A-Priori Direct Numerical Simulation Modelling of Co-variance Transport in Turbulent Stratified Flames

Abstract

Three-dimensional Direct Numerical Simulations of statistically planar turbulent stratified flames at global equivalence ratios = 0.7 and = 1.0 have been carried out to analyse the statistical behaviour of the transport of co-variance of the fuel mass fraction Y F and mixture fraction ξ (i.e. \(\widetilde{Y_F^{\prime\prime} \xi ^{\prime\prime}}={\overline {\rho Y_F^{\prime\prime} \xi^{\prime\prime}} } \Big/ {\overline \rho })\) for Reynolds Averaged Navier Stokes simulations where \(\overline q \), \(\tilde{q} ={\overline {\rho q} } \big/ {\overline \rho }\) and \(q^{\prime\prime}= q-\tilde{q}\) are Reynolds averaged, Favre mean and Favre fluctuation of a general quantity q with ρ being the gas density and the overbar suggesting a Reynolds averaging operation. It has been found that existing algebraic expressions may not capture the statistical behaviour of \(\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}\) with sufficient accuracy in low Damkohler number combustion and therefore, a transport equation for \(\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}\) may need to be solved. The statistical behaviours of \(\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}\) and the unclosed terms of its transport equation (i.e. the terms originating from turbulent transport T 1 , reaction rate T 4 and molecular dissipation \(\left( {-D_2 } \right))\) have been analysed in detail. The contribution of T 1 remains important for all cases considered here. The term T 4 acts as a major contributor in = 1.0 cases, but plays a relatively less important role in = 0.7 cases, whereas the term \(\left( {-D_2 } \right)\) acts mostly as a leading order sink. Through an a-priori DNS analysis, the performances of the models for T 1 , T 4 and \(\left( {-D_2 } \right)\) have been addressed in detail. A model has been identified for the turbulent transport term T 1 which satisfactorily predicts the corresponding term obtained from DNS data. The models for T 4 , which were originally proposed for high Damkohler number flames, have been modified for low Damkohler combustion. Predictions of the modified models are found to be in good agreement with T 4 obtained from DNS data. It has been found that existing algebraic models for \(D_2 =2\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} \) (where D is the mass diffusivity) are not sufficient for low Damkohler number combustion and therefore, a transport equation may need to be solved for the cross-scalar dissipation rate \(\widetilde{\varepsilon }_{Y\xi } ={\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} } \big/ {\overline \rho }\) for the closure of the \(\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}\) transport equation.