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.