#### Abstract

The statistical behaviours of the instantaneous scalar dissipation rate of reaction progress variable 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 . The statistical behaviours of 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 is positive, whereas the molecular dissipation term acts as a leading order sink. The mean contribution of the strain rate term is predominantly negative for the cases considered here. The mean reaction rate contribution is positive (negative) towards the unburned (burned) gas side of the flame, whereas the mean contribution of the diffusivity gradient term assumes negative (positive) values towards the unburned (burned) gas side. The local statistical behaviours of , , , , , and have been analysed in terms of their marginal probability density functions (pdfs) and their joint pdfs with local tangential strain rate and curvature . Detailed physical explanations have been provided for the observed behaviour.

#### 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 of a reaction progress variable is directly related to the Favre mean/filtered value of SDR [1–4], 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 [5–16]. 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] where

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., ) denotes molecular diffusion of SDR. The second term on the right hand side of (1a) (i.e., ) 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., ) represents the effects of fluid-dynamic straining, whereas the fourth term (i.e., ) denotes the reaction rate contribution to the SDR transport. The penultimate term on the right hand side of (1a) (i.e., ) 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, 13–16]:
where is the th component of flame normal vector and 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., ).

It is often necessary to solve a transport equation for in the context of Reynolds averaged Navier-Stokes (RANS) simulations and LES [17–30]. The transport equation for can be obtained by Reynolds averaging or LES filtering of (1a) and (1b) as

The terms , , , and are unclosed and therefore it is important to understand the statistical behaviours of , , , , , and (since , , , , , and , where 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., , , , and ) 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 (for the above definition of , the flame elements convex towards the reactants has a positive curvature) dependences of , , , , , and 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., ) and the terms of its transport equation , , , , and ;(2)to explain the observed strain rate and curvature dependences of , , , , , 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.

#### 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 , which can be defined in terms of a suitable reactant (product) mass fraction as , 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 is considered, where is the thermal flame thickness with , , and 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 is found to be (i.e., ), where is the mass diffusivity in the unburned gas.

The simulation domain for cases P1–P5 is discretised using a uniform Cartesian grid of . 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., , where 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 [4–17, 22–32].

The initial values of root-mean-square turbulent velocity fluctuation normalised by unstrained laminar burning velocity , integral length scale to flame thickness ratio , turbulent Reynolds number , Damköhler number and Karlovitz number , heat release parameter , and Zel’dovich number for cases P1–P5 are provided in Table 1, where and are the unburned gas density and viscosity, respectively, and is the activation temperature. As scales as [35], the change in turbulent Reynolds number in cases P1–P5 is brought about by modifying and independently of each other (e.g., () 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 , where is the initial eddy turn over time and is the chemical time scale. In all cases, statistics were extracted after one chemical time scale , which corresponds to a time equal to in case P4, in cases P1, P3, and P5, and 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 , , and 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 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 , where for the thermochemistry used in these cases. A Cartesian grid of 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 and has an approximate radius . 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 , Karlovitz number , and Damköhler number based on the root-mean-square turbulent velocity fluctuation 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 before data were collected for analysis, where 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 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 after the initial flow-through time. Standard values have been taken for Prandtl number and ratio of specific heats, . The global Lewis number is taken to be unity for all cases considered in this analysis.

The grid spacing for all cases ensures 10 grid points within . As Karlovitz number can be scaled as , the grid spacing can be taken to be . This indicates that assumes the smallest value in case P5 amongst the cases considered here as the value is the highest in case P5. For case P5, remained 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 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.

#### 3. Results and Discussion

##### 3.1. Flame-Turbulence Interaction

The contours of in the central 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 . Turbulent eddies penetrate into the preheat zone in the thin reaction zones regime combustion (), but the reaction zone remains unperturbed because the Kolmogorov length scale is larger than the reaction zone thickness. The isosurfaces of representing the preheat zone (i.e., ) show more distortion than the isosurfaces representing the reaction zones (i.e., ) due to penetration of turbulent eddies within the preheat zone. However, this tendency is more prevalent for high values of and (e.g., cases P3, P4, P5, and V3) but the isosurfaces of remain mostly parallel to each other for small values of and (e.g., cases P1, P2, V1, and V2) indicating that the internal flame structure is weakly affected by turbulence in these cases.

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

The variations of with for statistically planar flames and V-flames are shown in Figures 2(a) and 2(b), where indicates the mean value of , which is obtained by ensemble averaging the quantity in question on a given 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 should not be confused with either Reynolds averaging or conventional conditional averaging operation in the context of RANS simulations because is evaluated using all the samples for a given value over the whole domain. Figures 2(a) and 2(b) show that the variations of 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 is skewed slightly towards the burned gas side of the flame (i.e., ). The peak magnitude of 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 values for the cases considered here for both statistically planar and V-flame configurations. In order to understand the distribution of across the flame front, the variations of the mean values of the terms , , 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, remains positive throughout the flame. By contrast, assumes negative values throughout the flame in all cases as dictated by (1a). Expressing for low Mach number, unity Lewis number flames give rise to an alternative expression for [3, 17, 25, 28, 29]:

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As dilatation rate is predominantly positive in premixed flames, for all values of is positive across the flame and vanishes on both ends of the flame.

