Asymptotic Analysis of the Orr–Sommerfeld Problem for Boundary-Layer Flows
The Quarterly Journal of Mechanics and Applied Mathematics, (1982)DOI: 10.1093/qjmam/35.1.69
Abstract
This paper is concerned with the derivation of ‘first approximations’ to the solutions of the Orr—Sommerfeld equation and to the eigenvalue relation which are uniformly valid on the unbounded interval 0IJz<∞. As in the case with secondorder equations, uniformity at infinity can be achieved by a suitable choice of the large parameter and this leads to the introduction of a new Langer variable. On re-expansion of the results for bounded values of z we recover the results given previously (1, 2) for symmetrical flows in a channel. Detailed calculations of the curve of marginal stability for the asymptotic suction profile, both with and without cross-flow, are in good agreement with the results of direct numerical calculations. A comparison is also made with one of the well-known heuristic approximations to the eigenvalue relation.