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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 control. As actuation, we consider two arrays of point sinks/sources located a certain distance above and below the vortex sheet and subject to the constraint that their mass fluxes separately add up to zero. First, we demonstrate using analytical computations that the Birkhoff--Rott equation linearized around the flat-sheet configuration is in fact controllable when the number of actuator pairs is sufficiently large relative to the number of discrete degrees of freedom present in the system, a result valid for generic actuator locations. Next, we design a state-based linear-quadratic regulator 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 that is overcome by performing computations with a suitably increased arithmetic precision. Analysis of the linear closed-loop system reveals exponential decay of the perturbation energy and the corresponding actuation energy in all cases. Computations performed for the nonlinear closed-loop system demonstrate that initial perturbations of non-negligible amplitude can be effectively stabilized when a sufficient number of actuators is used. We also thoroughly analyse 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 for relatively large initial perturbations, provided that the actuation has sufficiently many degrees of freedom. [ABSTRACT FROM AUTHOR]

This is a review article of recent research developments on the motion of a polygonal ring configuration of vortex structures with singular vorticity distributions in incompressible and inviscid flows on a non-rotating sphere. Numerical computation of a single vortex sheet reveals that the Kelvin-Helmholtz instability gives rise to the formation of a polygonal ring arrangement of rolling-up spirals. An application of methods of Hamiltonian dynamics to the N-vortex problem on the sphere shows that the motion of the ring configuration of homogeneous point vortices, which is a simple model for the rolling-up spirals, becomes chaotic after a long time evolution. Some remarks on an extension of the present research and a generic non-self-similar collapse are also provided. [ABSTRACT FROM AUTHOR]

We study the motion of N point vortices with N∈ℕ on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with N degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the N -ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N=5 n or 6 n with n∈ℕ, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable N-ring. Using a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable N-ring is shown to be chaotic. [ABSTRACT FROM AUTHOR]