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

A weakly nonlinear theory is presented for investigating the initial nonlinear development of Goertler vortices in the neighborhood of the neutral position and for deriving two coupled evolution equations. The latter makes it possible to determine whether the vortices are decaying or growing depending on the sign of a controlling constant. It is found that if the vortices are growing, the mean flow correction becomes so large as to affect the basic state of distances downstream of the neutral position. A fully nonlinear theory concerning the further downstream development of these large-amplitude Goertler vortices is also presented. The upper and lower boundaries of the region of vortex activity are determined by a free-boundary problem involving the boundary layer equations. Attention is also given to the secondary instability of the flow in the transition layers centered at the upper and lower boundaries of the region of vortex activity. It is found that the superimposed perturbations are spanwise periodic traveling waves, which are pi/2 radians out of phase with the fundamental.