  We investigate 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 M = 1, and with moderate Reynolds numbers, R_lambda ~
100. The turbulent dynamics is analyzed using compressible and incompressible
versions of the dynamic spectral transfer (ST) and the Karman-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 k l ~ 3^1/2. 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 owing 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 ...
