On the instability of Görtler vortices to nonlinear travelling waves

Recent theoretical work by Hall & Seddougui (1989) has shown that strongly nonlinear high-wavenumber Görtler vortices developing within a boundary layer flow are susceptible to a secondary instability which takes the form of travelling waves confined to a thin region centred at the outer edge of the vortex. This work considered the case in which the secondary mode could be satisfactorily described by a linear stability theory, and in the current paper our objective is to extend this investigation of Hall & Seddougui (1989) into the nonlinear regime. We find that, at this stage, not only does the secondary mode become nonlinear, but it also interacts with itself so as to modify the governing equations for the primary Görtler vortex. In this case, then, the vortex and the travelling wave drive each other, and indeed the whole flow structure is described by an infinite set of coupled nonlinear differential equations. We undertake a Stuart-Watson type of weakly nonlinear analysis of these equations and conclude, in particular, that on this basis there exist stable flow configurations in which the travelling mode is of finite amplitude. Implications of our findings for practical situations are discussed, and it is shown that the theoretical conclusions drawn here are in good qualitative agreement with available experimental observations.