IUTAM Symposium on 150 Years of Vortex Dynamics: Proceedings of the IUTAM Symposium “150 Years of Vortex Dynamics” held at the Technical University of Denmark, October 12-16, 2008 9048185831, 9789048185832

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Hassan Aref Editor

IUTAM Bookseries

Symposium “150 Years of Vortex Dynamics” Proceedings of an IUTAM Symposium held at the Technical University of Denmark, October 12–16, 2008

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150 Years of Vortex Dynamics

IUTAM BOOKSERIES Volume 20 Series Editors G.M.L. Gladwell, University of Waterloo, Waterloo, Ontario, Canada R. Moreau, INPG, Grenoble, France Editorial Board J. Engelbrecht, Institute of Cybernetics, Tallinn, Estonia L.B. Freund, Brown University, Providence, USA A. Kluwick, Technische Universität, Vienna, Austria H.K. Moffatt, University of Cambridge, Cambridge, UK N. Olhoff, Aalborg University, Aalborg, Denmark K. Tsutomu, IIDS, Tokyo, Japan D. van Campen, Technical University Eindhoven, Eindhoven, The Netherlands Z. Zheng, Chinese Academy of Sciences, Beijing, China

Aims and Scope of the Series The IUTAM Bookseries publishes the proceedings of IUTAM symposia under the auspices of the IUTAM Board.

For other titles published in this series, go to www.springer.com/series/7695

Hassan Aref Editor

150 Years of Vortex Dynamics

Previously published in Theoretical and Computational Fluid Dynamics Volume 24, Issues 1–4.

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Editor Hassan Aref Virginia Polytechnic Institute & State University Dept. Engineering Science & Mechanics Blacksburg VA 24061 USA [email protected]

ISBN: 978-90-481-8583-2 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010924731 © Springer Science+Business Media B.V. 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Printed on acid-free paper. Springer is part of Springer Science+Business Media (www.springer.com)

Contents

EDITORIAL

150 Years of vortex dynamics H. Aref 1 ORIGINAL ARTICLES

The persistence of spin H.K. Moffatt 9 Why study vortex dynamics? R.J. Donnelly 17 A new calculus for two-dimensional vortex dynamics D. Crowdy 25 On relative equilibria and integrable dynamics of point vortices in periodic domains M.A. Stremler 41 Collapse and concentration of vortex sheets in two-dimensional flow K.A. O’Neil 55 Self-propulsion of a free hydrofoil with localized discrete vortex shedding: analytical modeling and simulation S.D. Kelly · H. Xiong 61 Vortex dynamics on a domain with holes M.d.C. Lopes Filho · H.J. Nussenzveig Lopes

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Background current concept and chaotic advection in an oceanic vortex flow E. Ryzhov · K. Koshel · D. Stepanov 75 Singular vortices in regular flows G. Reznik · Z. Kizner 81 Cosmic vortices in hot stars and cool disks E.A. Spiegel 93 2D vortex interaction in a non-uniform flow X. Perrot · X. Carton 111 Localized dipoles: from 2D to rotating shallow water Z. Kizner · G. Reznik 117 Barotropic elliptical dipoles in a rotating fluid R. Trieling · R. Santbergen · G. van Heijst · Z. Kizner 127

On instability of elliptical hetons M. Sokolovskiy · J. Verron · X. Carton · V. Gryanik

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Explosive instability of geostrophic vortices. Part 1: baroclinic instability X. Carton · G.R. Flierl · X. Perrot · T. Meunier · M.A. Sokolovskiy 141 Explosive instability of geostrophic vortices. Part 2: parametric instability X. Carton · T. Meunier · G.R. Flierl · X. Perrot · M.A. Sokolovskiy 147 The N-vortex problem on a sphere: geophysical mechanisms that break integrability P.K. Newton 153 From generation to chaotic motion of a ring configuration of vortex structures on a sphere T. Sakajo 167 Vortex motion on a sphere: barrier with two gaps R.B. Nelson · N.R. McDonald 173 Asymmetric vortex merger: mechanism and criterion L.K. Brandt · T.K. Cichocki · K.K. Nomura 179 Numerical simulations of falling leaves using a pseudo-spectral method with volume penalization D. Kolomenskiy · K. Schneider 185 High-performance computing techniques for vortex method calculations T.K. Sheel · S. Obi 191 Stability of elliptical vortices from “Imperfect–Velocity–Impulse” diagrams P. Luzzatto-Fegiz · C.H.K. Williamson 197 Chaotic streamlines in the flow of knotted and unknotted vortices O. Velasco Fuentes 205 Falling cards and flapping flags: understanding fluid–solid interactions using an unsteady point vortex model S. Michelin · S.G. Llewellyn Smith 211 Swimming in an inviscid fluid E. Kanso 217 Vorticity generation during the clap–fling–sweep of some hovering insects D. Kolomenskiy · H.K. Moffatt · M. Farge · K. Schneider 225 Kinetic theory of stellar systems, two-dimensional vortices and HMF model P.-H. Chavanis 233 Vortices for computing: the engines of turbulence simulation N. Kevlahan 257 Vorticity dynamics in turbulence growth P. Orlandi · S. Pirozzoli 263 Sharp vorticity gradients in two-dimensional turbulence and the energy spectrum E.A. Kuznetsov · V. Naulin · A.H. Nielsen · J. Juul Rasmussen 269 Super-rotation flow in a precessing sphere S. Kida · N. Nakazawa 275 Vortex dynamics of turbulence–coherent structure interaction D.S. Pradeep · F. Hussain 281 Statistical theory applied to a vortex street generated from meander of a jet K. Iga 299

Vortex ring velocity and minimum separation in an infinite train of vortex rings generated by a fully pulsed jet P.S. Krueger 307 Topology of vortex creation in the cylinder wake M. Brøns · A.V. Bisgaard 315 Bifurcations in the wake of a thick circular disk F. Auguste · D. Fabre · J. Magnaudet 321 Vortices in time-periodic shear flow R. Kunnen · R. Trieling · G.J. van Heijst

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Structure of a bathtub vortex: importance of the bottom boundary layer S. Yukimoto · H. Niino · T. Noguchi · R. Kimura · F.Y. Moulin 339 Separation vortices and pattern formation A. Andersen · T. Bohr · T. Schnipper 345 Global time evolution of viscous vortex rings Y. Fukumoto 351 Viscous ring modes in vortices with axial jet S. Le Dizès · D. Fabre 365 Short-wave stability of a helical vortex tube: the effect of torsion on the curvature instability Y. Hattori · Y. Fukumoto 379 On the motion of thin vortex tubes A. Leonard 385 Dynamics of vortex line in presence of stationary vortex V.E. Zakharov 393 A locally induced homoclinic motion of a vortex filament M. Umeki 399 Self-similar collapse of 2D and 3D vortex filament models Y. Kimura 405 Applications of 2D helical vortex dynamics V.L. Okulov · J.N. Sørensen 411 Coaxial axisymmetric vortex rings: 150 years after Helmholtz V.V. Meleshko 419 Dynamics of vortex rings in viscous fluids R.J. Donnelly 449 Author Index

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Theor. Comput. Fluid Dyn. (2010) 24:1–7 DOI 10.1007/s00162-009-0178-6

E D I TO R I A L

Hassan Aref

150 Years of vortex dynamics

Received: 12 October 2009 / Accepted: 4 November 2009 / Published online: 18 December 2009 © Springer-Verlag 2009

Abstract An IUTAM symposium with the title of this paper was held October 12-16, 2008, in Lyngby and Copenhagen, Denmark, to mark the sesquicentennial of publication of Helmholtz’s seminal paper on vortex dynamics. This volume contains the proceedings of the Symposium. The present paper provides an introduction to the volume. Keywords Vortex dynamics · Helmholtz sesquicentennial · IUTAM symposium 1 Overview of the symposium In 1858, interleaved with work on physiology for which he may today be better known, Hermann Ludwig Ferdinand Helmholtz – the titulation “von” was added later – published “Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen” [4]. This paper may rightfully be seen as the initiation of the subfield of fluid mechanics that we today refer to as vortex dynamics. Thus, in 2008 it seemed fitting to commemorate and celebrate the sesquicentennial of Helmholtz’s seminal work, and the birth of vortex dynamics, by holding an IUTAM symposium on the subject. Logically, 150 Years of Vortex Dynamics became the title of the symposium. The symposium was held in the facilities of the Technical University of Denmark (DTU), at Kongens Lyngby, a suburb of Copenhagen, from October 12 to 16, 2008. The main auditorium used is known as Oticon salen (see Fig.1). The International Scientific Committee of the Symposium consisted of M. Brøns (Denmark), G. J. F. van Heijst (The Netherlands), S. Kida (Japan), V. V. Meleshko (Ukraine), H. K. Moffatt (UK, IUTAM representative), P. K. Newton (USA) and H. Aref (Denmark and USA) as the chair. The Local Organizing Committee at DTU consisted of A. Andersen, T. Bohr, M. Brøns (Treasurer), H. Bruus, D. Glass, and again H. Aref as chair. We had great help from three of our PhD students J. Rønby Pedersen, T. Schnipper and L. Tophøj. IUTAM has recognized the field of vortex dynamics through a number of symposia in the past. The apt characterization by Küchemann [2] of vortices as the “sinews and muscles of fluid motions” stems from his summary of the symposium on Concentrated Vortex Motions in Fluids (Ann Arbor, 1964). Later IUTAM symposia on vortices and vortex dynamics include Fundamental Aspects of Vortex Motion (Tokyo 1988) [1], Dynamics of Slender Vortices (Aachen, 1997), Tubes, Sheets and Singularities in Fluid Dynamics (Zakopane, 2001), Elementary Vortices and Coherent Structures: Significance in Turbulence Dynamics (Kyoto, 2004) and Hamiltonian Dynamics, Vortex Structures and Turbulence (Moscow, 2006). This 2008 symposium was H. Aref Center for Fluid Dynamics, Technical University of Denmark, Lyngby, DK-2800, Denmark H. Aref (B) Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061, USA E-mail: [email protected] Reprinted from the journal

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Fig. 1 Attendees assembled in the lecture hall, Oticon salen at DTU, for the IUTAM symposium 150 Years of Vortex Dynamics

Fig. 2 Group photo of attendees at the IUTAM symposium 150 Years of Vortex Dynamics

intended as a broad state-of-the-art survey of the field. It concerned vortices and vortex dynamics in all manner of flows, from laminar to turbulent, terrestrial to cosmic, geophysical to biological, addressed through analysis, experiment and computation. Classical themes and theories are still of interest, e.g., relative equilibria of point vortices, albeit now applied to Bose-Einstein condensates, or kinetic theory now applied to systems of vortices as a prototype of turbulence. The variety of topics will be evident from the program listed below and the papers presented in this volume. The role of vortices in the drag and propulsion of swimming and flying animals was explored by a number of attendees and would, undoubtedly, have pleased Helmholtz whose contributions to physiology are well known. The symposium was attended by 108 registered participants and 10 accompanying persons. A group photo of attendees appears as Fig.2. Some students and researchers at the DTU also attended off and on. The official scientific participants came from 19 nations: Algeria (1), Belgium (1), Brazil (2), Canada (1), Czech Republic (2), Denmark (14), France (11), Germany (1), Greece (1), Israel (1), Italy (4), Japan (14), Lithuania (3), Mexico (2), The Netherlands (2), Poland (4), Russia (8), Ukraine (4), UK (6) and USA (26). A participant from Bangladesh was unable to secure a visa on time and so did not attend. The oral part of his seminar/poster presentation was given by the chair assisted by Prof. A. Leonard (Caltech) based on slides he had sent, and his poster was displayed in the appropriate session. The social events included a get-together party at Hotel Strand in Copenhagen on the evening of Sunday, October 12, a Reception at the Carlsberg Academy, the former Honorary Residence primarily known for its illustrious resident Niels Bohr, on Monday, October 13, where two special lectures were presented, and a banquet at Charlottenlund Travbane (the sulky race track) north of Copenhagen on Wednesday, October 15. The two lectures given at the reception, cf. Fig.3, were

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Fig. 3 Top left Lunch time, Glassalen at DTU; top right Keith Moffatt lecturing at the reception at the Carlsberg Academy; bottom left Ed Spiegel speaking at the banquet at Charlottenlund Travbane; bottom right making sure the AV is set up correctly

Keith Moffatt, “The persistence of spin” Russell Donnelly, “What goes around comes around: Why study vortex dynamics?” Both were designed for a somewhat general audience although the assembled specialists in vortex motion listened most attentively! The written versions of these lectures are included in the present volume but were not included in the accompanying Special Issue of the journal Theoretical and Computational Fluid Dynamics. Tomas Bohr made some remarks about the history and layout of the remarkable building we were in, and added some recollections from his childhood when his grandfather was the honorary resident. Several speeches were made at the banquet. Vladimir Zakharov recited one of his poems in Russian and then provided an impromptu translation into English. The main banquet speaker was Ed Spiegel (see Fig.3). The technical program of the Symposium consisted of 5 keynote lectures, 39 contributed lectures, and 50 seminar/poster presentations. An additional 26 papers had been submitted for consideration. Some of these had been declined by the International Scientific Committee, which reviewed all submissions and decided on the list of keynote speakers, others had been accepted but the author was ultimately unable to attend. As will be seen from the detailed program listing, the papers were distributed over four days with a keynote lecture to start off the morning. This was followed by contributed lectures. The seminar/poster sessions were grouped around the lunch breaks with 10 papers in each and an hour for short oral presentations and discussion. The posters were on display around the periphery of the lecture hall for the full four days of the Symposium so that continued discussion of them could and did take place. Paul Newton generously offered a copy of his book on the N -vortex problem [3] as a prize for the seminar/poster presentation that garnered the most votes from attendees. The winner was Philip du Toit for his seminar/poster presentation “Visualizing mixing in geophysical vortices”. Financial support for the Symposium was provided by: The Center for Fluid Dynamics at the Technical University of Denmark, Fluid•DTU, Florida State University, Virginia Tech Foundation, and IUTAM. All funds received from IUTAM were used for the support of younger participants in the Symposium.

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2 The program The detailed program of the Symposium is given below. Only the presenting author is indicated. An asterisk before an entry indicates that a written version of the paper is included in the proceedings. The author list and title may deviate from the corresponding paper presented at the Symposium. The papers in this volume are arranged with the two special lectures immediately following this introduction, and then the papers presented at the Symposium proper in the sequence in which they were presented. All lecture presenters were encouraged to submit papers for a Special Issue of Theoretical and Computational Fluid Dynamics (TCFD), papers that are then re-published in the present volume. Seminar/poster presenters were encouraged to submit very brief papers on their work for possible inclusion. However, depending on the number and lengths of papers received, these papers might not all be published in the Special Issue. Since all papers submitted were subject to standard reviewing practices of TCFD, any paper could be published subsequently with an annotation that the work was first presented at the Symposium. In the end, a total of 48 manuscripts were received, reviewed and accepted. Of these only 7 were based on seminar/poster presentations. The Special Issue, and the current volume, thus give a particularly good impression of the work presented as lectures at the Symposium. These provide a good overview of the directions of the field that are currently active. The detailed program follows: Sunday, October 12, 2008 Meet and greet attendees at Copenhagen’s Kastrup airport; Get-together party at Hotel Strand. Monday, October 13, 2008 Registration Opening: 09:00 - 09:30: Welcome by Hassan Aref, Chair of the Symposium; Welcome on behalf of IUTAM by Keith Moffatt, Vice President of IUTAM; Welcome to DTU by Kristian Stubkjær, Dean of Research, DTU. Keynote lecture 1: 09:30 - 10:20 (Hassan Aref, chair) Crowdy, “A new calculus for two-dimensional vortex dynamics”

∗ Darren

Lecture session 1: 10:20 - 11:20 (Slava Meleshko, chair) ∗ Mark Stremler, “On the statics and dynamics of point vortices in periodic domains” ∗ Kevin ONeil, “Collapse and concentration of vortex sheets in two-dimensional flow” ∗ Scott Kelly, “Self-propulsion of a free hydrofoil with localized discrete vortex shedding: analytical modeling and simulation” Break: 11:20 - 11:45 Seminar session and poster viewing I: 11:45 - 12:45 (Keith Moffatt, chair) Vaclav Kolar, “On the relationship between a vortex and vorticity” Elena Meshcheryakova, “New exact solutions in vortex dynamics” ∗ Milton Lopes Filho, “Vortex dynamics in a two-dimensional domain with holes” Hassan Aref, “Relative equilibria of point vortices” Peter Clarkson, “Polynomials and soliton equations” Laust Tophøj, “Chaotic scattering of two identical point vortex pairs” Yuko Matsumoto, “A moment model for the motion of a dipole in two-dimensional incompressible fluid” Irene Gned, “The self-induced dynamics of vortex patches” Eugene Wayne, “A generalization of the Helmholtz-Kirchhoff model” Alexey Borisov, “Chaotic vibration of the liquid ellipsoid filled with vortical fluid” Lunch, Glassalen, DTU Cantine: 12:45 - 13:45 (see Fig.3) Seminar session and poster viewing II: 13:45 - 14:45 (GertJan van Heijst, chair) Eduardo Ramos, “Vortices generated by fixed and free-moving magnets in shallow electrolyte layers” Pavlos Vlachos, “Kelvin-Helmholtz and Rayleigh-Taylor instabilities during accumulation and disperson of ferrofluid aggregates” Clara Velte, “Local helical symmetry of vortices by vortex generators in a low Reynolds number wall bounded flow”

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Georgi Sutyrin, “Dynamics of coherent vortices in rotation-dominated flows” Philip duToit, “Visualizing mixing in geophysical vortices” ∗ Konstantin Koshel, “Background current concept and chaotic advection in an oceanic vortex flow” ∗ Gregory Reznik, “Quasi-geostrophic singular vortices embedded in a regular flow” Oleg Derzho, “Rotating dipole and tripole vortices in polar regions” Tatyana Krasnopolskaya, “Modelling of Gulf Stream by the von Kármán vortex street” Dmytro Cherniy, “The vortex model of circulation flow in sea channel” Lecture session 2: 14:45 - 15:45 (Mikhail Sokolovskiy, chair) ∗ Ed Spiegel, “Cosmic vortices” ∗ Xavier Perrot, “Baroclinic vortex interaction in a time-varying flow” ∗ Ziv Kizner, “Localized shallow-water dipoles” Break: 15:45 - 16:15 Photograph of all attendees (see Fig.2). Lecture session 3: 16:15 - 17:15 (Gregory Reznik, chair) ∗ Ruben Trieling, “Elliptical barotropic f -plane dipoles in a rotating fluid” ∗ Mikhail Sokolovskiy, “On the instability of elliptic hetons” ∗ Xavier Carton, “Explosive instability of geostrophic vortices” Lectures and Reception at the Carlsberg Academy, 18:00 - 20:30 (see Fig.3). Tuesday, October 14, 2008 Keynote lecture 2: 09:30 - 10:20 (Alexey Borisov, chair) ∗ Paul Newton, “The N -vortex problem on a sphere” Lecture session 4: 10:20 - 11:20 (Ziv Kizner, chair) ∗ Takashi Sakajo, “Motion of a ring structure of coherent vortices on a sphere with pole vortices” ∗ Rhodri Nelson, “Finite area vortex motion on a sphere with impenetrable boundaries” ∗ Keiko Nomura, “Asymmetric vortex merger: mechanism and criterion” Break: 11:20 - 11:45 Seminar session and poster viewing III: 11:45 - 12:45 (Paul Newton, chair) Stefanella Boatto, “Vortices on closed surfaces” Vitalii Ostrovskyi, “Platonic and Archimedean solid based point vortex equilibria on the sphere” George Chamoun, “Von Kármán streets on the sphere” Ivan Mamaev, “New integrable problem of motion of point vortices on a sphere” Adas Jakubauskas (for A. J. V. Milyus), “On a question of definition of potential by the vortex motion of a liquid” Xinyu He, “An example of finite-time singularities in the 3D Euler equations” Valery Klyatskin, “Sound radiation by vortex motions” E. Milyute, “A dynamics of a substance in an isolated spherical vortex and its relationship with radiation” Sergio Pirozzoli, “Vortical structures in turbulence growth” Osamu Sano, “Collision of a vortex ring on granular layer” Lunch, Glassalen, DTU Cantine: 12:45 - 13:45. Seminar session and poster viewing IV: 13:45 - 14:45 (Mark Stremler, chair) Anders Andersen, “The vortices in the wake of a falling paper card” ∗ Kai Schneider, “Numerical simulation of falling leaves using a pseudo-spectral method with volume penalization” Makoto Iima, “Robustness of insects free-flight in terms of flapping motion and vortex patterns” Vasileios Vlachakis, “Vortex dynamics of wakes: analysis of the Domm system” Takeshi Watanabe, “Study of vortex flows behind the circular cylinder” Hamid Oualli, “Vortex flow eddy street behind a circular cylinder superimposed to simultaneous rotation and cross-section variation in uniform flow” Grégoire Winckelmans, “Redistribution on hierarchically refined grids for Lagrangian vortex element methods” ∗ Tarun Kumar Sheel, “High performance computing technique for vortex method calculations” Monika Nitsche, “High order quadratures for the boundary integrals governing axisymmetric interface motion” Robert Krasny, “Lagrangian panel method for vortex sheet motion in 3D Flow” Reprinted from the journal

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Lecture session 5: 14:45 - 15:45 (Charles Williamson, chair) Ian Eames, “Ghost vortices and disappearing bodies: the concept of momentum and impulse” ∗ Paolo Luzzatto-Fegiz, “Determining the stability of steady inviscid flows through preferred bifurcation diagrams” ∗ Oscar Velasco Fuentes, “Chaotic streamlines in the flow of knotted and unknotted vortices” Break: 15:45 - 16:15 Lecture session 6: 16:15 - 17:15 (Anders Andersen, chair) ∗ Sebastien Michelin, “Falling, flying, swimming, flapping: understanding fluid-solid interactions using a vortex shedding model” ∗ Eva Kanso, “Low-order models of swimming in an inviscid fluid” ∗ Dmitry Kolomenskiy, “Vorticity generation during the clap-fling-sweep of hovering insects” Banquet, Charlottenlund Travbane 18:00 (see Fig.3) Wednesday, October 15, 2008 Keynote lecture 3: 09:30 - 10:20 (Ed Spiegel, chair) ∗ Pierre-Henri Chavanis, “Kinetic theory of two dimensional point vortices from a BBGKY-like hierarchy” Lecture session 7: 10:20 - 11:20 (Anthony Leonard, chair) ∗ Nicholas Kevlahan, “Vortices for computing: the engines of turbulence simulation” ∗ Paolo Orlandi, “Vorticity dynamics in turbulence growth” ∗ Jens Juul Rasmussen, “Sharp vorticity gradients in two-dimensional turbulence and the energy spectrum” Break: 11:20 - 11:45 Lecture session 8: 11:45 - 12:45 (Jens Nørkær Sørensen, chair) Roman Lagrange, “Dynamics of a fluid inside a precessing cylinder” ∗ Shigeo Kida, “Super-rotation flow in a precessing sphere” ∗ Fazle Hussain, “Mechanisms of core perturbation growth in vortex-turbulence interaction” Lunch, DTU Cantine: 12:45 - 13:45 Seminar session and poster viewing V: 13:45 - 14:45 (Morten Brøns, chair) Charles Williamson, “Stability of classical flows and new vortical solutions from preferred bifurcation diagrams” Leonid Shirkov, “Statistical mechanics of shear layers” ∗ Keita Iga, “Statistical theory applied to a vortex street generated from meander of a jet” Jens Nørkær Sørensen, “Onset of three-dimensional flow structures in rotating flows” Bo Hoffmann Jørgensen, “Control of vortex breakdown in a closed cylinder with a rotating lid” Ziemowitz Malecha, “Bursting phenomena of boundary layer induced by 2D vortex patch” ∗ Paul Krueger, “Vortex ring velocity and minimum separation in an infinite train of vortex rings generated by a fully-pulsed jet” Pawel Regucki, “Investigation of vortex ring with finite-amplitude Kelvin waves using 3D VIC method” Chris Weiland, “The role of vortex ring formation on the development of impulsively induced supercavitation” V. Milyuvene, “A new look at a vortical dynamics of a substance in the Universe” Lecture session 9: 14:45 - 15:45 (Fazle Hussain, chair) ∗ Morten Brøns, “Topology of vortex creation in the wake of a circular cylinder” ∗ David Fabre, “Bifurcations in the wake of axisymmetric objects” Luca Zannetti, “About finite area wakes past bluff bodies and growing vortex patches” Break: 15:45 - 16:15 Lecture session 10: 16:15 - 17:15 (Oscar Velasco Fuentes, chair) ∗ GertJan van Heijst, “Behavior of a vortex in a time-periodic shear flow” ∗ Hiroshi Niino, “Structure of a bathtub vortex: importance of the bottom boundary layer” ∗ Tomas Bohr, “Separation vortices and surface shapes” Thursday, October 16, 2008 Keynote lecture 4: 09:30 - 10:20 (Shigeo Kida, chair) ∗ Yasuhide Fukumoto, “Global time evolution of viscous vortex rings”

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Lecture session 11: 10:20 - 11:20 (Russell Donnelly, chair) ∗ Stéphane Le Dizés, “Viscous ring modes in vortices with jet” ∗ Yuji Hattori, “Short-wave stability of a helical vortex tube: The effect of torsion on the curvature instability” ∗ Anthony Leonard, “On the motion of thin vortex tubes” Break: 11:20 - 11:45 Lecture session 12: 11:45 - 12:45 (Yasuhide Fukumoto, chair) ∗ Vladimir Zakharov, “Dynamics of vortex line in presence of stationary vortex” ∗ Makoto Umeki, “A locally induced homoclinic motion of the vortex filament” ∗ Yoshifumi Kimura, “Self-similar collapse of 2D and 3D vortex filament models” Lunch, Glassalen, DTU Cantine: 12:45 - 13:45 Lecture session 13: 13:45 - 14:45 (Yoshifumi Kimura, chair) ∗ Valery Okulov, “Applications of 2D helical vortex dynamics” ∗ Slava Meleshko, “Coaxial axisymmetric vortex rings – 150 years after Helmholtz” ∗ Russell Donnelly, “Dynamics of vortex rings in viscous fluids” Keynote lecture 5: 14:45 - 15:35 (Tomas Bohr, chair) Thomas Leweke, “Vortex pairs” Closing: 15:35 - 15:45 In view of the variety of topics covered, the fields of science included, the multitude of analytical, numerical and experimental techniques used, the areas of application, and the generally very high level of scientific discourse at the Symposium, one has to conclude that on its 150th birthday the field of vortex dynamics, Helmholtz’s brainchild of 1858, is alive, vibrant and thriving. Acknowledgements The author is deeply indebted to Dorte Glass for her excellent work on all facets of organizing the Symposium. He greatly appreciates the collegiality of the international network of vortex dynamicists who took part in this celebration of their field. The author’s work with arranging the Symposium, including the editing of the present volume, was supported by a Niels Bohr Visiting Professorship at the Technical University of Denmark provided by the Danish National Research Foundation.

References 1. Aref, H. & Kambe, T. 1988 Report on the IUTAM Symposium: Fundamental aspects of vortex motion. Journal of Fluid Mechanics 190, 571–595. 2. Helmholtz, H. von 1858 Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. Journal für reine und angewandte Mathematik 55, 25–55. English translation by Tait, P. G. 1867 On integrals of the hydrodynamical equations, which express vortex-motion. Philosophical Magazine (4) 33, 485–512. 3. Küchemann, D. 1965 Report on the I.U.T.A.M. symposium on concentrated vortex motions in fluids. Journal of Fluid Mechanics 21, 1–20. 4. Newton, P. K. 2001 The N-vortex Problem: Analytical Techniques. Applied Mathematical Sciences, vol. 145 (Springer-Verlag, New York).

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O R I G I NA L A RT I C L E

H. K. Moffatt

The persistence of spin

© Springer-Verlag 2010

Abstract Three spinning toys that illustrate fundamental dynamical phenomena are discussed: the rising egg, prototype of dissipative instability; the shuddering Euler’s disc, prototype of finite-time singularity; and the reversing rattleback, prototype of chiral dynamics. The principles underlying each phenomenon are discussed, and attention is drawn to analogous phenomena in the vorticity dynamics of nearly inviscid fluids. Keywords Euler’s disc · Rattle back · Chiral dynamics · Spin · Vorticist PACS

1 Introduction This informal evening lecture, delivered at the Carlsberg Academy, was concerned with dynamical principles illustrated by the behaviour of three mechanical toys. The first of these, the tippe-top, was the subject of a famous 1954 photograph (Fig. 1) of Niels Bohr (then honorary resident of the Carlsberg Academy) and Wolfgang Pauli, who were evidently intrigued by the tippe-top’s ability to turn upside-down as it spins, a vivid example of a dissipative (or slow) instability driven by slipping friction at the point of contact with the floor. The rising egg (the spinning hard-boiled egg that rises to the vertical if spun rapidly enough) is a similar phenomenon, which may be understood in terms of minimising the spin energy subject to conservation, not of angular momentum, but rather of the ‘adiabatic Jellett invariant’ (see Sect. 2). My second toy is ‘Euler’s disc’ (see www.eulersdisc.com), that rolls on its bevelled edge on a plane table, and settles to rest (like a spun coin) with a rapid shudder, a beautiful table-top example of a finite-time singularity. Controversy surrounds the question of what is the dominant contribution to the rate of dissipation of energy for this toy; however, whatever mechanism is invoked, the singularity is resolved in the final split second of the motion, most probably by loss of contact between the disc and the table and consequent release of the no-slip constraint. My third toy is the rattleback, a canoe-shaped object that spins reasonably smoothly in one direction, but which, when spun in the opposite direction, exhibits a pitching instability leading to spin reversal. The rattleback is in fact very slightly deformed giving it a chiral (i.e. non-mirror-symmetric) mass distribution. It is this chiral property in conjunction with spin that leads to the intriguing behaviour, which may be regarded as providing a prototype of ‘chiral dynamics’. The nature of the dissipative effects that damp the motion are as yet ill-understood; but a semi-empirical choice of linear damping coefficients provides a model that agrees well, at least qualitatively, with observed behaviour. H. K. Moffatt (B) Trinity College, Cambridge, CB2 1TQ, UK E-mail: [email protected]

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Fig. 1 Niels Bohr and Wolfgang Pauli investigate the behaviour of the tippe-top; Photograph (1954) by Erik Gustafson, courtesy AIP Emilio Segre Visual Archives, Margrethe Bohr Collection

The fluid counterpart of spin is, of course, vorticity, and it is, therefore, perhaps appropriate that, at this Symposium commemorating Helmholtz’s seminal 1858 article on the laws of vortex motion, the phenomenon of spin may be taken as a natural starting point. Just as friction plays a key role in each of the above toy examples, so frictional (i.e. viscous) effects in fluids are nearly always important no matter how small the viscosity may be, a fact that should be constantly borne in mind (as Helmholtz was himself well aware) when adopting inviscid modelling techniques.

2 The rising egg Spin a hard-boiled egg sufficiently rapidly on a table, and it will rise to the vertical, a parlour trick that invariably provokes a startled response: what is it that makes the centre-of-mass rise in this way? The key to understanding this phenomenon was provided by Moffatt and Shimomura [15] (and in the comprehensive treatment of this problem by Moffatt et al. [17], Shimomura et al. [18], Branicki et al. [5] and Branicki and Shimomura [4]; see also Bou-Rabee et al. [3]) on the basis of the ‘gyroscopic approximation’, which is applicable when the spin is large and the slipping friction at the point of contact with the table weak. These conditions are the counterparts of the conditions of small Rossby number and large Reynolds number in the mechanics of rotating fluids. They imply a state of ‘balance’ in which (at leading order) coriolis forces dominate the behaviour. Nevertheless, the frictional force at the point of slipping contact with the table exerts a torque relative to the centre-of-mass of the egg, causing this centre-of-mass to rise on a slow (dissipative) time-scale. Figures 2, 3 and 4 show three posters prepared for the Summer Science Exhibition held at the Royal Society, London, in July 2007. These posters were designed by Andrew Burbanks (University of Portsmouth). The purpose of these posters was to make the scientific principles that govern the behaviour of such spinning toys accessible to a wide public. The first poster (Fig. 2) was based on photographs provided by Yutaka Shimomura of the rising egg phenomenon, including side reference to the analogous role of coriolis forces in large scale atmospheric dynamics. Under the gyroscopic approximation, the equations governing the spin of the egg admit an invariant J = h, where  is the component of angular velocity about the vertical, and h is the height of the centre-of-mass above the table. This invariant similarly exists for the case of any convex axisymmetric body spinning with slipping friction on a horizontal plane boundary. For the particular case of a body that is part spherical, this invariant was found by Jellett [10] (and so is appropriately called the Jellett invariant), and, in this case, it is exact (without need to make the gyroscopic approximation). This is the case for the tippe-top (a mushroom-shaped spherical-cap top) which so intrigued Niels Bohr and Wolfgang Pauli; the manner in which the tippe-top turns upside-down when it spins was successfully explained by Hugenholtz [9].

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Fig. 2 Poster prepared for the Royal Society Summer Science Exhibition, 2007; graphic design by Andrew Burbanks. The rising egg (photos courtesy of Y. Shimomura). These posters were designed to attract the attention of a wider public to the dynamics of spin and analogous behaviour in fluid dynamical contexts

Fig. 3 As for Fig. 2; poster for Euler’s disc (photo courtesy of J. Bendick, inventor)

The existence of the Jellett invariant provides a reasonably transparent explanation for the rise of the egg. The slipping friction causes a slow loss of energy (to heat), and the system seeks a minimum energy state compatible with the ‘prescribed’ and constant value of J . At high spin rates (as required for the gyroscopic approximation), the energy is predominantly the kinetic energy of spin (the potential energy being much smaller). Consider the case of a uniform prolate spheroid with semi-axes a, b with a > b, the kinetic energy when the axis is horizontal is E 1 = (5/2)J 2 /(a 2 + b2 )b2 ,

(1)

and the kinetic energy when the axis is vertical is E 2 = (5/4)J 2 /a 2 b2 .

(2)

Hence, since a > b, it follows that E 2 < E 1 , i.e the vertical state has lower energy (for the same value of J ) and is, therefore, the preferred stable equilibrium. Note that for the case of an oblate spheroid (a < b), the 11

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Fig. 4 As for Fig. 2; poster for the rattleback

opposite conclusion holds: if spun sufficiently rapidly on the table about its axis of symmetry, it will rise to spin on its rim. A ‘go’-stone, or indeed a mint imperial, provides a suitably oblate object on which to experiment. The above argument is reminiscent of that used to explain why a body rotating freely in space will tend, under the effects of weak internal friction, to rotate about its axis of greatest inertia. For a prolate spheroid, this is an axis perpendicular to the axis of symmetry, a conclusion quite the opposite of that obtained above for the spinning egg problem. So why the difference? It is because in the case of the freely rotating body, angular momentum is conserved, and energy is minimised subject to this constraint. For the spinning egg, angular momentum is not conserved, but the Jellett invariant takes its place; this makes all the difference. The moral is: for any weakly dissipative system, first determine what is conserved (even in some approximate sense), then minimise energy subject to this constraint. The rise of the egg is only one ‘filtered’ aspect of the fifth-order nonlinear dynamical system that governs the non-holonomic rigid body dynamics. Superposed on the ‘slow’ rise are rapid oscillations amplitudes of which are controlled by the current state of spin. For sufficiently large spin (well above that required to make the egg rise), these oscillations can cause the normal reaction between egg and table to fluctuate to such an extent that the egg momentarily loses contact with the table, a prediction verified experimentally by Mitsui et al. [12]. The analogous oscillations in the geophysical context are Rossby waves superposed on the mean large-scale atmospheric circulation. Dissipative instabilities have been set in an abstract geometric context by Bloch et al. [2]. The rising egg provides a prototype dissipative instability, brought to life by this simple table-top demonstration.

3 Euler’s disc The toy known as Euler’s disc is a heavy steel disc that can be rolled on its edge on a horizontal surface (or on the slightly concave dish that is supplied with it, see Fig. 3). It works well on a glass-topped table. This toy exhibits a ‘finite-time singularity’ in the final stage of its motion before it comes to rest on the table: the point of rolling contact of the disc on the table describes a circle with angular velocity  that increases ‘to infinity’ as the angle α between the plane of the disc and the table decreases to zero, as it obviously must in this final stage of motion. I place ‘to infinity’ in single quote because, in reality, this potential singularity must be ‘resolved’ in some way by physical processes during the final split second when the motion is arrested. Much interest focuses on the nature of this resolution. As in the case of the rising egg, it is the weak dissipation of energy that induces the singular behaviour, which again has a startling effect when observed for the first time. According to analytical dynamics (with all dissipative effects neglected) a steady precessing state is possible in which 2 sin α = 4g/a,

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(3)

where a is the radius of the disc and g is the acceleration of gravity. The energy E of the system (kinetic plus potential) is proportional to sin α, and in practice, dissipative effects, no matter how weak, must lead to a slow decrease of α to zero, and hence, for so long as the quasi-static condition (3) persists, to an unlimited increase of . Much controversy surrounds the question of what is the dominant mechanism of energy dissipation for this system. One quantifiable mechanism is that due to viscous dissipation in the thin layer of rapidly sheared layer of air in the decreasing gap between the disc and the table [1,14]. (I am frequently asked what happens if the experiment is performed in a vacuum; the answer is that at the low pressures attainable in laboratory vacuum systems, the viscosity μ of air is very little different from its value at atmospheric pressure, as discovered by Maxwell [11], and so this dissipative mechanism persists unaffected by decrease of pressure!). Rolling friction due to plastic deformation at the point of contact is also important, as evidenced by the fact that the disc behaves differently on different surfaces (e.g. on glass, polished steel, polished wood, …). Whatever the dominant mechanism may be, what matters is the dependence of the rate of dissipation of energy  on E. For the viscous mechanism, and small α, this has a power law character  ∼ E −λ ,

(4)

where λ > 0. This rate of dissipation increases as E → 0 for the simple reason that the rate of shearing obviously increases as α → 0 and  → ∞. Since dE/dt = −,

(5)

E ∝ (t0 − t)1/(λ+1) ,

(6)

it follows immediately from (4) that

where t0 is determined by the value of E at the initial instant t = 0. Hence, E (and so α) goes to zero at the finite time t0 , and apparently, from (3),  becomes infinite at this same instant. The resolution of this infinity is not hard to find. The downward acceleration of the centre of mass of the disc increases without limit according to the above description. When this downward acceleration equals g, the normal reaction at the point of contact with the table vanishes, so that (presumably) the disc loses contact with the table, and the rolling condition (on which (3) is based) no longer applies. This breakdown of the ‘adiabatic condition’ (3) occurs literally a split second (∼ 0.03 s for the toy Euler disc) before the singularity time t0 , and we move into a different dynamical phase of ‘free-fall’ during this final split second. Much attention is currently devoted to the ‘finite-time singularity problem’ in fluid mechanics. Roughly paraphrased, this may be stated as follows: at high enough Reynolds number, can the vorticity become infinite within a finite time, starting from smooth finite-energy initial conditions? Even in the inviscid limit, for which the laws discovered by Helmholtz [7] are applicable, this problem remains unsolved, and is also the subject of considerable controversy (see, for example [6,8]). In these circumstances, Euler’s disc provides a reassuring table-top demonstration that finite-time singularities (with an appropriate resolution mechanism) do occur even in dissipative systems (and, indeed in this case, occur only by virtue of the dissipative mechanism!). 4 The rattleback My third example is the toy known as the celt, or more popularly the ‘rattleback’, a canoe-shaped object that spins smoothly in one direction, but which, when set spinning in the opposite direction, becomes unstable and reverses direction. This occurs because the axis of the rattleback is very slightly deformed into an S-shape, so slight as to be hard to detect with the naked eye, but sufficient, nevertheless, to induce this striking behaviour. Attention was first drawn to the phenomenon by Walker [20] (later Sir Gilbert Walker, oceanographer, after whom the circulation of the Southern Ocean is named), and has been analysed afresh from time to time, most recently by Moffatt and Tokieda [16], who describe the celt as a ‘prototype of chiral dynamics’. The object is chiral, in the sense that it is not mirror-symmetric: it cannot be brought into coincidence with its mirror image. When spun in a clockwise sense, the rattleback is subject to a ‘pitching’ instability which extracts energy from the spin, ultimately inducing the reversal. This instability is then stabilised, but a new weaker ‘rolling’ instability develops, which is then potentially capable of causing a second reversal for the same reason. In fact, in ideal (frictionless) circumstances, total energy is conserved, and this behaviour becomes periodic in time. 13

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Fig. 5 Spin reversals of the rattleback: time evolution of spin N (t) (blue) and the amplitudes A(t) of pitching instability (red) and B(t) rolling instability (green); the mathematical model is as derived in Moffatt and Tokieda [16]

Weak friction (whose precise nature is again as obscure as for the Euler disc) damps the energy, and only a finite number of reversals can occur. Figure 5 shows the sort of behaviour that can result from these considerations. The blue curve shows the spin N (t) as a function of (dimensionless) time, while the red and green curves show the amplitudes A and B of the pitching and rolling modes of instability, respectively. Weak linear damping of each mode has here been chosen in such a way that four reversals of spin occur before the rattleback comes to rest. I have achieved such behaviour in practice with a carefully machined massive rattleback for which frictional forces have been minimised relative to inertial acceleration and gravity. Note how the pitching mode (red) is destabilised when N > 0 and stabilised when N < 0, while precisely the opposite happens for the rolling mode (green). Note also that the rolling mode has a weaker growth rate than the pitching mode, as required to deliver the asymmetric spin behaviour. In fluid mechanics, the simplest measure of chirality in a fluid flow is the helicity  H = u · ω dV, (7) where u and ω are the velocity and vorticity fields, respectively, and the integral is taken over the whole fluid domain. This helicity is invariant under precisely those circumstances for which the Helmholtz laws apply and ‘vortex lines are frozen in the fluid’. It is evident from the rattleback example that very weak chirality can have a profound effect on the observed dynamics of a system in which dissipative effects are small. Similarly in fluid mechanics, helicity can have profound consequences; in particular, it is the mean helicity of a turbulent flow that is responsible for the growth of magnetic fields in conducting fluids due to dynamo instability [13,19], i.e. for the very existence of magnetic fields generated in the interiors of planets, stars and galaxies. And what could be more profound than that?

References 1. Bildsten, L.: Viscous dissipation for Euler’s disc. Phys. Rev. E 66, 056309 (2002) 2. Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Ratiu, T.S.: Dissipation induced instabilities. Ann. Inst. H. Poincare´ Anal. Nonlin. 11, 37–90 (1994) 3. Bou-Rabee, N.M., Marsden, J.E., Romero, L.: A geometric treatment of Jellett’s egg. Z. Angew. Math. Mech. 85, 1–25 (2005) 4. Branicki, M., Shimomura, Y.: Dynamics of an axisymmetric body spinning on a horizontal surface. IV. Stability of steady states and the ‘rising egg’ phenomenon for convex axisymmetric bodies. Proc. R. Soc. A 462, 3253–3275 (2006) 5. Branicki, M., Moffatt, H.K., Shimomura, Y.: Dynamics of an axisymmetric body spinning on a horizontal surface. III. Geometry of steady states for a general axisymmetric body. Proc. R. Soc. A 462, 371–390 (2006) 6. Bustamante, M.D., Kerr, R.M.: 3D Euler about a 2D symmetry plane. Physica D 237, 1912–1920 (2008) 7. von Helmholtz, H.: 1858 Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55 (1858) [English translation by Tait, P.G.: On integrals of the hydrodynamical equations, which express vortex-motion. Phil. Mag. 33(4), 485–512 (1867)]. 8. Hou, T.Y., Li, R.: Blowup or no blowup? The interplay between theory and numerics. Physica D 237, 1937–1944 (2008) 9. Hugenholtz, N.M.: On tops rising by friction. Physica 18, 515–527 (1952) 10. Jellett, J.H.: A Treatise on the Theory of Friction. Macmillan, London (1872) 11. Maxwell, J.C.: On the viscosity or internal friction of air and other gases. Proc. R. Soc. A 15, 14–17 (1866)

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12. Mitsui, T., Aihara, K., Terayama, C., Kobayashi, H., Shimomura, Y.: Can a spinning egg really jump. Proc. R. Soc. A 460, 3643–3672 (2006) 13. Moffatt, H.K.: Turbulent dynamo action at low magnetic Reynolds number. J. Fluid Mech. 41, 435–452 (1970) 14. Moffatt, H.K.: Euler’s disk and its finite-time singularity. Nature 404, 833–834 (2000) (see also Nature 408, 540) 15. Moffatt, H.K., Shimomura, Y.: Spinning eggs—a paradox resolved. Nature 416, 385–386 (2002) 16. Moffatt, H.K., Tokieda, T.: Celt reversals: a prototype of chiral dynamics. Proc. R. Soc. Edinb. 138, 361–368 (2008) 17. Moffatt, H.K., Shimomura, Y., Branicki, M.: Dynamics of an axisymmetric body spinning on a horizontal surface. I. Stability and the gyroscopic approximatiuon. Proc. R. Soc. A 460, 3643–3672 (2004) 18. Shimomura, Y., Moffat, H.K., Branicki, M.: Dynamics of an axisymmmetric body spinning on a horizontal surface. II. Self-induced jumping. Proc. R. Soc. Lond. A 461, 1753–1774 (2005) 19. Steenbeck, M., Krause, F., Rädler, K.-H.: Berechnung der mittleren Lorentz–Feldstärke v × B für ein elektrisch leitendes Medium in turbulenter, durch Coriolis–Kräfte beeinflusster Bewegung. Z. Naturforsch. 21a, 369–376 (1966) 20. Walker, G.T.: On a dynamical top. Q. J. Pure Appl. Math. 28, 175–184 (1896)

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O R I G I NA L A RT I C L E

Russell J. Donnelly

Why study vortex dynamics?

© Springer-Verlag 2010

Abstract Most scientists have a very strong reason to love the field in which they work. This informal evening talk was my own answer as to why a condensed matter physicist working in fluid mechanics should be interested in the subfield of vortex dynamics. Keywords Vortices · Quantized vortices · Vortex rings · Vortex collisions PACS 47.32,Cc Vortex dynamics is at the heart of the turbulence problem. Vortices have been described by Dietrich Küchemann as the “sinews and muscles of fluid motion and the “voice of the flow” (since sound can be generated by vortex motion). Vortices play a role in the development, maintenance, and interaction of ordered flow structures which transport quantities like mass momentum and energy in an efficient way. Vortex phenomena span an incredible range of scales: quantized vortices in helium II have a core size of an Angstrom, tornados and waterspouts are of ordinary human scale, phenomena such as Jupiter’s red spot have planetary scales, and vortex motions are evident in entire galaxies. Hans Lugt [1] has offered a partial list of vortex motions in order of increasing scale. Quantized vortices in superfluid helium Smallest turbulent eddies Vortices generated by insects Vortex rings of squids Bathtub vortex Dust whirl on streets Vortices in machinery Whirls behind weirs Wing-tip vortices Whirlpools in turbine intakes and in tidal currents Dust devils Tornadoes and waterspouts Steam rings from volcanoes Vortices shed from the Gulf Stream hurricanes High- and low-pressure systems Ocean circulations General circulation of the atmosphere Convection cells inside the earth R. J. Donnelly (B) Department of Physics, University of Oregon, Eugene, OR 97403, USA 17

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Planetary atmospheres Great Red Spot of Jupiter Sun spots Rotation inside of stars Galaxies Some examples of these phenomena are contained in Figs. 1, 2, 3, 4, 5, and 6.

Fig. 1 Velocity and pressure distribution near a quantized vortex line in superfluid helium as a function of the radius in Angstroms. The pressure gradient attracts negative ion bubbles to the vortex core enabling one to probe the dynamics of quantized vortices. The pressure gradient in viscous fluids attracts bubbles also, as we see in Figs. 4 and 9

Fig. 2 Historic photograph of a tornado in Manhattan, Kansas, May 31, 1949. Courtesy: NOAA

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Fig. 3 Sixteenth century drawing by Olaus Magnus showing tidal vortices near the Lofoten Islands in Norway

Fig. 4 Steam ring ejected from Mount Etna in Sicily. These have a radius of about 100 m. Courtesy: Marco Fulle, www.stromboli. net.

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Fig. 5 Two typhoons spinning around each other in the north Pacific, shown in a satellite photograph by NOAA. Typhoons have a range of scales mostly in hundreds of kilometers radius

Fig. 6 The whirlpool galaxy (Messier 51) with satellite galaxy in constellation Canes venatic. The bright disc has a radius of about 38,000 light years. (Photograph from Palmar Observatory/California Institute of Technology)

How can one study such complicated flows as we see in these pictures? One way is to try to find the simplest flow one can think of, and try to understand it in detail. It has been recognized by many investigators that the vortex rings (e.g., smoke rings, volcanic steam rings) are one of the most fundamental and fascinating phenomena in vortex dynamics. Vortex rings have important roles in superfluids as well as classical fluids. To use a (somewhat overworked) analogy, vortex rings are the “hydrogen atom” of topics in fluid mechanics such as turbulence.

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Fig. 7 Sketch of a vortex ring gun used at the University of Oregon to make vortex rings in water. Visualization of the rings is made possible by an electrochemical technique. Color appears at the electrode on the exit of the gun

Fig. 8 Photo of a vortex ring traversing the tank, taken from above. The core of the vortex appears clearly

How do we make vortex rings in water? We simply make a piston plunger. The vortex gun has an exit diameter of D0 = 2.54 cm. Embedded in the wall at the exit is one electrode for the Baker visualization technique, an electrochemical technique using the pH indicator, thymol blue. The piston is actuated by a rack and pinion system powered by a servo motor which can independently adjust strokes L up to about 4 cm, and stroke times T from about 40–160 ms (Figs.7, 8, 9). Systematic measurements have allowed us to develop expressions for the velocity V , radius R, core size a, and circulation  in terms of the stroke length L and stroke time T . This opens the way to study the collisions of vortex rings and all-important subjects such as vortex collisions and reconnections. We have constructed a Vortex Ring Spectrometer which allows us to study collisions of vortex rings whose characteristics are set separately for each gun. A good example of the kind of collision experiment that can be performed is shown in Fig. 10. So why study vortex dynamics? I trained in low-temperature physics, specializing in superfluidity. It turns out that most experiments revealing the properties of superfluids are really fluid mechanics experiments, although under rather extreme conditions. It is only natural that this interest should lead one to do experiments in ordinary fluids as well. It is very satisfying to do experiments and make theories about phenomena on a laboratory scale, which might be relevant to structures such as we show in Figs.1, 2, 3, 4, 5, and 6 spanning more than 30 orders of magnitude in scale. I should mention here that it is a particular honor to speak here tonight. My late wife, Marian Card Donnelly was an expert in the architecture of the Scandinavian countries, and we visited Denmark many times, including a six month sabbatical in 1972 at the Niels Bohr Institute. We visited this site a number of times when it was a very busy brewery. This building, the Jacobsen House, was designed by the master brewer J. C. Jacobsen in collaboration with architect N. S. Nebelong in 1853. For a number of years it served as the “house of honor” for an outstanding Danish scholar, and was the home of Niels Bohr from 1932 to his death in 1962. It is the setting for the distinguished play “Copenhagen” by Michael Frayn, attempting to describe a meeting between Bohr and Heisenberg during the second world war. Tomas Bohr gave us this evening a charming insight of 21

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Fig. 9 Photograph of an aging vortex ring with well-developed instabilities on its core. Some hydrogen bubbles deliberately introduced are seen above, on the left

Fig. 10 An example of vortex ring collisions (time runs horizontally and downward). You can see the collision of two rings forming an intermediate state which is unstable and fissions. Each product ring carries material from each projectile ring .

visiting here as a grandchild of Niels. The brewery was the last site discussed in Marian’s book “Architecture in the Scandinavian countries”. [2] “Perhaps it may seem surprising that a history of architecture in the Scandinavian countries should conclude with a factory. Let us remember, however, that building in the Nordic countries began, as at Ulkestrup, with the most elementary shelters for human activity. Fragments of tools and weapons, made of less perishable materials than was clothing, indicate the production of goods for use at the very outset of human habitation in

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this region. To the shelters for meeting individual physical needs were added shelters for individual and communal social and spiritual needs, and these fundamental human requirements have not changed over 10,000 years.” (The talk ended with a showing of dolphins making and playing with bubble rings in water. The pictures can be retrieved by searching for “Dolphin bubble rings” on Google. The bubbles are attracted to the core by the pressure gradient illustrated in Fig. 1. Swimmers can also make bubble rings.) References 1. Lugt, H.J.: Introduction to Vortex Theory, p. 5. Vortex Flow Press, Potomac (1996) 2. Donnelly, M.C.: Architecture in the Scandinavian Countries, p. 341. The MIT Press, Cambridge, MA (1992)

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Theor. Comput. Fluid Dyn. (2010) 24:9–24 DOI 10.1007/s00162-009-0098-5

O R I G I NA L A RT I C L E

Darren Crowdy

A new calculus for two-dimensional vortex dynamics

Received: 7 November 2008 / Accepted: 23 February 2009 / Published online: 30 April 2009 © Springer-Verlag 2009

Abstract This article provides a user’s guide to a new calculus for finding the instantaneous complex potentials associated with point vortex motion in geometrically complicated planar domains, with multiple boundaries, in the presence of background flows. The key to the generality of the approach is the use of conformal mapping theory together with a special transcendental function called the Schottky–Klein prime function. Illustrative examples are given. Keywords Two dimensional flow · Complex potential · Multiply connected PACS 47.15.ki, 47.15.km, 47.32.C 1 Introduction A first course in theoretical fluid dynamics usually includes, quite early on, a study of ideal fluids where viscosity is neglected and, by an appeal to Kelvin’s circulation theorem which guarantees the persistence of an initially irrotational flow in an ideal fluid, goes on to introduce the notion of a complex potential—an analytic function of a complex variable z = x + i y—from which the student can build up various flows of interest. These include uniform flows, irrotational straining flows, flows with fluid sources or sinks and those involving a collection of point vortices whose dynamical evolution can be computed and studied using a system of ordinary differential equations. A canonical solution that every student sees, not least because of its importance in basic aerofoil theory, is for steady uniform potential flow past a circular obstacle. Often, conformal mapping is then introduced to resolve the flow around more realistic aerofoil shapes. All this has arguably found its place in the canon of theoretical fluid dynamics. Treatments of it appear in standard monographs such as Acheson [1], Batchelor [2] and Milne-Thomson [22]. But there is one obvious extension of this basic theory that is scarcely mentioned in any of the aforementioned textbooks and which is of fundamental importance: how does the theory change when the flow domain contains more than one cylinder (or obstacle)? What is the complex potential for uniform flow past two cylinders? Or three? These are natural questions but the student will struggle to find the answers in the extant fluid dynamics literature. It turns out, as we show here, that it is possible to present a rather complete mathematical theory—which we go so far as to call a “calculus”—which fills this lacuna in the classical fluid dynamical canon. It is a natural extension of what is already well-known (when just one obstacle is present) and, in this author’s view, should be known to anybody interested in theoretical fluid dynamics. Flows in geometrically complex domains are ubiquitous in fluid dynamics: aerodynamicists are interested in multi-aerofoil configurations, civil engineers Communicated by H. Aref In remembrance of Philip Geoffrey Saffman (1931–2008). D. Crowdy Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK E-mail: [email protected] Reprinted from the journal

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D. Crowdy

compute the forces on an array of bridge supports in a laminar flow, geophysical fluid dynamicists study the motion of oceanic eddies around island clusters and even in the resurgent field of biofluid dynamics there is much interest in modelling, for example, the motion and behaviour of schools of fish interacting both with each other and with their wake structures. All the basic mechanisms in such problems can be studied, at least within a two-dimensional context, using the calculus to be presented here. The fluid dynamical research literature is, at best, patchy on the subject of planar flows in multiply connected domains. Isolated special results have been reported, mostly for the doubly connected case when just two obstacles are present (in early aeronautics, this was the famous “biplane problem”—Sedov [27] includes a chapter on it). Indeed, as evidence of the incoherence of the literature on this topic, the results for doubly connected domains appear to have been rediscovered more than once over recent decades (e.g. [19,15,18]). Analyses of the doubly connected case invariably use the theory of elliptic functions. For triply and higher connected regions, few analytical results exist (before the work of the present author and collaborators). Our strategy is to build the special calculus we need using a single special function called the Schottky–Klein prime function. The calculus incorporates the doubly connected situation as a special case and it thus captures all prior results, albeit from a novel perspective. We refer to the framework described here as a calculus because, as in standard calculus, the introduction of just a few basic special functions, together with their defining properties, provides the means to construct a rich variety of more complicated functions and solutions to more difficult problems. The same is true of the new calculus here. Because this paper is supposed to be a user’s guide, we adopt a practical “how to” approach and omit all unnecessary (albeit important) subsidiary mathematical details and proofs—the reader should refer to the author’s original papers for those supporting facts. This approach can be likened to the student of standard calculus making liberal and unhindered use of the fact that sin θ is a 2π-periodic function without worrying too much about why that is true, or indeed, how to show it. Standard calculus is best learned by the execution of specific examples. We therefore include illustrative examples indicating the scope and versatility of the methods. It will be possible to systematically derive analytical expressions for the complex potentials for general two-dimensional inviscid flows in general multiply connected fluid domains. The presence of point vortices and solid boundaries is incorporated. There are no restrictions on the number of such solid boundaries (provided there is a finite number) nor, indeed, on their shapes. Conformal mapping theory takes care of that. 2 Riemann mapping theorem First, we describe the setting in which the new calculus is couched. It is founded on one of the most important achievements of 19th century mathematics: the Riemann mapping theorem and its extensions. The Riemann mapping theorem, in its original form, states that any simply connected region of the plane is conformally equivalent to a unit disc. Expressed another way, the theorem guarantees the existence of a conformal mapping from the interior of the unit disc in, say, a parametric ζ -plane, to any given simply connected fluid region in the z-plane. Let this conformal mapping be z(ζ ). If the fluid region is unbounded, there must be some point ζ = β inside or on the boundary of the unit ζ -disc which maps to the point at infinity. Indeed, z(ζ ) must have a simple pole at ζ = β. There are three real degrees of freedom in the specification of this conformal map. One way to pin these down is to choose the value of β arbitrarily. Then, the map is determined up to a single rotational degree of freedom. Often this is set by insisting that, near ζ = β, the map has the local behaviour a z(ζ ) = + analytic (1) ζ −β where a is taken to be real. The aim is to build a calculus for finding complex potentials associated with flows that are irrotational apart from a set of isolated singularities. The first step is to acknowledge that, owing to the conformal invariance of the boundary value problems to be solved, we can equally well find the required complex potentials as functions of the parametric variable ζ (rather than as functions of z). For geometrically complicated domains, this observation provides a major simplification. 3 A motivating example To illustrate the ideas, consider the unbounded fluid region exterior to a unit radius cylinder centred at z = 0. This region is the conformal image of a unit ζ -disc, in a parametric ζ -plane, under the simple conformal mapping

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Reprinted from the journal

A new calculus for two-dimensional vortex dynamics

3

3

2

2

1

1 z(ζ)

0

0

−1

−1

−2

−2

−3 −3

−2

−1

0

1

2

−3 −3

3

−2

−1

0

1

2

3

Fig. 1 Conformal mapping z(ζ ) = 1/ζ from the interior to the exterior of the unit disc

z(ζ ) =

1 . ζ

(2)

Clearly, ζ = 0 maps to z = ∞ (Fig. 1). 3.1 A point vortex outside a cylinder Suppose a point vortex of unit circulation is situated at some point z α outside the unit-radius cylinder. Let ζ = α be the preimage of this point under the mapping z(ζ ) so that 1 . (3) zα The complex potential for a point vortex of unit circulation at ζ = α existing in free space is well-known to be given by α=

i log(ζ − α). (4) 2π This function does not, however, satisfy the boundary condition that the unit circle |ζ | = 1 is a streamline. Since the streamfunction is the imaginary part of the complex potential, we must find a complex potential which is, say, purely real on |z| = 1 so that it has constant imaginary part equal to zero there. One function, built from (4), which is certainly real on |ζ | = 1 is w(ζ ) = −

w(ζ ) + w(ζ )

(5)

but it is not an analytic function of ζ so it is not a candidate for the required complex potential. It is also real everywhere in the complex ζ -plane while we only need it to be real on |ζ | = 1. To fix all this, we exploit the fact that on |ζ | = 1 it is true that ζ = 1/ζ so we can replace ζ in the final term of (5) to give w(ζ ) + w(1/ζ ).

(6)

Two things then happen: first, this function is now an analytic function of ζ (it is no longer a function of ζ ); second, it is still purely real on |ζ | = 1 (but not necessarily off this boundary circle). Also, owing to the presence of the term w(ζ ), it has the required logarithmic singularity at ζ = α reflecting the presence of a point vortex there. In this way, we have constructed a solution to our problem. Substitution of (4) into (6) produces   i ζ −α − log , (7) 2π 1/ζ − α or, on rearrangement, it can be written as   i ζ −α i log − log ζ + constant. − 2π |α|(ζ − 1/α) 2π

(8)

The construction we have just presented is basically the content of the so-called Milne-Thomson circle theorem (see, for example, Acheson [1] or Milne-Thomson [22]). Reprinted from the journal

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3.2 Circulation around the obstacle or island Now we define the function

  ζ −α i G 0 (ζ, α) ≡ − log . 2π |α|(ζ − 1/α)

(9)

For any value of α inside the unit disc, G 0 (ζ, α) is purely real on the boundary |ζ | = 1 of the circular object; it also has logarithmic singularities at ζ = α and ζ = 1/α corresponding to point vortices at these points. It is easy to check that this complex potential leads to a circulation equal to −1 around the cylinder. Actually, by the conformal invariance of the problem, (9) is the complex potential for a point vortex exterior to an object of arbitrary shape: to complete the solution, it is only necessary to know the form of the function z(ζ ) mapping the unit ζ -disc to the region exterior to the object. Now, the solution (8) can be written as G 0 (ζ, α) + G 0 (ζ, 0) + constant.

(10)

(10) is only one of many possible solutions to the problem as stated: other possible solutions to the same problem are G 0 (ζ, α) + γ G 0 (ζ, 0) + constant

(11)

where γ is an arbitrary real constant. Because the term γ G 0 (ζ, 0) corresponds to a point vortex of circulation γ at ζ = 0 (corresponding to the point at infinity), all we have done in (11) is to change the net circulation at infinity (without changing the circulation of the point vortex at ζ = α). Equivalently, we have changed the total circulation around the cylinder to −1 − γ . Clearly, to pin down a unique solution to the problem it is necessary to additionally specify the required value of the circulation around the cylinder in the original problem statement. If we want, say, the total circulation around the cylinder to be zero then we need to take γ = −1 which corresponds to placing a point vortex of circulation γ = −1 at physical infinity. The idea behind the calculus is to find the appropriate generalizations of expression (9) for a point vortex at some position α outside a collection of obstacles. We will also need to find a way to independently control the circulations around the various obstacles. 4 Multiply connected conformal mapping The calculus we are developing applies to fluid domains involving any finite number of obstacles (or objects, or islands) in the flow. An unbounded fluid domain containing just one obstacle is simply connected, one containing two obstacles is doubly connected, and so on. It is therefore important to understand something about the conformal mapping of multiply connected domains. The generalization of the Riemann mapping theorem to multiply connected regions was given in the early twentieth century by Koebe [16]: any multiply connected domain (with finite connectivity) is conformally equivalent to some multiply connected circular domain. A canonical class of such domains consists of the unit disc in a ζ -plane with M smaller circular discs excised. Let this domain be Dζ and let C j , for j = 1, . . . , M, denote the circular boundary of the jth excised circular disc. Also, let us denote the unit circle |ζ | = 1 by C0 . The only geometrical parameters needed to uniquely specify such a domain are the centres {δ j | j = 1, . . . , M} and the radii {q j | j = 1, . . . , M} of the circles {C j | j = 1, . . . , M}. We shall refer to the data {δ j , q j | j = 1, . . . , M} as the conformal moduli of Dζ . The values of the conformal moduli cannot be picked arbitrarily; rather, they are determined (up to the usual three real degrees of freedom of the mapping theorem mentioned earlier) by the target domain in the z-plane. 4.1 A three cylinder example This is best illustrated by example. Consider the unbounded fluid region exterior to three equal circular obstacles as shown in Fig. 2. The obstacles all have radius s and are centred on the real axis at −d, 0 and d. From the geometrical symmetries of this domain, it is reasonable to seek a circular preimage region Dζ which shares these symmetries. We therefore pick that β = 0 maps to infinity and consider the unit ζ disc with two smaller

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A new calculus for two-dimensional vortex dynamics

Fluid region

circular region D

ζ

z(ζ)=s/ζ

s

s

q

s

q

obstacles

δ

d

Fig. 2 An example conformal map from a circular domain Dζ to the exterior of three circular obstacles

circular discs, each of radius q, centred at ±δ (see the rightmost diagram in Fig. 2). Let us introduce the conformal mapping z(ζ ) =

s . ζ

(12)

This takes |ζ | = 1 to the circle |z| = s and it also maps ζ = 0 to z = ∞. It is also easy to verify that if we pick q and δ such that q=

s2 sd , δ= 2 , 2 2 d −s d − s2

(13)

then the circle |ζ − δ| = q will map to |z − d| = s while |ζ + δ| = q maps to |z + d| = s. Clearly, the conformal moduli depend on the geometry of the domain in the physical plane. Once the conformal moduli are known, we can also define a set of M Möbius maps to be θ j (ζ ) ≡ δ j +

q 2j ζ 1 − δjζ

.

(14)

Knowledge of the domain Dζ , its conformal moduli {δ j , q j | j = 1, . . . , M} and the maps {θ j (ζ )| j = 1, . . . , M}, together with the conformal mapping z(ζ ), are the only things needed to devise the new calculus. 5 A fact from function theory We seek to generalize the expression (9) to the multiple obstacle case. We will now do something which, at first, seems pedantic: to introduce the notation ω(ζ, α) ≡ (ζ − α) for the simple monomial function (ζ − α) so that we can rewrite (9) as   i ω(ζ, α) G 0 (ζ, α) = − log . 2π |α|ω(ζ, 1/α)

(15)

(16)

Recall that this is just the complex potential for a point vortex of circulation +1 around a single obstacle of arbitrary shape and giving a circulation equal to −1 around that obstacle. Suppose now that we have a point vortex of unit circulation exterior to a collection of M + 1 obstacles of arbitrary shape. For M > 0 the fluid region, which we will call Dz , is multiply connected. By the multiply connected Riemann mapping theorem, we know that there is a conformal mapping z(ζ ) from a conformally equivalent circular domain Dζ , with some choices of the centres and radii {q j , δ j | j = 1, . . . , M} (the conformal moduli) and with some point β in Dζ mapping to z = ∞. Reprinted from the journal

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We can now ask the natural question: what is the (generalized) complex potential associated with a point vortex of circulation +1 at a point α in Dz and having circulation −1 around the obstacle whose boundary is the image of C0 ? The following remarkable fact lies at the heart of the new calculus: the answer is again given by formula (16)! 5.1 The Schottky–Klein prime function What do we mean by the last statement? It turns out that there exists a special function, which we will continue to denote by ω(ζ, α) (even though it is only given by the simple formula (15) when M = 0), such that (16) remains the formula for the complex potential for the flow generated by a point vortex of circulation +1 at position α in the multiply connected region Dζ . This complex potential produces circulation −1 around the obstacle whose boundary is the image of C0 and happens to produce zero circulation around all other obstacles. It is noteworthy, and convenient, that the required formula (16) for the required fluid dynamical complex potential remains the same; all that changes is what we mean by ω(ζ, α). The function ω(ζ, α) is called the Schottky–Klein prime function and it plays a fundamental role in complex function theory extending far beyond the realm of fluid dynamics. When M = 0 (the simply connected case) ω(ζ, α) is defined by (15); for M > 0 its functional form is more complicated. Much more will be said about this function, and how to compute it, later. Since it is a function of two complex variables, henceforth we write it as ω(., .). Here are the only two facts that are important: (1) ω(ζ, α) has a simple zero at ζ = α. (2) The properties of ω(., .) are such that G 0 (ζ, α) has constant imaginary part on all the boundary circles of Dζ (this means that all the obstacle boundaries are streamlines). Let us assume for now that ω(., .) is a known (and computable) special function associated with a circular domain Dζ that is conformally equivalent to some multiply connected fluid domain Dz of interest. Let z(ζ ) be the conformal map taking Dζ to Dz . 6 Building the new calculus We know that G 0 (ζ, α) ≡ −

  ω(ζ, α) i log 2π |α|ω(ζ, 1/α)

(17)

is the complex potential for a unit circulation point vortex outside any number of objects. It produces a circulation −1 around the particular obstacle which is the image of C0 and circulation 0 around the other M obstacles. 6.1 Adding circulation around the obstacles But suppose we want to be able to specify that there is a circulation γ j  = 0 around the obstacle whose boundary is the image of C j for j = 1, . . . , M? How do we construct the relevant complex potential? To do this, more functions must be constructed within our calculus but, importantly, they are also built from ω(., .). Of course, we need M additional functions because we have M other obstacles which may possibly have circulation around them. We therefore define the M functions   i ω(ζ, α) G j (ζ, α) = − log , j = 1, . . . , M. (18) 2π |α|ω(ζ, θ j (1/α)) Note the appearance of θ j (ζ ) which, for each j, is one of the M Möbius maps (14) introduced earlier. G j (ζ, α) has the following significance: it is the complex potential corresponding to a point vortex of unit circulation at the point α but now with circulation −1 around the obstacle whose boundary is the image of C j . It produces zero circulation around all the other obstacles. Our choice of notation is helpful: the subscripts reflect which

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A new calculus for two-dimensional vortex dynamics

obstacle has circulation −1 around it. Again, it is a consequence of the peculiar properties of the function ω(., .) that imply that G j (ζ, α) has constant imaginary part on all the circles {C j | j = 0, 1, . . . , M}. Now, adding the required circulations around the obstacles is easy: the required complex potential is just −

M 

γ j G j (ζ, β).

(19)

j=0

This analytical result for the complex potentials associated with specifying the circulations around multiple obstacles was first described by Crowdy [6] in the context of adding circulations around stacks of aerofoils in ideal flow.

6.2 Uniform flow past multiple objects The complex potential for uniform flow past multiple obstacles can also be constructed from the function G 0 (ζ, α) (hereafter denoted simply by G 0 ). We seek a complex potential which looks, as z → ∞, like U e−iχ z and has constant imaginary part on all the boundaries of Dζ . This corresponds to uniform potential flow with speed U at angle χ to the x axis. Let α = αx + iα y . Consider the two functions 1 (ζ, α) =

∂G 0 ∂G 0 , 1 (ζ, α) = . ∂αx i∂α y

(20)

Owing to the identities 1 ∂G 0 = ∂αx 2



∂G 0 ∂G 0 + ∂α ∂α

 ,

1 1 ∂G 0 = i ∂α y 2



∂G 0 ∂G 0 − ∂α ∂α

 ,

(21)

it is simple to check that both 1 (ζ, α) and 1 (ζ, α) have a simple pole, each with residue i/(4π), at α. Since the imaginary part of G 0 is constant on all boundaries of Dζ , so are the imaginary parts of 1 (ζ, α) and i 1 (ζ, α) (this is because we take parametric derivatives with respect to the real and imaginary parts of α, not derivatives with respect to ζ ). We can now take real linear combinations of the two functions 1 (ζ, α) and i 1 (ζ, α) in order to give us the required singularity at infinity. Indeed the analytic function − 4πU cos χ [i 1 (ζ, α)] − 4πU sin χ [1 ]

(22)

where U and χ are real constants has a simple pole, with residue U e−iχ at α. It also has constant imaginary part on all the boundaries of Dζ . On use of (21), (22) can be written   ∂G 0 ∂G 0 − e−iχ . (23) 2πU i eiχ ∂α ∂α If, as ζ → β, we have z(ζ ) =

a + analytic ζ −β

for some constant a, then the required complex potential for the uniform flow is given by   ∂G 0 ∂G 0  − e−iχ . 2πU ai eiχ ∂α ∂α α=β

(24)

(25)

It is important, in this formula, to take derivatives with respect to α and α before letting α = β. This derivation of the complex potentials associated with uniform flow past multiple obstacles was first described in [7]. It was used in Crowdy [6] for the computation of the lift and interference forces on stacks of aerofoils in uniform flow and with non-zero circulations around them. Reprinted from the journal

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D. Crowdy

6.3 Straining flows around multiple objects We can even build the complex potentials for higher order flows using G 0 . Let us seek a complex potential which looks, as z → ∞, like eiλ z 2 (where  and λ are some real constants) and which has constant imaginary part on all the boundaries of Dζ . Therefore, consider the three second parametric derivatives given by 2 (ζ, α) =

∂2G0 ∂2G0 ∂2G0 ,

(ζ, α) = (ζ, α) = . ,  2 2 ∂αx2 ∂αx ∂(iα y ) ∂α 2y

(26)

The functions 2 (ζ, α), i 2 (ζ, α) and 2 (ζ, α) have constant imaginary part on all the boundaries of ∂ Dζ . What about their singularities? By virtue of (21), we have   1 ∂2G0 ∂2G0 ∂2G0 2 (ζ, α) = + +2 , 4 ∂α 2 ∂α∂α ∂α 2  2  1 ∂ G0 ∂2G0 −

2 (ζ, α) = , 4 ∂α 2 ∂α 2   1 ∂2G0 ∂2G0 ∂2G0 + − 2 . 2 (ζ, α) = − 4 ∂α 2 ∂α∂α ∂α 2

(27)

Now the quantity ∂2G0 ∂α∂α

(28)

has a δ-function singularity at ζ = α that we must eliminate. It is therefore natural to consider the combination ˆ 2 (ζ, α) ≡ 

1 1 (2 (ζ, α) − 2 (ζ, α)) = 2 4



∂2G0 ∂2G0 + 2 ∂α ∂α 2

 .

(29)

ˆ 2 (ζ, α) and 2 (ζ, α) each have a This function also has constant imaginary part on all the boundaries of Dζ .  ˆ 2 (ζ, α) and i 2 (ζ, α) second order pole, with strength i/(8π), at α. Now we find real linear combinations of  giving the required singularity at infinity. The required function is ˆ 2 (ζ, α). − 8π cos λ[i 2 (ζ, α)] + 8π sin λ

(30)

This function has a second order pole of strength eiλ at ζ = α. On making use of (21) it can be written as   2 2 −iλ ∂ G 0 iλ ∂ G 0 2πi e −e . ∂α 2 ∂α 2

(31)

Then if z(ζ ) has the behaviour given in (24) at ζ = β then the required complex potential for the quadratic straining flow is  2

2πa i e

−iλ ∂

2G

0 ∂α 2

−e

iλ ∂

2G

0 ∂α 2

   

.

(32)

α=β

Such a flow will also generally involve a uniform flow at infinity but this can be subtracted off using the uniform flow solution of the previous section. This appears to be the first time that the complex potential for a general irrotational straining flow past multiple objects has been explicitly written down (although the possibility of doing so was advertised in Crowdy [7]).

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A new calculus for two-dimensional vortex dynamics α

α1

2

γ0

γ

2

γ

1

s α3

d

Fig. 3 Three point vortices moving around three circular islands. There are non-zero circulations around each of the islands

6.4 Moving objects What if the obstacles in the flow are moving with some prescribed velocity? (so far we have assumed that the solid objects in the flow are stationary). The calculus of the functions we have already introduced can be adapted to this circumstance too. The complex potentials associated with such motion can be written down using the prime function ω(., .). This time, there is a slight difference that the expressions involve integrals. It is established in Crowdy [10] that the complex potential in which the jth obstacle is moving with complex velocity U j for j = 0, 1, . . . , M is given by 

  1  WU (ζ ) = Re[−iU0 z(ζ  )] d log ω(ζ  , ζ ) − d log ω(ζ −1 , ζ −1 ) 2π C0

 M

   1  − Re[−iU j z(ζ  )] + d j d log ω(ζ  , ζ ) − d log ω(θ j (ζ −1 ), ζ −1 ) , 2π j=1

(33)

Cj

where the constants {d j | j = 1, . . . , M} solve a linear system that is recorded in [10] (we refer the reader there for full details). The subscript on WU (ζ ) is a vector U ≡ (U0 , U1 , . . . , U M ) of the complex velocities of the M + 1 obstacles.

7 Examples To illustrate the flexibility of the new calculus, and how to use it, we will consider some illustrative examples. There are just a few keys steps in analyzing any given problem. They are as follows: (1) Analyze the geometry and determine the conformal moduli {q j , δ j | j = 1, . . . , M} and the mapping z(ζ ). Well-known numerical methods from the theory of multiply connected conformal mapping may be necessary here. For circular objects the required conformal map is just a Möbius map and everything is explicit. (2) Construct the Möbius maps {θ j (ζ )| j = 1, . . . , M} and compute the Schottky–Klein prime function ω(., .). (3) Do calculus with the functions {G j (ζ, α)| j = 0, 1, . . . , M} to solve the given fluid problem.

7.1 Three point vortices near three circular islands Problem 1 Suppose that there are three circular islands, all of radius s, centred at (−d, 0), (0, 0) and (d, 0) and each having a circulation γ around it. The fluid also contains 3 point vortices at positions α1 , α2 and α3 each of which have circulation . What is the instantaneous complex potential? (Fig. 3). Reprinted from the journal

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D. Crowdy

γ U

d

s

γ

Fig. 4 Uniform flow past two equal cylinders

1. The geometry: This fluid region has already been considered in an earlier example (Fig. 2). The circular region Dζ is the unit ζ -disc with two smaller discs excised, each of radius q and centred at ±δ. The point β = 0 maps to infinity. 2. Möbius maps: There are two Möbius maps in this case given by θ1 (ζ ) = δ +

q 2ζ 1 − δζ

, θ2 (ζ ) = −δ +

q 2ζ 1 + δζ

.

(34)

3. Do calculus: The complex potential in this case is w1 (ζ ) =

3 

G 0 (ζ, αk )

← point vortices

k=1

−3G 0 (ζ, 0) −

2 

← make all round-obstacle circulations zero

γ j G j (ζ, 0)

← add in required round-obstacle circulations.

(35)

j=0

7.2 What is the lift on a biplane? Problem 2 Consider two circular aerofoils stacked vertically both of radius s and centred at ±id. Far away, the flow is uniform with speed U parallel to x-axis. Suppose there is a circulation γ around each of them. If, as shown in Fig. 4, this circulation is negative, one can generally expect there to be a lift force on each aerofoil in the vertical direction. We must find the relevant complex potential. 1. The geometry: The fluid domain in this case is doubly connected so there is a conformal mapping to it from a concentric annulus ρ < |ζ | < 1 (which is the domain Dζ for this case). The required conformal mapping is a Möbius map with the form  √  ζ− ρ z(ζ ) = i A , (36) √ ζ+ ρ where ρ=

123

1 − (1 − (s/d)2 )1/2 , A=d 1 + (1 − (s/d)2 )1/2 34



1−ρ 1+ρ

 =

 d 2 − s2.

(37) Reprinted from the journal

A new calculus for two-dimensional vortex dynamics

α

U

1

α

2

d

s

δ1

δ

2

Fig. 5 Generalized Föppl flows with two cylinders

√ Note that ζ = − ρ maps to infinity with local behaviour a z(ζ ) = √ + analytic ζ+ ρ where

√ a = −2i A ρ.

(38)

(39)

2. Möbius maps: There is only a single Möbius map in this case given by θ1 (ζ ) = ρ 2 ζ. 3. Do calculus: The complex potential is then given by   ∂G 0  ∂G 0 w2 (ζ ) = 2πU ai − α=−√ρ ∂α ∂α −

1 

√ γ G j (ζ, − ρ)

(40)

← uniform flow (χ = 0)

← round obstacle circulations

(41)

j=0

To compute the lift distribution, w2 (ζ ) can be inserted into the usual Blasius integral formula for the hydrodynamic force on an object [1]. The forces (and torques) on any number of aerofoils can be computed similarly and various calculations of this kind can be found in [6]. 7.3 Generalized Föppl flows with two cylinders Problem 3 Consider the same two cylinders as in Example 2, but we suppose there is now zero circulation around the cylinders and two point vortices in the wake of each cylinder. Each pair of point vortices are supposed to be of equal and opposite sign, say ±. This is a generalization of the classic Föppl steady vortex pair behind a cylinder [26] to the situation where two cylinders are present (see Fig. 5 where the case  < 0 is shown). The complex potential in this case is   ∂G 0 ∂G 0  w3 (ζ ) = 2πU ai − α=−√ρ ← uniform flow (χ = 0) ∂α ∂α + G 0 (ζ, α1 ) − G 0 (ζ, α2 ) + G 0 (ζ, δ1 ) − G 0 (ζ, δ2 ) ← point vortices

(42)

Although we have placed 4 point vortices in the flow (which, as we have seen from Example 1, can be expected to require an additional point vortex at infinity) note that their total circulation is zero. This means that there is no need for any additional point vortex at infinity. This finite parameter space can, in principle, be investigated to find generalized Föppl equilibrium configurations. Reprinted from the journal

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D. Crowdy Γ

−Γ

d

U=−i

Fig. 6 A moving cylinder, with point vortex wake, approaching a stationary wall at constant speed

7.4 Cylinder with wake approaching a wall Problem 4 Suppose a cylinder, of unit radius, is positioned such that its lowest point is at a height of d units above an infinite straight wall along the real axis. Suppose the wall is stationary but that the cylinder it is moving at complex speed U = −i towards the wall. Furthermore, to model the wake behind the cylinder, we place two point vortices at symmetrical positions behind the moving cylinder: a point vortex of circulation  is at position z = α while another, of circulation −, is at −α. What is the instantaneous complex potential? (Fig. 6). 1. The geometry: The domain is again doubly connected so we take Dζ as ρ < |ζ | < 1. The conformal mapping from this annulus to the fluid domain is again a Möbius map of the form z(ζ ) =

i(1 − ρ 2 ) 2ρ



ζ +ρ ζ −ρ

 (43)

where d=

(1 − ρ)2 . 2ρ

(44)

The circle |ζ | = 1 maps to the boundary of the cylinder while |ζ | = ρ maps to the infinite plane wall. 2. Möbius maps: There is only a single Möbius map in this case given by θ1 (ζ ) = ρ 2 ζ.

(45)

w4 (ζ ) = G 0 (ζ, α) − G 0 (ζ, −α) ← point vortices +WU (ζ ) ← flow due to moving cylinder

(46)

3. Do calculus:

where U = (−i, 0). The dynamics of the vortices in this problem was studied by Crowdy, Surana and Yick [11] using Kirchhoff–Routh theory.

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A new calculus for two-dimensional vortex dynamics

U Point vortices

U2

U1

U

0

Moving objects

Fig. 7 Three moving ellipse-like bodies in an ambient uniform flow together with an array of point vortices

7.5 Model of school of swimming fish Suppose we have three objects, with ellipse-like shapes, each travelling at some complex speed U j for j = 0, 1, 2 in a uniform ambient flow of speed U parallel to the x-axis. Assume there is no circulation around any of the objects. In the fluid around them, at some instant, there are six point vortices having circulations k (for k = 1, . . . , 6) and situated at positions αk (for k = 1, . . . , 6). This configuration might, for example, model a school of three fish swimming in a uniform ambient flow with the point vortices modelling their instantaneous wakes. What is the instantaneous complex potential for this flow? 1. The geometry: The domain is triply connected. For the particular geometry shown in Fig. 7, Dζ is the unit disk with two circular discs excised, each of radius q and centred at ±δ. The functional form of the mapping happens to be given by   ∂  ∂  z(ζ ) = −a G 0 (ζ, α) + c, +b (47) ∂α α=0 ∂α α=0 where a, b and c are real constants. We will not explain this formula but will simply mention that such shapes happen to be exact solutions of a free boundary problem involving bubbles in Hele-Shaw flows [13]. It is interesting to note, though, that it is built from the same function G 0 that we have used to construct the calculus (it turns out that G 0 has intimate connections with conformal slit maps, a fact used to study the motion of point vortices through gaps in walls by Crowdy & Marshall [8]). The key point is that the vast literature of conformal mapping can be imported into our calculus to deal with multiply connected geometries of (in principle) any shape. In the map (47), the point β = 0 maps to infinity and, locally, z=

a + analytic. ζ

(48)

2. Möbius maps: There are two Möbius maps in this case given by θ1 (ζ ) = δ +

q 2ζ 1 − δζ

, θ2 (ζ ) = −δ +

q 2ζ 1 + δζ

.

(49)

3. Do calculus: Again we construct the complex potential by adding together the appropriate components: w5 (ζ ) =

6 

k G 0 (ζ, αk )

k=1



−

6 

← point vortices

 k G 0 (ζ, 0) ← make round-obstacle circulations zero

k=1

+2πU ai +WU (ζ )



 ∂G 0 ∂G 0  ← uniform flow (χ = 0) − ∂α ∂α α=0 ← flow generated by moving bodies

(50)

where U = (U0 , U1 , U2 ). Reprinted from the journal

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D. Crowdy

8 How to compute the SK prime function? With this powerful calculus at hand, just one question remains: how does one evaluate the SK prime function? One possibility is to use a classical infinite product formula for it as recorded, for example, in Baker [3]. It is given by ω(ζ, α) = (ζ − α)

 (θk (ζ ) − α)(θk (α) − ζ ) θk

(θk (ζ ) − ζ )(θk (α) − α)

,

(51)

where the product is over all compositions of the basic maps {θ j , θ −1 j | j = 1, . . . , M} excluding the identity and all inverse maps. In the doubly connected case, if we take Dζ to be the concentric annulus ρ < |ζ | < 1 with 0 < ρ < 1 then there is just a single Möbius map given by θ1 (ζ ) = ρ 2 ζ . The infinite product (51) is then over all mappings of the form {θ j (ζ ) = ρ 2 j ζ | j ≥ 1}.

(52)

Using this in (51) leads to the expression ω(ζ, α) = − where C =

∞

k=1 (1 − ρ

2k )

α P(ζ /α, ρ), C

(53)

and the function P(ζ, ρ) ≡ (1 − ζ )

∞ 

(1 − ρ 2k ζ )(1 − ρ 2k ζ −1 ).

(54)

k=1

Since P(ζ, ρ) is analytic in the annulus ρ < |ζ | < 1 it also has a convergent Laurent series there. It is given by the (rapidly convergent) series P(ζ, ρ) = A

∞ 

(−1)n ρ n(n−1) ζ n ,

(55)

n=−∞

where A=

∞ 

(1 + ρ 2n )2

n=1

∞ 

ρ n(n−1) .

(56)

n=0

This Laurent series converges everywhere in the fundamental annulus ρ < |ζ | < ρ −1 and proves a much faster way of evaluating P(ζ, ρ) than a method based on (54). It is interesting to point out that the function P(ζ, ρ) is closely connected to the first Jacobi theta function 1 [29] which is one way to see the connection between the approach here and the approach to doubly connected problems using elliptic function theory that usually appears in the literature [15,18,19]. For the case of general multiply connected domains, it is not known if the product (51) converges for all choices of the parameters {q j , δ j | j = 1, . . . , M} and, even if it does, its rate of convergence can be so slow as to make use of (51) impractical in many circumstances. It can, however, be safely used in some cases, especially of small connectivity. It is then natural to ask: is there a Laurent series representation, analogous to (55) in the M = 1 case, for cases where M > 1?. Such a representation will obviate the need to use the infinite product (51). Recently, Crowdy and Marshall [14] have devised a novel numerical algorithm based on precisely such Laurent series representations. It can be used to evaluate ω(ζ, α), with great speed and accuracy, for broad classes of domains and without resorting to use of the infinite product (51). The algorithm works by writing X (ζ, α) = (ζ − α)2 Xˆ (ζ, α),

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A new calculus for two-dimensional vortex dynamics

where X (ζ, α) = ω2 (ζ, α) and then computing the coefficients in the following Laurent expansion of Xˆ (ζ, α):   M  M  ∞ ∞ (k) m   cm qk dm(k) Q m k ˆ X (ζ, α) = A 1 + . (58) + (ζ − δk )m (ζ − δk )m k=1 m=1

k=1 m=1

It is important to note that the algorithm in [3] does not depend on a sum or product over a Schottky group. This renders this method of evaluating the prime function much faster in practice than making use of (51). Full details of this numerical algorithm can be found in [14] and freely downloadable MATLAB M-files will soon be available at the website: http://www.ma.ic.ac.uk/~dgcrowdy/SKPrime.

9 Other considerations The purpose of this user’s guide has been to show how to write down analytic expressions for the instantaneous complex potentials for any given ideal flow in two dimensions. These results can be applied in various circumstances and extended in different directions. For completeness, we will give a brief overview of these additional matters.

9.1 Kirchhoff–Routh theory Here, we have been solely concerned with finding the instantaneous complex potential associated with distributions of point vortices in geometrically complicated domains, possibly with additional background flows (such as uniform or straining flows) and with possible circulations around any obstacles. But what about the dynamics of the point vortices? The new calculus can help us with the dynamical problem too. To see how, we invoke an important result from 1941 due to C.C. Lin [20]. He showed that the problem of point vortex motion in general multiply connected domains is a Hamiltonian system and wrote down a formula for the governing Hamiltonians in terms of a “special Green’s function” associated with the domain in which the vortices were moving. Lin did not provide any explicit way to construct this Green’s function but derived his results based purely on its existence. Actually, the special Green’s function discussed by Lin has the explicit expression, in multiply connected circular domains Dζ , given by the explicit formula for G 0 (ζ, α) introduced in (17); indeed, G 0 (ζ, α) is precisely Lin’s special Green’s function. This crucial fact was first pointed out by Crowdy and Marshall [4]. Furthermore, in a second 1941 paper, Lin [21] went on to show how the Hamiltonian transforms under conformal mapping: if H (ζ ) is the Hamiltonian for N -vortex motion in Dζ , then the Hamiltonian H (z) in any domain Dz that is the conformal image of Dζ under the map z(ζ ) is just H

(z)

({z k }) = H

(ζ )

  N   dz  k2 ({αk }) + log   , 4π dζ αk

(59)

k=1

where z k = z(αk ). This second result completes our theory: it means that it is enough to know the functional form of the special Green’s function in multiply connected circular domains (and we already do) because (59) then gives the Hamiltonians in any other conformally equivalent domain. Crowdy and Marshall [8] have explored this theory in the context of single vortex motion around circular islands [5] and through gaps in walls.

9.2 Contour dynamics The calculus built around the Schottky–Klein prime function can help in more sophisticated situations too. Perhaps the next most popular model of vorticity, after the point vortex model, is the vortex patch model [26] where vortex structures are modelled as finite-area regions of uniform vorticity. Owing to the fact that, in two dimensions, vorticity is convected with the flow, it is enough to follow just the boundaries of any such uniform vortex regions thereby reducing the dynamical model to the tracking of a set of contours in the fluid. Such numerical methods are known collectively as contour dynamics methods [25]. In recent work, Crowdy and Reprinted from the journal

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Surana [12] have shown how to generalize such methods to finding the evolution of vortex patches in geometrically complicated, multiply connected domains. Again, these methods rely on combined use of conformal mapping methods with the properties of the Schottky–Klein prime function. The use of such methods is in its infancy but they appear to be highly effective. 9.3 Vortex motion on a sphere It turns out that our theoretical development generalizes quite naturally to the flows on the surface of a sphere once it is endowed with a complex analytic structure by means of a stereographic projection. Surana and Crowdy [28] have shown how to generalize the calculus both to the case of point vortex motion and vortex patch motion in complicated domains on a spherical surface. Acknowledgments DGC acknowledges an EPSRC Advanced Research Fellowship. He is grateful to J. S. Marshall for helpful discussions.

References 1. Acheson, D.J.: Elementary Fluid Dynamics. Oxford University Press, Oxford (1990) 2. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967) 3. Baker, H.F.: Abelian functions: Abel’s theorem and the allied theory of theta functions. Cambridge University Press, Cambridge (1897) 4. Crowdy, D.G., Marshall, J.S.: Analytical formulae for the Kirchhoff–Routh path function in multiply connected domains. Proc. R. Soc. A. 461, 2477–2501 (2005) 5. Crowdy, D.G., Marshall, J.S.: The motion of a point vortex around multiple circular islands. Phys. Fluids 17, 056602 (2005) 6. Crowdy, D.G.: Calculating the lift on a finite stack of cylindrical aerofoils. Proc. R. Soc. A. 462, 1387–1407 (2006) 7. Crowdy, D.G.: Analytical solutions for uniform potential flow past multiple cylinders. Eur. J. Mech. B/Fluids 25(4), 459–470 (2006) 8. Crowdy, D.G., Marshall, J.S.: The motion of a point vortex through gaps in walls. J. Fluid Mech. 551, 31–48 (2006) 9. Crowdy, D.G., Marshall, J.S.: Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6(1), 59–76 (2006) 10. Crowdy, D.G.: Explicit solution for the potential flow due to an assembly of stirrers in an inviscid fluid. J. Eng. Math. 62(4), 333–344 (2008) 11. Crowdy, D.G., Surana, A., Yick, K.-Y.: The irrotational flow generated by two planar stirrers in inviscid fluid. Phys. Fluids 19, 018103 (2007) 12. Crowdy, D.G., Surana, A.: Contour dynamics in complex domains. J. Fluid Mech. 593, 235–254 (2007) 13. Crowdy, D.: Multiple steady bubbles in a Hele–Shaw cell. Proc. R. Soc. A. 465, 421–435 (2009) 14. Crowdy, D.G., Marshall, J.S.: Computing the Schottky–Klein prime function on the Schottky double of planar domains. Comput. Methods Funct. Theory 7(1), 293–308 (2007) 15. Ferrari, C.: Sulla trasformazione conforme di due cerchi in due profili alari, Memoire della Reale Accad. della Scienze di Torino, Ser. II 67 (1930) 16. Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable. American Mathematical Society, Providence (1969) 17. Hejhal, D.A.: Theta Functions, Kernel Functions and Abelian Integrals, vol. 129. American Mathematical Society, Providence (1972) 18. Johnson, E.R., McDonald, N.R.: The motion of a vortex near two circular cylinders. Proc. R. Soc. Lond. Ser. A 460, 939–954 (2004) 19. Lagally, M.: Die reibungslose Strmung im Aussengebiet zweier Kreise. Z. Angew. Math. Mech. 9, 299–305, 1929 (English translation: The frictionless flow in the region around two circles, N.A.C.A., Technical Memorandum No 626, (1931)) 20. Lin, C.C.: On the motion of vortices in two dimensions I: existence of the Kirchhoff–Routh function. Proc. Natl. Acad. Sci. 27, 570–575 (1941) 21. Lin, C.C.: On the motion of vortices in two dimensions II: some further investigations on the Kirchhoff–Routh function. Proc. Natl. Acad. Sci. 27, 576–577 (1941) 22. Milne-Thomson, L.M.: Theoretical Hydrodynamics. Dover, New York (1996) 23. Nehari, Z.: Conformal Mapping. Dover, New York (1952) 24. Prosnak, W.J.: Computation of fluid motions in multiply connected domains. Wissenschaft & Technik, Frankfurt (1987) 25. Pullin, D.I.: Contour dynamics methods. Ann. Rev. Fluid Mech. 24, 89–115 (1992) 26. Saffman, P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992) 27. Sedov, L.I.: Two-Dimensional Problems in Hydrodynamics and Aerodynamics. Interscience Publishers, New York (1965) 28. Surana, A., Crowdy, D.G.: Vortex dynamics in complex domains on a spherical surface. J. Comput. Phys. 227(12), 6058–6070 (2008) 29. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1927)

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Theor. Comput. Fluid Dyn. (2010) 24:25–37 DOI 10.1007/s00162-009-0156-z

O R I G I NA L A RT I C L E

Mark A. Stremler

On relative equilibria and integrable dynamics of point vortices in periodic domains

Received: 29 June 2009 / Accepted: 13 August 2009 / Published online: 10 October 2009 © Springer-Verlag 2009

Abstract The motion of two point vortices defines an integrable Hamiltonian dynamical system in either singly or doubly periodic domains. The motion of three point vortices in these domains is also integrable when the net circulation is zero. The relative vortex motion in both domains can be reduced to advection of a passive particle by fixed vortices in an equivalent Hamiltonian system. A survey of the solutions for vortex motion in these systems is discussed. Some initial conditions lead to relative equilibria, or vortex configurations that move without change of shape or size. These configurations can be determined as stagnation points in the reduced problem or through explicit solution of the governing equations. These periodic point-vortex systems present a rich collection of interesting solutions despite the few degrees of freedom, and several questions on this subject remain open. Keywords Point vortices · Vortex dynamics · Vortex equilibria PACS First, Second, More 1 Introduction The point vortex representation provides a mathematically tractable approach to reduced order modeling of vortex-dominated flows. As discussed by Charney [12], physical understanding of a flow system is often best achieved by considering as few degrees of freedom as possible; for flows with motion dominated by regions of isolated vorticity, the point vortex approximation achieves this reduction while continuing to conserve important physical properties such as mass, energy, and momentum. Here I give an overview of relative equilibria and integrable dynamics of point vortex systems in singly and doubly periodic domains, with a focus on the cases of two or three ‘base’ vortices per period. Ever since Helmholtz [16] introduced the concept of what we now call a point vortex, it has been known that the dynamics of two such vortices in the unbounded plane produces only relative equilibria, with the separation between the vortices remaining constant for all time. If the net strength of the vortices is zero, they translate uniformly in a direction perpendicular to their separation; if instead the net strength is non-zero, they rotate about the center of vorticity, which lies on the line passing through both vortices. When the system consists of three vortices in the plane, the dynamics remains integrable but the vortex trajectories are more complex, with most initial conditions leading to time dependent vortex separations. The general three-vortex system in the plane has been investigated quite thoroughly, starting with the work of Gröbli [15] and including independent investigations by numerous authors; see [4] for an overview. Relative equilibrium configurations in the unbounded plane have also received considerable attention in the literature [6]. Communicated by H. Aref M. A. Stremler (B) Department of Engineering Science and Mechanics, Virginia Tech, Mail Code 0219, Blacksburg, VA 24061, USA E-mail: [email protected] Reprinted from the journal

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The consideration of singly periodic arrays of point vortices began with von Kármán’s [28,29] model of the laminar wake behind a bluff body. This model can be viewed as consisting of two infinite rows of periodically spaced vortices in the unbounded plane, or of two ‘base’ vortices in a strip of fluid that is periodic in the x direction and infinite in the y direction; the latter viewpoint will generally be adopted here. In von Kármán’s model, the base vortex pairs have strengths that are equal in magnitude but opposite in sign, in which case every choice of vortex separation leads to a relative equilibrium. If instead the net circulation is taken to be non-zero, most initial conditions do not lead to relative equilibria, although the behavior is still integrable and relatively straightforward [23]. When there are three base vortices in a periodic strip, the dynamics are integrable when the sum of the vortex strengths is zero [2]. The analysis of this problem is quite recent relative to Gröbli’s analysis of three vortices on the unbounded plane or von Kármán’s analysis of vortex wakes. Some motivation for considering systems with three vortices per period comes from experimental observations of the ‘P+S’ mode in the wake behind an oscillating cylinder [31], in which three vortices are generated per shedding cycle. As in the unbounded plane, most initial conditions lead to time-dependent vortex separations, even under the requirement of zero net circulation. When there are more than three base vortices, this motion is almost always non-integrable. However, specific conditions can be determined for the existence of relative equilibrium configurations of three or more base vortices in a singly periodic domain [18,21,23]. Doubly periodic vortex configurations were first considered by Tkachenko [26], who examined the simple lattice consisting of only one base vortex and its periodic images. This vortex configuration forms a relative equilibrium that rotates uniformly with constant angular velocity. Some motivation for considering equilibrium configurations of point vortices in a doubly periodic domain come from the broader problem of Coulomb interactions, including vortex configurations in superfluids [32] and superconductors [1]. As a result, several investigations, including [11,13,27], have considered the minimum energy and stability of these simple lattice configurations. As with the singly periodic domain, cases with two or three base vortices lead to vortex motion for most initial conditions, and this motion is integrable [20,25]. The dynamics of three (or more) vortices in a doubly periodic domain can be related to reduced order modeling of two-dimensional turbulence [3,5]. This manuscript consists of a review of previous work on vortex equilibria and dynamics in periodic domains, a presentation of some new results for vortex equilibria in doubly periodic domains, and a connection between examples for the various cases. The equations of motion for N point vortices in singly and doubly periodic domains are presented in a unified way in Sect. 2. Relative equilibria and dynamics of two point vortices in both domains are discussed in Sect. 3. Equilibrium configurations with N = 3 are presented in Sect. 4, with the doubly periodic results appearing here for the first time, and vortex motion with N = 3 is reviewed in Sect. 5.

2 Governing equations Let the position of a point vortex be given in the complex plane by z β = xβ + i yβ , and let the strength of that vortex be β . The velocity induced at a point z by a collection of N base vortices in a singly or doubly periodic planar domain with no other boundaries is given through superposition by N dz ∗ 1  β W p (z − z β ), = dt 2πi

(1)

β=1

where the asterisk on z denotes complex conjugation and W p gives the form of the velocity field generated by each vortex, with the index p indicating the periodicity of the domain. The well-known representation on the unbounded plane ( p = 0) is given by taking W0 (z) = 1/z; the representations for the singly and doubly periodic domains are given below in (7) and (8), respectively. The velocity of a vortex at z α is then found by taking lim z→z α dz ∗ /dt and removing the infinite self-induced velocity, which gives N dz α∗ 1  β W p (z α − z β ), = dt 2πi

(2)

β=1

where the prime on the sum indicates omission of the singular term α = β. The symmetry of the vortex interaction considered here requires that W p (z) be an odd function of z. Thus, multiplying (2) by α and summing over α shows that the linear impulse,

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 = X + iY =

β z β ,

(3)

β=1

is a constant of the motion for each of the systems considered here. Note that in the periodic domains  is not well defined, as its value depends on which vortices in the system are chosen as the base vortices. The angular impulse, I =

N  α=1

α z α z α∗ ,

(4)

which is a constant of the motion on the unbounded plane, is not a constant of the vortex motion for the singly and doubly periodic domains. Two other quantities that play a role in the analysis are the first symmetric function of the base vortex strengths, S1 =

N 

α ,

(5)

α=1

and the second symmetric function of the vortex strengths, S2 =

N  

α β .

(6)

α,β=1

In a singly periodic domain ( p = 1) with period L ∈ R, the form of the vortex interaction can be written as [2,14] W1 (z; L) = (π/L) cot(π z/L).

(7)

This interaction can be viewed as that due to an infinite row of vortices with equal strengths arranged at periodic intervals n L, or directly as a simple pole (i.e., a base vortex) with residue π/L in a singly periodic domain (or periodic strip) of width L. When (7) is substituted into (2), the resulting equations of motion give the interaction of N base vortices in a strip, together with their periodic images. In a doubly periodic domain ( p = 2) with half-periods ω1 , ω2 ∈ C, the vortex interaction can be represented as [25] W2 (z; ω1 , ω2 ) = ζ (z; ω1 , ω2 ) + [(πω1∗ )/( ω1 ) − η1 /ω1 ] z − π z ∗ / .

(8a)

Without any loss of generality it can be assumed that the lattice is oriented so that ω1 is purely real, in which case the vortex interaction is given by [24]1 W2 (z; ω1 , ω2 ) = ζ (z; ω1 , ω2 ) − (η1 /ω1 ) z + (2πi/ ) y.

(8b)

In (8), ζ (z) = ζ (z; ω1 , ω2 ) is the Weierstrass zeta function [22,30], ηi = ζ (ωi ) are the quasi-periods, and = 2i(ω1 ω2∗ − ω2 ω1∗ )

(9)

is the area spanned by the periods (2ω1 , 2ω2 ). Determining the simplified form of W2 (z) in (8b) requires use of Legendre’s relation [22,30], η1 ω2 − η2 ω1 = iπ/2.

(10)

The basic form of the equations in a doubly periodic domain was first discussed by Tkachenko [26], who established the form of W2 (z) for the simple lattice. This result was extended by O’Neil [20] to the general lattice, with the second term in W2 (z) given as an infinite sum. Substituting (8) into (2) gives the equations of motion

1

Note that there is an error (a missing factor of −i) in the final term of Eq. 28 in [24].

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for N base vortices in a parallelogram with periodic boundary conditions. Upon making this substitution, the final term in (8b) becomes N    1  i  − S1 z α∗ − ∗ . β z α∗ − z β∗ = 2 i 2

(11)

β=1

When S1  = 0, (11) corresponds to rigid rotation with angular velocity S1 /2 about the point z = /S1 , which can be interpreted as being in a frame of reference that rotates with the vortex system. This term makes it possible for a system with non-zero net circulation to be represented with a doubly periodic function. When S1 = 0, (11) becomes a constant that ensures the vortex lattices translates with the physically correct velocity. As in the unbounded plane, the vortex system governed by (2) can be written in Hamiltonian form as α

dxα ∂H , = dt ∂ yα

α

dyα ∂H , =− dt ∂ xα

(12)

with the Hamiltonian given by H =−

N N   1   α β p z α − z β . 2π

(13)

α=1 β=α+1

The form of p (z) depends on the domain. In the unbounded plane, 0 (z) = log |z|. In the periodic strip, 1 (z) = (π/L) ln |sin (π z/L)| ,

(14)

and in the periodic parallelogram [24,25],   2 (z) = ln |σ (z)| − Re η1 z 2 /(2ω1 ) − π y 2 / ,

(15)

where σ (z) is the Weierstrass sigma function [22,30]; ζ (z) is the logarithmic derivative of σ (z). The Hamiltonian in the doubly periodic domain can be directly related to the Fourier representation for the energy of the fluid through an identity for phase-modulated lattice sums [24]. As is typical of special functions, numerical evaluation of the Weierstrass ζ - and σ -functions requires the calculation of infinite sums and/or products. It is convenient to express both of these functions in terms of the Jacobian ϑ1 -function, which is well understood and easily computed. The necessary formulas can be found, for example, in [22] or [30], and numerous computational resources (such as Mathematica and the GNU Scientific Library) include the Weierstrass and Jacobian functions. 3 Relative equilibria and dynamics in periodic domains with N = 2 When there are only two base vortices, the equations of motion (2) become simply dz 1∗ 2 = W p (z 1 − z 2 ), dt 2πi

dz 2∗ 1 = W p (z 2 − z 1 ). dt 2πi

(16)

If S1 = 0, then 2 = −1 = , and every (non-singular) two-vortex configuration is a relative equilibrium configuration in which both vortices translate uniformly with constant velocity V = i W p∗ (z 1 − z 2 ) /2π. If the domain in this case is a periodic strip of width L and the vortex separation is z 1 − z 2 = a + ib, then taking sinh(πb/L) = sin(πa/L)

(17)

gives vortex street configurations that are stable to infinitesimal perturbations in the vortex positions that preserve the linear impulse and energy of the system [17]. The special case b = L sinh−1 (1)/π ≈ 0.28 L gives von Kármán’s staggered vortex street [29]. The general choices for (a, b) from (17) give rise to obliquely translating vortex streets; some exploration of the streamline patterns in these flows can be found in [8]. For the case with N = 2 and S1 = 0 in a doubly periodic domain, the vortex separation z 1 − z 2 = a + ib is again a constant of the motion for all (non-singular) choices of (a, b). The stability of these configurations remains an open question.

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I

II I

I

II

II

I

I

I

I I I

III

(a)

I

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I

I

(b)

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III I

(c)

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(d)

Fig. 1 Representation in the w3 plane with N = 2 and √ S1  = 0 for a the singly periodic √ domain and b–d the doubly periodic domain with ω1 ∈ R and b ω2 = i ω1 , c ω2 = (1 + i 3) ω1 /2, and d ω2 = (1/ 3 + i) ω1 . Solid circles mark the vortex locations; open circles give the positions of stable fixed points in the co-moving frame. The boundary of the periodic domain is marked by dashed lines, separatrices are shown with heavy lines, and other representative streamlines are shown with light lines. Roman numerals label various regimes of motion discussed in the text.

If the two base vortices have S1  = 0, then taking the difference of the equations in (16) gives an equation for the evolution of the vortex separation w3 = z 1 − z 2 , dw3∗ S1 = W p (w3 ). dt 2πi

(18)

This equation can be interpreted as the velocity field due to a single (fixed) base vortex at w3 = 0 with strength S1 . The system governed by (18) is also Hamiltonian. Streamlines in this reduced system correspond to trajectories of the vortex at z 1 relative to the vortex at z 2 in the original two-vortex system. Reproducing the full vortex motion is achieved by numerical integration of one of Eq. 16 with w3 (t) given by solutions of (18). In the singly periodic system, streamlines of the flow in the w3 plane are as shown in Fig. 1a. Changing S1 ( = 0) does not alter the structure of this flow in the w3 plane. Stagnation points occur only at w3 = (2n + 1)L/2, and these stagnation points are joined by streamlines, or separatrices, that delineate the regimes of vortex motion labeled I, II, and III in Fig. 1a. Almost all initial conditions give rise to relative motion of the two vortices. For initial conditions chosen in regime I, the two base vortices orbit one another; in regimes II and III, the base vortices separate in time and interact with the periodic images of the other vortex [23]. Note that, for this case of N = 2 and S1  = 0, this behavior depends only on the relative positions of the vortices, not on their relative strengths, and thus not on the linear impulse of the system. However, the scale, speed, and net displacement of the two-vortex motion does depend on the individual vortex strengths. The fixed points in the w3 plane correspond to relative equilibrium configurations in the two-vortex system. Since these fixed points are saddle points, the corresponding relative equilibria are unstable to perturbations in the vortex positions. In the doubly periodic system, the w3 -plane representation gives advection of a passive particle by a simple lattice of vortices with strength S1 in a frame of reference that rotates with the lattice. The streamline patterns in the w3 plane depend on the chosen values of the half-periods, an additional parameter that is not present in the singly periodic system. Three examples are shown in Fig. 1. For the square lattice, there are two regimes of motion delineated by the separatrices joining the stationary points at the half-periods ω1 and ω2 (and their periodic images). In regime I, streamlines form closed curves around the vortex at the origin, and the corresponding base vortices orbit one another similar to those in regime I in the singly periodic domain. In regime II, streamlines form closed curves around a third fixed point at ω3 = ω1 + ω2 . The corresponding base vortices move on circular paths, but they no longer orbit each other. The fixed points again correspond to relative equilibria of the base vortices. The points at ω1 and ω2 correspond to unstable relative equilibrium configurations that are analogous to those in the singly periodic domain. These two configurations are identical under a rotation of the coordinate system by π/2. In contrast to the singly periodic system, the w3 -plane representation can also contain stable fixed points. The corresponding equilibrium configurations are (neutrally) stable to perturbations in the relative vortex positions. For the square domain in Fig. 1b, taking w3 = ω3 gives a stable equilibrium configuration. The rhombic domain in Fig. 1c contains unstable configurations corresponding to the fixed points at w3 = ω1 , ω2 , and ω3 , and stable configurations corresponding to the fixed points at w3 = ω3 ± ω3 /3. Finally, note that doubly periodic domains with |ω2 | > |ω1 | contain regimes of motion in which the two base vortices separate in time and interact with the periodic images of the other vortex, such as regimes II and III in Fig. 1d. Reprinted from the journal

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4 Relative equilibria in periodic domains with N = 3 Consider the case in which all of the vortices in a particular system move with the same translational velocity or the same angular velocity. In these relative equilibria, the vortex positions are fixed with respect to the appropriate moving frame of reference. On the unbounded plane, relative vortex equilibria can either translate or rotate. Net translation in a spatially periodic domain is clearly allowable. However, the equations of motion for the singly periodic domain do not allow for any rotation, while the equations for the doubly periodic domain allow only for the rotation that comes from the co-rotating frame when S1  = 0. This restriction on rotation is a straightforward result of the fact that the W p (z) are periodic functions. Thus, the analysis here need only consider relative equilibria in periodic domains governed by (2), together with (7) or (8), that translate with a uniform velocity V , so that the equation of motion for a vortex at z α becomes V∗ =

N 1  β W p (z α − z β ). 2πi

(19)

β=1

Multiplying (19) by α and summing over α shows that, since W p (z) is odd in z and S1 is real, having a relative equilibrium configuration in a periodic domain requires that S1 V = 0.

(20)

For N = 3, it is convenient to introduce the notation w1 = z 2 − z 3 , w2 = z 3 − z 1 , and w3 = z 1 − z 2 ,

(21)

so that the equations for equilibria become 2πi V ∗ = 2 W p (w3 ) − 3 W p (w2 ) = 3 W p (w1 ) − 1 W p (w3 ) = 1 W p (w2 ) − 2 W p (w1 ).

(22)

Relative equilibrium configurations (with N = 3) can thus be determined by solving (22) subject to the constraint (20). O’Neil [19] proves that there exist two such solutions for a generic choice of strengths. The W p (wi ) terms in (22) are not independent. For the singly periodic domain, they are related through the cotangent identity, which gives W1 (w1 ) = −

(π/L)2 − W (w ) W (w ) π π π 1 2 1 3 cot (z 1 − z 2 ) + (z 3 − z 1 ) = . L L L W1 (w2 ) + W1 (w3 )

(23)

For the doubly periodic domain, the W p (wi ) are related through the addition formula for the Weierstrass ζ -function [22] as W2 (w1 ) + W2 (w2 ) + W2 (w3 ) = ζ (w1 ) + ζ (w2 ) + ζ (w3 ) = −

1 ℘  (w3 ) − ℘  (w2 ) , 2 ℘ (w3 ) − ℘ (w2 )

(24)

where ℘ (z) = ℘ (z; ω1 , ω2 ) = −ζ  (z) is the Weierstrass ℘ -function and ℘  (z) its derivate with respect to z. 4.1 The case S1 = 0 When S1 = 0, the difference between any two equations in (22) gives W p (w1 ) + W p (w2 ) + W p (w3 ) = 0

(25)

for both the singly and doubly periodic domains. Given one of the vortex separations, combining (25) with the appropriate addition formula for either the cotangent (23) or the Weierstrass ζ -function (24) gives the other two vortex separations in an equilibrium configuration. These equilibrium configurations are independent of the individual vortex strengths as long as S1 = 0. For the singly periodic domain, combining (23) and (25) gives a quadratic equation for any one of the W1 (wi ). If one assumes that w3 is specified, then equilibrium configurations are given by solutions of  W1 (w3 ) i 3 W12 (w3 ) + 4 (π/L)2 , (26) ± (∓) W1 (w1(2) ) = − 2 2

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(a)

(b) 1

2

1

–2

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–3

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(c) 3

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1

1

–4

–3

Fig. 2 Singly periodic vortex configurations and representative streamlines in the co-moving frame for the case S1 = 0. Vortex positions are labeled according to the vortex strength. The configurations in a–c all have identical vortex positions; for the assignment of vortex strengths 1 ≥ 2 > 0, these configurations have w3 ≈ (−0.354 + i 0.062)L. The configurations in b and d correspond to the labeled stagnation points in Fig. 5.

where the signs are chosen so that W1 (w1 )  = W1 (w2 ). Avoiding singular solutions in which vortices lie on top of each other requires that W1 (wi )  = 0, ±i. Two different example configurations are shown in Fig. 2, with one configuration repeated for several different choices of strength ratio. Instead of specifying the vortex separation w3 , one may wish to determine singly periodic configurations that translate with a given speed. In this case, substitution of (26) into (22) leads to the equation for w3 

L V∗ 2 L 2 −  1 L V ∗ 3 3 − 1, (27) W1 (w3 ) = i ± π s s s s √ where s= |S2 |/2. The vortex positions in panels a–c of Fig. 2 were chosen using (27) so that the configuration in b translates only in the x-direction. Such configurations may be considered simple models of ‘exotic’ vortex wakes in which three vortices are shed per cycle from a bluff body [7], although more work is needed to understand the connection between these point vortex systems and the exotic wakes observed in experiments. For the doubly periodic domain, combining (24) and (25) shows that equilibrium configurations with S1 = 0 are given by ℘  (w2 ) = ℘  (w3 )

(28)

as long as ℘ (w2 )  = ℘ (w3 ). As with the singly periodic system, the vortex positions that give an equilibrium configuration are determined without regard for the individual vortex strengths. The function ℘  (z) has zeros at ω1 , ω2 , and ω3 = ω1 + ω2 , so (28) will generally have three solutions; however, one of these solutions will give ℘ (w2 ) = ℘ (w3 ) and thus is not valid. The simplest of these solutions is to take each of the w j ’s to be integer multiples of the half periods; these cases are discussed in Sect. 4.2. It appears that this is essentially as far as the mathematical analysis can be taken for doubly periodic domains, and other relative equilibria are determined by numerical solution of (28) with, say, w3 given. Two examples of such equilibria are shown in Fig. 3 for the same choice of w3 as in panels a–c of Fig. 2. Configuration a in Fig. 3 has almost the same vortex positions as configuration b in Fig. 2; the value of w3 is taken to be the same in both cases, but the solutions for w2 differ slightly. This doubly periodic configuration can be viewed as an array of interacting parallel vortex streets that are moving at a slightly oblique angle with respect to the real axis. 4.2 The case V = 0 If the vortex configuration is stationary, then (22) reduces to W p (w1 ) W p (w2 ) W p (w3 ) = = . 1 2 3 Reprinted from the journal

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(29)

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(a)

(b)

Fig. 3 Doubly periodic vortex configurations and representative streamlines in the co-moving frame for the case S1 = 0 with ω1 ∈ R and ω2 = i ω1 . Vortex positions are labeled according to strength. These configurations are the two solutions for w3 ≈ 2(−0.354 + i 0.062)ω1 with 1 ≥ 2 > 0, which is the same value of w3 used for configurations a–c in Fig. 2.

For the singly periodic domain, combining (29) with the addition formula for the cotangent (23) gives the solutions  W1 (w j ) = ± j / S2 /2, j = 1, 2, 3, (30) where S2 cannot be zero. In contrast, stationary vortex configurations can exist in the unbounded plane only if S2 = 0. As discussed in [23], the structure of these stationary states depends on the sign of S2 . If S2 > 0, each w j must be real, and the only possible equilibrium configurations consist of all the vortices lying on a line parallel to the real axis, with the relative positions of the vortices depending on the relative vortex strengths. If S2 < 0, the vortex separations must have an imaginary component, and the real component of these separations must be an integer multiple of L/2. Thus, two of the vortices always lie on a vertical line, and the third vortex is either on this same line or offset by L/2. Examples of these different configurations can be found in [23]. For the doubly periodic domain, combining (29) with the addition formula for the Weierstrass ζ -function (24) leads to the system of equations (for S1  = 0) W2 (w2 ) W2 (w3 ) 1 ℘  (w3 ) − ℘  (w2 ) W2 (w1 ) = = =− . 1 2 3 2S1 ℘ (w3 ) − ℘ (w2 )

(31)

If the w j ’s are taken to be integer multiples of the half-periods, then (31) is satisfied for any choice of the vortex strengths, including those with S1 = 0 as mentioned in Sect. 4.1. Two such examples are shown in Fig. 4. Figure 4c shows a rotating equilibrium configuration of identical vortices for a non-trival solution to (31) that forms a hexagonal-like array. Previous considerations of lattice configurations of identical vortices focus on the rectangular and triangular lattices (see, e.g., [11,26]), and the configuration shown here appears to be a new result. It would be interesting to examine the stability of such non-triangular lattices of identical vortices and compare the energy of these configurations with the known results for rectangular and triangular lattices [11].

5 Vortex dynamics in periodic domains with N = 3 The dynamics of three point vortices in either a singly or doubly periodic domain is integrable when S1 = 0 [2,25]. In this case, all of the vortex separations can be written in terms of w3 as w1 = (1 w3 − ) / 3 ,

w2 = (2 w3 + ) / 3 ,

(32)

and the equations of motion (2) can be combined to give  

 

 dw3∗ 1 2   −3 = W p [w3 ] + W p w3 − + Wp w3 + . dt 2πi 3 1 3 2

(33)

This system describing the evolution of w3 = ξ + i η is integrable and can be written in Hamiltonian form as

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(a)

(c)

(b)

Fig. 4 Doubly periodic vortex configurations and representative streamlines for the case V = 0 with ω1 ∈ R and ω2 = i ω1 . Vortex positions are shown with solid circles and are labeled according to strength; stable fixed points are shown with open circles. The vortex separations are, for a and b, w2 = ω2 , w3 = ω1 − ω2 for (1 , 2 , 3 ) = (1, 2, ±3) and, for c, w2 ≈ 0.895 ω2 , w3 ≈ ω1 + 0.552 ω2 .

∂H dξ = , dt ∂η

dη ∂H =− , dt ∂ξ

with the Hamiltonian given by   



 3   2 1 3 3 p [w3 ] + p w3 − + w3 + . H= 2π 1 3 1 2 3 2

(34a)

(34b)

For given values of α and , level curves of H give the relative vortex separation w3 = z 1 − z 2 for a specific initial condition, and thus by (32) the remaining vortex separations. The full motion of the original three vortices can then be determined by numerical integration of one of the equations of motion (2) together with the solution for w3 . The Hamiltonian system in the w3 plane can be interpreted as advection by an arrangement of stationary point vortices, similar to (18) for the N = 2 case. Let  represent the period of the system, so that  = L in the periodic strip and  = mω1 + nω2 (m, n ∈ I) in the periodic parallelogram. The three terms on the right-hand-side of (33) each have period , (1 + 2 ) / 1 , and (1 + 2 ) / 2 , respectively. Let2 γ = 2 / 1 ,

(35)

and assume without any loss of generality that γ ≤ 1. If γ is rational, then there exists a commensurate period on which all three of the terms in (33) are periodic. In this case, γ can be written in lowest terms as γ = p/q, where p and q are integers. A commensurate period for the three terms is found by requiring pq M1 = p( p + q) M2 = q( p + q) M3 ,

(36)

where M1 , M2 , and M3 are integers. This requirement is satisfied by taking M1 = p+q, M2 = q, and M3 = p, so that the commensurate period in w3 -space is ( p + q) . Each of the terms in (33) can now be written in terms of this commensurate period by using the fact that one vortex in a periodic domain is equivalent to superposition of identical vortices in a larger periodic domain. These stationary vortex systems are, in fact, relative equilibrium configurations [2,25], which can be determined by substituting the vortex positions and strengths given below into the equations of motion (2). In the singly periodic domain, this equivalence of different-size domains can be stated as W1 (z; L) =

 M−1  M−1  1  z − mL W1 W1 [z − mL; ML] . ;L = M M m=0

2

(37)

m=0

Note that this definition of γ differs from that in [2,25]; the definition used here makes the analysis somewhat simpler.

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I

VI 2 1

II

−3 b

I

d

III

II

2 1 −3

III

IV

3L

1

IV

VII

2 −3

IV

1 2 −3

L

Fig. 5 Dynamics of three point vortices in a periodic strip with (1 , 2 , 3 ) = (2, 1, −3) and  ≈ L(0.050 + i 1.040)/π. In the w3 -plane representation of the vortex motion (top right), solid lines are separatrices delimiting regimes of motion and dotted lines are streamlines corresponding to the real space motions in panels I–IV. The equilibrium configurations corresponding to stagnation points b and d are shown in Fig. 2. In panels I–IV, vortex trajectories are shown for one period of the relative motion, initial and final locations of the base vortices are marked with solid circles, and the initial positions are labeled according to the vortex strength. In panel IV, initial and final locations of periodic image vortices are shown with open circles.

Note that the residue of W1 (z/M) is M. Using (37), the equation of motion for w3 (33) can be written in the singly periodic domain as ⎧ p+q−1 −3 ⎨  W1 [w3 − m L; ( p + q)L] = dt 2πi ⎩

dw3∗

m=0

+

3 1

q−1  m=0

⎫  ⎬   p−1   − m L 3 3  + m L 3 W 1 w3 − ; ( p + q)L + W 1 w3 + ; ( p + q)L . (38) ⎭ 1 2 2 m=0

This equation can be interpreted as advection of a passive particle at w3 due to the following relative equilibrium configuration of 2( p + q) vortices: p + q vortices with strength −3 at the locations m L , q vortices with strength −32 / 1 at the locations ( − m L 3 )/ 1 , and p vortices with strength −32 / 2 at the locations −( + m L 3 )/ 2 . When identifying base vortices, it is useful to choose the values of m ∈ I so that these vortices are all in the ‘fundamental strip’ 0 ≤ w3 < ( p + q)L. As with N = 2 in Sect. 3, separatrices in the w3 plane delineate different regimes of motion in the original three vortex system. The stagnation points at which the separatrices intersect correspond to relative equilibrium configurations of the original three vortices. An example system is shown in Fig. 5 with 1 = 2 and 2 = 1, for which γ = 1/2. The value of  ≈ L(0.050 + i 1.040)/π was determined numerically by requiring that the stagnation point at b correspond to a relative equilibrium configuration that translates only in the x direction. In this case, the commensurate period in the w3 plane is 3L, there are a total of six vortices per period, and there are 13 different regimes of motion. In seven of these regimes, the three original base vortices move together, with at least two of the vortices orbiting each other, and the relative vortex motion is periodic in time, although there is a net displacement of the configuration; representative examples are shown in Fig. 5 for regimes I–III. In the other six regimes, the base vortices separate, and the time-periodic reoccurrence of the initial base vortex configuration involves periodic images of the original base vortices; a representative example is shown in Fig. 5 for regime IV. In the doubly periodic system, the identity [22]

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ζ (z; ω1 , ω2 ) = Mζ (M z; Mω1 , Mω2 )

(39)

allows the point vortex interaction to be represented in a larger domain using the relationship W2 (z; ω1 , ω2 ) =

M−1 

W2 (z − mn ; Mω1 , Mω2 ) ,

(40)

m,n=0

in which mn = mω1 + nω2 . Substitution of (40) into (33) gives the equation of motion for w3 in a doubly periodic domain as ⎧ p+q−1 dw3∗ −3 ⎨  = W2 [w3 − mn ; ( p + q)ω1 , ( p + q)ω2 ] dt 2πi ⎩ m,n=0

3 + 1 +

3 2

q−1  m,n=0

   − 3 mn W 2 w3 − ; ( p + q)ω1 , ( p + q)ω2 1

p−1  m,n=0

⎫  ⎬  + 3 mn . W 2 w3 + ; ( p + q)ω1 , ( p + q)ω2 ⎭ 2

(41)

Similar to the singly periodic case, this system can be interpreted as advection of a passive particle at w3 due to a relative equilibrium configuration consisting of ( p + q)2 + p 2 + q 2 vortices: ( p + q)2 vortices with strength −3 at the locations mn , q 2 vortices with strength −32 / 1 at the locations ( − 3 mn )/ 1 , and p 2 vortices with strength −32 / 2 at the locations −( + 3 mn )/ 2 . Two example representations of vortex positions and separatrices in w3 -space are shown in Fig. 6 for the doubly periodic case. Taking γ = 1/2 and  = 0 gives a relatively simple system. Some of the w3 -plane vortices coincide, so there are only 12 vortices instead of 14, giving 13 regimes of motion. In every regime, motion along a phase-space trajectory gives |w3 | < 6|ω3 | for all time. Taking a generic value of   = 0 complicates the system, as can be anticipated by the results in Fig. 5 for the singly periodic system; examples in the doubly periodic system with   = 0 can be found in [25]. The system is also complicated by changing the value of γ , as seen by comparing the examples in Fig. 6. Both cases have similar underlying structure, but

(a)

(b)

i

i

Fig. 6 Vortex positions and separatrices in the w3 plane corresponding to the original three-vortex system with ω1 ∈ R, ω2 = √ i ω1 ,  = 0, and the vortex strength ratios (a) γ = 1/2 and (b) γ = 2(1 − 5/3)√≈ 0.51. With γ = 1/2 the w3 -plane representation has spatial half-periods (3ω1 , 3ω2 ), while the irrational value γ = 2(1 − 5/3) gives a spatially aperiodic vortex system in the w3 plane. Both of these phase space representations are symmetric about the lines ξ = 0, η = 0, and ξ = η due to taking  = 0. Reprinted from the journal

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taking γ to be irrational breaks the spatial periodicity in the w3 plane and generates numerous thin regimes of motion that wind around several of the w3 -plane vortices. In most of these thin regimes, the original three base vortices will separate over distances much greater than 6|ω3 | and will interact with multiple image vortices before coming back to their original arrangement. Since  = 0, all of the w3 -plane trajectories in Fig. 6 are closed and the corresponding base vortex systems are always time-periodic. Based on the number of degrees of freedom, it is expected that passive particles advected by the three base vortices in these periodic systems will have chaotic trajectories. A stronger statement about chaotic transport can be based on the occurrence of ‘topological chaos’ [9], or exponential stretching and folding of the fluid that is guaranteed by the topology of the time-dependent vortex interactions. The separation of base vortices across several spatial periods, and the ensuing interactions with image vortices, can produce the necessary non-trivial topology in these spatially periodic systems [10]. Some analysis of topological chaos has been done for the singly periodic domain, but a similar analysis has not yet been conducted for the doubly periodic domain. Furthermore, the effects of this topological chaos on the details of scalar transport and mixing in these vortex systems remains an open question. Acknowledgements This work was supported in part by the National Science Foundation under Grant No. CBET-0442845. I thank Hassan Aref for introducing me to this subject, for providing many years of guidance and ongoing collaboration, and for organizing this enjoyable symposium on vortex dynamics.

