Integrable, Chaotic, and Turbulent Vortex Motion in Two-Dimensional Flows

Abstract

Divi sion of En gineering, Brown Un iver sity, Providence, Rh ode Is land 02912It is indeed rather astonishing how little practical value scientific knowledge has for ordinary men, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to its reputed utility. G. H. Hardy, A Mathematician's Apology INTRODUCTORY REMARKSVortex dynamic s would appear to be exempt from Hardy 's pe ssimi stic verdict. On one hand, the evolution of vorticity, and thu s the motion s of vortice s, are esse ntial ingredients of virtually any real flow. He nce vortex dynamic s is of profound practical importance. On the other hand, vortex motion ha s alway s con stituted a mathematically so phi sticated branch of fluid mechanics that continue s to invite the application of novel analyti cal technique s. Indeed it is ne it her dull nor co mmonplace .Thi s central role of vorticity in fluid mechanic s is not difficult to understand. As we know, any velocity field, v, can be sp lit into a su m of two field s, one with the sa me divergence as v, and no curl, and one with the sa me cu rl as v and vani shing divergence. Th is important re sult is due to Stoke s and to Helmholtz (1858; se e Sommerfeld 1964). In incompre ssi ble flow , as we deal with exclu si vely here, the fir st part is irrotational and diver gence-free and thu s lead s to the linear problem of po tential flow. , The se cond part, however , derive s directly from the vorticity of the field ' v. In the dynamic s of thi s part lie s the esse nce of the problem. 345 0066-4189/8 3/011 5-03 45$ 02.00 Annu. Rev. Fluid Mech. 1983.15:345-389. Downloaded from www.annualreviews.org by NORTH CAROLINA STATE UNIVERSITY on 10/20/12. For personal use only. Quick links to online content Further ANNUAL REVIEWS

Studies of perturbed three vortex dynamics

“…Three vortex dynamics, which we summarize here, can be described in a very efficient manner using trilinear coordinates, which have their roots in algebraic geometry.. Our approach follows that of Synge [52], Tavantzis & Ting [53], and Ting et al [56], and we refer the reader to these sources and Aref [1,2] for further details. We begin with a dynamic formulation introduced by Gröbli [28] in terms of the sides of the (possibly degenerate) triangular configuration denoted as R 1 := |z 2 − z 3 |, R 2 := |z 1 − z 3 |, and R 3 := |z 1 − z 2 |, which can be expressed as…”

“…A local stability analysis at E (or E * ) shows that there are three distinct types of dynamical behavior corresponding to the value of K (2) := K (2) 3 : These are the elliptic case when K (2) > 0, the hyperbolic case when K (2) < 0, and the exceptional parabolic case when K (2) = 0. When the system is elliptic, E (and E * ) is a center; E (and E * ) is a saddle point in the hyperbolic case; and when the system is parabolic, E (and E * ) is no longer isolated -it lies on the curve of fixed points C on T (with copy C * on T * ) defined as…”

“…Although not essential for solving the three vortex problem -as demonstrated in [28], [52], [53], and [56] -presumably one could use the integrals in involution to reduce it to a solvable one-degree-of-freedom Hamiltonian system along the lines indicated by Borisov and his collaborators in such papers as [15], [16], and [17]. On the other hand, the symplectic structure underlying Hamiltonian dynamics has proven to be extraordinarily effective in resolving a wide variety of vortex problems such as the formulation of point vortex dynamics on the sphere by Bogomolov [14], the proof of complete integrability of the three vortex problem on the sphere by Kidambi & Newton [34] (see also [15], [17], [35] and [44]), verification of non-integrability of the general n vortex problem and (n − 1) coaxial vortex ring problem for n > 3 by Bagrets & Bagrets [6] (see also Ziglin [58]), and several other important results such as in [2], [3], [23], [37], [40], [49], and [57].…”

Vortex Motion and the Geometric Phase. Part I. Basic Configurations and Asymptotics

“…We analyse the angle holonomy for 2 (the phase object) as it moves primarily under the influence of the field of 1 (the parent vortex), with 3 (the farfield vortex) providing the superimposed field. The equations governing the point vortex motion can be written compactly in complex form as [39], [38],ż *…”

“…This is due to our inability to invert the orbit-dependent relation φ(τ ) and hence to get an explicit expression for G as defined in Section 2.3. In principle, this relation can be obtained from the exact solutions of the vortex motion (37) and (38). Specifically, φ(τ ) = tan −1 (−y vor /x vor ) where x vor and y vor are the inverse tangent and inverse hyperbolic tangent functions (respectively) defined by the exact solutions.…”