Perturbation dynamics in viscous channel flows

Abstract

Plane viscous channel flows are perturbed and the ensuing initial-value problems are investigated in detail. Unlike traditional methods where travelling wave normal modes are assumed as solutions, this work offers a means whereby arbitrary initial input can be specified without having to resort to eigenfunction expansions. The full temporal behaviour, including both early-time transients and the long-time asymptotics, can be determined for any initial small-amplitude three-dimensional disturbance. The bases for the theoretical analysis are: (a) linearization of the governing equations; (b) Fourier decomposition in the spanwise and streamwise directions of the flow; and (c) direct numerical integration of the resulting partial differential equations. All of the stability criteria that are known for such flows can be reproduced. Also, optimal initial conditions measured in terms of the normalized energy growth can be determined in a straightforward manner and such optimal conditions clearly reflect transient growth data that are easily determined by a rational choice of a basis for the initial conditions. Although there can be significant transient growth for subcritical values of the Reynolds number, it does not appear possible that arbitrary initial conditions will lead to the exceptionally large transient amplitudes that have been determined by optimization of normal modes when used without regard to a particular initial-value problem. The approach is general and can be applied to other classes of problems where only a finite discrete spectrum exists (e.g. the Blasius boundary layer). Finally, results from the temporal theory are compared with the equivalent transient test case in the spatially evolving problem with the spatial results having been obtained using both a temporally and spatially accurate direct numerical simulation code.