References 1. Abo-Shaer, J.R., Raman, C., Vogels, J.M., Ketterle, W.: Observation of vortex vattices in Bose Einstein condensates. Science 292, 476–479 (2001) 2. Aref, H., Stremler, M.A.: On the motion of three point vortices in a periodic strip. J. Fluid Mech. 314, 1–25 (1996) 3. Aref, H., Stremler, M.A.: Point vortex models and the dynamics of strong vortices in the atmosphere and oceans. In: Lumley, J.L. (ed.) Fluid Mechanics and the Environment: Dynamical Approaches, pp. 1–17. Springer Verlag, Lecture Notes in Physics (2001) 4. Aref, H., Rott, N., Thomann, H.: Gröbli’s solution of the three-vortex problem. Ann. Rev. Fluid Mech. 24, 1–20 (1992) 5. Aref, H., Boyland, P.L., Stremler, M.A., Vainchtein, D.L.: Turbulent statistical dynamics of a system of point vortices. In: Gyr, A., Kinzelbach, W., Tsinober, A. (eds.) Fundamental Problematic Issues in Turbulence, pp. 151–161. Birkhäuser– Verlag, Basel (1999) 6. Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., Vainchtein, D.L.: Vortex crystals. Adv. Appl. Mech. 39, 1–79 (2003) 7. Aref, H., Stremler, M.A., Ponta, F.L.: Exotic vortex wakes—point vortex solutions. J. Fluids Struct. 22, 929–940 (2006) 8. Aref, H., Brøns, M., Stremler, M.A.: Bifurcation and instability problems in vortex wakes. J. Phys. Conf. Ser. 64, 012015 (2007) 9. Boyland, P.L., Aref, H., Stremler, M.A.: Topological fluid mechanics of stirring. J. Fluid Mech. 403, 277–304 (2000) 10. Boyland, P.L, Stremler, M.A., Aref, H.: Topological fluid mechanics of point vortex motions. Physica D 175, 69–95 (2003) 11. Campbell, L.J., Doria, M.M., Kadtke, J.B.: Energy of infinite vortex lattices. Phys. Rev. A 39(10), 5436–5439 (1989) 12. Charney, J.G.: Numerical experiments in atmospheric hydrodynamics. In: Experimental Arithmetic, High Speed Computing and Mathematics. Proc. Symp. Appl. Math, vol. 15, pp. 289–310. Am. Math. Soc., Providence, R. I. (1963) 13. Fetter, A.L., Hohenberg, P.C., Pincus, P.: Stability of a lattice of superfluid vortices. Phys. Rev. 147(1), 140–152 (1966) 14. Friedmann, A., Poloubarinova, P.: Über fortschreitende Singularitäten der ebenen Bewegung einer inkompressiblen Flüssigkeit. Recueil de Géophysique, Tome V (Fascicule II, Leningrad), pp. 9–23 (1928) 15. Gröbli, W.: Spezielle Probleme über die Bewegung geradliniger paralleler Wirbelfäden. Zürcher und Furrer, Zürich (1877) Also in: Vierteljschr. Naturf. Ges. Zürich 22, 37–81, 129–165 (1877) 16. Helmholtz, H.: Über die Integrale der hydrodynamischen Gleichungen, welche Wirbelbewegungen entsprechen. J. reine angew. Math. 55, 25–55 (1858). Transl. by P.G. Tait: On integrals of the hydrodynamical equations which express vortexmotion. Phil. Mag. 33, 485–512 (1867) 17. Maue, A.W.: Zur Stabilität der Kármánschen Wirbelstrasse. Z. Angew. Math. Mech. 20, 129–137 (1940) 18. Montaldi, J., Souliére, A., Tokieda, T.: Vortex dynamics on a cylinder. SIAM J. Appl. Dyn. Sys. 2(3), 417–430 (2003) 19. O’Neil, K.A.: Symmetric configurations of vortices. Phys. Lett. A 124(9), 503–507 (1987) 20. O’Neil, K.A.: On the Hamiltonian dynamics of vortex lattices. J. Math. Phys. 30(6), 1373–1379 (1989) 21. O’Neil, K.A.: Continuous parametric families of stationary and translating periodic point vortex configurations. J. Fluid Mech. 591, 393–411 (2007) 22. Sansone, G., Gerretsen, J.: Lectures on the theory of functions of a complex variable. P. Noordhoff, Groningen (1960) 23. Stremler, M.A.: Relative equilibria of singly periodic point vortex arrays. Phys. Fluids 15(12), 3767–3775 (2003) 24. Stremler, M.A.: Evaluation of phase-modulated lattice sums. J. Math. Phys. 45, 3584–3589 (2004) 25. Stremler, M.A., Aref, H.: Motion of three point vortices in a periodic parallelogram. J. Fluid Mech. 392, 101–128 (1999) 26. Tkachenko, V.K.: On vortex lattices. Sov. Phys. JETP 22, 1282–1286 (1966) 27. Tkachenko, V.K.: Stability of vortex lattices. Sov. Phys. JETP 23, 1049 (1966) 28. von Kármán, T.: Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfärt. 1. Teil. Nachr. Ges. Wiss. Göttingen. Math.-Phys. Kl., 509–517 (1911). Reprinted in: Collected works of Theodore von Kármán, vol.1, pp. 324–330. Butterworth, London (1956)

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29. von Kármán, T.: Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfärt. 2. Teil. Nachr. Ges. Wiss. Göttingen. Math.-Phys. Kl., 547–556 (1912). Reprinted in: Collected works of Theodore von Kármán, vol. 1, pp. 331–338. Butterworth, London (1956) 30. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis (4th ed.). Cambridge University Press, Cambridge (1927) 31. Williamson, C.H.K., Roshko, A.: Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2, 355–381 (1988) 32. Yarmchuck, E.J., Gordon, M.J.V., Packard, R.: Observation of stationary vortex arrays in rotating superfluid Helium. Phys. Rev. Lett. 43, 214–217 (1979)

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Theor. Comput. Fluid Dyn. (2010) 24:39–44 DOI 10.1007/s00162-009-0106-9

O R I G I NA L A RT I C L E

Kevin A. O’Neil

Collapse and concentration of vortex sheets in two-dimensional flow

Received: 12 November 2008 / Accepted: 6 January 2009 / Published online: 15 May 2009 © Springer-Verlag 2009

Abstract Numerical evidence is presented for the existence of collapse configurations of vortex sheets (onedimensional singular distributions of vorticity) in a two-dimensional ideal fluid. Point vortices are used to approximate the vortex sheets. These and related motions cause a significant concentration of vorticity, with possible relevance to the concentration seen in the evolution of turbulent flows. Keywords Vortex sheet · Point vortex · Relative equilibrium · Periodic solution PACS 47.32.C · 47.15.ki · 47.27.De

0 Introduction Singular distributions of vorticity provide good models of two-dimensional fluid flow in some circumstances, despite being only weak solutions to the Euler equations of fluid motion. These singular distributions take the form of point vortices and vortex sheets. In a rotating superfluid, for example, coherent vortices emerge that are essentially identical point vortices, and energy-minimizing configurations of such vortices have been observed. These are termed relative equilibria, because the vortex velocities vanish in a certain rotating reference frame [2,13]. Similarly the evolution of a periodic vortex sheet, through its development of a curvature singularity and subsequent roll-up, has been used as a model of the roll-up of a thin shear layer. Moving in the other direction, certain point vortex relative equilibrium configurations have been proved to have analogs in nonsingular vortex patch configurations. Since point vortices form a finite-dimensional dynamical system, they are more accessible to study than vortex sheets. There is now an extensive literature on existence and forms of relative equilibria of identical point vortices [1]. For certain systems with circulations of both signs, there are also unusual collapse configurations that have been shown to exist on the plane, sphere, cylinder and torus [1,6,8–11]. The question may be posed: are there collapsing configurations of vortex sheets? This paper gives preliminary evidence supporting an affirmative answer to the question, in that several collapsing configurations of vortex sheets and point vortices are approximated numerically. The vortex sheets are replaced by arrays of many (e.g. n = 100) identical point vortices; the convergence of the configurations with n increasing from 50 to 300 is good. Some aspects of these configurations, and their possible relevance to the evolution of turbulent flows, are also discussed. Communicated by H. Aref K. A. O’Neil (B) Department of Mathematical and Computer Sciences, The University of Tulsa, 800 S. Tucker Dr., Tulsa, OK 74104, USA E-mail: [email protected] Reprinted from the journal

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1 Periodic motion of vortex sheets We begin with a brief review of the governing equations of motion. A vortex sheet is a distribution of singular vorticity along a curve in the fluid plane. The simple smooth regular curve may be parametrized as z(s), with parameter s ∈ (0, 1), and assumed to have derivative z  (s) bounded away from zero so that the curvature is bounded. Let  denote the total circulation of the sheet, distributed along the curve with linear density /|z  |. The fluid velocity induced by this distribution is discontinuous along the curve. The motion of the vorticity itself, the average of the velocity limits on either side of the curve, is given by the Birkhoff–Rott equation [12]: d 2πi z¯ (s) = p.v. dt

1 0

 dσ. z(s) − z(σ )

(1)

The vortex sheet will rotate uniformly around the origin with angular velocity ω > 0—forming a relative equilibrium—when the velocity of the vorticity at z(s) is iωz(s), i.e. when the relation 1 0 = 2π ω¯ ¯ z (s) − p.v. 0

 dσ z(s) − z(σ )

(2)

1 is satisfied for all s. Integrate (2) over s to obtain the necessary condition 0 = ω 0 z(s) ds, showing that the center of vorticity must be at the center of rotation. Multiply both sides of (2) by z and integrate to get 1 a second necessary condition 4πω 0 |z|2 ds =  2 ; this shows the connection between the rotation rate and the scale of the distribution. A uniform distribution of vorticity along a circle centered at 0 satisfies (2), as does the elliptic distribution of vorticity on a line segment (−a, a) that is obtained as the limit of a Kirchhoff elliptic vortex patch when width is reduced to zero. These are the only solutions to (2) that have appeared in the literature. Equation (2) is easily modified to accommodate vorticity that is distributed among several vortex sheets and/or point vortices. Given M vortex sheets parametrized by nonintersecting simple curves z 1 (s), . . . , z M (s) and total circulations 1 , . . . ,  M , as well as point vortices of circulations γk at positions ζk , 1 ≤ k ≤ m, the functional equations satisfied by a relative equilibrium configuration of this vorticity distribution are 0 = 2π ω¯ ¯ z j (s) −



1 p.v.

k

0 = 2π ω¯ ζ¯ j −

0

1  k

0

 k γk dσ − z j (s) − z k (σ ) z j (s) − ζk

(3a)

k

 γk k . dσ − ζ j − z k (σ ) ζ j − ζk

(3b)

k = j

(Note that only the integral in (3a) having j = k is singular.) The associated necessary conditions (assuming ω  = 0) are 0=

 j

1

 j z j (s) ds +



γjζj

(4a)

j

0

⎞ ⎛ 1    2    j |z j (s)|2 ds + γ j |ζ j |2 ⎠ = γj − γ j2 4πω ⎝ j +

(4b)

0

2 Collapse configurations of vortex sheets It is well known that point vortices can be arranged into configurations that collapse (have a collision singularity) in finite time [1,8]. A necessary condition is that the total circulation m 1 and the sum of squared circulations m 2 satisfy m 21 = m 2 . These configurations are similar to relative equilibria in that the ratio of velocity to position has the same value iω for all vortices. The difference is that ω = α + iβ has nonzero

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Fig. 1 A collapsing configuration of one point vortex with circulation −1 (disk) and one positive vortex sheet with total circulation 2; the sheet has been discretized into 100 point vortices of circulation 2/99 (circles) (the linear vorticity density may be deduced from the spacing of the point vortices along the sheet). The collapse parameter is ω = 1 + 0.5i. The center of vorticity is the origin, indicated by a cross

imaginary part, with β/α > 0. Thus one component of the velocity is directed toward the center of the configuration, causing the size of the configuration to be proportional to (tc − t)1/2 , with collision occurring at time tc . Individual vortices move along logarithmic spirals: letting (r, θ ) be the polar coordinates of one of the vortices, one has the relation  β r (t)/r (t0 ) = exp − (θ (t) − θ (t0 )) . (5) α The question may be asked, can vortex sheets and point vortices also be arranged into collapsing configurations? That is, can solutions be found to Eqs. (3) with ω = α + iβ and β/α > 0? Equations (4) are still necessary conditions; since the right side of (4b) is real, two requirements for such a configuration are   2  0= j + γj − γ j2 (6a) 0=



1

 j |z j (s)|2 ds +



γ j |ζ j |2

(6b)

0

The first condition involves circulations only; in the simplest case of one vortex sheet and one point vortex, it reduces to 1 /γ1 = −2. Solutions to Eqs. (3) with nonreal ω can be investigated numerically by discretizing the integrals; this amounts to approximating the one-dimensional vorticity distribution on each vortex sheet by a curvilinear array of point vortices. The difficulties posed by the singularity of the integrals responsible for each sheet’s self-induced motion can be minimized by a symmetric discretization [4,7,14,15], which amounts to making the point vortices have equal circulations. Thus a collapse configuration of point vortices and vortex sheets can be approximated by a collapse configuration of point vortices. Equations (3a) and (3b) are replaced by (3b) with no integrals at all. These equations can be solved numerically by a simple scheme based on Newton’s method. The initial configuration used should resemble the presumed collapse configuration, with the many identical weak point vortices closely spaced along a curve or curves and the strong point vortices well separated from them. As the solution is approached, the matrix of partial derivatives used in Newton’s method will develop a small singular value due to the invariance of solutions to (3b) under rotations around the origin, ζ j  → ζ j eiθ . The inversion of the matrix for the last few iterations may be accomplished through the use of a truncated singular value decomposition. This method was used for all the examples presented below; for each configuration the vortex Reprinted from the journal

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Fig. 2 Same as Fig. 1 except β/α equal to 0.3 (top) and 0.1 (bottom)

positions all satisfy Eqs. (3b) with a residual not greater than 10−15 . In all cases the solutions were found to exhibit reasonable convergence as the number of point vortices used to approximate each sheet was increased. In Fig. 1, a collapse configuration with one sheet and one point vortex is shown. The point vortex has circulation −1, and by condition (6a) the sheet therefore must have circulation 2. The sheet is approximated here by 100 weak point vortices, so that the collapse equations are satisfied by all 101 point vortices in the approximating configuration. Because of the necessary condition m 21 = m 2 for point vortex collapse, each weak point vortex in the sheet has circulation 2/99—the approximating array has slightly greater circulation than the sheet it is approximating. The collapse parameter ω here has the value 1 + 0.5i, and the center of vorticity (the site of the eventual collapse) is marked by a cross. The linear vorticity density of the sheet can be seen to vary, as the point vortices are less closely spaced near the ends of the sheet. Figure 2 shows the same vortex system in collapse configurations with different values of the collapse parameter; the value ω = 1 + 0i yields a relative equilibrium configuration (not shown) that has a reflection symmetry. More complex configurations that collapse in finite time also exist. Figure 3 shows two configurations having two point vortices. In the upper configuration, there is a single vortex sheet, approximated by 100

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Collapse and concentration of vortex sheets in two-dimensional flow

Fig. 3 Collapse configurations with two negative point vortices. Top rapidly collapsing single-sheet configuration with β/α = 1.0; point vortices have circulation −1, −0.98. Bottom an asymmetric collapse configuration of two sheets; β/α = 0.4; point vortex circulations −0.8 (right), −1.0 (left)

point vortices. The value of ω is 1 + i, so the component of velocity directed toward the origin is as large as the rotational component. This configuration can be imagined to be the result of joining together two of the configurations shown in the first figure. The lower figure shows another way to “combine” two collapse configurations to obtain a collapse. The two sheets (with 50 point vortices in each) have the same circulation, even though the two point vortices do not. Many other compound collapse configurations can be found. 3 Discussion Among the different types of vorticity distributions, vortex sheets are unique in that the magnitude of the linear vorticity density can change as the fluid flow evolves in time. For very weak vortex sheets, the motion of the vorticity is mainly determined by the external flow, and the vorticity may be concentrated (the sheet shortened) in the same was as a linear distribution of passive markers in the flow may be concentrated by shear. For instance, a weak vortex sheet near a strong point vortex will be shortened or lengthened depending on the orientation of the sheet relative to the point vortex. Any concentration is quickly followed by dilution however as the sheet is rotated by the flow. The novel aspect of the configurations discussed above is that the circulation of the vortex sheet is strong enough to keep the point vortex in its “proper” location for continuing collapse. The speed at which this concentration takes place is also surprising. As noted in Eq. (5), the collapse configurations revolve around the center of vorticity following logarithmic spirals. If the ratio β/α has the value 0.45 then in the time it takes the configuration to revolve 90◦ around the origin, the vortex sheet is compressed to half its original length and the vorticity density doubles (the total circulation of the sheet stays the same, of course). For ratio β/α = 0.7 and 0.9, the concentration factor is 3 and 4, respectively! The configuration shown in Fig. 3 with β/α = 1.0 (top) would, in theory, disappear very quickly. What is the relevance of these configurations to nonsingular vorticity distributions? When the vorticity is nonsingular, no actual concentration can occur so that collapse configurations are impossible. Nonetheless one can get arbitrarily good approximations to them, that (at least initially) display some of the properties of a collapse. For example, collapse configurations of three point vortices have been approximated by vortex patches and used as initial conditions for flow that induces vortex merger [5]. It would be interesting to search for the occasional spontaneous appearance of an approximate sheet collapse configuration in turbulent flows, where filamentary vorticity is always present. These configurations could be a mechanism for pushing vorticity into a smaller region that would both cancel the negative vorticity and roll up the remaining positive into a Reprinted from the journal

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coherent vortex. Another interesting but unknown aspect is the interaction between the collapse motion and the inherent sheet instability, which may be made more severe by the shear from the accompanying negative vortex [3,5]. References 1. Aref, H., Newton, P., Stremler, M., Tokieda, T., Vainchtein, D.: Vortex crystals. Adv. Appl. Mech. 39, 1–79 (2002) 2. Campbell, L., Ziff, R.: Vortex patterns and energies in a rotating superfluid. Phys. Rev. B 20, 1886–1902 (1979) 3. Elhmaidi, D., Provenzale, A., Lili, T., Babiano, A.: Stability of two-dimensional vorticity filaments. Phys. Lett. A 333, 85–90 (2004) 4. Hou, T., Lowengrub, J., Krasney, R.: Convergence of a point vortex method for vortex sheets. SIAM J. Numer. Anal. 28, 308–320 (1991) 5. Kevlahan, N.K.-R., Farge, M.: Vorticity filaments in two-dimensional turbulence: creation, stability and effect. J. Fluid Mech. 346, 49–76 (1997) 6. Kidambi, R., Newton, P.: Motion of three point vortices on a sphere. Phys. D 116, 143–175 (1998) 7. Liu, J.-G., Xin, Z.: Convergence of the point vortex method for 2D vortex sheet. Math. Comput. 70, 595–606 (2000) 8. Novikov, E., Sedov, B.: Vortex collapse. Sov. Phys. JETP 22, 297–301 (1979) 9. O’Neil, K.: Stationary configurations of point vortices. Trans. Am. Math. Soc. 302, 383–425 (1987) 10. O’Neil, K.: Collapse of point vortex lattices. Phys. D 37, 531–538 (1989) 11. O’Neil, K.: Relative equilibrium and collapse configurations of four point vortices. Regul Chaotic Dyn 12, 117–126 (2007) 12. Saffman, P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992) 13. Thuneberg, E.V.: Introduction to the vortex sheet of superfluid 3 He-A. Phys. B 210, 287–299 (1995) 14. Van de Vooren, A.: A numerical investigation of the rolling-up of vortex sheets. Proc. R. Soc. Lond. Ser. A 373, 67–91 (1980) 15. Wu, S.: Mathematical analysis of vortex sheets. Commun. Pure Appl. Math. 59, 1065–1206 (2005)

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Theor. Comput. Fluid Dyn. (2010) 24:45–50 DOI 10.1007/s00162-009-0174-x

O R I G I NA L A RT I C L E

Scott David Kelly · Hailong Xiong

Self-propulsion of a free hydrofoil with localized discrete vortex shedding: analytical modeling and simulation

Received: 10 November 2008 / Accepted: 24 September 2009 / Published online: 2 March 2010 © Springer-Verlag 2010

Abstract We present a model for the self-propulsion of a free deforming hydrofoil in a planar ideal fluid. We begin with the equations of motion for a deforming foil interacting with a pre-existing system of point vortices and demonstrate that these equations possess a Hamiltonian structure. We add a mechanism by which new vortices can be shed from the trailing edge of the foil according to a time-periodic Kutta condition, imparting thrust to the foil such that the total impulse in the system is conserved. Simulation of the resulting equations reveals at least qualitative agreement with the observed dynamics of fishlike locomotion. We conclude by comparing the energetic properties of two distinct turning gaits for a free Joukowski foil with varying camber. Keywords Aquatic locomotion · Noncanonical Hamiltonian systems

1 Introduction Recent study by the authors and others has addressed two complementary problems in fluid-body interactions from the standpoint of analytical mechanics. The self-propulsion of a deformable body in an ideal fluid devoid of vorticity is treated as a problem in Lagrangian mechanics in [1–3] and related articles. Parallels between this problem and locomotion at low Reynolds number are detailed in [2,3]; the addition of liftlike forces due to circulation is described in [4]. Hamiltonian models for the interactions of free rigid bodies with discrete vortex structures are presented, meanwhile, in [5–9]. In the present article, a summary of work detailed in [10], we merge these lines of research to provide a model for the self-propulsion of a deformable body in a planar ideal fluid—specifically, a hydrofoil defined by a time-varying conformal map—which is able to shed vorticity discretely from a single point on its surface in accordance with a periodically applied Kutta condition. The shedding of each vortex is accompanied by the application of an impulsive force to the hydrofoil to conserve the total impulse in the system. Between vortex shedding events, the equations of motion possess a Hamiltonian structure which extends that underpinning the interaction of a free rigid body with a system of vortices. Computational experiments with this model demonstrate its qualitative fidelity to biological and robotic fishlike locomotion. Communicated by H. Aref S. D. Kelly(B) Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, Charlotte, NC, USA E-mail: [email protected] H. Xiong Quantitative Risk Management Inc., Chicago, IL, USA E-mail: [email protected] Reprinted from the journal

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Undulatory fishlike locomotion has inspired designs for manmade aquatic vehicles since the eighteenth century or before [11], yet the control of such devices remains largely an empirical science due to the scarcity of mathematical models that exhibit sufficient fidelity to the relevant physics, under a sufficient diversity of circumstances, without exceeding the formal scope of model-based control theory. High-fidelity models for fishlike locomotion are almost universally computational in nature; contemporary approaches to control design require analytical models. The present article addresses this divide by framing a mechanism central to fishlike locomotion—the development of thrust through localized vortex shedding from the caudal fin—in a Hamiltonian setting. Although rooted in conservative mechanics, the model described herein embodies the same principle of momentum balance that governs aquatic locomotion in a truly viscous fluid, and illuminates the structure of propulsive wakes corresponding to diverse swimming gaits. 2 Modeling We model the contour of a hydrofoil with time-varying shape as the image of a circle in the complex plane under a conformal map z = x + i y = F(ζ ) with time-varying parameters s j . We require the area within the foil to remain constant in time to avoid an infinite term in the kinetic energy of the resulting fluid–foil system. We express the dynamics of the moving foil relative to the foil-fixed z-frame. In between vortex shedding events, the flow resulting from the motion of the foil and vortices—assuming the fluid to be at rest infinitely far away, and excluding small domains around the vortices themselves—is determined by a potential function of the form   W (z) = w(ζ ) = U w1 (ζ ) + V w2 (ζ ) + w3 (ζ ) + s˙ j ws j (ζ ) + wvk (ζ ), (1) j

k

where U and V are the x- and y-components of the foil’s translational velocity,  is its angular velocity, and the complex potentials wvk represent the contributions of the vortices to the flow. Relative to the foil-fixed frame, the total linear and angular impulse in the fluid–foil system may be expressed in the form ⎡ ⎤ Lx  ⎣ L y ⎦ = I (s)ξ + B(s)˙s + γk K k (s, z k ), (2) P k where s is the vector of shape parameters for the foil, γk and z k are the strength and location of the kth vortex, I (s) and B(s) are shape-dependent matrices with appropriate dimensions, and ξ = [U V ]T . The motion of the foil is governed by Kirchhoff’s equations   d dP ¯ + × L = 0, + U × L = 0, (3) dt dt ¯ = [0 0 ]T , L = [L x L y 0]T , P = [0 0 P]T , and U = [U V 0]T . The motion of the kth vortex is where  determined using Routh’s rule [12] such that ⎞ ⎛  ∂F dW 1 k ζ˙k = ⎝ s˙ j ⎠  − (U + i V + iz k ) − , (4) dz ∂s j F (ζk ) j

where Wk (z) = W (z) − iγk log(z − z k ). Observing that ⎡ ⎤ ⎡ ⎤  

 yk Lx )ds l × (n × ∇φ v  2 ⎦ −xk I (s)ξ = ⎣ L y ⎦ − B(s)˙s − (−2πγk ) ⎣ − 1 (n × ∇φ )ds − l 1 v 2 2 2 P − 2 (xk + yk ) k

(5)

in the notation of [7], we may define the Hamiltonian function H (L x , L y , P, x1 , y1 , . . . , x N , y N ) =

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1 T ξ I (s)ξ − 2π H1 2

(6) Reprinted from the journal

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when N wake vortices are present, where      1  2   H1 = γk s˙ j ψs j (xk , yk ) − γk log ζk ζk − rc2  + log  F  (ζk ) 2 k

+

j

1  2

k

k

γk γ j

    log ζk − ζ j  − log ζk ζ j − rc2  .



(7)

j =k

If μ = [L x L y P]T , then the motion of the foil is governed in between vortex shedding events by the Lie– Poisson equations μ˙ = ad∗δ H/δμ μ

(8)

on se∗ (2) [13], while the positions of the vortices evolve such that (−2πγk )

dxk dyk ∂H ∂H , (−2πγk ) . = =− dt ∂ yk dt ∂ xk

(9)

We introduce new vortices to the wake near the trailing edge of the foil in simultaneous accord with the conservation of impulse and the Kutta condition. This leaves flexibility in the way in which we select the (coupled) position and strength of each new vortex. We provide a comparison of different methods for doing so in [10], and settle on a method described in [14] for the present discussion. According to this method, we situate each new vortex along a contour interpolated from the trailing edge of the foil to the current position of the last vortex shed, and determine the strength of each new vortex as a function of its initial position. Vortices shed at different times may have different strengths; the strength of each shed vortex is assumed to remain constant thereafter. Enforcing the constraints described above, we introduce each vortex with strength γk at the image of the point ζk such that  dw  = 0, (10) dζ ζ =ζT where ζT is the pre-image of the trailing edge of the foil under the transformation defining the foil’s shape, and such that I (s) ξ + γk K k (s, z k ) = 0,

(11)

where ξ refers to the impulsive change in the foil’s linear and angular velocity as a result of shedding. We note that this method for introducing new vortices relies on the fact that the complex transformation defining the foil’s shape has zero derivative at a particular point on the foil. We also note that shed vortices are guaranteed to appear outside the foil, and not inside, provided the contour of the foil is not deformed to such an extreme that it crosses itself. 3 Simulation Figure 1 shows results from two different simulations based on our model. The top row portrays the acceleration from rest of a von Mises foil undulating according to periodic variations in two shape parameters which, roughly speaking, prescribe the cyclic propagation of traveling waves along the length of the foil from front to back. Wake vortices are depicted in the snapshot on the left as tiny colored circles; red circles correspond to clockwise vortices and blue circles to counterclockwise vortices. We observe the roll-up of wake vorticity into staggered coherent structures; over time, these assume the form of an inverse Kármán vortex street, consistent with experimental observations of the wakes trailing oscillating foils [15]. The plot on the right compares the predicted x- and y-displacement of the foil-fixed frame over time to that which would be measured were the mechanism for vortex shedding disabled (“ns”), underscoring the role played by vortex shedding in thrust development. The bottom row depicts the execution of a snap turn by a Joukowski foil, the camber of which is varied rapidly about zero through a single sinusoidal period, providing sufficient momentum for the foil to coast in an oblique direction thereafter. In the context of fishlike robotic locomotion, our model provides a platform not only for algorithmic motion planning but also for evaluating the energetics of individual maneuvers. The agility, energy efficiency, Reprinted from the journal

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2

1.5

1

0.5

0

−0.5 0

1

2

3

4

5

6

7

8

9

10

Fig. 1 Top left: An undulating von Mises foil sheds vortices as it accelerates from rest. Top right: The displacements of the same foil in the x and y directions as functions of time, contrasted with the displacements that would be achieved if the mechanism for vortex shedding were disabled (“ns”). Bottom row: A Joukowski foil executes a snap turn via an aggressive change in camber

and stealth of marine animals all motivate the study of biological locomotion, but the relationships among these performance metrics are nontrivial. We illustrate a few key distinctions here by contrasting two different turning gaits for a free Joukowski foil with a single internal degree of freedom. The top two rows of Fig. 2 depict initial, intermediate, and final frames from simulations of these two gaits; the corresponding trajectories of the origin of the foil-fixed frame are depicted at the lower left. The first turn results from eight periodic oscillations in camber superposed with a nonzero bias, the second from two abrupt flicks of the tail to the foil’s left. The Joukowski profile depicted in Fig. 2 is realized as the image of a circle with radius rc in the ζ plane under a transformation of the form z = ζ + ζc +

a2 , (ζ + ζc )

(12)

where rc , ζc , and a are chosen to vary with time, parametrized by a single shape parameter s(t), to realize area-preserving variations in camber. Neglecting the infinite self-energy of the vortices, the total kinetic energy in the system admits the decomposition KE = KEdue to body motion + KEdue to vortices , where KEdue to body motion and KEdue to vortices = −

 k



+π

k

123

 

1  T T I B ξ = ξ s˙ T C s˙ B 2

(14)

       πγk γ j log ζk − ζ j  + πγk γ j log ζk − ζ j∗ 

j =k



(13)

γk

 



γk log

k

64

|ζk | . rc

k

j

(15) Reprinted from the journal

Self-propulsion of a free hydrofoil

2.5

60

2

50

1.5

40

1

30

0.5

20

0

10 0

−0.5 0

0.5

1

1.5

2

2.5

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3.5

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Fig. 2 Top two rows: Initial, intermediate, and final snapshots of different turning gaits for a Joukowski foil. Bottom left: The trajectories determined in the z plane by these two gaits. Bottom right: The fluid energy as a function of time for these two gaits

The plot at the lower right of Fig. 2 depicts the fluid energy—excluding the self-energy of individual vortices—as a function of time through the two different turns shown above. The control effort  1 tfinal E= s˙ (t)2 dt (16) 2 tinitial associated with each of these maneuvers, which may be understood as a measure of economy of motion, is roughly 50% greater in the case of the second turn, which is completed in three-eighths the time required for the first. The energy imparted to the fluid during the second turn, however, is significantly greater. The first turn would arguably be the better choice for a robotic vehicle attempting to escape acoustic detection, the latter the better choice for retreat after being detected. Acknowledgments Support for this study was provided in part by NSF grants CMMI 04-49319 and ECCS 05-01407.

References 1. Kanso, E., Marsden, J.E., Rowley, C.W., Melli-Huber, J.B.: Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15, 255–289 (2005) 2. Kelly, S.D.: The mechanics and control of robotic locomotion with applications to aquatic vehicles. Ph.D. thesis, California Institute of Technology (1998) Reprinted from the journal

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3. Kelly, S.D., Murray, R.M.: The geometry and control of dissipative systems. In: Proceedings of the IEEE Control and Decision Conference (1996) 4. Kelly, S.D., Hukkeri, R.B.: Mechanics, dynamics, and control of a single-input aquatic vehicle with variable coefficient of lift. IEEE Trans. Robot. 22(6), 1254–1264 (2006) 5. Shashikanth, B.N.: Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and N point vortices: the case of arbitrary smooth cylinder shapes. Regul. Chaotic Dyn. 10(1), 1–14 (2005) 6. Borisov, A.V., Mamaev, I.S., Ramodanov, S.M.: Motion of a circular cylinder and n point vortices in an ideal fluid. Regul. Chaotic Dyn. 8(4), 449–462 (2003) 7. Shashikanth, B.N., Marsden, J.E., Burdick, J.W., Kelly, S.D.: The Hamiltonian structure of a 2-D rigid circular cylinder interacting dynamically with N point vortices. Phys. Fluids 14, 1214–1227 (2002) 8. Shashikanth, B.N., Sheshmani, A., Kelly, S.D., Marsden, J.E.: Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape. Theor. Comput. Fluid Dyn. 22(1), 37–64 (2008) 9. Shashikanth, B.N., Sheshmani, A., Kelly, S.D., Wei, M.: Hamiltonian Structure and dynamics of a neutrally buoyant rigid sphere interacting with thin vortex rings. J. Math. Fluid Mech. (2008). doi:10.1007/s00021-008-0291-0 10. Xiong, H.: Geometric mechanics, ideal hydrodynamics, and the locomotion of planar shape-changing aquatic vehicles. Ph.D. thesis, University of Illinois at Urbana-Champaign (2007) 11. Richie, J.: Weapons: Designing the Tools of War. Oliver Press, Minneapolis, MN (2000) 12. Saffman, P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992) 13. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer, New York (1999) 14. Streitlien, K., Triantafyllou, M.S.: Force and moment on a Joukowski profile in the presence of point vortices. AIAA J. 33(4), 603–610 (1995) 15. Triantafyllou, M.S., Triantafyllou, G.S.: An efficient swimming machine. Sci. Am. 272, 64–70 (1995)

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Theor. Comput. Fluid Dyn. (2010) 24:51–57 DOI 10.1007/s00162-009-0141-6

S P E C I A L I S S U E O R I G I NA L A RT I C L E

Milton da Costa Lopes Filho · Helena J. Nussenzveig Lopes

Vortex dynamics on a domain with holes

Received: 26 October 2008 / Accepted: 2 July 2009 / Published online: 30 August 2009 © Springer-Verlag 2009

Abstract We explore the relationship between the hydrodynamic Green’s function and the Dirichlet Green’s function in a bounded domain with holes. This relationship is expressed using the harmonic measures on the domain, following the work of Flucher and Gustafsson (Vortex motion in two-dimensional hydrodynamics, energy renormalization and stability of vortex pairs, TRITA preprint series, 1997). The explicit form of this relation expresses velocity in terms of vorticity in a way which turns out to be very convenient, especially for analysis. We explain the advantages and describe some applications. Keywords Green’s function · Hydrodynamics · Ideal flow · Vorticity PACS 47.32.C 1 Introduction Two-dimensional vortex dynamics amounts to the coupling of the vorticity transport equation, or Helmholtz Law, with the Biot–Savart law appropriate for the domain under consideration. For topologically complicated domains, such as a bounded domain in the plane with several holes, difficulties appear in writing the Biot–Savart law. In 1941, Lin gave an answer to this problem, by using a modified Green’s function, which was later called hydrodynamic Green’s function by Flucher and Gustafsson (see [5,11]). In his work, Lin used the hydrodynamic Green’s function to construct vortex dynamics, relying on previous work by Koebe (see [10]), for existence and uniqueness and proving symmetry and conformal invariance properties. However, Lin’s construction is abstract, giving little clue on how to compute, even approximately, in concrete examples. More recently, two approaches have become available regarding the explicit treatment of vortex dynamics in topologically complicated domains. The first appeared in an unpublished paper by Flucher and Gustafsson (see [5]), and it amounts to a way to express the hydrodynamic Green’s function in terms of the usual Green’s function of the domain. The second approach has appeared in a series of recent articles by Crowdy and Marshall, especially [2,3], and it is based on a special function associated with circle domains and conformal mappings. Our purpose in this paper is to present a simplified formulation of Flucher and Gustafsson’s approach, describe a pair of applications, and draw a brief comparison between the two approaches. We begin with a precise description of both Green’s functions. Let  be a bounded domain in the plane, with smooth boundary ∂, consisting of the simply connected bounded domain with boundary 0 , with simply Communicated by H. Aref M. da Costa Lopes Filho’s research supported in part by CNPq grant #303.301/2007-4 and H. J. Nussenzveig Lopes’s research supported in part by CNPq grant #302.214/2004-6. M. d. C. Lopes Filho (B) · H. J. Nussenzveig Lopes IMECC, Universidade Estadual de Campinas-UNICAMP, Caixa Postal 6065, Campinas, SP 13083-970, Brazil E-mail: [email protected] E-mail: [email protected] Reprinted from the journal

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connected bounded domains with boundaries i , i = 1, . . . , n removed. In particular, this means that ∂ is the disjoint union of the i , i = 0, 1, . . . , n. Our point of departure is to consider the Dirichlet Green’s function G = G(x, y), defined on the closure of  × , as the unique solution of the equation  x G = δ(x − y) in  (1) G=0 on ∂, for each fixed y ∈ . We call a vector field which is divergence-free, curl-free and tangent to ∂ a harmonic vector field. The harmonic vector fields form a vector space, which we denote by H . A classical result, called Hodge Theorem implies that the vector space H has dimension  n, the number of holes in . The basic idea behind Hodge Theorem is that the Dirichlet functional X  →  |X |2 dx is strictly convex when restricted to the affine space of divergence-free vector fields tangent to ∂, with prescribed circulation on inner boundary components. An instance of Kelvin’s variational principle implies that minimizers of the Dirichlet functional must be harmonic, and strict convexity implies uniqueness of these minimizers, giving precisely one harmonic vector field in each prescribed circulation class. In fact, Hodge Theorem is much more general, applying to spaces of differential forms on arbitrary manifolds. The reader may consult [1,6] for details. We introduce the vector fields X i and i = 1, . . . , n, as the unique divergence-free, curl-free vector fields tangent to the boundary of  such that  X j · ds = δi j , i, j = 1, . . . , n, i

where δi j is the Kroneker delta symbol, equal to 1 if i = j, zero otherwise. These vector fields form a basis of H . We will also introduce the harmonic measures φi , as the unique solutions of the boundary value problem: ⎧ ⎨ φi = 0 φi = 0 on  j , j  = i . (2) ⎩φ = 1 on i i We denote the Biot–Savart operator K by K = K [ω] ≡ ∇ ⊥ ψ, with  ψ(x) = G(x, y)ω(y)dy, 

and (a, b)⊥ = (−b, a). Given smooth, compactly supported vorticity ω in , any velocity u which is divergence-free, tangent to ∂ and has curl equal to ω can be uniquely written in the form u = K [ω] +

n

ai X i ,

(3)

i=1

where a i are constants. This statement is an application of Hodge Theory to this simple situation, and it amounts to using Hodge’s Theorem, together with basic theory of Laplace’s equation on a bounded domain, see the Hodge–Kodaïra Decomposition Theorem in [1]. Given a smooth initial velocity u 0 in , there exists a unique smooth solution of the incompressible Euler equations u = u(x, t) in , defined for all time, with initial velocity u 0 . This is a classical result of Kato (see [9]). Using (3), we can write u(x, t) = K [ω(·, t)](x) +

n

a i (t)X i .

i=1

The reduction of Euler equations to vortex dynamics takes place by taking the curl of the conservation of momentum equations, arriving at the transport equation ωt + u · ∇ω = 0. If  is simply connected, the

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reduction is finished by coupling this equation with the expression u = K [ω]. However, if  is not simply connected, we require, in addition, some way of evolving the harmonic part of the flow, expressed by the {a i }. Let us now introduce the hydrodynamic Green’s function. For each y ∈  fixed, let G h = G h (x, y) be the unique solution of the boundary value problem: ⎧ x G h = δ(x − y) in  ⎪ ⎪ ⎨ G = α (y) on i , i = 1, . . . n h i (4) G = 0 on 0 h ⎪ ⎪ ⎩  ∇ ⊥ G ds = 0, for i = 1, . . . , n i

h

In the problem above, the (y-dependent) constants αi are like Lagrange multipliers, unknown a priori. We refer the reader to [5] for a proof of Koebe’s and Lin’s results on existence, uniqueness, and symmetry of G h . In analogy to the Dirichlet case, for ω smooth and compactly supported in , we introduce a hydrodynamic Biot–Savart operator K h = K h [ω] = ∇ ⊥ ψ, with  ψ(x) = G h (x, y)ω(y)dy. 

It is easy to see that the condition about the circulation of ∇ ⊥ G h on the boundary components implies that K h [ω] has vanishing circulation on i , for i = 1, . . . , n. If we choose an initial velocity u 0 , with initial vorticity curl u 0 = ω0 , we can write u 0 = K h [ω0 ] +

n

γi X i ,

i=1

but now γi is precisely the circulation of u 0 around i . Therefore, if u is the solution of the Euler equations with initial velocity u 0 and ω its vorticity, we can write, for each time t > 0, u = K h [ω] + γi X i . Expressed in this way, the harmonic part of the flow is independent of time, because it is determined by the circulation of the solution around boundary components, and these are conserved quantities by Kelvin’s circulation theorem. Therefore, the scalar problem  ωt + (K h [ω] + γi X i )∇ω = 0 , (5) ω(x, 0) = ω0 is fully equivalent to the 2D Euler system, and may be called its vorticity formulation. 2 Relation between Dirichlet and hydrodynamic Green’s functions The reason to be unhappy with the vorticity formulation presented in the previous section is that we do not know much about the operator K h . From the mathematical point of view, there is a vast toolbox of harmonic analysis and PDE theory available for K , whereas, at least in principle, nothing is known about K h . In fact, the whole difficulty with K h is contained in the behavior in y of the functions αi . Indeed, if one subtracts the equations for G and G h we can use the linearity of Laplace’s equation to derive G h (x, y) = G(x, y) +

n

αi (y)φi (x),

(6)

i=1

and the harmonic measures φi are rather pleasant analytically. In [13], one of the authors managed to close the vortex dynamics equation using K itself, by proving the following result: Lemma 1 Let u be a smooth, divergence-free vector field in , tangent to ∂, with ω = curl u. Let ai be such that u = K [ω] +

n

ai X i .

i=1

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Then, for each i = 1, . . . , n we have





ai =

φi ωdx + 

uds. i

In other words, for a general vector field, one may express the harmonic part of the flow in terms of harmonic moments of vorticity plus the circulations around boundary components, which are conserved in time. As it turns out, this result is contained in the analysis done in [5], and it deserves emphasis just because it condenses, in simple terms, much of [5]. The proof is also very simple, and we refer the interested reader to [13]. Lemma 1 is also true for exterior domains, if one defines the harmonic potentials in the exterior domain as the unique bounded harmonic function in the exterior domain which is equal to 1 in one of the boundary components, vanishing on all the others. For example, in the exterior of a single bounded obstacle, the harmonic potential is φ(x) ≡ 1, and the extension of Lemma 1 says that the coefficient of the harmonic part is equal to the integral of vorticity plus the circulation of velocity around the boundary. For u a solution of the Euler equations, both these quantities are conserved, which implies that the harmonic part of the decomposition  u = K [ω] + a X is constant in time, and that the circulation of K [ω] around the boundary is − ω, both known facts (see [7]). Next, we use Lemma 1 to compute the functions αi . We fix y ∈  and we take ∇ ⊥ of (6), computing the circulation of the resulting identity around i . This yields  0= i

∇ ⊥ G h (x, y)ds =



∇ ⊥ G(x, y)ds +

n j=1

i

 α j (y)

∇ ⊥ φ j (x)ds.

i

and we use Lemma 1 with ω = δ(x − y) to show that the circulation of ∇ ⊥ G around i is −φi (y). We conclude that  n 0 = −φi (y) + α j (y) ∇ ⊥ φ j (x)ds. (7) j=1

i

If we define the n × n matrix M = (m i j ) by  mi j ≡

∇ ⊥ φ j (x)ds.

i

The matrix M is always invertible. Indeed, we observe that the vector fields ∇ ⊥ φi are linearly independent, since a linear combination βi ∇ ⊥ φi vanishes if the gradient of = βi φi vanishes, which in turn implies

= c, with c a constant. Evaluating in 0 , we see that c = 0, and evaluating in i we conclude that βi = 0, as we wished. Therefore, vector fields {∇ ⊥ φ j } form a basis of H . The matrix M is precisely the change-of-basis matrix between {X i } and {∇ ⊥ φ j }, and therefore, it must be invertible. pi j φ j (y), and therefore, If we denote M −1 = ( pi j ), we can use (7) to deduce that αi (y) = n

G h (x, y) = G(x, y) +

pi j φ j (y)φi (x).

(8)

i, j=1

Moreover, it is easy to see that the symmetry of G h and G imply that M −1 is symmetric. Note that the argument above fails for exterior domains. Indeed, for the exterior of a single obstacle, the harmonic measure is constant, and therefore its ∇ ⊥ does not span the appropriate space. Finally, we note that we can write the vortex dynamics equations in the form ⎛ ⎞ ⎡ ⎤  n ⎣ φi ωdx + γi ⎦ X i .⎠ ∇ω = 0, ωt + ⎝ K [ω] + (9) i=1

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and comparing this expression with (5) we have that ⎡ ⎤  n ⎣ φi (x)ω(x)dx ⎦ X i , K h [ω] = K [ω] + i=1



which provides an easy way of extending analytic properties of K to K h . 3 Applications Next, we would like to provide a couple of applications of the description of vortex dynamics presented above. The first one is the main result in [13]. The basic problem is to study the limiting behavior of certain sequences of solutions of the incompressible 2D Euler equations in domains with a small hole. At first, the authors, together with Iftimie, considered flows in the exterior of a single bounded obstacle in the plane. More precisely, we considered flows u ε = u ε (x, t), satisfying the Euler equations in the exterior domain R2 − ε, with fixed initial vorticity ω0 , smooth and compactly supported in R2 − ε, and with initial velocity u 0 = K ε [ω0 ] + α X ε , for some α ∈ R. Here, K ε is the Dirichlet Biot–Savart operator in R2 − ε, and X ε is the generator of the harmonic vector fields in the same domain. With extensive use of conformal maps and their properties, this problem got reduced to the case of the exterior of the disk, where we have explicit formulas for the Biot–Savart kernel which could be used to obtain estimates. We proved that there exists a subsequence of the u ε , which we do not rename, which converges in L 2 to a flow u. This limit flow, and the corresponding vorticity ω are a weak solution of a modified vorticity equation with the form: ⎧ ω + u · ∇ω = 0 ⎪ ⎨ t div u = 0 , (10) ⎪ ⎩ curl u = ω + γ δ0 ω(x, 0) = ω0  where γ = α − ω0 dx, and δ0 is the Dirac delta function at x = 0. The constant γ is the circulation of u 0 around the small obstacle. For details, see [7]. In [13], the objective was to examine the same problem for a bounded domain with a finite number of holes, one of them small. For this result, we could not use conformal maps in the way it was done before, because we could not reduce the problem to one where the Biot–Savart kernel was explicitly known. Therefore, a new technique had to be used, which depended crucially on the precise description of vortex dynamics provided by Lemma 1 plus many of the fine properties of the Biot–Savart operator K . In particular, the conformal map technique used in [7] restricted our main results to small obstacles which were disappearing self-similarly, where the new technique used in [13] allowed for small obstacles of arbitrary shape, as long as they were contained in a small disk. The coefficients of the harmonic parts of the velocity are not constant in time in this case, and the coefficient associated with the small hole becomes constant in the limit, decoupling from the rest of the flow as the size of the hole becomes small. What is left in the limit is a point-vortex perturbation on the velocity with strength equal to the coefficient of the harmonic generator associated with the small hole. The limit flow v, with vorticity ω, satisfies the following equation: ⎧ ωt + v · ∇ω = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j ⎪ ⎨ v = K  [ω + αk,0 δ(x − P)] + k−1 j=1 (β j (t) + αk,0 φ j (P))X   ⎪ ⎪ β j (t) =  φ j (x)ω(x, t)d x + α j,0 −  φ j (x)ω0 (x)d x, j = 1, . . . , k − 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ω(x, 0) = ω0 (x) and β j (0) = α j,0 . Here, P marks the location where the small obstacle disappeared, αk,0 is the circulation of the harmonic part of the initial velocity around the small obstacle. The second application concerns vortex sheet initial data flow. Our point of departure was extending Delort’s Theorem (existence of solutions with vortex sheet initial data with distinguished sign, (see [4])) to general exterior domains. This turns out to be rather easy. In fact, Delort’s existence result concerns weak Reprinted from the journal

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solutions of the velocity formulation of the equations, which does not see the topology of the domain. Our next question was to look for existence of a solution to the weak vorticity formulation of the equations, and then, both the bounded domain and the exterior domain cases were interesting. For smooth ideal flows, the circulation of the velocity around connected components of the boundary are constants of motion. It is a reasonable question, interesting both mathematically and physically, whether weak solutions also satisfy Kelvin’s circulation theorem in this restricted sense. In fact, for integrable vorticities the answer is affirmative, but at the level of vortex sheets, there is a possibility that the circulation around a boundary component may change as the boundary sheds or absorbs vorticity. This is interesting because it opens the possibility of weak solutions which capture the observed behavior of actual high Reynolds number flows in the presence of boundaries. We develop a new notion of weak solution which includes, and evolves, inviscid flows with vorticity production at the boundary and we prove existence of weak solutions in this sense for flows with distinguished sign vortex sheet initial data. This result depends, in an essential manner, on the precise form of the vortex dynamics equations, as expressed in (9). The results described above are joint research between the authors, Iftimie and Sueur, and it is currently in preparation.

4 Conclusions We will discuss briefly the relation between the treatment of vortex dynamics in domains with topology as presented here with the remarkable progress obtained in the same problem by Crowdy, Marshall, and others in a series of recent papers. Broadly, Crowdy et al. begin with expressions for the hydrodynamic Green’s function on circular domains, i.e., bounded, connected domains whose boundary is a finite number of disjoint circles. In such domains, the hydrodynamic Green’s function can be expressed in terms of a special function called the Schottky–Klein prime function, whose deep group-theoretical properties enables an infinite product representation. This produces both explicit formulas for the hydrodynamic Green’s function for the circular domains and approximations by truncation of the product formula which can be used in computations. Furthermore, any reasonable multiply connected domain may be conformally mapped into a suitable circular domain, so that, at least in principle, this approach leads to a general, fairly explicit account of vortex dynamics in complicated domains which are computationally usable. For details, see [2]. Even more closely related with the subject of the present article, in [3], Crowdy and Marshall used their method to obtain explicit formulas for the Dirichlet Green’s function as well, first using the prime function to construct variants of the hydrodynamic Green’s function, including expressions for the αi , and then using (6). This produces new formulas for the Dirichlet Green’s function on circular domains, which can then be propagated by conformal mapping to other domains. Their approach does not require the use of the relation expressed in Lemma 1. The approach by Crowdy et al. uses sophisticated machinery based on conformal mapping to build a detailed, nearly explicit picture of vortex dynamics in complicated domains. From a computational point of view, this has great advantages in precision over an approach based on Lemma 1. However, constructing the hydrodynamic Green’s function, and therefore vortex dynamics beginning from the Dirichlet Green’s function in the manner provided by Lemma 1 builds on a broad installed base of finite element solvers for the Poisson problem in complicated domains, and it is easier to implement, because it does not require computational conformal mapping. Leaving computations aside, our main point is that the description of vortex dynamics in complicated topologies provided by Lemma 1 is mathematically useful. For example, it would be difficult to study the asymptotic behavior of the prime function and of the conformal maps involved in treating the small hole problem. One natural question associated with the small hole problem is the problem of homogenization— studying the limiting behavior associated with flow past a periodic or random array of holes which becomes finer. Results in this direction have been obtained by Lions and Masmoudi using two-scale convergence in [12], but further work, especially regarding vortex dynamics, would be quite interesting. Another interesting problem associated with the small hole asymptotics would be to obtain rigorous error estimates for the point island approximation, recently introduced by Johnson and Robb McDonald in [8]. Acknowledgments We would like to thank F. Sueur and D. Iftimie for many useful discussions on this subject. This work is supported in part by FAPESP grant # 2007/51490-7.

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References 1. Aubin, Th.: Nonlinear Analysis on Manifolds. Monge–Ampère equations.. Springer, Berlin (1982) 2. Crowdy, D. Marshall J.: Analytical formula for the Kirchhoff–Routh path function in multiply connected domains. Proc. R. Soc. London Ser. A 461, 2477–2501 (2005) 3. Crowdy, D., Marshall, J.: Green’s function for Laplace’s equation in multiply connected domains. IMA J. Appl. Math. 72, 278–301 (2007) 4. Delort., J-M.: Existence de nappes de tourbillon en dimension deux. J. Am. Math. Soc. 4, 553–586 (1991) 5. Flucher, M., Gustafsson, B.: Vortex motion in two-dymensional hydrodynamics, energy renormalization and stability of vortex pairs. TRITA preprint series (1997) 6. Hodge, W.V.D.: The theory and applications of harmonic integrals, 2nd edn. Cambridge University Press, Cambridge (1952) 7. Iftimie, D., Lopes Filho, M.C., Nussenzveig Lopes, H.J.: Two dimensional incompressible ideal flow around a small obstacle. Commun. Partial Diff. Equ. 28, 349–379 (2003) 8. Johnson, E.R., Robb McDonald, N.: The point island approximation in vortex dynamics. Geophys. Astrophys. Fluid Dyn. 99, 49–60 (2005) 9. Kato, T.: On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Ration. Mech. Anal. 25, 188– 200 (1967) 10. Koebe, P.: Abhandlungen zur theorie der konformen abbildung. Acta Math. 41, 305–344 (1914) 11. Lin, C.C.: On the motions of vortices in two dimensions I and II. Proc. Natl. Acad. Sci. 27, 570–575, 575–577 (1941) 12. Lions, P.-L., Masmoudi, N.: Homogenization of the Euler system in a 2D porous medium. J. Math. Pures Appl. 84, 1–20 (2005) 13. Lopes Filho, M. C.: Vortex dynamics in a two dimensional domain with holes and the small obstacle limit. SIAM J. Math. Anal. 39, 422–436 (2007)

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Theor. Comput. Fluid Dyn. (2010) 24:59–64 DOI 10.1007/s00162-009-0170-1

O R I G I NA L A RT I C L E

E. Ryzhov · K. Koshel · D. Stepanov

Background current concept and chaotic advection in an oceanic vortex flow

Received: 31 October 2008 / Accepted: 24 September 2009 / Published online: 27 October 2009 © Springer-Verlag 2009

Abstract The concept of background currents is offered in the examples of barotropic and baroclinic quasigeostrophic models. The background currents are characterized by constant value of potential vorticity, which minimize the energy of a system. Hamiltonian character of motion equations of fluid particles allows to apply such models to study chaotic advection. Keywords Background currents · Chaotic advection · Topographic vortex PACS 47.32.Ef · 47.52.+j · 92.10.-c 1 Introduction The equations of fluid particles’ motion take a very simple form from the Lagrangian point of view dr dt = V(r, t) V(r, t) is a given Eulerian velocity field. The latter field can be obtained by solving the dynamical motion equations (the dynamical approach) or due to some kinematical speculations or measurements. The dynamics of fluid particles (or tracers) are described in a nontrivial case by a set of nonlinear differential equations with fully deterministic right-hand sides (the Eulerian velocity field is regular). It is well known from the theory of dynamical systems that the solutions of deterministic equations can be chaotic in the sense of the exponential sensitivity to small variations of initial conditions and parameters. The term ’chaotic advection’ was introduced by Aref [1]. The chaotic properties can arise in plane fluid flows, but only in unsteady case [5]. For example, the velocity components of incompressible planar flows are known to be expressed in terms of a streamfunction ψ (x, y, t): x˙ = u = −ψ y ,

y˙ = v = ψx ,

(1)

and vorticity ω = vx − u y ≡ ψ. The advection equations (1) are now the Hamiltonian equations with the streamfunction ψ (x, y, t) playing the role of a Hamiltonian. Thus, two-dimensional advection in an incompressible fluid is equivalent to the Hamiltonian dynamics of a system with one and a half degrees of freedom in the nonstationary case and with one degree of freedom in the stationary one. If the streamfunction ψ is specified disregarding the laws of fluid motion, then (1) is a kinematic model. The problem of dynamical compatibility lies in the fact that ψ must satisfies relations following from dynamical equations. As promising models for studying chaotic advection, we consider a class of simple dynamically consistent models of geophysical hydrodynamics based on the concept of background currents proposed by Kozlov [6]. Communicated by H. Aref E. Ryzhov(B) · K. Koshel · D. Stepanov Pacific Oceanological Institute of the RAS, Baltiiskaya 43, Vladivostok, Russia E-mail: [email protected] Reprinted from the journal

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2 Background currents Work [6] gives one possible constructive definition of a background current. We illustrate the main ideas of the proposed method on the simple examples. Potential vorticity  is one of the basic characteristics of quasi-two-dimensional geophysical currents. It combines the contributions of the relative vorticity ω and planetary–topographic interactions; the latter are generally complicated by stratification. Since  and ω are linearly related, fixing either of them allows determining the velocity field unambiguously for the corresponding boundary conditions. Both  and ω can be used to distinguish between coherent structures such as vortices, jets, and fronts. Traditionally, it is preferable to use the relative vorticity ω. In contrast, the potential vorticity in the case of infinitesimally small viscosity is a Lagrangian invariant, i.e., satisfies the equation [7] d (2) ≡ t + ux + v y = 0. dt Owing to its invariance, the potential vorticity  is a more convenient quantity for determining the current structure. The current with horizontally homogeneous steady state potential vorticity  satisfying (2) can be reasonably accepted as a background current. However, the result will depend on the magnitude of . From ¯ provides a global minimum for the mechanical this set of solutions, we single out the one for which  =  energy (kinetic and sensible potential energy) of the system. Using the simplest quasi-geostropic barotropic model [6], the algorithm for constructing of background current is presented. The potential vorticity takes the form [7]  = ω + F,

F = f (y) + ( f 0 /H )h(x, y),

(3)

where f (y) is the Coriolis parameter with a reference value f 0 , H is the mean basin depth, and h(x, y) is the topographic bottom evaluation. Under the rigid lid boundary condition at the surface, the current in a basin D with boundary ∂ D described by (2) and (3), and with the function ψ (b) (l, t) specifying the flux through the boundary(l is the coordinate along the boundary) ψ|∂ D = ψ (b) (l, t) is considered.  From (3), we can obtain the following expression for the kinetic energy E = 21 D (∇ψ)2 dD. The nec   ¯ dD = 0, which essary condition of global minimum for the mechanical energy ∂ E/∂ = D F −    −1  ¯ = F ≡ , where weighting functions  in the simplest case is the implies  D FdD D dD Green’s function for the Laplace operator. Thus, a background current is determined unambiguously by a given distribution of the background relative vorticity ω¯ as a solution of the Dirichlet problem  ψ¯ = ω¯ ≡ F − F, ψ¯ ∂ D = ψ (b) (l, t) . (4) A formal solution of (4) can be written in an explicit form [6]    ∂  ψ (r, t) = ψ0 (r) + ψ1 (r, t) , ψ0 (r) =  (r, ρ) ω¯ (ρ) dDρ , ψ1 (r, t) = dlρ . (5) ψ¯ lρ , t ∂n ρ ∂ Dρ

D

Consider the construction of background current in a quasi-geostrophic approximation of a two-layer fluid under the rigid lid boundary condition at the surface. Stream functions in layers are ψ1 = ψ − (1 − δ)ψ ∗ , ψ2 = ψ + δψ ∗ ,

(6)

where δ = H1 /H and barotropic and baroclinic stream functions ψ and ψ ∗ satisfy equations ψ =  − f −

1 f0 f0 h, ψ ∗ |∂ D = ψ ∗(b) (l, t), h, ψ|∂ D = ψ (b) (l, t), ψ ∗ − ψ ∗ = ∗ − H a H2

(7)

where H1 , H2 are the thicknesses of undisturbed layers, a = [g ∗ H δ(1 − δ)]1/2 / f 0 is the internal deformation radius, g ∗ is the reduced acceleration due to gravity,  is the horizontal Laplace operator,  = δ1 + (1 − δ)2 , ∗ = 2 − 1 and the potential vorticity in layers 1 and 2 satisfy the conservation law di /dt = 0.  The total mechanical energy is E = 21 D [H1 (∇ψ1 )2 + H2 (∇ψ2 )2 + g ∗ ζ 2 ]dD, where ζ = ( f 0 /g ∗ )ψ ∗ is ∂E ∂E = ∂ the elevation of the interface. The condition of global minimum for the mechanical energy ∂ ∗ =0 f0 f0 ∗ ¯ ¯ gives the expressions for background potential vorticities  = < f >0 + H 0 and  = H2 k where   −1

k = D P F (k) dD F (k) dD is the operation of weighting averaging. The weighting functions F (0) and F (k) are the solutions of equations F (k) −k 2 F (k) = 1, F (k) |∂ D = 0, at the k = 0, and k = 1/a, respectively.

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Fig. 1 Steady flow ( = 0) streamlines θ = 0. Separatrix (bold line) and the boundary of a stability circle (dotted line). The dependence of the transport time of markers (middle) and accumulated Lyapunov time (right) on their initial position  = 0.1, ν = 0.35. Each class of markers contains 1/3 of their total number

3 Chaotic advection According to the concept of background currents [6], the streamfunction has the structure of Eq. (5), where the stationary planetary component ψ0 is vortical (ψ0 = ω in the region D and ψ0 |∂ D = 0) and the incident-flow nonstationary component ψ1 is nonvortical (ψ1 = 0 in D and ψ1 is set at the boundary ∂ D). In that case we may expect to find chaotic properties of particle trajectories [2]. In the simplest case of an unbounded f -plane with the topographic bottom elevation h (x, y), we have for the background relative vorticity ω¯ = −( f 0 /H )h. The only incident flow bounded on the entire plane is a spatially uniform flow W (t) directed at an angle θ (t) to the x-axis, with a streamfunction ψ1 = W (t) (x cos θ − y sin θ ), W > 0. The simplest shape of the topographic bottom elevation is an axially symmetric h (r ), r 2 = x 2 + y 2 . Using characteristic scales of length L ∗ , velocity V ∗ , time t ∗ = L ∗ /V ∗ , and bottom elevation h ∗ = h (0) and introducing dimensionless variables, we have by the Stokes formula r r V = −σ

ρh (ρ) dρ,

(8)

0

where σ = h ∗ /H Ro is the topographic parameter and Ro = V ∗ / f 0 L ∗ = 1/ f 0 t ∗ is the Rossby number. Quasigeostrophic approximation requires h ∗ /H = O (Ro), i.e., σ = O (1) [4]. Besides, obviously V  (1) = 0, i.e., horizontal scale is determinated by the distance from the center at which azimuthal velocity V (r ) = dψ0 /dr 2 reaches its extreme value. Applying (8) to a sea-mount of Gaussian form h (r ) = h(0)e−αr , we get [4]  2 σ e−αr − 1 , whence using condition V  (1) = 0 we can find α ≈ 1.256..., as the single positive V = 2αr root of equation 1 + 2α = eα , hence Vm = V (1) ≈ −0.3285σ . Consider the stationary incident flow with constant velocity W (θ = 0). There are no critical points in total flow when W + Vm > 0, while when W + Vm < 0 there are two such points, elliptic and hyperbolic ones (see Fig. 1). A separatrix, intersecting itself at the hyperbolic point, divides the flow region into two parts: one with closed streamlines surrounding the elliptic point inside a homoclinic loop and the other with unbounded trajectories outside it. A homoclinic picture presented in Fig. 1 shows a typical structurally unstable portrait, such that superposition of harmonic perturbation of W upon it breaks down topological equivalence of corresponding maps and generates chaotic manifestations [5]. We will consider the simplest unidirectional oscillating flow as an incident flow W = W0 [1 +  sin(νt)]. Figure 1 shows a typical picture of the chaotic character and destruction of the vortex core. The important quantitative characteristic of the divergence of the initially close trajectories is the accumulated Lyapunov exponent, which can be calculated on a finite time interval with the help of a simple algorithm [5] proposed in Reprinted from the journal

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Fig. 2 Boundary of trajectories escaping from the vortex domain in the bottom layer of the model system, as determined using (dashed curves) equtions (10) and (solid curves) simulations for: (1) ν = 0.15, (2) ν = 1.3 (optimum), (3) ν = 4.4—left. Fragments of Poincare sections for the bottom layer at ν = 1.3 and  = 0.025 (a) and 0.0290 (b). Lobes schematically show the fragments of intersection of the stable and unstable manifolds (see text for explanation). Plots of the stochastic layer thickness versus perturbation frequency ( = 0.1) determined by: (1) total thickness, (2) analytical estimation, (3) escape in one turn, (4) fraction N of escaping markers relative to their initial total number—right

[8]. The useful information is also provided by ‘accumulated Lyapunov time‘, which is defined as a quantity inverse to the corresponding accumulated Lyapunov exponent. The dependence of the time of marker transport from vortex region into the free-flow region and of its accumulated Lyapunov time on its initial position are presented in Fig. 1. In both cases, the regions from which markers are not carried and undergo the evolution are distinguishable. A good qualitative correspondence of the regions, from which markers have not been carried out up to certain time moment and of the it outlined according to a certain value of the accumulated Lyapunov time, can be noted [5]. Let us consider the constructing of background current under seamount in a two-layer ocean on f -plane for a simple case of delta-shape seamount h(x, y) = τ∞ δ(x, y), where τ∞ is a volume of seamount and the eastward incident flow U on boundaries. Solutions of (7) in the lower layer [5] are ψ2 = −U y − f0Hτ∞ (ln(ar ) − γ K 0 (ar )), where K 0 (r ) is a Macdonald function, γ = H1 /H2 . Passing to the dimensionless variables in accordance with the relations (x, y) = a(x ∗ , y ∗ ), t = a/U t ∗ , ψ = ψ ∗ and omitting the primes, we finally obtain ψ2 = −W y − σ (ln(r ) − γ K 0 (r )),

dx dy y x = W y + σ (1/r + γ K 1 (r )), = −σ (1/r + γ K 1 (r )), dt r dt r (9)

where W = W0 (1 + W sin(νt)). In the steady flow (i.e., for W (t) = W0 ) a phase portrait of the system (9) has a typical homoclinic structure [5]. At the periodic perturbation of the external flow, the trajectories exhibit chaotic behavior that makes possible exchange between the vortex domain and the external flow. In order to evaluate the stochastic layer thickness, we use a variant of the Mel’nikov theory proposed by Gledzer [3]. According to a procedure proposed in [3], the condition, that the Oy axis crosses a trajectory, imposes the following limitation on the bottom layer [9],

  r2 1 1 δcr = 2(W r2s + γ r2s K 0 (r2s )) F(ξ )dξ , F(ξ ) = (10) + γ K 1 (ξ )W (νT0 (ξ ) W ξ r0 where T0 (r ) =

r r0

ξ dξ (W ξ )2 −(ψ2 +ln ξ −γ K 0 (ξ ))2

√

and r2s is a coordinate of the hyperbolic point in the unperturbed

case. It should be noted that these estimates coincide with one obtained using the Mel’nikov integral with allowance for the topology of lobes (regions bounded by stable and unstable manifolds of the hyperbolic point). Figure 2 shows the stochastic layer thickness δcr as a function of the perturbation amplitude W calculated using formula (10) and the results of direct modeling of the trajectories. A comparison of these curves shows

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the estimation (10) is inapplicable at small perturbation frequencies, as for the large and intermediate frequencies, it is valid when the perturbation amplitude are quiet small. These calculations confirm the fact the role of a small parameter in evaluation of the stochastic layer thickness is the ratio W /ν [10]. Another feature is a stepwise character of variation of the stochastic layer thickness depending on the perturbation amplitude. This behavior is readily explained as follows: estimates obtained using the Mel’nikov theory predict only the envelope of the stochastic layer, whereas escaping trajectories correspond to initial positions in the outer lobes of the homoclinic structure. This situation is illustrated in Fig. 2, which shows two fragments of the Poincare sections at a frequency ν = 1.3 and a perturbation amplitude W = 0.0250 (a) and 0.0290 (b). Points A and B indicate the initial positions of trajectories leaving the vortex region in one turn, which define the boundary of the stochastic layer. We show as well the regions of stable and unstable manifolds, the intercepts of which with the ordinate axis (x = 0) corresponding to the boundary of the stochastic layer. As can be seen, the indicated trajectory is inside the outer lobe (Fig. 2a, point A). As the perturbation amplitude grows, the next lobe increases and, when it crosses the axis x = 0, the layer thickness exhibits a jump-like increase (Fig. 2b, point B). The jump magnitude is equal to the distance between points A and B. The jump-like changes in the layer thickness explain why our estimations fail to be valid at low frequencies. Indeed, the lobe dimensions at a small perturbation frequency are very large, while the stochastic layer thickness is determined by the lobe topology rather than by its dimensions predicted by the perturbation theory. At sufficiently large frequencies, the lobes determining the layer thickness turn out to be rather small, and the deviation of estimates from the results of calculations is not very large. Our calculations show that estimate (10) in practice allows only establishing the fact that a stochastic layer is present and, probably, indicate a change in the layer thickness depending on the perturbation frequency. If the perturbation amplitudes are sufficiently small, but still large enough for the Mel’nikov theory to fail, it is also possible to introduce a notion of the stochastic layer thickness. In this case, it is necessary to take into account the trajectories leaving the vortex domain during more than one turn. Then, the stochastic layer boundary is determined by the position of a trajectory that is closest to the hyperbolic point and never leaves the vortex domain. Figure 2 shows the results of calculations of the stochastic layer thickness determined in this way as a function of the perturbation frequency. As can be seen, estimate (10) provides a quite well qualitative (but not quantitative) description of the total stochastic layer thickness at low and high frequencies. In the vicinity of a frequency value that is optimal from the standpoint of stochastization, the deviations are quite significant. Nevertheless, the obtained frequency dependence of the stochastic layer thickness can be used for roughly evaluating the interval of frequencies that are optimal for the development of chaos [5]. In order to confirm this conclusion, Fig. 2 shows the degree of stochastization N (ν) of the vortex domain boundary, which was calculated as the fraction of escaping markers relative to their total number, 104 , uniformly distributed in the vortex domain [5]. A significant feature of the total stochastic layer thickness  is the noisy character at higher frequencies. This behavior is due to fractal character of the nonlinear resonances distribution. It should be also noted that the frequency dependence of the degree of exchange exhibits, in addition to the aforementioned high-frequency noise, some local maxima and minima well correlated with analogous structure on the layer thickness determined using the trajectory leaving the vortex domain in one turn. No such correlation is observed for greater perturbations. 4 Conclusion The concept of background currents is a way to construct a class of simple dynamically consistent models to study chaotic advection. The concidered physical examples are illustrations of such models. It is a limited concept, but we hope that a simple and constructive method will be useful in a qualitative investigation of realistic oceanographic situation. We show that the stochastic layer thickness in the vicinity of a hyperbolic point exhibits a jump-like change with increasing perturbation amplitude. This behavior is explained by the topology of regions bounded by stable and unstable manifolds: as the perturbation amplitude increases, more and more distant (from the hyperbolic point) regions of manifolds begin to intersect, thus determining the stochastic layer boundary. It is also demonstrated that the estimate obtained using the perturbation theory can be used to evaluate the interval of perturbation frequency, which are optimal for chaos development, although quantitative estimates of the stochastic layer thickness cannot be obtained in this way. It is established that the stochastic layers of vortex structures depend on the perturbation parameters in a more complicated manner than it is predicted by the Mel’nikov theory. Reprinted from the journal

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Acknowledgements The work was supported by the RFBR (Grants No. 08-05-00061 and partly 07-05-92210), FEB RAS (Grants No. 09-II-CO-07-002, 09-1P17-07) and by the Presidential Grant No. MK–1364.2008.5.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Aref, H.: Stirring by chaotic advection. J. Fluid Mech. 143, 1–21 (1984) Aref, H., El Naschie, M.S.: Chaos Applied to Fluid Mixing. Pergamon, London (1994) Gledzer, A.E.: Mass entrainment and release in ocean eddy structures. Izv. Atmos. Ocean. Phys. 35, 759–766 (1999) Izrailsky, Yu.G., Kozlov, V.F., Koshel’, K.V.: Some specific features of chaotization of the pulsating barotropic flow over elliptic and axisymmetric sea-mounts. Phys. Fluids 16, 3173–3190 (2004) Koshel’, K.V., Prants, S.V.: Chaotic advection in the ocean. Phys. Uspekhi 49, 1151–1178 (2006) Kozlov, V.F.: Background flows in geophysical fluid dynamics. Izv. Akad. Nauk FAO 31, 245–250 (1995) Pedlosky, J.: Geophysical Fluid Dynamics. 2nd edn. Springer-Verlag, New York (1987) Pierrehumbert, R.T., Yang, H.: Global chaotic mixing on isentropic surfaces. J. Atmos. Sci. 50(15), 2462–2480 (1993) Ryzhov, E.A., Koshel’, K.V., Stepanov, D.V.: Chaotic advection induced by a topographic vortex in baroclinic ocean. Tech. Phys. Lett. 34(6), 531–534 (2008) Zaslavskii, G.M.: Physics of Chaos in Hamiltonian Systems. Computer Research Institute, Izhevsk (2004)

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Theor. Comput. Fluid Dyn. (2010) 24:65–75 DOI 10.1007/s00162-009-0150-5

O R I G I NA L A RT I C L E

Gregory Reznik · Ziv Kizner

Singular vortices in regular flows

Received: 13 November 2008 / Accepted: 14 July 2009 / Published online: 11 August 2009 © Springer-Verlag 2009

Abstract A review of the theory of quasigeostrophic singular vortices embedded in regular flows is presented with emphasis on recent results. The equations governing the joint evolution of singular vortices and regular flow, and the conservation laws (integrals) yielded by these equations are presented. Using these integrals, we prove the nonlinear stability of a vortex pair on the f -plane with respect to any small regular perturbation with finite energy and enstrophy. On the β-plane, a new exact steady-state solution is presented, a hybrid regular-singular modon comprised of a singular vortex and a localized regular component. The unsteady drift of an individual singular β-plane vortex confined to one layer of a two-layer fluid is considered. Analysis of the β-gyres shows that the vortex trajectory is similar to that of a barotropic monopole on the β-plane. Non-stationary behavior of a dipole interacting with a radial flow produced by a point source in a 2D fluid is examined. The dipole always survives after collision with the source and accelerates (decelerates) in a convergent (divergent) radial flow. Keywords Quasigeostrophic singular vortices · Regular flow · f -plane · β-plane PACS 47.32.C-, 47.35.Fg, 92.10.ak, 98.54.Cm

1 Introduction As a rule, intense localized mesoscale eddies in the atmosphere and ocean are highly nonlinear, their dynamics being affected by a number of physical factors, such as the planet’s rotation and sphericity (the so-called β-effect), background large-scale flows, density stratification, and topography. Although nowadays numerical models are capable of incorporation of a great number of factors, understanding of the major mechanisms of the vortex dynamics is often achieved by invoking simplified analytical models that, however, correctly reproduce the underlying physics. One of such approaches is based on the approximation of distributed eddies by singular vortices [23]. The form of a singular vortex is not universal, and can be chosen based on physical considerations. We define the singular vortices as ‘elementary particles’ which make up any steadily translating localized vortex on the f - or β-plane. Generally, this approach results in that the streamfunction of a singular vortex is a linear combination of exponentially decaying at infinity modified Bessel functions, while the intrinsic vorticity of Communicated by H. Aref G. Reznik (B) P.P. Shirshov Institute of Oceanology, 36 Nakhimovskiy Pr., 117997 Moscow, Russia E-mail: [email protected] Z. Kizner Departments of Mathematics and Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel E-mail: [email protected] Reprinted from the journal

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the vortex, in addition to the delta-function component, may contain some term that exponentially decays at infinity and has a logarithmic singularity in the point where the vortex is located. In the absence of this additional term, a singular vortex is a conventional point vortex. Evolution of the dynamical system consisting of an ensemble of singular vortices and a regular flow is described by a set of interrelated equations: a partial differential equation for the regular component and a set of ordinary differential equations for the coordinates of the vortices. Such a theory was first suggested by Reznik [23] for the barotropic case (one-layer fluid); here we present recent results related to both one- and two-layer fluids. The paper is organized as follows. The equations that govern the cooperative evolution of singular vortices and a regular background flow appear in Sect. 2. Invariants of motion are examined in Sect. 3. Stability of f -plane vortex pairs is considered in Sect. 4. An exact β-plane two-layer hybrid modon solution comprising of one upper-layer singular vortex and a localized regular component is presented in Sect. 5. Unsteady evolution of an individual singular vortex confined to the upper layer in a two-layer fluid on the β-plane is examined in Sect. 6. The interaction of a logarithmic point-vortex dipole with a two-dimensional radial flow is considered in Sect. 7. The main results are summarized in Sect. 8.

2 Basic equations 2.1 Conservation of potential vorticity in layered models We start from the conventional equations of conservation of quasigeostrophic potential vorticity in a two-layer fluid, ∂i + J (ψi , i ) = 0, ∂t

i = qi + βy, i = 1, 2,

(2.1a)

Subscripts 1 and 2 are indices of the first (upper) and second (lower) layer; ψi , i , and qi are the streamfunction, potential vorticity (PV), and intrinsic vorticity in layer i, respectively; J (,) is the Jacobian operator. The intrinsic vorticities are given by the following equations: q1 = ∇ 2 ψ1 + 1 (s1 ψ2 − ψ1 ),

q2 = ∇ 2 ψ2 + 2 (s2 ψ1 − ψ2 ).

(2.1b,c)

Constants 1 , 2 , s1 , and s2 are defined below using the following notations: f 0 is the reference value of the Coriolis parameter and β = const its northward gradient; g  is the reduced gravity; and Hi and ρi are the depth and fluid density of layer i. Depending on parameters 1 , 2 , s1 , and s2 , (2.1) describe a hierarchy of models, from the 1-layer to 2 21 -layer model. Recall that the term ‘n 21 -layer model’ is commonly used to refer to an (n + 1)-layer fluid, where the upper n layers are assumed to be active and the infinitely deep lower layer n + 1, motionless. In the 2 21 -layer model, constants 1 , 2 , s1 , and s2 are: 1 =

f 02 , g1 H1

2 =

f 02 ρ3 − ρ1 , g3 H2 ρ2 − ρ1

s1 =

ρ2 , ρ1

s2 =

ρ1 (ρ3 − ρ2 ) , ρ2 (ρ3 − ρ1 )

(2.2)

ρ2 − ρ1 . ρ2

(2.3)

where g1 = g(ρ2 − ρ1 )/ρ1 , g3 = g(ρ3 − ρ2 )/ρ2 , ρ3 > ρ2 > ρ1 . In a two-layer fluid with a free upper surface, 1 , 2 , s1 , s2 , and g  are: 1 =

f 02 , g  H1

2 =

f 02 , g  H2

s1 = 1,

s2 =

ρ1 , ρ2

g = g

In the two-layer rigid-lid model, parameters 1 , 2 , s1 , and g  are the same as in (2.3), and s2 = 1. In the 1 21 -layer model, (2.1) describe the motion in only one layer 1, so the parameter 1 is the same as in (2.3), and 2 = 0, ψ2 ≡ 0, s1 = 1. Finally, the one-layer (2D) model follows from (2.1) when s1 = 1, s2 = 1, ψ1 = ψ2 .

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2.2 Two-layer modons and singular vortices First, we consider localized steadily translating vortical solutions to (2.1), referring to such solutions as to modons. The layer streamfunctions of a modon propagating along the x-axis depend only on the arguments x − U t and y, so that ψi = ψi (x − U t, y), i = 1, 2, where U is the constant translation speed of the modon (the translation can only be zonal if β  = 0). Therefore, in the co-moving frame of reference, (2.1a) can be rewritten as:   β J ψi + U y, qi − ψi = 0, i = 1, 2. (2.4) U In each layer, a bounded interior domain Di and an infinite exterior domain exist, filled with closed and open streamlines (contours of the co-moving streamfunction ψi + U y), respectively [4]. Correspondingly, (2.4) can be reduced to  β Z i (ψi + U y)  = 0, r ∈ Di qi − ψi = , i = 1, 2, (2.5) 0, r∈ / Di U where Z i is some differentiable function. Any solution to (2.5) can be presented as:     u   l ψi = Z 1 [ψ1 (r ) + U y ]ψi,s (|r − r |)dr + Z 2 [ψ2 (r ) + U y  ]ψi,s (|r − r |)dr , i = 1, 2, D1

(2.6)

D2

u , ψ u , ψ l , and ψ l obey the equations: where functions ψ1,s 2,s 1,s 2,s u u q1,s − p 2 ψ1,s = δ(x)δ(y), l q1,s

−p

2

l ψ1,s

= 0,

l q2,s

u u q2,s − p 2 ψ2,s = 0,

−p

2

l ψ2,s

(2.7a)

= δ(x)δ(y),

(2.7b)

and p 2 = β/U . Equations (2.7a) and (2.7b) determine the upper- and lower-layer singular vortices, respectively. Hereafter, δ() is Dirac’s delta-function; variables marked by subscript s are associated with singular vortices; superscript u (or l) indicates that the corresponding fields are induced by the singular vortex located in the upper (lower) layer. Solutions to the linear system of (2.7a) and (2.7b) have the form [27]: u u ψ1,s = c(α − K 0 ( p−r ) − α + K 0 ( p+r )), ψ2,s = c(K 0 ( p+r ) − K 0 ( p−r )), − +

l ψ2,s

−

(2.8)

+

= cα α (K 0 ( p−r ) − K 0 ( p+r )), = c(α K 0 ( p+r ) − α K 0 ( p−r )), (2.9)  where c = 1/[2π(α + − α − )]; r = x 2 + y 2 is the polar radius; and K m () is the modified m-order Bessel function. Parameters p± are defined as   1 2 (2.10) = p2 + (1 + 2 ) ± (1 − 2 )2 + 41 2 s1 s2 ≥ 0, p± 2 l ψ1,s

while coefficients α ± (real numbers) are explicitly expressed via parameters 1 , 2 , s1 , and s2 (see [27] for details). The inequality in (2.10) means that that the modon translation speed lies outside the range of the u (or ψ l ) has a logarithmic singularity phase speeds of the linear Rossby waves. Note that streamfunction ψ1,s 2,s u l at r = 0, while streamfunction ψ2,s (or ψ1,s ) is regular throughout the (x, y)-plane and represents the motion induced in the lower (upper) layer by the singular part of the vortex confined to the upper (lower) layer. Formulas (2.4)–(2.10) allow us to interpret any two-layer modon as a superposition of a continuum of singular vortices of the form (2.7) packed in the finite interior domains. In what follows, the singular part of a mixed singular/regular solution to (2.1) is a sum of a finite number of singular vortices (2.7). Parameter p 2 will be set arbitrarily, with the only restriction imposed by the inequality in (2.10), which assures decay of the vortex velocity field at infinity. In the absence of a background regular flow and the β-effect, the evolution of an ensemble of point vortices is governed by ordinary differential equations for the vortex coordinates. However, if the background regular flow is present, then a regular velocity field—in addition to the velocity field due to the vortices themselves— affects the motion of the vortices and, in turn, undergoes transformations. Thus, a system of coupled differential equations describing the combined evolution of the regular flow and singular vortices, should be considered. Reprinted from the journal

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2.3 Mixed singular-regular vortical systems Let streamfunction ψi in each layer be a superposition of the streamfunctions ψi,r and ψi,s of the regular and singular flows, respectively: ψi = ψi,r + ψi,s ,

i = 1, 2.

(2.11)

We assume the singular part of the flow to consist of N1 upper-layer and N2 lower-layer singular vortices of the type (2.7): ψi,s =

N1

u A1,n 1 ψi,s (|r − rn 1 |)+

n1

N2

l A2,n 2 ψi,s (|r − rn 2 |), i = 1, 2,

(2.12)

n2

where Ai,n i is the amplitude of vortex n i in layer i, and r = rn i (t) is its trajectory. Ai,n i is necessarily constant [23]. By substituting (2.11) and (2.12) into (2.1a), and separately equating the regular and singular parts to zero, we get: ∂ (qi,r + p 2 ψi,s + βy) + J (ψi,r + ψi,s , qi,r + p 2 ψi,s + βy) = 0, ∂t mi (x˙m , y˙m ) = k × ∇(ψi,r + ψi,s )r=rm i , i = 1, 2.

(2.13a) (2.13b)

Here k is the vertical ort, and the regular-component intrinsic vorticity in the layers is q1,r = ∇ 2 ψ1,r + 1 (s1 ψ2,r − ψ1,r ),

q2,r = ∇ 2 ψ2,r + 2 (s2 ψ1,r − ψ2,r ),

(2.14a,b)

mi whereas ψi,s is the streamfunction field induced by all singular vortices but vortex m i . Equation (2.13b) imply that the motion of a singular vortex is induced by other singular vortices, and by the regular-flow component as well. The most complex (2.13a) describes the evolution of the regular-flow streamfunction ψi,r ; thus the quantity qi,r + p 2 ψi,s + βy (which can be referred to as the regular PV) is conserved in each fluid element distinct from the singular vortices. This conservation involves that the regular intrinsic vorticity qi,r in the element depends not only on the meridional coordinate y of the element, but also on the disposition of the singular vortices.

3 Invariants of motion Equation (2.13) yield a few conservation laws which were derived in our earlier publications [23,27]; here we provide only the expressions for the enstrophy and energy (for the mass and momentum integrals see [23,27]). The enstrophy integral can be written in the form: 

Sr + p 2 K S − β d = 1 s1 /2 s2 . A1,m 1 ym 1 + d A2,m 2 ym 2 = const, (3.1) m1

m2

Functional K S in (3.1) is: KS = −

⎧ 1⎨

2⎩

u A1,m 1 A1,n 1 ψ1,s (rm 1 ,n 1 ) + d

m 1  =n 1

+ 2d



l A2,m 2 A2,n 2 ψ2,s (rm 2 ,n 2 )

m 2  =n 2

m 1 ,m 2

⎫ ⎬ u A1,m 1 A2,m 2 ψ2,s (rm 1 ,m 2 ) , ⎭

(3.2)

where rm i ,n i = |rm i − rn i | is the distance between vortices m i and n i . At β = p = 0, with the use of (2.7), expression (3.2) reduces to the Kirchhoff function for a two-layer ensemble of conventional f -plane point vortices.

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Owing to the inequality s1 s2 ≤ 1, functional Sr defined as     1 s1 2 2 2 2 2 2 2 2 2 [q1,r + p (∇ψ1,r ) ] + d[q2,r + p (∇ψ2,r ) ] + p 1 ψ1,r + ψ2,r − 2s1 ψ1,r ψ2,r dxdy, Sr = 2 s2 is positive-definite. According to (3.1) and (3.2), both the changes in the distances between the singular vortices and non-zonal displacements of the vortices will cause changes in Sr . The energy integral can be written as:   Er − p 2 E s,r + K S − A1,m 1 ψ1,r r=r − d A2,m 2 ψ2,r r=r = const, (3.3) m1

m1

m2

m2

where functional K S is given by (3.2), and Er and E s,r are defined as:  1 2 2 Er = + (s1 /s2 )ψ2,r − 2s1 ψ1,r ψ2,r )]dxdy, [(∇ψ1,r )2 + d(∇ψ2,r )2 + 1 (ψ1,r 2   1 2 2 + dψ2,s )dxdy. (ψ1,s E s,r = (ψ1,s ψ1,r + dψ2,s ψ2,r )dxdy + 2

(3.4) (3.5)

Functional Er is positive-definite and represents the energy of the regular component. The last two terms in the left-hand side of (3.3) and the first term in the right-hand side of (3.5) represent the energy of interaction between the singular and regular components. Functional K S and the second term in the right-hand side of (3.5) represent the self-interaction energy of the singular mode. 4 Stability of a point-vortex pair on the f -plane The enstrophy and energy invariants (3.1), (3.3) provide a way for the examination of nonlinear stability of an arbitrary f -plane point-vortex pair in the free-surface models described in Sect. 2. When the regular component is absent, the dynamics of a vortex pair with given amplitudes A1 and A2 (e.g. [6]) is determined by the separation between the vortices, r1,2 , only. When ψi,r = 0 the separation r1,2 is a constant of motion, therefore, the vortex pair is stable relative to small perturbations of r1,2 : small changes in r1,2 result in small changes of the center of mass and the speed of the pair only. In principle, even a small regular perturbation imposed on a vortex pair can result in significant changes in the separation between the vortices and in their velocities [6]. However in a recent work [27] we have shown that if, at the initial moment, the perturbation is small, and the enstrophy and energy of the regular component of the perturbation are finite, then (i) the perturbation remains small for all subsequent times, and (ii) it causes only small changes in the separation between the vortices. This implies that any point-vortex pair, either rotating or translating, is stable relative to small perturbations that contain regular components. The proof of the above statement is based on the enstrophy and energy conservation. For a vortex pair on the f -plane, where p = 0, the conservation laws (3.1), (3.3) become:  1 2 2 Sr = + dq2,r )dxdy =  = const, (4.1) (q1,r 2   Er + K S − A1,m 1 ψ1,r r=r − d A2,m 2 ψ2,r r=r = E 0 = const. (4.2) m1

m1

m2

m2

Relationship (4.1) means that the enstrophy of the regular component is conserved. Using (4.1) and (2.14), we can estimate in terms of  the absolute value of the regular streamfunction in layer i, and √ the energy Er of the regular component of the flow. These estimates are just the inequalities |ψi,r | < Ci  and Er < Eˆ r , where Ci and Eˆ r are constants that depend on the model parameters only. If enstrophy  is sufficiently small, then Er and the energy of interaction between the regular and singular components [the last two terms in the left-hand side of (4.2)] are small compared to the total energy E 0 . This implies that a sufficiently small regular perturbation can cause only small changes in K S . In the case of a vortex pair, K S is a continuous and monotonic function of only one argument, the separation r1,2 between the two vortices (see 3.2). Therefore, small changes in K S are possible only if changes in r1,2 are small, i.e., only when the vortex pair is stable. In more details this proof appears in [27]. Reprinted from the journal

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5 Steadily translating systems on the β-plane 5.1 Stationary systems of singular vortices We now consider the simplest steady solutions to (2.13) for which the regular component ψi,r is zero. It is readily seen that, in this case, (2.13) is satisfied only if mi (x˙m i , y˙m i ) = k × (∇ψi,s )r=rm i = (U, 0), i = 1, 2,

(5.1a,b)

i.e. when all the vortices move along the x-axis with the same speed U = β/ p 2 . Constraints (5.1) on the vortex coordinates and amplitudes can be rewritten as: N1

am i ,n 1 A1,n 1 +

n1

N2

N1

am i ,n 2 A2,n 2 = −U,

n2

bm i ,n 1 A1,n 1 +

n1

N2

bm i ,n 2 A2,n 2 = 0,

(5.2a,b)

n2

where u  (am i ,n 1 , bm i ,n 1 ) = (ym i − yn 1 , xm i − xn 1 )(ψi,s )r =rm

i ,n 1

/rm i ,n 1 ,

(5.3a)

l  (am i ,n 2 , bm i ,n 2 ) = (ym i − yn 2 , xm i − xn 2 )(ψi,s )r =rm

i ,n 2

/rm i ,n 2 ;

(5.3b)

the tag denoting differentiation by r . It can be shown (see e.g. [8,23]) that given amplitudes A1,n 1 , A2,n 2 , the left-hand sides of (5.2) are linearly dependent, and the solvability condition for system (5.2) is N1

A1,n 1 + d

n1

N2

A2,n 2 = 0.

(5.4)

n2

Condition (5.4) is equivalent to the momentum conservation of the vortex system. It is analogous to the conditions derived in [23] for the barotropic case, and in [8] for the rigid-lid two-layer model. In (5.2), there are 2N1 + 2N2 − 2 linearly independent equations with respect to 2N1 + 2N2 − 2 unknowns, the latter being the coordinates x¯m i = xm i − x1 and y¯m i = ym i − y1 of N1 + N2 − 1 vortices relative to a chosen vortex with the coordinates x1 , y1 . All steadily translating singular vortex ensembles on the β-plane suggested up to date are solutions to (5.2) and (5.4). These are the barotropic dipole singular modons [3,7,22], and a more complex three-vortex Rossby modon [22]. Analogous vortex systems traveling steadily along a ‘parallel’ were obtained for barotropic flows on a rotating paraboloid and sphere [17,22]. In a rigid-lid two-layer fluid, two singular vortices placed in different layers can form a hetonic pair similar to the point-vortex hetons on the f -plane which were found and analyzed in mid 80s [6,9,10]. A pair of aligned hetons with specially fitted intensities and separations can make up a β-plane hetonic quartet, a steadily translating collinear ensemble of four discrete vortices; properties and stability of such a quartet were studied recently [13,14]. Periodic vortex streets comprised of the β-plane singular vortices were derived and examined for the barotropic models and two-layer rigid-lid model [7,8]. 5.2 Hybrid modon Another type of steadily translating solutions is a hybrid modon that incorporates both singular and regular components. Here we briefly present a recently found exact solution of this kind [28] for the rigid-lid two-layer model. The modon travels along the x-axis at a constant speed U . Its singular component is represented by an upper-layer singular vortex proportional to that given by (2.7a), and (2.8), and located in the modon centre. The modon streamfunction in the layers can be written as u ψi = ψi,r (x − U t, y) + Aψi,s (x − U t, y), i = 1, 2,  u is given by (2.8) with r = (x − U t)2 + y 2 . where A is the singular vortex amplitude, and ψi,s

(5.5)

(0)

The regular part of solution, ψi,r , is made up of a circularly symmetric monopole ψi,r (r ), or rider, and a (d)

dipole ψi,r (r ) sin θ antisymmetric relative to the x-axis: (0)

(d)

ψi,r = ψi,r (r ) + ψi,r (r ) sin θ.

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(5.6) Reprinted from the journal

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Fig. 1 Rider component of a hybrid modon. Profiles of the rider streamfunctions at H1 : H2 = 1 : 4 and A = 1. a Upper-layer (0) (0) singular streamfunction ψ1,s , b lower-layer singular streamfunction ψ2,s , c regular streamfunctions ψ1,r and ψ2,r in the layers (after [28])

Fig. 2 Dipole component of the hybrid modon. Contours of the dipole streamfunctions in the layers at H1 : H2 = 1 : 4. Solid (d) lines positive contours (anticyclone), dashed lines negative contours (cyclone). a Upper layer, max |ψ1,r | ≈ 0.3, contour interval (d)

0.05; b lower layer, max |ψ1,r | ≈ 2.18, contour interval 0.4 (after [28])

Amplitude A of the singular vortex is arbitrary, and the amplitude of the circularly symmetrical regular rider depends on A. In Fig. 1, the radial profiles of the singular and regular components of the circularly symmetric rider are shown in the case A > 0 (i.e., when the singular vortex is a cyclone). In this case, the regular-component rider must have an anticyclonic rotation in both layers to equilibrate the singular vortex, i.e. to yield u , zero total angular momentum of the modon (e.g. [4,5]). In the upper layer, the singular component, ψ1,s

(0) dominates over the regular rider, whereas in the lower layer, the regular component ψ2,r is dominant. The dipole component of the hybrid modon is shown in Fig. 2. An important feature of the dipole component is that the amplitude of the lower-layer streamfunction is much larger (almost by an order of magnitude) than the amplitude of the upper-layer streamfunction. Another important property of the hybrid modon solution is its smoothness in the sense that the streamfunction of the regular field in each layer is continuous up to the second derivatives. Our special interest in a solution with a continuous regular PV is motivated by the fact that, in regular dipole-plus-rider solutions, the modons with smoothly matched exterior and interior fields (first suggested by Kizner [11,12] for any stratification) are much more durable [15,16] than the modons with non-smooth riders described in [4].

6 Non-stationary evolution of an individual singular vortex In this section, we consider the non-stationary dynamics of an individual singular vortex on a β-plane assuming the regular field to be absent at the initial moment. At subsequent times the nonlinearity and β-effect result in Reprinted from the journal

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the development of a regular flow component forcing the vortex to move. Understanding of this process is quite good in barotropic models. Bogomolov [2] investigated the initial stage of motion of a point vortex in a thin rotating spherical layer and showed that a cyclone (anticyclone) point vortex placed in an initially immovable fluid starts moving northwest (southwest). Reznik [23] considered the evolution of a point vortex in the β-plane 1 21 -layer model for longer times and showed that the beta-effect and nonlinearity create a regular dipole circulation in the vortex vicinity, the so-called beta-gyres. The beta-gyres advect the vortex along the dipole axis and, in turn, are advected by the flow associated with the vortex itself. The latter causes turning of the dipole axis counterclockwise for a cyclone and clockwise for an anticyclone. As a result, the cyclone (anticyclone) moves north-westward (south-westward) along some curved trajectory. Analysis of the drift of an individual singular vortex in the non-divergent barotropic model generally confirms this scheme [29]. The difference, however, is that, with increasing time, the divergent point vortex tends to move zonally with the Rossby drift speed, whereas the non-divergent singular vortex tends to move meridionally with increasing speed. The evolution of a non-divergent non-localized vortex on a barotropic β-plane was examined by Llewellyn Smith [19]. Non-stationary evolution of a singular vortex in a stratified fluid on a β-plane depends significantly on the vertical structure of the vortex. This is because of the possible interaction between the vortex elements residing at different depths. Reznik et al. [25] examined the evolution of a purely baroclinic singular vortex in a two-layer fluid under the rigid lid condition. Initially such a vortex is a pair of coaxial oppositely signed singular vortices placed in different layers, the total angular momentum of the pair being zero. Due to the β-effect, the vortex axis inclines; the upper and lower vortices move apart and start interacting, which causes the eastward drift of the vortex pair. Numerical experiments with distributed baroclinic vortices [15,20,21] show that, eventually, the separation between the vortices stabilizes, so that the vortex pair transforms into an eastward-traveling modon. Most recent in this context is the study of the motion of a singular vortex residing in one layer of a two-layer fluid under rigid lid [28]. Such a vortex is not purely baroclinic but rather contains both the baroclinic and barotropic modes comparable in magnitude. The presence in the initial state of a substantial barotropic component (along with the baroclinic one) makes the upper and lower flows co-rotating, which changes the vortex evolution considerably compared to that considered in [25]. Below we provide the results of this study (for details see [28]). Assume that the vortex moves along some trajectory r = r0 = (x0 (t), y0 (t)) with the speed U = (U, V ), where U = x˙0 (t) and V = y˙0 (t). In a moving frame of reference with the coordinates (x  , y  ) = (x −x0 , y−y0 ), the layer streamfunctions are: u ψi = ψi,r (x  , y  , t) + Aψi,s (x  , y  ), i = 1, 2.

(6.1)

Here ψi,r is the streamfunction of the regular flow induced by the β-effect and nonlinearity. Substitution of (6.1) into (2.13) and calculation of ψi,r reveals that, in the leading order, the regular flow in each layer appears as a dipole, i.e. a pair of β-gyres, whose structure and magnitude vary with time [28]. Once function ψ1,r is known, the translation speed of the vortex can be calculated based on (2.13b). The results of these calculations are conveniently analyzed in terms of the barotropic and baroclinic β-gyres, with the barotropic and baroclinic streamfunctions, ψBT and ψBC , determined by the following linear relationships: ψ1,r = ψBT + α2 ψBC ,

ψ2,r = ψ BT − α1 ψBC ;

αi = Hi /(H1 + H2 );

(6.2a,b)

(specific expressions for ψBT and ψBC appear in [28]). The development of the β-gyres is shown in Fig. 3 for the case A > 0 (cyclonic singular vortex). As may be seen from Fig. 3, the regular flow in each layer is a dipole that induces the north-westward drift of the singular vortex. The strength of the dipoles increases with time and, therefore, the total translation speed also grows with time, and the drift of the monopole accelerates. An important feature of this evolution is that, with the passage of time, the barotropic β-gyre enhances much faster than the baroclinic one: by t equal to 50 advective time units the barotropic component exceeds the baroclinic one in the order of magnitude. It can be shown that, at some moment, the growth of the baroclinic component ceases; thereafter, this component decays [28]. The main effect of the baroclinic β-gyres is apparent in tilting the vortex axis (which is vertical at t = 0), but this effect attenuates with time, while the effect of the barotropic β-gyres enhances. Thus, the evolution of an individual singular vortex confined to one layer, being different from that of a purely baroclinic singular vortex, becomes similar to the drift of a barotropic monopole: a cyclone moves roughly northwestward, and an anticyclone southwestward. We believe this behavior is due to the presence of a sufficiently strong barotropic component in the individual singular vortex.

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Fig. 3 Barotropic and baroclinic β-gyres. Contours of streamfunctions at H1 : H2 = 1 : 4. Solid and dashed lines as in Fig. 2. Upper panel t = 1, lower panel t = 50. a ψBT , max |ψBT | ≈ 0.26, contour interval 0.05; b ψBC , max |ψBC | ≈ 0.15, contour interval 0.05; c ψBT , max |ψBT | ≈ 10.97, contour interval 1; d ψBC , max |ψBC | ≈ 1.01, contour interval 0.1 (after [28])

7 2D vortex pair in a radial flow The following example of non-stationary behavior of point vortices interacting with a regular flow applies to the astrophysical problem of obscuring tori in the active galaxy nuclei [1]. The singular flow induced by a number of logarithmic vortices is given by the sum 1 ψs = Am ln |r − rm |, (7.1) 2π m while the regular field is a radial flow produced by a point source, i.e. ψr = −Qϕ,

(7.2)

where r and ϕ are the polar coordinates with the origin in the source. The source strength Q is set to be constant for simplicity; the radial flow diverges at Q > 0 and converges at Q < 0. With the regular field given by (7.2), (2.13a) is satisfied identically, and the motion of the vortices is described in a Hamiltonian form by the ordinary differential equations:   ∂H ∂H xm 1 Am (x˙m , y˙m ) = − , Am An ln rm,n − Q Am arc cot . (7.3) , H= ∂ ym ∂ xm 4π ym m m =n

We now consider a dipole moving towards the source so that the dipole axis (the x-axis in Fig. 4a) always passes through the origin. In this case, due to the obvious symmetry, (7.3) can be reduced to two equations for one vortex of the strength A whose y-coordinate is, for example, positive: x˙ = −∂ H /∂ y, Reprinted from the journal

y˙ = ∂ H /∂ x,

H = (A/4π) ln y − Qarc cot(x/y). 89

(7.4)

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Fig. 4 Dipole in a divergent radial flow. a Dipole moving towards the source. b Vortex trajectories in a divergent flow at different values of L = 2π Q/A and E = 0. The trajectory marked with L = 1 separates the regimes with and without reverse motion of the dipole; arrows indicate the direction of the vortex motion (after [1])

 Here A = A2 > 0, r = x 2 + y 2 . It readily follows from (7.4) that y˙ > 0 at Q > 0 and y˙ < 0 at Q < 0, i.e. the distance between the vortices monotonically increases (decreases) in a divergent (convergent) radial flow. Clearly, the Hamiltonian H , or energy, of system (7.4) is an invariant of motion. Therefore, the possible vortex trajectories can be determined from the equation H = E = const. Typical trajectories are given in Fig. 4b, they depending on the ratio of the source strength to the dipole amplitude, L = 2π Q/A. In a divergent radial flow (Q > 0) vortices in the dipole gradually move apart, get around the source and, after all, the dipole propagates in the same direction but with a smaller speed. The separation between the dipole vortices at the initial and final stages are d+∞ = 2 exp (4π E/A) and d−∞ = 2 exp [4π(E + π Q)/A] ,

(7.5)

respectively. Thus, the dipole always survives but, after passing the source vicinity, the separation increases and the translation speed decreases exp(4π 2 Q/A) times compared to the initial values. The condition L < 1 (‘weak’ source) implies that x = x(t) is a monotonic function of time, and the corresponding trajectory does not have inflection points. At L = 1, there is one point at the trajectory in which x˙ = 0, and at L > 1 there are two such points. Therefore, at L > 1 (‘strong’ source) there is some period of time when x˙ < 0 and, therefore, the dipole executes a reverse motion. In a convergent radial flow, where Q < 0, the trajectories of the dipole vortices have the same shape as in the previous case, but the sequence of events is opposite: in a dipole moving towards a sink, the vortex separation decreases and the dipole speed grows. Due to the exponential dependence of the final translation speed on the sink strength |Q|, the dipole can accelerate considerably, increasing its speed by many orders of magnitude after passing the sink region. It is useful to note also that the above evolution of the dipole models the behavior of a vortex ring in a spherical radial flow. 8 Summary and discussion This paper presents a review of the key ideas and main results of the theory of quasigeostrophic singular vortices in one- and two-layer fluids. We focused on the problem of interaction of singular vortices with the regular flow. We presented the equations governing the joint evolution of singular vortices and regular flow, and the conservation laws yielded by these equations, and discussed a few problems that can be treated by means of this theory. Using the energy and enstrophy integrals we proved the nonlinear stability of an arbitrary point vortex pair on the f -plane with respect to any small regular perturbation with finite energy and enstrophy. We analyzed steadily translating systems of singular vortices on β-plane and presented a hybrid modon – a superposition of an upper-layer singular vortex and a regular localized flow consisting of a dipole component and a circularly symmetric rider. We discussed the problem of non-stationary drift of a singular vortex on a β-plane in barotropic and stratified fluids. This drift depends strongly on the vertical structure of the vortex. Special consideration was given to the situation when a singular vortex resides in one layer of a two-layer fluid and, therefore, contains both the barotropic and baroclinic modes that are comparable in magnitude. The main result

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of our analysis is the conclusion that, with the passage of time, the influence of the baroclinic component of the β-gyres on the vortex drift attenuates, and the β-gyres tend to become barotropic. Accordingly, the trajectory of the vortex appears qualitatively similar to the track of a barotropic monopole on a β-plane. Finally, in connection with the problem of motion of obscuring tori in active galaxy nuclei, we considered the evolution of a logarithmic point-vortex pair in a two-dimensional radial flow produced by a point source, and showed that the dipole always survives after collision with the source. In a divergent background flow, the separation between the vortices increases, while the dipole translation speed decreases after the collision. In a convergent flow, the vortices get closer, and the translation speed increases with increasing time. Acknowledgments We thank the organizers of the IUTAM Symposium, in particular, Hassan Aref and Dorte Glass for their hospitality and care. This research was supported by the Russian Foundation for Basic Research grant 08-05-00006. Z.K. acknowledges the support from the Israel Science Foundation grant 628/06.

References 1. Bannikova, E., Kontorovich, V., Reznik, G.M.: Dynamics of a vortex pair in radial flow. J. Exp. Theor. Phys. 105(3), 542–548 (2007) 2. Bogomolov, V.A.: On the vortex motion on a rotating sphere. Izv. Akad. Nauk SSSR, Phys. Atmos. Okeana 21, 391 (1985) 3. Flierl, G.R.: Isolated eddy models in geophysics. Ann. Rev. Fluid Mech. 19, 493–530 (1987) 4. Flierl, G.R., Larichev, V.D., McWilliams, J.C., Reznik, G.M.: The dynamics of baroclinic and barotropic solitary eddies. Dyn. Atmos. Oceans. 5, 1–41 (1980) 5. Flierl, G.R., Stern, M.E., Whitehead, J.A.: The physical significance of modons: Laboratory experiments and general integral constraints. Dyn. Atmos. Oceans. 7, 233 (1983) 6. Gryanik, V.M.: Dynamics of singular geostrophic vortices in a two-layer model of the atmosphere (ocean). Izv., Atmos. Ocean Phys. 19, 227–240 (1983) 7. Gryanik, V.M.: Singular geostrophic vortices on the β-plane as a model for synoptic vortices. Oceanology 26, 126–130 (1986) 8. Gryanik, V.M., Borth, H., Olbers, D.: The theory of quasi-geostrophic von Karman vortex streets in the two-layer fluids on a beta-plane. J. Fluid Mech. 505, 23 (2004) 9. Hogg, N.G., Stommel, H.M.: The heton, an elementary interaction between discrete baroclinic geostrophic vortices, and its implications concerning eddy heat-flow. Proc. Roy. Soc. London A. 397, 1–20 (1985) 10. Hogg, N.G., Stommel, H.M.: Hetonic explosions: the breakup and spread of warm pools as explained by baroclinic point vortices. J. Atmos. Phys. 42, 1465–1476 (1985) 11. Kizner, Z.I.: Rossby solitons with axisymmetrical baroclinic modes. Doklady Akad. Nauk SSSR 275, 1495–1498 (1984) 12. Kizner, Z.I.: Solitary Rossby waves with baroclinic modes. J. Mar. Res. 55, 671–685 (1997) 13. Kizner, Z.: Stability and transitions of hetonic quartets and baroclinic modons. Phys. Fluids. 18(5), 056601 (2006) 14. Kizner, Z.: Hetonic quartet: Exploring the transitions in baroclinic modons. In: A.V. Borisov et al. (eds.), IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, IUTAM Bookseries, 6. Springer, 125–133 (2008) 15. Kizner, Z., Berson, D., Khvoles, R.: Baroclinic modon equilibria on the beta-plane: stability and transitions. J. Fluid Mech. 468, 239 (2002) 16. Kizner, Z., Berson, D., Khvoles, R.: Non-circular baroclinic modons: Constructing stationary solutions. J. Fluid Mech. 489, 199–228 (2003) 17. Klyatskin, K., Reznik, G.M.: On point vortices on a rotating sphere. Oceanology 29(1), 21–27 (1989) 18. Larichev, V.D., Reznik, G.M.: On the two-dimensional solitary Rossby waves. Dokl. Akad. Nauk SSSR 231(5), 1077– 1079 (1976) 19. Llewellyn Smith, S.G.: The motion of a non-isolated vortex on the beta-plane. J. Fluid Mech. 346, 149 (1997) 20. McWilliams, J.C., Flierl, G.R.: On the evolution of isolated, nonlinear vortices. J. Phys. Oceanogr. 9, 1155 (1979) 21. Mied, R.P., Lindemann, G.J.: The birth and evolution of eastward-propagating modons. J. Phys. Oceanogr. 12, 213– 230 (1982) 22. Reznik, G.M.: Point vortices on a β-plane and Rossby solitary waves. Oceanology 26, 165–173 (1986) 23. Reznik, G.M.: Dynamics of singular vortices on a β-plane. J. Fluid Mech. 240, 405–432 (1992) 24. Reznik, G.M., Dewar, W.: An analytical theory of distributed axisymmetric barotropic vortices on the beta-plane. J. Fluid Mech. 269, 301–321 (1994) 25. Reznik, G.M., Grimshaw, R.H.J., Sriskandarajah, K.: On basic mechanisms governing two-layer vortices on a beta-plane. Geoph. Astrophys. Fluid. Dyn. 86, 1–42 (1997) 26. Reznik, G.M., Grimshaw, R., Benilov, E.: On the long-term evolution of an intense localized divergent vortex on the beta-plane. J. Fluid. Mech. 422, 249–280 (2000) 27. Reznik, G.M., Kizner, Z.: Two-layer quasigeostrophic singular vortices embedded in a regular flow. Part I: Invariants of motion and stability of vortex pairs. J. Fluid Mech. 584, 185–202 (2007) 28. Reznik, G.M., Kizner, Z.: Two-layer quasigeostrophic singular vortices embedded in a regular flow. Part II: Steady and unsteady drift of individual vortices on a beta-plane. J. Fluid Mech. 584, 203–223 (2007) 29. Reznik, G.M., Kravtsov, S.: Dynamics of a barotropic singular monopole on the beta-plane. Izv., Russian Acad. Sci., Atm., Ocean Phys. 32(6), 762–769 (1995)

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Theor. Comput. Fluid Dyn. (2010) 24:77–93 DOI 10.1007/s00162-009-0172-z

O R I G I NA L A RT I C L E

Edward A. Spiegel

Cosmic vortices in hot stars and cool disks

Received: 18 March 2009 / Accepted: 13 August 2009 / Published online: 8 November 2009 © Springer-Verlag 2009

Abstract The radiation that permits us to observe cosmic bodies also plays a role in their structure and evolution. While the thermal aspects of the radiation are familiar to fluid dynamicists, at least qualitatively, the dynamical effects of the radiation are perhaps less so, though these effects are becoming quite important in current astrophysical studies. This subject, which I have provisionally been calling photofluiddynamics after some discussion with the late James Lighthill, has a number of applications to cosmic objects. The most massive stars known are very hot and are the sites of vigorous fluid dynamical activity. The processes involved are of interest, not only in themselves, but also in the way they affect the observed features of the hottest stars by forming coherent vortices and magnetic flux tubes. Similar structures in accretion disks, particularly in protoplanetary systems, arise and play important roles in the evolution of those objects. Here, we shall consider only disks that, like the primitive solar nebula, are relatively cool and in which vortices may participate in the formation of planets. Keywords Macroturbulence · Fluidized beds · Photon bubbles · Accretion · Eddington limit · η Carinae PACS 97.10.Gz · 47.20.Ky · 47.20.Bp · 47.55.dd · 47.55.Lm · 95.30.Lz · 47.70.-n Prologue Cosmic bodies are at such great distances from us that we do not see what is happening on them very well. What we do see is external and at low resolution and is usually consequent on the inner workings that we are trying to learn about. However, a meeting on vortices makes an astrophysicist perk up and wonder about how a discussion of these ubiquitous structures may advance our understanding of distant objects that are fluid dynamically very active. In modern fluid dynamics, and especially in astrophysical fluid dynamics, much reliance is placed on numerical experiments. Modern means of visualizing the output of such experiments are helpful in giving the researcher some understanding of the workings of processes that are not so easily observed directly. However, in this article, I want to lay stress on the advantages of using laboratory experiments to gain insight into unresolved fluid processes in cosmic bodies. There are many experiments that have been performed that may be of use in this endeavor and even some that are easily performed and that also may be helpful. My aim here is to discuss some aspects of cosmic vorticity on the basis of laboratory experiments bolstered a little by physical intuition and, in passing, by mathematical arguments. I believe that in this way one can go quite far in uncovering processes in stars and disks that might not otherwise have been anticipated. Several of the experiments, I will call upon have been done by others while a few are so simple that I have done them myself or with friends. I hope this can show how astrophysical fluid dynamics can be advanced by simple experiments and that laboratory astrophysics need not be so largely confined to the realm of atomic physics. Of course, I realize that it is not really possible in the lab to duplicate the fluid processes that occur in cosmic Communicated by H. Aref E. A. Spiegel(B) Astronomy Department, Columbia University, New York, NY 10027, USA E-mail: [email protected] Reprinted from the journal

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conditions. The laboratory phenomena that I call on here are merely analogs of the astrophysical ones that I will propose. They are selected because they manifest physical features that I consider central to the problem at hand. 1 Dond’escono quei vortici? What is a vortex? For some, structures such as the rolls of two-dimensional convection are vortices. To me, that is too inclusive a characterization. I prefer the definition offered by Saffman in his book on vortex dynamics [43]. For him, a vortex is an isolated structure. In two dimensions it is “a finite region in the plane with non-zero vorticity surrounded by irrotational fluid.” In three dimensions he makes a similar if more elaborate distinction—in short, a vortex is a (vortex) filament. However, we cannot stay within the strict confines of vortex dynamics in an astrophysical story; we must include magnetic fields. Therefore, I shall include discrete magnetic flux tubes under the rubric of cosmic vortices, if only because it makes the general discussion easier to manage than if I treated them separately. That being said, we may assert that every rotating, turbulent, astronomical object that can be observed with sufficient resolution is seen to display what we may call coherent vortices, structures that last much longer than their turnover times. In fact, the prevalence of vortices throughout the universe is becoming increasingly apparent and this brief sketch is intended to suggest some ideas about the topic that I hope will interest students of vortex dynamics. I have described other aspects elsewhere [47], but both here and there the discussion is frankly phenomenological. In order to support my arguments, I will call upon results from laboratory experiments (mostly of others) unlike the more popular approaches based on numerical experiments. In what follows, I focus on a pair of examples to illustrate the roles that vortices may play in the dynamics of their host bodies. If this entails more astrophysics than you care to know, try the second example (Chapter 3); it is the lighter one. Unfortunately, in neither example, am I able to properly respond to the question that forms the title of this chapter of the article and that I owe to Lorenzo Da Ponte, the founder of the Italian Department at the university where I work. 2 Stellar vortices 2.1 Sunspots1 The sun usually displays a certain number of spots that are somewhat darker than the surrounding surface. These can be seen with the naked eye (but do not look!), so their existence has been known for millenia. When Hale measured magnetic fields on the sun a century ago, he noticed that they were quite strong in the sunspots (a few kilogauss) [27]. Much later, in the thirties, L. Biermann proposed that the relative darkness in the spots was a consequence of the local inhibition of convection by the protruding magnetic flux tubes. Hale [27] and others [35,58] observed “whirls” related to sunspots though he was cautious in accepting the belief of several theoreticians that the observed fields were generated by vortices and that sunspots were analogs of terrestrial storms. An extensive development of the vortex model was given by Bjerknes [8], an early user of the term “cosmic vortices.” The attempt to seek the origin of the observed magnetic fields in vortices may perhaps be seen as an early (if unsuccessful) attempt to fashion a solar dynamo theory. There is however not much evidence for the prevalence of vortices on the sun’s surface. Admittedly, there is an occasional report of what may be a strong vortex on the sun, as in the twisted sunspot pointed out by Akasofu [2]. Yet, there is nothing like the level of vortical activity that might be expected to develop in turbulent convection in a rotating system, though there are signs of small scale vorticity on the solar surface [9]. These dimly seen plunging plumes may descend from the vertical vorticity modes of convection theory [34] boosted by rotation to resemble the helical structures calculated by Veronis [55], pictures of which grace the cover of the Dover edition of Chandrasekhar’s book on hydrodynamic stability. Though such helices may contribute to dynamo action in the solar convection zone, they are not the well-defined objects that are of interest here. The powerful vortices that were invoked in the early theories to produce sunspots are not often seen at the solar surface. However, the tachocline, a transition layer between the outer convective layer of the sun and the inner sun, is rather like a weather layer in planetary atmospheres and may be banded and even form vortices [48]. Here, again there are suitable experiments on rotating convection that may provide guidance for 1

This brief mention of sunspots is provided by the way of orientation for those new to standard astrophysics.

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such thinking [28]. Such a notion engaged Steve Meacham and me some years ago and he made simplified simulations in which the putative tachoclinic vortices wound up and extruded magnetic fields so that they floated upward to emerge (we presumed) to produce spots. At some stage, helioseismology may be able to test this possibility. It may not be appropriate to speak of magnetic flux tubes in a meeting on vortices, but the predominance in the sun of the former over the latter presents an interesting question. Perhaps the main physical difference between the solar magnetic activity and the Jovian vortical activity may be traced to the difference in the magnetic Prandtl number as some recent (as yet) unpublished simulations by Annalisa Bracco suggest. As the magnetic Prandtl number is tuned, we go from the magnetic regime to the vortical regime. The crossover might then take place in the brown dwarfs, stars intermediate in mass between Jupiter and the sun. Investigation of this dichotomy may help in understanding the formation of both kinds of vector tubes.

2.2 Hot stellar atmospheres 2.2.1 Vigorous activity I opened this discussion with sunspots to recall how a relatively familiar object fits into the story. However, the real interest here is in another class of stars. For stars, as for other entities, classification begins with just two categories—good/bad, big/small and so on. Further refinement normally follows and new categories may be introduced, but let’s settle for two here. Thus, we may classify stars as hot or cool (relatively speaking). In order to classify a star at a glance, we may look at its surface temperature, TS , and its luminosity, L, the rate at which it emits radiant energy. A plot (log–log, say) of these two properties against each other reveals a dominant narrow strip called the main sequence, running monotonically through this parameter plane—the Hertzsprung–Russell diagram. The theory of stellar structure, supported by extensive measurement, tells us that the main sequence is essentially a one-parameter family and that parameter is the star’s mass. The story of how this understanding was developed is classical astrophysics and is to be found in many basic texts. In brief, if we know the mass of a star and we assume that it is in near hydrostatic equilibrium, we can gauge its central pressure, hence its central temperature and its internal energy. We may then observe the rate of energy emission, and so find the thermal lifetime, called the Kelvin–Helmholtz time. Since the K–H lifetimes are far longer than dynamical times, this justifies the hydrostatic assumption that we just made. And we see that a more massive a star needs a hotter core to provide enough pressure to keep its outer layers from collapsing in on it. Thus, the hot/cool dichotomy is a high-mass/low-mass distinction. In the stars of low mass, of which the sun is a good example (M = 2 × 1033 g), the outer layers (one-third of the radius in the solar case) are fully convective. The cause is the partially ionized hydrogen, the most abundant element by far, which renders the medium very opaque. Thus, a steep temperature gradient is needed to drive the photons through the outer layers of the sun and that promotes convective instability. Moreover, the partial ionization raises the specific heat and hence lowers the adiabatic temperature gradient (g/C p ), and so favors the onset of convection. By contrast, the hot stars are almost fully ionized and thermal convection does not play a role in the outer layers of the typical main sequence hot stars. On the other hand, while the cores of hot stars are convecting, the cores of the cool stars have little or no convective activity. In gross, the difference is caused by the different chains of nuclear reactions in those stellar cores. The reactions that occur in the cores of cool stars are not so very sensitive to temperature, whereas the processes in the cores of hot stars are quite temperature sensitive. In the latter case, then, the hot cores develop very steep temperature gradients and they have powerful convection. You might think from this description that the atmospheres of hot stars are not of great interest to the fluid dynamicist. Yet, the opposite is true. The vigorous fluid dynamical activity of the atmospheres of the hottest stars is one of the two main themes of this narrative and I believe that it involves strong vortices. I will attempt to justify this claim on physical grounds, but first I need to describe an early problem that hot stellar atmospheres posed. (Hot here means surface temperatures, TS , upwards of a few×104 K.) The profiles of spectral lines observed from stars indicate the dispersions of velocities in their atmopheres, where the spectral lines are formed. Random motions of particles contributing to the line formation produce something like a Gaussian line profile whose dispersion reveals the rms value of the component of the velocity along the line of sight. O. Struve and others reported long ago that, in very hot stars, the rms speeds are apparently supersonic, according to this detection method. The generally received interpretation of this result is that very hot stellar atmospheres are in supersonic turbulent motion. Without going into what (if anything) is Reprinted from the journal

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meant by “turbulent” in this context, let me mention some suggestions that have been made about what could be causing such lively motions. •

•

•

Ledoux [33] and others found that the cores of hot stars are unstable to growing oscillations for reasons related to the energy generating mechanisms in the stellar cores. For massive stars, there is a Hopf bifurcation (overstability, in Eddington’s terminology) and the stellar pulsations that develop may become vigorous and excite many modes with rich structures. A mixture of such modes might cause the line broadening. Stars form by condensing out of the dusty gas of the interstellar medium. The protostars will generally have a net angular momentum and, as they contract down to stellar size, their rotation rates increase, typically reaching rotation rates at which their centrifugal accelerations are comparable to the gravitational accelerations at the surfaces of those stars. In the likely conditions in such stars, the medium is baroclinic. That, combined with rapid rotation, has led people to seek rotational instabilities to rationalize the vigorous motions in hot stellar atmospheres [59]. A third possibility, the one I will pursue here, begins with this observation cum suggestion of Huang and Struve [30]: turbulent motions are favored in regions of low density in hot stars and...radiation pressure is probably important in creating turbulence.

None of these proposed causes of intense fluid dynamical activity is sufficiently explicit. It is unlikely that any of them has enough fluid dynamical theory behind it as yet to convince readers of these proceedings of their validity. (If such readers decide to look further into these issues, so much the better.) In any case, it need not be that only one of these mechanisms is operative nor is it excluded that other mechanisms may yet be proposed. My aim here is to suggest how the third of them may lead to vortex formation in the atmospheres of very hot stars. 2.2.2 Photon bubbles A vision of the nature of vigorous radiative stirring of the matter in hot stellar atmospheres was revealed to me in the late sixties at a GFD meeting in Newcastle. I had my epiphany in a lecture on fluidized beds by A.A. Mills [39] when he demonstrated what happens when a gas is forced upward through a bed of particles at a sufficiently high-flow rate to levitate the particles. When that happens, the bed expands in a sort of phase transition. Since the drag per particle decreases when the number of particles per unit volume decreases, this expansion does not continue indefinitely and a two-fluid state is achieved consisting of the gas traversing the bed and the newly formed gas of levitated particles. The gas flowing through the layer forms bubbles (or particle voids) that produce a vigorous stirring of the fluidized bed. The phenomenon is easily reproduced in the laboratory, especially if you have friends like Jack Whitehead who allow you to use their lab facilities and to enlist the help of their technicians (see Fig. 1). The dependence of the drag per particle on the particle number density is at the origin of instabilities in the bed and, in a way, is analogous to the so-called κ-mechanism of stellar pulsation theory, in which the opacity of the medium varies in a suitable manner with changes in thermodynamic quantities. Those instabilities lead to waves that propagate vertically and that have been nicely recorded by Guazzelli [26]. The waves in turn are subject to secondary instabilities and those give rise to particle voids or bubbles [26]. In the stellar analog [41], radiation flowing out of a star, like the fluid flowing through fluidized beds, exerts an upward force on the stellar material. The matter is ionized and the principal source of opacity is from electron scattering, which has a known cross-sectional area, σ , per electron. If the outward flux of radiant energy is F, then the momentum flux is F/c where c is the speed of light. The outward flowing radiation is partially blocked by the intervening matter, which feels an outward force expressed, per unit mass, as an equivalent acceleration, σ F/(mc), ˆ where mˆ is of the order of the mass of a proton in typical stellar chemical compositions. When the radiative acceleration is just equal to the gravitational acceleration at the surface, g, we have an analog of the onset of fluidization, called the Eddington limit in astrophysics. Since the stellar material was a gas to begin with, this limit corresponds only to the onset of levitation. In the early seventies, radiatively driven instabilities had already been found to excite waves in hot stellar atmospheres. In a lively meeting in Nice [14] several papers on the radiative wave instabilities in hot stellar atmospheres were presented and their physical consequences debated. Even now there remain issues that need to be better understood about these instabilities. In particular, it has been found that acoustic waves become

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Fig. 1 Bubbling in a gas-fluidized bed of glass beads

unstable in stratified atmospheres with heat conduction without the benefit of radiative forces [36,54]. Similar instabilities are also found when there are prevalent magnetic fields [37]. The importance of the additional contribution that is made by the radiative forces may then become qualitatively significant in the nonlinear regime and this feature also has an analog in the case of fluidized beds. A discussion of these issues would take me too far from my main subject so I reluctantly leave them here. In the sixties, hypothetical supermassive stars were under discussion as the possible energy sources of quasars. If these stars existed, they would be held up mainly by radiative forces and they would be convectively unstable [44]. This situation led Kahn and and Stanton [50] to consider that the thermals or convective bubbles with their low-mass densities would have high radiation pressures. This (photo)convection would be driven by buoyancy forces and, in this respect, it differs from the bubbling of fluidized beds, in which buoyancy is not a primary influence. Perhaps, the photoconvective bubbles have something in common with bubbles that are thought to form in the presence of magnetic fields [3,7,23], in which magnetic buoyancy may play a role. As in fluidized beds [26], the pulsation that arises in the primary radiative instability may give way to a secondary (or parametric) instability that produces bubbles. An appropriate instability in a simple pulsating star model has been isolated [40], even with no radiative forces, though complete justification of that remark awaits a full nonlinear study. The bursting bubbles seen in Fig. 1 are then surrogates for the postulated photon bubbles that may be a source of the vigorous dynamics of hot stellar atmospheres. It may also be worth interjecting here that the geometry of the flow can play a role in these processes. About 35 years ago in the Woods Hole GFD Program, Mike Frese and I decided to test this notion. We fluidized a layer of sand with water in a wedge of the sort that is used in the so-called Jeffery–Hamel flow. Later, I tried it with nitrogen (at 80 psi) as the fluidizing fluid. Things get quite lively in the fluidized wedges as seen front on in Fig. 2. The figure also shows that not every interesting dynamical structure that forms in fluidized beds need have sharp boundaries and that some of them may resemble convective plumes. For clear-cut edges on structures in the photofluid case, we would expect the photon mean free paths to be rather smaller than the sizes of the structures. For the stellar case, it is natural to introduce a nondimensional Eddington number E=

σF , mcg ˆ

(1)

whose value at the Eddington limit is unity. For a global evaluation, we may adopt the surface value of E . If we then multiply numerator and denominator by 4π R 2 , where R is the stellar radius, we find that E= Reprinted from the journal

σL , 4π mcG ˆ M 97

(2)

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Fig. 2 Fluidized sand in a wedge seen face on

where G is Newton’s constant. Thus, the Eddington number is measurable from outside a star, although the value found by this approach will have been modified by the action of the bubbles. The value of E for the sun and other cool stars, where radiative forces are generally negligible, is very small, especially when almost all the energy flux is carried by convection. This global form of E is useful for the hottest stars on the main sequence because of the mass–luminosity relation—in the high mass range, L varies like M to a power between 3 and 4. This means that, for the main sequence hot stars, E is a known function of M near the hot end of the main sequence. For those stars then, roughly, E ≈ 10−5 M7/3 where M is the ratio of the mass of the star to the mass of the sun. The Eddington limit occurs for main sequence stars with M ≈ 130. It has been suggested that there are almost no stars with masses greater than this because their matter would be blown away by the outward radiative force. Indeed, matter is observed to leave those stars in powerful winds. The mass loss process in the hottest stars is time dependent and episodic. Therefore, the question that has to be answered is, what is the mean rate at which mass is blown away by the radiative forces? We want to know whether very massive stars can lose enough mass to lower their values of E and so survive for a longish time. If enough mass is lost, we may see what was once the core of a massive star, made predominantly of helium (say), the hydrogen that was in the core having been used up in nuclear reactions. Indeed, helium stars are known (they provided my first encounter with this subject when I was a student) and this may be one way to make them. A suggestive rule of thumb for fluidized beds is that the throughput of the gas in the beds in excess of that required to cause fluidization escapes in the bubbles. If an analogous rule applies in hot stars, it would regulate the outflow so that any radiative flux in excess of that required to make E  1 would also escape in bubbles or radiative plumes. Moreover, just as bubbles in fluidization carry away particles with high cross-sections in a process called elutriation, we may look for something similar that would cause unusual chemical abundances in some stars. There are other possibilities to be explored such as the collapse a bubble when it comes near to the surface of the star. The collapse may produce a shaped charge effect or the sort of splash shown in Fig. 2. There is a lot to this problem and we are far from doing it justice. And then those profligate stars often fool us by being multiple stars, though astronomers know something of their tricks. An example of a star that is thought to be close to the Eddington limit (or perhaps just under) is η Carinae. It lies only 7, 500 light years away in a rich nebulosity where massive stars are being formed and it has a companion of about the same mass. Figure 3 shows the present appearance of the material around η Carinae as seen by Nathan Smith et al. using the Hubble Space Telescope [45]. The double-lobed cloud of material was blown out in an eruption that took place in the 1840s and some students of such behavior suggest

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Fig. 3 The hot star, η Carinae, whose outburst around 1842 produced this remnant (Courtesy Nathan Smith, U. Calif., Berkeley & NASA)

Fig. 4 Small hydrogen bubbles captured by vortices in a rotating tank [29]

that η Carinae will blow itself up any time now, say within the next million years. Then again, it may reduce its mass and save itself. The question is whether it can do this. 2.2.3 Vortices at last Once we accept the premise that bubbles are bursting at the surfaces of hot stars, we should ask how the rapid rotation of those stars may affect the fluid dynamics. The mass seen to flow from these stars at a great rate draws out magnetic fields that produce torques back on the stellar surfaces. These slow down the rotation rates after a time. However, at the rates at which radiant energy pours from the hot stars, they do not last very long before they disappear from our view one way or another. Therefore, for the observed hot stars, we are probably safe in accepting that they are young and so they are still rotating rapidly, as observed. Turning back to experiments, we consider Fig. 4 showing turbulence in a large rotating tank of water studied in Grenoble [29]. The rotation axis was vertical. In order to visualize the background flow, small hydrogen bubbles were introduced as markers. Figure 4 is a top view showing the vortices that did form. (Side views included in the original figure confirm this interpretation.) The bubbles used were so small that they did not Reprinted from the journal

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significantly affect the fluid motion and they are attracted right into the vortices. For the problem at hand, this fits in with another relevant experimental result: in bubbling fluidized beds, colliding bubbles coalesce and so form ever larger bubbles up to some limiting size at which they become unstable [17,18]. In a rotating bed, as the bubbles grow larger and congegrate, we may expect a strong vortex to form. Such a phenomenon was produced in a rotating fluidized bed by Dan Goldman and Chen Li at Georgia Tech. More accessible, if less germane, experimental configurations involve bubbling in rotating tanks of water. For example, Turner [53] has documented vortex formation in a rotating tank when dynamically important bubbles are introduced by dissolving Alka Seltzer tablets (see also [25]). At Woods Hole, Helfrich, Whitehead and I used a device for aerating fish tanks to produce bubbles in a rotating tank. In these cases central vortices formed. They were analogous to those seen in the rotating fluidized beds of Li and Goldman, though the central vortex in the rotating fluidized bed collapsed rather violently when the rotation was stopped. 2.2.4 What good is a vortex? In order to get an idea of why I think all this is relevant to stellar fluid dynamics, consider the very simple example of a vortex in the atmosphere of a hot star. We may approximate the gas pressure and the radiation pressure, respectively, as pg =

k m¯

ρT and pr =

1 E, 3

(3)

where the energy density of radiation is E = aT 4 ,

(4)

a is a Sefan’s constant and m, ¯ the mean particle mass, is hardly different from m. ˆ Atmospheric layers are geometrically thin and we may neglect their curvature and treat the gravitational acceleration as constant. In the especially simple case of a nonrotating, hot stellar atmosphere we may write the hydrostatic condition as ∂P = −gρ, ∂z

(5)

where z is the vertical coordinate and P = pg + pr is the total pressure. We adopt the case of a circular vortex and let r be the distance from the central axis and v be the circular velocity around the axis. In the r -direction, the static balance is ∂P v2 =ρ , ∂r r

(6)

and we shall assume that the centrifugal force is derivable from a potential. Then (5) and (6) may be combined to yield the condition ∇ P × ∇ V = 0,

(7)

where V is the total potential energy (gravitational plus centrifugal). Hence, P is constant on surfaces of constant V . In this pared down version of a model with no radiative viscosity or drag or other dissipative effects, v is not determined although, in real life, it may change on a long-time scale that we are ignoring here. For this illustration, we assume that v is independent of time and z and that it is given by the standard potential vortex with v = v0 r/r0 for r/r0 < 1 and v = v0 r0 /r for r/r0 > 1, where r0 and v0 are constants. Then, ⎧ 2 ⎨ r 2 − 1, if r < r0 ; 2 V = +gz − v0 2r0r 2 (8) ⎩− 0 , if r > r . 0 2 2r For a description of the flow of radiation through the star based on these considerations, we assume pure Thomson scattering so that the radiant energy is conserved. Hence, ∇ · F = 0,

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Fig. 5 Level lines of the total pressure gradient in a radiating atmosphere. The orthogonal curves suggest the radiative stream function

where F is the (total) radiative flux vector. We further adopt the simple approximation that the radiative flux is proportional to the gradient of the radiation pressure. Then, F =−

mc ˆ ∇ E = −K ∇ T, 3ρσ

(10)

where K =

4a mcT ˆ 3 3ρσ

(11)

may be interpreted as a radiative conductivity and the quantity m/ρσ ˆ is the mean free path of a photon through the atmosphere. It is significant that K is proportional to pr / pg . The message of this result is that K is large where pr >> pg so that, where there is comparatively little matter, the radiation flows out quite freely. However, when the density gets quite low, as it does in the vortex core, this treatment of the radiative transfer is inadequate and we need to elaborate the physics. This is not the place to do that, so we settle for the qualitative implications of Fig. 5 in which the quasi-horizontal curves show the level curves on which P (or V ) is constant and the somewhat vertical curves are the normals to them. The latter provide a qualitative description of the streamlines of the outflow [20] as focused by the vortex. As the surfaces of constant pressure are warped by the vortex, the radiation emerges from the star inhomogeneously as suggested by Fig. 5. However, what is the role of the global rotation, which has so far been omitted? It is a source of the angular momentum that is concentrated into the vortices by the incoming bubbles. This is another topic that requires elucidation and it will play a role in the following section on disks. When the medium is rotating differentially, there is a sharp difference between cyclonic and anticyclonic vortices and the issue of which ones form has been of some concern in the planetary context [57]. There will be a difference in the resultant density inhomogeneities that will be reflected in the distribution of the emerging radiative flux. However, this issue is less central here because, in hot stars, we are likely to be in the case of magnetic flux tubes rather than vortices. The story as I just told it is qualitatively reasonable because, in a magnetic flux tube, the matter density is going to be low in the core of a magnetic tube. The idea behind this suggestion is that the total pressure tends to homogenize and where the magnetic pressure is large the gas pressure will be small. The conclusion then is that magnetic flux tubes will produce strong localized inhomogeneities in the radiative outflows that provide safety valves for escaping radiation. Whatever their immediate dynamical effects, magnetic fields will allow stars to ameliorate the results of exceeding the Eddington limit, as some stars seem to be doing.

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3 Disks 3.1 Protoplanetary disks Normal stars are nearly spherical. They display very interesting phenomena and their study still has much to teach us. Nevertheless, disks are the cosmic bodies that are of greatest current astrophysical interest (if we overlook the entire universe). These are objects that rotate so rapidly that they take on rather flat shapes. Disks may themselves be thought of as vortices, though I shall not pursue that image here. In cases where the radiation pressure is not important, a simple axisymmetric model serves to give an impression of the structure of a disk. If a disk is axisymmetric and barotropic, it may be seen from the hydrostatic equations that the circular speed around a very thin disk is nearly independent of z, the coordinate measured from the plane of symmetry. This is like a vortex in which the centrifugal force is balanced by gravitational attraction toward the symmetry axis. The radial pressure gradient is generally negligible in rapidly rotating disks such as spiral galaxies. In the disks of spiral galaxies, the source of the force that may balance the centrifugal force produced by the circular motion is not known, but is generally taken to be a gravitational field produced mainly by dark matter that apparently has not been seen nor heard from. We do not know much about it beyond its name, which was conferred about 75 years ago. Here, we are concerned with the disks that are currently great centers of attentive study, the accretion disks that form as matter flows into important centers of gravitational attraction. In one example of disk formation, one member of a binary star receives matter from its companion and that matter forms a disk as it flows onto to its recipient. A more impressive example is provided by massive black holes in the centers of galaxies that accrete material. The dissipation in the resulting disks produces energy prodigiously and this is the mechanism of the brilliance of the quasars. Those examples involve too many nagging details to be discussed here, so I shall concentrate in the rest of this discussion on another category of disks, those that are to be found around stars of modest mass in the latter stages of their formation. It is generally accepted that planetary systems form in rotating disks of gas and dust that are left behind as protostars condense out of the interstellar medium. This vision had its beginnings in the nebular hypothesis of Kant and its more detailed description by LaPlace [12]. In the modern version, the dust and gas in the disk is supposed to coalesce somehow into planetesimals, the bodies that ultimately merge become planets and or perhaps morph into other objects in planetary systems. My concern in this discussion is with the role that vortices may play in that process. The general idea is that the disk around a newly formed star is thin and axially symmetric in the large, though deviations from this idealized picture are significant as I am arguing. The mass in the disk is much less than that of the central star and, in first approximation, the gravitational field in the disk is that provided by the star. The mean motion of the material in the disk is then (nearly) circular and Keplerian, with velocity 

2G M v= r

1/2 ,

(12)

where M is the mass of the central protostar and r is the distance from its center. The Reynolds number in such a disk is extremely large and, until recent years, it was generally taken for granted that such a disk would be turbulent. The eddies of this turbulence were called vortices by Weizsäcker [56] who visualized the flow (in a suitable frame) as rather regular as in Fig. 6. At the time of his work, there was some belief in a regular spacing of planetary orbits and, perhaps, Weizsäcker was inspired by Bohr’s atomic model to seek the explanation along those lines. In von Weizsäcker’s discussion of the role of turbulence in planetary formation and in the rediscussion of his work by Chandrasekhar [15], the word vortex was freely used. With those credentials, Weizsäcker’s ideas must be admitted to our discussion, though I would not call Weizsäcker’s image a portrait of vortices but rather of cellular motion. In any case, there is a more serious issue that arises. Keplerian flow is linearly stable according to Rayleigh’s criteria for stability of circular motion [49]. Therefore, the question of whether protoplanetary disks are turbulent is not easily answered. If no external driving mechanism is provided to start up turbulence or cells or some other type of fluid dynamical activity one needs some destabilizing effect to get some eddies going. This might come from nonlinear terms, baroclinic effects, magnetic fields, non-normal modes or initial conditions. Though all of these have been proposed, as far as I know, the only one of these effects that has been shown to promote instability heretofore, is the magnetic field [4]. And even the consequences of the magnetically caused instability, called magnetorotational instability, are under discussion.

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Fig. 6 Weizsäcker’s notion of the turbulent eddies in the primitive solar nebula

However, there is reason to believe that the primitive solar nebula, for example, was quite cool and not ionized in its early days, so that it is not clear how much of a role magnetic fields can play in the dynamics of the disks like the primitive solar nebula that preceded our own solar system. This is an interesting technical issue and one can devise various scenarii that favor one or the other vision of the process, but that matter is too involved for discussion here. Instead, let me mention a destabilizing effect that may work in a cool accretion disk.

3.2 Suction-induced shear instability If you introduce suction at the center of a circular motion, you get a swirling flow with an inward velocity component as in accretion flows. This is the sort of flow discussed in Batchelor’s book for reducing drag on airfoils and it also shares some features with the earth’s polar vortex. In order to study the instabilities in this class of swirling (or accreting) flows Gallet, Doering and I [21,22] have focused on the case of a stable Taylor–Couette flow between concentric cylinders to which suction is added. So far, we have treated only a differential rotation that is not Keplerian and is incompressible, so we are still not in the case of an accretion disk. It is of interest however that suction can destabilize a stable circular shear flow as it does in the case of plane Couette flow [19]. In the base flow whose stability we have studied, the outer cylinder rotates and the inner one is static. This produces a flow that is known to be stable. When we impose the same radially inward fluxes on both cylinders, as indicated in Fig. 7, we introduce a global inward component to the velocity field and consequently an inward mass flux. This flow has an inflection point in its circular velocity. The linear stability theory of the resulting flow shows that there is robust instability for a wide range of conditions with modes of instability as shown in Fig. 8. The energy stability theory of this swirling flow suggests (but does not establish) that there is also nonlinear (subcritical) instability. Therefore, the stately cellular motion that was envisioned by Weizsäcker is not likely to remain very orderly as one may anticipate from the existence of strong cellular motion. That is why, in an article on turbulence intended for twelve-year olds [46], I described the Weizsäcker vision with discrete vortices as illustrated in Fig. 9. Even this is too regular, but I did not want to further trouble the artist who had produced the figure for the article. On the opposite page of that encyclopedia I showed a picture from a simulation of geophysical vortices, hoping that an enterprising twelve-year old would make the connection and show up in my office with some original simulations. When, after a time, no one showed up, Dowling and I offered the suggestion that coherent vortices should be seen on disks explicitly into a paper about vortices Reprinted from the journal

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R2

Rotating cylinder

Injection angle

R1

Fig. 7 The model for Taylor–Couette flow with suction from [21]

1 0.98

Y

0.96 0.94 0.92 0.9 0.88 -0.2

-0.1

0 X

0.1

0.2

Fig. 8 A mode of instability in the swirling flow of Fig. 7 after [21]

Fig. 9 Artist’s conception of vortex field in a Keplerian velocity field with an exaggerated regularity as in von W.’s vision

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Fig. 10 The vortex field in a Keplerian velocity field arising from random initial conditions with no driving. Light spots are anticyclonic; yellow is an artist’s conception of the sun (After [10])

in stars and planets2 [20]. Finally, Philip Yecko appeared in my office wanting to study vortices in disks. He was no longer twelve, but still a graduate student nonetheless. Around that time, the possible role of vortices in the growth of planetesimals in protoplametary disks was being raised [6,52]. A vivid survey of the subject has been provided by Chavanis [16] who managed to cite even Descartes in his review.

3.3 Vortical attraction For a time, objections to the notion that there could be vortices in disks were expressed on the grounds that the Keplerian shear would quickly destroy such structures. In fact, this objection does apply to cyclonic vortices as several simulations in Keplerian shears have revealed [10,11,16,24]. Indeed, anticyclonic vortices decay only on viscous times unless or until they are swallowed by a larger vortex. Figure 10 shows an example of the vorticity field that forms after a few rotations in a maintained Keplerian flow with superposed, random velocity fluctuations in the initial conditions. The anticyclonic vortices form very quickly even in a stable-base flow if there is sufficient vorticity initially.3 The study of vortices in disks has been developing apace in recent years (e.g., [5,31,38]) and there is no general agreement on the development of these structures. Of special interest is the loss of vorticity (by radiation of Rossby waves, say) whose consequence may be the radial migration of vortices as they try to preserve their potential vorticity and, in the process, transport angular momentum. In the case of Keplerian disks, the anticyclonic vortices live many rotation periods of the disk. These vortices enter into the process of the formation of planetesimals by drawing dust particles together. In Fig. 4 we saw how the injected bubbles gather in the vortices, though they tend not to be found in the inner cores. Barge and Sommeria [6] and Tanga et al. [52] have suggested that, in a turbulent rotating disk, dust particles will be drawn into the anticylonic vortices that form. The locally enhanced particle density increases the rate of interparticle collision and thus favors their coalescence. In order to provide a better idea of this process, simultaneous calculations of the fluid dynamics and the particle motion in the resulting velocity field were performed [11]. The initial vorticity field was that of a Keplerian flow plus a superposed random vorticity field. The particles were initially distributed uniformly throughout the disk with the local Eulerian velocity of the fluid. The basic Keplerian flow component was maintained throughout the calculation. The simultaneous vorticity field and particle distribution are shown at 2 M. Abramowicz produced observational evidence that could be interpreted as caused by a vortex on an accretion disk in a quasar [1]. I forbear from discussing quasars here—they are too complicated. 3 In the hydromagnetic case, magnetic vortices form quickly also [10].

