Effects of Lewis number on vorticity and enstrophy transport in turbulent premixed flames

The effects of Lewis number Le on both vorticity and enstrophy transport within the flame brush have been analysed using direct numerical simulation data of freely propagating statistically planar turbulent premixed flames, representing the thin reaction zone regime of premixed turbulent combustion. In the simulations, Le was ranged from 0.34 to 1.2 by keeping the laminar flame speed, thermal thickness, Damköhler, Karlovitz, and Reynolds numbers unchanged. The enstrophy has been shown to decay significantly from the unburned to the burned gas side of the flame brush in the Le ≈ 1.0 flames. However, a considerable amount of enstrophy generation within the flame brush has been observed for the Le = 0.34 case and a similar qualitative behaviour has been observed in a much smaller extent for the Le = 0.6 case. The vorticity components have been shown to exhibit anisotropic behaviour within the flame brush, and the extent of anisotropy increases with decreasing Le. The baroclinic torque term has been shown to be principally responsible for this anisotropic behaviour. The vortex stretching and viscous dissipation terms have been found to be the leading order contributors to the enstrophy transport for all cases, but the baroclinic torque and the sink term due to dilatation play increasingly important role for flames with decreasing Le. Furthermore, the correlation between the fluctuations of enstrophy and dilatation rate has been shown to play an important role in determining the material derivative of enstrophy based on the mean flow in the case of a low Le. C 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4939795]


I. INTRODUCTION
2][3] The presence of heat release, density variation, and flame normal acceleration in turbulent flames significantly affects the underlying turbulent flow structure and is manifested in flame-generated turbulence 4 and counter-gradient scalar transport 5,6 to name a few.While these issues have been the focus of turbulent combustion research for decades (as reviewed elsewhere), 7 relatively limited effort has been directed to the analysis of the statistical behaviour of vorticity ⃗ ω and enstrophy Ω transports in turbulent reacting flows.In non-premixed flames, the alignment of the vorticity vector with local principal strain rates was analysed by Nomura and Elghobashi, 8 Boratov et al., 9 and Jaberi et al., 10 These analyses demonstrated that vorticity vector ⃗ ω aligns with the intermediate principal strain rate in non-premixed flames similar to that in non-reacting turbulent flows, but the vorticity vector in non-premixed flames also shows appreciable probabilities of local alignment with the most extensive principal strain rate.The non-premixed flame Direct Numerical Simulation (DNS) data by Boratov et al. 9 have demonstrated that the extent of vorticity alignment with the most extensive principal strain rate increases in regions where the magnitude of strain rate dominates over the vorticity magnitude.The analysis by Jaberi et al. 10 indicated that the alignment of vorticity with the intermediate (most extensive) principal strain rate decreases (increases) due to chemical heat release in non-premixed flames, whereas the vorticity vector ⃗ ω remains mostly perpendicular to the most compressive principal strain rate in both reactive and non-reactive regions of non-premixed turbulent combustion.
In premixed flames, the alignment of vorticity with local principal strain rates has been numerically analysed by Hamlington et al., 11 who addressed the thin reaction zones regime combustion.These authors have revealed that vorticity alignment with local principal strain rates in the thin reaction zones regime flames is qualitatively similar to previous findings in the context of non-premixed combustion (i.e., predominant alignment with the intermediate principal strain rate; negligible alignment with the most compressive principal strain rate, and an increased alignment with the most extensive principal strain rate in the heat releasing zone).It was further shown by Hamlington et al. 11 that vorticity magnitude decays significantly in the burned gas across the flame brush, whereas Treurniet et al. 12 demonstrated that vorticity magnitude increases in the burned gas for the flames with high density ratio (or heat release parameter).Lipatnikov et al. 13 analysed the terms of enstrophy and vorticity transport equation for weakly turbulent premixed flames representing the corrugated flamelets regime.While Hamlington et al., 11 Treurniet et al., 12 and Lipatnikov et al., 13 dealt with DNS data, Steinberg et al. [14][15][16][17] experimentally investigated the enstrophy field in turbulent premixed flames using cinema-stereoscopic Particle Image Velocimetry (PIV) measurements of rim-stabilised turbulent premixed flames.
Recently, Chakraborty 18 revealed that the global Lewis number Le can significantly affect the vorticity statistics in premixed turbulent combustion.1][19][20][21][22][23][24][25][26][27][28] For example, in the corrugated flamelets regime, and for the cases with high Karlovitz number and low Le, where the most extensive principal strain rate is controlled by the local dilatation rate, 18 the vorticity vector ⃗ ω predominantly aligns with the intermediate and the most compressive principal strain rates.][21][22][23][24][25][26][27][28] While each individual species j has its own Lewis number Le j , in simplified models of molecular transport, the Lewis number of the deficient reactant (fuel or oxidant) is often taken to be the characteristic global Lewis number Le 29 as was done in the aforementioned analysis by Chakraborty. 18It is worth noting here that alternative methods of assigning a characteristic Lewis number have been proposed based on heat release measurements 30,31 and mole fractions of the mixture constituents. 325][46][47][48][49][50][51][52][53] Various concepts, which have been developed in order to explain such effects in turbulent flames, are reviewed elsewhere. 54,55However, the influences of Le on vorticity ⃗ ω and enstrophy Ω transport are yet to be analysed in detail in the existing literature.In this respect, the main objectives of the present analysis are as follows.
