A Kolmogorov-like exact relation for compressible polytropic turbulence

Abstract

AbstractCompressible 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 $\mathcal{S}$ which may act as a source or a sink for the mean energy transfer rate. At subsonic scales, we predict dimensionally that the isotropic $k^{-5/3}$ energy spectrum for the density-weighted velocity field ($\rho ^{1/3} \boldsymbol {v}$), previously obtained for isothermal turbulence, is modified by a polytropic contribution, whereas at supersonic scales $\mathcal{S}$ 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.