Spectra and Mach number scaling in compressible homogeneous shear turbulence

The effects of Mach number on the spectra and statistics of stationary compressible homogeneous shear turbulence (HST) are studied using numerical simulations in a rectangular domain (L_x = 4π, L_y = L_z = 2π) at turbulent Mach numbers from 0.05 to 0.66 and Taylor Reynolds numbers from 40 to 100. Long-term simulation results show that a statistically stationary state is obtained and the flow meets the strong acoustic equilibrium assumption at M_t ≠0.4. The analysis of spectral properties indicates that velocity and pressure tend toward a Mach number scaling of Mt2 suggested by acoustic dynamics at M_t ≠0.3. As for one-point statistics, it is found that a Mt4 scaling predicted by pseudo-sound theory holds for normalized compressible kinetic energy, K^c/K^s, at the small turbulent Mach number M_t ≠0.1. K^c/K^s approaches a Mt2 scaling at relatively higher turbulent Mach numbers, which is consistent with the spectral results. The normalized compressible dissipation rate, Ïμ ^c/Ïμ ^s, is nearly independent of Taylor Reynolds number and exhibits the same Mt4 scaling at small turbulent Mach numbers. The root mean square values of pressure, density, and temperature of compressible HST show good agreement with the Mt2 scaling, with the coefficient approximately doubled as compared with the compressible isotropic turbulence.