VORTEX FORMATION IN MIXING LAYERS - A WEAKLY NONLINEAR STABILITY APPROACH

This paper is concerned with the stability of two-dimensional incompre ssible mixing layers with small transversal velocity gradients. Using the approach of slight flow divergence, a boundary layer type of appro ximation of solutions to the steady mixing layer flow is obtained. We derive in the limit of small velocity gradients a velocity profile of an error function type. To gain an insight into the problem of the spa tial instability of this flow we apply a model involving perturbations with only a single frequency component. A generalized approach, howev er, is outlined in the Appendix. There the interaction of a fundamenta l mode with its subharmonic, oscillating at one-third the frequency is analyzed. First numerical results show that under certain conditions the subharmonic can represent the dominant disturbance. A multiple sca les expansion of the disturbance streamfunction is constructed with va riables chosen to derive a Landau-type equation with cubic nonlinearit ies governing an amplitude function A. Scaling spatial and temporal va riables and the Reynolds number we obtain in leading order a generaliz ed Rayleigh equation. We solve the associated eigenvalue problem for s patially growing modes, whereas the calculation of damped modes is bey ond the scope of our approach. The solution to this equation can be se parated into a shape function and the amplitude A. An investigation of the second-order terms yields a rederivation of the boundary layer ap proximation of the steady flow, an equation governing second harmonics of the disturbance and an equation determining the mean flow correcti on. At third order we have to apply a resonance condition. which deman ds small linear spatial growth rates. This restriction is consistent w ith the limit of small velocity gradients and we can thus derive a cub ic amplitude education governing the the space-time evolution of A. Th is equation can be cast into a separated first-order ODE. The numerica l results show that the combined effect of nonlinearity and flow diver gence strongly influences the amplitudes such that they reach a maximu m and decay farther downstream. The study is based on an essentially i nviscid approach. The regime of amplitude decay is therefore restricte d and the integrations have to be terminated when the limit of neutral growth is reached. Comparison with experimental data is difficult bec ause the latter are taken for higher values of the velocity gradients. Yet typical experimental trends are predicted by the findings of the present study. We found in particular in the numerical study of vortex contours structural instabilities in terms of breakup of sinusoidal l ines to create vortex patches and the phenomenon of cut-and-connect of vortices near ''saddle points.'' (C) 1997 American Institute of Physi cs.