INITIAL-VALUE PROBLEMS FOR SPOT DISTURBANCES IN INCOMPRESSIBLE OR COMPRESSIBLE BOUNDARY-LAYERS

This theoretical study of spot development in boundary layers is motivated by the need for some basic understanding of nonlinear spots, on which there appears to be little or no previous acceptable theory. The eventual aim is to be able to describe theoretically the transitional and/or turbulent spots which are often investigated experimentally. Here, as a starting point to guide possible nonlinear studies, we concentrate in particular on relatively long-scale inviscid spot disturbances in the context of the unsteady Euler equations, although short-scale with respect to Tollmien-Schlichting lengths for example. Both the incompressible and the compressible ranges are examined, for small disturbances, with the corresponding initial-value problems being treated computationally and analytically. The spatial spreading rate of the resulting spot is affected significantly by compressibility, and in fact tends to zero in the hypersonic extreme. The typical amplitudes provoked downstream and their decay lengths also vary considerably with the free-stream Mach number, producing two distinct structures in the transonic regime for example and an elongated structure in the hypersonic regime. The downstream behaviour found at comparatively large times, for finite Mach numbers, is used to provide guidance for nonlinear theory. There are also some potentially useful comparisons and links found with experiments, in both laminar and turbulent conditions.