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

TL;DR

This paper derives a transport equation for scale-space energy density in compressible inhomogeneous turbulence, enhancing understanding of energy transfer and non-local interactions influenced by variable density and dilatation effects.

Contribution

It introduces a detailed derivation of the scale-space energy density transport equation tailored for compressible inhomogeneous flows, highlighting effects of density variations and dilatation.

Findings

Identifies the impact of variable density on turbulence energy transfer.

Highlights the role of dilatation in energy dynamics.

Provides a foundation for improved turbulence modeling in compressible flows.

Abstract

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.