Enduring two-dimensional perturbations with significant non-modal growth

Laminar shear flows can display large non-modal perturbation growth, often through the lift-up mechansm, and can undergo subcritical transition to turbulence. The process is three-dimensional. Two-dimensional (2D) spanwise-independent perturbations are often considered less important as they typically undergo modest levels of transient growth and are short-lived. Strikingly, we show the existence of 2D non-modal perturbations that get amplified significantly and survive for long periods of time. Two-layer and three-layer viscosity stratified plane shear flows are taken to be the mean states. We show that while the two-layer flow is always modally stable, the three-layer flow supports exponential growing instabilities only when the middle layer is the least viscous. The non-modal stability analysis is performed only for the modally stable configurations of these flows. At later times, the non-modal perturbations feature strongly confined vortical structures near the interface in the two-layer flow. For the three-layer flow, similar observations are noted when all the three layers have different shear rates with the vortices prominently seen in the vicinity of the interface between the least viscous and middle layers. For the three-layer flow configuration with the outer layers having equal shear rates, the perturbation structure shows symmetry about the middle layer and evolves such that the Orr mechanism can repeatedly occur in a regenerative manner resulting in the perturbation energy evolving in a markedly non-monotonic fashion. When these same perturbations are introduced in a uniform plane shear flow, the extent of non-modal transient growth is shown to be significantly smaller.