Transient growth in Stratified Shear Flows

Stability of fluid flow has traditionally been studied by modal stability analysis. Modal stability analysis gives the asymptotic nature (long time behavior, t→∞) of how the perturbations to a fluid flow grow. The modal or the linear stability analysis predicts that the plane poiseuille flow (channel flow) becomes unstable at Reynold’s number, Re=5772.22 but experiments have shown that flow becomes unstable at a much lower Re. Similarly the pipe poiseuille flow is predicted, by modal stability theory, to be stable at any Re but becomes unstable around Re=2000. It was generally accepted that neglecting the nonlinearities in the sta-bility equations is reason for this discrepancy. But recently using Non-modal stability the transient behavior of perturbations has been studied by various peo-ple, i.e. the nature of growth of energy of perturbation at very short times. If the energy of perturbations grow at short times, it gives rise to the possibility for nonlinearities to kick in and cause the instability in the fluid flow. There is a natural gradient of density in the atmosphere and flow in atmosphere is generally modeled to be that of a simple shear. As a first approximation th