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Fig. 11 The simultaneous distributions of vorticity and of dust at three times during a simulation of vortex formation and dust agglomeration

three times after the beginning of the process in Fig. 11. The redistribution of the dust particles by the flow was computed from the equation of motion for individual particles. In doing this, we need to allow for the attraction of the Sun (or central mass) on the particles and the frictional drag on them when their velocities v differ from the velocity u of the ambient gas. We neglect the gravitational field produced by the dust itself while ∗2 the gravitational force of the Sun, − Gr M 3 r, is expressed in terms of the local Keplerian vorticity as −4ω K r. The feedback of the particles on the motion of the gas is also neglected. The frictional drag has the general expression −ξ(v − u), where ξ is the friction parameter appropriate to the location of Jupiter. (See [11] for details.) The behavior seen here is consistent with the previous studies in which vortices were introduced by hand, so to say, onto a rotating disk [6,52]. We work in a rotating frame of reference with angular velocity so that a particle feels the Coriolis force −2 × v as well as the centrifugal force 2 r from the global rotation. The equation of motion for a dust particle is therefore   d2 r dr dr = −2 × (13) − ξ − u(r, t) + [ 2 − 4ω∗K 2 (r )] r. dt 2 dt dt In unusual circumstances, additional terms must be introduced (e.g., [42,52]) but they are not of immediate interest here. In order to interpret the behavior of dust particles around a vortex in a disk, we go into a cylindrical coordinate system, (r, θ ), centered on the vortex axis as in [52] with θ = 0 in the direction of the local disk velocity. At the position of the vortex, presumed fixed in the disk, the local disk angular velocity is . We neglect the size of the vortex and the disturbance it produces in the ambient flow so that 2 = G M/R 3 , where R is the distance of the vortex from disk center. Then, in the neighborhood of the vortex, the radial acceleration of a particle with respect to the vortex axis is approximately d2 r ≈ r θ˙ 2 + 2r θ˙ − ξ(˙r − u r )θ˙ + 3r 2 sin θ dt 2

(14)

The first two terms on the right side of (14) combine into r θ˙ (θ˙ + 2 ) and reveal the essence of the process. By definition, > 0. For a cyclonic vortex, θ˙ > 0, so that particles are driven outward from the vortex. For an

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anticyclonic vortex, θ˙ < 0 and, just outside the vortex core, |θ˙ | is not as large as the ambient vorticity, so that the full term is negative and particles are sucked in. Well inside the core, depending on the vortex structure, we may have (θ˙ + 2 ) < 0 and particles may be pushed away from the vortex center. The other two terms in (14) play some role as well but are typically small and are not part of the main story. In the present approach, the vortices result from a dynamical calculation that shows how the evolution of vorticity in the disk has important consequences for the time scale of the capture process [11]. Turbulent fluctuations initially present in the disk already begin to drain particles into regions of negative vorticity while cyclonic regions rapidly depopulate. Therefore, the early evolution of the system is marked by the formation of packets of dust which render the disk inhomogeneous. This segregation, caused by the cyclone/anticyclone asymmetry, is efficient. At t = 2 (the time unit is essentially a few years), the density of the dust particles in the region ω˜ < − ω˜ 0 has already been increased by a factor 2. When the coherent vortices form, a large number of dust particles are already present nearby and the capture rate is increased. Therefore, the intensity of turbulent fluctuations present at the early stage of the disk evolution may play a significant role in the time scale of planet formation. And, in this meeting about vortices, it may be appropriate to add that we may owe our existence to the action of vortices. 4 The end In this discussion, I have been guided by laboratory analogs in trying to unravel some of the astrophysical phenomena that I have been describing. Laboratory experiments are perhaps more vivid than numerical ones, but it is easier to control the parameters in the latter. Each is helpful if used well but neither is quite able to reproduce the extremes of astrophysical conditions so there is room for argument. There is so much room that I hardly know how to begin this ending and will not even attempt a summary of what I have written but will simply add a few remarks to all that in an attempt to exit gracefully. An issue that I do not yet know how to resolve is whether a field of vortices forms in a bubbling rotating object (star or disk) or whether one central radiative column (phortex?) prevails. Newton used the fact that there were stars to conclude that the universe is infinite. Reversing his argument, I would imagine that the distinction in the present problem will turn on the ratio of the horizontal extent of the system to the depth of the bubbling region. 4.1 Hot stars I have not gone far into the astrophysical details of the hottest stars beyond mentioning their vigorous activity. This includes mass loss and variability, phenomena that are seen in milder versions in the sun. In the solar case, the driving mechanism is the magnetic activity manifested most directly in sunspots. This connection has led to the conjecture that the hottest stars have spots [13], on observational grounds. This fits in with the picture of formation of (generalized) vortices by coalescence of photon (or photomagnetic) bubbles under the influence of the rapid rotation of these stars. Whether this points a moral for those who study vortex formation in general is a question that might bear consideration. The immediate problem in that part of the discussion is how much the vortical escape valves for radiation can reduce the rate of mass loss and so permit the most massive stars to settle into a comfortable old age before they do themselves a serious mischief. A closely related problem is how stars with E  1 manage to form in the first place. Recent simulations suggest that radiation can punch through the inflowing matter and allow hot stars to form [32]. Other astrophysical minutiae clearly may be brought up at a time like this, but I have to desist because this is not the place. However, the possibility of vortices as escape valves in rotating fluidization may also have implications for problems of a practical nature [51]. 4.2 Cool disks Back in 1982, André Brahic wrote that one of the two great difficulties faced in forming a theory of the origin of the solar system is that we know only one planetary system [12]. However, we now know many more planetary systems but they differ considerably from our own. The theory of planetary formation is in great ferment and I have merely indicated one issue that arises in that lively subject. That vortices can make themselves felt in this complicated process is a nice surprise and, as one can imagine, they are very likely to make an impact on Reprinted from the journal

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the theory of hot disks as well. If I omit these latter it is because this article has already exceeded the length that the editor had in mind for it. The suggestion that vortices may exist on disks [20] (the ideas of Weizsäcker having been forgotten by and large) was greeted with great skepticism. One complaint was that there was no instability that could lead to vortex forming conditions short of introducing a magnetic field into the disk. Since it is not clear that the early planet forming disks are warm enough to produce the ionization needed to lend dynamical importance to the magnetic fields, the negativity was perhaps well founded. As I reported above, if there is an inflow in an otherwise stable differential rotation, instability occurs. (Though the disk case has not been explicitly treated from this point of view, the mechanism seems generic.) The instability will produce angular momentum transport and so engender the very inflow that gave rise to it. This is clearly a scenario for nonlinear instability. Acknowledgements I am grateful to P.-H. Chavanis for reminding me that Weizsäcker used the term vortices in his theory of planetary formation, to W.W. Sargent for telling me about the photoconvective work of Kahn and Stanton Thorne, to C. Doering for reading the ms and to H. Aref for his editorial advice about succinctness that came too late. And I am happy to thank Antonello Provenzale for his emendations though I regret that he withheld them till after the ms was submitted. Jack Whitehead took the photograph shown in Fig. 1, Nathan Smith gave me permission to use Fig. 3, Mike Frese participated in the fluidized wedge experiments, Dan Goldman and Chen Li kindly rotated their fluidized bed. I am happy to record here my appreciation to them all.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Abramowicz, M.A., Lanza, A., Spiegel, E.A., Szuszkiewicz, E.: Vortices on accretion disks. Nature 356, 41–43 (1991) Akasofu, S.-I.: Vortical distribution of sunspots. Planet. Space Sci. 33, 275–277 (1985) Arons, J.: Photon bubbles—overstability in a magnetized atmosphere. Astrophys. J. 388, 561–578 (1992) Balbus, S.A., Hawley, J.F.: Instability, turbulence, and enhanced transport in accretion disks. Rev. Mod. Phys. 70, 1–53 (1998) Baranco, P., Marcus, P.S.: Three-dimensional vortices in stratified protoplanetary disks. Astrophys. J. 623, 1157–1170 (2005) Barge, P., Sommeria, J.: Did planet formation begin inside persistent gaseous vortices?. Astron. Astrophys. 295, L1–L4 (1995) Begelman, M.C.: Nonlinear photon bubbles driven by buoyancy. Astrophys. J. 636, 995–1001 (2006) Bjerknes, V.: Solar hydrodynamics. Astrophys. J. 64, 93–107 (1926) Bonet, J.A., Márquez, I., Sánchez Almeida, J., Cabello, I., Domingo, V.: Convectively driven vortex flows in the sun. Astrophys. J. Lett. 687, L131–L134 (2008) Bracco, A., Provenzale, A., Spiegel, E.A., Yecko, P.A. : Spotted disks. In: Abramowicz, M., Bjornsen, G., Pringle, J. (eds.) Theory of Black Hole Accretion Disks, pp. 254–372. Cambridge University Press, Cambridge (1998) Bracco, A., Chavanis, P.-H., Provenzale, A., Spiegel, E.A.: Particle aggregation in Keplerian flows. Phys. Fluids 11, 2280–2287 (1999) Brahic, A.: (ed.) Formation of Planetary Systems, p. 15. CEPADUES Editions, France (1982) Cassinelli, J.P.: Origin of nonradiative heating/momentum in hot stars. In: Underhill, A.B., Michalitsianos, A.G. (eds.) Evidence for Non-Radiative Activity in Hot Stars, pp. 2–23. NASA, Washington (1985) Cayrel, R., Steinberg, M. (eds.): Physique des Mouvements dans les Atmospheres Stellaires. Colloques Internationaux du C.N.R.S., No. 250 (1976) Chandrasekhar, S.: On a new theory of Weizscker on the origin of the solar system. Rev. Mod. Phys. 18, 94–102 (1946) Chavanis, P.H.: Trapping of dust by coherent vortices in the solar nebula. Astron. Astrophys. 356, 1089–1111 (2000) Darton, R.C., LaNauze, R.D., Davidson, J.F., Harrison, D.: Chemical Engineering Research and Design, ICHemE (1977) Davidson, J.F., Harrison, D.: Fluidized Particles. Cambridge University Press, Cambridge (1963) Doering, C.R., Worthing, R.A., Spiegel, E.A.: Energy dissipation in a shear layer with suction. Phys. Fluids 12, 1955–1968 (2000) Dowling, T.E., Spiegel, E.A.: Stellar and Jovian vortices. In: Gottesman, S., Buchler, J.R. (eds.) Fifth Florida Workshop on Nonlinear Astrophysics, Ann. N.Y. Acad. Sci., vol. 617, pp. 190–216 (1990) Gallet, B.: Instability theory of swirling flows with suction. In: Cenedesi, C., Whitehead, J. (eds.) 2007 Program of Study: Boundary Layers, vol. 210, pp. 1–20. WHOI-2008-05, Woods Hole Oceanographic Inst. http://www.whoi.edu/page.do? pid=19276 (2008) Gallet, B., Doering, C.R., Spiegel, E.A.: Radial inow destabilizes the circular Taylor–Couette velocity. Phys. Fluids (in press) (2009) Gammie, C.F.: Photon bubbles in accretion discs. MNRAS 297, 929–935 (1998) Godon, P., Livio, M.: Vortices in protoplanetary disks. Astrophys. J. 523, 350–356 (1999) Gough, G.O., Lynden-Bell, D.: Vorticity expulsion by turbulence: astrophysical implications of an Alka-Seltzer experiment. J. Fluid Mech. 32, 437–447 (1968) Guazzelli, E.: Fluidized beds: from waves to bubbles. In: Hinrichsen, H., Wolf, D.E. (eds.) The Physics of Granular Media, pp. 213–232. Wiley GmbH & Co. KGaA, Weinheim (2004) Hale, G.E.: Solar vortices. Astrophys. J. 28, 100–117 (1908) Heimpel, M., Aurnou, J.: Turbulent convection in rapidly rotating spherical shells: A model for equatorial and high latitude jets on Jupiter and Saturn. Icarus 107, 540–557 (2007) Hopfinger, E.J., Browand, F.K.: Vortex solitary waves in a rotating, turbulent flow. Nature 295, 393–395 (1982) Huang, S.-S., Struve, O. : Stellar rotation and atmospheric turbulence. In: Greenstein, J.L. (ed.) Stellar Atmospheres, pp. 321–343. University of Chicago Press, Chicago (1960)

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31. Johnson, B.M., Gammie, C.F.: Vortices in thin, compressible, unmagnetized disks. Astrophys. J. 635, 149–156 (2005) 32. Krumholz, M.R., Klein, R., McKee, C.F., Offner, S.S.R., Cunningham, A.J.: The formation of massive star systems by accretion. Science 323, 754–759 (2009) 33. Ledoux, P.: On the vibrational stability of gaseous stars. Astrophys. J. 94, 537 (1941) 34. Ledoux, P., Schwarzschild, M., Spiegel, E.A.: On the spectrum of turbulent convection. Astrophys. J. 133, 184–197 (1961) 35. Lewis, E.P: Solar vortices and magnetic fields. Nature 78, 569–570 (1908) 36. Lou, Y.: A stability study of nonadiabatic oscillations in background polytropes. Astrophys. J. 361, 527–539 (1990) 37. Lou, Y: On the subadiabatic magnetohydrodynamic overstability in background polytropes. Astrophys. J. 367, 367–378 (1990) 38. Lovelace, R.V.E., Li, H., Colgate, S.A., Nelson, A.: Rossby wave instability of Keplerian accretion disks. Astrophys. J. 513, 805–810 (1999) 39. Mills, A.A.: Fluidization phenomena and possible simplications for the origin of lunar craters. Nature 224, 863–866 (1969) 40. Poyet, J.-P., Spiegel, E.A.: The onset of convection in a radially pulsating star. Astron. J. 84, 1918–1931 (1979) 41. Prendergast, K.H., Spiegel, E.A.: Photon bubbles. Comments Astrophys. Space Phys. 5, 43–49 (1973) 42. Provenzale, A., Babiano, A., Bracco, A., Pasquero, C., Weiss, J.B.: Coherent Vortices and Tracer Transport. Lecture Notes in Physics, vol. 744, 101 ff. Springer, Berlin (2008) 43. Saffman, P.G.: Vortex Dynamics. pp. 63 Cambridge University Press, Cambridge (1992) 44. Shapiro, S.L., Teukolsky, S.A.: Black Holes, White Dwarfs, and Neutron Stars. Wiley-Interscience Publication, New York (1983) 45. Smith, N. et al.: Kinematics and ultraviolet to infrared morphology of the inner homunculus of? Carinae. Astrophys. J. 605, 405–424 (2004) 46. Spiegel, E.A.: Currents in Chaos. Science Year, pp. 126–141 (1979) 47. Spiegel, E.A.: Phenomenological photofluiddynamics. In: Rieutord, M., Dubrulle, B. (eds.) Stellar Fluid Dynamics and Numerical imulations: From the Sun to Neutron Stars, pp. 127–142. EDP Sciences, Les Ulis (2006) 48. Spiegel, E.A.: Reflections on the tachocline. In: Hughes, D., Rosner, R., Weiss, N. (eds.) The Solar Tachocline, pp. 31–50. Cambridge University Cambridge, Cambridge (2007) 49. Spiegel, E.A., Zahn, J.P.: Instabilities of differential rotation. Comments Astrophys. Space Phys. 2, 178–183 (1974) 50. Stanton, V.A.: Convection in a model of a quasar. Thesis, Manchester University (1970) 51. Taib, M.R., Swithenbank, J., Nasserzadeh, V., Ward, M., Cottam, D.: Investigation of sludge waste incineration in a novel rotating fluidized bed incinerator. Trans. Inst. Chem. Eng. 77B, 298–304 (1999) 52. Tanga, P., Babiano, A., Dubrulle, B., Provenzale, A.: Forming planetesimals in vortices. Icarus 121, 158–170 (1996) 53. Turner, J.S.: The constraints imposed on tornado-like vortices by the top and bottom boundary conditions. J. Fluid Mech. 25, 377–400 (1966) 54. Umurhan, O.M.: Conducting sound. Thesis, Columbia University (1998) 55. Veronis, G.: Cellular convection with finite amplitude in a rotating fluid. J. Fluid Mech. 5, 401–435 (1959) 56. von Weizscker, C.F.: ber die Entstehung des Planetensystems. Zts. Astrophys. 22, 319–355 (1943) 57. Williams, G.P., Wilson, R.J.: The stability and genesis of Rossby vortices. J. Atmos. Sci. 45, 207–241 (1988) 58. Woltjer, J. Jr.: Note on the circular vortex in the theory of sunspots. Bull. Astron. Inst. Neth. 7, 164 (1934) 59. Zahn, J.-P.: Rotational instabilities and stellar evolution. In: Stellar Instability and Evolution. Proc. Symp. Canberra, Australia, pp. 185–194. D. Reidel Pub. Co., Dordrecht (1974)

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Theor. Comput. Fluid Dyn. (2010) 24:95–100 DOI 10.1007/s00162-009-0127-4

O R I G I NA L A RT I C L E

Xavier Perrot · X. Carton

2D vortex interaction in a non-uniform flow

Received: 30 October 2008 / Accepted: 24 June 2009 / Published online: 29 July 2009 © Springer-Verlag 2009

Abstract In a two-dimensional incompressible fluid, we study the interaction of two like-signed Rankine vortices embedded in a steady shear/strain flow. The numerical results of vortex evolutions are compared with the analytical results for point vortices. We show the existence of vortex equilibria, and of merger for initial distances larger than those without external flow. The evolutions depend on the initial orientation of the vortices in the external flow. Keywords Two-dimensional incompressible flows · Vortex merger · Pseudo-spectral model PACS 47.32.cb, 47.27.ek, 47.10.Fg, 47.11.Kb

1 Introduction Vortices play an essential role in 2D turbulent fluids where they achieve the energy cascade towards larger scales, the enstrophy cascade towards small scales being carried out via filaments. These vortices and filaments are often the end-product of the merger of smaller vortices. The merging process has often been studied, in particular with piecewise-constant (Rankine) vortices [6,9,13,17]. In 2D incompressible and inviscid flows, the merger of two equal Rankine vortices occurs if the two vortex centers are initially closer than 3.2 radii. This critical distance is modified by other factors, such as unequal size of vortices [12,18], flow divergence [1], barotropic instability of the vortices [4] or beta effect [2]. Recently, theoretical studies considered the interaction of two point vortices with steady or unsteady external flow [5,10,11,14,15]. Here, we use their results to interpret numerical simulations of the interaction of two identical Rankine vortices, in steady deformation fields. Firstly we recall the analytical results for point vortices (Sect. 2). In Sect. 3, the interaction of two Rankine vortices in a steady external flow is modeled with a spectral model. Finally conclusions are drawn.

2 The analytical results In an inviscid, incompressible and homogeneous 2D fluid, motions are governed by the vorticity equation: ∂t ζ + J (ψ, ζ ) = 0 where ζ = ∇ 2 ψ is relative vorticity, ψ is stream function and J is the Jacobian operator. Communicated by H. Aref X. Perrot (B) · X. Carton LPO, UEB/UBO, Brest, France E-mail: [email protected] Reprinted from the journal

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−0.09

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Fig. 1 The different regimes for vortices initially on the neutral point axis; grey circle merger, black cross fixed point, black square anticlockwise rotation, black/grey star oscillation with initial anticlockwise/clockwise motion, black diamond expulsion after destructive interaction, black triangle straining out, grey diamond expulsion, solid black line positions of the neutral point, as determined by the analytical study of point vortices, grey dashed line limit of merger when there is no external flow

For the analytical study, we consider two point-vortices with circulation , and an external flow composed of global rotation (with rate (t)) and of strain (with rate S(t)). The external flow and the vortex pair have central symmetry. The vortex positions with respect to the center of the plane are given by their radius ρ and by their angle θ (resp. θ + π). The equations of motion are, for any of the two vortices: 1 ∂H = −s(t)ρ sin(2θ ) ρ ∂θ

(1a)

∂H 1 = + ρω(t) − s(t)ρ cos(2θ ) ∂ρ 4ρ

(1b)

vr∗ = ρ˙ = − vθ∗ = ρ θ˙ =

where we have set vr∗ = πvr / , vθ∗ = πvθ / , s(t) = π S(t)/  = s0 (1 + εδ cos(σ t)), ω(t) = π(t)/  = ω0 (1 + εδ cos(σ t)), |ε|  1, |δ| ∼ 1. where ε is a small parameter for the expansion and δ represents the relative contribution of the unsteady part of the flow. A normalized Hamiltonian corresponds to this system H (ρ, θ, t) = 41 [ln(2ρ)−2s(t)ρ 2 cos(2θ )+2ω(t)ρ 2 ] In the expansion of the equations of motion in ε, the zeroth order ∂t ρ0 = ∂t θ0 = 0 provides the equilibria (at most four, related by symmetry). Their existence and position depend on the values of the steady strain and rotation rates (s0 and ω0 ): 1 θ0 = nπ/2, ρ0 = √ 2 (−1)n s0 − ω0 The stability of the equilibria is given by the first-order equations: – If −ω0 > |s0 |, the equilibria are two neutral points and two saddle-points. – If |s0 | > |ω0 |, they are two saddle-points. – And if ω0 > |s0 |, there is no equilibrium.

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0τ

1.43τ

2.86τ

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5.73τ

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8.59τ

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Fig. 2 The time evolution of two vortices initially located near the neutral points and near the separatrices (black solid line); the two red crosses on the separatrices are the saddle points and the two others correspond to the neutral points; the colored contours are vorticity contours (color figure online)

Adding an unsteady component to the external flow leads to the appearance of a modulation in the oscillation around the neutral point. This modulation can be studied by a multiple time scale expansion. This addition also leads to the appearance of Hamiltonian chaos in the vicinity of the heteroclinic curves (due to their mutual crossing, as shown by Melnikov’s method and via Poincaré sections). Chaos extends in the Poincaré section when δ increases [15]. 3 Interaction of two finite-area vortices in a steady external flow Our numerical study focuses on vortex evolutions when −ω0 > |s0 | (case of neutral point existence). A pseudo-spectral model of 2D incompressible flows is initialized with two circular vortices with radius r , unit vorticity and intercentroid distance d. The vortices can be located either on the axis of the neutral points, or on that of the saddle points. We set ω0 = 2s0 . 3.1 Vortices initially on the axis of the neutral points In this case, the vortices can remain near the neutral points. We explore the dynamical regimes for d/r ∈ [2.3, 10], and ω0 ∈ [−0.09, −0.01]. The results are summarized in Fig. 1. Reprinted from the journal

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0τ

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Fig. 3 The straining out of a vortex pair; the two lines correspond to the strain axes strain (extension axis with stars, compression axis with circles. The black ellipses, the red crosses and the colored contours are as in Fig. 2 (color figure online)

For weak external flow and small d/r , the regimes are the same as without external flow: merger and anticlockwise rotation. When the external flow amplitude increases, it acts against merger at small d/r . Merger is then replaced by vortex interaction with filament shedding, followed by strain-induced expulsion when the vortices have decreased in size. For moderate external flow and larger d/r , vortices can reach stationary positions, which agree with the neutral positions of point vortices, determined analytically. In the (d/r, ω0 ) plane, the vicinity of this stationary case corresponds to vortex oscillations around the neutral point position. For values of d/r smaller than those of the fixed point, these oscillations start anticlockwise because co-rotation dominates. On the contrary, for d/r larger than that of the fixed point, they start clockwise because the external flow dominates. For still larger external flow and d/r , vortex interaction leads to filament shedding, erosion and eventually, the vortices are expelled along the extension axes of the external strain. Merger is again observed for ω ∼ −0.08 and large d/r . To the best of our knowledge, this is the first instance where such merger is reported. Figure 2 shows that, in this case, vortices move along the separatrices, thereby reducing their distance and finally, merger is allowed. When the external strain is intense, vortices are elongated until they are destroyed, as Kida [7] showed, for an elliptical vortex in a strong external shear flow. As in Kida’s study, we observe stationary vortex positions and vortex rotation for weak external strain. But we do not observe vortex nutation. Note that here, contrary to Kida’s study, vortices are not centered in the strain, and a finite value of strain-induced velocity always exists at the vortex center. For strong external strain, the induced vortex deformation dominates their tendency to merge and leads to vortex destruction, as seen in Fig. 3. The elliptical vortex model, which assumes that vortices remain elliptical at all times [9], is used for comparison with the pseudo-spectral model results. In the elliptical model, the merging criterion is based on vortex contact. The elliptical model results shown in Fig. 4 show satisfactory agreement with the pseudo-spectral model results for vortex merger.

3.2 Vortices initially along the axis of the saddle points In this case, the vortices cannot remain near their initial positions because the saddle-points are unstable. The dynamical regimes are investigated for d/r ∈ [3.2, 8] and for ω0 ∈ [−0.09, −0.01]. The results are summarized in Fig. 5.

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−0.09 −0.08 −0.07

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−0.06 −0.05 −0.04 −0.03 −0.02 −0.01 3

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Fig. 4 Dynamical regimes for elliptical vortices: in black, regions of vortex merger (vortices get in contact); in white, distant interactions −0.09

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Fig. 5 The different regimes for vortices initially along the axis of the saddle points; grey circle merger, black/grey square anticlockwise/clockwise rotation, black triangle straining out, black diamond expulsion, solid black line analytical positions of the saddle point

For weak external flow and for small d/r , the dynamical regimes are merger and co-rotation, as for vortices initially near the neutral points. Again for small d/r , an increase in external flow first favors merger and then acts against it. There is also a reversal in the initial direction of rotation of the vortices. Indeed, for close vortices, increasing the external strain first tends to bring them closer to the center and to each other. But for large external strain, the vortices become located beyond the saddle-points, and are then expelled. The position of the saddle-points thus found numerically agrees with the analytical value (for point-vortices) when it is farther than 3.3r from the center. The direction of rotation is governed by vortex interaction at small distances and by external strain at large distances. For stronger external flow, the straining out of vortices leads to their destruction. In summary, this case is more simple than the former, due to the absence of steady vortex position. Reprinted from the journal

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4 Conclusion Numerical simulations of two equal vortex interaction under external strain and rotation confirm the analytical study for point vortices : the fixed points, both stable and unstable, are influential on the interaction of finitearea vortices. If the vortices are initially located on the axis of neutral points, the dynamical regimes in the parameter plane are varied and well explained by the relative influence of their companion vortex (influence of d/r ) and of the external flow (influence of s and ω). An essential result obtained here is the possibility for two initially distant vortices to merge, a process allowed by the external flow. If the vortices lie initially on the axis of the saddle points, the regime diagram is simpler, but again, evidences the competition between these two influences. An analysis of vortex deformation with the Okubo–Weiss criterion (see previous work by [3,8,16]), distinguishing vortex-induced strain and external strain, will be achieved to better understand the influence of vortex erosion during their interaction. A generalization of this study to stratified flows will be undertaken for oceanic applications. References 1. Cushman-Roisin, B.T.: Geostrophic turbulence and the emergence of eddies beyond the radius of deformation. J. Phys. Oceanogr. 20, 97–113 (1990) 2. Bertrand, C., Carton, X.: Vortex merger on the β-plane. C. R. Acad. Sci. Paris 316(II), 1201–1206 (1993) 3. Brandt, L.K., Nomura, K.K.: The physics of vortex merger: further insight. Phys. Fluids 18, 051701.1–051701.4 (2006) 4. Carton, X.: On the merger of shielded vortices. Europhys. Lett. 18, 697–703 (1992) 5. Carton, X., Legras, B., Maze, G.: Two-dimensional vortex merger in an external strain field. J. Turbul. 3, 045 (2002) 6. Dritschel, D.G.: The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95–134 (1985) 7. Kida, S.: Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520 (1981) 8. Kimura, Y., Herring, J.R.: Gradient enhancement and filament ejection for nonuniform elliptic vortex in 2D turbulence. J. Fluid Mech. 439, 43–56 (2001) 9. Legras, B., Dritschel, D.G.: The elliptical model of two-dimensional vortex dynamics. Phys. Fluids A3, 845–869 (1991) 10. Liu, Z.: Vortices in deformation background flow—a sensitivity source of the atmosphere. PhD Thesis, The University of Wisconsin, Milwaukee (2006) 11. Maze, G., Lapeyre, G., Carton, X.: Dynamics of a 2d vortex doublet under external deformation. Reg. Chaot. Dyn. 9, 179– 263 (2004) 12. Melander, M.V., Zabusky, N.J., McWilliams, J.C.: Asymmetric vortex merger in two dimensions: which vortex is “victorious”? Phys. Fluids 30, 2604–2610 (1987) 13. Melander, M.V., Zabusky, N.J., McWilliams, J.C.: Symmetric vortex merger in two dimensions. J. Fluid Mech. 195, 303–340 (1988) 14. Perrot, X., Carton, X.: Vortex interaction in an unsteady large-scale shear-strain flow. Proc. IUTAM Symp. Hamiltonian Dyn. Vortex Struct. Turbul. 6, 373–382 (2008) 15. Perrot, X., Carton, X.: Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discr. Cont. Dyn. Syst. B 11, 971–995 (2009) 16. Trieling, R.R., Velasco-Fuentes, O.U., van Heijst, G.J.F.: Interaction of two unequal corotating vortices. Phys. Fluids 17, 087103.1–087103.17 (2005) 17. Waugh, D.: The efficiency of symmetric vortex merger. Phys. Fluids A4, 1745–1758 (1992) 18. Yasuda, I., Flierl, G.R.: Two-dimensional asymmetric vortex merger: merger dynamics and critical merger distance. Dyn. Atmos. Oceans 26, 159–181 (1997)

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Theor. Comput. Fluid Dyn. (2010) 24:101–110 DOI 10.1007/s00162-009-0131-8

O R I G I NA L A RT I C L E

Ziv Kizner · Gregory Reznik

Localized dipoles: from 2D to rotating shallow water

Received: 1 November 2008 / Accepted: 24 June 2009 / Published online: 29 July 2009 © Springer-Verlag 2009

Abstract The progress made in the theory of localized dipoles over the course of the past century is overviewed. The dependence between the dipole shape, on the one hand, and the vorticity–streamfunction relation in the frame of reference co-moving with the dipole, on the other hand, is discussed. We show that, in 2D non-divergent and quasi-geostrophic dipoles, circularity of the trapped-fluid region and linearity of the vorticity–streamfunction relation in this region are equivalent. The existence of elliptical dipoles of high smoothness is demonstrated. A generalization of the dipole theory to the rotating shallow water model is offered. This includes the construction of localized f -plane dipole solutions (modons) and demonstration of their soliton nature, and derivation of a necessary condition for an eastward-traveling β-plane modon to exist. General properties of such ageostrophic modons are discussed, and the fundamental dissimilarity of fast and/or large dipoles in the rotating shallow water model from quasi-geostrophic dipoles is demonstrated and explained. Keywords Dipole vortex · Vorticity–streamfunction relation · f -plane · β-plane · Quasi-geostrophic approximation · Rotating shallow water · Modon · Soliton PACS 47.27.De, 47.32.C-, 47.35.Fg, 92.10.ak 1 Introduction The first regular dipole solution to Euler equations in two dimensions was independently suggested about a century ago by Lamb [19] and Chaplygin [5] (these classical works were reproduced and discussed by Meleshko and van Heijst [21], see also the translation of Chaplygin’s original work into English [6] supplied with comments of Meleshko and van Heijst). These solutions are valid also in rotating fluids in the f -plane model, where the flow is two-dimensional and non-divergent, and the Coriolis parameter f is assumed constant. Seven decades later, similar localized 2D solutions on the f - and β-plane, termed modons, were suggested, representing a standing [25] and propagating [20] highly localized dipoles in the quasi-geostrophic (QG) approximation. The β-plane approximation is based on the assumption that the Coriolis parameter f is not constant, but rather depends linearly on the northward coordinate y, i.e. f = f 0 + βy; quasi-geostrophicity in this context means that slight divergence of the flow due to the elevation of the free surface is allowed (see also below); while two-dimensionality implies that the horizontal velocity is depth-independent. The interest in modon solutions was stimulated by the extensive research of mesoscale oceanic and atmospheric Communicated by H. Aref Z. Kizner (B) Departments of Physics and Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel E-mail: [email protected]; [email protected] G. Reznik P.P. Shirshov Institute of Oceanology, RAS, 36 Nakhimovsky Prosp., 117997 Moscow, Russia Reprinted from the journal

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eddies in the 70s and 80s. The importance of translating dipoles was evidenced, for example, by the discovery of the so-called mushroom structures, quite durable dipole-like patterns common in a wide range of space scales of ocean currents [8]; vortical structures in magnetized plasma represent another field of application of dipoles. During the past three decades, a considerable progress has been made in developing the dipole/modon theory in application to Euler equations in two dimensions, and to rotating and stratified fluids in the QG approximation (both on the f - and β-plane). Analytical solutions for QG two-layer [7,10,13,15,24] and 3D and three-layer [11–13] modons added a stratification facet to the dipole theory, whereas invoking numerical methods has considerably extended the variety of the dipole/modon shapes [2,9,10,13–16]. The quasi-geostrophic approximation is applicable to flows that are weak relative to the fluid rotation. More specifically, in a quasi-geostrophic flow, the Rossby number is small and the relative deviations of isopycnal surfaces (or of the layer interfaces in a layered model) from their static levels are also small. In intense atmospheric and oceanic vortices, both conditions of the applicability of the QG approximation are normally violated. The rotating shallow water (RSW) model is free of the above restrictions. This model can be interpreted as either a free-surface single layer of a rotating fluid acted upon by gravity, or a two-layer fluid, where only one layer is active, while the other is infinitely deep and, therefore, passive. The variation in the thickness of the (active) fluid layer is allowed to be comparable to the total layer thickness. In this paper we give a brief overview of the theory of localized 2D non-divergent and QG dipoles on the f - and β-plane emphasizing the key ideas and the progress made over the course of the past century. Starting with the Lamb–Chaplygin dipole, through relatively new elliptical dipoles (Sect. 2), we proceed to the most recent solutions for localized f -plane dipoles in rotating shallow water, describe their properties and explain the distinction of intense RSW dipoles from QG dipoles (Sect. 3). Conditions for the existence of the β-plane RSW modons are briefly discussed.

2 Circularity and linearity of 2D and QG dipoles It is common knowledge that, in 2D non-divergent flows, a streamfunction ψ can be defined as u = −∂ψ/∂ y, v = ∂ψ/∂ x (where u and v are the velocity x- and y-components), and that vorticity ζ = ∂v/∂ x − ∂u/∂ y = ψ (where  is the Laplacian) is conserved in any fluid particle. For a structure that propagates steadily in the x-direction at a constant speed U , the vorticity conservation manifests itself in the existence of a functional relation ζ = F(ψ + U y) between vorticity and the co-moving streamfunction ψ + U y (termed also streakfunction). In the classical Lamb–Chaplygin solution, the dipole edge, or separatrix, is taken to be a circle. This contour separates the (x, y)-plane into the dipole interior, i.e. the trapped-fluid region where the co-moving streamlines (the level curves of function ψ + U y) are closed, and the exterior region where the streamlines are open (Fig. 1). At infinity, the flow is uniform, hence, irrotational. Due to the vorticity conservation, this implies that everywhere outside the separatrix the vorticity is zero, so that the exterior flow is governed by Laplace’s equation ψ = 0. Regarding the interior region, both Lamb and Chaplygin postulated linearity of functional F by setting ζ = −k 2 (ψ + U y); constant k was determined by matching the exterior and interior streamfunction, velocity and vorticity fields at the separatrix. Although the total energy of this dipole is finite, the solution is weakly localized: the streamfunction drops off at infinity as r −1 , where r is the polar radius. In geophysical applications, more realistic appears the quasi-geostrophic f -plane approximation, in which slight divergence is allowed and the conserved vorticity is defined as q=ζ−

123

f2 f2 ψ = ψ − ψ, g H0 g H0 118

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Localized dipoles: from 2D to rotating shallow water

Fig. 1 Lamb–Chaplygin circular dipole. Schematic of the co-moving streamlines; arrows indicate flow directions; shaded area zero vorticity (exterior) region

where g and H0 are acceleration due to gravity and the unperturbed (mean, or ambient) fluid layer depth, respectively. Here again steady propagation at a constant speed U in the x-direction is associated with the existence of a functional relation q = F(ψ + U y). As with the Lamb–Chaplygin dipole, a QG dipole solution can be achieved analytically by setting the separatrix to be a circle and assuming a linear relationship q = −k 2 (ψ + U y) in the dipole interior. Since vorticity is zero outside the separatrix, i.e. because the exterior flow is governed by a Helmholtz equation ψ −

f2 ψ = 0, g H0

this time, a higher degree of localization is achieved compared with the Lamb–Chaplygin dipole: at infinity the streamfunction drops off exponentially. A similar approach allowing analytical construction of circular dipole solutions works in the QG approximation with inhomogeneous background vorticity distribution in two and three dimensions, and in layered models. Analytical solutions of this type on a β-plane, γ -plane (where f = f 0 − 21 γ r 2 and γ = const), and in spherical geometry have been suggested in a number of publications [7,11–13,15,20,22–24,26,27]. Of course, in different models, the conserved vorticity is defined differently (see, e.g., [17]), but in all these solutions the linearity of the vorticity–streamfunction relation (in a co-moving frame of reference) inside the separatrix is postulated. In the QG β-plane approximation, the conserved vorticity is q=ζ−

f2 ψ + βy. g H0

Importantly, on a β-plane, steady propagation can only be zonal (eastward or westward), and the relation between q and ψ + U y outside the separatrix of a localized dipole, is necessarily linear, q=

β (ψ + U y). U

This is easily seen when the far-field vorticity βy is correlated with the far-field co-moving streamfunction U y [20]. Such a relation again leads to a Helmholtz equation   β f2 ψ − + ψ =0 U g H0 that has solutions decaying at infinity either at U > 0, or at U < −βg H0 / f 2 . Thus a β-plane modon propagates at a speed that lies outside the range of phase speeds of Rossby waves, and ψ drops off exponentially at infinity. These facts categorize a β-plane modon as a two-dimensional soliton. Solutions for dipoles propagating with the speeds of Rossby waves, can also be constructed [1,3]. However, such dipoles have infinite total energy, i.e. are not localized; that is the reason why they are termed non-local modons. Non-local modons are out of the scope of this paper, where only localized dipoles are considered. Reprinted from the journal

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Fig. 2 2D non-divergent elliptical dipoles on an f -plane. Upper panel vorticity contours for various ellipticities (shaded areas zero vorticity regions; + and –, vorticity signs); lower panel vorticity versus ψ + U y relations corresponding to the dipoles shown in the upper panel

In the above-referenced works devoted to circular dipoles, no wherefores of the choice of linear vorticity versus co-moving streamfunction relations in the dipole interior were discussed. This possibly caused a delusory opinion that functional F inside the separatrix could be chosen arbitrarily (or almost arbitrarily). What actually happens is that, once the conditions of continuous matching of the streamfunction and velocities at the separatrix are imposed, the form of the vorticity–streamfunction relation (in a co-moving frame of reference) is a function of the separatrix shape. This idea was first outlined by Boyd and Ma [2] and was later strengthened by Kizner et al. [14]. Fourier analysis shows that, once the separatrix is a circle, the linearity of the 2D non-divergent and QG vorticity operators ψ, ψ − f 2 /g H0 ψ, and ψ − f 2 /g H0 ψ + βy involves the linearity of the relation of ζ or q versus ψ + U y [14]. Regarding the opposite statement, it can be shown analytically that, for elliptical separatrices, the vorticity versus co-moving streamfunction relation cannot be linear, whereas numerical tests convince: any non-circularity always implies nonlinearity [2,9,14]. Thus, circularity of the separatrix and linearity of functional F inside the separatrix are equivalent properties of 2D non-divergent and QG dipoles. This holds also in multi-layer and 3D quasi-geostrophic dipoles [14]. Below we will see that ageostrophicity changes the situation radically causing the interior vorticity versus co-moving streamfunction relation to be nonlinear even in circular dipoles. Dipoles with elliptical separatrices were obtained by a combination of analytical and numerical methods in single-layer and two-layer QG frameworks [2,9,14,16]. As can be concluded from Fig. 2 (adapted after [16]), the stronger the separatrix ellipticity, the stronger the nonlinearity of the interior vorticity versus co-moving streamfunction relation. One more observation is that, on the f -plane, a unique ellipse aspect ratio exists, at which the vorticity versus co-moving streamfunction graph is tangent to the ψ + U y axis. As readily follows from the relation ζ = F (ψ + U y) or q = F (ψ + U y), this provides the high-smoothness of the dipole solution, that is, the continuity of the first and second derivatives of vorticity at the separatrix; we term such a dipole supersmooth [9,16]. The set of supersmooth single-layer β-plane modons is wider: for each value of parameter β S/U , where S is the area of the interior domain, there exists a unique ellipse aspect ratio that involves supersmoothness (for details see [9]). Nonlinear (i.e. noncircular) and, in particular, supersmooth dipole solutions are important since in laboratory and numerical experiments, where the bottom and lateral friction are present, dipoles evolve towards nonlinear quasi-elliptical and, eventually, supersmooth states. In [10], two-layer supersmooth f - and β-plane dipoles are presented characterized by non-coincidence of the separatrices in the layers. Numerical tests show that supersmooth modons (whether single- or two-layer) are remarkably stable [9,10].

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3 Dipoles in rotating shallow water 3.1 Steadily translating RSW structures on an f -plane As noted in the Introduction, in intense oceanic and atmospheric vortices, ageostrophic effects are significant. This motivated our special interest in dipoles, or modons, in the rotating shallow water model. Consider a single- or two-layer fluid on an f -plane in the gravity field. Let H = H0 + h be the thickness of the active layer; here, as in the QG approximation, H0 is the unperturbed (mean, or ambient) thickness, and h is the deviation of the layer thickness from H0 . The motion is three-dimensional, but the horizontal scale is assumed to be much larger than the vertical one, so that the hydrostatic approximation is valid. Under these conditions, the horizontal velocity can be assumed depth-independent, and integration of the continuity equation over z leads to three equations in two dimensions, known as the rotating shallow water model. Aiming at the construction of RSW dipoles, we consider a vortical structure that propagates steadily along the x-axis at a constant speed U . In a frame of reference co-moving with the structure, equations of the RSW model are: ∂u ∂u +v − ∂x ∂y ∂v ∂v +v + (u − U ) ∂x ∂y ∂ [(u − U )H ] + ∂x

(u − U )

∂h = 0, ∂x ∂h f u + g = 0, ∂y ∂ (v H ) = 0, ∂y f v + g

(1) (2) (3)

where g  is either the acceleration due to gravity (in the single-layer case), or the ‘reduced gravity’ associated with gravity and the fractional density difference between the fluid layers (in the two-layer case). Using (3) an integral (over the depth) streamfunction  can be defined as: (u − U )H = −

∂ ∂ , vH = . ∂y ∂x

(4)

Equations (1)–(4) yield the conservation of the Bernoulli function, B, and potential vorticity (PV), P, in a fluid particle (column): B=

 1 (u − U )2 + v 2 + g  h + U f y = (), 2 d () ζ+ f = F() = . P= H0 + h d

(5) (6)

Markedly, functional F in (6) is a derivative of functional that appears in (5). 3.2 Exterior and interior flows: soliton nature of the f -plane RSW dipoles To construct a dipole solution, we start with fixing a circular separatrix of some radius L. Because the far-field flow is uniform, h and ζ go to zero at infinity and, accordingly, P goes to a constant value f /H0 . Thus, due to the PV conservation, P is identically constant all over the exterior region, i.e. F = d /d = const and, accordingly, B is a linear function of : B=

f U2 . + H0 2

(7)

Now, that functionals and F for the exterior region are specified, using (5)–(7) we can write the equations that govern the flow outside the dipole separatrix:   1 f (∇)2 2 −U −  + g  h + U f r sin θ = 0, (8) 2 2 (H0 + h) H0 h  ∇h · ∇ f − − = 0. (H0 + h)2 (H0 + h)3 H0 H0 + h Reprinted from the journal

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(9)

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The boundary conditions at infinity and at the separatrix are: H → H0 and  → U H0 r sin θ as r → ∞, |r =L = C.

(10)

Here, polar coordinates r and θ are used, because this facilitates the computations (see below). According to (10), far away from the dipole, the layer depth H and streamfunction  approach their ambient asymptotics H0 and U H0 y = U H0 r sin θ that represent the uniform flow in which the dipole is embedded. At the separatrix, which is a streamline, the streamfunction must assume a constant value C. Given L, U and C, this nonlinear problem is solved numerically using a Fourier–Chebyshev expansion of  and h in θ and r and applying an iterative procedure of successive linearization of equations (8) and (9). The iterations are initialized by a circular QG dipole solution discussed in Sect. 2; to solve the linear problems arising at every iteration step, collocations are applied. Constant C is arbitrary at the moment. Based on Eqs. (8) and (9) we can investigate the far-field asymptotics of the solution sought. For this purpose,  is conveniently split into a sum  = φ + U H0 y,

(11)

where φ is the deviation of  from the uniform-flow streamfunction pattern. Note that, according to (10), φ and h vanish as r → ∞. This allows the linearization of equations (8) and (9) about φ and h, which, after elimination of h, yields:   2   2 f U2 ∂ φ ∂ 2φ 1−  + − φ = 0. g H0 ∂ x 2 ∂ y2 g  H0

(12)

Solutions to (12) that decay at infinity exist only if the coefficient in brackets at the first term is positive, i.e. when the dipole translational speed is smaller than the phase speed of gravity waves Ug : |U | ≤ Ug =



g  H0 .

(13)

So an RSW dipole possesses the typical soliton properties: (i) its localization is exponential, and (ii) its translation speed lies outside the range of phase speeds of inertia-gravity waves, the only waves allowable in the model (for details see [18]). Notice that, in discussing the soliton nature of RSW dipoles, the circularity of the separatrix is never used; moreover, the vortical structure must not necessarily be a dipole. In fact, equations (7)–(9) and (12) apply to the exterior of any steadily translating localized RSW vortical structure. In the interior domain, the original momentum equations are used because here no specific form of functional is known: 

 1 ∂ ∂ f ∂ 1 ∂ ∂h Jr,θ , − − + f Ur cos θ + g  = 0, (14) H ∂r H ∂r ∂θ H ∂θ ∂θ       ∂h 1 f ∂ 1 ∂ 2 1 ∂ + + − f U sin θ − g  = 0, (15) Jr,θ , rH r H ∂θ H ∂r H ∂r ∂r where Jr,θ is the Jacobian in r and θ . Equations (14) and (15) are supplemented with the boundary conditions      ∂ (I n)  ∂ (E x)     H (I n)  = H (E x)  ,  (I n)  = C, = ,     r =L r =L r =L ∂r ∂r r =L r =L

(16)

that assure continuous matching of the exterior and interior layer thickness, streamfunction and velocity at the separatrix. In accordance with (14) and (15), conditions (16) afford also the continuity of gradients of h. The continuity of PV at the separatrix is not guaranteed: the PV jump is a function of constant C. However, C can be specially fitted so as to furnish a smooth solution, i.e., with a continuous PV and smooth B fields [18]. The numerical procedure for solving the interior problem given by (14)–(15) is similar to that used in the exterior. The exterior and interior iterative procedures are enveloped in a loop aimed at the determination of the specific value of C that provides a smooth solution. Below only smooth dipoles are considered.

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3.3 Smooth RSW dipole solutions: distinction from QG dipoles As obvious from (8)–(10) and (14)–(16), smooth circular f -plane RSW dipoles form a two-parameter family, the parameters being the dipole’s size L and translation speed U . Conventional scaling of Eqs. (1)–(3) with L and U chosen as length and velocity scales results in the emergence of two independent non-dimensional  parameters. As such can be chosen, e.g., parameters Ro = U/L f and = L/L Ro , where L Ro = g  H0 / f . Parameter Ro is the conventional Rossby number, and is the dipole’s size relative to the Rossby radius of deformation L Ro . In the light of the ‘soliton constraint’ (13), an alternative Rossby number Ro can be introduced as the ratio of the dipole translation speed to the phase speed of gravity waves: Ro = U/Ug = U/L Ro f . At a fixed dipole size, the QG limit is achieved as Ro or Ro go to zero, i.e., when U → 0. In Fig. 3 (adapted after [18]), two non-dimensional smooth shallow-water dipoles of the same size = 1 are presented. The first dipole propagates much slower than inertia-gravity waves (Ro = Ro = U/Ug =0.04). As seen from Fig. 3a, the dipole’s streamfunction  and the layer-thickness deviation h are nearly symmetric, and the total layer depth differs from a constant just a little. In the second example (Ro = Ro = U/Ug =0.20), the translation speed is five times stronger, and the asymmetry between the cyclonic and anticyclonic vortices is obvious. Importantly, the minimal layer thickness achieved in the cyclone is rather close to zero. The PV distribution and the relation between PV and the streamfunction for the same two solutions are shown in Fig. 3b. Again, the first dipole appears almost symmetric, and, because the flow in this dipole is close to the quasi-geostrophic balance (U 0, the coefficient at the third term in (21) is positive, thus the condition for a solution decaying at infinity  to exist is (13). In other words, a necessary condition for the existence of eastward-traveling modons is U < g  H0 . As for the westward propagation, at any U < 0, there exists a critical value y = YC R (U ) < 0 that nullifies the third term in (21), so that the type of equation changes when y passes YC R (U ). Therefore, the existence of truly localized RSW modons on an unbounded β-plane is questionable. However, when evolution of westward-propagating vortical dipoles is studied in a numerical model, a bounded domain is normally considered, and no question of strict localization arises. For example, reasonably localized westward-propagating RSW dipoles on the equatorial β-plane ( f 0 = 0) emerged in the numerical simulations of Boyd [4]. He solved an initial-value problem by starting the computations from some dipolar state and marching until the flow had become approximately steady in a frame of reference moving with the vortex pair; the computations were run in a finite-size box, with the boundary conditions being periodicity in x and vanishing of u, v, and h at some y = ±ymax . 4 Conclusion Considering localized 2D non-divergent and QG dipoles on the f - and β-plane we have shown that circularity of the separatrix and linearity of the vorticity vs. co-moving streamfunction relation inside the separatrix are equivalent properties of dipoles. An f -plane QG dipole can propagate in any direction and at any speed, because this model does not allow any waves; β-plane modons can only propagate zonally with speeds that are outside the range of phase speeds of Rossby waves. Ellipticity of the separatrix implies nonlinearity of the vorticity versus co-moving streamfunction relation, and the stronger the separatrix ellipticity, the stronger the nonlinearity of this relation. On the f -plane, a unique ellipse aspect ratio exists providing supersmoothness of the dipole (that is, smoothness of the dipole vorticity field), while β-plane supersmooth elliptic modons make up a one-dimensional manifold in the two-dimensional parameter space of all elliptical modons. In numerical simulations, supersmooth modons prove to be remarkably stable. Using a combination of analytical and numerical methods, we have shown the existence of ageostrophic, shallow-water f -plane dipoles and demonstrated their soliton nature. In essentially ageostrophic dipoles, the cyclonic and anticyclonic vortices exhibit considerable asymmetry even at speeds that are relatively small compared to the phase speed of gravity waves. The requirement that the active-layer thickness must be positive imposes a restriction on the allowable size and speed of a smooth dipole. We have determined the domain in the parameter space, where smooth dipoles do exist, and found that the bigger the dipole, the smaller the allowed speed and vice versa. A necessary condition has been derived for the existence of eastward-propagating β-plane RSW modons, according to which modons must propagate slower than gravity waves. We believe that the results presented in this paper can serve as a starting point for future research in different lines. Firstly, the stability of the RSW modons found can now be studied numerically. Secondly, an examination Reprinted from the journal

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of the response of the f -plane RSW dipoles to small β-perturbations might be beneficial. Even slowly propagating mid-latitude mesoscale RSW vortices (i.e. where ∼ 1) can be quite intense, hence, highly nonlinear (Sect. 3.3). In such modons, the β-effect is expected to be significant in the far field (where the motion is weak), remaining for a long time negligible in the dipole itself. Therefore one might anticipate that a slightly β-perturbed intense dipole with an eastward component in its translation speed will gradually transform into a steady eastward-traveling β-plane modon. A consideration of the behavior of a generally westward-traveling f -plane RSW modon subjected to a small β-perturbation could also be of interest. Thirdly, in the context of the investigation of soliton properties of RSW dipoles, numerical experiments on dipole scattering appear to be promising. Fourthly, minor modifications of the technique presented in Sects. 3.1–3.3 would make possible direct computations of mid-latitude and equatorial β-plane RSW modons. Acknowledgments We thank the organizers of the IUTAM Symposium, in particular, Hassan Aref and Dorte Glass, for their hospitality and care. We are also thankful to Slava Meleshko, GerJan van Heijst, and Ruben Trieling for helpful discussions. This research was supported by the Israel Science Foundation grant 628/06. G.R. acknowledges the support from the Russian Foundation for Basic Research grant 08-05-00006.

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Theor. Comput. Fluid Dyn. (2010) 24:111–115 DOI 10.1007/s00162-009-0133-6

O R I G I NA L A RT I C L E

Ruben Trieling · Rudi Santbergen · GertJan van Heijst · Ziv Kizner

Barotropic elliptical dipoles in a rotating fluid

Received: 15 December 2008 / Accepted: 24 June 2009 / Published online: 30 July 2009 © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract Barotropic f -plane dipolar vortices were generated in a rotating fluid and a comparison was made with the so-called supersmooth f -plane solution which—in contrast to the classical Lamb–Chaplygin solution—is marked by an elliptical separatrix and a doubly continuously differentiable vorticity field. Dyevisualization and high-resolution particle-tracking techniques revealed that the observed dipole characteristics (separatrix aspect ratio, cross-sectional vorticity distribution and vorticity versus streamfunction relationship) are in close agreement with those of the supersmooth f -plane solution for the entire lifespan of the dipolar vortex. Keywords Supersmooth dipole · Separatrix · Rotating fluid · f -Plane · Laboratory experiments PACS 47.32.C-, 47.32.Ef 1 Introduction Dipolar vortices play an important part in large-, meso- and small-scale geophysical flows due to their selfpropelling motion and robustness. The long lifespan and stability of these coherent structures have motivated researchers to look for explicit stationary solutions relevant to ideal fluid. In a comoving frame of reference, these solutions are characterized by the existence of a closed streamline (separatrix) that demarcates the interior region, where the streamlines are closed, from the exterior region, where the streamlines are open. One of the most popular dipole solutions is the Lamb–Chaplygin dipole [1,2] which is marked by a circular separatrix. This solution was suggested about a century ago and has been widely used in various studies. Dipolar vortices in geophysical flows, however, are often slightly compressed in the direction of translation which has been confirmed by laboratory experiments [3,4] and numerical simulations [5,6]. For that reason, in more recent theoretical studies [7,8], the concept of Lamb and Chaplygin was generalized for barotropic f - and β-plane dipoles with elliptical separatrices. Among the possible solutions there exist the so-called supersmooth solutions whose first and second vorticity derivatives are continuous at the separatrix. Such solutions are characterized by a specific separatrix aspect ratio. Because of the uniqueness of the supersmooth solution and its remarkable stability, the present research aims at finding such structures in the laboratory. Although a β-plane can be easily established in the laboratory, we restrict ourselves to the f -plane for which only a single (to within scaling) supersmooth solution exists. Communicated by H. Aref R. Trieling (B) · R. Santbergen · G. J. van Heijst Fluid Dynamics Laboratory, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands E-mail: [email protected] Z. Kizner Departments of Mathematics and Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel E-mail: [email protected] Reprinted from the journal

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Fig. 1 Family of elliptical barotropic f -plane dipoles with separatrix aspect ratios ε = 1 (Lamb–Chaplygin solution), ε = 1.10, ε = 1.16 (supersmooth f -plane solution) and ε = 1.30 (after [7]). Upper panels, vorticity fields. Lower panels, vorticity versus (comoving) streamfunction relationships in the vortex interior. The contours are plotted with a 20% interval of the maximum vorticity. Thin solid and dashed contours correspond to positive and negative isolines of vorticity, respectively. The thick solid lines indicate the zero isolines of vorticity (outside the separatrix, the vorticity is identically zero)

2 Theory Elliptical barotropic f -plane dipoles, when properly scaled, represent a one-parameter family of solutions which can be parametrized by the ellipse aspect ratio ε = r y /r x . Here, r x and r y are the ellipse radii in the x- and y-axes, respectively, where the assumption is made that the dipole translates in the x-direction. Figure 1 shows four different solutions corresponding to ε = 1, ε = 1.1, ε = 1.162 and ε = 1.30. The vorticity fields are displayed in the upper panels, whereas the vorticity versus (comoving) streamfunction relationships are shown in the lower panels. The Lamb–Chaplygin dipole is recovered as a special case, with ε = 1, in which the separatrix is a circle that encloses a region of nonzero vorticity, with the vorticity versus streamfunction relationship being linear. The main effects of the extension of the separatrix in the y-direction (for a fixed √ dipole translation speed U and geometrical mean ellipse radius a = r y r x ) are sharpening of the vorticity distribution, the corresponding growth of the peak vorticities, and the change of slope of the graph of vorticity versus streamfunction function at the origin. For ε = 1 the slope is positive, while for ε = 1.30 the slope is negative. Since this slope is a monotonic function of ε, there exists only a unique aspect ratio, ε ≈ 1.162, at which the slope is zero; this specific aspect ratio also corresponds to the only elliptic f -plane dipole solution with continuous first and second normal vorticity derivatives at the separatrix [7,8]. For this reason, in order to highlight the higher degree of smoothness compared to other elliptical solutions, this dipole solution is referred to as the supersmooth solution whose characteristics may be close to those of the dipoles observed in the laboratory.

3 Method The laboratory experiments were performed in a tank with horizontal dimensions 1.0×1.5 m that was mounted on top of a turntable. The tank was filled with water to a depth of 20 cm and the flow was allowed to adjust to a solid-body rotation for at least half an hour. The angular velocity of the turntable was set to 0.7 rad s−1 . A dipolar vortex was created by dragging a vertical plate, with horizontal size d = 25 cm, through the fluid in the horizontal direction while simultaneously lifting the plate in the vertical direction. After the plate was removed, a well-defined quasi-steadily translating dipolar vortex emerged within a few rotation periods. The initial flow characteristics were controlled by variation of the drag speed V and the drag distance s. To obtain a quasi-two-dimensional flow, the drag speed was chosen such that the Rossby number Ro = V / d was smaller than unity. Also the drag distance was limited since otherwise a chain of dipoles was produced, much like a Von Kármán vortex street. Dye-visualization and high-resolution particle-tracking techniques were used to obtain qualitative and quantitative information about the horizontal flow field.

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4 Results Figure 2 shows the time evolution of a typical dipolar vortex. The interior region of the vortex was visualized by low-concentration fluorescent dye which was originally released at the front of the plate. In addition, a line of high-concentration dye was introduced at some distance from the initial position of the plate. This line of dye eventually demarcates the interior region of the dipole and facilitates the identification of the separatrix. For several initial forcing parameters and different times, the separatrix aspect ratio ε = r y /r x was determined by measuring the ellipse major and minor semiaxes from dye distributions as shown in Fig. 2. In order to overcome the lack of information at the rear part of the separatrix, the dye-distribution images were reflected symmetrically about the apparent axis of symmetry y, being defined as the line passing through the local vortex cores. It is seen from Fig. 3 (left panel) that the separatrix aspect ratio fluctuates around a mean value ε = 1.28 ± 0.07 which is higher than, but close to the value ε ≈ 1.162 associated with the supersmooth elliptical dipole solution. The fluctuations are likely due to the inaccuracy of the measurement procedure, as well as surface waves induced by the plate. As a second approach, the separatrix aspect ratio was determined from the streamfunction field. The latter was computed from the measured horizontal velocity field and corrected for the instantaneous translation speed of the dipole. The dipole translation speed was obtained from the observed displacements of the local vortex cores. In order to estimate the separatrix aspect ratio, the separatrix was least-square fitted with an elliptic contour. Figure 3 (right panel) shows the time evolution of the separatrix aspect ratio for two experimental dipoles as obtained from the comoving streamfunction, with the mean value being ε = 1.26 ± 0.08, which is close to that obtained for the dye-visualization experiments. The measured horizontal velocity field also allows for the computation of the vorticity and the streamfunction distribution. Both quantities are plotted in Fig. 4 for a dipolar vortex with the separatrix aspect ratio ε = 1.18. Also shown, in the same figure, is the elliptic contour that was least-square fitted to the dipole separ-

Fig. 2 Dye visualization of a dipole propagating through a line of dye. The dye eventually demarcades the interior region of the dipole providing an indication of the separatrix aspect ratio 1.8

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Fig. 4 Contour plots of vorticity (left panel) and comoving streamfunction (right panel) for a dipolar vortex with separatrix aspect ratio ε = 1.18. On either side of the dipole translation axis, the contours are plotted with a 20% interval of the local extrema of vorticity and streamfunction, respectively. Thin solid and dashed isolines of vorticity correspond to positive and negative values of vorticity, respectively, the zero isoline being omitted. Also shown is the elliptic contour (thick solid line) that was least-square fitted to the separatrix whose approximate location is indicated by the thin solid contours (zero streamfunction isolines) in the right panel. Forcing parameters: V = 3.0 cm s−1 and s = 10 cm 1

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Fig. 5 Comparison of experimental results with the supersmooth dipole solution. Left panel, cross-sectional vorticity distribution along a straight line passing through the dipole vorticity extrema. Right panel, vorticity versus comoving streamfunction relationship for the interior dipole region. The supersmooth f -plane solution is indicated by the solid lines. On either side of the dipole symmetry axis, both the vorticity and the streamfunction have been normalized by their local extremal values. The distance along the dipole symmetry axis y has been normalized by half the distance between the local extrema of the comoving streamfunction. Forcing parameters as in Fig. 4

atrix whose approximate location is indicated by the solid streamlines. The apparent sepratrix splitting is due to the presence of a 10%–20% deviation from non-divergence in the velocity field as well as the inaccuracy in the estimated dipole translation speed. The vorticity of the dipole appears to be concentrated in two well-separated vortex cores, characteristic of sufficiently extended elliptic dipoles, and the dipole’s separatrix is close to elliptic. Figure 5 (left panel) shows the corresponding cross-sectional vorticity distribution along a straight line passing through the local extrema of the comoving streamfunction. Also shown is the corresponding scatter plot of the vorticity versus comoving streamfunction for the interior dipole region (right panel). In both cases, close agreement with the supersmooth elliptical solution (solid lines) is obvious. Especially note the zero slope at which the vorticity versus streamfunction graph crosses the origin. This agreement was monitored for the entire lifespan of the dipolar vortex.

5 Discussion and conclusions In the present study we have investigated the temporal evolution of concentrated barotropic f -plane dipoles whose characteristics were observed to be close to that of the supersmooth dipole solution. This agreement was obtained for the entire life span of the vortex and for a wide range of forcing parameters. The apparent robustness of the observed dipoles seems to confirm previous numerical simulations [8] in which the ( f - and β-plane) supersmooth dipoles were found to be the most stable among all the elliptical

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dipoles in the nearly inviscid limit. In contrast, in a related numerical study on concentrated vortex dipoles [9] it was shown that various initial vorticity distributions evolve, through viscous effects, towards a specific family of dipoles parametrized by the dipole aspect ratio a/b (not to be confused with the separatrix aspect ratio), where a is the radius of the local vortex cores based on the polar momentum in half a plane (with single-signed vorticity) and b is the separation distance between the vortex core centroids. In this respect, the apparent stability of the observed dipoles in rotating-tank experiments may be merely due to the relatively large Reynolds number, which was within the range 2500–8750 based on the size and the translation speed of plate, while it was initially of the order of 104 using the ratio of single-signed circulation and kinematic viscosity. The vorticity distribution associated with the supersmooth dipole is different, however, from the initial vorticity distributions considered in the aforementioned study involving concentrated vortex dipoles. It therefore still needs to be verified whether the supersmooth dipole should indeed evolve, through viscous effects, towards the specific family of dipoles or will it preserve its characteristics for a long time, as in the nearly inviscid limit. In this respect, it should be beard in mind that the bottom friction (Ekman layer) can have a stronger effect upon an experimental dipole than the lateral friction considered in the numerical simulations [9]. In any case, one of the effects of viscosity is smoothing of the vorticity profile in any vortical structure. Therefore, it can be expected that if viscosity makes dipoles to tend towards a specific family of shapes, this family must be supersmooth, although, not necessarily exactly elliptical. Considering the closeness of the observed dipolar vortices to the supersmooth f -plane dipole, the latter seems to provide a better model for the observed elliptic f -plane dipoles than the Lamb–Chaplygin model. Acknowledgments We thank the organizers of the IUTAM Symposium, in particular Hassan Aref and Dorte Glass for their effort and hospitality. Z.K. acknowledges the support from the Israel Science Foundation grant 628/06, and the financial support from the Netherlands Organization for Scientific Research (NWO) for his working visit at TUE. We are grateful to Enric Pallàs-Sanz for his contribution to the laboratory experiments. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References 1. Lamb, H.: Hydrodynamics, 6th edn. Cambridge University Press, Cambridge, 738 p. (1932) 2. Meleshko, V.V., van Heijst, G.J.F.: On Chaplygin’s investigations of two-dimensional vortex structures in an inviscid fluid. J. Fluid Mech. 272, 157–182 (1994) 3. Flór, J.B., van Heijst, G.J.F.: An experimental study of dipolar vortex structures in a stratified fluid. J. Fluid Mech. 279, 101–133 (1994) 4. Velasco Fuentes, O.U., van Heijst, G.J.F.: Experimental study of dipolar vortices on a topographic beta-plane. J. Fluid Mech. 259, 79–106 (1994) 5. Hesthaven, J.S., Lynov, J.P., Nielsen, A.H., Rasmussen, J.J., Schmidt, M.R., Shapiro, E.G., Turitsyn, S.K.: Dynamics of a nonlinear dipole vortex. Phys. Fluids 7, 2220–2229 (1995) 6. Nielsen, A.H., Rasmussen, J.J.: Formation and temporal evolution of the Lamb dipole. Phys. Fluids 9, 982–991 (1997) 7. Kizner, Z., Khvoles, R.: Two variations on the theme of Lamb–Chaplygin: supersmooth dipole and rotating multipoles. Regul. Chaotic Dyn. 9, 509–518 (2004) 8. Khvoles, R., Berson, D., Kizner, Z.: The structure and evolution of elliptical barotropic modons. J. Fluid Mech. 530, 1–30 (2005) 9. Sipp, D., Jacquin, L., Cosssu, C.: Self-adaptation and viscous selection in concentrated two-dimensional vortex dipoles. Phys. Fluids 12, 245–248 (2000)

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Theor. Comput. Fluid Dyn. (2010) 24:117–123 DOI 10.1007/s00162-009-0107-8

O R I G I NA L A RT I C L E

Mikhail Sokolovskiy · Jacques Verron · Xavier Carton · Vladimir Gryanik

On instability of elliptical hetons

Received: 3 November 2008 / Accepted: 18 March 2009 / Published online: 6 May 2009 © Springer-Verlag 2009

Abstract Using the method of contour surgery, we examine the evolution of an initially vertically aligned elliptical heton. A classification of quasi-stable and unstable regimes for the case of two-layered vortex structure is suggested. Keywords Elliptical vortex · Vortex instability · Heton PACS 47.32.C , 47.32.cd , 47.20.Cq

1 Introduction The classic Kirchhoff [1] solution for the elliptical vortex patch with semi-axes a and b and vorticity ω, which rotates as a solid body with a constant angular velocity  = ωab/(a + b)2 is known from 1876. Seventeen years later Love [2] showed that for χ = a/b > 3 this stationary solution is unstable. Recently, Mitchell and Rossi [3] have given a complete quantification of the regimes of linear and non-linear instability of the elliptical vortex with respect to the parameter χ . Kirchhoff solution had multiple generalizations. In particular, – Chaplygin [4,5], and later Kida [6], Dritschel [7], Dhanak and Marshall [8], Legras and Dritschel [9] and others have shown that the introduction of an ambient velocity field, linearly dependent on the coordinates (this is analogous to the affine transformation of coordinates, converting ellipses into ellipses), allows to Communicated by H. Aref Investigation was conducted within the framework of the European Research Group “Regular and Chaotic Hydrodynamics”. M. Sokolovskiy (B) Water Problems Institute of RAS, 3 Gubkina Str., 119333 Moscow, Russia E-mail: [email protected] J. Verron Laboratoire des Ecoulements Géophysiques et Industriels, UMR 5519, CNRS, BP53 X, 38041 Grenoble Cedex, France E-mail: [email protected] X. Carton Laboratoire de Physique des Océans, UFR Sciences, UBO, 6 Av. Le Gorgeu, 29200 Brest, France E-mail: [email protected] V. Gryanik Alfred-Wegener-Institute for Polar and Marine Research, Bremerhaven, Germany E-mail: [email protected] Reprinted from the journal

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construct a solution, defining the behaviour of a pulsating vortex, which rotates with variable velocity, and it is a time-dependent value of χ . – Polvani and Flierl [10] have introduced a notion of a generalized Kirchhoff vortex for a system of N embedded one into other elliptical patches. They have studied the stability of such a solution, and constructed diagrams of stable and unstable states in the space of external geometric parameters. – Kozlov [11] has summarized the problem of an elliptical vortex when the effect of the “entrainment” due to the introduction of an “effective” bottom friction is taken into account. This mechanism initiates cyclonicanticyclonic asymmetry (when ω changes its sign), which is observed in ocean and atmosphere conditions. In [11] the author provides the conditions under which the model gives the particular cases of Kirchhoff, Chaplygin and Kida. The above works investigate the dynamics of elliptical vortex patches in a homogeneous non-rotating fluid. These results may be applied also to the case of rotating fluid under the condition of geostrophic balance (when the hydrodynamic pressure plays the role of the stream function). In this paper, in the framework of quasi-geostrophic approximation [12] we examine the problem on the behaviour of two vortex patches in a stratified rotating fluid, which consists of two non-mixing layers of equal thickness. It is supposed also that in the initial moment elliptical vortices of the upper and bottom layers are kinematically identical, and are located strictly one over the other, and they have opposite signs of ω. Such a two-layer vortex represents a heton [13]. In [14], a stationary solution in the form of an axially symmetrical heton has been investigated to be stable with respect to small harmonic disturbances of the boundary at χ = 1. There is shown that at γ = R/Rd > 1.705 (R is a characteristic horizontal scale, Rd is the internal deformation Rossby radius [12]) may appear unstable modes. Thus, it is established that even the circular heton may be unstable with respect to the degree of the stratification. Let us note that similar results have been obtained independently in [15] and [16].

2 Numerical modelling of evolution For elliptical vortex patches (χ > 1) it is possible to carry out only numerical study of stability. It should be noted, that the combination of the vertically aligned non-circular vortex patches of opposite cyclonic vorticity in two-layer fluid, strictly speaking, does not satisfy a stationary state. In this case the quasi-elliptical pulsating vortex patches rotate in different directions, and each of them induces an external field for its partner, which changes periodically both in time and in space. Therefore, we will interpret the initial state represented as the combination of two vertically aligned ellipses as some instantaneous disturbed quasi-stationary state. Below, we give the results of numerical experiments for investigating the evolution of such vortices. We used the two-layer version of the Contour  Dynamics Method (surgery) [14,17]. Before the discussion of the results we would like to note that Rd = g  h/2 f (g  = gρ/ρ0 , g is the acceleration of gravity, ρ—difference of fluid densities in the bottom and upper layers, h is characteristic vertical scale, ρ0 the mean value of the density, f is Coriolis parameter equal to a doubled rotation velocity of the whole plane). Thereby, the parameter γ may be treated in two ways: as a stratification parameter (γ  1 and γ  1 being the limits of the strong and weak stratification, correspondingly), and as a geometrical characteristic of the typical size of the vortex with respect to the radius of deformation.

2.1 General results In Fig. 1, a diagram of states of the initially elliptical heton in the space of (γ , χ ) variables represents the result of numerous calculations. One can see areas of existence both of stable quasi-periodic solutions, and of the solutions with different types of instability. Summary results are the following: – In case of a strong stratification (or small relative sizes of vortex patches) the barotropic type of interaction is predominant, when the vortices from the different layers weakly interact one with the other, and their evolution occurs essentially by the self-interaction. – When the eccentricity of the initially elliptical vortices grows, in the frames of this class of motions, there takes place a consecutive transition from the quasi-stable states of the system to the unstable ones with non-symmetrical division of the vortex patches; and further to symmetrical division into two parts. In the same time the area of the quasi-barotropic motions shrinks.

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Fig. 1 Diagram of possible states of an elliptical heton with the vertical axis in a rectangular area γ ∈ (0, 5], χ ∈ [1, 20]. Symbols for sub-areas: S1 , stable states; S2 , quasi-stable states after a partial loss of the mass because of dropping of vortex threads; U1bt , decay of the elliptical patches into non-symmetric parts because of the barotropic instability; U2bt , decay of elliptical patches into symmetric parts because of the barotropic instability; U1bc , decay of the elliptical patches into two hetons with tilted axes running away in the opposite directions because of the barotropic instability; U2bc , cascade baroclinic instability with running away of a series of hetons along the main ellipse axis; U3bc , cascade baroclinic instability with running away of hetons along both ellipse axes. Notches on the solid straight line χ = 1 divide the following areas for axially symmetrical heton: stability, and unstable mode with m = 2, m = 3 and m = 4 (from left to right)

– In case of a moderate stratification, a baroclinic type of interaction predominates with the realization of the vortex structure decay into two running in the opposite directions two-layer pairs with tilted axes. With the growth of ellipticity, the domain of existence with respect to the parameter γ of this class of motions also significantly shrinks. – In case of a strong stratification, the main mechanism of the development of the cascade instability, resulting in the further running away of more than two two-layer pairs.

2.2 Analysis of realizations of different regimes During our computation, the potential vorticity of the upper-layer vortex patch was taken to be positive, and of the lower-layer one is negative. The absolute value of ω was defined in each of the experiments by in such a way that a half of the rotational period of a fluid particle on the contour of a circular heton at the given value of γ would correspond to a dimensionless unit of time. Figure 2 shows the examples of calculations of the elliptical hetons’ evolution in the vicinity of the boundary of areas S1 and U1bc at the fixed values of the parameter χ = 2. In the first case a quasi-stable rotation of pulsating vortex patches in the upper and lower layers (barotropic type of interaction) occurs in the opposite directions. Vortex patches take a shape close to the elliptical one in the moment of the mutual overlapping (when the angular velocity of rotation is maximum) and the dumbbell shape when the conventional semi-axes have normal position (angular velocity is minimum in those moments). Note, that these effects—irregularities of rotation and deformation of contours—are expressed the stronger, the closer is the correspondent point of the plane (γ , χ ) to the boundary of the stability and instability areas. In the second case, due to the mechanism Reprinted from the journal

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Fig. 2 Synchronous configurations of the quasi-stable (left at γ = 1.56) and unstable (right at γ = 1.57) elliptical hetons at χ = 2 at the indicated moments of dimensionless time. Solid and dashed lines depict contours of the vortex patches in the upper and bottom layer, correspondingly

of baroclinic instability, the initial structure decays into two running away two-layer pairs with tilted axes. Let us note that there exists qualitative analogy of these variants with the case of two aligned point hetons [13], when the distance between them is either small, or large, correspondingly. Figure 3 demonstrates the evolution on the non-linear stage of the barotropic instability for the non-symmetrical (U1bt , left column) and symmetrical (U2bt , central and right columns) types at a fixed value of γ and different values of χ at the initial stage (before the separation of the vortex patches) and at the quasi-stationary stage of rotating movements of the vortices, that have being formed as a result of the decay of the initial vortex structure. The fact of the non-symmetrical decay of elliptical barotropic vortices at relatively small values of χ has been established for the first time in the numerical calculations by Kozlov and Makarov [18]. In Fig. 4 we illustrate the possibility of transition between the regimes of the cascade instability U2bc and U3bc , when the parameter γ increases. Note, that the initial evolution in the both variants runs identically, but

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Fig. 3 Synchronous configurations of the non-symmetrical (left at χ = 5.5) and symmetrical (central at χ = 10 and right at χ = 14.5) unstable elliptical hetons when γ = 0.25 at the indicated dimensionless times

at the instant t = 2, the central part in the second case begins by now to stretch in the vertical direction, what ensure the further running away of small vortex patches in the opposite directions of the y axis. This is the manifestation of the fourth mode formation. Reprinted from the journal

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Fig. 4 Synchronous configurations of unstable elliptical hetons with vertical axes at χ = 2 and γ = 4.55 on the left (type U2bc ) and γ = 4.57 on the right (type U3bc )

Note that Figs. 2 and 4 demonstrate a “robust” character of the vortex structures behaviour in dependence on the value of the parameter γ : in the vicinity of the boundary between the different regimes an insignificant change of the stratification parameter causes the transition to the qualitatively another type of the vortex patches’ interaction. 3 Summary In the present work, the classification of stable and unstable regimes of elliptical hetons is given. We have shown that the possibility exists of two types of instability—barotropic and baroclinic ones, and that a weak stratification stimulates the conditions for cascading instability. Acknowledgments We thank Hassan Aref and Dorte Glass for their care and hospitality during IUTAM Symposium in Lygby (October 12–16, 2008). MS was supported by Russian Foundation for Basic Research (projects 07-05-92210, 07-05-00452 and 08-05-00061).

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Kirchhoff, G.: Vorlesungen über matematische Physik: Mechanik. Taubner, Leipzig (1876) Love, A.E.H.: On the stability of certain vortex motion. Proc. Lond. Math. Soc. 25, 18–42 (1893) Mitchell, T.B., Rossi, L.F.: The evolution of Kirchhoff elliptical vortices. Phys. Fluids 20, 054103 (2008) Chaplygin, S.A.: On a pulsating cylindrical vortex. Trans. Phys. Sect. Imperial Moscow Soc. Frends Nat. Sci. 10, 13–22 (1899) Meleshko, V.V., van Heijst, G.J.F.: On Chaplygin’s investigations of two-dimensional vortex structures in an inviscid fluid. J. Fluid Mech. 272, 157–182 (1994) Kida, S.: Motion of an elliptical vortex in an uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520 (1981) Dritschel, D.G.: The stability of elliptical vortices in an external straining flow. J. Fluid Mech. 210, 223–261 (1990) Dhanak, M.R., Marshall, M.P.: Motion of an elliptical vortex under applied periodic strain. Phys. Fluids A5, 1224– 1230 (1993) Legras, B., Dritschel, D.G.: The elliptical model of two-dimensional vortex dynamics. I. The basic state. Phys. Fluids A3, 845– 854 (1991) Polvani, L.M., Flierl, G.R.: Generalized Kirchhoff vortices. Phys. Fluids 29, 2376–2379 (1986) Kozlov, V.F.: Model of two-dimensional vortex motion with an entrainment mechanism. Fluid Dyn. 27, 793–798 (1991) Pedlosky, J.: Geophysical Fluid Dynamics, 2nd edn. Springer, Berlin (1987) Hogg, N.G., Stommel, H.M.: The heton, an elementary interaction between discrete baroclinic geostrophic vortices, and its implications concerning eddy heat-flow. Proc. R. Soc. Lond. A397, 1–20 (1985) Kozlov, V.F., Makarov, V.G., Sokolovskiy, M.A.: A numerical model of baroclinic instability of axially symmetric vortices in a two-layer ocean. Izvestiya Atmos. Ocean. Phys. 22, 868–874 (1986) Pedlosky, J.: The instability of continuous heton clouds. J. Atmos. Sci. 42, 1477–1486 (1985) Helfrich, K.R., Send, U.: Finite-amplitude evolution of two-layer geostrophic vortices. J. Fluid Mech. 197, 331–348 (1988) Makarov, V.G.: Computational algorithm of the contour dynamics method with changeable topology of domains under study. Model. Mech. 5(22), 83–95 (1991) (in Russian) Kozlov, V.F., Makarov, V.G.: Evolution modelling of unstable geostrophic eddies in a barotropic ocean. Oceanology 24, 737– 743 (1984)

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Theor. Comput. Fluid Dyn. (2010) 24:125–130 DOI 10.1007/s00162-009-0140-7

S P E C I A L I S S U E O R I G I NA L A RT I C L E

Xavier Carton · Glenn R. Flierl · Xavier Perrot · Thomas Meunier · Mikhail A. Sokolovskiy

Explosive instability of geostrophic vortices. Part 1: baroclinic instability

Received: 17 October 2008 / Accepted: 24 June 2009 / Published online: 8 August 2009 © Springer-Verlag 2009

Abstract In a quasi-geostrophic model, we study the baroclinic instability of a two-layer vortex. The singular unstable modes for potential vorticity anomalies are compared with the classical normal modes. Short-time singular modes are explosively unstable and, at short times, depend only on the baroclinic component of the flow. As time progresses, they evolve towards the normal modes and their sensitivity to flow parameters is explored. Asymptotic solutions are provided. Keywords Quasi-geostrophic equations · Inviscid flows · Vortex stability · Normal or singular modes · Parametric resonance PACS 92.05.Bc, 92.10.ak, 92.10.ei, 47.15, 47.20.Cq

1 Introduction In stratified rotating turbulence and in planetary fluids, baroclinic processes have been recognized as essential in the generation and evolution of vortices. Indeed, such vortices can be generated by the instability of intense surface jets, or via mechanical or thermal forcing at the fluid surface. Once formed, these vortices most often reach a cyclogeostrophic balance in stratified, rotating fluids. Up to now, the stability of circular geostrophic vortices has been mostly studied with normal-mode perturbations [1–7]. These studies have shown that the nonlinear evolution of such unstable vortices, perturbed with normal modes, can lead to more complex vortices (multipoles). Explosive growth of perturbations on atmospheric flows has often been observed. Such explosive growth is usually related to singular modes, which are the temporarily fastest growing perturbation of the linearized dynamical equations [8]. Singular modes are a linear combination of the normal modes when these latter are not orthogonal (i.e. when the linear operator is not self-adjoint). The present study considers the nature and Communicated by H. Aref X. Carton (B) LPO, UMR 6523, UEB/UBO, Brest, France E-mail: [email protected] X. Perrot · T. Meunier LPO, UEB/UBO, Brest, France G. R. Flierl EAPS, MIT, Cambridge, MA, USA M. A. Sokolovskiy Institute for Water Problems of the RAS, Moscow, Russia Reprinted from the journal

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properties of singular modes in the baroclinic instability of a circular vortex in a two-layer model, and relates them to normal modes. 2 Model equations, linear instability of the baroclinic vortex and normal modes The two-layer quasi-geostrophic equations describe the conservation of layerwise potential vorticity in the absence of forcing and of dissipation dq j = 0, with q j = ∇ 2 ψ j + F j (ψk − ψ j ), dt where q j is layerwise potential vorticity (the subscripts j = 1, 2 denote upper and lower layers, respectively, and k = 3 − j), F j are the layer coupling coefficients (F j = f 02 /g  H j ), H j is layer thickness, and  √ H = H1 + H2 . The internal deformation radius is Rd = g  H1 H2 / f 0 H and γ is its inverse. In the two-layer quasi-geostrophic model, we study the instability of a circular baroclinic vortex, composed of two superimposed disks of constant potential vorticity Q j and of unit radius. This mean vortex is steady and has potential vorticity Q j (r ) and streamfunction ψ j (r ). Hereafter, the perturbation is assumed to have a wave-like dependence in angle, but not always an exponential variation with time (i.e. the perturbation is not always a normal mode): its streamfunction is ψ j (r, θ, t) = A j (t)φ j (r ) exp(ilθ ). The perturbation is a displacement of the vortex boundary which becomes. This vortex boundary is then r j = 1 + η j (θ, t). The vorticity contour displacement is written η j (θ, t) = η0j (t) exp(ilθ ). The linear instability equations are obtained by applying three conditions : continuity of normal and tangential velocities at the boundary for the total flow, kinematic condition for the contour displacement (see again [1– 3,5]). For convenience, the linear equations are written in terms of barotropic and baroclinic flow components. These components are defined by Q t = h 1 Q 1 + h 2 Q 2 , Q c = Q 1 − Q 2 with h j = H j /H . The mean flow has only tangential velocity given in barotropic and baroclinic components by Vt (r ) = Q t r/2, Q t /2r (inside and √ outside the vortex) and Vc (r ) = Q c I1 (γ r )K 1 (γ ), Q c I1 (γ )K 1 (γ r ). We also write δ = H1 /H2 , ξ = (1−δ)/ δ and V0 = Vt (1)/Vc (1). With these notations, the linear instability is described in vertical modes by the equations       1 1 0 0 ∂t ηt = −il Vt (1) 1 − ηt + Vc (1) 1 − ηc0 , l 2l I1 (γ )K 1 (γ )       Il (γ )K l (γ ) Il (γ )K l (γ ) ηt0 + Vt (1)(1 − 2Il (γ )K l (γ ))ηc0 + ξ Vc (1) 1 − ηc0 . ∂t ηc0 = −il Vc (1) 1− I1 (γ )K 1 (γ ) I1 (γ )K 1 (γ ) This problem can be set in vector form as ∂t X = AX with X (ηt0 , ηc0 ) and    a 0 b0 . A = −il c0 d0 Normal modes are obtained by setting ∂t X = σ X where σ contains the growth rate s and the rotation rate ω of the perturbation (σ = s − ilω). They are computed via σ± =

−il(a0 + d0 ) il √ ± 2 2

with = (a0 − d0 )2 + 4b0 c0 . Instability occurs when < 0. These growth rates are plotted in the (γ , V0 ) plane in Fig. 1 (upper left panel) for l = 2 and equal layer thicknesses (δ = 1); one can clearly see that the largest growth rates are obtained for deformation radii smaller than the vortex radius (γ > 1) and for moderate barotropic potential vorticity of the mean state. Vertical symmetry of the physical configuration explains the invariance of results with respect to a sign reversal in

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V0 . For a shallower upper layer (H1 /H2 = 0.2, upper right panel of Fig. 1), this symmetry is broken and maximal growth rates are displaced towards negative components of barotropic potential vorticity of the mean state. Such vortices intensified at depths are not typical of the ocean. For very thin upper layers (not shown), the maximal growth rates are not substantially modified in amplitude, but they are shifted towards strongly negative V0 . Finally, for higher modes, maximal growth rates are similar in amplitude to those of l = 2, but are found at larger deformation radii, as usual for baroclinic instability (see [3]). 3 Singular modes definition and results and asymptotic formulations For the linear equation ∂t X = AX with X (ηt0 , ηc0 ), the solution will be X (t) = M(t)X (0),

M(t) = exp(At)

if A is stationary. M is called the resolvent. Singular modes are defined as those providing the largest amplification rate at given time t; this rate is λ(t) =

|X (t)|2 |M(t)X (0)|2 = . |X (0)|2 |X (0)|2

Therefore, λ(t) is the largest eigenvalue of M ∗ (t)M(t) or exp(A∗ t) · exp(At). If A is self-adjoint (Hermitian), then its eigenvectors (the normal modes) are orthogonal and identical to the singular modes, because then M ∗ M = exp(2 At). Singular modes differ from normal modes when A is not Hermitian. The anti-Hermitian part of the matrix is related to the baroclinic component of the mean state and to the different Green’s functions for the Poisson and Helmholtz problems. Detailed calculation procedures for the singular modes and their associated eigenvectors are provided in [9,10]. These formulae have been adapted to the present problem. For comparison with normal mode growth rates, we associate singular growth rates to the singular amplification rates via σs (t) = Log(λ(t))/2t. Singular growth rates are plotted in Fig. 2 at different times. At short times, these rates do not depend on the barotropic component of the flow. Indeed, the onset of baroclinic instability depends on the sign reversal of the mean potential vorticity gradient (Charney–Stern criterion) and on the proper phase relation between layerwise perturbations. This involves only the baroclinic component of the flow. One can note also that singular growth rates are much larger then than those of normal modes (twice as large). This result can also be obtained via an asymptotic expansion of M ∗ M at short times (as shown by [8,9]): M ∗ M ∼ I d + (A∗ + A)t where I d is the identity matrix. The amplification rates thus obtained are identical to those obtained by the complete computation of eigenvalues of M ∗ M at short time (see again upper left panel of Fig. 2). This expansion also explains mathematically the independence of λ on V0 in this case: the extradiagonal terms involve Reprinted from the journal

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only Vc and not Vt . Now, as time goes, the diagonal terms of M ∗ M will come into play and sensitivity to V0 will appear. This is shown in Fig. 2, upper right and lower panels, for increasing time. A decrease in amplification rate accompanies this evolution. At long times, the singular growth rates become comparable to the normal mode growth rates. Indeed, as shown by [10], and calling η+ (1, a+ ) and η− (1, a− ) the eigenvectors associated to σ+ and to σ− , the amplification rate at long times can be expressed as λ(t) =

(1 + |a+ |2 )(1 + |a− |2 ) exp(2Re(σ+ t)) |a+ − a− |2

and thus σs → Re(σ+ ) (see again [9]). One can also note that the associated eigenvector is then the bi-orthogonal of η+ and is η++ (a+ , −1). Finally, the sensitivity of singular modes to physical parameters is explored. For higher modes than l = 2, the time-evolution of singular modes is very similar to that shown in Fig. 2. For unequal layer thicknesses, the parity bias of growth rates with respect to V0 grows with time as shown in Fig. 3, but the global evolution is similar to that of equal layer thicknesses. 4 Conclusions In this paper, we have shown how a baroclinic vortex can be unstable both to normal and to singular modes in a two-layer quasi-geostrophic model. The sensitivity of baroclinic instability with normal modes to the barotropic component of the mean flow, to stratification and to the perturbation wavenumber have been explored. Singular modes can grow in regions of parameter space where normal modes are stable. Singular modes are

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due to the non-Hermiticity of the matrix associated with linear dynamics (or in other words to the absence of orthogonality of normal modes). Singular modes are much more unstable than normal modes, on short times. On the long run, their growth rates converge towards those of the normal modes. Acknowledgments XC and XP acknowledge fruitful discussions with Dr Bach Lien Hua during the course of this work.

References 1. Pedlosky, J.: The instability of continuous heton clouds. J. Atmos. Sci. 42, 1477–1486 (1985) 2. Kozlov, V.F., Makarov, V.G., Sokolovskiy, M.A.: A numerical model of baroclinic instability of axially symmetric vortices in a two-layer ocean. Izv. Atmos. Ocean. Phys. 22, 868–874 (1986) 3. Flierl, G.R.: On the instability of geostrophic vortices. J. Fluid Mech. 197, 349–388 (1988) 4. Helfrich, K.R., Send, U.: Finite-amplitude evolution of two-layer geostrophic vortices. J. Fluid Mech. 197, 331–348 (1988) 5. Sokolovskiy, M.A.: Numerical modelling on nonlinear instability of axially symmetric two-layered vortices. Izv. Atmos. Ocean. Phys. 24, 735–743 (1988) 6. Carton, X.J., Mc Williams, J.C.: Barotropic and baroclinic instabilities of axisymmetric vortices in a QG model. In: Nihoul, J.C.J., Jamart, B.M. (eds.) Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, Liège 1988 International Colloquium on Ocean Hydrodynamics, vol. 50, pp. 225–244. Elsevier Oceanographic Series, Amsterdam (1989) 7. Carton, X.J., Corréard, S.M.: Baroclinic tripolar vortices: formation and subsequent evolution. In: Sorensen, J.N., Hopfinger, E.J., Aubry, N. (eds.) Simulation and Identification of Organized Structures in Flows, IUTAM/SIMFLOW 1997 Symposium in Lyngby, pp. 181–190. Kluwer, Dordrecht (1999) 8. Farrell, B.F., Ioannou, P.J.: Generalized stability theory. Part I: autonomous operators. J. Atmos. Sci. 53, 2025–2040 (1996) Reprinted from the journal

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9. Riviere, G., Hua, B.L., Klein, P.: Influence of the beta-effect on non-modal baroclinic instability. Q. J. R. Meteorol. Soc. 127, 1375–1388 (2001) 10. Fischer, C.: Linear amplification and error growth in the 2D Eady problem with uniform potential vorticity. J. Atmos. Sci. 55, 3363–3380 (1998)

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Theor. Comput. Fluid Dyn. (2010) 24:131–135 DOI 10.1007/s00162-009-0139-0

S P E C I A L I S S U E O R I G I NA L A RT I C L E

Xavier Carton · Thomas Meunier · Glenn R. Flierl · Xavier Perrot · Mikhail A. Sokolovskiy

Explosive instability of geostrophic vortices. Part 2: parametric instability

Received: 17 October 2008 / Accepted: 24 June 2009 / Published online: 8 August 2009 © Springer-Verlag 2009

Abstract In a two-layer quasi-geostrophic model, a baroclinic vortex is submitted to a periodic forcing of its mean baroclinic azimuthal velocity. It is shown that parametric effects could stabilize a vortex which is baroclinically unstable in the absence of forcing. Conversely, parametric resonance can destabilize a baroclinically stable vortex, under conditions on the vortex parameters, on the ratio of layer thicknesses or on the forcing frequency. Keywords Quasi-geostrophic equations · Inviscid flows · Vortex stability · Parametric resonance PACS 92.05.Bc, 92.10.ak, 92.10.ei, 47.15, 47.20.Cq

1 Introduction In stratified rotating turbulence and in planetary fluids, vortices play an essential role in the transport of momentum, heat and tracers. The baroclinic instability of vortices has been studied with normal-mode perturbations [1–3]. The nonlinear evolution of these vortices, perturbed with normal modes, usually leads to multipolar vortices. Part 1 of this paper has compared the properties of linear baroclinic instability of vortices with piecewise-constant potential vorticity, perturbed with normal modes or with singular modes. In a time-varying flow, the resonance of baroclinically neutral waves and of the forcing can lead to parametric instability, in particular for parallel flows [4]. Parametric instability has not yet been studied for vortex flows. Theoretical elements underlying this instability have been developed in [5]. Here, we consider the parametric resonance of neutral waves with a periodic external forcing which modifies the baroclinic velocity of this vortex. We study the properties of this parametric instability. Communicated by H. Aref X. Carton (B) LPO, UMR 6523, UEB/UBO, Brest, France E-mail: [email protected] T. Meunier · X. Perrot LPO, UEB/UBO, Brest, France

G. R. Flierl EAPS, MIT, Cambridge, MA, USA M. A. Sokolovskiy Institute for Water Problems of the RAS, Moscow, Russia Reprinted from the journal

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2 Model equations and parametric instability of an oscillating baroclinic vortex The forced two-layer quasi-geostrophic equations describe the evolution of layerwise potential vorticity dq j = E j (r, t), q j = ∇ 2 ψ j + F j (ψk − ψ j ) dt where q j is the layerwise potential vorticity (the subscripts j = 1, 2 denote upper and lower layers respec2 /g  H ), H is layer thickness, and tively, and k = 3 − j), F j are the layer coupling coefficients j j  (F j = f 0√  H = H1 + H2 . The internal deformation radius is Rd = g H1 H2 / f 0 H and γ is its inverse. The term E j (r, t) is a time periodic forcing of the baroclinic mean flow. If neutral waves traveling on the vortex periphery resonate with this forcing (like a pendulum whose axis of rotation oscillates vertically), they can be amplified via the process of parametric instability. Part 1 of the paper has shown that the linear instability of a two-layer vortex with piecewise-constant potential vorticity can be described in vertical modes by the equation ∂t X = AX with X (ηt , ηc ) and    ab A = −i cd Calling Vt (1), Vc (1) the barotropic and baroclinic velocities of the mean flow, at the vortex boundary, we have a = lVt (1)(1 − 1/l), b = lVc (1)(1 − 1/(2l I1 (γ )K 1 (γ )), c = lVc (1)(1 − (Il (γ )K l (γ ))/(I1 (γ )K √ 1 (γ )) and d = lVt (1)(1 − 2Il (γ )K l (γ )) + ξlVc (1)(1 − (Il (γ )K l (γ ))/(I1 (γ )K 1 (γ )), where ξ = (1 − δ)/ δ with δ = H1 /H2 . 3 Parametric instability near marginality of baroclinic instability We assume that the mean baroclinic velocity is close to that at marginality of baroclinic instability (called Vc0 at the vortex boundary). The time periodic forcing adds a weak unsteady component to this velocity to allow parametric instability. The baroclinic azimuthal velocity at the vortex boundary is then written Vc (1) = Vc0 [1 + εh(t)] with ε  1, h(t) is a time periodic function. Under these conditions, the linearized dynamics matrix is A = A0 + εh(t)A1 A0 is given by the expression of A hereabout with Vc (1) = Vc0 . Note that marginality of baroclinic instability is defined by (a0 −d0 )2 +4b0 c0 = 0. The four terms of A1 are given by a1 = 0, b1 = Vc0 (1−1/(2l I1 (γ )K 1 (γ )), c1 = Vc0 (1 − (Il (γ )K l (γ ))/(I1 (γ )K 1 (γ )) and d1 = ξ Vc0 (1 − (Il (γ )K l (γ ))/(I1 (γ )K 1 (γ )). It is easily shown that matrices A0 and A1 do not commute, if Vt  = 0, and therefore parametric instability is possible [5]. The calculations developed in [4] are adapted to our case. The contour perturbation X is expanded in powers of ε as X = X 0 +ε X 1 +ε2 X 2 +· · ·. The fast time is t0 = lb0 t, slow times are defined by t1 = εt0 , t2 = ε2 t0 . . .. First, we assume that h(t) = H cos(ωt0 ) + εG. H is the amplitude of the oscillatory part of the mean flow, G is the supercriticality of the mean flow (if positive, or the subcriticality if negative) and ω is the pulsation of the oscillatory mean flow. This will allow parametric resonance of neutral Rossby waves, associated with the potential vorticity jump at the vortex boundary, with the oscillatory baroclinic mean flow. To simplify the linear instability equations, we define the new variables Y j = X j ex p(il(a0 + d0 )t0 /2),

B0 = A0 +

il (a0 + d0 )I d, 2

where I d is the identity matrix. The linear equations are now written ∂t0 Y0 = B0 Y0 ∂t0 Y1 + ∂t1 Y0 = B0 Y1 + H cos(ωt0 )A1 Y0 ∂t0 Y2 + ∂t1 Y1 + ∂t2 Y0 = B0 Y2 + H cos(ωt0 )A1 Y1 + G A1 Y0

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Calling (y jt , y jc ) the components of Y j , and setting α = (d0 − a0 )/(2b0 ), β = d1 /b0 , the equations at zeroth order in ε are ∂t0 y0t = iαy0t − i y0c ∂t0 y0c = −iαy0t + iα 2 y0c By differentiating the equations in t0 , we can show that the neutral waves are described by ∂t0 Y0 = 0 and by y0c = αy0t . At first order, the equations are ∂t0 y1t + ∂t1 y0t = iαy1t − i y1c − i H cos(ωt0 )y0c ∂t0 y1c + ∂t1 y0c = −iαy1t + iα 2 y1c + iα 2 H cos(ωt0 )y0t − iαβ H cos(ωt0 )y0t We differentiate each equation in t0 to find ∂t20 Y1 , we make use of the fact that ∂t0 Y0 = 0, and then we integrate ∂t20 y1t to find y1t = −

H [α(2α − β) cos(ωt0 ) + iαω sin(ωt0 )] y0t ω2

and y1c is readily obtained via the first equation of this first order system, as y1c = i∂t1 y0t +

iH [ωα(α − β) sin(ωt0 ) + iα 2 (2α − β) cos(ωt0 )] y0t ω2

Finally, at second order in ε, the equations are ∂t0 y2t + ∂t1 y1t + ∂t2 y0t = iαy2t − i y2c − i H cos(ωt0 )y1c − i Gy0c ∂t0 y2c +∂t1 y1c +∂t2 y0c = −iαy2t + iα 2 y2c + iα 2 H cos(ωt0 )y1t − iαβ H cos(ωt0 )y1t + iα 2 GY0t − iβGy0c The term ∂t2 Y0 participates in the evolution at longer times and therefore is not kept here. The equations are time-averaged in t0 to isolate the wave interaction with the forcing, so that all linear terms in cos(ωt0 ), sin(ωt0 ) vanish. The value of Y1 found above is substituted in the remaining expressions. This yields the slow-time variation of the contour, due to the interaction of neutral waves with the forcing   ∂ 2 y0t α2 H 2 2 − α (1 − χ ) G − (1 − χ /2) y0t = 0 ω2 ∂t12 where χ = β/α. The same equation holds for y0c . This equation is now physically interpreted: For a supercritical steady flow (G > 0, H = 0, χ = 0), adding an unsteady mean flow (H  = 0 or χ  = 0) can have several effects: – either χ < 1 and the unsteady component of the flow can stabilize the vortex flow when α 2 H 2 /ω2 > G/(1 − χ /2) (the case χ = 0 is the case described in [4]), – or χ = 1, and there is linear growth of y0t with the slow time t1 , – or χ > 1 and stabilization will occur if (1 − χ /2)α 2 H 2 /ω2 < G. Note that if χ > 2, this condition is automatically satisfied. For a subcritical steady flow (G < 0, H = 0, χ = 0), adding an unsteady mean flow (H  = 0 or χ  = 0) can have several effects: – either χ < 1, and the flow remains stable in the presence of an oscillatory component of the baroclinic velocity (this includes the equal layer thickness case ξ = 0 for which χ = 0; a specific study of this case is given below), – or χ = 1, and there is linear growth of y0t with the slow time t1 , – or 1 < χ ≤ 2 and then the flow becomes unconditionally unstable on long times, – or finally 2 < χ , and the flow will be unstable on long times if 2|G| > (χ − 2)α 2 H 2 /ω2 . Reprinted from the journal

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Contours of Phi and of marginal baroclinic instability, l=2, delta=0.2 5 1.5

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Fig. 1 Contours of φ (with values) in the (γ , V0 ) plane, superimposed on the marginal curve for baroclinic instability (solid black line without value)

To relate χ to the vortex parameters, we write χ = β/α = 2d1 /(d0 − a0 ) = 2/(1 + φ), where   Vt (1) 1l − 2Il (γ )K l (γ )

φ= )K l (γ ) Vc0 ξ 1 − II1l (γ (γ )K 1 (γ ) This formulation excludes the equal layer case ξ = 0. We can now state that the condition χ = 1 is φ = 1, and the condition χ = 2 is φ = 0. For l = 2, δ = 0.2 (ξ ∼ 1.8), the values of φ are plotted in the (γ , V0 ) plane (where V0 = Vt (1)/Vc0 , see Fig. 1). The marginal curve of baroclinic instability is superimposed. Since the isolines φ = 0 and φ = 1 cross the marginal curve, stabilization of supercritical flows and destabilization of subcritical flows can occur. In the case of equal layer thicknesses, ξ = 0 (and thus χ = 0); then, destabilization of a subcritical flow cannot occur with the forcing used above. But Pedlosky and Thomson [4] showed that a weak, low frequency forcing of the type h(t) = ε2 (G + H cos(ωt1 )) can lead to such a destabilization. Note that ωt1 = εωt0 , hence the low frequency forcing. We adapt their calculation to the present case. At zeroth order in ε, the equations are unchanged from the previous case. At first order in ε, they are ∂t0 y1t + ∂t1 y0t = iαy1t − i y1c ∂t0 y1c + ∂t1 y0c = −iαy1t + iα 2 y1c . The same calculation as for zeroth order shows that ∂t0 Y1 = 0. Finally, at second order, the equations are ∂t0 y2t + ∂t1 y1t + ∂t2 y0t = iαy2t − i y2c − i H cos(ωt1 )y0c − i Gy0c ∂t0 y2c + ∂t1 y1c + ∂t2 y0c = −iαy2t + iα 2 y2t + iα 2 H cos(ωt1 )y0t + iα 2 Gy0t Again, we exclude the term ∂t2 Y0 which participates in the evolution at longer times, we time-average the equations in t0 , we differentiate the first order equations in t1 and we substitute ∂t1 Y1 from the second order equations. This leads to ∂ 2 y0t − 2α 2 [G + H cos(ωt1 )] y0t = 0 ∂t12

√ This equation is a Mathieu equation which is integrated numerically. The resonant pulsation ω = 2 −2α 2 G is chosen. The results are shown on Fig. 2 for H = 0 (non amplified oscillation) and for H = 0.1 (parametrically amplified oscillation). Thus, contour perturbations on a baroclinically stable vortex can be amplified via resonance with a low frequency forcing.

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Perturbation amplitude versus time, parametric instability near marginality 0.06 0.04 0.02

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0

−0.02 −0.04 −0.06 −0.08 −0.1 −0.12 0

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Fig. 2 Parametric amplification of the contour perturbation for γ = −2.0, G = −0.1, H = 0.1, y0t (t1 = 0) = 0.01, dy0t t1 (t1 = 0) = 0.0 and at marginality of baroclinic instability. The sinusoidal variation corresponding to the steady case H = 0.0 is shown as reference

4 Conclusions For a baroclinic vortex in a two-layer quasi-geostrophic flow, the interaction of neutral Rossby waves, associated with the vorticity jump at the vortex periphery, with the oscillatory component of the mean baroclinic velocity, can lead to parametric resonance, near marginality of baroclinic instability. Under given conditions on the steady and oscillatory mean velocities, parametric effects can stabilize vortex flows, which would otherwise be baroclinically unstable. Conversely, parametric resonance can destabilize subcritical baroclinic flows, if layer thicknesses are different. If layer thicknesses are equal, it was shown that a low frequency oscillation of the baroclinic mean flow can destabilize a subcritical vortex flow. For application to the ocean, the present study should be extended to more complex flows. Acknowledgments XC acknowledges fruitful discussions with Dr Francis Poulin during the course of this work. The authors thank a referee for a thorough analysis of the paper and for fruitful suggestions.

References 1. Pedlosky, J.: The instability of continuous heton clouds. J. Atmos. Sci. 42, 1477–1486 (1985) 2. Kozlov, V.F., Makarov, V.G., Sokolovskiy, M.A.: A numerical model of baroclinic instability of axially symmetric vortices in a two-layer ocean. Izv. Atmos. Ocean. Phys. 22, 868–874 (1986) 3. Flierl, G.R.: On the instability of geostrophic vortices. J. Fluid Mech. 197, 349–388 (1988) 4. Pedlosky, J., Thomson, J.: Baroclinic instability of time dependent currents. J. Fluid Mech. 490, 189–215 (2003) 5. Farrell, B.F., Ioannou, P.J.: Generalized stability theory. Part II: Non-autonomous operators. J. Atmos. Sci. 53, 2041–2053 (1996)

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Theor. Comput. Fluid Dyn. (2010) 24:137–149 DOI 10.1007/s00162-009-0109-6

O R I G I NA L A RT I C L E

Paul K. Newton

The N-vortex problem on a sphere: geophysical mechanisms that break integrability

Received: 4 November 2008 / Accepted: 31 March 2009 / Published online: 7 June 2009 © Springer-Verlag 2009

Abstract We describe the dynamical system governing the evolution of a system of point vortices on a rotating spherical shell, highlighting features which break what would otherwise be an integrable problem. The importance of the misalignment of the center-of-vorticity vector associated with a cluster of point vortices with the axis of rotation is emphasized as a crucial factor in the interpretation of dynamical features for many flow configurations. We then describe two important physical mechanisms which break what would otherwise be an integrable problem—the interactions between the local center-of-vorticity vectors of more than one region of concentrated vorticity, and the coupling between the center-of-vorticity vector and the background vorticity field which supports Rossby waves. Focusing on the Polar vortex splitting event of September 2002, we describe simple (i.e., low dimensional) mechanisms that can trigger instabilities whose subsequent development cause the onset of chaotic advection and global particle transport. At the linear level, eigenvalues that oscillate between elliptic and hyperbolic configurations initiate the pinch-off process of a passive patch representing the Polar vortex. At the nonlinear level, the evolution and topological bifurcations of the streamline patterns are responsible for its further splitting, stretching, and subsequent transport over the sphere. We finish by briefly describing how to incorporate conservation of potential vorticity and the development of a model governing the probability density function associated with the point vortex system. Keywords N -vortex problem on sphere · Nonintegrable Hamiltonian systems · Antarctic polar vortex PACS 47.32.C-, 47.10.Fg, 45.50.Jf, 92.60.-e 1 Introduction and derivation of equations In this article, we describe models based on point vortex discretizations of the vorticity field for incompressible flow on a sphere. Our context is the modeling of atmospheric events and our primary focus is on identifying physical mechanisms that complicate the clean, integrable aspects of the core problem, which is now well understood and whose literature can be found in Newton [23]. This paper summarizes the author’s Keynote lecture presented at the IUTAM Symposium ‘150 Years of Vortex Dynamics’ which took place at the Danish Technical University, Copenhagen, 12–16 Oct 2008. In the paper, I attempt to: (i) set the stage for the geophysical context for the models presented; (ii) summarize and review a coherent theme developed in recent papers by the author and co-workers; (iii) highlight where and how the types of low-dimensional discrete vortex models presented are useful in gaining insights into how much more complex geophysical processes ‘break-integrability’; (iv) point to important extensions of the current hierarchy of models that are key towards making the models more useful with respect to understanding atmospheric flows. Communicated by H. Aref P. K. Newton University of Southern California, Los Angeles, USA E-mail: [email protected]; [email protected] Reprinted from the journal

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Fig. 1 a A (frictionless) spinning disc on a rotating sphere has been studied as a simple mechanistic model of a geophysical vortex. Original figure can be found in Ripa [33]. b Yet even the most highly concentrated vorticity fields (Hurricane Katrina 23 August 2005) present additional modeling complications due to internal deformations, filamentation, background coupling, and vertical stratification (http://www.nnvl.noaa.gov/)

To set the stage, we mention recent work on the modeling and dynamics of a frictionless spinning disc on the surface of a rotating sphere, as shown in Fig. 1a, which presents a clean barebones model of some of the key features encountered when attempting to model the evolution of a concentrated geophysical vortex on a rotating planet. The initial work of Nycander [28] established the analogy between the steady westward drift of a distributed geophysical vortex and the steady precession of a rigid body. This was followed by McDonald [21] who further pointed out that the nutation of a rigid body is analogous to unsteady inertial oscillations of geophysical vortex trajectories. Then Ripa [32–34] systematically compared the errors due to a β-plane treatment as opposed to a full spherical approach, documenting the subsequent errors on the order O(R −1 ), where R is the earth’s radius. From this work, it is clear that even the most highly concentrated vorticity field on the sphere, such as Hurricane Katrina, shown in Fig. 1b, presents significant additional modeling challenges due to the distributed vorticity fields associated with hurricanes, as well as the additional ‘background’ vorticity field spread across the spherical surface with which it interacts. In addition, traditional β-plane models, while unquestionably useful, do not paint a complete picture of the kinds of dynamical and kinematic mechanisms that can occur on the full sphere. Two of the main features we emphasize in this paper are (i) the nonlinear coupling between distinct concentrated clusters of vorticity, as shown in Fig. 2, and (ii) the nonlinear coupling of these clusters with the background field which supports Rossby waves, as shown in Fig. 3. The first can be modeled staying within the context of finite-dimensional systems (N < ∞), while the second requires that the point vortices be placed in a continuous background field and thus be viewed as an ‘embedded’ dynamical system.

1.1 Interacting particle system Consider a charged point-particle of strength β ∈ R located at the North pole on a unit sphere. It generates 2 a velocity field √ that is independent of the azimuthal coordinate φ. Assume its influence decreases like 1/L , where L = 2(1 − cos θ ) is the chord distance, as shown in Fig. 4. This gives rise to a velocity field: θ˙ = 0, φ˙ =

β β 1 1 = . 2π L 2 4π (1 − cos θ )

(1) (2)

In Cartesian coordinates, with: xβ = (0, 0, 1); x = (sin θ cos φ, sin θ sin φ, cos θ ),

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(a)

(b)

Fig. 2 a Satellite photo of Hurricane Andrew (http://jpl.nasa.gov/), a category 5 hurricane formed on 16 August 1992. The cloud formations provide a snapshot of the instantaneous vorticity field (approximate) on a two-dimensional spherical shell. b Each concentrated vorticity region has a local center-of-vorticity vector (shown are J1 and J2 ) which is misaligned with the axis of rotation. Also shown is the background field represented as solid-body rotation and distributed point vortices throughout. The embedded dynamical system will evolve from this initial state via a complex evolution involving all of these key ingredients

Fig. 3 Rossby waves are the main physical mechanism that couple to the concentrated vorticity clusters and destroy rotational symmetry about the polar axis breaking conserved quantities. Polar and subtropical jetstreams provide symmetry-breaking perturbations to the underlying solid-body velocity field. Download from http://www.daukas.com

it is straightforward to show that (1), (2) are equivalent to x˙ =

β x β × x , 4π (1 − x · xβ )

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(5)

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Fig. 4 a A particle with strength  located at the North Pole generates a velocity field on the sphere which is azimuthally independent (i.e., moves on constant latitudinal curves) and decays with distance. An elliptic fixed point at the antipode is a topological necessity as a result of the Poincaré index theorem. b The angular velocity from the North Pole decays monotonically with distance

By linear superposition, a collection of N particles, each with their own strength β ∈ R (β = 1, . . . , N ), generates the velocity field: x˙ =

N  β x β × x . 4π (1 − x · xβ )

(6)

β=1

This equation expresses the velocity on the sphere in terms of the vorticity distribution, which we recognize as a discrete inversion of the vorticity–velocity relation ω = ∇ × u. In fact, (6) is the discrete Biot-Savart law on the unit sphere [23]. Finally, invoking Helmholtz’ [14] assumption that each particle moves with the local fluid velocity, we replace x with xα to obtain the N -vortex dynamical system on the sphere: x˙ α =

N  xβ × xα  β , (α = 1, . . . , N ) 4π (1 − xα · xβ )

(7)

2 = |xα − xβ |2 = 2(1 − xα · xβ ). lαβ

(8)

β=1

The prime on the summation indicates that we exclude the term β = α so as to avoid the singularity. It is worth pointing out that (7) can also be written as: x˙ α =

N  ˆ β × (xα − xβ )  β n , (α = 1, . . . , N ) 2 2π lαβ β=1

(9)

where nˆ β is the unit normal vector on the sphere at the vortex location, i.e., nˆ β ≡ xβ . The planar N -vortex equations are obtained from (9) by using the unit normal in the plane, i.e., nˆ β ≡ eˆz . Thus, the spherical problem contains an extra coupling term arising from the unit normal to the surface that the planar problem does not contain, as the normal vector on the plane is constant. Figure 4 brings out another important difference between the spherical and planar problems. Since the velocity field is invariant to rotations around the North-South polar axis, it is clear that an elliptic fixed point at the South Pole is forced upon us by topological considerations. Thus, if c represents the number of centers of the vector field (which in this case is two) and s represents the number of saddles (which in this case is zero) then c−s = 2. This simple and intuitive fact is a consequence of a much deeper and more general result, known as the Poincaré index theorem which states that the index I f (S) of a two-dimensional surface S relative to any C 1 vector field f on S with at most a finite number of critical points, is equal to the Euler–Poincaré characteristic

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of S, denoted χ (S), i.e., I f (S) = χ (S). It is a standard topological fact (see Katok and Hasselblatt [17]) that for a sphere, χ (S) = 2. The index for a center is +1, while that for a saddle is −1. Hence, if c denotes the number of centers present (point vortices plus other centers), and s denotes the number of saddles, then one must have c − s = 2. These facts were exploited by Kidambi and Newton [19] to categorize all integrable streamline patterns on the sphere. A surprising result is that for N ≤ 3, there are only 12 topologically distinct ‘primitive’ patterns, each of which can be homotopically deformed to form a catalogue of 35 kinematically distinct patterns from which all ‘snapshots’ of an integrable flowfield can be built. 1.2 Spherical coordinates While the Cartesian formulation is, in some ways the most transparent, for the purposes of studying integrability, the spherical coordinate system is more useful. With xα = (sin θα cos φα , sin θα sin φα , cos θα ), xβ = (sin θβ cos φβ , sin θβ sin φβ , cos θβ ),

(10) (11)

N 1   β sin(θβ ) sin(φα − φβ ) , 2π 2(1 − cos(γαβ ))

(12)

in (7), the equations become: θ˙α = −

β=1

sin(θα )φ˙ α =

N 1   β (sin(θα ) cos(θβ ) − cos(θα ) sin(θβ ) cos(φα − φβ )) , 2π 2(1 − cos(γαβ ))

(13)

β=1

where the denominator is the chord distance (squared) between vortex α and vortex β: 2 2(1 − cos(γαβ )) = 2(cos(θα ) cos(θβ ) + sin(θα ) sin(θβ ) cos(φα − φβ )) ≡ lαβ .

These equations have a Hamiltonian structure, where 1  2 H=− α β log(lαβ ) 4π

(14)

(15)

α 0) velocity field: uv(δ) (x) =

N xα × x   , 4π 1 + δ 2 − x · xα

us(δ) (x) = −

α=1

M  α=1

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(28)  ∂Xα dθ. log |x − Xα |2 + δ 2 ∂θ 159

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Fig. 6 The embedded dynamical system. Point vortices arranged in ring formation are embedded in a background field of continuous vorticity. The background field is made up of strips of constant vorticity whose strengths are chosen to represent solid-body rotation. The contours which separate the strips are discretized and tracked along with the point vortices making up the ring T:0.00

T:12.00

(b)

(a)

Fig. 7 a Initial orientation of four-point vortex ring embedded in a background field consisting of ten strips of constant vorticity. b Subsequent fully coupled evolution of the ring with the background. Note the wrapping of the vorticity strips around the point vortices effectively increasing their strength. See Newton and Sakajo [25] for full details (δ)

Here, uv (x) represents the regularized velocity field due to each of the N equal strength () point vortices, (δ) while us (x) represents the velocity field generated by each constant vorticity strip (ωα ), whose contours are parametrized by Xα . When the contours are discretized, Xk , (k = 1, . . . , M), one obtains a dynamical system of the form: ∂x j = uv(δ) (x j ) + us(δ) (x j ), j = 1, . . . , N , ∂t ∂Xk = us(δ) (Xk ) + uv(δ) (Xk ), k = 1, . . . , M. ∂t

(29) (30)

The fully coupled system evolves as shown in Fig. 7 for a ring comprised of four point vortices and ten constant vorticity strips. Note the wrapping of the contours around the point vortices effectively increases their strength relative to the background. In Newton and Sakajo [25] we show in detail how integrability is lost (for the ring) via the following clear sequence of events: 1. The ring triggers an instability in the background field and sets Rossby waves in motion. 2. These waves, which move retrograde to the solid-body rotation (as pointed out in [10]), cause the ring to alter its initial rotational direction. 3. The wrapping of contours around point vortices effectively increases the strength of the ring. 4. Eventually, the point vortices are strong enough to overcome the effect of the Rossby waves, thus the ring reverts back to its original rotational direction.

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5. Energy is continually exchanged between the ring and the background field breaking the conservation of both H and J (associated with the ring). 6. The stability and integrity of the ring is compromised. The long-time evolution of the N -vortices constituting the ring are expected to be chaotic, although there are significant computational challenges for producing accurate long-time measures of nonintegrability (e.g., Lyapunov exponents) due to the need to accurately track the contours whose length grows continuously requiring frequent re-meshing. 3 Polar vortex splitting event A nice paradigm for studying global transport on the sphere due to a ‘local’ geophysical event is the much studied Polar vortex splitting event of 2002. The event is thoroughly documented in a devoted journal issue— Journal of Atmospheric Sciences: Special Issue on the Antarctic Stratospheric Sudden Warming and Split Ozone Hole of 2002, 62(3) March 2005. In late September of 2002, over a period of a few days, the Antarctic ozone hole split in two. This is clearly evident in the ozone concentrations shown in Fig. 8. A split ozone hole implies a split vortex which subsequently initiated a sudden stratospheric warming. These types of warmings occur on a regular basis in the Arctic (although not annually) and are thought to be produced by the dynamical momentum forcing resulting from the breaking and dissipation of planetary-scale Rossby waves in the stratosphere. Prior to 2002, however, no stratospheric sudden warming had been observed in the Antarctic, and it was widely believed to be impossible. Figure 8 shows three panels of the Antarctic ozone hole in September 2001 (one year before the split), September 2002 (split ozone hole), and September 2003 (one year after the split). The goal of modeling such an event are first, to capture the basic features of this complex vortex splitting process, then to use a simple model to analyze its effect on the global transport of atmospheric tracer particles. The ingredients for the model can be captured, most simply, in a linear setting. We imagine that the Polar vortex is a passive patch being pulled and stretched at the Pole by a perturbed ring configuration representing the outer edge (shear layer) of the vortex, as shown initially in Fig. 10a. Since the Polar patch is centered at an elliptic point generated by the point vortices, one might first consider a linearized problem around the Pole. First, we construct a time-dependent Hamiltonian system with a fixed point (the center of the Polar vortex) that oscillates between hyperbolic and elliptic through one period: H(x, y; t) =

1 1 a(t)x 2 + b(t)y 2 , 2 2

hence

 x ; x˙ = Ax; x = y

A=

0

b(t)

−a(t) 0

(31)

.

(32)

Fig. 8 The splitting of the Antarctic ozone hole (http://www.gse-promote.org/), September 2002, as evident in total column ozone concentration. September 2001 and September 2003 are shown for comparison. The splitting event in 2002 is also evident in the stratospheric vorticity or potential vorticity fields Reprinted from the journal

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Fig. 9 Eigenvalue evolution of the time-dependent matrix A through one period as a linear mechanism capable of pinching off a passive patch. a First half period 0 ≤ t ≤ π/ω in which the eigenvalues lie on the imaginary axis; b Second half period π/ω ≤ t ≤ 2π/ω in which they lie on the real axis

√ The instantaneous eigenvalues of A are λ = ± −ab, where we let: a(t) = α sin(ωt/2), b(t) = α cos(ωt/2),

(33) (34)

√ so that λ(t) = ± √α − sin(ωt). The evolution of the eigenvalues, through one period, is shown in Fig. 9. For 2 half the period, the hyperbolic structure elongates and begins to tear the patch apart, whereas for the other half period, the elliptic structure rotates and wraps the patch on itself. We further transform coordinates to a rotating frame by letting: xr = Mx,

(35)

where

M=

cos t − sin t sin t cos t



, xr =

xr yr

 .

(36)

Then x˙ = M˙ T xr + M T x˙ r , where M −1 ≡ M T . In the new coordinates, this gives rise to the time-dependent Hamiltonian: H(xr , yr ; t) =

1 (b(t) sin2 t + a(t) cos2 t − )xr2 + (a(t) − b(t)) cos t sin t xr yr 2 1 + (a(t) sin2 t + b(t) cos2 t − )yr2 . 2

(37)

To see the effect of such a construction in a full nonlinear setting, we focus on a perturbed vortex ring formation. The outer edge of the polar vortex constitutes a shear layer, as represented in our model by the initial placement of point vortices while their subsequent evolution represents the deformation of the Polar vortex region. A prototypical splitting event can be created by a perturbed latitudinal 4-vortex ring, when the perturbation amplitude (a sub-harmonic perturbation in which the initial position of every other vortex is shifted by a small amount in order to initiate the pairing process) exceeds a certain threshold. The grey patch in the middle region of the ring shown in Fig. 10 pinches in the middle to form two elliptic regions, which subsequently stretch and fold due to a blinking back and forth between a local elliptic region and a hyperbolic region created by the perturbed ring. Measures of ergodicity and transport of these kinds of perturbed ring models and the connection with topological bifurcations of global streamline patterns can be found in Newton and Ross [24].

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Fig. 10 A perturbed four-vortex ring demonstrating its ability to split a single passive patch into two. The streamlines blink back and forth between and elliptic region and a hyperbolic region which is ultimately responsible for the stretching and pinch-off of the passive mass

4 Extensions and discussion We finish by pointing to two important extensions to the current hierarchy of models described in the paper that are key extensions from the point of view of understanding and modeling atmospheric dynamics. Both of these, alone, would break the integrability of the basic model for N = 3 by destroying the underlying conserved quantities. First, one must conserve potential vorticity Q, defined as:

 ω+ f Q= = const., (38) H N α , f = 2 p sin φ (Coriolis term), and H = 1 (layer depth). To achieve this, one can where ω = α=1 allow the point-vortex intensities α to depend on position φα : α (φα ) + 2 p sin φα = α(0) + 2 p sin φα(0) , (0)

(39)

(0)

where α and φα represents their intensity and position initially. The vortex intensity as a function of position is then given as: α (φα ) = 2 p (− sin φα + sin φα(0) ) + α(0) .

(40)

With the center-of-vorticity vector defined as in (24), clearly we lose these important conserved quantities and it is hard to imagine how integrability is maintained, or even relevant. Second, because of uncertainty in data measurements as well as incomplete modeling assumptions, it is desirable to construct a probabilistic version of the models presented in this paper. A natural way to achieve this is shown in Fig. 11. Instead of working with the deterministic point-vortex positions, one should work with the equations that govern their probability density functions (pdf’s). In isolation, as shown in Fig. 11a, the pdf associated with an individual point-vortex is the solution to the heat-equation on the surface of a sphere, hence the probabilistic ‘position’ (i.e., the ‘support’ of the probability density function) of the point-vortex diffuses self-similarly until in the limit t −→ ∞, it is equally likely to be located anywhere on the sphere’s surface. However, since the collection of point-vortices interact nonlinearly according to the evolution equations (7), we replace the deterministic position variables, xα in this formulation with the mean value µα . The probability density function for each point-vortex would then be governed (approximately) by: p(sα , t; µα ) ∼

1 exp(−|sα − µα |2 /4Dt), 4πDt

(41)

where µα is governed by the system (7). Here sα is the geodesic distance from the mean µα , and D is a diffusion constant related to the time-scale at which uncertainty infiltrates the deterministic system. Thus, the full system will unfold dynamically carrying a mixture of a ‘deterministic’ timescale (say, the positive Lyapunov exponents in the case N > 3) associated with the basic model (22) along with a timescale associated with Reprinted from the journal

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(b)

Fig. 11 a The support of the probability density function associated with an isolated point-vortex on a sphere of radius R grows self-similarly according to the diffusion equation on the surface of a sphere. b In addition, the collection of point-vortices interact according to the N -vortex dynamical system (7). The combination of these two effects remain to be explored

the rate at which uncertainty pervades the system (i.e., the diffusion constant D). The future development of any initial configuration intended to model a specific atmospheric event must depend on the interplay of these important timescales. Extensions such as these remain to be developed further within the context of the models presented in this paper. Low dimensional and ‘embedded’ models of specific atmospheric events, of the type discussed in this article, offer the possibility of obtaining detailed quantitative results in simplified settings based on analysis and simulation which ultimately can be validated or invalidated with more realistic numerical models and atmospheric data. The emphasis here has been on pointing out the importance of (i) the misalignment of the J vector with the axis of rotation; (ii) coupling with the background field which, at the very least, means full nonlinear interaction with Rossby waves; (iii) periodic oscillation of the local eigenvalue structure from hyperbolic to elliptic as a linear mechanism capable of splitting and stretching a passive patch; (iv) subsequent deformation and transport of the patch due to finite-amplitude oscillations of the shear layer constituting the outer layer of the Polar vortex. Within the context of these models, potential vorticity conservation and probabilistic ingredients are currently being developed. Acknowledgments Funding for this work was provided by the National Science Foundation through grants NSF-DMS-0504308 and NSF-DMS-0804629. I would like to thank R. Kidambi, M. Jamaloodeen, H. Shokraneh, and G. Chamoun whose Ph.D. theses, along with the postdoctoral work of Shane Ross at USC helped develop aspects of what is described in this article. I would also like to thank colleagues from whom I have learned much including Hassan Aref, Alexey Borisov, Darren Crowdy, David Dritschel, Eva Kanso, Yoshi Kimura, Chjan Lim, Phil Marcus, Jerry Marsden, Takashi Sakajo, Mark Stremler and Tamas Tel.

References 1. Baines, P.: The stability of planetary waves on a sphere. J. Fluid Mech. 73, 193–213 (1976) 2. Boatto, S., Pierrehumbert, R.T.: Dynamics of a passive scalar in a velocity field of four identical point vortices. J. Fluid Mech. 394, 137–174 (1999) 3. Bogomolov, V.A.: Two-dimensional fluid dynamics on a sphere. Izv. Atm. Ocean. Phys. 15(1), 18–22 (1979) 4. Bogomolov, V.A.: Dynamics of vorticity at a sphere. Fluid Dyn. 6, 863–870 (1977) 5. Borisov, A.V., Pavlov, A.E.: Dynamics and statics of vortices on a plane and a sphere I. Regul. Chaotic Dyn. 3(1), 28–38 (1998) 6. Borisov, A.V., Lebedev, V.G.: Dynamics and statics of vortices on a plane and a sphere II. Regul. Chaotic Dyn. 3(2), 99–114 (1998) 7. Bowman, K.P., Mangus, N.J.: Observations of deformation and mixing of the total ozone field in the antarctic polar vortex. J. Atmos. Sci. 50(17), 2915–2921 (1993) 8. Charlton, A.J., O’Neil, A., Lahoz, W.A., Berrisford, P.: The splitting of the stratospheric polar vortex in the southern hemisphere, September 2002: Dynamical evolution. J. Atmos. Sci. 62, 590–602 (2005) 9. Cushman-Roisin, B.: Introduction to Geophysical Fluid Dynamics. Prentice-Hall, NJ (1994) 10. DiBattista, M.T., Polvani, L.M.: Barotropic vortex pairs on a rotating sphere. J. Fluid Mech. 358, 107–133 (1998)

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11. Dritschel, D.G., Polvani, L.M.: The roll-up of vorticity strips on the surface of a sphere. J. Fluid Mech. 234, 47–69 (1992) 12. Haynes, P.: Transport, stirring and mixing in the atmosphere. In: Chaté, H., Villermaux, E. (eds.) Proceedings of the NATO Advanced Study Institute on Mixing, Chaos and Turblence, Cargese Corse, France, 7–20 July 1996, pp. 229–272. Kluwer, Dordrecht (1999) 13. Haynes, P.: Stratospheric dynamics. Ann. Rev. Fluid Mech. 37, 263–293 (2005) 14. Helmholtz, H.: On integrals of the hydrodynamical equations, which express vortex-motion. Phil. Mag. (Ser. 4) 33, 485–510 (1858) 15. Hobson, D.D.: A point vortex dipole model of an isolated modon. Phys. Fluids A 3, 3027–3033 (1991) 16. Joseph, B., Legras, B.: Relation between kinematic boundaries, stirring, and barriers for the Antarctic polar vortex. J. Atmos. Sci. 59, 1198–1212 (2002) 17. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, London (1995) 18. Kidambi, R., Newton, P.K.: Motion of three point vortices on a sphere. Physica D 116, 143–175 (1998) 19. Kidambi, R., Newton, P.K.: Streamline topologies for integrable vortex motion on a sphere. Physica D 140, 95–125 (2000) 20. Marcus, P.S.: Jupiter’s Great Red Spot and other vortices. Ann. Rev. Astron. Astrophys. 31, 523–573 (1993) 21. McDonald, N.R.: The time-dependent behavior of a spinning disc on a rotating planet: a model for geophysical vortex motion. Geo. Astro. Fluid Dyn. 87, 253–272 (1998) 22. McDonald, N.R.: The motion of geophysical vortices. Phil. Trans. R. Soc. Lond. A 357, 3427–3444 (1999) 23. Newton, P.K.: The N-vortex problem: analytical techniques. In: Applied Mathematical Science, vol. 145. Springer, New York (2001) 24. Newton, P.K., Ross, S.D.: Chaotic advection in the restricted four-vortex problem on a sphere. Physica D (1) 223, 36– 53 (2006) 25. Newton, P.K., Sakajo, T.: The N-vortex problem on a rotating sphere: III. Ring configurations coupled to a background field. Proc. R. Soc. A 463, 961–977 (2007) 26. Newton, P.K., Shokraneh, H.: The N-vortex problem on a rotating sphere: I multi-frequency configurations. Proc. R. Soc. A 462, 149–169 (2006) 27. Newton, P.K., Shokraneh, H.: Interacting dipole pairs on a rotating sphere. Proc. R. Soc. A 464(2094), 1525–1541 (2008) 28. Nycander, J.: Analogy between the drift of planetary vortices and the precession of a spinning body. Plasma Phys. Rep. 22, 771–774 (1996) 29. Pierrehumbert, R.T.: Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves. Geo. Astro. Fluid. Dyn. 58, 285–319 (1991) 30. Pierrehumbert, R.T., Yang, H.: Global chaotic mixing on isentropic surfaces. J. Atmos. Sci. 50(15), 2462–2480 (1993) 31. Polvani, L.M., Dritschel, D.G.: Wave and vortex dynamics on the surface of a sphere. J. Fluid Mech. 255, 35–64 (1993) 32. Ripa, P.: Inertial oscillations and the β-plane approximation(s). J. Phys. Oceanogr. 27, 633–647 (1997) 33. Ripa, P.: Effect of the Earth’s curvature on the dynamics of isolated objects Part I: the disk. J. Phys. Oceanogr. 30, 2072–2087 (2000) 34. Ripa, P.: Effect of the Earth’s curvature on the dynamics of isolated objects Part II: the uniformly translating vortex. J. Phys. Oceanogr. 30, 2504–2514 (2000) 35. Saffman, P.G.: Vortex Dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, London (1992) 36. Sakajo, T.: The motion of three point vortices on a sphere. Jpn. J. Ind. Appl. Math. 16, 321–347 (1999) 37. Sakajo, T.: Integrable four-vortex motion on a sphere with zero moment of vorticity. Phys. Fluids 19(1), 017109 (2007) 38. Simmons, A., Mortal, M., Kelly, G., McNally, A., Untch, A., Uppala, S.: ECMWF analysis and forecasts of stratospheric winter polar vortex break-up: September 2002 in the southern hemisphere and related events. J. Atmos. Sci. 62 (2005) 39. Waugh, D.W.: Contour surgery simulations of a forced polar vortex. J. Atmos. Sci. 50, 714–730 (1992) 40. Waugh, D.W., Plumb, R.A., Atkinson, R.J., Schoeberl, M.R., Lait, L.R., Newman, P.A., Loewenstein, M., Tooney, D.W., Avallone, L.M., Webster, C.R., May, R.D.: Transport out of the lower stratospheric Arctic vortex by Rossby wave breaking. J. Geo. Res. 99(D1), 1071–1088 (1994)

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Theor. Comput. Fluid Dyn. (2010) 24:151–156 DOI 10.1007/s00162-009-0116-7

O R I G I NA L A RT I C L E

Takashi Sakajo

From generation to chaotic motion of a ring configuration of vortex structures on a sphere

Received: 30 October 2008 / Accepted: 17 February 2009 / Published online: 2 June 2009 © Springer-Verlag 2009

Abstract This is a review article of recent research developments on the motion of a polygonal ring configuration of vortex structures with singular vorticity distributions in incompressible and inviscid flows on a non-rotating sphere. Numerical computation of a single vortex sheet reveals that the Kelvin-Helmholtz instability gives rise to the formation of a polygonal ring arrangement of rolling-up spirals. An application of methods of Hamiltonian dynamics to the N -vortex problem on the sphere shows that the motion of the ring configuration of homogeneous point vortices, which is a simple model for the rolling-up spirals, becomes chaotic after a long time evolution. Some remarks on an extension of the present research and a generic non-self-similar collapse are also provided. Keywords Flows on sphere · Vortex sheet · Point vortices · Saddle-center equilibria PACS 47.10A-, 45.15.ki

1 Introduction We consider the incompressible and inviscid flows on a non-rotating sphere. In particular, we focus on a polygonal ring configuration of vortex structures with singular vorticity distributions such as vortex sheets and point vortices. A vortex sheet is a one-dimensional discontinuous curve of the velocity filed, on which the vorticity distributes, while a point vortex corresponds to the pointwise vorticity distribution like Dirac’s δ-function. According to Kelvin’s circulation theorem, the vortex sheet and the point vortex evolve as if they were a material line and a point. Thus we have only to track the evolution of the line and the point, which makes it easy to deal with the problem by mathematical and numerical means. However, on the other hand, the intrinsic circulation is no longer conserved, when the sphere is rotating. As was shown in [2], the interaction between point vortices and the background rotation of the sphere is complicated, which makes it much harder to consider the problem mathematically. In the present review, as an external forcing, we introduce two point vortices that are fixed at the north and the south poles of the sphere, which we call the pole vortices. The strengths of the pole vortices are parameters we change. Communicated by H. Aref I would like to show my gratitude to School of Mathematics at the University of Sheffield for giving me nice research environments from September 2008 through March 2009. Partially supported by JSPS grant 19654014. T. Sakajo (B) Department of Mathematics, Hokkaido University, Sapporo, Japan E-mail: [email protected] T. Sakajo PRESTO, Japan Science and Technology Agency, Tokyo, Japan Reprinted from the journal

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(a) t=9.0

(b) t=18.0

(c) t=23.0

(d) t=35.0

Fig. 1 Snapshots of the vortex-sheet evolutions for various strengths of pole vortices. a 1 = 2 = −π; b 1 = 0.5π, 2 = −2.5π; c 1 = 1.5π, 2 = −3.5π; d 1 = 2.5π, 2 = −4.5π

In Sect. 2, we give numerical results on the evolution of a single vortex sheet corresponding to a line of latitude of the sphere [10] to show that the Kelvin-Helmholtz instability results in the generation of many rolling-up spirals whose centers are aligned along the line of latitude. Then in Sect. 3, we approximate the long-time evolution of the rolling-up spiral structure by a set of identical point vortices. The motion of the point vortices is described by a Hamiltonian dynamical system, to which we use methods of Hamiltonian dynamics in order to show the existence of a chaotic orbit. In the last section, we give some concluding remarks on an extension of the results and an example of a collapse of inhomogeneous point vortices. 2 Generation of ring configuration of vortex structure The vortex sheet is one of the simplest mathematical models of shear flows when the shear region is assumed to be infinitely thin. The vortex sheet is represented by a one-dimensional curve (θ (α, t), ϕ(α, t)) in the spherical coordinates, in which 0 ≤ α < 2π is a Lagrangian parameter along the curve and t is time. The evolution of the vortex sheet on the sphere is derived from that of the three-dimensional vortex sheet by assuming the flow is confined in the surface of the sphere [8], which is given as follows: 1 θt = − 4π

2π 0

1 ϕt = − 4π sin θ

2π 0

sin θ  sin(ϕ − ϕ  ) dα  , 1 − cos θ cos θ  − sin θ sin θ  cos(ϕ − ϕ  ) + δ 2

cos θ cos θ  cos(ϕ − ϕ  ) − sin θ sin θ  1 dα  +    2 1 − cos θ cos θ − sin θ sin θ cos(ϕ − ϕ ) + δ 4π



(1)

1 2 − 1 − cos θ 1 + cos θ

 . (2)

θ

in which = θ (α  , t) and ϕ  = ϕ(α  , t). The second term in the right-hand side of (2) represents the velocity field induced by the pole vortices, whose strengths are denoted by 1 and 2 , respectively. The parameter δ > 0 is a regularization parameter. Note that when δ = 0, the integrals in (1) and (2) are defined in the sense of Cauchy’s principal value integral. The regularization is required to compute the long-time evolution of the vortex sheet, since, without the regularization, the vortex sheet along the line of latitude θ0 , i.e., (θ, ϕ) = (θ0 , α), is a linearly unstable steady solution for (1) and (2) due to the Kelvin-Helmholtz instability, in which small sinusoidal disturbances on the sheet grow exponentially at a rate proportional to their wavenumber. The instability gives rise to the formation of a curvature singularity in finite time regardless of the strengths of the pole vortices, which has been reported in detail in [10]. Figure 1 shows some snapshots of the vortex-sheet evolutions for various 1 and 2 with δ = 0.1. The initial configuration is given by θ (α, 0) = π4 + 0.01 sin α and ϕ(α, 0) = α + 0.01 sin α. The vortex sheet evolves into a structure with many rolling-up spirals whose centers are aligned along a line of latitude of the sphere. The number of the rolling-up spirals depends on the strengths of the pole vortices. Let us note that the number can vary depending on the value of δ in general, but we have checked that the number is unchanged in the range of 0.09 ≤ δ ≤ 0.3. 3 Chaotic motion of ring configuration of homogeneous point vortices In order to consider the further long-time evolution of the ring structure of rolling-up spirals, we assume that their motion is approximated by that of point vortices that are set at the center of the spirals. Their strengths are

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determined so that the circulation contained in the rolling-up spirals concentrates in their centers. We further assume that the point vortices are equally spaced along the line of latitude and the strengths are unity. This polygonal ring configuration of homogeneous point vortices is called the N -ring. In what follows, based on the papers [5–7,9], we describe how the N -ring evolves when it becomes unstable. Let (m , m ) for m = 1, . . . , N denote the position of N point vortices in the spherical coordinate. The motion of the N point vortices on the sphere with the pole vortices is described by a Hamiltonian dynamical system with N degrees of freedom [1], whose Hamiltonian is given by H =−

N N 1  log(1 − cos m cos  j − sin m sin  j cos( m − j )) 8π m=1 j =m

−

N N 2  1  log(1 − cos m ) − log(1 + cos m ). 4π 4π m=1

m=1

The N -ring is a relative equilibrium for the Hamiltonian system. It follows from the linear stability analysis that we have 2N eigenvalues for the linearized equation including multiplicity, which are denoted by λ± 0 =0 and λ± for m = 1, . . . , M = [N /2] in which the symbol [x] denotes the largest integer less than x. They m  ± 2  ± 2  ± 2 < λj for 0 < i < j and so the N -ring is neutrally stable when λ M < 0; satisfy an order relation λi   2 2  2 < · · · < λ± < 0 < λ± otherwise it is linearly unstable [7]. In particular, when λ± 1 M−1 M , the unstable N -ring is called a saddle-center equilibrium. On the other hand, Yagasaki [11] gave a mathematical theory for Hamiltonian dynamical systems with saddle-center equilibria, which is briefly summarized here. The detailed description of the theory and its proof are provided in [11]. Let us consider the Hamiltonian dynamical systems with two degrees of freedom, x˙ = J1 Dx H (x, y), y˙ = J1 D y H (x, y), (x, y) ∈ R2 × R2 ,

(3)

where H is a given smooth Hamiltonian and J1 is the 2 × 2 symplectic matrix. We make three assumptions on the flow (3); (A1) the x-plane {(x, y) ∈ R2 × R2 | y = 0} is invariant under the flow (3); (A2) there exist saddle-centers in the invariant plane; (A3) the unstable and stable manifolds to the saddle-centers are connected by either a homoclinic or a heteroclinic orbit. According to the Lyapunov center manifold theorem, there exists a family of unstable periodic orbits in the neighborhood of the saddle-center equilibrium. Existence of chaotic orbit can be proven if the unstable and stable manifolds to these unstable periodic orbits intersect transversely. It is equivalent to say that a function, called the Melnikov function, that measures the distance between the unstable and the stable manifolds has a simple zero. We can further reduce the condition of the existence of the simple zero for the Melnikov function to the following simple inequality, which can be checked by numerical means. 2 2 2 2 ˆ = bˆ12  m 1 m 1 + bˆ11 m 2 m 1 + bˆ22 m 1 m 2 + bˆ21 m 2 m 2 − m 1 m 2 − m 1 m 2 > 0,

(4)

in which m i and m i are the eigenvalues for the linearized equation of (3) in the neighborhood of the saddle-centers in (A2), and bˆi j is the (i, j)-component of a matrix defined by the fundamental solutions to the variational equations with respect to the saddle-centers and the heteroclinic (or the homoclinic) orbit. See the papers [5,11] for the precise definition of the matrix. As a matter of fact, the above abstract theory is not applicable to the N -vortex problem as it is, since the N -vortex problem has N degrees of freedom. Thus, we reduce the N -vortex problem to low-dimensional dynamical systems by introducing two discrete transformations σ p and πe : (m , m )  → (m , m ) on point-vortex configurations; The transformation σ p represents the rotation around the z-axis by the degree 2π p/N , m =  N − p+m , m = N − p+m + 2π p/N for m = 1, . . . , p; m = m− p , m = m− p + 2π p/N for m = p + 1, . . . , N .

