Spectra and statistics in compressible isotropic turbulence

Spectra and one-point statistics of velocity and thermodynamic variables in isotropic turbulence of compressible fluid are examined by using numerical simulations with solenoidal forcing at the turbulent Mach number M-t from 0.05 to 1.0 and at the Taylor Reynolds number Re-lambda from 40 to 350. The velocity field is decomposed into a solenoidal component and a compressible component in terms of the Helmholtz decomposition, and the compressible velocity component is further decomposed into a pseudosound component, namely, the hydrodynamic component associated with the incompressible field and an acoustic component associated with sound waves. It is found that the acoustic mode dominates over the pseudosound mode at turbulent Mach numbers M-t >= 0.4 in our numerical simulations. At turbulent Mach numbers M-t <= 0.4, there exists a critical wave number k(c) beyond which the pseudosound mode dominates while the acoustic mode dominates at small wave numbers k < k(c). In the pseudosound-mode-dominated region, the compressible velocity is fully enslaved to the solenoidal velocity, and its spectrum scales as M(t)(4)k(-3) in the inertial range. It is also found that in the inertial range, the spectra of pressure, density, and temperature exhibit a k(-7/3) scaling for M-t <= 0.3 and a k(-5/3) scaling for M-t >= 0.5.