The success of Prandtl's boundary layer theory and the desire to extend it to higher orders led to the formulation in the 1950s of a systematic perturbation procedure known as the method of matched asymptotic expansions. Not only is this method important in formulating approximate theories, but its understanding is essential in order to achieve success in any computational study where the dependent variables change rapidly within a small distance. Before describing matched asymptotic expansions, a brief review will be given of asymptotic methods for evaluating integrals, such as the saddle point method. The method of matched asymptotic expansions will then be introduced and applied to ordinary differential equations containing a small parameter. It will be shown how to determine the rescaling of both dependent and independent variables in the thin layers where rapid variations can occur. Next, the application to various partial differential equations will be illustrated concluding with the derivation of the boundary layer equations. Their failure in applications such as a shock-boundary layer interaction, where rapid variations occur in more than one direction, will be discussed briefly and the triple deck theory formulated to deal with that situation will be introduced.
Transition to turbulence in shear flows is generally viewed as the outcome of a hydrodynamic instability. In most applications, the Reynolds number is large. A consequence is that stiffness problems occur in the equations governing linear stability such as the Orr-Sommerfeld equation. Ignoring viscosity would seem to be justifiable in such cases, but singularities then arise for neutral or for weakly amplified perturbations. An understanding of internal boundary layers permits the development of numerical schemes that are able to deal with such situations. The basic ideas underlying some of these schemes will be outlined.
Lagerstrom, P. A., Matched Asymptotic Expansions: Ideas and Techniques, Springer-Verlag, 1988.