Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 1. An asymptotic theory valid for small amplitude disturbances

Abstract

A strict distinction is made between the two fundamental assumptions in the Stuart-Watson theory of nonlinear stability, one of which is that the amplitude of disturbance is sufficiently small, while the other is that the damping or amplification rate for an infinitesimal disturbance is small. This distinction leads to classification of the nonlinear stability theory into two asymptotic theories: the theory based on the first assumption can be applied to subcritical flows with Reynolds numbers away from the neutral curve, even to flows with no neutral curve, such as plane Couette flow or pipe Poiseuille flow, while the theory based on the second assumption is available only for Reynolds numbers and wavenumbers in the neighbourhood of the neutral curve. In the theory based on the first assumption the concept of trajectories in phase space, together with the method of eigenfunction expansion, is introduced in order to display nonlinear behaviour of the disturbance amplitude and to provide the most rational definition of the Landau constant available for classification of the behaviour patterns.