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

The statistical behaviours of the instantaneous scalar dissipation rateN c 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 Re t . The statistical behaviours of N c 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 T 1 is positive, whereas the molecular dissipation term (−D 2 ) acts as a leading order sink. The mean contribution of the strain rate term T 2 is predominantly negative for the cases considered here. The mean reaction rate contribution T 3 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 N c , T 1 , T 2 , T 3 , (−D 2 ), and f(D) have been analysed in terms of their marginal probability density functions (pdfs) and their joint pdfs with local tangential strain rate a T and curvature k m . Detailed physical explanations have been provided for the observed behaviour.


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 ẇ of a reaction progress variable  is directly related to the Favre mean/filtered value of SDR Ñ = ∇ ⋅ ∇/ [1][2][3][4], where  is the fluid density and  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 |∇| statistics in turbulent premixed flames [5][6][7][8][9][10][11][12][13][14][15][16].Thus, strain rate and curvature are expected to have appreciable influences on local statistics of SDR   and its transport.The transport equation of the instantaneous SDR of reaction progress variable   is given as [3,17] ( 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.,  1 = ∇⋅(∇  )) denotes molecular diffusion of SDR.The second term on the right hand side of (1a) (i.e.,  1 = −2∇ ⋅ ∇[ ẇ + ∇ ⋅ (∇)]/) 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.,  2 = −2(/  )(  /  )(/  )) represents the effects of fluid-dynamic straining, whereas the fourth term (i.e.,  3 = 2( ẇ /  )(/  )) denotes the reaction rate contribution to the SDR transport.The penultimate term on the right hand side of (1a) (i.e., − 2 = −2 2 ( 2 /    )( 2 /    )) denotes molecular dissipation of   , and the terms involving temporal and spatial gradients of diffusivity are collectively referred to as () (see (1b)).
Although the statistical behaviours of |∇| and the terms of its transport equation were analysed earlier, the terms of   transport equation are fundamentally different from the terms of the |∇| transport equation, which can be written for a given c isosurface in the following manner [11,[13][14][15][16]: where   = −(/  )/|∇| is the th component of flame normal vector and   = ( ẇ + ∇ ⋅ (∇))/|∇| is the local flame displacement speed.It is evident from (1a) and (1b) and (2) that the statistical behaviour of   transport is likely to be different from |∇| transport although the quantities   and |∇| are closely related to each other (i.e.,   = |∇| 2 ).It is often necessary to solve a transport equation for Ñ in the context of Reynolds averaged Navier-Stokes (RANS) simulations and LES [17][18][19][20][21][22][23][24][25][26][27][28][29][30] The terms  1 ,  2 ,  3 , (− 2 ), and () are unclosed and therefore it is important to understand the statistical behaviours of   ,  1 ,  2 ,  3 , (− 2 ), and () (since lim Δ → 0 Ñ =   , lim Δ → 0  1 =  1 , lim Δ → 0  2 =  2 , lim Δ → 0  3 =  3 , lim Δ → 0 (− 2 ) = (− 2 ), and lim Δ → 0 () = (), 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   and the terms of its transport equation (i.e.,  1 ,  2 ,  3 , and (− 2 )) are yet to be analysed in detail in the existing literature.This paper aims to address this gap by analysing local tangential strain rate   = (  −     )  /  and curvature   = 0.5(  /  ) (for the above definition of   , the flame elements convex towards the reactants has a positive curvature) dependences of   ,  1 ,  2 ,  3 , (− 2 ), and () at different locations within the flame using direct numerical simulations (DNS) data of turbulent premixed freely propagating statistically planar flame and turbulent Vflame configurations.In this respect, the main objectives of this study are as follows: (1) to analyse local statistical behaviours of instantaneous SDR (i.e.,   ) and the terms of its transport equation  1 ,  2 ,  3 , (− 2 ), and (); (2) to explain the observed strain rate and curvature dependences of   ,  1 ,  2 ,  3 , (− 2 ), and (); (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.

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, threedimensional 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 , which can be defined in terms of a suitable reactant (product) mass fraction , 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 36.1th × 24.1 th × 24 th is considered, where  th = ( ad −  0 )/ Max |∇ T|  is the thermal flame thickness with  ad ,  0 , and T being the adiabatic flame, unburned reactant, and instantaneous dimensional temperatures, respectively, and the subscript "" refers to the unstrained laminar flame quantities.For the thermochemistry used in cases P1-P5, the thermal flame thickness  th is found to be 1.785 0 /  (i.e.,  th = 1.785 0 /  ), where  0 is the mass diffusivity in the unburned gas.The simulation domain for cases P1-P5 is discretised using a uniform Cartesian grid of 345 × 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 timeadvancement 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., Δ ≤ , where Δ and  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][5][6][7][8][9][10][11][12][13][14][15][16][17][22][23][24][25][26][27][28][29][30][31][32].
The initial values of root-mean-square turbulent velocity fluctuation normalised by unstrained laminar burning velocity   /  , integral length scale to flame thickness ratio / th , turbulent Reynolds number Re  =  0   / 0 , Damköhler number Da =   /   th and Karlovitz number Ka = (  /  ) 3/2 (/ th ) −1/2 , heat release parameter  = ( ad −  0 )/ 0 , and Zel' dovich number  =  ac ( ad −  0 )/ 2 ad for cases P1-P5 are provided in Table 1, where  0 and  0 are the unburned gas density and viscosity, respectively, and  ac is the activation temperature.As Re  scales as Re  ∼ Da 2 Ka 2 [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  sim ≥ Max(  ,   ), where   = /  is the initial eddy turn over time and   =  th /  is the chemical time scale.In all cases, statistics were extracted after one chemical time scale   , which corresponds to a time equal to 2.0  in case P4, 3.0  in cases P1, P3, and P5, and 4.34  for case P2.It is worth noting that the chemical time scale   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   /  , / th , and  th / 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  th is greater than the Kolmogorov length scale  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.
