On the Nonlinear Initial Value Problem for Vortex–Wave Interactions in Shear Flows

Small-amplitude wave systems interacting nonlinearly can produce 0(1)amplitude streamwise vortex structures through the vortex–wave interaction mechanism described, for example, by 1–3]. The key feature of the interaction is that the spanwise velocity component of a vortex is small as compared to the streamwise component so that a nonlinear wave system driving the spanwise velocity component through Reynolds stresses can provoke a 0(1) response of the vortex. The wave system can correspond to either a Rayleigh or Tollmien–Schlichting wave disturbance, but previous work on the initiation of the process has been confined to Rayleigh waves (see, for example, 5, 6]). Here, we address the nonlinear initial value problem for Tollmien–Schlichting wave–vortex interactions in channel flows. The evolution of the disturbances is accounted for using the phase equation approach of 7]. We determine the circumstances, if any, under which the finite amplitude vortex–wave equilibrium states of 4] are generated. Our discussion of the nonlinear evolution of a wave system points toward a possible mechanism for the experimentally observed breakup of three-dimensional instabilities into shorter streamwise scales.