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

The amplitude-dependent neutral stability properties, mainlyof an accelerating boundary-layer flow, are studied theoreticallyfor large Reynolds numbers when the disturbance size is sufficientlylarge to provoke a strongly non-linear critical layer withinthe flow field. The theory has a rational basis aimed at a detailedunderstanding of the delicate physical balances controllingstability. It shows that when the fundamental disturbance size rises to O(R^-1/3, where R is the Reynolds number based on theboundary-layer thickness, the neutral wavelength shortens andthe wavespeed increases in such a way that they become comparablewith the typical thickness and speed, respectively, of the basicflow. In this Rayleigh-like situation a new (previously negligible)feature emerges, that of a substantial pressure variation acrossthe critical layer, which strongly affects the jump conditionson the Rayleigh solutions holding outside the critical layer.As a result of the strong non-linearity the total velocity jumpis affected non-linearly by the critical layer vorticity, whilein contrast the phase shift remains linearly dependent on thevorticity. Furthermore, it is shown that the phase shift, notthe total velocity jump, dictates the neutral stability criteria. Also, flow reversal occurs near the wall where the disturbanceis greater than the basic flow. The link between the viscouseffects in the wall layers and in the critical layer fixes theamplitude-dependence of the neutral modes throughout. As thedisturbance amplitude increases the critical layer with vorticitytrapped within it moves toward the edge of the boundary layerand is forced to leave the boundary layer when exceeds O(R^-1/3,if neutral stability is to be maintained. This departure israther abrupt, involving a dependence on (scaled amplitude)^–12.A study of the more practical application to temporally growingdisturbances should be interesting.