Instability of tilted shear flow in a strongly stratified and viscous medium

TL;DR

This paper performs a linear stability analysis of tilted shear flows in strongly stratified, viscous media, revealing how low-Froude-number modes behave and are stabilized with increasing Froude number, aligning with experimental observations.

Contribution

It introduces a reduced Orr-Sommerfeld model for low-Froude-number modes and analyzes their stability, providing new insights into stratified shear flow instabilities and their experimental validation.

Findings

Low-Froude-number mode is a horizontal inflectional instability.

Increasing Froude number stabilizes the mode.

Small vertical velocities proportional to Froude number squared influence stability.

Abstract

A linear stability analysis is performed on a tilted parallel wake in a strongly stratified fluid at low Reynolds numbers. A particular emphasis of the present study is given to the understanding of the low-Froude-number mode observed by the recent experiment (Meunier, J. Fluid Mech., vol. 699, 2012, pp. 174-197). In the limit of low Froude number, the linearised equations of motion can be reduced to the Orr-Sommerfeld equation on the horizontal plane, except the viscous term that contains vertical dissipation. Based on this equation, it is proposed that the low-Froude-number mode would be a horizontal inflectional instability, and should remain two dimensional at small tilting angles. To support this claim, the asymptotic regime where this equation would be strictly valid is subsequently discussed in relation to previous arguments on the proper vertical length scale. Furthermore, the absolute and convective instability analysis of parallel wake is performed, revealing qualitatively good agreement with the experimental result. The low-Froude-number mode is found to be stabilised on increasing Froude number, as is in the experiment. The emergence of small vertical velocity at finite Froude number, the size of which is proportional to the square of Froude number, plays the key role in the stabilisation. It modifies the inflectional instability and is also associated with the paradoxically stabilising buoyancy on increasing Froude number. Lastly, we proposed some possible behaviours of the base flow when the tilting angle changes, and they may provide a better approximation to produce the behaviour consistent with the experiment.