as achem~τf(Ka)SL/δth, where f(Ka) is expected to decrease with increasing Ka [43]. Following Meneveau and Poinsot [44], aturb can be scaled as aturb~u′/l, which gives rise to achem/aturb~τf(Ka)SLl/u′δth~τf(Ka)Da~τf(Ret1/2/Da)Da. Alternatively, turbulent straining can be scaled as [45] aturb~u′/λ (where λ is the Taylor microscale), which yields achem/aturb~τf(Ka)SLλ/u′δth~τf(Ka)Da/Ret1/2~τf(Ret1/2/Da)Da/Ret1/2~τf(Ka)/Ka. The above scaling relations suggest that achem strengthens with respect to aturb with increasing Da for a given value of Ret. Previous analyses [22–24, 28, 30] demonstrated that ∇c predominantly aligns with eα for Da≫1 flames, whereas ∇c aligns with eγ in Da<1 flames for comparable values of Ret. Both achem/aturb~τf(Ka)Da and achem/aturb~τf(Ret1/2/Da)Da/Ret1/2~τf(Ka)/Ka indicate that an increase in Ka~Ret1/2/Da for a given value of Da (e.g., cases P1, P3, and P5) gives rise to weakening of achem in comparison to aturb. This increases the extent of ∇c alignment with eγ with increasing Ka when Da is held constant as in cases P1, P3, and P5. In cases P1 and P3, ∇c predominantly aligns with eα; however the extent of this alignment decreases from P1 to P3. This predominant alignment of ∇c with eα in cases P1 and P3 leads to a negative contribution of 〈T2〉 in these cases. In case P5, ∇c predominantly aligns with eγ in the unburned and fully burned gases but achem overcomes aturb in the regions of intense heat release close to the middle of the flame and as a result ∇c aligns with eα in the reaction zone. Thus the mean value of 〈T2〉 in case P5 assumes positive values towards both the unburned and burned gas sides, whereas the mean contribution of 〈T2〉 remains negative close to the middle of the flame. The relation achem/aturb~τf(Ka)Da/Ret1/2 indicates that achem weakens in comparison to aturb with decreasing τDa/Ret1/2. The quantity τDa/Ret1/2 assumes values equal to 0.96, 0.55, and 0.49 for cases P2, P3, and P4, respectively, when the statistics were extracted. This leads to larger extent of ∇c aligning with eγ in case P4 (case P3) than in case P3 (case P2). This leads to predominantly negative contribution of 〈T2〉 in cases P2 and P3, whereas 〈T2〉 assumes positive values towards the unburned and burned gas sides of the flame in case P4. However, achem overcomes aturb in the regions of intense heat release at the middle of the flame and ∇c starts to align with eα in the reaction zone giving rise to negative values of 〈T2〉 in case P4. In cases V1 and V2, the values of achem/aturb~τf(Ka)Da/Ret1/2 are larger than the corresponding value in case V3 (see the parameters in Table 1). Thus, the extent of ∇c alignment with eα(eγ) decreases (increases) from case V1 to case V3. This gives rise to positive values of 〈T2〉 towards both unburned and burned gas sides of the flame in case V3. This tendency is less prevalent in cases V1 and V2 due to smaller extent of ∇c alignment with eγ than in case V3. However, the mean contribution of 〈T2〉 is negative in the middle of the flame for cases V1–V3 due to the alignment of ∇c with eα in the heat releasing zone.The contribution of〈T3〉 remains positive (negative) towards the unburned (burned) gas side of the flame with the transition from positive to negative value taking place close to c≈0.85. In order to explain this behaviour, T3 can be rewritten as
(6)T3=-2Dni∂w˙∂xi|∇c|=-2D∂w˙∂n|∇c|,
where n is the spatial coordinate in the local flame normal direction and the flame normal vector n→ points towards the unburned gas side of the flame. For single step chemistry considered here, the maximum w˙ occurs close to c≈0.85 [10, 14]. This suggests that the probability of finding negative (positive) values of ∂w˙/∂n is significant for c<0.85 (c>0.85), which gives rise to positive (negative) value of 〈T3〉 towards the unburned (burned) gas side of the flame.Figure3 shows that 〈f(D)〉 is weakly negative towards the unburned gas side before becoming positive towards the burned gas side in all the cases. The magnitude of the mean contribution of 〈f(D)〉 remains comparable to that of 〈T1〉 in all cases indicating that 〈f(D)〉 cannot be neglected even for cases P1–P5, where ρD is considered to be constant. In cases P1–P5, ρNc[∂D/∂t+uj∂D/∂xj] can be expressed using ρ=ρ0/(1+τc) for globally adiabatic Le=1.0 flames as T1/2=ρNc∂uj/∂xj (i.e., ρNc[∂D/∂t+uj∂D/∂xj]=ρNc(∂uj/∂xj) for constant ρD) and the first two terms on the right hand side of (1b) vanish for constant values of ρD. The contributions of 〈(TD3+TD4)〉 are responsible for the change in sign of 〈f(D)〉 in cases P1–P5. These terms are also principally responsible for sign change of 〈f(D)〉 in cases V1–V3.