The 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  = 29.7th , where  th = 3.563 0 /  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 timeintegration in cases V1-V3 are similar to those used for cases P1-P5.The flame holder centre is located at  1 = 3.48 th and has an approximate radius  = 1.2 th .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 Re ,inlet =  0   inlet / 0 , Karlovitz number Ka = (  inlet /  ) 3/2 (/ th ) −1/2 , and Damköhler number Da =   /  inlet  th based on the rootmean-square turbulent velocity fluctuation   inlet at the inlet are provided in Table 1 along with the values of  and .
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 time   = / in before data were collected for analysis, where  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 14.9 th ≤  1 ≤ 29.1 th 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 0.2  after the initial flow-through time.Standard values have been taken for Prandtl number (Pr = 0.7) and ratio of specific heats,  = 1.4.The global Lewis number is taken to be unity for all cases considered in this analysis.
The grid spacing Δ for all cases ensures 10 grid points within  th .As Karlovitz number can be scaled as Ka ∼  2 th / 2 , the grid spacing Δ can be taken to be Δ ≤  th /10 ∼  √ Ka/10.This indicates that /Δ ∼ 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, Δ remained Δ ∼ /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 (), dynamic viscosity (), and density-weighted mass diffusivity () 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.thickness.The isosurfaces of  representing the preheat zone (i.e.,  ≤ 0.5) show more distortion than the isosurfaces representing the reaction zones (i.e., 0.7 ≤  ≤ 0.9) due to penetration of turbulent eddies within the preheat zone.However, this tendency is more prevalent for high values of   /  and Ka (e.g., cases P3, P4, P5, and V3) but the isosurfaces of  remain mostly parallel to each other for small values of   /  and Ka (e.g., cases P1, P2, V1, and V2) indicating that the internal flame structure is weakly affected by turbulence in these cases.In order to understand the distribution of ⟨  ⟩ ×  th /  across the flame front, the variations of the mean values of the terms ⟨ 1 ⟩, ⟨ 2 ⟩, ⟨ 3 ⟩, ⟨(− 2 )⟩, and ⟨()⟩ conditional on  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, ⟨ 1 ⟩ remains positive throughout the flame.By contrast, ⟨(− 2 )⟩ assumes negative values throughout the flame in all cases as dictated by (1a).Expressing  =  0 /(1 + ) for low Mach number, unity Lewis number flames give rise to an alternative expression for  1 [3,17,25,28,29]:

Statistical
As dilatation rate   /  is predominantly positive in premixed flames, ⟨ 1 ⟩ for all values of  is positive across the flame and vanishes on both ends of the flame.
The quantity ⟨ 2 ⟩ assumes negative values throughout the flame front for cases P1 and V1.Although ⟨ 2 ⟩ 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  2 can be expressed in the following manner [3, 22-24, 28, 30]: where   ,   , and   are the most extensive, intermediate, and most compressive principal strain rates and , , and  are the angles of these principal strain rates with ∇.Equation ( 5) demonstrates that the predominant alignment of   (  ) with ∇ leads to a negative (positive) contribution to  2 .It has been discussed in the previous analyses [23,24,28,30] that the alignment of ∇ with   and   is determined by relative strengths of the strain rate induced by flame normal acceleration  chem and turbulent straining  turb .It has been demonstrated earlier that ∇ preferentially aligns with   (  ) when  chem ( turb ) dominates over  turb ( chem ).The strain rate induced by flame normal acceleration due to chemical heat release can be scaled as  chem ∼ (Ka)  / th , where (Ka) is expected to decrease with increasing Ka [43].Following Meneveau and Poinsot [44],  turb can be scaled as  turb ∼   /, which gives rise to  chem / turb ∼ (Ka)  /   th ∼ (Ka)Da ∼ (Re 1/2  /Da)Da.Alternatively, turbulent straining can be scaled as [45]  turb ∼   / (where  is the Taylor microscale), which yields ∼ (Ka)/Ka.The above scaling relations suggest that  chem strengthens with respect to  turb with increasing Da for a given value of Re  .Previous analyses [22-24, 28, 30] demonstrated that ∇ predominantly aligns with   for Da ≫ 1 flames, whereas ∇ aligns with   in Da < 1 flames for comparable values of Re  .Both  chem / turb ∼ (Ka)Da and  chem / turb ∼ (Re 1/2  /Da)Da/Re 1/2  ∼ (Ka)/Ka indicate that an increase in Ka ∼ Re 1/2  /Da for a given value of Da (e.g., cases P1, P3, and P5) gives rise to weakening of  chem in comparison to  turb .This increases the extent of ∇ alignment with   with increasing Ka when Da is held constant as in cases P1, P3, and P5.In cases P1 and P3, ∇ predominantly aligns with   ; however the extent of this alignment decreases from P1 to P3.This predominant alignment of ∇ with   in cases P1 and P3 leads to a negative contribution of ⟨ 2 ⟩ in these cases.In case P5, ∇ predominantly aligns with   in the unburned and fully burned gases but  chem overcomes  turb in the regions of intense heat release close to the middle of the flame and as a result ∇ aligns with   in the reaction zone.Thus the mean value of ⟨ 2 ⟩ in case P5 assumes positive values towards both the unburned