A DNS Study of Closure Relations for Convection Flux Term in Transport Equation for Mean Reaction Rate in Turbulent Flow

The present work aims at modeling the entire convection flux \(\overline {\rho \mathbf {u}W}\) in the transport equation for a mean reaction rate \(\overline {\rho W}\) in a turbulent flow, which (equation) was recently put forward by the present authors. In order to model the flux, several simple closure relations are developed by introducing flow velocity conditioned to reaction zone and interpolating this velocity between two limit expressions suggested for the leading and trailing edges of the mean flame brush. Subsequently, the proposed simple closure relations for \(\overline {\rho \mathbf {u}W}\) are assessed by processing two sets of data obtained in earlier 3D Direct Numerical Simulation (DNS) studies of adiabatic, statistically planar, turbulent, premixed, single-step-chemistry flames characterized by unity Lewis number. One dataset consists of three cases characterized by different density ratios and is associated with the flamelet regime of premixed turbulent combustion. Another dataset consists of four cases characterized by different low Damköhler and large Karlovitz numbers. Accordingly, this dataset is associated with the thin reaction zone regime of premixed turbulent combustion. Under conditions of the former DNS, difference in the entire, \(\overline {\rho {u}W}\), and mean, \(\tilde {u}\overline {\rho W}\), convection fluxes is well pronounced, with the turbulent flux, \(\overline {\rho u^{\prime \prime }W^{\prime \prime }}\), showing countergradient behavior in a large part of the mean flame brush. Accordingly, the gradient diffusion closure of the turbulent flux is not valid under such conditions, but some proposed simple closure relations allow us to predict the entire flux \(\overline {\rho \mathbf {u}W}\) reasonably well. Under conditions of the latter DNS, the difference in the entire and mean convection fluxes is less pronounced, with the aforementioned simple closure relations still resulting in sufficiently good agreement with the DNS data.