The nonlinear interaction of Tollmien-Schlichting waves and Taylor-Goertler vortices in curved channel flows

The development of Tollmien-Schlichting waves (TSWs) and Taylor-Goertler vortices (TGVs) in fully developed viscous curved-channel flows is investigated analytically, with a focus on their nonlinear interactions. Two types of interactions are identified, depending on the amplitude of the initial disturbances. In the low-amplitude type, two TSWs and one TGV interact, and the scaled amplitudes go to infinity on a finite time scale; in the higher-amplitude type, which can also occur in a straight channel, the same singularity occurs if the angle between the TSW wavefront and the TGV is greater than 41.6 deg, but the breakdown is exponential and takes an infinite time if the angle is smaller. The implications of these findings for external flow problems such as the design of laminar-flow wings are indicated. It is concluded that longitudinal vortices like those observed in the initial stages of the transition to turbulence can be produced unless the present interaction mechanism is destroyed by boundary-layer growth.