Effects of bulk viscosity, heat capacity ratio and Prandtl number on the dispersion relationship of the compressible Navier-Stokes equation

TL;DR

This paper investigates how bulk viscosity, heat capacity ratio, and Prandtl number influence the dispersion characteristics of the 3D compressible Navier-Stokes equations, revealing significant effects on acoustic and entropic modes and proposing an empirical relation for disturbance evolution.

Contribution

It provides a detailed analysis of dispersion variations with key parameters and introduces an empirical relation for bulk viscosity ratio affecting disturbance evolution.

Findings

Acoustic modes are dispersive up to a bifurcation wavenumber.

Dispersion function deviation scales with wavenumber squared at low wavenumbers.

Empirical relation for bulk viscosity ratio predicts significant disturbance evolution.

Abstract

Here, variation of the dispersion characteristics of 3D linearised compressible Navier-Stokes equation with respect to bulk viscosity ratio $\kappa/\mu$, specific heat ratio $\gamma$ and Prandtl number $Pr$ is presented. The 3D compressible NSE supports two vortical, one entropic and two acoustic modes. While the vortical and entropic modes are non-dispersive in nature, the acoustic modes are dispersive only up to a certain bifurcation wavenumber. The characteristics and variation of relative diffusion coefficient for entropic and acoustic modes and a specially designed dispersion function for acoustic modes with depressed wavenumber $\eta$ is presented which depend on bulk viscosity ratio, $\gamma$ and $Pr$. At lower wavenumber components, the deviation of the dispersion function from the inviscid and adiabatic case is proportional to $\eta^2$ at the leading order and the relative diffusion coefficients increase linearly with $\kappa/\mu$ and $\gamma$ while varying inversely with $Pr$. When the bulk viscosity ratio is increased, the shape and extent of the dispersion function is altered significantly and the change is more significant for higher wavenumber components. The relative diffusion coefficient for entropic and acoustic modes show contrasting variation with wavenumber depending upon $\kappa/\mu$, $\gamma$ and $Pr$. We show by solving linearised compressible NSE that relatively significant evolution and radiation of acoustic and/or entropic disturbances are noted when the bulk viscosity ratio is close to the corresponding critical value for which the bifurcation wavenumber is maximum. Based on this criterion, we have presented an empirical relation to obtain $\kappa/\mu$ depending upon $\gamma$ and $Pr$ which would indicate the range of bulk viscosity ratio for obtaining relatively significant disturbance evolution.