Effects of spanwise confinement on stratified shear instabilities

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 (2018)]. Using a normal mode streamwise and temporal expansion for flows in ducts with various aspect ratios B and nontrivial 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 relations obtained for spanwise invariant perturbations, for different aspect ratios B and bulk Richardson numbers Rib. In this limited parameter space, the presence of lateral walls has a stabilizing effect, in that the 1D growth-rate predictions are almost systematically an upper bound to the 2D growth rates, which in turn decrease monotonically as lateral walls are brought together with increased spanwise confinement (B→0). 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 dispersion relation obtained for oblique waves, characterized by a continuously varying spanwise wavenumber in addition to the streamwise wavenumber. 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.