WEAKLY NONLINEAR STABILITY OF VISCOUS VORTICES IN 3-DIMENSIONAL BOUNDARY-LAYERS

Recently it has been demonstrated that three-dimensionality can play an important role in dictating the stability of any Gortler vortices which a particular boundary layer may support. According to a linearized theory, vortices within a high Gortler number flow can take one of two possible forms within a two-dimensional flow supplemented by a small crossflow of size O(Re-1/2G3/5), where Re is the Reynolds number of the flow and G the Gortler number. Bassom & Hall (1991) showed that these forms are characterized by O(1)-wavenumber inviscid disturbances and larger O(G1/5)-wave-number modes which are trapped within a thin layer adjacent to the bounding surface. Here we concentrate on the latter, essentially viscous, vortices. These modes are unstable in the absence of crossflow but the imposition of small crossflow has a stabilizing effect. Bassom & Hall (1991) demonstrated the existence of neutrally stable vortices for certain crossflow/wavenumber combinations and here we describe the weakly nonlinear stability properties of these disturbances. It is shown conclusively that the effect of crossflow is to stabilize the nonlinear modes and the calculations herein allow stable finite-amplitude vortices to be found. Predictions are made concerning the likelihood of observing some of these viscous modes within a practical setting and asymptotic work permits discussion of the stability properties of modes with wavenumbers that are small relative to the implied O(G1/5) scaling.