An alternative approach to linear and nonlinear stability calculations at finite Reynolds numbers

Abstract

An extended version of the interactive boundary-layer approach which has been used widely in steady-flow calculations is applied here to the linear and nonlinear stability properties of channel flows and boundary layers in the moderate-to-large Reynolds-number regime. This is the regime of most practical concern. First, for linear stability the agreement found between the interactive approach and Orr-Sommerfeld results remains fairly close even at Reynolds numbers as low as about$\frac{1}{10}$of the critical value for plane Poiseuille flow, or$\frac{1}{5}$for Blasius flow. Secondly, nonlinear unsteady calculations and comparisons with full solutions obtained by enlarging the same method are also presented. Overall the work suggests that, at the finite Reynolds numbers where real interest lies, the dominant physical processes of instability in channel flow and boundary layers are of boundary-layer form, with interaction, and it suggests also an alternative numerical technique for determining those processes. This alternative technique uses the interactive boundary-layer method as the central means for obtaining full unsteady Navier-Stokes solutions.