## 3.3. Local Behaviour ofNc and Its Curvature and Strain Rate Dependences

The marginal probability density functions (pdfs) of normalisedNc+ (i.e., Nc×δth/SL) for different c isosurfaces across the flame are shown in Figures 4(a) and 4(b) in log-log scale for cases P3 and V2, respectively. The pdfs of Nc in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not explicitly shown here. The pdfs for c<0.5 are not shown in Figures 4(a) and 4(b), as Nc assumes small values in the preheat zone of the flame due to small magnitude of scalar gradient ∇c. It is evident from Figures 4(a) and 4(b) that the pdfs of Nc are qualitatively similar for statistically planar and V-flames and in both cases the probability of finding high values of Nc is most prevalent in the middle of the flame with slight skewness towards the burned gas side (i.e., c≈0.7) and the probability of finding high values of Nc decreases on both unburned and burned gas sides of the flame front. This is consistent with the observed behaviour of the mean values of Nc conditional on c shown in Figure 2. It can be seen in Figure 4 that a log-normal distribution captures the qualitative behaviour of the pdf of Nc although there are some disagreements in the pdf tails. This is consistent with several previous experimental [46–52] and numerical [53–55] studies investigating the scalar dissipation rate pdf of a passive scalar. An approximate log-normal distribution of SDR in turbulent premixed flames has also been reported in a previous analysis [56].The marginal pdf of normalisedNc+ (i.e., Nc×δth/SL) and the log-normal distribution in log-log scale for c=0.5, 0.7, and 0.9 across the flame for cases (a) P3 and (b) V2.
(a)
(b)The joint pdfs ofNc and tangential strain rate aT for cases P1, P5, V1, and V3 are shown in Figure 5(a) for c=0.8 isosurface, which is close to the most reactive region for the present thermochemistry. It can be seen from Figure 5(a) that Nc and aT are positively correlated on c=0.8 isosurface for cases P1, P5, V1, and V3 and similar qualitative behaviour has been observed also for other c isosurfaces in all cases considered here. This positive correlation between Nc and aT can be explained in the following manner.(i)
The dilatation rate∇·u→ can be expressed as ∇·u→=aT+an, where an=ninj∂ui/∂xj is the normal strain rate. For unity Lewis number flames, ∇·u→ can be scaled as ∇·u→~achem~τf(Ka)SL/δth, whereas aT can be taken to scale with turbulent strain rate aturb (i.e., aT~aturb~u′/l according to Meneveau and Poinsot [44] and aT~aturb~u′/λ according to Tennekes and Lumley [45]).(ii)
Above scalings indicate that∇·u→/aT scales as ∇·u→/aT~τf(Ret1/2/Da)Da and ∇·u→/aT~τf(Ret1/2/Da)Da/Ret1/2~τf(Ka)/Ka according to the scaling arguments by Meneveau and Poinsot [44] and Tennekes and Lumley [45], respectively. Both ∇·u→/aT~τf(Ret1/2/Da)Da and ∇·u→/aT~τf(Ret1/2/Da)Da/Ret1/2~τf(Ka)/Ka suggest that the magnitude of aT is likely to supersede the magnitude of ∇·u→ in most locations within the flame for small values of Da and high values of Ka.(iii)
It has been shown in several previous analyses [10, 30] that both ∇·u→ and aT assume predominantly positive values and thus a higher magnitude of aT than ∇·u→ induces a negative (i.e., compressive) normal strain rate an. Thus, an increase in aT often leads to a decrease in an=∇·u→-aT for small (high) values of Da (Ka). Thus, the isoscalar lines come close to each other under the action of decreasing an, which leads to increase in the magnitude of scalar gradient ∇c. This is reflected in the positive correlation between Nc and aT.(a) Joint pdfs betweenNc×δth/SL and normalised tangential strain rate aT×δth/SL on c=0.8 isosurface for cases P1, P5, V1, and V3. (b) Joint pdf between Nc×δth/SL and normalised curvature κm×δth on c=0.8 isosurface for cases P1, P5, V1, and V3.
(a)
(b)The joint pdfs betweenNc and curvature κm for cases P1, P5, V1, and V3 are shown in Figure 5(b) for c=0.8 isosurface. Cases P2 and P3 (case V2) are not explicitly shown here due to their similarities to cases P1 and P5 (case V1), respectively. It can be seen from Figure 5(b) that the joint pdf between Nc and κm exhibits both positive and negative correlating branches on c=0.8 isosurface for cases P5 and V3, and as a result of this, the net correlation between Nc and κm remains weak. The positive correlation branch between Nc and κm remains weak for small values of u′/SL in statistically planar flames (see Figure 5(b) for case P1) and this branch disappears completely in the V-flames with small values of u′/SL (see Figure 5(b) for case V1). Similar behaviour is observed for other c isosurfaces in all cases considered here and the correlation between Nc and κm is weak throughout the flame for high values of u′/SL (e.g., cases P3–P5 and V3). However, the disappearance of the positive correlating branch in the joint pdf of Nc and κm in Figure 5(b) indicates that Nc and κm are negatively correlated with each other throughout the flame for small values of u′/SL (e.g., cases P1, P2, V1, and V2). The observed behaviour can be explained based on the following physical mechanisms.(i)
Previous analyses (e.g., [57]) demonstrated that both aT and ∇·u→ remain negatively correlated with κm in turbulent premixed flames, and thus the behaviour of an at locations with large positive curvature is principally determined by aT since ∇·u→ is small in these zones due to defocusing of heat. Small values of aT are associated with high values of κm at these locations, which lead to small values of Nc at high values of positive κm due to positive correlation between Nc and aT. This leads to a negative correlating branch between Nc and κm at the positively curved zones.(ii)
The dilatation rate∇·u→ is large in the negatively curved locations due to strong focussing of heat and the magnitude of ∇·u→ can locally be high enough to supersede the magnitude of aT, which leads to a positive value of an. This tendency strengthens with decreasing κm, especially in the zones with large negative curvature, which gives rise to an increase in an with decreasing curvature. As the distance between the isoscalar lines increases with increasing an, the magnitude of scalar gradient ∇c decreases with decreasing κm in the negatively curved zones. This leads to the positive correlating branch in the joint pdf of Nc and κm (see Figure 5(b) for cases P5 and V3).(iii)
The relative strengths of the positive and negative correlating branches ultimately determine the net correlation betweenNc and κm in the high u′/SL cases. The probability of finding high negative curvature remains small for small values of u′/SL and as a result the probability of finding high values of ∇·u→, which locally overcomes aT, to induce a positive value of an, becomes rare (e.g., cases P1 and V1). Thus the combination of positive correlations between Nc and aT and negative correlations between aT and κm leads to a predominantly negative correlating branch between Nc and κm in the low u′/SL cases (e.g., cases P1 and V1; see Figure 5(b)).The strain rate and curvature dependences ofNc discussed above, in turn, affect the local statistical behaviours of T1, T2, T3, (-D2), and f(D) in response to aT and κm. The curvature and strain rate dependences of T1, T2, T3, (-D2), and f(D) are discussed next.