and burned gas sides, whereas the mean contribution of ⟨ 2 ⟩ remains negative close to the middle of the flame.The relation  chem / turb ∼ (Ka)Da/Re 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 ∇ aligning with   in case P4 (case P3) than in case P3 (case P2).This leads to predominantly negative contribution of ⟨ 2 ⟩ in cases P2 and P3, whereas ⟨ 2 ⟩ assumes positive values towards the unburned and burned gas sides of the flame in case P4.However,  chem overcomes  turb in the regions of intense heat release at the middle of the flame and ∇ starts to align with   in the reaction zone giving rise to negative values of ⟨ 2 ⟩ in case P4.In cases V1 and V2, the values of  chem / turb ∼ (Ka)Da/Re 1/2  are larger than the corresponding value in case V3 (see the parameters in Table 1).Thus, the extent of ∇ alignment with   (  ) decreases (increases) from case V1 to case V3.This gives rise to positive values of ⟨ 2 ⟩ 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 ∇ alignment with   than in case V3.However, the mean contribution of ⟨ 2 ⟩ is negative in the middle of the flame for cases V1-V3 due to the alignment of ∇ with   in the heat releasing zone.The contribution of ⟨ 3 ⟩ remains positive (negative) towards the unburned (burned) gas side of the flame with the transition from positive to negative value taking place close to  ≈ 0.85.In order to explain this behaviour,  3 can be rewritten as where  is the spatial coordinate in the local flame normal direction and the flame normal vector ⃗  points towards the unburned gas side of the flame.For single step chemistry considered here, the maximum ẇ occurs close to  ≈ 0.85 [10,14].This suggests that the probability of finding negative (positive) values of  ẇ / is significant for  < 0.85 ( > 0.85), which gives rise to positive (negative) value of ⟨ 3 ⟩ towards the unburned (burned) gas side of the flame.
Figure 3 shows that ⟨()⟩ 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 ⟨()⟩ remains comparable to that of ⟨ 1 ⟩ in all cases indicating that ⟨()⟩ cannot be neglected even for cases P1-P5, where  is considered to be constant.In cases P1-P5,   [/ +   /  ] can be expressed using  =  0 /(1+) for globally adiabatic Le = 1.0 flames as  1 /2 =     /  (i.e.,   [/ +   /  ] =   (  /  ) for constant ) and the first two terms on the right hand side of (1b) vanish for constant values of .The contributions of ⟨( 3 + 4 )⟩ are responsible for the change in sign of ⟨()⟩ in cases P1-P5.These terms are also principally responsible for sign change of ⟨()⟩ in cases V1-V3.

Local Behaviour of 𝑁 𝑐 and Its Curvature and Strain Rate
Dependences.The marginal probability density functions (pdfs) of normalised  +  (i.e.,   ×  th /  ) for different  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   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   assumes small values in the preheat zone of the flame due to small magnitude of scalar gradient ∇.It is evident from Figures 4(a) and 4(b) that the pdfs of   are qualitatively similar for statistically planar and V-flames and in both cases the probability of finding high values of   is most prevalent in the middle of the flame with slight skewness towards the burned gas side (i.e.,  ≈ 0.7) and the probability of finding high values of   decreases on both unburned and burned gas sides of the flame front.This is consistent with the observed behaviour of the mean values of   conditional on  shown in Figure 2. It can be seen in Figure 4 that a log-normal distribution captures the qualitative behaviour of the pdf of   although there are some disagreements in the pdf tails.This is consistent with several previous experimental [46][47][48][49][50][51][52] and numerical [53][54][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 joint pdfs of   and tangential strain rate   for cases P1, P5, V1, and V3 are shown in Figure 5(a) for  = 0.8 isosurface, which is close to the most reactive region for the present thermochemistry.It can be seen from Figure 5(a) that   and   are positively correlated on  = 0.8 isosurface for cases P1, P5, V1, and V3 and similar qualitative behaviour has been observed also for other  isosurfaces in all cases considered here.This positive correlation between   and   can be explained in the following manner.
(i) The dilatation rate ∇ ⋅ ⃗  can be expressed as ∇ ⋅ ⃗  =   +   , where   =       /  is the normal strain rate.For unity Lewis number flames, ∇⋅ ⃗  can be scaled as ∇ ⋅ ⃗  ∼  chem ∼ (Ka)  / th , whereas   can be taken to scale with turbulent strain rate  turb (i.e.,   ∼  turb ∼   / according to Meneveau and Poinsot [44] and   ∼  turb ∼   / according to Tennekes and Lumley [45]).∼ (Ka)/Ka according to the scaling arguments by Meneveau and Poinsot [44] and Tennekes and Lumley [45], respectively.Both (iii) It has been shown in several previous analyses [10,30] that both ∇ ⋅ ⃗  and   assume predominantly positive values and thus a higher magnitude of   than ∇ ⋅ ⃗  induces a negative (i.e., compressive) normal strain rate   .Thus, an increase in   often leads to a decrease in   = ∇ ⋅ ⃗  −   for small (high) values of Da (Ka).Thus, the isoscalar lines come close to each other under the action of decreasing   , which leads to increase in the magnitude of scalar gradient ∇.This is reflected in the positive correlation between   and   .
