"On Control of vortex interaction"
D.L. Vainchtein (IKI RAN; Dept. of Mechanical and Environmental Engineering, University of California Santa Barbara, USA)
Abstract:
In the present talk I look at two different methods of controlling the interaction of two 2D vortices.
In the first part of my talk I discuss how the behavior of two identical vortex patches can be controlled by putting a point vortex of time-varying strength in the center of vorticity. In particular, one can be interested in forcing or preventing the merging. I consider two possible approaches. The first one is to use the method of flat coordinates. This method allows us, for example, to move the patches from a merger-bound initial state to a co-rotating steady state and this transition can be made in an arbitrarily short time. The setback of this method is that it destroys original flow structures and is very energy consuming, as it must overcome the original dynamics. Another approach involves using only low actuator impact control fields, that can be considered as perturbations of the original velocity field. We call this method "adiabatic control". Such an approach preserves the general structure of the original flow, is power-efficient and still moves patches arbitrarily close to a target location in finite time. Using this problem as an example I compare these approaches stressing their relative advantages and disadvantages.
In the second part I consider the problem of finding an
optimal control for changing the separation between two point vortices
using a single source/sink. The cost function is chosen to be the total
flow through the source(sink). I use Pontryagin's Maximum principle to
obtain the optimal control. I prove that the optimal control is a set of
pulses that are in phase or out of phase with the internal rotation of
vortices depending on whether the objective is to increase or decrease the
vortex separation.