Dynamics of circular arrangements of vorticity in two dimensions

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 corotating 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 preannular 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. Five or fewer vortices initially elongate radially, and then reorient their long axis closer to the azimuthal direction so as to form an annulus. With six or more vortices, the initial alignment is already azimuthal. 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 generalized Lamb-Oseen vortex (GLO), and diffuses inward 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. Here again, the odd mode causes an early collapse of the annulus via decaying turbulence into a single central vortex, whereas the even mode disturbance causes a more orderly progression into a single vortex. Results from linear stability analysis agree with the nonlinear simulations, and predict the frequencies of the most unstable modes better than they predict the growth rates. 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.

Compressibility effects on quasistationary vortex and transient hole patterns through vortex merger

“…In this work 8 it was observed that the sustained vortex pattern preceded with the formation of hole capture and transient hole patterns within the patch vortex. Similar observation was later reported by Swaminathan et al 11 in the incompressible limit. The direct parallel between such an arrangement of fluid vortices and magnetised plasma column under strong axial magnetic field had been exploited in the past 8 .…”

“…Also the results qualitatively match with the recent observation by Swaminathan et al 11 addressed using Fourier pseudo-spectral technique. A detailed and complete description of the merging process is depicted in these 8,11 earlier communications in incompressible limit (M ≡ 0).…”

Clustering of heavy particles in vortical flows: a selective review

“…Similarly, in an N-vortex configuration, when N identical point vortices are placed at the vertices of a regular polygon, we have N fixed points. The behaviour of N vortices started out thus is interesting in itself, and the reader is referred to [30]. In the present article we ask how stable the fixed points are, i.e., up to what Stokes number particles can they attract to themselves.…”

Coarsening in the two-dimensional incompressible Toner-Tu equation: Signatures of turbulence

“…Similar to defect dynamics in the GL equation [30,33,34], each vortex gets attracted to the saddle due to the opposite topological charge; (ii) The two vortices rotate around each other similar to convective merging in NS [31,32]; and (iii) The flexure of the vortex trajectory depends on the Re (see Supplemental Material). Thus a vortex merger event in the two-dimensional ITT equation has ingredients both from the NS and GL equations.…”

“…On the other hand, for high Re = 2π × 10 4 , vorticity snapshots resemble 2D turbulence. In particular, similar to vortex merger events in 2D [31,32], it is easy to identify a pair of co-rotating vortices undergoing a merger and the surrounding filamentary structure.…”