Secondary Instability in Boundary-Layer Flows

Abstract

The stability of a secondary Tollmien–Schlichting wave, whose wavenumber and frequency are nearly one half those of a fundamental Tollmien–Schlichting instability wave, is analyzed using the method of multiple scales. Under these conditions, the fundamental wave acts as a parametric exciter for the secondary wave. The results show that the amplitude of the fundamental wave must exceed a critical value to trigger this parametric instability. This value is proportional to a detuning parameter which is the real part of k − 2K, where k and K are the wavenumbers of the fundamental and its subharmonic, respectively. For Blasius flow, the critical amplitude is approximately 29% of the mean flow, and hence many other secondary instabilities take place before this parametric instability becomes significant. For other flows where the detuning parameter is small, such as free‐shear layer flows, the critical amplitude can be small, thus the parametric instability might play a greater role.