1. To demonstrate the effects of characteristic Lewis number Le on the statistical behaviour of the transport of vorticity ⃗ ω and enstrophy Ω in turbulent premixed flames.2. To provide physical explanations for the observed behaviours of the various terms in the vorticity ⃗ ω and enstrophy Ω transport equation.
The above objectives are met by extracting vorticity ⃗ ω and enstrophy Ω statistics from DNS data of freely propagating statistically planar turbulent premixed flames with characteristic Lewis number ranging from Le = 0.34 to 1.2.
The rest of the paper is organised as follows.The mathematical background and numerical implementation pertaining to this analysis are presented in Secs.II and III of this paper.Following this, the results are presented and subsequently discussed.The main findings are summarised and conclusions are drawn in Sec.V of this paper.

II. MATHEMATICAL BACKGROUND
The momentum conservation equation for the ith direction is given by where u i is the ith component of velocity, ρ is the gas density, and is the component of stress tensor, µ = ρν and ν are dynamic and kinematic viscosities, respectively, and the summation convention applies for the repeated index k.Taking curl of Eq. (1) yields the transport equation of the ith component of vorticity . (2) The term t 1i on the right hand side of Eq. ( 2) is the ith component of the vortex-stretching term.The ith component of the viscous torque term t 21i arises due to the misalignment between the gradients of viscous stress and density and vanishes in constant-density flows.The ith component of term t 22 is responsible for the diffusion of vorticity and is equal to ν∂ 2 ω i /∂ x j ∂ x j in constant-density flows or in the case of a constant µ (as assumed in the current analysis).The fourth term on the right hand side of Eq. (2) (i.e., ith component of term t 3 ) is responsible for vorticity destruction by dilatation, whereas the last term on right hand side of Eq. (2) (i.e., ith component of term t 4 ) is responsible for baroclinic effects arising from the misalignment of the density and pressure gradients.Both term t 3 and t 4 vanish in constant-density flows.
Multiplying ω i both sides of Eq. (2) yields the transport equation of enstrophy On Reynolds averaging Eq. (3) provides (Ref.13) where Q indicates the Reynolds averaged value of a general quantity Q.The term T I is the vortex stretching contribution to the mean enstrophy Ω transport, whereas the term T II is the average of the scalar product of two vectors, the vorticity and the viscosity torque.The term T III , which reads the dynamic viscosity is constant, represents the combined action of molecular diffusion and dissipation of the mean enstrophy Ω.These two sub-terms can be of the same order of magnitude for small values of turbulent Reynolds number, whereas the dissipation sub-term (i.e., −ν ∂ω i /∂ x j ∂ω i /∂ x j ) dominates for high values of turbulent Reynolds number.The term T IV is responsible for the dissipation of enstrophy due to dilatation.The term T V is the baroclinic torque term which arises due to misalignment between pressure and density gradients.The statistical behaviour of these terms will be discussed in detail in Section IV of this paper.

III. NUMERICAL IMPLEMENTATION
5][46][47][48][49][50][51][52] A well-known compressible code called SENGA 56  of mass, momentum, energy, and species are solved in non-dimensional form.The dimensionless forms of the conservation equations are presented in Appendix A. The simulation domain is taken to be a cube of size 24.1δ th × 24.1δ th × 24.1δ th , where δ th = (T ad − T 0 )/ max ∇ ⌢ T L is the thermal flame thickness with T 0 , T ad , and T being the unburned gas, adiabatic flame, and instantaneous dimensional temperatures, respectively, and the subscript "L" refers to the unstrained laminar flame condition.A uniform Cartesian grid of 230 × 230 × 230 has been used to discretise the simulation domain, which ensures about 10 grid points within δ th .The spatial derivatives for the internal grid points are evaluated using the 10th order central difference scheme and the order of differentiation drops gradually to a one-sided 2nd order scheme at the non-periodic boundaries.A low storage third order explicit Runge-Kutta scheme 57 is used for explicit time advancement.The turbulent velocity fluctuations are initialised using a pseudo-spectral method 58 using the Bachelor-Townsend spectrum. 59he scalar field is initialised by an unstrained planar laminar flame solution.The initial values of root-mean-square value of turbulent velocity fluctuation normalised by the unstrained laminar burning velocity u ′ /S L , integral length scale normalised by the unstrained laminar flame thickness ratio l/δ th , Damköhler number Da = lS L /u ′ δ th , and Karlovitz number Ka = (u ′ /S L ) 1.5 (l/δ th ) −0.5 are 7.5, 2.45, 0.33, and 13.2, respectively.These values of u ′ /S L , l/δ th , Da, and Ka represent the thin reaction zones regime combustion according to the regime diagram by Peters. 60A single value of heat release parameter τ = (T ad − T 0 )/T 0 = 4.5 was set in all studied cases, whereas the Lewis number was varied, i.e., Le = 0.34, 0.6, 0.8, 1.0, and 1.2, with S L and δ th being kept unchanged by varying the pre-exponential factor in the expression for the reaction rate.The five cases characterized with these five Le will be referred to as cases A-E, respectively.Standard values are taken for Prandtl number Pr, ratio of specific heats γ = c p /c v and the Zeldovich number β = T ac (T ad − T 0 )/T 2 ad (i.e., Pr = 0.7, γ = 1.4,β = 6.0),where T ac is the activation temperature.The flame Mach number Ma = S L /(γRT 0 ) 0.5 is taken to be 0.014 for all cases with R being the gas constant.All the simulations have been carried out for a chemical time scale t chem = δ th /S L , which corresponds to about 3.34 initial integral eddy turnover times (i.e., t chem = 3.34l/u ′ ) for the cases considered here.The value of u ′ /S L decayed by 50% ahead of the flame, whereas l/δ th increased by a factor of 1.7 when the statistics were extracted.The simulation time used in the current analysis remains comparable to several previous analyses, [61][62][63][64][65][66][67] which have contributed significantly to the fundamental understanding of turbulent reacting flows in the past.
