Energy cascade and scaling in supersonic isothermal turbulence

TL;DR

This paper uses simulation data to verify an exact fourth-order relation in supersonic isothermal turbulence, supporting a Kolmogorov-like energy cascade and identifying universal features across Mach numbers.

Contribution

It reconstructs and verifies an exact fourth-order relation from Navier-Stokes equations for supersonic turbulence, revealing universal properties and deriving a new approximate relation.

Findings

Support for a Kolmogorov-like inertial energy cascade in supersonic turbulence

Existence of two compressible analogues of the four-fifths law

Identification of a universal fourth-order relation across Mach regimes

Abstract

Supersonic turbulence plays an important role in a number of extreme astrophysical and terrestrial environments, yet its understanding remains rudimentary. We use data from a three-dimensional simulation of supersonic isothermal turbulence to reconstruct an exact fourth-order relation derived analytically from the Navier-Stokes equations (Galtier and Banerjee, Phys. Rev. Lett., vol. 107, 2011, p. 134501). Our analysis supports a Kolmogorov-like inertial energy cascade in supersonic turbulence previously discussed on a phenomenological level. We show that two compressible analogues of the four-fifths law exist describing fifth- and fourth-order correlations, but only the fourth-order relation remains `universal' in a wide range of Mach numbers from incompressible to highly compressible regimes. A new approximate relation valid in the strongly supersonic regime is derived and verified. We also briefly discuss the origin of bottleneck bumps in simulations of compressible turbulence.