© 1969 by Oxford University Press
COMPOSITE SERIES IN THE ORR-SOMMERFELD PROBLEM FOR SYMMETRIC CHANNEL FLOW
( Department of Mathematics, The City University London E.C.1. )
A new approach is made to the asymptotic theory (for large values of the Reynolds number) of the Orr-Sommerfeld problem for the stability of symmetric channel flow. The well-known Lin-Heisenberg approximation is shown to need a rationalization, because certain asymptotic series are used in regions where their validity is open to question. This point is discussed in terms of ‘inner’, ‘intermediate’ and ‘outer’ limiting processes.
The method adopted is to first define certain exact solutions of the Orr-Sommerfeld differential equation by using well-known convergent series, treated in more detail than usual. These series are not uniformly useful for asymptotic analysis, so asymptotic approximations to the exact solutions are obtained in the form of composite series by the use of the matching principle described by Van Dyke (1), as modified by Fraenkel (2).
The present work is quite different from the rigorous discussions of Wasow (3, 4). The emphasis here is on obtaining useful explicit approximations to those exact solutions needed for the eigenvalue problem. While no claim is made to rigour, the analysis does permit a systematic discussion of asymptotic error-estimates, and application to the asymptotic theory of the neutral curves is fairly straightforward. This leads to a new derivation of Lin's results, and to a number of new results of some interest. These include a discussion of the transition from ‘viscous’ to ‘inviscid’ instability as the basic velocity profile acquires a point of inflexion, a second-order analysis of the upper branch of the neutral curve when the velocity profile has no inflexion, and a first-order analysis when the velocity profile has an inflexion. The asymptotic neutral curves are shown with numerically calculated curves for various profiles. The paper is concluded by a detailed discussion of the Lin-Heisenberg approximations to the eigenvalue relation.