The Reynolds/Favre averaged values have been calculated by ensemble averaging the relevant quantities in transverse directions (i.e., x 2 − x 3 planes).The statistical convergence of the averaged quantities has been assessed by comparing the corresponding values obtained using half of the sample size in the transverse directions using a distinct half of the domain, with those obtained based on full sample size.Both the qualitative and quantitative agreements between these sets of values are found to be satisfactory, and only the results obtained based on full sample size will be presented here for the sake of conciseness.
In premixed flames, the species field is often characterised in terms of a reaction progress variable c, which increases from 0.0 in unburned gases to 1.0 in fully burned products.The reaction progress variable c can be defined in terms of a suitable reactant (product) mass fraction Y R (Y P ) in the follow- The statistically planar flames propagate in the negative x 1 -direction for all cases considered here so the Favre averaged reaction progress variable c remains a unique function of x 1 .Thus, all the Reynolds averaged quantities are plotted as a function of c for all cases considered here in Sec.IV of this paper.

IV. RESULTS AND DISCUSSION
The distributions of normalised vorticity magnitude  Local increases in burning rate and self-acceleration of upstream-pointing bulges in turbulent flames with Le < 1 are qualitatively similar to those in the corresponding laminar premixed flames due to the imbalances between the reactant and heat fluxes, which manifest themselves in the form of thermo-diffusive instability with respect to weak perturbations.9][70][71][72] The linear stability analysis of thermo-diffusive instability for planar [33][34][35][36] and spherical 73,74 laminar premixed flames resulted in analytical expressions for the instability growth rate and largest wavenumber (smallest wavelength) of a perturbation that could trigger the instability.Interested readers are referred to Refs.75-79 and the reviews conducted in Refs.70-72 for the latest developments in the linear stability analysis of laminar premixed flames with Le < 1.
Several analyses [44][45][46][47][48][49][50][51][52][53] attributed large values of A T /A L or R T /R L for turbulent premixed flames with Le < 1 to the thermo-diffusive instability of laminar flamelets which separate the unburned and burned gases.An alternative concept 54,55,69,80 of the Lewis number effects in premixed turbulent combustion emphasizes the propagation of highly stretched leading reaction zones into the unburned gas (the so-called leading edge concept).However, a comparison of the thermo-diffusive instability and leading point concepts is beyond the scope of the present study.
It was previously demonstrated by Chakraborty et al. 52 that the augmented rate of burning and strong flame normal acceleration for Le ≪ 1 flames (e.g., Le = 0.34 flame considered here) can lead to significant flame-generated turbulence within the flame brush.For instance, Fig. 1(a) shows that the vorticity magnitude √ ω i ω i × δ th /S L gets significantly augmented towards the burned gas side of the flame front in the Le = 0.34 case.The same tendency can be discerned in some locations for the Le = 0.6 case, but the effects of flame generated turbulence (i.e., vorticity generation) are much weaker than for the Le = 0.34 flame.For the Le ≈ 1.0 (e.g., 0.8, 1.0, and 1.2 cases) flames, the distribution of the normalised vorticity magnitude √ ω i ω i × δ th /S L is significantly different.It can be seen from Fig. 1 that the probability of finding large magnitudes of √ ω i ω i × δ th /S L decreases from the unburned to the burned gas side of the flame front for flames with Le ≈ 1.0 (e.g., 0.8, 1.0, and 1.2 cases).The above difference in the vorticity magnitude distribution in response to Le can further be seen from the variation of the Reynolds averaged normalised vorticity magnitude (ω i ω i ) 1/2 × δ th /S L with Favre averaged reaction progress variable c shown in Fig. 2(a) for the different Lewis number cases considered here.It can be seen from Fig. 2(a) that (ω i ω i ) 1/2 × δ th /S L decays monotonically from unburned to burned gas side of the flame brush for flames with Le ≈ 1.0 (e.g., Le = 0.8, 1.0, and 1.2) cases considered here.The Le = 0.6 flame shows a behaviour which is qualitatively similar to the Le ≈ 1.0 cases.However, the vorticity decay within the flame brush in the Le = 0.6 case is weaker than in the Le = 0.8, 1.0, and 1.2 cases.A similar vorticity decay has been observed for the enstrophy transport for low Damköhler number (i.e., Da < 1) unity Lewis number combustion analysed by Hamlington et al. 11 The decay of (ω i ω i ) 1/2 × δ th /S L across the flame brush was also observed for the corrugated flamelets regime flames by Treurniet et al. 12 and Lipatnikov et al. 13 However, in the present simulations, the quantity (ω i ω i ) 1/2 × δ th /S L increases from the unburned gas side to the middle of the flame brush before decaying towards the burned gas side in the Le = 0.34 case.A similar trend was observed in some of the high Damköhler number (i.e., Da > 1) unity Lewis number flames with high values of τ in previous analyses. 12,13e variations of the rms values of the normalised Favre averaged vorticity for all the different Lewis number cases considered here.Figures 2(b)-2(f) indicate that there is a difference in the magnitudes of the rms values of Favre averaged vorticity components between the direction of mean flame propagation and in the transverse directions for all cases, and this anisotropy is particularly strong in the Le = 0.34 case.This is consistent with previous analyses [11][12][13] which revealed that the presence of the flame makes the vorticity field substantially anisotropic.It can be seen from Figs. 2(b

