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A numerical study of supersonic turbulence

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Abstract

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 τac. In two dimensions eddies merge over many τac. In three dimensions vortex sheets break-up into short vortex filaments within two τac. Entropy fluctuations, produced by strong shocks, are stretched into filaments over several τac. 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.

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Communicated by M.Y. Hussaini

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.

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Porter, D.H., Pouquet, A. & Woodward, P.R. A numerical study of supersonic turbulence. Theoret. Comput. Fluid Dynamics 4, 13–49 (1992). https://doi.org/10.1007/BF00417962

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