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

TL;DR

This paper derives a new second-order PDE for the probability density function of turbulent vorticity, depending only on conditional mean, offering potential new closure schemes for turbulence modeling.

Contribution

It introduces a novel PDF evolution equation for turbulent vorticity that depends solely on the conditional mean, differing from traditional quadratic-statistics-based models.

Findings

Derived a mixed-type second-order PDE for vorticity PDF

Proposed new closure schemes based on conditional linear statistics

Contrasted with Reynolds' mean flow equations

Abstract

The vorticity random field of turbulent flow is singled out as the main dynamical variable 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 (first order) of the underlying turbulent flow, which is in contrast with Reynolds' mean flow equation which relies on a quadratic statistics. Therefore the new PDF PDE may provide new closure scheme based on the conditional linear statistics, and some of them will be described in the present paper too.