Statistical Analysis and a-priori Modelling of Flame Surface Density Transport in Turbulent Stratified Flames: A Direct Numerical Simulation Study

Statistically planar turbulent premixed and stratified flames for different initial intensities of decaying turbulence have been simulated for global equivalence ratios < I center dot > = 0.7 and < I center dot > = 1.0 using three-dimensional simplified chemistry based Direct Numerical Simulations (DNS). The simulation parameters are chosen such that the thin reaction zones regime combustion is realised in all cases and a random bi-modal distribution of equivalence ratio I center dot is introduced in the unburned gas ahead of the flame to account for the mixture inhomogeneity for stratified flames. The modelling of the unclosed terms (i.e. the turbulent transport term T (1), the tangential strain rate term T (2), the propagation term T (3), and the curvature term T (4)) of the generalised FSD transport equation has been addressed in the context of RANS simulations. It has been found that the turbulent transport term T (1) remains small in comparison to the leading order contributions of the tangential strain rate and curvature terms (i.e. T (2) and T (4), respectively) in the globally stoichiometric cases, but T (1) begins to play a more important role in the globally fuel-lean cases. The strain rate term T (2) remains positive throughout the flame brush and acts as a leading order source term for all the flames considered in this analysis. It is has been found that the magnitude of T (2) decreases with decreasing root-mean-square velocity fluctuations u (') (< I center dot >) for a given value of < I center dot > (u (')). The contribution of the propagation term T (3) remains generally positive towards the unburned gas side of the flame brush but assumes generally negative values towards the burned gas side of the flame brush. Moreover, whilst the order of magnitude of the propagation term T (3) is comparable in all cases, T (3) remains small in comparison to the leading order contributors (i.e. T (2) and T (4)) in the globally stoichiometric cases however it plays a more important role in the globally fuel-lean cases. The curvature term T (4) acts as a leading order sink term in all cases except towards the unburned gas side of the flame brush in low u (') globally stoichiometric (i.e. < I center dot > = 1.0) flames. Furthermore, it has been demonstrated that the magnitude of T (4) decreases with decreasing u (') (< I center dot >) for a given value of < I center dot > (u (')). Appropriate model expressions have been identified for T (1), T (2), T (3) and T (4) based on an a-priori analysis of the DNS data.