On the transport equation for probability density functions of turbulent vorticity fields.

Jiawei Li,Zhongmin Qian,Mingrui Zhou

doi:10.1098/rspa.2021.0534

Abstract

Vorticity random fields of turbulent flows (modelled over the vorticity equation with random initial data for example) are singled out as the main dynamic variables for the description of turbulence, and the evolution equation of the probability density function (PDF) of the vorticity field has been obtained. This PDF evolution equation is a mixed type partial differential equation (PDE) of second order which depends only on the conditional mean (which is a first-order statistics) of the underlying turbulent flow. This is in contrast with Reynolds mean flow equation which relies on a quadratic statistics. The PDF PDE may provide new closure schemes based on the first-order conditional statistics, and some of them will be described in the paper. We should mention that the PDF equation is interesting by its own and is worthy of study as a PDE of second order.

Highlights

In statistical fluid mechanics, the velocity U(x, t) of a turbulent flow is promoted to a random field, cf. [2], indexed by space variable x ∈ R3 and time parameter t
These observations are valuable in modelling turbulent flows via the vorticity, which are already applied in vortex methods
We propose the heat flow method to model statistical quantities needed for closing the probability density function (PDF) partial differential equation (PDE) and obtain concrete PDF examples

Summary

Introduction

In statistical fluid mechanics (cf. [1]), the velocity U(x, t) of a turbulent flow is promoted to a random field, cf. [2], indexed by space variable x ∈ R3 and time parameter t. From this point of view, the turbulence problem, if there is one, seeks for a description of the distribution of the velocity field. There are good evidences which demonstrate that some sort of superposition property of vorticity may be maintained, not exactly due to highly nonlinear and nonlocal nature of turbulence These observations are valuable in modelling turbulent flows via the vorticity, which are already applied in vortex methods (cf [10,11]).

Probability density function equation for the vorticity

Inviscid fluid flows

Weakly homogeneous and weakly isotropic flows

Modelling probability density function of weakly isotropic flows

Heat flow method