Symmetry transformations are one of the most fundamental features of differential equations and illuminate the axiomatic properties of classical mechanics. Therefore we performed a symmetry analysis, also called Lie group analysis (Bluman and Kumei 1989) of the Navier-Stokes equations in the high Reynolds number limit for compressible turbulent flow. The Navier-Stokes equations for compressible flows admit different symmetries than the equations of incompressible flows. These symmetries have to be taken into account for developing models describing compressible turbulence.
An approach to derive turbulent scaling laws based on symmetry analysis is presented in this paper. Starting from the governing equations of compressible turbulent flows we derived their symmetry properties. In the case of isotropic turbulence four symmetry groups were combined. Invariants were computed from the symmetries. These invariants constitute turbulent scaling laws or similarity solutions of the governing equations. Scaling laws were derived for the turbulent kinetic energy in a compressible isotropic flow. These laws describe the decay of turbulence. The analytical solution are compared to results from Direct Numerical Simulation (DNS).