The Stability of Time-Periodic Flows

Abstract

The stability of periodic states of mechanical systems has long been an object of study. Dynamic stabilization and destabilization can lead to dramatic modifications of behavior depending on the proper tuning of the amplitude and frequency of the modulation. It has only been in the recent past that attention has been focused on such possibilities in hydrodynamics. The interest lies not only with the mechanics of this new class of problems but with the possibilities for applications. If an imposed modulation can destabilize an otherwise stable state, then there can be a major enhancement of heat/mass/momentum transport. If an imposed modulation can stabilize an otherwise unstable state, then higher efficiencies can be attained in various processing techniques. The aim here is to give reviews of three prototypal problems having sinusoidal time variation: parallel shear flows, convective instabilities, and centrifugal insta­ bilities.1 These will be used as vehicles for a discussion of scale analysis, a procedure which is crucial to the understanding of these as well as more general flows. Before proceeding with the examination of periodic basic states, a word must be said in reference to the definition of stability. Since the basic state is unsteady, it seems natural to compare the disturbance growth rate with the rate of change of the basic state (Shen 1961). However, in periodic states the repeating sequence of basic-state acceleration followed by basic-state deceleration leads to ambiguities in interpretation. As a result, there is fairly common agreement to follow Rosenblat (1968) and term a basic periodic state unstable if there exists a disturbance that experiences net growth over each modulation cycle. A state on which every disturbance decays at every instant is called stable. namely, monotonically stable. It may happen that a state is neither unstable nor stable, i.e. the basic state is subject