ON THE BEHAVIOR OF 3-DIMENSIONAL WAVE-PACKETS IN VISCOUSLY SPREADING MIXING LAYERS

We consider analytically the evolution pf a three-dimensional wave pac ket generated by an impulsive source in a mixing layer. The base flow is assumed to be spreading due to viscous diffusion. The analysis is r estricted to small disturbances (linearized theory). A suitable high-f requency ansatz is used to describe the packet; the key elements of th is description are a complex phase and a wave action density. It is fo und that the product of this density and an infinitesimal material vol ume convecting at the local group velocity is not conserved: there is a continuous interaction between the base flow and the wave action. Th is interaction is determined by suitable mode-weighted averages of the second End fourth derivatives of the base-flow velocity profile. Alth ough there is some tendency for the dominant wavenumber in the packet to shift from the most unstable value towards the neutral value, this shift is quite moderate. In practice, wave packets do not become local ly neutral in a diverging base flow (as do instability modes), therefo re, they are expected to grow more suddenly than pure instability mode s and do not develop critical layers. The group velocity is complex; t he full significance of this is realized by analytically continuing th e equations for the phase and wave action into a complex domain. The i mplications of this analytic continuation are discussed vis-a-vis the secondary instabilities of the packet: very small-scale perturbations on the phase can grow very rapidly initially, but saturate later becau se most of the energy in these perturbations is convected away by the group velocity. This remark, as well as the one regarding critical lay ers, has consequences for nonlinear theories.