Dynamics of circular arrangements of vorticity in two dimensions
Dynamics of circular arrangements of vorticity in two dimensions
Date
2016-06
Authors
Vilasur Swaminathan, Rohith
Ravichandran, S.
Perlekar, Prasad
Govindarajan, Rama
Ravichandran, S.
Perlekar, Prasad
Govindarajan, Rama
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Abstract
The merger of two like-signed vortices is a well-studied problem, but in a turbulent
ow, 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
ow 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 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. 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.
Description
Author Posting. © The Author(s), 2016. This is the author's version of the work. It is posted here by permission of American Physical Society for personal use, not for redistribution. The definitive version was published in Physical Review E 94 (2016): 013105, doi:10.1103/PhysRevE.94.013105.