Nonlinear Theories for Shear Flow Instabilities: Physical Insights and Practical Implications

Abstract

This article reviews the nonlinear stability theories that have been developed to explain laminar–turbulent transition processes in boundary and free shear layers. For such spatially developing shear flows, a high–Reynolds number approach is necessary to account for, in a systematic and self-consistent manner, multiple competing physical factors, such as nonlinearity, nonparallelism, nonequilibrium, and viscosity. While the basic ideas and fundamental mechanisms are rooted in the classical weakly nonlinear theory, which was formulated primarily for exactly parallel flows and on the basis of finite Reynolds number, the high–Reynolds number formulations lead to low-dimensional evolution systems, which differ significantly from the finite–Reynolds number counterparts and better describe the observations. Owing to efforts in the past 30 years or so, nonlinear evolution systems have been derived for inviscid Rayleigh modes, viscous Tollmien–Schlichting waves, (first and second) Mack modes, and cross-flow vortices. Theories have also been developed for nonlinear intermodal interactions, including oblique mode interaction, subharmonic resonance, phase-locked interactions, and fundamental resonance; these underpin many intriguing behaviors in the three-dimensional stages of transition. These theories and results explain several key nonlinear features observed and should play a useful role in guiding future experimental and numerical investigations.