## 3.4. Local Behaviour ofT1 and Its Curvature and Strain Rate Dependences

The marginal pdfs ofT1 for different c isosurfaces across the flame are shown in Figures 6(a) and 6(b) for cases P3 and V2, respectively. The pdfs of T1 in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not explicitly shown here. It is evident from Figures 6(a) and 6(b) that the pdfs of T1 are qualitatively similar for statistically planar and V-flames and in both cases T1=2ρ(∇·u→)Nc assumes predominantly positive values throughout the flame. As dilatation rate ∇·u→ is principally positive due to thermal expansion in premixed flames [10, 30], the contribution of T1=2ρ(∇·u→)Nc is predominantly positive throughout the flame. Moreover, Figures 6(a) and 6(b) demonstrate that the probability of finding high values of T1 is most prevalent in the middle of the flame with slight skewness towards the burned gas side (i.e., c≈0.7) and the probability of finding high values of T1 decreases on both unburned and burned gas sides of the flame. This is consistent with the observed behaviour of the mean values of T1 conditional on c shown in Figure 3. The probability of finding large magnitudes of ∇·u→ is the highest at a location which is slightly skewed towards the burned gas side of the flame [30]. As the distributions of Nc and ∇·u→ are slightly skewed towards the burned gas side of the flame, the probability of finding large values of T1=2ρ(∇·u→)Nc becomes high around c≈0.7.The marginal pdfs ofT1×δth2/ρ0SL2 for c=0.1, 0.3, 0.5, 0.7, and 0.9 for cases (a) P3 and (b) V2. Joint pdf between T1×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL on c=0.8 isosurface for cases (c) P3 and (d) V2. Joint pdf between T1×δth2/ρ0SL2 and normalised curvature κm×δth on c=0.8 isosurface for cases (e) P3 and (f) V2.
(a)
(b)
(c)
(d)
(e)
(f)The joint pdfs betweenT1 and aT for cases P3 and V2 are shown in Figures 6(c) and 6(d), respectively, for c=0.8 isosurface. It can be seen from Figures 6(c) and 6(d) that T1 and aT are positively correlated on c=0.8 isosurface for cases P3 and V2 and similar qualitative behaviours have been observed for other c isosurfaces in all cases considered here. Both ∇·u→ and aT are positively correlated for all flames considered here, which along with positive correlation between Nc and aT (see Figure 5) gives rise to a positive correlation between T1=2ρ(∇·u→)Nc and aT.The joint pdfs betweenT1 and κm for cases P3 and V2 are shown in Figures 6(e) and 6(f), respectively, for c=0.8 isosurface. It can be seen from Figures 6(e) and 6(f) that the joint pdf between T1 and κm exhibits a negative correlation on c=0.8 isosurface for cases P3 and V2, and similar qualitative behaviour has been observed for other c isosurfaces in all cases considered here. In all cases, the net correlation between Nc and κm is weak (see Figure 5(b)), but ∇·u→ assumes high (small) values at negatively (positively) curved locations because of focussing (defocusing) of heat. This leads to a predominantly negative correlation between ∇·u→ and κm [57]. The negative correlation between ∇·u→ and κm is principally responsible for the negative correlation between T1=2ρ(∇·u→)Nc and κm.


## 3.5. Local Behaviour ofT2 and Its Curvature and Strain Rate Dependences

The marginal pdfs ofT2 for different c isosurfaces across the flame are shown in Figures 7(a) and 7(b) for cases P3 and V2, respectively. The pdfs of T2 in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not explicitly shown here. Figures 7(a) and 7(b) show that the probability of finding negative values of T2 supersedes the probability of finding positive values. The probability of finding negative values of T2 increases as the heat releasing zone (see the pdfs for c=0.7 isosurface) is approached. It has been discussed earlier that the effects of achem overcome the effects of aturb in the heat releasing zone to give rise to a preferential alignment of ∇c with eα even for small values of Da. This preferential alignment of ∇c with eα in these zones gives rise to negative values of T2 according to (5). The extent of ∇c alignment with eα(eγ) decreases (increases) towards both unburned and burned gas sides of the flame due to diminishing effects of achem.The marginal pdfs ofT2×δth2/ρ0SL2 for c=0.1, 0.3, 0.5, 0.7, and 0.9 for cases (a) P3 and (b) V2. Joint pdf between T2×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL on c=0.8 isosurface for cases (c) P3 and (d) V2. Joint pdf between T2×δth2/ρ0SL2 and normalised curvature κm×δth on c=0.8 isosurface for cases (e) P3 and (f) V2.
(a)
(b)
(c)
(d)
(e)
(f)The contours of joint pdfs betweenT2 and aT for c=0.8 are shown in Figures 7(c) and 7(d) for cases P3 and V2 and the correlation coefficients between T2 and aT for different c isosurfaces across the flame for all cases are shown in Table 2. It is evident from Figures 7(c) and 7(d) and Table 2 that T2 and aT are positively correlated for high u′/SL cases (e.g., cases P5 and V3) although the strength of the correlation changes through the flame. However, T2 and aT are weakly correlated with each other within the flame, where the effects of heat release are significant for cases with small and moderate values of u′/SL (see Table 2). In order to explain this behaviour, it is useful to rewrite T2 in the following manner:
(7)T2=-2ρanNc.Table 2
Correlation coefficients betweenT2 and aT and between T2 and κm on c= 0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces.
Case
T
2
-
a
T
T
2
-
κ
m
c
=
0.1
c
=
0.3
c
=
0.5
c
=
0.7
c
=
0.9
c
=
0.1
c
=
0.3
c
=
0.5
c
=
0.7
c
=
0.9
P1
0.642
−0.098
−0.217
−0.092
0.685
0.141
0.509
0.720
0.614
−0.250
P2
0.673
−0.090
−0.208
−0.084
0.676
0.116
0.506
0.719
0.616
−0.227
P3
0.751
0.376
0.263
0.263
0.648
0.544
0.252
0.412
0.423
−0.065
P4
0.802
0.593
0.616
0.689
0.827
0.052
0.196
0.235
0.198
−0.027
P5
0.783
0.662
0.614
0.616
0.787
0.028
0.137
0.223
0.242
−0.014
V1
0.245
0.367
0.749
0.816
0.971
0.612
0.623
0.354
0.135
−0.043
V2
0.591
0.335
0.495
0.699
0.920
0.345
0.565
0.497
0.328
−0.067
V3
0.767
0.721
0.738
0.768
0.888
−0.009
0.057
0.077
0.028
−0.155Based on (7) the strain rate dependences of T2 can be explained in the following manner.(i)
It has already been demonstrated thatNc and aT are positively correlated with each other (see Figure 5(a)). The quantity (-an)=aT-∇·u→ tends to increase with increasing aT in the regions where the effects of ∇·u→ are weak. This along with positive correlation between Nc and aT leads to a positive correlation between T2 and aT for both unburned and burned gas sides of the flame for all cases.(ii)
The magnitudes of∇·u→ and aT increase with decreasing κm, and thus (-an)=aT-∇·u→ might not increase (even decrease) with increasing aT in the heat releasing zone of the flame where the effects of ∇·u→ are strong. The aT dependences of (-an) and Nc ultimately determine the nature of the correlation between T2 and aT. The strain rate and curvature dependences of ∇·u→ weaken with increasing u′/SL [58], so (-an)=aT-∇·u→ increases with increasing aT, which leads to a positive correlation between T2 and aT for the major portion of the flame for cases with high values of u′/SL (see Table 2).The joint pdfs betweenT2 and κm for cases P3 and V2 are shown in Figures 7(e) and 7(f), respectively, for c=0.8 isosurface and the correlation coefficients between T2 and κm for different c isosurfaces across the flame are shown in Table 2 for all cases considered here. It is evident from Figures 7(e) and 7(f) and Table 2 that T2 and κm remain weakly positively correlated except the burned gas side of the flame. The observed curvature dependence of T2 could be explained based on the following physical mechanisms.(i)
The effects of dilatation rate∇·u→ and thermal expansion are particularly strong in the negatively curved regions due to focussing of heat. By the same token, the effects of heat release are weak in the positively curved zones due to defocusing of heat. Thus, the effects of achem are more likely to dominate over the effects of aturb in the negatively curved zones, which increase the extent of ∇c alignment with eα as demonstrated earlier by Hartung et al. [58]. Weakening of the heat release effects at positively curved zones due to defocusing of heat leads to a greater (lesser) extent of ∇c alignment with eγ(eα) in the positively curved zones. The extent of ∇c alignment with eα increases in the negatively curved zones, which in turn makes T2 increasingly negative (see (5)) and the magnitude of the negative contribution of T2 decreases for positive curvature locations. This gives rise to a positive correlation between T2 and κm, as observed from Figures 7(e) and 7(f) and Table 2.(ii)
However, the effects ofaturb are more likely to dominate over the effects of achem towards the burned gas side and thus the extent of ∇c alignment with eγ is determined by local turbulent flow conditions. The effects of flame-generated turbulence become stronger at the negatively curved zones due to stronger thermal expansion effects resulting from focussing of heat especially in the heat releasing zone. The straining induced by flame-generated turbulence may overcome relatively weak effects of ∇·u→ towards the burned gas side, which can give rise to an increasing extent of ∇c alignment with eγ increases in the negative curved zones. This in turn gives rise to an increase in T2 (see (5)) with decreasing κm towards the burned gas side and leads to a negative correlation between T2 and κm (see Table 2).


