Reinitiation and feedback in global instabilities of subsonic spatially developing mixing layers.

Abstract

Two-dimensional numerical simulations of compressible, spatially evolving, unforced, planar shear flows are used to investigate the role of feedback in the reinitiation of vortex roil up. The calculated pattern of coherent structures shows global self-sustaining instabilities in which new vortex roll ups are triggered in the initial shear layer by pressure disturbances originating in the fluid accelerations downstream. This reinitiation mechanism, absent in the linear treatments of stability, is demonstrated conclusively here. Characterizing the local nature of the free-shear-flow instabilities and their global, nonlinear development in space and time is of fundamental importance for practical shear-flow control. Linear stability analyses have shown that the spatially evolving, subsonic mixing layer is convectively unstable with respect to vertical fluctuations.‘*2 Thus, except in rare configurations with global-absolute instabilities, turbulent mixing layers are expected to be driven by environmental disturbances,3 and self-sustained instabilities in free mixing layers are not expected.4 Mechanisms influencing reinitiation of the Kelvin-Helmholtz (KH) instability and vortex roll up in free-shear flows are (1) disturbances in the free streams, (2) disturbances due to boundary layers, wakes, small recirculation zones, or acoustical environmental disturbances, and (3) disturbances fed back from downstream. In laboratory experiments, these mechanisms are difficult to isolate because turbulence in free streams and boundary layers cannot be eliminated. In numerical simulations of spatially evolving shear flows, (1) and (2) can be essentially eliminated, but uncertainties in implementing inflow and outflow boundary conditions potentially modify or mask upstream effects due to downstream events. Analysis of incompressible numerical simulations of a spatially evolving mixing layer has shown that even though the flow is locally convectively unstable everywhere, global resonances may be triggered in simulations due to spurious reflections at the boundaries.’ An important question is whether a free mixing layer can be globally unstable with upstream self-excitation induced by pressure disturbances generated by finite-amplitude fluid accelerations downstream. In this paper we address this question with a fully compressible numerical simulation of a spatially evolving mixing layer, where the propagation of acoustic disturbances can be resolved and boundary effects are negligible. The idea of a feedback mechanism through which the downstream events influence the upstream flow was noted for mixing layers6 and for free jets.’ Based on experiments, a feedback loop for free and forced shear flows was proposed that involved growing finite-amplitude vortex roll ups propagating downstream and pressure waves from these changing predominantly rotational flows propagating upstream.*-” The resulting feedback formula describes the distribution of merging locations in weakly excited experimental’ and numerically simulated jets. ’ ’ However, extensive experimental investigations have not been able to conclusively establish the existence of this feedback loop because of unavoidably facilitydependent background disturbances. I2