AN APPLICATION OF FEEDBACK CONTROL TO STABILISE A SPECIFIC CASE OF THE NEGATIVE BUOYANCY FLOW INSTABILITY

This paper is one that David Maull would have enjoyed savaging. Firstly, because he always enjoyed lively discussions about hare-brained schemes for using feedback control to produce unnatural states of motion, and secondly, because he was never happier than when probing details of imaginative undergraduate projects. This paper results from such a project in the group where David spent most of his professional life, a project so far-fetched that it would have attracted his most critical attention. It would also have attracted his characteristically encouraging support. The idea that instabilities of inviscid vortex sheets, which prevent them from being naturally steady, might be controlled through suitably excited actuators is being tentatively advanced, and some peculiar features of the controlled state also attract attention. The project that produced this paper addressed one peculiarity that turned out to be an error [see Ffowcs Williams (1982)], the solution to the actively controlled sheet turning out to be different and much simpler; this solution is published in Ffowcs Williams (2001). The project went on to consider whether another planar interface that is naturally unstable might be made stable by the action of an adjustable nearby surface. The situation envisaged is an unstable density stratified arrangement where heavy fluid lies initially at rest above a lighter fluid, the interface between the two fluids being planar. Above the interface is an impervious structural surface, which can be moved by a small amount by actuators designed to induce in the fluid a motion that will prevent the growth of any gravitational mode of instability. We show that it is indeed possible to define a controller with the required performance, but its implementation would be a very demanding task. A finite body of fluid could, on the other hand, be held suspended against gravity by relatively straightforward control arrangements. A particular example is worked out in detail and the control actuators defined, as is the transfer function between small oscillations of the interface and the actuator that is needed to implement the control system. An example is given to show how the unstable (but controlled) interface wobbles but settles down to rest after suffering an externally induced disturbance.