On the Catalytic Effect of Resonant Interactions in Boundary Layer Transition

This paper is concerned with a fascinating phenomenon in boundary layer transition, namely, three-dimensional disturbances undergo rapid amplification despite that they have smaller linear growth rates than two-dimensional ones. Physical mechanisms are sought by considering two types of nonlinear interactions between oblique and planar instability modes. The first is the well-known subharmonic resonance. The relevant mathematical theory and its main predictions are briefly summarised. This mechanism, however, operates only among a very restrictive set of modes, and hence is unable to explain the broadband nature of the amplifying disturbances observed in experiments. The second mechanism involves the interaction between a planar and a pair of oblique Tollmien—Schlichting (T—S) waves which are phase-locked in that they travel with (nearly) the same phase speed. It is a more general type of interaction than subharmonic resonance since no further restriction is imposed on the frequencies. Yet similar to subharmonic resonance, this interaction also leads to super-exponential growth of the oblique modes, while the planar mode remains to follow linear stability theory. The dominant planar mode therefore plays the role of a catalyst, the implications of which for the e^N-method and for transition control are discussed.