The Rate of Change of the Kinetic Energy Spectrum of Flow in a Compressible Fluid

Abstract

The variance spectrum of velocities in a non-homogeneous, compressible fluid does not represent the wave-number distribution of kinetic energy, as it does in incompressible, homogeneous (constant density) fluids. Use of a truncated Fourier transform and the assumption that the flow occurs in a finite area show that the kinetic energy spectrum in the former case is the co-spectrum between the velocity and the momentum. The Navier-Stokes equations are used to study the time rates of change of the kinetic energy spectrum produced by the various physical effects contained in those equations. Introduction of the assumption of homogeneity and incompressibility in the equations derived here gives the same qualitative results as Batchelor's (1953) study of the time rate of change of the spectrum of turbulent flow. Kinetic energy in a compressible, non-homogeneous fluid can draw on internal and potential energy, but these energy sources are not available to flow in incompressible, homogeneous fluids. It is shown that compressibility effects are not important in the action of the inertial or viscous effects on the total kinetic energy.