The Fokker–Planck closure for turbulent molecular mixing: Passive scalars

Abstract

A turbulent-molecular-mixing closure for passive scalar mixing is derived based on the theory of diffusion in layerlike lamellar structures. The closure is formulated in terms of the Fokker–Planck (FP) equation (or an equivalent stochastic differential equation), and is to be employed in conjunction with the probability density function (pdf) balance equation appearing in the pdf methods for modeling turbulent reactive flows. Like the mapping closure, the FP closure predicts a limiting Gaussian pdf for the passive scalar concentration in isotropic turbulence. In addition, the FP closure models the joint pdf of the scalar concentration and the scalar gradient and thus predicts the scalar dissipation rate. The closure predictions for the scalar rms concentration, the marginal pdf, and the joint pdf as well as other relevant statistics have been studied by Monte Carlo simulation. The shape and temporal evolution of the marginal pdf for the scalar concentration compare favorably with published results found from direct numerical simulations and from the mapping closure. Finally, unlike the mapping closure, the FP closure is shown to easily extend to the case of multiple scalar mixing.