Nonlinear evolution of a first mode oblique wave in a compressible boundary layer: I. Heated/cooled walls

Abstract

The nonlinear stability of an oblique mode propagating in a two-dimensional compressible boundary layer is considered under the long wavelength approximation. The growth rate of the wave is assumed to be small so that the ideas of unsteady 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 show clearly the effects of wall heating and cooling, and in particular how heating destabilizes the boundary layer for these long wavelength inviscid modes at O(1) Mach numbers. A generalized expression for the linear growth rate is obtained and is shown to compare very well for a range of frequencies and wave angles at moderate Mach numbers with full numerical solutions of the linear stability problem. The numerical solution of the nonlinear unsteady critical layer problem using a novel method based on Fourier decomposition and Chebyshev collocation is discussed and some results are presented