A new density variance - Mach number relation for subsonic and supersonic, isothermal turbulence

TL;DR

This paper introduces a new linear relation linking density variance and Mach number in isothermal turbulence, validated through high-resolution simulations across subsonic to supersonic regimes, and accounts for different forcing mechanisms.

Contribution

It proposes a novel linear relation between density variance and compressible velocity, reducing dependence on forcing type, supported by extensive hydrodynamical simulations.

Findings

Density variance scales linearly with Mach number, with a forcing-dependent proportionality.

A new relation links density variance to compressible velocity fluctuations, independent of forcing.

Results span Mach numbers from 0.1 to 15, covering subsonic to supersonic turbulence.

Abstract

The probability density function (PDF) of the gas density in subsonic and supersonic, isothermal, driven turbulence is analyzed with a systematic set of hydrodynamical grid simulations with resolutions up to 1024^3 cells. We performed a series of numerical experiments with root mean square (r.m.s.) Mach number M ranging from the nearly incompressible, subsonic (M=0.1) to the highly compressible, supersonic (M=15) regime. We study the influence of two extreme cases for the driving mechanism by applying a purely solenoidal (divergence-free) and a purely compressive (curl-free) forcing field to drive the turbulence. We find that our measurements fit the linear relation between the r.m.s. Mach number and the standard deviation of the density distribution in a wide range of Mach numbers, where the proportionality constant depends on the type of the forcing. In addition, we propose a new linear relation between the standard deviation of the density distribution and the standard deviation of the velocity in compressible modes, i.e. the compressible component of the r.m.s. Mach number. In this relation the influence of the forcing is significantly reduced, suggesting a linear relation between the standard deviation of the density distribution and the standard deviation of the velocity in compressible modes, independent of the forcing, ranging from the subsonic to the supersonic regime.