Excitation of secondary instabilities in boundary layers

Abstract

The receptivity to fundamental and subharmonic secondary instabilities is analysed for two-dimensional boundary layers. Fundamental modes are excited by the direct scattering of Tollmien–Schlichting (TS) waves over surface variations. The excitation of subharmonic modes stems from the combined scattering of acoustic free-stream disturbances and TS waves over surface variations. The surface variations are localized in their streamwise extent and are the result of roughness or suction. The velocity field is expanded in terms of small parameters characterizing the acoustic disturbance and the surface variation. The TS wave is included as part of the base flow leading to a non-homogeneous system with periodic coefficients governing the receptivity. The receptivity amplitudes show a strong dependence on the TS-wave amplitude, and for subharmonic modes a strong dependence on the TS-wave phase at the location of the surface variation. The receptivity analysis shows a significant bias toward fundamental modes of secondary instability for larger TS-wave amplitudes – except for conditions of extremely high free-stream sound level. A combination of receptivity results and stability results suggests a bias toward subharmonic modes for TS-wave amplitudes below 0.5% and toward fundamental modes for TS-wave amplitudes above 0.5% (normalized by the local edge velocity).