## 3.6. Local Behaviour ofT3 and Its Curvature and Strain Rate Dependences

The marginal pdfs of normalisedT3 for different c isosurfaces across the flame are shown in Figure 8 for cases P3 and V2, respectively. The pdfs of T3 in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not explicitly shown here. The pdfs for c<0.5 are not shown in Figure 8 because T3 assumes negligible value in the preheat zone of the flame due to negligible magnitude of w˙. It is evident that T3 assumes positive values for the major portion of the flame for both statistically planar and V-flames and the probability of finding high positive values increases towards the most reactive zone (e.g., c=0.7 in Figure 8) of the flame front. However, T3 assumes negative values only towards the burned gas side (e.g., c=0.9) of the flame front for both planar and V-flames. This is consistent with the behaviour of 〈T3〉 shown in Figure 3. The physical mechanism behind the transition from positive to negative values of the mean contribution of T3 (see (6)) is also responsible for obtaining negative (positive) values of T3 towards the burned (unburned) gas side of the flame.Figure 8
The marginal pdfs ofT3×δth2/ρ0SL2 for c=0.5, 0.7, and 0.9 for cases P3 and V2.The contours of joint pdfs betweenT3 and aT for c=0.5, 0.7, and 0.9 isosurfaces are shown in Figures 9(a)–9(f) for cases P3 and V2 and similar qualitative behaviour has been observed for other cases considered here. It is evident from Figures 9(a)–9(f) that T3 and aT remain positively correlated for the part of the flame where finding positive values of T3 is prevalent. On the other hand, T3 and aT are negatively correlated with each other towards the burned gas side of the flame where T3 is predominantly negative. The observed aT dependence of T3 can be explained in the following manner.(i)
It has been demonstrated earlier thatNc and aT are positively correlated with each other which suggests that |∇c|=|∂c/∂n| increases with increasing aT. For low Mach number, unity Lewis number flames w˙ depend only on c and thus high values of |∂w˙/∂n| are associated with high values of |∇c|=|∂c/∂n| and Nc.(ii)
AsNc and aT are positively correlated with each other, the magnitude of reaction rate contribution |T3|=|2D(∂w˙/∂n)|∇c|| is positively correlated with tangential strain rate aT. Thus, T3 is positively (negative) correlated with aT, where T3 assumes positive (negative) values.Joint pdfs betweenT3×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL on (a) c=0.5, (b) 0.7, and (c) 0.9 isosurfaces for case P3. Joint pdfs between T3×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL on (d) c=0.5, (e) 0.7, and (f) 0.9 isosurfaces for case V2.
(a)
(b)
(c)
(d)
(e)
(f)The joint pdfs betweenT3 and κm for cases P3 and V2 are shown in Figure 10 for c=0.5, 0.7, and 0.9 isosurfaces and similar qualitative behaviour has been observed for other cases considered here. It is evident from Figure 10 that the joint pdf of T3 and κm exhibits both positive and negative correlating branches and the net correlation is weak throughout the flame. The physical explanations for the observed κm dependence of T3 can be summarised in the following manner.(i)
The term|T3|=|2D(∂w˙/∂n)|∇c|| is expected to be positively (negatively) correlated with curvature κm at negatively (positively) curved locations for high values of u′/SL, as in the case of Nc (see cases P5 and V3 in Figure 5(b)), because high values of |∂w˙/∂n| are associated with high values of Nc and |∇c|=|∂c/∂n|.(ii)
As a result of the aforementioned physical mechanisms, the termT3 and κm remain positively (negatively) correlated with curvature κm at negatively (positively) curved locations in the planar flames where T3 assumes positive values. By contrast, the joint pdfs of T3 and κm exhibit negative (positive) correlation with curvature κm at negatively (positively) curved locations within the flame where T3 assumes negative values for the planar flames considered here (see Figure 10(c)). However, Nc remains predominantly negatively correlated with κm for V-flame cases (see Figure 5(b)) and thus T3 shows positive (negative) correlation with curvature where T3 assumes negative (positive) values (see Figures 10(d)–10(f)).Joint pdfs betweenT3×δth2/ρ0SL2 and normalised curvature κm×δth on (a) c=0.5, (b) 0.7, and (c) 0.9 isosurfaces for case P3. Joint pdfs between T3×δth2/ρ0SL2 and normalised curvature κm×δth on (d) c=0.5, (e) 0.7, and (f) 0.9 isosurfaces for case V2.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.7. Local Behaviour of(-D2) and Its Curvature and Strain Rate Dependences

