30 Numerical Methods of Solution

The first numerical solution of the Orr-Sommerfeld equation was obtained by Thomas (1953) in an effort to resolve the controversies which existed at that time concerning the asymptotic methods of approximation, and his results fully confirmed the conclusions of Heisenberg (1924) and Lin (1945) that plane Poiseuille flow is indeed unstable. Since then a number of numerical methods have been developed for dealing with various aspects of the Orr-Sommerfeld problem, and in this section therefore we shall discuss briefly some of these methods. Among them we can distinguish those which are applicable (a) to the determination of the curve of marginal stability or, more generally, the curves on which c _i= constant, (b) to the determination of the eigenvalue spectrum for fixed values of ? and R, and (c) to the determination of the associated eigenfunctions. Numerical methods are particularly helpful in studying the stability characteristics of flows for which the relevant values of the Reynolds number are low and in studying, for example, the dependence of ? _c and R _c on other secondary parameters of the problem. A full discussion of the stability of compressible boundary layers lies outside the scope of this book but this important problem is one for which asymptotic methods, similar to those discussed in 27, have thus far proved to be inadequate (Lees & Reshotko 1962) whereas a direct numerical attack on the problem has been quite...