Scale dependence and cross-scale transfer of kinetic energy in compressible hydrodynamic turbulence at moderate Reynolds numbers

We investigate the properties of the scale dependence and cross-scale transfer of kinetic energy in compressible three-dimensional hydrodynamic turbulence by means of two direct numerical simulations of decaying turbulence with initial Mach numbers M=1/3 and 1, and with moderate Reynolds numbers, Rλ∼100. The turbulent dynamics is analyzed using compressible and incompressible versions of the dynamic spectral transfer (ST) and the Kármán-Howarth-Monin (KHM) equations. We find that the nonlinear coupling leads to a flux of the kinetic energy to small scales where it is dissipated; at the same time, the reversible pressure-dilatation mechanism causes oscillatory exchanges between the kinetic and internal energies with an average zero net energy transfer. While the incompressible KHM and ST equations are not generally valid in the simulations, their compressible counterparts are well satisfied and describe, in a quantitatively similar way, the decay of the kinetic energy on large scales, the cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. There exists a simple relationship between the KHM and ST results through the inverse proportionality between the wave vector k and the spatial separation length l as kl≃3. For a given time, the dissipation and pressure-dilatation terms are strong on large scales in the KHM approach, whereas the ST terms become dominant on small scales; this is due to the complementary cumulative behavior of the two methods. The effect of pressure dilatation is weak when averaged over a period of its oscillations and may lead to a transfer of the kinetic energy from large to small scales without a net exchange between the kinetic and internal energies. Our results suggest that for large-enough systems, there exists an inertial range for the kinetic energy cascade. This transfer is partly due to the classical, nonlinear advection-driven cascade and partly due to the pressure dilatation-induced energy transfer. We also use the ST and KHM approaches to investigate the properties of the internal energy. The dynamic ST and KHM equations for the internal energy are well satisfied in the simulations but behave very differently with respect to the viscous dissipation. We conclude that ST and KHM approaches would better be used for the kinetic and internal energies separately.