2(f) that
 ρ(ω 1 − ω1 ) 2 / ρ 1/2 monotonically decays from unburned to burned gas side of the flame brush for all cases, including the Le = 0.34 case.However,  ρ(ω i − ωi ) 2 / ρ 1/2 in the Le = 0.34 case shows augmentation of its magnitude from the unburned gas side to the middle of the flame brush before decreasing again towards the burned gas side of the flame brush, which is similar to the This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.240.225.107On: Fri, 29 Jan 2016 09:39:17 variation of (ω i ω i ) 1/2 within the flame brush as shown in Fig. 2(a).It is evident from Fig. 2(b) that the augmentation of vorticity magnitude within the flame for the Le = 0.34 case originates principally due to vorticity components in the directions normal to the mean direction of flame propagation (e.g., within the flame brush for the Le = 0.34 case).For other cases, all the components of rms of Favre averaged vorticity (i.e., × δ th /S L decay from the unburned to the burned gas side of the flame brush. The variations of (ω i ω i ) 1/2 , (ω n ω n ) 1/2 (here the repeated indices n do not indicate summation) and (ω t ω t ) 1/2 (here the repeated indices t indicate summation over two tangential directions) with c are shown in Figs.3(a)-3(e) for the Le = 0.34, 0.6, 0.8, 1.0, and 1.2 cases, respectively, where ω n ω n = N i N j ω i ω j and ω t ω t = (δ ij − N i N j )ω i ω j are the flame normal and flame tangential vorticity components, respectively, with N i = −(∂c/∂ x i )/|∇c| being the ith component of the local flame normal vector.It is evident from Figs. 3(a)-3(e) that (ω n ω n ) 1/2 decays from the unburned to burned gas side of the flame for all cases.By contrast, (ω t ω t ) 1/2 decays from the unburned gas side to the middle of flame before rising again and assuming the maximum value close to the burned gas side of the flame for the Le = 0.34 case.The augmentation of (ω t ω t ) 1/2 within the flame is principally responsible for the generation of (ω i ω i ) 1/2 within the flame for the Le = 0.34 case.The quantity (ω t ω t ) 1/2 decays from the unburned to the burned gas side of the flame for the Le = 0.6, 0.8, 1.0 and 1.2 cases considered here.Assuming two tangential directions are statistically similar, the variations of (ω t ω t /2) 1/2 are also compared with the distributions of (ω n ω n ) 1/2 in Figs.3(a)-3(e).It can be seen from Figs. 3(a)-3(e) that there are significant differences in the distributions of (ω t ω t /2) 1/2 and (ω n ω n ) 1/2 within the flame brush for all cases but the degree of anisotropy between (ω t ω t /2) 1/2 and (ω n ω n ) 1/2 increases with decreasing Le.The anisotropic behaviour of vorticity components in Figs.3(a)-3(e) is consistent with the behaviour of Favre mean vorticity components shown in Figs.2(b)-2(f), respectively.The flame normal direction for statistically planar flames predominantly coincides with the mean direction of flame propagation (i.e., x 1 -direction), and thus the variation of (ω n ω n ) 1/2 ((ω t ω t /2) 1/2 ) has been found to be qualitatively similar to ).Furthermore, a comparison between Figs. 1-3 reveals that the decay of vorticity magnitude from the unburned to the burned gas side weakens with decreasing global Lewis number Le even though flames are subjected to statistically similar turbulent flow field on the unburned gas side of the flame.
It is instructive to investigate the statistical behaviour of the terms of the vorticity and enstrophy transport equations (i.e., Eqs.(2) and (3)) in order to understand the influences of global Lewis number Le on the vorticity and enstrophy transports.The variations of the normalised values of (t 1t t 1t ) 1/2 , (t 21t t 21t ) 1/2 , (t 22t t 22t ) 1/2 , (t 3t t 3t ) 1/2 , and (t 4t t 4t ) 1/2 with c are shown in Figs.4(a)-4(e) for the Le = 0.34, 0.6, 0.8, 1.0, and 1.2 cases, respectively, where with q = 1, 21, 22, 3, and 4. The quantities t qn t qn 1/2 and t qt t qt 1/2 could be interpreted as the Reynolds averaged values of the magnitudes of the components of the vector ⃗ t q (where q = 1, 21, 22, 3, and 4) in local flame normal and tangential directions, as used in a previous analysis. 