On the Instability of Gortler Vortices to Nonlinear Travelling Waves

Author to whom correspondence should be addressed Recent theoretical work by Hall & Seddougui (1989) has shownthat strongly nonlinear high-wavenumber Görtler vorticesdeveloping within a boundary layer flow are susceptible to asecondary instability which takes the form of travelling wavesconfined to a thin region centred at the outer edge of the vortex.This work considered the case in which the secondary mode couldbe satisfactorily described by a linear stability theory, andin the current paper our objective is to extend this investigationof Hall & Seddougui (1989) into the nonlinear regime. Wefind that, at this stage, not only does the secondary mode becomenonlinear, but it also interacts with itself so as to modifythe governing equations for the primary Görtler vortex.In this case, then, the vortex and the travelling wave driveeach other, and indeed the whole flow structure is describedby an infinite set of coupled nonlinear differential equations.We undertake a Stuart-Watson type of weakly nonlinear analysisof these equations and conclude, in particular, that on thisbasis there exist stable flow configurations in which the travellingmode is of finite amplitude. Implications of our findings forpractical situations are discussed, and it is shown that thetheoretical conclusions drawn here are in good qualitative agreementwith available experimental observations.