A Kolmogorov-like exact relation for compressible polytropic turbulence

Compressible hydrodynamic turbulence is studied under the assumption of a polytropic closure. Following Kolmogorov, we derive an exact relation for some two-point correlation functions in the asymptotic limit of a high Reynolds number. The inertial range is characterized by: (i) a flux term implying in particular the enthalpy; and (ii) a purely compressible term l which may act as a source or a sink for the mean energy transfer rate. At subsonic scales, we predict dimensionally that the isotropic k(-513) energy spectrum for the density-weighted velocity field (rho(1/3)nu), previously obtained for isothermal turbulence, is modified by a polytropic contribution, whereas at supersonic scales 9 may impose another scaling depending on the polytropic index. In both cases, it is shown that the fluctuating sound speed is a key ingredient for understanding polytropic compressible turbulence.