13It is worth noting that t qn t qn 1/2 and t qt t qt 1/2 are not the Reynolds averaged magnitudes of the terms of the transport equation of Chakraborty, Konstantinou, and Lipatnikov Phys.Fluids 28, 015109 (2016) respectively, under general conditions.Instead, t qn t qn 1/2 and t qt t qt 1/2 are associated with terms of the transport equation of N i N j Dω j /Dt and (δ ij − N i N j )Dω j /Dt, respectively.Interested readers are referred to Appendix B for further discussion in this regard.The quantities t qt t qt 1/2 and t qn t qn 1/2 can only be interpreted as the terms of the transport equation of ω t = (ω i − ω j N i N j ) and ω n = N i ω j N j under the strong assumption that the rotation of the moving frame of reference is ignored, which amounts to ω j D(N i N j )/Dt = 0. Nevertheless, the difference between Dω n /Dt = D(N i N j ω j )/Dt and in Appendix B).Accordingly, t qn t qn 1/2 and t qt t qt 1/2 represent magnitudes of the leading order contributors to the transport equation of ω n = N i ω j N j and ω t = (ω i − ω j N i N j ), respectively.A comparison between Figs. 4 and 5 reveals that the magnitude of the baroclinic torque contribution (t 4t t 4t ) 1/2 remains much greater than the magnitude of (t 4n t 4n ) 1/2 in the Le = 0.34 case.Nevertheless, the latter quantity does not vanish in the low Le flames, because vectors ∇ρ and ∇c are not exactly parallel in this case, contrary to an adiabatic flame with Le = 1.0,where ∇ρ and ∇c are exactly parallel, and, hence, the normal component of ∇ρ × ∇p and the baroclinic term (t 4n t 4n ) vanish.This behaviour can be explained in the following manner.The mixture density ρ can be expressed as ρ = ρ 0 /(1 + τT) for flames with constant molecular weight (as in the present DNS), where T = ( T − T 0 )/(T ad − T 0 ) is the non-dimensional temperature. 81The non-dimensional temperature T can be equated to c for globally adiabatic, low Mach number Le = 1.0 flames, which leads to ∇ρ = −τ ρ 2 ∇c/ρ 0 = τ ρ 2 |∇c| ⃗ N/ρ 0 .Thus, the vectors ∇ρ and ∇c are parallel (alternatively ∇ρ × ∇p and ⃗ N = −∇c/|∇c| are mutually perpendicular) in the Le = 1.0 flame considered here.It is worth noting that c T for non-unity Lewis number flames and the quantities increasingly deviate from each other with decreasing Le.As a result, ∇ρ = −τ ρ 2 ∇T/ρ 0 τ ρ 2 |∇c| ⃗ N/ρ 0 for flames with Le 1.0 and thus ∇ρ × ∇p and ⃗ N = −∇c/|∇c| are not mutually perpendicular to each other.This gives rise to non-zero values of (t 4n t 4n ) 1/2 in the non-unity Lewis number flames.It is evident from Fig. 4 that the baroclinic torque contribution (i.e., (t 4t t 4t ) 1/2 ) dominates over the magnitudes of other contributions (i.e., (t 1t t 1t ) 1/2 , (t 21t t 21t ) 1/2 , (t 22t t 22t ) 1/2 , and (t 3t t 3t ) 1/2 ) for the Le = 0.34 and 0.6 cases, whereas the magnitude of (t 4n t 4n ) 1/2 remains smaller than the magnitudes of (t 1n t 1n ) 1/2 , (t 22n t 22n ) 1/2 , and (t 3n t 3n ) 1/2 and comparable to (t 21n t 21n ) 1/2 .Thus the baroclinic term t 4 is principally responsible for the anisotropy of the vorticity components within the flame brush in the Le = 0.34 and 0.6 cases.For Le ≈ 1.0 cases, the contribution of baroclinic torque (t 4t t 4t ) 1/2 remains comparable to (t 1t t 1t ) 1/2 , (t 21t t 21t ) 1/2 , (t 22t t 22t ) 1/2 , and (t 3t t 3t ) 1/2 , whereas the baroclinic term (t 4n t 4n ) 1/2 remains negligible in comparison to (t 1n t 1n ) 1/2 , (t 22n t 22n ) 1/2 , and (t 3n t 3n ) 1/2 .Under conditions of the present DNS, the viscous diffusion term t 22 plays an important role in the Reynolds averaged vorticity transport (see Eq. ( 2)).For instance, the viscous diffusion term (t 22t t 22t ) 1/2 is significantly greater than the vortex stretching term (t 1t t 1t ) 1/2 in all five cases, see Fig. 4, while the magnitudes of the normal components of these two terms (i.e., (t 1n t 1n ) 1/2 and (t 22n t 22n ) 1/2 ) are comparable in the major part of the flame brush in each case, see Fig. 5.These observations are associated with relatively moderate values of turbulent Reynolds number Re t for the cases considered here, whereas the vortex stretching term is expected to dominate at high values of Re t .It can be seen from Fig. 3 that a strong augmentation of the vorticity magnitude in the transverse direction takes place within the flame brush, which also sets up a strong vorticity gradient within the flame in the Le = 0.34 case.This gives rise to an increase in the magnitudes of the vortex-stretching (i.e., ω j (∂u i /∂ x j )) and the dissipation (i.e., −ν ∂ω i /∂ x j ∂ω i /∂ x j ) contributions to the components of vorticity transport terms in flame tangential direction, and thus (t 1t t 1t ) 1/2 and (t 22t t 22t ) 1/2 rise from the unburned gas side and assume peak values within the flame brush before decreasing again on the burned gas side for the Le = 0.34 case.The vorticity magnitude in the transverse direction decreases monotonically from the unburned to the burned gas side of the flame brush for the other cases, and thus (t 1t t 1t ) 1/2 and (t 22t t 22t ) 1/2 decrease monotonically from the unburned to the burned gas side of the flame brush for the other (i.e., Le = 0.6, 0.8, 1.0, and 1.2) cases.
It can be seen from Figs. 4 and 5 that the contributions of dilatation and baroclinic torque in both flame normal and tangential directions (i.e., (t 3n t 3n ) 1/2 , (t 4n t 4n ) 1/2 , (t 3t t 3t ) 1/2 , and (t 4t t 4t ) 1/2 ) vanish both in the unburned and burned gas sides of the flame brush as the effects of density variation and dilatation rate diminish both in the unburned and burned gas sides of the flame brush.Furthermore, Figs. 4 and 5 indicate that the relative contributions of viscous torque due to density variation, dilatation, and baroclinic terms (i.e., t 21 ,t 3 , and t 4 ) weaken, and their magnitudes decrease, with increasing Le.The vorticity transport for the Le ≈ 1.0 (i.e., Le = 0.8, 1.0, and 1.2) cases considered here is principally determined by the vortex stretching and viscous diffusion (i.e., t 1 and t 22 in Eq. ( 2)), which is similar to the vorticity transport for non-reacting flows.
