Studies of perturbed three vortex dynamics

Blackmore, Denis; Ting, Lu; Knio, Omar M.

doi:10.1063/1.2428272

Cited by 18 publications

(13 citation statements)

References 53 publications

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“…4 See also Ref. 5 for a recent account of Synge's approach. However, the only systematic analysis of the linear stability problem for the collinear relative equilibria, of which the author is aware, is the 1988 paper by Tavantzis and Ting.…”

Section: Introductionmentioning

confidence: 99%

Stability of relative equilibria of three vortices

Aref

2009

Physics of Fluids

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Three point vortices on the unbounded plane have relative equilibria wherein the vortices either form an equilateral triangle or are collinear. While the stability analysis of the equilateral triangle configurations is straightforward, that of the collinear relative equilibria is considerably more involved. The only comprehensive analysis available in the literature, by Tavantzis and Ting ͓Phys. Fluids 31, 1392 ͑1988͔͒, is not easy to follow nor is it very physically intuitive. The symmetry between the three vortices is lost in this analysis. A different analysis is given based on explicit formulas for the three eigenvalues determining the stability, including a new formula for the angular velocity of rotation of a collinear relative equilibrium. A graphical representation of the space of vortex circulations is introduced, and the resultants between various polynomials that enter the problem are used. This approach adds considerable transparency to the solution of the stability problem and provides more physical understanding. The main results are summarized in a diagram that gives both the stability and instability of the various collinear relative equilibria and their sense of rotation.

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Section: Introductionmentioning

confidence: 99%

Stability of relative equilibria of three vortices

Aref

2009

Physics of Fluids

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“…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.…”

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confidence: 99%

Dynamics of planar vortex clusters with binaries

Ting

Knio

Blackmore

2007

Proc Appl Math and Mech

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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...

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“…In the direct method, we obtain a time-dependent system of differential equations for the motion of a general passive fluid particle by assuming that the dynamics of the pair of coaxial rings has already been determined; i.e., we have the solution (s 1 (t), x 1 (t)), (s 2 (t), x 2 (t)) of (17) and (18), which means that we also know θ 1 (t) := θ (s 1 (t)) and θ 2 (t) := θ (s 2 (t)) owing to (15).…”

Section: Direct Formulation Of Kinematicsmentioning

confidence: 98%

A coaxial vortex ring model for vortex breakdown

Blackmore

Brøns

Goullet

2008

Physica D: Nonlinear Phenomena

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A simple -yet plausible -model for B-type vortex breakdown flows is postulated; one that is based on the immersion of a pair of slender coaxial vortex rings in a swirling flow of an ideal fluid rotating around the axis of symmetry of the rings. It is shown that this model exhibits in the advection of passive fluid particles (kinematics) just about all of the characteristics that have been observed in what is now a substantial body of published research on the phenomenon of vortex breakdown. Moreover, it is demonstrated how the very nature of the fluid dynamics in axisymmetric breakdown flows can be predicted and controlled by the choice of the initial ring configurations and their vortex strengths. The dynamic intricacies produced by the two ring + swirl model are illustrated with several numerical experiments.

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Studies of perturbed three vortex dynamics

Cited by 18 publications

References 53 publications

Stability of relative equilibria of three vortices

Stability of relative equilibria of three vortices

Dynamics of planar vortex clusters with binaries

A coaxial vortex ring model for vortex breakdown

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