Singular eigen functions in hydrodynamic stability : The roles of rotation, stratification and elasticity

Abstract

Hydrodynamic stability, an important branch of fluid mechanics, owes its popularity to the successful prediction of the transition of (unstable) laminar states in a wide class of flows. While the traditional approach has been reliant on an analysis within a modal framework, the merits of a nonmodal approach have been recognized in the last two and a half decades particularly in the context of shearing flows. Although the non-modal approach typically involves the solution of an initial value problem, the resulting temporal response, for short times and at large Reynolds numbers (Re), may also be understood in terms of the dynamics of the underlying inviscid continuous spectra (CS). The equations governing the evolution of small-amplitude perturbations in shearing flows in this limit are usually singular, and the continuous spectra owe their origin to such singular points. The thesis is mainly concerned with the structure of the singular eigenfunctions comprising such inviscid continuous spectra in rotating flows, and to a lesser extent, with the singular eigenfunctions in homogeneous and stratified parallel shearing flows, and rotational flows in the presence of elasticity. The manner in which such eigenfunctions may be superposed to obtain a solution of the initial value problem is also considered. The detailed analysis is devoted to the singular modes of a Rankine vortex both in two and three dimensions; an analytically soluble problem that nevertheless offers insight into the singular eigenfunctions associated with more general vorticity profiles. The Rankine analysis is then extended to smooth vortices. In three dimensions, such an extension is made by drawing an analogy with (stably) stratified shear flows and the associated continuous spectra. The final part of the thesis discusses the continuous spectrum of an elastic vortex column in the limit of high Reynolds and Deborah numbers. Further, a novel two-dimensional instability of an elastic vortex column, that arises from the resonant interaction of elastic shear waves, is analyzed in detail both numerically and via an asymptotic analysis valid in the limit of weak elasticity.