It is evident from Eq. ( 2) that the dilatation contribution destroys all vorticity components irrespective of the direction due to predominantly positive dilatation rate ∂u i /∂ x i values in premixed flames.The rate of burning diminishes with increasing Le, which is reflected in the decrease in the mean value of normalised dilatation rate (∂u i /∂ x i ) × δ th /S L magnitude with increasing Le, as shown in Fig. 6(a).This increase in the magnitude of dilatation rate ∂u i /∂ x i for small values of Lewis number is responsible for increased magnitudes of t 3 = −(∂u k /∂ x k ) ω i with decreasing Le.However, the magnitude of t 3i = −(∂u k /∂ x k ) ω i does not change in proportion to (∂u k /∂ x k ) because an increase in the dilatation term reduces the magnitude of ω i in the term t 3i .It is also worth noting that (t 3t t 3t ) 1/2 is comparable with (t 1t t 1t ) 1/2 and (t 21t t 21t ) 1/2 , but is substantially smaller than (t 4t t 4t ) 1/2 in the low Lewis number flames in Figs.4(a) and 4(b).This difference between (t 3t t 3t ) 1/2 and (t 4t t 4t ) 1/2 is also associated with the dependence of t 3t on the relevant vorticity components, i.e., due to an important role played by vorticity diffusion under conditions of the present DNS, the magnitude of vorticity components is insufficient for the dilatation term to counterbalance the baroclinic torque term.
It is shown elsewhere 48,50 that the flame thickness decreases, though the probability of finding high temperature spots (including super-adiabatic temperature values) increases with decreasing Le in turbulent flames, because the molecular diffusion of reactants into the reaction zone overwhelms conductive heat flux out from the zone for small values of Le.Accordingly, the magnitude of density gradient |∇ρ| increases with decreasing Le, which can be confirmed from Fig. 6(b) where the variation of |∇ρ| × δ th /ρ 0 with c is shown for all cases considered here.36]68,69 The high magnitude of ∇ρ and the particular nature of misalignment between this vector and the divergence of the viscous stress tensor also lead to relatively large magnitudes of the effects of viscous torque due to density variation (i.e., t 21 ) in the vorticity transport equation.It can be seen from Figs. 4 and 5 that both (t 21t t 21t ) 1/2 and (t 21n t 21n ) 1/2 play non-negligible role in vorticity transport for small values of Le (e.g., Le = 0.34 case considered here).The previous analyses by Hamlington et al. 11 and Treurniet et al. 12 did not report any significant influences of t 21 but Lipatnikov et al. 13 reported considerable influences of t 21 for unity Lewis number weakly turbulent flames with high values of τ, where ∇ρ ≈ −τ ρ 2 ∇T/ρ 0 is expected to assume large magnitudes.
As discussed above, for low Mach number unity Lewis number flames, the non-dimensional temperature T can be equated to c, and thus ∇ρ can be expressed as ∇ρ = τ ρ 2 |∇c| ⃗ N/ρ 0 , which leads to 6(b) reveals that these quantities are close to each other even for Le 1.0 flames, thus, implying that (−∇T.⃗ N) remains close to |∇c| in all simulated cases.This suggests that ∇ρ can be taken to scale as ∇ρ ∼ τ ρ 2 |∇c| ⃗ N/ρ 0 and thus ∇ρ mostly aligns with the flame normal direction.This suggests that the baroclinic torque ρ −2 ∇ρ × ∇p is expected to have weak contributions in the flame normal direction but its contribution to vorticity transport in tangential directions is likely to be strong, as indicated by Figs. 4 and 5. e 1 is likely to be small, as can be inferred from Fig. 1(a).
Moreover, an increase in turbulent burning rate due to a decrease in Le (see Table I) results in an increasing magnitude of mean pressure gradient |(∇ p) 1 | normal to the mean flame brush, which can be substantiated from Fig. 7(f).This effect also contributes to the aforementioned increase in |(∇p) t | with decreasing Le, because the probability of finding a substantial angle between (∇ p) 1 and ⃗ N is sufficiently large in the Le = 0.34 case.As a result, the magnitude of ρ −2 ∇ρ × ∇p is high in this case.As the extent of flame wrinkling and the magnitude of |(∇ p) 1 | diminish with increasing Le, the mean magnitude of the tangential pressure gradient and relative contribution of baroclinic torque weakens with an increase in Lewis number.Interested readers are referred to Ref. 51 for further discussion on the effects of Le on |(∇ p) 1 |, which is not repeated here for the sake of conciseness.
It is worth noting that baroclinic torque not only generates vorticity but also damps vorticity, depending of an angle between the vectors ⃗ ω and ρ −2 ∇ρ × ∇p.This angle is characterized by cos . The variation of cos θ p with c for all cases considered here are shown in Fig. 8(b).It can be seen that the directions of ⃗ ω and ρ −2 ∇ρ × ∇p are completely independent of each other for leading and trailing edges of the flame brush for the Le = 0.6, 0.8, 1.0, and 1.2 flames.However, in the Le = 0.34 flame, the directions of ⃗ ω and ρ −2 ∇ρ × ∇p are related on the burned gas side due to significant density variation caused by temperature inhomogeneity in the burned gas.Within the flame brush, cos θ p assumes relatively high magnitudes and this effect is particularly strong for the Le = 0.34 case where the tangential components of ρ −2 ∇ρ × ∇p are principally responsible for the augmentation of (ω t ω t ) 1/2 within the flame brush.
