Universal Scaling of Pressure Dilatation in Homogeneous Compressible Turbulence

Compressible turbulence plays a key role in phenomena ranging from star formation to hypersonic flight. However, our understanding of these flows remains challenging due to the coupling of thermodynamic and hydrodynamic processes and an expanded parameter space. A key challenge is the modelling of pressure dilatation — the reversible transfer of energy between internal energy and turbulent kinetic energy due to dilatational fluctuations — for which little is known theoretically beyond its mean. We analyze the scaling of both the mean and the much-less studied variance of pressure dilatation using a large database of direct numerical simulations of statistically steady, homogeneous compressible turbulence spanning a wide range of Reynolds and turbulent Mach numbers. In addition to these traditional parameters, we consider the recently proposed parameter δ (the ratio of dilatational to solenoidal velocity fluctuations) and find that it collapses both mean and variance behavior. We evaluate existing models, particularly pressure-equipartition and pseudo-sound theory, and show how they break down outside their expected ranges. In these regions, alternative scaling laws are proposed and compared with DNS data. Our results further support δ as a key parameter in identifying universal scaling laws in compressible turbulence.