An initial-value problem for fully three-dimensional inflectional boundary layer flows

In a recent paper the present authors considered the effects of small cross-flow upon the nonlinear evolution of two oblique waves of unequal amplitude. The present analysis extends this work in two ways: (i) the cross-flow is increased to O(1) making it of comparable magnitude to the streamwise component and (ii) by taking the long-wavelength limit of Rayleigh's equation a whole spectrum of wave numbers can now be catered for. Thus (ii) allows a fairly general initial disturbance to be accommodated. Another significant effect due to the presence of a whole spectrum of wave numbers is that the critical layer jump is now forced by a quadratic, as opposed to the usual cubic, nonlinearity. Numerical solutions of the new nonlinear amplitude evolution equation are presented for a special class of initial disturbance.