Three-dimensional mode growth in boundary layers with tuned and detuned subharmonic resonance

Abstract

This work involves the use of controlled periodic disturbances to excite a plane Tollmien-Schlichting (TS) wave at one frequency ( f 2D) along with pairs of oblique waves with equal but opposite wave angles at a different frequency ( f 3D) in order to study the resonant growth of 3D modes in a Blasius boundary layer. In our earlier work (Corke & Mangano 1989; Corke 1990), the frequency of the oblique modes was exactly the subharmonic of the plane Tollmien-Schlichting (TS) mode. These modes were also phase-speed locked so that in terms of their streamwise wave numbers, α2D=1/2 α3D. This so-called ‘tuned’ subharmonic resonance leads to the enhanced growth of the otherwise linearly damped oblique waves, as well as the growth of higher harmonic 3D modes with frequencies and wave numbers: (3/2 f2d, 3/2 α2D, ±β3D), (5/2 f2d, 5/2 α2D, ±β3D), (f2d, α2D, ±2β3D) and (0, 0, ±2β3D). Even when the initial 3D oblique waves have frequencies which are close to the TS subharmonic frequency, a ‘detuned’ subharmonic resonance leads to the enhanced growth of the 3D mode. In addition, it promotes the growth of numerous discrete modes produced by successive sum and difference interactions. These interacted modes are also three dimensional, with higher amplification rates that increase with the interaction order. The growth of these modes accounts for the rapid spectral filling, and low-frequency modulation commonly observed in natural subharmonic transition. Starting from a ‘tuned' resonance, this scenario then provides a mechanism for the generation of a broad spectrum at the later stages of subharmonic mode transition. However, the results also suggest that with ‘natural’ transition, starting from low-amplitude broadband disturbances, the most likely 2D/3D resonance will be ‘detuned’.