27 Heuristic Methods of Approximation

In studying the behaviour of the solutions of the Orr-Sommerfeld equation for large values of ?R, our ultimate goal is to obtain asymptotic approximations which are uniformly valid in a bounded domain of the (complex) z-plane containing not only the turning point z _c but also the boundary points z ₁ and z ₂ and which can then be used to derive approximations to the eigenvalue relation. In this section, however, we will consider the heuristic methods of approximation which were developed mainly by Heisenberg (1924), Tollmien (1929, 1947) and Lin (1945, 1955). Although the resulting approximations are not uniformly valid, they are adequate for many computational purposes and they do provide valuable insight into the general structure of the problem. In addition, these approximations continue to play an important role in some of the more recent developments, as discussed in Chapter 5.

The heuristic theory can perhaps best be understood from the point of view of the method of matched asymptotic expansions. The essential elements of the theory include the derivation of first approximations of inner and outer type, the matching of these approximations, and finally the combining of them to form composite approximations. To simplify the present discussion we shall now restrict attention to the case of marginal stability for which c and hence is real. It is then convenient to suppose that U(z) is monotone increasing so that is positive. To insure that...