PERTURBATION DYNAMICS IN VISCOUS CHANNEL FLOWS

Plane viscous channel flows are perturbed and the ensuing initial-valu e problems are investigated in detail. Unlike traditional methods wher e travelling wave normal modes are assumed as solutions, this work off ers a means whereby arbitrary initial input can be specified without h aving to resort to eigenfunction expansions. The full temporal behavio ur, 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) lineariza tion of the governing equations; (b) Fourier decomposition in the span wise and streamwise directions of the flow; and (c) direct numerical i ntegration of the resulting partial differential equations. All of the stability criteria that are known for such flows can be reproduced. A lso, optimal initial conditions measured in terms of the normalized en ergy growth can be determined in a straightforward manner and such opt imal 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 va lues of the Reynolds number, it does not appear possible that arbitrar y initial conditions will lead to the exceptionally large transient am plitudes that have been determined by optimization of normal modes whe n used without regard to a particular initial-value problem. The appro ach is general and can be applied to other classes of problems where o nly a finite discrete spectrum exists (e.g. the Blasius boundary layer ). Finally, results from the temporal theory are compared with the equ ivalent transient test case in the spatially evolving problem with the spatial results having been obtained using both a temporally and spat ially accurate direct numerical simulation code.