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Properties of the turbulent cascade of kinetic energy are studied usingdirect numerical simulations of three-dimensional hydrodynamic decayingturbulence with a moderate Reynolds number and the initial Mach number $M=1$.Compressible and incompressible versions of the Karman-Howarth-Monin (KHM) andlow-pass filtering/coarse-graining approaches are compared. In the simulationthe total energy is well conserved; the scale dependent KHM and coarse-grainedenergy equations are also well conserved; the two approaches show similarresults, the system does not have an inertial range for the cascade of kineticenergy, the region where this cascade dominates also have a non-negligiblecontribution of the kinetic-energy decay, dissipation, and pressure-dilatationeffects. While the two approaches give semi-quantitatively similar results forthe kinetic energy cascade, dissipation and pressure-dilatation rates, theydiffer in the increment separation and filtering scales; these scales are notsimply related. The two approaches may be used to find the inertial range andto determine the cascade/dissipation rate of the kinetic energy.

On cascade of kinetic energy in compressible hydrodynamic turbulence