The variations of the normalised values of the terms of the right hand side of the enstrophy transport equation (i.e., T I ,T II ,T III ,T IV and T V ) with c are shown in Figs.9(a)-9(e) for Le = 0.34, 0.6, 0.8, 1.0, and 1.2 cases, respectively.It can be seen that the mean contribution of vortex-stretching term T I remains positive throughout the flame brush for all cases.The vortex-stretching term T I can be expressed as where e α , e β , and e γ are the most extensive, compressive, and the most compressive principal strain rates, and α, β, and γ are the angles between ⃗ ω and the principal strain rate directions associated with e α , e β , and e γ , respectively.It was previously shown by Chakraborty 18 that Le significantly affects the alignment of ⃗ ω with the most extensive and compressive principal strain rates and the extent of alignment with the most extensive strain rate decreases with decreasing Le but the vortex-stretching term T I acts to generate enstrophy for all Le cases irrespective of the nature of the alignment between ⃗ ω and the principal strain rates.Interested readers are referred to Ref. 18 for further information in this regard.The correlation between density variation and viscous action T II remains small in magnitude in comparison to the other terms.The viscous dissipation term T III acts as a major sink term for all cases.It is worth remembering that T III includes the contributions from the viscous diffusion and dissipation of enstrophy, with the latter contribution being always negative.Under conditions of the present DNS, the magnitudes of the viscous diffusion and dissipation of enstrophy are comparable, but the latter mechanism is expected to dominate at high Reynolds numbers.The dilatation rate term T IV assumes non-zero negative values only within the flame brush.However, the magnitude of the dilatation rate term T IV remains small in comparison to the viscous dissipation term T III for the Le ≈ 1.0 cases considered here but the magnitude of T IV becomes comparable to T III for the low Le flames (e.g., Le = 0.34 and 0.6 cases considered here).The baroclinic torque term T V generates enstrophy within the flame brush but vanishes both in the unburned and burned gas sides.It is evident from Figs. 9(a)-9(e) that the relative magnitude of baroclinic torque term T V with respect to the magnitude of viscous dissipation term T III increases with decreasing Le.Figures 9(a)-9(e) further reveal that the vortex-stretching and viscous dissipation terms remain the leading order contributors in all cases considered here but the dilatation and baroclinic terms play leading order roles only in the low Le flames (e.g., Le = 0.34 and 0.6 cases considered here).Furthermore, it has been found that the magnitudes of the normalised values of the terms T I ,T II ,T III ,T IV , and T V decrease with increasing Le.Equation ( 4) can be rewritten as where D ( ) / Dt = ∂ ( ) /∂t + ūk ∂ ( ) /∂ x k is the material derivative associated with the mean flow.The last term on the right hand side can be rewritten as The normalised values of (T I + T II + T III + T IV + T V ), T VI , T VI (i), and T VI (ii) for all cases are shown in Figs.10(a)-10(e).Figure 10(a) indicates that the behaviour of T VI is principally determined by T VI (ii).
It is evident from Fig. 10 that T VI (ii) remains positive for cases with small Le, i.e., flame normal acceleration gives rise to positive correlation between fluctuations of enstrophy and dilatation rate, and this term plays an increasingly important role for flames with small values of Le (e.g., Le = 0.34 and 0.6 cases considered here).The term T VI (ii) partially eclipses the sink contribution of the dilatation term T IV in flames with small values of Le (e.g., Le = 0.34 and 0.6 cases considered here).The fluctuations of enstrophy and dilatation rate are not strongly correlated in flames with Le ≈ 1.0 and thus the term T VI (ii) assumes small magnitude throughout the flame brush.Figure 10(a) further indicates that the net contribution of (T I + T II + T III + T IV + T V + T VI ) assumes positive values in some locations within the flame brush for the Le = 0.34 case.By contrast, (T I + T II + T III + T IV + T V + T VI ) assumes predominantly negative values within the flame brush for the Le = 0.6, 0.8, 1.0, and 1.2 cases.This suggests that a fluid particle moving with the mean flow from unburned to burned gas side experiences a monotonic drop of Ω (i.e., D Ω/Dt < 0) for the Le = 0.6, 0.8, 1.0, and 1.2 flames, whereas the fluid particle moving with mean flow locally experiences an increase in Ω (i.e., D Ω/Dt > 0) for the Le = 0.34 flame.This is consistent with the observations from Fig. 2(a) which show a decay of (ω i ω i ) 1/2 × δ th /S L from unburned to burned gas side of the flame brush in the Le = 0.6, 0.8, 1.0, and 1.2 flames, but (ω i ω i ) 1/2 × δ th /S L increases within the flame brush for the Le = 0.34 flame.

