Merger of multiple vortices

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.