(5)

and πe replaces the north pole with the south pole by rotating them around the x-axis, 1 = π − 1 , 1 = 1 , m = π −  N −m+2 , m = 2π + 2 1 − N −m+2 for m  = 1. Reprinted from the journal

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(a)

(b)

(c) P12

P6

P6(πe)

P12(σ6)

P6(σ3)

P12(σ3)

P6(σ2) P6(σ3πe)

P12(πe)

P12(σ4)

P12(σ6πe)

P12(σ3πe)

P12(σ2)

(d) P12(σ4πe)

Fig. 2 A tree structure of invariant dynamical systems in the N -vortex problem for a N = 6 and b N = 12. The gray hatched region highlights the existence of two-degree-of-freedom Hamiltonian system embedding one-degree-of-freedom Hamiltonian system. Panels c and d show the plots of contour lines in the reduced system P6 (σ3 πe ) for  = 1.25 and  = 1, respectively. The saddle-center 6-ring is represented by the black solid circles 0.008

(a)

(b)

(c)

(d)

(e)

0.006

Δ^

0.004

0.002

0.4

0.6

0.3 0.4 0.2

x2

x2

0

0.1

-0.002

0.2

0

0 0.5

1

Γ

1.5

0.8

0.9

1

x1

1.1

0 0

0.2

0.4

0.6

0.8

1

1.2

x1

ˆ for the saddle-center 6-ring for 1 ≤  < 3/2, which is always positive. Panels b–e show the chaotic orbits in Fig. 3 a Plot of  the neighborhood of the 6-ring and the unstable and stable manifolds to unstable periodic orbits near the 6-ring for b, d  = 1.25 and c, e  = 1, which clearly demonstrate the existence of transversal intersections

We can mathematically prove the existence of low-degree-of-freedom Hamiltonian dynamical systems that are invariant with respect to σ p if N has a factor p, to πe if N is even and 1 = 2 (≡ ) and to their composition σ p πe , which are symbolized by PN (σ p ), PN (πe ) and PN (σ p πe ), respectively [7]. Applying it to the case of N = 6n, n ∈ Z, we obtain a tree structure of invariant dynamical systems as in Fig. 2. It indicates that there exists a two-degree-of-freedom Hamiltonian system PN (σ6 πe ) that embeds an invariant one-degree-of-freedom Hamiltonian PN (σ3 πe ). We check the assumptions (A1)–(A3); (A1) is certainly satisfied. We can easily check (A2) since we find many saddle-center equilibria including the 6-ring in the reduced system for a certain range of . Here, we pay attention to the 6-ring to show how the theory is applied. (see [5] for more examples) Regarding the assumption (A3), we have only to plot contour lines of the reduced Hamiltonian of PN (σ3 πe ). Figure 2c,d shows the contour lines of the invariant plane for  = 1.25 and  = 1, respectively. The black solid circles represent the saddle-center 6-ring, while the gray circles are the other saddle-center equilibria, which indicates that the 6-ring is connected by homoclinic orbits for  = 1.25 and by heteroclinic orbits for  = 1. In fact, the 6-ring is a saddle-center equilibria with the homoclinic or the heteroclinic connections for 1 ≤  < 3/2, ˆ is positive for the range of . In for which the criterion (4) is confirmed numerically. Figure 3a shows that  order to support the present results, we show in Fig. 3b,c the orbits of the unstable 6-ring for  = 1.25 and 1, respectively, which are chaotic. Moreover, we give in Fig. 3d,e the unstable and stable manifolds to unstable

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(a)

(b) θ0 3 4 2

1

θ0 Fig. 4 The collapse of inhomogeneous point vortices with inhomogeneous strengths 1 = 3, 2 = 2, 3 = 2 and 4 = −1. a Initial configuration of the point vortices, the three points is aligned along the line of latitude θ0 . b The finite-time collapsing solution for θ0 = 0.2π. The triple collapse is non-self-similar, partial and generic, which has never been reported so far. See [3] for more information

periodic orbits in the neighborhood of the 6-ring for the same s. We observe clearly that the orbits are chaotic and the unstable and stable manifolds intersect transversely.

4 Concluding remarks We have shown the existence of chaotic motion in the perturbed N -ring for the special case of N = 6n. However, as a matter of fact, it is also possible to demonstrate numerically that the saddle-center N -ring becomes chaotic for N = 5n, N = 7n and N = 8n with a direct numerical computation of the unstable and stable manifolds to unstable periodic orbits and tori when the strengths of the pole vortices are identical [4,5]. We have assumed that the ring configuration of rolling-up spirals is approximated by the homogeneous N -ring. On the other hand, as we see in Fig. 1b, the sizes of spirals differ from each other and thus we have to consider the ring structure of inhomogeneous point vortices in general. However, because of lack of any particular symmetry in the inhomogeneous N -vortex system, few theories for the Hamiltonian dynamical systems are available. Furthermore, the situation is even worse, since the inhomogeneous point vortices may collapse in finite time and thus we are unable to define the N -vortex problem as a dynamical system globally in time. An example of a collapse recently reported in [3] illustrates the difficulty. Figure 4a shows an initial configuration of four point vortices with strengths 1 = 3, 2 = 2, 3 = 2 and 4 = −1, in which three point vortices are aligned as the 3-ring at the line of latitude θ0 . The orbit of the four point vortices for θ0 = 0.2π is shown in Fig. 4b, which indicates that the vortex triple 234 collapses to one point in finite time and the other remains away from the collapsing point. The collapse is observed continuously on the initial data for a wide range of 0 < θ0 < c ≈ 0.355π, which means the collapse is a generic phenomenon. Thus, we are unable to define the dynamical systems global in time for this 3-ring system.

References 1. Newton, P.K.: The N-vortex problem, analytical techniques. Springer, New York (2001) 2. Newton, P.K., Sakajo, T.: The N-vortex problem on a rotating sphere: III. Ring configurations coupled to a background field. Proc. Roy. Soc. A. 463, 961–977 (2007) 3. Sakajo, T.: Non self-similar, partial and robust collapse of four point vortices on sphere. Phys. Rev. E. (2008). doi:10.1103/ PhysRevE.78.016312 4. Sakajo, T., Yagasaki, K.: Chaotic motion in the N-vortex problem on a sphere: II. Saddle-centers in three-degree-of-freedom Hamiltonians. Phys. D. 237, 2078–2083 (2008) 5. Sakajo, T., Yagasaki K.: Chaotic motion in the N-vortex problem on a sphere: I. Saddle-centers in two-degree-of-freedom Hamiltonians. J. Nonlinear Sci. 18, 485–525 (2008). doi:10.1007/s00332-008-9019-9 6. Sakajo, T.: Erratum: Invariant dynamical systems embedded in the N-vortex problem on a sphere with pole vortices. Phys. D 225, 235–236 (2007) 7. Sakajo, T.: Invariant dynamical systems embedded in the N-vortex problem on a sphere with pole vortices. Phys. D 217, 142–152 (2006) Reprinted from the journal

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8. Sakajo, T.: Equation of vortex sheet on sphere with a rotating effect and its long time evolution. In: Shi, Z.-C., Okamoto, H. (eds.) Proceedings of 7th China–Japan seminar on numerical mathematics, pp. 254–264. Science Press, Beijing (2005) 9. Sakajo, T.: Transition of global dynamics of a polygonal vortex ring on a sphere with pole vortices. Phys. D 196, 243– 264 (2004) 10. Sakajo, T.: Motion of a vortex sheet on a sphere with pole vortices. Phys. Fluids 16, 717–727 (2004) 11. Yagasaki, K.: Horseshoes in two-degree-of-freedom Hamiltonian systems with saddle-centers. Arch. Rat. Mech. Anal. 154, 275–296 (2000)

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Theor. Comput. Fluid Dyn. (2010) 24:157–162 DOI 10.1007/s00162-009-0097-6

O R I G I NA L A RT I C L E

Rhodri B. Nelson · N. Robb McDonald

Vortex motion on a sphere: barrier with two gaps

Received: 19 November 2008 / Accepted: 6 January 2009 / Published online: 25 April 2009 © Springer-Verlag 2009

Abstract Techniques based on the conformal mapping and the numerical method of contour dynamics are presented for computing the motion of a finite area patch of constant vorticity on a sphere in the presence of a thin barrier with two gaps. Finite area patch motion is compared with exact point vortex trajectories and good agreement is found between the point vortex trajectories and the centroid motion of finite area patches when the patch remains close to circular. Patch centroids are, in general, closely constrained to follow point vortex trajectories. However, Kelvin’s theorem constrains the circulation about the barrier to be a constant of the motion, thus, forcing a time-dependent volume flux through the gaps. More exotic motion is observed when the through-gap flow forces the vortex patch close to an edge of a barrier, resulting in the vortex splitting with only part of the patch passing through the gap. As the gap width is decreased this effect becomes more dramatic. Keywords Vorticity · Geophysical · Sphere · Boundaries PACS 47.32.C-, 47.15.ki, 47.32.cb, 92.10.A1 Introduction Vortex motion in the presence of impenetrable boundaries is an important problem in vortex dynamics due to its relevance in modeling geophysical flows, especially oceanographic flows. Oceanic eddies frequently interact with topography such as ridges and coastlines and the resulting interactions can play an important role in ocean circulation and various other ocean processes. With such motivation in mind Pedlosky [1] models the flow of the stratified abyssal ocean in the presence of a partial meridional (north–south) barrier, Sheremet [2] the passage of a western boundary current across a gap and Nof [3] the rotating exchange flow through a narrow gap between large-scale ocean basins. Because deep ocean vortices, observed to exist at depths up to 4 km (McWillams [4]), can propagate large distances they will frequently encounter mid-ocean ridges occurring at similar depths (e.g. the Walvis Ridge in the South Atlantic). Owing to their role in ocean circulation and the transport of tracers, understanding the circumstances in which they are able to penetrate gaps in ridges is important. In addition to deep ocean vortex-topography interactions, surface trapped vortex structures also frequently interact with complex topography. One such example is the collision of North Brazil Current Rings with the Lesser Antilles [5]. Motivated by such vortex interactions with complex topographic features, a number of recent studies have been conducted into the motion of both point (line) and finite area vortices in bounded domains. For problems Communicated by H. Aref R. B. Nelson (B) · N. R. McDonald Department of Mathematics, University College London, London, UK E-mail: [email protected] E-mail: [email protected] Reprinted from the journal

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involving vortex motion in planar bounded domains governed by the two-dimensional Euler equations with the vorticity distribution chosen such that it is a singular point in the domain (a point vortex), progress can be made using conformal mapping techniques. Provided a conformal mapping exists between the domain and a relatively simpler domain in which a vortex Hamiltonian (Kirchhoff–Routh path function) [6] can be constructed, this Hamiltonian along with information from the mapping gives the Hamiltonian in the original domain from which the vortex trajectory can be deduced. Such point vortex systems can give a good indication of the dynamics of a vortex patch with constant vorticity, especially when the boundary of a vortex remains close to circular since the velocity field exterior to a circular vortex patch is identical to that of a point vortex with the same circulation. Crowdy and Marshall [7] derived explicit formulas to compute point vortex trajectories in the presence of multiple cylinders in domains of arbitrary connectivity. Vortex trajectories about circular islands off a coast line and in the open ocean are computed. An analogous study of point vortex motion near a wall with multiple gaps is presented in Crowdy and Marshall [8]. In many geophysical systems, it is of interest to consider cases in which the vorticity distribution is not singular, but a finite region of vorticity which is able to deform, and when in close proximity to solid objects or other regions of vorticity can filament or even split. The conformal mapping method is not directly applicable to such cases as a finite area vortex patch is governed by Poisson’s equation which is not invariant under conformal mapping. Johnson and McDonald [9–11] present techniques to compute finite area vortex motion in the presence of boundaries. In these studies, exact point vortex trajectories and finite area vortex motion is computed about a barrier with a single gap, two circular islands and a barrier with two gaps, respectively. In a recent study, Crowdy and Surana [12] presented a method for implementing contour dynamics in planar domains of arbitrary connectivity based on the constructing the Green’s function in a pre-image circular domain. Various examples are presented including the motion of a patch near an infinite wall and a gap in a wall. For planetary scale geophysical flows where the curvature of the Earth can play a significant role in the evolution of the system, it is clearly of interest to consider such problems on the surface of a sphere. Hence, it is desirable to develop methods that accurately compute vortex behavior on the surface of a sphere. Moving from the plane to the sphere introduces new effects into the problem. The periodicity of the sphere introduces a ‘feed back’ effect and the curvature of the sphere can introduce ‘shielding’ effects. In the absence of topography on the sphere, it is required that the total vorticity on the surface of the sphere is equal to zero: the Gauss integral constraint. Point vortex motion on the surface of the sphere in the presence of impenetrable boundaries is considered in the studies of Kidambi and Netwon [13] and Crowdy [14]. Kidambi an Newton [13] use the method of images to derive the exact point vortex trajectories for a spherical cap, longitudinal wedge, half-longitudinal wedge, channel and rectangle. Crowdy [14] presents a formulation based on a generalization of the Kirchhoff–Routh path function to the surface of a sphere, abandoning the need for special symmetries demanded by the method of images. The analog of the ‘gap in a wall’ problem of Johnson and McDonald [9] on the surface of a sphere is also considered. Dritschel [15] extends the method of contour dynamics to the surface of a sphere. This method has been used to model waves and vortices on a sphere (in the absence of boundaries) and compute steadily rotating vortex structures [16,17]. In a recent study by Surana and Crowdy [18], the method of Crowdy and Surana [12] is adapted to study the motion of vortex patches on a spherical surface in possibly arbitrarily connected domains. Examples considered include a polar cap, a barrier with a single gap and a channel. Recently, Nelson and McDonald [19] presented an alternative method of computing vortex patch motion in singly connected domains on a sphere based on the stereographic projection and using methods similar to Johnson and McDonald [9]. In this work, Nelson and McDonald [19] is extended to consider the doubly connected case of a thin barrier on a sphere with two gaps.

2 Point vortex and vortex patch motion The system consists of a thin layer of constant depth, incompressible and inviscid fluid in a doubly connected bounded domain D on the surface of the unit sphere. Let the barrier lie along the great circle corresponding to (G ) (G ) φ = 0, 2π (where φ ∈ [0, 2π) is the azimuthal angle) except at two gaps G1 and G2 between θ0 1 and θ1 1 and (G2 ) (G2 ) between θ0 and θ1 , respectively (where θ ∈ [0, π] is the latitudinal angle). Introduce the stereographic projection into the complex z-plane such that

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z = cot

  θ eiφ , 2

(1)

which conformally maps D to Dz with the north pole being mapped to ∞ and the south pole to the origin of the z-plane. The barrier is projected to the line z = 0 with gaps lying between z 0(G1 ) and z 1(G1 ) and between z 0(G2 ) (G ) and z 1 2 . Following Johnson and McDonald [11], the domain is further decomposed into two subdomains D1,2 , with D1 denoting the upper half of the stereographically projected z-plane (z > 0) and D2 the lower half. Let the streamfunction induced by the vortices in the region D1 satisfying the no-normal flow condition along z = 0 be ψ1 so that ψ1 = 0,

on z = 0.

(2)

Similarly, let the streamfunction induced by the vortices in the region D2 satisfying the no-normal flow boundary condition on z = 0 be ψ2 , so ψ2 = 0,

on z = 0.

(3)

Denote the total streamfunction induced by all vortices in D = D1 ∪ D2 by  and introduce and irrotational flow field with streamfunction ψ0 such that  ψ1 + ψ0 in D1 = (4) ψ2 + ψ0 in D2 . Thus  is continuous across the gaps and ψ1,2 vanish there, so ψ0 is continuous across the gaps. Mathematical details outlining further considerations and method of solution to this problem are detailed in [11] and [19]. Johson and McDonald [11] discusses the planar analog of this problem while Nelson and McDonald [19] discusses the appropriate adjustments required to take into account the geometry of the sphere. Briefly, for the case of finite area patches, the appropriate velocities to determine ψ1 and ψ2 are computed on the surface of the sphere using a standard contour dynamics algorithm and then the boundary value problem is stereographically projected into the complex z-plane as described in [19]. The irrotational flow field ψ0 is then computed using conformal mapping to a periodic rectangle and involves the use of elliptic functions(see [11]). Velocities are then projected back to the surface of the sphere using the procedure in [19]. Importantly, in this case, the image of the vortex patch is explicitly included when determining ψ1 and ψ2 . Thus, the difference here to the examples considered in [19] is that when projecting velocities back to the sphere the velocity field arising due to the Gauss constraint (see eq. (9) in [19]) is not added back on to the total velocity field. 2.1 Point vortex trajectories The vortex Hamiltonian for the planar analog of this problem is derived in Johnson and McDonald [11]. The relationship between a Hamiltonian on the sphere to a Hamiltonian in the complex z-plane is given by [18] H s ({φαi }, {θαi }) = H z ({z αi }) +

N 1  2 1  j log , 4π (1 + z(α j )¯z (α j ))

(5)

j=1

where αi denotes the position of the ith vortex in a pre-image ζ domain. For the case of a single point vortex with circulation 1 =  located at z = z α1 and when the two gaps in the barrier are collinear with G1 is located between (π, π/2) and (π, θ1 ) and G2 located between (0, π/2) and (0, θ1 ) where θ1 ∈ (π/2, π) such that the resulting ‘island’ centered about the south pole corresponds to |z| < k where k = cot(θ1 /2), the vortex Hamiltonian is given by H s (z, z¯ ) = ( 2 /4π) log |(k 2 − z 2 )1/2 (1 − z 2 )1/2 ϑ1 [iπ(sn−1 (z/k) − K  )/2K ]| +( 2 /4π) log |1/(1 + z z¯ )|,

(6)

 where, K is the complete elliptic √ integral of the first kind with modulus k, K the complete elliptic integral of 2 the first kind with modulus 1 − k , ϑ1 denotes Jacobi’s ϑ1 -function and sn the elliptic function of modulus k. Note that in the formulation presented here the circulation around the ‘island’ is not necessarily zero, the

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circulation around the barrier extending to ∞ in the z-plane will, however, be zero. This is due to the fact that unlike in planar cases, a vortex cannot initially be placed at ‘infinity’ such that its induced velocity field at the ‘island’ is zero. Thus to ensure a net zero circulation around the ‘island’, it is required to place a point vortex of circulation − in the τ = sn−1 (z/k) plane such that it maps to the center of the island (i.e. the South Pole) in the z-plane. Vortex trajectories (and also finite area vortex motion) for the case when the ‘island’ lies asymmetrically between the barrier spanning half a great circle can be deduced through mapping to the symmetric case. An island in the stereographically projected zˆ -plane located between x 0 < ˆz < x1 (with −1 < x0 < x1 < 1) within a gap corresponding to −1 < ˆz < 1 is mapped back to the symmetric problem in the z-plane via the Mobius map given by z = (−α zˆ + 1)/(ˆz − α),

(7)

leaving the gap of width 2 unchanged but moving the island to lie along |z| ≤ k < 1 where α = exp(cosh−1 [(x0 x1 + 1)/(x0 + x1 )]),

k = exp(cosh−1 [(x0 x1 − 1)/(x0 − x1 )]).

(8)

This mapping along with the transformation H zˆ ({ˆz α }) = H z ({z α }) + ( 2 /4π) log |dˆz /dz|zˆ α ,

(9)

and Eq. (5) gives the appropriate vortex Hamiltonian in the zˆ -plane. Figure 1a and b shows the point vortex trajectories about an island with z ≤ |0.4| located symmetrically between a barrier spanning half a great circle. In contrast to the planar case [11] and also single gap case on the sphere [19], the Hamiltonian exhibits an elliptic point located to either side of the island indicating the possibility of steady vortex structures residing in either or both of these regions. Away from the island vortex trajectories resemble those of the single gap case [19]. Figure 1c gives an example of point vortex trajectories when the island is placed asymmetrically within the gap. The vortex Hamiltonian for the case when G1 and G2 are located at arbitrary locations on the barrier is obtained through mapping to the case of two collinear gaps and applying the appropriate transformation to the (L ) (L ) (L ) Hamiltonian (6). Consider two thin islands L1 and L2 in the z˜ -plane with end points z˜ 0 1 , z˜ 1 1 and z˜ 0 2 (L ) (L ) (L ) (L ) (L ) and z˜ 1 2 where z˜ 0 1 < z˜ 1 1 < z˜ 0 2 < z˜ 1 2 . Denote the centres and half-widths of L1 and L2 by c1 , r1 and c2 , r2 respectively. The Mobius map that maps the z˜ -plane to the zˆ plane considered above is given by zˆ = r1 /(˜z − c1 ).

(10)

This maps H1 to the line ˆz = 0 except for a single gap located between −1 < ˆz < 1. H2 is mapped to an island lying within this gap. The domain Dzˆ is then mapped to the domain Dz such that the gaps are now collinear as outlined above. In addition, to ensure zero circulation around the islands a point vortex of circulation − must be added in the τ = sn−1 (z/k) plane such that it is mapped to the point at infinity in the z˜ plane (see [10,8]).

(a)

(b)

(c)

Fig. 1 a, b Show point vortex trajectories for the case when one barrier spans half a great circle and the other barrier is located symmetrically within the gap between z < |0.4| in the stereographic z-plane. a View centered on the south pole and b is centered the north pole. c View centered on (φ, θ) = (0, 0) of point vortex trajectories about an island located between 0.2 < z < 0.7 located asymmetrically between a barrier spanning half a great circle

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(a)

(b)

Fig. 2 a View centered at (φ, θ) = (0, 0) of the motion of a vortex patch with  = 1.5376 and ω = 5 with its centroid initially located at (φco , θco ) = (π/4, π/5). One barrier spans half a great circle and the island is placed symmetrically within the barrier such that |z| ≤ 0.7. The dashed line represents the exact point vortex trajectory and the + marks the patch centroid location. The patch is shown at times t = 0, 2, 4 and 6. b View centered at (φ, θ) = (0, 3π/4) of the motion of a vortex patch with  = 0.3867 and ω = 5 with its centroid initially located at (φco , θco ) = (π/6, 3π/5). One barrier spans half a great circle and the island is placed asymmetrically within the barrier such that −0.1 ≤ ˆz ≤ 0.7. The dashed line and + have the same meanings as in a. The patch is position is shown in increments of 5 time units

2.2 Patch motion Computations of patch motion were carried out using a time step of dt = 0.01 and resolution parameter between boundary nodes set to 0.01. Note that no surgery is implemented in the following examples. Figure 2a and b gives examples of patch motion about a symmetrically placed and an asymmetrically placed island respectively. In Fig. 2a, the through gap flux forces the patch close to the island and it is ‘pinched’ against the island edge. Up until the point where the patch is distorted by its interaction with the topography, as expected, the patch centroid follows the point vortex very closely. After being squeezed against the island edge, part of the patch passes through the gap while part does not. The through gap volume flux induced by the vortex then the results in the vortex being ‘torn apart’. Such motion is typical in the planar case when the patch is pinched again an edge of the island [11]. In Fig. 2b, the patch remains relatively undistorted and thus the motion of its centroid follows that of a point vortex very closely. 3 Closing remarks A model has been presented to compute the motion of a vortex patch in a doubly connected domain, namely a thin barrier with two gaps, on the surface of the sphere that utilizes the invariance of the harmonic problem exterior to the vortex patch under conformal mapping. Good agreement is seen between the motion of point vortices and patches that undergo little distortion. Behavior typical to that seen in the planar case is seen resulting from the through gap fluxes generated by the vortex. New features, such as elliptic points of the Hamiltonian residing to either side of the island, are also observed. The method presented here can be applied to other doubly connected domains, indeed, the method can in principle be applied to any domain for which a conformal mapping to an annulus exists. It is also in principle possible to apply the method to domains of higher connectivity by mapping to the unit disk with smaller circular disks excised and solving the corresponding Dirichlet problem to obtain the appropriate irrotational flow. References 1. 2. 3. 4. 5. 6. 7.

Pedlosky, J.: Stratified abyssal flow in the presence of fractured ridges. J. Phys. Oceanogr. 30, 403–417 (1994) Sheremet, V.A.: Hysteresis of a western boundary current leaping across a gap. J. Phys. Oceanogr. 31, 1247–1259 (2001) Nof, D.: Choked flows from the Pacific to the Indian Ocean. J. Phys. Oceanogr. 25, 1369–1383 (1995) McWilliams, J.C.: Submeoscale, coherent vortices in the ocean. Rev. Geophys. 23, 165–182 (1985) Fratantoni, D., Johns, W., Townsend, T.: Rings of the North Brazil current. J. Geophys. Res. 100, 10633–10654 (1995) Saffman, P.G.: Vortex Dynamics. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge (1992) Crowdy, D., Marshall, J.S.: The motion of a point vortex around multiple circular islands. Phys. Fluids 17, 056602-1– 056602-13 (2005) 8. Crowdy, D., Marshall, J.S.: The motion of a point vortex through gaps in walls. J. Fluid. Mech. 551, 31–48 (2006)

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9. Johnson, E.R., McDonald, N.R.: The motion of a vortex near a gap in a wall. Phys. Fluids 16, 462–469 (2004) 10. Johnson, E.R., McDonald, N.R.: The motion of a vortex near two circular cylinders. Proc. R. Soc. Lond. A 460, 939–954 (2004) 11. Johnson, E.R., McDonald, N.R.: Vortices near barriers with multiple gaps. J. Fluid Mech. 531, 335–358 (2005) 12. Crowdy, D., Surana, A.: Contour dynamics in complex domains. J. Fluid Mech. 593, 235–254 (2008) 13. Kidambi, R., Newton, P.K.: Point vortex motion on a sphere with solid boundaries. Phys. Fluids 12, 581–588 (2000) 14. Crowdy, D.: Point vortex motion on the surface of a sphere with impenetrable boundaries. Phys. Fluids 18, 036602–036602-7 (2006) 15. Dritschel, D.G.: Contour dynamics/surgery on the sphere. J. Comput. Phys. 79, 477–483 (1988) 16. Dritschel, D.G., Polvani, L.M.: The roll-up of vorticity strips on the surface of a sphere. J. Fluid Mech. 234, 47–69 (1992) 17. Polvani, L.M., Dritschel, D.G.: Wave and vortex dynamics on the surface of a sphere. J. Fluid Mech. 225, 35–64 (1993) 18. Surana, A., Crowdy, D. Vortex dynamics in complex domains on a spherical surface. J. Comput. Phys. 12, 6058 (2008). doi:10.1016/j.jcp.2008.02.027 19. Nelson, R.B., McDonald, N.R.: Finite area vortex motion on a sphere with impenetrable boundaries. Phys. Fluids 21, 016602 (2009)

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Theor. Comput. Fluid Dyn. (2010) 24:163–167 DOI 10.1007/s00162-009-0154-1

O R I G I NA L A RT I C L E

L. K. Brandt · T. K. Cichocki · K. K. Nomura

Asymmetric vortex merger: mechanism and criterion

Received: 26 December 2008 / Accepted: 2 June 2009 / Published online: 11 September 2009 © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract The merging of two unequal co-rotating vortices in a viscous fluid is investigated. Two-dimensional numerical simulations of initially equal sized Lamb-Oseen vortices with differing relative strengths are performed. Results show how the disparity in deformation rates between the vortices alters the interaction. Key physical mechanisms associated with vortex merging are identified. A merging criterion is formulated in terms of the relative timing of core detrainment and destruction. A critical strain parameter is defined to characterize the establishment of core detrainment. This parameter is shown to be directly related to the critical aspect ratio in the case of symmetric merger. Keywords Vortex interactions · Merger · Unequal vortices PACS 47 · 47.32C · 47.32cb 1 Introduction Vortex merger is a fundamental flow process which plays an important role in the transfer of energy and enstrophy across scales in transitional and turbulent flows. It also has practical significance, e.g., in the nearfield wake dynamics of an aircraft. Yet despite its elementary nature, the physical mechanisms of two-dimensional vortex merger have not been fully resolved [1,2,12]. In the idealized interaction of two equal size and strength (symmetric) co-rotating vortices, merger occurs if the aspect ratio, a/b (core size/separation distance), exceeds a critical value, (a/b)cr . The determination of (a/b)cr has been the focus of a number of studies, e.g., [11,10,9]. The physical mechanisms of symmetric merger have also been considered, e.g., [7,8,4,13,1,2]. In the more general interaction of two unequal size and/or strength (asymmetric) vortices, there is a greater range of flow behavior and the interaction may result in the destruction of the smaller/weaker vortex. Previous inviscid flow studies have identified distinct flow regimes based on the efficiency of the interaction [5,12]. Elastic interaction occurs when there are only small deformations and essentially no change in circulation of the vortices. Partial and complete straining-out are associated with a reduction or destruction, respectively, of the smaller vortex, with no increase in the larger vortex. Partial and complete merger are associated with increased circulation of the initially larger vortex. Flow regime maps have been presented in terms of the initial core size (or strength) ratio and initial separation distance. Trieling et al. [12] attempted to normalize the separation distance by an averaged core size, defined in terms of the second moment of vorticity [9]. However, this did not yield a universal critical aspect ratio delimiting the merging regimes. It is unclear if (a/b)cr can be generalized in this way. Furthermore, there has been limited consideration of viscous flows. Communited by H. Aref L. K. Brandt · T. K. Cichocki · K. K. Nomura (B) University of California, San Diego, La Jolla, CA 92093-0411, USA E-mail: [email protected] Reprinted from the journal

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(a)

(b)

(c)

Fig. 1 Vorticity contour plots showing time evolution of flows: a o,2 /o,1 = 1.0, b o,2 /o,1 = 0.8, c o,2 /o,1 = 0.6. Taken from [3]

This paper summarizes the main findings of our recent study [3] in which we investigate the interaction of two unequal co-rotating vortices in a viscous fluid. The study follows from our previous work on equal vortices [2] identifying the primary physical mechanisms associated with vortex merging and develops a more generalized description and criterion for asymmetric merger.

2 Numerical simulations Numerical simulations of two-dimensional, incompressible, viscous flow are performed. The initial flow consists of two Lamb-Oseen co-rotating vortices of equal size and unequal strength. The initial aspect ratio is ao /bo = 0.157, where ao is defined based on the second moment of vorticity. The Reynolds number of the stronger vortex (vortex 1) is Re,1 = o,1 /ν = 5000, where o is the initial circulation and ν is the kinematic viscosity. The circulation of the weaker vortex (vortex 2) is varied in the range 0.4 ≤ o,2 /o,1 ≤ 1.0. A convective time scale is the approximate rotational period of the system, T = 2π 2 bo 2 / o , where  o = 0.5(o,1 + o,2 ). Nondimensional time is tc ∗ = t/T . Details are in [3].

3 Physical mechanisms Figure 1 shows representative flow evolutions for several different o,2 /o,1 . Initially, the two vortices rotate about each other due to their mutually induced velocity. As in previous studies, it is useful to consider the flow structure in the co-rotating frame of reference. Figure 2 shows the instantaneous co-rotating streamlines (bold lines) indicating the inner core regions and the exchange band where fluid is advected around both vortices [7,2]. In time, all the flows result in a single vortex. Figure 1a corresponds to symmetric vortices (o,2 /o,1 = 1.0). Our previous analysis of symmetric merger describes the merging process in terms of four phases of development [2]. In the diffusive/deformation phase, the separation distance remains relatively constant while the cores grow by diffusion. The cores also begin to deform. The deformation can be described in terms of the interaction of the vorticity gradient, ∇ω, and the mutually induced rate of strain, S. As shown in Fig. 2a, each vortex exhibits a quadrapole structure of

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Fig. 2 Vorticity contours (thin lines) with gray shading corresponding to ω production term, Ps = −(∇ω T S∇ω)/|∇ω|2 (light gray scale: Ps > 0, dark gray scale: Ps < 0), and instantaneous streamlines (bold lines) in the co-rotating frame at tc∗ = 0.32. Taken from [3]

Ps = −(∇ω T S∇ω)/|∇ω|2 , which indicates alternate regions of gradient amplification/attenuation by compressive/extensional straining associated with its elliptic deformation. During this time, a distinct functional relation between ω and streamfunction exists suggesting quasi-equilibrium conditions. However, in the vicinity of the hyperbolic points, and in particular the central hyperbolic (CH) point where mutual interaction strengthens ∇ω amplification (Fig. 2—light shading near center is Ps > 0), the dynamics of ∇ω and S eventually produces a tilt in ω contours [1]. At the outer hyperbolic points, this initiates filamentation. During the convective/deformation phase, the induced flow by the filaments acts to advect the vortices towards each other and enhances the mutually induced S but does not drive the merger to completion. The enhanced tilting and diffusion of ω near the CH point causes ω to be detrained from the core region and enter the exchange band where it is advected away. This leads to the departure from quasi-equilibrium conditions. In the convective/entrainment phase, the vortex cores erode significantly. At some point, the integrity of the vortices is sufficiently diminished. The cores are then mutually entrained into the exchange band region, whose induced flow becomes dominant and transforms the flow into a single compound vortex. A critical aspect ratio, associated with the start of the convective/entrainment phase, is determined for a range of flow conditions [2]: (a/b)cr = 0.235 ± 0.006. It is also noted that this time is comparable to the time at which the core size, a 2 (t), deviates from viscous (linear) growth [2]. In the final diffusive/axisymmetrization phase, the flow evolves towards axisymmetry by diffusion [4]. In the case of asymmetric vortex pairs (Fig. 1b, c), the difference in vortex strengths alters the flow structure and interaction. The vortices may no longer experience the flow processes simultaneously. As in the symmetric case, the vortices initially grow by diffusion. However, the deformation rates and Ps (Fig. 2b) are stronger at the weaker vortex due to the difference in induced S, and the tilt of ω contours and subsequent core detrainment occurs earlier than in the stronger vortex. However, the dominant attracting motion occurs only when, and if, core detrainment is established by the stronger vortex. If this occurs, then there will be some extent of mutual (reciprocal), but unequal, entrainment. This is observed in the present simulations for 0.7 ≤ o,2 /o,1 ≤ 0.9 and is illustrated here in Fig. 1b (o,2 /o,1 = 0.8). In these cases, the stronger vortex ultimately dominates and entrains ω from the weaker vortex. Thus, the process is considered as vortex merger since the result is an enhanced compound vortex. If core detrainment is not established by the stronger vortex before significant erosion occurs in the weaker vortex (Fig. 1c, o,2 /o,1 = 0.6), the weaker vortex is destroyed while the stronger vortex remains relatively unaffected. In this case, convective merger does not occur.

4 Merging criterion From the above description, we consider a critical state for a given vortex to be associated with the establishment of core detrainment. If both vortices reach this state, there will be some degree of mutual entrainment which Reprinted from the journal

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Fig. 3 Time development of the strain parameter, γi for a vortex 1 and b vortex 2. Symbols: (circle): o,2 /o,1 = 1.0, (box): o,2 /o,1 = 0.9, (triangle): o,2 /o,1 = 0.8, ∗: o,2 /o,1 = 0.7, ×: o,2 /o,1 = 0.6. Taken from [3]

results in an enhanced vortex, i.e., convective merger will occur. Based on these ideas, a merging criterion is developed. We consider the onset of the core detrainment process to be associated with the flow achieving a sufficiently high strain rate, with respect to some characteristic ω, for the process to proceed. Recall that the process is initiated by the tilting of ω contours in the vicintiy of the CH point. We therefore consider one characteristic quantity to be the strain rate at the CH point, SCH . A characteristic core vorticity is the maximum, ωvi . In order to relate the strain rate at the CH point to the maximum vorticity of the vortex, we normalize each quantity by a characteristic local (initial) strain rate. This introduces the appropriate scaling. We define the nondimensional ∗ = S /S ∗ strain rate, SCH CH CH,o and nondimensional vorticity, ωvi = ωvi /Svi ,o . A strain parameter for vortex i is then defined as,  ∗ ∗ 1/2 S (t ) ∗ γi (t ) ≡ 1 CH , (1) ∗ ∗ 2 ωvi (t ) which measures the relative strength of the induced strain rate at the CH point to the vortex strength. Figure 3 shows the strain parameter for the stronger and weaker vortex, γ1 (t ∗ ) and γ2 (t ∗ ), respectively, evaluated from the simulations. We consider the critical value of the vortex strain parameter to be the value at the critical ∗ , when core detrainment (and entrainment into exchange band) is established, i.e., γ ∗ time, tcr,i cr,2 = γ2 (tcr,2 ) ∗ and γcr,1 = γ1 (tcr,1 ). From our simulation results, we find: γcr,1 ≈ 0.249 ± 0.003 and γcr,2 ≈ 0.245 ± 0.005 ∗ ∗ and tcr,1 are determined from the behavior of a 2 (t)). Since the values are within the range of (where tcr,2 uncertainty, we obtain a single value for the critical strain parameter, γcr,1 ≈ γcr,2 ≈ γcr = 0.247 ± 0.007. In the case of a symmetric vortex pair, through scaling analysis [3], it is shown that,    ∗ (t ∗ ) 1/2 ∗ ) 1/2 SCH SCH (tcr cr γcr = γcr,2 = 1 = ∗) ∗ ∗ 4ωvi (tcr 2 ωv2 (tcr ) a  ∗) aω∗ (tcr ω = f . (2) = f ∗ ∗) 2d|CH−V (t b cr | cr Thus, for symmetric merger, the critical strain parameter is directly related to the critical aspect ratio. From the simulation results, the proportionality factor is evaluated to be f ≈ 1.05 ± 0.03, and using the computed values of γcr,2 , (aω /b)cr = γcr,2 / f ≈ 0.233 ± 0.005. This compares well with the previously determined value, (aω /b)cr = 0.235 ± 0.006 [2]. The merging criterion is formulated in terms of the timing of the key physical processes: weaker vortex core ∗ , stronger vortex core detrainment t ∗ , and weaker vortex destruction t ∗ . For the purposes detrainment tcr,2 cr,1 de,2 ∗ is estimated using the velocity gradient tensor [3]. We consider the classifications developed of this study, tde,2

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for inviscid asymmetric vortex interactions [12,5] and modify the descriptions for viscous flow. Based on our analysis, we classify the observed interactions and merging regimes as follows, ∗ = t ∗ < t ∗ ): detrainment from both vortices, mutual entrainment of the cores – Complete merger (tcr,2 cr,1 de,2 transforms the flow into a single vortex (present results: o,2 /o,1 = 1.0), ∗ < t ∗ < t ∗ ): detrainment from both vortices, weaker vortex is destroyed and en– Partial merger (tcr,2 cr,1 de,2 trained by the stronger vortex (present results: o,2 /o,1 = 0.9, 0.8, 0.7), ∗ ∗ ∗ – Strained-out (tcr,2 < tde,2 < tcr,1 ): detrainment from weaker vortex only, weaker vortex is destroyed (present results: o,2 /o,1 ≤ 0.6).

All interactions eventually result in a single vortex. In complete merger, the circulation of the final compound vortex is greater than that of either original vortex. This increase is due to the mutual entrainment of both vortices and the transformation of the flow into a single vortex. In partial merger, the stronger vortex dominates and is enhanced by the entrained vorticity from the weaker vortex. In this case, vorticity is detrained from both vortices, however, the weaker vortex is destroyed before the stronger vortex is significantly eroded. When the weaker vortex is strained out, it is eventually destroyed and the stronger vortex remains with its circulation relatively unchanged. There is no mutual entrainment and the interaction does not yield a compound/enhanced vortex, i.e., merger does not occur. 5 Summary This paper summarizes the main findings of our investigation of the interaction and merging of two unequal co-rotating vortices in a viscous fluid [3]. The key physical processes are identified and described. A merging criterion, based on the relative timing of core detrainment and core destruction, is developed. The establishment of core detrainment is characterized by a critical strain parameter, which can be directly related to the critical aspect ratio in the case of symmetric merger.

Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References 1. Brandt, L.K., Nomura, K.K.: The physics of vortex merger: further insight. Phys. Fluids 18, 1–4 (2006) 2. Brandt, L.K., Nomura, K.K.: The physics of vortex merger and the effects of ambient stable stratification. J. Fluid Mech. 592, 413–446 (2007) 3. Brandt, L.K., Nomura, K.K.: Characterization of the interactions of two unequal co-rotating vortices. J. Fluid Mech. (Submitted) 4. Cerretelli, C., Williamson, C.H.K.: The physical mechanism for vortex merging. J. Fluid Mech. 475, 41–77 (2003) 5. Dritschel, D.G., Waugh, D.W.: Quantification of the inelastic interaction of unequal vortices in two-dimensional vortex dynamics. Phys. Fluids 4, 1737–1744 (1992) 6. Melander, M.V., McWilliams, J.C., Zabusky, N.J.: Asymmetric vortex merger in two dimensions: which vortex is “victorious”?. Phys. Fluids 30, 2610–2612 (1987) 7. Melander, M.V., Zabusky, N.J., McWilliams, J.C.: Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 305–340 (1998) 8. Meunier, P.: Etude experimentale de deux tourbillons co-rotatifs. Ph.D. Dissertation, Universite d’Aix-Marseille I, France (2001) 9. Meunier, P., Ehrenstein, U., Leweke, T., Rossi, M.: A merging criterion for two-dimensional co-rotating vortices. Phys. Fluids 14, 2757–2766 (2002) 10. Overman, E.A., Zabusky, N.J.: Evolution and merger of isolated vortex structures. Phys. Fluids 25, 1297–1305 (1982) 11. Saffman, P.G., Szeto, R.: Equillibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23, 2339–2342 (1980) 12. Trieling, R.R., Velasco Fuentes, O.U., van Heijst, G.J.F.: Interaction of two unequal corotating vortices. Phys. Fluids 17, 1–17 (2005) 13. Velasco Fuentes, O.U.: Vortex filamentation: its onset and its role on axisymmetrization and merger. Dyn. Atmos. Oceans 40, 23–42 (2005)

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Theor. Comput. Fluid Dyn. (2010) 24:169–173 DOI 10.1007/s00162-009-0171-0

O R I G I NA L A RT I C L E

Dmitry Kolomenskiy · Kai Schneider

Numerical simulations of falling leaves using a pseudo-spectral method with volume penalization

Received: 3 December 2008 / Accepted: 24 September 2009 / Published online: 31 October 2009 © Springer-Verlag 2009

Abstract The dynamics of falling leaves is studied by means of numerical simulations. The two-dimensional incompressible Navier–Stokes equations, coupled with the equations governing solid body dynamics, are solved using a Fourier pseudo-spectral method with volume penalization to impose no-slip boundary conditions. Comparison with other numerical methods is made. Simulations performed for different values of the Reynolds number show that its decrease stabilizes the free fall motion. Keywords Fluid–solid interaction · Free fall · Volume penalization method PACS 47.63.-b

1 Introduction The dynamics of falling leaves, being most remarkable for its aesthetics, is at the same time a phenomenologically rich and practically important subject. Generally speaking, it may be regarded as an example of a dynamical system exhibiting both regular and apparently chaotic behavior. Scientific interest in this phenomenon dates back to the nineteenth century—Maxwell’s paper is the earliest reference (see [8]). Kelvin, Tait and Kirchhoff considered motion of a solid body in an inviscid fluid. This Hamiltonian system has been studied since then by different authors (see, e.g., [4]). The free fall of thin plates in real fluids, more difficult for rigorous analysis, has been considered in experiments and in numerical simulations (see [9,1,2] and references therein). An important manifestation of viscous effects are the cusp-like turning points of the trajectory, where the centre of mass of the plate elevates [1]. This motion may be regarded as ‘passive’ flight for its similarity with flapping of insect wings. The dynamics of a falling plate is characterized by three dimensionless numbers: the Reynolds number Re, the dimensionless moment of inertia I ∗ , and, in a particular case of elliptical cross-section, its eccentricity e [1]. The Reynolds number is of special interest, since it gives an idea about the range of scales where winged animals can take advantage of the centre of mass elevation to facilitate flapping of their wings. In this paper we study the influence of the Reynolds number in the range from 10 to 1100. We apply the fluid-structure interaction model reported in [11,10,7] to perform numerical simulations of falling leaves, Communicated by H. Aref D. Kolomenskiy(B) M2P2-CNRS, Universités d’Aix-Marseille, 38 rue Joliot-Curie, 13451, Marseille Cedex 20, France E-mail: [email protected] K. Schneider M2P2-CNRS and CMI, Universités d’Aix-Marseille, 39 rue Joliot-Curie, 13453, Marseille Cedex 13, France E-mail: [email protected] Reprinted from the journal

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A=AsUAf Af : =0

As : =1 As

Fig. 1 The physical domain A containing the fluid domain A f and the solid obstacle As with its boundary ∂ As .

and show a stabilizing effect of decreasing Re. This result agrees with experimental observations made by Willmarth et al. [13] for falling circular disks, and then by Smith [12] for wings. 2 Physical model and numerical method We consider interaction between viscous incompressible fluid and a solid body moving in it. The two-dimensional Navier–Stokes equations are written in the vorticity–stream function (ω-) formulation. Moving solid obstacles of arbitrary shape are taken into account using the volume penalization method. The penalized equations   χ 2 ∂t ωη + uη · ∇ωη − ν∇ ωη + ∇ × (1) (uη − us ) = 0, η ∇ 2  = ωη ,



uη = ∇ ⊥  + U∞ ,

(2)

are solved in the domain A = A f As containing both the fluid and the solid obstacle (see Fig. 1). The viscosity of the fluid is ν, η is the penalization parameter, χ is the mask function describing the shape of the solid, us is the velocity field of the solid, U∞ is the free-stream velocity, ωη = ∇ × u η is the vorticity, and ∇ ⊥ = (−∂ y , ∂x ). The density of the fluid is normalized to unity, ρ = 1. The volume penalization method is motivated by an idea of modelling solid obstacles as porous media with vanishing permeability. When η in Eq. (1) is tending to zero, the penalized problem converges to the no-slip boundary problem [3]. The fluid and the solid are thus considered as one medium with permeability varying in space and in time. This allows to implement efficiently and in a relatively straightforward manner such features as arbitrary shape and number of obstacles. The motion of the solid is governed by Newton’s second law, which yields ODEs for the center of gravity position xcg and for the angle of incidence θcg .  d2 xcg χ mb = (3) (uη − us )d A + m b g, 2 dt η A

d2 θcg = Jb dt 2

 A

χ (x − xcg ) × (uη − us )d A, η

(4)

where m b and Jb are, respectively, the buoyancy corrected mass and moment of inertia of the solid body. The integrals, as well as the buoyancy correction, represent the action of fluid forces calculated using the penalized model. Note that the integration is performed over the volume of the domain, and not over the surface of the solid body. This is more convenient for numerical evaluation. For the spatial discretization of (1)–(2) we use a classical Fourier pseudo-spectral method. The temporal integration schemes are an adaptive second order Adams-Bashforth [10] for (1), with exact integration of the Laplacian, and a first-order explicit scheme for (3)–(4). The motion of the obstacle is modelled with a shift of the mask function. For its translation we rotate the phase of its Fourier coefficients: χ (x, t) = χ0 (x − δx)

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χ (k, t) = e−ikδx χ 0 (k),

(5) Reprinted from the journal

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20

y, [chord lengths]

y, [chord lengths]

15

10

5

Θ

0

−5 0

5

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40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

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t=2.8 t=3.6 t=4.0 t=3.2

0 2 4 6 8 10 12 14 16 18 20 22 24 26

x, [chord lengths]

x, [chord lengths]

Fig. 2 Left: trajectory of the tumbling plate at Re = 1100. Right: trajectories of the plate dropped edge-on at three different Reynolds numbers (colour online). Table 1 Averaged translational and angular velocities, and descent angle of the falling plate at Re = 1100

Present computation Andersen, Pesavento and Wang [1] Jin and Xu [6]

U (cm/s)

V (cm/s)

z (1/s)

(deg)

11.3 15.6 15.3

−7.2 −7.4 −11.2

19.4 18.0 16.9

32.5 25.3 36.2

where k is the wavenumber. Solid body rotation at an angle θ is decomposed into three skewing operations:       1 − tan(θ/2) 1 0 1 − tan(θ/2) cos θ − sin θ . (6) = R(θ ) = 0 1 sin θ 1 0 1 sin θ cos θ The mask function is smoothed in order to avoid Gibbs oscillations. This is done by applying a Gaussian filter to the discontinuous mask function. For further details on the numerical scheme we refer the reader to [7].

3 Results As a matter of validation, we made a comparison with numerical results reported in [1] and [6]. A solid plate having an elliptical cross-section is considered for that purpose. Its eccentricity equals e = b/a = 0.125, and its dimensionless moment of inertia is I ∗ = 0.5e(e2 + 1)ρsolid /ρ = 0.17, where ρ√ solid is the density of the solid. The Reynolds number is based on its size 2a and its terminal velocity, u t = πbg(ρsolid /ρ − 1), and it is equal to Re = 1100. The fluid is initially at rest. The plate is released from rest at an initial angle of incidence θ0 = 0.2. The periodic domain width and height are, respectively, L x = 10 and L y = 20 times the chord length of the plate. The domain is discretized with N x × N y = 1024 × 2048 grid points. The penalization parameter is η = 10−3 . Figure 2 (left) shows the trajectory of the tumbling plate, while a comparison of the average translational and angular velocities and the angle of descent with available values in [1,6] is presented in Table 1. The agreement between the three methods is rather reasonable, taking into account the sensitivity of the problem to perturbations, and the accuracy is satisfactory at least to make qualitative conclusions. More details on the validation of the numerical method can be found in [7]. To study the influence of viscosity on the dynamics of the plate, simulations at three different Reynolds numbers are performed: Re = 10,100 and 1000. The plate, having the same elliptical cross-section as before, is released in a fluid at rest, with zero initial velocity and with its longer axis oriented almost vertically, θ0 = π/2 + 0.01 (a small deviation is needed to provoke instability). The periodic domain size equals L x × L y = 10 × 40 chord lengths, the number of grid points is N x × N y = 1024 × 4096, and η = 10−3 . Reprinted from the journal

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Fig. 3 Vorticity at Re = 10 (a), Re = 100 (b), and Re = 1000 (c) (colour online).

Fig. 4 Pressure field near the plate at Re = 1000, corresponding to Fig. 3c (colour online).

Figure 2 (right) shows three trajectories of the plate, corresponding to the three values of the Reynolds number. The vertical orientation is unstable in the range of Re concerned. At Re = 10 the plate falls broadsideon, after a short transient. At Re = 100 the same steady-state establishes, but the transient motion is oscillatory. A similar transient is reported in [13] for falling disks. At Re = 1000 the broadside-on state is no more stable, and the plate is moving in an apparently chaotic manner, rocking from side to side and occasionally overturning. This behaviour is consistent with the bifurcation diagram obtained experimentally in [12], where the point Re = 1000, I ∗ = 0.17 in parameter space lies on the border between the ‘rocking motion range’ and the ‘autorotation range’. Note that Re = 1000 is just slightly lower with respect to the computation shown in the previous section, but, together with a different initial condition, this results in a qualitatively different behaviour: the plate did not reach a tumbling state. Figure 3 displays corresponding vorticity snapshots for the same Reynolds numbers. The wake undergoes a transition between Re = 100 and Re = 1000. At higher Re it contains distinct vortical structures formed due to hydrodynamic instabilities. Interactions between the intensive vortices and the plate are complex (see an example in Fig. 3c), and extremely sensitive to perturbations. In contrast, at lower Re the wake is stable and the vorticity field is much smoother (see Fig. 3 a, b). Figure 4 shows the pressure field near to the plate at Re = 1000 at consecutive time instants. It indicates depression in the separated vortices, as intensive as it is inside of the boundary layer. This gives rise to strong and complex interactions between the plate and its wake, resulting in an aperiodic rocking motion. For computation of fluid forces it is therefore important to resolve with good accuracy not only the boundary layer, but also the vortical flow around the plate.

4 Conclusions and perspectives A numerical method has been developed to solve the Navier–Stokes equations coupled with the equations which govern the free fall of a solid body. The two-dimensional Navier–Stokes equations in vorticity–stream

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function formulation are discretized using a Fourier pseudo-spectral method with an adaptive second order Adams–Bashforth time stepping. The volume penalization method is used to impose the no-slip boundary condition on the boundary of the solid body. Solids of arbitrary shape can be modelled by simply changing the mask function in the penalization term, and it is also straightforward to generalize this approach to study flows past multiple solid bodies. A numerical simulation of a tumbling plate at Re = 1100 has been compared with similar results in [1,6]. The agreement is adequate for a qualitative study. Numerical simulations at Re = 10,100 and 1000 have shown that decreasing Re has a stabilizing effect on the free fall dynamics, an observation which agrees with experimental results [12,13,5]. It is important to note that the number of grid points required to resolve the flow in the boundary layer is increasing as N x × N y ∝ Re. Hence, numerical simulations at Reynolds numbers up to a few thousands are feasible with the resolution of 102.4 grid points per chord length of the plate and a sufficiently large size of the periodic domain. Perspectives for future work include a precise computation of the critical Reynolds number corresponding to the transition between steady descent and oscillatory motion, as well as its comparison with the critical Reynolds number for the flow past a fixed plate. Possibly the freely-falling plates can adapt their attitude in such a way as to delay the onset of oscillations, as it was recently suggested in [5] in the context of freely rising three-dimensional axisymmetric bodies. In this connection we are currently working to increase the order of accuracy of the numerical method, which would allow this kind of computations. The extension of the model to three spatial dimensions is also planned. This will make possible a direct comparison with experiments, where the aspect ratio of the plate is finite. Acknowledgements The authors thank the Deutsch-Französische Hochschule, project ‘S-GRK-ED-04-05’, for financial support. This work was performed using HPC resources from GENCI-IDRIS (Project 91664).

References 1. Andersen, A., Pesavento, U., Wang, Z.J.: Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 65–90 (2005a) 2. Andersen, A., Pesavento, U., Wang, Z.J.: Analysis of tranitions between fluttering, tumbling and steady descent of falling cards. J. Fluid Mech. 541, 91–104 (2005b) 3. Angot, P., Bruneau, C.H., Fabrie, P.: A penalisation method to take into account obstacles in viscous flows. Numer. Math. 81, 497–520 (1999) 4. Borisov, A.V., Kozlov, V.V., Mamaev, I.S.: On the fall of a heavy rigid body in an ideal fluid. Proc Steklov Inst Math 253(SUPPL. 1), S24–S47 (2006) 5. Fernandes, P.C., Risso, F., Ern, P., Magnaudet, J.: Oscillatory motion and wake instability of freely rising axisymmetric bodies. J. Fluid Mech. 573, 479–502 (2007) 6. Jin, C., Xu, K.: Numerical study of the unsteady aerodynamics of the freely falling plates. Commun. Comput. Phys. 3(4), 834–851 (2008) 7. Kolomenskiy, D., Schneider, K.: A Fourier spectral method for the Navier-Stokes equations with volume penalization for moving solid obstacles. J. Comput. Phys. 228, 5687–5709 (2009) 8. Maxwell, J.C.: On a particular case of the decent of a heavy body in a resisting medium. Camb. Dublin Math. J., 9, 145–148 (1853); also Scientific Papers, 1, 115–118 (Cambridge University, 1890) 9. Pesavento, U., Wang Z.J.: Falling paper: Navier-Stokes solutions, model of fluid forces, and center of mass elevation. Phys. Rev. Lett. 93, 144501 (2004) 10. Schneider, K.: Numerical simulation of the transient flow behaviour in chemical reactors using a penalization method. Comput. Fluid. 34, 1223–1238 (2005) 11. Schneider, K., Farge, M.: Numerical simulation of the transient flow behaviour in tube bundles using a volume penalization method. J. Fluids Struct. 20, 555–566 (2005) 12. Smith, E.H.: Autorotating wings: an experimental investigation. J. Fluid Mech. 50(3), 513–534 (1971) 13. Willmarth, W.W., Hawk, N.E., Harvey, R.L.: Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 197–208 (1964)

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Theor. Comput. Fluid Dyn. (2010) 24:175–179 DOI 10.1007/s00162-009-0149-y

O R I G I NA L A RT I C L E

Tarun Kumar Sheel · Shinnosuke Obi

High-performance computing techniques for vortex method calculations

Received: 29 October 2008 / Accepted: 14 July 2009 / Published online: 7 August 2009 © Springer-Verlag 2009

Abstract A fast vortex method has been developed by using a special-purpose computer, MDGRAPE-3. MDGRAPE-3, successor of MDGRAPE-2, has been applied to the same calculations (Sheel et al. in Comput. Fluids 36:1319–1326, 2007) and the improvement in speed was 1000 times faster when compared with the ordinary PC Xeon 5160 (3.0 GHz) for N = 106 . The simultaneous use of the fast multipole method, Pseudoparticle multipole method with the special-purpose computers gives further acceleration of vortex method calculations. In addition, performances and efficiency have been investigated carefully. Further possibility has been investigated carefully and compared the present result with the previous one. Keywords Vortex method · Special-purpose computer · Fast multipole method PACS 40

1 Introduction There has always been a strong relationship between progress in vortex methods and advancements in acceleration techniques that utilize this method. When the classical vortex methods became popular nearly 30 years ago, the calculation cost of the N -body solver was O(N 2 ) for N particles. Due to this enormous calculation cost, the intention at that time was not to fully resolve the high Reynolds number fluid flow, but to somewhat mimic the dominant vortex dynamics using discrete vortex elements. One of the main difficulties of vortex methods to be accepted in the mainstream of computational fluid dynamics is the numerical complexity of calculating the velocity using the Biot–Savart law, which is in fact analogous to an “N -body problem” and hence requires O(N 2 ) operations for N vortex elements. The large numbers of elements are required for accurate vortex methods calculation at high Reynolds number flows that is very high computation cost. Therefore, significant acceleration techniques are necessary to reduce the computation cost of N -body interaction calculation for millions of particles having the cost of O(N 2 ) with growing N . There are two techniques to reduce the force calculation cost of an N -body simulation which hardware and software. In the hardware techniques, there are two techniques, one is a parallel computer and the other is a special-purpose computer. To accelerate the vortex methods calculation, parallel calculation has been widely Communicated by H. Aref T. K. Sheel (B) Department of Mathematics, Shahjalal University of Science and Technology, Sylhet 3114, Bangladesh E-mail: [email protected]; [email protected] S. Obi Department of Mechanical Engineering, Keio University, Yokohama, Japan Reprinted from the journal

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used in previous studies [6]. Even though it accelerates the calculation significantly; there are some difficulties to use parallel computations for longer calculations. It has limitations with parallelization according to hardware specifications. The memory bandwidth is a big problem to calculate for a large number of vortex elements, which required special consideration. Power consumption and heat dissipation interrupt the longer time calculations. These problems are becoming serious for advanced scientific computation. Shortcomings of parallel computers, the special-purpose approach can solve parallelization limit thoroughly. It has relaxed power consumption according to hardware specification. The cost-performance is minimum ∼100 times better than that of parallel computation using ordinary cluster computers [10]. The special-purpose computer has been used in the present calculations to avoid the difficulties of parallel computations with higher speed. Hardware accelerators such as MDGRAPE-2 [9], and MDGRAPE-3 [5,10] have been developed and successfully applied to Molecular Dynamics (MD) simulations [9]. In this paper, we will focus on MDGRAPE-3 and the simultaneous use of the FMM and MDGRAPE-3 to accelerate the present vortex method calculation at high Reynolds number flows. 2 Vortex methods Vortex methods are a powerful tool for the simulation of incompressible flows at high Reynolds number. In order to accurately compute the viscous transport of vorticity, gradients of velocity need to be well resolved. Vortex methods are part of a wider class of methods: the Lagrangian methods used to simulate unsteady, convection-dominated, problems. The three-dimensional incompressible flow of a viscous fluid has been studied here. The evolution equation for vorticity is Dωi (1) = (ωi · ∇) u + ν∇ 2 ωi Dt where ωi is vorticity defined as ω = ∇ × u, u is the velocity of vortex element, (ω · ∇)u is called stretching term and represents the rate of change of vorticity by deformation of vortex lines and the term ν∇ 2 ω represents the change of vorticity by viscous diffusion. The velocity field on three-dimensional problem is,        x − x × ω x 1 u (x) = − dV x (2) 3 4π |x − x | where x, and x are positions of vortex elements and d V is the volume of element. Using the Winckelmans model [11] as a cutoff function, Biot–Savart law is as follows ui = −

N 2 2 1  ri j + (5/2) σ j  5/2 ri j × γ j 4π j=1 ri2j + σ j2

(3)

where ri j = ri − r j , σ j and γ j are distance of the position vector, core radius and strength of element. The subscript i stand for the target elements, while j stands for the source elements. When the stretching term of Eq. (1) can be discretized as follows: dωi = (ωi · ∇) u dt

(4)

if I put vortex strength γi = ωi d 3 xi in Eq. (4), then it becomes dγi (5) = (γi · ∇) u i dt Hence, the vortex strength of an individual element is expressed by Eq. (3) in a discretized formulation as ⎫ ⎧ 2 2 ⎪ ⎪ 2 2 ⎨  ri j + (5/2) σ j ri j + (7/2) σ j   ⎬ 1 dγi (6) − = 7/2 γi · ri j ri j × γ j ⎪ 5/2 γi × γ j + 3  ⎪  2 dt 4π ⎭ ri j 2 + σ 2 j ⎩ ri j + σ j2 j where all notations denotes the same meaning as of Eq. (3). In this calculation, the viscous diffusion was calculated using the core-spreading method developed by Leonard [3]. For convection of the particles, second order accurate Adams–Bashforth numerical method was used for calculation of time advance.

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3 The MDGRAPE-3 The block diagram of an MDGRAPE-3 board can be seen in [5]. It consists of twelve MDGRAPE-3 chips, and each chip is connected in serial to send/receive the data. One small MDGRAPE-3 board (consists of 2 chips) has the peak performance of 330 GFlops. In order to communicate with the host computer, a field-programmable gate array (FPGA, Xilinx XC2VP30) is installed on the board. The present calculations, Xeon 5160 (3.0 GHz) duel core processor have been used as a host PC. To be used MDGRAPE-3, it is necessary to modify the program in order to call the application programming interface (API) of this system. The MDGRAPE-3 board can perform force calculations on many i-particles without any control of the host computer. Therefore, the host computer can perform other calculations, while the board performs the force calculations. One major problem in this sense is that the MDGRAPE-3 chips can only handle two types of calculations. The Coulomb potential i =

  2 b j g a ri j ,

(7)

  2 b j g a ri j ri j

(8)

N  j=1

and the Coulomb force fi =

N  j=1

where g( ) is an arbitrary function equivalent to an intermolecular force, which must be defined prior to the calculation same as of MDGRAPE-2. a, and b j are arbitrary constants, which can be used for scaling. To apply these libraries to the calculation of a vortex method, Biot–Savart law in Eq. (3) is expressed as follows. ui =

N 

   2 B j g A ri j + εi2j ri j

(9)

j=1

where A, B j are arbitrary constants. The details mathematical formulations are introduced in [7]. 3.1 Performances Figure 1a shows the calculation time against the number of vortex elements with and without the use of MDGRAPE-3. The calculation time has been obtained for the calculation of the impingement of two identical inclined vortex rings. It is clearly seen that the computation time is reduced by a factor of 1000 for N ∼ 106 . The efficiency and breakdown of overhead communication for the host PC have been calculated using following equations. Nstep = nmd(xmd + ymd + zmd) (a)

MDGRAPE-3 Xeon 5160(3.0GHz)

1

η

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(b) 1.5

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Where xmd, ymd, and zmd represent the number of pairwise interactions in x-direction, y-direction, z-direction. Nstep CPU Time(sec /step) NAPPL η(efficiency) = NGRAPE

NAPPL =

(11) (12)

Where NGRAPE represent the total number of pair wise interactions for one second (peak performance of MDGRAPE-3) for Coulomb force calculation which is NGRAPE = 1010 . Result of the efficiency measurements is shown in Fig. 1b. The N in x-axis represents the number of pair wise interactions particles for one second (Eq. 11) and the y-axis represents the maximum efficiency of corresponding N . The solid line represents the peak performance of MDGRAPE-3 board for molecular dynamics calculations and circle represents the maximum efficiency (η) of vortex method calculations. From the results of this measurement, it is proved that the efficiency is improved with increase of number of particles and whatever the host is, the board provides high performance if the number of particles is large. It is clearly observed from the figure that the efficiency of the present calculation is 98% compared with the peak performance of MDGRAPE-3(NGRAPE = 1010 ) for the largest N of VM calculations. 4 FMM on MDGRAPE-3 This section focuses on the simultaneous use of the FMM and MDGRAPE-3. The details mathematical formulations and calculation algorithm can be found in [8]. In the present calculations, the FMM by Cheng et al. [2] has been used. The most time consuming parts of the FMM are the multipole to local (M2L) translation and the direct calculation. The balance between these two steps is dependent on the level of box divisions. These two steps must be balanced by changing the level of box divisions according to the number of particles being calculated. The multipole and local expansions and their translations are impossible to calculate on the GRAPE architecture. Therefore, in a straightforward implementation of the FMM, MDGRAPE-3 can only be used for the final step of the FMM where the direct interaction of the particles is calculated. It is possible to calculate both hot-spots of the FMM on MDGRAPE-3 if the multipole to local translation is converted into an N -body interaction. This requires the use of two independent methods—the Poisson integral method by Anderson [1], and the pseudo-particle method by Makino [4]. 4.1 Tests for CPU-time and error The calculation cost and accuracy are the important issue for any numerical simulation. In this calculation these two factors have been investigated carefully. The calculation has been accelerated retained the accuracy at an acceptable level. The cpu-time has been compared with different acceleration techniques at one time step by changing the number of particles. The |L 2 | norm error is defined as the difference in the induced velocity of the same particles between the host and MDGRAPE-3 for the same time step as follows.   (u host − u md3 )2 + (vhost − vmd3 )2 + (whost − wmd3 )2 2  L (norm error) = (13)  2 2 2 + whost u host + vhost where the suffices md3 and host represent with and without the use of MDGRAPE-3, respectively. The CPU-time for all methods (when optimized) are plotted in Fig. 2a. Xeon 5160(3 GHz), MDG3, FMM, FMM-MDG3, and PPM-MDG3 represent the calculation without FMM or MDGRAPE-3, with MDGRAPE-3, with FMM, with FMM and MDGRAPE-3, and with the pseudo-particle method and MDGRAPE-3. The direct calculation without MDGRAPE-3 has a high asymptotic constant and an order of O(N 2 ). All calculations were performed on a dual core Xeon 5160 (3.0 GHz) processor. The direct calculation on MDGRAPE-3 has a lower asymptotic constant but still has a scaling of O(N 2 ). On the contrary, the FMM without MDGRAPE-3 has a high asymptotic constant, but its complexity is O(N ). The combination of the FMM and MDGRAPE-3 results in a calculation with a low asymptotic constant and O(N ) complexity. The pseudo-particle method on MDGRAPE-3 has a speed comparable to the FMM on MDGRAPE-3. However, it appears from Fig. 2a

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that the pseudo-particle method on MDGRAPE-3 does not quite scale as O(N ). At N = 106 the FMM on MDGRAPE-3 is approximately four times faster than the FMM. The |L 2 | norm error from the direct calculation without MDGRAPE-3 is shown for all other methods in Fig. 2b. The MDGRAPE-3 contains errors of its own, which stem from the partially single precision calculation, and use of interpolation for the calculation of the cut-off function. This error is constantly lower than the FMM errors for all N. The error of the pseudo-particle method is lower than that of MDGRAPE-3 for N < 4 × 104 , thus the PPM-MDG3 matches MDG3 until this value. 5 Conclusions The vortex method calculation has been accelerated using MDGRAPE-3, combination with the FMM and PPM. The efficiency of this special-purpose computer has been achieved up to 98% compared with the Coulomb force calculation. It is observed that MDGRAPE-3 has reduced computation cost 1000 times compared with Xeon 5160(3.0GHz) PC for N = 106 . The FMM is about 13 times faster and the PPM is about 48 times faster when calculated on the MDGRAPE-3. The FMM is approximately three times faster than the PPM without the MDGRAPE-3. However, the PPM becomes faster than the FMM when it is calculated on the MDGRAPE-3. Consequently, PPM with MDGRAPE-3 is the fastest technique among others discussed above. References 1. Anderson, C.R.: An implementation of the fast multipole method without multipoles. SIAM J. Sci. Stat. Comput. 13, 923– 947 (1992) 2. Cheng, H., Greengard, L., Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys 155, 468–498 (1999) 3. Leonard, A.: Vortex Methods for Flow Simulations. J. Comput. Phys. 37, 289–335 (1980) 4. Makino, J.: Yet another fast multipole method without multipoles- pseudoparticle multipole method. J. Comput. Phys. 151, 910–920 (1999) 5. Narumi, T., Ohno, Y., Okimoto, N., Koishi, T., Suenaga, A., Futatsugi, N., Yanai, R., Himeno, R., Fujikawa, S., Ikei, M., Taiji, M.: A 55 TFLOPS simulation of amyloid-forming peptides from yeast prion Sup35 with the special purpose computer system MDGRAPE-3. In: Proceedings of the SC06 (High Performance Computing, Networking, Storage and Analysis), CDROM, Tampa, USA, Nov 11–17 (2006) 6. Sbalzarini, I.F., Walther, J.H., Bergdorf, M., Hieber, S.E., Kotsalis, E.M., Koumoutsakos, P.: PPM—a highly efficient parallel particle-mesh library for the simulation of continuum systems. J. Comput. Phys. 215, 566–588 (2006) 7. Sheel, T.K., Yasuoka, K., Obi, S.: Fast vortex method calculation using a special-purpose computer. Comput. Fluids 36, 1319–1326 (2007) 8. Sheel, T.K., Yokota, R., Yasuoka, K., Obi, S.: The study of colliding vortex rings using a special-purpose computer and FMM. Transactions of the Japan Society for Computational Engineering and Science, vol. 2008, No. 20080003 9. Susukita, R., Ebisuzaki, T., Elmegreen, B.G., Furusawa, H., Kato, K., Kawai, A., Kobayashi, Y., Koishi, T., McNiven, G.D., Narumi, T., Yasuoka, K.: Hardware accelerator for molecular dynamics: MDGRAPE-2. Comput. Phys. Commun. 155, 115– 131 (2003) 10. Taiji, M., Narumi, T., Ohno, Y., Futatsugi, N., Suenaga, A., Takada, N., Konagaya, A.: Protein explorer: a petaflops specialpurpose computer system for molecular dynamics simulations. Proc. Supercomputing, in CD-ROM (2003) 11. Winckelmans, G.S., Leonard, A.: Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows. J. Comput. Phys. 109, 247–273 (1993)

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Theor. Comput. Fluid Dyn. (2010) 24:181–188 DOI 10.1007/s00162-009-0151-4

O R I G I NA L A RT I C L E

Paolo Luzzatto-Fegiz · Charles H. K. Williamson

Stability of elliptical vortices from “Imperfect–Velocity– Impulse” diagrams

Received: 17 January 2009 / Accepted: 9 June 2009 © Springer-Verlag 2009

Abstract In 1875, Lord Kelvin proposed an energy-based argument for determining the stability of vortical flows. While the ideas underlying Kelvin’s argument are well established, their practical use has been the subject of extensive debate. In a forthcoming paper, the authors present a methodology, based on the construction of “Imperfect–Velocity–Impulse” (IVI) diagrams, which represents a rigorous and practical implementation of Kelvin’s argument for determining the stability of inviscid flows. In this work, we describe in detail the use of the theory by considering an example involving a well-studied classical flow, namely the family of elliptical vortices discovered by Kirchhoff. By constructing the IVI diagram for this family of vortices, we detect the first three bifurcations (which are found to be associated with perturbations of azimuthal wavenumber m = 3, 4 and 5). Examination of the IVI diagram indicates that each of these bifurcations contributes an additional unstable mode to the original family; the stability properties of the bifurcated branches are also determined. By using a novel numerical approach, we proceed to explore each of the bifurcated branches in its entirety. While the locations of the changes of stability obtained from the IVI diagram approach turn out to match precisely classical results from linear analysis, the stability properties of the bifurcated branches are presented here for the first time. In addition, it appears that the m = 3, 5 branches had not been computed in their entirety before. In summary, the work presented here outlines a new approach representing a rigorous implementation of Kelvin’s argument. With reference to the Kirchhoff elliptical vortices, this method is shown to be effective and reliable. Keywords Vortex dynamics · Stability · Kirchhoff elliptical vortices · Kelvin’s argument PACS 47.10.Fg, 47.20.Ky, 47.32.cd

1 Introduction More than a century ago, Lord Kelvin proposed that a steady vortical flow realizes an extremum of the kinetic energy, for a given impulse [22]. It appears that Kelvin regarded this proposition as being self-evident, as he provided no proof for it; the first analytical confirmation is instead due to Benjamin in 1976 [1]. The argument can be illustrated as follows, with reference to a vortex configuration rotating at a rate . Under Communicated by H. Aref P. Luzzatto-Fegiz (B) · C. H. K. Williamson Sibley School of Mechanical and Aerospace Engineering, Cornell University, Upson Hall, Ithaca, NY 14853-7501, USA E-mail: [email protected] E-mail: [email protected] Reprinted from the journal

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vorticity-preserving perturbations, a steady two-dimensional flow is associated with a stationary point of the functional H : H = E − J,

(1)

where E and J are the excess kinetic energy and angular impulse, respectively, given by:  1 ω(x) ω(x ) log |x − x | dx dy dx  dy  4π  1 J =− ω(x) |x|2 dx dy. 2

E =−

(2) (3)

In the above, ω is the vorticity, while the integrals are taken over all space. Since E and J are conserved in an inviscid fluid, while  is treated as a fixed parameter, H is also a conserved quantity. If the stationary point is a maximum or a minimum in the solution space (implying that the second variation δ 2 H is positive or negative definite), then a displacement away from the solution would lead to a change in H , which is impossible; hence the solution must be stable to isovortical perturbations, thus yielding a sufficient condition for stability. Similarly, a necessary condition of instability is that the stationary point is a saddle [22]. The second variation δ 2 H can therefore be used, in principle, to assess stability; unfortunately, computing δ 2 H is often unfeasible, since solutions of practical interest are usually known only numerically. The implementation of Kelvin’s argument has thus been the subject of extensive debate. Saffman and Szeto [21], having numerically found steady solutions for two co-rotating vortices, circumvented this difficulty as follows. Equation 1 can be interpreted as establishing extrema of E under the constraint that J = const., with  taking the role of a Lagrange multiplier. A plot of E versus J for their flow then shows that, for a given J , there exist two E branches, joined at a fold point. The top branch was interpreted as a maximum (and hence stable), while the lower branch was speculated to be a saddle (possibly unstable). However, Dritschel [4] later pointed out that there seems to be no necessary link between the shape of a plot of E versus J and the curvature of the H surface. Furthermore, he stated that even if such correspondence could be established, additional changes of stability could occur away from folds in E and J , by means of bifurcations to new families of solutions. As a consequence of these arguments, the method proposed by Saffman and Szeto has been considered unreliable. In a forthcoming paper [13], we address both these issues by proposing a new approach to this problem. By building on ideas from dynamical systems theory, we show that changes of stability are associated with extrema in a velocity–impulse diagram (instead of an impulse–energy diagram). We deal with the second issue raised by Dritschel [4] by exploiting the fact that bifurcations are not structurally stable [19]; hence by introducing a small imperfection and re-computing the steady states, we obtain distinct solution branches, thus uncovering any bifurcations. All changes of stability are therefore apparent in an “Imperfect–Velocity–Impulse” (IVI) diagram. Let us consider, as a schematic example of a typical construction of an IVI diagram, a possible scenario involving the detection of a subcritical bifurcation for a family of equilibrium solutions of the Euler equations (see Fig. 1). First, the steady base flows are computed, and the associated velocity–impulse diagram is plotted (Fig. 1a). Introducing a small imperfection in the governing equations, and re-computing the steady states, breaks the original curve into two distinct branches, revealing an extremum in J (Fig. 1b). Since the extremum consists of a local minimum, it can be shown that stability is lost as the curve is traversed from right to left [13,12]. Therefore, if we suppose, for example, that the right-most portion of the branch is stable (marked by ‘S’ in the figure), the portion of the branch to the left of the minimum in J will have one unstable mode (denoted by ‘1U’). Finally, by taking the strength of the imperfection to zero, we recover the underlying bifurcated solution branch (shown in red in Fig. 1c). As an example of the practical use of the IVI diagram approach, we consider the stability of the family of elliptical vortices discovered by Kirchhoff [8]. While the elliptical family can be characterized analytically, detecting bifurcations through an IVI diagram requires finding equilibrium vortices numerically; the computational procedure is outlined in Sect. 2. In Sect. 3.1, we describe the unfolding of the IVI diagrams for the elliptical vortices. A large body of work exists regarding the stability of the Kirchhoff ellipses (see for example [10,16]); in Sect. 3.2, these results are employed to verify the efficacy of the IVI diagram approach.

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(a)

(b)

(c) 1U

New branch

S

S

1U

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1U

Basic branch W

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1U

Basic branch

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W

Fig. 1 Typical construction of an Imperfect–Velocity–Impulse (IVI) diagram. a shows a velocity–impulse diagram for a basic branch of steady solutions. Introducing a small perturbation and re-computing the steady states breaks the basic branch into two distinct branches (shown in b), revealing a change of stability. In (c), by bringing the perturbation to zero, we recover the underlying new solution branch (shown in red)

2 Computational method To ensure that all solutions are captured, we must employ a numerical method that is capable of resolving arbitrary vortex shapes; this is particularly important in the light of the fact that bifurcations commonly lead to new families of solutions having fewer (if any) symmetries relative to the original solution. Furthermore, it is known that families of uniform vortices usually terminate with limiting vortex states, the boundaries of which exhibit one or more corners (e.g. [25,17]); computing such shapes calls for a particularly efficient numerical approach. We intend to submit the full details of the method for publication shortly [11]; in what follows, we briefly outline the approach. A patch of uniform vorticity corresponds to a steady state if, as the vortex translates or rotates, its boundary does not deform. This implies that, in a frame of reference moving with the vortex, the component of the velocity field normal to the boundary must vanish. For a rotating configuration, we can therefore write: uc · n = 0

(4)

on the vortex boundary, where uc = (u c , vc ) = (u, v) + (y, −x) is the velocity in the co-rotating frame, and n is the unit-normal vector to the boundary. For a given vorticity distribution, the velocity field u is obtained by inverting the Poisson equation ∇ 2 ψ = −ω by means of the appropriate Green function (see e.g. [20]). The vorticity field can then be iteratively adjusted until Eq. 4 is satisfied. Since Deem and Zabusky [3] first employed the method of contour dynamics to solve Eq. 4, a large variety of approaches has been developed to find equilibrium uniform vortices. These techniques make use of either Newton iteration (e.g. [6]) or of a relaxation approach to obtain new solutions (e.g. [18]). Unfortunately, each approach is limited in the possible range of vortex shapes it can resolve. While Newton iteration guarantees convergence and can therefore, in principle, capture arbitrary shapes, it may become prohibitively expensive for vortices exhibiting fine-scale features. On the other hand, relaxation methods are much less computationally intensive, and can thus affordably resolve details, such as corners; however, as convergence is not assured, they usually fail to compute shapes having fewer than two planes of symmetry (e.g. [4]). We address these issues by introducing a methodology that overcomes the large computational expenses associated with Newton iteration. Given a guess for a new solution, we compute the correction normal to the vortex shape (say, γ (s)) as a function of arc-length along the boundary s. We then expand γ in a Fourier series in terms of a modified arc-length parameter s˜ , constructed according to local curvature, in order to ensure fast convergence in the series. This allows us to affordably and reliably compute vortices of arbitrary shape. We choose  as the control parameter, and employ generalized continuation to ensure that we can negotiate any turning points; new solutions are obtained by employing an Euler predictor and a Newton corrector. The vortex is taken to have unit vorticity and area. The accuracy in the solution typically depends on the number of modes M used in the Fourier expansion and on the size of the step δ taken in the solution space; the algorithm uses a fixed value of M and adjusts δ to ensure that the highest-order terms in the Fourier series are negligibly small. The method was initially tested against the exact elliptical solutions, while selected bifurcated branches were checked by changing M and δ; Reprinted from the journal

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for all of the results presented here, the shapes were verified to be accurate to at least seven significant figures. All calculations were performed on a laptop computer with a 1.83 GHz processor, which is an indication of the modest computational power required to implement the approach.

3 Stability of elliptical vortices 3.1 Number of unstable modes and bifurcated branches from IVI diagrams The Kirchhoff vortices represent a rare case of a vortical solution of the Euler equation that can be represented analytically ([8]; see also [9,20]). Once again considering, without loss of generality, patches with unit vorticity and area, it can be show that the impulse is given by: J () =

2 − 1 , 8π

(5)

where  is related to the axis ratio λ = b/a by  = λ(λ + 1)−2 . Since J is singular as λ → 0, and J = −(4π)−1 at λ = 1, we choose to plot −(4π J )−1 instead of J (see Fig. 2; notice that the value of  for any extrema would be unchanged). Since J () is monotonic, any changes of stability have to occur through bifurcations. The family of solutions begins with a circular vortex (top-right-hand corner in Fig. 2) and terminates into a vortex sheet (located at the origin in the figure). The circular vortex can be argued to be stable, since any deformation of the boundary would tend to ‘spread out’ its vorticity. This, in turn, would lead to a decrease in the flow’s kinetic energy, which is not possible in an inviscid flow; thus the circular vortex must be stable (as already noted by Lord Kelvin [23]). Alternatively, a straightforward linear stability analysis (see [24,20]) can be used to obtain the same conclusion. (Incidentally, the circular vortex has also been shown to be nonlinearly stable [5].) By starting with a near-circular vortex and using an IVI diagram to detect the introduction of unstable modes along the family, we can therefore determine the stability properties of the elliptical vortices. The imperfection is constructed as follows. Examining the flow field in the co-rotating frame (for which streamlines are shown in Fig. 3), one can find four stagnation points, two residing in the recirculation regions above and below the vortex in the figure (marked by a cross, ×), and two taking the form of saddle points (denoted by the bull’s eyes, ). We either introduce a weak point vortex in each of the recirculation regions, or place a point source and a sink at each of the saddle nodes. In the results presented here, we switch between these two imperfections depending on which construction yields the clearest example of bifurcation breaking, as we shall explain in further detail below. The addition of these flow elements is accommodated by a small modification to the code outlined in Sect. 2; this involves counting the contribution to u from the flow elements, while solving for the position of the stagnation points in the co-rotating flow. The strength of each imperfection is treated as a fixed parameter.

–(4π J )–1

1

0.5

0 0

0.05

0.1

0.15

0.2

0.25

Fig. 2 Velocity–impulse plot for the Kirchhoff ellipses. Since no extrema are immediately apparent, any exchanges of stability have to develop through bifurcations

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Fig. 3 Streamlines in a frame of reference rotating with the elliptical vortex. The shaded region is occupied by uniform vorticity. The imperfection can be constructed by either placing a point vortex at each of the stagnation points marked by a cross, or by introducing a source and a sink of equal strengths at the stagnation points denoted by a bull’s eye

We begin to develop the IVI diagram for the elliptical vortices by considering a near-circular vortex, introducing a source–sink combination, and re-computing the steady states. As a fraction of the circulation of the elliptical vortex, each source/sink is chosen to have strength m/ = ±10−4 . We gradually reduce  and seek . new steady states. A local minimum in impulse J is reached at  = 0.1877405, indicating that stability is lost at this location (as shown in the left-hand diagram in Fig. 4a), while revealing the presence of a bifurcation. The second imperfect branch in Fig. 4a is computed by introducing the same imperfection to elliptical vortices further along the family (i.e. with lower ) and seeking steady states for progressively increasing . By subsequently letting the strength of the source and sink approach zero, we recover the underlying bifurcated branch. This family is found to end with a limiting shape (shown in the right-hand part of Fig. 4a), which exhibits a 90◦ corner. By visual inspection of these non-elliptical shapes, we can establish that this bifurcation is associated with a perturbation with azimuthal wavenumber m = 3. The IVI diagram indicates that this family of solutions has one unstable mode. The next bifurcation is investigated by introducing two point vortices, each having strength PV / = 10−4 ; more will be said on this choice of imperfection below. Restarting our search for equilibria past the m = 3 . bifurcation, for decreasing , we eventually encounter a new local minimum in J at  = 0.1480637, thus detecting a second loss of stability at an additional bifurcation (shown in the inset of Fig. 4b). Note that, in order to clearly see this minimum, we have had to magnify the plot by a factor of 2,000! In a manner similar to the m = 3 case, this branch exhibits a turning point in . However, in contrast to the m = 3 bifurcation, the adjacent branch now reveals a local maximum in J , after which the left-hand branch moves away from the right-hand one, instead of approaching it (second inset in Fig. 4b). Removing the imperfection, and computing the underlying family, indicates that these solutions connect to the elliptical vortices at a point of tangency in the velocity–impulse diagram, through what appears to be a transcritical bifurcation. For increasing , the new bifurcated solution branch (shown in red) terminates with a shape resembling a cat’s eye. With decreasing , we are led to a state consisting of two identical vortices connected at a point (see the left-most shape in Fig. 5). This branch can be continued into a family of two identical co-rotating vortices (see [2,14]). In this case, we find that this bifurcation is associated with modes with azimuthal wavenumber m = 4. It should be pointed out that the use of a source/sink combination as an imperfection was also found to successfully break the m = 4 bifurcation into two branches. Although these two branches do not share any steady states, they are found to overlap at a point in the IVI diagram. While the resulting diagram is somewhat less clear for the purposes of illustration, stability properties can still be correctly inferred in a straightforward manner. We employ again a source/sink combination for the next bifurcation. The imperfect steady states display a . minimum at  = 0.1197174, leading to a further loss of stability (left-hand plot in Fig. 4c). The structure of this bifurcation is qualitatively similar to the m = 3 case. While the underlying solution branch also ends in a limiting shape with a 90◦ corner (right-hand plot in Fig. 4c), this bifurcation is found to be associated with an m = 5 perturbation. Reprinted from the journal

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(a)

0.6038

0.645

1U m=3

–(4πJ )–1

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S

1U 1U

0.5988

0.595 0.1873

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–(4πJ)–1

2U 1U

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m=4

× 2000

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m=5

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0.319

2U 3U

3U 0.31478 0.11971

0.313 0.11972

0.11973

0.1192

0.1203

0.1214

0.1225

Fig. 4 Construction of the IVI diagram for the first three bifurcations of the elliptical vortices. The left-hand column shows the branches that were computed after introducing the imperfection; on the right-hand side, the underlying bifurcated branch (computed by taking the imperfection to zero) is shown in red. From top to bottom, the bifurcations shown in (a), (b) and (c) are associated with perturbations having azimuthal wavenumber m = 3, 4 and 5, respectively. Open and filled circles denote changes of stability and limiting shapes, respectively

For any bifurcations detected through the IVI diagram approach, the precise location crit of the change of stability is established by computing the steady state connecting two branches; a comparison with classical results is provided in Sect. 3.2.

3.2 Stability from classical results There is a large body of results regarding the stability of the elliptical vortices (see [15] for a recent review). In 1893, Love [10] obtained an expression for the eigenvalue σ associated with a perturbation of wavenumber m, as a function of the axis ratio λ:  2    1 1 − λ 2m 2mλ 2 σ =− , (6) −1 − 4 (1 + λ)2 1+λ where a real value of σ yields an instability. The values of  for each bifurcation m can then be calculated by setting σ = 0 in Eq. 6, and using the expression  = λ(λ + 1)−2 . We verify that, as one should expect, the

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–(4π J )–1

S m=4

0.5

1U m=3

2U

rti o vo Tw

ces

3U m=5

0 0

0.05

0.1

0.15

0.2

0.25

Fig. 5 Overall view of the resulting IVI diagram, showing the first three branches bifurcating from the elliptical states

locations of the changes of stability from our methodology match Love’s prediction to at least seven significant figures. Indeed, since the changes of stability in our study are found by determining the intersection between the bifurcated and the basic solution branches, the precision in the stability boundaries is limited only by the numerical accuracy with which the steady states are computed. Love’s analysis formed the basis of part of the work of Kamm [7], who computed the beginning of the bifurcated branches presented here; the m = 4 branch was later explored in its entirety by Cerretelli and Williamson [2], who approached the problem by initially considering two co-rotating vortices with lower . Due to the large computational cost associated with previous numerical methods [7], it appears that the m = 3, 5 families (including the limiting shapes) had not been computed before. It should be pointed out that the stability of all of these bifurcated branches was previously unknown. Furthermore, while attempting to apply the approach of Saffman and Szeto, involving a (J, E) plot, to determine stability, Kamm found that all of the new equilibria had the same energy and impulse as a member of the elliptical family (to numerical accuracy). The bifurcated branches were therefore indistinguishable from the elliptical family in a plot of E versus J , preventing Kamm from reaching any conclusions regarding stability. This appears to be a further problem that can be associated with the use of an impulse–energy plot, which should be considered in addition to the theoretical objections previously posed by Dritschel [4]. In contrast, the use of a velocity-impulse diagram does not appear to suffer from the same issues. In spite of presenting a significant amount of fine-scale details in the velocity–impulse plot, we must note that the bifurcations were easily detected with the imperfection approach presented here. As noted in Sect. 2, the step change in the control parameter was automatically adjusted to preserve accuracy; we did not need to pose further restrictions to ensure that bifurcations were detected. Therefore an IVI diagram (coupled with a suitable numerical method) is found to be reliable in revealing bifurcations. Once a new branch is detected, the turning points can be carefully mapped by employing progressively smaller step sizes. Finally, we should remark that, as a part of a separate work, the authors examine the m = 4 branch through a linear stability analysis [14], finding accurate agreement with the results presented here.

4 Conclusions In this paper, we successfully employ the “Imperfect–Velocity–Impulse” (IVI) diagram methodology to determine the stability of elliptical vortices, providing a detailed example of the application of this approach to a classical flow. The imperfection used to reveal bifurcations is constructed by placing either point vortices, or sources and sinks, at the stagnation points of the co-rotating flow; the steady states are then computed using a novel numerical approach capable of accurately resolving vortex shapes of lesser symmetry. The first three bifurcations for the family of Kirchhoff elliptical vortices (corresponding to instabilities with azimuthal wavenumber m = 3, 4 and 5) are revealed; their detection appears to be insensitive to the numerical parameters employed. Inspection of the IVI diagram gives the stability properties for the elliptical Reprinted from the journal

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vortices, as well as for the bifurcated families. Each non-elliptical branch is found to terminate with limiting shapes exhibiting one or more 90◦ corners. While the locations of these changes of stability precisely match available results from linear analysis [10], the stability of the non-elliptical shapes represents a new result. It also appears that the existence of limiting states for the m = 3, 5 branches had not been established before. It should be pointed out that, in this study of the Kirchhoff elliptical vortices, all changes of stability are revealed by the use of imperfection theory. However, in general, one might expect to also find changes of stability occurring at extrema in impulse for the base flow. In conclusion, we demonstrate that the new IVI-diagram approach outlined here, which represents a rigorous implementation of Kelvin’s argument, accurately and reliably detects changes of stability for the family of Kirchhoff elliptical vortices. Acknowledgments We would like to thank Prof. S. Leibovich and Prof. P. H. Steen for several useful discussions.

References 1. Benjamin, T.B.: The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. In: Applications of Methods of Functional Analysis to Problems in Mechanics, pp. 8–29. Springer (1976) 2. Cerretelli, C., Williamson, C.H.K.: A new family of uniform vortices related to vortex configurations before merging. J. Fluid Mech. 493, 219–229 (2003) 3. Deem, G.S., Zabusky, N.J.: Vortex waves: stationary “V States”, interactions, recurrence, and breaking. Phys. Rev. Lett. 40, 859–862 (1978) 4. Dritschel, D.G.: The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95–134 (1985) 5. Dritschel, D.G.: Nonlinear stability bounds for inviscid, two-dimensional, parallel or circular flows with monotonic vorticity, and the analogous three-dimensional quasi-geostrophic flows. J. Fluid Mech. 191, 575–581 (1988) 6. Elcrat, A., Fornberg, B., Miller, K.: Stability of vortices in equilibrium with a cylinder. J. Fluid Mech. 544, 53–68 (2005) 7. Kamm, J.R.: Shape and stability of two-dimensional uniform vorticity regions. Ph.D. thesis, California Institute of Technology, Pasadena (1987) 8. Kirchhoff, G.: Vorlesungen über mathematische Physik: Mechanik. Teubner, Leipzig (1876) 9. Lamb, H.: Hydrodynamics, 6th edn. Cambridge University Press (1932) 10. Love, A.E.H.: On the stability of certain vortex motions. Proc. London Math. Soc. 25, 18–42 (1893) 11. Luzzatto-Fegiz, P., Williamson, C.H.K.: An accurate and efficient method for computing uniform vortices. J. Comp. Phys. (2009) (In preparation) 12. Luzzatto-Fegiz, P., Williamson, C.H.K.: Determining the stability of steady inviscid flows through “imperfect–velocity– impulse” diagrams. J. Fluid Mech. (2009) (In preparation) 13. Luzzatto-Fegiz, P., Williamson, C.H.K.: A new approach to obtain stability of vortical flows. Phys. Rev. Lett. (2009) (In preparation) 14. Luzzatto-Fegiz, P., Williamson, C.H.K.: On the stability of a family of steady vortices related to merger. J. Fluid Mech. (2009) (In preparation) 15. Mitchell, T.B., Rossi, L.F.: The evolution of Kirchhoff elliptic vortices. Phys. Fluids 20, 054103 (2008) 16. Moore, D.W., Saffman, P.G.: Structure of a line vortex in an imposed strain. In: Olsen, J.H., Goldburg, A., Rogers, M. (eds.) Aircraft Wake Turbulence, pp. 339–354. Plenum, New York (1971) 17. Overman, E.A.: Steady-state solutions of the Euler equations in two dimensions II. Local analysis of limiting V-states. SIAM J. Appl. Math. 46, 765–800 (1986) 18. Pierrehumbert, R.T.: A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129–144 (1980) 19. Poston, T., Stewart, I.: Catastrophe theory. Dover, New York (1978) 20. Saffman, P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992) 21. Saffman, P.G., Szeto, R.: Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23, 2239–2342 (1980) 22. Thomson, W.: Vortex statics. Math. Phys. Pap. IV, 115–128 (1875) 23. Thomson, W.: On maximum and minimum energy in vortex motion. Math. Phys. Pap. IV, 166–185 (1880) 24. Thomson, W.: Vibrations of a columnar vortex. Math. Phys. Pap. IV, 152–165 (1880) 25. Wu, H.M., Overman, E.A., Zabusky, N.J.: Steady-state solutions of the Euler equations in two dimensions: Rotating and translating V-states with limiting cases. I. Numerical algorithms and results. J. Comp. Phys. 53, 42–71 (1984)

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Theor. Comput. Fluid Dyn. (2010) 24:189–193 DOI 10.1007/s00162-009-0132-7

O R I G I NA L A RT I C L E

Oscar Velasco Fuentes

Chaotic streamlines in the flow of knotted and unknotted vortices

Received: 29 December 2008 / Accepted: 2 June 2009 / Published online: 24 July 2009 © Springer-Verlag 2009

Abstract This paper describes the motion and the flow induced by a thin tubular vortex coiled on a torus. The vortex is defined by the number of turns, p, that it makes round the torus symmetry axis and the number of turns, q, that it makes round the torus centerline. All toroidal filamentary vortices are found to progress along and to rotate round the torus symmetry axis in an almost steady manner while approximately preserving their shape. The flow, observed in a frame moving with the vortex, possesses two stagnation points. The stream tube emanating from the forward stagnation point and the stream tube ending at the backward stagnation point transversely intersect along a finite number of streamlines. This produces a three-dimensional chaotic tangle whose geometry depends primarily on the value of p. Inside this chaotic shell there are two major stability tubes: the first one envelopes the vortex whereas the second one runs parallel to it and possesses the same topology. When p > 2 there is an additional stability tube enveloping the torus centerline. Keywords Chaotic Streamlines · Filamentary vortices · Toroidal knots PACS 47.32.C-, 47.32.cd, 47.52.+j

1 Introduction Smoke rings have been observed for over 200 years in man-made situations (e.g. firing cannons and puffing steam-pipes) and possibly longer in natural situations (e.g. exhalations of volcanoes and geysers). Already at the dawn of the XIX century it was recognised that they are whirling masses of fluid, and it was even suggested that their motion is responsible for their stability and capacity to carry fluid [1]: the “quick rotation of the ring, from within outwards (…) seems, in some manner or another, as if it kept the parts together.” Fifty years later, in the epoch-making paper celebrated by this symposium, Helmholtz [2] gave a mathematical explanation of the motion of ring vortices and provided instructions on how to generate them in a real fluid. In 1867, Kelvin [7] placed the ring vortex in a prominent position with his hypothesis that matter consists of vortex atoms whirling in an all-pervading ideal fluid. While pursuing this hypothesis, Kelvin [9] speculated that thin tubular vortices coiled on a torus could exist as steady structures and that they would be stable “provided only the core is sufficiently thin.” He characterized the steady motion of the “outer form of the core” (i.e. the surface of the tubular vortex) as the sum of a progression along and a rotation round the axis of the torus. Kelvin described the motion inside the vortex as a solid body rotation round the sinuous axis of the filament and characterised the motion outside the vortex invoking the electromagnetic analogy because this analogy “does much to promote a clear understanding of the still somewhat strange fluid-motions with which we are at present occupied”. In this Communicated by H. Aref O. Velasco Fuentes Departamento de Oceanografía Física, CICESE, Ensenada, Mexico E-mail: [email protected] Reprinted from the journal

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paper we study the motion and the velocity field of toroidal filamentary vortices. Our purpose is to determine how they depend on the topology of the vortex. 2 Vortex motion Following Kelvin’s description we assume that a thin tubular vortex of strength  is uniformly coiled on a torus whose centerline has a radius r0 and whose cross-section is a circle of radius r1 . In Cartesian coordinates the axis of the filamentary vortex is given as follows: x = (r0 + r1 cos φ) cos θ, y = (r0 + r1 cos φ) sin θ, z = r1 sin φ. where φ is the angle round the torus centerline and θ is the angle round the torus symmetry axis. They are given by φ = qs and θ = ps, where p and q are co-prime integers and s is a parameter in the range 0 − 2π. Hence, before closing on itself, the toroidal filamentary vortex V p,q makes p turns round the torus symmetry axis and q turns round the torus centerline. These numbers determine the topology of the vortex, as follows: when p > 1 and q > 1 the vortex forms a toroidal knot, when either p = 1 or q = 1 the vortex forms a toroidal unknot. In the latter situation, however, it is useful to make a distinction between toroidal helices (when p = 1 and q > 1) and toroidal loops (when p > 1 and q = 1). We compute the induced velocities with the Rosenhead–Moore approximation to the Biot-Savart law [6]:   [x − r(s)] × ds u(x) = − , (1) 4π [|x − r(s)|2 + μ2 a 2 ]3/2 where a is the radius of the filament’s cross section and μ is a constant with value 21 e−3/4 . The filament is represented by a finite number of material markers, say n, whose value depends on a and the filaments’s length L. We found that for n > 2L/a the speed of a circular ring is computed within 0.5 % of the analytical value. We assume the same criterion to represent a toroidal filamentary vortex and thus n varies in the range 150–1500 in the simulations discussed here. Once chosen, the value of n is kept constant during a simulation because the filaments evolve without undergoing appreciable changes in length or shape. The evolution is computed with a fourth-order Runge–Kutta scheme with fixed time step. As predicted by Kelvin [9], all vortices are observed to progress along and to rotate round the torus symmetry axis. Both components of the vortex motion are approximately uniform. The linear speed, U , is proportional to p and is almost unaffected by the value of q (Fig. 1, left panel). This behaviour is easily explained as follows: since r1 /r0 = 0.1 the progression speed behaves as if there was a single ring with circulation p instead of p loops of a filament with circulation . As a matter of fact U ≈ 3/4 pU0 , where U0 is the speed of a circular ring of strength  and radius r0 [8]:    1 8r0 U0 = − (2) log 4πr0 a 4 The angular speed, , grows with increasing p and decreasing q (Fig. 1, right panel). Note that toroidal helices ( p = 1) rotate in the opposite sense and at a much lower rate than toroidal loops (q = 1); and toroidal knots rotate in the same sense and at lower rates than toroidal loops. 3 Fluid flow Toroidal filamentary vortices are almost stationary and of fixed shape when observed in a system which progresses with speed U and rotates with angular speed ; therefore we will describe the fluid motion in this co-moving frame. Since r1 /r0 = 0.1 the velocity field, away from the immediate vicinity of the vortex, can be considered as a perturbation of the velocity field of a ring vortex of strength p and radius r0 . The cross section of this virtual ring vortex should be closer to the cross section of the torus (πr12 ) than to the cross section of the toroidal vortex (πa 2 ). Thus whereas the toroidal vortex is thin, the virtual vortex is fat. This has important consequences because the speed of fat rings is smaller than that of thin rings (see Eq. 2), and the

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Fig. 1 The motion of toroidal filamentary vortices as a function of p and q. Left U , the speed of linear motion along the torus axis (scaled by /4πr0 ). Right , the angular speed of rotation round the torus axis (scaled by /2πr0 r1 ). The black dots indicate the only points where toroidal vortices exist. The contours of U and  were drawn by interpolation to show how these speeds change in the parameter space

ring speed determines the geometry of the flow, as was qualitatively described by Kelvin [7] (see, e.g. [6] for a quantitative analysis). Let us thus describe the flow geometry of a fat ring vortex. The velocity field possesses two stagnation points, both located on the ring’s symmetry axis. The forward one, P, has a linear attractor and a planar repellor; the backward one, Q, has a linear repellor and a planar attractor. There is one streamline that starts infinitely far upstream and ends at P, and one streamline that starts at Q and ends infinitely far downstream. The two stagnation points are connected by one streamline starting at Q and ending at P, and by an infinite number of streamlines starting at P and ending at Q. These lines form a surface with the shape of a flattened spheroid. This stream surface is called separatrix, because the streamlines located inside it are qualitatively different from those located outside it. While all streamlines are plane curves, the former are closed whereas the latter are open and of infinite length. From a more physical point of view, the separatrix is the surface that divides the ambient fluid and the fluid carried by the vortex; that is to say, the separatrix is the boundary of the vortex “atmosphere”, as Kelvin [7] called the irrotational fluid permanently trapped by the vortex. The flow just described is perturbed when we substitute back the toroidal filamentary vortex in place of the fat ring vortex. In such case the stagnation points survive but the separatrix disappears: instead of a single surface starting at P and ending at Q, there are now two surfaces. The first one, called the unstable manifold, starts at P and ends infinitely far downstream; the second one, called the stable manifold, starts infinitely far upstream and ends at Q. These surfaces must necessarily intersect but, because of the uniqueness of streamlines, they do so along a finite number of streamlines which start at P and end at Q. Figure 2 shows a meridional cross section of the stable and unstable manifolds of the toroidal helix V1,5 . None of the curves shown is a streamline: the flow induced by toroidal vortices always has swirl and thus streamlines are not plane curves. The black curve, which represents the intersection of the unstable manifold, starts at P and smoothly moves downstream. As it approaches Q it starts to oscillate about the grey curve, which represents the intersection of the stable manifold. The inverse occurs as the grey curve moves from P to Q. The existence of this geometric structure, known as heteroclinic tangle, implies that streamlines are chaotic (see e.g. [10]). The heteroclinic tangle also provides, through the lobe-dynamics mechanism [5], a template for the wandering of streamlines around different flow regions. Figure 3 shows meridional cross-sections of the unstable manifold for all toroidal filamentary vortices in the range 1 < p < 5 and 1 < q < 5. The shape of the manifolds is mainly determined by p, whereas the value of q is important only for toroidal helices ( p = 1). Note, for example, that the shape of the manifold of the toroidal knot V2,5 is more similar to that of the toroidal loop V2,1 than to the toroidal knot V3,5 . As p grows the oscillations of the unstable manifold start closer to the backward stagnation point Q. When p > 3 the unstable and stable manifolds differ very little (except in the immediate neighbourhood of P and Q) so that the separatrix is practically restored and the vortex atmosphere looks like that of a fat ring vortex. Reprinted from the journal

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Fig. 2 Meridional cross section of the stable and unstable manifolds (grey and black lines, respectively) for the toroidal helix V1,5 . The circles indicate the intersections of the vortex filament, the black and grey crosses indicate the forward and backward stagnation points, respectively

5

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Fig. 4 Poincaré sections of streamlines induced by toroidal loops: a V3,1 , b V4,1 and c V5,1 . The black circles represent the intersection of the filamentary vortex with the Poincaré section

In the vicinity of the filamentary vortex the flow is always very different from that of a ring vortex (fat or otherwise). We explore the behaviour of the flow in this region with the aid of Poincaré sections. These are constructed by numerically computing streamlines that start on the meridional semi-plane θ = 0 and plotting each intersection of these streamlines with the semi-plane (Fig. 4). There are two conspicuous tubes of stability: the first one envelopes the filamentary vortex whereas the second one runs parallel to it and possesses the same topology, that is to say, it is a thin tube that closes on itself after the same number of turns, p, round the torus symmetry line and the same number of turns, q, round the torus centerline. When p = 1, 2 these stability tubes are embedded in the unbounded chaotic sea generated by the intersections of the manifolds. When p > 2 the tubes of stability, together with a third stability tube that envelopes the torus centerline, are embedded in a chaotic sea that is itself bounded by a KAM torus.

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4 Concluding remarks Our numerical results support Kelvin’s [9] speculation about the existence of toroidal filamentary vortices which are steady and stable. Since Kelvin did not explicitly mentioned it, one qualification needs to be added to his description: the filament must lie on a torus with a small aspect ratio. Indeed, the vortex becomes unstable when the aspect ratio r1 /r0 is only slightly larger than the value used in this paper (0.1). On the other hand, the value of the ratio a/r0 has no influence on the steadiness or stability of the vortices, as long as the thin-filament assumption remains valid. Some changes do occur as a/r0 decreases from 0.05, the value used in this paper, to 0.01: the progression speed U increases and though the stability tubes survive their volumes significantly decrease. Hence, there most certainly exist toroidal filamentary vortices which are steady and stable solutions of the Euler equations. Finding the actual analytical form is, however, still an open problem. The opposite is true for the “localized induction approximation” (LIA). In this case the analytic form of steady solutions has been found [3] and their stability properties have been determined [4]: they are stable when q > p and unstable when p > q. But if these LIA vortices are made to evolve under the Biot-Savart law they exhibit a different quasi-steady motion and are stable for both q > p and p > q [4]. These contradicting results stem from the limited physical validity of LIA. Acknowledgments This work was supported by CONACyT (México) under grant number 90116.

References 1. Briggs, R.: On phosphoric rings. J. Nat. Phil. Chem. Arts 7, 64 (1804) 2. Helmholtz, H.: Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. Journal für die reine und angewandte Mathematik 55, 25–55 (1858). English translation in Philos. Mag. 33, 485–512 (1867) 3. Kida, S.: A vortex filament moving without change of form. J. Fluid Mech. 112, 397–409 (1981) 4. Ricca, R., Samuels, D., Barenghi, C.: Evolution of vortex knots. J. Fluid Mech. 391, 29–44 (1999) 5. Rom-Kedar, V., Leonard, A., Wiggins, S.: An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394 (1990) 6. Saffman, P.: Vortex Dynamics. Cambridge University Press, Cambridge (1995) 7. Thomson, W. (Lord Kelvin): On vortex atoms. Philos. Mag. 34, 15–24 (1867). Also in Mathematical and Physical Papers (MPP), vol. 4, pp. 1–12 8. Thomson, W. (Lord Kelvin): The translatory velocity of a circular vortex ring. Philos. Mag. 33, 511–512 (1867). Also in MPP, vol. 4, pp. 67–68 9. Thomson, W. (Lord Kelvin): Vortex statics. Proc. R. Soc. Edinburgh 9, 59–73 (1875). Also in MPP, vol. 4, pp. 115–128 10. Wiggins, S.: Chaotic Transport in Dynamical Systems. Springer, Berlin (1992)

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Theor. Comput. Fluid Dyn. (2010) 24:195–200 DOI 10.1007/s00162-009-0117-6

O R I G I NA L A RT I C L E

Sébastien Michelin · Stefan G. Llewellyn Smith

Falling cards and flapping flags: understanding fluid–solid interactions using an unsteady point vortex model

Received: 30 October 2008 / Accepted: 17 February 2009 / Published online: 6 June 2009 © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract A reduced-order model for the two-dimensional interaction of a sharp-edged solid body and a highReynolds number flow is presented, based on the inviscid representation of the solid’s wake as point vortices with unsteady intensity. This model is applied to the fall of a rigid card in a fluid and to the flapping instability of a flexible membrane forced by a parallel flow. Keywords Fluid–solid interaction · Point vortex · Vortex shedding PACS 47.15.ki, 47.63.mc

1 Introduction The fluttering and tumbling motions of falling cards or disks in air [3,5,11,18,24,29] and the flapping of flexible membranes in parallel flows [4,10,28,30] are two examples of high Reynolds number (Re) fluid-solid interactions, where the full coupling between the solid and fluid dynamics leads to complex and highly unsteady behaviors. High-Re fluid–solid interactions are also essential in understanding locomotion techniques used by insects and fishes to move in fluids [26,27]. The coupling between fluid and solid equations occurs on moving boundaries whose position is itself the result of the coupled dynamics. To deal with this difficulty in numerical simulations, different techniques have been proposed, including coupled fluid–solid solvers using fitted moving grids [8,22] or immersed boundary methods [31]. The computational cost and complexity of these simulations justify parallel efforts to develop reduced-order models able to capture the relevant physics of the interaction. In most of the applications detailed above, the solid is thin and sharp-edged. At high Re, strong vortices are shed from the corners by separation of the boundary layers. The present work describes an inviscid model for two-dimensional fluid–solid interactions based on a simplified representation of the vortical wake of the solid. The strong unsteadiness of the problem is retained. This approach differs significantly from empirical models [2,25] which are based on generalization of steady results (typically the Kutta-Joukowski lift theorem) and do not contain any unsteady effect other than added inertia. In the following sections, the general point vortex model will be presented and applied to the motion of a rigid sharp-edged body and of a flexible membrane. Communicated by H. Aref S. Michelin (B) · S. G. Llewellyn Smith Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, La Jolla, CA 92093-0411, USA E-mail: [email protected] S. Michelin Ecole Nationale Supérieure des Mines de Paris, 60-62 Boulevard Saint Michel, 75272 Paris Cedex 06, France Reprinted from the journal

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2 Coupled vortex-solid problem We focus here on two-dimensional problems. At sufficiently high Re, the pressure is dominant in the fluid forces acting on the solid and viscosity’s influence is limited to the shedding of vorticity. In an inviscid model and in the absence of vortex shedding, the relative motion of the solid sharp corner and the fluid leads to a singularity in the velocity field at the solid’s edge. To enforce regularity of the flow velocity in the present inviscid model, one point vortex is introduced for each shedding corner. At each time step, the intensity of this vortex is adjusted to cancel exactly the velocity singularity at the corresponding corner, rather than creating a new element of vorticity at each time step as in a continuous vortex sheet approach [1,13,14]. To include the irreversible nature of the vortex sheet roll-up in this model, when a vortex reaches a maximum intensity, its intensity becomes frozen and a new vortex is started from the generating corner [9]. The equation of motion of these unsteady point vortices, the Brown–Michael equation [7], is obtained by enforcing the conservation of momentum around the vortex and the associated branch cut [23]: z˙ n + (z n − z n,0 )

˙ n = w˜ n , n

(1)

where z n and n are, respectively, the vortex position and intensity, z n,0 the position of the shedding corner and w˜ n the desingularized flow velocity at the vortex position. Failing to include the corrective term in ˙ n / n would result in an unbalanced force on the branch cut linking the vortex to its generating corner [23]. Assuming the system is started from rest, the circulation at infinity must remain zero at all time. This condition allows one to solve uniquely for the flow potential given the position and intensity of the vortices as well as the solid velocity and deformation. The fluid–solid problem can be reformulated as a vortex-solid coupled problem by computing the pressure distribution on the solid (or, for a rigid body, the total force and torque applied by the fluid) as an explicit function of the vortex characteristics. 3 Motion of a rigid body: Maxwell’s problem For a simply connected rigid body, the time-independent geometry of the solid can be obtained by conformal mapping of a circle. The outside of the circle of radius a in the mapped ζ -plane is mapped onto the outside of the solid C in the physical z-plane using the mapping z = c + eiθ g(ζ ),

(2)

with c the center of mass position and θ the solid’s orientation. The function g entirely describes the solid geometry, and in particular, corners on C correspond to g  (ζ = aeiφ p ) = 0 on the mapped circle boundary. The complex potential is then obtained in the mapped plane. The regularity condition and (1) can then be written as a system of equations for n and ζn , the vortex position in the mapped plane. Using an unsteady generalization of Blasius’ theorem, the force and torque on the solid are obtained as a sum of complex integrals on the solid boundary, which can be computed exactly in terms of c, θ , n and ζn , as well as their time derivatives. The reader is referred to [20] for more details and for derivation of the full equations. The vortex model is used to study the fall of a thin card of half chord l and mass M per unit length in the third dimension, in a fluid of density ρ. This problem is also known as Maxwell’s problem [19]. Experimental results have reported that above a critical Re value, the broadside-on fall of the card (horizontal position) becomes unstable and either large scale fluttering or tumbling regimes are observed [11,24,29]. In a purely inviscid representation with no vortex shedding, fluid effects are limited to added inertia; from a non-zero initial angle, the solid experiences fluttering oscillations with decreasing amplitude and increasing frequency [6,16,17]. The asymptotic motion is a free fall in a reduced gravitational field and the broadside-on position is stable [16,17]. The introduction of vortex shedding destabilizes this position, and a plate released with a small angle to the horizontal first displays an unstable fluttering motion before flipping and entering rotating regimes (Fig. 1). A similar exponential growth of the plate’s oscillations was also observed in a previous study using a vortex sheet approach [14]. In both the point vortex and the vortex sheet methods, the integration eventually stops when a new element of vorticity must be created at the leading edge under small angle of attack, as such shedding conditions cannot be handled with an inviscid method [13,14]. This limits the validity of the method once the plate enters rotating regimes. We, therefore, concentrate on the application of this method to the initial fluttering phase where the angle of attack is large.

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Fig. 1 Fall of a rigid card with M/2ρl 2 = 0.7 released with an initial angle θ0 = π/2048 with the horizontal. a The position of the plate is plotted every t = 0.4. b Orientation of the plate θ (solid) and horizontal position of the center of mass (dashed). c Intensities of the successive vortices shed by the plate’s corners. d Snapshot of the streamlines for t = 28, shortly before the flip of the plate. Positive (resp. negative) vortices are represented by upward- (resp. downward-) pointing triangles. All quantities have been non-dimensionalized using the half-chord l, the fluid density ρ and gravity g as reference quantities

The low-order representation of the wake using point vortices provides some physical insight on the destabilization process. Several vortices are shed once the amplitude of the plate’s angular motion becomes large. However, during the initial small-amplitude fluttering regime, the wake can be represented to a good approximation as a horizontal pair of vortices of opposite intensities. In the absence of the plate, the momentum of these vortices is vertical and oriented downward. The plate (because of the image vorticity and its tilted orientation) slightly deflects this vortex momentum in the direction opposite to the plate’s lateral velocity. During one half oscillation, horizontal momentum is transferred to the vortices in the opposite direction to the plate motion, thereby creating an accelerating force on the solid and destabilizing the broadside-on position [20].

4 Motion of a flexible body: flapping of a flexible membrane We consider here the motion of a two-dimensional flexible membrane forced by an imposed parallel flow U∞ of density ρ. L is the length of the membrane in the streamwise direction, ρs its mass per unit area and B its bending rigidity per unit length in the third dimension. All quantities are non-dimensionalized using L, U∞ and ρ as reference quantities. Though still valid in theory, conformal mapping techniques become very difficult to use because of the time-dependence of the mapping. To study the flow over an infinitely thin membrane, the solid is instead represented as a bound-vorticity distribution κ(s, t) (with 0 ≤ s ≤ 1 the curvilinear coordinate) solution of a singular integral equation obtained from the normal flow condition on the membrane [1,13,21]: 1 2π

1 0

    1  n ∂ ζ¯ eiθ (s0 ) κds = Im eiθ U∞ + , Re − ζ (s0 ) − ζ (s) 2πi ζ − zn ∂t 

(3)

with ζ (s) the position of the membrane and θ (s) its orientation. The complex flow velocity is obtained as the superposition of U∞ and the contribution from the vorticity (bound-vorticity distribution κ for the solid, discrete vortices (z n , n ) for the wake), and the pressure jump is obtained from Bernoulli’s theorem: ±

s0

[ p] (s0 ) =

κ˙ ds + κu p ,

(4)

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x

Fig. 2 (Left) Critical stability curve of the rest position of the membrane in the (M ∗ , U ∗ )-plane (solid) compared to the linear stability analysis results in [10,15] (dashed). (Right) Shape of the flapping mode observed for (a) M ∗ = 0.5 and U ∗ = 9, (b) M ∗ = 3 and U ∗ = 11, (c) M ∗ = 10 and U ∗ = 11 and (d) M ∗ = 20 and U ∗ = 11. The position of the membrane is plotted every t = 0.08

where u p is the principal value of the relative tangential fluid velocity. The internal solid dynamics is described using a large displacement inextensible Euler–Bernoulli beam model with clamped-free boundary conditions and is solved together with (3)–(4) using a second-order semi-implicit finite difference scheme and Chebyshev spectral methods [21]. √ The problem is characterized by two non-dimensional parameters (μ, η) with μ the inertia ratio and η ∗ the time-scale ratio of the aerodynamic destabilization to the restoring elastic force (or alternatively (M , U ∗ ) with U ∗ the imposed flow non-dimensionalized by the flag properties): μ=

ρs B 1 1 = . , η = ∗ ∗2 = 2 M∗ ρL M U ρ L 3 U∞

(5)

We are interested in the stability of the (straight) state of rest of the flag. The system is started from rest and a small vertical perturbation is added to the parallel flow at infinity. For a given mass ratio, the flag state of rest becomes unstable when the elastic response is too slow compared to the pressure forcing (small η or large U ∗ ). The critical curve Uc (M ∗ ) is obtained as the minimum value of U ∗ above which the flag does not return to its flat position after being perturbed, and compares well with previous linear stability results (see Fig. 2). Above the critical curve and for intermediate values of U ∗ , a flapping regime develops with a structure determined by the most linearly unstable mode, as observed in experimental results [10]. If the flow velocity is increased further, a transition to a chaotic regime is observed. The reader is referred to [21] for more details on the model and flag instability results. The model can be generalized to study the coupled flapping of N membranes. For N = 2, and given M ∗ and U ∗ , in-phase (resp. out-of-phase) flapping modes and wake structures are observed for small (resp. large) separation D/L (Fig. 3) in agreement with experimental observations [12,30]. 5 Conclusions The unsteady point vortex method presented here allows one to reduce significantly the computational complexity and cost of two-dimensional fluid–solid interactions, while still retaining most of the physical effects, in particular the strong unsteadiness of vortex shedding. It allows one to obtain qualitative (destabilization of a falling card by the shed vorticity) and quantitative (stability curve and flapping modes shape of a flexible flag) insights into several important fluid–solid interactions. The method is particularly suited to describe the shedding of vorticity from the solid’s trailing edge or at high angle of attack, as in both cases the vortex wake is advected away from the body. As for the vortex sheet approach [13,14], the point vortex method’s ability to describe leading-edge shedding at small angle of attack is limited, as viscous effects and interaction of the newly shed vortex with the boundary layer on the solid are expected to become dominant. We, therefore, restricted our application of this method to the initial fluttering

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Falling cards and flapping flags

(a)

(b)

2

2

1

1

y

y 0

0

−1

−1

−2

−2 0

1

2

3

4

5

6

7

0

x

1

2

3

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Fig. 3 Instantaneous streamlines in the flapping regime with two identical parallel membranes with M ∗ = 0.5 and U ∗ = 8, positioned at a distance a D = 0.15L and b D = 0.7L. Positive (resp. negative) wake vortices are represented with upward- (resp. downward-) pointing triangles. For small distances, the membranes flap in phase and the wake is similar to the one-membrane problem. For larger distances, the membranes flap in opposition of phase, and the wake consists of vortex pairs arranging on two symmetric diverging tracks

regime of the falling card or flapping membranes shedding vortices from the trailing edge only. In the latter case, the effect of leading-edge shedding is expected to be negligible in a first approximation, as the angle of attack remains small at all time. By representing the wake as a discrete distribution of point vortices, this method provides a theoretical tool to understand the fluid forces as transfer of momentum to the vortex wake. Because of its reduced computational cost (compared to direct numerical simulations or even the vortex sheet approach), this reduced-order model can be of particular interest for problems where the cost of other techniques is prohibitive (e.g. optimization in locomotion studies). Acknowledgements This work was supported by NSF award CTS-0133978 and the Human Frontier Science Program Research Grant RGY 0073/2005. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References 1. Alben, S., Shelley, M.J.: Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos. Phys. Rev. Lett. 100, 074301 (2008) 2. Andersen, A., Pesavento, U., Wang, Z.J.: Analysis of transitions between fluttering, tumbling and steady descent of falling cards. J. Fluid Mech. 541, 91–104 (2005) 3. Andersen, A., Pesavento, U., Wang, Z.J.: Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 65–90 (2005) 4. Balint, T.S., Lucey, A.D.: Instability of a cantilevered flexible plate in viscous channel flow. J. Fluids Struct. 20, 893–912 (2005) 5. Belmonte, A., Eisenberg, H., Moses, E.: From flutter to tumble: inertial drag and Froude similarity in falling paper. Phys. Rev. Lett. 81, 345–348 (1998) 6. Borisov, A.V., Kozlov, V.V., Mamaev, I.S.: Asymptotic stability and associated problems of dynamics of falling rigid body. Reg. Chaotic Dyn. 12, 531–565 (2007) 7. Brown, C.E., Michael, W.H.: Effect of leading edge separation on the lift of a delta wing. J. Aero. Sci. 21, 690–694, 706 (1954) 8. Connell, B.S.H., Yue, D.K.P.: Flapping dynamics of a flag in uniform stream. J. Fluid Mech. 581, 33–67 (2007) 9. Cortelezzi, L., Leonard, A.: Point vortex model of the unsteady separated flow past a semi-infinite plate with transverse motion. Fluid Dyn. Res. 11, 263–295 (1993) 10. Eloy, C., Lagrange, R., Souilliez, C., Schouveiler, L.: Aeroelastic instability of a flexible plate in a uniform flow. J. Fluid Mech. 611, 97–106 (2008) 11. Field, S.B., Klaus, M., Moore, M.G., Nori, F.: Chaotic dynamics of falling disks. Nature 388, 252–254 (1997) 12. Jia, L.B., Li, F., Yin, X.Z., Yin, X.Y.: Coupling modes between two flapping filaments. J. Fluid Mech. 581, 199–220 (2007) 13. Jones, M.A.: The separated flow of an inviscid fluid around a moving plate. J. Fluid Mech. 496, 405–441 (2003) 14. Jones, M.A., Shelley, M.J.: Falling cards. J. Fluid Mech. 540, 393–425 (2005) 15. Kornecki, A., Dowell, E.H., O’Brien, J.: On the aeroelastic instability of two-dimensional panels in uniform incompressible flow. J. Sound Vib. 47, 163–178 (1976) Reprinted from the journal

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16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

Kozlov, V.V.: Heavy rigid body falling in an ideal fluid. Mech. Solids 24, 9–17 (1989) Mahadevan, L.: Tumbling of a falling card. C. R. Acad. Sci. Ser. IIb 323, 729–736 (1996) Mahadevan, L., Ryu, W.S., Samuel, A.D.T.: Tumbling cards. Phys. Fluids 11, 1–3 (1999) Maxwell, J.C.: On a particular case of the decent of a heavy body in a resisting medium. Camb. Dublin Math. J. 9, 145–148 (1854) Michelin, S., Llewellyn Smith, S.G.: An unsteady point vortex method for coupled fluid-solid problems. Theor. Comp. Fluid Dyn. (2009, in press). doi:10.1007/s00162-009-0096-7 Michelin, S., Llewellyn Smith, S.G., Glover, B.J.: Vortex shedding model of a flapping flag. J. Fluid Mech. 617, 1–10 (2008) Pesavento, U., Wang, Z.J.: Falling paper: Navier–Stokes solutions, model of fluid forces, and center of mass elevation. Phys. Rev. Lett. 93, 144501 (2004) Rott, N.: Diffraction of a weak shock with vortex generation. J. Fluid Mech. 1, 111–128 (1956) Smith, E.H.: Autorotating wings: an experimental investigation. J. Fluid Mech. 50, 513–534 (1971) Tanabe, Y., Kaneko, K.: Behavior of a falling paper. Phys. Rev. Lett. 73, 1372–1375 (1994) Triantafyllou, M.S., Triantafyllou, G.S., Yue, D.K.P.: Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32, 33–53 (2000) Wang, Z.J.: Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183–210 (2005) Watanabe, Y., Suzuki, S., Sugihara, M., Sueoka, Y.: An experimental study of paper flutter. J. Fluids Struct. 16, 529–542 (2002) Willmarth, W.W., Hawk, N.E., Harvey, R.L.: Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 197–208 (1964) Zhang, J., Childress, S., Libchaber, A., Shelley, M.: Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature 408, 835–839 (2000) Zhu, L., Peskin, C.: Simulation of flapping flexible filament in a flowing soap film by the immersed boundary method. J. Comp. Phys. 179, 452–468 (2002)

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Theor. Comput. Fluid Dyn. (2010) 24:201–207 DOI 10.1007/s00162-009-0118-5

O R I G I NA L A RT I C L E

Eva Kanso

Swimming in an inviscid fluid

Received: 30 October 2008 / Accepted: 17 February 2009 / Published online: 17 June 2009 © Springer-Verlag 2009

Abstract We present a set of equations governing the motion of a body due to prescribed shape changes in an inviscid, planar fluid with nonzero vorticity. The derived equations, when neglecting vorticity, reduce to the model developed in Kanso et al. (J Nonlinear Sci 15:255–289, 2005) for swimming in potential flow, and are also consistent with the models developed in Borisov et al. (J Math Phys 48:1–9, 2007), Kanso and Oskouei (J Fluid Mech 800:77–94, 2008), Shasikanth et al. (Phys Fluids 14(3):1214–1227, 2002) for a rigid body interacting dynamically with point vortices. The effects of cyclic shape changes and the presence of vorticity on the locomotion of a submerged body are discussed through examples. Keywords Swimming · Dynamics · Vorticity PACS 47, 47.10.Fg, 47.32.C-, 47.63.-b, 47.63.mc

1 Introduction The net locomotion of a deformable body submerged in an infinite volume of fluid depends critically on the dynamic coupling between the body shape deformations and the unsteady motion of the surrounding fluid. A mathematical description of this coupling at finite Reynolds numbers would require taking into account the detailed effects of viscosity which are primarily manifested in the dynamics of the thin shear layers around the body that separate at the body tail to create vortical structures. The classical studies of Wu and Lighthill addressed this problem in two different ways. Wu considered an idealized model of a deformable plate swimming in an inviscid fluid and used the assumption of small shape amplitudes which enables one to solve for the trailing vortex sheet analytically and investigate the problem of optimum shape deformations in the sense of minimizing the energy lost in creating the trailing wake [20,21]. Lighthill, on the other hand, studied the swimming of a slender body due to large amplitude deformations and avoided solving for the complex wake dynamics by considering the momentum balance in a control volume containing the deformable body and bounded by a plane attached at its trailing edge [10]. In this note, the classical balances of linear and angular momenta of the solid–fluid system are used to obtain the equations governing the swimming of a deformable body in response to prescribed (actively controlled) shape deformations and the effect of vorticity [6]. The underlying balance of momenta, though classical in nature, provide a unifying framework for the swimming of planar and three-dimensional bodies, therefore bridging the gap between the models developed by Wu and Lighthill, and they may hold even in the presence of viscosity. When neglecting vorticity, the derived equations reduce to a known model for the locomotion Communicated by H. Aref E. Kanso University of Southern California, Los Angeles, CA, USA E-mail: [email protected] Reprinted from the journal

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Fig. 1 The centerline’s deformations and those of boundary of the body are prescribed a priori as functions of time t in the bodyfixed frame (b1 , b2 ). The net displacement (xo , yo ) and rotation β with respect to the inertial frame (e1 , e2 ) are then computed using the governing equations of motion

of an articulated body in potential flow [4]. The equations are also consistent with the models developed in [2,5,15] for the case of the dynamics of a rigid body in a fluid with point vortices. We examine the locomotion of a planar submerged body through three examples. The first example studies the effect of cyclic shape deformations, both oscillatory and undulatory, on the locomotion in potential flow [4,6]. This example suggests that the ‘reactive’ forces (proportional to the added mass) play a central role in the locomotion due to transverse shape deformations, contrary to a common belief that the forces due to shed vorticity are solely responsible for locomotion. The second example examines the effect of the ambient vorticity, modeled using point vortices, on the net locomotion of a rigid body [5]. This example is inspired by Beal et al. [1] where dead trout was reported to recover sufficient energy from the surrounding flow to allow it to passively swim upstream. Indeed, we identify configurations where even a rigid body (no elasticity) can swim passively in the direction opposite to the motion of vortices at no energy cost. The last example is concerned with the locomotion of a deformable body due to a self-generated wake vorticity modeled using pairs of point vortices shed periodically from the tail of the deformable body [6]. 2 Equations of motion Consider a planar deformable body B immersed in an infinitely large volume of fluid F , see Fig. 1. Let the body undergo prescribed planar shape deformations, not necessarily small, but satisfying an inextensibility condition (the length of the centerline or spinal column remains constant) and an area-preservation condition (the total area enclosed by the boundary remains constant). The net locomotion of the body and its shape deformations can be expressed with respect to an orthonormal inertial frame {e1,2,3 } where {e1 , e2 } span the plane of deformations and e3 is the unit normal to this plane. The net locomotion of the body B is identified with a rotation β about e3 and a translation xo (= xo e1 + yo e2 ) in the {e1 , e2 } directions, say of the geometric center of B. It is convenient to introduce a body-fixed frame {b1,2,3 } where b3 ≡ e3 , which is a frame undergoing the same rigid rotation β as the body and whose origin is placed at the geometric center xo . The angular and translational velocities can be expressed in the fixed inertial frame as β˙ e3 and v = x˙o e1 + y˙o e2 ˙ (the dot is used to denote ∂/∂t). Alternatively, one could write in the body frame  =  b3 , where  = β, and V = V1 b1 + V2 b2 , where V1 = x˙o cos β + y˙o sin β and V2 = −x˙o sin β + y˙o cos β. Similarly, the velocity associated with the shape deformations can be expressed in inertial frame as vshape or in body-fixed frame as Vshape (where Vshape and vshape are related by the rigid rotation β). The equations governing the net locomotion are derived based on the following assumptions: (i) there are no external forces or moments applied on the solid–fluid system; and (ii) the body is neutrally buoyant and the fluid density is normalized to unity. The circulation around the body need not be necessarily zero as long as the circulation is zero at infinity (fluid at rest at infinity). The balance laws of linear and angular momenta (or impulse) of the body-fluid system read as (see [18]), ⎡ ⎤   d ⎣ mv + x × (n × u) ds + x × ω da ⎦ = 0, dt ⎡

∂B

d ⎣ 1 ˙ 3− J βe dt 2

123

F

 x2 (n × u) ds − ∂B

218

1 2



⎤

(1)

x2 ω da ⎦ = 0,

F

Reprinted from the journal

Swimming in an inviscid fluid

where m and J are the bodies’ mass and moment of inertia, respectively, n is the normal unit vector to the boundary ∂ B pointing into the fluid, x denotes the position vector with respect to the inertial frame, u denotes the spatial velocity field of the fluid, ω = ∇ × u is the vorticity of the fluid, da and ds are standard area and line elements. These balances of momenta are valid for planar and three-dimensional flows and may be generalized to hold even in the presence of viscosity. In unbounded two-dimensional flows, the balance of linear momentum remains true in the presence of viscosity while that of angular momentum is true only when there is no net circulatory flow at infinity (see [18], Sections 3.6, 3.10). In this note, we consider the case of an inviscid incompressible fluid undergoing only planar flows. It turns out that it is more convenient to study the dynamics in body-fixed frame. It is easier to prescribe the shape deformations using the body-fixed coordinates X. Also, when expressed in body-frame, (1) can be shown to have a familiar form, namely, the same form as Kirchhoff’s equations for the motion of a submerged rigid body (see e.g. [6]). P˙ = P × ,

˙ = P × V, 

(2)

where P = Ploc + Pshape + Pw and  = loc + shape + w are the total linear and angular momenta expressed in body-fixed frame. The momenta Ploc and loc denote the linear and angular momenta associated with the net locomotion  Ploc = mV +

˙ β + V1 ϕ1 + V2 ϕ2 )nds, (βϕ

loc

˙ 3+ = J βb

∂B



˙ β + V1 ϕ1 + V2 ϕ2 )X × nds, (βϕ

(3)

∂B

while Pshape and shape are the linear and angular momenta imparted to fluid by the prescribed shape deformations,  Pshape =

 ϕshape nds, shape =

∂B

ϕshape X × nds,

(4)

∂B

and Pw and w denote the linear and angular momenta associated with the presence of vorticity,  Pw =

 X × ωda +

F

∂B

1 X × (n × uw ) ds, w = − 2



1 X ωda − 2

 X2 (n × uw ) ds.

2

F

(5)

∂B

In (3, 4), ϕ− are solutions to Laplace’s equation subject to  ∂ϕβ  = X × n · b3 , ∂n ∂ B

 ∂ϕ1  = n · b1 , ∂n ∂ B

 ∂ϕ2  = n · b2 , ∂n ∂ B

 ∂ϕshape  = Vshape · n ∂n ∂ B

(6)

and zero velocity at infinity. In (5), uw is the vortical component of the fluid velocity with uw · n|∂ B = 0. It is important to emphasize that in (3–6), the vectors n, ω, and uw are expressed in body frame (i.e., rotated by the angle β). To close the model in (2), one needs to compute Pw and w , which represent the contribution of vortex shedding and vorticity to the momentum balance. This closure can be obtained in a variety of ways the simplest of which is to consider an irrotational (potential) fluid model. The effect of the fluid is then completely encoded using the added mass effect and the terms Pw and w are set to zero. This case was studied in Kanso et al. [4] for a system of articulated rigid bodies using Hamilton’s variational principle and a Lagrangian function equal to the kinetic energy of the solid–fluid system. Another approach would be to model the wake using discrete vortex structures as done in [2,5,15] for a rigid body interacting with point vortices. Indeed, equations (2) are consistent with those derived in [2,5,15] where Pshape = shape = 0 and an additional set of equations governing the dynamic behavior of the point vortices is obtained. Finally, note that Pw and w can be computed by coupling (2) to a numerical solver based on vortex shedding models such as those presented in [3,17]. Reprinted from the journal

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3 Examples of locomotion due to shape changes and the effect of vorticity 3.1 Deformable body in potential flow Assume that the shape deformations do not generate vorticity in the fluid such that, starting with zero vorticity, the flow remains irrotational for all time and Pw = w = 0. This assumption is physically meaningful only when the actual flow does not separate at the tail, i.e., when the velocity of the fluid at the trailing edge is tangent to the spatial trajectory described by the trailing edge. Two types of shape deformations are considered: oscillatory and undulatory, see Fig. 2 (left). A deformation is imposed first on the inextensible centerline, whose length is normalized to unity. The oscillatory deformations are described by the relative angles θ1 and θ2 associated with two joints placed at 1/2 and 3/4 from the head. The undulatory deformations are described by a traveling wave along the inextensible centerline of the form Ycenterline = A cos(ωt − κ X centerline ). The boundary points are then obtained using the orthogonality assumption: normal cross-sections to the centerline remain normal at all time, i.e., the boundary points are specified along the (rotated) normal directions at an offset corresponding to the thickness of the body at that point. In Fig. 2, the deformable body is assumed to start from rest (zero momentum). The velocity potentials ϕ− are computed at each time step using a panel method (see [7]). A standard fourth-order Runge–Kutta scheme is used to integrate (2) for both the oscillatory and undulatory cases. The obtained trajectories are depicted in Fig. 2 (center and right). Although the undulatory deformations appear to be more efficient in the sense that the body travels almost twice the distance traveled by the oscillating body, it is far less efficient in terms of energy required to maintain these undulatory deformations. Indeed, the time average of the energy input by the shape deformations (Tshape = 21 ∂ B ϕshape (∂ϕshape /∂n)ds) is almost four times more for the undulatory case. Swimming in potential flow can be interpreted as swimming due to ‘reactive forces’ (proportional to the added mass) which is also discussed in Lighthill’s slender body theory, see [10], where it is argued that, for fish moving at large Reynolds numbers, the added or virtual mass of fluid which acquires momentum through shape changes of the animal far exceeds the ‘resistive’ forces due to boundary layer and vortex shedding. Finally, note that swimming in potential flow can be thought of in terms of gauge-theoretic methods of geometric mechanics and optimal control, as in the ‘falling cat’ problem (see [12]). Gauge-theoretic methods were pioneered by Shapere and Wilczek [14] for swimming at low Reynolds numbers and have only been applied sparingly since, see, e.g., [4,8,11,13].

