Control action for stabilizing free shear layers

Abstract

The possibility of acoustic control of the two-dimensional instabilities of a lossless plane shear layer of vanishing thickness is studied. The shear layer is formed from a body of incompressible fluid sliding over another fluid at rest. It is unstable through the generation of Kelvin–Helmholtz waves. We consider the possibility of adding to this linearly unstable flow a simple source, driven in such a way that its field interferes destructively with the instability to render the flow stable. The required strength of the unsteady control source is determined in terms of the fluctuating velocity at some fixed position in the moving fluid. We show that no unstable Kelvin–Helmholtz wave could survive the action of such a source. Next, we examine the scope for constructing the control signal from a measurement of the flow velocity at some fixed position. The source is a linear functional of the monitored velocity and we give the transfer function that would be required for the instabilities to be controlled. We prove that such control action would completely stabilize the otherwise unstable vortex sheet, and that other alternative sensor/actuator arrangements could also be effective. We go on to show that our particular very simple arrangement could not in fact be realized because, if required to work at all frequencies, it would not be causal. If we insisted on causality the vortex sheet would then only be stabilized over most frequencies. That would of course make the controlled flow completely different from the vortex sheet whose instabilities are so well known – and troublesome. We conjecture that there will exist some variations of the basic control arrangement described here that are both physically realizable and effective over the required frequency range. From our study of the initial value problem we have concluded that short perturbations would be attenuated very rapidly.