Nonlinear evolution of modes in a compressible boundary layer

The nonlinear stability of an oblique mode propagating in a 2D compressible boundary layer is considered under the long wave-length approximation. The growth rate of the wave is assumed to be small so that the ideas of nonlinear critical layers can be applied. It is shown that the spatial/temporal evolution of the mode is governed by a pair of coupled unsteady nonlinear equations for the disturbance vorticity and density. Expressions for the linear growth rate shows clearly the effects of wall heating and cooling, and in particular how heating destabilizes the boundary layer for these long wavelength inviscid modes at 0(1) Mach numbers. The numerical solution of the nonlinear problem is discussed and some results are presented.