3.2 Rigid body interacting with point vortices Beal et al. [1] showed that dead trout can recover sufficient energy from the surrounding flow to allow it to passively swim upstream. Inspired by this observation, we propose a simple model to emulate the motion of a body in an externally generated vortex street. The model consists of a rigid body (no shape deformations, Pshape = shape = 0) interacting dynamically with surrounding point vortices where new vortex pairs are introduced in the flow at the leading end of the ellipse and old vortex pairs are removed from the flow as they move far from the ellipse. The introduction of vortex pairs emulates the vortices shed periodically by a source external to the body (which would be in a staggered configuration in contrast to the case considered in Fig. 3 where the vortices are introduced in symmetric pairs for simplicity). The removal of vorticity emulates 0.5

1 0.5

−0.5

0 −0.5

−1.5 −8

−6

−4

−2

0

−1

0

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Fig. 2 Top and bottom left oscillatory and undulating shape deformations. The stretched-out length is normalized to unity. The thickness at the head is chosen to be 0.03 and it decays linearly towards the tail at a slope of 0.0175. Center and right the trajectories shown in solid lines correspond to an oscillatory deformation given by θ1 = 0.5 cos(2πt) and θ2 = 0.5 cos(2πt − π/4) while those in dashed lines correspond to an undulatory deformation given by Ycenterline = 0.1 cos(2π(t − X centerline /1.5)). The integration time is 10

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5

0.4 0.3

0

0.2 0.1 0 0

50

100 150 200 250

−5 −5

0

5

10

15

20

25

30

35

40

45

Fig. 3 The motion of the elliptic cylinder due to an external vortex street. At any given time t, the ellipse is assumed to interact dynamically with three neighboring vortex pairs: a vortex pair is introduced periodically at the leading end of the ellipse while another is removed as it moves far away from the trailing end of the ellipse. The parameter values are set at a = 3, b = 1 and = ±10. The point vortices are initially located at (X 1,2 , Y1,2 ) = (20, ±4.5), (X 3,4 , Y3,4 ) = (6, ±3.7) and (X 5,6 , Y5,6 ) = (−10.5, ±3.6), while the ellipse is initially placed at (0, 0) with zero initial velocity and zero orientation. The total integration time is 250. The ellipse acquires a nonnegative velocity (shown on the left) which causes it to move in the positive e1 -direction while the vortex pairs move in the opposite direction. A snapshot of the (x, y) plane together with the instantaneous flow streamlines are shown on the right

the diminishing effect of the vortices far from the body. For concreteness, let the rigid body have an elliptic cross-sectional area with major and minor axes a and b respectively and, at a given time, be interacting

n dynamically with N point vortices of strength k (k = 1, . . . , N ) such

that the sum of their strength k=1 k is zero. That is, the vorticity ω is given by a delta distribution ω = k δ(X − Xk )b3 , where Xk = (X k , Yk ) denote the position of the vortices in body frame. The potential and stream functions can be obtained by conformally transforming the domain bounded internally by the elliptic body into that bounded by the unit circle. Consequently, a Kirchhoff–Routh function W (Xk ) can be constructed such that the velocity of the kth vortex is given by (see [5] for details): ˙ k + V +  × Xk ) = u|X = ∂ W b1 − ∂ W b2 .

k (X k ∂Yk ∂ Xk

(7)

Equations (2) and (7) form a closed system for the set of 2N + 3 unknowns P,  and Xk . Figure 3 shows that, by properly choosing the initial location of the vortex pairs and their strength, the body swims passively in the direction opposite to the motion of the vortex pairs. Here, the ellipse itself spends no energy (it does not generate any force) and its motion is due entirely to energy exploited from the presence of the point vortices. 3.3 Deformable body shedding vortex pairs We propose a simple model to emulate the motion of a deformable body under the effect of its own vortex wake. The proposed model is not intended to study the details of vortex shedding (which is a complicated event involving flow separation at the tail) but focuses on the effects of the shed vortices on the net locomotion. The model consists of a body allowed to undergo shape deformations such that a vortex pair (a dipole) is ‘shed’ from the tail (i.e., introduced into the flow) each time the tail reaches its minimum and maximum flapping angle as shown in Fig. 4 (left). The dipole model (consisting of two vortices of equal and opposite strengths placed a finite distance apart) is chosen for two reasons: (i) there is experimental evidence that fish and flapping airfoils shed a dipole every half cycle of their oscillation (see e.g. [19]) and (ii) it guaranties that no circulation is introduced into the fluid domain (thus, satisfying Kelvin’s circulation theorem). We assume that the total momentum of the body-fluid system remains constant during vortex shedding, which takes place instantaneously when the tail reaches its maximum flapping amplitude, that is, at the shedding time tshedding , one has P|t +

shedding

= P|t −

shedding

,

|t +

shedding

= |t −

shedding

.

(8)

Between two consecutive shedding events, the motion of the body and the point vortices is governed by (2) and (7). We integrate these equations in time while periodically introducing new vortex pairs in the flow at the tail and removing old vortex pairs as they move away from the tail. In order to isolate the effect of the vortex shedding on the net locomotion, we consider shape deformations where only the tail is allowed to flap such that no net displacement can occur in the absence of vortex shedding. This is due to the reversibility of motion Reprinted from the journal

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0.5 0 −0.5 −1 −7

−6

−5

−4

−3

−2

−1

0

1

Fig. 4 Left a vortex pair (a dipole) is shed periodically when the tail reaches its maximum and minimum flapping angle. The stretched-out length is normalized to unity. The thickness at the head is chosen to be 0.03 and it decays linearly towards the tail at a slope of 0.0175. The length of the tail is set to 1/4 and its oscillatory motion is given by θ = 0.5 cos(2πt). Right the motion of a deformable body due to shed vortex pairs. The dipole strength is set to = ±1.5. The total integration time is 3.5

in potential flow when reversing the boundary conditions, that is, time reversibility of the fluid model. This reversibility property is broken when vorticity is introduced. Figure 4 (right) is a depiction of the trajectory of the body in inertial frame together with the trajectories traced by the vortex pairs from the time they are shed or introduced into the flow to the time when they are removed from the flow. Figure 4 shows that, for the used set of parameters, the body undergoes a swimming motion similar to that observed in biological fish. The shown behavior changes depending on the parameter values, especially the strength of the vortices (results not shown here). The vortices could be too weak to move away from the body or too strong to allow the body to swim away (the body gets absorbed by the vortices). Biological fish are also able to manipulate the intensity of the shed vortices up to a limit dictated by their muscles strength.

4 Summary A set of equations governing the free swimming of a deformable body due to prescribed shape changes and the effect of vorticity is presented. These equations are used to discuss three examples of: (i) locomotion in potential flow due to ‘reactive forces’ (the added mass effect), (ii) passive locomotion due to ambient vorticity and (iii) locomotion due to self-generated vorticity. Future extensions will include generalizing these methods to study the coupled dynamics of multiple submerged bodies and their wakes in an effort to model fish schooling. Acknowledgments The author would like to thank Prof. Paul Newton and her graduate student Babak Oskouei for their valuable input. This work is partially supported by NSF through the award CMMI 06-44925.

References 1. Beal, D.N., Hover, F.S., Triantafyllou, M.S., Liao, J.C., Lauder, G.V.: Passive propulsion in vortex wakes. J. Fluid Mech. 549, 385–402 (2006) 2. Borisov, A.V., Mamaev, I.S., Ramodanov, S.M.: Dynamic interaction of point vortices and a two-dimensional cylinder. J. Math. Phys. 48, 1–9 (2007) 3. Jones, M.A.: The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405–441 (2003) 4. Kanso, E., Marsden, J.E., Rowley, C.W., Melli-Huber, J.: Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15, 255–289 (2005) 5. Kanso, E., Oskouei, B.: Stability of a coupled body-vortex system. J. Fluid Mech. 800, 77–94 (2008) 6. Kanso, E.: Swimming due to transverse shape deformations. J. Fluid Mech. (2009, in press) 7. Katz, J., Plotkin, A.: Low-Speed Aerodynamics. Cambridge Aerospace Series, Cambridge (2001) 8. Kelly, S.D.: The mechanics and control of robotic locomotion with applications to aquatic vehicles. Ph.D. thesis, California Institute of Technology (1998) 9. Lamb, H.: Hydrodynamics. Dover, New York (1932) 10. Lighthill, J.: Mathematical Biofluiddynamics. Society for Industrial and Applied Mathematics. Philadelphia (1975) 11. Miloh, T., Galper, A.: Self-propulsion of general deformable shapes in a perfect fluid. Proc. R. Soc. Lond. A 442, 273–299 (1993) 12. Montgomery, R.: Isoholonomic problems and some applications. Commun. Math. Phys. 128, 565–592 (1990) 13. Radford, J.: Symmetry, Reduction and swimming in a perfect fluid. Ph.D. thesis, California Institute of Technology (2003) 14. Shapere, Wilczek: Self-propulsion at low Reynolds number. Phys. Rev. Lett. 58(2), 2051–2054 (1987) 15. Shashikanth, B.N., Marsden, J.E., Burdick, J.W., Kelly, S.D.: The Hamiltonian structure of a 2D rigid circular cylinder interacting dynamically with N Point vortices. Phys. Fluids 14(3), 1214–1227 (2002) (see also Erratum, Phys. Fluids 14(11), 4099)

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16. Shashikanth, B.N.: Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and N point vortices: the case of arbitrary smooth cylinder shapes. Reg. Chaos. Dyn. 10(1), 110 (2005) 17. Shukla, R.K., Eldredge, J.E.: An inviscid model for vortex shedding from a deforming body. Theor. Comput. Fluid Dyn. 21(5), 343–368 (2007) 18. Saffman, P.G.: Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics (1992) 19. Tytell, E.D., Lauder, G.V.: The hydrodynamics of eel swimming: I. Wake structure. J. Exp. Biol. 207, 1825–1841 (2004) 20. Wu, T.: Hydrodynamics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid. J. Fluid Mech. 46(2), 337–355 (1971) 21. Wu, T.: Hydrodynamics of swimming propulsion. Part 2. Some optimum shape problems. J. Fluid Mech. 46(3), 521–544 (1971)

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Theor. Comput. Fluid Dyn. (2010) 24:209–215 DOI 10.1007/s00162-009-0137-2

O R I G I NA L A RT I C L E

Dmitry Kolomenskiy · Henry Keith Moffatt · Marie Farge · Kai Schneider

Vorticity generation during the clap–fling–sweep of some hovering insects

Received: 13 November 2008 / Accepted: 2 July 2009 / Published online: 31 July 2009 © Springer-Verlag 2009

Abstract Numerical simulations of the Lighthill–Weis-Fogh mechanism are performed using a Fourier pseudo-spectral method with volume penalization. Single-winged and double-winged configurations are compared, and the vortex shedding patterns are related to the lift generated in both cases. The computations of the lift coefficient are validated against the results reported previously by Miller and Peskin (J Exp Biol 208:195–212, 2005). Keywords Lighthill–Weis-Fogh mechanism · Insect flight · Vortex flows · Volume penalization method PACS 47.63.-b, 47.63.M-

1 Introduction The Lighthill–Weis-Fogh clap–fling–sweep mechanism is a movement of wings used by some insects to improve their flight performance. In 1973 Weis-Fogh [1] studied the flight of different hovering insects, such as some species of moths, flies or wasps, in particular the Chalcid wasp Encarsia formosa whose wing span is less than 2 mm. It has two pairs of wings, with a wing chord of about 0.2 mm, which move as a single unit. Weis-Fogh showed that the observed lift coefficient is much too high to be compatible with steady-state aerodynamics and, by taking movies at frequency 7150 s−1 , he decomposed each downstroke, whose frequency is about 400 s−1 , into three phases: the wings clap at the end of upstroke, ‘fling open’ like a book, then separate and sweep horizontally until the end of downstroke. Although the motion is three-dimensional, Lighthill showed the same year [2] that the lift generation can be explained using only two-dimensional inviscid fluid dynamics. Communicated by H. Aref D. Kolomenskiy (B) M2P2-CNRS, Universités d’Aix-Marseille, 38 rue Joliot-Curie, 13451 Marseille Cedex 20, France E-mail: [email protected] H. K. Moffatt Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK E-mail: [email protected] M. Farge LMD-IPSL-CNRS, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 5, France E-mail: [email protected] K. Schneider M2P2-CNRS and CMI, Universités d’Aix-Marseille, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France E-mail: [email protected] Reprinted from the journal

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We propose to consider Lighthill’s two-dimensional model, but using instead viscous fluid dynamics to study how such an unsteady motion generates vorticity, circulation, and lift on the wings. Numerical solution of the Navier–Stokes equations gives us a an insight into both the vortex dynamics and the lift generation. In addition, we make a comparison of our results with those obtained by Miller and Peskin [3] using a different numerical method. 2 Fluid–structure interaction model The fluid–structure interaction model used in this work is an extension of the one described in [4,5] to the case of two symmetrically moving wings. We consider solid obstacles, e.g., wings, placed in a viscous incompressible fluid. The motion of the fluid is governed by the Navier–Stokes equations, completed with a no-slip condition at the solid–fluid interface and a suitable initial condition. The flow is two-dimensional, therefore we use the vorticity-stream function formulation of the equations. The no-slip boundary condition is modelled using the volume penalization method [6]. The wings are assumed to be slightly permeable, and the flow is governed by the penalized equation   χ ∂t ωη + uη · ∇ωη − ν∇ 2 ωη + ∇ × (1) (uη − us ) = 0, η where the penalization parameter η is a small number which controls the permeability of the obstacle. The equation is written for dimensionless parameters and variables. Units are chosen in order to set the dimensionless density ρ = 1. The vorticity ωη is unknown. The velocity is determined as a sum uη = ∇ ⊥  + U∞ , with U∞ being the free-stream velocity and  being the stream function, satisfying ∇ 2  = ωη ,

(2)

where ∇ ⊥  = (−∂ y , ∂x ) stands for the orthogonal gradient of the stream function. The parameter ν is the kinematic viscosity. Equations (1) and (2) are valid in a computational domain A which incorporates both the solid wings As and the fluid A f (see Fig. 1). Geometry and kinematics of the wings are given by the mask function  1 for x ∈ As χ= (3) 0 for x ∈ A f and the velocities of the wings us (x, t) at each point. The penalized flow field converges towards the no-slip boundary flow field when the penalization parameter η tends to zero [6]. For the spatial discretization of (1) and (2) we use a classical Fourier pseudo-spectral method in a periodic domain A [7]. The vorticity field is dealiased at each time step using the 2/3 rule. The solution is advanced in

Fig. 1 Sketch of the flapping wing configuration

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time with an adaptive second-order Adams–Bashforth scheme for the non-linear term, while the viscous term is integrated exactly. An attractive feature of the volume penalization method is that it offers the possibility of moving solid obstacles while keeping unchanged the computational mesh. This motion implies changing in time the mask function χ (x, t). Translation of the obstacle in each direction is implemented by turning the phase of the corresponding Fourier coefficients, and rotation is decomposed into three skewing operations [5]. The volume penalization method also enables us to compute the fluid forces acting on the wing via volume integration, which is convenient for numerical implementation:  χ F= (4) (uη − us )d A + Vc u˙ c , η A

where Vc is the volume of the wing, and u˙ c is the acceleration of its center of gravity (assuming that its density is uniform).

3 Numerical experiment 3.1 Physical and numerical parameters Two configurations are considered: double-winged and single-winged. The single-winged configuration consists of a rectangular wing of chord c = 1 and thickness h = 1/32. The wing is hinged at its lower-right corner (x0 , y0 ), as shown in Fig. 1. We decompose the flapping motion into three phases. Clap: at time t = 0 the wing is arranged vertically. Fling: it starts rotating with a linearly increasing angular velocity z . At time t = 0.1 the angular velocity mounts to its maximum z = max = 4 and then starts decreasing as z = max exp(−15.5[t − 0.1]2 ). Sweep: at t = 0.275, when the angle of incidence mounts to approximately α = 57◦ , the wing starts translating in the negative-x direction. The velocity is changing as Vx = Vmax (exp(−100[t − 0.275]2 ) − 1), and we consider two cases: Vmax = 1 and Vmax = 4. The smooth transient between fling and sweep is introduced to avoid high accelerations and associated added mass effects. Figure 2 displays these kinematics. It shows x—the horizontal coordinate of the hinge, Vx —horizontal component of its velocity, α—angle of the wing, measured between the vertical and the chord, and z —its angular velocity. The double-winged configuration has its left wing identical to the single-winged configuration, and its right wing symmetric with respect to the vertical. The hinge points coincide during the fling phase, such that initially there is no gap between the wings. The surrounding fluid is initially at rest. Its dimensionless density is ρ = 1 and its dimensionless kinematic viscosity is ν = 0.05. These parameters yield dimensionless numbers which are in agreement with what can be observed in nature. Thus, the Reynolds number is Refling = 80, when based on the chord and the maximum

Fig. 2 Kinematics of the flapping motion: horizontal position of the hinge (top-left), horizontal velocity (top-right), angle of incidence (bottom-left), and angular velocity (bottom-right) of the left wing Reprinted from the journal

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tip-speed during fling, and Resweep = 20 or 80, when based on the chord and the translation speed Vmax during sweep. The periodic domain is A = [0, 5] × [0, 5]. It is discretized with N x × N y = 1024 × 1024 grid points, which yields 204.8 points per chord and 6.4 points per thickness. The edge of the wing is smoothed with a transitional ‘porous’ layer covering two points inside and two points outside, which models the hairs covering Encarsia formosa. The penalization parameter is η = 5 · 10−4 . 3.2 Results Figure 3 (left) shows the vorticity plots of the double-winged configuration at three successive time instants. We observe that strong vortices are formed at the tips of the wings (leading edges), reflecting the fact that the air rushes in the opening. But during sweep these vortices form a pair which remains localized between the wings, generating a downward jet which is not observed for the single wing, see Fig. 3 (right). Another important feature is that the trailing-edge vortices, formed when the wings separate, are of the same sign as the leading-edge vortices, in contrast to [3], which followed a slightly different scenario. This increases the downward air flow through the opening gap, supporting Lighthill’s idea of high lift generation, and showing how the circulation persists during the sweep phase. Figure 3 (right) presents vorticity plots for the single wing following the same protocol. Comparison with Fig. 3 (left) shows the importance of the topology change involved in sweep. Overall, the single-winged configuration exhibits a typical unsteady airfoil flow, which is radically different from the flow generated by the clap–fling–sweep mechanism where topology is changing. Figure 4 shows the time evolution of the lift coefficient produced by both configurations, c L = 2FL /ρ 2 c3 , where FL is the lift force. Only the force acting on the left wing is shown for the double-winged configuration. The latter creates at least double the lift per wing during the fling motion. When the wing stops rotating the

Fig. 3 Vorticity plots of the double-winged (left) and the single-winged (right) configurations, combined with vector plots of the velocity. Snapshots correspond to the sweep motion at times t = 0.32 (top), 0.42 (middle) and 0.52 (bottom)

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Fig. 4 Lift coefficient per wing of the double-winged and the single-winged configurations

Fig. 5 Flow in the neighbourhood of the trailing edges of the double-winged (left) and the single-winged (right) configurations at t = 0.52

lift drops rapidly for both configurations, which is an added mass effect. The lift recovers after the transition and becomes higher again for the double-winged configuration than for the single-winged. At this stage the difference of the lift coefficient c L is approximately proportional to ( 2 − 1 )V , where 1 is the circulation around the wing and its free vortices for the single-winged configuration, 2 is the circulation corresponding to one wing of the double-winged configuration, and V is the velocity of the hinge. The positive c L supports the idea that the high circulation, created during the fling of the two wings, survives after the change of topology and results in an instantaneous lift. The strength and the sign of the separated vortices are, indeed, crucial for the lift generation at the beginning of sweep (cf. [2,8]). The leading edge vortices, when they are near to the wing surface, increase the circulation and the lift. In case of the single-winged configuration this is counteracted by the upward flow past the trailing edge (see Fig. 5, which shows a zoom on this region). The situation is different for the double-winged configuration, since the flow field in the neighbourhood of the hinge is due to the action of two mechanisms: (1) a downward jet of the two leading edge vortices, and (2) local low Reynolds number effects, since Relocal = (r/c)2 Refling is tending to zero in the vicinity of the hinge, i.e. for vanishing radii r . The straight iso-vorticity lines radiating from the hinge point, which can be observed in Fig. 3 (left-top), are characteristic for the low Re flow. Therefore local solutions of the Stokes equation derived in [9] are applicable. Both of the above mechanisms result in a negative pressure jump across the hinge, which acts to suppress the possible upflow through the opening between the wings during the initiation of sweep. A more precise analysis of this phenomenon, and especially the low Re effects, will be reported in another paper. Reprinted from the journal

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Fig. 6 Comparison of the lift coefficient obtained in the present simulations (solid and dashed lines) with the corresponding curve in [3] (dash-dot). The spacing between the wing centerlines equals c/6 during the ‘near’ clap–fling. During the ‘full’ clap–fling there is no spacing between the wings

4 Comparison with the results of Miller and Peskin Numerical simulations of the Lighthill–Weis-Fogh mechanism were recently performed by Miller and Peskin [3], who studied influence of the Reynolds number on the lift coefficient. They also compared the single and the double wings. However, they actually considered a so-called ‘near’ clap–fling, since the wings remained separated by a distance of 1/6 chord length (possibly due to restrictions of the numerical method). Therefore the downward jet between the separating wings is not present in their simulations. They also report a smaller increase in lift, compared to our result shown in Sect. 3. For the purpose of validation we have applied the fluid–structure interaction code, presented in Sect. 2, to a similar configuration at Re = 128, with the same kinematic parameters as described in [3]. The wings are rectangular, their thickness to chord ratio is 1/32. Unlike in our previous simulations, the hinges are situated at the centerlines of the wings, and not at their corners. This is more consistent with the model of Miller and Peskin, which is based on the immersed boundary method. The periodic domain size is L x × L y = 10 × 10, it is discretized with N x × N y = 2048 × 2048 grid points. The permeability parameter is η = 5 · 10−4 . Five point smoothing of the mask function is applied. The result of this simulation is shown in a solid line in Fig. 6. The lift coefficient is calculated as c L = 2FL /ρV 2 c, where V = 1.108 max c, max is the maximum angular velocity during fling. The reference curve is shown in dash-dot. Time is shown as a fraction of the stroke τ/τmax . Both simulations predict qualitatively the same evolution of the lift coefficient. The discrepancy is of order 10% or less. It can be explained as follows: in [3] theoretically the wings are thin, and the gap (c/6) between them is the gap between their centerlines; however in practice the regularized delta function has a support of several grid cells, and the effective gap between the wings becomes smaller, yielding higher lift. To check the influence of spacing between the wings, or in other words, the difference between the ‘near’ and the ‘full’ clap–fling, we also performed a computation with initially zero spacing between the hinge points. The result is shown in a dashed line in Fig. 6, and it displays 15% higher lift peaks. Note the significant increase in lift during the initial portion of sweep, which is due to stronger vorticity generation in the preceding fling motion. However, the lift drops more rapidly as the wings move further apart. 5 Conclusions A numerical method has been developed for simulations of solid obstacles moving through a viscous incompressible fluid. It is reasonably efficient, easy to implement and to use even when the topology of the obstacles is changing in time, as in the case of the clap–fling–sweep mechanism where the two wings break apart. The model has been validated against the results reported in [3]. The Lighthill–Weis-Fogh clap–fling–sweep mechanism has been studied. Our numerical simulations confirm that the clap–fling–sweep mechanism enhances the lift, as a consequence of higher circulation round each wing. For the latter, flow in the neighbourhood of the hinge point is of major importance, and its more detailed analysis is envisaged.

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Acknowledgments DK, KS and MF acknowledge Trinity College, Cambridge, for hospitality. DK thanks Tony Maxworthy and Jane Wang for useful discussions. Some of the numerical simulations presented in this paper were conducted using NEC SX-8 computer at IDRIS. DK and KS thank the Deutsch-Französische Hochschule, project ‘S-GRK-ED-04-05’, for financial support.

References 1. Weis-Fogh, T.: Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol. 59, 169–230 (1973) 2. Lighthill, M.J.: On the Weis-Fogh mechanism of lift generation. J. Fluid Mech. 60(1), 1–17 (1973) 3. Miller, L.A., Peskin, C.S.: A computational fluid dynamics of ‘clap and fling’ in the smallest insects. J. Exp. Biol. 208, 195–212 (2005) 4. Schneider, K., Farge, M.: Numerical simulation of the transient flow behaviour in tube bundles using a volume penalization method. J. Fluids Struct. 20, 555–566 (2005) 5. Kolomenskiy, D., Schneider, K.: A Fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles. J. Comput. Phys. 228, 5687–5709 (2009) 6. Angot, P., Bruneau, C.H., Fabrie, P.: A penalisation method to take into account obstacles in viscous flows. Numer. Math. 81, 497–520 (1999) 7. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988) 8. Maxworthy, T.: Experiments on the Weis-Fogh mechanism of lift generation by insects in hovering flight. Part 1. Dynamics of the ‘fling’. J. Fluid Mech. 93(1), 47–63 (1979) 9. Moffatt, H.K.: Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18(1), 1–18 (1963)

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Theor. Comput. Fluid Dyn. (2010) 24:217–239 DOI 10.1007/s00162-009-0129-2

O R I G I NA L A RT I C L E

Pierre-Henri Chavanis

Kinetic theory of stellar systems, two-dimensional vortices and HMF model

Received: 7 November 2008 / Accepted: 24 June 2009 / Published online: 11 August 2009 © Springer-Verlag 2009

Abstract We discuss the kinetic theories of stellar systems, two-dimensional vortices and Hamiltonian mean field model, stressing their analogies and differences. We describe the evolution of the system as a whole and discuss the timescale of relaxation towards the Boltzmann distribution predicted by statistical mechanics. We also consider the relaxation of a “test” particle in a bath of “field” particles and analyze it with the aid of a Fokker–Planck equation involving a term of diffusion counterbalanced by a friction or a drift. Keywords Long-range interactions · Kinetic theory · Stellar systems · Two-dimensional vortices PACS 05.20.-y, 05.20.Dd, 98.10.+z, 47.32.C-

1 Introduction Self-gravitating systems display a sort of organization revealed in the Hubble classification or in de Vaucouleur’s law for the surface brightness of elliptical galaxies [1]. Similarly, two-dimensional flows with high Reynolds numbers have the striking property of organizing spontaneously into coherent structures that dominate the dynamics [2–5]. These structures correspond to jets and vortices in the atmosphere of giant planets (like Jupiter’s great red spot) or in the oceans (like the Gulf Stream or the Kuroshio Current). There exists numerous analogies between stellar systems and 2D vortices despite their very different nature [4]. In fact, it is possible to understand their structure and organization in terms of similar ideas of statistical mechanics and kinetic theory. Stellar systems As understood early by Hénon [6], self-gravitating systems display two successive types of relaxation. On a short timescale, of the order of a few dynamical times t D , the system is governed by the Vlasov equation (or collisionless Boltzmann equation) [1]. Starting from a generically unstable or unsteady initial condition, the Vlasov–Poisson system undergoes a process of violent relaxation leading to a quasistationary state (QSS). This QSS is a steady state of the Vlasov equation on the coarse-grained scale. This is a virialized state that is in mechanical equilibrium but not in thermodynamical equilibrium (in the usual sense). As a result, it is described by a non-Boltzmannian distribution. Lynden-Bell [7] has developed a statistical theory of the Vlasov equation in order to describe these QSSs. Unfortunately, the power of prediction of his theory is limited by the problem of incomplete relaxation and additional arguments must be advocated to understand the structure of galaxies [8,9]. On longer timescales, stellar encounters (sometimes referred to as “collisions” by an abuse of language) must be taken into account and the system is governed by the gravitational Communicated by H. Aref P.-H. Chavanis Laboratoire de Physique Théorique, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France E-mail: [email protected] Reprinted from the journal

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Vlasov–Landau equation [10–18]. This equation “tends” to approach the Boltzmann distribution predicted by statistical mechanics [19]. The timescale of this “collisional” process is given by the Chandrasekhar relaxation time t R ∼ (N /ln N )t D where N is the number of stars [11,1]. However, the complete relaxation of the system towards the Boltzmann distribution is hampered by the escape of high energy stars [20–25] and by the gravothermal catastrophe [26–31]. The first stage of the collisional evolution is driven by evaporation. Due to a series of weak encounters, the energy of a star can gradually increase until it reaches the local escape energy and leaves the system. During the collisional process, a stellar system passes through a succession of QSS that are steady states of the Vlasov equation slowly evolving due to the cumulative effect of encounters. This quasi stationary distribution function is close to the Michie–King model [1] and the system has a “core–halo” structure. Due to evaporation, the halo expands while the core shrinks as required by energy conservation. At some point of the evolution, when the energy passes below a critical value, the system undergoes an instability related to the Antonov instability [26] and the gravothermal catastrophe [27] sets in. This instability is due to the negative specific heat of the inner system that evolves by losing energy and thereby growing hotter. This leads to core collapse [32] and, ultimately, to the formation of binary stars [33]. Therefore, stellar systems are at most metastable states (local entropy maxima) [34] before core collapse takes place. At the present epoch, small groups of stars such as globular clusters (N ∼ 105 , t D ∼ 105 years, age ∼ 1010 years) are in the collisional regime. They are either in long-lived metastable statistical equilibrium states (described by the Michie–King model) or experiencing core collapse [1]. In contrast, large clusters of stars like elliptical galaxies (N ∼ 1011 , t D ∼ 108 years, age ∼ 108 years) are still in the collisionless regime and their apparent organization is a result of an incomplete violent relaxation [7–9] occurring in the first stage of the gravitational dynamics [1]. Two-dimensional turbulence Geophysical and astrophysical flows, like the oceans and the atmospheres of giant planets, are characterized by very large Reynolds numbers so these flows are described by the Navier–Stokes equation with a very small viscosity. Furthermore, due to the rotation of the planet, the stratification and the thin depth of the upper layer, the dynamics is mostly two-dimensional [35]. In the strict inviscid, incompressible and two-dimensional limit, these flows are described by the 2D Euler equation [2–4]. Starting from a generically unstable or unsteady initial condition, the 2D Euler–Poisson system undergoes a process of violent relaxation leading to a QSS that is a steady state of the 2D Euler equation on the coarse-grained scale. These QSSs correspond to large-scale vortices like Jupiter’s great red spot and other jovian vortices [36]. Miller [37] and Robert and Sommeria [38] have developed a statistical theory of the 2D Euler equation in order to explain these QSSs. This theory has been used to describe the vortices of 2D turbulence (monopoles, dipoles, tripoles, etc.) [39–41], the large-scale flows observed in the oceans [42] and the jovian vortices [43–45]. Large-scale vortices may also be present in galactic disks in relation with the emission of spiral density waves [46] and, presumably, in Keplerian disks where they might have played a major role in the process of planet formation [47,48]. On longer timescales, viscosity comes into play and drives the system to a state of rest in the absence of forcing. Interestingly, the QSSs of the 2D Euler–Poisson system (vortices) [37,38] and the QSSs of the Vlasov–Poisson system (galaxies) [7] can be understood with the same ideas of statistical mechanics [4,49]. Point vortices To make progress in the theoretical description of two-dimensional turbulent flows, it can be useful to consider the evolution of a system of N point vortices [50–53] instead of a continuous vorticity field. The statistical mechanics of two-dimensional point vortices was first studied by Onsager [54] in a seminal paper. This point vortex gas is an N -body Hamiltonian system [50,52] that shares some analogies with electric charges in a plasma and stars in a stellar system [4]. Like stellar systems, the point vortex gas displays two successive types of relaxation [4,55,56]. On a short timescale, of the order of a few dynamical times t D , the evolution of the smooth vorticity field is governed by the 2D Euler equation [55]. As we have already indicated, the 2D Euler–Poisson system can develop a process of violent relaxation leading to a QSS on the coarse-grained scale [37,38,57–59]. On longer timescales, “distant collisions” (i.e. correlations or finite N effects) between vortices must be taken into account. Using projection operator technics [55] or a BBGKYlike hierarchy [60], we have derived a kinetic equation at the order O(1/N ) which describes the “collisional” evolution of the system on a timescale N t D . However, this kinetic equation does not lead to the Boltzmann distribution predicted by statistical mechanics [61,62]. Indeed, for an axisymmetric distribution, the evolution stops when the profile of angular velocity becomes monotonic [63]. This means either that the evolution of the point vortex gas is not ergodic (in which case it will never reach the Boltzmann distribution) or that the relaxation towards statistical equilibrium occurs on a very long timescale larger than N t D [60,63]. To settle this issue, we would have to develop the kinetic theory at the order 1/N 2 , 1/N 3 , . . . by taking into account three-body, four-body, etc. correlation functions, which is a formidable task.

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Hamiltonian mean field (HMF) model One fundamental feature of stellar systems and 2D vortices in that they involve long-range interactions. Recently, there has been a great activity in physics to study the dynamics and thermodynamics of systems with long-range interactions [64]. For example, a small community of researchers has considered a simple toy model, called the HMF model, which consists in particles moving on a circle and interacting via a cosine potential [65,66]. Interestingly, this model displays several analogies with stellar systems and two-dimensional vortices, in particular a rapid collisionless relaxation towards a QSS and a slower collisional relaxation towards the Boltzmann distribution [66–73]. In this paper, we discuss the analogies (and the differences) in the kinetic theories of stellar systems, 2D vortices and other systems with long-range interactions like the HMF model. We study the evolution of the system “as a whole” and the relaxation of a “test” particle (tagged particle) in a bath of “field” particles. We develop a formalism that can take into account the spatial inhomogeneity of the system that is specific to systems with long-range interactions. Our presentation follows [18] for stellar systems, [60] for two-dimensional vortices and [74] for other systems with long-range interactions including the HMF model as a particular case. Other kinetic theories of these systems are described in the text and in the references.

2 Stellar systems 2.1 The N -stars system We consider an isolated system of stars (with identical mass m) in Newtonian interaction whose dynamics is fully described by the Hamilton equations [1]: dri dvi ∂H ∂H m , m , = =− dt ∂vi dt ∂ri

 1 1 2 H= mvi − Gm 2 . 2 |ri − r j | N

i=1

(1)

i< j

This  Hamiltonian system conserves the energy E = H , the mass M = N m and the angular momentum L = i mri × vi (for sake of brevity, we will assume that the system is non-rotating so that L = 0. Then, it can be shown that this constraint can be ignored). For t → +∞, we expect this system to reach a statistical equilibrium state.1 Since the system is isolated (fixed energy E), the proper statistical ensemble is the microcanonical ensemble. At statistical equilibrium, assuming ergodicity, the N -body distribution is given by the microcanonical distribution expressing the equiprobability of the accessible configurations [28–31]: PN (r1 , v1 , . . . , r N , v N ) =

1 δ(E − H (r1 , v1 , . . . , r N , v N )). g(E)

(2)

  The normalization factor g(E) = δ(E − H ) i dri dvi is the density of states with energy E. The microcanonical entropy of the system is defined by S(E) = ln g(E) and the microcanonical temperature by β(E) = 1/T (E) = ∂ S/∂ E (we take the Boltzmann constant k B = 1). We define the thermodynamic limit as N → +∞ in such a way that the normalized energy  = E R/G M 2 and the normalized temperature η = βG Mm/R are of order unity (we assume that the system is enclosed within a spherical box of radius R) [27–31]. We can renormalize the parameters so that the gravity constant G ∼ 1/N while m ∼ 1, β ∼ 1, E/N ∼ 1 and V ∼ 1. Using another normalization of the parameters, the thermodynamic limit is such that the mass of the particles behaves like m ∼ 1/N while G ∼ 1, β ∼ √ N, E ∼ 1 and V ∼ 1. In this scaling, the total mass M ∼ N m is of order unity. The dynamical time t D ∼ 1/ Gρ ∼ 1 (where ρ is the density) [1] is also of order unity in each normalization. In the thermodynamic limit N → +∞ defined previously, it can be rigorously shown that the N -body distribution function factorizes in a product of N one-body distribution functions [75,76]: PN (r1 , v1 , . . . , r N , v N ) = N i=1 P1 (ri , vi ). Therefore, the mean field approximation becomes exact in the thermodynamic limit N → +∞ (except possibly close to a critical point [34]). Furthermore, the one-body distribution P1 (r, v), or equivalently 1 In order to have a strict statistical equilibrium state for self-gravitating systems, it is necessary to enclose the system within a box (to avoid evaporation) and regularize the gravitational potential at small distances (to avoid singular effects arising from gravitational collapse) [28–31]. Metastable gaseous states (local entropy maxima) are obtained by introducing a small-scale regularization , developing the statistical mechanics and finally taking  → 0.

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the smooth distribution function f (r, v) = N m P1 (r, v), is obtained by solving the maximization problem [26–28,30,31]: max{S B [ f ]|E[ f ] = E, f

M[ f ] = M},

(3)

where  SB [ f ] = −

f f ln drdv, m m

 E=

f

v2 1 drdv + 2 2



 ρ dr,

M=

f drdv,

(4)

are the Boltzmann entropy, the mean field energy and the mass. The energy includes the kinetic energy   2 K = f v2 drdv and the potential energy W = 21 ρ dr where  is the mean field gravitational potential produced by the smooth distribution of stars ρ = f dv according to the Poisson equation  = 4π Gρ.

(5)

The mean force by unit of mass acting on a star is then F = −∇. Fundamentally, the Boltzmann entropy can be written S = ln W , where W is the number of microstates (defined by the exact position and velocity (ri , vi ) of the particles) associated with a given macrostate (defined by the smooth distribution function f (r, v), i.e. by the number of particles in phase space cells of size ( r, v), irrespectively of their precise position in the cells). Using the Stirling formula for N 1, the expression (4) of the entropy is readily obtained [19]. Note that this expression does not depend whether the particles interact or not (the Boltzmann entropy (4) has the same form as for the classical perfect gas). Assuming ergodicity, the statistical equilibrium state corresponds to the most probable distribution of particles, i.e. the macrostate that is the most represented at the microscopic level. It is obtained by maximizing the Boltzmann entropy while conserving the total mass and the total energy. Introducing Lagrange multipliers and writing the variational principle in the form δS B − βδ E − αδ M = 0,

(6)

we find that the critical points of constrained entropy are given by the mean field Maxwell–Boltzmann distribution f = Ae−βm(

v2 2 +)

.

(7)

The spatial density is then given by the mean field Boltzmann distribution ρ = A e−βm . Substituting this relation in the Poisson equation (5), we obtain the Boltzmann–Poisson equation  = 4π G A e−βm .

(8)

The statistical equilibrium state is then obtained by solving this equation and relating the Lagrange multipliers β and α (or A ) to the constraints E and M. Then, we have to make sure that the resulting distribution is a (local) maximum of S B at fixed mass and energy, not a minimum or a saddle point [26–28,31,77]. Note that for systems with long-range interactions, the statistical ensembles are generically inequivalent. It is therefore crucial to consider the microcanonical ensemble when describing an isolated Hamiltonian system of stars. Ensemble inequivalence was first encountered in astrophysics (see reviews in [28,30,31]) where it was realized that configurations with negative specific heats are allowed in the microcanonical ensemble but not in the canonical ensemble [78,79].

2.2 BBGKY-like hierarchy and 1/N expansion To settle whether a stellar system will reach the Boltzmann distribution (7) predicted by statistical mechanics and determine the timescale of the relaxation, we need to develop a kinetic theory. Basically, the evolution of the N -body distribution function PN (r1 , v1 , . . . , r N , v N , t) is governed by the Liouville equation  N  ∂ PN  ∂ PN ∂ PN + Fi · + vi · = 0, ∂t ∂ri ∂vi

(9)

i=1

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where Fi = −

 ri − r j  ∂ = −Gm = F( j → i), ∂ri |ri − r j |3 j =i

(10)

j =i

is the gravitational force by unit of mass experienced by the ith star due to the interaction with the other stars. Here,  denotes the exact gravitational potential produced by the discrete distribution of stars and F( j → i) denotes the exact force created by particle j on particle i. The Liouville equation, which is equivalent to the Hamilton equations (1), contains too much information to be of practical use. In practice, we are only interested in the one-body or two-body distributions. Let us introduce the reduced probability distributions  P j (x1 , . . . , x j , t) = PN (x1 , . . . , x N , t) dx j+1 . . . dx N , (11) where the notation x stands for (r, v). From the Liouville equation (9), we can construct the complete BBGKY hierarchy for the reduced distribution functions (11). It reads [74]: j j j  j  ∂ Pj  ∂ Pj   ∂ Pj ∂ P j+1 + vi · + F(k → i) · + (N − j) dx j+1 = 0. F( j + 1 → i) · ∂t ∂ri ∂vi ∂vi i=1 k=1,k =i

i=1

i=1

(12) This hierarchy of equations is not closed since the equation for the one-body distribution P1 (x1 , t) involves the two-body distribution P2 (x1 , x2 , t), the equation for the two-body distribution P2 (x1 , x2 , t) involves the three-body distribution P3 (x1 , x2 , x3 , t), and so on. The idea is to close the hierarchy by using a systematic expansion of the solutions in powers of 1/N in the thermodynamic limit N → + ∞. We first decompose the two- and three-body distributions in the suggestive form P2 (x1 , x2 ) = P1 (x1 )P1 (x2 ) + P2 (x1 , x2 ), P3 (x1 , x2 , x3 ) = P1 (x1 )P1 (x2 )P1 (x3 ) + P2 (x1 , x2 )P1 (x3 ) +P2 (x1 , x3 )P1 (x2 ) + P2 (x2 , x3 )P1 (x1 ) + P3 (x1 , x2 , x3 ).

(13) (14)

Considering the scaling of the terms in each equation of the hierarchy, we argue that there exists solutions of the whole BBGKY hierarchy such that the correlation functions P j scale like 1/N j−1 at any time [18]. This implicitly assumes that the initial condition has no correlation, or that the initial correlations respect this scaling (if there are strong correlations in the initial state, like “binary stars”, the kinetic theory will be different from the one developed in the sequel). If this scaling is satisfied, we can consider an expansion of the solutions of the equations of the hierarchy in terms of the small parameter 1/N . This is similar to the expansion in terms of the plasma parameter made in plasma physics. However, in plasma physics the systems are spatially homogeneous (due to Debye shielding which restricts the range of interaction) while, in the present case, we must take into account spatial inhomogeneity. This brings additional terms in the kinetic equations that are absent in plasma physics. If we introduce the notations f = N m P1 (distribution function) and g = N 2 P2

(two-body correlation function), we get at the order 1/N [18]:  ∂ f1 N −1 ∂ f1 ∂ ∂ f1 + = −m · F(2 → 1)g(x1 , x2 ) dx2 , (15) + v1 · F1 · ∂t ∂r1 N ∂v1 ∂v1  ∂g ∂g 1 ∂ f1 ∂ f1 ∂g +F1 · + 2 F (2 → 1) f 2 + · F(3 → 1)g(x2 , x3 , t) dx3 +(1 ↔ 2) = 0, +v1 · ∂t ∂r1 ∂v1 m ∂v1 ∂v1 m (16) where we have introduced the abbreviations f 1 = f (r1 , v1 , t) and f 2 = f (r2 , v2 , t). We have also introduced the mean force (by unit of mass) created in r1 by all the particles  f2 F1 = F(2 → 1) dr2 dv2 = −∇1 , (17) m and the fluctuating force (by unit of mass) created by particle 2 on particle 1: 1 F (2 → 1) = F(2 → 1) − F1 . (18) N Equations (15)–(16) are exact at the order O(1/N ). They are closed because the three-body correlation function P3 , of order O(1/N 2 ), can be neglected at the order 1/N . Reprinted from the journal

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2.3 The limit N → + ∞: the Vlasov equation (collisionless regime) In the limit N → +∞ for a fixed time t, the correlations between stars can be neglected (mean field approximation) so the N -body distribution function factorizes in N one-body distribution functions, i.e. N PN (x1 , . . . , x N , t) = i=1 P1 (xi , t). Substituting this result in Eq. (12), we find that the smooth distribution function f (r, v, t) = N m P1 (r, v, t) is solution of the Vlasov equation [74]: ∂ f1 ∂ f1 ∂ f1 + v1 · + F1 · = 0, F1 = −∇1 ,  = 4π G ∂t ∂r1 ∂v1

 f dv.

(19)

This equation also results from Eq. (15) if we neglect the correlation function g in the r.h.s. The Vlasov equation describes the collisionless evolution of the system for times smaller than N t D (where t D is the dynamical time). In practice N 1 so that the domain of validity of the Vlasov equation is huge (for example, in stellar systems N ∼ 106 –1012 stars). When the Vlasov equation is coupled to a long-range potential of interaction, like the gravitational potential, it can develop a process of phase mixing and violent relaxation leading to a QSS on a very short timescale, of the order of a few dynamical times t D . These QSSs correspond to galaxies in astrophysics [1]. Lynden-Bell [7] has developed a statistical mechanics of the Vlasov equation to try to describe this process of violent relaxation. Kinetic theories of violent relaxation are developed in [49,80–84]

2.4 The order O(1/N ): a general kinetic equation (collisional regime) If we neglect the contribution of the integral in Eq. (16), an approximation that is commonly made in stellar dynamics,2 we can formally solve this equation with the Green function constructed with the smooth field F. Substituting the resulting expression in Eq. (15), we get [18]:  t ∂ f1 ∂ f1 N −1 ∂ f1 ∂ + = μ dτ dr2 dv2 F μ (2 → 1, t)G(t, t − τ ) + v1 · F1 · ∂t ∂r1 N ∂v1 ∂v1 0

∂ ∂ f × F ν (2 → 1) ν + F ν (1 → 2) ν f (r1 , v1 , t − τ ) (r2 , v2 , t − τ ). ∂v1 ∂v2 m

(20)

This kinetic equation can also be obtained from a more abstract projection operator formalism [14] or from a quasilinear theory based on the Klimontovich equation [84]. The kinetic equation (20) is valid at the order 1/N so that it describes the “collisional” evolution of the system (ignoring collective effects) on a timescale of order N t D . Equation (20) is a non-Markovian integro-differential equation. The Markov approximation is not rigorously justified for self-gravitating systems because the force auto-correlation function decreases algebraically, like 1/t [88], instead of exponentially. Equation (20) is the general kinetic equation of stellar dynamics taking into account delocalizations in space and time.

2.5 The Landau equation for stellar systems Self-gravitating systems are spatially inhomogeneous but, in many studies, the collisional current is calculated by making a local approximation [1] replacing f (r2 , v2 , t) by f (r1 , v2 , t) and F (i → j) by F(i → j). This local approximation is motivated by the work of Chandrasekhar and von Neumann [89] who showed that the distribution of the gravitational force is a Lévy law (called the Holtzmark distribution) that is dominated by the contribution of the nearest neighbor. If, furthermore, we make a Markovian approximation and extend the 2 This integral takes into account collective effects between particles. In plasma physics, these collective effects are responsible for Debye shielding leading to the Lenard–Balescu equation [85,86]. They regularize the divergence at large scales appearing in the Landau equation. In stellar dynamics, their role is less clear and is difficult to investigate [87] because the system is spatially inhomogeneous (if we assume naively that the system is homogeneous, this integral leads to strong divergences related to the Jeans instability).

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time integral to infinity in Eq. (20), the integrals on τ and r2 can be calculated explicitly [74]. We then find that the evolution of the distribution function is governed by the Landau equation [10]:    ∂ f1 ∂ f1 ∂ ∂ f1 ∂ f2 ∂ f1 + F1 · = μ (21) + v1 · K μν f 2 ν − f 1 ν dv2 , ∂t ∂r1 ∂v1 ∂v1 ∂v2 ∂v1   wμ wν μν 21 μν K = 2πmG ln  δ − , (22) w w2 where now f 1 = f (r1 , v1 , t) and f 2 = f (r1 , v2 , t) denote the local distribution functions, w = v1 − v2 is the  kmax relative velocity of the stars involved in an encounter and ln  = kmin dk/k is the Coulomb factor that has to be regularized with appropriate cut-offs (see, e.g., [10,14,18] for a more complete discussion). Note that the spatial inhomogeneity of the gravitational system is taken into account in the advective term of Eq. (21), so this kinetic equation will be referred to as the Vlasov–Landau equation. This is the fundamental kinetic equation of stellar dynamics [1,15,16]. The Landau equation can be derived from the Boltzmann equation in the limit of weak deflections | v| 1 [10,28], from the Fokker–Planck equation by evaluating the first and second velocity increments  v μ  and  v μ v ν  due to a succession of two-body encounters [11–13,90], from projection operator techniques [14,91], from the BBGKY hierarchy [18,74,92] or from a quasilinear theory [84]. The Vlasov–Landau–Poisson system conserves the total mass and the total mean field energy (kinetic + potential) of the system. It also increases the Boltzmann entropy monotonically: S˙ B ≥ 0 (H -theorem). The mean field Maxwell–Boltzmann distribution (7) is the only stationary solution of this equation, cancelling the advective term and the collision term individually. Therefore, the system tends to reach the Boltzmann distribution on a timescale tR ∼

N tD , ln N

(23)

where the N t D scaling comes from the fact that the Vlasov–Landau equation is valid at the order O(1/N ) and the logarithmic correction is due to the divergence3 of the Coulombian factor with N , as ln  ∼ ln N . In practice, the strict relaxation towards the Boltzmann distribution is hampered by the problems of evaporation and gravothermal catastrophe [1,15,16,28,31] discussed in Sect. 1. 2.6 Test particle in a thermal bath: the Fokker–Planck equation We now consider the relaxation of a “test” star (tagged particle) evolving in a steady distribution of “field” stars. Let us call P(r, v, t) the probability density of finding the test star at position r with velocity v at time t. The evolution of P(r, v, t) can be obtained from the Landau equation (21) by considering that the distribution function of the field stars f 2 is fixed. Therefore, if we replace f 1 = f (r1 , v1 , t) by P = P(r, v, t) and f 2 = f (r1 , v2 , t) by f 0 = f (r, v0 ), where f (r, v) is any stable stationary solution of the Vlasov equation, we get [17,74]:    ∂ f0 ∂P ∂P ∂P ∂ dP ∂P (24) ≡ +v· + F · = μ K μν f 0 ν − P ν dv0 . dt ∂t ∂r ∂v ∂v ∂v ∂v0 Equation (24) can be written in the form of a Fokker–Planck equation   dP ∂ μν ∂ P μ − Pη , = μ D dt ∂v ∂v ν

(25)

3 In plasma physics and stellar dynamics, the Landau equation (21)–(22) and the Fokker–Planck equation (32) present a logarithmic divergence at small scales if we assume in the derivation that the particles follow linear trajectories [10]. In fact, this approximation is not correct for encounters with small impact parameters that lead to large deflections | v| 1. In that case, we must take into account discrete interactions between particles and regularize the logarithmic divergence at the Landau length [10]. This length appears naturally in Chandrasekhar’s calculations [11,12] that fully take into account the hyperbolic nature of the particles’ trajectories. On the other hand, the Landau and Fokker–Planck equations present a divergence at large scales. In plasma physics, it is solved by the Debye shielding in the Lenard–Balescu treatment. In stellar dynamics, it is solved by the finite extension of the system using the Jeans length as an upper length. With these regularizations, it can be shown that ln  ∼ ln N leading to the ln N factor in the relaxation time (see, e.g., [18]).

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involving a diffusion term D μν =



and a friction term μ

ημ ≡ Fpol =

K μν f 0 dv0 , 

K μν

∂ f0 dv0 . ∂v0ν

(26)

(27)

The diffusion coefficient is due to the fluctuations of the gravitational force and it can be derived directly from the Kubo formula [84,93]. The friction term Fpol results from a polarization process and it can be derived directly from a linear response theory [84,93]. In the present case, the coefficients of diffusion and friction depend on the velocity v of the test star. Hence, it is more appropriate to write Eq. (24) in a form that is fully consistent with the general Fokker–Planck equation     dP 1 ∂2 ∂  v μ v ν   v μ  = − , (28) P P dt 2 ∂v μ ∂v ν t ∂v μ t with  v μ v ν  = D μν , 2 t

 v μ  ∂ D μν μ + ημ ≡ Ffriction . = t ∂v ν

(29)

The two expressions (25) and (28) have their own interest. The expression (28) where the diffusion coefficient μ is placed after the two derivatives ∂ 2 (D P) involves the total friction force Ffriction =  v μ / t and the expression (25) where the diffusion coefficient is placed between the derivatives ∂ D∂ P isolates the part of the μ friction ημ = Fpol due to the polarization [84]. Using ∂ K μν /∂v ν = −∂ K μν /∂v0ν and an integration by parts, we easily derive ∂ D μν /∂v ν = ημ so that Ffriction = 2Fpol : the two forces differ by a factor 2 [17,84]. In stellar dynamics, the diffusion coefficient and the friction force were first computed by Chandrasekhar [11,12] by directly evaluating  v μ v ν  and  v μ . For the Boltzmann distribution   v2 βm 3/2 f 0 (r, v) = ρ(r) e−βm 2 , (30) 2π corresponding to the statistical equilibrium state (thermal bath), using the property K μν w ν = 0, we easily find from Eqs. (26) and (27) that [17,74]: μ

Fpol = −βm D μν v ν .

(31)

The friction tensor is given by a generalized Einstein relation: ξ μν = βm D μν . The Fokker–Planck equation (25) then becomes [12,16,17,74]:  

∂P dP ∂ ν + βm Pv = μ D μν (v) , (32) dt ∂v ∂v ν μ ν

where the diffusion tensor is given by Eqs. (26) and (30). It can be put in the form D μν = (D − 21 D⊥ ) v v 2v + 1 μν where D and D are the diffusion coefficients in the directions parallel and perpendicular to  ⊥ 2 D⊥ δ the velocity of the test star. The friction force can then be written as Fpol = −D βmv (Chandrasekhar’s √ dynamical friction [12]) with a friction coefficient ξ = D βm (Einstein relation). Setting x = βm/2v and 2 /3t )G μν (x), the Fokker–Planck equation (32) becomes D μν (v) = (2vm R  

∂P 1 ∂ dP μν ν (x) + 2P x = G . (33) dt tR ∂ x μ ∂xν The diffusion tensor can be calculated analytically [11,12,16,17]. It is given by 1 xμxν 1 G μν = (G  − G ⊥ ) 2 + G ⊥ δ μν , 2 x 2

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where G =

2π 3/2 2π 3/2 G(x), G ⊥ = [erf(x) − G(x)], x x

(35)

with 2 1 G(x) = √ 2 πx

x

t 2 e−t dt = 2

0



1 2x −x 2 e erf(x) − . √ 2x 2 π

(36)

The local relaxation time is given by 1 tR = 3



2π 3

1/2

3 vm , ρmG 2 ln 

(37)

2 = 3/(βm) is the mean square velocity of the field particles. Of course, the relaxation time is smaller where vm in regions of high density and larger in regions of low density. Introducing the dynamical time t D = R/vm and 1/2 N t where estimating the density by ρ = 3M/(4π R 3 ), we get the average relaxation time t R = 4π( 2π 3 ) η2 ln  D η = βG Mm/R. The Fokker–Planck equation (32) has been used by Chandrasekhar [12,22] to determine the rate of escape of stars from a cluster. It is similar to the Kramers equation in Brownian theory. However, in the present case, the diffusion coefficient D μν (v) is anisotropic and depends on the velocity v of the test particle. For t → +∞, the velocity distribution of the test particle P(r, v, t) relaxes towards the Maxwellian distribution (30) of the bath (thermalization). This takes place on the typical Chandrasekhar relaxation time

tR ∼

N tD . ln N

(38)

3 Two-dimensional point vortices 3.1 The N -vortex system We consider an isolated system of point vortices (with identical circulation γ ) on an infinite plane. Their dynamics is fully described by the Kirchhoff–Hamilton equations [50,52]: γ

dxi ∂H ∂H dyi = =− , γ , dt ∂ yi dt ∂ xi

H =−

γ2  ln |ri − r j |, 2π

(39)

i< j

where the positions (x, y) of the point vortices are canonically conjugate.This Hamiltonian system conserves  the energy E = H , the circulation  = N γ , the angular momentum L = i γ ri2 and the impulse P = i γ ri . We take the origin at the center of vorticity so that P = 0 (we shall ignore this constraint in the following). For t → +∞, we expect this system to reach a statistical equilibrium state described by the microcanonical distribution [54,60,62]:  1 PN (r1 , . . . , r N ) = γ ri2 ), (40) δ(E − H (r1 , . . . , r N ))δ(L − g(E, L) 

i





where g(E, L) = δ(E − H )δ(L − i γ ri2 ) i dri is the density of states. The microcanonical entropy of the system is defined by S(E, L) = ln g(E, L). The microcanonical temperature and the angular velocity are given by β = 1/T = (∂ S/∂ E) L and  L = (2/β)(∂ S/∂ L) E . As first realized by Onsager [54], the temperature of the point vortex gas can be negative. At negative temperatures β(E) < 0, corresponding to high energy states, point vortices of the same sign group themselves in “supervortices” similar to large-scale vortices in the atmosphere of giant planets. We define the thermodynamic limit as N → +∞ in such a way that the normalized energy  = E/  2 and the normalized temperature η = βγ are of order unity [60]. We can renormalize the parameters so that the circulation of the vortices behaves like γ ∼ 1/N while β ∼ N , E ∼ 1 and V ∼ 1. In this scaling, the total circulation  ∼ N γ is of order unity. The dynamical time t D ∼ 1/ω ∼ R 2 /  ∼ 1 is also of order unity. Reprinted from the journal

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In the thermodynamic limit N → +∞ defined previously, it can be rigorously shown [94–97] that the N -body distribution function factorizes in a product of N one-body distribution functions: PN (r1 , . . . , r N ) = N i=1 P1 (ri ). Therefore, the mean field approximation becomes exact in the thermodynamic limit N → +∞. Furthermore, the one-body distribution function P1 (r) or equivalently the smooth density of point vortices n(r) = N P1 (r) or the smooth vorticity field ω(r) = N γ P1 (r) is obtained by solving the maximization problem [4,61]: max{S B [ω]|E[ω] = E, [ω] = ,

L[ω] = L},

ω

where

 S B [ω] = −

ω ω ln dr, γ γ

1 E= 2





(41) 

ωψ dr,  =

ω dr,

L=

ωr 2 dr,

(42)

are the Boltzmann entropy, the mean field energy, the circulation and the angular momentum. Here ψ is the mean field stream function produced by the smooth distribution of vortices ω according to the Poisson equation − ψ = ω.

(43)

The mean velocity of a point vortex is then V = −z × ∇ψ where z is a unit vector normal to the flow. Fundamentally, the Boltzmann entropy can be written S = ln W , where W is the number of microstates (complexions) corresponding to a given macrostate specified by the smooth vorticity field ω(r). Using the Stirling formula for N 1 leads to expression (42). Introducing Lagrange multipliers and writing the variational principle in the form δS B − βδ E − αδ − β 2L δL = 0, we obtain the mean field Boltzmann distribution ω = Aγ e

  L −βγ ψ+ r2 2

,

(44)

where A is a positive constant. Substituting this relation in the Poisson equation (43), we obtain the BoltzmannPoisson equation − ψ = Aγ e

  L −βγ ψ+ r2 2

.

(45)

The statistical equilibrium state is then obtained by solving this equation and relating the Lagrange multipliers β, α (or A) and  L to the constraints E,  and L. Then, we have to make sure that the resulting distribution is a maximum of S B at fixed circulation, energy and angular momentum (most probable state), not a minimum or a saddle point. This problem is studied in [4,62,94,98].

3.2 BBGKY-like hierarchy and 1/N expansion To settle whether the point vortex gas will reach the Boltzmann distribution (44) predicted by statistical mechanics and determine the timescale of the relaxation, we need to develop a kinetic theory. Basically, the evolution of the N -body distribution of the point vortex gas is governed by the Liouville equation ∂ PN  ∂ PN Vi · = 0, + ∂t ∂ri N

(46)

i=1

where Vi = −z ×

  ri − r j ∂ψ γ = = V( j → i), z× ∂ri 2π |ri − r j |2 j =i

(47)

j =i

is the velocity of point vortex i due to the interaction with the other vortices. Here, ψ denotes the exact stream function produced by the discrete distribution of point vortices and V( j → i) denotes the exact velocity induced by vortex j on vortex i. The Liouville equation (46) is equivalent to the Hamiltonian system

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(39). From the Liouville equation(46), we can construct a complete BBGKY-like hierarchy for the reduced distributions P j (r1 , . . . , r j , t) = PN (r1 , . . . , r N , t) dr j+1 . . . dr N . It reads [60]: j j j   ∂ Pj ∂ Pj   ∂ P j+1 V(k → i) · + (N − j) dr j+1 = 0. + V( j + 1 → i) · ∂t ∂ri ∂ri i=1 k=1,k =i

(48)

i=1

We can close the hierarchy of equations using a systematic expansion of the solutions in powers of 1/N in the thermodynamic limit N → +∞. In this limit, the correlation functions P j scale like 1/N j−1 . If we introduce the notations ω = N γ P1 (smooth vorticity field) and g = N 2 P2 (two-body correlation function), we get at the order 1/N [60]:  ∂ω1 ∂ω1 ∂ N −1 = −γ · V(2 → 1)g(r1 , r2 ) dr2 , + V1 · ∂t N ∂r1 ∂r1  ∂g ∂ω1 ∂g 1 ∂ ω1 + 2 V (2 → 1)ω2 + · V(3 → 1)g(r2 , r3 , t) dr3 + (1 ↔ 2) = 0, + V1 · ∂t ∂r1 γ ∂r1 ∂r1 γ

(49) (50)

where we have introduced the abbreviations ω1 = ω(r1 , t) and ω2 = ω(r2 , t). We have also introduced the mean velocity in r1 created by all the vortices  V1 =

V(2 → 1)

ω2 dr2 = −z × ∇ψ1 , γ

(51)

and the fluctuating velocity created by point vortex 2 on point vortex 1: V (2 → 1) = V(2 → 1) −

1 V1 . N

(52)

These equations are exact at the order O(1/N ). They are closed because the three-body correlation function P3 , of order O(1/N 2 ), can be neglected at the order 1/N .

3.3 The limit N → + ∞: the 2D Euler equation (collisionless regime) In the limit N → +∞ for a fixed time t, the correlations between point vortices can be neglected (mean field approximation) and the N -body distribution factorizes in N one-body distributions, i.e. PN (r1 , . . . , r N , t) = N i=1 P1 (ri , t). Substituting this result in Eq. (48), we find that the smooth vorticity field ω(r, t) of the point vortex gas is solution of the 2D Euler equation [60]: ∂ω1 + V1 · ∇ω1 = 0, V1 = −z × ∇ψ1 , ψ = −ω. ∂t

(53)

This equation also results from Eq. (49) if we neglect the correlation function g in the r.h.s. The 2D Euler equation describes the collisionless evolution of the point vortex gas for times smaller than N t D (where t D is the dynamical time). The 2D Euler–Poisson system can undergo a process of mixing and violent relaxation towards a QSS on a very short timescale, of the order of a few dynamical times t D . These QSSs have been observed experimentally in magnetized pure electron plasmas which are isomorphic to point vortex systems [57]. Miller [37] and Robert and Sommeria [38] have developed a statistical mechanics of the 2D Euler equation to predict these QSSs. The 2D Euler equation is analogous to the Vlasov equation in stellar dynamics and the Miller–Robert–Sommeria theory is analogous to the Lynden–Bell theory (see [4,49]). Kinetic theories of violent relaxation are developed in [49,60,99–101]. Reprinted from the journal

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3.4 The order O(1/N ): a general kinetic equation (collisional regime) If we neglect the contribution of the integral in Eq. (50), we can formally solve this equation with the Green function constructed with the smooth velocity field V. Substituting this result in Eq. (49), we obtain [60]:  t ∂ω1 N −1 ∂ω ∂ + V1 · = μ dτ dr2 V μ (2 → 1, t)G(t, t − τ ) ∂t N ∂r1 ∂r1 0

∂ ∂ ω ν ν × V (2 → 1) ν + V (1 → 2) ν ω(r1 , t − τ ) (r2 , t − τ ). ∂r1 ∂r2 γ

(54)

This kinetic equation can also be obtained from a more abstract projection operator formalism [55] or from a quasilinear theory based on the Klimontovich equation [60]. This is a non Markovian integro-differential equation. It is valid at the order 1/N so that it describes the “collisional” evolution of the point vortex gas (ignoring collective effects) on a timescale of order N t D . Equation (54) is the general kinetic equation of point vortices taking into account delocalizations in space and time. This kinetic equation is the vortex analogue of the kinetic equation (20) for stellar systems. 3.5 Kinetic equation for axisymmetric flows Let us consider an axisymmetric distribution of point vortices that is stable with respect to the 2D Euler equation. We want to investigate its collisional evolution due to finite N effects according to Eq. (54). If we make a Markovian approximation and extend the time integral to infinity, the integrals on τ and θ2 in Eq. (54) can be calculated explicitly [55,63]. We then find that the evolution of the smooth vorticity field ω(r, t) is governed by a kinetic equation of the form [55,60]: ∂ω1 1 ∂ = 2π 2 γ ∂t r1 ∂r1 with

  +∞ 1 ∂ω1 1 ∂ω2 r2 dr2 χ (r1 , r2 )δ(1 − 2 ) ω2 − ω1 , r1 ∂r1 r2 ∂r2

(55)

0

  2   +∞ 1  1 r< 2m 1 r< , χ (r1 , r2 ) = = − 2 ln 1 − 2 8π m r> 8π r>

(56)

m=1

r where (r, t) = Vθ (r, t)/r = r12 0 ω(s, t)s ds is the angular velocity and r> = max(r1 , r2 ), r< = min(r1 , r2 ). We have introduced the abbreviations ω12 = ω(r12 , t) and 12 = (r12 , t). This equation, which ignores collective effects,4 is the vortex analogue of the Landau equation (21) in plasma physics and stellar dynamics. It has been derived by Chavanis [55,60] using projection operator techniques, a BBGKY-like hierarchy or a quasilinear theory. It conserves the circulation , the energy E and the angular momentum L of the system. It also increases the Boltzmann entropy monotonically: S˙ B ≥ 0 (H -theorem) [63]. The change of the vorticity distribution in r1 is due to a condition of resonance (encapsulated in the δ-function) between vortices located in r1 and vortices located in r2  = r1 which rotate with the same angular velocity (r2 , t) = (r1 , t). Clearly, this condition can be satisfied only when the profile of angular velocity is non-monotonic. The collisional evolution of the point vortices is thus truly due to long-range interactions since the current in r1 is caused by “distant collisions” with vortices located in r2  = r1 that can be far away. This is different from the case of plasma physics and stellar dynamics where the collisions are assumed to be local in space [1]. The mean field Boltzmann distribution (44) is a particular steady state of Eq. (55) but it is not the only one [63]. In fact, all the vorticity profiles ω(r ) associated with a monotonic profile of angular velocity (r ) are steady states of the 4 Collective effects can be taken into account by keeping the contribution of the integral in Eq. (50). For axisymmetric flows, this leads to the kinetic equation derived by Dubin and O’Neil [102] from a quasilinear theory of the Klimontovich equation. This is the vortex analogue of the Lenard–Balescu equation (76) in plasma physics. In plasma physics, the collective effects accounting for Debye shielding in the Lenard–Balescu equation regularize the logarithmic divergence at large scales arising in the Landau equation. Since the kinetic equation (55) does not present any divergence the neglect of collective effects may not be crucial for point vortices.

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0.5

0.4 t=0

Ω(r,t)

0.3

0.2

0.1

0

0

0.25

0.5

0.75

1

r

Fig. 1 Evolution of the profile of angular velocity obtained by solving numerically the kinetic equation (55)–(56). Time increases from bottom to top. The evolution stops when the profile of angular velocity becomes monotonic so that there is no resonance anymore. The final profile does not correspond to the Boltzmann distribution (44). The kinetic equation (55)–(56) is valid on a timescale N t D . Therefore, on this timescale the vortex gas does not reach statistical equilibrium but remains blocked in a QSS with a monotonic profile of angular velocity. The Boltzmann distribution (44) may be reached on a longer timescale (due to higher order correlations) but this timescale of relaxation is not known. It is not even clear whether the point vortex gas will ever relax towards the statistical equilibrium distribution (44)

kinetic equation (55) since the δ-function is zero for these profiles. Therefore, the collisional evolution of the point vortex gas described by Eq. (55) stops when the profile of angular velocity becomes monotonic (so that there is no resonance) even if the system has not reached the Boltzmann distribution (see Fig. 1). In that case, the system settles on a QSS that is not the most mixed state predicted by statistical mechanics. This “kinetic blocking” has been illustrated numerically in [63]. On the timescale N t D on which the kinetic theory is valid, the collisions tend to create a monotonic profile of angular velocity. Since the entropy increases monotonically, the vorticity profile tends to approach the Boltzmann distribution (the system becomes “more mixed”) but does not attain it in general because of the absence of resonance. The Boltzmann distribution may be reached on longer timescales, larger than N t D . To describe this regime, we need to determine the terms of order N −2 or smaller in the expansion of the solutions of the BBGKY hierarchy for N → +∞. This implies in particular the determination of the three-body correlation function, which is a formidable task. At the moment, we can only conclude from the kinetic theory that the relaxation time satisfies [60]: tR > N tD .

(57)

The relaxation towards the Boltzmann distribution (44) is therefore a very slow process. The possible slow timescale of mixing was pointed out by Onsager in a letter to Lin [103]: “I still have to find out whether the processes anticipated by these considerations are rapid enough to play a dominant role in the evolution [...].” and it is now confirmed by the kinetic theory.5 On the other hand, up to now, there is no rigorous proof coming from the kinetic theory that the point vortex gas will ever relax towards the Boltzmann distribution predicted by statistical mechanics. Indeed, the dynamics may not be ergodic [62,104] and the final convergence of the vorticity profile towards the Boltzmann distribution (44) still remains an open problem. This is at variance with the Landau [10] and Lenard-Balescu [85,86] equations of plasma physics which always converge towards the Boltzmann distribution.6 In these equations, the collisional evolution of the system is also due to a condition of resonance k · v1 = k · v2 [see Eq. (76)] but, in d = 3 dimensions, this condition can always be satisfied. Therefore, the system relaxes towards the Boltzmann distribution which is the only steady state of the Landau and Lenard–Balescu kinetic equations. 5 Note that the point vortex gas can undergo a violent collisionless relaxation towards a QSS [37,38] on a very short timescale (see Sect. 3.3). As far as we know, this possibility was not considered by Onsager [54] who only focused on the ordinary statistical equilibrium state. 6 In stellar dynamics, the evolution is more complex due to the evaporation and the gravothermal catastrophe [1,15,16,28,31].

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3.6 Relaxation of a test vortex in a thermal bath: the Fokker–Planck equation We now consider the relaxation of a “test” vortex (tagged particle) evolving in a steady distribution of “field” vortices. Let us call P(r, t) the probability density of finding the test vortex at position r at time t. For simplicity, we consider axisymmetric distributions (see [60] for generalizations). The evolution of P(r, t) can be obtained from the kinetic equation (55) by considering that the distribution of the field vortices ω2 is fixed [55,63]. If we replace ω1 = ω(r1 , t) by P = P(r, t) and ω2 = ω(r2 , t) by ω0 = ω(r0 ) where ω(r) is any stable stationary solution of the 2D Euler equation, we get [60,55]:   +∞ 1 ∂ 1 ∂ r0 dr0 χ (r, r0 )δ( − 0 ) − P(r, t)ω(r0 ). r ∂r r0 ∂r0

∂P 1 ∂ = 2π 2 γ ∂t r ∂r

(58)

0

Equation (58) can be written in the form of a Fokker–Planck equation  

∂P 1 ∂ ∂P = r D − Pη , ∂t r ∂r ∂r

(59)

involving a diffusion coefficient 2π 2 γ D= r2

+∞ r0 dr0 χ (r, r0 )δ( − 0 )ω(r0 ),

(60)

0

and a drift term η≡

pol Vr

2π 2 γ = r

+∞ dω dr0 χ (r, r0 )δ( − 0 ) (r0 ). dr

(61)

0

The diffusion coefficient is due to the fluctuations of the velocity field and it can be derived directly from the Kubo formula [55,60,105]. The drift results from a polarization process and it can be derived directly from a linear response theory [60,106]. Expressions (60) and (61) for the diffusion coefficient and the drift term can also be obtained directly from the Hamiltonian equations, by making a systematic expansion of the trajectories of the point vortices in powers of 1/N in the limit N → +∞ as shown in Appendix C of [63]. In the present case, the coefficients of diffusion and drift depend on the position r of the test vortex. Hence, it is more appropriate to write Eq. (58) in a form which is fully consistent with the general Fokker–Planck equation  

  ∂P 1 ∂ ∂ ( r )2  1 ∂  r  = r P − rP , (62) ∂t 2r ∂r ∂r t r ∂r t with ( r )2  = D, 2 t

 r  ∂D = + η ≡ Vrdrift . t ∂r

(63) pol

The two expressions (59) and (62) involve, respectively, the drift due to the polarization Vr or the total drift Vrdrift [60]. If the profile of angular velocity of the field vortices (r ) is monotonic,7 we can use the identity δ(−0 ) = δ(r − r0 )/| (r )| and we find that [60,63]: D(r ) =

1 γ ln  ω(r ), 4 |(r )|

(64)

7 According to Eq. (57), such a distribution is steady on a timescale of order N t or longer. On the other hand, according to D Eq. (72), the relaxation time of a point vortex in a bath is of order (N / ln N )t D . Therefore, we can assume that a distribution of field vortices with a monotonic profile of angular velocity is steady on the timescale on which the Fokker–Planck approach is valid, although this distribution is not the statistical equilibrium state (i.e. not a thermal bath).

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and γ 1 dω ln  (r ), (65) 4 |(r )| dr  1 where (r ) = r  (r ) is the local shear and ln  ≡ +∞ m=1 m is a Coulomb factor that has to be regularized with appropriate cut-offs as discussed in [63,105]. It is then found that ln  ∼ 21 ln N in the thermodynamic limit N → +∞. The expressions (64) and (65) of the diffusion coefficient and of the drift, proportional to the inverse of the shear, were first obtained by Chavanis [55,106]. Similar results have been discussed by Dubin [105]. Comparing Eq. (65) with Eq. (64), we find that the drift velocity is related to the diffusion coefficient by the relation [55,60]: pol

Vr

=

d ln ω , (66) dr which can be viewed as a generalized form of Einstein relation for an out-of-equilibrium distribution of field vortices. Combining the previous results we find that, for a distribution of field vortices with a monotonic profile of angular velocity, the Fokker–Planck equation (59) can be written  

∂P ∂P 1 ∂ d ln ω = r D(r ) −P , (67) ∂t r ∂r ∂r dr pol

Vr

=D

with a diffusion coefficient given by Eq. (64). If the field vortices are at statistical equilibrium (thermal bath), their vorticity profile is the Boltzmann distribution L 2

ω(r ) = Aγ e−βγ ψ (r ) , ψ (r ) = ψ(r ) + (68) r . 2 Then, we have dω dψ

(r0 ) = −βγ ω(r0 ) (r0 ) = βγ ω(r0 )((r0 ) −  L )r0 , dr dr

(69)

where we have used (r ) = −(1/r )dψ/dr . Substituting this relation in Eq. (61), using the δ-function to replace (r0 ) by (r ), using (r ) −  L = −(1/r )dψ /dr and comparing the resulting expression with Eq. (60), we finally find that [63,60]: pol

Vr

= −Dβγ

dψ

. dr

(70)

The drift is perpendicular to the relative mean field velocity V  = −(dψ /dr )eθ and the drift coefficient (or mobility) satisfies an Einstein relation ξ = Dβγ [55,106]. We recall that the drift coefficient and the diffusion coefficient depend on the position r of the test vortex and that the temperature is negative in cases of physical pol interest [54]. We also stress that the Einstein relation is valid for the drift Vr due to the polarization only, not for the total drift Vrdrift . We do not have this subtlety for the usual Brownian motion where the diffusion coefficient is constant. For a thermal bath, using Eq. (70), the Fokker–Planck equation (59) can be written  

1 ∂ ∂P dψ

∂P = r D(r ) + βγ P , (71) ∂t r ∂r ∂r dr where D(r ) is given by Eq. (60) with Eq. (68). Of course, if the profile of angular velocity of the Boltzmann distribution is monotonic, we find that Eq. (67) with Eq. (68) returns Eq. (71) with a diffusion coefficient given by Eq. (64) with Eq. (68). Note that the systematic drift Vpol = −Dβγ ∇ψ of a point vortex [106] is the counterpart of the dynamical friction Fpol = −D βmv of a star [12] and the Fokker–Planck equation (71) is the counterpart of the Kramers–Chandrasekhar equation (32). The Fokker–Planck equations (67) and (71) have been studied in detail by Chavanis and Lemou [63] for different types of bath distribution. The distribution of the test vortex P(r, t) relaxes towards the distribution of the bath ω(r) on a typical timescale tR ∼ Reprinted from the journal

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where the logarithmic correction comes from the divergence8 of the Coulombian factor with N as ln  ∼ 1 2 ln N . However, the relaxation process is very peculiar and differs from the usual exponential relaxation of Brownian particles. In particular, the evolution of the front profile in the tail of the distribution is very slow (logarithmic) and the temporal correlation function r (0)r (t) decreases algebraically, like ln t/t (for a thermal bath), instead of exponentially [63]. This is due to the rapid decay of the diffusion coefficient D(r ). Similar results were established earlier for the HMF model by Bouchet and Dauxois [69] and Chavanis and Lemou [107]. 4 Spatially homogeneous systems: the Lenard–Balescu equation 4.1 Statistical equilibrium states We consider an isolated system of material particles in interaction whose dynamics is fully described by the Hamiltonian equations m

dri dvi ∂H ∂H , m , = =− dt ∂vi dt ∂ri

 1 2 mvi + m 2 u(|ri − r j |), 2 N

H=

i=1

(73)

i< j

where u i j = u(|ri − r j |) is a non-singular binary potential of weak long-range interactions. The thermodynamic limit corresponds to N → +∞ in such a way that the normalized energy  = E/(u ∗ N 2 m 2 ) and the normalized temperature η = β N m 2 u ∗ are of order unity where u ∗ represents the typical value of the potential of interaction (see [108] for more details). In the thermodynamic limit, the N -body distribution factorizes in a product of N one-body distributions. At statistical equilibrium, the one-body distribution function f = N m P1 is given by the mean field Maxwell–Boltzmann distribution  v2 f = Ae−βm( 2 +) , (r) = ρ(r )u(|r − r |) dr , (74) which maximizes the Boltzmann entropy at fixed mass and energy given by Eq. (4). Integrating on the velocity, we get the density ρ = A e−βm , where (r) is given by  Eq. (74). The equilibrium density profile is solution of the integro-differential equation ∇ ln ρ = −βm∇ ρ(r )u(|r − r |) dr . In the following, we consider spatially homogeneous systems so that  = 0 and ρ = M/V . The equilibrium state is the d-dimensional Maxwell distribution   v2 βm d/2 f (v) = ρ e−βm 2 . (75) 2π The HMF model is a toy model of systems with long-range interactions [65] which presents properties similar to self-gravitating systems and 2D vortices [66]. This is a one-dimensional (d = 1) model in which particles of unit mass are restrained to move on a circle. They are characterized by their angle θi with respect to k a reference axis and by their velocity vi . They interact via a potential u i j = − 4π cos(θi −θ j ) which is restricted to one Fourier mode. The thermodynamic limit corresponds to N → +∞ in such a way that the normalized energy  = E/(k N 2 ) and the normalized temperature η = βk N are fixed. We can renormalize the parameters in such a√way that the coupling constant k ∼ 1/N while β ∼ 1, E/N ∼ 1 and V ∼ 1. The dynamical time t D ∼ 1/ kρ is also of order unity [108]. The spatially homogeneous phase (75) is thermodynamically stable M for T > Tc = k4π and unstable for T < Tc (in that case it is replaced by an inhomogeneous phase). The phase transition at T = Tc is second order [65,66]. There is a huge bibliography about the HMF model reviewed in [109]. 8 In point vortex dynamics, the kinetic equation (55)–(56) governing the evolution of the system as a whole does not present any divergence (this differs from the Landau equation (21)–(22)) so that this equation is valid on a timescale ∼ N t D . However, a logarithmic divergence occurs in the Fokker–Planck equation (67) when we make the bath approximation so that this equation is valid on a timescale t R ∼ (N / ln N )t D . This divergence comes from the assumption made in the derivation that the point vortices follow the streamlines produced by the mean flow [55,60]. This is not correct for encounters with small impact parameters that lead to more complex trajectories [105]. In that case, we must take into account discrete interactions between point vortices and regularize the logarithmic divergence appropriately [63,105]. With this regularization it can be shown that ln  ∼ 21 ln N leading to the ln N factor in the relaxation time (72) of a test vortex in a bath. Since the kinetic equation (55)–(56) only takes into account distant collisions, we do not have any divergence for the evolution of the system as a whole so there is no ln N factor in the collisional timescale (57).

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4.2 Evolution of the system as a whole At the order 1/N , the evolution of the Hamiltonian system (73) is governed by the coupled kinetic equations (15)–(16). Collective effects can be taken into account by keeping the contribution of the integral in Eq. (16). For spatially homogeneous systems, Eq. (16) can be solved by using Laplace–Fourier transforms [110,111] and the two-body correlation function g(v1 , v2 , t) can be expressed in terms of the distribution function f (v, t). Substituting this result in Eq. (15) yields the Lenard–Balescu equation [85,86]: ∂ f1 ∂ = π(2π)d m μ ∂t ∂v1



dv2 dkk μ k ν

u(k) ˆ 2 δ[k · (v1 − v2 )] |(k, k · v2 )|2

 f2

∂ f1 ∂ f2 − f1 ν ∂v1ν ∂v2

 ,

(76)

where (k, ω) is the dielectric function  ˆ (k, ω) = 1 − (2π) u(k) d

k · ∂ f /∂v dv. k·v−ω

(77)

If we neglect collective effects, which amounts to making |(k, k · v2 )|2 = 1 in Eq. (76), we recover after some simple transformations [112] the Landau equation (21). The Lenard–Balescu equation can be derived from the Bogoliubov integral equation [85], from diagram techniques [86], from a quasilinear theory [113] or from the BBGKY hierarchy [110,111]. The Lenard–Balescu equation conserves mass and energy (reducing to the kinetic energy for a spatially homogeneous system) and monotonically increases the Boltzmann entropy (H -theorem). The collisional evolution is due to a condition of resonance between the orbits of the particles. For spatially homogeneous systems, the condition of resonance encapsulated in the δ-function corresponds to k · v1 = k · v2 with v1  = v2 . For d > 1, the only stationary solution of the Landau and Lenard–Balescu equations is the Maxwell distribution. These equations relax towards the Maxwell distribution (75) for t → +∞. Since the collision term in Eq. (76) is valid at the order O(1/N ), the relaxation time scales like t R ∼ N t D , (d > 1).

(78)

This scaling, predicted in [74], has been numerically observed for a 2D Coulombian plasma [114]. For one-dimensional systems, like the HMF model, the situation is different [66,69,74]. For d = 1, the kinetic equation (76) reduces to ∂ ∂ f1 = 2π 2 m ∂t ∂v1



k2 u(k) ˆ 2 δ(v1 − v2 ) dv2 dk |k| |(k, kv2 )|2



∂ f1 ∂ f2 f2 − f1 ∂v1 ∂v2

 = 0.

(79)

Therefore, the collision term C N [ f ] vanishes at the order 1/N because there is no resonance. The kinetic equation reduces to ∂ f /∂t = 0 so that the distribution function does not evolve at all on a timescale ∼ N t D . This implies that, for one-dimensional homogeneous systems, the relaxation time towards statistical equilibrium is larger than N t D . We thus expect that t R > N t D , (d = 1).

(80)

For the HMF model, when the system remains spatially homogeneous, it is found that the relaxation time scales like t R ∼ e N t D [72,73]. When the statistical equilibrium state is inhomogeneous, other scalings like t R ∼ N t D [67] or t R ∼ N 1.7 t D [68] have been reported. The fact that the Lenard–Balescu collision term vanishes in 1D is known for a long time in plasma physics [80]. This result is also similar to the one found for 2D vortices [55] (Sect. 3.5).

4.3 Test particle in a thermal bath: the Fokker–Planck equation We now consider a “test” particle (tagged particle) evolving in a steady distribution of “field” particles. Let us call P(v, t) the probability density of finding the test particle with velocity v at time t. The evolution of P(v, t) can be obtained from the Lenard–Balescu equation (76) by considering that the distribution f 2 of the field particles is fixed. Thus, we replace f 1 = f (v1 , t) by P = P(v, t) and f 2 = f (v2 , t) by f 0 = f (v0 ) where Reprinted from the journal

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f (v) is any stable stationary solution of the Vlasov equation. This procedure transforms the integro-differential equation (76) into a differential equation [74]:    ∂ ∂P u(k) ˆ 2 ∂ ∂ d μ ν δ[k · (v − v0 )] − ν f (v0 )P(v, t), (81) = π(2π) m μ dv0 dkk k ∂t ∂v |(k, k · v)|2 ∂v ν ∂v0 where  ˆ (k, ω) = 1 + (2π)d u(k)

k · ∂∂vf dv, ω−k·v

(82)

is the dielectric function corresponding to the fixed distribution function f (v). For example, for the Maxwellian distribution function (75), the dielectric function can be expressed as [110,111]:

 ω (k, ω) = 1 + (2π)d u(k)βmρW ˆ βm , (83) k where 1 W (z) = √ 2π



x 2 e−x /2 dx, x −z

(84)

is the W -function of plasma physics (the integral has to be performed along the Landau contour). For any complex z, we have the analytical formula [110,111]: W (z) = 1 − ze

−z 2 /2

z e

y 2 /2

 dy + i

π −z 2 /2 . ze 2

(85)

0

Equation (81) can be written in the form of a Fokker–Planck equation (25) involving a diffusion term  u(k) ˆ 2 μν d δ[k · (v − v0 )] f (v0 ), D = π(2π) m dv0 dkk μ k ν |(k, k · v)|2

(86)

and a friction term ημ = π(2π)d m



dv0 dkk μ k ν

∂f u(k) ˆ 2 δ[k · (v − v0 )] ν (v0 ). 2 |(k, k · v)| ∂v0

(87)

In plasma physics, these expressions for the diffusion and the friction were first derived by Hubbard [115] by directly evaluating  v μ v ν  and  v μ . This was done independently from the works of Lenard and Balescu (see discussion in [84]). If we ignore collective effects, these expressions can also be obtained directly from the Hamiltonian equations by making a systematic expansion of the trajectory of the particles in powers of 1/N in the limit N → +∞ as shown in Appendix A of [84]. The properties of the Fokker–Planck equation (81) are studied in [17,74] depending on the dimension of space d. For d > 1, P(v, t) relaxes for t → +∞ towards a distribution function that is different from the distribution of the bath f (v) except if f (v) is the Maxwell distribution (this is because for d > 1 the Maxwellian distribution is the unique steady state of the Lenard–Balescu equation). Since the Fokker–Planck equation is valid at the order O(1/N ), the timescale of relaxation is N t D . However, if the distribution f of the bath is different from the statistical equilibrium state (Maxwellian), it will change precisely on the timescale N t D (see Sect. 4.2) due to “collisions” (finite N effects). Therefore, a non-isothermal bath is not steady on the timescale on which the relaxation of the test particle proceeds and the bath approach is not self-consistent. As a result, for d > 1, only a thermal bath should be considered. For the Maxwellian distribution (75), we find that the coefficients of diffusion and friction are related to each other by the Einstein relation ημ = −βm D μν v ν . Therefore, the Fokker–Planck equation takes the form (32) with a diffusion coefficient given by Eq. (86) where the dielectric function is given by Eqs. (83) and (85) (see [74] for μ ν more details). The diffusion coefficient can be decomposed in the form D μν = (D − 21 D⊥ ) v v 2v + 21 D⊥ δ μν so that Fpol = −βm D v. Explicit expressions of the diffusion coefficients D and D⊥ for d = 3 (well-known) and d = 2 (new) are given in [17] when collective effects are neglected.

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In d = 1, the situation is different [66,69,74]. The Fokker–Planck equation (81) reduces to  

∂P ∂P ∂ d ln f = D(v) −P , ∂t ∂v ∂v dv

(88)

where D(v) is given by +∞ k u(k) ˆ 2 D(v) = 4π m f (v) dk . |(k, kv)|2 2

(89)

0

Equation (88) is similar to the Smoluchowski equation describing the motion of a Brownian particle in a “potential” U (v) = − ln f (v). In d = 1, the distribution function of the test particle P(v, t) always relaxes towards the distribution of the bath f (v) for t → +∞. The typical timescale governing the relaxation of the test particle towards the bath distribution is of order N t D . On this timescale, the distribution of the bath is steady because the collision term of order O(1/N ) vanishes in d = 1 (see Sect. 4.2). Therefore, the bath approach is valid for any Vlasov stable distribution f (v) of the field particles. The particularity of the dimension d = 1 is due to the fact that the Lenard–Balescu collision operator (79) cancels out for any distribution function f (v) while in d = 2 and d = 3 the cancellation of the collision operator (76) occurs only when the distribution is Maxwellian. These results are similar to those found for 2D vortices (see Sect. 3.6) and, indeed, Eq. (88) is the counterpart of Eq. (67). For the Maxwellian distribution in d = 1, using Eqs. (83) and (85), the Fokker–Planck equation (88) and the diffusion coefficient (89) can be rewritten [74]:  

∂P 1 ∂ ∂P = G(x) + 2P x , (90) ∂t tR ∂ x ∂x with +∞ G(x) = 2L dk

kη(k)2 e−x

2

0

2

[1 − η(k)B(x)]2 + πη(k)2 x 2 e−2x

2

,

(91)

and t R = 4(2π)1/2

ρ L2 . mvm

(92)

√ 2 x 2 2v 2 In the foregoing formulae, we have set x = βm/2v, D(v) = t Rm G(x), B(x) = 1 − 2xe−x 0 e y dy and η(k) = −2π u(k)βmρ. ˆ We have also introduced the size of the domain L in order to make Eq. (91) dimen2 x 2 sionless. The function B(x) can be written B(x) = 1 − 2x D(x) where D(x) = e−x 0 e y dy is Dawson’s integral. It has the asymptotic behaviors B(x) = 1 − 2x 2 + · · · for x → 0 and B(x) ∼ − 2x1 2 for x → +∞. Finally, t R is a typical relaxation time constructed with the r.m.s. velocity vm = (1/βm)1/2 . The relaxation time is equal to t R = 4(2π)1/2 N t D where we have introduced the dynamical time t D = L/vm and the particle number N = ρ L/m. As indicated previously, the Fokker–Planck collision term scales like 1/N in the thermodynamic limit N → +∞, so the relaxation time scales like tR ∼ N tD .

(93)

For periodic potentials, like in the HMF model, the integral   over k in Eq. (91) must be replaced by a discrete summation over the different modes using dk ↔ 2π n . The kinetic theory of the HMF model has been L developed by Bouchet [116], Bouchet and Dauxois [69] and Chavanis et al. [66,70,74]. The kinetic equations can be readily obtained as a particular case of the general Fokker–Planck equation (81) for a one dimensional k system (d = 1) in which the potential of interaction is truncated to one Fourier mode u i j = − 4π cos(θi − θ j ) k of period L = 2π. The Fourier transform of the potential is uˆ n = − 4π (δn,1 + δn,−1 ). In that case, the Fokker–Planck equation (81) reduces to Eq. (88) where D(v) is given by D(v) = Reprinted from the journal

k 2 f (v) . 4 |(1, v)|2 251

(94)

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On the other hand, for a thermal bath with Maxwellian distribution (75), the Fokker–Planck equation (81) can be written as Eq. (90) with a diffusion coefficient G(x) =

8π 2 e−x

2

(T /Tc − B(x))2 + π x 2 e−2x

where we have used ηn = η(δn,1 + δn,−1 ) with η =

kM 4π T

=

Tc T

2

,

(95)

(see Sect. 4.1).

5 Conclusion In this paper, we have stressed the analogies and the differences between the kinetic theories of stellar systems and 2D vortices (and other systems with long-range interactions like the HMF model). We have developed a common formalism [18,60,74], based on an expansion of the BBGKY hierarchy in powers of 1/N in a proper thermodynamic limit N → +∞, which allows one to draw a close parallel between the two systems (see Table 1). In particular, this formalism takes into account the spatial inhomogeneity of the system that is specific to systems with long-range interactions (another possibility to describe inhomogeneous systems is to introduce angle-action variables [117]). This contrasts with more traditional kinetic theories applying to neutral gases and plasmas where the system is spatially homogeneous due to short-range interactions or Debye shielding. In the case of stellar systems, it is well-known from the work of Chandrasekhar [12] that the kinetic evolution of the system is due to a process of diffusion (in velocity space) counterbalanced by a dynamical friction, like in Brownian theory. In the case of point vortices, we have found [55,60,106] that the kinetic evolution of the system is due to a process of diffusion (in position space) counterbalanced by a systematic drift. The kinetic theory of stars and 2D vortices allows us to determine relevant timescales in the evolution of the system. In stellar dynamics, the timescale of two-body relaxation is t R ∼ (N / ln N )t D . However, the complete relaxation towards the Boltzmann distribution is hampered by the problems of evaporation and gravothermal catastrophe [1,15,16,28,31]. The relaxation time of a test star in a cluster of field stars (bath) is also t R ∼ (N / ln N )t D . In vortex dynamics, the timescale of two-body relaxation is t R ∼ N t D . However, the collisional evolution stops as soon as the profile of angular velocity becomes monotonic even if the system has not reached the Boltzmann distribution. Therefore, the system can remain frozen in a stationary solution of the 2D Euler equation for a long time larger than N t D . Only non-trivial correlations between vortices (three-body, four-body, etc.) can induce further evolution of the system. Therefore, the relaxation towards the Boltzmann distribution, if it really occurs (!), takes a time larger than N t D . In contrast, the relaxation time of a test vortex in a “sea” of field vortices (bath) scales like t R ∼ (N / ln N )t D . For systems with longrange interactions and non-singular potential, the relaxation time of a spatially homogeneous system towards the Boltzmann distribution is t R ∼ N t D for d ≥ 2 and longer for d = 1 (scalings like N t D , N 1.7 t D and e N t D have been reported for the HMF model). The relaxation time of a test particle in a bath scales like tR ∼ N tD . Table 1 Analogies between the different kinetic equations for stellar systems and 2D vortices Domain of validity N -body problem Exact kinetic equation for PN (x1 , ..., x N , t) Kinetic equation for P1 (x, t) when N → +∞ Violent relaxation: kinetic equations for the coarse-grained field General kinetic equation for P1 (x, t) at the order O(1/N ) Simplified kinetic equation for P1 (x, t) at the order O(1/N ) Test particle in a (thermal) bath Second moment ( x)2  First moment ( x)

123

Stellar systems Newton equations (1) Liouville equation (9)

2D vortices Kirchhoff equations (39) Liouville equation (46)

Vlasov equation (19)

2D Euler equation (53)

Maximum Entropy Production Principle [49] or quasilinear theory [80–82,84] Kinetic equation (20)

Maximum Entropy Production Principle [49,99,100] or quasilinear theory [60,101] Kinetic equation (54)

Landau equation (21)–(22) with local approximation Fokker–Planck equation (24) or (32) Diffusion coefficient (26) or (34)-(36) Dynamical friction (27) or (31)

Kinetic equation (55)–(56) for axisymmetric flows Fokker–Planck equation (58) or (67) or (71) Diffusion coefficient (60) or (64) Systematic drift (61) or (66) or (70)

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For simplicity, we have described the case of a single species system of stars or point vortices, but the generalization of the kinetic theory to a multi-species system is possible: for stellar systems, we refer to [11,12,16,17], for the HMF model to [66] and for point vortices to [63,105,118–121] for various approaches. Acknowledgments I am grateful to H. Aref, E. Spiegel, G.J.F. van Heijst and P. Newton for encouragements and interesting discussions during the IUTAM symposium “150 years of vortex dynamics” (Lyngby, 12–16 october 2008). I am also grateful to S. Coleman for bringing to my attention the paper of Khanin [104].

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118. 119. 120. 121.

Marmanis, H.: Proc. R. Soc. Lond. A 454, 587 (1998) Sire, C., Chavanis, P.H.: Phys. Rev. E 61, 6644 (2000) Newton, P.K., Mezic, I.: J. Turbul. 3, 52 (2002) Nazarenko, S., Zakharov, V.E.: Physica D 56, 381 (1992)

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Theor. Comput. Fluid Dyn. (2010) 24:241–245 DOI 10.1007/s00162-009-0115-8

O R I G I NA L A RT I C L E

Nicholas Kevlahan

Vortices for computing: the engines of turbulence simulation

Received: 28 October 2008 / Accepted: 4 February 2009 / Published online: 12 June 2009 © Springer-Verlag 2009

Abstract Vortices have been described as the “sinews of turbulence”. They are also, increasingly, the computational engines driving numerical simulations of turbulence. In this paper, I review some recent advances in vortex-based numerical methods for simulating high Reynolds number turbulent flows. I focus on coherent vortex simulation, where nonlinear wavelet filtering is used to identify and track the few high energy multiscale vortices that dominate the flow dynamics. This filtering drastically reduces the computational complexity for high Reynolds number simulations, e.g. by a factor of 1000 for fluid–structure interaction calculations (Kevlahan and Vasilyevvon in SIAM J Sci Comput 26(6):1894–1915, 2005). It also has the advantage of decomposing the flow into two physically important components: coherent vortices and background noise. In addition to its computational efficiency, this decomposition provides a way of directly estimating how space and space–time intermittency scales with Reynolds number, Reα . Comparing α to its non-intermittent values gives a realistic Reynolds number upper bound for adaptive direct numerical simulation of turbulent flows. This direct measure of intermittency also guides the development of new mathematical theories for the structure of high Reynolds number turbulence. Keywords Turbulence · Vortices · Wavelets PACS 42.27.De, 42.27.Ek, 42.27.Jv, 47.32.C

1 Introduction von Helmholtz [12] attempted to address the failure of Euler’s inviscid equations to accurately describe fluid flow by introducing the concept of vorticity, and idealized singular flow structures such as vortex filaments and vortex sheets [4]. Although viscosity does not appear explicitly in Helmholtz’s vorticity equation, Helmholtz thought of vorticity as being introduced into the flow via internal friction at solid boundaries. In fact, he used the concept of the unsteady vortex sheet to accurately calculate the tones of an organ pipe. Helmholtz ignored the details of vorticity production, which remains a difficult theoretical problem to this day. Thus, from the very beginning, vorticity and vortices have been used to model and compute fluid flow. Since Helmholtz’s pioneering work, vorticity-based descriptions have proven to be the simplest way of understanding and computing a wide variety of compressible and incompressible flows. This is due to the power of vorticity-based theorems such as Helmholtz’s theorem, Kelvin’s circulation theorem and the helicity Communicated by H. Aref N. Kevlahan Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada E-mail: [email protected] Reprinted from the journal

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Fig. 1 DNS of homogeneous isotropic turbulence at Reλ = 1217 at increasingly small scales (left to right) (from Yokokawa et al. [15]). The flow has spontaneously organized into many tube-like coherent vortices. Note the fractal-like structure of the vorticity field

conservation theorem. For example, the force on an obstacle is due to vortex motion,  1 d F (t) = − x × ω dV 2 dt V∞

as is far-field sound emitted from a localized source, such as a jet, pF = −

xi x j xi x j xk ε(1) (tr ) ρ0 (4) ρ0 (3) − 2 Q i j (tr ) 3 + 3 Q i jk (tr ) 4 + · · · 15πc2 r c r c r

where the Q’s are moments of the vorticity distribution. The lift generated by a wing, and other aerodynamic flows, can be explained by systems of vortex lines. Not surprisingly, rotational flow of superfluids can be described extremely well by collections of (quantized) vortex lines. Coherent vortices are the viscous flow structures corresponding to the vortex filaments of ideal flow. In other words, they are localized regions of intense vorticity that persist for a significant period of time. Coherent vortices are most obvious in inhomogeneous flows, such as jets and mixing layers, but they are also the dominant flow structures of statistically homogeneous and isotropic turbulent flows (see Fig. 1). Coherent vortices are not just ubiquitous features of turbulent flows, they are also believed to control many aspects of turbulence dynamics. In the same way that Helmholtz used the vortex sheet as a simplified model of the driven flow in an organ pipe, coherent vortices could form the basis of a reduced theoretical model of turbulence. It is important to distinguish between statistically representative coherent structures and instantaneous coherent vortices. Proper orthogonal decomposition (POD) decomposes a turbulent velocity field into a set of orthogonal eigenfunctions which are solutions of the eigenvalue problem  (n) (n) Ri j (x, x  ) φ j (x  ) dx  = λ(n) φi (x), where Ri j (x, x  ) = u i (x)u j (x  ) is the two-point correlation tensor. If a flow is periodic in time, the first eigenmode few eigenmodes represent typical coherent structures. However, it is important to remember that since the POD modes correspond to eigenfunctions of energy (not enstrophy), they do not correspond precisely to vortices. In addition, if the turbulence is not periodic in time, the POD modes do not represent actual instantaneous flow structures. In fact, if the turbulence is only statistically stationary in time and homogeneous in space the POD modes are simply Fourier modes (i.e. sines and cosines), which are certainly not coherent vortices! Nevertheless, the first few POD modes provide significant insight into the flow structure, and can form the basis of a reduced dynamical model of certain flows (e.g. jets). Coherent vortices, in contrast, are defined based on the analysis of an individual flow realization at an instant in time. Coherent vortices were introduced as a way of visualizing flows. The criterion for identifying a coherent vortex may be based simply on vorticity thresholding, or on more robust measures, such as the eigenvalues of the symmetric tensor S 2 + 2 [6]. We focus on an alternative definition, based on de-noising the turbulent flow (i.e. coherent vortices are what is left after the turbulence has been de-noised). As in the case of POD modes, coherent vortices can also form the basis of a reduced dynamical model of the flow. It is therefore natural to try to use vorticity or coherent vortices as the basis for numerical simulations of fluid flow.

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Vortices for computing: the engines of turbulence simulation

Fig. 2 Vorticity field of two-dimensional turbulence at Re = 40400 (from Kevlahan et al. [8]). a Computed from 263169 Fourier modes using the pseudo-spectral method, b computed using 7895 coherent wavelet modes, c energy spectra: dashed line wavelet, solid line pseudo-spectral

2 Vortices for computation Vortex methods for the numerical simulation of fluid dynamics date back over 80 years to Prager [10] and Rosenhead [11]. However, they were not used commonly (or justified mathematically) until the advent of powerful electronic computers and the mathematical justification provided by the work of Chorin [2], Beale and Majda [1] and others. Although they were originally developed for two-dimensional flows, a highly accurate three-dimensional vortex method for wake flows was introduced by Winckelmans and Leonard [13]. In vortex methods the circulation of the initial vorticity field is first discretized onto a set of N particles (or point vortices). The particles move by Lagrangian advection due to the velocity field of all other particles. Vorticity diffusion is modelled either by adding Brownian noise, or by re-distributing circulation amongst nearby particles at the end of each time step. Since there is no explicit boundary condition for the vorticity equations, vorticity production at solid boundaries is included computationally at each time step by adding the vorticity flux necessary to cancel velocity at the boundary [3]. This new vorticity is then advected away from the boundary. Following Helmholtz, vortex methods are used primarily for fluid–structure interaction problems. Although the vortex particles are indeed the “engines” of the numerical simulation, they do not correspond to the coherent vortices seen in direct numerical simulations (DNS) and laboratory experiments. In other words, this method does not cut away the noisy “fat” from the coherent “sinews” of turbulence. In order to reduce the computational complexity of numerical simulations of high Reynolds number turbulence Farge et al. [5] proposed coherent vortex simulation (CVS) where the computational elements represent precisely the coherent vortices of the flow. To avoid the long-standing problem of how to define the coherent vortices in a turbulent flow we used the following computationally inspired ansatz: the coherent vortices are defined to be what remains once the (Gaussian) noise has been removed. Since noise is by definition incoherent, the remainder should correspond to what we intuitively think of as coherent vortices. These coherent vortices form the computational elements of the simulation: the wavelet modes are adapted at each time step to identify and track the coherent vortices. The effect of the noise is either modelled simply, or neglected entirely. De-noising is achieved by nonlinear wavelet filtering, since this procedure optimally removes additive Gaussian noise (see Fig. 3). Figure 2 shows an example of this approach applied to two-dimensional turbulence. It demonstrates that all scales of the coherent vortices are captured by only three of the total possible wavelet modes. The coherent vortices represented by the significant wavelet modes are both the “sinews” of the turbulence and the “engines” of the numerical simulation (since they are the computational elements, equivalent to the point vortices of vortex methods).

3 Vortices and intermittency Coherent vortex simulation is an efficient and accurate numerical scheme for solving the Navier–Stokes equations. More importantly, it is also a coherent vortex model of turbulence. A CVS in space (i.e. coherent vortex filtering in space at each time step) estimates the number of spatial degrees of freedom, while a CVS in space–time (i.e. coherent vortex filtering in both space and time simultaneously) estimates the total number Reprinted from the journal

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Fig. 3 PDF of vorticity for nonlinear wavelet filtering of 2-D turbulence. The PDF of coherent vorticity ω< matches the PDF of the total vorticity, while the PDF of incoherent vorticity ω< is Gaussian, i.e. it is noise

of dynamical degrees of freedom. Note that space–time CVS ensures the vortices are truly coherent: they necessarily exist for a significant period of time. Therefore, using CVS, the number of active wavelet modes is an upper bound on the number of degrees of freedom N required to represent a turbulent flow at a given Reynolds number. A sequence of CVS at different Reynolds numbers allows us to estimate the exponent α in the relation N ∼ Reα . A simple non-intermittent estimate gives α = 3 (or even α = 4 [14]) for space–time modes and α = 9/4 for spatial modes in threedimensional turbulence (or 3/2 and 1, respectively, for two-dimensional turbulence). Comparing the computed α to their equivalent non-intermittent values directly measures turbulence intermittency. If the active regions of the flow are assumed to have a fractal structure, α can be used to calculate the fractal dimension [9]. Note that a commonly used indicator of intermittency, the anomalous scaling of structure function exponents, measures intermittency only indirectly, and would suggest, contrary to observation, that two-dimensional turbulence is not intermittent. Our definition directly measures the proportion of the turbulent flow that is active in both space and time. Kevlahan et al. [8] were able to estimate, for the first time, that the number of space–time modes of a twodimensional turbulent flow scales like Re0.9 (compared with the usual estimate of Re1.5 ), while the number of spatial coherent modes scales like Re0.7 (compared with the usual estimate of Re1 ). These scaling exponents are a direct measure of the intermittency of the flow, and hence of the space-fillingness of the coherent vortices. The next step in this programme is to measure the space and space–time intermittency of three-dimensional flows. It is commonly believed that three-dimensional turbulence is more intermittent than two-dimensional turbulence (perhaps due to the vortex stretching term), although the range of active scales is larger. Threedimensional turbulence is a more challenging problem for adaptive methods like CVS, due to the complicated geometry and topology of coherent vortices in three dimensions (see Fig. 1). A more serious challenge is the interpretation of dynamics in the four-dimensional space–time domain. Can we find an (approximate) reduced three- or two-dimensional model of the full four-dimensional dynamics? The degree of intermittency (as measured by the exponent α) should answer this question (Fig. 4).

4 Conclusions Helmholtz originally introduced vorticity (and vortex lines and filaments) to improve the ability of Euler’s equations to compute real flows. Since then vorticity and vortex-based methods have led to better computational schemes and a deeper understanding of turbulent flows. We propose that CVS should be used not just to compute turbulent flows, but also to analyse their structure. The way the size of the CVS scales with Reynolds number provides a direct estimate of the intermittency of the flow (and hence the space-fillingness of its active regions). A space–time CVS calculation gives a numerical estimate of the size of the dynamical system governing the turbulent, and hence gives upper bounds on the dimension of a reduced dynamical model of turbulence. It is likely that the usual non-intermittent estimates

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Fig. 4 Adaptive wavelet grids at Re = 40400 (half the time domain, i.e. 9 time levels, is shown) (from Kevlahan et al. [8]). a Space–time grid: first time interval t ∈ [0, 2.1]. b Space–time grid: final time interval t ∈ [123.8, 126.0]. c Spatial grid only at t = 126.0. Note the strong time intermittency of the solution: the smallest time step is strongly localized in space

of computational degrees of freedom (Re3 or even Re4 in three dimensions) will prove unduly pessimistic if adaptive coherent vortex methods are used. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Beale, J., Majda, A.: Vortex method I: convergence in three dimensions. Math. Comput. 39, 1–27 (1982) Chorin, A.: Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–796 (1973) Cottet, G., Koumoutsakos, P.: Vortex Methods: Theory and Practice.Cambridge University Press, London (2000) Darrigol, O.: Worlds of Flow, chap. 4. Oxford University Press, Oxford (2005) Farge, M., Schneider, K., Kevlahan, N.K.R.: Non-gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis. Phys. Fluids 11, 2187–2201 (1999) Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995) Kevlahan, N., Vasilyev, O.: An adaptive wavelet collocation method for fluid–structure interaction at high Reynolds numbers. SIAM J. Sci. Comput. 26(6), 1894–1915 (2005) Kevlahan, N.K.R., Alam, J., Vasilyev, O.: Scaling of space–time modes with Reynolds number in two-dimensional turbulence. J. Fluid Mech. 570, 217–226 (2007) Paladin, G., Vulpiani, A.: Degrees of freedom of turbulence. Phys. Rev. A 35(4), 1971–1973 (1987) Prager, W.: Die druckverteilung an körpern in ebener potentialströmung. Phys. Z. 29, 865–869 (1928) Rosenhead, L.: The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond. Ser. A 134, 170–192 (1931) von Helmholtz, H.: Über integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entspreschen. JRAM 55, 25–55 (1858) Winckelmans, G., Leonard, A.: Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows. J. Comput. Phys. 109, 247–273 (1993) Yakhot, V., Sreenivasan, K.: Anomalous scaling of structure functions and dynamic constraints on turbulence simulations. J. Stat. Phys. 121(5/6), 823–841 (2005) Yokokawa, M., Itakura, K., Uno, A., Ishihara, T., Kaneda, Y.: 16.4-Tflops direct numerical simulation of turbulence by a fourier spectral method on the earth simulator. In: Proceedings of the IEEE/ACM SC2002 Conference, p. 50. IEEE Computer Society (2002)

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Theor. Comput. Fluid Dyn. (2010) 24:247–251 DOI 10.1007/s00162-009-0100-2

O R I G I NA L A RT I C L E

Paolo Orlandi · Sergio Pirozzoli

Vorticity dynamics in turbulence growth

Received: 4 November 2008 / Accepted: 18 March 2009 / Published online: 28 April 2009 © Springer-Verlag 2009

Abstract The mechanisms of vorticity amplification in the formation of turbulence are investigated by means of direct numerical simulations of the Navier–Stokes equations with different initial conditions and Reynolds numbers. The simulations show good universality of the enstrophy evolution, that occurs in two stages. The first stage is dominated by the effect of vortex stretching, and it finishes with a k −3 power-law energy spectrum. The second stage is dominated by the action of viscosity on the small scales, and it finishes with a Kolmogorov k −5/3 energy spectrum. Keywords Turbulence · Energy spectra · Vortex dynamics PACS 47.27.eK, 47.32.C-, 47.27.Cn

1 Introduction The statistical and structural properties of isotropic turbulence are today relatively well understood. The occurrence of a constant energy flux in wavenumber space giving a k −5/3 spectrum has been very well assessed both in experiments and computations. However, much less is known about the mechanisms of formation of turbulence starting from arbitrary deterministic initial conditions, and, in particular, about the physical phenomena underlying the onset of an inertial range scaling. This is, perhaps, due to the fact that the events leading to the transition to turbulence may sensitively depend upon the initialization details, and, therefore, may not be expected to be universal [4]. Brachet et al. [2] numerically simulated the early time evolution of the Taylor–Green vortex, and showed inviscid, laminar dynamics at the early times, and the formation of a k −n power-law spectrum (with n  1.6 − 2.2) around the peak enstrophy time (t ). Highly symmetric initial conditions, corresponding to the so-called Kida–Pelz flow, were considered in the study of Boratav and Pelz [1]. Those authors found an initial (at-least) exponential growth of the enstrophy, followed by a breakup of the initially smooth flow field, associated with sharp peaks of the velocity derivative skewness and flatness. During this period of intense events, the maximum vorticity was found to scale approximately as (tc − t)−1 , and all flow statistics were found to be far from Kolmogorov-type scaling laws (with spectra evolving from k −4 to k −3 ); however, after the peak enstrophy is attained, a k −5/3 spectral range was found. Direct numerical simulations (DNS) of the Navier–Stokes equations at low Reynolds numbers starting from random initial conditions (with energy concentrated at the smallest wavenumbers) were performed by Holm and Kerr [3]. Those authors analyzed the evolution of the global enstrophy, and were able to identify two time scales, one Communicated by H. Aref P. Orlandi · S. Pirozzoli (B) Dipartimento di Meccanica e Aeronautica, Università di Roma “La Sapienza”, Via Eudossiana 18, 00184 Rome, Italy E-mail: [email protected] E-mail: [email protected] Reprinted from the journal

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associated with the enstrophy production (which is independent of Re), and one associated with the saturation of enstrophy, which was shown to depend strongly on Re. In this paper, we present results of initial value problems, with different orderly initial conditions and different Reynolds numbers, with the objective to investigate the possible occurrence of a universal mechanism of transition to turbulence. For this purpose we analyze the statistical properties of the flows in terms of the global enstrophy budget and of the energy spectra. 2 Numerical method and initial conditions The three-dimensional incompressible Navier–Stokes equations are solved in a cubic box of size 2π L (where L is the reference length) with the assumption of periodicity in the three coordinate directions. The numerical method is described in detail in Orlandi and Carnevale [8]. The initial conditions for the numerical tests are as follows. 2.1 Colliding Lamb dipoles The Lamb dipole is an exact two-dimensional form preserving propagating solution of the Euler equations [6]) having continuous velocity field and continuous first spatial derivatives. In polar coordinates centered in the dipole, the associated vorticity distribution is  (K r ) −2U K JJ01(K a) sin(θ − θ0 ), r < a , ω= (1) 0 r ≥a where, U is the propagation speed of the dipole, a is its radius (a = L is assumed here), and K is a constant such that K a/L ≈ 3.8317 is the first strictly positive zero of the Bessel function J1 . Eq. 1 is a steady solution of the Euler equations in the co-moving reference frame. The initial velocity conditions for this test case have been enforced by placing three dipoles of the type given by Eq. 1 with vorticity parallel to the x3 direction, with centers at x1 /L = π/3, π, 5π/3, x2 /L = π/2, and moving in the positive-x2 direction, and three dipoles with vorticity parallel to the x1 direction, centered at x3 /L = π/3, π, 5π/3, x2 /L = 3π/2, and translating in the negative-x2 direction. With such arrangement, collision of the dipoles nominally occurs at tU/L ≈ 4. The initial energy spectrum for this flow field is sharply decreasing as k −6 [8]. 2.2 Taylor–Green flow The initial conditions for this test case are given by u 1 = U sin(x1 ) cos(x2 ) cos(x3 ), u 2 = U cos(x1 ) sin(x2 ) cos(x3 ), u 3 = 0.

(2)

This velocity field gives an energy spectrum with only one non-zero mode corresponding to k L = 1. 2.3 Kida–Pelz flow The initial conditions for this test case are given by u 1 = U sin(x1 ) (cos(3x2 ) cos(x3 ) − cos(x2 ) cos(3x3 )), u 2 = U sin(x2 ) (cos(3x2 ) cos(x3 ) − cos(x2 ) cos(3x3 )), u 3 = U sin(x3 ) (cos(3x1 ) cos(x2 ) − cos(x1 ) cos(3x2 )).

(3)

This flow was first considered by Kida and √ Murakami [5], and it gives an initial energy spectrum with only one non-zero mode corresponding to k L = 11.

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All calculations have been performed for three values of the Reynolds number Re = U L/ν = 1,000, 3,000, 5,000, on a 3843 grid, which allows sufficient resolution of the flow up to the smallest scales. In the following, all quantities are made non-dimensional with respect to the reference velocity U and the reference lengthscale L.

3 Results   2 The time evolution of the total enstrophy  = 1/2 i ωi is reported in Fig. 1. For representation purposes, in the figure, we report the time derivative of  as a function of  itself, both properly scaled with respect to the initial value 0 . This type of representation is very useful since it allows to easily detect power-law ˙ scalings of . Indeed, (t) ∼ bα implies  1 1−α (t) ∼ (tbtc − t) , α  = 1 , (4) α=1 e , with subsequent blow-up at finite time tc if α > 1. If α = 1, exponential growth of  follows. The figure clearly indicates a roughly universal, two-stage evolution of the enstrophy. In the first stage, a super-exponential growth of  ∼ (tc − t)−2 occurs up to  ≈ 5 0 , corresponding to a time t = t ∗ , which depends upon the initial conditions. In the second stage, exponential growth of  occurs, up to a time t = t at which enstrophy attains its maximum value, which depends upon both the initial conditions and the Reynolds number. To interpret these results, it is useful to consider the enstrophy evolution equation   d  ∂ 2 ωi , (5) = ωi ω j Si j + ν ωi dt ∂x j∂x j  

 

P D

where, P and D denote the enstrophy production due to stretching, and the enstrophy dissipation, respectively. The evolution of P and D is reported in Figs. 2 and 3, respectively. The figures show a good deal of universality in the selected representation, and indicate that the enstrophy evolution is inviscid-dominated in the first stage, up to the time t ∗ , at which viscous dissipation starts to counteract vortex stretching. After t ∗ , the enstrophy production continues to increase as 3/2 , but the dissipation term also increases as 3/2 , thus making the overall enstrophy growth only exponential. The results are consistent with the observations made by Orlandi and Carnevale [8] for the inviscid interaction of orthogonal Lamb dipoles. They found that positive correlation of the stretching rate with the vorticity modulus is related to the alignment of the vorticity vector with the intermediate eigenvector of the rate of strain tensor,  and  proportion 2 2 ality between s2 and ω2 . Observing that the vortex stretching term can be cast as P = s ω ≈ s2 ω2 , it follows that P ∼ 3/2 . ˙ with  indicates that the effect of viscosity is During the second stage, the diminished (linear) growth of  to inhibit the positive correlation between vorticity and stretching rate, and the flow evolves if it was subjected Reprinted from the journal

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to a constant, positive σ . It is also interesting to observe that in the second stage, the scaling of the dissipation as 3/2 implies ν|∇ω| ∼  2

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(6)

where, is a characteristic lengthscale for vorticity fluctuations. Equation 6 indicates that, since ∼ η, a Kolmogorov scaling establishes in the flow well before the enstrophy peak occurs. For all test cases, the energy spectra exhibit a negative power-law scaling during the phase of enstrophy growth, with a slope that decreases with time, and that depends upon the initial conditions. The energy spectra at the end of the first stage and around the peak enstrophy time are reported in Fig. 4. At the end of the first stage, a k −3 power law emerges for all test cases. This is in agreement with the findings of Orlandi [7] for Lamb dipole collision and for the Taylor–Green vortex in the inviscid case. In the inviscid case, the k −3 power law occurs right before the supposed finite-time singularity. A Kolmogorov k −5/3 scaling (with Kolmogorov constant C ≈ 1.64) is attained for all test cases around t ≈ t , and it persists for some time after the enstrophy peak.

4 Conclusions The evolution of enstrophy in smooth initial value problems for the Navier–Stokes equations has been analyzed at different Reynolds numbers. The simulations indicate that the enstrophy evolves according to a roughly universal, two-stage mechanism. The first stage is dominated by the action of the vortex stretching mechanism, that would lead to blow-up of enstrophy in finite time, and it ends with strong events leading to the formation of a k −3 energy spectrum. During the second stage, viscosity becomes important starting from the smallest scales, and the rate of enstrophy growth is reduced, in a manner that seems to rule out the possibility of a

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finite-time singularity for the Navier–Stokes equations. The presence of a finite viscosity seems also to be decisive for the formation of a Kolmogorov k −5/3 energy spectrum, that establishes around the peak enstrophy time. References 1. Boratav, O.N., Pelz, R.B.: Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. Phys. Fluids 6, 2757–2784 (1994) 2. Brachet, M.E., Meiron, D.I., Orszag, S.A., Nickel, B.G., Morf, R.H., Frisch, U.: Small scale structure of the Taylor–Green vortex. J. Fluid Mech. 130, 411–452 (1983) 3. Holm, D.D., Kerr, R.: Development of enstrophy and spectra in numerical turbulence. Phys. Fluids A 5, 2792–2798 (1993) 4. Holm, D.D., Kerr, R.: Transient vortex events in the initial value problem for turbulence. Phys. Rev. Lett. 88, 244501 (2002) 5. Kida, S., Murakami, Y.: Kolmogorov’s spectrum in a freely decaying turbulence. J. Phys. Soc. Jpn. 55, 9–12 (1986) 6. Lamb, H.: Hydrodynamics. Cambridge University Press, London (1932) 7. Orlandi, P.: Energy spectra power laws and structures. J. Fluid Mech. 623, 353–374 (2009) 8. Orlandi, P., Carnevale, G.F.: Nonlinear amplification of vorticity in inviscid interaction of orthogonal Lamb dipoles. Phys. Fluids 19, 057106 (2007)

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Theor. Comput. Fluid Dyn. (2010) 24:253–258 DOI 10.1007/s00162-009-0135-4

O R I G I NA L A RT I C L E

E. A. Kuznetsov · V. Naulin · A. H. Nielsen · J. Juul Rasmussen

Sharp vorticity gradients in two-dimensional turbulence and the energy spectrum

Received: 11 January 2009 / Accepted: 24 June 2009 / Published online: 21 July 2009 © Springer-Verlag 2009

Abstract Formation of sharp vorticity gradients in two-dimensional (2D) hydrodynamic turbulence and their influence on the turbulent spectra are considered. The analog of the vortex line representation as a transformation to the curvilinear system of coordinates moving together with the di-vorticity lines is developed and compressibility of this mapping appears as the main reason for the formation of the sharp vorticity gradients at high Reynolds numbers. In the case of strong anisotropy the sharp vorticity gradients can generate spectra which fall off as k −3 at large k, which appear to take the same form as the Kraichnan spectrum for the enstrophy cascade. For turbulence with weak anisotropy the k dependence of the spectrum due to the sharp gradients coincides with the Saffman spectrum: E(k) ∼ k −4 . Numerical investigations of decaying turbulence reveal exponential growth of di-vorticity with a spatial distributed along straight lines. Thus, indicating strong anisotropy and accordingly the spectrum is close to the k −3 -spectrum. Keywords Turbulence · Di-vorticity · Euler equations · Kraichnan and Saffman spectra PACS 52.30.Cv, 47.65.+a, 52.35.Ra

1 Introduction We present investigations of the relation between vortical structures and the energy spectrum in 2D turbulent flows with particular attention to the formation of sharp vorticity gradients and their influence on the spectrum in the large wavenumber regime. There are two types of well-known spectra for 2D turbulence. The first one, derived by Kraichnan [1–3], corresponds to an enstrophy cascade toward the small-scale regime, where viscous dissipation becomes essential. The Kraichnan spectrum has, up to a logarithmic factor, a power law dependence in the inertial range: E(k) ∼ k −3 . (Recall that 2D turbulence also possess an inverse energy cascade toward large scales leading to the Kolmogorov dependence E(k) ∼ k −5/3 [1–3]). The other spectrum, obtained by Saffman [4], has a different power dependence: E(k) ∼ k −4 . This is ascribed to sharp vorticity gradients forming in decaying 2D turbulence at high-Reynolds numbers. Under the assumption of isotropy and a dilute distribution of the localized regions of sharp gradients Saffman constructed the energy spectrum at large k as a superposition of the spectra from the gradients. Communicated by H. Aref E. A. Kuznetsov P.N. Lebedev Physical Institute, 53 Leninsky Ave., 119991 Moscow, Russia E-mail: [email protected] V. Naulin · A. H. Nielsen · J. Juul Rasmussen (B) Risø National Laboratory for Sustainable Energy, Technical University of Denmark, P.O. Box 49, 4000 Roskilde, Denmark E-mail: [email protected] Reprinted from the journal

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Many numerical experiments for 2D turbulence (see, e.g., [5] and references therein for a detailed discussion of 2D turbulence simulations) show that the Saffman spectrum is formed with a good accuracy in the initial stage, before the excitation of the large-scale coherent vortices. Here, we would like to point to the interesting work by Ohkitani [6], where by means of the Weiss decomposition [7] it was shown that the so-called h-regions (h-hyperbolic, i.e., regions where straining is dominating over vorticity) give the spectrum k −3 , i.e., coinciding up to a logarithmic factor with the Kraichnan spectrum; the contribution from the e-regions (e-elliptic, i.e., vorticity dominated regions) yields a behavior like the Saffman spectrum ∼ k −4 . We present qualitative arguments for the formation of sharp vorticity gradients in 2D flows for smooth initial conditions. The main idea is to apply an analog of the so-called vortex line representation (VLR) introduced for three-dimensional vortical flows [8–10]. This representation is based on a mixed Lagrangian–Eulerian description and connected with movable vortex lines. The VLR is a mapping to a curvilinear system of coordinates and turns out to be compressible, which appear to be a mechanism for enhancement of vorticity and may lead to formation of singularities. For 2D flows the vorticity is a Lagrangian invariant quantity and cannot be locally enhanced. However, the so-called di-vorticity, B = ∇ × ωˆz , represents a frozen-in field and satisfies an equation similar to the vorticity equation for 3D Euler flows. Thus, we apply the VLR to the di-vorticity field. The di-vorticity lines are compressed leading to a local enhancement of the di-vorticity and hence the vorticity gradient, which may grow very large, but cannot become infinite in finite time [11–13]. Considering the effect of these sharp vorticity gradients on the energy spectrum we follow Saffman [4]. Using the stationary phase method we demonstrate that the contribution from one discontinuity is very anisotropic: it has a sharp angular peak along the direction perpendicular to the discontinuity. In the peak the energy spectrum falls-off like k −3 . Assuming a dilute distribution of the sharp gradients and averaging over all angles in the case of isotropic turbulence the spectrum becomes ∼ k −4 , in full agreement with the Saffman spectrum. However, in the case of strong anisotropy the angle averaging results in a spectrum: ∼ k −3 [5]. To support the arguments above and reveal the connection between the sharp vorticity gradients and the tail of the energy spectrum, we have performed a numerical study of the evolution of decaying 2D turbulence (for details see [5]). We found that the di-vorticity growth is close to an exponential increase, that is far from rigorous estimations with a double exponential dependence [11–14]. Secondly, we observed that the di-vorticity maxima are distributed very sharply in space concentrated on a random net of lines that can be interpreted in favor of the Saffman mechanism for the formation of 2D turbulent spectra due to vorticity discontinuities. 2 Two-dimensional analog of the vortex line representation Consider 2D flow of an ideal fluid described by the Euler equation for the vorticity ω(x, y, t), dω ∂ω ≡ + (v · ∇)ω = 0, ∇ · v = 0, dt ∂t

(1)

where the velocity v defines the vorticity: ω = (∇ × v) · zˆ . According to Eq. (1) the vorticity is a Lagrangian invariant advected by the fluid being constant along a fluid particle trajectory. Let us introduce the divergence-free vector field B ≡ ∇ × ωˆz . It is easily seen that this vector is tangent to the line ω(x, y) = const. The equation of motion for B can be obtained from Eq. (1) after differentiating with respect to coordinates: ∂B = ∇ × [v × B]. (2) ∂t Thus, the vector B (often called the di-vorticity (see e.g. [15])) constitutes a frozen-in quantity. In terms of the substantial derivative Eq. (2) can be rewritten as  dB 1 ˆ = (B · ∇)v ≡ ωˆz × B + SB. (3) dt 2 Here the first term on the r.h.s. describes the rotation of the di-vorticity vector with the angular velocity −ω/2 and the second term is responsible for stretching of the di-vorticity lines where Sˆik = (∂vk /∂ xi + ∂vi /∂ xk )/2 is the stress tensor. Hence the divorticity length |B| will locally increase due to stretching when 1 dB2 ˆ > 0. = (B · SB) 2 dt

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The same effect takes place for three dimensional turbulence, where the stretching of vorticity lines enhances the vorticity. Increasing (or decreasing) the di-vorticity field, based on Eq. (4), is not sufficient to clarify the physical mechanism for its growth. As is seen from Eq. (2) only one velocity component, vn , normal to the vector B changes the field B. In this case the tangential component vτ (parallel to B) plays a passive role providing the incompressibility condition: ∇ · vτ + ∇ · vn = 0. This observation is the key point for introducing the vortex line representation (VLR) for the three-dimensional Euler equations [8–10]. To construct the analog of VLR for the 2D Euler equations we consider new Lagrangian trajectories, given by vn , dr = vn (r, t), r|t=0 = a. dt

(5)

The solution of these ODE’s defines the mapping r = r(a, t).

(6)

In terms of this mapping the di-vorticity equation (2) can easily be integrated: B(r, t) =

(B0 (a) · ∇a )r(a, t) , J

(7)

where B0 (a) is the initial di-vorticity, J is the Jacobian of the mapping (6): J = ∂(x, y)/∂(ax , a y ). According to the definition of the mapping the Jacobian is not fixed, it may change in time and space, i.e., the mapping r = r(a, t) represents a compressible mapping. This means that the di-vorticity lines can be compressed. In this approach the velocity of motion of di-vorticity lines is simply the normal velocity vn (it should be noted that this approach in slightly different form was suggested in [16]). As well-known from gas-dynamics compressibility of the mapping is the main cause for steepening and ultimately breaking, resulting in the formation of sharp gradients for the velocity and density of the gases. This happens in finite time and in the general situation the singularity first appears at one separate point. In gas-dynamics this process is completely characterized by the mapping determined by the transition from the Eulerian to the Lagrangian description. Vanishing of the Jacobian corresponds to the emergence of a singularity, i.e., a shock wave. The Jacobian in the denominator of the expressions (7) can tend to zero implying the increasing of the di-vorticity due to compressible character of the mapping. This leads to compression of di-vorticity lines corresponding to the formation of sharp gradients for the vorticity. However, the breaking process for 2D flows of ideal fluids can only take place in infinite time [11–13].

3 Two-dimensional spectra We will suppose that the formation of sharp gradients of the vorticity in 2D Euler flows is possible and consider how this process can effect the form of the turbulent spectrum. Let the vorticity have a jump  = (x) along the line y = 0 with both negligible jump width δ  L (L is the characteristic jump length) and weak bending of the line. Then we may write: ∂ω/∂ y = δ(y). Hence it is immediately seen that the Fourier transform of ω will have a power-law fall-off at large k, i.e., inversely proportional to k y multiplied by some function of k x due to the dependence of  on x. If we neglect the dependence on k x replacing it by some constant, we immediately obtain an energy spectrum with a power dependence, which appears to be similar to the Kraichnan spectrum: E(k) ∼ k −3 . This is an important conjecture demonstrating that a spectrum similar to the Kraichnan spectrum, which is often observed in high resolution numerical simulations, may be related to vorticity jumps. To find the spectrum one needs to calculate the Fourier transform of the pair correlation function F(r) = ω(x)ω(x + r), where angle brackets means average over the ensemble of jumps. Hence the energy density spectrum ε(k) is given by the standard expression: ε(k) =

Fk |ωk |2 = 2k 2 8π 2 Sk 2

where ωk is the Fourier transform of the vorticity ω(r) and the over-bar denotes average with respect to random variables of the jumps, i.e. their coordinates, directions, etc., and S is the average area, which is assumed to be sufficiently large. Reprinted from the journal

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To simplify the problem we will suppose that all jumps are concentrated on finite intervals with the vorticity jump vanishing at the interval endpoints x1,2 . In this case the Fourier amplitude k at large k, L −1  k  δ −1 , has a narrow angular peak which can be found by means of the stationary phase method. Then the results of average with respect to the coordinates of jumps and their lengths can be written as follows (before angle averaging!):  2 n  l , at θk ≤ θ0 ≡ (k L)−1 , (8) 2 4 8π k N ( )2  , at θk  (k L)−1 . (9) ε2 (k) ≈ 2 8 2 4π k cos θk sin4 θk x Here n = N /S is the density of jumps, l = x12 (x)d x, l = L the mean value of l = x2 − x1 , θk the angle between the jump normal n and the wave vector k,  denoting the derivative of  taken at the endpoints xi . These formulas demonstrate singular behavior for ε(k) for angles θk close to 0 and π/2 (as well as, to π and −π/2). At small angles θk ≤ (k L)−1 the expression (9) is matched with Eq. (8). For the angle range close to π/2 Eq. (9) is modified due to the bending of the line of the jump. Thus, the energy density distribution ε(k) has a very narrow angle maximum at θk near zero with decay at large wave numbers as ∼ k −4 , this results in the energy spectrum E(k) ∼ k −3 . For all other angles ε(k) decays proportionally to k −6 at large k. The final result for the energy spectrum can be obtained after averaging Eqs. (8) and (9) with respect to angles. For the isotropic case the angle average gives    2 n 2L 4

2 E(k) ≈ (10)  l  + ( )  , 2π 2 k 4 L 3 ε1 (k) ≈

which coincides with the spectrum obtained by Saffman [4]. In the anisotropic case the spectrum form depends on the ratio between θ0 = (k L)−1 and the width of the angle distribution function θ . If θ will be larger than θ0 then the spectrum fall-off will be the same as for the isotropic case, i.e., the Saffman spectrum (10). For the strong enough angle ordering (which can be conditioned by box boundaries as well as by anisotropy of the pumping of turbulence), when θ < θ0 , the energy spectrum E(k, θ ) will have very anisotropic behavior of the jet type: the fall-off ∝ k −3 at the jet −4 in perpendicular direction to the jet in accordance with (9). maximum with a decrease like ∝ k⊥ 4 Numerical modelling To support the arguments of the previous sections we have performed a numerical study of the evolution of decaying 2D turbulence. The Euler equations (1) with a hyperviscosity term to keep the integration scheme stable and avoid the so-called bottle-neck instability are integrated numerically on a double periodic domain by employing a high resolution fully de-aliased spectral scheme. The domain size was taken to be unity, i.e., determining the spatial scale, and the resolution was 2048 × 2048. For the time integration we employ a third order stiffly-stable scheme, and the time scale corresponds to ω0−1 , where ω0 is the maximum vorticity. Figure 1 shows the evolution of the vorticity from an initial distribution of randomly placed vortices all with amplitude ω0 = 1, Gaussian profiles and different sizes. At time = 100 the vorticity field has the typical structure for 2D turbulence; it consist of large scale structures (vortices) with concentrated vorticity and strongly filamented structures between the vortices. Corresponding to the vorticity field we show the instantaneous one-dimensional energy spectrum E(k) in Fig. 1b. For t = 0E(k) is the spectrum of superimposed Gaussian vortices, and at t = 95 a k −3 spectrum has developed at high wave numbers. Thus, with reference to the discussion of the Saffman spectra above this corresponds to the spectrum expected in the anisotropic regime where the stripes of vorticity gradients are near straight lines. In Fig. 2a, which depicts the structure of the di-vorticity field, we observe stripes of strongly amplified—up to thousand times—di-vorticity, and indeed the stripes are close to straight lines. Comparing the di-vorticity field with the high pass filtered vorticity field in Fig. 2b we observe a similar structure suggesting that the vorticity gradients are responsible for the large-k part of the spectrum, i.e., the k −α part. 2 , i.e., the square of maximum In Fig. 3a we show the time evolution of the maximum value of |B|2 , Bmax value of the vorticity gradient. The function is not smooth since we have plotted the absolute maximum within the domain at each time, and jumps in the first derivative appear when the maximum is appearing at a new

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(a)

(b) T= 0 T=15 T=95

k**3*E(k)

0.001

0.0001

1e-05

1e-06

1

10

100

1000

k

Fig. 1 a Vorticity field at time 100 corresponding to ≈ 8 vortex turnover times. b Compensated energy spectrum k 3 E(k)

(a)

(b)

Fig. 2 a The squared length of the di-vorticity vector |B|2 at time 100. The maximum value is 2.5 · 105 . b High pass filtered vorticity field from Fig. 1a, i.e., the vorticity at mode numbers k > 10 is plotted. The amount of energy in the filtered field is about 1% of the total energy

(a) 10

x 10

(b)

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9 5

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|

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|2

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Fig. 3 a The temporal evolution of the squared maximum of the di-vorticity equal to the squared maximum vorticity gradient |∇ω|2 . b Close up of the growth of the maximum in two cases

position. Each smooth part of the curve relates to the development of the maximum value at one point. We observe a rapid growth of Bmax , which then saturates and decays. The highest value attained by Bmax during this simulation approach 1000, which corresponds to the width of the filaments δ  0.001. The growth of Bmax is evidently arrested by the hyperviscosity. The growth of Bmax in one point is monitored Fig. 3b for a couple Reprinted from the journal

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of cases. The growth is initially exponential, but for later times the growth saturates and becomes weaker than exponential. This evolution certainly respects the bound on the vorticity gradient, e.g., [11–13]. ˆ [7] demonstrated that the Finally, an analysis of the Weiss field, W = 41 (σ 2 − ω2 ), where σ 2 = − 41 det[ S] di-vorticity stripes are aligned with the contours of W in the strain dominated regions, W > 0, while W < 0 in the stripes, i.e., vorticity dominates [5]. This is in line with the original arguments of Weiss (see also [6]). 5 Conclusion We have performed a detailed investigation of the relation between turbulent spectra and sharp vorticity gradients in 2D turbulent flows: –

We demonstrated that the k-behavior of the spectra generated by sharp vorticity gradients, originating from compressible advection of di-vorticity, depends significantly on the anisotropy of the spectra. If the angular distribution in the spectrum has one or more very sharp peaks then the one-dimensional spectrum has a tail falling-off like k −3 at large k. This demonstrates that a spectrum with a similar exponent as for the Kraichnan spectrum may indeed be argued to arise from sharp vorticity gradients in the case where the fronts of these gradients are along almost straight line segments. In the opposite case of an isotropic smooth angular dependence the spectrum has the asymptotic behavior k −4 as for the Saffman spectra. Simulations of decaying turbulence reveal very strong vorticity gradients concentrated on narrow stripes aligned along straight line segments and accordingly the spectral exponent is close to −3. The strong amplification of the di-vorticity of up to thousand times is one of the main results of the simulations. Fitting of the temporal behavior for the maxima demonstrates an initial exponential increase with saturation at later times. This amplification has a natural explanation in the compressibility of the mapping (6) providing the transfer from the Eulerian description to the system of movable curvilinear di-vorticity lines.