The marginal pdfs of(-D2) for c isosurfaces representative of leading edge, reaction zone, and trailing edge of the flame (e.g., c=0.3, 0.7 and 0.9 isosurfaces) are shown in Figure 11(a) for cases P3 and V2. The pdfs of (-D2) in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to that in case P3 (case V2) and thus are not explicitly shown here. Figure 11(a) shows that (-D2) assumes negative values throughout the flame and the probability of finding high magnitude of (-D2) increases from unburned gas side towards a region of the flame which is severely skewed towards the burned gas side (e.g.,c = 0.9 isosurface). This behaviour is found to be consistent with the mean behaviour of (-D2) shown in Figure 3. It can further be seen from Figure 11(a) that the pdfs of (-D2) for statistically planar and V-flames are qualitatively similar to each other.The marginal pdfs of(-D2)×δth2/ρ0SL2 for c=0.3, 0.7, and 0.9 for cases (a) P3 and V2. Joint pdfs between (-D2)×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL on c=0.8 isosurface for cases (b) P3 and (c) V2.
(a)
(b)
(c)The contours of joint pdfs between(-D2) and aT for c=0.8 isosurface are shown in Figures 11(b) and 11(c) for cases P3 and V2 and the correlation coefficients between (-D2) and aT for different c isosurfaces across the flame for all cases considered here are shown in Table 3. Figures 11(b) and 11(c) and Table 3 show that (-D2) and aT are predominantly negatively correlated throughout the flame but the strength of this negative correlation weakens with increasing u′/SL and the correlation becomes weakly positive at the middle of the flame for high values of u′/SL (e.g., cases P4 and P5). This behaviour can be explained in the following manner.(i)
The instantaneous SDRNc and the molecular dissipation term (-D2) can be taken to scale as Nc~D/δ2 and (-D2)~(-ρD2/δ4)~(-ρNc2) (where δ is the typical local flame thickness) because in premixed flames. the gradient of progress variable is only existent within the flame thickness. Alternatively, (-D2) can be considered to be governed by small-scale eddies and thus the characteristic length scale can be taken to be the Kolmogorov length scale η. Thus one obtains (-D2)~(-ρD2/η4)~(-ρNc2)Ka2 when dissipation processes are taken to be governed by η but δ/η~Ka1/2 remains of the order of unity for all cases considered here (see Table 1) and both scalings of (-D2) (i.e., (-D2)~(-ρD2/δ4)~(-ρNc2) and (-D2)~(-ρD2/η4)~(-ρNc2)Ka2) suggest that high magnitudes of the dissipation term |-D2| are associated with high values of aT due to positive correlation between Nc and aT (see Figure 5(a)).(ii)
As(-D2) assumes negative values, the quantities (-D2) and aT are predominantly negatively correlated throughout the flame due to positive correlation between |-D2| and aT. However, the negative correlation between (-D2) and aT weakens with increasing u′/SL due to weakening of positive correlation between Nc and aT. Thus the correlation between (-D2) and aT becomes weakly positive at the middle of the flame for high values of u′/SL (e.g., cases P4 and P5).Table 3
Correlation coefficients between(-D2) and aT and between (-D2) and κm on c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces.
Case
(
-
D
2
)
-
a
T
(
-
D
2
)
-
κ
m
c
=
0.1
c
=
0.3
c
=
0.5
c
=
0.7
c
=
0.9
c
=
0.1
c
=
0.3
c
=
0.5
c
=
0.7
c
=
0.9
P1
−0.598
−0.809
−0.368
−0.183
−0.522
0.280
0.577
0.582
−0.020
0.261
P2
−0.581
−0.806
−0.389
−0.218
−0.488
0.272
0.554
0.600
0.023
0.225
P3
−0.546
−0.699
−0.028
0.050
−0.296
0.283
0.422
0.364
−0.234
−0.001
P4
−0.513
−0.652
0.221
−0.078
−0.483
0.231
0.338
−0.074
−0.412
−0.100
P5
−0.472
−0.581
0.111
−0.031
−0.378
0.205
0.288
0.174
−0.317
−0.107
V1
−0.712
−0.695
−0.069
−0.616
−0.679
0.700
0.825
−0.176
0.619
0.766
V2
−0.628
−0.525
0.089
−0.445
−0.703
0.475
0.696
0.033
0.159
0.633
V3
−0.452
−0.355
0.130
−0.312
−0.697
0.289
0.481
0.049
−0.104
0.260The joint pdfs of(-D2) and curvature κm for cases P3 and V2 are shown in Figure 12 for c=0.3, 0.7, and 0.9 isosurfaces and the correlation coefficients between (-D2) and κm for different c isosurfaces across the flame are shown in Table 3 for all cases considered here. The joint pdfs of (-D2) and κm in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not shown here. It can be seen from Figure 12 that the quantities (-D2) and κm are nonlinearly related to one another. The physical explanations behind the observed behaviour are provided below.(i)
The molecular dissipation term(-D2) can alternatively be expressed as
(8)(-D2)=-2ρD2{[∂|∇c|∂n]2+4κm|∇c|∂|∇c|∂n+4κm2|∇c|2+2[∂2c∂x1∂x2∂2c∂x1∂x2-∂2c∂x1∂x1∂2c∂x2∂x2]+2[∂2c∂x1∂x3∂2c∂x1∂x3-∂2c∂x1∂x1∂2c∂x3∂x3]+2[∂2c∂x2∂x3∂2c∂x2∂x3-∂2c∂x2∂x2∂2c∂x3∂x3]}.Joint pdfs between(-D2)×δth2/ρ0SL2 and normalised curvature κm×δth on (a) c=0.3, (b) 0.7. and (c) 0.9 isosurfaces for case P3. Joint pdfs between (-D2)×δth2/ρ0SL2 and normalised curvature κm×δth on (d) c=0.3, (e) 0.7, and (f) 0.9 isosurfaces for case V2.
(a)
(b)
(c)
(d)
(e)
(f)The above expression clearly indicates that the third term on the right hand side of (8) (i.e., -8ρD2κm2|∇c|2) induces nonlinear curvature dependence of the molecular dissipation term (-D2).(ii)
The quantity∂|∇c|/∂n remains negative (positive) towards the unburned (burned) gas side of the flame [10, 59, 60]; thus the second term on the right hand side is positively (negatively) correlated with κm towards the unburned (burned) gas side of the flame. The first term on the right hand side of (8) can be taken to scale with (-ρNc2) (i.e., -2ρD2(∂|∇c|/∂n)2~-2ρNc2). It has already been shown that the joint pdfs of Nc and κm exhibit both positive and negative correlating branches for high values of u′/SL (see cases P5 and V3 in Figure 5(b)) and thus the joint pdf of -2ρD2(∂|∇c|/∂n)2 and κm is also expected to show branches with both positive and negative correlations in these cases. The weak negative correlation between Nc and κm for small values of u′/SL (see cases P1 and V1 in Figure 5(b)) leads to weak positive correlation between -2ρD2(∂|∇c|/∂n)2~-2ρNc2 and κm. The last three terms on the right hand side vanish in the limit of small scale isotropy and for the present cases they remain weakly correlated with curvature.The relative strengths of the above mechanisms determine the net curvature dependence of(-D2). Thus, both positive and negative correlations between (-D2) and κm have been observed within the flame front in all cases considered here.


