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

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. $widetilde{Y_F^{primeprime} xi ^{primeprime}}={overline {rho Y_F^{primeprime} xi^{primeprime}} } Big/ {overline rho })$ for Reynolds Averaged Navier Stokes simulations where $overline q $ , $tilde{q} ={overline {rho q} } big/ {overline rho }$ and $q^{primeprime}= q-tilde{q}$ 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 $widetilde{Y_F^{primeprime} xi^{primeprime}}$ with sufficient accuracy in low Damköhler number combustion and therefore, a transport equation for $widetilde{Y_F^{primeprime} xi^{primeprime}}$ may need to be solved. The statistical behaviours of $widetilde{Y_F^{primeprime} xi^{primeprime}}$ 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 Damköhler number flames, have been modified for low Damköhler 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 =2overline {rho Dnabla Y_F^{primeprime} nabla xi^{primeprime}} $ (where D is the mass diffusivity) are not sufficient for low Damköhler number combustion and therefore, a transport equation may need to be solved for the cross-scalar dissipation rate $widetilde{varepsilon }_{Yxi } ={overline {rho Dnabla Y_F^{primeprime} nabla xi^{primeprime}} } big/ {overline rho }$ for the closure of the $widetilde{Y_F^{primeprime} xi^{primeprime}}$ transport equation.