Algebraic instability and transient growth in stratified shear flows

Abstract

In the present work we study the asymptotic and short time stabihty of stratified shear flow analytically and numerically. The algebraic growth and instability of stably stratified inviscid couette flow is studied analytically. A transient growth analysis has been carried out on both stable and unstable stratification for viscous Couette and Poiseuille flow. Stratification in viscosity has also been considered in the study of Poiseuille flow. To begin with we tried to understand the nonnormality and transient growth through a simple two dimension nonnormal system. With this 2D system we demonstrate that nonnormality is a necessary but not a sufficient condition for transient growth. We derive the limits of nonnormality in which the system can exhibit transient growth. The dependence of transient growth on the initial condition has also been studied and the initial condition which gives the maximum possible growth at a given time is derived. The stability of stratified Couette flow is a well studied problem in literature, yet the problem is not well understood. The mechanisms by which the flow becomes unstable, e.g. how atmospheric turbulence sets in is still a problem of open research. Traditionally the main interests lay in obtaining the large time exponent for the flow perturbations to make predictions on the stability of the flow. It was not until the 1990's that the transient aspect of the problem was looked at. We have carried out work on the bounded inviscid Couette flow to address both short time as well long time aspect of the stabiltiy. To understand and explain the problem we first use a toy problem. We show in the toy problem that there is a linear instability for singular initial condition in temperature and an algebraic growth for smooth initial conditions. The asymptotic exponents we obtain for the full problem are in agreement with the literature. Based on the toy problem we argue for the full problem that there will be transient algebraic growth for a smooth initial condition, although this still has to be numerically verified. In the viscous analysis we study the effect of stratification on the transient growth. In the density stratified Couette flow we find that a stable stratification decreases the maximum transient growth and an unstable stratification increases it. It was also found that stable stratification increases the spanwise dependence of disturbances causing the maximum transient growth. The results for the Poiseuille are qualitatively similar to the Couette flow case for both stable and unstable stratification. Including viscosity stratification leads to very large transient growth. When density stratification is considered along with viscosity stratification it has very little effect on the transient growth and there is no qualitative change in the nature of transient growth.