Detailed Balance and Exact Results for Density Fluctuations in Supersonic Turbulence

TL;DR

This paper develops a probabilistic framework to analyze density fluctuations in supersonic turbulence, deriving exact statistical relations and providing criteria for numerical code validation and insights into numerical diffusion effects.

Contribution

It introduces exact statistical relations for density fluctuations in supersonic turbulence and offers a method to assess numerical diffusion effects on the density PDF.

Findings

Exact relations: $ abla imes {s u} | s angle=0$ and $ abla imes {s u} abla s| s angle=0$

Objective criteria for numerical code validation based on density statistics

Method to estimate numerical diffusion effects on the density PDF

Abstract

The probabilistic approach to turbulence is applied to investigate density fluctuations in supersonic turbulence. We derive kinetic equations for the probability distribution function (PDF) of the logarithm of the density field, $s$, in compressible turbulence in two forms: a first-order partial differential equation involving the average divergence conditioned on the flow density, $\langle \nabla \cdot {\bs u} | s\rangle$, and a Fokker-Planck equation with the drift and diffusion coefficients equal to $-\langle {\bs u} \cdot \nabla s | s\rangle$ and $\langle {\bs u} \cdot \nabla s | s\rangle$, respectively. Assuming statistical homogeneity only, the detailed balance at steady state leads to two exact results, $\langle \nabla \cdot {\bs u} | s \rangle =0$, and $\langle {\bs u} \cdot \nabla s | s\rangle=0$. The former indicates a balance of the flow divergence over all expanding and contracting regions at each given density. The exact results provide an objective criterion to judge the accuracy of numerical codes with respect to the density statistics in supersonic turbulence. We also present a method to estimate the effective numerical diffusion as a function of the flow density and discuss its effects on the shape of the density PDF.