Dynamical Instability of Laminar Axisymmetric Flow of Perfect Fluid with Stratification

TL;DR

This paper investigates the instability of non-homogeneous, axisymmetric perfect fluid flows with stratification, revealing how entropy gradients influence surface gravity and sound wave instabilities through numerical analysis.

Contribution

It provides new insights into how entropy gradients affect the stability of stratified fluid flows, especially regarding surface gravity and sound wave modes, using numerical integration.

Findings

Entropy gradients can induce surface gravity modes in non-homogeneous flows.

Internal gravity modes grow only in flows unstable to axisymmetric perturbations.

Incorrect entropy distribution assumptions violate boundary conditions in free boundary problems.

Abstract

The instability of non-homoentropic axisymmetric flow of perfect fluid with respect to non-axisymmetric infinitesimal perturbations was investigated by numerical integration of hydrodynamical differential equations in two-dimensional approximation. The non-trivial influence of entropy gradient on unstable sound and surface gravity waves was revealed. In particular, both decrease and growth of entropy against the direction of effective gravitational acceleration $g_{eff}$ give rise to growing surface gravity modes which are stable with the same parameters in the case of homoentropic flow. At the same time increment of sound modes either grows monotonically while the rate of entropy decrease against $g_{eff}$ gets higher or vanishes at some values of positive and negative entropy gradient in the basic flow. The calculations have showed also that growing internal gravity modes appear only in the flow unstable to axisymmetric perturbations. At last, the analysis of boundary problem with free boundaries uncovered that's incorrect to set the entropy distribution according to polytropic law with polytropic index different from adiabatic value, since in this case perturbations don't satisfy the free boundary conditions.