The relation between turbulent burning velocity and flame surface area for turbulent stratified flames has been analysed using a direct numerical simulation database. The simulations have been carried out for different initial turbulence intensities and initial integral length scales of equivalence ratio fluctuation for a given root-mean-square value in a globally stoichiometric mixture. Additionally, statistically similar turbulent premixed flames have been considered for comparison. The turbulent burning velocity and flame surface area for stratified flames are found to be significantly smaller than in premixed flames for slow mixing cases (large scalar integral length scale, low turbulence intensity), though this trend weakens in fast mixing cases (low scalar integral length scale, high turbulence intensity). In slow mixing cases, the increase of burning rate occurs at a smaller proportion than the increase of flame surface area.The contributions of the components arising from tangential diffusion of displacement speed and cross-scalar dissipation rate to the turbulent burning velocity are found to be negligible in comparison to that arising from the combined reaction and normal diffusion component for all cases. The increased probability of finding fuel-lean and fuel-rich mixture affects the contribution of turbulent burning velocity arising from the combined reaction and normal diffusion component of displacement speed for slow mixing cases. Front-supported flame elements dominate over back-supported flame elements for large scalar length scale cases, but this behaviour reverses for small scalar length scale cases. These findings suggest that increases of burning rate and flame surface area do not take place in the same proportions for turbulent stratified combustion.

The second-order structure functions and their components conditioned upon various events have been analysed for un-weighted and density-weighted velocities using a Direct Numerical Simulation (DNS) database. The heat release due to combustion has been shown to have significant influences on the structure functions and their components conditioned on different mixture states. The use of densityweighted velocities changes the relative magnitudes of differently conditioned structure functions but does not reduce the scatter of these magnitudes. The structure functions conditioned to constant-density unburned reactants at both points and normalized using the root-mean-square velocity conditioned to the reactants are larger at higher values of mean reaction progress variables ̅ (deeper within the flame brush), with this trend being not weakened with increasing turbulence intensity ′⁄. These results indicate that, contrary to a common belief, combustion-induced thermal expansion can significantly affect the incoming constant-density turbulent flow of unburned reactants even at ′⁄ and Karlovitz number as large as 10 and 18, respectively. The statistical behaviours of the structure functions reveal that the magnitude of the flame normal gradient of the velocity component tangential to the local flames can be significant and it increases with increasing turbulence intensity. Moreover, the structure functions conditioned on both points in the heat release zone bear the signature of the anisotropic effects induced by baroclinic torque for the flames belonging to the wrinkled flamelets and corrugated flamelets regimes. These anisotropic effects weaken with increasing turbulence intensity in the thin reaction zones regime.

The second-order velocity structure function statistics have been analysed using a DNS database of statistically planar turbulent premixed flames subjected to unburned gas forcing. The flames considered here represent combustion for moderate values of Karlovitz number from the wrinkled flamelets to the thin reaction zones regimes of turbulent premixed combustion. It has been found that the second-order structure functions exhibit the theoretical asymptotic scalings in the dissipative and (relatively short) inertial ranges. However, the constant of proportionality for the theoretical asymptotic variation for the inertial range changes from one case to another, and this value also changes with structure function orientation. The variation of the structure functions for small length scale separation remains proportional to the square of the separation distance. However, the constant of proportionality for the limiting behaviour according to the separation distance square remains significantly different from the theoretical value obtained in isotropic turbulence. The disagreement increases with increasing turbulence intensity. It has been found that turbulent velocity fluctuations within the flame brush remain anisotropic for all cases considered here and this tendency strengthens towards the trailing edge of the flame brush. It indicates that the turbulence models derived based on the assumptions of homogeneous isotropic turbulence may not be fully valid for turbulent premixed flames.

