Exact relation for correlation functions in compressible isothermal turbulence

TL;DR

This paper derives an exact relation for correlation functions in compressible isothermal turbulence, revealing a fundamental difference from incompressible turbulence and proposing a modified energy spectrum scaling.

Contribution

It introduces an exact relation for two-point correlation functions in compressible turbulence, highlighting the role of a new term affecting energy transfer.

Findings

A new term acts as a source or sink in energy transfer.

The relation simplifies to a form similar to incompressible turbulence under isotropy.

A $k^{-5/3}$ spectrum may be preserved using density-weighted velocity.

Abstract

Compressible isothermal turbulence is analyzed under the assumption of homogeneity and in the asymptotic limit of a high Reynolds number. An exact relation is derived for some two-point correlation functions which reveals a fundamental difference with the incompressible case. The main difference resides in the presence of a new type of term which acts on the inertial range similarly as a source or a sink for the mean energy transfer rate. When isotropy is assumed, compressible turbulence may be described by the relation, $- {2 \over 3} \epsilon_{\rm{eff}} r = {\cal F}_r(r)$, where ${\cal F}_r$ is the radial component of the two-point correlation functions and $\epsilon_{\rm{eff}}$ is an effective mean total energy injection rate. By dimensional arguments we predict that a spectrum in $k^{-5/3}$ may still be preserved at small scales if the density-weighted fluid velocity, $\rho^{1/3} \uu$, is used.