On spikes and spots: strongly nonlinear theory and experimental comparisons

Abstract

Spikes and spots are discussed mostly for incompressible boundary layers, with the emphasis towards strong nonlinearity. The distinction between forced and free disturbances then becomes blurred, as spikes and spots reproduce each other. First, the forced case is concentrated on the start of spikes. The theory used is that of the two- or three-dimensional interacting boundary layer, capturing nonlinear Tollmien-Schlichting waves, for example, or following a vortex-wave interaction. Finite-time breakup produces shortened time and length scales, yielding agreement with computations and experiments on the first spike in transition, with subsequent spot formation. After the breakup, normal pressure gradients and vortex wind-up become significant locally. Second, the free case concerns initial-value problems for spots containing a wide band of three-dimensional nonlinear disturbances. The theory points to successive nonlinear stages starting at the wing tips near the spot trailing edge but gradually entering the middle as the amplitudes increase downstream. This effect combined with shortening scales produces a spread angle near 11°, very close to the experimental observations. The overall spot structure is described briefly, including also the leading edge. Viscosity arises later in two ways; for the case mentioned above with spikes originating near the surface and also through a novel interaction influencing the global spot.