Exact relation for correlation functions in compressible isothermal turbulence

Compressible isothermal turbulence is analyzed under the assumption of homogeneity and in the asymptotic limit of a high Reynolds number. An exact relation is derived for some two-point correlation functions which reveals a fundamental difference with the incompressible case. 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, $- {2 \over 3} \epsilon_{\rm{eff}} r = {\cal F}_r(r)$, where ${\cal F}_r$ is the radial component of the two-point correlation functions and $\epsilon_{\rm{eff}}$ 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, $\rho^{1/3} \uu$, is used.