## 3.8. Local Behaviour off(D) and Its Curvature and Strain Rate Dependences

The marginal pdfs off(D) for c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces across the flame front are shown in Figure 13(a) for cases P3 and V2. The pdfs of f(D) in cases P1, P2, P4, and P5 (cases V1 and V3) are qualitatively similar to those in case P3 (case V2) and thus are not explicitly shown here. It is evident from Figure 13(a) that f(D) predominantly assumes negative (positive) values towards the unburned (burned) gas side of the flame (see Figure 3). The density-weighted diffusivity ρD is considered to be constant in cases P1–P5 and thus TD1 and TD2 are identically zero in these cases. The marginal pdfs of TD3 and TD4 for case P3 are shown in Figures 13(b) and 13(c), which show that both TD3 and TD4 predominantly assume positive (negative) values towards burned (unburned) gas side of the flame. As TD5=T1/2 in cases P1–P5, the pdfs of TD5 are qualitatively similar to those of T1 and thus are not shown here. This indicates that TD5 shows predominant probability of finding positive values throughout the flame (see Figure 6). The pdfs of TD1,TD2,TD3,TD4, and TD5 for case V2 are shown in Figures 13(d)–13(h), respectively. It is evident from Figures 13(f) and 13(g) that both TD3 and TD4 assume positive (negative) values towards burned (unburned) gas side of the flame, whereas TD5 assumes positive values throughout the flame, which is qualitatively similar to the behaviour of the corresponding term in case P3, where ρD is assumed to be constant. Figures 13(d) and 13(e) show that both TD1 and TD2 assume predominantly positive (negative) values towards the unburned (burned) gas side of the flame.The marginal pdfs off(D)×δth2/ρ0SL2 for c=0.1, 0.3, 0.5, 0.7, and 0.9 for cases (a) P3 and V2. The marginal pdfs of (b) TD3×δth2/ρ0SL2 and (c) TD4×δth2/ρ0SL2 for c=0.1, 0.3, 0.5, 0.7, and 0.9 across the flame for case P3. The marginal pdfs of (d) TD1×δth2/ρ0SL2, (e) TD2×δth2/ρ0SL2, (f) TD3×δth2/ρ0SL2, (g) TD4×δth2/ρ0SL2, and (h) TD5×δth2/ρ0SL2 for c= 0.1, 0.3, 0.5, 0.7, and 0.9 across the flame for case V2.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)The contours of joint pdfs betweenf(D) and aT (κm) for c=0.1, 0.5, and 0.7 isosurfaces are shown in Figure 14 (Figure 15) for cases P3 and V2, and the correlation coefficients between f(D) and aT (κm) for different c isosurfaces across the flame for cases P3 and V2 are shown in Figures 16(a) and 16(b) (Figures 16(c) and 16(d)), respectively. Both Figures 14 and 16 indicate that f(D) and aT are negatively (positively) correlated with each other towards the unburned (burned) gas side of the statistically planar flame (i.e., case P3), whereas f(D) and aT are positively correlated with each other from the middle to the burned gas side in the V-flame case (i.e., case V2), but this correlation remains weak towards the unburned gas side. It is evident from Figure 15 that f(D) and κm remain weakly correlated for both statistically planar and V-flames, which is consistent with the correlation coefficient between f(D) and κm shown in Figures 16(c) and 16(d). However, there are qualitative differences in the joints pdfs between f(D) and κm for cases P3 and V2.Joint pdfs betweenf(D)×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL for case P3 on (a) c=0.1, (b) 0.5, and (c) 0.7 isosurfaces. Joint pdfs between f(D)×δth2/ρ0SL2 and normalised tangential strain rate aT×δth/SL for case V2 on (d) c=0.1, (e) 0.5, and (f) 0.7 isosurfaces.
(a)
(b)
(c)
(d)
(e)
(f)Joint pdfs betweenf(D)×δth2/ρ0SL2 and normalised curvature κm×δth on (a)c = 0.1, (b) 0.5, and (c) 0.7 isosurfaces for case P3. Joint pdfs between f(D)×δth2/ρ0SL2 and normalised curvature κm×δth on (d)c= 0.1, (e) 0.5, and (f) 0.7 isosurfaces for case V2.
(a)
(b)
(c)
(d)
(e)
(f)(a) Correlation coefficients for theTD3-aT, TD4-aT, TD5-aT, and f(D)-aT correlations on c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces for case P3. (b) Correlation coefficients for the TD1-aT, TD2-aT, TD3-aT, TD4-aT, TD5-aT, and f(D)-aT correlations on c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces for case V2. (c) Correlation coefficients for the TD3-κm, TD4-κm, TD5-κm, and f(D)-κm correlations on c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces for case P3. (d) Correlation coefficients for the TD1-κm, TD2-κm, TD3-κm, TD4-κm, TD5-κm, and f(D)-κm correlations on c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces for case V2.
(a)
(b)
(c)
(d)In order to explain the observed strain rate dependence off(D), the correlation coefficients between TD3, TD4, and TD5 (TD1,TD2,TD3,TD4, and TD5) with aT, for c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces, are also shown in Figure 16(a) (Figure 16(b)) for case P3 (case V2). It is evident from Figures 16(a) and 16(b) that both TD3 and TD4 remain negatively (positively) correlated with aT towards the unburned (burned) gas side of the flame.The strain rate dependences ofTD1,TD2,TD3,TD4, and TD5 can be explained in the following manner.(i)
The magnitudes ofTD3 and TD4 can be taken to scale as |TD3|~ρDNc/δ2~ρNc2 and |TD4|~ρDNc/δ2~ρNc2, which indicates that |TD3|~ρNc2 and |TD4|~ρNc2 remain positively correlated with aT due to positive correlation between Nc and aT (see Figure 5(a)). This suggests that the negative (positive) values of TD3 and TD4 (see Figures 13, 16(a), and 16(b)) lead to negative (positive) correlations of these terms with aT due to positive correlations between Nc and aT (also due to positive correlation between |TD3|~ρNc2 (|TD4|~ρNc2) and aT).(ii)
