Nonlinear evolution of secondary instabilities in boundary-layer transition

Methods of nonlinear stability theory are applied to analyze the evolution of disturbances in the three-dimensional stage immediately preceding the breakdown of a laminar boundary layer. A perturbation scheme is used to solve the nonlinear equations and to develop a dynamical model for the interaction of primary and secondary instabilities. The first step solves for the two-dimensional primary wave in the absence of secondary disturbances. Once this finite-amplitude wave is calculated, it is decomposed into a basic-flow component and an interaction component. The basic-flow component acts as a parametric excitation for the three-dimensional secondary wave, while the interaction component captures the resonance between the secondary and primary waves. Results are presented in two principal forms: amplitude growth curves and velocity profiles. Our results agree with experimental data and the few available results of transition simulations and, moreover, reveal the origin of the observed phenomena. The method described establishes the basis for physical transition criteria in a given disturbance environment.This work has been supported by the Air Force Office of Scientific Research under Contract F46920-87-K-0005 and Grant AFOSR-88-0186 (TH) and by an ONT Postdoctoral Fellowship (JDC).