–

– – –

Acknowledgments This work was supported by the Danish Center for Scientific Computing and partially by INTAS. The work of E.K. was also supported by the RFBR (grants no. 06-01-00665 and no. 07-01-92165), the Program of RAS “Fundamental problems of nonlinear dynamics” and Grant NSh 7550.2006.2.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Kraichnan, R.H.: Inertial ranges in 2D turbulence. Phys. Fluids 11, 1417 (1967) Kraichnan, R.H.: Inertial range transfer in two and three-dimensional turbulence. J. Fluid Mech. 47, 525 (1971) Kraichnan, R.H.: Kolmogorovs inertial-range theories. J. Fluid Mech. 62, 305 (1974) Saffman, P.G.: On the spectrum and decay of random 2D vorticity distributions at large Reynolds number. Stud. Appl. Math. 50, 377 (1971) Kuznetsov, E.A., Naulin, V., Nielsen, A.H., Juul Rasmussen, J.: Effects of sharp vorticity gradients in two-dimensional hydrodynamic turbulence. Phys. Fluids 19, 105110 (2007) Ohkitani, K.: Wave number space dynamics of enstrophy cascades in a forced two-dimensional turbulence. Phys. Fluids A 3, 1598 (1991) Weiss, J.: The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Phys. D 48, 273 (1991) Kuznetsov, E.A., Ruban, V.P.: Hamiltonian dynamics of vortex lines in hydrodynamic-type systems. JETP Lett. 67, 1076– 1081 (1998) Kuznetsov, E.A., Ruban, V.P.: Hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems. Phys. Rev. E 61, 831–841 (2000) Kuznetsov, E.A.: Vortex line representation for flows of ideal and viscous fluids. JETP Lett. 76, 346–350 (2002) Wolibner, W.: Un théorème sur l’existence du movement plan d’un fluide parfait, homogène, incompressible, pedant un temp infiniment long. Math. Z. 37, 698 (1933) Kato, T.: On classical solutions of 2-dimensional non-stationary Euler equation. Arch. Ration. Mech. Anal. 25, 189 (1967) Yudovich, V.I.: Nonstationary flow of an ideal incompressible liquid. J. Math. Numer. Phys. Math. 6, 1032–1066 (1965) Rose, H.A., Sulem, P.L.: Fully developed turbulence and statistical mechanics. J. Phys. 39, 441–484 (1978) Kida, S.: Numerical simulations of two-dimensional turbulence with high-symmetry. J. Phys. Soc. Jpn. 54, 2840 (1985) Kuznetsov, E.A., Passot, T., Sulem, P.L.: Compressible dynamics of magnetic field lines for incompressible nagnetohydrodynamic flows. Phys. Plasmas 11, 1410 (2004)

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Theor. Comput. Fluid Dyn. (2010) 24:259–263 DOI 10.1007/s00162-009-0147-0

S P E C I A L I S S U E O R I G I NA L A RT I C L E

Shigeo Kida · Nobuyuki Nakazawa

Super-rotation flow in a precessing sphere

Received: 30 January 2009 / Accepted: 24 June 2009 / Published online: 7 August 2009 © Springer-Verlag 2009

Abstract The super rotation here means that the majority of fluid inside a precessing sphere rotates around the precession axis with angular velocity larger than that of the precession rotation itself. This phenomenon observed experimentally and numerically is explained to be driven by a cooperative interplay between the Coriolis force, the pressure gradient and the spherical geometry in the boundary layer. Keywords Precessing sphere · Super-rotation flow PACS 47.15 1 Precessing sphere We consider the motion of an incompressible viscous fluid confined in a precessing spherical cavity of which both the spin and precession angular velocity vectors are constant in magnitude and orthogonal to each other at all the time (Fig. 1). In the precession frame the fluid motion is described by the Navier–Stokes equation with the Coriolis term, the continuity equation and the non-slip boundary condition (given by the spin rotation on the spherical surface). Apart from the initial condition the flow dynamics in this system is characterized by only two parameters, namely, the Reynolds number Re = as2 /ν and the Poincaré number Γ = p / s . Here, a is the sphere radius, s is the spin angular velocity, p is the precession angular velocity, and ν is the kinematic viscosity of fluid. A variety of flow states, either steady, periodic or chaotic, are realized depending on these two control parameters [1–3]. In this paper, we focus our attention on the flow at large values of the Reynolds and Poincaré numbers, and present numerical simulation results at Re = 1000 and Γ = 10 as well as an asymptotic analysis for Re  1 and Γ  1. 2 Super-rotation flow in the interior Since neither the governing equations nor the boundary condition have explicit time dependence in the precession frame, steady flows are possible to be realized relative to this frame. The details of the numerical method are described elsewhere [2], only stressing that the boundary layer discussed below is numerically well resolved. In Fig. 2, we show a schematic flow structure, observed by direct numerical simulation at Re = 1000 and Γ = 10. The gray zone attached to the spherical surface is the boundary layer of thickness typically 0.05a (but the thickness is exaggerated in this figure). The rest, i.e. the white zone which occupies Communicated by H. Aref S. Kida (B) · N. Nakazawa Department of Mechanical Engineering and Science, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan E-mail: [email protected] Reprinted from the journal

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Fig. 1 A precessing sphere with the Cartesian and spherical polar coordinates

Fig. 2 Boundary layer and Taylor–Proudman region

most volume of the spherical cavity, is the Taylor–Proudman region, where the flow field is uniform along the precession (z) axis. The three closed lines are streamlines drawn by using the actual numerical simulation data. Fluid rotates counter-clockwise around the precession axis, meaning that fluid particles swirl around this axis with angular velocity larger than that of the precession rotation. This is the super rotation. An almost solid-body rotation flow in the interior of the sphere is clearly seen in Fig. 3. In Fig. 3a the streamlines on the equatorial (x, y) plane are drawn by many particle trajectories projected on this plane. We see that they are on concentric circles centred at the origin inside the dotted circle. The curved arrow indicates the flow direction. The region between the dotted circle and the boundary circle is the boundary layer. In Fig. 3b superposed is the y component of velocity on lines parallel to the spin (x) axis on four different planes parallel to the equatorial plane. Circles indicate the boundary for each plane. It is seen that except in the vicinity of the boundary the four velocity profiles coincide with each other, implying that the flow is uniform in the z direction.

3 Eastward drift in the boundary layer In the vicinity of the boundary (0.95 < ∼ r ≤ 1) the interactions between the Coriolis and viscous forces together with the spherical geometry of the boundary make non-trivial flow structures. In Fig. 4a, b we show

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(a)

(b)

Fig. 3 Flow in the Taylor–Proudman region. a Streamlines on the equatorial (x, y) plane. b The y component of velocity u y on a line parallel to the x axis is drawn on four planes of z = 0, 0.25, 0.5 and 0.75

Fig. 4 Streamlines on spheres of a r = 1 and b r = 0.97. Here, θ = 21 π − θ is the latitude, and ϕ is the longitude (see Fig. 1). Curved arrows show the flow direction. Solid circles indicate the centres of swirl motion. Straight arrows in b indicate shift of the swirl centres. The radial velocity is positive in the gray region, and negative in the white

the streamlines on the spherical boundary and on a sphere of r = 0.97 which is in the boundary layer, respectively. Because of the non-slip boundary condition the fluid exhibits a simple solid-body rotation around the spin axis in Fig. 4a. In the boundary layer, however, the flow is a bit complicated through the interplay between the Coriolis force and the spherical boundary. As seen in Fig. 4b, the swirl motion still prevails, but the flow is much distorted from the solid-body rotation and the swirl centre (shown with a solid circle) shifts westward significantly (by angle α ≈ 20◦ , say) from the spin axis. This is caused by the Coriolis force, that is, a fluid element is tilted towards to smaller longitude. Moreover, we observe a small latitudinal shift of the swirl centre either to the North or to the South (see Sect. 5). This latitudinal shift induces the eastward drift of fluid motion in the boundary layer, as discussed below. The distribution of the radial component of velocity (shown by gray scale in Fig. 4b) is also noteworthy. The radial velocity takes positive values roughly in the northern hemisphere (θ > 0) around the right swirling centre (|ϕ −α| < 90◦ ), and in the southern hemisphere (θ < 0) around the left swirling centre (|ϕ −α −180◦ | < 90◦ ); it is negative otherwise. This is due to the latitudinal variation of the radial component of the Coriolis force. For example, in the white region in the northern hemisphere (θ < 0 and |ϕ − α − 180◦ | < 90◦ ) the fluid moves to the right with speed increasing as the latitude, so that the Coriolis force which points downward (to lower latitude) increases as the latitude. Thus, the fluid in this area receives compressive forcing, which leads to the negative radial flow. The sign of the radial velocity in the other areas is explained in exactly the same way. This drives the axial flow (along the z axis) in the interior (see Eq. (8)). The flow at the edge of the boundary layer drives the fluid motion in the interior. In Fig. 5, we show a typical fluid particle trajectory in the boundary layer by the Mercator projection (Fig. 5a) and by a perspective view (Fig. 5b). Here, the solid lines represent the trajectory and the thin line in Fig. 5a is the mirror image of its southern (z < 0) part with respect to the equator. It is seen that the particle drifts slowly, O(1/Γ ), eastward Reprinted from the journal

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z (a)

(b) 45

_ θ

0

-45

x 90

phi ϕ

180

Fig. 5 Particle trajectory. a Mercator projection. θ ≡ π/2 − θ. b Perspective view. The particle moves from left to right in a, whereas from right to left in b

(increasing ϕ) while swirling clockwise. Observe that the northern part of the orbit is always slightly higher in latitude than the the mirror image of the southern part. Since the fluid motion is faster in higher latitude, this asymmetry with respect to the equator leads to the eastward drift of fluid particles. 4 Asymptotic analysis As seen in the preceding sections, a thin boundary layer and the Taylor–Proudman regions are developed in the velocity field for large values of Re and Γ . This flow can be derived by asymptotic analysis in the limit of Re  1 and Γ  1. Here, we summarize the √ results and the detailed analysis is reported elsewhere. In the boundary layer of thickness δ = 1/ Γ Re the latitudinal and longitudinal components of velocity are, in the leading order of the 1/Γ expansion, written as     1 1/2 1/2 1/2 u θ (s, μ, ϕ) = − cos(|μ| s) sin ϕ + |μ| sin(|μ| s) cos ϕ exp[−|μ| s] + O δ, , (1) Γ     1 , (2) u ϕ (s, μ, ϕ) = sgn(μ) sin(|μ|1/2 s) sin ϕ − |μ| cos(|μ|1/2 s) cos ϕ exp[−|μ|1/2 s] + O δ, Γ respectively, where μ = cos θ, r = 1 − δs, and sgn(μ) stands for the sign of μ. These two components of velocity decrease exponentially to zero as s increases; that is, they are negligibly small at the edge of the boundary layer. In the first order, however, the longitudinal velocity remains a finite non-zero value,  1 1 − μ2 (457μ2 + 68|μ| − 15) u ϕ (∞, μ, ϕ) = (3) Γ 7200μ2 at the edge of the boundary, which drives the super-rotation flow in the Taylor–Proudman region (see Eq.(7)). The radial component of velocity is obtained as   δ sgn(μ) 1 − μ2 u r (s, μ, ϕ) = − sin ϕ − 3|μ| cos ϕ 4|μ|3/2 

− exp[−|μ|1/2 s] (1 + 2|μ|1/2 s) cos(|μ|1/2 s) − sin(|μ|1/2 s) sin ϕ   

1 , (4) − 3|μ| cos(|μ|1/2 s) + (3|μ| − 2|μ|1/2 s) sin(|μ|1/2 s) cos ϕ + O δ 2 , Γ  δ sgn(μ) 1 − μ2 u r (∞, μ, ϕ) = − (sin ϕ − 3|μ| cos ϕ) 4|μ|3/2 in the limit of s → ∞. This drives the axial flow of O(δ) in the Taylor–Proudman region (see Eq. (8)).

which leads to

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In the Taylor–Proudman region the velocity field is independent of z so that the cylindrical polar coordinate (ρ, ϕ, z) with x = ρ cos ϕ and y = ρ sin ϕ is convenient to represent the field. The velocity field in the Taylor–Proudman region can be derived as vρ (ρ, ϕ) = 0,

 1 + 68 1 − ρ 2 (> 0), vϕ (ρ, ϕ) = ρ Γ 7200(1 − ρ 2 )    δρ 3 1 − ρ 2 cos ϕ − sin ϕ , vz (ρ, ϕ) = 2 5/4 4(1 − ρ ) 442 − 457ρ 2

(6) (7) (8)

by making use of Eqs. (3) and (5) as the boundary condition for vϕ and vz , respectively. These analytical results agree well with the numerically obtained velocity field shown in the preceding sections. The positivity of the longitudinal velocity (7) shows the super rotation. 5 Interpretation of super rotation In this section, we consider the driving mechanism of super-rotation flow in the interior of a precessing sphere. For the sake of explanation let us take the area 90◦ < ϕ < 270◦ in the boundary layer (see Fig. 3a), where the fluid rotates predominantly clockwise around the negative pole of the spin axis (θ , ϕ) = (0◦ , 180◦ ). In the northern (or southern) hemisphere of this area the fluid moves eastward (or westward) and receives the Coriolis force directed towards (or away from) the spherical boundary. The radial component of the Coriolis force must be balanced with the pressure gradient. It is shown, on the other hand, that the pressure is uniform in the leading order of the 1/Γ expansion throughout the boundary layer. Therefore, the pressure in the next order should be lower (or higher) in the northern (or southern) hemisphere, which causes the northward shift to the fluid. The same argument can be applied to the other half area to show the southward shift of fluid motion. Such a slight asymmetry in the latitudinal direction induces the eastward drift of fluid elements as seen in Fig. 5, which drives the super-rotation flow in the interior. Acknowledgments The authors would like to express their cordial gratitude to Professor H. Niino for his invaluable suggestion. The numerical calculations were carried out on SX8 at YITP in Kyoto University.

References 1. Goto, S., Ishii, N., Kida, S., Nishioka, M.: Turbulence generator using a precessing sphere. Phys. Fluids 19, 061705 (2007) 2. Kida, S., Nakayama, K.: Helical flow structure in a precessing sphere. J. Phys. Soc. Jpn. 77, 054401 (2008) (9 p) 3. Kida, S., Nakayama, K., Honda, N.: Streamline tori in a precessing sphere at small Reynolds numbers. Fluid Dyn. Res. 41 11401 (2009) (16 p)

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Theor. Comput. Fluid Dyn. (2010) 24:265–282 DOI 10.1007/s00162-009-0177-7

O R I G I NA L A RT I C L E

D. S. Pradeep · F. Hussain

Vortex dynamics of turbulence–coherent structure interaction

Received: 22 January 2009 / Accepted: 29 October 2009 / Published online: 5 January 2010 © Springer-Verlag 2010

Abstract We study the interaction between a coherent structure (CS) and imposed external turbulence by employing direct numerical simulations (DNS) designed for unbounded flows with compact vorticity distribution. Flow evolution comprises (i) the reorganization of turbulence into finer-scale spiral filaments, (ii) the growth of wave-like perturbations within the vortex core, and (iii) the eventual arrest of production, leading to the decay of ambient turbulence. The filaments, preferentially aligned in the azimuthal direction, undergo two types of interactions: parallel filaments pair to form higher-circulation “threads”, and anti-parallel threads form dipoles that self-advect radially outwards. The consequent radial transport of angular momentum manifests as an overshoot of the mean circulation profile—a theoretically known consequence of faster-than-viscous vortex decay. It is found that while the resulting centrifugal instability can enhance turbulence production, vortex decay is arrested by the dampening of the instability due to the “turbulent mixing” caused by instability-generated threads. Ensemble-averaged turbulence statistics show strong fluctuations within the core; these are triggered by the external turbulence, and grow even as the turbulence decays. This surprising growth on a normal-mode-stable vortex results from algebraic amplification through “linear transient growth”. Transient growth is examined by initializing DNS with the “optimal” modes obtained from linear analysis. The simulations show that the growth of transient modes reproduces the prominent dynamics of CS-turbulence interaction: formation of thread-dipoles, growth of core fluctuations, and appearance of bending waves on the column’s core. At the larger Reynolds numbers prevailing in practical flows, transient growth may enable accelerated vortex decay through vortex column breakdown. Keywords Coherent structure · Turbulence · Direct numerical simulation · Vortex dynamics 1 Introduction Turbulent flows of practical interest often feature organized vortices (coherent structures, CS). The influence of ambient turbulence on CS evolution is of interest in problems such as boundary layer drag/heat transfer, vortex-induced vibration of structures, aeroacoustic noise, and turbulent mixing. Vortex-turbulence interaction is important in aircraft wake prediction, where the strength and decay rate of wake vortices must be accurately determined under a variety of environmental conditions to optimize aircraft separation. Existing models are predominantly empirical [21], reflecting our inadequate understanding of this complex flow. The present study considers the simplest case of vortex–turbulence interaction: a rectilinear vortex column (without mean axial flow) embedded in a sea of (initially) incoherent fluctuations. The problem has been Communicated by H. Aref D. S. Pradeep (B) · F. Hussain Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA E-mail: [email protected] Reprinted from the journal

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most extensively studied in the context of aircraft trailing vortices (see [21] for a review). The focus has been on determining how the mean flow and turbulence quantities (intensities and Reynolds stresses) evolve. Hoffman and Joubert [6] invoked a mixing-length type hypothesis and postulated a “universal” inner core having logarithmic dependence of circulation on radius (similar to the log law of the turbulent boundary layer). Govindaraju and Saffman [4] used different arguments, based on a flow self-similarity hypothesis, but arrived at the same conclusion. The mechanisms are yet to be conclusively established. Saffman [19] developed a theory of turbulent line vortices by assuming that the flow evolves to a self-similar state, but the existence of such a state is in doubt because of the rapid decay of turbulence, as inferred in high-Re experiments [3] and observed in moderate-Re numerical simulations [22]. Implicit in these theories is the existence of turbulence fluctuations in the vortex core, associated with significant Reynolds stresses. Experiments [14] and recent numerical studies [7,17] show such core turbulence, but only when there is also a large mean axial flow (which destabilizes the vortex). Modeling predictions are unclear if turbulence can be sustained in the absence of mean axial flow. For example, Zeman [24] suggests that the production mechanisms in the flow are so weak that turbulence has insignificant impact on vortex decay. In recent years, computational studies have provided valuable insights into the physical mechanisms operating in the flow. Melander and Hussain [12] modeled a CS as a columnar vortex embedded in initially fine-scale and unstructured turbulence. They found that the CS organizes the turbulence into numerous coherent filaments. These filaments, wraping around the column, contain predominantly azimuthal vorticity. The preferential alignment of turbulent vorticity results in dynamics analogous to two-dimensional turbulence. The filaments pair and grow in size, contributing to energy backscatter. Simultaneously, velocity perturbations induced in the vortex core by the filaments generate bending waves on the column. Excitation of bending waves is significant because it can lead to the breakdown of the vortex core, especially if there is background shear [8]. In such a case, it might be important to predict the dominant wavelength excited by the turbulence on the CS, since it determines the instability growth rate and hence the longevity of the vortex. In turbulence modeling, flows with mean swirl remain a challenge and a subject of active research. It is well known that the classical k − ε model grossly overestimates turbulence levels and the decay rate of the large-scale vortex [24]. This is because the effects of mean vorticity do not appear in the turbulence kinetic energy equation. Disagreements are seen between different closure schemes (developed in the context of nonrotating flows). For instance, while some closures predict near-total suppression of turbulence both within and outside the vortex core and consequent viscosity-determined decay of the vortex [24], other closures predict significant turbulence intensities and Reynolds stress outside the core and therefore faster-than-viscous vortex decay [23]. A comparison of experimental data and various turbulence models in [10] illustrates current modeling deficiencies. The broad objective of this study is to numerically investigate vortex-turbulence interaction through direct numerical simulation (DNS), focusing particularly on possible mechanisms of core perturbation growth. Section 2 presents the numerical method for DNS and initial condition for our simulations. We discuss DNS results of flow evolution in Sect. 3 along with ensemble-averaged statistics showing surprisingly strong core fluctuations induced by the external turbulence. Section 4 focuses on the role of optimal transient modes in causing core fluctuation growth. Concluding remarks appear in Sect. 5.

2 Numerical method We simulate an isolated, temporally evolving vortex column periodic in the axial (z) direction. DNS uses a modified version of the Fourier pseudo-spectral method developed by Rennich and Lele [18]. The use of this method was motivated by our observation that vortex-turbulence simulations using the classical, triply-periodic Fourier method showed an unexpectedly strong dependence on the computational domain size (see [15] for a detailed discussion). This is caused by the Fourier method’s inability to represent a net nonzero circulation within the computational domain. Thus, the numerically represented circulation of an isolated vortex goes to zero at the edge of the domain, instead of asymptoting to a constant value. The vortex is rendered centrifugally unstable; when the vortex is embedded in ambient turbulence, this artificial instability can amplify fluctuations and cause the vortex column to undergo transition. This artifact of triply-periodic boundary conditions is illustrated in Fig. 1, where we compare simulations performed using the triply-periodic scheme and the Rennich–Lele method. (In the figure and hereafter, T ≡ r1 /v1 , where r1 is the radius at which peak mean velocity v1 is attained.)

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Fig. 1 Comparison of unbounded (top row) and triply periodic (bottom row) flow evolutions with identical initial condition. Vorticity contours with increment 0.1ω0 are plotted in a meridional plane at T = 0, 30, 60 and 90

The present numerical scheme applies to flows with vorticity that is compact along the two unbounded directions (say x and y), and periodic in the third (z). An initial vorticity field ω is prescribed and the vorticity equation ∂ω + ∇ × (ω × u) = ν∇ 2 ω ∂t

(1)

is integrated forward in time. This involves calculation of the induced velocity field u. In the standard Fourier method, the (divergence-free) velocity field is obtained from the Fourier-transformed vorticity field,  u(k) = ik × ω/k 2 where tildes denote Fourier-transformed quantities and k is the wavevector. In addition to the velocity induced by vorticity within the computational domain,  u contains contributions from the infinite array of image flows in neighboring domains in x and y. To eliminate the influence of neighboring domains, u is written as the sum of rotational and irrotational components: u = uω + ∇φ + U . Here uω represents the rotational part of the velocity, i.e., ∇ × uω = ω; ∇φ is the irrotational component of velocity, which ensures that the boundary conditions at r → ∞ are satisfied; and U is the mean velocity, due to the k = 0 component of the vorticity field. Each of these three components is solenoidal. The velocity potential φ satisfies ∇ 2 φ = 0, which is solved using the boundary conditions that φ is finite at r = 0 and as r → ∞. To solve the Laplace equation for φ, we expand in terms of Fourier modes, resulting in     mk mk mk = 0, r 2φ + rφ − k 2r 2 + m 2 φ (2) , three computational domains are where m and k are the azimuthal and axial wavenumbers. To determine φ constructed, as shown in Fig. 2a. The main computational domain has a square z-cross-section (a–d in the √ figure) of edge length L. It is circumscribed by a cylindrical domain of radius Re = L/ 2 and inscribed with a cylindrical domain of radius Ri = L/2. It is required that ω = 0 at r = Ri . We solve for φ separately in r < Ri and Ri < r < Re . Regularity of φ at r = 0 and the finiteness of φ at r → ∞ are imposed to obtain separate solutions within and outside the cylindrical domain Ri . These solutions are then matched at r = Ri to obtain the total velocity field. It is convenient to calculate the potential velocity on a cylindrical-polar mesh, whereas the rest of the computation uses Cartesian meshing. Interpolations between the two coordinates are performed using the polynomial method. These operations lead to a loss of the pseudospectral method’s exponential convergence property. However, as shown in [18], the problem can be alleviated by increasing the order of the interpolation scheme. Figure 2c illustrates the decay in the global error (here for an Oseen vortex) with increasing mesh size and interpolation order. The calculations reported in this article have been conducted with fourth-order interpolation. Reprinted from the journal

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Y

(b)

(a) y

1

Z

X

0.8

b

c 0.6

uθ uz

Re Ri

0.4

0.2

x a

0

d

−0.2

L

2

4

r

6

3

(d) 10

(c) 10 −2

Re = 72.78 (initial) Re = 15.68 Mansour & Wray

102 O(1) O(2)

101

Ε/(εν 5)1/4

global error

10 −4

10 −6 O(3) O(4)

10 −8

100 10 −1

O(5) O(6)

10−10

10 −2

O(7)

10

10 −3

−12

4

5

6

7

8

9

10−2

10−1 3

1/4

100

κ(ν /ε)

log2 (N)

Fig. 2 a Computational domains for calculating the potential velocity correction. b Axial and azimuthal velocity profiles calculated within the code (symbols) compared with known analytical form (solid lines) for the q-vortex (inset, vorticity magnitude contours). c Global velocity error in the q-vortex velocity field for varying interpolation order and mesh size N . d Comparison of calculated energy spectrum with that of Mansour & Wray

With the correct velocity field in hand, the remainder of the algorithm follows along the standard lines of the Fourier pseudospectral method. We compute the nonlinear term ω × u in physical space, and transform to Fourier space to compute its curl. Note that since ω has a compact distribution, so does ω × u. Therefore, we can use the Fourier transform for computing ∇ × (ω × u) without introducing any errors due to the imposition of periodicity. Figure 2b shows the calculated azimuthal and axial velocity profiles associated with a ‘q’ vortex (an Oseen-like vortex with axial velocity) placed off-center of the cylindrical domain: note that the smooth variation of the velocity across the domain boundary shows the successful calculation of the correct potential velocity. The numerical code has been implemented to run on parallel processors. All aspects of the algorithm are trivially parallelized, except the three-dimensional Fourier transforms. The computational domain is divided into equal slices along a coordinate direction and the slices are distributed across processors. Transforms in two directions are local and carried out first. The transform in the third direction is preceded by a global matrix transposition to render the data local.

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Table 1 Growth rates σ for q-vortex computed via DNS m

q

k/m

σ

σ |MP

1 2 3 4

0.458 0.693 0.779 0.815

0.811 0.591 0.555 0.544

0.2413 0.3132 0.3539 0.3758

0.2424 0.3138 0.3556 0.3777

σ |MP are values from inviscid analysis; k and m are the axial and azimuthal wavenumbers, respectively

The code was validated in two stages. In the first, the standard Fourier pseudospectral kernel was validated via calculations of low-Taylor microscale Reynolds number (Reλ ) decaying isotropic turbulence, checked against those reported in the literature. The boundary conditions are triply periodic. Turbulence was initialized with random phases and with the energy spectrum     q2 1 σ σ κ 2 E(κ, 0) = , κ exp − 2 A κpσ +1 2 κp ∞ where κp = 15 is the peak-energy wavenumber, A = 0 κ σ exp(−σ κ 2 /2) dκ, and σ = 6. The numerical value of viscosity was ν = 0.002 and the turbulence intensity was set to q 2 = 3. The computational domain size was (2π)3 and the resolution 1283 . The initial field is unphysical and has zero velocity derivative skewness, but time evolution leads to truly isotropic turbulent flow. The spectrum becomes fuller and Reλ decays drastically. In the course of evolution, the velocity derivative skewness peaked at −0.47, which agrees well with the values reported in the literature. Figure 2d compares the energy spectrum calculated with that obtained in [9] at Reλ = 15.7. The full code was then validated by computing the linear instability growth rates for the q-vortex: U (r, θ, z) = 0, V (r, θ, z) =

  q 1 − exp −r 2 , W (r, θ, z) = 0. r

DNS was compared with the inviscid results of [11], who calculated the peak growth rate σ , the corresponding axial wavenumber k, and q as functions of azimuthal wavenumber m. The vortex is given a small amplitude perturbation of the form ωz = αqe−r cos(mθ + kz), ωθ = −αq 2

k −r 2 re cos(mθ + kz), m

where α is the perturbation amplitude. The evolution of three-dimensional energy is monitored, and the modal growth rate is calculated once the energy begins to grow at an exponential rate. The simulations were performed using 96 × 96 × 16 grid points at a vortex Reynolds number Re = /ν = 104 . Table 1 lists the computed growth rates σ and the inviscid values σ |MP predicted by [11]. The growth rates agree to within 0.5% in all cases. 2.1 Initial condition The numerical simulations are initialized by superposing the vorticity fields corresponding to a laminar vortex column and a radially compact region of random, fine-scale fluctuations.   The vortex column has the vorticity distribution, ωz = ω0 exp −ξ 2 , where ξ = r/r0 ; r0 , ω0 are the vortex size and the peak axial vorticity magnitude, respectively. We use the cylindrical-polar coordinate system (r, θ, z), with total velocity components (u, v, w) and vorticity components (ωr , ωθ , ωz ); in the following, primed quantities (e.g., u  , v  , w  ) denote fluctuations and upper-case quantities (e.g., V ) the mean. The turbulence is initialized by first prescribing a homogeneous, isotropic turbulent velocity field, characterized by its three-dimensional energy spectrum. To achieve radial compactness, the velocity field is multiplied in physical space by a compact filtering function F(r ) (see Fig. 3a). The velocity field is no longer divergence-free; therefore, the filtered velocity field’s curl is taken and inputted to the Navier–Stokes solver as the turbulence vorticity field. Calculation of velocity from this vorticity field inside the solver automatically ensures that the velocity field remains solenoidal. Reprinted from the journal

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(a)

(b)

x 0.01

6 F

1

F3

F

2

TKE

0.8

0.4

4

F2 F1

2

F

3

0

0 0

2

4

0

6

r

2

4

r

Fig. 3 Effect of spatial filtering of turbulence: a Plots of three different spatial filters. b TKE profiles at T = 20

The main parameters describing the flow are the vortex Reynolds number Re = 0 /ν, where ν is the kinematic viscosity, and the normalized turbulence intensity

u max /v1 , where ⎛ ⎜

u (r ) = ⎝

1 2π L z

⎞1/2 L z 2π ⎟ (u 2 + v 2 + w 2 ) dθ dz ⎠ 0

(3)

0

is the r.m.s. velocity fluctuation, u  , v  , w  are the velocity fluctuations and L z is the axial periodicity length. A note is in order on the effect of spatial filtering of the turbulence field on flow evolution. The filtering being unavoidable in this computational method, its effect cannot be rigorously analyzed. However, by applying different filters, we find that the effect is not significant. To illustrate this, we consider simulations using three different filters shown in Fig. 3a. Both the filter shape and its width are varied and all other flow parameters kept fixed. The profiles of turbulence kinetic energy (K (r ) =

u 2 /2) after 20 turnover times are plotted in Fig. 3b. Turbulence kinetic energy (TKE) profiles at T = 0 follow closely the filter shapes in Fig. 3a. Flow structure at T = 0 and 20 is shown in terms of radial fluctuation velocity u  contours for each of the three filter-cases in Fig. 4. It can be seen that despite the large differences in the filters, the TKE profiles are quite similar after some evolution. Turbulence far away from the vortex core decays rapidly (note, e.g., that TKE is nearly zero for r > 4 for F3 at t = 20, whereas initial TKE extends to r ≈ 5.5). Close to the core periphery, where the mean strain is the largest and thus can sustain turbulence through production, turbulence intensities remain significant at T = 20. Note that turbulence levels are the largest within the core; this is remarkable because there the mean flow is in near solid-body rotation and, therefore, production is negligible. This trend is observed even for the extreme case of filter F2 , in which there are no initial fluctuations in the core. The growth of core fluctuations is due to the induced perturbations from external structures; the physics of this interaction—significant from the point of view of possible vortex core transition—is analyzed in greater detail later in this text (see Sect. 4). All results in this article have been obtained with filters of type F3 .

3 Flow evolution In the simulation described below with core radius r0 = π/8, turbulent annulus width rt = 0.9π, vortex circulation 0 = 1, and wavenumber of turbulence spectrum peak k p = 16 (which means that the energy containing eddies have a characteristic length-scale ∼ r0 ). The circulation-based Reynolds number is Re ≡

0 /ν = 12, 500 and the initial turbulence intensity

u max /v1 is 15%. The flow is simulated using 1922 ×288 2 grid points in an initial domain size of (2π) × 4π. The vortex spreads radially outward with time, and at T = 60 the flow field is re-meshed to a domain of size (3π)2 × 4π. The initial fluctuating field is unphysical, but the turbulence adjusts to the presence of the mean flow within a few vortex turnover times, T . In this transient period, a large fraction of the three-dimensional energy rapidly cascades to smaller scales, where it is dissipated. Vorticity magnitude contours in a meridional plane are plotted at three stages of the evolution in Fig. 5. Initially, the turbulence is unstructured and nearly homogeneous in the vicinity of the vortex core. The mean strain field stretches and tilts fluctuation vorticity, producing the growth

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(a)

(c)

(b)

(e)

(f)

(d)

Fig. 4 Contours of u  in a meridional plane for filtering function cases F1 (a, b), F2 (c, d) and F3 (e, f) at times T = 0(a, c, e) and T = 20(b, d, f). Shaded regions denote negative u  ; u  /V1 levels plotted are [−0.25, 0.25, 0.03]. Also shown is the ωz = 0.1 contour indicative of the vortex core

of fine-scale enstrophy (Fig. 5b). Following this initial period of dissipation, the dominant turbulence length scale is then observed to grow and organized fine scales begin to appear (T = 30). By T = 120 the turbulence external to the core is dominated by relatively well-organized vortices—with core radii considerably larger than the initial turbulence length scale. These organized filaments (“threads”) are arranged into numerous dipoles (marked d1–d8 in Fig. 5c). Note that the vorticity outside the vortex core is significantly weaker than within. There are clearly two distinct regions in the flow. In the “core” region (where mean vorticity is large and the flow is in near-solid body rotation), the flow appears to be nearly laminar by T = 120. It is well known that a vortex column supports a spectrum of axially propagating waves, neutrally stable in the inviscid limit and damped in the viscous flow. The external turbulence excites these waves, manifesting as undulations of core vorticity in Fig. 5c. Visualization at closely space intervals (not shown) reveals oscillatory motion in the vortex core, as does the application of helical wave decomposition [13]. Outside the core (where vorticity is nearly zero and the strain rate is large), the flow is fully turbulent and dominated by well-organized threads. Both the size of the vortex filaments and the width of this turbulent region increase with time (discussed in the following). 3.1 Thread vortex dynamics The preferential alignment of turbulent vorticity results in vortex dynamics analogous to two-dimensional turbulence. Threads evolve under the combined influence of (a) their self-induced motion due to curvature, (b) mutually induced motion among threads, and (c) the vortex column’s velocity field, which includes the meridional velocity associated with the vortex waves. The combined effect of (a) and (b) leads to a reorganization of threads via two types of events. First, threads with parallel ωθ that are sufficiently close to one another undergo pairing. An example of pairing is shown in Fig. 6, where the vortices labeled A and B are seen to merge. Such events contribute to the growing turbulence length scale (compare panels (a) and (b) in Fig. 5), although viscous diffusion is also significant in this moderate Re flow. The self-induced motions of a pair of threads with anti-parallel ωθ causes them to approach one another, and thus to form dipoles (Fig. 7). Several such dipoles (d1–d8 in Fig. 5c) can be seen at T = 120. These dipoles are similar to colliding vortex rings, the difference of course being that the threads are not axisymmetric. The sign of ωθ in each thread of the dipole is such that the dipole advects itself radially outward. (Note that if the ωθ signs of the two threads were to be interchanged, their self-induced motions would cause them move away from each other in the axial Reprinted from the journal

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(b)

(a)

(c)

d5

d1 d2

d4

d3

d8 d6

d7

Fig. 5 Vorticity magnitude contours in a meridional plane at a T = 0, b T = 30, and c T = 120. Contour levels used are: a: [0.1ω0 , 2ω0 , 0.1ω0 ]; b: [0.5ω0 , ω0 , 0.1ω0 ] in the core region and [0.05ω0 , 0.5ω0 , 0.05ω0 ] outside, and c: [0.4ω0 , ω0 , 0.1ω0 ] in the core and [0.02ω0 , 0.4ω0 , 0.02ω0 ] outside. Several dipoles d1–d8 are indicated in c. Only half the z-domain length is shown in a and b

direction, and the two threads would not form a dipole.) Dipole formation and radial advection are the main inviscid mechanisms for the growth of the outer turbulent region. An important aspect of the ωθ stretching responsible for the threads is that the mean strain rate increases with decreasing r . As a vortex filament with nonzero ωr is advected around the column, filament segments at small r are stretched more than those at larger r . As a result, a vortex tube with initially unform cross-sectional area subsequently has nonuniform core area: thread core area decreases with decreasing r . When thread vorticity becomes sufficiently strong, the local velocity field swirls around the thread axis. The differential rotation along each vortex line within the thread core causes the line to coil around the filament axis, i.e., threads are “polarized” [13]. Polarization has implications for thread stability because coiled vortex lines induce flow along the thread’s axial direction; this axial flow, if sufficiently intense, destabilizes the threads. At sufficiently high Reynolds numbers (based on thread circulation), the threads will not survive indefinite stretching, but instead break up as the result of instability. This polarization-induced instability is illustrated for an isolated, fully polarized vortex in Fig. 8. The thread Re here is 1,000 and the vortex is perturbed with random, small-amplitude

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Fig. 6 Pairing of two azimuthal vorticity filaments as visualized by ωθ contours in a meridional plane. Shaded regions correspond to negative ωθ

Fig. 7 Dipole formation of two azimuthal vorticity filaments as visualized by ωθ contours in a meridional plane. Shaded regions correspond to negative ωθ

(a)

(b)

(c)

Fig. 8 The instability of a fully polarized thread: a t = 0, b t = 20, c t = 40

perturbations. The perturbations amplify in the form of helical vortex filaments in the column’s core, which will later break-up into finer-scale vortices. The dynamics within the turbulence is thus seen to be a combination of anti-cascading (pairing) and cascading events (thread stretching and instability). In the present simulation of vortex-turbulence interaction, the turbulence Reynolds number (say, based on the Taylor microscale) is quite small (∼ 30), and therefore, the thread instability is damped. At larger Re, it can be expected that the dominant turbulence length-scale will be set by the balance between the competing effects of forward cascade by stretching and thread instability, and inverse cascade via thread pairing.

3.2 Circulation overshoot The formation of thread dipoles and their radially outward transport of mean angular momentum accelerates the decay of the coherent vortex column. This has interesting implications for the mean azimuthal velocity Reprinted from the journal

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(a)

(b) V

Viscous torque

Turbulence

t0 t1

^u ____

Γ

u’v’

r

V r

r Vortex core

Γ

t0

t1

circuit

C

r Fig. 9 Mechanics of overshoot development in a turbulent vortex. a Retardation of angular momentum by viscous forces. Profiles of V, , turbulence intensity

u and Reynolds stress u  v  also shown. b Schematics of V and profiles at initial time t0 and at a subsequent time t1

profile. Govindaraju & Saffman [4] derived the following “circulation defect integral” from the azimuthal velocity equation: 1 J (t) ≡ r1 (t)2

∞ 0

0 − 2ν(t − t0 ) r dr = ,

0 r1 (t)2

(4)

where = 2πr V, 0 is the circulation at infinity and t0 is the virtual time-origin of the flow. Note that a laminar vortex grows as (νt)1/2 and hence preserves J . On the other hand, if r1 (t) grows at a rate faster than that set by viscous decay, J (t) decreases with time. It is shown that if the decay is faster-than-viscous, the integrand in (4) must be negative somewhere, i.e., the circulation profile must develop an overshoot (having a region with > 0 ). The mechanics causing the overshoot can be understood in terms of the angular momentum equation [21]. Let us consider first a laminar Oseen vortex and compute the angular momentum contained within a circuit at a very large radius (where mean vorticity, which decays exponentially at large r , is negligible). The angular momentum of the mass of fluid contained within the circuit C (at radius R) (Fig. 9a) is diminished by the torque generated by the surfaces forces at C. The rate of change of angular momentum is given by d dt

R

R2 d d vr dr =

(R) − π 2 dt dt

R ωz r 3 dr

2

0

0

R2 d =

− 2ν 2 dt

(5)

Now, the rate of change of (the circulation at R) is given in terms of the line integral of vorticity:  d

= −ν ∇ × ω · dl. dt

(6)

C

Thus the decay of is negligibly small (since ωz is negligibly small) at large R, whereas the decay of angular momentum (5) occurs at a viscous rate. Now, in a turbulent vortex, the decrease of mean strain rate with increasing r implies that turbulence far away from the vortex core experiences very little stretching. At sufficiently large r the turbulence cannot be sustained against dissipation and therefore the flow is laminarized. The typical turbulence intensity profile of a “mature” turbulent vortex is shown in Fig. 9a. Quantitative arguments are given in [4] for why the turbulence intensity

u and, consequently, the Reynolds stress u  v  must both decay to zero more rapidly with increasing r than the mean strain rate S (which decays as r −2 ). Contrarily, near the vortex core, the mean strain rate is large,

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(a) 1.2

(b) 0.46

0.8 1.05

Γ

J(t) 0.42

0.4 0.95 1

3

T=0 30 90 0.38

0 0

1

2

3

4

0

200

r

400

T

Fig. 10 a Mean circulation = 2π V r profiles at three different stages of evolution; inset shows the overshoot region. b Evolution of J (t)

0.24

Turbulent decay

Peak Velocity

Laminar decay 0.2

0.16

0.12 0

200

400

600

T Fig. 11 Evolutions of peak azimuthal velocity for a turbulent and a corresponding laminar (i.e., with the same vorticity profile, but with turbulence removed) vortex. Here Re = 6,000

and so are

u and u  v  . Turbulence can therefore be expected to diffuse the mean momentum (on a convective time scale) close to the vortex core. This would result in the decay of the mean velocity V and circulation profiles as shown in Fig. 9b. Note, however, that at r = R the retardation is still purely viscous. Therefore, over the short convective time-scale of turbulent diffusion, the decay of angular momentum within the C is very small (since angular momentum decay occurs on a viscous timescale). The loss of angular momentum at small r must therefore be compensated for by a gain at larger r (V profile at t1 in Fig. 9b). The corresponding

profile would then display an overshoot. The mean circulation profile for the present simulation is plotted in Fig. 10, along with J (t). There is indeed some decay of J (t), of approximately 6% over a time period of 250 turnover times. Consistent with this decay, the circulation profile—shown at three stages of the evolution in Fig. 10a develops an overshoot, seen at T = 30. The overshoot decays with time and its radial location shifts to larger r . The overshoot implies that the flow is now centrifugally unstable. Thus, fine-scale turbulence has the effect of inducing a large-scale inviscid instability, absent initially. The overshoot in the present simulation is very modest, because the turbulence Re is quite small. Further the location of the overshoot is remote from the vortex’s core, and any enhancement of turbulence production through this mechanism might be expected not to accelerate significantly the decay of the turbulent vortex. Indeed a direct comparison of a turbulent and the corresponding laminar vortices shows this to be the case. Figure 11 plots the peak azimuthal velocity for the two vortices. Apart from an abrupt initial drop in the velocity at T ≈ 0, possibly a consequence of the numerical construction of the initial vorticity field, the two vortices decay at nearly identical rates. However, the mechanism underlying the circulation overshoot is inviscid and may strengthen with increasing Re. The eigenmodes of the centrifugal instability resemble an array of rings, of alternating sign, around the column. An interesting positive feedback mechanism between the threads and the centrifugal instability is thus possible, in which the centrifugal instability energizes the threads, which in turn, maintain the instability via the transport of mean angular momentum. Reprinted from the journal

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In order to investigate this possibility, we briefly consider a model flow comprising a laminar vortex column with an artificially imposed circulation overshoot of 15% at the vortex core periphery. The initial circulation profile is shown in Fig. 12a. The centrifugal instability implied by the overshoot causes perturbation energy growth (Fig. 12a) when the vortex column is perturbed with a random, small-amplitude perturbation. Flow visualization (not shown) shows the development of spiral vortex filaments wrapping the column’s core, very similar to threads. The transport of mean angular momentum by these threads, however, leads to a net dampening of the overshoot magnitude, as seen in Fig. 12a. This can be understood in terms of the radial transport of mean vorticity z by the threads. The initial overshoot implies two adjacent layers of opposite-signed z . The rate of viscous annihilation of vorticity in these layers is enhanced by the radially outward and inward transport of z by the radial motions induced by the threads. The net effect is to dampen the overshoot amplitude. This experiment suggests an equilibrium level at which an overshoot can be maintained: if the overshoot is weaker, dipole-transport will amplify the circulation defect; contrarily, if the overshoot is stronger, mean vorticity annihilation through thread-induced mixing will weaken. It remains to be seen whether the circulation overshoot effect is a significant aspect of vortex decay at higher Re; at the present Re (∼ 104 ), the overshoot appears to be too weak to cause significant acceleration of vortex decay. 3.3 Turbulence statistics To quantify the effect of turbulence on the vortex column’s decay we consider some turbulence statistics. It can be shown that the growth of core radius is related to the Reynolds stress at r1 (the location of peak mean azimuthal velocity) [24]:   V 1 dr1   u v +ν . (7) ∼ dt v1 r r =r1 Figure 13 plots results for u 2 , v 2 , w 2 and the Reynolds stress u  v  for an ensemble of 20 simulations. The ensemble is generated with different random initial conditions for the turbulence, but with all parameters held constant. The simulations are at a reduced Re of 3000, and statistics are compiled at the same phase of evolution for each realization. Note that at this low Re turbulence survives only for a short period of time; cf. Fig. 5, where fluctuations are still apparent at T = 120. The stresses u  w  and v  w  should be zero by symmetry, and are seen (not shown) to be much smaller than u  v  , indicating that a reasonably large sample size has been considered. Figure 13 shows that there is strong growth of core TKE with little variation of turbulence intensity amplitudes within the vortex core across realizations. In contrast, the Reynolds stress u  v  shows a sharp distinction between the flow inside the core and that outside. Whereas outside the core, there is little variation in the Reynolds stress levels that develop in the different realizations, there is very large scatter (with nearly zero mean when averaged across all the realizations) within the core. There are two striking aspects to these distributions. First, note the peak in u  v  outside the column’s core (at r > 1). This is caused by the presence of the organized thread-dipoles whose radial motion contributes a

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positive u  , v  cross-correlation, necessary for turbulence production. The stress rapidly drops to zero as the core edge is approached (r ∼ 1), which by (7) would imply an insignificant acceleration of vortex decay. The second striking aspect is the presence of intense turbulence fluctuations within the core. This may perhaps be unexpected given stabilizing effects of mean flow rotation. The simultaneous occurrence of large fluctuation intensities and negligible Reynolds stress within the core is due to the dominance of wave-like motions therein. The velocity perturbations induced by the external threads trigger a set of Kelvin waves within the core. Each component wave oscillates with its own frequency, and the contributed Reynolds stress associated with each wave is zero when averaged over the oscillation time-period. Further, it is now well known [2] that in turbulence subjected to solid-body rotation the nonlinear interactions (hence cascade) amongst different waves are considerably weakened. The net result of the induced wave motions and their predominantly linear-regime-like behavior is that Reynolds stress is negligible. While the origin of the core fluctuations is clear, we find that these fluctuations can be amplified by the external turbulence. The amplification occurs even as the turbulence decays and the fluctuation levels in the core can exceed the external turbulence levels. We examine these physics of core perturbation amplification in the following section.

4 Core perturbation growth A set of lower-resolution simulations was examined to understand the nature of core fluctuations. The profiles of turbulent kinetic energy K (r ) are plotted at different stages of flow evolution in Fig. 14a. Averages are taken over θ and z, which are the homogeneous directions of the flow. The initial turbulence intensity

u max /v1 is approximately 10%, the energy containing scales of the turbulence are approximately 1/5 the vortex core size and Re = 6,000. Reprinted from the journal

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It is seen the evolution of turbulence intensity is significantly different outside the vortex core and within. Turbulence production outside the core is the largest in regions close to the vortex core edge, where the strain rate due to the column’s velocity is the largest. However, at this moderate Re , dissipation dominates production and there is net decay of K . Production by the column’s strain causes the local maxima of K (seen at T = 10 in Fig. 14a); turbulence self-transport is responsible for the radially outward motion of this turbulence intensity peak (see profiles at T ≥ 20). Inside the vortex core the fluctuations grow even as the external turbulence decays. Peak values of K occur at the vortex axis (r = 0); at T = 10, K (r = 0) is approximately three times larger than the peak value of k outside the core. After T = 10, turbulence within the core also decays; this decay is due to the weakening induced perturbations from threads—both due to dissipation-induced decay of the threads as well as dipole advection away from the vortex. The growth of core fluctuations is reliant upon the perturbations induced by the threads. Thread-induced velocity, although irrotational, can excite and force vortical (Kelvin) waves—manifesting as amplifying velocity fluctuations in the column’s core. To show this we compare two simulations initialized with identical turbulence intensities: in the first (Fig. 14b) turbulence is present outside the vortex core, whereas in the second turbulence is confined to the region r < r0 (Fig. 14c). The comparison clearly shows that in the absence of external excitation core turbulence cannot be sustained, whereas with external turbulence, core fluctuations can be amplified. With increasing Re the dissipation rate decreases whereas production by the vortex’s strain field is unaffected. Thus, there is slower decay of the external turbulence and consequently greater forcing of turbulence within the core. Core fluctuation intensities therefore increase with increasing Reynolds number (Fig. 14d). It is conceivable that at sufficiently large Re, the core fluctuation intensities will attain levels sufficiently large to cause vortex core transition—although such levels are not attained for the Reynolds numbers currently accessible to our DNS.

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4.1 Transient growth of core perturbations In order to understand the mechanisms of core fluctuation growth, we have previously studied the (linear) evolution of small-amplitude perturbations in an Oseen vortex [16]. The Oseen vortex is normal-mode stable (i.e., all perturbations that have exponential-in-time behavior are decaying) but supports substantial algebraic growth of perturbations, that contribute vorticity outside the vortex core. The linear analysis is framed as an initial value problem for perturbations of the form        u , v , w , p (r, θ, z, t) = Re {i u , v, w ,  p } (r, t) exp (imθ + ikz) , (8) where u  , v  , w  are the radial, azimuthal and axial velocity perturbations and p  the perturbation pressure; m and k are the azimuthal and axial wavenumbers, respectively, of the Fourier-decomposed perturbation. Note that, unlike normal-mode stability analysis, no exponential-in-time behavior is assumed. The linear analysis established that (algebraically) growing perturbations exist and extracted those perturbations (“optimal modes”) which maximize energy amplification, over a given time period of amplification t. “Globally-optimal” perturbations are defined as the optimal perturbations that maximize the amplification (referred to as the gain, G(t) ≡ E(t)/E(0), where E(t) is perturbation energy at time t) over all t, for a given set of parameter values: m, k and Re; here Re is the vortex Reynolds number, defined as circulation/viscosity. We summarize here some key results from linear analysis. Figure 15a plots the globally maximal gains G max (k) for axisymmetric (m = 0) and bending wave (m = 1) modes. G max for m = 0 increases monotonically with decreasing k, attaining its largest value in the k → 0 limit. The largest-growing modes are, however, also those with the smallest growth rates. Hence peak amplification is attained at very large t for such modes. This is expected to limit the physical significance of these modes in vortex-turbulence interaction, where large-growth m = 0 modes can be overwhelmed by faster growing perturbations. Figure 15b plots the radial perturbation vorticity ωr profiles for m = 0 and m = 1 global optimal perturbations. For both modes, most of the initial vorticity is in the radial component; the growth mechanism involves the tilting of ωr to ωθ and concomitant vortex stretching. It can be seen that the m = 1 mode is localized at smaller r than the axisymmetric mode. The Oseen vortex’s mean strain rate increases with decreasing r , implying the stretching of perturbation vorticity is faster for m = 1. This causes the bending wave modes to grow faster than the axisymmetric mode. With increasing Re, m = 0 modes migrate to progressively larger r (and have progressively decreasing growth rates), whereas m = 1 modes remain localized at small r where they resonantly excite a bending wave in the vortex core. While for most k values, m = 0 modes have larger gains than m = 1 modes (Fig. 15a), the bending wave optimals posses two features that give these modes primary significance in vortex–turbulence interaction: first, as seen above, m = 1 feature large growth rates (compared to m = 0) and large gains (compared to m > 1 modes); second, bending wave modes grow through resonance with core waves, thereby exciting large fluctuations in the vortex core. We evaluate the effects of nonlinearity by initializing the globally optimal m = 1 perturbation at an amplitude of 6.6%. The simulation is performed at vortex Reynolds number Re ≡ 0 /ν = 5, 000, where is the √ 0 √ vortex circulation and ν the kinematic viscosity. Computational domain size L x × L y × L z = 10 2 × 10 2 × 2π/1.4 and the simulations employ 512 × 512 × 128 grid points. Mode evolution is shown in terms of vorticity contours in a meridional plane in Fig. 16 and in terms of vorticity isosurfaces in Fig. 17. Initially the vorticity perturbation is located only outside the core (Fig. 16a, h), with ωθ being small in comparison with ωr and ωz . The strain field of the vortex column’s swirl tilts ωr into ωθ and stretches the vorticity. The consequent growth of ωθ is seen in Fig. 16b–d. The external vorticity is organized into a spiral vortex filament (“thread”) (Fig. 17a). The optimal mode’s external ωθ is localized at a radius where the thread’s-induced velocity perturbation in the column’s core can resonate with a bending wave eigenmode of the Oseen vortex. The resonance leads to significant growth of core perturbation vorticity, whose magnitude soon exceeds that of the thread (Fig. 16d). Note that the bending wave is associated with nearly equal magnitudes in all three vorticity components. In the meridional x − z plane of Fig. 16, the growth of ωx (not shown, but having core vorticity magnitudes comparable to ω y )—which coincides here with ωr —causes appreciable deflection of the vortex column’s axis; the deformation of the initially rectilinear column into a helical bending wave can be discerned in Fig. 16d, k, and more clearly in Fig. 17a, b. The preceding mechanisms are essentially linear, but nonlinear effects strengthen progressively and are clear at larger times, t > 30. The most significant nonlinear effect is the roll-up of the external vorticity patches into vortex-like filaments, associated with closed streamline patterns. The self-induced motion of each segment of the spiral vortex filament advects the segment axially (along z). Further, the weaker ωθ cell located initially Reprinted from the journal

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Fig. 17 Isosurfaces of total vorticity magnitude ω at a t = 30, b t = 60, and c t = 90. Light-shaded surfaces are ω = 1 and depict the vortex column core; dark-shaded surfaces are ω = 0.3 in (a,b) and ω = 0.15 in (c) and depict the external vortex filament

at smaller r is wrapped around the stronger ωθ cell (Fig. 16d). The detailed vortex dynamics of these vortex filaments is complex, but the net effect of the self-induced motions of the threads (negligible in the linear limit) is that opposite-signed cells are organized into dipole-like structures. The mutually induced motions within the dipole carry the dipole radially outwards. Such radially outward motion of the external filaments is seen by comparing Fig. 16c, g, and more clearly in the isosurfaces in Fig. 17a, c. The cumulative effect of transient growth is that flow appears to be nearly turbulent at late times. The development of a turbulence-like flow in a perturbed, stable vortex is remarkable. While no core transition is observed in the present case, the isosurfaces in Fig. 17c show the significant distortion—with large-amplitude “twist” waves—of the initially cylindrical column. With increasing Re, core break-up into finer-scale filaments may very well occur since the transient amplifications increase with increasing Re (note that practical flows such as the aircraft wake feature Re up to four orders of magnitude higher than simulated here). In practical flows, where Re are about four orders of magnitude higher than in the simulation considered herein, core transition through transient excitation of bending waves appears to be a distinct possibility.

5 Concluding remarks Modeling of turbulent vortices—important in many practical applications—remains a challenge because of the complex effects of vorticity in suppressing turbulence fluctuations, in addition to the role of mean strain in promoting vortex instability and perturbation growth. Direct numerical simulations have been pursued to better understand the physical mechanisms of vortex decay, as well as to provide benchmarking data to turbulence models. Simulation results show that the primary effect of the vortex is to organize turbulence into azimuthal threads, which in turn form outward traveling vortex dipoles. Turbulence close to the vortex column is amplified by the mean strain and spreads due to dipole motion. The inviscid transport of mean angular momentum by the dipoles is confirmed by the appearance of an overshoot in the mean circulation profile. The overshoot, although modest (≈ 5%), makes possible increased turbulence production through a centrifugal-instability-like mechanism. Our numerical experiment on a vortex with an artificially enhanced overshoot, however, suggests that the azimuthal threads produced by the instability suppress the overshoot through mixing, and subsequent vorticity annihilation, of the adjacent positive and negative mean vorticity layers present in the region of the overshoot. This self-limiting mechanism may explain why large circulation overshoots have not been observed in experiments. Perhaps the most striking observation in our simulations is that core fluctuations, while reliant upon external turbulence, can continue to grow even as the turbulence decays and attain intensities exceeding those outside the core. The typical mechanisms of turbulence growth via production and transport are absent in the column core, both effects of mean flow rotation; this makes the growth of core fluctuations enigmatic. A possible explanation comes from recent results showing that the vortex supports algebraic perturbation amplification, through the mechanisms of transient growth [1,16]. In particular, transient bending-wave modes (i.e., modes with azimuthal wavenumber m = 1) localize the perturbation energy in the core and exhibit strong growth through a resonance mechanism in which external threads force a vortex core (Kelvin) wave. These predictions Reprinted from the journal

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from transient-growth analysis are consistent with prior observations [12] of a vortex in turbulence exhibiting core bending waves. We have further pursued the transient-growth mechanism by initializing DNS with an optimal bending-wave mode obtained from linear analysis. Simulations show that a moderate-amplitude perturbation can experience sufficient amplification leading to a turbulent-like flow, outside the core, and cause accelerated vortex decay. Concomitant with the amplification and spreading of external turbulence is the growth of a bending wave, causing significant core distortion. Transient growth eventually ceases, perhaps with nonlinear effects (the self-transport of perturbations away from the “critical” radius where resonant forcing of the core wave is possible) enhancing the (inviscid) linear mechanisms of transient growth arrest. These observations raise questions deserving further investigation. First, can transient-growth-generated structures trigger a self-sustaining cycle of turbulence production? We note that evidence for a self-sustaining mechanism has been found in wall-bounded shear flows [5], where transient growth first produces streamwise vortices, which in turn generate or enhance low-speed streaks and thereby facilitate a new cycle of transient growth [20]. An analogous mechanism may be possible around a vortex column, in which thread dipoles induce radial vorticity perturbations—necessary for transient growth—by tilting the mean vortex lines. A second question of interest is whether the vortex can be sufficiently distorted to cause the coherent core to break up (i.e., transition) into finer-scale structures—a phenomenon that may manifest as an abrupt or rapid decay of the mean velocity profile. No transition has been observed in the range of Re (up to ∼ 104 ) accessible to our DNS. Noting, within our moderate-Re simulations, that peak perturbation levels rise with increasing Re, we believe that core transition can perhaps be induced in practical flows (such as the aircraft wake, where Re is typically three to four orders of magnitude higher than studied here). Incorporating the effects of transient growth in turbulence models may lead to more accurate prediction of vortex decay in turbulent fields—a problem of interest in air traffic control at busy airports. References 1. Antkowiak, A., Brancher, P.: Transient growth for the Lamb-Oseen vortex. Phys. Fluids 16(1), L1–L4 (2004) 2. Cambon, C., Scott, J.F.: Linear and nonlinear models of anisotropic turbulence. Ann. Rev. Fluid Mech. 31, 1–54 (1999) 3. Davenport, W.J., Rife, M.C., Liapis, S.I., Follin, G.J.: The structure and development of a wing-tip vortex. J. Fluid Mech. 312, 67–106 (1996) 4. Govindaraju, S.P., Saffman, P.G.: Flow in a turbulent trailing vortex. Phys. Fluids 14(10), 2074–2080 (1971) 5. Hamilton, J.M., Kim, J., Waleffe, F.: Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348 (1995) 6. Hoffman, E.R., Joubert, P.N.: Turbulent line vortices. J. Fluid Mech. 16(3), 395–411 (1963) 7. Jacquin, L., Pantano, C.: On the persistence of trailing vortices. J. Fluid Mech. 471, 159–168 (2002) 8. Kerswell, R.R.: Elliptical instability. Ann. Rev. Fluid Mech. 34, 83–113 (2002) 9. Mansour, N.N., Wray, A.A.: Decay of isotropic turbulence at low Reynolds number. Phys. Fluids 6(2), 808–814 (1994) 10. Marshall, J.S., Beninati, M.L.: Turbulence evolution in vortex dominated flows. In: Debnath, L., Riahi, D.N. (eds.) Advances in Fluid Mechanics, vol. 25 (Nonlinear instability, chaos and turbulence II, p. 1), pp. 1–40. WIT Press, Southampton, England (2000) 11. Mayer, E.W., Powell, K.G.: Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91–114 (1992) 12. Melander, M.V., Hussain, F.: Coupling between a coherent structure and fine-scale turbulence. Phys. Rev. E 48(4), 2669– 2689 (1993a) 13. Melander, M.V., Hussain, F.: Polarized vortex dynamics on a vortex column. Phys. Fluids A 5, 1992–2003 (1993b) 14. Phillips, W.R.C., Graham, J.A.H.: Reynolds-stress measurements in a turbulent trailing vortex. J. Fluid Mech. 147, 353– 371 (1984) 15. Pradeep, D.S., Hussain, F.: Effects of boundary condition in numerical simulations of vortex dynamics. J. Fluid Mech. 516, 115–124 (2004) 16. Pradeep, D.S., Hussain, F.: Transient growth of perturbations in vortex column. J. Fluid Mech. 550, 251–288 (2006) 17. Ragab, S., Sreedhar, M.: Numerical simulation of vortices with axial velocity deficits. Phys. Fluids 7(3), 549–558 (1995) 18. Rennich, S.C., Lele, S.K.: Numerical method for incompressible vortical flows with two unbounded directions. J. Comput. Phys. 137, 101–129 (1997) 19. Saffman, P.G.: Structure of turbulent line vortices. Phys. Fluids 16(8), 1182–1188 (1973) 20. Schoppa, W., Hussain, F.: Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108 (2002) 21. Spalart, P.: Aircraft trailing vortices. Ann. Rev. Fluid Mech. 30, 107–138 (1998) 22. Sreedhar, M., Ragab, S.: Large eddy simulation of longitudinal stationary vortices. Phys. Fluids 6(7), 2501–2514 (1994) 23. Wallin, S., Girimaji, S.S.: Evolution of an isolated turbulent trailing vortex. AIAA J. 38(4), 657–665 (2000) 24. Zeman, O.: The persistence of trailing vortices: a modeling study. Phys. Fluids 7(1), 135–143 (1995)

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Theor. Comput. Fluid Dyn. (2010) 24:283–289 DOI 10.1007/s00162-009-0120-y

O R I G I NA L A RT I C L E

Keita Iga

Statistical theory applied to a vortex street generated from meander of a jet

Received: 31 October 2008 / Accepted: 21 April 2009 / Published online: 20 June 2009 © Springer-Verlag 2009

Abstract Statistic features of a vortex street formed by instability of a jet are investigated by numerical calculation and statistic theory. A formation process of a vortex street is numerically calculated using a simple barotropic quasi-geostrophic system: a jet in the initial state begins to meander owing to its instability and vortices are formed in both flanks of the jet and become a steady vortex street. Statistic theory of vorticity mixing for two-dimensional fluid, which describes the statistically steady equilibrium state based on the maximum entropy assumption, is applied to the numerically obtained features of the steady vortex street. The theoretically derived relation between stream function and potential vorticity explains the results in the numerical calculation very well. However, in the numerical calculation, there remain regions where the fluid is not mixed well. By calculating mixing process of another scalar, the unmixed region is clearly shown on the physical plane. Keywords Vortex street · Meander of jet · Barotropic instability · Statistic theory · Mixing entropy PACS 47.32.ck, 92.10.Lq

1 Introduction In winter in East Asia Region, cold airmass often outbreaks from the Continent. Under such weather conditions, in the lee side of an island with a high mountain, a weak wind region is made and a vortex street called Kármán vortex is often generated and its flow pattern is visualized by clouds. Similar vortices are also found in the oceans: the Kuroshio current, after leaving the coast of Japan, which is called Kuroshio Extension, usually meanders strongly and generates eddies on both its sides. In this way, jet or weak flow region often meanders because of its instability and generates a vortex street, in the atmosphere and in the oceans. Such a formation process of vortex street is relatively easily simulated under two-dimensional Euler equation or in barotropic quasi-geostrophic equation (aka CHM equation), and has been investigated by many researchers (e.g., [1]). In many cases, such a vortex street generated by the meander of a jet reaches a steady state, where aligned vortices move with a constant velocity. To describe such a steady state, the statistic theory for vorticity, which was proposed by Robert and Sommeria [2], is a very powerful tool. According to their theory, the final steady state is determined by a simple principle familiar in statistic physics: in the final equilibrium state, the ‘mixing entropy’ must be maximum under the conservation restriction for several quantities such as energy, momentum and so on. They derived a relation which the vorticity and the stream function must satisfy. As an example of the application of this Communicated by H. Aref K. Iga Ocean Research Institute, The University of Tokyo, Tokyo 164-8639, Japan E-mail: [email protected] Reprinted from the journal

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theory, Sommeria et al. [3] calculated the simplest situation consisting of only two uniform vorticity regions, which describes a shear flow, and showed the validity of their theory. A jet-shaped current situation demands at least three kinds of uniform vorticity regions. This situation was investigated by Thess et al. [4], and they showed that this ‘maximum mixing entropy principle’ is also applicable for a trapezoidal jet situation. According to their result, the vortices broaden wide to the side boundaries of the fluid region, in the final state with maximum mixing entropy. The width of the region crucially affects the forms of the vortices in the steady state; the vortices spread according to the width of the region. However, when we perform such numerical calculations as models of those in the atmosphere or in the oceans, we usually make the region sufficiently wide in order to avoid the influence of boundaries, and implicitly assume that the numerical result scarcely depends on the width of the region if it is large enough. In this article, we will investigate the features of vortex street formed from a jet in a sufficiently wide region and consider whether we can directly apply these statistic theory to such situations. 2 Statistic theory on mixing of vortices In this section, we will briefly summarize the statistic theory on mixing of vortices by Robert and Sommeria [2]. Although their original theory is described based on two-dimensional Euler system, we will summarize it in the form of barotropic quasi-geostrophic system version, in order to apply it to numerical results with quasi-geostrophic equation. The quasi-geostrophic equation considering here is ∂ (1) (∇ 2 ψ − λ2 ψ) + J (ψ, ∇ 2 ψ) = 0, ∂t which describes large-scale flows in the atmosphere and in the oceans, where ψ is the stream function, and λ is the reciprocal of Rossby’s radius of deformation. In this system, a quantity q called potential vorticity which is defined as q ≡ ∇ 2 ψ − λ2 ψ

(2)

is conserved following the motion of each fluid particle. Therefore, the potential vorticity distribution given in the initial state is only re-distributed by time evolution without changing its value for each fluid particle. We consider, as an initial state, a situation where the whole region is covered by ‘patches’ of fluids with uniform potential vorticity of finite kinds Q i i = 1, 2, . . . , n). The areas of the regions with potential vorticity Q i occupy Ri of the whole region. Since each fluid particle keeps its potential vorticity, the fluid patches are stirred mosaically as the process goes on, and the fluid does not reach a steady state in a strict sense. However, as the mixing proceeds, the mosaic pattern becomes finer; viewing the potential vorticity distribution macroscopically, an equilibrium state is possible. Considering such a final macroscopic equilibrium state, let the ratio of the region where the fluid with potential vorticity Q i occupies in the vicinity of the point (x, y) be ri (x, y). Then, the macroscopic potential vorticity q is expressed as q(x, y) =

n 

Q i ri (x, y),

(3)

i=1

and the fluid in this equilibrium state is described by ri (x, y). Now, we will consider the principle to determine ri (x, y) in the equilibrium state. For this purpose, we define mixing entropy S using ri expressed as  n     S≡− dxdy (4) ri log ri . i=1

In the analogy of statistic physics, we require that the mixing entropy S should be maximum in the equilibrium state as the principle to determine ri : the variation of the entropy (4)   n  (5) δS = − dxdy (log ri + 1) δri i=1

should be zero.

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We must remember here that there are several conserved quantities in this system. The mixing entropy should be maximum, but it is under the condition that these quantities are conserved. It is also expressed as follows: the variation of S should be zero under the constraint conditions that the variations of these conserved quantities keep zero.   We will next list up such conserved quantities and also their variations. The absolute momentum M ≡ dxdy(yq) is conserved; its variation is as follows:  n       δM = dxdy (yδq) = dxdy y (6) Q i δri . i=1

 1

The energy E ≡ − 2

dxdy(ψq) is also conserved, and the variation is   n      Q i δri . δE = − dxdy (ψδq) = − dxdy ψ

(7)

i=1

Since the fluid patches are only re-distributed from the initial   state, the sum of ri (x, y) throughout the whole region should be equal to Ri in the initial state, i.e., Ri = dxdyri is conserved. The variation of Ri is   dxdyδri . (8) δ Ri = In the vicinity of each point, the sum of area ratio of all kinds of patches should be unity: and the variation is n 

δri (x, y).

n

i=1 ri (x,

y) = 1,

(9)

i=1

We should obtain the condition that (5) becomes zero under the constraint conditions that (6)–(9) are all zero. In order to execute this calculation, we have only to determine ri so that the variation   n n   δ F ≡ δS + αδ M + βδ E + γi δ Ri + dxdy (x, y) δri (x, y) (10) i=1

i=1

should be zero, introducing Lagrange multipliers. From the condition that the δ F in (10) should be zero for any variation δri , we obtain the relation log ri + 1 − Q i (αy − βψ) − γi − = 0 (i = 1, 2, . . .).

(11)

Solving this relation for ri , we obtain exp[Q i (αy − βψ) + γi ] . r i = n i=1 exp[Q i (αy − βψ) + γi ] The potential vorticity q is expressed by stream function ψ as [2] n n  i=1 Q i exp[Q i (αy − βψ) + γi ] q= . Q i ri =  n i=1 exp[Q i (αy − βψ) + γi ]

(12)

(13)

i=1

In order to construct a jet-like flow as an initial state, at least three kinds of potential vorticity patches are necessary. We will here apply the general theory with n kinds of potential vorticity patches to the situation with n = 3. We will consider an initial state of the simplest symmetric jet, which consists of three kinds of potential vorticity values Q 1 = Q 0 , Q 2 = 0 and Q 3 = −Q 0 . If we assume that the symmetry between positive and negative vortices in the equilibrium state as well as the initial state, we can consider γ1 = γ3 (≡ γ ), and thus obtain the relation [4] q= Reprinted from the journal

2Q 0 exp(γ ) sinh[Q 0 (αy − βψ)] . 2 exp(γ ) cosh[Q 0 (αy − βψ)] + 1 301

(14)

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K. Iga

3 Numerical calculations In this section, we will show the result of numerical calculations and examine the theory shown in the previous section. The equation numerically integrated is quasi-geostrophic equation (1) adding a small hyperviscosity term: ∂ (∇ 2 ψ − λ2 ψ) + J (ψ, ∇ 2 ψ) = −ν∇ 6 ψ. ∂t

(15)

We can regard the hyperviscosity as a kind of macroscopic-viewing in the theory. We performed the numerical calculation by pseudo-spectral method using sine and cosine expansion with cut-off wavenumber 21 in the x-direction, and sine expansion with cut-off wavenumber 85 in the y-direction. The conversion grids are 64 and 128 in the x- and y-directions, respectively. The system size is L = 2π with periodic boundary in the x-direction, and 2L = 4π with v = 0 boundaries in the y-direction. We set the reciprocal of Rossby’s radius of deformation λ = 0.5 and hyperviscosity coefficient ν = 4.0 × 10−6 . We used leap-frog scheme for timeintegration, with a Heun scheme inserted every 20 time-steps, and the time-step interval is t = 5.0 × 10−3 . The initial state is given so as the potential vorticity distribution to be ⎧ ⎨ −4.0 (−0.75 < y < 0) 4.0 (0 < y < 0.75) (16) q= ⎩ 0 (|y| > 0.75), superposing a small disturbance ψ  (x, y, 0) = 0.02 exp(−4y 2 ) × sin x. The stream function, velocity, and potential vorticity used in the numerical calculation as the initial state are shown in Fig. 1, which has a jet-like distribution. In order to avoid Gibbs’ phenomena, the potential vorticity q is not given as a precise step function; rather somewhat smoothened. The time evolution from this initial state is shown in Fig. 2. By time t = 40, the flow reaches almost steady state, where the stream function keeps its form proceeding in the x-direction. Figure 3 shows the scatter plot of stream function (a uniform flow is subtracted) versus potential vorticity at every conversion grid at t = 800, when we can consider the flow sufficiently steady. In this figure, the theoretical relation (14) is also shown; we can see that the theory well explains the numerical result. However, the scatter plot of the numerical calculation has another ‘branch’ along q = 0 axis which does not fit this theoretical curve. As we can presume from the fact that this second branch lies along q = 0, this branch corresponds to a region away from the jet, where the mixing process does not extend and the potential vorticity remains zero. We can see that the flow region is divided into two areas; well-mixed region where the distribution of the stream function and that of the potential vorticity obey the theoretical relation and unmixed region which is not affected by the mixing process. In order to show the border between them in the physical space, we have only to show on which branch each grid point 6.3

-4.0

0.0 0.0

T= 0.0

4.0

-6.3 Fig. 1 Stream function (solid line), velocity (dashed line) and potential vorticity (dash-dotted line) in the initial state. Potential vorticity takes only three values +4.0, 0.0 and −4.0

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Statistical theory applied to a vortex street generated from meander of a jet

T= 0.0

T= 4.0

T= 8.0

T= 12.0

T= 40.0

T= 800.0

-1.00

-1. 00

-1.0 0

.000

.000

00

.0

0

0 1.

.0 00

1.00

.0

1.00

.000

.000

00

.000

.000

.0

00

00 1.

.000

00

.0

-1.00

.000

.000

.000

.000

Fig. 2 Time evolution of stream functions. The jet meanders to form a vortex street. At t = 40, the flow reaches almost steady state

T= 800.0 5.0

0.0

-5.0 -2.0

0.0

2.0

Fig. 3 Scatter plot of stream function subtracted uniform flow ψ − αy/β (abscissa) versus potential vorticity q (ordinate) at t = 800. The relation (14) for α/β = −0.11, β = 1.1 and γ = −4.5 is shown by dashed line

is located. However, as we can see in Fig. 3, the first branch corresponding to the well-mixed region and the second branch corresponding to the unmixed region are closely located around the origin of the scatter plot; it is difficult to distinguish these two regions only from this figure. Here, let us consider an imaginary passive scalar c which satisfies c = |q| at the initial state, and calculate how this scalar c is mixed by t = 800. Since q has three values of +4, 0 and −4 in the initial state, c ≡ |q| has only two values 0 and 4: c = 4 around the jet axis and c = 0 away from it. Regions with q ∼ 0 exist also in the well-mixed region: if the neighborhood of a point contains almost the same ratio of the fluid with q = +4 and that with q = −4, macroscopically mixed potential vorticity q becomes almost zero. However, passive scalar c keeps non-zero value even at such a point, since it contains patches with c = 4 (q = ±4). Therefore, we can identify unmixed region as a set of points where not only q but also c are almost zero. The scatter plot between stream function and the passive scalar c is shown in Fig. 4, where we can distinguish the branches more clearly than Fig. 3. The relation between ψ and c can be theoretically derived from (12) as c=

2Q 0 exp(γ ) cosh[Q 0 (αy − βψ)] , 2 exp(γ ) cosh[Q 0 (αy − βψ)] + 1

(17)

and it is also shown by a dashed line using the same parameters as those in Fig. 3. Now, we can show the region corresponding to the second branch, since we can clearly distinguish the two branches in Fig. 4. The corresponding region is shown in Fig. 5 as shaded areas, away from the jet-axis. Reprinted from the journal

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T= 800.0 5.0

0.0

-5.0 -2.0

0.0

2.0

Fig. 4 Scatter plot at t = 800 of stream function subtracted uniform flow ψ − αy/β (abscissa) versus a passive scalar c whose value is c = |q| at the initial state (ordinate). The relation (17) for the same parameter as those for Fig. 3 is shown by dashed line

-1.00

T= 800.0

00

1.00

.0

.0

00

.000

Fig. 5 The stream functions at t = 800. Region corresponding to the second branch in Figs. 3 and 4 are shaded, which indicates the unmixed region

In order to determine the forms of the vortex streets in the final state, we must have information how far the mixing of the vortices extend, but this estimation is unknown from the above discussions, and we need other principles or criterions. Chavanis and Sommeria [5] discussed this problem in isolated vortices, and considered the equilibrium state using additional kinetic constraint which determines the area of the well-mixed region. It may become such a principle also in our case. 4 Conclusions We performed numerical calculation of the process where a vortex street is formed from a jet owing to its instability, and applied a statistic theory to the obtained potential vorticity distribution. The following results are obtained: 1. The relation between potential vorticity and stream function predicted from the statistic theory well explains the potential vorticity distribution of the vortex street.

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2. The scatter plot of stream function versus the potential vorticity have two branches, which means that the mixing does not spread far enough. In order to determine theoretically the final state of the vortex street, we need more information about how far the mixing extends. References 1. Flierl, G.R., Malanotte-Rizzoli, P., Zabusky, N.J.: Nonlinear waves and coherent vortex structures in barotropic β-plane jets. J. Phys. Oceanogr. 17, 1408–1438 (1987) 2. Robert, R., Sommeria, J.: Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310 (1991) 3. Sommeria, J., Staquet, C., Robert, R.: Final equilibrium state of two-dimensional shear layer. J. Fluid Mech. 233, 661– 689 (1991) 4. Thess, A., Sommeria, J., Jüttner, B.: Inertial organization of a two-dimensional turbulent vortex street. Phys. Fluids 6, 2417– 2429 (1994) 5. Chavanis, P.H., Sommeria, J.: Classification of robust isolated vortices in two-dimensional hydrodynamics. J. Fluid Mech. 356, 259–296 (1998)

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Theor. Comput. Fluid Dyn. (2010) 24:291–297 DOI 10.1007/s00162-009-0130-9

O R I G I NA L A RT I C L E

Paul S. Krueger

Vortex ring velocity and minimum separation in an infinite train of vortex rings generated by a fully pulsed jet

Received: 7 November 2008 / Accepted: 24 June 2009 / Published online: 22 July 2009 © Springer-Verlag 2009

Abstract A pulsed jet with a period of no flow between pulses (i.e., a fully pulsed jet) produces a multiplicity of vortex rings whose characteristics are determined by the jet pulsing parameters. The present study analyzes the case of impulsively initiated and terminated jet pulses in the limit of equal pulse duration and period to determine the minimum possible vortex ring separation obtainable from a fully pulsed jet. The downstream character of the flow is modeled as an infinite train of thin, coaxial vortex rings. Assuming inviscid flow and matching the circulation, impulse, kinetic energy, and frequency of the jet and vortex ring train allow the properties of the vortex ring train to be determined in terms of the ratio of jet slug length-to-diameter ratio (L/D) used for each pulse. The results show the minimum ring separation may be made arbitrarily small as L/D is decreased and the corresponding total ring velocity remains close to half the jet velocity for L/D < 4, but the thin-ring assumption is violated for L/D > 1.5. The results are discussed in the context of models of pulsed-jet propulsion. Keywords Vortex rings · Pulsed jets PACS 47.32.cf

1 Introduction Jet pulsation engenders the formation of a vortex ring with each jet pulse leading to a train of (initially) coaxial vortex rings. The effect is most pronounced in fully pulsed jets where a period of no-flow appears between pulses so that each pulse is similar to a starting jet. The vortex ring formation process is responsible for a number of interesting characteristics of pulsed jets including enhanced entrainment [1,15] and thrust augmentation [2,6]. The role of vortex rings as dominant structures in pulsed jets has led to 2D [13] and axisymmetric [16] models of pulsed jets for aquatic propulsion based on infinite trains of equally spaced vortex pairs (2D) or rings (axisymmetric). These models, however, do not constrain the characteristics of the vortex train by the characteristics of the generating jet, allowing the circulation and/or spacing of the vortices to take on arbitrary values. In particular, the analysis of Weihs [16] predicts the thrust generated by a fully pulsed jet can be enhanced by an increase in the velocity of the vortex ring train compared to that of an isolated vortex ring if the ring separation is reduced below about three ring radii for a given ring circulation, , core radius, e, and ring radius b. Weihs notes that ring separation may be reduced by increasing the pulsing frequency, but no analysis of the minimum possible ring separation for a given , e, and b is provided. Communicated by H. Aref P. S. Krueger Department of Mechanical Engineering, Southern Methodist University, P.O. Box 750337, Dallas, TX 75275, USA E-mail: [email protected] Reprinted from the journal

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The present analysis develops the relationship between the downstream properties of a full-pulsed round jet and the pulsing parameters of the jet in the limit of no separation between pulses using simple models for both the jet and resulting train of vortex rings.

2 Flow model In the idealized case, the pulses of a fully pulsed jet initiate and terminate instantaneously as shown in the model jet velocity time trace (U J (t)) shown in Fig. 1. As t p → T , the separation between pulses decreases and the separation between the rings formed by each pulse also decreases. In the limit of t p = T , the ring separation is minimal. In this limit, the jet can be viewed as steady with periodic, finite—but instantaneous—disturbances that break the jet into discrete chunks. Under this idealization, the over-pressure associated with pulsation [2,5,6] may assumed to be negligible and the circulation, impulse, and kinetic energy expelled with each pulse can be determined by the slug model [4,9]. As each pulse rolls up into a vortex ring, the asymptotic downstream state is modeled as an infinite series of equally spaced coaxial vortex rings. Assuming thin-core vortex rings with the coordinate systems shown in Fig. 2, the simplest model for the proposed vorticity distribution is ∞ 

ωθ = δ(r − b)

δ(z − na)

(1)

n=−∞

where  is the circulation of the rings, a is the ring separation, and b is the radius of the rings. This is sufficient for determining the impulse associated with each ring and the mutually induced ring velocity, to leading order, but determining the self-induced ring velocity and the ring kinetic energy requires specification of the core vorticity distribution. To that end a circular vortex core of radius e 1.5, so the results for large L/D violate the thin ring assumption. Second, Fig. 4a shows that 0.14 < e/a < 0.18 over the range of conditions considered. Finite e/a violates the point vortex assumption used for neighboring vortices. The associated error in E i is not expected to be large, however, because ψn depends on (e/a)3 to leading order. Finally, a coaxial vortex ring train is an unstable equilibrium configuration. Levy and Forsdyke [8] proved that a coaxial train of thin vortex rings is unstable to perturbations in the ring radii and axial locations. In particular, periodic axial and radial displacements of vortex rings can lead to mutual slip-through or ‘leapfrogging’ [11,12,17] of adjacent vortex ring pairs. The flow model could, in principle, be extended to account for longitudinally periodic leapfrogging in the limit of thin, inviscid vortex rings by allowing a and b to be functions of time and vortex ring location, but that is beyond the scope of the present study. More significantly, a coaxial train of vortex rings is also unstable to perturbations in the inclination of the rings relative to the jet axis, causing them to wander off axis (which cannot be accounted for by an extension of the present axisymmetric model). This instability was exploited by Reynolds et al. [10] to create ‘blooming’ jets and Krueger and Gharib [6] observed this as the dominant instability in a nominally unperturbed pulsed-jet. Reprinted from the journal

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12 10

WT / W0

8 6 4 2

0

0

1

2

3

4

L /D

Fig. 5 Total ring velocity compared to the velocity of an isolated ring

Nevertheless, several vortex rings typically remain on axis before instability disintegrates the train and the infinite sums in Eqs. (3) and (4) may be expected to provide a reasonable approximation to the real situation of multiple coaxial vortex rings.

5 Conclusion Using an infinite train of thin, coaxial vortex rings as a model for the flow resulting from a fully pulsed jet, the characteristics of the vortex ring train for the case of minimum ring separation were determined as functions of L/D by enforcing the kinematic and kinetic constraints dictated by the jet for impulsively started and terminated jet pulses in the limit t p → T under the assumption of inviscid flow. The results show that a/b scales with L/D, so arbitrarily small ring separation is theoretically achievable with sufficiently short pulse duration. In addition, WT /U0 remains less than one and approaches the slug model value of 1/2 as L/D is decreased. Even though WT /W0 increases as L/D decreases due to induction from surrounding vortices, WT is not amplified above the jet velocity (contrary to the model proposed by Weihs [16]) because  also decreases with L/D. While these results are expected to be representative of real jet flows, the analysis is only approximate because for L/D > 1.5 the thin ring assumption is violated and instability of the vortex ring train makes coaxial trains longer than a few vortex rings unlikely. Acknowledgments This material is based upon work supported by the National Science Foundation under Grant No. 0347958.

References 1. Bremhorst, K., Hollis, P.G.: Velocity field of an axisymmetric pulsed, subsonic air jet. AIAA J. 28, 2043–2049 (1990) 2. Choutapalli, I.M.: An experimental study of a pulsed jet ejector. Ph.D. Dissertation, Florida State University, Tallahassee, FL (2006) 3. Fraenkel, L.E.: Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51, 119–135 (1972) 4. Gharib, M., Rambod, E., Shariff, K.: A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121–140 (1998) 5. Krueger, P.S., Gharib, M.: The significance of vortex ring formation to the impulse and thrust of a starting jet. Phys. Fluids 15, 1271–1281 (2003) 6. Krueger, P.S., Gharib, M.: Thrust augmentation and vortex ring evolution in a fully pulsed jet. AIAA J. 43, 792–801 (2005) 7. Lamb, H.: Hydrodynamics, pp. 236–239. Dover, New York (1932) 8. Levy, H., Forsdyke, A.G.: The stability of an infinite system of circular vortices. Proc. R. Soc. London A 114, 594–604 (1927) 9. Mohseni, K., Gharib, M.: A model for universal time scale of vortex ring formation. Phys. Fluids 10, 2436–2438 (1998) 10. Reynolds, W.C., Parekh, D.E., Juvet, P.J.D., Lee, M.J.D.: Bifurcating and blooming jets. Annu. Rev. Fluid Mech. 35, 295– 315 (2003) 11. Shariff, K., Leonard, A., Ferziger, J.H.: Dynamics of a class of vortex rings. NASA TM-102257 (1989) 12. Shariff, K., Leonard, A.: Vortex rings. Annu. Rev. Fluid Mech. 24, 235–279 (1992) 13. Siekmann, J.: On a pulsating jet from the end of a tube, with application to the propulsion of certain aquatic animals. J. Fluid Mech. 15, 399–418 (1963)

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14. Taylor, G.I.: Formation of a vortex ring by giving an impulse to a circular disk and then dissolving it away. J. Appl. Phys. 24, 104 (1953) 15. Vermeulen, P.J., Rainville, P., Ramesh, V.: Measurements of the entrainment coefficient of acoustically pulsed axisymmetric free air jets. J. Eng. Gas Turbul. Power 114, 409–415 (1992) 16. Weihs, D.: Periodic jet propulsion of aquatic creatures. Fortschritte der Zoologie 24, 171–175 (1977) 17. Yamada, H., Matsui, T.: Mutual slip-through of a pair of vortex rings. Phys. Fluids 22, 1245–1249 (1979)

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Theor. Comput. Fluid Dyn. (2010) 24:299–303 DOI 10.1007/s00162-009-0110-0

O R I G I NA L A RT I C L E

Morten Brøns · Anders V. Bisgaard

Topology of vortex creation in the cylinder wake

Received: 6 November 2008 / Accepted: 16 February 2009 / Published online: 2 June 2009 © Springer-Verlag 2009

Abstract We analyze the topology of the two-dimensional flow around a circular cylinder at moderate Reynolds numbers in the regime where the vortex wake is created. A normal form for the stream function close to the cylinder is presented and used to predict the streamline pattern both in the steady and the periodic regime, where two different vortex shedding scenarios are identified. The theoretical predictions are verified numerically. For the vorticity, a very different topology occurs with infinite nested sequences of iso-curves moving downstream. General equations of motion for critical points are derived. Keywords Wakes · Topology · Bifurcation · Streamline patterns · Vortex dynamics PACS 47.15.Tr · 47.20.Ky · 02.40.Pc · 47.32.C1 Introduction The flow around a circular cylinder is one of the cornerstones of fluid mechanics, and a huge literature with a wealth of information on the flow is available. A prominent feature of the flow is the vortex wake, which exists robustly in various configurations depending on the Reynolds number Re = U D/ν, where U is the free-stream velocity, D is the diameter of the cylinder, and ν is the viscosity. The basic route to the creation of the wake is as follows: for Re below 5–7, the flow is steady and attached to the cylinder. As Re is increased the flow separates and a symmetric pair of steady vortices are created behind the cylinder. This flow is stable until Re ≈ 45–49, where a Hopf bifurcation makes the flow periodic, vortex shedding appears, and a vortex wake is created. At still higher Re, the flow becomes three-dimensional and turbulent [4–6,9]. In the present paper, we discuss systematically the topological changes which occur in the flow as the vortex wake is created. We will consider both the velocity and the vorticity fields. 2 Streamline topology For a two-dimensional velocity field (u, v), the instantaneous streamlines at t = t0 are the trajectories for the system x˙ = u(x, y, t0 ),

y˙ = v(x, y, t0 ).

(1)

Communicated by H. Aref M. Brøns (B) Department of Mathematics, Technical University of Denmark, 2800 Kongens Lyngby, Denmark E-mail: [email protected] A. V. Bisgaard NKT Flexibles I/S, Priorparken 510, 2605 Brøndby, Denmark E-mail: [email protected] Reprinted from the journal

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Fig. 1 Local streamline topologies near the aft point of symmetry for the normal form Eq. (2). Left the topologies in the symmetric case, c0,2 = c0,3 = 0. a c1,2 > 0, b c1,2 < 0, c bifurcation diagram for the full normal form for c1,2 < 0

The complete set of trajectories is the phase portrait of the velocity field. As external parameters, and, in the case of unsteady flows, time, are varied, the phase portrait may bifurcate and change qualitatively. An understanding of such bifurcations can be obtained systematically by the use of normal forms. A normal form is a simplification of the velocity field which still represents the flow qualitatively correct [3]. Brøns et al. [4] show that the relevant normal form for the stream function the flow behind a circular cylinder as vortices are created is ψ = y 2 (c0,2 + c1,2 x + c0,3 y + x y + x 3 ).

(2)

Here, the coordinate system is chosen such that the x-axis follows the cylinder wall, and the origin is placed at the aft point of separation. The factor y 2 occurs due to the no-slip condition at the wall. In the steady regime, the flow has mirror symmetry with respect to the line x = 0, and hence both c0,2 and c0,3 must be zero. Varying the remaining parameter c1,2 across zero gives rise to a bifurcation where a symmetric pair of recirculation zones are created (see Fig. 1a, b). When the flow becomes periodic, the symmetry is broken, and the full normal form is relevant. For c1,2 > 0 no bifurcations occur, so only the half-space c1,2 < 0 is relevant. A typical slice of the parameter space is shown in Fig. 1c. The space is partitioned by bifurcation curves into regions of different topologies. Note in particular the curve marked III, where the streamline topology is that of the steady separated flow, but with an asymmetry except at the origin (c0,2 , c0,3 ) = (0, 0). In the steady regime, the topological state must then be associated with that point. As the flow turns periodic in the Hopf bifurcation, small amplitude oscillations occur. Since the normal form parameters are smooth functions of the physical parameters (here time and Re), the instantaneous streamline topology right after the bifurcation must be represented by a small closed orbit encircling the origin. Such an orbit is schematically shown as the small gray ellipse in Fig. 1c. This must be placed in the regions a, a  , crossing from one region to another through the curve III at two time instances during a period. As Re is increased, the amplitude of the oscillations grow. At a certain stage, the orbit in the parameter space may become so large that it crosses other bifurcation curves and into new topologies. Such a closed curve is shown as the large gray ellipse in Fig. 1c. This qualitative analysis is confirmed by numerical simulations. We find a transition of the topology in the steady domain from that of Fig. 1a and b at Re = ReS = 6.29. A theoretical determination of this value could possibly be obtained from an analysis of the boundary layer structure such as performed by Proudman

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( (b)

(a) III

(a)

(b)

(c)

a

(c)

(d)

III

a

(e)

(d) III

II

(f)

a b

II

II

a

(l)

(k)

(j) II

b

(i)

(h)

(g) III

a

a

Fig. 2 Numerical simulations showing the two sequences of instantaneous streamline topologies in the cylinder wake in the periodic regime. Left Re = 45.6, corresponding to the small gray ellipse in Fig. 1c. Right Re = 100, corresponding to the large gray ellipse in Fig. 1c

and Johnson [8]. If an analytical expression for c1,2 were derived, the bifurcation point could immediately be found. We are not aware of any studies of flow topology bifurcations from this point of view. A Hopf bifurcation is found at Re = ReH = 45.08. For ReH < Re < ReT = 45.68, the flow topology does indeed follow the first periodic scenario. It only persists in this very narrow interval of Re, and the second scenario is present for ReT < Re < 200. We have not examined the flow beyond Re = 200 as three-dimensional effects become important at this stage. For further details, including the numerics (see [4]). It is interesting to see that the streamline topology does not exhibit a vortex wake. It is tempting to identify the critical points of center type, shown as isolated dots in Figs. 1 and 2, as vortex centers. However, even if vortices from this point of view are shed from the cylinder in the second periodic scenario, they collapse and disappear again not far downstream. This occurs when the bifurcation curves II and II are crossed. A very different picture appears for the vorticity, as we will now show. 3 Vorticity topology The streamlines are iso-curves of the stream function. In two-dimensional flow the vorticity is likewise a scalar field, and the iso-curves can be determined. In numerical simulations, vorticity is often color-coded, and regions of high-positive or low-negative vorticity can be identified and with good reason be denoted vortices. To understand the dynamics of such regions, it seems fruitful to identify the local minima and maxima of the vorticity and obtain information about the dynamics of these points. Bifurcations of these points can be studied by the same methods as those used for the topological bifurcation analysis of the stream function which yields the streamline pattern, and we proceed here to initiate such an analysis. Finding the vorticity from the normal form Eq. (2) with the symmetry condition c1,2 = c1,3 = 0 it is easy to show that a saddle point of the vorticity is created simultaneously with the symmetric recirculation zone as c0,2 passes through zero. The saddle point is not the same as the saddle point for the stream function, and no centers are created for the vorticity. One easily finds from Eq. (2) that the stream function saddle is 2 ). This is confirmed numerically located at y = −c1,2 , while the vorticity saddle is at y = −c1,2 /3 + O(c1,2 in Fig. 3. While this transition clearly marks a change in the vorticity topology, no obvious physical interpretation presents itself, and in general it seems to be difficult to assess the physical importance of saddle points of vorticity. The streamline topology of the steady separated flow (Fig. 1b) is structurally unstable, as a small perturbation may destroy the connections between the off-wall saddle point and the separation points on the wall. This is exactly what happens in the first periodic scenario. In contrast, the vorticity topology of Fig. 3b is structurally stable, so the transition to periodic flow does not imply any change in the vorticity topology close to the cylinder. However, several cylinder diameters downstream significant topology changes occur in the periodic regime. Preliminary numerical simulations for a Re slightly above ReH are shown in Fig. 4. For further details Reprinted from the journal

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Fig. 3 Numerical solution showing the vorticity topology in the steady regime. a Re = 1.54, b Re = 20

Fig. 4 Numerical simulations showing the periodic sequence of vorticity topologies a → Il → a  → Iu → a in the periodic regime, here for Re = 45.6. The lower right figure is a qualitative sketch of the vorticity contours, including for clarity only a single loop encircling a vorticity extremum

see [2]. A typical configuration (Fig. 4a) consists of a double array of vorticity extrema, nested within a loop connected to a saddle point and further upstream to the cylinder surface at two points, one close to the front stagnation point of the cylinder, one close to the point where the separatrix of the saddle close to cylinder meets the cylinder. The further downstream the loop extends, the closer it meets the cylinder to the front stagnation point where ω = 0. Hence, at the array of saddles the vorticity tends to zero in the direction downstream from the cylinder. This configuration moves downstream with the flow, and at two time instants during a period a new vorticity extremum and a corresponding saddle point are created in a simple saddle-center or cusp bifurcation (Il and Iu ). Since the flow is periodic, the vorticity pattern has to return to its original state after one period, and this can only happen if the arrays of critical points of the vorticity are infinitely long. The bifurcation scenario is that of Hilbert’s Hotel: the hotel has infinitely many rooms, but even if it is full, there is always room for a new guest. The occupant of room 1 moves to room 2, the occupant of room 2 moves to room 3, etc., leaving room 1 for the new guest. It is interesting to compare this scenario with the flow complexification in turbulent flow discussed by Moffatt [7]. Here, more and more complex flow patterns, both in velocity and vorticity, are found through generic bifurcations of which the saddle-center bifurcation is the simplest one. This scenario appears right after the Hopf bifurcation and hence describes the first stage of the vortex wake. Work to describe the further topological development at higher Re is in progress.

4 Dynamics of critical points Since the critical points of a scalar field are the key to understand the topology of the iso-curves, it is of interest to consider equations of motion for such points. Starting with the vorticity field ω(x, y, t), a critical point (x(t), y(t)) fulfills ωx (x(t), y(t), t) = 0, ω y (x(t), y(t), t) = 0,

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where subscripts denote differentiation. Differentiating these relations with respect to t and using the Navier–Stokes equations ωt + (v · ∇)ω = νω

(4)

where ν is viscosity one obtains after a little algebra ω yy ωx − ωx y ω y , H ωx x ω y − ωx y ωx , y˙ = v − ν H

(5)

H = ωx x ω yy − ω2x y .

(6)

x˙ = u − ν

with the Hessian determinant

For inviscid flow, the critical points of vorticity follow the velocity field and are hence material points. Viscosity introduces a slip relative to the fluid velocity. One would expect this slip velocity to vanish in the far wake as the velocity field approaches the unperturbed field. This agrees with the alignment of streaklines with the vorticity field some distance from the cylinder [1]. However, from a mathematical point of view some care should be taken, as both numerators and denominators in Eqs. (5) tend to zero in the far wake. The same approach can be applied for the stagnation points, the critical points of the stream function, u(x(t), y(t), t) = 0, v(x(t), y(t), t) = 0.

(7)

Differentiating and using the Navier–Stokes equations in the form 1 vt + (v · ∇)v = − ∇ p + νv ρ

(8)

yields x˙ = y˙ =

1 ρ ( px v y

− p y u y ) − ν(v y u − u y v)

J 1 ( p u − p v ) − ν(u x v − vx u) y x x x ρ J

, (9) ,

with the Jacobian J = u x v y − vx u y .

(10)

Compared to the equations of motion for the vorticity extrema, Eq. (9) are quite complex, involving both the velocity field itself and the pressure gradient, even in the case of vanishing viscosity. Further research is needed to clarify if these equations can provide any useful insights. References 1. Anagnostopoulos, P.: Computer-aided flow visualization nad vorticity balance in the laminar wake of a circular cylinder. J. Fluids Struct. 11, 33–72 (1997) 2. Bisgaard, A.V.: Structures and Bifurcations in Fluid Flows with Applications to Vortex Breakdown and Wakes. PhD thesis, Department of Mathematics, Technical University of Denmark (2005) 3. Brøns, M.: Streamline topology—patterns in fluid flows and their bifurcations. Adv. Appl. Mech. 41, 1–43 (2007) 4. Brøns, M., Jakobsen, B., Niss, K., Bisgaard, A.V., Voigt, L.K.: Streamline topology in the near-wake of a circular cylinder at moderate Reynolds numbers. J. Fluid Mech. 584, 23–43 (2007) 5. Coutanceau, M., Defaye, J.-R.: Circular cylinder wake configurations: flow visualisation survey. Appl. Mech. Rev. 44(6), 255–305 (1991) 6. Jackson, C.P.: A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45 (1987) 7. Moffatt, H.K.: The topology of scalar fields in 2D and 3D turbulence. In: Kambe, T., Nakano, T., Miyauchi, T. (eds.) IUTAM Symposium on Geometry and Statistics of Turbulence, pp. 13–22. Kluwer, Dordrecht (2001) 8. Proudman, I., Johnson, K.: Boundary-layer growth near a rear stagnation point. J. Fluid Mech. 12((2), 161–224 (1962) 9. Williamson, C.H.K.: Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539 (1996)

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Theor. Comput. Fluid Dyn. (2010) 24:305–313 DOI 10.1007/s00162-009-0144-3

S P E C I A L I S S U E O R I G I NA L A RT I C L E

Franck Auguste · David Fabre · Jacques Magnaudet

Bifurcations in the wake of a thick circular disk

Received: 19 December 2008 / Accepted: 9 June 2009 / Published online: 26 August 2009 © Springer-Verlag 2009

Abstract Using DNS, we investigate the dynamics in the wake of a circular disk of aspect ratio χ = d/w = 3 (where d is the diameter and w the thickness) embedded in a uniform flow of magnitude U0 perpendicular to its symmetry axis. As the Reynolds number Re = U0 d/ν is increased, the flow is shown to experience an original series of bifurcations leading to chaos. The range Re ∈ [150, 218] is analysed in detail. In this range, five different non-axisymmetric regimes are successively encountered, including states similar to those previously identified in the flow past a sphere or an infinitely thin disk, as well as a new regime characterised by the presence of two distinct frequencies. A theoretical model based on the theory of mode interaction with symmetries, previously introduced to explain the bifurcations in the flow past a sphere or an infinitely thin disk (Fabre et al. in Phys Fluids 20:051702, 2008), is shown to explain correctly all these results. Higher values of the Reynolds number, up to 270, are also considered. Results indicate that the flow encounters at least four additional bifurcations before reaching a chaotic state. Keywords Wake instabilities · Bifurcation theory PACS 47.15. Fe Stability of laminar flows, 47.15. Tr Laminar wakes, 47.10. Fg Dynamical systems methods 1 Introduction Bodies moving within a viscous fluid may display a large variety of dynamical behaviours. A famous example, described five centuries ago by Leonardo da Vinci is the rise of small bubbles which may follow zigzagging or spiraling motions [13]. Another example, which puzzled Maxwell since 1857, is the falling of a paper card, which may display various motions such as fluttering, tumbling, chaotic oscillations, etc. [15,17]. It is now recognized [10] that such path instabilities are directly linked to an instability of the recirculating region in the near wake of the body. Therefore, to understand these movements it is useful to consider first the related and simpler problem of bodies held fixed within a uniform incoming flow. In a recent paper [5], we used DNS to investigate the bifurcation scenarios in the wake of two reference bodies, namely a sphere and an infinitely thin disk held normal to the stream. For both bodies the Reynolds number is defined as Re = Uν0 d , where U0 is the incoming velocity, d is the diameter of the body, and ν is the kinematic viscosity of the fluid. The case of the sphere is well known from the literature [3,9,12,14,16]. A first bifurcation occurs for Re ≈ 210 resulting in a wake characterized by the presence of a steady pair of streamwise vortices and the occurrence of a constant lift force on the body. This wake configuration, hereinafter Communicated by H. Aref F. Auguste · D. Fabre (B) · J. Magnaudet Institut de Mécanique des Fluides de Toulouse, University of Toulouse, Toulouse, France E-mail: [email protected] E-mail: [email protected]; [email protected] E-mail: [email protected] Reprinted from the journal

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referred to as the “Steady-State” (SS) mode, breaks the axisymmetry but retains a reflectional symmetry with respect to an azimuthal plane. A second bifurcation is observed for Re ≈ 272, leading to a periodic state which retains the symmetry plane and is associated with a lift force oscillating around a non-zero mean value. In experiments, this mode leads to the shedding of hairpin vortices all oriented in the same direction [14] ; hence we will refer to it as the “Zig-Zig” mode (Z z, to be distinguished with the “Zig-Zag” mode (Z Z ) which will be introduced below). In the case of a flat disk, a different bifurcation sequence is observed. A first bifurcation is observed for Re = 115.5, leading to a SS mode with a reflectional symmetry. Then a Hopf bifurcation is found for Re = 121.5. Unlike the case of the sphere, the mode observed after this bifurcation does not preserve the reflectional symmetry. The resulting mode is characterized by a lift force oscillating in a direction around a mean orientation; it can be described as a “Yin-Yang” (Y Y ) mode, owing to the characteristic pattern observed just behind the body (see Fig. 4 of the present paper). A third bifurcation is observed for Re = 139.4, leading to a recovery of the planar symmetry. The resulting flow is characterized by a lift force oscillating along a given direction with a zero mean. This mode is referred to as a “Standing-Wave” (SW ) mode by analogy with other related problems. It can also be called the “Zig-Zag” (Z Z ) mode, as in experiments it would be associated with the shedding of hairpin vortices in alternating directions. To explain these results, Fabre et al. [5] introduced a system of nonlinear amplitude equations describing the interaction between the two dominant unstable modes. This model allowed us to explain the differences observed in the wake of the two bodies and accurately reproduces the evolution of the lift forces. For the case of an infinitely thin disk, these conclusions were recently confirmed by Meliga et al. [11], who obtained the same system of equations using a weakly nonlinear global stability approach. In the present paper, we continue the numerical exploration of the wake dynamics of axisymmetric bodies by considering a geometry which is somehow intermediate between that of a sphere and that of a thin disk, namely a thick disk with an aspect ratio χ = d/w = 3, where d is the diameter and w the thickness. This geometry was chosen because an extensive experimental study was carried out in our team with freely moving bodies of this type rising in salted water [6,7]. In Sect. 2 we describe the numerical method we use to investigate the flow past this body. In Sect. 3 we describe the various flow regimes we observe as the Reynolds number is increased in the range Re ∈ [150, 218]. In Sect. 4 we show that the theoretical model introduced by Fabre et al. [5] can be adapted to explain these computational results. In Sect. 3, we briefly address the transition to chaos occurring in the range Re ∈ [218, 270]. We finally provide some conclusions in Sect. 6.

2 Numerical method and strategy of investigation The numerical code used in the present study is similar to that used in [5] and was described in [1,10]. The code solves the three-dimensional Navier–Stokes equations for an incompressible and homogeneous fluid. Temporal evolution is discretized by a third-order Runge–Kutta scheme. The divergence-free condition is satisfied using a projection method. The flow is described using a cylindrical grid with 118(x) × 70(r ) × 32(θ ) nodes (the x-axis corresponds to the symmetry axis). A nonuniform grid distribution is used near the body to properly capture the boundary layer and near wake. The characteristic grid size is 0.015d near the body, and is about 0.1d in the near wake (down to x = 2d). The grid extends to a distance of roughly 10 diameters in all directions. The boundary conditions are (i) a no-slip condition on the body itself, (ii) a kinematic condition u x = U0 in the inlet plane and on the lateral boundaries, and (iii) a non-reflecting outlet condition in the outlet plane. The grid used here provides results in good agreement with those found during the convergence study performed for a thin disk by Auguste et al. [1]. The range of Reynolds numbers Re ∈ [150, 220] was scrutinized with steps of Re = 1 in most of the interval, and not larger than Re = 2. Computations were run on the Altix server computer of CICT in Toulouse (parallelized on two processors). Typical runs consisted of 50000 time steps (corresponding to a dimensionless time tU0 /d about 800); however, much longer computational times were required to achieve convergence close to some of the bifurcations. Typically, computations were started with an initial velocity field originating from a former computation performed with a neighbouring value of Re; close to the bifurcations, the Reynolds number was varied both upwards and downwards to detect an eventual subcritical behaviour. Most of the bifurcations described here were localized within an interval Re = 1; however, in some cases, observing the transient behavior and fitting using theoretical results, as done for instance in [3], allowed a more accurate estimate of the threshold values.

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To describe the efforts experienced by the body, we introduce the drag coefficient C x , the side force coefficients C y and C z , and the torque coefficients Cmx , Cmy and Cmz defined as follows:     Fx , Fy , Fz = C x , C y , C z ρU02 π(d/2)2 /2.     Mx , M y , Mz = Cmx , Cmy , Cmz ρU02 π(d/2)3 To describe the various flow regimes, we draw two types of “phase diagrams”,  based respectively on side forces (C y − C z diagram) and drag/lift forces (C x − C L diagram, where C L = C y2 + C z2 is the lift coefficient). Finally, note that the exact or average reflectional symmetry planes existing in some of the regimes described in the next section are arbitrary and are selected by the initial conditions of the simulation. To make the discussion simpler, a convenient rotation was applied to the numerical results to make the symmetry directions coincide with the y- or z-axis. 3 The sequence of bifurcations in the range Re ∈ [150,217] As for the reference cases of a sphere and a flat disk, the flow remains steady and axisymmetric at low enough values of the Reynolds number. In this range, the flow is characterised by a toroidal recirculation region (except at very low values of Re where the latter vanishes). Such a flow, hereinafter referred to as the “trivial” state (T S) is illustrated in Fig. 1a, which displays the vorticity and streamlines in a transverse plane for Re = 150. In this regime the hydrodynamic force on the body reduces to a drag force. As the Reynolds number increases, a steady bifurcation occurs. In agreement with the reference cases of a sphere and a disk, it leads to a “Steady-State” mode (SS) characterised by a reflectional symmetry plane and an associated lift force. The threshold value associated with this bifurcation was estimated to Rec1 ≈ 159.4, in agreement with the previous experimental estimate Rec1 = 116.5(1 + χ −1 ) proposed by Fernandes et al. [6]. The flow in this regime is illustrated in Fig. 1b which displays iso-surfaces of the streamwise vorticity component (through two orthogonal views). In this case, the symmetry plane is the x − z plane and the lift force is along the positive z direction. The next bifurcation is of Hopf type and occurs for a threshold Reynolds number estimated to Rec2 ≈ 179.8. This value also reasonably matches the previous estimate Rec2 = 125.6(1 + χ −1 ) [6]. Above this threshold, the flow within the wake and the forces experienced by the body become unsteady and periodic. A Strouhal number characterising this flow can be defined as St = U0dT0 , where T0 is the shedding period. Right at the threshold, this Strouhal number is St = 0.109, a value comparable to but slightly smaller than those found for the two reference bodies (Stsphere ≈ 0.133 , Stdisk ≈ 0.119). The regime resulting from this bifurcation is illustrated in Fig. 2. It can be observed that the wake retains the symmetry plane selected by the previous bifurcation. This allows us to identify this mode as the “Zig-zig” one already observed in the wake of a sphere. Figure 2a shows the structure of the flow at two instants of the shedding cycle. The C y − C z diagram (Fig. 2b) indicates that the lift force is oscillating about a non-zero mean value along a line located within the symmetry plane of the flow. The C x − C L diagram (Fig. 2c) reveals a single loop. This loop travels in the anti-clockwise (a)

(b)

1.2

ωθ

2 1 0.5 0.1 -0.1 -0.5 -1 -2 -3 -4 -5 -10

1

r

0.8 0.6 0.4 0.2 0 -0.5

0

0.5

1

1.5

2

x

Fig. 1 Steady flow regimes: a axisymmetric flow for Re = 150 depicted by azimuthal vorticity (colors online) and streamlines, b non-axisymmetric steady state (SS) for Re = 165 depicted by iso-surfaces of the streamwise vorticity (two orthogonal views) Reprinted from the journal

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(a)

(b)

(c)

0.08 0.07 0.068

0.06

CL

Cz

0.066 0.04

0.064 0.062

0.02

0.06 0.058

0 -0.04

-0.02

0

0.02

1.006

0.04

1.008

1.01

1.012

Cx

Cy

Fig. 2 Periodic, reflectional-symmetry-preserving mode (or “Zig-Zig” mode) for Re = 182. a Iso-surfaces of streamwise vorticity (at two time instants of the cycle corresponding to the maximum and the minimum of the lift force, respectively), b C y − C z diagram, c C x − C L diagram

(a)

(b)

(c) 0.08 0.075

0.06 0.07 0.065

CL

0.04

Cz

0.06 0.055

0.02

0.05 0 -0.06

-0.04

-0.02

0

Cy

0.02

0.04

0.06

0.045 0.996 0.998

1

1.002 1.004 1.006 1.008 1.01

Cx

Fig. 3 Quasi-periodic pulsating mode (or “Knit-Knot” mode) for Re = 187. a Iso-surfaces of the streamwise vorticity at two instants of time, b C y − C z diagram (the end of the time series is represented with a thick line), c C x − C L diagram

direction, indicating that extrema in the drag force are encountered slightly before those of the lift force. Note that the Strouhal number characterising this flow regime (and actually also the three next regimes) is remarkably constant, and only varies by a few percents around the value St = 0.109 selected at Re = Rec2 . A third bifurcation is found to take place for a threshold value Rec3 ∈ [184, 185]. The flow observed beyond this bifurcation is illustrated in Fig. 3 (for Re = 187). This flow is of a totally new type, and is characterised by the breaking of the reflectional symmetry (as can be seen in Fig. 3a which shows the structure of the streamwise vorticity at two sample instants) and the occurrence of a secondary frequency. The C y − C z diagram (Fig. 3b) reveals an attractor with a complicated structure evoking the shape of a wool ball. This is why we term this regime the “Knit-Knot mode” (K K ). The motion along the attractor can be understood as the superposition of two movements, namely a rapid motion along an elliptic path, with a period T0 very close to that found in the previous regime, and a slow pulsation of this ellipse around a mean direction (which corresponds to the z direction in the figure). To help understand this motion, the last two elliptic oscillations are displayed with thick lines in the figure. The C x − C L diagram shows another projection of the attractor which also reveals a two-period motion. In this diagram the pulsation is seen as a slight jitter of the main cycle. Note that the oscillation period T p greatly varies in the range of existence of this mode, from T p ≈ 96T0 at Re = 185, down to T p ≈ 48T0 at Re = 187, and then up to T p ≈ 54T0 at Re = 190. On the other hand the main period remains remarkably constant. The “Knit-knot” mode described above is observed up to a threshold value of Rec4 ∈ [190, 191] where a fourth bifurcation is detected. This bifurcation is characterised by the disappearance of the above slow pulsation, and the flow then comes back to a purely periodic state. The resulting flow regime is illustrated in Fig. 4 (for Re = 195). The structure of the streamwise vorticity (Fig. 4a) allows us to identify this mode with the mode existing in the wake of a thin disk in the range Re ∈ [121, 139] [5]. The C y − C z diagram (Fig. 4b)

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(a)

(b)

(c)

0.06 0.04

Cz

0.02 0 -0.02 -0.04 -0.06 -0.06 -0.04 -0.02

0

0.02 0.04 0.06

Cy

Fig. 4 Periodic, reflectional-symmetry-breaking mode (or “Yin-Yang” mode) for Re = 195. a Iso-surfaces of the streamwise vorticity at two instants, b C y − C z diagram, c iso-levels of the streamw vorticity in the x = 0.5 plane

(a)

(b)

(c) 0.08 0.06

0.05 0.04

Cy

Cz

0.02

0

0 -0.02 -0.04

-0.05 -0.06

-0.05

0

Cy

0.05

-0.08 0.979 0.98 0.981 0.982 0.983 0.984 0.985 0.986 0.987

Cx

Fig. 5 Periodic mode with reflectional symmetry and zero mean lift (or “Zig-Zag” mode) for Re = 216. a Iso-levels of the streamwise vorticity in the plane located at x = 0.5, b C y − C z diagram, c C x − C L diagram

shows that the lift force is oscillating back and forth along a closed path about a mean direction (here chosen to be the z direction), while the C x − C L diagram (not shown) reveals a single loop. This mode was called the “reflectional-symmetry-breaking” (R S B) mode in [5]. It can also be termed as the “Yin-Yang mode” owing to the characteristic shape of the streamwise vorticity contours in a cross-section of the wake (see Fig. 4c). The next (i.e. fifth) bifurcation is found for Rec5 ≈ 215. This bifurcation is in all respects similar to that occurring in the wake of a thin disk for Re ≈ 139, as it is associated with a recovery of a symmetry plane in the wake, the orientation of which (here y) is orthogonal to that selected by the initial bifurcation (here z). The resulting flow is illustrated in Fig. 5. The C y − C z diagram (Fig. 5b) indicates that the lift force is contained within the symmetry plane of the flow and is oscillating around a zero mean value. The C x − C y drag-lift diagram (Fig. 5c) shows a butterfly-like attractor, revealing that the drag passes twice through a maximum (and then a minimum) during one period of oscillation of the lift force. The structure of the flow is illustrated in Fig. 5a at two instants of time corresponding, respectively, to a positive extremum (upper plot) and a negative extremum (lower plot) of C y . The plots show that in this case the body sheds symmetrical hairpin-like structures during each half-period of oscillation, thus justifying the denomination of this regime as a “Zig-Zag mode” (Z Z ). A synthetic view of all the regimes described so far is given in Fig. 6 which presents the maximum and minimum values of force coefficients (Fig. 6a) and torque coefficients (Fig. 6b) as a function of the Reynolds number in the range Re ∈ [150, 220]. As can be noticed, all quantities are continuous across the bifurcations (but generally have discontinuous slopes), indicating that all bifurcations are regular and supercritical. The first bifurcation at Rec1 is associated with the onset of a lift force and a lateral torque and also with an increase of the drag force compared to that associated with the axisymmetric solution (displayed with a dotted line in Fig. 6a). Such a discontinuity in the slope of the C x curve has also been reported in the case of a sphere [3]. At the second bifurcation (i.e. for Re = Rec2 ), the lift and drag forces and the side torque become unsteady Reprinted from the journal

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(a)

Rec1 Axi

Rec2 SS

Rec3 Zz

Rec4 KK

Rec6

Rec5 YY

ZZ

(b) 0.15

0.01

Rec1 Axi

Rec2 SS

Rec3 Zz

Rec4 KK

Rec5 YY

Rec6 ZZ

1.04 0.008

0.96 0.05

0.0003

0.006 0.0002 0.004

Cmx,max

Cx,max, Cx,min

0.98

Cml,min, Cml,max

0.1

1

CL,min, CL,max

1.02

0.0001

0.94 0.002 0.92 0.9 160

180

200

0 220

0 160

180

200

0 220

Re

Re

Fig. 6 Variations of force coefficients (a) and torque coefficients (b) in the range Re ∈ [150, 220]

and oscillate between the maximum and minimum values plotted in the figure. The third bifurcation at Rec3 is characterised by the occurrence of an axial torque which oscillates about zero with a maximum amplitude represented by the Cm,x curve. The presence of such an axial torque is characteristic of regimes in which the reflectional symmetry is broken. Consequently, this torque is non-zero within the range of existence of the “Knit-knot” and “Ying-yang” modes and dies out at the fifth bifurcation corresponding to Re = Rec5 .

4 Theoretical modelling In a previous study [5], we introduced a theoretical model which successfully reproduces the first steps of the bifurcation sequence observed in the cases of a sphere and a thin disk. In this section we review this model and show how it can be adapted to the present case. The model is based on a velocity field with the following expansion:     u = U0 (r, x) + Re a0 (t)e−iθ uˆ s (r, x) + Re a1 (t)e−iθ uˆ h,−1 (r, x) + a2 (t)eiθ uˆ h,+1 (r, x) + · · · ,

(1)

where U0 (r, x) denotes the axisymmetric solution of the Navier–Stokes equations for a given value of Re, uˆ s (r, x) is the most amplified mode (associated with an azimuthal wavenumber m = 1 and a real eigenvalue λs ), and uˆ h,m (r, x) is the next most amplified mode (associated with an azimuthal wavenumber m = 1 and a complex eigenvalue λh + iωh ), whereas a0 , a1 , a2 are three complex amplitudes. Starting from expansion Eq. (1), the central manifold theorem states that if the leading modes are simultaneously nearly neutral, the whole problem can be reduced to a system of ordinary differential equations (ODE) governing their amplitudes [4,8]. This ODE system has the generic form, known as the normal form of the problem:     a˙0 = λs a0 + l0 |a0 |2 a0 + l1 |a1 |2 + |a2 |2 a0 + il2 |a2 |2 − |a1 |2 a0 + l3 a¯0 a¯2 a1 ,   a˙1 = (λh + iωh )a1 + B|a1 |2 + (A + B)|a2 |2 a1 + C|a0 |2 a1 + Da02 a2 ,   a˙2 = (λh + iωh )a2 + B|a2 |2 + (A + B)|a1 |2 a2 + C|a0 |2 a2 + D a¯0 2 a1 ,

(2) (3) (4)

where l0 to l3 are real coefficients, while A, B, C, D are complex. Interestingly, this system is also relevant to the Taylor–Couette flow problem where it describes the interaction between “Taylor vortices” and “spiral vortices”. A classification of the solutions up to secondary bifurcations is available in [8]. A refined investigation of the possible solutions up to ternary bifurcations, and of their relevance to the present problem, is in progress (Fabre and Knobloch, in preparation). In Ref. [5] the coefficients involved in (4) were fitted using the computational results obtained for the sphere and the thin disk, leading to a good agreement between the predictions of the model and the DNS

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Y−Y ??

K−K Z−z

Z−Z

S−S H−H

Spiral

TS

150

160

170

180

190

200

210

220

Fig. 7 Theoretical bifurcation diagram. Arbitrary bifurcation parameter as function of the Reynolds number. Full (resp. dashed) lines represent stable (resp. unstable) branches; circles represent the quasi-periodic pulsating “Knit-Knot” branch

results. We have repeated this approach in the present case. The corresponding fitting led to the following set of coefficients: λs = 0.05(Re − 159.8), λh = 0.04(Re − 190), ωh = 0.685, l0 = −100, l1 = −1386, l2 = 0, (5) l3 = −1800, A = 100, B = −225, C = 22, D = 16.8 + 50i. This set of parameters leads to a sequence of bifurcations in close agreement with our numerical results. In particular, the predicted threshold Reynolds number values are as follows: Rec1 = 159.8,

Rec2 = 179.9,

Rec3 = 184.7,

Rec4 = 190.4,

Rec5 = 215.2.

(6)

The theoretical bifurcation diagram obtained with this set of parameters is plotted in Fig.  7. The figure displays an arbitrary measure of the amplitude of the different states (roughly proportional to |a0 |2 + |a1 |2 + |a2 |2 ) as function of the Reynolds number; following the usual convention, full (resp. dashed) lines correspond to stable (resp. unstable) branches and circles to quasi-periodic solutions. This diagram correctly reproduces the sequence of bifurcations revealed by the numerical simulations up to Re = 217, and accurately predicts the threshold values of the successive bifurcations, thus confirming the relevance of the model. Note that it also predicts the existence of two additional branches, namely a “spiral” mode and a quasi-periodic mode of a different nature (termed H − H in the figure), which are both unstable and thus not numerically observed. A more quantitative fitting of the lift forces, as performed in Ref. [5] for the thin disk, was not tried. Therefore, the set of coefficients given above is only indicative. A direct and rigorous determination of the coefficients of the normal form using a weakly nonlinear expansion of the Navier–Stokes equations, as done by Meliga et al. [11] in the case of a thin disk, would certainly be preferable. We finally point out that the present model is unable to account for the subsequent bifurcations to be described in the next section, which are most likely associated with the emergence of new leading modes, in addition to the two modes already included in the expansion (1). 5 The route to chaos in the range Re ∈ [217, 270] We now turn to the next features that occur at higher Reynolds number. To this end, we investigated the range of Reynolds numbers [217, 270] in a less exhaustive way than reported in the previous section. Figure 8 displays sample phase diagrams of the regimes we detected. The “Zig-Zag” mode, which was the last of the sequence of bifurcations documented in Sect. 3, has a limited range of existence, as a new bifurcation occurs for Rec6 ∈ [217, 218]. The mode observed beyond this Reprinted from the journal

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(b)

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Fig. 8 Force diagrams for selected values of Re on the route to chaos: C x − C y diagram for Re = 220 (a) and Re = 230 (b); C y − C z diagram for Re = 245 (c) and Re = 270 (d)

threshold is characterised by the persistence of a reflectional symmetry plane and the occurrence of a second frequency. The corresponding attractor, depicted in Fig. 8a through its C x − C y projection (for Re = 220), is typical of a quasi-periodic system. The leading frequency f 0 is very close to the one existing in the previous regimes, while the secondary frequency f 1 is close to one-third of the leading frequency.1 For Re = 230, the C x − C y diagram (Fig. 8b) shows that the attractor becomes again a closed loop, indicating that the flow has come back to a periodic state. This can be explained as the result of a phase locking of the two main frequencies which, in this case, get into an exact 1:3 resonance.2 This exact resonance is lost when the Reynolds number is further increased and, for Re = 235, the attractor (not shown) becomes again quasi-periodic and similar to the one plotted in Fig. 8a. The planar symmetry is eventually lost for Re ≈ 240. The last two plots (Fig. 8c, d) illustrate the shape of the attractor through its C y − C z projection for Re = 245 and Re = 270, respectively. In the first case the trajectory seems to retain some symmetries as it does not explore the whole domain. This suggests that this regime is not yet completely chaotic but rather multi-periodic. On the other hand, in the second case a true chaotic state seems to be reached, as the trajectory makes larger excursions in the whole domain. We did not attempt to localise more precisely the threshold value of Re associated with the onset of chaos, since computations are very time-consuming in this range of parameters. Moreover, a simple glance at the force diagrams is not sufficient to discriminate true chaos from multi-periodic solutions and more powerful means of investigation are required, such as the computation of Lyapunov exponents. Future work will be devoted to this issue.

6 Summary In this paper, we used DNS to investigate the wake dynamics of a thick disk with an aspect ratio χ = 3, held fixed in an imposed upstream flow parallel to its symmetry axis. In the range Re ∈ [150, 216], an original sequence of bifurcations was evidenced, which is somehow intermediate between the reference cases of a sphere and a thin disk, respectively. In the first two steps of the sequence, the flow successively encounters a steady, non-axisymmetric state (SS) associated with a constant lift force, and a periodic, reflectional-symmetry preserving mode associated with a lift force oscillating about a non-zero mean value (“Zig-zig” mode). This sequence is identical to what happens in the wake of a sphere. However, a third bifurcation occurs, leading to an original type of flow which breaks the reflectional symmetry and oscillates in a quasi-periodic manner (“Knitknot” mode). Two additional bifurcations lead successively to a periodic, reflectional-symmetry-breaking mode (“Yin-Yang” mode) and to a periodic mode with reflectional symmetry and a lift force oscillating about zero (“Zig-Zag” mode). This part of the sequence is identical to what happens in the wake of a thin disk, except that in the latter case the “Yin-Yang” mode takes place after the second bifurcation of the sequence, immediately after the steady, non-axisymmetric state (SS). In line with previous results obtained for the reference cases of a sphere and a thin disk, this whole sequence is fully explained by a theoretical model describing the interaction between the two leading modes. However, this model is unable to explain the dynamics encountered in the 1 This nearly harmonic relation between the frequencies led us to call this regime a “Honky-Tonky” mode (as if it were a music instrument that would sound slightly out of tune). 2 We thus propose to call this retuned mode a “Boogie-Woogie” mode.

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range Re ∈ [217, 270]. In this range, at least three additional bifurcations exist, leading eventually to a fully chaotic state. Additional work is still required to completely describe this route to chaos. This work is currently continued in the case where the objects are freely moving within the fluid under the effect of buoyancy and hydrodynamic forces. Preliminary results [2] have demonstrated an excellent agreement with the experiments conducted in our team [6,7]. They have also revealed the existence of several new kinds of trajectories, including steady oblique paths, periodic and quasi-periodic trajectories and weakly chaotic regimes. The cartography of these regimes is in progress. The application of normal form theory to this case is also a promising issue. Acknowledgments The numerical simulations were performed on the Altix server computer of CICT in Toulouse, under grant P0727. The authors acknowledge Edgar Knobloch for fruitful discussions on the theoretical model.

References 1. Auguste, F., Fabre, D., Magnaudet, J.: Ecoulement de fluide visqueux autour d’un disque en incidence frontale. 18ème Congrès Français de Mécanique, Grenoble (France) (2007) 2. Auguste, F., Fabre, D., Magnaudet, J.: Numerical study of solid cylinders moving freely in a viscous fluid. EFMC7 Conference, Manchester (UK) (2008) 3. Bouchet, G., Mebarek, M., Ducek, J.: Hydrodynamic forces acting on a rigid fixed sphere in early transitional regimes. Eur. J. Mech. B 25, 321–336 (2006) 4. Crawford, J.D., Knobloch, E.: Symmetry and symmetry-breaking bifurcations in fluid dynamics. Ann. Rev. Fluid Mech. 23, 341–387 (1991) 5. Fabre, D., Auguste, F., Magnaudet, J.: Bifurcations and symmetry breaking in the wake of axisymmetric bodies. Phys. Fluids 20, 051702 (2008) 6. Fernandes, P.C., Risso, F., Ern, P., Magnaudet, J.: Oscillatory motion and wake instability of freely-rising axisymmetric bodies. J. Fluid Mech. 573, 479–502 (2007) 7. Fernandes, P.C., Ern, P., Risso, F., Magnaudet, J.: Dynamics of axisymmetric bodies rising along a zigzag path. J. Fluid Mech. 606, 209–223 (2007) 8. Golubitsky, M., Stewart, I., Schaeffer, D. M. (1988) Singularities and groups in bifurcation theory, vol. II. Applied Mathematical Sciences. Springer, Berlin 9. Johnson, T.A., Patel, V.C.: Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70 (1999) 10. Magnaudet, J., Mougin, G.: Wake instability of a fixed spheroidal bubble. J. Fluid Mech. 572, 311–337 (2007) 11. Meliga, P., Sipp, D., Chomaz, J.M.: Unsteadiness in the wake of the sphere: receptivity and weakly non-linear global stability analysis. 5th BBVIV Conference, Costa do Sauipe (Brazil) (2007) 12. Mittal, R.: Planar symmetry in the unsteady wake of a sphere. AIAA J. 37, 388–390 (1999) 13. Mougin, G., Magnaudet, J.: Path instability of a rising bubble. Phys. Rev. Lett. 88, 14502 (2002) 14. Ormières, D., Provansal, M.: Vortex dynamics in the wake of a sphere. Phys. Rev. Lett. 83, 80–83 (1999) 15. Pesavento, U., Wang, Z.J.: Falling paper: Navier–Stokes solutions, model of fluid forces and center of mass elevation. Phys. Rev. Lett. 93, 14451 (2004) 16. Tomboulides, A.G., Orzag, S.A.: Numerical investigation of transitional and weak turbulent flow past a sphere. J. Fluid Mech. 416, 45–73 (2000) 17. Wilmarth, W.W., Hawk, N., Harvey, R.: Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 197– 208 (1964)

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Theor. Comput. Fluid Dyn. (2010) 24:315–322 DOI 10.1007/s00162-009-0099-4

O R I G I NA L A RT I C L E

R. Kunnen · R. Trieling · G. J. van Heijst

Vortices in time-periodic shear flow

Received: 8 January 2009 / Accepted: 18 March 2009 / Published online: 8 May 2009 © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract Vortices emerging in geophysical turbulence may experience deformations due to the non-uniform ambient flow induced by neighbouring vortices. At first approximation this ambient flow is modeled by a linear shear flow. It is well known from previous studies that the vortex may be (partially) destructed through removal of weak vorticity at the vortex edge—a process referred to as ‘stripping’. While most previous studies considered a stationary external shear flow, we have examined the behaviour of the vortex embedded in a linear shear flow whose strength changes harmonically in time. Aspects of the vortex dynamics and the (chaotic) transport of tracers have been studied by both laboratory experiments and numerical simulations based on a simple kinematical model. Keywords Vortex dynamics · Rotating fluids · Shear flow · Lobe dynamics · Laboratory experiments · Contour kinematics PACS 47.32.C- · 47.32.F.f 1 Introduction Background rotation and/or density stratification in a fluid tend to suppress one velocity component, often giving the flow a quasi-two-dimensional (quasi-2D) character. Such conditions are commonly encountered in large-scale geophysical flows. As a result of the inverse energy cascade active in 2D turbulence, such flows (even when quasi-2D) show a tendency of self-organisation, usually observed in the emergence of vortex structures (see e.g. [6]). The vortices of 2D turbulence are embedded in a non-uniform flow field, and hence they generally undergo time-dependent straining deformations. A natural question arising is: how is a vortex affected by the strain imposed by the exterior flow? Only a very few analytical studies have been carried out for vortices in a strain or shear flow: the case of steady vortex patches in a linear shear flow was studied by Moore and Saffman [10], while this work was extended by Kida [4] to include unsteady solutions. In fact, work on this topic was also carried out by S. A. Chaplygin and published (in Russian) already in 1899, see the review by Meleshko and van Heijst [8]. Much later, Legras and Dritschel [5] performed a numerical study of the behaviour of nested patches of uniform vorticity in a shear flow by using the method of contour dynamics. In experimental studies on 2D vortices one often applies background rotation, see the recent review by van Heijst and Clercx [3]. In this line of approach Trieling et al. [16] performed an experimental study of a single vortex in an annular shear flow. While most previous studies were aimed at vortices in a steady strain/shear flow, in the present paper we will focus on time-dependent external shear flow. In order to study the tracer transport in such unsteady flows we introduce an extremely simple kinematical model of the flow, consisting of a point Communicated by H. Aref R. Kunnen · R. Trieling · G. J. van Heijst (B) Fluid Dynamics Laboratory, Department of Physics, Eindhoven University of Technology, Eindhoven, The Netherlands E-mail: [email protected] Reprinted from the journal

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vortex of constant strength placed in a linear shear flow whose amplitude is perturbed harmonically. Such an approach was also used by Perrot and Carton [13,14] in their study of vortex interactions in an unsteady ambient shear flow. In fact, this approach is closely related to earlier studies by Rom-Kedar et al. [15] and Velasco Fuentes et al. [17,18] on modulated point vortices, in which the vortex strength was made (or allowed) to be time dependent. In addition, a laboratory experiment is carried out in a rectangular fluid tank rotating about its vertical axis. A cyclonic vortex is generated by siphoning, while the time-periodic shear flow is produced by a modulation of the rotation speed of the turntable. The model results show very good agreement with the dye-visualised vortex flow. The kinematical model and the technique of the numerical computations based on it will be briefly reviewed in Sect. 2, while the experimental arrangement and the flow visualization results will be described in Sect. 3. The main conclusions will be drawn in Sect. 4. 2 Theoretical model The trajectory (x(t), y(t)) of a fluid particle in an incompressible two-dimensional flow can be described by the following equations: dx ∂ = , dt ∂y

dy ∂ =− , dt ∂x

(1)

where  is the stream function. We can interpret (1) as a Hamiltonian system with Hamiltonian function . If  is independent of time this Hamiltonian system has one degree of freedom, while for time-dependent  it has two degrees of freedom. When  is periodic in time this system is said to have ‘one and a half’ degrees of freedom [11]. The dynamics of systems of various degrees of freedom differ considerably. In steady, onedegree-of-freedom flows only linear dynamics can occur. Aref [1] showed that time-periodic flows can produce chaotic particle paths, thus enabling complicated fluid mixing in these flows. The stream function of the flow induced by a point vortex in a linear shear flow is 1 γ (2) α0 y 2 − ln(x 2 + y 2 ), 2 4π with α0 the shear strength and γ the strength of the point vortex. For the remainder of this section√we restrict ourselves to positive values of α0 and γ . In that case there exist √ two stagnation points p± at (0, ± γ /2πα0 ) in the flow. We denote this characteristic radius by r p = γ /2πα0 . The corresponding streamline pattern is shown in Fig. 1a. The dotted streamlines divide the flow into several separate regions; therefore they are known as separatrices. =

a

b p+

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Fig. 1 a Streamline pattern of the flow due to a point vortex in an unperturbed linear shear flow. The hyperbolic points are indicated by p+ and p− . b In the perturbed case, the unstable manifold W+u of p+ intersects with the stable manifold W−s of p− . The intersection of W+s and W−u is omitted here for clarity. The symbols D, D  , E, and E  indicate detrainment and entrainment lobes

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The behaviour of this Hamiltonian system changes drastically when a perturbation is introduced. Here we insert a small harmonic variation in the shear strength α α(t) = α0 [1 +  cos(ω p t)],

(3)

in which α0 is the basic-state shear strength,  is the (small) amplitude of perturbation, and ω p is the perturbation ˜ = /α0 r 2p : frequency. We can now non-dimensionalise the stream function with x˜ = x/r p , t˜ = α0 t, and  ˜ = 

   ωp 1 1 2 t˜ − ln(x˜ 2 + y˜ 2 ). y˜ 1 +  cos 2 α0 2

(4)

(The scaling of the logarithmic term is accounted for in the arbitrary constant that can always be added to a stream function.) Henceforth we drop the tildes for convenience. Two dimensionless numbers determine the flow: (i) the perturbation amplitude , and (ii) the ratio of ω p and α0 , which we will from now on refer to as the timescale ratio σ = ω p /α0 . The equations of motion (1) thus become y dx , = y[1 +  cos(σ t)] − 2 dt x + y2 x dy = 2 . dt x + y2

(5)

When the trajectory of a given particle with initial position X is given by the function t (X) for all t, we can introduce a Poincaré mapping F for the periodic flow as xn+1 = F(xn ) = T (xn ),

(6)

with xn the position of a particle after a time nT , and T being the period of perturbation. The stable and unstable manifolds W s and W u of a hyperbolic point p are defined as W s (p) = {∀ X ∈ R N |F(X) → p ⇔ t → ∞}, W u (p) = {∀ X ∈ R N |F(X) → p ⇔ t → −∞}.

