Spectra and excitation modes of turbulence in a compressible fluid

A random-force method is developed for the statistical description of homogeneous and isotropic turbulence in a compressible fluid in the case where acoustic turbulence is excited by an external force with large-amplitude potential and vortex components. A closed equation is derived for the characteristic turbulence functional of a compressible fluid, and turbulence excitation by a random force is analyzed. Spectral characteristics of the turbulence are investigated in the region of wavenumber space where dissipative factors and temporal changes in entropy can be neglected. It is shown that the Cauchy problems for the characteristic functionals of incompressible and compressible fluids with energy pumping by a large-amplitude external force possess group invariance, which leads to a similarity theorem for the functionals. Assuming an equilibrium region in the turbulence spectrum, this invariance property is employed to determine the turbulence energy spectra in the equilibrium region, spectral characteristics for two-point semi-invariants of arbitrary order, and the kinetic-energy spectrum of the turbulence.