Nonlinear development and secondary instability of large-amplitude Goertler vortices in hypersonic boundary layers

The weakly nonlinear development of large-amplitude Goertler vortices in a small neighborhood of the neutral stability position given by the linear theory is considered. It is shown that if the basic state and the wall curvature satisfy a certain condition, the Goertler vortices will grow until they become so large as to drive a mean flow correction as large as the basic state. Attention is then given to the further downstream evolution of these large-amplitude vortices, and it is shown that the vortices have a triple layer structure which consists of a region of vortex activity bounded by two transition layers over which the amplitude of the harmonic part of the vortex decays to zero exponentially. The positions of the two transition layers are found to be governed by a free boundary problem, which is solved for a number of curvature distribution cases. The secondary instability of the two transition layers is investigated with respect to traveling waves which are pi/2 out of phase in the spanwise direction with the steady Goertler vortices.