The quantity assumes negative values throughout the flame front for cases P1 and V1. Although 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 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 .

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 and turbulent straining . It has been demonstrated earlier that preferentially aligns with when dominates over . The strain rate induced by flame normal acceleration due to chemical heat release can be scaled as , where is expected to decrease with increasing [43]. Following Meneveau and Poinsot [44], can be scaled as , which gives rise to . Alternatively, turbulent straining can be scaled as [45] (where is the Taylor microscale), which yields . The above scaling relations suggest that strengthens with respect to with increasing for a given value of . Previous analyses [22–24, 28, 30] demonstrated that predominantly aligns with for flames, whereas aligns with in flames for comparable values of . Both and indicate that an increase in for a given value of (e.g., cases P1, P3, and P5) gives rise to weakening of in comparison to . This increases the extent of alignment with with increasing when 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 in these cases. In case P5, predominantly aligns with in the unburned and fully burned gases but overcomes 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 in case P5 assumes positive values towards both the unburned and burned gas sides, whereas the mean contribution of remains negative close to the middle of the flame. The relation indicates that weakens in comparison to with decreasing . The quantity 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 in cases P2 and P3, whereas assumes positive values towards the unburned and burned gas sides of the flame in case P4. However, overcomes 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 in case P4. In cases V1 and V2, the values of 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 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 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 remains positive (negative) towards the unburned (burned) gas side of the flame with the transition from positive to negative value taking place close to . In order to explain this behaviour, 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 occurs close to [10, 14]. This suggests that the probability of finding negative (positive) values of is significant for (), which gives rise to positive (negative) value of 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 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 for globally adiabatic flames as (i.e., for constant ) and the first two terms on the right hand side of vanish for constant values of . The contributions of 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.

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

The marginal probability density functions (pdfs) of normalised (i.e., ) 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 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., ) 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 [46–52] 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].

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The joint pdfs of and tangential strain rate for cases P1, P5, V1, and V3 are shown in Figure 5(a) for 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 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 , whereas can be taken to scale with turbulent strain rate (i.e., according to Meneveau and Poinsot [44] and according to Tennekes and Lumley [45]).(ii)Above scalings indicate that scales as and according to the scaling arguments by Meneveau and Poinsot [44] and Tennekes and Lumley [45], respectively. Both and suggest that the magnitude of is likely to supersede the magnitude of in most locations within the flame for small values of and high values of .(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 (). 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 .

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The joint pdfs between and curvature for cases P1, P5, V1, and V3 are shown in Figure 5(b) for 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 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]) 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 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 , , , , and in response to and . The curvature and strain rate dependences of , , , , and are discussed next.

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

The marginal pdfs of for different isosurfaces across the flame are shown in Figures 6(a) and 6(b) 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. It is evident from Figures 6(a) and 6(b) that the pdfs of are qualitatively similar for statistically planar and V-flames and in both cases assumes predominantly positive values throughout the flame. As dilatation rate is principally positive due to thermal expansion in premixed flames [10, 30], the contribution of is predominantly positive throughout the flame. Moreover, Figures 6(a) and 6(b) demonstrate that 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., ) and the probability of finding high values of decreases on both unburned and burned gas sides of the flame. This is consistent with the observed behaviour of the mean values of 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 becomes high around .

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The joint pdfs between and for cases P3 and V2 are shown in Figures 6(c) and 6(d), respectively, for isosurface. It can be seen from Figures 6(c) and 6(d) that and are positively correlated on 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 and .

The joint pdfs between and for cases P3 and V2 are shown in Figures 6(e) and 6(f), respectively, for isosurface. It can be seen from Figures 6(e) and 6(f) that the joint pdf between and exhibits a negative correlation on 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 and .

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

The marginal pdfs of for different isosurfaces across the flame are shown in Figures 7(a) and 7(b) 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. Figures 7(a) and 7(b) show that the probability of finding negative values of supersedes the probability of finding positive values. The probability of finding negative values of increases as the heat releasing zone (see the pdfs for isosurface) is approached. It has been discussed earlier that the effects of overcome the effects of in the heat releasing zone to give rise to a preferential alignment of with even for small values of . This preferential alignment of with in these zones gives rise to negative values of 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 .

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The contours of joint pdfs between and for are shown in Figures 7(c) and 7(d) for cases P3 and V2 and the correlation coefficients between 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 2 that and are positively correlated for high cases (e.g., cases P5 and V3) although the strength of the correlation changes through the flame. However, 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 this behaviour, it is useful to rewrite in the following manner:

Based on (7) the strain rate dependences of 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 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 and . The strain rate and curvature dependences of weaken with increasing [58], so increases with increasing , which leads to a positive correlation between and for the major portion of the flame for cases with high values of (see Table 2).

