On the Stability of a Time Dependent Boundary Layer

The aim of this article is to determine the stability characteristics of a Rayleigh layer, which is known to occur when the fluid above a flat plate has a velocity imparted to it (parallel to the plate). This situation is intrinsically unsteady, however as a first approximation we consider the instantaneous stability of the flow. The Orr-Sommerfeld equation is found to govern fixed downstream wavelength linear perturbations to the basic flow profile. By the solution of this equation we can determine the Reynolds numbers at which the flow is neutrally stable; this quasi-steady approach is only formally applicable for infinite Reynolds numbers. We shall consider the large Reynolds number limit of the original problem and use a three deck mentality to determine the form of the modes. The results of the two calculations are compared, and the linear large Reynolds number analysis is extended to consider the effect of weak nonlinearity in order to determine whether the system is sub or super critical.