Effects of critical layer structure on the nonlinear evolution of waves in free shear layers

A theoretical study is made of the nonlinear development of a slowly modulated wavetrain of the symmetric neutral mode of a tanhy shear layer. The shear layer Reynolds number is assumed to be large and the flow in the critical layer is taken to be both nonlinear and viscous. The complex wave amplitude A is shown to satisfy a nonlinear evolution equation, in which the nonlinearity takes an unusual form, depending on a realvalued function O of A, representing the phase shift in the stream function, or jump in streamwise fluctuating velocity, across the critical layer. The function f) is determined by a study of the critical layer structure, and surprisingly it is found to be identical with Haberman’s phase shift function for a critical layer away from a point of inflexion of the basic profile. However, the structure of the critical layer depends on the rate of change of A as well as on A itself, and this is essential to the form of the evolution equation. Some solutions of the evolution equation, illustrating both temporal and spatial variation of A , are computed. In the particular case of spatially varying waves, the growth rate reaches a maximum after which, at large distances, it gradually decreases to zero and the wave amplitude only grows algebraically as the f power of the distance.