The effects of spanwise confinement on stratified shear instabilities

TL;DR

This study investigates how transverse confinement affects stratified shear instabilities in inclined ducts, revealing stabilizing effects of lateral walls and the emergence of multiple unstable modes influenced by aspect ratio and stratification.

Contribution

It introduces a comprehensive analysis of 2D dispersion relations considering spanwise variations, extending previous 1D models and highlighting the impact of lateral walls and aspect ratio on instability modes.

Findings

Lateral walls have a stabilizing effect on instabilities.

Increasing aspect ratio leads to more unstable modes with odd-even spatial structures.

Most unstable modes tend to oscillate minimally in the spanwise direction.

Abstract

We consider the influence of transverse confinement on the instability properties of velocity and density distributions reminiscent of those pertaining to exchange flows in stratified inclined ducts, such as the recent experiment of Lefauve et al. (J. Fluid Mech. 848, 508-544, 2018). Using a normal mode streamwise and temporal expansion for flows in ducts with various aspect ratios $B$ and non-trivial transverse velocity profiles, we calculate two-dimensional (2D) dispersion relations with associated eigenfunctions varying in the 'crosswise' direction, in which the density varies, and the spanwise direction, both normal to the duct walls and to the flow direction. We also compare these 2D dispersion relations to the so-called one-dimensional (1D) dispersion relation obtained for spanwise invariant perturbations, for different aspect ratios $B$ and bulk Richardson numbers $Ri_b$. In this limited parameter space, the presence of lateral walls has a stabilizing effect. Furthermore, accounting for spanwise-varying perturbations results in a plethora of unstable modes, the number of which increases as the aspect ratio is increased. These modes present an odd-even regularity in their spatial structures, which is rationalized by comparison to the so-called one-dimensional oblique (1D-O) dispersion relation obtained for oblique waves. Finally, we show that in most cases, the most unstable 2D mode is the one that oscillates the least in the spanwise direction, as a consequence of viscous damping. However, in a limited region of the parameter space and in the absence of stratification, we show that a secondary mode with a more complex `twisted' structure dominated by crosswise vorticity becomes more unstable than the least oscillating Kelvin-Helmholtz mode associated with spanwise vorticity.