Mathematical framework for analysis of internal energy dynamics and spectral distribution in compressible turbulent flows

The objective of this work is to develop the framework and governing equations needed to analyze the interactions and spectral distribution of internal energy in a manner similar to that of a kinetic energy analysis. To emulate the role of velocity in a kinetic energy analysis, a variable ($\varphi \sim \sqrt{p}$, where $p$ is pressure) is introduced to enable the examination of internal energy dynamics. Evolution equations for the mean and fluctuating components of $\varphi$ are derived. These equations enable the precise examination of mean-turbulent flow internal energy interactions, internal-kinetic energy exchange, and the spectral distribution of internal energy.

Figure 1

Interactions between kinetic and internal energies in the nonlinear regime using Eqs. (10), (13), (31), and (36).