  Scale-space energy density function, $E(\mathbf{x}, \mathbf{r})$, is defined
as the derivative of the two-point velocity correlation. The function E
describes the turbulent kinetic energy density of scale r at a location x and
can be considered as the generalization of spectral energy density function
concept to inhomogeneous flows. We derive the transport equation for the
scale-space energy density function in compressible flows to develop a better
understanding of scale-to-scale energy transfer and the degree of non-locality
of the energy interactions. Specifically, the effects of variable-density and
dilatation on turbulence energy dynamics are identified. It is expected that
these findings will yield deeper insight into compressibility effects leading
to improved models at all levels of closure for mass flux, density-variance,
pressure-dilatation, pressure-strain correlation and dilatational dissipation
processes.

Fluid Dynamics