The termTD5 remains positively correlated with aT throughout the flame, which is consistent with the positive correlation between T1 and aT shown in Figure 6(c), as TD5=0.5T1 in statistically planar cases P1–P5 considered here (see Figure 16(a)). Even though ρD increases with increasing temperature in cases V1–V3, TD5 can still be taken to scale with T1 (i.e., TD5~T1~ρτNcSL/δ) and the positive correlation between T1 and aT leads to a positive correlation between TD5 and aT in case V2 and also in cases V1 and V3 (see Figure 16(b)).(iii)
The magnitudes of the termsTD1 and TD2 can be scaled as |TD1|~ρD2/δ4~ρNc2 and |TD2|~ρD2/δ4~ρNc2, respectively, which suggests that |TD1| and |TD2| are expected to be positively correlated with aT due to positive correlation between Nc and aT (see Figure 5(a)). As TD1 and TD2 assume predominantly positive (negative) values towards the unburned (burned) gas side of the flame, these terms scale with TD1~TD2~ρNc2 (TD1~TD2~-ρNc2) towards the reactant (product) side of the flame. Thus the positive correlation between Nc and aT leads to positive (negative) TD1-aT and TD2-aT correlations towards the unburned (burned) gas side of the flame (see Figure 16(b)).(iv)
In cases P1–P5, the termsTD3,TD4, and TD5 remain positively correlated with aT towards the burned gas side of the flame (see Figure 16(a)) and these positive correlations result in a net positive correlation between f(D) and aT towards the burned gas side of the flame. On the other hand, TD3 and TD4 remain negatively correlated with aT towards the unburned gas side of the flame (see Figure 16(a)) and these correlations dominate over the positive correlation between TD5 and aT to result in a net negative correlation between f(D) and aT towards the unburned gas side of the flame.(v)
In cases V1–V3, the termsTD3,TD4, and TD5 remain positively correlated with aT towards the burned gas side of the flame (see Figure 16(b)) and these positive correlations overcome the negative TD1-aT and TD2-aT correlations to give rise to a net positive correlation between f(D) and aT towards the burned gas side of the flame. By contrast, TD3 and TD4 remain negatively correlated with aT towards the unburned gas side of the flame (see Figure 16(b)) and these correlations oppose the positive TD1-aT,TD2-aT, and TD5-aT correlations to result in a weak correlation between f(D) and aT towards the unburned gas side of the flame.The correlation coefficients betweenTD3, TD4, and TD5 (TD1,TD2,TD3,TD4, and TD5) with κm for c=0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces are also shown in Figure 16(c) (Figure 16(d)) for case P3 (case V2). It is evident from Figures 16(c) and 16(d) that f(D) and κm are weakly correlated throughout the flame for both planar and V-flame cases. The curvature κm dependences of TD1,TD2,TD3,TD4,TD5, and f(D) can be explained in the following manner.(i)
BothTD3 and TD4 remain negatively (positively) correlated with aT towards the unburned (burned) gas side of the flame for both planar and V-flame cases (see Figures 16(a) and 16(b)), whereas aT and κm are negatively correlated throughout the flame. Thus, high (low) values of TD3 and TD4 are associated with high positive values of κm towards the unburned (burned) gas side of the flame, which gives rise to positive (negative) correlations of TD3 and TD4 with κm towards the unburned (burned) gas side.(ii)
AsTD5=0.5T1 for statistically planar flame cases (i.e., case P1–P5), a strong negative correlation between TD5 and κm has been observed nearc= 0.7 isosurface, which is consistent with the negative correlation between T1 and κm shown in Figure 6(e). Even though ρD increases with increasing temperature in V-flame cases (i.e., cases V1–V3), TD5 can still be taken to scale with T1 (i.e., TD5~T1~ρτNcSL/δ) and the negative correlation between T1 and κm (see Figure 6(f)) leads to a weak negative correlation between TD5 and κm in case V2 and also in cases V1 and V3.(iii)
BothTD1 and TD2 remain positively (negatively) correlated with aT towards the unburned (burned) gas side of the flame in case V2 (see Figure 16(b)), whereas aT and κm are negatively correlated throughout the flame. Thus, small (high) values of TD1 and TD2 are associated with high positive (negative) values of κm towards the unburned gas side of the flame, which gives rise to negative correlations of TD1 and TD2 with κm towards the unburned gas side in case V2. The combination of negative correlations of TD1 and TD2 with aT towards the burned gas side, as well as the strengthening of negative correlation between aT and κm towards the burned gas side of the flame, leads to the weakening of negative correlations of TD1 and TD2 with κm, as the burned gas side is approached and the correlation between TD2 with κm eventually becomes positive towards the burned gas side of the flame (see Figure 16(d)).(iv)
The termsTD3,TD4, and TD5 remain negatively correlated with κm towards the burned gas side of the flame and these negative correlations dominate over weak positive TD1-κm and TD2-κm correlations to result in a net negative correlation between f(D) and κm towards the burned gas side of the flame in case V2. The terms TD3 and TD4 remain positively correlated with κm towards the unburned gas side of the flame (see Figure 16(d)) and these correlations are opposed by the negative TD1-κm, TD2-κm, and TD5-κm correlations to result in a weak correlation between f(D) and κm towards the unburned gas side of the flame front in case V2 and other V-flame cases.


