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Scale-space energy density function, $E(\mathbf{x}, \mathbf{r})$, is definedas the derivative of the two-point velocity correlation. The function Edescribes the turbulent kinetic energy density of scale r at a location x andcan be considered as the generalization of spectral energy density functionconcept to inhomogeneous flows. We derive the transport equation for thescale-space energy density function in compressible flows to develop a betterunderstanding of scale-to-scale energy transfer and the degree of non-localityof the energy interactions. Specifically, the effects of variable-density anddilatation on turbulence energy dynamics are identified. It is expected thatthese findings will yield deeper insight into compressibility effects leadingto improved models at all levels of closure for mass flux, density-variance,pressure-dilatation, pressure-strain correlation and dilatational dissipationprocesses.

Detailed derivation of scale-Space Energy Density Transport Equation for Compressible Inhomogeneous Turbulent Flows