Uniform asymptotic approximations to the solutions of the Dunn-Lin equations

Abstract

In this paper, we develop a uniform asymptotic theory for the Dunn-Lin equations which govern the stability of compressible boundary layers at moderate Mach numbers. “First approximations” to the solutions are derived when the Prandtl number is equal to unity, in which case the structure of the approximations is especially simple. The rapidly varying parts of the approximations can be expressed in terms of certain generalized Airy functions and the slowly varying parts can be expressed in terms of quantities all but one of which are well-known from the older heuristic theories. The approximations obtained in this way are uniformly valid in a full neighborhood of the turning point. Because of the simplicity of the present theory, it is expected that the techniques developed in this paper can also be applied to other more general stability equations for compressible boundary layers, such as the Lees-Lin equations.