On the nonlinear evolution of Görtler vortices in non{parallel boundary layers

The non-linear development of finite amplitude Gortler vortices in a non-parallel boundary layer on a curved wall is investigated using perturbation methods based on the smallness of e, the non-dimensional wavelength of the vortices. The crucial stage in the growth or decay of the vortices takes place in an interior viscous layer of thickness O(e') and length 0(e). In this region the downstream velocity component of the perturbation contains a mean flow correction of the same order of magnitude as the fundamental which is driving it. Moreover, these functions satisfy a pair of coupled non-linear partial differential equations which must be solved subject to some initial conditions imposed at a given downstream location. It is found that, depending on whether the boundary layer is more or less unstable downstream of this location, the initial disturbance either grows into a finite amplitude GSrtlcr vortex or decays to zero. For the Blasius boundary layer on a concave wall it is found that GSrtler vortices can only develop if the rate of increase of curvature of the wall is sufficiently large. In this case the finite amplitude solution which develops initially in an e-neighbourhood of the position where the disturbance is introduced changes it