Abstract: The weak nonlinear instability of a nonparallel boundary layer is studied by nonlinear parabolized stability equations(PSE). The PSEs are solved by employing a first-order backward difference scheme in streamwise direction and a fourth-order central difference scheme in the wallnormal direction. The nonlinear terms are acquired by the predictorcorrector and iterative approach. The evolution of some small amplitude disturbances and their high-order harmonic waves are studied. The results show that stable harmonic waves may become unstable with the existence of strong basic disturbance, and the critical positions at which they become unstable tend to move upwind with the increasing amplitude of the basic disturbance. The results also show that large vortices are likely to break into small vortices. The nonlinear nonparallel results are compared with data of direct numerical simulations (DNS) based on Navier-Stokes equations, which indicated the correctness of the PSE approach.

Key words: parabolized stability equation, nonlinear, nonparallel flow, supersonic boundary layer