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 eN-method and for transition control are discussed.
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Wu, X., Stewart, P.A., Cowley, S.J. (2008). On the Catalytic Effect of Resonant Interactions in Boundary Layer Transition. In: Bonilla, L.L., Moscoso, M., Platero, G., Vega, J.M. (eds) Progress in Industrial Mathematics at ECMI 2006. Mathematics in Industry, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71992-2_10
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