Harnessing the Kelvin-Helmholtz Instability: Feedback Stabilization of an Inviscid Vortex Sheet.

In this investigation we use a simple model of the dynamics of an inviscid vortex sheet given by the Birkhoff-Rott equation to obtain fundamental insights about the potential for stabilization of shear layers using feedback. First, we demonstrate using analytical computations that the Birkhoff-Rott equation linearized around the flat-sheet configuration is in fact controllable when a pair of point vortices located on both sides of the sheet is used as actuation. On the other hand, this system is not controllable when the actuation has the form of a pair of sinks/sources with zero net mass flux. Next we design a state-based LQR stabilization strategy where the key difficulty is the numerical solution of the Riccati equation in the presence of severe ill-conditioning resulting from the properties of the Birkhoff-Rott equation and the chosen form of actuation, an issue which is overcome by performing computations with a suitably increased arithmetic precision. Analysis of the linear closed-loop system reveals a range of different behaviors varying from a purely exponential decay of the perturbation energy to its significant transient growth as the number of actuators is reduced. Computations performed for the nonlinear closed-loop system demonstrate that initial perturbations of nonnegligible amplitude can be effectively stabilized when a sufficient number of actuators is used in an aligned arrangement. We also thoroughly analyze the sensitivity of the closed-loop stabilization strategies to the variation of a number of key parameters. Subject to the known limitations of inviscid vortex models, our findings indicate that, in principle, it may be possible to stabilize shear layers, provided the actuation has sufficiently many degrees of freedom and the initial perturbations are not too large.