V. CONCLUSIONS
The effects of Lewis number Le on the transport of vorticity and enstrophy within the flame brush have been analysed using DNS data of freely propagating statistically planar turbulent premixed flames with Le ranging from 0.34 to 1.2.The investigated flames propagate in intense small-scale turbulence, characterized by Ka > 1 and Da ∼ O(1), and are associated with the thin reaction zones regime of premixed turbulent combustion.It has been found that, under conditions of the present study, enstrophy decreases significantly from the unburned to the burned gas side of the flame brush in the Le ≈ 1.0 flames.However, a considerable amount of enstrophy augmentation within the flame brush has been observed for the Le = 0.34 case and a similar, but less pronounced behaviour has been observed in the Le = 0.6 case.The vorticity components have been shown to exhibit anisotropic behaviour within the flame brush and the extent of anisotropy increases with decreasing Le.It has been demonstrated that the baroclinic torque term is principally responsible for this anisotropic behaviour.The vortex stretching and viscous dissipation terms have been found to be the leading order contributors to the enstrophy transport for all cases; however, the baroclinic torque and the sink term due to dilatation play an increasingly important role for small values of Le.In the case of a low Le, it has been demonstrated that the correlation between the fluctuations of enstrophy and dilatation rate plays an important role in determining the material derivative of enstrophy based on mean flow.The qualitative nature of the findings of the current paper is unlikely to be modified in the presence of detailed chemistry, but three-dimensional DNS data for high values of turbulent Reynolds number are definitely required for deeper understanding of enstrophy transport in premixed turbulent flames.Furthermore, the present analysis does not address the near wall effects on vorticity dynamics in turbulent reacting flows, which are likely to have significant influences on the vorticity transformation mechanisms discussed in this paper.Some of the aforementioned issues will form the basis of future investigations in this regard.

ACKNOWLEDGMENTS
The first two authors (N.C. and I.K.) are grateful to EPSRC and N8/ARCHER for financial and computational support, respectively.The third author (A.L.) is grateful to Chalmers Transport Area of Advance for financial support.

APPENDIX A: NON-DIMENSIONAL FORM OF CONSERVATION EQUATIONS
The non-dimensional mass, momentum, energy, and progress variable transport equations are presented below, where the non-dimensional quantities are given by Eq. (A4) that Le comes into play directly through the species conservation equation, which involves Sc = Pr • Le.The effects of Le are reflected in the density and pressure gradient fields which in turn affect the vorticity transport.

APPENDIX B: DECOMPOSITION OF VORTICITY TRANSPORT EQUATION
The vorticity transport equation (i.e., Eq. ( 2)) can be written in the following form: where . (B2) Equations ( B1) and (B2) can be manipulated as follows.One the one hand, and and therefore, one can write the following: It is worth stressing, however, that Dω n /Dt N i N j K j and Dω t /Dt K i − K j N i N j , i.e., Dω n /Dt K n and Dω t /Dt K t .
On the other hand, Therefore, The variations of ω j D(N 1 N j )/Dt and ω j D(N 2 N j )/Dt (ω j D(N 3 N j )/Dt is statistically similar to ω j D(N 2 N j )/Dt and is thus not explicitly not shown here) with c for all cases considered here are presented in Fig. 11.It can be seen from Figs. 11 and 5 that the magnitude of ω j D(N i N j )/Dt remains much smaller than the magnitudes of the leading order terms of t qn t qn 1/2 , e.g., (t 22n t 22n ) 1/2 .
Accordingly, the magnitudes of the terms of the conservation equation of N i N j ω j are expected to be close to those of t qn t qn 1/2 = ( ( , where q = 1, 21, 22, 3, and 4. FIG. 1. Distribution of (ω i ω i ) 1 2 × δ th /S L in the central x 1 − x 3 plane at time t = t chem for the Le = (a) 0.34, (b) 0.6, (c) 0.8, (d) 1.0, and (e) 1.2 cases.
u ref , P + = P/ρ ref u 2 ref , τ + k i = τ k i /ρ ref u 2 ref , E + = E/C p T 0 , (A5) ẇ+ = ẇ L ref /ρ ref u ref , ρ + = ρ/ρ ref , λ + = λ/λ ref , D + = D/D ref , Le = λ ref /ρ ref D ref , with P is the pressure, E = C v T + u k u k /2 + H(1 − c)is the specific internal energy, and H is the heat of reaction per unit mass of reactants consumed.Therefore,E + = 1 γ (1 + τT + ) + 1 2 (γ − 1)Ma 2 u + k u + k + τ(1 − c).(A6) In Eqs.(A1)-(A4), Re = ρ ref u ref L ref /µ ref is the nominal Reynolds number, Ma = u ref /a ref is the Mach number, γ = C p /C v isthe ratio of specific heats, Pr is the Prandtl number, and Sc = Pr • Le is the Schmidt number with ρ ref , λ ref , D ref , u ref , L ref , a ref , and µ ref are the reference values of density, thermal conductivity, mass diffusivity, velocity scale, length scale, acoustic velocity, and viscosity, respectively.Here the density, thermal conductivity, mass diffusivity, viscosity, and acoustic speed of the unburned gas are taken to be ρ ref , λ ref , D ref , µ ref , and a ref , respectively, while S L and 10δ th are considered to be u ref and L ref , respectively.The gas is assumed to follow the ideal gas law P = ρR T which takes the following non-dimensional form:P + = 1 γ Ma 2 ρ(1 + τT).(A7)Equations (A1)-(A4) are solved in conjunction with Eq. (A7) in the compressible DNS code called SENGA.56 Interested readers are referred to Ref. 56 for further information.It can be seen from This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.240.225.107On: Fri, 29 Jan 2016 09:39:17 has been used for the DNS simulations where the standard conservation equations This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.240.225.107On: Fri, 29 Jan 2016 09:39:17

TABLE I .
The effects of Lewis number on normalised volume-integrated reaction rate of progress variable R T /R L and normalised flame surface area A T / A L after 3.34 initial eddy turn over times.