On cascade of kinetic energy in compressible hydrodynamic turbulence

TL;DR

This study compares compressible and incompressible turbulence models using direct numerical simulations, revealing the absence of a clear inertial range for kinetic energy cascade at Mach 1 and highlighting differences in measurement scales.

Contribution

It provides a comparative analysis of KHM and coarse-graining methods in compressible turbulence, emphasizing their similarities, differences, and applicability in identifying energy cascade characteristics.

Findings

No inertial range observed for kinetic energy cascade at Mach 1.

Both approaches conserve total energy and show similar results for energy transfer rates.

Differences in increment separation and filtering scales between methods.

Abstract

Properties of the turbulent cascade of kinetic energy are studied using direct numerical simulations of three-dimensional hydrodynamic decaying turbulence with a moderate Reynolds number and the initial Mach number $M=1$. Compressible and incompressible versions of the Karman-Howarth-Monin (KHM) and low-pass filtering/coarse-graining approaches are compared. In the simulation the total energy is well conserved; the scale dependent KHM and coarse-grained energy equations are also well conserved; the two approaches show similar results, the system does not have an inertial range for the cascade of kinetic energy, the region where this cascade dominates also have a non-negligible contribution of the kinetic-energy decay, dissipation, and pressure-dilatation effects. While the two approaches give semi-quantitatively similar results for the kinetic energy cascade, dissipation and pressure-dilatation rates, they differ in the increment separation and filtering scales; these scales are not simply related. The two approaches may be used to find the inertial range and to determine the cascade/dissipation rate of the kinetic energy.