The governing parameters in compressible turbulence

Incompressible turbulence is commonly characterized by a single non-dimensional parameter, the Reynolds number ($R_\lambda$). However, when compressible effects are present other parameters are needed to characterize the flow, in part, because of the coupling between hydrodynamics and thermodynamics, and the type of external forcing used to sustain
fluctuations against dissipation. It is common to use the turbulent Mach number $M_t=u'/c$
($u'$ is the rms velocity and $c$ the mean speed of sound) as a measure of compressibility.
However, using a large database of Direct Numerical Simulations of stationary compressible isotropic turbulence forced with both solenoidal and dilatational forcing at a range $R_\lambda$ (38-450) and $\mt$ (0.1-0.6) we show that these two parameters alone fail to describe the state of the flow uniquely. This is illustrated through fluctuations of both
thermodynamic as well as hydrodynamic variables. We propose a new set of governing parameters which can indeed scale all the data successfully regardless of the levels of dilatations introduced by the forcing and allow us to distinguish different statistical regimes the flow could be in. Our proposal is also compared successfully against other results in the literature.