Some years ago Hall and Smith in a number of papers developed a theory governing the interaction of vortices and waves in shear flows. In recent years immense interest has been focused on so-called self-sustained processes in turbulent shear flows where the importance of waves interacting with streamwise vortex flows has been elucidated in a number of; see for example the work of Waleffe and colleagues, Kerswell, Gibson, etc. These processes have a striking resemblance to coherent structures observed in turbulent shear flow and for that reason they are often referred to as exact coherent structures. It is shown that the structures associated with the so-called 'lower branch' state, which has been shown to play a crucial role in these self-sustained process, is nothing but a Rayleigh wave vortex interaction with a wave system generating streamwise vortices inside a critical layer. The theory enables the reduction of the 3D Navier Stokes equations to a coupled system for a steady streamwise vortex and an inviscid wave system. The reduced system for the streamwise vortices must be solved with jump conditions in the shear across the critical layer and the position of that layer constitutes a nonlinear pde eigenvalue problem. Remarkable agreement between the asymptotic theory and numerical simulations is found thereby demonstrating the importance of vortex-wave interaction theory in the mathematical description of coherent structures in turbulent shear flows. The theory offers the possibility of drag reduction in turbulent shear flows by controlling the flow to the neighborhood of the lower branch state. The relevance of the work to more general shear flows is also discussed.