The joint pdfs between and for cases P3 and V2 are shown in Figures 7(e) and 7(f), respectively, for isosurface and the correlation coefficients between 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 and remain weakly positively correlated except the burned gas side of the flame. The observed curvature dependence of 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 are more likely to dominate over the effects of 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 increasingly negative (see (5)) and the magnitude of the negative contribution of decreases for positive curvature locations. This gives rise to a positive correlation between and , as observed from Figures 7(e) and 7(f) and Table 2.(ii)However, the effects of are more likely to dominate over the effects of 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 (see (5)) with decreasing towards the burned gas side and leads to a negative correlation between and (see Table 2).

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

The marginal pdfs of normalised for different isosurfaces across the flame are shown in Figure 8 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 are not shown in Figure 8 because assumes negligible value in the preheat zone of the flame due to negligible magnitude of . It is evident that 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., in Figure 8) of the flame front. However, assumes negative values only towards the burned gas side (e.g., ) of the flame front for both planar and V-flames. This is consistent with the behaviour of shown in Figure 3. The physical mechanism behind the transition from positive to negative values of the mean contribution of (see (6)) is also responsible for obtaining negative (positive) values of towards the burned (unburned) gas side of the flame.

The contours of joint pdfs between and for , 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 and remain positively correlated for the part of the flame where finding positive values of is prevalent. On the other hand, and are negatively correlated with each other towards the burned gas side of the flame where is predominantly negative. The observed dependence of can be explained in the following manner.(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 depend only on and thus high values of are associated with high values of and .(ii)As and are positively correlated with each other, the magnitude of reaction rate contribution is positively correlated with tangential strain rate . Thus, is positively (negative) correlated with , where assumes positive (negative) values.

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The joint pdfs between and for cases P3 and V2 are shown in Figure 10 for , 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 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 can be summarised in the following manner.(i)The term 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 are associated with high values of and .(ii)As a result of the aforementioned physical mechanisms, the term and remain positively (negatively) correlated with curvature at negatively (positively) curved locations in the planar flames where assumes positive values. By contrast, the joint pdfs of and exhibit negative (positive) correlation with curvature at negatively (positively) curved locations within the flame where assumes negative values for the planar flames considered here (see Figure 10(c)). However, remains predominantly negatively correlated with for V-flame cases (see Figure 5(b)) and thus shows positive (negative) correlation with curvature where assumes negative (positive) values (see Figures 10(d)–10(f)).

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##### 3.7. Local Behaviour of and Its Curvature and Strain Rate Dependences

The marginal pdfs of for isosurfaces representative of leading edge, reaction zone, and trailing edge of the flame (e.g., , 0.7 and 0.9 isosurfaces) are shown in Figure 11(a) for cases P3 and V2. The pdfs of 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 assumes negative values throughout the flame and the probability of finding high magnitude of 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 shown in Figure 3. It can further be seen from Figure 11(a) that the pdfs of for statistically planar and V-flames are qualitatively similar to each other.

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The contours of joint pdfs between and for isosurface are shown in Figures 11(b) and 11(c) for cases P3 and V2 and the correlation coefficients between 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 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.(i)The instantaneous SDR and the molecular dissipation term can be taken to scale as and (where is the typical local flame thickness) because in premixed flames. the gradient of progress variable is only existent within the flame thickness. Alternatively, 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 . Thus one obtains when dissipation processes are taken to be governed by but remains of the order of unity for all cases considered here (see Table 1) and both scalings of (i.e., and ) suggest that high magnitudes of the dissipation term are associated with high values of due to positive correlation between and (see Figure 5(a)).(ii)As assumes negative values, the quantities and are predominantly negatively correlated throughout the flame due to positive correlation between and . However, the negative correlation between and weakens with increasing due to weakening of positive correlation between and . Thus the correlation between and becomes weakly positive at the middle of the flame for high values of (e.g., cases P4 and P5).

The joint pdfs of and curvature for cases P3 and V2 are shown in Figure 12 for , 0.7, and 0.9 isosurfaces and the correlation coefficients between and for different isosurfaces across the flame are shown in Table 3 for all cases considered here. The joint pdfs of 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 and are nonlinearly related to one another. The physical explanations behind the observed behaviour are provided below.(i)The molecular dissipation term can alternatively be expressed as

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The above expression clearly indicates that the third term on the right hand side of (8) (i.e., ) induces nonlinear curvature dependence of the molecular dissipation term .(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 (i.e., ). 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 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 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 . Thus, both positive and negative correlations between and have been observed within the flame front in all cases considered here.

##### 3.8. Local Behaviour of and Its Curvature and Strain Rate Dependences

The marginal pdfs of for , 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 and are identically zero in these cases. The marginal pdfs of and for case P3 are shown in Figures 13(b) and 13(c), which show that both and predominantly assume positive (negative) values towards burned (unburned) gas side of the flame. As in cases P1–P5, the pdfs of are qualitatively similar to those of and thus are not shown here. This indicates that shows predominant probability of finding positive values throughout the flame (see Figure 6). The pdfs of , and for case V2 are shown in Figures 13(d)–13(h), respectively. It is evident from Figures 13(f) and 13(g) that both and assume positive (negative) values towards burned (unburned) gas side of the flame, whereas 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 and assume predominantly positive (negative) values towards the unburned (burned) gas side of the flame.

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