Towards modeling inhomogeneous compressible turbulence using a two-scale statistical theory

Abstract

Turbulence models for compressible flows are investigated using a statistical theory called the two-scale direct-interaction approximation. Inertial-range spectra for velocity and density variances are assumed to derive models for several correlations systematically; they include the dilatation dissipation, mass flux, Reynolds stress, and pressure–dilatation correlation. Model expressions are shown to contain two important parameters: the turbulent Mach number and the density variance normalized by the mean density. Typical terms are material derivatives of the turbulent kinetic energy and its dissipation rate as well as the mean velocity divergence. The statistical theory is also applied to a two-time velocity correlation to derive a transport equation for the eddy viscosity. The equation is combined with the turbulent kinetic energy equation to derive a model equation for the dissipation rate. Direct numerical simulation data of isotropic and homogeneous shear turbulence are used to examine models for the dilatation dissipation and the pressure dilatation. The normalized density variance is shown to be useful to explain results from two runs of isotropic turbulence with different initial conditions.