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Compressible turbulence is still a subject poorly understood despite the number of applications ranging from aeronautics to astrophysics. In order to better understand recent direct numerical simulations made in the context of interstellar turbulence which reveal intermittency with anomalous scalings we have investigated isothermal hydrodynamics under the assumption of homogeneity and in the asymptotic limit of a high Reynolds number. An exact relation has been derived for some two-point correlation functions which reveals a fundamental difference with the incompressible case for which we have the classical Kolmogorov (4/5 or 4/3) law. The main difference resides in the presence of a new type of term which acts on the inertial range similarly as a source or a sink for the mean energy transfer rate. When isotropy is assumed, compressible turbulence may be described by the relation :
−2εeffr = Fr(r), where Fr is the radial component of a two-point correlation functions and εeff is an effective mean total energy injection rate. By dimensional arguments, we predict that a spectrum in k−5/3 may still be preserved at small scales if the density-weighted fluid velocity ρ1/3u is used. A steeper power law is expected at the largest scales if the effective mean total energy exhibits a scale dependence. The theoretical predictions are in relatively good agreement with the most recent direct numerical simulations.
Author(s):
Sebastien Galtier
Universite Paris-Sud
France
Supratik Banerjee
Universite Paris-Sud
France