## 3.9. Modelling Significances

A modelled transport equation ofN~c may need to be solved alongside other modelled conservation equations in RANS/LES simulations, when the rate of generation of scalar gradients does not remain in equilibrium with its destruction rate. The local strain rate and curvature dependences of SDR are expected to play important roles in LES simulations, as the necessity of capturing local behaviour of N~c is particularly important in the context of LES. As N~c approaches Nc with decreasing filter width Δ (i.e., limΔ→0N~c=Nc) in the context of LES, the local resolved-scale strain rate and curvature dependences of N~c and the terms of its transport equation (i.e., T¯1,T¯2,T¯3,(-D¯2), and f(D)¯) are likely to be qualitatively similar to the local strain rate and curvature dependences of Nc and the terms of its transport equation (i.e., T1,T2,T3,(-D2), and f(D)), respectively. The above discussion suggests that the models for N~c,T¯1,T¯2,T¯3,(-D¯2), and f(D)¯ in LES should be developed in such a manner so that they are capable of capturing the resolved strain rate and curvature dependences of N~c,T¯1,T¯2,T¯3,(-D¯2), and f(D)¯ for a range of different filter widths and approach local strain rate and curvature dependences of Nc,T1,T2,T3,(-D2), and f(D) for small filter widths.It can further be observed from Figures5–16 that the local strain rate and curvature dependence of Nc and the terms of its transport equation for V-flames are found to be qualitatively similar to the behaviour observed for the statistically planar flame cases. As the SDR statistics are principally governed by the small-scale molecular processes, the local statistics of Nc and the terms of its transport equation are largely independent of the flow configuration. Thus, the models for N~c transport developed based on data extracted from a canonical configuration might broadly be applicable to different geometries.Moreover, it is worth noting thatρD is assumed to be constant in cases P1–P5 whereas ρD is taken to be temperature dependent in cases V1–V3. However, the statistical behaviours of Nc and the terms of its transport equation are found to be broadly similar qualitatively in all cases indicating that the models, which have been developed based on DNS databases with constant ρD, should at least be able to capture the qualitative trends of N~c transport.


## 4. Conclusions

The statistical behaviours of the instantaneous SDRNc and the terms of its transport equation have been analysed using simple chemistry DNS databases of statistically planar and V-flames for a range of different values of turbulent Reynolds number. It has been found that the mean and local behaviours of Nc and the terms of its transport equation are similar for both statistically planar and V-flames for the range of turbulent Reynolds number explored here. In all the cases, Nc is positively correlated with tangential strain rate aT throughout the flame. By contrast, Nc and local curvature κm remain negatively correlated for small values of u′/SL but the joint pdf of Nc and κm shows branches with both positive and negative correlation, and the net correlation becomes weak for high values of u′/SL. It has been found that the mean contributions of the density-variation term T1 and the molecular dissipation term (-D2) in the transport equation of Nc are the leading order source and sink, respectively. The mean value of the strain rate contribution T2 to the SDR transport is predominantly negative for the major part of the flame due to predominant ∇c alignment with the most extensive principal strain rate eα, although positive contributions of T2 were observed towards the unburned gas side for high u′/SL cases where ∇c preferentially aligns with the most compressive principal strain rate eγ. The mean value of the reaction rate contribution to the SDR transport T3 remains positive for the major part of the flame before assuming negative values towards the burned gas side. The mean contribution of the term originating due to the diffusivity gradient in the SDR transport f(D) remains negative towards the unburned gas side before assuming positive values towards the burned gas side of the flame.It has been found that the density variation termT1 remains positively correlated with tangential strain rate aT, whereas the correlation between T1 and the local curvature κm is negative throughout the flame. The strain rate term T2 is predominantly positively correlated with both aT and κm for the major part of the flame front. The qualitative nature of the local strain rate and curvature dependences of T3, (-D2), and f(D) change across the flame. The reaction rate contribution T3 and the tangential strain rate aT remain positively (negatively) correlated towards the unburned (burned) gas side of the flame. However, T3 is weakly correlated with curvature κm throughout the flame for all cases considered here. The molecular dissipation term (-D2) is negatively correlated with aT throughout the flame, whereas the joint pdf of (-D2) and κm shows branches with both positive and negative correlation and their qualitative behaviours change across the flame. The diffusivity gradient term f(D) and aT are found to be negatively (positively) correlated with each other towards the unburned (burned) gas side of the flame, whereas the joint pdfs of f(D) and κm show a weak positive (negative) correlation in the unburned (burned) gas side of the flame. Detailed physical explanations have been provided for the strain rate and curvature dependences of Nc and the terms of its transport equation. The qualitative nature of these statistics has been found to be unaltered for the range of turbulent Reynolds number Ret considered here, but the strength of the correlations is affected by Ret. Moreover, the local strain rate and curvature dependence of Nc and the terms of its transport equation for V-flames are found to be broadly similar qualitatively to the behaviour observed for the statistically planar flame cases. Moreover, the assumption of mass diffusivity variation with temperature has been shown not to affect the qualitative behaviour of SDR and its transport statistics.In the context of single step chemistry, the reaction progress variablec can be uniquely defined, but c can be defined based on different species mass fractions in the presence of detailed chemistry. However, the conclusions drawn in this analysis are unlikely to change if c is defined based on the mass fraction of a major reactant/product, which is closely correlated with density change and heat release. The present single-step Arrhenius type chemistry qualitatively captures the statistics of |∇c| transport obtained using detailed chemistry based simulations for the flames with global Lewis number close to unity. This can be substantiated by qualitative similarities between the strain rate and curvature dependence of the terms of |∇c| transport equation obtained from detailed chemistry [14] and single-step chemistry [15] based DNS simulations. Given the close relation between |∇c| and SDR, it can be expected that the conclusions drawn regarding SDR transport will at least be qualitatively valid for detailed chemistry based simulations.It is worth noting that the effects of differential diffusion of heat and mass are not addressed in the present analysis, and the presence of differential diffusion may have influences on the local strain rate and curvature dependences ofNc and the different terms of its transport equation. As the SDR statistics are principally governed by the small-scale molecular processes, the local statistics of Nc and the terms of its transport equation obtained from this analysis are expected to be qualitatively similar for higher values of Ret than the values of turbulent Reynolds number considered here. However, this analysis has been carried out for moderate values of Ret; thus further experimental and computational studies at large values of Ret in the presence of detailed chemistry are needed for further confirmation and deeper understanding of the statistics of SDR transport in turbulent premixed flames.



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*Source: 280671-2014-04-03.xml*