(7)

Under perturbation the separatrices degrade to a structure of intersecting stable and unstable manifolds of the hyperbolic points, called the heteroclinic tangle, see Fig. 1b. The hyperbolic points, however, remain as fixed points of the Poincaré map. In the heteroclinic tangle the stable and unstable manifolds enclose socalled lobes. The lobes display the fluid exchange between the interior region and the exterior flow, across the previously impervious separatrices. For a detailed treatment of lobe dynamics, see e.g. Wiggins [19]. The locally chaotic character of the transport induced by this time-dependent flow system is nicely illustrated by the Poincaré map presented in Fig. 2. This map shows the positions of 2 × 1,631 tracer particles initially positioned in the entrainment lobes near the hyperbolic points, after 50 cycles of the harmonic perturbation. One clearly observes three ‘empty’ regions, in which no particles have penetrated: one central region, in which no particles can penetrate due to the elliptic nature of the vortex core, and two curved, elongated regions on either side of the elliptical central region. The distribution of the particles clearly reveals the entrainment near the hyperbolic points in the form of lobes that become very elongated when closer to the hyperbolic points. Calculations with different perturbation frequencies ω p have revealed that for higher frequencies the size of the empty central region increases, while the dispersion band of the tracer particles becomes narrower. Also, the lobes become more elongated and narrower, while covering a decreasing area for increasing frequency. Melnikov [9] derived an analytical technique that measures the distance between the stable and unstable manifolds for small perturbation amplitudes. This expression is now known as the Melnikov function. Using this function one can calculate the area of a lobe, and hence derive a quantity that provides a measure of the net exchange of mass between the vortex and its exterior. In order to study the transport properties of the perturbed vortex for larger perturbation amplitudes, we adopted the so-called contour kinematics method, which monitors in time a contour defined by passive markers connected by small line segments. When sufficient markers are used, the contour appears smooth. Initially the markers form a circle. Integrating the advection equation (1) for each marker results in the evolution of the contour in time. The contour will get stretched and folded by the Reprinted from the journal

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Fig. 2 Poincaré map showing the positions of 2 × 1,631 passive tracer particles, initially placed in the entrainment lobes close to the hyperbolic points, after 50 cycles of the perturbation. Perturbation frequency 1.6π

flow. All time integrations are performed using a variable-order Runge–Kutta scheme. The contour kinematics method has been tested extensively, and for a more detailed description, the reader is referred to Meleshko and van Heijst [7]. The mechanism of stretching and folding, clearly present near the hyperbolic points, see Fig. 1b, leads to rapid distortions of closed material contours released close to these points. This effect has been investigated by monitoring the length stretch of a contour initially placed around the fixed point p+ for different values of the frequency f s = ω p /2π. The length stretch λ is defined as λ = l(t)/l(0), where l(t) is the contour length at time t and l(0) is the initial length at t = 0. The results of the calculations are shown in Fig. 3. At first, one observes an exponential stretch when the contour is still near the hyperbolic point. From t = 2.5 onwards, the contour stretch is different for different frequencies. The broken line represents the unperturbed case, in which the contour length increases linearly. In the perturbed case, however, fluid particles may separate at an exponential rate. The stretch curves for t > 2.5 are roughly linear, thus revealing exponential stretching. For higher perturbation frequencies f s , the stretching and folding mechanism active near the stagnation points is more effective and results in enhanced stretching of the contours, as visible in the steeper slopes of the stretch curves in Fig. 3. 3 Laboratory experiments 3.1 Description of the experimental set-up The experiments were performed in a rectangular plexiglass tank of dimensions length × width × height = 200 × 40 × 30 cm. This tank was mounted on a turntable, centred on the axis of rotation. It was filled with

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Fig. 3 Length stretch l(t)/l(0) of a contour initially placed around the fixed point p+ . The broken line represents the unperturbed case (ε = 0). The solid lines represent perturbed cases, for different values of the frequency f s . Other parameter values: γ = 8π, a0 = 1, ε = 0.1

water to a depth of H = 20 cm. A co-rotating camera was mounted about 1 m above the water surface. A perforated plexiglass tube with 32 holes, distributed uniformly over a length of 15 cm, was placed vertically along the axis of rotation. By virtue of the Taylor–Proudman theorem (see e.g. [12]), rotation causes the flow in the tank to behave as a two-dimensional flow. Furthermore, a vortex can be generated under rotation by siphoning fluid through the perforated tube. The inward flow towards the tube is deflected by the Coriolis force, resulting in a cyclonic vortex. The circulation of this vortex is given by  D 2q = −2 ( × v) · ds = − , (8) Dt H C with D/Dt the material derivative, C a material contour enclosing the sink tube,  the rotation vector, ds an infinitesimal segment of C, and q the (negative) volume flux. An instantaneous increase of the rotation speed of the tank leaves the fluid lagging behind, and thus an anticyclonic flow is observed in a co-rotating frame of reference. van Heijst et al. [2] studied the spin-up process and the resulting flow in the rotating frame. When using a large-aspect-ratio tank this flow creates to very good approximation a linear shear in the central region of the tank. The magnitude of the shear constant α is then equal to twice the value of the change in rotation velocity . Oscillations of the shear are made by constantly changing the computer-controlled rotation speed of the turntable. The anticyclonic flow enacted by a stepwise change of  is dampened through Ekman dynamics. The characteristic timescale associated with this process is the so-called Ekman decay time TE = H/(ν)1/2 , with ν being the kinematic viscosity. An experiment in such a shear flow should therefore preferably be finished well within this time. In our experiments TE was typically 240 s. Qualitative experiments were performed in which the flow was visualised with a fluorescent dye. The dye, a passive tracer, is evidently suitable for visualisation of the unstable manifolds of the hyperbolic points. A blob of dye released around a hyperbolic point will be advected away from this point along the unstable manifold, thereby marking it. Quantitative experiments were also performed (by using a high-resolution combined PIV–PTV technique), but the results of these will be reported elsewhere. 3.2 Results In order to obtain qualitative information about the evolution of the vortex in the perturbed shear flow, dye visualization experiments were carried out. In one particular series of experiments a small blob of fluorescent dye was introduced at the free surface of the fluid, near one of the stagnation points in the flow. The evolution of the dye pattern was recorded from above by a co-rotating camera. Four snapshots of this dye-visualised experiment are displayed in Fig. 4 (left column), on which the structure of the unstable manifolds is clearly visible. In Reprinted from the journal

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Fig. 4 Time evolution of the vortex in an oscillating shear flow: dye visualization in the laboratory experiment (left column), numerical contour kinematics simulation (right column). The snapshots correspond with t/T = 0, 0.4, 0.6, 1.0 and thus span a full perturbation period T . Parameter values:  = 0.7 rad s−1 ,  = 0.05 rad s−1 , = 6.3 × 10−3 m2 s−1 ,  = 0.4, f s = 0.1 s−1 (in the numerical simulation: γ0 = 6.3 × 10−3 , α0 = 0.1, f s = 0.1, ε = 0.4)

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particular one observes the process of repeated stretching and folding taking place near the stagnation points. Furthermore, the lobes are elongated along the unstable manifolds of the hyperbolic points, and hence leaving the vortex region. It is instructive to make a comparison with the plot of the unstable manifolds, see Fig. 1b. Additionally, see Fig. 4 (right column), a contour kinematics simulation was made for the same flow conditions, with the initial, circular contour placed around the upper stagnation point. It is obvious that the distortion of the contour is very similar to the dye patterns observed in the experiment: even the details of the repeated stretching and folding near the lower stagnation point seem to be well captured by the contour kinematics simulations. This stresses the value of this simple kinematical flow model, based on a potential vortex. An equally good agreement between experimental dye-visualisations and a (modulated) point-vortex model was obtained by Velasco Fuentes et al. [17,18] in their studies of topography-perturbed dipolar and tripolar vortices. A very careful comparison between the dye visualization and the contour kinematics simulations reveals a few small differences, however: the dye visualizations indicate that the stagnation points are not exactly positioned on a line perpendicular to the basic shear flow direction, while in the kinematical model they are (by definition). The explanation lies in the continuous, finite-sized vorticity distribution present in the ‘real’ flow, i.e. in the laboratory experiment. Hence, the shape of the vorticity distribution may change during the experiment and besides, the vortex is continuously eroded by stripping of low-amplitude vorticity at its edge. As a result, the experimental vortex shrinks and decays in the course of the experiment, while that of the kinematical model has a fixed strength. Of course, such effects may also be incorporated in the model, but this lies beyond the scope of the present paper.

4 Conclusion The behaviour of tracer transport due to a vortex in a time-periodic shear flow was modeled in the simplest possible way, viz. by a point vortex in a harmonically perturbed linear shear flow. The two stagnation points appearing in this flow play a key role in the tracer transport: the intersecting stable and unstable manifolds show lobes, the area of which is directly related to entrainment/detrainment between the vortex interior and exterior. For small perturbation amplitudes the lobe area—and hence the exchange rate of material—can be determined by using Melnikov’s theory. For larger amplitudes, one is forced to use numerical techniques like the contour kinematics method. Simulations with an initially circular contour placed around one of the hyperbolic points show the rapid advection away from this point and the subsequent substantial deformation when reaching the other stagnation point. Such contour advection simulations nicely reveal the stretching and folding mechanism present near the stagnation points in the perturbed flow. The length stretch of the contour is found to behave exponentially in time, the growth rate increasing with higher frequency. A laboratory experiment has been performed in which the transport properties of the sheared vortex were studied qualitatively. The vortex flow was created in a homogeneous fluid contained in a long rectangular tank rotating about its vertical axis by removing some fluid by siphoning. A shear flow could be established for some time in the central part of the tank by accelerating or decelerating the rotating table slightly. By adding a sinusoidal modulation on top of this increased or decreased rotation speed, the shear flow was perturbed harmonically. The resulting flow was visualized by locally adding neutrally buoyant dye to the fluid. When introduced near one of the stagnation points, the evolution of the dye patch clearly shows the repeated stretching and folding taking place near the stagnation points, thus visualizing the structure of the stable and unstable manifolds. The agreement between the contour kinematics simulations based on the simple point-vortex model and the visualized flow in the laboratory experiment illustrates how powerful such a simple kinematical model may be for the purpose of describing tracer transport. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References 1. Aref, H.: Stirring by chaotic advection. J. Fluid Mech. 143, 1–21 (1984) 2. van Heijst, G.J.F., Davies, P.A., Davis, R.G.: Spin-up in a rectangular container. Phys. Fluids A 2, 150–159 (1990) Reprinted from the journal

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3. van Heijst, G.J.F., Clercx, H.J.H.: Laboratory modelling of geophysical vortices. Annu. Rev. Fluid Mech. 41, 143–164 (2009) 4. Kida, S.: Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Jpn 50, 3517–3520 (1981) 5. Legras, B., Dritschel, D.G.: Vortex stripping and the generation of high vorticity gradients in two-dimensional flows. Appl. Sci. Res. 51, 445–455 (1993) 6. McWilliams, J.C.: The emergence of isolated, coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43 (1984) 7. Meleshko, V.V., van Heijst, G.J.F.: Interacting two-dimensional vortex structures: point vortices, contour kinematics and stirring properties. Chaos Solitons Fractals 4, 977–1010 (1994) 8. Meleshko, V.V., van Heijst, G.J.F.: On Chaplygin’s investigations of two-dimensional vortex structures in an inviscid fluid. J. Fluid Mech. 272, 157–182 (1994) 9. Melnikov, V.K.: On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12, 1–57 (1963) 10. Moore, D.W., Saffman, P.G.: Structure of a line vortex in an imposed strain. In: Olsen, J.H., Goldburg, A., Rogers, M. (eds.) Aircraft Wake Turbulence and its Detection, pp. 339–354 (1971) 11. Ottino, J.M.: The Kinematics of Mixing: Stretching, Chaos and Transport. Cambridge University Press, London (1989) 12. Pedlosky, J.: Geophysical Fluid Dynamics. Springer, Heidelberg (1979) 13. Perrot, X., Carton, X.: Vortex interaction in an unsteady large-scale shear/strain flow. In: Borisov, A.V., Kozlov, V.V., Mamaev, I.S., Sokolovskiy, M.A. (eds.) Proceedings IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, pp. 373–381. Springer, Heidelberg (2008) 14. Perrot, X., Carton, X.: Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discr. Cont. Dyn. Syst. Ser. B 11, 971–995 (2009) 15. Rom-Kedar, V., Leonard, A., Wiggins, S.: An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394 (1990) 16. Trieling, R.R., Linssen, A.H., van Heijst, G.J.F.: Monopolar vortices in an irrotational annular shear flow. J. Fluid Mech. 360, 273–294 (1998) 17. Velasco Fuentes, O.U., van Heijst, G.J.F., Cremers, B.E.: Chaotic transport by dipolar vortices on a β-plane. J. Fluid Mech. 291, 139–161 (1995) 18. Velasco Fuentes, O.U., van Heijst, G.J.F., van Lipzig, N.P.M.: Unsteady behaviour of a topography-modulated tripole. J. Fluid Mech. 307, 11–41 (1996) 19. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, Heidelberg (1990)

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Theor. Comput. Fluid Dyn. (2010) 24:323–327 DOI 10.1007/s00162-009-0128-3

O R I G I NA L A RT I C L E

Shinji Yukimoto · Hiroshi Niino · Takashi Noguchi · Ryuji Kimura · Frederic Y. Moulin

Structure of a bathtub vortex: importance of the bottom boundary layer

Received: 1 November 2008 / Accepted: 2 June 2009 / Published online: 24 July 2009 © Springer-Verlag 2009

Abstract A bathtub vortex in a cylindrical tank rotating at a constant angular velocity  is studied by means of a laboratory experiment, a numerical experiment and a boundary layer theory. The laboratory and numerical experiments show that two regimes of vortices in the steady-state can occur depending on  and the volume flux Q through the drain hole: when Q is large and  is small, a potential vortex is formed in which angular momentum outside the vortex core is constant in the non-rotating frame. However, when Q is small or  is large, a vortex is generated in which the angular momentum decreases with decreasing radius. Boundary layer theory shows that the vortex regimes strongly depend on the theoretical radial volume flux through the bottom boundary layer under a potential vortex : when the ratio of Q to the theoretical boundary-layer radial volume 1 flux Q b (scaled by 2π R 2 (ν) 2 ) at the outer rim of the vortex core is larger than a critical value (of order 1), the radial flow in the interior exists at all radii and Regime I is realized, where R is the inner radius of the tank and ν the kinematic viscosity. When the ratio is less than the critical value, the radial flow in the interior nearly vanishes inside a critical radius and almost all of the radial volume flux occurs only in the boundary layer, resulting in Regime II in which the angular momentum is not constant with radius. This criterion is found to explain the results of the laboratory and numerical experiments very well. Keywords Vortex dynamics · Rotating and swirling flows · Boundary layers · Bathtub vortex · Potential vortex PACS 47.32C, 47.32Ef, 47.15Cb Communicated by H. Aref S. Yukimoto · H. Niino (B) Ocean Research Institute, The University of Tokyo, Nakano 164-8639, Japan E-mail: [email protected] Present address: S. Yukimoto Mitsubishi UFJ NICOS Co. Ltd., Chiyoda, Tokyo 101-8960, Japan T. Noguchi Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan R. Kimura The Open University of Japan, Chiba 261-8586, Japan F. Y. Moulin Université de Toulouse, IMFT, Allée du professeur Camille Soula, 31400 Toulouse, France Reprinted from the journal

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1 Introduction Strong natural vortices such as tornadoes and dustdevils are often modeled by a Rankine vortex in which the angular momentum outside the vortex core is constant. Recent observations of tornadoes using a portable Doppler radar, however, show that angular momentum outside the core decreases with decreasing radius (see [1]). In order to clarify the mechanism by which the velocity distribution of a strong vortex is determined, we have performed laboratory and numerical experiments on a bathtub vortex in a rotating tank. There have been a number of studies on bathtub vortices (see, e.g. [2–7]). The effects of the bottom boundary layer on geostrophic vortices have been well studied by Mory and Yurichenko [5] and Andersen et al. [7], and the structure of non-linear boundary layers by Burggraf et al. [8] and Andersen et al. [9]. It is worth noting that trailing vortices behind planes also exhibit such non-constant angular momentum regions near their core, a pattern that influences strongly their dynamics and stability properties (see, e.g. [10,11]). However, for natural vortices like tornadoes, the mechanism by which the velocity distribution of an ageostrophic vortex outside the vortex core is determined has not been fully clarified and in this particular context, the effects of the bottom boundary layer on the vortex structure have not been properly considered. 2 Laboratory experiment The schematic of the experimental set-up is shown in Fig. 1a. A bathtub vortex is generated in a cylindrical tank of 40 cm diameter which rotates about its vertical central axis at a constant angular velocity . The working fluid, fresh water, is drained at a volume flux Q through a circular hole of 2.5 cm diameter at the bottom center, and the same amount of water is returned to the tank through its sidewall the upper part of which is made of sponge. The mean water depth of the tank is 18 cm. Horizontal velocity field measurements presented here were made at z =10 cm via a PIV technique. Figure 1b shows the radial distributions of angular momentum which demonstrates that two regimes of vortices are realized in the experiment: when Q is large and  is small (solid line), the angular momentum (defined in the non-rotating frame) in the steady state is constant outside the vortex core (Regime I); when Q is small or  is large (dashed line), on the other hand, the angular momentum decreases with decreasing radius (Regime II). For Regime II, a flow visualization in the radial-vertical plane shows that a dye introduced near the top of the sidewall travels along the sidewall, goes into the boundary layer, moves radially inward and goes out of the drain hole (discussed in [12]). 3 Numerical experiment The axisymmetric numerical model in cylindrical coordinates which is used in the present study consists of two prognostic equations for the vorticity in the r –z plane and angular momentum per unit mass M(r ) = vr . Here, r and z are the radial and vertical coordinates, respectively, v is the tangential velocity and the vorticity is expressed in terms of the stream function in the r –z plane through the continuity equation. Finite difference and leap-frog schemes are used to solve the equations. The boundary conditions are stress free at the central

Fig. 1 a Schematic of the experimental apparatus and b measured radial profiles of the angular momentum per unit mass M = vr for  = 0.1 rad/s and Q = 100 cm3 /s (solid line), and  = 0.4 rad/s and Q = 33 cm3 /s (dashed line)

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M (cm2/s)

z (cm)

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0 40

0 160

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Fig. 2 The streamlines in the r –z plane (a and c) and radial profiles of angular momentum per unit mass at the mid-depth (b and d) as obtained from the numerical experiment. For a and b,  = 0.1 rad/s and Q = 100 cm3 /s, and for c and d,  = 0.4 rad/s and Q = 33 cm3 /s. The vertical thick line on the right side of the sidewall in a and c denotes the inflow region, and the thick horizontal line below the horizontal axis in each panel the drain hole

axis, free slip at the upper lid and no slip at the bottom and sidewall which are rotating at the angular velocity . Note that the deformation of the free surface was neglected for simplicity. The drain hole occupies the central one-eighth radius of the bottom, and the inflow region the upper one-quarter of the sidewall. The calculation is started from an initial state in which the whole system is in a solid-body rotation of angular velocity , the vorticity in the r –z plane is zero and the volume flux from the sidewall and through the drain hole is Q. It takes about 30-60 min to attain a steady state depending on the values of the external parameters  and Q. For various combinations of  and Q, time evolutions of the angular velocity in the laboratory experiment are well reproduced by the corresponding numerical experiment (see [13]). Figure 2a, b shows streamlines in the r –z plane and radial profiles of angular momentum in the steady state at the height of 10 cm from the bottom, respectively, for Regime I, and Fig. 2c, d those for Regime II. A comparison between Figs. 1b and 2b, d shows that the numerical experiment well reproduces the steady-state radial distribution of the angular momentum in the laboratory experiment. For Regime I, a majority of streamlines that start from the sidewall reaches the outer rim of the vortex core through the interior: this indicates that a non-zero radial flux towards the vortex core exists everywhere in the interior flow, transporting circulation from the external rotating wall and leading to an angular momentum distribution constant with radius (Fig. 2b). By contrast, for Regime II, all of the streamlines go into the bottom boundary layer (Fig. 2c) for r > 14 cm, and no radial flow is maintained in the interior for r < 14 cm, yielding an angular momentum that decreases with decreasing radius for r < 14 cm (see Fig. 2d). It is worth noting that when the bottom boundary condition is artificially changed to free-slip, the angular momentum outside the vortex core becomes radially constant for the two regimes, demonstrating the leading role of the bottom boundary layer in generating non-constant angular momentum profiles. 4 Boundary layer theory Since the numerical experiment has shown that the structure of the vortex is governed by the bottom boundary layer, the characteristics of the bottom boundary layer under a potential vortex is studied by a boundary layer theory. The structure of the boundary layer below a potential vortex was studied by Burggraf et al. [8], but the radial flux through the boundary layer, which is important in the present problem, was not reported. Here, the axisymmetric boundary layer equations (see [13]) were integrated with time to obtain a steady-state solution, with the side and bottom boundaries rotating at the angular velocity , the inner boundary chosen open, and the horizontal velocity components approaching those of the potential vortex at the upper boundary. The boundary layer structure thus obtained coincides well with the one obtained for Regime I in the numer1 ical experiment (see [13]). The radial volume flux through the bottom boundary layer scales as 2π R 2 (ν) 2 , and depends on a non-dimensional function F(r ) that increases monotonously with decreasing radius (Fig. 3). Here, R is the inner radius of the tank, ν the kinematic viscosity and r the non-dimensional radial coordinate. The maximum radial volume flux Q bmax through the bottom boundary layer is attained at the radius of the 1 drain hole, and is given by Q bmax = 0.88 × 2π R 2 (ν) 2 (as indicated in Fig. 3). Reprinted from the journal

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1.0 0.88

F (r )

0.8 0.6 0.4 0.2 0

0.125

0

0.2

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Fig. 3 Non-dimensional radial volume flux F(r ) through the bottom boundary layer. The thick solid curve shows the one obtained from the boundary-layer equation, and the dashed curve that obtained from the numerical experiment in Sect. 3. The thin vertical line denotes the non-dimensional radius of the drain hole (0.125) and the thin horizontal line the corresponding non-dimensional radial volume flux (0.88) 120

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Fig. 4 The angular momentum at r = 4 cm scaled by that at the sidewall on the  − Q plane as obtained from the numerical experiment. The solid curve shows the criterion given by Eq. (1)

Thus, since a potential vortex forms only when a non-zero radial flux exists in the whole interior, the radial volume flux in the boundary layer must remain everywhere smaller than the volume flux Q prescribed at the drain hole. As a consequence, a simple criterion for the realization of a potential vortex might be given by 1

Q > 0.88 × 2π R 2 (ν) 2 = 0.88 × 2π R · R ·

 ν 1 2



(1)

In Fig. 4, results from numerical experiments are plotted in the  − Q parameter plane, along with a solid line corresponding to the criterion (1) proposed above. As shown in the figure, potential vortices (those with an angular momentum at r = 4 cm roughly equal to the angular momentum prescribed by the external wall rotation) form only when (1) is satisfied. This criterion also nicely explains the results of the laboratory experiments (see [13]). 5 Summary and conclusions The mechanism through which the radial distribution of the tangential velocity in a bathtub vortex is determined is studied by means of laboratory and numerical experiments and a boundary layer theory. Two different regimes of the vortex are found to exist : In the first regime (Regime I), a potential vortex in which the angular momentum is constant outside the vortex core is found both experimentally and numerically. In the second regime (Regime II), on the other hand, the angular momentum decreases with decreasing radius. Which regime occurs depends on the relative magnitudes of the volume flux Q and the maximum value Q bmax of the radial volume flux Q b in the boundary layer below a potential vortex. When Q > Q bmax , Regime I is realized, while Regime II is realized when Q < Q bmax .

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The present study clearly shows that the radial distribution of the tangential velocity in a bathtub vortex is strongly controlled by the bottom boundary layer. It has often been assumed that the radial distribution of the tangential velocity in strong vortices in nature and in engineering flows in a container is approximated by that of a Rankine vortex. However, the present study demonstrates that the angular momentum outside the vortex core is not necessarily constant. These results may give some insight in the recent observation by Wurman and Gill [1] of the velocity distribution in violent atmospheric vortices. The present paper summarizes the essence of the first author’s doctor thesis [13]. More detailed results will be soon reported elsewhere. References 1. Wurman, J., Gill, S.: Finescale radar observations of the Dimmitt, Texas (2 June 1995), Tornado. Mon. Wea. Rev. 128, 2135– 2164 (2000) 2. Lewellen, W.S.: A solution for three-dimensional vortex flows with strong circulation. J. Fluid Mech. 14, 420–432 (1962) 3. Turner, J.S.: The constraints imposed on tornado-like vortices by the top and bottom boundary conditions. J. Fluid Mech. 25, 377–400 (1966) 4. Lundgren, J.: The vortical flow above the drain-hole in a rotating vessel. J. Fluid Mech. 155, 381–412 (1985) 5. Mory, M., Yurichenko, N.: Vortex generation by suction in a rotating tank. Eur. J. Mech. B/Fluids 12, 729–747 (1993) 6. Echavez, G., McCann, E.: An experimental study on the free surface vertical vortex. Exp. Fluids 33, 414–421 (2002) 7. Andersen, A., Bohr, T., Stenum, B., Rasmussen, J.J., Lautrup, B.: The bathtub vortex in a rotating container. J. Fluid Mech. 556, 121–146 (2006) 8. Burggraf, O.R., Stewartson, K., Belcher, R.: Boundary layer induced by a potential vortex. Phys. Fluids. 14, 1821–1833 (1971) 9. Andersen, A.T., Lautrup, B., Bohr, T.: An averaging method for nonlinear laminar Ekman layers. J. Fluid Mech. 487, 81–90 (2003) 10. Jacquin, L., Fabre, D., Geffroy, P., Coustols, E.: The properties of a transport aircraft wake in the extended near field—an experimental study. In: Proceedings of the AIAA, Aerospace Sciences Meeting and Exhibit, 39th, Reno (2001) 11. Jacquin, L., Fabre, D., Sipp, D., Coustols, E.: Unsteadiness, instability and turbulence in trailing vortices—comptes rendusPhysique (2005) 12. Noguchi, T., Yukimoto, S., Kimura, R., Niino, H.: Structure and instability of a sink vortex. In: Proceedings of PSFVIP-4 (2003) 13. Yukimoto, S.: Structure of a Suction Vortex: Importance of the Bottom Boundary Layer. Doctoral Dissertation, Dep. Earth Planet. Sci., The University of Tokyo, 77 p. (2008)

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Theor. Comput. Fluid Dyn. (2010) 24:329–334 DOI 10.1007/s00162-009-0102-0

O R I G I NA L A RT I C L E

Anders Andersen · Tomas Bohr · Teis Schnipper

Separation vortices and pattern formation

Received: 6 January 2009 / Accepted: 18 March 2009 / Published online: 29 May 2009 © Springer-Verlag 2009

Abstract In this paper examples are given of the importance of flow separation for fluid patterns at moderate Reynolds numbers—both in the stationary and in the time-dependent domain. In the case of circular hydraulic jumps, it has been shown recently that it is possible to generalise the Prandtl–Kármán–Pohlhausen approach to stationary boundary layers with free surfaces going through separation, and thus obtain a quantitative theory of the simplest type of hydraulic jump, where a single separation vortex is present outside the jump. A second type of jump, where an additional roller appears at the surface, cannot be captured by this approach and has not been given an adequate theoretical description. Such a model is needed to describe “polygonal” hydraulic jumps, which occur by spontaneous symmetry breaking of the latter state. Time-dependent separation is of importance in the formation of sand ripples under oscillatory flow, where the separation vortices become very strong. In this case no simple theory exists for the determination of the location and strengths of separation vortices over a wavy bottom of arbitrary profile. We have, however, recently suggested an amplitude equation describing the long-time evolution of the sand ripple pattern, which has the surprising features that it breaks the local sand conservation and has long-range interaction, features that can be underpinned by experiments. Very similar vortex dynamics takes place around oscillating structures such as wings and fins. Here, we present results for the vortex patterns behind a flapping foil in a flowing soap film, which shows the interaction and competition between the vortices shed from the round leading edge (like the von Kármán vortex street) and those created at the sharp trailing edge. Keywords Separation · Vortex formation · Hydraulic jump · Sand ripples · Vortex street PACS 47.54.-r, 47.32.ck, 47.32.Ff

1 Introduction Flow separation is at the basis of many interesting structures at Reynolds numbers, where both inertia and viscosity are important. In this paper we shall give examples of such flows as they occur in hydraulic jumps, sand ripples and wakes—areas where progress has been made recently although there still is a strong need of better understanding. Separation typically occurs in flows past a sharp edge or rapid flows along a solid boundary, when the friction becomes so large that part of the flow starts moving backwards. When separation develops, it typically occurs in a finite region of the flow, called a separation vortex—or bubble. The theory of separation is notoriously difficult. The reason is that separation occurs in connection with boundary layers Communicated by H. Aref A. Andersen · T. Bohr (B) · T. Schnipper Department of Physics and Center for Fluid Dynamics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark E-mail: [email protected] Reprinted from the journal

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and the standard boundary layer theory due to Prandtl becomes singular at separation points. Since most structurally interesting problems occur at moderate Reynolds number, scaling approaches (such as “Triple deck” expansions designed for Re → ∞) do not work very well. 2 Circular and polygonal hydraulic jumps In a circular hydraulic jump, a jet of fluid impinges on a flat solid surface. As shown in Fig. 1a, the rapid flow in the thin layer (thickness around 1 mm) close to the jet switches suddenly to a slow, thick layer (thickness around 1 cm) at a well-defined, circular boundary—the circular hydraulic jump. This is an example of stationary separation [1–3] as shown in Fig. 1b. The standard modelling of such flows is made by shallow water theory in which the jump appears as a shock between supercritical (rapid) and subcritical (slow) flows [4]. It has, however, recently been shown [5] that it is possible to compute the flow fairly accurately by a simple system of ODEs derived from boundary layer theory, modelling the radial velocity profile u(r, z) as a cubic polynomial, allowing a separated profile. Thus, for a flow with height profile h(r ) we assume that u(r, z)/v(r ) = a(r )η + b(r )η2 + c(r )η3 where η = z/ h(r ) and v(r ) is the average velocity through the layer. Due to the boundary conditions (kinematic and dynamic) on the free surface, the coefficients a, b, and c can be expressed in terms of a single shape parameter λ(r ) as, for example a = λ + 3, b = −(5λ + 3)/2, and c = 4λ/3. The separation condition is ∂u(r ∗ , z)/∂z = 0 at the bottom (z = 0), i.e., a = 0, or λ = −3. The dynamics is governed by the boundary layer equations and to obtain a closed system of ODEs with the above ansatz we need to satisfy two moments. In [5] they were chosen as the average of the momentum equation through the layer and the limit of the momentum equation on the bottom z = 0. We then obtain two coupled non-autonomous ODEs for h and λ of the form h = −

5λ + 3 4r λ h 4 − (5λ + 3)   ; G (λ)λ = + G(λ) r h3 h r h4

(1)

where 1 G(λ) = h

h  2 u 6 λ λ2 dz = − + . v 5 15 105

(2)

0

Here, the flux q = r h(r )v(r ) has been normalised to unity by re-scaling the variables. This model does become singular, but only on the lines h = 0 and λ = 7/2. It is therefore possible to describe a flow with separation (λ < −3). Similar approaches are possible for hydraulic jumps in channels [6]. The type of jump described so far typically occurs when the jet is impinging on a (not too large) flat plate and simply allowed to exit by falling over the rim. If the outflow of the fluid is partly blocked by placing a weir at the rim an interesting transition is observed as the weir height, and thereby the external fluid height h ext , is increased [7]. Crossing a threshold of h ext , the jump becomes unstable and acquires a new structure with a larger jump region between two concentric circular loci, and in this jump region the surface flow is backwards. (a)

(b)

Fig. 1 a Circular hydraulic jump in a “kitchen sink”. b Schematic drawing of the flow in a circular hydraulic jump. The grey region is a separation vortex

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(b)

(a)

Fig. 2 a Schematic drawing contrasting the flows in type I and type II circular hydraulic jumps. b A heptagonal hydraulic jump with ethylene glycol. Courtesy of Clive Ellegaard

Thus a “roller”, i.e., a separation vortex at the surface has now appeared as sketched in Fig. 2a. This type II state is typically unstable and the circular symmetry is spontaneously broken [8], leading, for sufficiently viscous fluids with sufficient surface tension [9], to a stationary polygonal structure as shown in Fig. 2b. Close to the transition there can be many corners (up to 13 has been observed) but as h ext is increased further, the number of corners gradually diminish, although several polygonal structures can be stable simultaneously. The outer circle remains and is still marked by a (weak) separation vortex on the bottom. No theoretical description exists for the type II hydraulic jump—even as regards the circular state, and thus only phenomenological models exist for the polygons. 3 Sand ripples under oscillatory flow Another case in which separation—in this case time dependent—plays a major role, is the formation and dynamics of the sand ripples that form under oscillatory flow. Such structures are readily observed in the shallow water near the beach and the pioneering works on this type of ripples were made by Ayrton [10], Bagnold and Taylor [11] many years ago. From their work as well as many more recent (e.g., [12–14]), the key role played by separation vortices is evident. This implies also that the wave length selection is simple: like the maximal size of the separation vortex, the wavelength is roughly proportional to the amplitude of the horizontal water displacement. The full description of sand ripples is very complicated. The flow is turbulent and the precise modelling of the granular transport is thus very demanding. But still, the most important overall features would be predictable if a general theory of separation of laminar flow along a wavy bottom was known. Short of this, we have recently succeeded in constructing an amplitude equation, which can describe many of the pertinent longtime features of the dynamics, without actually solving for the separation vortex pattern [15]. The amplitude equation describes the sand ripples entirely in terms of the local height h (the elevation of the sand-bottom). It does not resolve the periodic drive, but only time scales much longer than this period [16]. The flat bed is unstable [17], but a sufficient inclination (i.e., one larger than the angle of repose) will quench the instability. The lowest order equation with these properties has the structure     ¯ + (h x )2 − 1 h x x − h x x x x + δ (h x )2 h t = −(h − h) (3) xx where h¯ is the average height over the one-dimensional spatial variable x, and  and δ are coefficients depending on external conditions. The first term (proportional to ) in this equation is unexpected and lack of appreciation of this has delayed the development of an appropriate amplitude equation [18]. It violates two basic principles, which are usually taken for granted: local sand conservation and locality. Sand must of course be locally conserved in the bulk and one might thus off-hand assume that the equation should have the form h t = −qx with some appropriate flux q. This presumes, however, that sand is only transported along the surface of the ripples. In fact it is known that sand is transported both into the bulk [19] and carried by the flow, and therefore this assumption is not necessarily true. Even including the -term, sand is globally conserved (with periodic boundary conditions), i.e., the inclusion of the average height h¯ explicitly into the equation ensures that this Reprinted from the journal

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Fig. 3 Dynamics of the doubling transition. As shown by the arrows, two new satellite ripples are initially formed in each trough. After a short time one satellite wins (here the left one) and this choice is obeyed in all the troughs. T is the time in oscillation periods. The length scale in centimetres can be seen from the ruler

quantity is independent of time. But this means that the growth of an individual ripple depends on the average ripple height—a global property. In a sufficiently large system, this would not be possible, but, as we shall see below, the hydrodynamical interactions are surprisingly long-ranged. √ The initial instability from a flat bed occurs in the model around the most unstable wavenumber k = 1/ 2 and subsequently the wavelength increases by coarsening. For  = 0 the coarsening would proceed without bound, until only one giant ripple was present [18,20,21]. For  > 0, however, the process stops at a finite wavelength as in the experiments. When the amplitude or the frequency of the tray is suddenly changed, a well-defined set of bifurcations typically take place [22,23]. Increasing the frequency (which does not affect the wavelength much) leads to a “pearling” transition, where a checkerboard of small ripples (“pearls”) appear in the troughs of the old ones in a reversible transition. Increasing the amplitude above a threshold value leads to coarsening in the form of “bulging” where the ripples undergo varicose undulations leading to dilations and compressions and subsequent disappearance of ripples at compression points. Finally, a sufficient decrease in amplitude introduces new ripples in the system. But this can apparently only happen by initially doubling the number of ripples by placing a new one in each trough—although the number of ripples thereby becomes too large. This transition has been called “doubling”, and a new ripple emerges in each trough. In the final stages superfluous ripples are subsequently removed by “bulging”. Of these transitions, only the doubling transition can be realised in a quasi one-dimensional system (a narrow channel). For this to occur in our amplitude Equation (3), it is crucial that δ > 0, since δ = 0 leads to up-down symmetric ripples where creation of new ripples cannot take place exclusively in the troughs. The short-time dynamics of the doubling transition reveals an interesting detail, showing the long-range character of the interaction. Figure 3 shows a sequence of height profiles from an experimental doubling transition. The initial instability creates two new satellite ripples in each trough, i.e., a kind of ripple “tripling”. Only one of these satellites, however, survives. This can be the left or the right satellite, but, surprisingly, the same satellite wins in all troughs. We observe both left and right winners, so we believe that this selection is not due to lack of symmetry in our experiment, but is due to spontaneous symmetry breaking. The effect is so strong that if we stop the system and move one of the ripples to the other side of the trough, the system will pull it back in line when it is restarted. Our simple amplitude equation does not capture this effect, and indeed we do not expect Equation (3) to be accurate on time scales of the order of the driving period. But it does show that the ripples are strongly coupled and that a more refined theory is needed, which includes the time-dependent separation vortices explicitly.

4 Wakes in a soap film Vortices can be beautifully visualised in soap films and thus dynamic separation processes can be explored effectively in soap films flowing around solid bodies [24–26]. In Fig. 4 we show visualisations of vortices behind a flapping foil in a vertically flowing soap film. By varying frequency and amplitude of the periodic oscillation we obtain a variety of wakes of relevance for fish swimming and with great aesthetic appeal. The

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Separation vortices and pattern formation

Fig. 4 Wakes generated by a flapping foil in a flowing soap film. a 2P wake at St D = 0.085, A D = 1.2. b 2P+2S wake at St D = 0.074, A D = 1.5. c 4P wake at St D = 0.069, A D = 1.6. The wake notation follows [27] where “S” signifies a single vortex and “P” signifies a pair of vortices of opposite signs. d Magnification of the area highlighted in (c) showing clearly the vortex formation process while the foil moves from left to right

vorticity in the wake is correlated to thickness variations of the film, which we visualise using a sodium lamp to produce interference patterns. Our experimental setup allows us to study both the vortex shedding at the trailing edge and the wake evolution far downstream. The foil is 1 mm wide and 6 mm long, it has a round leading edge and a sharp trailing edge, and it is driven with symmetric pitching oscillations about the 1/12 chord point. The experiment is characterised by three dimensionless parameters, i.e., the Strouhal number St D = D f /U , the dimensionless amplitude A D = 2 A/D, and the Reynolds number Re = DU/ν, where D is the width of the foil, f is the oscillation frequency, U is the free-stream speed, A is the flapping amplitude, and ν is the kinematic viscosity of the soap film. In the experiment, we vary St D and A D while keeping Re constant at approximately 2 · 102 and in this way we can obtain a number of vortex wake patterns behind the foil, including normal (von Kármán) and inverted (von Kàrmàn) vortex streets. As has been noted earlier for a cylinder oscillated transversely to a free stream [27], the shedding frequency is locked to the imposed frequency f . The “leading edge” vortices generated at the rounded upstream edge of the foil travel down-stream and interact at the tip with the vortices formed at the trailing edge due to the flapping motion. As shown in Fig. 4 the resulting patterns can be rather complicated with several vortices or pairs forming in each cycle. Patterns analogous to Fig. 4a have also been visualised in water tunnels [28,29]. These patterns are of importance for swimming and flying animals, which move by flapping their fins or wings. In particular, we would like to understand the thrust generated and for that purpose a simple model of the vortex generation would be important—analogous to other separating flows.

5 Conclusion We have shown that separation plays a large role in the formation of flow structure in many systems, either with free surfaces, with interfaces or simply near solid bodies where boundary layers can form. It is a challenge to develop theories that can account for separation under quite general conditions, e.g., along a wavy boundary or near a free surface—and in both stationary and time-dependent settings. Reprinted from the journal

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

Tani, I.: Water jump in the boundary layer. J. Phys. Soc. Japan 4, 212 (1949) Watson, E.J.: The radial spread of a liquid jet over a horizontal plate. J. Fluid Mech. 20, 481 (1964) Bohr, T., Dimon, P., Putkaradze, V.: Shallow-water approach to the circular hydraulic jump. J. Fluid Mech. 254, 635 (1993) Rayleigh, L.: On the theory of long waves and bores. Proc. R. Soc. Lond. A 90, 324 (1914) Watanabe, S., Putkaradze, V., Bohr, T.: Integral methods for shallow free-surface flows with separation. J. Fluid Mech. 480, 233 (2003) Bonn, D., Andersen, A., Bohr, T.: Hydraulic jumps in a channel. J. Fluid Mech. 618, 71 (2009) Bohr, T., Ellegaard, C., Espe Hansen, A., Haaning, A.: Hydraulic jumps, flow separation and wave breaking: an experimental study. Physica B 228, 1 (1996) Ellegaard, C., Espe Hansen, A., Haaning, A., Hansen, K., Marcussen, A., Bohr, T., Lundbek Hansen, J., Watanabe, S.: Creating corners in kitchen sinks. Nature 392, 767 (1998) Bush, J.W.M., Aristoff, J.M., Hosoi, A.E.: An experimental investigation of the stability of the circular hydraulic jump. J. Fluid Mech. 558, 33 (2006) Ayrton, H.: The origin and growth of ripple-mark. Proc. R. Soc. Lond. A 84, 285 (1910) Bagnold, R.A.: Motion of waves in shallow water. Interaction between waves and sand bottoms. Proc. R. Soc. Lond. A 187, 1 (1946) Stegner, A., Wesfreid, J.E.: Dynamical evolution of sand ripples under water. Phys. Rev. E 60, R3487 (1999) Scherer, M.A., Melo, F., Marder, M.: Sand ripples in an oscillating annular sand–water cell. Phys. Fluids 11, 58 (1999) Rousseaux, G., Stegner, A., Wesfreid, J.E.: Wavelength selection of rolling-grain ripples in the laboratory. Phys. Rev. E 69, 031307 (2004) Schnipper, T., Mertens, K., Ellegaard, C., Bohr, T.: Amplitude equation and long-range interactions in underwater sand ripples in one dimension. Phys. Rev. E 78, 047301 (2008) Cross, M., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 852 (1993) Blondeaux, P.: Sand ripples under sea waves. Part 1. Ripple formation. J. Fluid Mech. 218, 1 (1990) Krug, J.: Coarsening of vortex ripples in sand. Adv. Complex Syst. 4, 353 (2001) Rousseaux, G., Caps, H., Wesfreid, J.-E.: Granular size segregation in underwater sand ripples. Eur. Phys. J. E 13, 213 (2004) Politi, P.: Kink dynamics in a one-dimensional growing surface. Phys. Rev. E 58, 281 (1998) Politi, P., Misbah, C.: Nonlinear dynamics in one dimension: a criterion for coarsening and its temporal law. Phys. Rev. E 73, 036133 (2006) Hansen, J.L., van Hecke, M., Haaning, A., Ellegaard, C., Andersen, K.H., Bohr, T., Sams, T.: Instabilities in sand ripples. Nature 410, 324 (2001) Hansen, J.L., van Hecke, M., Haaning, A., Ellegaard, C., Andersen, K.H., Bohr, T., Sams, T.: Stability balloon for twodimensional vortex ripple patterns. Phys. Rev. Lett. 87, 204301 (2001) Couder, Y., Basdevant, C.: Experimental and numerical study of vortex couples in two-dimensional flows. J. Fluid Mech. 173, 225 (1986) Gharib, M., Derango, P.: A liquid film (soap film) tunnel to study two-dimensional laminar and turbulent shear flows. Physica D 37, 406 (1989) Zhang, J., Childress, S., Libchaber, A., Shelley, M.: Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature 408, 835 (2000) Williamson, C.H.K., Roshko, A.: Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2, 355 (1988) Koochesfahani, M.M.: Vortical patterns in the wake of an oscillating airfoil. AIAA J. 27, 1200 (1989) Lai, J.C.S., Platzer, M.F.: Jet characteristics of a plunging airfoil. AIAA J. 37, 1529 (1999)

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Theor. Comput. Fluid Dyn. (2010) 24:335–347 DOI 10.1007/s00162-009-0155-0

O R I G I NA L A RT I C L E

Y. Fukumoto

Global time evolution of viscous vortex rings

Received: 11 January 2009 / Accepted: 9 June 2009 / Published online: 8 October 2009 © Springer-Verlag 2009

Abstract This article gives an overview of growing knowledge of translation speed of an axisymmetric vortex ring, with focus on the influence of viscosity. Helmholtz–Lamb’s method provides a shortcut to manipulate the translation speed at both small and large Reynolds number, for a vortex ring starting from an infinitely thin core. The resulting asymptotics significantly improve Saffman’s formula (1970) and give closer lower and upper bounds on translation speed in an early stage. At large Reynolds numbers, Kelvin–Benjamin’s kinematic variational principle achieves a further simplification. At small Reynolds numbers, the whole life of a vortex ring is available from the vorticity obeying the Stokes equations, which is closely fitted, over a long time, by Saffman’s second formula. Keywords Vortex ring · Helmholtz–Lamb’s method · Variational principle · Viscous decay PACS 47.32.cf · 47.15.ki · 47.15.G-

1 Introduction Vortex rings are ubiquitous coherent structures in high Reynolds number flows, and are of fundamental importance in fluid mechanics as indicated by the fact that visualized cross-section of a vortex ring is put on the cover of Batchelor’s textbook [1].1 Okabe [2] recollected that such a beautiful pattern was gained only once or twice among a 100 trials, and an interval of 10 or 20 min between trials were required to wait for the water in a tank becoming clean and still. Vortex rings are used for producing thrust and lift by insects, fishes, and animals. Vortex rings are capable of transporting neutrally buoyant materials. Recently they find their utility for creating virtual reality in the field of entertainment. There is an attempt to use an air cannon, as a means of olfactory display, to deliver smells encapsulated in a vortex ring to a targeted person. In a theater, virtual reality contents are created solely by image and sound. Reality is enhanced if we appeal to tactile display. A mini-theater is planned in which air cannons are designed to produce vortex rings, in synchronization with the image and the sound, so that the audience experiences direct impact and freshness [3]. These applications to entertainment necessitate controlled vortex rings, and raise questions pertaining to an inverse problem. When does a vortex ring arrive at a specified point? How far does the ring travel? How large the vortex ring has grown at the moment of impact? The purpose of this article is to give a possible 1 G. K. Batchelor knew Okabe-Inoue’s photographs through the annual report of their Institute (private communication with J. Okabe).

Communicated by H. Aref Y. Fukumoto (B) Faculty of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan E-mail: [email protected] Reprinted from the journal

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answer to these questions, under restricted situations, while giving a brief survey of the growth of knowledge of traveling speed of a vortex ring. Study on motion of vortex rings started simultaneously with the birth of the field of vortex dynamics when Helmholtz introduced the vorticity and proved its property of being frozen into the fluid in his seminal paper a century and a half ago [4]. By an elaboration from the Euler equations, now being widely known through Lamb’s textbook [5], Helmholtz had reached an identity for U of a thin axisymmetric vortex ring, steadily translating in an inviscid incompressible fluid of infinite extent. Helmholtz–Lamb’s method is expounded in Sect. 3. Ignoring constant terms compared with a logarithmically large term, Helmholtz related the translation speed U to the total kinetic energy H and the hydrodynamic impulse Pz , and made a crude estimation of this relation for a thin core, of core radius σ and ring radius R0 carrying the circulation , as      8R0 U ≈ H/(2Pz ) ≈ + const. . (1) log 4π R0 σ Continuing Helmholtz’s analysis, Kelvin (1867) determined the constant to be −1/4 in the above formula, for a distribution of vorticity, in the core, proportional to the distance from the axis of symmetry. Only the resulting expression, without derivation, was recorded in an appendix to Tait’s English translation of Helmholtz’s article [4]. On those days, vortex rings were hot as possible entities of atoms embedded in the ether. The implication of Helmholtz’ laws, invariance in time of the circulation and linkages of vortex lines, led Kelvin to this belief. Thomson [6] pursued the idea of the vortex atoms. To derive the translation speed, he employed a straightforward approach of taking the boundary of the core as a free boundary coincident with a streamline, but Kelvin’s formula was unattained; the constant that he gave was −1 rather than −1/4. This discrepancy was traced back to his insufficient treatment of the Biot–Savart law for deriving the velocity field around the vortex core, and was rescued by Hicks [7]. An interpretation of this difference was explained in Sect. 2. By adapting his technique for calculating the gravity potential around the Saturn ring, Dyson [8] contrived an ingenious systematic perturbation method evaluating the Biot–Savart law, and thereby overcame the difficulty to proceed to third (virtually fourth) order in ε = σ/R0 .         8R0 8R0  1 3σ 2 5 4 U= log (2) − − − + O(ε log ε) . log 4π R0 σ 4 8R02 σ 4 The same result was reached, in a thin limit, by transforming the free boundary-value problem of the Euler equations into an integral equation [9,10]. This integral equation was solved for the whole family of axisymmetric vortex rings with vorticity in the core being proportional to the distance from the symmetric axis. This is referred to as Fraenkel–Norbury’s family [11]. Fraenkel [10] pointed√out that, by a suitable renormalization of thickness parameter ε with which the fat limit corresponds to ε = 2, √ (2) is applicable, with an error no more than 5%, to the translation speed of Hill’s vortex, the fat limit (ε = 2). This agreement has inspired us to generalize Dyson’s formula to more realistic vortex rings [12–14]. To O(ε), Kelvin’s formula was extended to allow for an arbitrary distribution of vorticity as ⎧ ⎫    ⎨ 4π 2 r  r ⎬ 1  8R0 U0 = + A − + O (ε, ε log ε) ; A = lim , r  v0 (r  )2 dr  − log log r →∞ ⎩  2 4π R0 σ 2 σ ⎭ 0

(3) where v0 (r ) is the local velocity of circulatory motion of the fluid, in the cross-section, around the toroidal center circle, as a function only of the local distance, r , from the circle [9,15,16]. The functional form of v0 (r ) remains indeterminate, but, if the viscosity ν is called into play, a unique profile as√a function of time, t, is singled out once the initial profile is given. The small parameter gives way to ε = ν/  [12,17]. Suppose that, at time t = 0, the vorticity is concentrated on a circle of radius R0 , the leading-order terms of toroidal vorticity ζ0 and azimuthal velocity v0 are provided by the Oseen diffusing vortex ζ0 =

123

  −r 2 /4νt   2 e 1 − e−r /4νt . , v0 = − 4πνt 2πr 352

(4) Reprinted from the journal

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z φ

r

P θ

Z(t)

0

R(t)

ρ

Fig. 1 Cylindrical and local moving coordinates

The minus sign in v0 comes from our choice of local azimuthal coordinates (see Fig. 1). Saffman [15] showed that viscous diffusion of vorticity gets along with Helmholtz–Lamb’s identity and obtained the translation speed of a vortex ring in a viscous fluid, simply by inserting (4) into (3), as       8R0  1 νt νt νt U0 = log √ , (5) , log − (1 − γ + log 2) + O 4π R0 2 2 νt R02 R02 R02 √ 2 where γ = 0.57721566 . . . is √Euler’s constant. The radius of viscous core is σ ≈ 2 νt, and (5) is valid at early times when the core is thin νt  R0 . The same formula was derived via the method of matched asymptotic expansions [17,16]. Recently, Fraenkel–Saffman’s formula (3) is extended to O(ε3 ). In other words, Dyson’s formula (2) is generalized to accommodate a general distribution of vorticity. At the same time, an extension of Saffman’s formula to O(ε3 ) is achieved. A brief announcement of these results is given in [14]. It is worth emphasizing that Helmholtz–Lamb’s method is far more efficient than matched asymptotic expansions. The former leads us to the correction of O(ε3 ) to translation speed of a vortex ring without having to enter into the O(ε3 ) velocity field. The development of theories of vortex rings attained before the early 1990s is well recorded in [18–21]. This article supplements these by focusing on theoretical development made after that, with particular emphasis put on higher-order extension of velocity formula and on viscous vortex rings at both very high and very low Reynolds numbers. Dyson’s technique for asymptotic development of the Biot–Savart law is instrumental for deriving the expression of the velocity field near the core. Before going to a description of higher-order extension of the translation speed, we sketch the essence of this technique in Sect. 2. Thereafter, Sect. 3 gives an account of Helmholtz–Lamb’s method, and, resorting to this method, presents the third-order correction to translation speed in Sect. 4. A variational principle brings a further simplification in derivation for an inviscid vortex [18,22,23]. Take the density of fluid to be ρf = 1 and define the hydrodynamic impulse by

1 P = x × ωdV. (6) 2 The translation velocity U of a vortex ring is then calculable through the variation δ H − U · δP = 0,

(7)

under the constraint that, for any smooth Lagrangian displacement of fluid particles, the vorticity is frozen into the fluid. Section 5 touches upon this principle, which is the theme of [14]. We may view (7) as a refinement of the crude estimate (1). Behind (7) lies Kelvin’s variational principle [24,25], as generalized to make allowance for motion [26,27], that a stationary configuration of vorticity in an inviscid incompressible fluid, in a steadily moving frame, is realizable as an extremal of energy on an iso-vortical sheet. Intriguingly, the same principle 2

In (3) and (5), Saffman’s estimate of the remaining terms has been improved [12].

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encompasses motion of a vortex ring ruled by the cubic nonlinear Schrödinger equation, which serves as a model for superfluid liquid helium and a Bose–Einstein condensate, at zero temperature [28]. The rest of article is concerned with motion of a vortex ring at very low Reynolds numbers. There is no permanent vortex ring. Without unstable waves, a vortex ring dies away due to the action of viscosity while entraining surrounding irrotational flows [29–31]. The decaying laws of an axisymmetric vortex rings in a viscous fluid were handled separately in the literature. Recently a solution of an initial-value problem valid over the whole time range is found for an axisymmetric vortex ring at low Reynolds numbers [32,33] which enables us to view, in perspective, the early-time behavior (5), Saffman’s second law valid in the matured stage √ νt ≈ R0 [15], and the decaying law U=

Pz 7Pz ≈ 0.0037037954 , √ 3/2 (νt)3/2 240 2 (πνt)

(8)

√ at large times νt  R0 [20,34–36]. A concise description of the low Reynolds number solution is given in Sect. 6. The last section (Sect. 7) is devoted to a summary and conclusions. 2 Asympytotic development of Biot–Savart law: Dyson’s technique Dyson’s ingenious technique [8] is, in effect, indispensable for manipulating asymptotic expansions of the flow field around the vortical core to high orders. Here we delineate its essence as generalized to an arbitrary distribution, including a continuous one, of vorticity [12]. Consider an axisymmetric vortex ring of circulation  moving in an infinite expanse of fluid. Choose cylindrical coordinates (ρ, φ, and z) with the z-axis along the axis of symmetry and φ along the vortex lines as shown in Fig. 1. We consider an axisymmetric distribution of vorticity ω = ζ (ρ, z)eφ localized about the circle (ρ, z) = (R(t), Z (t)), where eφ is the unit vector in the azimuthal direction. The vector potential A(x) of the velocity field u(x) (u = ∇ × A) has azimuthal component only. We introduce the Stokes streamfunction ψ by A(x) = −(ψ/ρ) eφ . The requirement of vanishing the vector potential at infinity, that is |A| ∝ 1/|x|2 as |x| → ∞, facilitates the calculation of the total kinetic energy. With this requirement, the Biot–Savart law is represented for the Stokes streamfunction as ρ ψ(ρ, z) = − 4π

∞ 2π ∞ −∞ 0

0

ζ (ρ  , z  )ρ  cos φ  dρ  dφ  dz   . ρ 2 − 2ρρ  cos φ  + ρ 2 + (z − z  )2

(9)

We introduce, in the meridional plane, local Cartesian coordinates (x, ˆ zˆ ) = (ρ − R, z − Z ) centered at (R(t), Z (t)). Supposing a rapid decay of ζ (x) with the distance from the circle, we perform an asymptotic expansion of (9) valid near the core. The first of the key steps of Dyson’s technique is to utilize the shift operator to rewrite (9) as ρ ψ =− 4π

∞ −∞



∂ ∂ d xˆ ddˆz ζ (xˆ , zˆ ) exp xˆ − zˆ  ∂R ∂ zˆ 









 2π 0

R cos φ  dφ   . ρ 2 − 2ρ R cos φ  + R 2 + zˆ 2

(10)

The asymptotic form of (9) is automatically generated by expanding the exponential function of the operators as     

∞ 1 ∂ ∂ ∂ ∂ 2     d xˆ dˆz ζ (xˆ , zˆ ) 1 + xˆ  − zˆ  + xˆ  − zˆ  ψρ, z) = ∂R ∂z 2! ∂R ∂z −∞   3  4 1 1  ∂  ∂  ∂  ∂ + + + · · · ψm (ρ, z; R). (11) xˆ − zˆ xˆ − zˆ 3! ∂R ∂z 4! ∂R ∂z Here ρR ψm (ρ, z; R) = − 4π

2π 0

123

cos φ  dφ   , ρ 2 − 2ρ R cos φ  + R 2 + (z − Z )2 354

(12)

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is the streamfunction for the flow induced by a circular line vortex of unit strength placed at (R, Z ), or a delta-function core ζ (ρ, z) = δ(ρ − R)δ(z − Z ). Observe that the monopole field (12) is symmetric with respect to interchange between ρ and R. It follows from the connection between ψ and ζ that, except at the core (ρ, z) = (R, Z ), ψm obeys 

∂2 ∂2 + ∂ R2 ∂z 2

 ψm =

1 ∂ψm at (ρ, z)  = (R, Z ). R ∂R

(13)

The second of the key steps of Dyson’s technique is to invoke this identity to replace the combination of second derivatives by single first derivative. The importance of this step for promoting cancelation of terms cannot be overemphasized. Without the help from (13), a flood of terms become uncontrollable. We introduce local cylindrical coordinates (r, θ ) in the meridional plain by (x, ˆ zˆ ) = (r cos θ, r sin θ ). The radius, r , is the shortest distance from the given point x to the vortex loop. Integration of (12) is implemented, in terms of the first and the second complete elliptic integrals [5]. Use of the asymptotic formulas of the complete elliptic integrals for modulus close to unity leads us to the near field of ψm , valid for σ  r  R, as        8R 8R R r ψm = − log −2+ log − 1 cos θ 2π r 2R r          8R r2 8R + 1 − log − 2 cos 2θ + · · · , + 4 2 2 log 2 R r r

(14)

(see [8]). The exponential decrease of coefficients, that is, in increase power of 2−1 , makes the higher-order formula of translation speed applicable to fat cores.   (1) (2) We anticipate ζ (x, z) = ζ0 (r ) + ζ11 cos θ + ζ0(2) + ζ21 cos 2θ + · · · for the vorticity, compatible with the Euler and the Navier–Stokes equations. For the coefficients ζi(k) j , being functions of r , k designates the order of perturbation and i labels the Fourier mode with j = 1 and 2 corresponding to cos iθ and sin iθ , respectively. With this form, we perform integration with respect to xˆ  and zˆ  in (11) and simplify the resulting expression with the help of (13). Substitution from (14) yields the asymptotic form of the Biot–Savart law, whose expression is, if we retain to first order in ε = σ/R say, as ψ =−

where  = 2π

∞ 0

          R0 8R0  8R0 d1 log −2 + − log −1 r + cos θ + · · · , 2π r 4π r r

(15)

(1)

r ζ0 dr, and the strength d1 of the dipole is connected with ζ0 and ζ11 .

The asymptotic form (15) serves as the inner limit of the outer solution and thus supplies the matching (1) (2) (2) condition on the inner solution. Given ζ0 , the profiles of ζ11 , ζ0 , and ζ21 should be determined by solving the Navier–Stokes or the Euler equations in the inner region. When the vorticity is confined in the core, the expression (15) is validated to the edge of the core, and the translation speed is determined by imposing the condition that the boundary is coincident with a streamline. This was the approach taken by the successors of Kelvin [6–8]. To recover Kelvin’s formula, representation (15), valid to O(ε), is sufficient, but Thomson [6] overlooked the contribution from the local dipole field which includes d1 . This dipole field stems from an effective vortex pair generated by vortex-line stretching on the convex side and contraction on the concave side when a straight vortex tube is bent into a torus, which has ability to drive itself. For Kelvin’s vortex ring, d1 = −3σ 2 /(16π R0 ) and this is equivalent to the flow field around a cylinder of radius σ moving in the z direction with the speed /(4π R0 ) × 3/4 [12,13]. This contribution repairs Thomson’s results. For a general distribution of vorticity, to carry out the inner expansion along with the extension of (15) is a rather cumbersome task. The treatment initiated by Helmholtz sidesteps the inner solution to a great extent, which is the topic of the following section. We note in passing that Dyson’s technique has been extended to a helical vortex tube [37] and to a general three-dimensional vortex tube [38]. Reprinted from the journal

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3 Helmholtz–Lamb’s method Helmholtz–Lamb’s method is very efficient in that it allows us to reach the correction of O(ε3 ) to translation speed of a vortex ring without having to derive the O(ε3 ) velocity field. Rott-Cantwell [36] gave a lucid account of this method. Under the boundary condition |A| → 0 as |x| → ∞ on  the vector potential A, the total kinetic energy H of fluid filling an unbounded space, defined by H = 1/2 u2 dV, has a representation, for the axisymmetric flow, of  





1 H= ω · AdV = −π ζ ψ d A = −π ζ ψ dρdz . (16) 2 The hydrodynamic impulse (6) is reduced to P = Pz e z ;

Pz = π

ζρ 2 d A.

Remember that the impulse is a constant even in the presence of viscosity [1,5,19]. Helmholtz [4] introduced the vorticity centroid 

2 ζρ 2 d A Z= ζρ zd A

(17)

(18)

and thought of its time derivative as the traveling speed of the vortex ring. By virtue of constancy of (17) and of the vorticity flux across a material surface whose local form is ζ d A [25–27], the differentiation of (18) in time t immediately yields the traveling speed U = dZ /dt in the form: 

  2 U= u z ζρ + 2u ρ ζ zρ d A ζρ 2 d A. (19) It was verified that the viscous does not alter this form [15,36]. Two alternative repre diffusion of vorticity  sentations of energy H = ω · AdV /2 = u · (x × ω)dV reads, for the axisymmetric flow [4]



  1 − (20) ψζ d A = u z ρ 2 − u ρ zρ ζ d A. 2  This is used to eliminate the integral u z ζρ 2 d A from (19), leaving Helmholtz–Lamb’s identity



1 ψζ d A + 3 ρzu ρ ζ d A. (21) U ζρ 2 d A = − 2 It is noteworthy that the derivation does not depend much on the detail of the dynamics, and hence (21) is applicable to a wide class of solutions. Helmholtz–Lamb’s identity (21) and Rott–Cantwell’s identity (19) both require the knowledge of velocity field in the core or the inner solution. We recall the asymptotic solution of the Euler or the Navier–Stokes equations at large Reynolds numbers [12] in the following section and at small Reynolds numbers [33] in Sect. 6. 4 High Reynolds number vortex ring The inner solution for steady motion of a vortex ring, or quasi-steady motion in the presence of viscosity, is found by solving the Euler or the Navier–Stokes equations, subject to the matching condition (15), in powers of the small parameter ε [12]. This is then substituted into (21). In the sequel we give an outline of evaluating (21) to obtain the third-order correction to the translation speed. The detailed procedure of calculating integrals in (21) is presented in the forthcoming article [39]. To work out the inner solution, we introduce the relative velocity u˜ in the meridional plane by u = ˙ Z˙ ). Here a dot stands for differentiation with respect to time. Let us nondimensionalize the inner variu˜ + ( R, ables. The radial coordinate is normalized by the core radius ε R0 (= σ ) and the local velocity (u, v), relative to the moving frame, by the maximum velocity /(ε R0 ). In view of (2), the normalization parameter for the

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˙ ring speed ( R(t), Z˙ (t)), the slow dynamics, should be /R0 . The suitable dimensionless inner variables are thus defined as r ∗ = r/ε R0 , t ∗ = t/

R0 ψ   ˙ Z˙ )/  . ˜ , ζ ∗ = ζ / 2 2 , u˜ ∗ = u/ , ψ∗ = , ( R˙ ∗ , Z˙ ∗ ) = ( R,   R0 R0 ε R0 R0 ε (22)

The difference in normalization between the last two of (22) should be kept in mind. Correspondingly to (22), the kinetic energy (16) and the hydrodynamic impulse (17) are normalized as H ∗ = H/  2 R0 , Pz∗ = Pz /  R02 . Hereinafter we drop the superscript ∗ for dimensionless variables. Dimensionless form of the radial position R of the core center is R = 1 + ε2 R (2) + O(ε3 ). We can maintain the first term to be unity by adjusting disposable parameters, bearing with the origin of coordinates, in the first-order field [12]. The second-order correction ε2 R (2) is tied with the viscous expansion. A glance at the Euler or the Navier–Stokes equations shows that the dependence, on θ , of the solution in a power series in ε is   (1) (2) (2) ψ = ψ (0) (r ) + εψ11 (r ) cos θ + ε2 ψ0 (r ) + ψ21 (r ) cos 2θ + O(ε3 ), (23)   (1) (2) ζ = ζ (0) (r ) + εζ11 (r ) cos θ + ε2 ζ0(2) (r ) + ζ21 (r ) cos 2θ + O(ε3 ). (24) Upon substitution from (23) and (24), we obtain a representation, to O(ε2 ) in dimensionless form, H = H (0) + ε2 H (2) and Pz = P (0) + ε2 P (2) of the kinetic energy and the hydrodynamic impulse, as H

(0)

∞ = −2π

rζ

2

(0)

ψ

(0)

dr,

H

(2)



∞  1 (1) (1) (2) (2) = −2π r ζ11 ψ11 + ζ (0) ψ0 + ζ0 ψ (0) dr, 2 2

0

(25)

0

P (0) = π,

  P (2) = π 2R (2) − 4πd (1) ,

(26)

where d (1) = d1 /(σ 2 ) is the dimensionless strength of dipole. (2) (2) Evaluation of (25) and (26) is relatively easy as these do not include the quadrupole field ψ21 and ζ21 . Given ζ (0) to O(ε0 ), the azimuthal velocity to O(ε0 ) satisfies v (0) = −∂ψ (0) /∂r, and the Stokes streamfunction complying with (15) is, to O(ε0 ), ⎧ r ⎫    ⎬

r ⎨ 8 1 ψ (0) = − v (0) (r  )dr  + lim log −2 . (27) v (0) (r  )dr  − r →∞ ⎩ ⎭ 2π εr 0

0

Without viscosity, the vorticity profile ζ (0) may be taken to be arbitrary, but viscosity plays the role of selecting its functional form [17]. It is expedient to handle the streamfunction ψ˜ for the flow relative to the coordinates ˜ The moving with the same speed Z˙ as the vortex ring along the z-direction, namely, ψ = − Z˙ ρ 2 /2 + ψ. first-order solution comprises a dipole field. Denoting the dipole coefficient of the streamfunction for the flow, (1) (1) (1) relative to the moving frame, to be ψ˜ 11 = ψ11 + r Z˙ (0) , the coefficient function ψ˜ 11 is given by ⎧ ⎫ ⎪ ⎪

r   ⎨ r 2 r ⎬   2 dr (1) (1) (0)  (0)   ˜ + c11 v (0) , v ψ11 = −v r (r ) dr (28) + ⎪ ⎪ r  [v (0) (r  )]2 ⎩2 ⎭ 0

0

(1)

where c11 is a disposable parameter tied with choice of the origin r = 0 of the local coordinates. The vorticity (1) (1) (2) is found from ζ11 = a ψ˜ 11 + r ζ (0) with a(r, t) = −1/v (0) (∂ζ (0) /∂r ). The Fourier coefficient ψ˜ 0 (r ) of the (2) (2) monopole component of O(ε2 ), relative to the moving coordinate frame, defined by ψ˜ 0 = ψ0 + Z˙ (0)r 2 /4 (1) (2) (2) is written in terms of v (0) , ψ˜ 11 and ζ0 . The O(ε2 ) monopole component ζ0 of vorticity obeys a heatconduction equation with source terms [12]. Reprinted from the journal

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A steady inviscid vortex ring or a quasi-steady viscous vortex ring corresponds to a state of the maximum energy and this critical state favors core shape with back-to-fore symmetry [26,40]. This symmetry, ζ (ρ, −ˆz ) = −ζ (ρ, zˆ ) and u ρ (ρ, −ˆz ) = −u ρ (ρ, zˆ ), simplifies the last integral in (21) to

J=

2π ∞ ρzu ρ ζ d A = 0

0



cos θ ∂ ψ˜ ∂ ψ˜ + r sin θ sin θ ∂r r ∂θ

 ζ r dr dθ.

(29)

Substituting from (23) and (24), (29) becomes J = J (0) + ε2 J (2) , to O(ε2 ), where J

J

(2)

=−π

r

2

∞ = −π

r 2 v (0) ζ (0) dr =

1 , 8π

(30)

0



∞

(0)

v

0

(0)

  (2)  (1)     (1) (2) (2) ∂ ψ˜ 11 ∂ ψ˜ 0 ψ˜ 21 1 (2) 1 ∂ ψ˜ 21 1 ψ˜ 11 (2) (1) (0) dr. − ζ11 + + − ζ ζ0 − ζ21 + 2 4 r ∂r r 2 ∂r ∂r (31)

The leading-order term H (0) of energy is evaluated with ease, by introducing (27) into (25), which is expressed, in dimensional variables, as    1 8R0 2 + A−2 , (32) H0 /  = R0 log 2 σ where H0 =  2 R0 H (0) and A is defined by (3). This expression, along with P (0) and J (0) , gives rise to Fraenkel–Saffman’s formula (3), via (21). The third-order correction U2 to the translation speed of the vortex ring (2) (2) requires evaluation of H (2) and J (2) . Evaluation of (31) is rather involved as it includes ζ21 and ψ˜ 21 , the (2) and the relation quadrupole field of O(ε2 ). But (31) is somehow simplified by use of equation governing ψ˜ 21 (2) (2) 2 (2) between ψ˜ 21 and ζ21 . For an inviscid vortex ring in steady motion, R2 = R0 ε R ≡ 0 without loss of generality, and, after some manipulations, we arrive at ⎧ ⎫    

∞ ⎬ 8R0 π 1 ⎨ d1 log − 2 − π B + r 4 ζ0 v0 dr , (33) U2 = 3 ⎭ σ 2 R0 ⎩ 2 0

where v0 =

v (0) /σ

ζ (0) /σ 2

and ζ0 = are dimensional variables, and ⎧ ⎫ ⎨ 1 r r    r ⎬ 2  r d 1 (1) log r  v0 ψ˜ 11 dr  + +A + log . B = lim r →∞ ⎩  2 16π 2 σ 2π σ ⎭

(34)

0

This is an extension, to O(ε3 ), of Fraenkel–Saffman’s formula (3). The same formula was reached by way of the variational principle [14] as will be touched upon in the following section. Even if viscosity is switched on, the higher-order asymptotics U2 is not invalidated at a large Reynolds number. Taking, as the initial condition, a circular line vortex of radius R0 , ζ (ρ, z, 0) = δ(ρ − R0 )δ(z − Z ) at t = 0,

(35)

the leading-order vorticity ζ0 is given by (4) [15,17], and the inhomogeneous heat-conduction equation gov(2) (1) erning ζ0 becomes tractable, with an introduction of similarity variables. The parameters c11 in (28) and R2 , both being functions of t, play a common role of specifying the radial position of the ring at O(ε2 ) relative to (1) R0 . This redundancy is removed, for instance, by taking c11 ≡ 0. Thus we are led to an extension of Saffman’s formula (5) in the form     4R0  νt U≈ log √ (36) − 0.55796576 − 3.6715912 2 . 4π R0 νt R0

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Fig. 2 Variation of speed of a viscous vortex ring with time. The upper and lower solid lines are the high- and low-Reynolds number asymptotics (36) and (44), respectively, while the thick dashed line is the Saffman’s formula (5). The dashed lines are the values read off from the graph of numerical simulations [41]

Figure 2 displays the comparison of the asymptotic formula (36) with a direct numerical simulation of the axisymmetric Navier–Stokes equations [41]. The normalized speed U R0 /  of the ring is drawn as a function of normalized time νt/R02 for its small values. The upper thick solid line is our formula (36), and the thick broken line is the first-order truncation (5). The dashed lines are the results of the numerical simulations, attached with the circulation Reynolds number /ν, ranging from 0.01 to 200. Augmented only with a single correction term, (33) appears to furnish a close upper bound on the translation speed. Notably, the large Reynolds number asymptotic formula (36) compares fairly well with the numerical result of even moderate and small Reynolds numbers. Constancy of the hydrodynamic impulse (17), regardless of the presence of viscosity, provides us with a shortcut to reach the radial motion of the ring; the third-order motion R (2) is gained solely from the first-order velocity field [12]. For the initial δ-function core (35), the peak-vorticity circle of radius R p (t) and the vorticity centroid Rc in the radial direction expands, respectively, as [12]

νt π 3νt R p ≈ R0 + 4.5902739 , Rc = . (37) ρ 3 ζ r dr dθ ≈ R0 + R0 Pz R0 √ The third-order formulas (36) and (37), intended for νt  R0 , has a wider applicability than envisaged, but ceases to be valid when the core is so swollen as to touch itself and cancelation of vorticity takes place on the symmetry axis (ρ = 0). Saffman [15] made a judicious treatment of simplifying the Navier–Stokes equations for estimation of the traveling speed of a vortex ring valid after νt ≈ R02 , and obtained, with use of some constant k and k  , R 2 = R02 + k  νt, U =

−3/2 Pz  2 . R0 + k  νt k

(38)

This tends, at νt  R02 , to Rott–Cantwell’s decaying law (8). Saffman’s matured-stage formula (38) exhibits a good fit to an experimental measurement of using the DPIV [42]. The measurements of location of peak vorticity tells k  = 7.8. If the small-time asymptotics (37) is translated into (38), k  = 9.1805478. The agreement is acceptable. (2) Although Helmholtz–Lamb’s method (21) saves the labor, the integral J in (21), with including ψ21 and (2) , stands as an obstacle. The variational principle (7), which comprises only the total energy H and the ζ21 (2) (2) impulse Pz , dispenses with ψ21 and ζ21 . A further simplification is achieved by relying on the variational principle with kinematic constraints. 5 Kelvin–Benjamin’s variational principle It is well-known that a stationary configuration of vorticity, embedded in an inviscid incompressible fluid, is realizable as an extremal of energy on an iso-vortical sheet [24,25]. An iso-vortical sheet comprises volumepreserving diffeomorphisms, or smooth maps of fluid particles, with vorticity frozen into the fluid. Put it in Reprinted from the journal

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another way, the critical state is sought among the class of ω that is reached by rearrangement of the initial distribution. Extending this conditional variational principle to a moving state, Benjamin [26] stated that an axisymmetric vortex ring moving steadily in an inviscid incompressible fluid is realizable as the maximum state of the kinetic energy H on an iso-vortical sheet, subject to the constraint of constant hydrodynamic impulse (6). An upper bound of the kinetic energy, supplied by a topological invariant [27,43], guarantees the existence of the maximum state. When translated into three dimensions, Kelvin–Benjamin’s principle takes the form of (7) with constant vector U playing the role of the Lagrangian multipliers [40]. The restriction of axisymmetry can be lifted and (7) is extended to a stationary vorticity distribution in a steadily moving frame [14]. An iso-vortical sheet is of infinite dimension. A family of solutions of the Euler equations includes a few parameters. By imposing certain relations among these parameters, we can maintain the solutions on a single iso-vortical sheet, and the restricted family of the solutions constitutes a finite dimensional set on the sheet. Thus the traveling speed of a vortex ring may be calculable through (7). Dyson’s vortex ring (2) was dealt with in this framework [22]. The same is true of Saffman’s formula (5) [14], though excluded from the list of [18]. We pose, as a natural profile of local velocity field featuring a vortex ring, v0 (r ) = −

r    d r  f , ζ0 = f ; 2πr σ 2πr dr σ

f (ξ ) = O(ξ 2 ) as ξ → 0, f (ξ ) → 1 as ξ → ∞, (39)

where f is an arbitrary function, though subjected to the above boundary conditions. The parameter σ introduces the scale for the core thickness. Suppose that the fluid particles occupying a toroidal region of radius r around the center circle of radius R is mapped to another toroidal region of radius rˆ around the center ˆ To maintain these flow field on an iso-vortical sheet, it is necessary for the local circucircle of radius R. lation along any material loop to remain unchanged [25,27,43]. Preservation of material volume enforces ˆ 2π 2 σ 2 R = 2π 2 σˆ 2 R, ˆ from which follows r/σ = rˆ /σˆ . Consequently, the local circu2π 2 r 2 R = 2π 2 rˆ 2 R, r lation around the circle of radius r , (r ) = 2π 0 ζ0 (r  )r  dr  =  f (r/σ ), is made invariant: (r ) = (ˆr ). Under an infinitesimal perturbation of R → Rˆ = R + δ R, σ → σˆ = σ + δσ , with R = R0 + R2 , (5) demands that, at each order, σ 2 R0 = const and σ 2 R2 = const, and that 2δσ/σ = −δ R0 /R0 = −δ R2 /R2 .  therefore  We can show that, under this perturbation, Aˆ = A + O (δ R)2 . In view of these constraints, the variation of (32) with respect to an iso-vortical perturbation becomes     8R0 2 1 δ H0 = (40) log + A− δ R0 . 2 σ 2 The variation of the leading term of impulse P0 = π R02 is δ P0 = 2π R0 δ R0 , and application of (7) restores Fraenkel–Saffman’s formula (3). This principle is extensible to higher orders, whereby the O(ε3 ) corrections (33) and (36) are produced [14]. Mohseni [44] devised an efficient algorithm of combining (7) with the slug model to estimate the translation velocity of fat vortex rings.

6 Low Reynolds number vortex ring Saffman’s second law (38) well describes the time-wise variation of traveling √ √ speed after the matured stage ( νt ≥ R0 ), by an adjustment of the disposable parameters k and k  . For νt  R0 , (38) approaches Rott– Cantwell’s decaying law (8) for which the velocity field is given by Phillips’ spherical dipole [45], an exact solution of the Stokes equations. Given an initial delta function core (35), the early-time behavior (5) of the translation speed is common to O(ε), independently of the Reynolds number /ν. At low Reynolds numbers, there is a solution that is valid over the whole time range (t ≥ 0), illustrating how the early time behavior (5) of a thin core crosses over to (8) [32,33]. We suppose that the vorticity is governed by the Stokes equations. Their solution, subject to the initial condition ζ0 (z, ρ, 0) = 0 δ(z)δ(ρ − R0 ), with 0 being a constant, is     z 2 + ρ 2 + R02 R0 ρ 0 R 0 ζ = √ exp − I1 , (41) 4νt 2νt 4 π(νt)3/2

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where I1 is the first-kind modified Bessel function of order unity. This expression was first found by Kaltaev [46] and was then given an interpretation in the context of evolution ofa viscous vortex ring  [32,33]. The total vorticity in the half meridional plane (ρ ≥ 0) decreases as  = 0 1 − exp −R02 /4νt , and the hydrodynamic impulse is Pz = π R02 0 . The corresponding Stokes streamfunction is 0 R 0 ρ =− 4

   

∞  2 pνt + z 2 pνt − z e pz erfc + e− pz erfc J1 ( p R0 )J1 ( pρ)d p. √ √ 2 νt 2 νt

(42)

0

√ The behavior of (41) and (42) at large times ( νt  R0 ) coincides with that of Phillips’ solution. Upon substitution from (41) and (42), (21) gives rise to the desired formula for the translation velocity:      0 R02 R02 R02 3 3 5 3 5 36 7 U = √ , ; , 3; − − 2 F2 , ; 2, ; − 2 F2 2 2 2 2νt 5 2 2 2 2νt 96 2π(νt)3/2     R02 R2 72νt , (43) + 2 exp − 0 I1 4νt 4νt R0 where 2 F2 is the generalized hypergeometric function [32]. One of advantages of this representation is to use the expansion of 2 F2 at small arguments and to exploit the analytic continuation to derive asymptotic expansions at large arguments. Using the asymptotic form of 2 F2 for negative large and small values of the argument, we can deduce early- and long-time behavior of (43) as follows [33].         √ 4R0 0 4R0 νt log √ for νt  R0 , U ≈ − 0.55796576 − 4.5 log √ − 1.0579658 2 4π R0 νt νt R0

=

7Pz √ 240 2 (πνt)3/2

⎧ ⎛  ⎞⎫  2 3 ⎬ ⎨ 2 2 √ R02 R R 33 0 125 0 ⎠ 1− +O⎝ + for νt  R0 . ⎩ ⎭ 196 νt 6272 νt νt

(44) (45)

The translation speed (43) based on the Stokesian dynamics of vorticity poses the strict lower bound on U . Figure 2 confirms this at small values of t. Fair agreement is observed between (44) and the numerical result of the Reynolds number /ν = 0.01, the lowermost dashed line. Notice that the normalized traveling speed is not very sensitive to /ν. The large-time behavior (45) gives corrections to Rott–Cantwell’s formula (8). Comparison of the first two terms of (45) with those of Saffman’s second formula (38) as expanded in R02 /νt yields k = √ 1320 11π 3/2 /2401 ≈ 10.15, k  = 98/11 ≈ 8.909. With this choice, (38) furnishes an excellent interpolation formula between (44) and (45), and exhibits a fairly good approximation to (43) even at small values of νt/R02 .  t Another advantage of the representation (43) is that we can calculate, in a tidy form, the distance s(t) = 0 U (τ )dτ traversed over time t by the viscous vortex ring [33]. This expression gives a partial answer to the question of the inverse problem raised in Sect. 1. When does a vortex ring arrive at a specified point? The traveling speed slows down with t, and ultimately decreases to zero in proportion to t −3/2 as is seen from (8). This implies that the vortex ring cannot be freely sent to remote regions. Taking the limit t → ∞ of s(t) shows that the distance smax of the furthermost reach is 50 R0 5Pz 0 R 0 smax = = ≈ 0.066314560 . (46) 24πν 24π 2 R0 ν ν The maximum reach smax extends in inversely proportion to ν. Given the impulse Pz , smax is larger as the initial ring radius R0 is smaller. The vortex bubble is a region encircling the vortex core bounded by a surface of zero streamfunction of the flow relative to the ring motion. For an inviscid vortex ring, the volume of the vortex bubble is a function of the Stokes streamfunction for the relative flow and is called the vortex-ring signature [27,47]. The vortex-ring signature of our low Reynolds number solution bears, with an appropriate normalization, some resemblance with that of the direct numerical simulation of the axisymmetric Navier–Stokes equations [31]. Recently, a family of similarity solutions is found [48], that includes (41) and (42) as an extreme and may describe a turbulent vortex ring in the other extreme. Reprinted from the journal

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7 Summary Vortex rings stimulated development of mathematical machinery. This article has described a partial history of this development as exemplified by Helmholtz–Lamb identity for the movement of the vorticity centroid, Dyson’s technique for asymptotic expansions of the Biot–Savart law, and Kelvin’s variational principle as augmented with the effect of motion of a vortical region by Benjamin. Helmholtz’s seminal paper [4] illuminated the preservation of topology of the vorticity field. This topological idea has been rediscovered in various guises over and over again and has played a vital role in developing theories of vortex dynamics, including vortex-ring motion. Mathematical labor to reach the same formula for the speed of a vortex ring dramatically decreases in order of the method of matched asymptotic expansions [12], Helmholtz–Lamb’s method and the variational principle [14]. By appealing to these efficient methods, we have succeeded in achieving higher-order extension of Fraenkel–Saffman’s and Saffman’s formulas, which are applicable to fat cores. Hopefully these methods carry over to helical vortex tubes with allowance made for torsion and the rotation of the system ([37]). For low Reynolds number motion, exploiting formulas associated with the generalized hypergeometric functions is advantageous to extract rich information. Acknowledgments This article is based on collaboration with H. K. Moffatt and F. Kaplanski. I owe much to them. This work was partially supported by a Grant-in-Aid for Scientific Research from the JSPS (Grant No. 19540406).

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Theor. Comput. Fluid Dyn. (2010) 24:349–361 DOI 10.1007/s00162-009-0146-1

S P E C I A L I S S U E O R I G I NA L A RT I C L E

Stéphane Le Dizès · David Fabre

Viscous ring modes in vortices with axial jet

Received: 8 January 2009 / Accepted: 24 June 2009 / Published online: 11 August 2009 © Springer-Verlag 2009

Abstract In this paper, we show the existence of new families of linear eigenmodes in vortices with axial jet. These modes are viscous in nature and concentrated in a ring around the vortex at the critical radial location rc > 0 where mΩc + kWc = 0 where Ωc and Wc are the radial derivative at rc of the angular and axial velocity of the vortex. Using a large Reynolds-number asymptotic approach for an arbitrary axisymmetrical vortex with axial flow, both the complex frequency and the spatial structure of the eigenmodes are obtained for any azimuthal and axial wave number. The asymptotic predictions are compared to numerical results for the q-vortex and a good agreement is demonstrated. We show that for sufficiently large Reynolds numbers, a necessary and sufficient condition of instability of viscous ring modes is that there exists a location rc where Ωc Ωc [rc Ωc (2Ωc + rc Ωc ) + (Wc )2 ] < 0 and Wc  = 0, which also corresponds to the condition of inviscid instability obtained by Leibovich and Stewartson (J Fluid Mech 126:335–356, 1983). Keywords Vortex · Swirling jet · Instability · Viscosity · Breakdown · q-vortex · Ring mode PACS 47.15.Fe, 47.20.Gv, 47.32.Cd

1 Introduction Vortex stability is a domain of research as old as the general field of vortex dynamics. But, despite the enormous amount of works, the linear stability properties of simple vortices as the q-vortex model are still not fully known. For a long time, it has been believed that the q vortex was stable for swirl number q above 1.5 [1]. Recently, it was shown that not only inviscidly unstable modes exist up to q ≈ 2.35 [5], but unstable viscous center modes exist whatever the a swirl number if the Reynolds number is sufficiently large [3,6]. It was demonstrated in [4,6] that these unstable viscous modes are generic perturbations in vortices with axial jet. Far from the marginal stability curves, they have a complicated multi-layered structure localized in an O(Re−1/6 ) neighborhood of the vortex center, and a growth rate of the form σ ∼ σ0 Re−1/3 where σ0 is a simple function of the vortex parameters at the center [6,9]. The modes that will be described in this paper will be very similar in nature to these center modes. But, instead of being localized near the center, they will Communicated by H. Aref S. Le Dizès (B) Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS & Aix-Marseille Université, 49, rue F. Joliot-Curie, B.P. 146, 13384 Marseille cedex 13, France E-mail: [email protected] D. Fabre Institut de Mécanique des Fluides de Toulouse, allée du Prof. Soula, 31400 Toulouse, France Reprinted from the journal

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be spatially concentrated near a particular radius corresponding to a double critical point of the inviscid equations. The mode structure will then form a cylindrical ring around the vortex. We shall provide the asymptotic structure of these ring modes in the limit of large Reynolds numbers. Ring modes have already been obtained in an inviscid framework. Leibovich and Stewartson [7] and Stewartson [8] have been able to construct unstable ring modes by performing an asymptotic analysis in the limit of large azimuthal wave numbers m. Interestingly, the modes were also found to be localized near the double critical point mentioned above and the sufficient condition of instability which was derived [7] will turn out to be also a sufficient condition of instability of viscous ring modes. The paper is organised as follows. In Sect. 2, the framework of the analysis is presented. Section 3 is concerned with the asymptotic analysis. The solutions in the different regions around the double critical point are obtained and matched in order to obtain the characteristics of the ring modes. An approximation for the most unstable ring modes and a general instability criterion are obtained in Sect. 4. In Sect. 5, the results are applied to the q-vortex model. We demonstrate that both the frequency and the spatial structure of the most unstable ring modes obtained by numerical integration are well predicted by the theory. The main results are then summarized in the last section.

2 Basic flow characteristics and perturbation equations As in LDF07 [6], we consider a general axisymmetrical vortex with axial flow whose velocity field in cylindrical coordinates is U(r ) = (0, V (r ), W (r )),

(1)

where V (r ) and W (r ) are the azimuthal and the axial velocity respectively. We also define the angular velocity Ω(r ) and the axial vorticity Ξ (r ) of this flow: Ω(r ) =

V (r ) 1 d(r V ) , Ξ (r ) = . r r dr

(2)

The flow is assumed unbounded and the viscous diffusion of the base flow is neglected. However, the viscous effects on the perturbations which are responsible for the instability are considered. This is possible because the growth time scale of the unstable modes will be of order Re1/3 , that is smaller than the time scale O(Re1/2 ) associated with the viscous diffusion of the basic flow. Perturbations are searched in the form of normal modes (U, P) = (u, v, w, p)eikz+imθ −iωt ,

(3)

where k and m are axial and azimuthal wave numbers and ω is the frequency. Inserting these expressions in the Navier-Stokes equations leads to the following linear system for the velocity and pressure amplitudes (u, v, w, p):   ∂p 2imv 1 u iΦu − 2Ωv = − + Δu − 2 + 2 , (4) ∂r Re r r   2imu 1 v imp + Δv − 2 − 2 , (5) iΦv + Ξ u = − r Re r r 1 Δw, (6) iΦw + W  w = −ikp + Re 1 ∂(r u) imv + + ikw = 0, (7) r ∂r r where the prime denotes the derivative with respect to r, Δ = ∂r2 + (1/r )∂r − m 2 /r 2 − k 2 is the Laplacian operator, and Φ(r ) = −ω + mΩ(r ) + kW (r ). Perturbation amplitudes are also subject to boundary conditions: they must vanish at infinity and be bounded at the origin. The Reynolds number Re is constructed using the characteristic scales of the basic flow. Our goal is here to carried out an asymptotic analysis as Re → ∞.

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3 Asymptotic analysis and matching 3.1 Overview of the analysis In LDF07, the existence of viscous modes localized in the center of the vortex was demonstrated. The frequency of these modes was shown to be at leading order ω ∼ mΩ(0)+kW (0). Moreover, owing to the peculiar properties of the vortex center which always satisfies Ω  (0) = W  (0) = 0, the origin was also a double critical point for these modes. Here, we shall show that the condition of existence of a double critical point rc ( = 0) is sufficient to form a viscous eigenmode. As for viscous center modes, we shall demonstrate that there exist families of eigenmodes whose frequency expands in the limit Re → ∞ as ω ∼ ω0 + Re−1/3 ω1 + Re−1/2 ω2 + · · ·

(8)

ω0 = mΩc + kWc , mΩc + kWc = 0.

(9)

where

In the above equations, the subscript c indicates values taken at rc and the primes derivatives with respect to the radial coordinate. The existence of a double critical point different from the origin is not guaranteed for all profiles. For example, for a vortex without jet it requires Ωc = 0 that is an extremum of the angular velocity at a non-zero radial location. The structure of the viscous ring modes will be as described in Fig. 1. There are important similarities between ring modes and center modes. As for center modes, in the Outer region the solution is non-viscous and possesses an essential singularity at the critical point rc which requires the presence of a large O(Re−1/6 ) layer around rc to be smoothed. A similar O(Re−1/4 ) layer (corresponding to an intermediate layer for the center modes) is also present for ring modes: it is in that region that eigenmodes are discretized. However, contrarily to center modes, ring modes are perfectly regular within the Intermediate region so no O(Re−1/3 ) layer (Inner region) is needed. The analysis given below follows in large part the presentation of LDF07. The reader can refer to this paper for details. As in LDF07, the matching between the solutions in the different regions is performed by using the pressure amplitude. 3.2 Outer Non-Viscous regions (ONV± ) In the ONV± regions, the solution is assumed to be non-viscous at leading order. The pressure amplitude of the ONV± solution satisfies at leading order     k2Λ m2 2mkW  Ω 1 Λ d p 0 2m d 2 p0   (Ω + Λ − ΩΛ ) + − − − + p0 = 0, (10) dr 2 r Λ dr r2 r Φ (0) Λ (Φ (0) )2 r (Φ (0) )2 where Φ (0) (r ) = −ω0 + mΩ(r ) + kW (r ), Λ(r ) = 2Ξ (r )Ω(r ) − (Φ (0) (r ))2 .

(11)

Because Φ (0) (rc ) = 0 and ∂r Φ (0) (rc ) = 0, the solutions of (10) possess an essential singularity at rc . Two (1) (2) independent solutions p0 and p0 can be chosen such that they behave near the critical point rc as     β β (1) (2) p0 ∼ |r − rc |1/2 exp , p0 ∼ |r − rc |1/2 exp − , (12) |r − rc | |r − rc |

Outer Non−viscous region (ONV−)

O(Re −1/6 )

O(Re −1/4 )

Outer Viscous region (OV−)

rc Intermediate region

O(Re −1/6 )

Outer Viscous region (OV+)

r Outer Non−viscous region (ONV+)

Fig. 1 Asymptotic structure of viscous ring modes Reprinted from the journal

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with

where

√ −Hc β=2 such that − π/2 < arg(β) ≤ π/2, Kc   Wc Hc = 2Ωc k ωc k − m , rc



K c = Φc(0) = mΩc + kWc .

(13)

(14)

In each O N V ± region, the solution to (10) is therefore a combination (1)

(2)

± p ± ∼ A± ∞ p0 + B∞ p0 ,

(15)

± where A± ∞ and B∞ are O(1) constants. As the solution can be multiplied by any amplitude factor, the only − + + + important numbers are the ratios K − = A− ∞ /B∞ and K = A∞ /B∞ which depend on the integration of (10) from 0 to rc and from +∞ to rc , respectively. In the following, we shall assume that neither K − nor K + is zero or infinite. When |r − rc | = O(Re−1/6 ), pressure correction terms become of the same order as p0 which indicates that we enter the Outer Viscous region.

3.3 Outer Viscous regions (OV± ) The form of the solution in the two Outer Viscous regions is the same as for center modes. In these regions, the perturbation varies on the characteristic scale r¯ = Re1/6 (r − rc ). Viscous effects become important such that the pressure fluctuations now depend on six independent solutions which have the following WKBJ approximation:   ¯ r) . p¯ ∼ Re−1/3 p¯ 2 (¯r ) exp Re1/6 φ(¯ (16)  r¯ ¯ r) = where φ(¯ 0 μ with 2  r¯ 2 2 L(μ, ω1 , r¯ ) ≡ μ μ + iω1 − i K c − Hc = 0, 2 2

(17)

and the constants Hc and K c have been defined in (14). As for center modes, the first ring modes are obtained for ω1 such that r¯ = 0 is a double turning point of the WKBJ approximation, that is Lμ (μ(0), ω1 , r¯ = 0) = 0. This provides the value of ω1 :  1/3 Hc ω1 = 3i . (18) 4 As shown in LDF07, after rescaling, Eq. (17) with ω1 given by (18) defines 6 different cases (K c can be assumed positive), and as for center modes, 3 of them will provide eigenmodes. The six roots of (17) for the 6 different cases have been plotted in figure 2 of LDF07. Here, we shall focus on the three eigenmode cases which corresponds, as in LDF07, when K c > 0, to √   3( 3 + i) −Hc 1/3 Mode A : ω1 = , (19) 2 4   −Hc 1/3 , (20) Mode B : ω1 = −3i 4 when Hc < 0, and to

√   3( 3 − i) Hc 1/3 Mode C : ω1 = , 2 4

(21)

when Hc > 0.

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The different solutions of (17) have been defined, as in LDF07, as follows: the branches μ(1) and μ(2) = (1) −μ correspond to the non-viscous branches which match with the behaviour near rc of the ONV solutions p (1) and p (2) respectively. Contrarily to the 4 other viscous branches, the non-viscous branches go to zero as |¯r | goes to infinity. The branches μ(3) and μ(4) are such that they possess a negative real part for large |¯r |. Thus, they are associated with viscous subdominant solutions in the ONV+ region and dominant solutions in the ONV− region. The branch μ(3) is in addition assumed to be equal to one of the non-viscous branches at r¯ = 0. For the 3 cases leading to eigenmode, μ(3) (0) = μ(1) (0) while for the other cases μ(3) (0) = μ(2) (0). The two other viscous branches are μ(5) = −μ(3) and μ(6) = −μ(4) . The condition of matching with a non-viscous solution in the ONV± regions imposes that μ(5) and μ(6) can not be part of the solution in the OV+ region, and μ(3) and μ(4) can not be part of the solution in the OV− region. Moreover, the branches μ(4) and μ(6) are not connected to any other branch. So, if one of them is present in one the two regions, it is also present on the other Outer Viscous region, which is forbidden. For this reason, the general solution in the ONV± regions can be written as follows:   1/6 ¯ (1) 1/6 ¯ (2) 1/6 ¯ (3) (1) (2) (3) in O V + p¯ + ∼ Re−1/3 A¯ (1)+ p¯ 2 e Re φ + A¯ (2)+ p¯ 2 e Re φ + A¯ (3)+ p¯ 2 e Re φ (22)   1/6 ¯ (1) 1/6 ¯ (2) 1/6 ¯ (3) (1) (2) (5) in O V − p¯ − ∼ Re−1/3 A¯ (1)− p¯ 2 e Re φ + A¯ (2)− p¯ 2 e Re φ + A¯ (5)− p¯ 2 e Re φ As shown in LDF07, the amplitude p¯ 2 for each solution is obtained by considering the problem at higher order. We obtain a similar equation for p¯ 2 as for center modes:   1  1 d p¯ 2 + μ Lμμ + Lμ¯r + Lω1 ω2 + H p¯ 2 = 0, Lμ (23) d r¯ 2 2 where the functions Lμ , Lμμ , Lμ¯r and Lω1 denote partial derivatives of L [defined in (17)] with respect to the indexes and taken at (μ, ω1 , r¯ ) except for the operator H which is slightly different from the expression given in LDF07:   r¯ 2 μ. (24) H = −2i K c r¯ μ2 + iω1 − i K c 2 It follows that



1 exp − p¯ 2 (¯r ) = Lμ

r¯

 ω2 Lω1 + H , Lμ

(25) ring

such that if we compare with the pressure amplitude of the equivalent center mode we have p¯ 2 (¯r ) = √ center r¯ p¯ 2 . The behaviour of the solution towards the Intermediate region, that is as |¯r | → 0, is obtained as for center modes. The behaviour of the phase φ¯ for the eigenmodes satisfies as |¯r | → 0 φ¯ (1) (|¯r |) = −φ¯ (1) (−|¯r |) ∼ μ0 |¯r | + G 0 r¯ 2 /4

(26)

φ¯ (2) (|¯r |) − φ¯ (2) (−|¯r |) ∼ −μ0 |¯r | − G 0 r¯ 2 /4

(27)

φ¯ (3) (|¯r |) ∼ μ0 |¯r | − G 0 r¯ 2 /4

(28)

¯ (5)

φ where μ0 = μ(1) 0 (0) =



(−|¯r |) ∼ −μ0 |¯r | + G 0 r¯ /4 2

|Hc | 4

1/6

(1)

eiγ0 , G 0 = ∂r¯ μ(1) 0 (0)/2 =

(29) 

2|K c | 3

1/2 eiπ/4 .

(30)

(1)

The angle γ0 is 5π/6, −π/2 − π/3 for modes A, B and C respectively. The normalisation of the amplitude p¯ 2 can be chosen such that p¯ 2 (¯r ) expands as |¯r | → 0 as (1)

(2)

(2)

(1)

p¯ 2 (|¯r |) = p¯ 2 (−|¯r |) ∼ |¯r |−1/2−α p¯ 2 (|¯r |) = p¯ 2 (−|¯r |) ∼ |¯r |−1/2+α (3) p¯ 2 (|¯r |) ∼ |¯r |−1/2+α p¯ 2(5) (−|¯r |) ∼ |¯r |−1/2−α

Reprinted from the journal

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(31) (32) (33) (34)

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with ω2 e3iπ/4 . α= √ 6|K c |

(35)

Gathering all the results, we finally obtain the behaviour of the OV+ and OV− solutions as |¯r | → 0:      1/6 2 1/6 2 p¯ + ∼ Re−1/3r¯ −1/2+α A¯ (2)+ e Re −μ0 |¯r |−G 0 r¯ /4 + A¯ (3)+ e Re +μ0 |¯r |−G 0 r¯ /4 



1/6 2 + Re−1/3r¯ −1/2−α A¯ (1)+ e Re μ0 |¯r |+G 0 r¯ /4 ,      1/6 2 1/6 2 p¯ − ∼ Re−1/3r¯ −1/2+α A¯ (1)− e Re −μ0 |¯r |−G 0 r¯ /4 + A¯ (5)− e Re +μ0 |¯r |−G 0 r¯ /4

+ Re−1/3r¯ −1/2−α A¯ (2)− e Re

1/6



μ0 |¯r |+G 0 r¯ 2 /4



,

(36)

(37)

The matching with the ONV± solutions provides a condition between the coefficients. The two subdominant viscous branches μ(3) and μ(5) become negligible as |¯r | goes to infinity. The two other non-viscous branches are matched to the non-viscous solutions of the ONV± regions. For the normalisation chosen above, p¯ 2 is such that for large |¯r | we have (1) (2) (1) |¯r |, p2 (|¯r |) = p2 (−|¯r |) ∼ C¯ ∞ (1)

(2)

(1)

where C¯ ∞ and C¯ ∞ are constants. If we define φ¯ ∞

(2) (1) (2) p2 (¯r ) = p2 (−|¯r |) ∼ C¯ ∞ |¯r |,  ∞ (1) = 0 μ , the conditions of matching reduce to

(1)

(1)

1/6 ¯ (1) ¯ (1)+ C¯ ∞ A+ Re1/12 e Re φ∞ , ∞ = A

A− ∞

¯ (1)−

=A

1/6 ¯ + (2) B∞ = A¯ (2)+ C¯ ∞ Re1/12 e−Re φ∞ ,

(1)

1/6 ¯ (2) C¯ ∞ Re1/12 e−Re φ∞ ,

− B∞

¯ (2)−

=A

(1)

1/6 ¯ (1) C¯ ∞ Re1/12 e Re φ∞

.

(38)

(39) (40)

It follows that the ratios of the amplitudes of the two ONV± solutions are ¯ (1) A¯ (1)+ + C∞ (1) = K exp(−2Re1/6 φ¯ ∞ ), (2) A¯ (2)+ C¯ ∞

¯ (2) A¯ (1)− − C∞ (1) = K exp(2Re1/6 φ¯ ∞ ). (1) A¯ (2)− C¯ ∞

(41)

(1) As explained in LDF07, for the eigenmodes, e(φ∞ ) > 0 which implies that A¯ (1)+ is exponentially small compared to A¯ (2)+ , and A¯ (2)− is exponentially small compared to A¯ (1)− .

3.4 Intermediate region |r − rc | = O(Re−1/4 ) In this region, the pressure fluctuation varies on the scale r˜ = Re1/4 (r − rc ) = Re1/12 r¯ and can be written as:     (42) p˜ ∼ p˜ 4+ (˜r ) exp Re1/12 μ0 r˜ + p˜ 4− (˜r ) exp −Re1/12 μ0 r˜ . Using the same approach as in LDF07, we obtain an equation for p˜ 4± which reads:

where

˜ Δ( ˆ Δ˜ + i Φ) ˜ − Hc ] p˜ ± = 0, [(Δ˜ + i Φ) 4

(43)

  r˜ 2 ∂ ∂2 −1/6 ˜ Φ = ω1 + Re , Δ˜ = μ20 + Re−1/12 2μ0 + Re−1/6 2 . ω2 − K c 2 ∂ r˜ ∂ r˜

(44)

From this equation, we deduce at the order Re−1/6   2 iω2 ∂ r˜ 2 + − i K p˜ 4± = 0, c ∂ r˜ 2 3 6 which can be also written as

123



∂2 x2 +α− 2 ∂x 4 370



p˜ 4± = 0,

(45)

(46) Reprinted from the journal

Viscous ring modes in vortices with axial jet

if we use the above definition (35) of α and



2|K c | x= 3

1/4 eiπ/8r˜ .

(47)

Equation (45) is a parabolic cylinder equation [2] which possesses two independent solutions Dα−1/2 (x), and Dα−1/2 (−x). The condition of matching with the OV± regions imposes that only one of them is present in each solution such that we obtain 1/12 1/12 p˜ = A˜ + Dα−1/2 (−x)e+Re μ0 r˜ + A˜ − Dα−1/2 (x)e−Re μ0 r˜ ,

(48)

where A˜ + and A˜ − are constants. 3.5 Matching and frequency selection The matching between Intermediate and Outer Viscous regions is performed using the asymptotic expansions of the parabolic cylinder solution as |x| → ∞ [2]: Dν (x) ∼ x ν e−x

2 /4

, |arg(x)| < 3π/4, √ 2π iπ ν −ν−1 x 2 /4 ν −x 2 /4 Dν (x) ∼ x e − e , π/4 < arg(x) < 5π/4. e x Γ (−ν)

The intermediate solution behaves for large |˜r | as 

α−1/4  r¯ 2 r¯ 2 1/6 1/6 1/4 G 0 Re1/12 |¯r | p˜ ∼ −i A˜ + eiπ α e Re (μ0 |¯r |−G 0 4 ) + A˜ − e Re (−μ0 |¯r |−G 0 4 ) r˜ →+∞

p˜

√ −α−1/2

r¯ 2 2π 1/6 1/4 + A˜ + e Re (μ0 |¯r |+G 0 4 ) G 0 Re1/12 r¯ Γ (−α + 1/2) 

α−1/2  r¯ 2 r¯ 2 1/6 1/6 1/4 G 0 Re1/12 |¯r | ∼ A˜ + e Re (−μ0 |¯r |−G 0 4 ) − i A˜ − eiπ α e Re (μ0 |¯r |−G 0 4 )

r˜ →−∞

+

(49)

√

2π 1/4 1/12 −ν−1 ˜ − Re1/6 (μ0 |¯r |+G 0 r¯2 ) 4 |¯r | A e G 0 Re Γ (−ν)

(50)

where α and G 0 have been defined in (30) and (35), respectively. Comparing the above expansions with (36) and (37) leads to the relations: α/4−1/8 A˜ + G 0 Reα/12−1/24 = A¯ (1)− = i A¯ (3)+ e−iπ α ;

˜−

α/4−1/8 G0 Reα/12−1/24

¯ (2)+

¯ (5)− −iπ α

= iA e ; √ 2π −α/4−1/8 Re−α/12−1/24 = A¯ (1)+ ; A˜ + G 0 Γ (−α + 1/2) √ 2π − −α/4−1/8 −α/12−1/24 ˜ A G0 Re = A¯ (2)− , Γ (−α + 1/2) A

=A

(51) (52) (53) (54)

from which we deduce using (41) − 2π G −α K+ A¯ (1)+ A¯ (2) −α/3 0 Re = = 2 (Γ (−α + 1/2)) K− A¯ (2)+ A¯ (1)−



(1) C¯ ∞ (2) C¯ ∞

2 (1) exp(−4Re1/6 φ¯ ∞ ).

(55)

(1)

Since e(φ¯ ∞ ) > 0, the above equation implies that, for large Re, 1/Γ (−α + 1/2) is very small, which means that α − 1/2 is close to a positive integer n. Using the asymptotic expansion: 1 ∼ − (−1)n n!(ν − n) Γ (−ν) ν→n Reprinted from the journal

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we obtain α∼n+

1 (1) ) ± G n Ren/6+1/12 exp(−2Re1/6 φ¯ ∞ 2

where G n is a numerical constant: G 2n+1 1 Gn = √ n! 2π



(57)

(1) K + C¯ ∞ . (2) K − C¯ ∞

(58)

We finally deduce the value of the second order frequency ω2 :   1 −3iπ/4 n/6+1/12 1/6 ¯ (1) exp(−2Re φ∞ ) with n = 0, 1, 2, 3, . . . n + ± G n Re ω2 = 6|K c |e 2

(59)

4 Ring modes characteristics 4.1 Characteristics of the eigenmodes As for center modes, the above analysis provides three families of ring modes corresponding to the three different values of the first order frequency correction ω1 defined in (19)–(21). In fact, for given values of Hc and K c , there exist two families (modes A and modes B) when Hc < 0 and one family (modes C) when Hc > 0. Only the modes A for which m(ω1 ) > 0 are unstable. Contrarily to center modes, ring modes in each family come by pair. This duplicity is visible in formula (59) for ω2 : for each n, there are two second order frequency corrections defining two different modes. This in particular means that there are twice more ring modes that center modes. Formula (59) also tells us that the frequency separation of the two modes in each pair is a priori exponentially small and depends on the index n. It also depends on the characteristics of the ONV± solutions via the coefficients K + and K − in G n . These coefficients intervene also in the definition of the eigenmodes in the different regions, which renders the approximation of the eigenmode in principle more complicated. However, it was shown in LDF07 that center modes A are mainly localized in the Outer Viscous region and well described by a simple WKBJ approximation. The same result is true for ring modes A because they have the same structure as center modes in the Outer Viscous regions. As in LDF07, expression (22) can be reduced to a single term, and if we do not consider the amplitude term, it takes the form of a geometrical optics approximation:

 p ≈ exp Re1/6 ηΛ(2) (γ Re1/6 |r − rc |) (60) where



     2  Hc 1/3 |K c |  4 1/6 , γ = , η=3 |K c |  4  6  Hc 

(61)

and Λ(2) is an universal complex function plotted in Fig. 2. The function Λ(2) was already plotted in figure 5(a) of LDF07. This approximation a priori applies in the two OV± regions of the ring modes. However, in practice, it turns out that the modes are large in one of the two regions only. In principle, one could determine the region where each mode is the largest by computing the coefficients K + and K − and the amplitude factors. But for the comparisons which will be made below, we shall not try to do that but instead use the above approximation in the region where the mode is localized. 4.2 Instability criterion Unstable ring modes exist only if Hc < 0 and the leading order growth rate of the most unstable ring mode is   3  Hc 1/3 (62) σ = Re−1/3   . 2 4

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0.8

1

0.4

ℑ m(Λ(2))

1.1

(2)

|exp(Λ )|

Viscous ring modes in vortices with axial jet

0.9

0

0.8

−0.4

0.7

0

1

2

3

4

5

−0.8

s Fig. 2 Characteristics of the function

Λ(2) .

Solid line:

| exp(Λ(2) (s))|

versus s. Dashed line: m(Λ(2) (s)) versus s

The condition of instability together with the double critical point condition mΩc + kWc = 0 implies that we should have at rc : Ωc Ωc [rc Ωc (2Ωc + rc Ωc ) + (Wc )2 ] < 0.

(63)

Wc

 = 0 then, for any m  = 0, there exists Inversely, if (63) is satisfied at any point rc and if at that location a wavenumber k, given by k = −mΩc /Wc which guarantees that rc is a double critical point for the frequency ω = mΩc + kWc . Thus the above analysis can be performed which means that there exist unstable ring modes A if the Reynolds number is sufficiently large. Note in particular that unstable ring modes require axial flow and swirl. If W = 0, the double critical point condition requires Ωc = 0 which implies Hc = 0, thus stability. For inviscid ring modes, we have the same condition of instability for the large azimuthal wavenumbers [7]: we should have (63) and Wc  = 0. In particular, these modes are also stable if W = 0. However, the axisymmetric mode m = 0, associated with the centrifugal instability is unstable for W = 0 as soon as (63) is satisfied. This mode can also be considered as a ring mode in the limit of large k but instead of being localized near a double critical point, the most unstable centrifugal mode is localized near the minimum of the Rayleigh discriminant Φ = 2Ω(r )Ξ (r ). √ The leading order growth rate of inviscid ring modes is [7] σ ∼ rc2 /(m 2 + k 2 rc2 ) −Hc . As this growth rate does not depend on the Reynolds number, inviscid ring modes are then expected to be more unstable than viscous ring modes, when Hc < 0, if the Reynolds number is sufficiently large. However, this is not guaranteed for all Reynolds numbers. Indeed, the growth rate of inviscid ring modes is the largest for large m, but it is also for large m that the damping associated with viscosity is the largest. Close to the curve Hc = 0, the leading order growth rate of both types of ring modes vanishes. It is therefore near such a marginal curve that we expect the competition and the interplay of viscous and non-viscous modes to become important. The analysis of that marginal curve has been performed for center modes in Fabre and Le Dizès [4]. It was shown that near that curve, the initial three-zone structure can be reduced to a single one-zone problem. Unfortunately, such an analysis cannot be performed the same way for ring modes, because the structure of the asymptotic problem remains complex, with the same number of zones that in the present analysis. The analysis of the marginal curve and of the competition between viscous and inviscid ring modes is then left for future works. 5 Application to the q-vortex model The q-vortex model (also called Batchelor vortex) is defined by 1 − exp(−r 2 ) ; W (r ) = exp(−r 2 ), r2 where q is the so-called swirl parameter. Ω(r ) = q

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√ For this vortex, the condition (63) for the instability of ring modes is q < 2. In fact, for any negative azimuthal wavenumber m, ring modes are unstable in the (q, k) domain delimited by the curves k > −mq/2, q > 0, k < −m/q,

(65)

and which has been sketched in Fig. 3 for m = −1. The left and upper marginal curves correspond to the vanishing of Hc . On these curves the above analysis breaks down. The lower curve is associated with the transformation of ring modes into center modes: the point rc is at the origin on this line. On this line, Hc does not vanish but the analysis breaks down because K c = 0. Below this line, there is no double critical point different from the origin. In Fig. 3 is also indicated the instability domain of viscous center modes [6]. This domain includes and is larger than the ring mode instability domain. The leading order growth rate of both types of modes is given by Re−1/3 (3/2)(|H (r )|/4)1/3 with r = 0 for center modes and r = rc for ring modes where the function H (r ) is given by (1 − e−r ) −r 2 e (kq + m). r2 2

H (r ) = 4q

(66)

The function |H (r )| is strictly decreasing, which means that for the q-vortex model, ring modes are always less unstable than center modes. The spectrum obtained by numerical integration of the eigenvalue problem is plotted in Fig. 4 for a particular example. The code is the same as the one used in LDF07. The chosen parameters correspond to the point indicated by a diamond in Fig. 3. For these parameters, there are no unstable inviscid modes but there exist unstable viscous ring modes and unstable viscous center modes, close to the frequencies ω0 ≈ −0.026 and ω0 ≈ 2.3 respectively. For both modes, we have compared the numerical spectrum to the asymptotic predictions (see the close-up views). For center modes, we have used the asymptotic formula obtained in Le Dizès and Fabre [6]. For ring modes, we have used the formula obtained in the previous section but we 3

4

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Fig. 3 Left: Domain of instability of viscous ring modes (dark gray region) and viscous center modes (dark and light gray regions) in the limit of large Reynolds numbers, here plotted for m = −1. Right: The axial vorticity field of the most unstable ring mode (top) and the most unstable center mode (bottom) for the parameters indicated by a diamond on the left graph and for Re = 106 . The vortex core size is visualized by the dashed circle. More information on the spectrum and the modes for these parameters are also found in Figs. 4 and 5

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imag(ω)

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Fig. 4 Temporal spectrum of the q-vortex model for q = 0.2, Re = 106 , m = −1 and k = 2.5, obtained by a numerical spectral code with N = 600 polynomials. Close-up views: comparison with the theory for ring modes (left) and for center modes (right)

have taken G n = 0 in formula (59) for simplicity. This simplification means that we have not separated the two ring modes obtained for each value of n. For both center and ring modes, modes A and modes B have been obtained. We can notice that the agreement between theory and numerics is good for both center modes and ring modes. Note in particular that there are indeed twice more ring modes than center modes in each mode family. The main frequency correction ω1 is correctly predicted as well as the spacing between consecutive frequencies. The spatial structure of the 3 most unstable ring modes is displayed in Fig. 5a,b and c. These modes are modes A. As expected from the theory, the structure of these modes are similar and concentrated in the neighborhood of the critical point rc . Figure 5d demonstrates that this structure is very well described by the geometrical optics approximation (60) of mode A in the Outer Viscous region. It is worth mentioning that there are no free parameters in this comparison albeit the adimensionalisation: the structure is normalized such that the pressure is equals to 1 at its maximum amplitude. The above agreement for both the frequency and the spatial structure of the ring modes is a strong validation of the theory. 6 Conclusion In this article, we have demonstrated the existence of three families of viscous ring modes localized in the neighborhood of a double critical point. We have shown that one family (modes A) provides unstable modes for sufficiently large Reynolds numbers as soon as there exist a point rc where Ωc Ωc [rc Ωc (2Ωc + rc Ωc ) + (Wc )2 ] < 0 and Wc  = 0. We have obtained an explicit expression for the frequency of the modes as well as a simple approximation for the unstable modes. The frequency prediction and the mode approximation have been tested with respect to numerical results for the q-vortex model and a good agreement has been demonstrated. We have shown that for the q-vortex model, the viscous ring modes are always less unstable than center modes. This is not always the case for all vortices. For example, a vortex with the following angular and axial velocity profiles: Ω(r ) = qr 2 e−r , W (r ) = r 4 e−r , 2

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Fig. 5 Spatial structure (real part in dashed line, and modulus in solid line) of the most unstable viscous ring modes of the q-vortex model for q = 0.2, Re = 106 , m = −1 and k = 2.5. a,b,c numerical results for the three most unstable modes. d geometrical optics approximation (60) of the ring mode A. The vertical dashed line indicates the position of the critical point rc

does not possess unstable viscous center modes but exhibits unstable viscous ring modes. However, as mentioned above, these viscous ring modes are always in competition with inviscid modes. It will be interesting to analyse the competition between both modes for weakly unstable profiles or in the neighborhood of the marginal curves. Acknowledgments This work has been supported by the European community (Far-WAKE project) and the French research agency (VORTEX Project).

References 1. Ash, R.L., Khorrami, M.R.: Vortex stability. In: Green, S.I. (ed.) Fluid Vortices, chap. VIII, pp. 317–372. Kluwer, Dordrecht (1995) 2. Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York (1978) 3. Fabre, D., Jacquin, L.: Viscous instabilities in trailing vortices at large swirl numbers. J. Fluid Mech. 500, 239–262 (2004)

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4. Fabre, D., Le Dizès, S.: Viscous and inviscid centre modes in the linear stability of vortices: the vicinity of the neutral curves. J. Fluid Mech. 603, 1–38 (2008) 5. Heaton, C.: Centre modes in inviscid swirling flows, and their application to the stability of the Batchelor vortex. J. Fluid Mech. 576, 325–348 (2007) 6. Le Dizès, S., Fabre, D.: Large-Reynolds-number asymptotic analysis of viscous centre modes in vortices. J. Fluid Mech. 585, 153–180 (2007) 7. Leibovich, S., Stewartson, K.: A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335–356 (1983) 8. Stewartson, K.: The stability of swirling flows at large Reynolds number when subjected to disturbances with large azimuthal wavenumber. Phys. Fluids 25, 1953–1957 (1982) 9. Stewartson, K., Ng, T.W., Brown, S.N.: Viscous centre modes in the stability of swirling Poiseuille flow. Phil. Trans. R. Soc. Lond. A 324, 473–512 (1988)

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Theor. Comput. Fluid Dyn. (2010) 24:363–368 DOI 10.1007/s00162-009-0145-2

S P E C I A L I S S U E O R I G I NA L A RT I C L E

Yuji Hattori · Yasuhide Fukumoto

Short-wave stability of a helical vortex tube: the effect of torsion on the curvature instability

Received: 5 January 2009 / Accepted: 24 June 2009 / Published online: 7 August 2009 © Springer-Verlag 2009

Abstract In this paper, we study the short-wave stability of a helical vortex tube. The base flow field inside the helical vortex tube is obtained by perturbation expansion assuming that , the ratio of the core to curvature radius of the helical tube, be small. After reviewing our recent results obtained by the short-wavelength stability analysis, the normal-mode stability analysis is carried out to confirm the results and pursue further details. It is shown that the helical vortex tube suffers from curvature instability found for vortex rings. The growth rate of the curvature instability is obtained both analytically and numerically. The effect of torsion and rotation on the stability appears at the second order of . It is shown that the effects of rotation on the growth rate found by normal-mode stability analysis converges to those found by short-wavelength stability analysis in the short-wave limit. Keywords Helical vortex tube · Curvature instability · Torsion · Rotation PACS 47.15.ki, 47.20.Cq, 47.32.cd 1 Introduction Helical tip vortices commonly appear in the wakes of helicopter rotors, wind turbines and other devices which have rotating wings. Since the helical tip vortices have a vital influence on those devices as exemplified by the aerodynamic performance, noise emission and cavitation, it is undoubtedly important to study their basic properties. We can legitimately model them by a helical vortex tube which extends infinitely. There are many works on the motion of a helical vortex filament of which thickness is assumed to be very small. Widnall [1] considered a filament with non-zero thickness to obtain the self-induced flow field; but she studied the stability of the helical vortex filament with respect to the sinusoidal displacement of the centerline of the filament, which neglects the flow field inside the filament. In this sense the stability of the helical vortex filament studied so far is the long-wave stability. As we have shown in the case of vortex ring [2,3], however, Communicated by H. Aref Y. Hattori Department of Basic Sciences, Kyushu Institute of Technology, Kitakyushu, Japan E-mail: [email protected] Present address: Y. Hattori (B) Institute of Fluid Science, Tohoku University, Sendai, Japan E-mail: [email protected] Y. Fukumoto Graduate School of Mathematics, Kyushu University, Fukuoka, Japan E-mail: [email protected] Reprinted from the journal

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curvature effect of a helical tube could cause the short-wave instability. A helical vortex tube would be the simplest vortical structure featured not only by curvature but also by torsion. The effect of torsion on the stability in the short-wavelength regime has been left unaddressed, though vortical structures in real flow are more or less curved and twisted by various effects. Recently we have made a local stability analysis of a helical vortex tube in the short-wave limit [4]. The method we used is the short-wavelength stability analysis, in which the wavelength of the linear disturbance is assumed to be much smaller than the core size of the tube. It is shown that the helical vortex tube is subjected to curvature instability. Although the results obtained by the short-wavelength stability analysis are asymptotically valid and give good estimate for wavelength comparable to the size of the system, the normal-mode stability analysis, in which the linear disturbance is decomposed to eigenmodes, would reveal thorough natures of the instability. In this paper, we study the short-wave stability of a helical vortex tube by the normal-mode stability analysis. After reviewing our recent results obtained by the short wavelength stability analysis [4] (Sect. 2), the linear stability is investigated by the normal-mode stability analysis (Sect. 3).

2 Base flow and short-wave limit results In this section we briefly summarize our recent results on the linear stability of a helical vortex tube obtained by the geometric optics method [5], which gives asymptotic growth rates in the short wave limit. See Hattori and Fukumoto [4] for the details. We consider a helical vortex tube whose centerline is a helix of constant curvature and torsion in an inviscid and incompressible fluid (Fig. 1). We assume that the tube have a circular core of finite thickness; when we consider the helical vortex tube in an unbounded domain, this assumption is valid up to O() but is not the case at O( 2 ) [6]. Here we impose this assumption in order to focus on the effects of torsion by minimizing other effects like self-induced strain. It should be noted that as far as the curvature instability is concerned the deformation of the core boundary does not affect the results in the short wave limit [4]. There exist tubes which translate and rotate due to the self-induced velocity or external forcing but do not change the shapes. In other words, steady solutions can be found for the Euler equation in a rotating frame. We use a helical coordinate system along the tube (r, θ, s), where s is the arclength along the centerline of the tube (Fig. 1). We solve the equation by perturbation expansion. Substituting the flow field expanded as U = U 0 + U 1 +  2 U 2 + · · · into the Euler equation expressed in the helical coordinate system, we obtain equations at each order of . The leading-order solution is set to be a solid body rotation as in the simplest case. Then the solution reads

⎞ ⎛ ⎛ 5 ⎞  r r3 2 ⎛ ⎞ sin 2ϕ − 1 − r sin ϕ 8 8 8 ⎟ ⎜ 0



⎜ ⎟ ⎟ 2⎜ r r3 ⎟ 5 7 2 U = ⎝r ⎠ +  ⎜ +  cos 2ϕ − (1) ⎟ + O( 3 ), ⎜ 8 16 ⎝ 8 − 8 r cos ϕ ⎠ ⎠ ⎝

0 5 0 √ β r − 38 αr 3 cos ϕ 2 8α + 1+α

Fig. 1 Helical vortex tube

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0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000

1.150

α= 0 α= 1 α=-1

0

0.2

growth rate ratio

σ

Short-wave stability of a helical vortex tube

0.4

0.6

0.8

1.100 1.050 1.000 0.950 0.900 0.850

1

-90 -60 -30

r0

0 φ

30

60

90

Fig. 2 Growth rates obtained by short-wavelength stability analysis. a Dependence on r0 . α = 0, ±1, γ = 0,  = 0.05, φ = 10◦ . b Ratio of growth rates as a function of φ. α = 1, γ = 0,  = 0.05, r0 = 1

where α is the ratio of torsion to curvature of the centerline helix. The tube rotates without changing its shape around the axis of helical symmetry shown in Fig. 1 owing to the self-induced velocity or forced motion by the rotating wing; the rate of rotation 0 is measured by β = 0 / 2 , which is O(1) since the core size and the maximum velocity in the core are taken as the units of length and velocity, respectively. At O() the flow field is the same with that of a vortex ring except that the angle θ is replaced with ϕ = θ − αs; the first-order dipole flow is twisted due to torsion. At O( 2 ) torsion gives rise to the velocity component parallel to the centerline. An immediate consequence is that the curvature instability is present for the helical vortex tube since the equation and the base flow are the same with the case of a vortex ring up to O() (see also Sect. 3). In the geometric optics method the linear disturbance is assumed to be in a form a exp [i (x, t)/δ], where a and denote the amplitude and the phase function of the disturbance. Since the expansion parameter δ is assumed to be much smaller than unity, the local wavevector which is defined by k = ∇ /δ has large magnitude so that the disturbance is a sort of wavepacket which is localized around and moves with a fluid particle. By virtue of the short-wave assumption the linearized Euler equation which governs the disturbance is reduced to a system of ordinary differential equations. The growth rate of the curvature instability is then obtained by solving the system both analytically and numerically. The analytical expression is   45r0 75r0 75r0 σ =  − + cos 2φ + i sin 2φ 128 256 256      √ √  5 5 3 9 +  sin φ 15 − γ + (2) αr02 + i cos φ 15 γ+ αr02  + O( 3 ), 64 256 64 128 √ where r0 is the radius of the orbit and γ = β/ 1 + α 2 . The angle φ determines the direction of the wavevector by k ∝ sin χ cos φer + sin χ sin φeθ + cos χ es at the leading order of , where χ ≈ 0.42π for curvature instability. The effects of torsion are shown in Fig. 2. In Fig. 2a the growth rate of the curvature instability obtained numerically is plotted as a function of r0 , which is the leading-order radius of the fluid particle orbit X. The torsion parameter α is set to 0 (vortex ring), 1 (right-handed helical tube) and −1 (left-handed helical tube). Note that the growth rates are unchanged by (α, γ , φ) → (−α, −γ , −φ). The torsion increases the growth rate for α = 1 and decreases it for α = −1 in the present case of φ = 10◦ . In Fig. 2b the ratio of the growth rate of the helical vortex tube α = 1 to that of the vortex ring is shown. The ratio is largest around φ = 10◦ . In Fig. 2a, b the numerical values shown by the symbols are in good agreement with the analytical values shown by the lines. 3 Normal-mode stability analysis In this section we apply the normal-mode stability analysis. We start with the linearized Euler equation in a rotating frame ∂u + U · ∇u + u · ∇U + 2 × u = −∇ p  , ∂t where the centrifugal force term  × ( × x) is put into the pressure term. Reprinted from the journal

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Next we write down the linearized Euler equation in the present coordinate system. Then we substitute Eq. (1) for the base flow U and expand the equation with respect to . For example, its u-component reads   ∂u  V0 ∂u  2V0  ∂ p  ∂U1  V1 ∂u  1 ∂U1  2V1  ∂u  + − v + =  −U1 − u − − v + v ∂t r ∂θ r ∂r ∂r ∂r r ∂θ r ∂θ r   ∂U1  ∂U2  V2 ∂u  1 ∂U2  2V2  ∂u  ∂u  2   +  −U2 − u − − v + v − W2 − 2γ (w cos ϕ − αv ) − ∗ w , ∂r ∂r r ∂θ r ∂θ r ∂s ∂s (4) ∂v  ∂ V0  V0  V0 ∂v  1 ∂ p + u + u + + ∂t r r ∂θ r ∂θ ∂r  ∂v  ∂ V1  V1 ∂v  1 ∂ V1  U1  V1  ∂ V0  =  −U1 − u − − v − v − u − ∗w ∂r ∂r r ∂θ r ∂θ r r ∂s    ∂v ∂ V ∂v ∂v  V U V ∂ V 1 2 2 2 2 2 +  2 −U2 − u − − v − v − u  − W2 ∂r ∂r r ∂θ r ∂θ r r ∂s  ∂ V1  (5) − ∗ w − 2γ (−w  sin ϕ + αu  ) ∂s   V0 ∂w  ∂ p V1 ∂w  ∂ p ∂w  ∂w  + + =  −U1 − − V0 sin ϕw  − r cos ϕ ∂t r ∂θ ∂s ∂r r ∂θ ∂s     V ∂ W ∂w 1 ∂ W ∂w ∂w 2  2 2  +  2 −U2 − u − − v − W2 − 2γ (v  sin ϕ − u  cos ϕ) ∂r ∂r r ∂θ r ∂θ ∂s  ∂ p (6) + (U1 cos ϕ − V1 sin ϕ − V0 r sin ϕ cos ϕ) w  − r 2 cos2 ϕ ∂s   u 1 ∂v  ∂w  ∂w  ∂u    + + + =  −r cos ϕ + u cos ϕ − v sin ϕ ∂r r r ∂θ ∂s ∂s    2 2 2 ∂w 2   +  −r cos ϕ + r cos ϕu − r sin ϕ cos ϕv , (7) ∂s where s ∗ = s. We expand the disturbance u = [u  , v  , w  , p  ]T in Fourier series in ϕ and s  u =  j exp [i (−ωt + nϕ + ks)] u j,n ,

(8)

j,n

where ω = ω0 + ω1 +  2 ω2 + · · · , k = k0 + k1 +  2 k2 + · · · . Substituting the expansion above we obtain linear ordinary differential equations at each order of . At the leading order we write the equations in the following form L 0 (ω0 , k0 , n)u 0,n = 0,

(9)

where ⎡ ⎢ ⎢ L 0 (ω, k, n) = −iωIv + M(k, n), M(k, n) = ⎢ ⎢ ⎣

in

−2 0

2

in 0

0 d dr

+

1 r

d dr in r

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⎥ ⎥ ⎥, 0 in ik ⎥ ⎦ in ik 0 r

(10)

and Iv = diag(1, 1, 1, 0). In order to have non-trivial solutions or Kelvin waves the operator L 0 should be degenerate; this imposes a dispersion relation between ω0 and k0 . In seeking the curvature instability we set (ω0 , k0 ) to one of the cross points of the dispersion curves of Kelvin waves whose azimuthal wavenumbers are m and m + 1. In the following the amplitude of the two waves are set to A and B, respectively, and we can write u 0,m = Auˆ 0,m and u 0,m+1 = B uˆ 0,m+1 .

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At O() and O( 2 ) we have d A ˆ Iv u 0,m , L 0 (ω0 , k0 , m)u 1,m = B N1−1 (m + 1)uˆ 0,m+1 − dt1 dB ˆ L 0 (ω0 , k0 , m + 1)u 1,m+1 = AN1+1 (m)uˆ 0,m − Iv u 0,m+1 , dt1 dA d A ˆ L 0 (ω0 , k0 , m)u 2,m = N1−1 (m + 1)u 1,m+1 − Iv u 1,m + B N2−1 uˆ 0,m+1 − Iv u 0,m , dt1 dt2 dB dB ˆ L 0 (ω0 , k0 , m + 1)u 2,m+1 = N1+1 (m)u 1,m − Iv u 1,m+1 + AN2+1 uˆ 0,m − Iv u 0,m+1 , dt1 dt2

(11) (12) (13) (14)

where ti =  i t and

    5m d 5 7m 5 5 9r ± (m) = ±i + ∓ + r − (1 − r 2 ) , N1,12 − , 16r 16 8 16 dr 16r 16     5 5m d 21r 7m 1 5 ± ± (m) = ∓i (m) = ± N1,21 − , N1,22 + ∓ − r − (1 − r 2 ) , 16r 16 16r 16 8 16 dr   5m d 1 7m 1 5 ± ± (m) = ± kr, + ∓ − r − (1 − r 2 ) , N1,34 N1,33 (m) = ± 16r 16 2 16 dr 2 i 1 1 ± ± ± (m) = ± , N1,42 (m) = − , N1,43 (m) = ± kr, N1,41 2 2 2 ± (m) = ± N1,11

  5α ± = ∓k − + N2,11 16   5α ± + = ∓k − N2,22 16  5α γ ± = ±i − + N2,31 16 2

     5α  γ 3α 3 ± = ±i −γ + r+ r , N2,13 1 − r2 , 2 16 16     γ 3α 3 7 5 ± r+ r , N2,23 = γ + − + r 2 α, 2 16 16 16      5α γ 5α 9α 2 γ 3α 3α 3 ± ± = = ∓k − + r , N2,32 − − r 2 , N2,33 + r+ r , 16 16 2 16 16 2 16

while the other components of Nl±1 vanish. Here N1±1 is due to the dipole field; it leads to the curvature instability. The torsion and rotation effects on the curvature instability are represented by N2±1 . The terms which involve ddtAi or dB dti are included since degeneracy of L 0 imposes compatibility conditions on the right-hand side of the above equations. It turns out that the amplitudes evolve according to dA = (a1 +  2 a2 )B, dt

dB = (b1 +  2 b2 )A, dt

up to O( 2 ). The coefficients ai and bi are calculated as     uˆ 0,m |N1−1 (m + 1)uˆ 0,m+1 uˆ 0,m+1 |N1+1 (m)uˆ 0,m    , , b1 =  a1 = uˆ 0,m |Iv uˆ 0,m uˆ 0,m+1 |Iv uˆ 0,m+1       uˆ 0,m |N2−1 uˆ 0,m+1 + uˆ 0,m |N1−1 (m + 1)u 1,m+1 − a1 uˆ 0,m |Iv u 1,m   , a2 = uˆ 0,m |Iv uˆ 0,m       uˆ 0,m+1 |N2+1 uˆ 0,m + uˆ 0,m+1 |N1+1 (m)u 1,m − b1 uˆ 0,m+1 |Iv u 1,m+1   , b2 = uˆ 0,m+1 |Iv uˆ 0,m+1 where we defined the inner product by u1 |u2  = Reprinted from the journal

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0.40

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0.10

0.00

0

5

10

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k Fig. 3 Second-order correction of growth rate obtained by normal-mode stability analysis. First principal modes of 0 ≤ m ≤ 60. α = 0, γ = 1

Finally the growth rate is given by σ = σ1 +  2 σ2 ,

σ1 = (a1 b1 )1/2 , σ2 =

σ1 2



a2 b2 + a1 b1

 .

(19)

The first-order growth rate σ1 is equal to that of the curvature instability of the vortex ring confined in a torus. The second-order correction σ2 is due to torsion and rotation effects. In Fig. 3 we show the second-order correction to the growth rate for the first principal modes √ of m = 0, 1, 2, . . . , 60. We set (α, γ ) = (0, 1). As m becomes large, σ2 approaches the asymptotic value 5 15/64 (see Eq. (4)) in the short wave limit shown by the line. This confirms that the short-wavelength stability analysis is consistent with the full normal-mode stability analysis also for the present case. The case α = 0 requires some care since the torsion causes O() shift of dispersion relations; the results will be reported in the near future. 4 Summary The short-wave stability of a helical vortex tube is studied by both the short wavelength and normal-mode stability analyses. It is found that the helical vortex tube is subjected to the curvature instability found for vortex rings. Torsion and rotation of the helical vortex tube appears as second-order correction to the first-order growth rate. It would be expected that any vortex which is curved is subjected curvature instability. Future study is planned to show the existence of curvature instability for more general and wider classes of curved vortices. References 1. 2. 3. 4. 5. 6.

Widnall, S.E.: The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663 (1972) Fukumoto, Y., Hattori, Y.: Curvature instability of a vortex ring. J. Fluid Mech. 526, 77–115 (2005) Hattori, Y., Fukumoto, Y.: Short-wavelength stability analysis of thin vortex rings. Phys. Fluids 15, 3151–3163 (2003) Hattori, Y., Fukumoto, Y.: Short-wavelength stability analysis of a helical vortex tube. Phys. Fluids 21, 014104 (2009) Lifschitz, A., Hameiri, E.: Local stability conditions in fluid dynamics. Phys. Fluids A 3, 2644–2651 (1991) Fukumoto, Y., Okulov, V.L.: The velocity field induced by a helical vortex tube. Phys. Fluids 17, 107101 (2005)

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Theor. Comput. Fluid Dyn. (2010) 24:369–375 DOI 10.1007/s00162-009-0126-5

O R I G I NA L A RT I C L E

Anthony Leonard

On the motion of thin vortex tubes

Received: 6 January 2009 / Accepted: 2 June 2009 / Published online: 9 July 2009 © Springer-Verlag 2009

Abstract The motion of a tube of vorticity with a cross sectional radius that is everywhere small compared to local radius of curvature of the tube is considered. In particular, we determine the inviscid motion of the 3D space curve that traces the centerline of the tube for an arbitrary distribution of axial vorticity within the core. Keywords Vorticity dynamics · 3D vortex tubes PACS 47.32.C1 Introduction We consider the motion of a tube of vorticity with circulation  and a cross sectional radius σ that is everywhere small compared to local radius of curvature of the tube. In particular, we determine the motion of the space curve C that traces the centerline of the tube for an arbitrary distribution of axial vorticity within the core. Consideration of special case of a thin vortex ring of radius R begins with Helmholtz’ analysis [1] and Kelvin’s result [2] for the speed of a ring U R with a uniform distribution of vorticity within the core (see also [3]),    8R 1 UR = log − , (1) 4π R σ 4 with corrections of O((/R)(σ/R)2 log(R/σ )) [4]. Later Hicks [5] analyzed the case of a hollow vortex ring with the result that the factor 1/4 in the above formula is replaced by 1/2. Fraenkel [4] generalized these results to inviscid rings with an arbitrary distribution of vorticity within the core. Soon after, Saffman [6] showed that Fraenkel’s analysis could be simplified considerably by transforming a certain integrand in an expression for the energy into an alternate form. This transformation is derived from the Euler equations and was originally attributed to Lamb [3]. However, Shariff and Leonard [7] noted that the equivalent of this transformation, corrected for an algebraic error, is contained in Helmholtz’ analysis [1] of the speed of a thin-cored vortex ring. As in the case of a vortex ring, the straightforward application of the Biot–Savart law to the centerline of a 3D vortex tube leads to a logarithmic divergence for the velocity as σ → 0. Over the years various ad hoc procedures have been proposed to account for finite-core effects, typically employing a smoothing procedure with a free parameter that is adjusted to produce the asymptotically correct result for the speed of a vortex ring. One also obtains the correct dispersion relation for long bending waves on a rectilinear vortex tube to Communicated by H. Aref A. Leonard Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA E-mail: [email protected] Reprinted from the journal

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the same order if such an adjustment is made. For a discussion see [8]. In another approach, Fukumoto and Miyazaki [9] used matched asymptotic expansions and a cutoff Biot–Savart law to determine the evolution equation for a thin-cored vortex tube with axial flow, recovering earlier results of Moore and Saffman [10] who used a force-balance approach to derive their result. In this paper, we extend Saffman’s analysis [6] of the speed of a vortex ring to determine the equation of motion of the space curve of a thin 3D vortex tube. 2 Vorticity kinematics We take the vorticity field to have the representation [8]  p(|x − r(ξ, t)|/σ ) ∂r ω(x, t) =  dξ, σ3 ∂ξ

(2)

C

where r(ξ, t) defines the space curve C at time t with parameter ξ and p, with the normalization ∞ p(r )r 2 dr = 1,

4π

(3)

0

determines the vorticity distribution over the core of the tube with radius σ . We will consider the case where the tube radius is everywhere much smaller than the local radius of curvature R(ξ, t), i.e., σ  R(ξ, t). We also assume that σ is independent of ξ . This assumption is justified as follows. A more detailed approach would take into account nonuniform, time-dependent σ and would also have to allow for the nonuniform, time-dependent axial flow that would result. See [11,12]. However, the analysis of linear and nonlinear area-varying waves on rectilinear vortex tubes yields waves speeds of O(/σ ) [12] and, thus, a characteristic time of O(σ L/ ) to travel a significant distance along the space curve C, where L is a lengthscale associated with C. On the other hand, the time scale for the source term for these waves is related to the strain rate experienced by length elements along C, which is O(L2 / ). Thus, area variations would be smoothed out on time scales shorter than those of the space-curve dynamics. Essentially the same argument was made by Moore and Saffman [10]. For simplicity we will henceforth suppress the time dependence of r and therefore ω, u, and other related quantities. For an infinite rectilinear tube (2) gives 2 ωo (r ) = 2 σ

∞  r/σ

p(y)ydy y 2 − r 2 /σ 2

,

(4)

where ωo is the distribution of vorticity over the core of a rectilinear vortex tube. Inversion of (4) gives σ2 p(r ) = − π

∞

d dy ωo (yσ )dy



r

y2 − r 2

.

(5)

We are assuming inviscid motion of the tube so that p and ωo are time-independent. However, the vorticity distribution over the core of the tube will depend on the local geometry of the space curve as discussed in Sect. 5. The velocity field may be obtained from the Biot–Savart law,  1 (x − x  ) × ω(x  )dx  u(x) = − . (6) 4π |x − x  |3 Substituting (2) into (6) we find  u(x) = − 4π

 (x − r(ξ )) × ∂ r (ξ ) q(|x − r(ξ )|/σ )dξ ∂ξ C

123

|x − r(ξ )|3 386

,

(7)

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On the motion of thin vortex tubes

where q is given by y p(r )r 2 dr.

q(y) = 4π

(8)

0

Although (7) gives u(x) for all x, including within the core of the tube, it is not appropriate for evaluating the centerline velocity of the tube because (2) is not a sufficiently accurate representation of the core vorticity. The procedure followed in the next section allows us to avoid this difficulty. To compute the fluid energy E in terms of the space curve C and the distribution function p we use the fact that  ˆ 2 1 |ω| E= dk, (9) 2(2π)3 k2 where ωˆ is the Fourier transform of ω,

 ˆ ω(k) =

ei k ·x ω(x)dx.

