A Priori Direct Numerical Simulation Analysis of the Closure of Cross-Scalar Dissipation Rate of Reaction Progress Variable and Mixture Fraction in Turbulent Stratified Flames

Abstract

The cross-scalar dissipation rate of reaction progress variable and mixture fraction widetilde{{varepsilon_{cxi } }} plays an important role in the modelling of stratified combustion. The evolution and statistical behaviour of widetilde{{varepsilon_{cxi } }} 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 } and initial scalar integral length scale ell_{phi }. The explicitly Reynolds averaged DNS data suggests that the linear relaxation model for widetilde{{varepsilon_{cxi } }} is inadequate for most cases, but its performance appears to improve with increasing initial ell_{phi } and u^{prime } values. An exact transport equation for widetilde{{varepsilon_{cxi } }} has been derived from the first principle, and the budget of the unclosed terms of the widetilde{{varepsilon_{cxi } }} 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_{cxi } }} 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_{cxi } }} 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_{cxi } }} transport equation.