The joint pdfs between   and curvature   for cases P1, P5, V1, and V3 are shown in Figure 5(b) for  = 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   and   exhibits both positive and negative correlating branches on  = 0.8 isosurface for cases P5 and V3, and as a result of this, the net correlation between   and   remains weak.The positive correlation branch between   and   remains weak for small values of   /  in statistically planar flames (see Figure 5(b) for case P1) and this branch disappears completely in the V-flames with small values of   /  (see Figure 5(b) for case V1).Similar behaviour is observed for other  isosurfaces in all cases considered here and the correlation between   and   is weak throughout the flame for high values of   /  (e.g., cases P3-P5 and V3).However, the disappearance of the positive correlating branch in the joint pdf of   and   in Figure 5(b) indicates that   and   are negatively correlated with each other throughout the flame for small values of   /  (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]    and   .This leads to a negative correlating branch between   and   at the positively curved zones. (ii) The dilatation rate ∇ ⋅ ⃗  is large in the negatively curved locations due to strong focussing of heat and the magnitude of ∇ ⋅ ⃗  can locally be high enough to supersede the magnitude of   , which leads to a positive value of   .This tendency strengthens with decreasing   , especially in the zones with large negative curvature, which gives rise to an increase in   with decreasing curvature.As the distance between the isoscalar lines increases with increasing   , the magnitude of scalar gradient ∇ decreases with decreasing   in the negatively curved zones.This leads to the positive correlating branch in the joint pdf of   and   (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 between   and   in the high   /  cases.
The probability of finding high negative curvature remains small for small values of   /  and as a result the probability of finding high values of ∇ ⋅ ⃗ , which locally overcomes   , to induce a positive value of   , becomes rare (e.g., cases P1 and V1).Thus the combination of positive correlations between   and   and negative correlations between   and   leads to a predominantly negative correlating branch between   and   in the low   /  cases (e.g., cases P1 and V1; see Figure 5(b)).
The strain rate and curvature dependences of   discussed above, in turn, affect the local statistical behaviours of  1 ,  2 ,  3 , (− 2 ), and () in response to   and   .The curvature and strain rate dependences of  1 ,  2 ,  3 , (− 2 ), and () are discussed next.As dilatation rate ∇ ⋅ ⃗  is principally positive due to thermal expansion in premixed flames [10,30], the contribution of  1 = 2(∇ ⋅ ⃗ )  is predominantly positive throughout the flame.Moreover, Figures 6(a) and 6(b) demonstrate that the probability of finding high values of  1 is most prevalent in the middle of the flame with slight skewness towards the burned gas side (i.e.,  ≈ 0.7) and the probability of finding high values of  1 decreases on both unburned and burned gas sides of the flame.This is consistent with the observed behaviour of the mean values of  1 conditional on  shown in Figure 3.The probability of finding large magnitudes of ∇ ⋅ ⃗  is the highest at a location which is slightly skewed towards the burned gas side of the flame [30].As the distributions of   and ∇ ⋅ ⃗  are slightly skewed towards the burned gas side of the flame, the probability of finding large values of  1 = 2(∇ ⋅ ⃗ )  becomes high around  ≈ 0.7.The joint pdfs between  1 and   for cases P3 and V2 are shown in Figures 6(c) and 6(d), respectively, for  = 0.8 isosurface.It can be seen from Figures 6(c) and 6(d) that  1 and   are positively correlated on  = 0.8 isosurface for cases P3 and V2 and similar qualitative behaviours have been observed for other  isosurfaces in all cases considered here.Both ∇ ⋅ ⃗  and   are positively correlated for all flames considered here, which along with positive correlation between   and   (see Figure 5) gives rise to a positive correlation between  1 = 2(∇ ⋅ ⃗ )  and   .The joint pdfs between  1 and   for cases P3 and V2 are shown in Figures 6(e) and 6(f), respectively, for  = 0.8 isosurface.It can be seen from Figures 6(e) and 6(f) that the joint pdf between  1 and   exhibits a negative correlation on  = 0.8 isosurface for cases P3 and V2, and similar qualitative behaviour has been observed for other  isosurfaces in all cases considered here.In all cases, the net correlation between   and   is weak (see Figure 5(b)), but ∇ ⋅ ⃗  assumes high (small) values at negatively (positively) curved locations because of focussing (defocusing) of heat.This leads to a predominantly negative correlation between ∇⋅ ⃗  and   [57].The negative correlation between ∇⋅ ⃗  and   is principally responsible for the negative correlation between  1 = 2(∇ ⋅ ⃗ )  and   .  2 that  2 and   are positively correlated for high   /  cases (e.g., cases P5 and V3) although the strength of the correlation changes through the flame.However,  2 and   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   /  (see Table 2).In order to explain

Local
Based on (7) the strain rate dependences of  2 can be explained in the following manner.
(i) It has already been demonstrated that   and   are positively correlated with each other (see Figure 5(a)).The quantity (−  ) =   − ∇ ⋅ ⃗  tends to increase with increasing   in the regions where the effects of ∇ ⋅ ⃗  are weak.This along with positive correlation between   and   leads to a positive correlation between  2 and   for both unburned and burned gas sides of the flame for all cases.
(ii) The magnitudes of ∇ ⋅ ⃗  and   increase with decreasing   , and thus (−  ) =   − ∇ ⋅ ⃗  might not increase (even decrease) with increasing   in the heat releasing zone of the flame where the effects of ∇ ⋅ ⃗  are strong.The   dependences of (−  ) and   ultimately determine the nature of the correlation between  2 and   .The strain rate and curvature dependences of ∇ ⋅ ⃗  weaken with increasing   /  [58], so (−  ) =   − ∇ ⋅ ⃗  increases with increasing   , which leads to a positive correlation between  2 and   for the major portion of the flame for cases with high values of   /  (see Table 2).
