  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.
