A numerical study of supersonic turbulence

We report on simulations of nonstationary supersonic isotropic turbulent flow using the Piecewise-Parabolic Method on uniform grids of 2048² in two dimensions and 256³ in three dimensions. Intersecting shock waves initiate the transfer of energy from long to short wavelengths. Weak shocks survive for many acoustic times τ_ac. In two dimensions eddies merge over many τ_ac. In three dimensions vortex sheets break-up into short vortex filaments within two τ_ac. Entropy fluctuations, produced by strong shocks, are stretched into filaments over several τ_ac. These filaments persist and are mirrored in the density. We observe three temporal phases: onset, with the initial formation of shocks; quasi-supersonic, with strong density contrasts; and post-supersonic, with a slowly decaying root mean square Mach number. Compressive modes quickly establish a k^-2 velocity power spectrum. In three dimensions solenoidal modes build-up during the supersonic phase delineating through time a Kolmogorov k^-5/3 envelope and leaving a self-similarly decaying ∼k^-0.9 spectrum at lower wave numbers.