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 _ac. In two dimensions eddies merge over many _ac. In three dimensions vortex sheets break-up into short vortex filaments within two _ac. Entropy fluctuations, produced by strong shocks, are stretched into filaments over several _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.This work was supported at the University of Minnesota by grants DE-FG02-87ER25035 from the Office of Energy Research of the Department of Energy, AST-8611404 from the National Science Foundation, and by equipment grants from Sun Microsystems, Gould Electronics, Seagate Technology, and the Air Force Office of Scientific Research (AFOSR-86-0239). Partial support for this work has also come from the Army Research Office Contract Number DAAL03-89-C-0038 funding the Army High Performance Computing Research Center (AHPCRC) at the University of Minnesota. At Nice this work was supported under DRET Contract 500-276, under the GdR CNRS-SPI Mécanique des Fluides, and under two special grants from the Observatoire de la Côte d'Azur.