Dynamics of circular arrangements of vorticity in two dimensions

TL;DR

This paper investigates the complex dynamics of multiple co-rotating vortices in two-dimensional turbulence, revealing a new long-lived annular vortex stage and differences in merger behavior between even and odd numbers of vortices.

Contribution

It introduces the formation of a long-lived annular vortex stage during multiple vortex mergers and highlights the distinct behaviors based on the number of vortices and Reynolds number.

Findings

Long-lived annular vortices form during multi-vortex mergers.

Merger dynamics differ significantly between even and odd numbers of vortices.

At higher Reynolds numbers, instabilities lead to different nonlinear evolution stages.

Abstract

The merger of two like-signed vortices is a well-studied problem, but in a turbulent flow, we may often have more than two like-signed vortices interacting. We study the merger of three or more identical co-rotating vortices initially arranged on the vertices of a regular polygon. At low to moderate Reynolds numbers, we find an additional stage in the merger process, absent in the merger of two vortices, where an annular vortical structure is formed and is long-lived. Vortex merger is slowed down significantly due to this. Such annular vortices are known at far higher Reynolds numbers in studies of tropical cyclones, which have been noticed to a break down into individual vortices. In the pre-annular stage, vortical structures in a viscous flow are found here to tilt and realign in a manner similar to the inviscid case, but the pronounced filaments visible in the latter are practically absent in the former. Interestingly at higher Reynolds numbers, the merger of an odd number of vortices is found to proceed very differently from that of an even number. The former process is rapid and chaotic whereas the latter proceeds more slowly via pairing events. The annular vortex takes the form of a generalised Lamb-Oseen vortex (GLO), and diffuses inwards until it forms a standard Lamb-Oseen vortex. For lower Reynolds number, the numerical (fully nonlinear) evolution of the GLO vortex follows exactly the analytical evolution until merger. At higher Reynolds numbers, the annulus goes through instabilities whose nonlinear stages show a pronounced difference between even and odd mode disturbances. It is hoped that the present findings, that multiple vortex merger is qualitatively different from the merger of two vortices, will motivate studies on how multiple vortex interactions affect the inverse cascade in two-dimensional turbulence.