Two-point closures for weakly compressible turbulence

Abstract

The aim of the present paper is to address the problem of homogeneous isotropic compressible turbulence within the framework of two-point statistical closures. In order to simplify the description, weak compressibility assumptions are first introduced in the equations governing the fluctuating field: the fluid is supposed to be barotropic and nonlinear terms involving density fluctuations are neglected. The equations for the two-point two-time correlations are written. They are closed by extending the direct interaction approximation to (weakly) compressible fields. This work is then used as a ground to derive an extension of the eddy damped quasi-normal Markovian closure that accounts for compressibility effects. Both closures reflect the existence of acoustic waves and the complex nature of nonlinear interactions between modes. The EDQNM model is solved numerically for the case of an isotropic turbulent field maintained statistically stationary by injecting energy in the large scales. At low Mach number, the spectrum of the purely compressible velocity is found first to scale as K−5/3, then to evolve, at large time, towards an asymptotic K−11/3 scaling. The same behavior is found for the pressure spectrum, equipartition of energy between the two compressible modes being observed. Results concerning the dissipation show that its dilatational component is proportional to the square of the turbulent Mach number. At higher Mach number, the slope of the compressible spectra appears to be less steep, and discrepancies from the Mt2 scaling are detected. The influence of the Reynolds number is also analyzed and the ratio of the compressible energy to the solenoidal kinetic energy is found to scale as Mt2 Re for Mt2 Re smaller than 10. A simple spectral model reproducing the main features of the EDQNM results at low Mach number is proposed.