Energy exchange between solenoidal and dilatational motions in compressible isotropic turbulence

In compressible turbulence, the dilatational motions and thermodynamic fluctuations become increasingly significant as Mach number increases and shocklets form. Understanding the mechanism of the energy transfer between solenoidal and dilatational motions is crucial in the study of compressible turbulent flows. This work follows Helmholtz decomposition of the velocity field and derives the budgets of the solenoidal and dilatational velocity variances for compressible turbulence. All terms in the budget equations are investigated using the direct numerical simulation data from a set of solenoidally forced compressible isotropic turbulence at different turbulent Mach numbers (M_t) with Reynolds number up to 170. The result shows that at low M_t, the terms in dilatational velocity variance equation oscillate about zero. As M_t increases, specific terms develop a bias in their statistical distribution and thus represent net variance transfer or exchange. The exchange processes are analyzed in detail at different Mach numbers both in physical space and in their statistical distribution.