The joint pdfs between  2 and   for cases P3 and V2 are shown in Figures 7(e) and 7(f), respectively, for  = 0.8 isosurface and the correlation coefficients between  2 and   for different  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  2 and   remain weakly positively correlated except the burned gas side of the flame.The observed curvature dependence of  2 could be explained based on the following physical mechanisms.
(i) The effects of dilatation rate ∇ ⋅ ⃗  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  chem are more likely to dominate over the effects of  turb in the negatively curved zones, which increase the extent of ∇ alignment with   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 ∇ alignment with   (  ) in the positively curved zones.The extent of ∇ alignment with   increases in the negatively curved zones, which in turn makes  2 increasingly negative (see (5)) and the magnitude of the negative contribution of  2 decreases for positive curvature locations.This gives rise to a positive correlation between  2 and   , as observed from Figures 7(e) and 7(f) and Table 2.
(ii) However, the effects of  turb are more likely to dominate over the effects of  chem towards the burned gas side and thus the extent of ∇ alignment with   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 ∇ ⋅ ⃗  towards the burned gas side, which can give rise to an increasing extent of ∇ alignment with   increases in the negative curved zones.This in turn gives rise to an increase in  2 (see ( 5)) with decreasing   towards the burned gas side and leads to a negative correlation between  2 and   (see Table 2).

Local Behaviour of 𝑇 3 and Its Curvature and Strain
Rate Dependences.The marginal pdfs of normalised  3 for different  isosurfaces across the flame are shown in Figure 8 for cases P3 and V2, respectively.The pdfs of  3 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  < 0.5 are not shown in Figure 8 because  3 assumes negligible value in the preheat zone of the flame due to negligible magnitude of ẇ .It is evident that  3 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.,  = 0.7 in Figure 8) of the flame front.However,  3 assumes negative values only towards the burned gas side (e.g.,  = 0.9) of the flame front for both planar and Vflames.This is consistent with the behaviour of ⟨ 3 ⟩ shown in Figure 3.The physical mechanism behind the transition from positive to negative values of the mean contribution of  3 (see (6)) is also responsible for obtaining negative (positive) values of  3 towards the burned (unburned) gas side of the flame.
The contours of joint pdfs between  3 and   for  = 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  3 and   remain positively correlated for the part of the flame where finding positive values of  3 is prevalent.On the other hand,  3 and   are negatively correlated with each other towards the burned gas side of the flame where  3 is predominantly negative.The observed   dependence of  3 can be explained in the following manner.The joint pdfs between  3 and   for cases P3 and V2 are shown in Figure 10 for  = 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  3 and   exhibits both positive and negative correlating branches and the net correlation is weak throughout the flame.The physical explanations for the observed   dependence of  3 can be summarised in the following manner.

Local Behaviour of (−𝐷 2 ) and Its Curvature and Strain
Rate Dependences.The marginal pdfs of (− 2 ) for  isosurfaces representative of leading edge, reaction zone, and trailing edge of the flame (e.g.,  = 0.3, 0.7 and 0.9 isosurfaces) are shown in Figure 11(a) for cases P3 and V2.The pdfs of (− 2 ) 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 (− 2 ) assumes negative values throughout the flame and the probability of finding high magnitude of (− 2 ) 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 (− 2 ) shown in Figure 3.It can further be seen from Figure 11(a) that the pdfs of (− 2 ) for statistically planar and V-flames are qualitatively similar to each other.The contours of joint pdfs between (− 2 ) and   for  = 0.8 isosurface are shown in Figures 11(b) and 11(c) for cases P3 and V2 and the correlation coefficients between (− 2 ) and   for different  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 (− 2 ) and   are predominantly negatively correlated throughout the flame but the strength of this negative correlation weakens with increasing   /  and the correlation becomes weakly positive at the middle of the flame for high values of   /  (e.g., cases P4 and P5).This behaviour can be explained in the following manner.
The joint pdfs of (− 2 ) and curvature   for cases P3 and V2 are shown in Figure 12 for  = 0.3, 0.7, and 0.9 isosurfaces and the correlation coefficients between (− 2 ) and   for different  isosurfaces across the flame are shown in Table 3 for all cases considered here.The joint pdfs of (− 2 ) and   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 (− 2 ) and   are nonlinearly related to one another.The physical explanations behind the observed behaviour are provided below.
(i) The molecular dissipation term (− 2 ) can alternatively be expressed as The above expression clearly indicates that the third term on the right hand side of (8) (i.e., −8 2  2  |∇| 2 ) induces nonlinear curvature dependence of the molecular dissipation term (− 2 ).
(ii) The quantity |∇|/ 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   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 (− 2  ) (i.e., −2 2 (|∇|/) 2 ∼ −2 2  ).It has already been shown that the joint pdfs of   and   exhibit both positive and negative correlating branches for high values of   /  (see cases P5 and V3 in Figure 5(b)) and thus the joint pdf of −2 2 (|∇|/) 2 and   is also expected to show branches with both positive and negative correlations in these cases.The weak negative correlation between   and   for small values of   /  (see cases P1 and V1 in Figure 5(b)) leads to weak positive correlation between −2 2 (|∇|/) 2 ∼ −2 2   and   .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 (− 2 ).Thus, both positive and negative correlations between (− 2 ) and   have been observed within the flame front in all cases considered here.

