NONLINEAR DEVELOPMENT OF VISCOUS GORTLER VORTICES IN A 3-DIMENSIONAL BOUNDARY-LAYER

In many practical situations where Gortler vortices are known to arise , the underlying basic velocity profile is three-dimensional. Only in recent years have studies been made of the stability of vortices in th ree-dimensional flows, and it has been shown that only a small crossfl ow velocity component is required in order to stabilize the Gortler me chanism completely. For large Gortler number (G much greater than 1) f lows, the most unstable linear vortex within a two-dimensional boundar y layer has a wavenumber of O(G1/5) and a corresponding growth rate of O(G3/5). Imposition of a crossflow component of size O(R(e)-1/2G3/5) (where R(e) is the Reynolds number of the flow) is sufficient to cause these higher wavenumber Gortler modes to decay. Indeed, for certain c rossflow/vortex wavenumber combinations, the vortices can be made neut rally stable. A weakly nonlinear analysis of near neutral modes reveal s that this slight nonlinearity is stabilizing and so can lead to fini te amplitude equilibrium states. In the present work, we give a nonlin ear account of the fate of the O(G1/5) wavenumber vortices as they evo lve downstream. A study of the large wavenumber modes within a two-dim ensional boundary layer [5], has shown that the effect of this strong nonlinearity is destabilizing and leads to a finite distance breakdown in the flow structure. Here we include the influence of the crossflow component and demonstrate how the stabilizing effects of crossflow an d the destabilizing nature of nonlinearity compete. Our calculations c an also describe unsteadiness in the vortex structure and they allow u s to speculate upon the relative likelihoods of observing various memb ers of the nonlinear Gortler modes in practice.