On distributions of velocity random fields in turbulent flows

TL;DR

This paper derives a new PDE for the probability density function of velocity in turbulent flows, offering a novel approach to turbulence modeling through a precise PDF transport equation.

Contribution

It introduces a comprehensive PDE for the velocity PDF in turbulence, generalizing classical Reynolds equations and enabling new turbulence modeling methods.

Findings

Derived a non-linear parabolic-transport PDE for velocity PDFs

Demonstrated the PDE's application with an explicit example

Provided a new framework for turbulence modeling

Abstract

The purpose of the present paper is to derive a partial differential equation (PDE) for the single-time single-point probability density function (PDF) of the velocity field of a turbulent flow. The PDF PDE is a highly non-linear parabolic-transport equation, which depends on two conditional statistical numerics of important physical significance. The PDF PDE is a general form of the classical Reynolds mean flow equation, and is a precise formulation of the PDF transport equation. The PDF PDE provides us with a new method for modelling turbulence. An explicit example is constructed, though the example is seemingly artificial, but it demonstrates the PDF method based on the new PDF PDE.