CONTROL OF KIRCHHOFF VORTICES
L. Friedland, Phys. Rev. E59, 4106-4111 (1999). Abstract,PDF
 

SHORT INTRODUCTION TO 2D VORTICES

    Circular Vortex in Two Dimensional Fluid.
    Azimuthal Fluid Velocity Field:   V(x,y,t)=V(r)ef

    Vorticity:               w=rot V=w(r)ez

    Simplest vortex:     w=const,   r<R
                                w=0,         r>R
    Stokes's law:         V=wr/2,     r<R
                                V=wR/(2r), r>R


 
 

Kirchhoff Elliptic Vortex is a generalization of a circular vortex.
Kirchhoff Vortex has a constant elliptic shape (principal axis a and b) and rotates with angular frequency
W=wab/(a+b)2=wr/(r+1)2, r=a/b
Individual particles in the Kirchhoff vortex move on off-axis circular trajectories with angular frequency 2W.


QUESTION: How to create a Kirchhoff vortex?
ANSWER:   Start with a circular vortex and put it in a weak oscillating potential flow of form
Vx=ex, Vy=-ey, where e=e0cos[f(t)] oscillates with a slowly decreasing frequency w(t)=df/dt. As the frequency of these oscillations passes that of rotating fluid particles in the circular vortex (their rotation frequency is w/2) the system may enter autoresonant stage in which the circular vortex deforms into the elliptic shape, while twice the frequency of this Kirchoff vortex nearly follows that of the driving flow, 2wr/(r+1)2~w(t). At the same time, the area of the ellipse is conserved, i.e., ab=R2. Therefore, the axis ratio r=a/b is the only parameter defining the elliptical crossection of the vortex and one can autoresonantly control r by varying the frequency of the driving flow:

r(t)=(w/w)+[(w/w)2-2(w/w)]1/2-1

THE FIGURE on the left shows the evolution of the axis ratio r and phase mismatch F=q-f between the vortex rotation phase and the phase of the driving oscillation for the case of oscillating frequency w(t)=w/2+10sin(0.157t) of the driving flow. One can see repetitive autoresonant excitations during two periods of oscillations of w. One can also see the phase locking in the system in the autoresonance stages of evolution.