Spatial evolution of nonlinear acoustic mode instabilities on hypersonic boundary layers

We consider the effects of strong critical-layer nonlinearity on the spatial evolution of an initially linear ‘acoustic mode’ instability wave on a hypersonic flat-plate boundary layer. Our analysis shows that nonlinearity, which is initially confined to a thin critical layer, first becomes important when the amplitude of the pressure fluctuations becomes O (1/ M 4 In M 2 ), where M is the free-stream Mach number. The flow outside the critical layer is still determined by linear dynamics and therefore takes the form of a linear instability wave — but with its amplitude completely determined by the flow within the critical layer. The latter flow is determined by a coupled set of nonlinear equations, which we had to solve numerically.