A pseudo-sound constitutive relationship for the dilatational covariances in compressible turbulence

Abstract

The mathematical consequences of a few simple scaling assumptions regarding the effects of compressibility are explored using a singular perturbation idea and the methods of statistical fluid mechanics. Representations for the pressure–dilatation and dilatational dissipation appearing in single-point moment closures for compressible turbulence are obtained. The results obtained, in as much as they come from the same underlying procedure, represent a unified development for both dilatational covariances. While the results are expressed in the context of a statistical turbulence closure they provide, with very few phenomenological assumptions, an interesting and clear mathematical model for the ‘scalar’ effects of compressibility. For homogeneous turbulence with quasi-normal large scales the expressions derived are – in the small turbulent Mach number squared isotropic limit – exact. The expressions obtained contain constants that have a precise physical significance and are defined in terms of integrals of the longitudinal velocity correlation. The pressure–dilatation covariance is found to be a non-equilibrium phenomenon related to the time rate of change of the kinetic energy and internal energy of the turbulence; it is seen to scale with α²M²_t ε_s [P_k/ε−1] (Sk/ε_s)². Implicit in the scaling is a dependence on the square of a gradient Mach number, S[lscr ]/c. A new feature indicated by the analysis is the appearance of the Kolmogorov scaling coefficient, α, suggesting that large-scale quantities embodied in the well-established ε∼u˜³/[lscr ] relationship provide a link to the structural dependence of the effects of compressibility. The expressions for the dilatational dissipation are found to depend on the turbulent Reynolds number and scale as M⁴_t (Sk/ε_s)⁴R⁻¹_t. The scalings for the pressure–dilatation are found to produce an excellent collapse of the pressure–dilatation data from direct numerical simulation.