Kinetic and internal energy transfer in implicit large eddy simulations
  of forced compressible turbulence
  We revisit the problem of how energy transfer through the turbulent cascade
operates in compressible hydrodynamic turbulence. In general, there is no
conservative compressible cascade since the kinetic and internal energy
reservoirs can exchange energy through pressure dilatation. Moreover,
statistically stationary turbulence at high Mach number can only be maintained
in nearly isothermal gas, i.e., if excess heat produced by shock compression
and kinetic energy dissipation is continuously removed from the system. We
mimic this process by a linear cooling term in numerical simulations of
turbulence driven by stochastic forcing. This allows us to investigate
turbulence statistics for a broad range of Mach numbers. We compute the rate of
change of kinetic and internal energy in wavenumber shells caused by advective,
compressive, and pressure dilatation effects and constrain power-law fits to
compressible turbulence energy spectra to a range of wavenumbers in which the
total energy transfer is close to zero. The resulting scaling exponents are
significantly affected by the forcing. Depending on the root mean square Mach
number, we find a nearly constant advective component of the cross-scale flux
of kinetic energy at intermediate wavenumbers for particular mixtures of
solenoidal and compressive modes in the forcing. This suggests the existence of
a natural, Mach number dependent mixture of forcing modes. Our findings also
support an advection-dominated regime at high Mach numbers with specific
scaling exponents (Burgers scaling for the pure velocity fluctuation $u$ and
Kolmogorov scaling for the mass-weighted variable $v=\rho^{1/3}u$).
