Energy spectrum in compressible turbulence

The general scaling of spectra in compressible turbulence is not well understood,
in particular the circumstances under which different theoretical proposals such as
pseudo-sound and equipartition apply. The evidence supporting different scaling laws seems
inconsistent across the literature, especially at high turbulent Mach numbers ($M_{t}$).
From an asymptotic expansion based on two parameters characterizing compressibility (the traditional $M_{t}$ as a measure of scale separation between acoustic and turbulent processes, and ${u^{'}_{d}}/{u^{'}_{s}}$ the ratio of dilatation to solenoidal rms velocity
as a measure of compressibility levels). We identify the dominant processes in the governing equations and determine the behavior expected for the spectrum. Using a very large DNS database with different forcing modes and a wide range of governing parameters
(Taylor microscale Reynolds numbers, $Re_{\lambda}$ up to 170 and turbulent Mach numbers, $M_{t}$ up to 0.8) we support these predictions and reconcile the seemingly contradictory results in the literature. We will discuss the potential implications of these findings for compressible scalar mixing.