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Scalar forcing in the context of turbulent stratified flame simulations aims to maintain the fuel-air inhomogeneity in the unburned gas. With scalar forcing, stratified flame simulations have the potential to reach a statistically stationary state with a prescribed mixture fraction distribution and root-mean-square value in the unburned gas, irrespective of the turbulence intensity. The applicability of scalar forcing for Direct Numerical Simulations of stratified mixture combustion is assessed by considering a recently developed scalar forcing scheme, known as the reaction analogy method, applied to both passive scalar mixing and the imperfectly mixed unburned reactants of statistically planar stratified flames under low Mach number conditions. The newly developed method enables statistically symmetric scalar distributions between bell-shaped and bimodal to be maintained without any significant departure from the specified bounds of the scalar. Moreover, the performance of the newly proposed scalar forcing methodology has been assessed for a range of different velocity forcing schemes (Lundgren forcing and modified bandwidth forcing) and also without any velocity forcing. It has been found that the scalar forcing scheme has no adverse impact on flame-turbulence interaction and it only maintains the prescribed root-mean-square value of the scalar fluctuation, and its distribution. The scalar integral length scale evolution is shown to be unaffected by the scalar forcing scheme studied in this paper. Thus, the scalar forcing scheme has a high potential to provide a valuable computational tool to enable analysis of the effects of unburned mixture stratification on turbulent flame dynamics.

The cross-scalar dissipation rate of reaction progress variable and mixture fraction $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ plays an important role in the modelling of stratified combustion. The evolution and statistical behaviour of $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ have been analysed using a direct numerical simulation (DNS) database of statistically planar turbulent stratified flames with a globally stochiometric mixture. A parametric analysis has been conducted by considering a number of DNS cases with a varying initial root-mean-square velocity fluctuation $$u^{\prime }$$ u ′ and initial scalar integral length scale $$\ell_{\phi }$$ ℓ ϕ . The explicitly Reynolds averaged DNS data suggests that the linear relaxation model for $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ is inadequate for most cases, but its performance appears to improve with increasing initial $$\ell_{\phi }$$ ℓ ϕ and $$u^{\prime }$$ u ′ values. An exact transport equation for $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ has been derived from the first principle, and the budget of the unclosed terms of the $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ transport equation has been analysed in detail. It has been found that the terms arising from the density variation, scalar-turbulence interaction, chemical reaction rate and molecular dissipation rate play leading order roles in the $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ transport. These observations have been justified by a scaling analysis, which has been utilised to identify the dominant components of the leading order terms to aid model development for the unclosed terms of the $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ transport equation. The performances of newly proposed models for the unclosed terms have been assessed with respect to the corresponding terms extracted from DNS data, and the newly proposed closures yield satisfactory predictions of the unclosed terms in the $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ transport equation.

The influence of mixture stratification on the development of turbulent flames in a slot-jet configuration has been analysed using Direct Numerical Simulation data. Mixture stratification was imposed at the inlet by varying the equivalence ratio between 0.6 and 1.0 with different alignments to the reaction progress variable gradient: aligned gradients (back-supported), opposed gradients (front-supported) and misaligned gradients. An additional premixed case with a global equivalence ratio of 0.8 was simulated for comparison. The flame is shortest for the front-supported case, followed by the premixed flame, with the back-supported and misaligned gradient flames being the tallest and of comparable size. This behaviour has been explained in terms of the variations of the mean equivalence ratio within the flame and the volume-integrated reaction rate in the streamwise direction. The difference in mixture composition for these cases results in significant variations in the burning rate, flame area, flame wrinkling and flame brush thickness in the streamwise direction. The globally front-supported case has the highest volume-integrated burning rate and flame area, while the back-supported case has the lowest. The misaligned scalar gradient case exhibits qualitatively similar behaviour to that of the globally back-supported case. The burning intensity is unity for a major part of the flame length but assumes values greater than unity towards the flame tip where the effects of flame curvature become strong. All cases predominantly exhibit the premixed mode of combustion within the flamelet regime, so flamelet assumption-based reaction rate closures, originally proposed for premixed combustion, were evaluated using a priori analysis. The terms which require improved closures have been identified and existing closures have been improved where necessary. It was found that the global nature of mixture stratification does not influence the performance of the mean reaction rate closures or the parameterisation of marginal probability density functions of scalars in turbulent stratified mixture combustion.