Local Behaviour of 𝑓(𝐷) and Its Curvature and Strain
Rate Dependences.The marginal pdfs of () for  = 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 () 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 () predominantly assumes negative (positive) values towards the unburned (burned) gas side of the flame (see Figure 3).The density-weighted diffusivity  is considered to be constant in cases P1-P5 and thus  1 and  2 are identically zero in these cases.The marginal pdfs of  3 and  4 for case P3 are shown in Figures 13(b) and 13(c), which show that both  3 and  4 predominantly assume positive (negative) values towards burned (unburned) gas side of the flame.As  5 =  1 /2 in cases P1-P5, the pdfs of  5 are qualitatively similar to those of  1 and thus are not shown here.This indicates that  5 shows predominant probability of finding positive values throughout the flame (see Figure 6).The pdfs of  1 ,  2 ,  3 ,  4 , and  5 for case V2 are shown in Figures 13(d)-13(h), respectively.It is evident from Figures 13(f) and 13(g) that both  3 and  4 assume positive (negative) values towards burned (unburned) gas side of the flame, whereas  5 assumes positive values throughout the flame, which is qualitatively similar to the behaviour of the corresponding term in case P3, where  is assumed to be constant.Figures 13(d) and 13(e) show that both  1 and  2 assume predominantly positive (negative) values towards the unburned (burned) gas side of the flame.
The contours of joint pdfs between () and   (  ) for  = 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 () and   (  ) for different  isosurfaces across the flame for cases P3 and V2 are shown in Figures 16(a 15 that () and   remain weakly correlated for both statistically planar and V-flames, which is consistent with the correlation coefficient between () and   shown in Figures 16(c) and 16(d).However, there are qualitative differences in the joints pdfs between () and   for cases P3 and V2.In order to explain the observed strain rate dependence of (), the correlation coefficients between  3 ,  4 , and  5 ( 1 ,  2 ,  3 ,  4 , and  5 ) with   , for  = 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  3 and  4 remain negatively (positively) correlated with   towards the unburned (burned) gas side of the flame.
The strain rate dependences of  1 ,  2 ,  3 ,  4 , and  5 can be explained in the following manner.
(i) The magnitudes of  3 and  4 can be taken to scale as  5(a)).This suggests that the negative (positive) values of  3 and  4 (see Figures 13,16 (iii) The magnitudes of the terms  1 and  2 can be scaled as , respectively, which suggests that | 1 | and | 2 | are expected to be positively correlated with   due to positive correlation between   and   (see Figure 5(a)).As  1 and  2 assume predominantly positive (negative) values towards the unburned (burned) gas side of the flame, these terms scale with  1 ∼  2 ∼  2  ( 1 ∼  2 ∼ − 2  ) towards the reactant (product) side of the flame.Thus the positive correlation between   and   leads to positive (negative)  1 −   and  2 −   correlations towards the unburned (burned) gas side of the flame (see Figure 16(b)).
(iv) In cases P1-P5, the terms  3 ,  4 , and  5 remain positively correlated with   towards the burned gas side of the flame (see Figure 16(a)) and these positive correlations result in a net positive correlation between () and   towards the burned gas   The correlation coefficients between  3 ,  4 , and  5 ( 1 ,  2 ,  3 ,  4 , and  5 ) with   for  = 0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces are also shown in Figure 16(c strain rate and curvature dependence of   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   and the terms of its transport equation are largely independent of the flow configuration.Thus, the models for Ñ transport developed based on data extracted from a canonical configuration might broadly be applicable to different geometries.
Moreover, it is worth noting that  is assumed to be constant in cases P1-P5 whereas  is taken to be temperature dependent in cases V1-V3.However, the statistical behaviours of   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 , should at least be able to capture the qualitative trends of Ñ transport.

Conclusions
The statistical behaviours of the instantaneous SDR   and the terms of its transport equation have been analysed using simple chemistry DNS databases of statistically planar and Vflames for a range of different values of turbulent Reynolds number.It has been found that the mean and local behaviours of   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,   is positively correlated with tangential strain rate   throughout the flame.By contrast,   and local curvature   remain negatively correlated for small values of   /  but the joint pdf of   and   shows branches with both positive and negative correlation, and the net correlation becomes weak for high values of   /  .It has been found that the mean contributions of the density-variation term  1 and the molecular dissipation term (− 2 ) in the transport equation of   are the leading order source and sink, respectively.The mean value of the strain rate contribution  2 to the SDR transport is predominantly negative for the major part of the flame due to predominant ∇ alignment with the most extensive principal strain rate   , although positive contributions of  2 were observed towards the unburned gas side for high   /  cases where ∇ preferentially aligns with the most compressive principal strain rate   .The mean value of the reaction rate contribution to the SDR transport  3 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 () 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 term  1 remains positively correlated with tangential strain rate   , whereas the correlation between  1 and the local curvature   is negative throughout the flame.The strain rate term  2 is predominantly positively correlated with both   and   for the major part of the flame front.The qualitative nature of the local strain rate and curvature dependences of  3 , (− 2 ), and () change across the flame.The reaction rate contribution  3 and the tangential strain rate   remain positively (negatively) correlated towards the unburned (burned) gas side of the flame.However,  3 is weakly correlated with curvature   throughout the flame for all cases considered here.The molecular dissipation term (− 2 ) is negatively correlated with   throughout the flame, whereas the joint pdf of (− 2 ) and   shows branches with both positive and negative correlation and their qualitative behaviours change across the flame.The diffusivity gradient term () and   are found to be negatively (positively) correlated with each other towards the unburned (burned) gas side of the flame, whereas the joint pdfs of () and   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   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 Re  considered here, but the strength of the correlations is affected by Re  .Moreover, the local strain rate and curvature dependence of   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 variable  can be uniquely defined, but  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  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 |∇| 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 |∇| transport equation obtained from detailed chemistry [14] and singlestep chemistry [15] based DNS simulations.Given the close relation between |∇| 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 of   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   and the terms of its transport equation obtained from this analysis are expected to be qualitatively similar for higher values of Re  than the values of turbulent Reynolds number considered here.However, this analysis has been carried out for moderate values of Re  ; thus further experimental and computational studies at large values of Re  in the presence of detailed chemistry are needed for further confirmation and deeper understanding of the statistics of SDR transport in turbulent premixed flames.

