A Lagrangian two-time probability density function equation for inhomogeneous turbulent flows

Abstract

An exact equation for the Lagrangian two-time velocity joint probability density function (pdf) is derived from the Navier–Stokes equation. The pdf equation contains as an unknown the conditional expectation of the fluid acceleration. A linear Markov model is proposed which leads to a modeled equation that is consistent both with Kolmogorov’s theory in the inertial subrange and with Reynolds-stress models. The dissipation rate is obtained from the joint pdf in a way that is consistent with the modeled dissipation equation. A Monte Carlo method can be used to solve the modeled two-time pdf equation for inhomogeneous turbulent flows.