Abstract:
We study the merger of three or more identical co-rotating vortices initially arranged on the
vertices of a regular polygon, and compare it to the merger of two like-signed vortices. The
latter is a well-studied problem, with the merger process there consisting of four stages. In
the multiple (three or more) vortex case, we find a new stage in the merger process, where an
annular vortical structure is formed and is long-lived. We find that merger on the whole is slowed
down significantly as the number of vortices goes up, and the formation of the annular structure
is primarily responsible for the delaying of the merger. In the three-vortex case, the vortices
initially elongate radially, and then reorient their long axis closer to the azimuthal direction, and
then diffuse out to form an annulus. The inviscid case is similar at short times, but at longer
times, rather pronounced filaments are visible (in the three and four- vortex cases), which are
practically absent in the viscous case. We find a qualitative change in the tilt history as we
increase the number of vortices from three to six and more. In the six-vortex case, the vortices
initially itself align themselves azimuthally. The annular stage is in contrast to the ‘second
diffusive stage’ in two-vortex merger.
In addition to this, we find that at high Reynolds numbers, the vortices merge asymmetrically
and the annulus even undergoes instabilities. In order to further understand the physics behind
this, we perform a quasi-steady viscous linear stability analysis of an annular vortex. In other
words, we study the cylindrical equivalent of a parallel shear flow in an infinite domain. Assuming
azimuthal symmetry, we find a solution of the Navier-Stokes equation with the initial condition
being that of an infinitesimally thin cylindrical vortex sheet. We obtain a Generalized LambOseen vortex profile, which we input as the mean flow for our stability analyses. We find that
in the infinite Reynolds limit (inviscid case), there is an upper cut-off for the azimuthal modes
going unstable. In general, we find that viscosity has a stabilizing effect, tending to preferentially
stabilize the higher modes. We also find two modes going unstable for the azimuthal wavenumber
two, till a low Re limit, where only one mode goes unstable.