Figure 2 :
Figure 2: Variation of the mean value of ⟨  ⟩ ×  th /  conditional on  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.

Figure 4 :
Figure 4: The marginal pdf of normalised  +  (i.e.,   ×  th /  ) and the log-normal distribution in log-log scale for  = 0.5, 0.7, and 0.9 across the flame for cases (a) P3 and (b) V2.
1 and Its Curvature and Strain Rate Dependences.The marginal pdfs of  1 for different  isosurfaces across the flame are shown in Figures 6(a) and 6(b) for cases P3 and V2, respectively.The pdfs of  1 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  1 are qualitatively similar for statistically planar and V-flames and in both cases  1 = 2(∇ ⋅ ⃗ )  assumes predominantly positive values throughout the flame.
(i) It has been demonstrated earlier that   and   are positively correlated with each other which suggests that |∇| = |/| increases with increasing   .For low Mach number, unity Lewis number flames ẇ depend only on  and thus high values of | ẇ /| are associated with high values of |∇| = |/| and   .(ii) As   and   are positively correlated with each other, the magnitude of reaction rate contribution | 3 | = |2( ẇ /)|∇|| is positively correlated with tangential strain rate   .Thus,  3 is positively (negative) correlated with   , where  3 assumes positive (negative) values.
(i) The term | 3 | = |2( ẇ /)|∇|| is expected to be positively (negatively) correlated with curvature   at negatively (positively) curved locations for high values of   /  , as in the case of   (see cases P5 and V3 in Figure 5(b)), because high values of | ẇ /| are associated with high values of   and |∇| = |/|.(ii) As a result of the aforementioned physical mechanisms, the term  3 and   remain positively (negatively) correlated with curvature   at negatively (positively) curved locations in the planar flames where  3 assumes positive values.By contrast, the joint pdfs of  3 and   exhibit negative (positive) correlation with curvature   at negatively (positively) curved locations within the flame where  3 assumes negative values for the planar flames considered here (see Figure 10(c)).However,   remains predominantly negatively correlated with   for Vflame cases (see Figure 5(b)) and thus  3 shows positive (negative) correlation with curvature where  3 assumes negative (positive) values (see Figures 10(d)-10(f)).
(i) The instantaneous SDR   and the molecular dissipation term (− 2 ) can be taken to scale as   ∼ / 2 and (− 2 ) ∼ (− 2 / 4 ) ∼ (− 2  ) (where  is the typical local flame thickness) because in premixed flames.the gradient of progress variable is
(a), and 16(b)) lead to negative (positive) correlations of these terms with   due to positive correlations between   and   (also due to positive correlation between| 3 | ∼  2  (| 4 | ∼  2 ) and   ).(ii) The term  5 remains positively correlated with   throughout the flame, which is consistent with the positive correlation between  1 and   shown in Figure6(c), as  5 = 0.5 1 in statistically planar cases P1-P5 considered here (see Figure16(a)).Even though  increases with increasing temperature in cases V1-V3,  5 can still be taken to scale with  1 (i.e.,  5 ∼  1 ∼     /) and the positive correlation between  1 and   leads to a positive correlation between  5 and   in case V2 and also in cases V1 and V3 (see Figure16(b)).

Table 1 :
Initial values of simulation parameters and nondimensional numbers relevant to the DNS database considered here.
Behaviour of the Mean Values of   and the Unclosed Terms of Its Transport Equation.The variations of ⟨  ⟩ ×  th /  with  for statistically planar flames and  ⟩ ×  th /  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 ⟨  ⟩ ×  th /  is skewed slightly towards the burned gas side of the flame (i.e.,  ≈ 0.7).The peak magnitude of ⟨  ⟩ ×  th /  does not change significantly in response to   /  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 ⟨  ⟩ × th /  values for the cases considered here for both statistically planar and V-flame configurations.
