Dynamics of planar vortex clusters with binaries

Abstract

We study an N -vortex problem having J of them forming a cluster, which means the distances between the vortices in the cluster is much smaller by O( ) than the distances, O( ), to the N − J vortices outside of the cluster. With the strengths of N vortices being of the same order, the velocity and time scales for the motion of the J vortices relative to those of the N − J vortices are O( −1 ) and O( 2 ) respectively. We show that this two-time and two-length scale problem can be converted to a standard two-time scale problem and then the leading order solution of the N-vortex problem can be uncoupled to two problems, one for the motion of J vortices in the cluster relative to the center of the cluster and one for the motion of the N − J vortex plus the center of the cluster. For N = 3 and J = 2, the 3-vortex problem is uncoupled to two binary vortices problems in the length scales and respectively. When perturbed in the scale , say by a fourth vortex even of finite strength, the binary problem becomes a 3-vortex problem, admitting periodic solutions. Since 3-vortex problems are solvable, the uncoupling enables us to solve 3-cluster problems having at most three vortices in each cluster.Based upon the comprehensive analysis of the three vortex problem in trilinear coordinates by Synge [1], supplemented by Tavantzis and Ting [3], Knio et al. studied the recurrent motions for perturbed three point vortex dynamics by the direct method, i. e., by showing how or where to find the recurrent motions. They pointed out that there is one and only one critical point, which will remain a center for the elliptic K > 0, parabolic K = 0, and hyperbolic cases K < 0, i. e., for all K, which denotes the sum of the products of the strengths of pairs of vortices. The critical point, called Q 3 , represents the coalescence of two vortices of the same sign, say the first two vortices, with strengths k 1 ≥ k 2 > 0 while k 2 ≥ k 3 . Thus perturbations of periodic orbits around the center Q 3 will be quasi-periodic. This expectation is confirmed by numerical examples presented by Knio et al. in Section 23-3 at the annual GAMM-2006 meeting at Berlin. Details of the study and numerical examples are reproduced in a recent paper by Blackmore et al. [4], which also presents a proof via the KAM theory. Here we present another proof via asymptotics. We note that the three vortex problem near the center Q 3 corresponds to those of two vortices in proximity forming a binary relative to the distance O( ) to the third vortex. In the scale , the unperturbed problem reduces to a two vortex problem. With the addition of a fourth vortex or a body at distance O( ), the problem then becomes a solvable three vortex problem again and periodic solutions can be found, while the perturbation could be finite. Instead of carrying out asymptotic analysis for vortices with a binary vortex we do it for a cluster of vortices and show more general results.We follow the notation of Synge defining the strength k of a vortex as its circulation Γ divided by 2π and denot...

Critical Solutions of Three Vortex Motion in the Parabolic Case

“…See the article by Blackmore et al [7] and the references therein and recent studies by Ting et al [8] and Knio et al [9]. Several numerical examples were presented to show transitions from periodic and quasi-periodic to chaotic regimes in accordance with the theoretical results.…”

“…The latter requires that the configuration △ 1 is collinear lying on an edge of △ Q or at the vertices of △ Q , where two vortices coincide to one, see Eqs. (8). These critical points of R in space are also critical points q in plane P, or x in the trilinear coordinates, x 1 , x 2 , x 3 with p = 1, see Eqs.…”