Amplitude-dependent Stability of Boundary-layer Flow with a Strongly Non-linear Critical Layer

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Abstract
The amplitude-dependent neutral stability properties, mainly of an accelerating boundary-layer flow, are studied theoretically for large Reynolds numbers when the disturbance size δ is sufficiently large to provoke a strongly non-linear critical layer within the flow field. The theory has a rational basis aimed at a detailed understanding of the delicate physical balances controlling stability. It shows that when the fundamental disturbance size δ rises to O(R-1/3, where R is the Reynolds number based on the boundary-layer thickness, the neutral wavelength shortens and the wavespeed increases in such a way that they become comparable with the typical thickness and speed, respectively, of the basic flow. In this Rayleigh-like situation a new (previously negligible) feature emerges, that of a substantial pressure variation across the critical layer, which strongly affects the jump conditions on the Rayleigh solutions holding outside the critical layer. As a result of the strong non-linearity the total velocity jump is affected non-linearly by the critical layer vorticity, while in contrast the phase shift remains linearly dependent on the vorticity. Furthermore, it is shown that the phase shift, not the total velocity jump, dictates the neutral stability criteria. Also, flow reversal occurs near the wall where the disturbance is greater than the basic flow. The link between the viscous effects in the wall layers and in the critical layer fixes the amplitude-dependence of the neutral modes throughout. As the disturbance amplitude increases the critical layer with vorticity trapped within it moves toward the edge of the boundary layer and is forced to leave the boundary layer when δ exceeds O(R-1/3, if neutral stability is to be maintained. This departure is rather abrupt, involving a dependence on (scaled amplitude)−12. A study of the more practical application to temporally growing disturbances should be interesting.

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