) demonstrated that both   and ∇ ⋅ ⃗  remain negatively correlated with   in turbulent premixed flames, and thus the behaviour of   at locations with large positive curvature is principally determined by   since ∇ ⋅ ⃗  is small in these zones due to defocusing of heat.Small values of   are associated with high values of   at these locations, which lead to small values of   at high values of positive   due to positive correlation between (5)aviour of  2 and Its Curvature and Strain Rate Dependences.The marginal pdfs of  2 for different  isosurfaces across the flame are shown in Figures 7(a) and 7(b) for cases P3 and V2, respectively.The pdfs of  2 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  2 supersedes the probability of finding positive values.The probability of finding negative values of  2 increases as the heat releasing zone (see the pdfs for  = 0.7 isosurface) is approached.It has been discussedearlier that the effects of  chem overcome the effects of  turb in the heat releasing zone to give rise to a preferential alignment of ∇ with   even for small values of Da.This preferential alignment of ∇ with   in these zones gives rise to negative values of  2 according to(5).The extent of ∇ alignment with   (  ) decreases (increases) towards both unburned and burned gas sides of the flame due to diminishing effects of  chem .The contours of joint pdfs between  2 and   for  = 0.8 are shown in Figures7(c) and 7(d) for cases P3 and V2 and the correlation coefficients between  2 and   for different  isosurfaces across the flame for all cases are shown in Table 2.It is evident from Figures 7(c) and 7(d) and Table 2th / 0 2 and normalised tangential strain rate   ×  th /  on  = 0.8 isosurface for cases (c) P3 and (d) V2.Joint pdf between  1 × 2 th / 0  2  and normalised curvature   ×  th on  = 0.8 isosurface for cases (e) P3 and (f) V2.
2th / 0  2  for  = 0.1, 0.3, 0.5, 0.7, and 0.9 for cases (a) P3 and V2.The marginal pdfs of (b)  3 × 2 th / 0  2  and (c)  4 ×  2 th / 0 S 2  for  = 0.1, 0.3, 0.5, 0.7, and 0.9 across the flame for case P3.The marginal pdfs of (d)  1 ×  2 th / 0  2  , (e)  2 ×  2 th / 0  2  , (f)  3 ×  2 th / 0  2  , (g)  4 ×  2 th / 0  2  , and (h)  5 ×  2 th / 0  2  for  = 0.1, 0.3, 0.5, 0.7, and 0.9 across the flame for case V2.Both Figures 14 and 16 indicate that () and   are negatively (positively) correlated with each other towards the unburned (burned) gas side of the statistically planar flame (i.e., case P3), whereas () and   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 2  and | 4 | ∼   / 2 ∼  2  , which indicates that | 3 | ∼  2  and | 4 | ∼  2  remain positively correlated with   due to positive correlation between   and   (see Figure ) (Figure16(d)) for case P3 (case V2).It is evident from Figures16(c) and16(d)that () and   are weakly correlated throughout the flame for both planar and V-flame cases.The curvature   dependences of  1 ,  2 ,  3 ,  4 ,  5 , and () can be explained in the following manner.(i)Both  3 and  4 remain negatively (positively) correlated with   towards the unburned (burned) gas side of the flame for both planar and V-flame cases (see Figures 16(a) and 16(b)), whereas   and   are negatively correlated throughout the flame.Thus, high (low) values of  3 and  4 are associated with high positive values of   towards the unburned (burned) gas side of the flame, which gives rise to positive (negative) correlations of  3 and  4 with   towards the unburned (burned) gas side.(ii) As  5 = 0.5 1 for statistically planar flame cases (i.e., case P1-P5), a strong negative correlation between  5 and   has been observed near c = 0.7 isosurface, which is consistent with the negative correlation between  1 and   shown in Figure 6(e).Even though  increases with increasing temperature in V-flame cases (i.e., cases V1-V3),  5 can still be taken to scale with  1 (i.e.,  5 ∼  1 ∼     /) and the negative correlation between  1 and   (see Figure 6(f)) leads to a weak negative correlation between  5 and   in case V2 and also in cases V1 and V3.(iii) Both  1 and  2 remain positively (negatively) correlated with   towards the unburned (burned) gas side of the flame in case V2 (see Figure 16(b)), whereas   and   are negatively correlated throughout the flame.Thus, small (high) values of  1 and  2 are associated with high positive (negative) values of   towards the unburned gas side of the flame, which gives rise to negative correlations of  1 and  2 with   towards the unburned gas side in case V2.The combination of negative correlations of  1 and  2 with   towards the burned gas side, as well as the strengthening of negative correlation between   and   towards the burned gas side of the flame, leads to the weakening of negative correlations of  1 and  2 with   , as the burned gas side is approached and the correlation between  2 with   eventually becomes positive towards the burned gas side of the flame (see Figure 16(d)).(iv) The terms  3 ,  4 , and  5 remain negatively correlated with   towards the burned gas side of the flame and these negative correlations dominate over weak positive  1 −   and  2 −   correlations to result in a net negative correlation between () and   towards the burned gas side of the flame in case V2.The terms  3 and  4 remain positively correlated with   towards the unburned gas side of the flame (see Figure 16(d)) and these correlations are opposed by the negative  1 −   ,  2 −   , and  5 −   correlations to result in a weak correlation between () and   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 of Ñ 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 Ñ is particularly important in the context of LES.As Ñ approaches   with decreasing filter width Δ (i.e., lim Δ → 0 Ñ =   ) in the context of LES, the local resolved-scale strain rate and curvature dependences of Ñ and the terms of its transport equation (i.e.,  1 ,  2 ,  3 , (− 2 ), and ()) are likely to be qualitatively similar to the local strain rate and curvature dependences of   and the terms of its transport equation (i.e.,  1 ,  2 ,  3 , (− 2 ), and ()), respectively.The above discussion suggests that the models for Ñ ,  1 ,  2 ,  3 , (− 2 ), and () in LES should be developed in such a manner so that they are capable of capturing the resolved strain rate and curvature dependences of Ñ ,  1 ,  2 ,  3 , (− 2 ), and () for a range of different filter widths and approach local strain rate and curvature dependences of   ,  1 ,  2 ,  3 , (− 2 ), and () for small filter widths.It can further be observed from Figures 5-16 that the local