Transport equation for the joint probability density function of velocity and scalars in turbulent flow

TLDR: In this article, the transport equation for the joint probability density function of velocity and scalars is shown to provide a good basis for modeling turbulent reactive flows, and closed approximations are presented for the terms involving the fluctuating pressure and viscous and diffusive mixing.

Abstract: The transport equation for the joint probability density function of velocity and scalars is shown to provide a good basis for modeling turbulent reactive flows. As in the equation for the probability density function of the scalars alone, nonlinear reaction schemes can be treated without approximation. The advantage of considering the joint probability density function equation is that convection (by both the mean and fluctuating velocities) appears in closed form. Consequently, the gradient‐diffusion assumption for turbulent transport is avoided. Closure approximations are presented for the terms involving the fluctuating pressure and viscous and diffusive mixing. These models can be expected to be reliable since they are compatible with accurate and proven Reynolds‐stress models. The resulting modeled transport equation for the joint probability density function can be solved by the Monte‐Carlo method for inhomogeneous flows with complex reactions.