Diffusion limit of the simplified Langevin PDF model in weakly inhomogeneous turbulence

Abstract

In this work, we discuss the modelling of transport in Langevin probability density function (PDF) models used to predict turbulent flows [1]. Our focus is on the diffusion limit of these models, i.e. when advection and dissipation are the only active physical processes. In this limit, we show that Langevin PDF models allow for an asymptotic expansion in terms of the ratio of the integral length to the mean gradient length. The main contribution of this expansion yields an evolution of the turbulent kinetic energy equivalent to that given by a k − ε model. In particular, the transport of kinetic energy is given by a gradient diffusion term. Interestingly, the identification between PDF and \hbox{$\overline{k}-\overline{\varepsilon}$} k − ε models raises a number of questions concerning the way turbulent transport is closed in PDF models. In order to validate the asymptotic solution, several numerical simulations are performed, with a Monte Carlo solver and also with a deterministic code.