Using (2) in (10) we find

 ˆ ω(k) =  p(kσ ˆ )

e i k ·r

C

∂r dξ, ∂ξ

where pˆ is the transform of p. Using (11) in (9) we find that the fluid energy is then given by   2 f (|r(ξ ) − r(ξ  )|/σ ) ∂r ∂r E= · dξ dξ  , 8π |r(ξ ) − r(ξ  )| ∂ξ ∂ξ 

(10)

(11)

(12)

C C

where the function f is related to pˆ by 2 f (y) = π

∞

sin(ky) 2 pˆ (k)dk. k

(13)

0

The normalization (3) implies p(0) ˆ = 1 and therefore f (∞) = 1. Thus (12), for σ → 0, although divergent, reduces to the expected expression for the total energy in terms of the vorticity field. 3 Space curve dynamics Alternatively, the energy may also be written as E= or using (2)

  E = C

 u · (x × ω)dx

  ∂r p(|x − r(ξ )|/σ ) dxdξ. u(x) · x × ∂ξ σ3

(14)

(15)

Now we decompose the velocity field in the vicinity of the tube as follows, u(x) = U (ξ ) + u (x, ξ ),

(16)

where U (ξ ) is the velocity of the space curve C, i.e., ∂r = U (ξ ) ∂t Reprinted from the journal

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(17)

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and u is the velocity field relative to U . Thus, using (16) in the energy integral (15), we find that E decomposes as follows, E = IU + I 

(18)

with obvious definitions for IU and I  . The strategy is to determine U (ξ ) such that IU = −I  + E, where now E is given by (12). Note that IU reduces to    ∂r dξ IU =  U (ξ ) · r(ξ ) × ∂ξ

(19)

(20)

C

so that (19) represents an integral equation for U (ξ ). Referring to Fig. 1 and considering the integrand of I  [(14) with u replaced by u ] in the vicinity of r(ξ ), we see that I  may be expressed as    I = dφ ζ (ζ u  ω − ηv  ω)d A. (21) C

A

The part of the integrand involving ζ u  ωd A requires special care because the leading order term in σ/R is O(R/σ ), where R is the local radius of curvature of C, and so must and does vanish. Thus, to find its contribution we would have to determine the O((σ/R)/σ ) correction to u  and the O((σ/R)/σ 2 ) correction to ω. However we can avoid this complication by assuming locally axisymmetric flow, as in a vortex ring, and then using Lamb’s transformation (see [6]) to transform I  to   I  = −3 dφ ζ ηv  ωd A. (22) C

A

It is interesting that Helmholtz [1] employed nearly the same approach in his analysis of the speed of a thincored vortex ring. In that development he used (1) the concept of material derivatives, (2) the fact that the circulation of a differential material ring is constant, (3) the fact that the total impulse is constant, (4) a clever choice for the centroid of vorticity, and (5) the energy of the system given by (14). As a result, his Eq. (9b) (where the factor 5 needs to be replaced by a 6) for the speed of the ring also bypasses the need to deal with the ρ

A

θ

ζ, v’ R (ξ) dφ

η, u’

Fig. 1 Coordinate system corresponding to the integral of Eq.(21). The pair (ρ, θ) are local cylindrical coordinates centered at ξ and (η, ζ ) are local cartesian coordinates with the origin at the center of curvature

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troublesome part of the integrand in (21) In any case, the area integral can now be approximated with sufficient accuracy using η = ρ cos θ, d A = ρdρdθ, ζ = R, ω = ωo (ρ), and ⎡ ρ ⎤  1 v  = ⎣ ωo (ρ  )ρ  dρ  ⎦ cos θ. (23) ρ 0

Using these results we find that 3 2 I =− 8π 

 ds,

(24)

C

where s is the distance parameter along the tube. We anticipate that U (ξ ) will consist of two components, U (ξ ) = U 1 (ξ ) + U 2 (ξ )

(25)

IU1 = −I 

(26)

IU2 = E.

(27)

where we choose U 1 and U 2 to satisfy

and

Considering the result (24), we make the ansatz that U 1 is local. If we substitute for U the quantity U1 = α

∂r ∂ 2r × 2 ∂s ∂s

(28)

into (20) with ξ = s and integrate by parts, we find that  IU1 = α

ds.

(29)

C

To achieve the above result we have used the fact that ∂r/∂s = t, where t is the unit tangent vector. We have also used the Frenet–Serret formulas: ∂t/∂s = κn, ∂n/∂s = −κt + τ b, and ∂b/∂s = −τ n, where κ(s) and τ (s) are the curvature and torsion, respectively, and n and b are the unit normal and binormal, respectively, with b = t × n. Thus, from (24), (26) and (29) we find that α = 3/8π so that (28) yields U 1 (ξ ) =

3 κ(ξ )b(ξ ). 8π

(30)

For U 2 we assume the form  U 2 (ξ ) = − 4π

 (r(ξ ) − r(ξ  )) × ∂ r (ξ  ) g(|r(ξ ) − r(ξ  )|/σ )dξ  ∂ξ  C

|r(ξ ) − r(ξ  )|3

,

where the function g is to be determined, and substitute the above for u into (15) to find    2 ∂r ∂r g(|r(ξ ) − r(ξ  )|/σ ) IU2 = − · (r(ξ ) − r(ξ  )) · r(ξ ) 4π |r(ξ ) − r(ξ  )|3 ∂ξ ∂ξ  C C  ∂r ∂r −(r(ξ ) − r(ξ  )) · r(ξ ) ·  dξ dξ  . ∂ξ ∂ξ Reprinted from the journal

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(31)

(32)

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To put IU2 into its final form we (1) exchange variables ξ ↔ ξ  in the first group of terms and combine the result with the original expression and (2) integrate by parts in ξ the second group of terms to obtain ⎡ ⎤   ∞ 2  g(y)dy ⎥ ∂r ∂r  2 ⎢ g(|r(ξ ) − r(ξ )|/σ ) IU2 = − dξ dξ  . (33) − · ⎣ ⎦ 8π |r(ξ ) − r(ξ  )| σ y2 ∂ξ ∂ξ   C C |r (ξ )−r (ξ )|/σ Thus, referring to (12) we see that (27) is satisfied if ∞ g(y)dy g(x) f (x) =− +2 . (34) x x y2 x

Solving (34) for g we obtain 2 g(x) = x

x f (y)dy − f (x).

(35)

0

Collecting our results for U (ξ ), namely (25), (30) and (31), we find that the velocity of the space curve r(ξ, t) is given by  (r(ξ ) − r(ξ  )) × ∂ r (ξ  ) g(|r(ξ ) − r(ξ  )|/σ )dξ  ∂r 3  ∂ξ  , (36) = U (ξ ) = κ(ξ )b(ξ ) − ∂t 8π 4π |r(ξ ) − r(ξ  )|3 C

where the function g ultimately depends on ωo (r ), the distribution of vorticity over the core, through the relations (5), (13), and (35). 4 Application to a constant–vorticity core For a constant-vorticity core ωo (r ) = /(πσ 2 ) for r < σ and = 0 otherwise. The function p is then computed √ as p(r ) = 1/(π 2 1 − r 2 ) for r < 1 and = 0 otherwise, with a Fourier transform given by p(k) ˆ = 2J1 (k)/k. Combining (13) and (35) with the above result for pˆ we find that  2 ∞  J (k)dk 8 (1 − cos(kx)) g(x) = (37) − sin(kx) 1 3 . 2 π kx k 0

Application to a vortex ring of radius R yields the speed of the ring given by (1) plus higher order terms. We can also apply the results to investigate the evolution of small perturbations to a rectilinear vortex tube. For disturbances proportional to exp(it + imθ + ikx), where m is the azimuthal mode in cylindrical coordinates and k is the axial wave number, we find that for m = ±1,   ˆ  d G 3 ˆ ˆ =± + G(kσ )−k (kσ )2 − G(0) (kσ ) (38) 4πσ 2 2 dk with ˆ G(k) =2

∞

cos(kx)g(x)dx x3

0

4 = 3

  2 ∞   3 J 2 (k  )dk  k k 2  −3  +1 1  . k k k

(39)

k

For long waves, kσ 0, we have co-rotating case, if q < 0, the counter-rotating case takes place. Equation 3 is a Hamiltonian system ∂ψ ∂H , =i ∂t ∂ ψ¯

∞   |ψ|2  2 2 H= 2( 1 + |ψ | − 1 + c ) + q log dz. |ψ0 |2

(5)

−∞

→ |ψ0 |2 , |ψ  |2 → c2 . We assume that at z → Hamiltonian H is a constant of motion. Other constants of motion are the following: ∞, |ψ|2

∞ N=

∞ |ψ| dz, 2

P =i

−∞

¯  − ψ ψ¯  )dz. (ψψ

(6)

−∞

In the degenerate case q = 0, Eq. 5 is completely integrable. It is just another version of the Landau–Lifshitz (or local induction) equation. By the Hashimoto transformation [9], see also [10], it can be transformed to the focusing NLSE. The Lax pair for Eq. 3 exists if q = 0; it is presented in Appendix. After separation of amplitude and phase ψ = Aei , n = A2 , v = z  R = 1 + |ψ  |2 = 1 + A2z + A2 v 2 Equation 3 takes form ∂ nv ∂n +2 = 0, ∂t ∂z R



∂ v 2 A + ∂t R

 =

∂ Az q + ∂z R A

(7)

Equation 7 are Hamiltonian ∂n ∂H = , ∂t ∂ In the long wave semiclassical limit Az  v, A

123

∂ ∂H =− . ∂t ∂n

R→ 394

(8)

 1 + A2 v 2 Reprinted from the journal

Dynamics of vortex line in presence of stationary vortex

the second of Eq. 7 simplifies up to the form ∂ v 2 q + = . ∂t A n

(9)

Now (7), (9) is a system of hydrodynamic type. In the NLSE limit R = 1 and these equations turn to the gas dynamic equations with an exotic dependance of pressure on density: P = q log n.

2 Helix solutions and their stability Equation 3 has an exact helix solution ψ = ψ0 = Aeikz−i t , A z = 0,

v = k,

R=

 1 + A2 k 2 ,

k2 q = √ − 2. 2 2 A 1+ A k

(10)

It is a rotating helix. If k 2 A2 , q 0). In the opposite case the helix rotates in negative direction ( < 0). In the marginal case q=√

k 2 A2 1 + k 2 A2

< 1,

the helix is stationary. If q = 0, the Hashimoto transformation converts the rotating helix to a stationary monochromatic wave that is an exact solution of NLSE. Studying the rotating helix stability, let us assume ψ = ψ0 (1 + δψei pz−iωt ),

|δψ|  1

and linearize the equations. After solving the linearized equations, we end up with formula ω2p

 p4 2q k 4 A2 2 = +p − + . √ (1 + k 2 A2 )2 (1 + k 2 A2 )2 A2 1 + k 2 A2

(11)

If q ≤ 0, the helix is unstable. In the case q = 0, the helix is unstable if p 2 < k 4 A2 . Now ω2p =

2  p2 p − k 4 A2 . 2 2 2 (1 + k A )

This is just modulational instability of monochromatic wave in the focusing NLSE. For q > 0, the co-rotating helix is stable if A2 < x/k 2 , where x is the solution of equation x2 3

(1 + x 2 ) 2

= 2q.

In this case helixes, which are close to the steady vortex, are stable, while the “remote” helixes are unstable. If k = 0 and q < 0, the instability of helix is a generalization of Crow instability for two antiparallel vortices. Reprinted from the journal

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3 Self-similar collapse of counter-rotating vortices One can look for self-similar solutions of Eq. 3 1

ψ = (t0 − t) 2 +iξ f



z √ t0 − t

 ,

(12)

where ξ is some unknown real constant. Here, we implicitly state that in this system the Leray scaling takes 1 place, i.e., the domain of vortices interaction shrinks proportionally to (t0 − t) 2 , where t0 is the time of singularity formation. Let us denote self-similar variable η = √t z − t . We obtain the following equation for the self-similar 0 solution:   1 q f ∂  . (13) − iξ f − ( f − η f ) = i + 2 ∂η 1 + | f  |2 f¯ Here, ξ is an eigenvalue of the nonlinear boundary problem with f  (0) = 0,

f (η) → η1+2iξ

at

η → ∞.

Equation 13 has reasonable solutions in the counter-rotating case q < 0. The eigenvalue ξ is a function on q. This is a subject of determination from the numerical experiment. We applied periodic boundary conditions ψ(0, t) = ψ(2π, t) and used the Strang splitting algorithm to solve Eq. 3 numerically. We took 2 k = 0, ψ(z, 0) = 1.25 − 0.05e−(z−π ) cos(z − π). With this choice of parameters, ω p = p 4 + 2q p 2 ,

p = 1, 2, . . .

we studied development of instability for q < −1/2. The case q = −1/2 is a marginal one, however, the instability develops even in this case. On Fig. 1 are presented different shapes of instability development for q = −1. The instability ends up with merging of the vortices at the moment of time t = T = 1.4721. 4 Solitonic solutions Solitons are the following solutions of Eq. 3: ψ = ψ(z − ct)eiλt .

(14)

Fig. 1 Development of instability at q = −1

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Here c and λ are constants. One can use hydrodynamic version of Eq. 3. Then Eq. 7 can be integrated nv n 0 v0 = Q = −cn 0 + 2 − cn + 2 R R0 n → n 0 , v → v0 , R → R0 as |z| → ∞. One can find v and R (1 + A 2 )(Q + cn)2  v =  n 4n − (Q + cn)2

 2 A 1 + A 2

R=

2

4n − (Q + cn)2

1 .

(15)

(16)

2

and we end up with a pretty complicated second order nonlinear equation for A(z):  1 + A 2 q Q 2 − c2 A4 ∂ A + − λA =  1 . ∂z R A 2 A2 4 A2 − (Q + c A2 )2 2

(17)

This equation can be integrated as follows 1

 1 + A 2

  λ 2 = q log A − A + E . 1 2 (4 A2 − (Q + c A2 )2 ) 2 2A

(18)

Here, E is a constant of integration. If the soliton is steady (c = 0) and asymptotic is not a helix but a straight vortex (Q = 0), we get A λ 1  = q log − (A2 − A0 2 ) + 1. A0 2 1 + A 2 “Free” soliton (q = 0) is given by equation



2

A =

2

1 1 − λ2 A2

−1

A = 0, if A = 0 or A = ∞, if A = λ2 . In this case A0 = 0. The total, very rich family of solutions depends on three parameters: c, Q, and λ. Detailed description of solitons and their stability will be published separately. Acknowledgments The author expresses acknowledgement to A. Dyachenko and A. Isanin for performing the numerical experiment. This study was supported by NSF grant DMS 0404577.

Appendix Equation 13 at q = 0 is a compatibility condition for the following overdeterminated linear system of equations imposed on a complex 2 × 2 matrix function  x = λ A ,   A  t = 2 λ2 + λ B  R Here, λ is a spectral parameter



i   − −i  0 v B= −v 0



A=

R= v=−

 1 + |  |2

 i ∂  2 ∂z 1 + |  |2

(19) (20)

(21) (22)

Equations 19 and 20 form the "Lax pair" for (13). Equation 13 is the first non-trivial term in the infinite integrable hierarchy, generated by Eq. 19. This equation is the Gauge equivalent to the Nonlinear Schrodinger equation. Reprinted from the journal

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References 1. Crow, S.C.: Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179 (1970) 2. Zakharov, V.E.: Wave collapse. Phys.-Uspekhi 155, 529–533 (1988) 3. Zakharov, V.E.: Quasi-two-dimensional hydrodynamics and interaction of vortex tubes. In: Passot, T., Sulem, P.-L. (eds.) Lecture Notes in Physics, Vol. 536, pp. 369. Springer, Berlin (1999) 4. Klein, R., Maida, A., Damodaran, K.: Simplified analysis of nearly parallel vortex filaments. J. Fluid Mech. 288, 201–248 (1995) 5. Lions, P.L., Maida, A.J.: Equilibrium statistical theory for nealy parallel vortex filaments. Comm. Pure Appl. Math. 53(1), 76–142 (2000) 6. Maida, A.J., Bertozzi, A.L.: Vorticity and incompressible flow. Cambridge University Press, Cambridge, MA (2002) 7. Kerr, R.M.: Evidence for a singularity of the three-dimensional incompressible Euler equation. Phys. Fluids A Fluid Dyn. 5, 1725 (1993) 8. How, T.Y., Li, R.: Blowup or no blowup? The interplay between theory and numerics. Physica D 237, 1937–1944 (2008) 9. Hashimoto, H.: A solution on vortex filaments. Fluid Dyn. Res. 3, 1–12 (1972) 10. Zakharov, V.E., Takhtajan, L.A.: Equivalence of a nonlinear Schrodinger equation and a Geizenberg ferromagnet equation. Theor. Math. Phys. 38(1), 26–35 (1979)

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Theor. Comput. Fluid Dyn. (2010) 24:383–387 DOI 10.1007/s00162-009-0160-3

O R I G I NA L A RT I C L E

Makoto Umeki

A locally induced homoclinic motion of a vortex filament

Received: 3 January 2009 / Accepted: 9 June 2009 / Published online: 2 December 2009 © Springer-Verlag 2009

Abstract An exact homoclinic solution of the Da Rios–Betchov equation is derived using the Hirota bilinear equation. This solution describes unsteady motions of a linearly unstable helical or wound closed filament under the localized induction approximation. Keywords Vortex filament · Localized induction approximation · Homoclinic motion PACS 47.32.cb · 02.30.Ik

1 Introduction It has been remarked that the focusing nonlinear Schrödinger (NLS) equation has a spatially periodic and temporally localized solutions, i.e. homoclinic solutions [1,2]. The solutions are homoclinic to a linearly unstable plane wave. Owing to the equivalence of the NLS and Da Rios–Betchov equations, we expect homoclinic solutions of the latter equation. In [2], the solutions come from the breather solutions of the defocusing NLS equation with the spatial symmetry. The homoclinic solution with the symmetry corresponds only to the nply wound circular filament. Therefore, the direct application of the NLS solutions by [2] gives somewhat unphysical situations at least in the context of vortex filament. In this article, the solutions are extended to those homoclinic to the plane wave with the spatial dependence, which corresponds to a helical filament due to the localized induction theory. The prescription is as follows: (1) find a linearly unstable homoclinic-point solution, (2) factor out the homoclinic-point solution (a plane wave in the following case), and derive a set of bilinear equations for the remaining denoted by the fraction g/ f , and (3) solve the bilinear equations with pure imaginary wave numbers for homoclinic solutions. The Da Rios–Betchov equation appears in the localized induction approximation (LIA) of the three-dimensional motion of the thin vortex filament. A historical survey is given in [3]. The Da Rios–Betchov equation is equivalent to the focusing NLS equation through the Hasimoto transform [4]. Its bilinear equation is given in [5]. The N -soliton solutions are given explicitly in [6]. The above prescription to seek for homoclinic solutions is based on the idea given in [7]. Communicated by H. Aref M. Umeki (B) Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan E-mail: [email protected] Reprinted from the journal

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2 Bilinear equations for the homoclinic solutions of the locally induced vortex filament The Da Rios–Betchov equation is given by ∂ x = x˙ = κb = κt × n, ∂t

(1)

where x(t, s) denotes the position of the filament, t the rescaled time, s the arclength, ( t, n, b) the orthonormal triad of the tangential, normal and binormal vectors, and κ the curvature. Differentiating (1) with respect to s yields ˙t = ( t × t s )s .

(2)

Letting ta = t1 + it2 [t = (t1 , t2 , t3 )], Eq. 2 can be written as t˙a = i(Ds ta · t3 )s , i t˙3 = (Ds ta∗ · ta )s , 2 where Ds is the Hirota bilinear operator,



Ds f (s) · g(s) =

∂ ∂ −  ∂s ∂s



(3) (4)

f (s)g(s  )s  =s .

(5)

We introduce a complex-valued function v as ta =

2v 1 − |v|2 , t3 = , N N

(6)

where N = 1 + |v|2 . Substituting (6) into (3)–(4) and some calculations lead to an evolution equation for v as ivt + vss −

2 v ∗ vs2 = 0. 1 + |v|2

(7)

The bilinear form for soliton solutions was obtained by [5], where Eq. 7 was implicit. It should be noted that Eq. 7 is invariant under the transformation v → 1/v. Equation 7 has a plane-wave solution: 



v = v0 = aei(k s−ω t) ,

(8)

where the dispersion relation is ω =

1 − a 2 2 k . 1 + a2

(9)

The plane wave corresponds to a circle if a = 1, otherwise a helix with a pitch (1 − a 2 )/2a. Letting v = v0 v1 , we obtain an equation for v1 as (1 + a 2 |v1 |2 )(iv1t + v1ss ) + 2ik  (1 − a 2 |v1 |2 )v1s +

2a 2 k 2 2 (−1 + |v1 |2 )v1 − 2a 2 v1∗ v1s = 0. 1 + a2

(10)

Letting v1 = g/ f and using the formula used in [5], a final equation for the homoclinic solutions of the Da Rios–Betchov equation is obtained as f gG 1 − f 2 G 2 + a 2 g 2 G ∗2 + 2G 3 Ds g · f = 0,

(11)

which can be decomposed into three bilinear equations:

123

G 1 = F1 ( f ∗ · f − a 2 g ∗ · g) − c(| f |2 − |g|2 ) = 0, G 2 = F1 f ∗ · g = 0,

(12) (13)

G 3 = F2 ( f ∗ · f + a 2 g ∗ · g) + id(| f |2 − |g|2 ) = 0,

(14)

400

Reprinted from the journal

A locally induced homoclinic motion of a vortex filament

where F1 = i(Dt + bDs ) − Ds2 , F2 = Ds , 2(1 − a 2 )  k, b= 1 + a2 2a 2 k 2 c = dk  = . 1 + a2

(15) (16) (17) (18)

In the case k  = 0 and a = 1, Eqs. 12–14 reduce to the bilinear equations of soliton solutions [5]. 3 One homoclinic solution The one homoclinic solution is given by f = 1 + a1 eη1 + a2 eη2 + a12 eη1 +η2 , g = 1 + b1 eη1 + b2 eη2 + b12 eη1 +η2 ,

(19) (20)

and for ( j, j ∗ ) = (1, 2), (0)

ηj = kjs + jt + ηj , 2iψ j , a j = a ∗−1 j∗ = e ∗−1 2iφ j , b j = b j∗ = e ∗ ψ j ∗ = ψ j , (modπ) φ j ∗ = φ ∗j , (modπ)

sin 2ψ j + a 2 sin 2φ j − 2ia sin(ψ j − φ j ) = 0, id kj = (ψ j + φ j : real) sin(ψ j + φ j ), a c sin 2(ψ j + φ j ),  j + bk j = − 1 + a2 a j j ∗ = a j a j ∗ e2θ j j ∗ , e

b j j∗ 2θ j j ∗

= b j b j∗ e

2θ j j ∗

,

= sec (ψ j + φ j ). 2

(21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31)

The above solution can be rewritten as v1 (s, t) =

e−4ψ1i cosh ξb (1 + e−θ12 sech ξb cos ζb ) , cosh ξa (1 + e−θ12 sech ξa cos ζa )

(0) ξa(b) = 1r t + η1r − 2φ(ψ)1i + θ12 ,

ζa(b) = K 1 (s

(0) − bt) + η1i

+ 2φ(ψ)1r ,

(32) (33) (34)

where k1 = i K 1 , K 1 real, and the subscript r (i) denotes the real (imaginary) part. From (27), we have the long-wave instability K /k  < 2a/(1 + a 2 ). When a = 1, the filament closes if the total arclength is 2nπ/k  and n K 1 = mk  for integers n, m. It implies a closed curve approaching an n-ply wound circle. For a  = 1, this solution gives unsteady motions of the filament approaching a helix as t → ±∞. We note that there is a temporal phase shift of the plane-wave solution due to the ξa and ξb terms in (33). An example of the one homoclinic solution is shown for the following parameters. √ (35) a = 1/ 2, k  = 1, ψ1 = (1/2) tan−1 [(1 − 3i)/4], ∗ ∗ (36) φ1 = π/4 − ψ1 , ψ2 = ψ1 + π, φ2 = φ1 . Reprinted from the journal

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t

4

6

t

3 5

4

t

3

1

4

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z 2

z 3

z 2

1

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0

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Fig. 1 The temporal evolution of the one homoclinic helical vortex filament with parameters given by (35) and (36)

Since we have an explicit expression of the tangential vector t of the filament, a numerical integration of t with respect to s at each time is done to illustrate temporal changes of its shape. Figure 1 shows the projection of the three-dimensional shape of the filament at t = ±6, ±3, and ± 1, respectively. Similarly, we can construct two and N -homoclinic solutions of the Da Rios–Betchov equation. The generality of homoclinic solutions is still less obvious than solitons (e.g. the criticism on [7] by [8]) and it will be worth seeking for them in systems with the linear instability. Finally, a comment may be useful on a relation between the presented homoclinic solutions and the torus-knot solutions considered by [9,10] and [11,12]. It is noted that the n-ply wound solution does not imply the torus solution. The presented solution is expressed by a rational form of exponential and trigonometric functions, related to an unknotted filament, homoclinic to the simple wound circle or helical vortex, and changes its form as time passes. Kida’s solutions are given by elliptic functions, of travelling-wave type, periodic in space and time, and move without change of form. As the focusing NLS has both solitons, periodic travelling-wave solutions and homoclinic solutions, the presented solution shows intrinsically new features and locates itself outside the class of Hasimoto soliton and Kida torus solution. References 1. Akhmedieva, N.N., Eleonskii, V.M., Kulagin, N.E.: Generation of periodic trains of picosecond pulses in an optical fiber: exact results. Sov. Phys. JETP 62, 894–899 (1985) 2. Ablowitz, M.J., Herbst, B.M.: On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation. SIAM J. Appl. Math. 50, 339–351 (1990) 3. Ricca, R.L.: The contributioons of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics. Fluid Dyn. Res 18, 245–268 (1996) 4. Hasimoto, H.: A soliton on a vortex filament. J. Fluid Mech. 51, 477–485 (1972)

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5. 6. 7. 8. 9. 10. 11. 12.

Hirota, R.: Bilinearlization of soliton equations. J. Phys Soc. Jpn. 51, 323–331 (1982) Fukumoto, Y., Miyazaki, T.: N-solitons on a curved vortex filament. J. Phys. Soc. Jpn. 55, 4152–4155 (1986) Umeki, M.: Complexification of wave numbers in solitons. Phys. Lett. A 236, 69–72 (1997) Stahlhofen, A.A., Druxes, H.: Comment on Complexification of wave numbers in solitons by M. Umeki. Phys. Lett. A 253, 247–248 (1999) Kida, S.: A vortex filament moving without change of form. J. Fluid Mech. 112, 397–409 (1981) Ricca, R.L.: Torus knots and polynomial invariants for a class of soliton equations. Chaos 3, 83–91 (1993) Ricca, R.L.: Erratum: Torus knots and polynomial invariants for a class of soliton equations. Chaos 3, 83–91 (1993) Ricca, R.L.: Erratum: Torus knots and polynomial invariants for a class of soliton equations. Chaos 5, 346 (1995)

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Theor. Comput. Fluid Dyn. (2010) 24:389–394 DOI 10.1007/s00162-009-0175-9

O R I G I NA L A RT I C L E

Yoshifumi Kimura

Self-similar collapse of 2D and 3D vortex filament models

Received: 10 February 2009 / Accepted: 13 August 2009 / Published online: 25 November 2009 © Springer-Verlag 2009

Abstract In this article, a very simple toy model for a candidate blow-up solution of the Euler equation by Boratav and Pelz (vortex dodecapole) is investigated. In this model, vortex tubes are replaced with straight vortex filaments of infinitesimal thickness, and the entire motion is monitored by tracing the motion of a representative point on one vortex filament. It is demonstrated that this model permits a self-similar collapse solution which provides the time dependence of the length scale as (tc − t)1/2 , (t < tc ), where the collapse time tc depends on the initial configuration. From the conservation of circulation, this time dependence implies that vorticity ω scales as (tc − t)−1 , which agrees with the one observed in the direct numerical (pseudo spectral) simulations of the vortex dodecapole. Finally, possible modification of the model is considered. Keywords Finite-time singularity · Euler’s equation · Vortex filaments · Self-similar solution PACS 47.10.-g · 47.32.C 1 Introduction The problem of search for singularities for Euler’s equation (or the inviscid limit of the Navier–Stokes equations) has been one of the most challenging topics in mathematical fluid mechanics and fascinating a number of researchers for a long time. For a recent review, analyzing the problem and summarizing results from various viewpoints, the reader is referred to [1]. As potential blow-up solutions of the Euler or the Navier–Stokes equations, a number of candidates have been proposed [2–4] and numerical investigation of these solutions has been continued with modified numerical methods [5,6]. In this article, a very simple vortex filament model for one of the candidate blow-up solutions by Boratav and Pelz [3] is discussed and modified. As we will see, this model, even though it is based on a contradictory assumption that the filaments remain straight, provides a self-similar development for the length scale of the system, and thus can predict the same time scaling of vorticity with the full pseudo-spectral simulations. Boratav and Pelz’s study was aiming at finding a singularity for a model flow proposed by Kida and Murakami [7]. The initial condition of Kida and Murakami’s flow has the following form, ⎧ ⎨ u(x, 0) = sin x(cos 3y sin z − cos y sin 3z) v(x, 0) = sin y(cos 3z sin x − cos z sin 3x) , (1) ⎩ w(x, 0) = sin z(cos 3x sin y − cos x sin 3y) which was an extension of the Taylor–Green vortex flow with higher symmetry. Actually, this initial condition has the maximum symmetry that a solution of the Navier–Stokes equation can possess [8]. Making use of the Communicated by H. Aref Y. Kimura (B) Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan E-mail: [email protected] Reprinted from the journal

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representative point P (x , y , 0)

Y

X

x-quadrupole

Z

y-quadrupole z-quadrupole

Fig. 1 The filament dodecapole consists of three quadrupoles, x, y and z. The representative point P(x 0 , y0 , 0) is located on one vortex filament of the z-quadrupole. Because of the symmetry, the motion of P is restricted on the x y-plane, and the motion is described by Eqs. 4 and 5

maximum symmetry, Kida and Murakami integrated the Navier–Stokes equation at the high Reynolds number pseudo-spectrally and observed the Kolmogorov spectrum with accurate evaluation of the Kolmogorov constant. Boratav and Pelz [3,9,10] investigated this flow searching for singularity and found a phenomenon that six vortex dipoles (dodecapole) collapse toward the origin of the coordinates in a self-similar fashion. They reported that near the collapse, vorticity scales (tc − t)−1 for the flow. Later, Pelz [11,12] himself verified this scaling by investigating the motion of the vortex quadrupoles with the vortex method. A simple toy model for the vortex collapse of dodecapole has been proposed [13], and in this article, discussion is made for a possible modification of the model. In this model, all the vortex tubes that compose the dodecapole are replaced with straight vortex filaments of infinitesimal thickness, and the motion of the system is described by tracking the motion of a representative point on one vortex filament. In spite of the simpleness, this model succeeds in delineating the self-similar development of the system and provides some insight in understanding the singular behavior. In the subsequent sections, the equations of the model are presented in Sect. 2 and the self-similar solutions presented in Sect. 3. Finally, discussion and possible modification are presented in Sect. 4.

2 Model equation The filament dodecapole, shown in Fig. 1, has three orthogonal vortex quadrupoles parallel to the x, y and z axes, which we name x-,y- and z-quadrupole, respectively. We locate a representative point at the intersection of one of the z-quadrupole with the plane of z = 0 in the first quadrant of the x y-plane and call it P = (x0 , y0 , 0), (x0 > 0, y0 > 0). First, we write down the velocity at (x, y) in the x y-plane (z = 0) induced by z-quadrupole with filaments parallel to the z axis and go through (x0 , y0 ), (−x0 , y0 ), (x0 , −y0 ), (−x0 , −y0 ). Letting the strength  = ±2π, the horizontal and vertical velocity components are y − y0 (x − x0 )2 + (y − y0 )2 y + y0 + (x − x0 )2 + (y + y0 )2 x − x0 v(x, y, 0) = + (x − x0 )2 + (y − y0 )2 x − x0 − (x − x0 )2 + (y + y0 )2

u(x, y, 0) = −

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406

y − y0 (x + x0 )2 + (y − y0 )2 y + y0 − , (x + x0 )2 + (y + y0 )2 x + x0 − (x + x0 )2 + (y − y0 )2 x + x0 + . (x + x0 )2 + (y + y0 )2 +

(2)

(3)

Reprinted from the journal

Self-similar collapse of 2D and 3D vortex filament models

Using above equations together with the permutation symmetry u(x, y, z) = v(z, x, y) = w(y, z, x) by Kida [8], we obtain the following expression for the x and y components of the velocity at P, which can be defined as x˙0 and y˙0 , respectively, as 1 y0 2x0 2x0 u = x˙0 = + − − + , 2 2 2 2 2y0 2(x0 + y0 ) (y0 − x0 ) + x0 (y0 + x0 )2 + x02      from z -quadrupole from y -quadrupole 1 x0 2y0 2y0 + + − . v = y˙0 = − 2 2 2 2 2x0 2(x + y0 ) (y0 − x0 ) + y0 (y0 + x0 )2 + y02   0    from z -quadrupole from x -quadrupole

(4)

(5)

The above equations provide an autonomous dynamical system for the motion of x0 (t) and y0 (t), and hereafter we shall concentrate on these equations. Strictly speaking, the filaments cannot remain straight while moving, and thus the expression for the induced velocity fields is exact only initially and approximate for later times. Nevertheless, the above equations can well capture the self-similar development and the scaling law that the aforementioned numerical simulations presented.

3 Self-similar solution In order to seek the similarity solutions for (4) and (5), we assume that all the variables develop with the same time dependence, f (t), and substitute x0 (t) = f (t)ξ,

y0 (t) = f (t)η.

(6)

into Eqs. 4 and 5 [14]. Then, we obtain η 2ξ 2ξ 1 − − + , f f˙ ξ = 2 2 2 2 2η 2(ξ + η ) (η − ξ ) + ξ (η + ξ )2 + ξ 2 1 2η ξ 2η f f˙ η = − + − . + 2 2 2 2 2ξ 2(ξ + η ) (η − ξ ) + η (η + ξ )2 + η2

(7) (8)

Next, we separate variables into the time and space parts by setting f f˙ = c, where c is a real constant determined by ξ and η which satisfy

1 1 η 2ξ 2ξ − − + ξ 2η 2(ξ 2 + η2 ) (η − ξ )2 + ξ 2 (η + ξ )2 + ξ 2

2η 1 ξ 2η 1 − − + + . = η 2ξ 2(ξ 2 + η2 ) (η − ξ )2 + η2 (η + ξ )2 + η2

(9)

(10)

Setting r = η/ξ , (10) reduces to 4r 8 − 80r 6 + 17r 4 − 80r 2 + 4 = 0 which has the solutions

⎧ ⎪ 2 2 ⎪ ⎨ r1,2 = s+ ± s+ − 1, ⎪ ⎪ ⎩ r 2 = s− ± s 2 − 1, − 3,4

where s± = 5 ±

(11)

(12)

√ 391/4. The real positive solutions are r = 4.453812 . . . and its inverse.

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1

0.8

y0(t)

0.6

0.4

0.2

θ 0 0

0.2

0.4

0.6

0.8

1

x0(t) Fig. 2 Trajectories of (4) and (5) with different initial conditions (marked by ×). Each of them approaches the origin with the line of y = r x (r = 4.453812 . . .) as an asymptotes. Using this value, the asymptotic angle θ between this line and the x 0 -axis can be calculated as θ ≈ 77.342677◦ .

The time dependence can be found by solving (9) with an initial condition f (0) = 1, and we find f =

√ 2ct + 1 = 2c(t − tc ),

(13)

1 where tc = − 2c . According to the sign of c the system either contracts (if c < 0) or expands (if c > 0), respectively. Using r and ξ the constant c has the following form

c=

4 − 16r 2 − 15r 4 . 2ξ 2 r (r 2 + 1)(r 4 + 4)

(14)

From the analysis above, we can see that the trajectory of (x0 , y0 ) is a straight line of slope r which crosses the origin at t = tc , where the critical time tc is a function of ξ , the x component of the initial position and r . The length scale δ develops as (tc − t)1/2 , and according to the conservation of circulation  = ωδ 2 , the scaling of vorticity, ω ∼ (tc − t)−1 , is obtained. Figure 2 is a plot of trajectories of (4) and (5) with different initial conditions marked by ×. Each trajectory approaches the origin with the line of the slope of r = 4.453812 . . . as an asymptote. We have verified that the calculated collapse time from (14) provides a satisfactory estimate once the trajectory is close to the straight line. From the value of r , the angle θ between the x0 axis and the asymptotic trajectory can be calculated as tan−1 r ≈ 77.342677◦ . It is interesting if this angle is verified in numerical simulations.

4 Discussion for modification In this section, we present some ideas for modification of the model. We have already mentioned that the biggest assumption (or drawback) of the present model is that the vortex filaments remain straight, which cannot be validated if we look at the whole domain of computation. Still, the model provides the same scaling law with the simulation, and we expect that it captures some essential mechanism of development. Therefore, the main problem is how to amend the drawback that vortex filaments remain straight. For this point, we propose the following two ideas of modification.

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Fig. 3 Trial solutions for outer stationary solutions

4.1 Search for outer solutions The first idea is that we regard Eqs. 4, 5 and the self-similar solution as the inner equations and their solutions of unknown global equations and global solutions. If we can find suitable outer equations and solutions and by connecting the inner and outer solutions smoothly, we can perhaps construct a global solution. If we assume that the vortex tubes form “quasi” stationary state in the outer field, it is expected that the magnitude of vorticity scales |ω(r )| ∼ r −2 from the conservation of circulation. Then, the problem would be equivalent to finding a (stable) stationary configuration of point vortices (12 plus and 12 minus) on a sphere. Finding such solutions, in general, may be quite cumbersome, and thus it should be natural to seek symmetric solutions. Here, we propose two types of trial solutions: the first type is one that has three vortices (with the same sign) in an equilateral triangular shape on each face of the octahedron. Two parameters, r and θ , are introduced to determine the size and the rotation angle of the triangle (Fig. 3(a)), The second type has two concentric circles on the northern and the southern hemispheres, and vortices are distributed on the circles symmetrically as Fig. 3(b) shows. For this type, two parameters, r1 and r2 , are introduced to determine the radius of circles. The program to obtain the candidates for the stationary configuration is as follows: (1) Map the positions of vortices on the sphere.  (2) Calculate the values of the Hamiltonian, H = iN= j i  j ln(1 − cos ρi j ), where i = ±1 and ρi j is the spherical distance between the ith and the jth vortices [15,16] (3) Draw contour plot of H as a function of the two parameters corresponding to the two types. (4) If there is an elliptic point in the contour plot, it is a candidate for a stationary configuration. 4.2 Localized model The second idea of modification is searching for localized solutions instead of global ones. The enstrophy plots from the Navier–Stokes simulations by Boratav and Pelz showed that the vortex dipoles have finite length and

l(t)

Fig. 4 Localized model: A vortex tube is decomposed into a core part near the origin and fluffy parts around it. The model is equivalent to introducing a compact support on vorticity field. We assume that the support has an effective length scale l(t). Reprinted from the journal

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they collapse to the origin self-similarly, and this observation supports the localized model. The central idea of the model is to decompose a vortex tube into the core part near the origin and fluffy parts around it. Mathematically it is equivalent to introduce a compact support for vorticity, and let us assume the support has an effective length scale l(t) (Fig. 4). From the similarity picture, we expect that l(t) = l0 f (t), where f (t) is given by (13). By assuming that the vortex filaments remain straight inside the core part around the origin, the Bio-Savart integral with the integration limit given by ±l(t) can be evaluated analytically. If we use the fact that the velocity at r induced by a vortex stick with a finite length has a factor (cos α + cos β)/2, where 0 < α, β < π/2 are angles between the line from r to the endpoints of the vortex stick and the direction vector of the vortex stick [17]. Even with this self-similar finite length modification, we expect that the singular solution persists (with a different collapse time). We would like to present the analysis elsewhere. Acknowledgements This research was partially supported by the Grant-in-Aid for Scientific Research (B) No. 18340025 by Japan Society for the Promotion of Science and by the Grant-in-Aid for Nagoya University Global COE Program, “Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos”, from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

References 1. Gibbon, J.D.: The three-dimensional Euler equations: Where do we stand?. Physica D. 237, 1894–1904 (2008) 2. Kerr, R.M.: Evidence for a singularity of the three-dimensional incompressible Euler equations. Phys. Fluids A. 5, 1725– 1746 (1983) 3. Boratav, O.N., Pelz, R.B.: Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. Phys. Fluids 6, 2757–2784 (1994) 4. Grauer, R., Marliani, C., Germaschewski, K.: Adaptive mesh refinement for singular solutions of the incompressible Euler equations. Phys. Rev. Lett. 80, 4177–4180 (1998) 5. Hou, T.Y., Li, R.: Dynamic depletion of vortex stretching and non-blowup of the 3-d incompressible euler equations. J. Nonlinear Sci. 16, 639–664 (2006) 6. Grafke, T., Homann, H., Dreher, J., Grauer, R.: Numerical simulations of possible finite time singularities in the incompressible Euler equtions: comparison of numerical methods. Physica D. 237, 1932–1936 (2008) 7. Kida, S., Murakami, Y.: Kolmogorov similarity in freely decaying turbulence. Phys. Fluids 30, 2031–2039 (1987) 8. Kida, S.: Three-dimensiional periodic flows with high-symmetry. J. Phys. Soc. Jpn. 54, 2132–2136 (1985) 9. Boratav, O.N., Pelz, R.B.: On the local topology evolution of a high-symmetry flow. Phys. Fluids 7, 1712–1731 (1995) 10. Boratav, O.N., Pelz, R.B.: Locally isotropic pressure Hessian in a high-symmetry flow. Phys. Fluids 7, 895–897 (1995) 11. Pelz, R.B.: Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E. 55, 1617– 1626 (1997) 12. Pelz, R.B.: Symmetry and the hydrodynamic flow-up problem. J. Fluid. Mech. 444, 299–320 (2001) 13. Kimura, Y.: Self-similar collapse of a 3D straight vortex filament model. Geophys Astrophys Fluid. Dyn. 103, 135–142 (2009) 14. Kimura, Y.: Similarity solutions of two-dimensional point vortices. J. Phys. Soc. Jpn. 56, 2024–2030 (1987) 15. Kimura, Y.: Vortex motion on surfaces with constant curvature. Proc. R. Soc. Lond. A 455, 245–259 (1999) 16. Newton, P.K.: The N -Vortex Problem. Springer, New York (2001) 17. Kimura, Y., Koikari, S.: Particle transport by a vortex soliton. J. Fluid. Mech. 510, 201–218 (2004)

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Theor. Comput. Fluid Dyn. (2010) 24:395–401 DOI 10.1007/s00162-009-0136-3

O R I G I NA L A RT I C L E

Valery L. Okulov · Jens N. Sørensen

Applications of 2D helical vortex dynamics

Received: 19 January 2009 / Accepted: 24 June 2009 / Published online: 24 July 2009 © Springer-Verlag 2009

Abstract In the paper, we show how the assumption of helical symmetry in the context of 2D helical vortices can be exploited to analyse and to model various cases of rotating flows. From theory, examples of three basic applications of 2D dynamics of helical vortices embedded in flows with helical symmetry of the vorticity field are addressed. These included some of the problems related to vortex breakdown, instability of far wakes behind rotors and vortex theory of ideal rotors. Keywords Helical vortices · Vortex breakdown · Instability of far wakes · Vortex theory of rotors PACS 47.32.C

1 Introduction In contrast to the equations governing helical Beltrami flows with colinear velocity and vorticity fields [1], the class of 2D helical vortex flows introduced in [2] is characterized by a simple linear correlation between axial wz and azimuthal wϕ velocity components. This relation simply states that wz + r wϕ /l = const, where 2πl is the pitch of the helical symmetry of the vorticity field and r is the radial distance from the symmetry axis. The primary assumption of the 2D theory has been carefully tested in swirling flows generated by different kinds of swirlers and vortex generators and for a wide range of operating conditions in, e.g. vortex devises [3], wakes behind rotors [4] and in the boundary layer downstream of a wall-mounted vortex generator [5]. The theory of 2D helical vortex dynamics is developing fast. At present, the fundamentals are based on various analytical components, which holds true for all values of the helix pitch such as (i) the 2D Biot–Savart law for helical filaments represented by Kapteyn series [6] or in a form with singularity separation [2]; (ii) solutions of helical vortex tubes with finite core, governed by series expansion of helical multipoles [7]; (iii) relations between the induction of vortex filaments and the self-induced velocity of helical vortex tubes [8] resulting in a closed analytical solution of the helix motion [9]; (iv) analytical representation of Goldstein’s solution for the circulation of a helical vortex sheet in equilibrium [10]; (v) Kelvin’s N -gon stability problem of point vortices generalized to multiple helical vortices [9,11]. In the following, we will show three examples of how the theory can be exploited to explain various features of helical flows: a simple model explaining vortex breakdown, a study of the instability of the far wake behind rotors, and some new results on the theory of ideal rotors (propellers or wind turbines). The main goal of the current work is to demonstrate the possibilities of using the concept of helical symmetry and attract attention the thriving theory. Communicated by H. Aref V. L. Okulov (B) · J. N. Sørensen Department of Mechanical Engineering and Center for Fluid Dynamics, Technical University of Denmark, 2800 Lyngby, Denmark E-mail: [email protected] E-mail: [email protected] Reprinted from the journal

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2 Control volume analysis of vortex breakdown The phenomenon of vortex breakdown has being observed in numerous studies of slender vortices in pipe flows with high swirl (see, e.g. [12,13]). A chronological list of most of the theories can be found in [14]. In the following, we will describe a simple control volume (CV) analysis of vortex breakdown in pipe flows, in which helical symmetry is utilized to predict the velocity profiles behind the breakdown zone. Vortex breakdown is characterized by two specific features: (1) a change in a flow topology; (2) a change in axial velocity from a jet-like profile before breakdown to a wake-like profile after breakdown. The second feature permits to study vortex breakdown as a change in helical symmetry of the vorticity field. For analyzing experimental data in connection with vortex breakdown, the following empirical relations are widely used [15]: wϕ =

K  f (r, ε)  f (r, ε) (1 − exp(−αr 2 )) ≡ ; wz = W1 + W2 exp(−αr 2 ) ≡ U − r 2πr 2πr

(1)

where K , W1 , W2 and α are empirical constants, with the following physical interpretation: circulation √= 2π K ; helical pitch l = K /W2 ; advection velocity U = W1 + W2 ; effective radius of vortex core ε = 1/ α and f (r, ε) = 1 − exp(−r 2 /ε2 ) [16]. In this interpretation, the columnar vortex (1) has everywhere a dense distribution of helical filaments in the core (Fig. 1). The jet- and wake-like velocity profiles may be explained as flows induced by a right-handed helical vortex (with a “positive” pitch) or a left-handed helical vortex (with a “negative” pitch). The ability to induce reverse flow by the left-handed helical vortex explains the first breakdown property—the flow topology change. Good correlations between measured velocity profiles and the axisymmetric solution (1), established in [15], permit to apply the CV analysis to study the flow. We do not take into account the influence of friction and flux losses on the pipe-wall. The CV consideration leads to conservation of five characteristic quantities [16,17]: flow rate Q, velocity circulation , axial flux of angular momentum L, axial flux of momentum S and axial flux of energy E, This leads to the following relationship valid at the outflow of the pipe cross-section {Q, , L , J, E} = {Q 0 , 0 , L 0 , J0 , E 0 }

(2)

with the right part is defined by the flow parameters before breakdown at the inflow pipe cross-section. Substitution of the expressions (1) in the left part of (2) allows to write a system of five non-linear algebraic equations for the parameters , l, ε, U of the vortex structure (1) and the static pressure p∞ in the pipe. Using algebraic manipulations to eliminate , l, U and p∞ , the equation system (2) reduces to a non-linear energy equation than only depends on the vortex core radius ε: 

k2 (ε) U (ε) U (ε) −Q 0 + J0 + 2 2



0 l(ε)

    2  0 0 2 +k4 (ε)0 − +k5 (ε)02 = E 0 k1 (ε) p∞ (ε)+k6(ε) l(ε) l(ε) (3)

k 2 (ε)−π k (ε)

where  = 0 ; l(ε) = 02 π L10 −Q 0 02k1 (ε) ; U (ε) =

L 0 k1 (ε)−Q 0 0 k2 (ε) ; 0 (k12 (ε)−π k2 (ε))

and p∞ (ε) =

J0 π

−

1 π [U (ε)Q 0

−

L0 l(ε)

+ k3 (ε)02 ] are functions of ε only; and ki are defined by integrals of f (r, ε),

Fig. 1 Velocity profiles induced by vortex structures with different symmetry of vortex lines: a right-handed helical vortex and b left-handed helical vortex

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Fig. 2 Root of (3) and distributions of axial wz and swirl wϕ velocities: points and cir cles are the experimental data from [9]; lines are result of the modeling (solid before vortex breakdown and doted after breakdown)

1 k0 (ε) = f (1, ε), k1 (ε) = 2π

1 f (r, ε)r dr, k3 (ε) = 2π

0

0

1 k2 (ε) = 2π 0 1 

k5 (ε) = π 0

1 f (r, ε)r dr, k4 (ε) = π 2

f 3 (r, ε) dr + 2π r

0

0

f 2 (r, ε) dr, k6 (ε) = π r

0

1

⎞ ⎛ r  2 f (σ, ε) ⎝ dσ ⎠ r dr, σ3

⎞ ⎛ r  2 f (σ, ε) ⎝ dσ ⎠ f (r, ε)r dr. σ3

1 f 3 (r, ε)r dr , 0

0

The non-linear equation (3) has several roots in the full range of ε which makes possible the existence of several vortices with corresponding , l, U, p∞ to this roots and different helical symmetry under the same integral flow characteristics in the outflow cross-section (Fig. 2). The existence of a set of roots explains the possibility of the existence of several flow regimes at the same flow parameters undergoing a transition from a right handed to a left handed vortex structure, which is the experimental evidence of vortex breakdown [15]. Indeed the first root of (3) corresponds to a right-handed helical vortex with positive l and defines the initial flow in the inflow of the pipe cross-section with a jet-like axial velocity profile (seen as solid lines in Fig. 2 which are directly identical to experimental data [15]—points). The other roots correspond to lefthanded helical vortices with negative values of l and wake-like axial velocity profiles (seen as dotted lines in Fig. 2 with a good correlation between one of them and the experimental data behind the breakdown zone [15]—circles). Thus, explaining vortex breakdown as a transition in helical symmetry from a right-handed to a left-handed helical vortex has been supported by this simple CV-analysis of the experimental data [15]. Further evidence for this explanation can be found in numerical simulations [18].

3 Stability of tip vortex in far wake behind rotor In 1912, Joukowski [19] proposed a simple model for a two-bladed propeller that basically consists of two rotating horseshoe vortices, corresponding to the tip and root vortices created by the rotation of the propeller with two blades. Corresponding to this, a far wake model of an N -blades rotor may be introduced as infinitely long N -helical vortices of strength  with constant pitch and radius, and a root or hub vortex represented by an infinitely long axial hub vortex of strength 0 = −N  (see the sketch in Fig. 3). Assuming a first-order perturbation of the position of the k-helical vortex of the form, δrk = δ r˜ k exp(αt + 2πiks/N ), where δ r˜ k is the amplitude vector of the perturbations, s is the sub-harmonic wave number that takes values within the range Reprinted from the journal

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Fig. 3 Le f t Comparison of maximum amplification rate for γ = 0 (solidlines calculated by analytical solution (4) and dashedlines by numerical simulation of [14]). Right Neutral stability curves for helical tip vortices modeled far wake behind N -blade rotor as function of the circulation ratio γ (stable regions are located on the dashedside of the curve)

[1, N − 1], corresponding to N − 1 independent eigenfunctions, and α is the amplification rate. An analysis of the stability [9,11] leads to the following analytical solution for the non-dimensional amplification rate √ α(4πa 2 / ) = AB, (4) where √   s  N 1 + τ2 τ 4τ 2 − 3 − − E −ψ − A = s(N − s) ; τ 4 (1 + τ 2 )5/2 s N N −2 1 + 2τ 2 (1 + τ 2 )3/2 − 2N + 2 + B = −4N γ + s(N − s) τ3 τ2 τ (1 + τ 2 )1/2

   3/2    s N 1 + τ2 1 1 3 2 2 τ − + E +ψ − − + − 2τ − ln N δ τ (1 + τ 2 )3/2 4 N s 4 τ   3 τ 3 ς (3) + ; 2τ 4 − 6τ 2 + 2 9/2 (1 + τ ) 4 N2 and ε = σ/a is the dimensionless core size, τ = l/a is the dimensionless pitch, γ = 0 /N  is the circulation ratio, E = 0.577215 . . . is the Euler constant, ς (3) = 1.20206 . . . is the Riemann zeta function, and ψ(·) is the psi function. The value γ = −1 corresponds to Joukowski’s far wake model. The first term of B describes the effect of the hub vortex. The vortex system is unstable if AB ≥ 0 for any combination of s, τ, ε, γ . As an illustration of (4), in Fig. 3, we plot the absolute maximum amplification rate as function of helical pitch and number of tip vortices N for a far wake model without hub vortex (γ = 0). In the plot, results from our analytical model are compared to the numerical calculations by [20] which are a generalization of the single vortex theory [21]. The right plot of Fig. 3 shows neutral stability curves of the most unstable modes, when α(N , s ∗ , γ , τ ) = 0, for different N -plets as function of the circulation ratio γ and helical pitch τ . From the data we can conclude that for N < 7 stable states may exist for 0 > γ > −1, but for the important case γ = −1, where the total circulation of the vortex configuration is zero, the far wake described by the model of Joukowski is unconditionally unstable for all pitch values. Most experiments on rotor wakes, by means of flow visualizations as well as numerical simulations [20,22], support this conclusion. But some investigations, however, indicate that the wake under some conditions may be stable. The review [23] shows examples from the pertinent literature on visualizations of stable vortex behavior. An explanation for this apparent contradiction is that the model of Joukowski is too simple to describe the general behavior of rotor flows. Indeed, Joukowski’s model is based on the assumption that the circulation is

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constant over the blade span, and that the wake only consists of a hub vortex and trailing tip vortices. Considering the non-constant circulation that usually characterizes the operating range of a rotor, trailing vortices are created behind the rotor blades. These vortices form helical vortex sheets or screw surfaces along with the tip vortices. As it was shown in the experiments of [24], a visualization of this wake seemed to be stable on a long distance behind the rotor plane, whereas it clearly became unstable when the roll-up process of the vortex sheet formed concentrated tip vortices like in Joukowski’s model.

4 Maximum power of wind turbines with finite number of blades The determination of the ideal or theoretical maximum efficiency of a wind turbine rotor is today still a subject that has not been completely clarified. The most important upper limit was nearly a century ago derived by Betz [25] who, using axial momentum theory, showed that no more than 59% of the kinetic energy contained in a stream tube having same cross area as the rotor can be converted into useful work. Using general momentum theory Glauert [26] developed a model for the optimum rotor that included rotational velocities. In this approach, the rotor is treated as a rotating actuator disk, corresponding to a rotor with an infinite number of blades, and the maximum efficiency was shown to be a function of tip speed ratio, with values ranging from zero efficiency at zero tip speed ratio to the aforementioned 59% at high tip speed ratios. For a more realistic rotor with a finite number of blades, there have been many numerical or approximate models to determine the optimum rotor performance (see e.g. [27,28]). Using an analytical approach Betz [29] showed that the ideal efficiency is obtained when the distribution of circulation along the blade produces a rigidly moving helicoid wake that moves backward (in the case of a propeller) or forward (in the case of a wind turbine) in the direction of its axis with a constant velocity. Based on the criterion of Betz, a theory for lightly loaded propellers was developed by Goldstein [10] using infinite series of Bessel functions. The theory was later generalized by Theodorsen [30] to cover cases of heavily loaded propellers. Paradoxically, in his analysis, Theodorsen correctly used the unsteady Bernoulli equation but neglected the time-dependent term in the momentum equation. Our interest in the subject was stimulated by the recent paper by Wald [31], who developed a complete set of Theodorsen’s equations for determining the properties of an optimum heavily loaded propeller. However, when we tried to use the equations to analyze the optimum behavior of a wind turbine the theory always predicted a much lower efficiency than expected and what is known from experiments. Further, when extending the theory to the case of a rotor with infinitely many blades, the results did not comply with Betz limit and the general momentum theory. Based on the analytical solution to the induction of helical vortex filament developed recently by Okulov [2,9], we analyzed in detail the original formulation of Goldstein [10] and found that the model by a simple modification could be extended to handle heavily loaded rotors in a way that is in full accordance with the general momentum theory [32]. Figure 4 presents the optimum power coefficient and corresponding thrust coefficient as a function of tip speed ratio for different number of blades. From the figures, it is evident that the optimum power coefficient has a strong dependency on the number of blades.

Fig. 4 Power coefficient, C P , and thrust coefficient, CT , as function of tip speed ratio for different number of blades of an optimum rotor. Horizontal dashed lines original Betz limit; points general momentum theory; dashed and solid lines present theory Reprinted from the journal

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The curves in Fig. 4 are compared to C P -values obtained from the general momentum theory. The comparison shows that the results from the present theory for a rotor with infinite many blades are in excellent agreement with the values computed from general momentum theory.

5 Conclusions Three applications of 2D dynamics of helical vortices embedded in flows with helical symmetry of the vorticity field have been considered. The applications include a novel explanation of vortex breakdown, a stability analysis of wakes behind rotors, and a solution to the problem of the optimum rotor with a finite number of blades. The main findings of the analysis can be summarized as follows: 1. The hypothesis that the change in axial velocity from a jet-like profile to a wake-like profile during vortex breakdown is associated with a transition in helical symmetry of the vortex structure has been supported by a control volume analysis of a swirling flow in a pipe. 2. An analytical solution to the stability problem of an infinitesimal spatial displacement of a multiple of N helical vortices with a hub vorticity distribution has been found. The solution allows us to provide an efficient analysis of some of the experimentally observed stable vortex arrays in the far wake behind wind turbines. 3. An analytical method to determine the loading on an optimum wind turbine rotor has been developed. The method, which basically is a modification to the original model of Goldstein, is based on an analytical solution to the helical wake vortex problem. The model enables to determine the optimum circulation distribution and the theoretical maximum efficiency of rotors with an arbitrary number of blades at all operating conditions.

References 1. Dritschel, D.G.: Generalized helical Beltrami flows in hydrodynamics and magneto hydrodynamics. J. Fluid Mech. 222, 525–541 (1991) 2. Okulov, V.L.: Determination of the velocity field induced by vortex filaments of cylindric and weak conic shapes. Russ. J. Eng. Thermophys. 5(2), 63–75 (1995) 3. Alekseenko, S.V., Kuibin, P.A., Okulov, V.L., Shtork, S.I.: Helical vortices in swirl flow. J. Fluid Mech. 382, 195–243 (1999) 4. Troldborg, N.: Actuator Line Modelling of Wind Turbine Wakes. PhD thesis, Technical University of Denmark (2008) 5. Velte, C.M., Hansen, M.O.L., Okulov, V.L.: Helical structure of longitudinal vortices embedded in turbulent wall-bounded flow. J. Fluid Mech. 619, 167–177 (2009) 6. Hardin, J.C.: The velocity field induced by a helical vortex filament. Phys. Fluids 25, 1949–1952 (1982) 7. Fukumoto, Y., Okulov, V.L.: The velocity field induced by a helical vortex tube. Phys. Fluids 17(10), 107101(1–19) (2005) 8. Boersma, J., Wood, D.H.: On the self-induced motion of a helical vortex. J. Fluid Mech. 384, 263–280 (1999) 9. Okulov, V.L.: On the stability of multiple helical vortices. J. Fluid Mech. 521, 319–342 (2004) 10. Goldstein, S.: On the vortex theory of screw propellers. Proc. R. Soc. Lond. A 123(792), 440–465 (1929) 11. Okulov, V.L., Sørensen, J.N.: Stability of helical tip vortices in a rotor far wake. J. Fluid Mech. 576, 1–25 (2007) 12. Hall, M.G.: Vortex breakdown. Annu. Rev. Fluid Mech. 4, 195–218 (1972) 13. Leibovich, S.: The structure of vortex breakdown. Annu. Rev. Fluid Mech. 10, 221–246 (1978) 14. Lucca-Negro, O., O’Doherty, T.: Vortex breakdown: a review. Prog. Energy Combust. Sci. 27(4), 431–481 (2001) 15. Garg, A.K., Leibovich, S.: Spectral characteristics of vortex breakdown flow fields. Phys. Fluids 22, 2053–2964 (1979) 16. Okulov, V.L.: The transition from the right helical symmetry to the left symmetry during vortex breakdown. Tech. Phys. Lett. 22(10), 798–800 (1996) 17. Murakhtina, T., Okulov, V.: Changes in topology and symmetry of vorticity field during the turbulent vortex breakdown. Tech. Phys. Lett. 26(10), 432–435 (2000) 18. Okulov, V.L., Sørensen, J.N., Voigt, L.K.: Vortex scenario and bubble generation in a cylindrical cavity with rotating top and bottom. Eur. J. Mech. B 24(1), 137–148 (2005) 19. Joukowski, N.E.: Vortex theory of a propeller screw. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lyubitelei Estestvoznaniya 16, 1 (1912) (in Russian) 20. Gupta, B.P., Loewy, R.G.: Theoretical analysis of the aerodynamic stability of multiple, interdigitated helical vortices. AIAA J. 12(10), 1381–1387 (1974) 21. Widnall, S.E.: The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663 (1972) 22. Bhagwat, M.J., Leishman, J.G.: Stability analysis of helicopter rotor wakes in axial flight. J. Am. Helicopter Ass. 45, 165– 178 (2000) 23. Vermeer, L.J., Sorensen, J.N., Crespo, A.: Wind Turbine Wake Aerodynamics. Prog. Aerosp. Sci. 39, 467–510 (2003) 24. Felli, M., Guj, G., Camussi, R.: Effect of the number of blades on propeller wake evolution. Exp. Fluids 44, 409–418 (2008) 25. Betz, A.: Das Maximum der theoretisch möglichen Ausnützung des Windes durch Windmotoren. Zeitschrift für Das Gesamte Turbinenwesen 26, 307–309 (1920)

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26. Glauert, H.: Airplane propellers. In: Durand, W.F. (ed.) Division in Aerodynamic Theory, vol. IV, pp. 169–360. Springer, Berlin (1935) 27. Chattot, J.-J.: Optimization of wind turbines using helicoidal vortex model. Trans. ASME 125, 418–424 (2003) 28. Wood, D.H., Boersma, J.: On the motion of multiple helical vortices. J. Fluid Mech. 447, 149–171 (2001) 29. Betz, A.: Schraubenpropeller mit Geringstem Energieverlust, Dissertation, Gottingen Nachrichten, Gottingen (1919) 30. Theodorsen, T.: Theory of propellers. McGraw-Hill, New York (1948) 31. Wald, Q.R.: The aerodynamics of propellers. Prog. Aerosp. Sci. 42, 85–128 (2006) 32. Okulov, V.L., Sørensen, J.N.: Refined Betz limit for rotors with a finite number of blades. Wind Energy 11, 415–426 (2008)

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Theor. Comput. Fluid Dyn. (2010) 24:403–431 DOI 10.1007/s00162-009-0148-z

S P E C I A L I S S U E O R I G I NA L A RT I C L E

Viatcheslav V. Meleshko

Coaxial axisymmetric vortex rings: 150 years after Helmholtz

Received: 7 April 2009 / Accepted: 24 June 2009 / Published online: 11 September 2009 © Springer-Verlag 2009

Abstract This article addresses the fascinating 150 years history of the classical Helmholtz paper that laid the foundation of the vortex dynamics. Among general theorems on vortex motion, this memoir contains the special section on circular vortex filaments and axisymmetric vortex rings, in particular. The objective of this article is both to clarify some purely mathematical questions connected with the Dyson model of coaxial vortex rings in inviscid incompressible fluid and to provide a historical overview of achievements in experimental, analytical, and numerical studies of vortex rings interactions. The model is illustrated by several examples both of regular and chaotic motion of several vortex rings in an unbounded fluid. Keywords Vortex rings · Helmholtz on vortex motion · Dyson model of thin coaxial vortex rings PACS 47.32.cf · 47.15.ki · 47.15.G In the historical record of science, the name of Helmholtz stands unique in grandeur, as a master and leader in mathematics, biology, and in physics. His admirable theory of vortex rings is one of the most beautiful of all beautiful pieces of mathematical work hitherto done in the dynamics of incompressible fluids. Lord Kelvin [97, p. III]

1 Introduction The epigraph opens a preface to the English abridged translation of the monumental three volume German treatise [96]. These words were written by Sir William Thomson (1824–1907), later Lord Kelvin (or, more precisely, Baron Kelvin of Largs) almost at the end of his long life which was so largely devoted to fluid dynamics, when he definitely could compare inputs into fluid dynamics made by many scientists of the nineteenth century. The subject of vortex dynamics (and, in particular, a study of vortex rings) can fairly be said to have been initiated by the seminal paper [66] of Hermann Ludwig Ferdinand Helmholtz (1821–1894)1 that appeared 150 years ago in one of the leading scientific journals of that time (Fig. 1). Among the authors of 22 papers in this volume, there are names of Jacobi, Cayley, Christoffel, Clebsch, Bjerknes, and other prominent scientists. 1 In 1882 Helmholtz was elevated by the German Emperor Wilhelm I to the ranks of the hereditary nobility and added ‘von’ to his name. As this occurred later, he appears throughout this article simply as Helmholtz.

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Fig. 1 Helmholtz’s (1858) paper [66]. a Title page of the journal. b First page. c Hermann Helmholtz, circa 1858. From http://www. digizeitschriften.de/resolveppn/GDZPPN002150212 and http://www-history.mcs.st-and.ac.uk/PictDisplay/Helmholtz.html

Lord Kelvin continues [97, p. IV]: The professional career of Helmholtz was unparalleled in the history of professions. He was Military Surgeon in the Prussian army five years; Teacher of Anatomy in the Academy of Arts in Berlin one year; Professor of Pathology and Physiology in Königsberg six years; Professor of Anatomy in Bonn three years; Professor of Physiology in Heidelberg thirteen years; Professor of Physics in the University of Berlin about twenty years till he became Director of the new ‘Physikalisch-Technische Reichsanstalt’. He occupied this post during the last years of his life, still continuing to give lectures as Professor of Physics. The life and scientific work of this outstanding natural philosopher of the nineteenth century—upon his death obituary notices appeared in more than 50 scientific journals all over the world—has been highly praised by his contemporaries [49,84,95,137,179]. A man of tremendous energy and curiosity, he produced more than 200 articles and books, among them classic papers on the conservation of energy and the vortex motion, a great handbook of physiological optics, and a treatise on the physiology and psychology of sound. Besides fundamental biography [96,97], there exists many books and articles in several languages devoted to Helmholtz, e.g., [26,27,30,41,111,141,229]. (Oddly enough that in an extensive collection [25] of 15 essays that describe, analyze, and interpret virtually all areas of Helmholtz’s work in medicine, heat and nerve physiology, color and vision theory, physiological optics and acoustics, energy conservation, electrodynamics, mathematics, chemical thermodynamics, epistemology, philosophy of science, etc., his important results on hydrodynamics and meteorology have been omitted. On the other hand, in comprehensive studies [87,88,186], Helmholtz’s input in mechanics and fluid dynamics is fully addressed.) A short overview of Helmholtz’s life and scientific accomplishments are presented on http://www-history.mcs.st-and.ac.uk/Biographies/Helmholtz.html which also contains 14 pictures of Helmholtz in various stages of his life, including three ones on stamps. There exists a great number of editions in several languages (German, English, French, Italian, Russian, Slovac, Japanese) of Helmholtz scientific writings, treatises on acoustics and physiological optics, popular lectures and addresses, lectures on theoretical physics, etc. A rather complete list of these books can be found, e.g. in Deutsche Nationalbibliothek at the address http://d-nb.info/gnd/11854893X, Library of Congress at the address http://catalog.loc.gov/, and many other University libraries (e.g., http://katalog.ub.uni-heidelberg.de/) scattered all over the world. The digital libraries, e.g. Internet Archive at the address http://www.archive.org/, the Open Library at the address http://openlibrary.org/, the WorldCat at the address http://www.worldcat.org/, the German digital journal library at the address http://www.digizeitschriften.de/, and Max Planck Institute for the History of Science at the address http://vlp.mpiwg-berlin.mpg.de/library/ contain a free access to scans of books and some journal papers by and about Helmholtz. Helmholtz’s motivations for taking up this new research interest remain unclear, for at that time he was professor of physiology and anatomy at the University of Bonn, and the memoir appeared in the year of his

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Fig. 2 Other publications of Helmholtz’s (1858) paper [66]. a First page of English translation [67]. b Title page of the book [72]. From http://www.archive.org/details/zweihydrodynamis00helmuoft. c Title page of the Russian translation [73] of the book [72]

coming to the University of Heidelberg as a professor of physiology [229]. One motivation seems to have been his interest in frictional phenomena, carried over from his interest in an application to organ pipes [32, p. 10], [33, p. 149], another was his growing awareness of the power of Green’s theorem in hydrodynamics [218, p. 249]. In a speech, at a banquet, on the occasion of his 70th birthday—an event that brought together 260 friends and admirers at Kaiserhof in Berlin on November 2, 1891—Helmholtz [71] gave the following expanded account:2 I have been able to solve a few problems in mathematics and physics, including some that the great mathematicians had puzzled over in vain from Euler onwards: e.g., the question of vortex motion, and the discontinuity of motions in fluids, that of the motions of sound at the open ends of organ pipes, etc. But any pride I might have felt in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors. I am fain to compare myself with a wanderer on the mountains, who, not knowing the path, climbs slowly and painfully upwards, and often has to retrace his steps because he can go no farther—then, whether by taking thought or from luck, discovers a new track that leads him on a little, till at length when he reaches the summit he finds to his shame that there is a royal way, by which he might have ascended, had he only had the wits to find the right approach to it. In my works I naturally said nothing about my mistakes to the reader, but only described the made track by which he may now reach the same heights without difficulty. In nineteenth century, the English translation [67] made by Tait of Helmholtz’s paper [66] has been published (Fig. 2a); and there also exists a short Italian account [69] based upon a lecture by Betti at the University of Pisa. Later, in twentieth and even in twenty-first centuries, this paper was again published in English [75] and Russian [76]. Surprisingly [231], Helmholtz practically never directly continued his investigations of the topic of vortex motion established in his ground-breaking paper [66]: in the first volume of his collected writings [70], Helmholtz has only added the titles to all six sections of the paper and included three short notes related to the animated discussion between him and French Academician Bertrand (“an acrimonious public controversy” as 2 According to the translation [97, p. 180]. There exist three other versions [32, p. 4], [33, p. 145], [74, p. 473] of translation of this fascinating passage.

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Truesdell [217, p. 58] called it, which has even attracted attention of Maxwell, W. Thomson, and Tait [134]). However, as it was pointed in [33, p. 153], manuscript fragments No. 679 and 680 in his Nachlass, Helmholtz discussed various aspects of vortex motion, namely invariants, stability, and friction. Instead, he wrote another remarkable paper [68] on discontinuous motion of an inviscid fluid, in which he used the notion of a vortex sheet from [66] (see [32] and [33, pp. 159–166] for details). Together these two fundamental papers were published (Fig. 2b) as a separate book [72] in the well-known Ostwalds Klassiker der exakten Wissenschaften series edited by A. Wangerin, with his extensive comments and corrections of some small algebraical mistakes. In turn, the latter book was translated by S. A. Chaplygin (Fig. 2c) into Russian [73] (with a censorship approval [sic] dated 26 April 1902). In the very beginning of his paper (Fig. 1b), Helmholtz [66, S. 25] pointed out that already Euler had mentioned cases of fluid motion in which no velocity–potential exists, for example, the rotation of a fluid about an axis where every element has the same angular velocity. A minute sphere of fluid may move as a whole in a definite direction, and change its shape, all while rotating about an axis. This last motion is the distinguishing characteristic of vorticity. To describe these rotations, Helmholtz introduced two new terms: Wirbellinien (vortex lines), defined as lines the direction of which coincided everywhere with the local direction of the axis of rotation of the fluid and Wirbelfäden (vortex tubes), bundles of vortex lines emerging from infinitely small area elements transverse to the rotation axes. According to the monograph [217] which contains a rich and scholarly bibliography on the whole subject, the notion of vorticity had already appeared in earlier works by d’Alembert (1749), Euler (1752–1755), Lagrange (1760), Cauchy (1827), and Stokes (1848)—in Truesdell’s view [217, p.59], “all these early works are purely formal and somewhat mystifying”. Helmholtz was the first to elucidate the key properties of those portions of a fluid in which vorticity occurs in form of three important propositions. First, he proved that in such an ideal substance vortex motion could neither be produced from irrotational flow nor be destroyed entirely by any natural forces that have a potential. Second, those fluid particles constituting a vortex line at a given moment would constitute a vortex line for all times. Thus, it was possible to speak of “the same” vortex line moving along in the fluid. Third, the product of the area of a cross section of a vortex tube and the angular velocity of the rotation at that point (called the strength of the vortex tube) was constant along the tube and in time. From this last proposition, Helmholtz concluded that vortex tubes either had to run back into themselves or else end at the boundary of the space in which the whole fluid was contained. Not surprising that Laue [110] called Helmholtz “den Vater der Wirbel sätze”. (The hydrodynamical textbook tradition of the nineteenth century, e.g., [93, p. 169], [103, p. 149], [104], typically regarded Helmholtz’s third statement as a proved proposition. At the end of twentieth century, it appears [28, p. 27], [46, p. 313] that the assertion about the closing of vortex tubes requires further qualifications since topological complications like branching or aperiodic vortex lines may arise.) The detailed account of Helmholtz’s benchmark study is provided, e.g., in [32, pp. 10–16], [46, pp. 311– 316], [33, pp. 149–153], and [142, pp. 210–217]. (There even exists books [15,23] specially devoted to this article!) Short descriptions of this famous memoir can be found in practically all biographies of Helmholtz and accounts of his scientific works, e.g., [84], [95, pp. 104–105], [97, pp. 167–170], [111, pp. 225–233], [218, pp. 249–250]. In the following years, the Helmholtz theory of vortices was included in practically all classical general textbooks and monographs on fluid mechanics and aerodynamics. It was described and elaborated in great detail, in articles, in leading general and specialized encyclopedias in various countries, in major courses of theoretical physics, in several courses of general physics, treatises on the history of mathematics and physics, and found its place in review papers, chapters or sections of popular books, and dissertations. A rather complete list of studies where the results of Helmholtz’s paper [66] were used in one way or another is given in [142, pp. 219–220]. In the present article, we will discuss only the influence made by the last section [66, Sect. 6], namely an axisymmetric vortex flow of localized vorticity and more precisely, with one or several coaxial vortex rings in an inviscid incompressible fluid of constant density. Saffman [182] remarks that ‘one particular motion exemplifies the whole range of problems of vortex motion and is also a commonly known phenomenon, namely the vortex rings. < . . . > Their formation is a problem of vortex sheet dynamics, the steady state is a problem of existence, their duration is a problem of stability, and if there are several, we have the problem of vortex interactions’. The topic of vortex rings is presented as a section or even a chapter in several influential textbooks on fluid mechanics and general physics by, e.g., Acheson [1], Appel [7], Basset [13], Batchelor [14], Edser [43],

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Faber [47], Joukowski [86], Kambe [90], Lamb [103,105–109], Lichtenstein [118], Milne-Thomson [145], Ogawa [154], Poincaré [160], Prandtl and Tietjens [161], Ramsay [162], Saffman [183], Sommerfeld [200], Villat [225], Wien [230], Wu, Ma, and Zhou [234], in research monographs [4,5,37,121,130,143,211], review articles [8–10,78,84,125,127,128,163,168,193,198], popular books [36,48,50,129,203,223,224], and books on the history of fluid mechanics [33,112,216]. The special attention deserves applications of this topic to locomotion of insects, swimming of fishes, flight of birds, see, e.g., [34,38,45,83,149,164,165]. Vortex rings are easily produced experimentally by dropping drops of one liquid into another (by coincidence, the first observation has been made by Rogers [176] in the same year of publication of the Helmholtz paper; later such study was performed in much more detail by J.J. Thomson and Newall [212]), or by puffing fluid out of a hole [35,84,119,120,139,151,152,166–168,170] (to name only a few in order to show the time interval; many other beautiful pictures are reproduced in [14,39,184,190,232]), or by exhaling smoke [12,77,203,233]. Figures of various examples of vortex rings generators from some old papers are collected in [142, Figs. 1–4]. A nice collection of movies and many photographs of vortex rings and their interactions in an experimental setting may be found on the web page http://serve.me.nus.edu.sg/limtt. The important issue of modeling of such a process which is considered in a great amount of studies is out of the scope of this article. Searching Google for “Vortex rings” (only in English!) results in about 317,000 entries on Google Web, 381,000 entries on Google Images, 56,400 entries on Google Scholar, and 3,380 entries on Google Book Search (these include not only books but also old and recent articles in many leading scientific journals). Surprisingly, the entry “Vortex rings Helmholtz” results in numbers 10,500, 5,040, 790 and 729, respectively. (As usual, not all results are relevant.) In the following, we consider the case in which the ratio of azimuthal vorticity to cylindrical radius is constant within each vortex ring. Dyson [40] developed a simplified model of thin interacting coaxial vortex rings with circular cores of small radius (for which the core dynamics can be neglected). This model has an interesting Hamiltonian structure. After Dyson, this model was employed by Hicks [81] to study in detail the phenomenon of leapfrogging of two vortex rings; sometimes it is called [197] the “Dyson–Hicks model”. Much later, this model was rediscovered in [156], and since that time according to the Google Scholar, ISI, and Scopus databases it has been used in more than 40 articles on the analytical and numerical studies of vortex rings interactions in unbounded inviscid fluid [59,61–64,99,144,146,157,171–173,197,226,228], comparison of experimental observations of vortex rings interactions [29,155,236,237], nonintegrability and chaotic phenomena in Hamiltonian systems [11,18–21,98,158,197], mixing of passive surrounding fluid [185,192,194,195,219], experimental studies of interaction of vortex rings with planes and axisymmetric rigid bodies [122,180, 202,235,238], and even for the study the so-called ‘vortex sound’ [65,89,91,92,123,147,187,188,196,204– 206,222]. Based upon the previous studies of the author [59,63–65,100,144] and his students [61,62,98,99], several typical cases of the vortex rings interaction are presented below. Riley et al. [171–173,226–228] discuss another interesting samples of such an interaction based upon the Dyson model.

2 Helmholtz’s (1858) memoir and its first reception by contemporary scientists In the Introduction to his paper [66, S. 26], Helmholtz states:3 Hence it appeared to me to be of importance to investigate the species of motion for which there is no velocity-potential. The following investigation shows that when there is a velocity-potential the elements of the fluid have no rotation, but that there is at least a portion of the fluid elements in rotation when there is no velocity-potential. By vortex-lines (Wirbellinien) I denote lines drawn through the fluid so as at every point to coincide with the instantaneous axis of rotation of the corresponding fluid element. By vortex-filaments (Wirbelfäden) I denote portions of the fluid bounded by vortex-lines drawn through every point of the boundary of an infinitely small closed curve. The investigation shows that, if all the forces which act on the fluid have a potential, —1. No element of the fluid which was not originally in rotation is made to rotate. 3 Here and in what follows, all quotations from [66] are given according to English translation [67] made by Tait. Italics are from the original and translated text.

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2. The elements which at any time belong to one vortex-line, however they may be translated, remain on one vortex-line. 3. The product of the section and the angular velocity of an infinitely thin vortex-filament is constant throughout its whole length, and retains the same value during all displacements of the filament. Hence vortex-filaments must either be closed curves, or must have their ends in the bounding surface of the fluid. Helmholtz [66, S. 31] and earlier Stokes [201, p. 290] used the symbols ξ, η, ζ and ω , ω , ω , respectively, to denote the quantities: dv dw − = 2ξ = −2ω , dz dy

dw du − = 2η = −2ω , dx dz

du dv − = 2ζ = −2ω . dy dx

(1)

(The notation with straight d for partial derivatives that time were opposite to the modern one ∂.) Here u, v, w are the rectangular Cartesian components of the Eulerian velocity vector u. Helmholtz called these quantities Rotationsgeschwindigkeiten, Stokes [201] the angular velocities of the fluid, and later Lamb [103, Art. 38], [105, Art. 31], [106, Art. 30], [107, Art. 30], [109, Art. 30] the component angular velocities of the rotation. W. Thomson [215] and Basset [13] called these entities component rotations and molecular rotations, respectively. For the general case of motion in which ω does not vanish, Helmholtz [66, S. 33] used the term Wirbelbewegungen, which has been later translated into English [67, p. 491] as vortex-motions. The name vorticity was introduced by Lamb in the fourth edition of his famous treatise [107, Art. 30] (and not in the second edition [105, Art. 31], as stated by Truesdell [217, p. 58]) for the vector ω, whose Cartesian components, (ξ, η, ζ ), are given as ξ = 21 ω , η = 21 ω , ζ = 21 ω via expression (1). According to Lamb [109, p. 31], this definition avoids the intrusion of an unnecessary factor 2 in a whole series of formulae relating to vortex motion. In the second section, Helmholtz gives proofs of his three laws of vortex motion based upon kinematical considerations and ingenious transformations of the dynamical equations for incompressible, homogeneous, inviscid fluid into his now famous vorticity equations. Helmholtz derives from the Euler equations two equivalent systems of equations (Eqs. (3) and (3a) in the original text) in terms of velocity and rotation: ⎧ ⎧ δξ du du du δξ du dv dw ⎪ ⎪ ⎪ ⎪ = ξ + η + ζ , =ξ +η +ζ , ⎪ δt ⎪ ⎪ ⎪ dx dy dz δt dx dx dx ⎪ ⎪ ⎨ δη ⎨ dv dv dv δη du dv dw =ξ +η +ζ , or (2) =ξ +η +ζ , ⎪ ⎪ δt dx dy dz δt dy dy dy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dw dw dw δζ δζ du dv dw ⎪ ⎪ ⎩ ⎩ =ξ +η +ζ , =ξ +η +ζ . δt dx dy dz δt dz dz dz Here the symbol

δ δt

in the left hand sides of (2) denotes the full derivative dψ dψ dψ dψ δψ = +u +v +w δt dt dx dy dz

in term of notation of that time. We use here the notation from the English translation [67, p. 491] instead even more confusing one ∂t∂ in both the original paper [66, S. 33] and subsequent reprints [70,72,73]. It was pointed out in [33, p. 150] that these equations have been already known to D’Alembert and Euler. The main innovation of Helmholtz consists in its kinematic interpretation, stated above. In the third section, Helmholtz addresses the inverse problem of finding the components of the velocity u, v, w from the components of rotation ξ, η, ζ (up to a potential flow that covers the boundary conditions). He independently obtains the representations of Stokes for the classical problem of vector analysis of determining a vector field of known divergence (“hydrodynamic integrals of the first class” in his terminology) and curl (“hydrodynamic integrals of the second class”). Determination of the velocity field for incompressible fluid leads to the Biot–Savart law of electromagnetism, which in the present case reads that each rotating element of fluid at point (a, b, c) induces in every other element at point (x, y, z) a velocity with direction perpendicular to the plane through the second element that contains the axis of the first element. The magnitude of this induced velocity is directly proportional to the volume of the first element, its angular velocity, and the sine of the angle between the line that joins the two elements and the axis of rotation, and is inversely proportional to the square of the distance between the two

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elements. Helmholtz also establishes analogies between the induced velocity and the forces on magnetized particles. In the fourth section, Helmholtz derives an elegant expression for the kinetic energy K —“vis viva” in his terminology—of infinite fluid with a compact distribution of vorticity within it. In the fifth section, Helmholtz studies certain simple cases in which the rotation of the elements occurs only in a set of parallel rectilinear vortex-filaments. In particular, he considers several infinitely thin, parallel vortex-filaments each of which carries a finite, limiting value, m, of the product of the cross-sectional area and the angular velocity. This is the now-celebrated concept of a point vortex. Helmholtz considers simple cases of the dynamics of such vortices. He establishes the law of conservation of the center of vorticity of an assembly of point vortices. The discussion is phrased in terms of the “center of gravity” of the vortices (considering their values of m as the analog of “masses”): ‘The centre of gravity of the vortex-filaments remains, therefore, stationary during their motions about one another, unless the sum of the masses be zero, in which case there is no centre of gravity.’ Helmholtz also considers the reactions which two parallel straight vortex-filaments in an unbounded fluid would have exerted on one another. Finally, in the last sixth section which has a title “Circular vortex-filaments” Helmholtz addresses the axisymmetric motion of several circular vortex-filaments whose planes are parallel to the x y-plane, and whose centers are on the z-axis, all motion is symmetrical about the z-axis. Then by introducing the polar cylindrical coordinates, according to x = χ cos , y = χ sin , z = z, a = g cos e, b = g sin e, c = c, and taking into account that a circular vortex-filament of radius g and axial coordinate c has the single component of rotation σ (g, c) with the axis of rotation is perpendicular to direction of g and the z-axis with the rectangular coordinates (ξ, η, ζ ) of angular velocity at the point (g, e, c) ξ = −σ sin e, η = σ cos e, ζ = 0, after some transformation, Helmholtz obtains for the radial component τ in the direction of the radius χ and axial component w at the point (χ , , z), the following expressions τ =−

dψ ψ dψ , w= + , dz dχ χ

or τχ = −

d(ψχ ) d(ψχ ) , wχ = , dz dχ

(3)

with expression for ψ(χ , z) given by the Eq. [(7.)], see Fig. 3, the first line. Here and in what follows the equations from the original paper [66] are enclosed into the square brackets. It should be noted that Wangerin [72] corrected some arithmetical mistakes which went unnoticed both in [70] and English translation [67]. These corrections are presented in Fig. 3. Helmholtz also uses the notion of streamline (“Strömungslinien” or “current-lines” English translation [67]) which equation is ψχ = Const. By considering the expression [(7.)] for ψ for a vortex filament of indefinitely small section, putting σ (g, c)dgdc = m 1 = const, Helmholtz arrives to the expression (the second line in Fig. 3) for the stream function of a circular vortex filament and velocity components (the third and fourth lines in Fig. 3). These expressions are used now in every textbook of fluid dynamics. Next, by considering a finite number of infinitely thin vortex filaments Helmholtz obtains several interesting identities: besides expressions in the fifth and sixth lines in Fig. 3, he also writes  (m n ρn τn ) = 0 , where ρn and τn are the radius of the vortex filament n and the velocity which it receives from other filaments. Reprinted from the journal

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Fig. 3 Corrections of small errors in Sect. 6. In the left column—the original text [66], in the right column—Wangerin’s corrections [72]

Fig. 4 Vortex ring box generator. a From Tait [203, p. 292]. b From W. Thomson’s letter to Helmholtz [209, p. 514]

Finally, by considering the distributed vorticity inside a vortex ring with m = σ dρdλ (ρ and λ being the radial and axial distance of the elementary circular vortex filament), Helmholtz obtains (in addition to the two last lines in Fig. 3) the Eq. [(9.)]:   dρ σρ dρdλ = 0, dt or since the product σ dρdλ = Const, as the result   1 σρ 2 dρdλ = Const. 2 Helmholtz calls M the entire “mass” of a slice that a plane which passes through the z-axis cuts from the vortex ring   M= σρdρdλ. If R 2 is the mean value of ρ 2 , then

  σρ 2 dρdλ = M R 2 ,

and since the integral at the left and the value of M remain unchanged in time, the value of R also remains unchanged during the motion of the single vortex ring in an unbounded fluid. Based upon the identity [(9b.)] (see Fig. 3, the last line) where K is the (constant) kinetic energy of the infinite fluid including the vortex ring, h is the constant density, after some not very strict estimates for indefinitely thing vortex ring, Helmholtz arrives at the equation 2

d K K (M R 2 l) = − or 2M R 2 l = C − t, dt 2π h 2π h

where l denotes the value of λ for the center of gravity of the slice section of the vortex ring. From this equation Helmholtz provides the following consequence:

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Fig. 5 Single vortex ring. a Appendix by W. Thomson to the English translation of Helmholtz paper [67]. b Self-induced forward motion of a vortex ring (from [203, p. 296])

Hence in a circular vortex-filament of very small section in an indefinitely extended fluid, the centre of gravity of the section has, from the commencement, an approximately constant and very great velocity parallel to the axis of the vortex-ring, and this is directed towards the side to which the fluid flows through the ring. Indefinitely thin vortex-filament of finite radius would have indefinitely great velocity of translation. A schematic picture (Fig. 5b) of self-induced motion of a vortex ring has been sketched in [203, p. 296] and then reproduced in [57, p. 172], [84, p. 42], and [142, p. 216]. Helmholtz notes several important conclusions: We can now see generally how two ring formed vortex-filaments having the same axis would mutually affect each other, since each, in addition to its proper motion, has that of its elements of fluid as produced by the other. If they have the same direction of rotation, they travel in the same direction; the foremost widens and travels more slowly, the pursuer shrinks and travels faster, till finally, if their velocities are not too different, it overtakes the first and penetrates it. Then the same game goes on in the opposite order, so that the rings pass through each other alternately. If they have equal radii and equal and opposite angular velocities, they will approach each other and widen one another; so that finally, when they are very near each other, their velocity of approach becomes smaller and smaller, and their rate of widening faster and faster. If they are perfectly symmetrical, the velocity of fluid elements midway between them parallel to the axis is zero. Here, then, we might imagine a rigid plane to be inserted, which would not disturb the motion, and so obtain the case of a vortex-ring which encounters a fixed plane. At the very end of his paper Helmholtz suggests a simple experiment: In addition it may be noticed that it is easy in nature to study these motions of circular vortex-rings, by drawing rapidly for a short space along the surface of a fluid a half-immersed circular disk, or the nearly semicircular point of a spoon, and quickly withdrawing it. There remain in the fluid half vortex-rings whose axis is in the free surface. The free surface forms a boundary plane of the fluid through the axis, Reprinted from the journal

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and thus there is no essential change in the motion. These vortex-rings travel on, widen when they come to a wall, and are widened or contracted by other vortex-rings, exactly as we have deduced from theory. This last problem has attracted attention of Klein [94] and his qualitative description led to discussion [2,16,17] on the possibility of vortex generation in an inviscid fluid. Quantitative calculations were performed by Taylor [208]. These studies, however, are out of topic of this article.

2.1 Some examples of receptions of Helmholtz’s paper In Britain, a Scottish physicist Tait, a close friend of W. Thomson, read Helmholtz’s paper already in July 1858 and immediately made a complete English translation for his own use. In 1866. Maxwell set Helmholtz’s theorems as a question on the Cambridge Mathematical Tripos [132, p. 242]: If the motion of an incompressible homogeneous fluid under the action of such forces as occur in nature we put α=

dv dw dw du du dv − , β= − , γ = − , dz dy dx dz dy dx

shew that dα dα dα du du du dα +u +v +w =α +β +γ . dt dx dy dz dx dy dz P, Q are adjacent particles of the fluid, such that at a given instant the projections on the axes of x, y, z of the line joining them are proportional to α, β, γ , respectively: shew that, during the subsequent motion of the fluid, the projections of the line joining P and Q will remain proportional to α, β, γ . On 13 November 1867, Maxwell wrote to Tait [133, p. 321]: Dear Tait If you have any spare copies of your translation of Helmholtz on ‘Water twists’ I should be obliged to you if you could send me one. I set the Helmholtz dogma to the Senate house in ’66, and got it very nearly done by some men, completely as to the calculation, nearly as to the interpretation. The Helmholtz theory of vortex motion has been essentially developed in the classical study by W. Thomson [215]. (Though read on 29 April 1867 at the meeting of the Royal Society of Edinburgh, this memoir was published only in 1869 after it had been ‘recast and augmented’ in 1868 and 1869.) Helmholtz and Thomson had met for the first time during a stay in Bad Kreuznach, a health resort in the Rhineland, in the summer of 1856, and afterwards they developed a close personal relationship. Lord Kelvin’s biographer [209, p. 512] states that Thomson had read Helmholtz’s paper [66] already in the Fall of 1858, but without the enthusiasm it later inspired. In a letter of 30 August 1859, Helmholtz reported to Thomson that he had been working on hydrodynamic equations including friction, but did not mention his already published Wirbelbewegungen study. However, according to very extensive biography [199, p.417], no evidence exists that Thomson knew of Helmholtz’s paper prior to early 1862 when Tait mentioned it to him. In his remarkable paper [215], Thomson worked through the entire vortex theory, introduced the concept of circulation  around a closed curve C as the mean value of the tangential velocity to the curve C multiplied by its length. He greatly simplified Helmholtz’s proofs of the vortex theorems and added a few theorems of his own. He showed that for any material contour moving according to Euler’s equation for incompressible flow, the circulation is an integral of the motion, a result known today as Kelvin’s circulation theorem This theorem was considered by Einstein [44] among the most important scientific results of Lord Kelvin. Apparently, Lamb was the first author to treat the vortex dynamics in a separate chapter in a major textbook [103] on fluid mechanics. It had its origin, Lamb stated in his Preface ‘in a course of Lectures delivered in Trinity College, Cambridge in 1874, when the need for a treatise on the subject was strongly impressed on my mind.’ Taylor noted [207] that ‘Lamb, in his first course of lectures on hydrodynamics broke new ground when he gave an account of Helmholtz’s great work on vortex motion.’ Already in 1874 Lamb lectured on Helmholtz’s memoir which became the main part of Chap. VI in [103]. Glazebrook [55, p. 377] recalls:

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And so when Lamb was put on to lecture on hydrodynamics, our coaches—I was then beginning my third year—told us to go and hear what he had to tell us. And his lectures were a revelation. For most of us, Besant’s Hydromechanics contained all we knew about the subject. Helmholtz was an unknown name. Some inkling of Sir William Thomson’s suggestion that certain mysterious entities known as vortex rings were the only true atoms had reached us. Lamb, in his own inimitable manner, unveiled the mysteries and made the properties of a liquid in rotational motion clear to us. < . . . > When he arrived at the lecture room, a small room, in Trinity College—I went to look at it recently for the sake of old times—it was filled, but with the scholars and other honours men of the second and third years; our coaches had realized that Lamb would have a message for us. And he realized at once that his audience had not come to listen to the elements of hydrodynamics. He had read Helmholtz’s memoir in Crelle’s Journal for 1858 and Thomson’s Edinburgh paper (1867) On Vortex Motion. Tait [203, pp. 291–293] also devised some extremely clever experiments to illustrate the vortex theory using smoke vortex rings in air. He used a box with a circular hole on one side and a rubber diaphragm on the opposite side. Within the box, a chemical agent (magnesium sulfate) produced a thick, white smoke. When struck on the rubber diaphragm, circular smoke rings shot out of the hole (Fig. 4a). (A similar and even more sophisticated device has already been invented by Reusch [166].) Two such boxes could be placed in various positions, causing the smoke rings to interact just as Helmholtz had indicated. Tait in his lectures [203, pp. 292–298] for the audience consisted of ‘a number of my friends, mainly professional men, who wished to obtain in this way a notion of the chief advances made in Natural Philosophy since their student days’ gave the complete verbal description of these Helmholtz’s thought experiments. Tait [203, pp. 293] also notes: ‘Of course, in air, fluid friction, which depends upon diffusion, soon interferes with this state of things. But, in the experiment, the ring (of some six or eight inches in diameter) was not sensibly altered by such causes in the first 20 feet of its path.’ W. Thomson visited Tait in Edinburgh in mid-January 1867 and saw the smoke rings with his own eyes. On 22nd January, he wrote to Helmholtz [209, pp. 513–515]: Just now, however, Wirbelbewegungen have displaced everything else, since a few days ago Tait showed me in Edinburgh a magnificent way of producing them. Take one side (or the lid) off a box (any old packing-box will serve) and cut a large hole in the opposite side. Stop the open side AB loosely with a piece of cloth, and strike the middle of the cloth with your hand. If you leave anything smoking in the box, you will see a magnificent ring shot out by every blow. A piece of burning phosphorus gives very good smoke for the purpose; but I think nitric acid with pieces of zinc thrown into it, in the bottom of the box, and cloth wet with ammonia, or a large open dish of ammonia beside it, will answer better (Fig. 4b). The nitrite of ammonia makes fine white clouds in the air, which, I think, will be less pungent and disagreeable than the smoke from the phosphorus. We sometimes can make one ring shoot through another, illustrating perfectly your description; when one ring passes near another, each is much disturbed, and is seen to be in a state of violent vibration for a few seconds, till it settles again into its circular form. The accuracy of the circular form of the whole ring, and the fineness and roundness of the section, are beautifully seen. If you try it, you will easily make rings of a foot in diameter and an inch or so in section, and be able to follow them and see the constituent rotary motion. In spite of the great popularity of Tait’s smoke box for generating vortex rings in air, the first observation of vortex rings probably corresponds with the introduction of smoking tobacco! Northrup [151, p. 211] writes: It is not improbable that the first observer of vortex motions was Sir Walter Raleigh; if popular tradition may be credited regarding his use of tobacco, and probably few smokers since his day have failed to observe the curiously persistent forms of white rings of tobacco smoke which they delight to make. But some two hundred eighty years went by, after the romantic days of Raleigh and Sir Francis Drake, who made tobacco popular in England, before a scientific explanation of smoke rings was attempted. Using the smoke-box apparatus, Dolbear [36, pp. 28–31] carefully repeated all Tait’s experiments and produced a long list of characteristics of smoke vortex rings (see [142, p. 234] for detail). The extensive study by Northrup [151,152] should also be mentioned here. It contains a very detailed description of a “vortex gun”, including all the parameters, together with beautiful photos of interacting vortex rings and vortex rings interacting with rigid obstacles, e.g., with a small silver watch chain. Modern studies with nice color pictures and video can be found at http://media.efluids.com/galleries/vortex. On 18 February 1867, Sir William Thomson (he was recently knighted for the successful laying of the Atlantic cable in 1866) read before the Royal Society of Edinburgh, a talk entitled “Vortex Atoms” (with an Reprinted from the journal

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abstract printed next day in the local newspaper The Scotsman [209, pp. 517–519]). For the meeting, Tait also perfected his smoke ring experiments. Tait’s letter of 11 February 1867 captures nicely the spirit in which he was working [46, p. 323]: ‘Have you ever tried plain air in one of your two boxes. The effect is very surprising. But eschew NO5 and Zn. The true thing is SO3 +NaCl. Have the NO3 rather in excess—and the fumes are very dense + not unpleasant. NO5 is DANGEROUS. Put your head into a ring and feel the draught.’ The talk was published as a remarkable paper [213] that led at the third part of the nineteenth century to extensive studies on the vortex atom theory and vast literature, see, e.g., [6,101,198], [142, pp. 228–233] for further details and references. It is worth noting that Helmholtz’s comments on the vortex atom theory were scarce [6, pp. 34–35]. On contrary, this theory received considerable attention from Kirchhoff and Maxwell. As Schuster recounts [189, p. 34]: ‘Kirchhoff, a man of cold temperament, could be roused to enthusiasm when speaking about it. It is a beautiful theory, he once told me, because it excludes everything else.’ Maxwell [138], in his article on the Atom written for the 1878 edition of Encyclopedia Britannica, provided a detailed description of properties of vortices (and the vortex rings, in particular) in an inviscid fluid and strongly supported the idea of vortex atoms. The subject of vortex motion selected by the Examiners for the Adams Prize for 1882 year was “A general investigation of the action upon each other of two closed vortices in a perfect incompressible fluid.” In his winning essay [211], in addition to the set subject, J.J. Thomson have discussed some points which are intimately connected with it and has applied some of the results to the vortex atom theory of matter; see, also, the review by Reynolds [169]. Part of this essay was published as an extensive memoir [210] which, in our opinion, still deserves more close attention. On the whole, reviewing in 1882 the progress of hydrodynamics Hicks [78, p. 63] stated: ‘During the last forty years, without doubt, the most important addition to the theory of fluid motion has been to our knowledge of the properties of that kind of motion where the velocities cannot be expressed by means of a potential. Helmholtz first gave us clear conceptions in his well-known paper’. 3 Development of Helmholtz’s Sect. 6 for a single vortex ring The already cited Helmholtz’s statement on the velocity of a single circular vortex ring in an unbounded inviscid fluid led to the vast body of results. Theoretical studies of the motion of a circular vortex ring of closed toroidal shape with core radius a and radius R of the center line of the torus, where a  R, led to a formula for the self-induced translational velocity Vring , directed normally to the plane of the ring:   2 a  8R 8R Vring = ln ln −C + O . (4) 4π R a R2 a Here  is the (constant) intensity of the vortex ring, equal to the circulation along any closed path around the vortex core, and C is a constant. There was some disagreement in the literature concerning the value of C. The value C = 41 was given (without proof, Fig. 5a) by W. Thomson [214] and later by Hicks [79], Basset [13], Dyson [40], and Gray [58]. (Hicks [79] has even invented, independently of Neumann who had used them in another connection, the “toroidal functions”.) This corresponds to the case where the vorticity inside the core varies directly as the distance from the centerline of the ring. The value C = 1 was given by Lewis [116], J. J. Thomson [211], Joukovskii [85], and Lichtenstein [117] for a uniform distribution of vorticity within the core. Detailed discussion of both these hypotheses and also the circular form of the core is given [117,191] and later repeated by Fraenkel [51]. Lamb [105, p.260] provide an elegant proof of value C = 41 . This result was repeated in all further editions, namely, [106, p.227], [107, p.233], [109, p.241]; however, it was absent in the corresponding place of the first edition [103, Art. 143]. The proof is based upon the accurate calculation of the kinetic energy T of the fluid with a thin circular vortex ring in it and usage relation [109, Art. 162, Eq. (6)] which coincides with (corrected) Eq. [(9 b.)] (the last line in Fig. 3) of Helmholtz [66, S. 53].4 Such a concise approach was later extended by Saffman [181] and Rott [177,178]. For a hollow vortex core [79,159], or if one assumes the fluid inside the core is stagnant, the value C = 21 was obtained. A detailed 4 It was pointed out by one of the reviewers that it is worth emphasizing up front that the (corrected) Helmholtz’ Eq. [(9b.)] can be used directly to compute the ring velocity, including the O(a/R) correction. This formula, of course, supersedes Lamb’s “elegant proof”, Fraenkel’s analysis, and Saffman’s use of Lamb’s transformation to avoid Fraenkel’s difficult estimates, for Eq. [(9b.)] does not require such a transformation.

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discussion of various model is given by Donnelly [37, Chap. I]. (It is worth noting that Maxwell [135] in his letter to W. Thomson, written well later after Thomson’s short appendix [214], uses the original Helmholtz’s equations and after some algebraical transformation arrives to the correct value of the kinetic energy of the fluid, but the omission of the second term in Eq. [(9b.)] finally leads to the erroneous value C = 47 in (4).) In the following, we will use the simplest of all admissible vorticity distributions (the so-called “standard model” [37, p. 25]) in which the ratio of azimuthal vorticity to cylindrical radius is constant. For such a thin vortex ring up to the main terms the velocity is   8R 1 Vring = ln − , (5) 4π R a 4 the momentum P of an unbounded fluid (or more properly, impulse) and the total kinetic energy K of the fluid (including the vortex core) for such a ring are  1 2 2 7 8R 2 P = ρπ R , K = ρ R ln − . (6) 2 a 4 Note the relation, Vring =

∂K , ∂P

(7)

which is valid under the condition of constant volume of the vortex ring, a 2 R = Const. For important particular case of a sphere in which the azimuthal vorticity varies linearly with radius, Hill [82] discovered his famous spherical vortex, an exact solution in closed form for a steadily translating vortex ring in the thick core limit. Fraenkel [51,52] and Norbury [150] provided analytical analysis and numerical solutions for the family of a steadily propagating ring between the thin-core and Hill limits. The former author recovered and enlarged analytical results of Dyson [40] for steady vortex rings by applying general theory to compute the shape of the cross-section to one order less than Dyson, and the propagation velocity to the same order as he did, and obtained (see [53,54] for further explanation):   8R 7 3 a2 3 a2 1 2 2 8R 2 P = ρπ R 1 + ln − + . (8) , K = ρ R ln 4 R2 2 a 4 16 R 2 a When a vortex ring is moving steadily through an irrotational fluid, one can distinguish three different regions of fluid motion: (1) that of the ring which is the rotational core and which keep its identity, (2) the portion of fluid in potential motion surrounding the first, which also keeps its identity and volume and travels uniformly through the fluid like a solid, and (3) the potential motion outside the second region which remains at rest at infinity. The distinction between the rotational region (1) and irrotational (2) and (3) is well known. Less attention than it deserves, however, appears to have been devoted to the discussion of the relationship between regions (2) and (3). Similar to the two-dimensional case of vortex pair consisting of two rectilinear vortices of equal and opposite circulation which carries in its steady motion a certain oval, domain (2) (the “atmosphere” as it was called by W. Thomson [213], the “body” [161, Art. 90], or the “vortex bubble” [202]) of surrounding fluid the vortex ring also carries with it a certain body of irrotationally moving fluid in its carrier. This was already pointed out by Maxwell in a letter [135, p. 490], and later by Reynolds [167], Lodge [124], Lamb [106, p. 227], and Northrup [151,152]. In contrast to the vortex pair, the atmosphere of the vortex ring depends upon the ratio of its core radius and mean radius n = a/R. Different diameters of the core produce different types of the vortex ring atmosphere (Fig. 6a). For a rather thick circular vortex, n > 0.0116, the carried mass of fluid looks like 1 a solid body (Fig. 6b). For a critical value of n = 86 according to the Eq. (5) the velocity of translation of the vortex ring is equal to the velocity of fluid at its centre and we have a “figure eight”. If n < 0.0116, the velocity of translation of the vortex ring is greater than that of the fluid through its centre, the portion of fluid which travels with it is always a ring and not a simply connected body (Fig. 6c). These data are due to Hicks [80] who provided a detailed numerical analysis of different core thickness. His results for four typical cases are given in Table 1. The dividing closed line which separates the atmosphere from a surrounding potential fluid represents a separatrix in term of the dynamical systems theory. For n > 0.0116 the vortex ring has two hyperbolic fixed points (more correct, the two circular lines). It is known that the stable and unstable manifolds of dynamical Reprinted from the journal

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Fig. 6 The “atmosphere” traveling with a vortex ring. a Streamlines in a moving frame. From [161, Art. 90]. b Experimental observation for a smoke vortex ring (from [39, Fig. 77]). c The sketch of “atmosphere” boundary for different thickness of the vortex core Table 1 Distribution of the total kinetic energy between volumes of fluid n=

a R

0.0005 0.0116 0.05 0.1

Volring R3

Volatm R3

K1 (%) K

K2 (%) K

K3 (%) K

0.000005 0.002 0.05 0.2

0.56 1.36 1.68 2.18

0.3 0.7 2.6 5.2

70.4 72.4 67.0 62.8

29.3 26.9 30.4 32.0

systems theory provide a powerful tool for understanding Lagrangian aspects of time-periodic flows. The periodic perturbation of the atmosphere boundary leads to the lobes formed by intersections of the two manifolds can be used to quantify rates of entrainment and detrainment into the volume of fluid transported with the vortex rings, see recent papers [192,194,195] for further detail.

4 Interaction of coaxial vortex rings Helmholtz’s comment on the leapfrogging motion of two coaxial equal vortex rings cited above has started vast activity in this direction and constantly attracted the attention of scientists. However, it was extremely difficult that time to obtain a more detailed account of such a motion. The corresponding case of two coaxial pairs of two point vortices was worked out analytically and illustrated graphically by Gröbli [60, S. 152] and later, independently, by Hicks [81]. Also, Love [126] proposed ‘to imitate some of the circumstances of the problem by considering the case where there are present in an infinite fluid two pairs of cylindrical vortices of indefinitely small section, the circulations about the two vortices of each pair being equal and of opposite sign, the circulations about the four vortices being equal in absolute magnitude, and the line of symmetry for one pair coinciding with that for the other. A single pair of this kind moves parallel to the axis of symmetry with constant velocity. Two pairs with circulations in the proper directions influence each other’s motions in a manner analogous to that exhibited by thin rings.’ He provided the complete analytical treatment of the problem in terms of some elliptic integrals and found a condition that the motion may be periodic, the length of the period, and the form of the curve described by one vortex of one pair relative to the homologous vortex of the other pair. All these studies have been briefly mentioned by Lamb [105, p. 261], [106, p. 228], [107, p. 234], [109, p. 242].

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Fig. 7 Interaction of coaxial vortex rings. a Leapfrogging motion (from [9, S. 1065]). b A qualitative scheme of leapfrogging (from [200, Art. 22]). c Maxwell’s “Wheel of Life” with three vortex rings (from [136, p. 447])

Auerbach [9, S. 1065] gave a quantitative illustration (Fig. 7a) of leapfrogging by two vortex rings, however, the details of calculations remain unclear,5 while Sommerfeld [200, Art. 22] provided the qualitative description of the process (Fig. 7b). Experimentally, the most clear picture of visualization of the leapfrogging interactions was obtained much later [236,237]. These pictures were reproduced in many studies, e.g., [32], [33, p. 154], [39, Fig. 79], [194]. Lim [119] gives another nice experimental example of the vortex rings leapfrogging. Much earlier Maxwell has even created the so-called “Wheel of Life” to understand the behavior of three identical coaxial vortex rings (Fig. 7c) and on 6 October 1868, he wrote to W. Thomson [136, pp. 446–447]: H2 [Hermann Helmholtz]’s 3 rings do as the 2 rings in his own paper that is those in front expand and go slower those behind contract and when small go faster and thread through the others. I drew 3 to make the motion more slow and visible not that I have solved the case of 3 rings more than to get a rough notion about this case and to make the sum of the three areas const. I have made them fat when small and thin when big.

4.1 Dyson’s model of interaction of coaxial vortex rings At the very end of large memoir consisting of two parts (the second part was communicated by Professor J. J. Thomson, F. R. S., received March 16, read April 20, 1893) and mainly devoted to a gravitational potential of an anchor ring whose cross-section is elliptic being in connection with Saturn, Dyson [40, pp. 1091–1106]6 considered two problems of vortex rings motion in an inviscid fluid. These problems deal with a steady motion of a single vortex ring in an infinite fluid, and of several thin coaxial vortex rings. The first problem has been referred to by Lamb already in [105, Art. 163] and later briefly in [109, Art. 166] with further reference to Love [127]. Surprisingly, the second problem has not been mentioned at all. Thus, the Dyson model for interaction of several coaxial vortex rings remain unnoticed till the last quarter of twentieth century. Below we outline the main points of the model (with correction some misprints in the original paper [40, Eqs. (90), (97), (106)]). In short, the essence of Dyson model is that each circular vortex ring moves with its self-induced axial velocity (which preserves its ring radius) and in addition has a velocity due to interaction with another vortex rings considered as circular vortex filaments. It is supposed that a thin coaxial vortex ring (V R)i with i = 1, . . . , N of mean radius Ri and position Z i along axial z-direction in the cylindrical polar coordinate system (r, φ, z) (Fig. 8a) has always the circular cross-section of radius ai which remains small comparing to 5 This picture was repeated later [10, S. 139] without any change, in spite of detailed study by Hicks [81]. It seems this picture being qualitatively correct is quantitatively wrong, for the small vortex ring initially has much bigger self-induced velocity than the large one; see below. 6 Sir Frank Watson Dyson (1868–1939) is mainly known as an astronomer, for he occupied the post of Astronomer Royal at Greenwich since 1910 till 1933. Among other things, he introduced the ‘Six Pips’ in 1924 and organized in 1919, two expeditions to study the Sun eclipse—one from Greenwich to Sobral in Brazil, and one from Cambridge, under Sir Arthur Eddington, to Principle in West Africa. The successful results of these expeditions confirmed Einstein’s prediction of the deflection of starlight in the Sun’s gravitational field. Dyson died on 25 May 1939 during the homeward voyage from Australia, where he had been visiting one of his daughters, and was buried at sea. Eddington notes [42, p. 160]: ‘He seems to have been most interested in the application of his results to hydrodynamics (motion of vortex rings); but gentle persuasion from the Isaac Newton Electors, who “while approving his programme, think it important that, as far as possible, he should pay attention to the astronomical application of his method”, [Dyson in 1892 was awarded the Isaac Newton Studentship for research in astronomy] led him to deal also with a ring of strongly elliptical section, for which his formulae were well-suited, and so to the theory of Saturn’s rings.’

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Fig. 8 The Dyson model for coaxial vortex rings. a Geometry of the problem. b A vortex ring approaching a rigid plane. From [40]

Ri . The volume of each ring remains constant in time, 2π 2 ai2 Ri = Consti , i = 1, . . . , N .

(9)

The flow of incompressible fluid is the axisymmetric one with radial u(r, z) and axial w(r, z) components ∂w of the velocity vector and the single scalar component ω(r, z) = ∂u ∂z − ∂r of the vorticity vector along the φ-axis which can be expressed via the stream function (r, z) as  1 ∂ 1 ∂ 1 ∂ 2 ∂ 2 1 ∂ u=− , w= , ω=− . + − r ∂z r ∂r r ∂z 2 ∂r 2 r ∂r Suppose that for each vortex ring the single scalar component of the vorticity vector along the φ-axis varies proportional over r, ωi = Mi r , with Mi is constant. Then the Helmholtz vorticity equation written in the modern notation [14, Art. 7.1] D(ω/r ) =0 Dt is satisfied identically inside each vortex ring. The stream-function (r, z) satisfies the equation ∂ 2 ∂ 2 1 ∂ = 0 outside the rings, + − 2 2 ∂z ∂r r ∂r ∂ 2 1 ∂ ∂ 2 + − + Mi r 2 = 0 inside the ring (V R)i , 2 2 ∂z ∂r r ∂r while  and its normal derivative ddn are continuous everywhere. The value (r  , z  ) at any point (r  , z  ) outside the rings is given by    N  r 2 cos φ dz dr dφ Mi

 (r  , z  ) = r  , 4π  2 2  2 − z) + r − 2rr cos φ + r (z i=1

(10)

(11)

where the integrals are taken throughout the volume of each ring. Inside the ring (V R)i the solution for (r  , z  ) is more complicated. Dyson [40, Eq. (87)] presents it in a symbolic form, which has got an admiration [51,53,54] even in recent time. After some algebra and expansion into Taylor series on ai he comes to the expression for the stream-function at a point (Ri −ai cos χ , Z i +ai sin χ ) with 0 ≤ χ ≤ 2π, on the surface of the ring (V R)i 

  i 1 1 8Ri 8Ri (Ri − ai cos χ , Z i + ai sin χ ) = −2 − − Ri ln ln ai cos χ + . . . 2π ai 2 ai 4 

N  ∂ Iij ∂ Iij  j + ai sin χ − ai cos χ + . . . , (12) Iij + 2π ∂ Zi ∂ Ri j=1

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where symbol  means the omission of the term in the sum with j = i, i = π Mi ai2 Ri = Consti is the intensity of the vortex ring (V R)i (or its circulation, according to W. Thomson; here we use the notation  instead of the original m, remember that m = /2), and  

  π Ri Rj cos φdφ 2 2 Iij = = R R − k ) − E(k ) (13) K(k  i j ij ij ij  kij kij 0 2 2 2 (Z i − Z j ) + Ri − 2Ri Rj cos φ + Rj kij2 =

4Ri Rj . (Ri + Rj )2 + (Z i − Z j )2

Here K(k), E(k) are complete elliptic integrals of the first and second kind, respectively. On the other hand, let the ring (V R)i moves forward with velocity Z˙ i and its radius increases with the velocity R˙ i . The radius of the cross-section will change, but since ai2 Ri = Consti and a˙ i = − 21 Raii R˙ i , therefore (ai /Ri being small), a˙ i is negligible compared with R˙ i . Calculating the normal velocity of any point on the ring’s surface and remembering the meaning of the stream function, we obtain at the surface of the vortex ring (V R)i (Ri − ai cos χ , Z i + ai sin χ ) = Ci − Ri Z˙ i ai cos χ − Ri R˙ i ai sin χ + · · · .

(14)

Comparing this with the value already found for  at the surface of the ring, we obtain 

i Ri Z˙ i = 4π Ri R˙ i = −

N  j=1

1 8Ri − ln ai 4





N   j ∂ Iij , + 2π ∂ Ri j=1

j ∂ Iij , i = 1, . . . , N . 2π ∂ Z i

(15)

If we introduce the notation, N N    i j Iij , U= 2π

(16)

i=1 j=1

then the system of equations (15) can be written as  i2 8Ri ∂U 1 ˙ i R i Z i = ln + − , 4π ai 4 ∂ Ri ∂U , i = 1, . . . , N . i Ri R˙ i = − ∂ Zi

(17)

The system of equations (17) together with (9) and initial conditions (0)

Ri (0) = Ri ,

(0)

(0)

(0)

(0)

Z i (0) = Z i , ai (0) = n i Ri , n i

 1 i = 1, . . . , N

(18)

represent a nonlinear system of ordinary differential equations for determining the trajectory (Ri (t), Z i (t)) of the i-th vortex ring. Remarkably, the system (17) has two independent first integrals: N  i=1

i Ri2 ≡

P = Const, πρ

(19)

and  N  i2 7 T 8Ri − Ri ln +U ≡ = Const. 4π ai 4 2πρ

(20)

i=1

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These integrals are the equations expressing the constancy of the linear momentum P and of the kinetic energy T of the infinite fluid and can be obtained in the following manner. Since U in (16) is a function of only Z i − Z j then from summation on i from 1 to N of the second line of (17) N  ∂U = 0, ⇒ ∂ Zi

N 

i=1

i Ri R˙ i = 0.

i=1

Therefore, after integration, we obtain the equation (19). Again, multiply the first line in (17) by R˙ i and the second line by − Z˙ i and then add, therefore  N N    i2 1 ˙ ∂U ˙ 8Ri ∂U ˙ Ri + Z i = 0. − ln Ri + 4π ai 4 ∂ Ri ∂ Zi i=1

(21)

i=1

Remembering (9) we arrive at (20). For any two vortex rings, the invariants (19) and (20) are sufficient for complete integrability of the system (17). Dyson also writes the system of equations (17) in terms of the kinetic energy T i Ri Z˙ i =

1 ∂T 1 ∂T , −i Ri R˙ i = , i = 1, . . . , N , 2πρ ∂ Ri 2πρ ∂ Z i

(22)

which can be transformed into the Hamiltonian system q˙i =

∂H , ∂ pi

p˙ i = −

∂H , i = 1, . . . , N ∂qi

(23)

with the canonical variables qi = i Z i , pi = Z i2 , and the Hamiltonian H = T /πρ. This representation has been independently obtained in [24,153,174,175,220]. (0) We dimensioned all parameters in respect to the parameters of the first vortex ring, namely 1 and R1 ; the time is dimensioned by 1 /(R1(0) )2 . Time integration of the system (17), with the associated initial conditions (18) and condition (9) was performed using the fourth order Runge–Kutta scheme with a variable step. The accuracy of the computed trajectories has been thoroughly investigated by integrating the system of equations with various initial step sizes and local error values. The initial step t = 0.005 with maximum step tmax = 0.001 and a local error 10−13 were sufficiently demeaned to ensure proper accuracy. Double precision computations were found necessary. Besides, the invariants (19) and (20) are used to additionally control the accuracy of simulations. To understand the various rich possibilities, it is worth stated that there are four dimensionless parameters (including an initial separation distance) for the two vortex rings case and eight dimensionless parameters for the three vortex rings case. 4.2 Interaction of two vortex rings For the classical Helmholtz’s case of mutual threading, the initial condition at t = 0 is that the two identical (0) (0) (0) (0) vortex rings of strength 1 = 2 = 1.0, and radii R1 = R2 = 1.0 with n 1 = n 2 = 0.01 are located at some (0) (0) distance Z 0 = Z 1 − Z 2 > 0. The interaction of the vortices in this configuration is that of the well-known game of leapfrogging. In course of the interaction, the radius of the rear vortex ring initially decreases, while the radius of the front vortex ring increases. The interaction causes the translational speed of the rear vortex to become larger than the speed of the front one. Therefore, the rear vortex gradually catches up with the front one and threads through it. The invariants (19) and (20) provides the possibility to construct the phase space for such a dynamical system: the ring radius R1 against the vortex separation Z 1 − Z 2 . Such phase trajectories were obtained in several studies [22,56,91]. Figure 9a illustrates the classical Helmholtz leapfrogging interaction and presents the trajectories of rings (0) for this case of interaction with R2(0) = 1.0, Z 2(0) = 1.0, n (0) 1 = n 2 = 0.01. The trajectory of the first ring is

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Fig. 9 Interaction of two coaxial vortex rings with positive circulations, the Dyson model. The trajectories of the centroids for (0) (0) (0) mutual threading (a) and slip-through (c) motion. a Helmholtz vortex rings 1 = 1, 2 = 1, R1 = 1.0, R2 = 1.0, Z 1 = (0) (0) (0) (0) 0.0, Z 2 = 1.0. b Admissible initial values of R2 and Z 2 − Z 1 for the mutual treading of vortex rings for several values of (0) (0) (0) (0) 2 . c 1 = 1, 2 = 2, R1 = 1.0, R2 = 1.0, Z 1 = −2.0, Z 2 = 0.0

shown by solid line, the trajectory of the second ring is indicated by dashed line. Vortex positions via equal moment t = 2.0 is denoted by filled circles. Directions of vortex motion are indicated by arrows. Dyson [40, Art. 46] proved the following theorem concerning the possibility of permanent mutual threading of two equal vortex rings: √ √ If κ 2 sin θ0 and κ 2 cos θ0 be the radii of two coaxial vortex √ rings of equal strength and volume when the rings are in the same plane (θ0 being < π/4, so that κ 2 sin θ0 is the radius of the inner ring): then these two rings will continue to thread one another in turns, or will separate to an infinite distance according as θ0 is > or < β, where β is determined by the equation

  8 7 3 √ 8 7 3 √ cos β sin β ln − + ln( 2 sin β) + √ ln − + ln( 2 cos β) √ n 4 2 n 4 2 2 2  π sin 2β cos φdφ 1 8 7 +√ = ln − . √ n 4 (1 − sin 2β cos φ) 2 0

(24)

Dyson calculated the value of β for several values of n = a/R and found that roughly speaking, when n is between 0.125 and 0.01, the initial ratio R1(0) /R2(0) with Z 1(0) = Z 2(0) = 0 for the leapfrogging motion should stay between 13 and 25 . Addressing now to Auerbach’s picture (Fig. 7a), we can see that this ratio is out of the defined interval, and, consequently, these two vortex rings can not perform the leapfrogging motion. Hicks [81] proved that the mutual threading of two vortex rings of arbitrary circulation is possible (inde(0) (0) pendent of initial separation Z 1 − Z 2 ) only if their self-induced velocities at infinity are equal. In other case, either second vortex ring never reaches first vortex ring or if it were slight greater, it would ultimately reach it, and, since the trajectories are symmetrical, passes over to infinity at another side. This condition provides a rather complicated analytical formula [64], which is hardly necessary to write down in full. Instead, Fig. 9b shows the domain of admissible initial parameters for mutual threading for several typical values of 2 . This figure generalizes the results of Dyson [40] and Hicks [81]. The example of a single interaction of two vortex rings with 2  = 1 is presented in Fig. 9c. Here 2 = (0) (0) (0) (0) 0.5, R2 = 1.0, Z 2 = 1.0, n 1 = n 2 = 0.01. After a single interaction, the distance between vortex rings indefinitely increases. An isolated single vortex ring in a unbounded space moves stationary with a self-induced velocity. It appears that the system of two vortex rings has also a steady regime of motion. To find the parameters for this regime, it is necessary to satisfy three conditions R˙ 1 = 0, Reprinted from the journal

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Fig. 10 Interaction of two coaxial vortex rings with opposite circulations, the Dyson model. The trajectories of the centroids for (0) (0) (0) (0) direct scattering (a) and mutual trapping (c) motion. a 1 = 1, 2 = −0.6, R1 = 1.0, R2 = 0.7, Z 1 = 0.0, Z 2 = 5.0. b (0) (0) (0) (0) Domains for several types of interactions. c 1 = 1, 2 = −0.6, R1 = 1.0, R2 = 0.85, Z 1 = 0.0, Z 2 = 0.0

This provides a complicated relation between the intensity 2 of the second vortex and its (then constant) radius R2  

8 1 2(1 − R2 ) 2 ln − E(k2 ) − K(k2 ) − E(k2 ) + a1 4 Rmax Rmin . 

(25) 2 = 1 2R2 (R2 − 1) 2 8R2 1 − E(k2 ) − ln K(k12 ) − E(k2 ) + R2 a2 4 Rmax Rmin 2 = (1 − R )2 , k 2 = 4R /(1 + R )2 . 2 with Rmax = (1 + R2 )2 , Rmin 2 2 2 2 In most cases the value 2 is negative except for the region of small radii for very thin rings. If vortex rings have different sign of circulation, 2 < 0, then their self-induced velocities have opposite (0) directions. If 2  = −1 or R2  = 1, then a symmetry in vortex rings motions is absent. Vortex rings move one through another. We point out two examples of vortex interaction in this case. Let the first vortex ring, which moves in the direction of positive z, has greater own energy in comparison with the second ring. This (0) (0) (0) (0) case corresponds to the following initial conditions: 2 = −0.6, R2 = 0.7, Z 1 = 0.0, Z 2 = 5.0, n 1 = (0) n 2 = 0.01. The results of calculations are shown in Fig. 10a. It is seen that ring 1 increases its own radius and passes through another ring. Detailed analysis shows that vortices have analogous interaction in cases when values of initial radii have a small difference. Vortex interaction follows another scenario for initial conditions, which are defined by large enough differ(0) (0) (0) ence in radii of vortex rings. The trajectories of vortices for case 2 = −0.6, R2 = 0.5, Z 1 = 0.0, Z 2 = (0) (0) 5.0, n 1 = n 2 = 0.01 are similar to ones in Fig. 10a, but here the smaller vortex ring does not change considerably its radius and once passes through the first vortex ring. Both cases for 2 < 0 are characterized by a single interaction. Two vortex rings after the interaction continue their motion in the directions of their self-induced velocities. The character of trajectories of vortex interaction is determined by parameters of rings in the moment when vortex rings have the same axial position, Z 1 = Z 2 . A limiting case, when one type of interaction is changed by another, is defined by values of ring radii, when their self-induced velocities are equal for given 2 . This limiting regime of motion exists for arbitrary values 2 . Each state is characterized by the defined values of radii of rings at Z 1 = Z 2 . Approaching of vortices to the stationary motion of rings determines the type of their further interaction. For example, if rings have radii less than values corresponded to the stationary motion, their radial velocities at Z 1 = Z 2 change the signs. As a result, the second vortex ring passes through the first ring. If vortices achieve the plane Z 1 = Z 2 parameters in a neighborhood of the stationary regime, the radial components of vortex velocities do not change their sign. Consequently, we have an interaction, which is characterized by pass of the second vortex ring above the first one. Figure 10b presents the results of calculations for different values of intensities of the second vortical ring 2 = −0.2, −0.4, −0.6, −0.8, and −1.0. Regions of initial parameters for each possible type of interaction at

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2 = −0.6 are shaded in different way. The region A is characterized by pass of the first vortex ring through second one, the region B corresponds to the motion of the second ring through the first vortex ring. Investigation shows that there is a third closed region C of parameters (except the case 2 = −1). The boundary of this region crosses the vertical axis at two points, the bottom point determines the values of the second rings for the stationary motion of vortices. Our results show that the region C corresponds to periodic interaction of the two vortex rings with intensities of opposite signs. The example of this motion is shown in Fig. 10c for the following initial parameters (0) of the second vortex ring: 2 = −0.6, R2(0) = 0.85, Z 1(0) = Z 2(0) = 0.0, n (0) 1 = n 2 = 0.01. Vortex rings, participating in an leapfrogging interaction, have a certain displacement along the axis of symmetry. Vortices do not withdraw at large distances during the interaction in comparison with periodic motion for rings with the same signs of intensities. Thus, all possible situations of interactions of two coaxial vortex rings as they move in the same direction, or in opposite directions, are: leapfrogging or mutual threading (Fig. 9a), slip-through motion (Fig. 9c), direct scattering (Fig. 10a), and mutual trapping (Fig. 10c).

4.3 Interaction of a vortex ring with a rigid sphere Dyson [40, Art. 45] also considered the case of a single vortex ring passing over a fixed rigid sphere of radius k. He applies the method of images and states following Lewis [115] that if 2 , R2 , a2 , Z 2 are the intensity, the radius, the core radius, and the distance along Z -axis, respectively, then an imaginary coaxial vortex ring with parameters 1 , R1 , a1 , Z 1 , where  1 = −2

R1 R1 , a1 = a2 , R2 R2



R12 + Z 12

 

 R22 + Z 22 = k 2 ,

(26)

together with the original one provides the zero normal velocity at the surface of the sphere. The equation of energy (20) written as     22 R2 22 7 7 7 8R2 8R1 8R∞ 2 R2 2 − R1 ln − I12 = 2 R∞ ln − R2 ln + − 2 , 2 a2 4 2R1 a1 4 R1 a∞ 4

(27)

where R∞ and a∞ are the values of R2 and a2 , respectively, when the vortex ring and the sphere are a long way apart, together with relations (26) provides an implicit equation [40, Eq. (118)] for the path of the vortex ring from infinity up to the sphere. Dyson provided some numerics for several typical cases. Additional results can be found, e.g., in [100,158,180,238]. This model can be generalized on the case of an oscillating sphere with periodically shedding coaxial vortex rings [202].

4.4 Interaction of three vortex rings Three coaxial vortex rings need for the determination of all possible situations at least six independent determinative parameters (if the parameters of all rings are divided by the parameters of one ring). An accurate proof of the nonintegrability of the system (17) in general case, is given in [11,20,21]. It is based on the phenomenon of splitting of the separatrices of the Hamiltonian system (23), which leads to the absence of additional analytic integrals. At the same time, the trajectories of the vortex rings become very sensitive to possible changes in the initial conditions. Figure 11, the simplest example of such sensitiveness is shown. Here the trajectories of motion for the simplest case of initial conditions for three identical vortex rings are slightly different in the Z coordinate. One can conclude, that it is impossible to predict the process of the interaction even with such a little change of one of the co-ordinates. Essential changes in values of circulations of the three vortex rings lead to splitting the system into two leapfrogging vortex ring and a single one moving either faster or slowly the two others (Fig. 12). Additional results for another initial positions of three vortex rings can be found, e.g., in [98,99,144]. Reprinted from the journal

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(0)

Fig. 11 Interaction of three coaxial vortex rings, the Dyson model. The trajectories of the centroids. 1 = 1, 2 = 1, 3 = 1, R1 (0) (0) (0) (0) (0) (0) (0) (0) (0) = 1.0, R2 = 1.0, R3 = 1.0. a Z 1 = 0.0, Z 2 = 1.00, Z 3 = 2.00. b Z 1 = 0.0, Z 2 = 1.01, Z 3 = 2.00. c Z 1 = 0.0, (0) (0) Z 2 = 1.00, Z 3 = 2.01

Fig. 12 Interaction of three coaxial vortex rings, the Dyson model. The trajectories of the centroids. R1(0) = 1.0, R2(0) = 1.0, (0) (0) (0) (0) R3 = 1.0, Z 1 = 0.0, Z 2 = 1.0, Z 3 = 2.0. a 1 = 2, 2 = 1, 3 = 1. b 1 = 1, 2 = 2, 3 = 1. c 1 = 1, 2 = 1, 3 = 2

4.4.1 Fluid transport by two interacting vortex rings The problem of motion and interaction of three vortex rings has one important application. It concerns with Lagrangian analysis of fluid transport in leapfrogging motion of two vortex rings. This issue is naturally related to stable and unstable manifolds—the separatrix of an atmosphere of a “fat” vortex ring (Fig. 6c), the heteroclinic tangle of these manifolds under periodic perturbation, and the process of vortex ring fluid entrainment and detrainment. Recent papers [192,194] contain a solid background of this subject together with the history of lobe dynamics associated to coaxial vortex ring interaction. The main idea in study of motion of two vortex ring atmosphere is to consider their boundaries as lines (in the cross-section plane) consisting of some passive fluid particles. Each particle can be considered as a third “vortex ring” with zero circulation. The contour tracking algorithms or more simple a collection of passive particles along the boundary of each atmosphere provides reasonable correspondence with well-known experimental photographs [236] of two equal vortex ring interaction (Fig. 13). In spite of earlier discussion [140,237], this comparison supports the conclusion [194] ‘that the observed experimental pattern may be due merely to complex motion of tracer in irrotational or weakly vortical fluid, with vortex cores behaving in a simple, non-deforming and almost classical manner. It is only the tracer that appears to deform and roll up around the leading vortex.’ Additional experimental and numerical observations of the Lagrangian transport and a quantitative measures for estimation of the entrainment and detrainment of the surrounding fluid under vortex rings interaction can be found in [31,99,102,185,219].

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Fig. 13 Leapfrogging of two identical vortex rings. a Photographs are from [236]. b Contour tracking of the atmospheres, the Dyson model

4.5 Interaction of N > 3 vortex rings The chaotic behavior of that system is observed [98,99] more clearly. On the other hand, it was reported [3] about an effect of localization discrete vortex rings into groups of large eddies consisting of coaxial vortex rings in jet flows under harmonic forcing. Similar phenomena was observed [131] in more complicated case of coaxial vortex rings with a rectilinear line vortex along the z-axis. The infinite train of identical vortex rings evenly spaced along the Z -axis propagates with a constant velocity [221]. The interesting question of a (linear) stability and vibration of such an infinite system was addressed in [113]. The main conclusion is that the arrangement is definitely longitudinally unstable for all values of the ratio of the distances apart the rings to their radii. Another paper [114] deals with possible nonaxisymmetric waves on the vortex ring. In particular, ‘it was found that for any given ratio of radius of ring section to radius of ring there exists a critical ratio of ring spacing to radius, separating the region of stable oscillation from that of instability, a result in some respects closely analogous to that found by Kármán for the stability of two infinite parallel rows of rectilinear vortices.’ In our opinion, these remarkable papers still deserve attention. The Dyson model might appear in great help.

5 Conclusion In 1981 in his influential paper for the special issue devoted to the 25th anniversary of the Journal of Fluid Mechanics Saffman [182, p. 57] wrote: There can be no doubt that in any list of the most important papers in fluid mechanics, a prominent place will be held by Helmholtz’s great paper ‘Ueber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen’ whose appearance in 1858 laid the foundations of the study of rotational or vortex motion and together with the subsequent papers of Lord Kelvin provides the basis for the understanding and description of rotational fluid motion under conditions such that the direct effect of viscosity is not important. A measure of the quality and significance of the paper is that, although it is 123 years old, it is as good and clear an exposition as any, and is old-fashioned only in the use of Cartesian notation (like Lamb’s 1932 text) rather than the modern vector or tensor notation. Reprinted from the journal

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Looking back upon the 150 years since publication of Helmholtz’s great paper [66], we see that these words remain true, and the study of the dynamics of vortex rings and, generally, the legacy of Helmholtz and Kelvin [148] is still an active and important topic in fluid mechanics research. Acknowledgements My best thanks are due to Hassan Aref for various discussions on fascination of vortex dynamics during a long period of mutual work starting in August 1990 at the IUTAM Symposium on Stirring and Mixing (La Jolla, USA), for sharing my interest in the historical part of any scientific research, and for his hospitality and care during my attendance in October 2008 of the IUTAM Symposium on 150 Years of Vortex Dynamics (Copenhagen, Denmark). This visit was funded by Kiev National Taras Shevchenko University (Ukraine). I am grateful to G.J.F. van Heijst (Eindhoven, The Netherlands), L. Zannetti (Turin, Italy), A. Leonard (Pasadena, United States), T.S. Krasnopolskaya, A.A. Gourjii (both from Kiev, Ukraine) for their unfailing kindness in coming to my aid with various suggestions and advice.

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Theor. Comput. Fluid Dyn. (2010) 24:433–435 DOI 10.1007/s00162-009-0134-5

O R I G I NA L A RT I C L E

Russell J. Donnelly

Dynamics of vortex rings in viscous fluids

Received: 5 January 2009 / Accepted: 2 June 2009 / Published online: 24 July 2009 © Springer-Verlag 2009

Abstract As part of a long range study of vortex rings, their dynamics, interactions with boundaries and with each other, we present the results of experiments on thin core rings generated by a piston gun in water. We characterize the dynamics of these rings by means of the traditional equations for such rings in an inviscid fluid suitably modifying them to be applicable to a viscous fluid. We develop expressions for the radius, core size, circulation, and bubble dimensions of these rings. Keywords Vortex rings · Vortex dynamics PACS 47.32.Cc The purpose of this short contribution is to summarize progress in developing expression for vortex ring propagation in viscous fluids. Our main paper on this subject is now in print [1]. The velocity V of a vortex ring of radius R, core radius a and circulation  in an ideal fluid can be written V = (/4π R)[ln(8R/a) − β]

(1)

where a is generally taken to be small with respect to R and information about the core structure is contained in the core structure parameter β as detailed in section 1.1 of [1]. A common approach to dealing with vortex motion in a viscous fluid is to take the term in square brackets (which we call ) as a constant, and replace the ring radius R by the radius of the gun R0 . This coupled with the “slug model” for the circulation (s = L 2 /2T ) allows one to deduce all sorts of expressions for various desired quantities by dimensional analysis. Here L is the stroke length and T the stroke time of the piston used to produce the ring. However it does not take much experimentation to note that vortex rings can have radii both bigger and smaller than the gun radius R0 . Neglect of the term means that information on the core size a and the vortex core structure parameter β has been lost. Equation (1) is the result for a perfect fluid, and it has been experimentally verified by Rayfield and Reif in superfluid helium-4 at 0.28 K [2]. The key question we have worked on is what modifications to (1) are needed to describe the velocity of a vortex ring in a viscous medium such as water? Indeed such an expression has been introduced by Saffman [3] for a Gaussian vorticity distribution in the core:    8R V = ln −β (2) 4π R a and where he finds a=

√

4νT , β = 0.558

(3)

Communicated by H. Aref R. J. Donnelly Department of Physics, University of Oregon, Eugene, OR 97403, USA E-mail: [email protected] Reprinted from the journal

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2a Rb

R

R

Rb

(a)

(b)

Fig. 1 a Sketch of a thin vortex ring of core radius a, ring radius R and bubble radius Rb . The eccentricity γ is the ratio of semi-minor to semi-major axes. The direction of motion is horizontal. b Photograph of a ring that has just passed through a sheet of tracer, which causes the apparent jet on the right. The comparison with a is clear

where T is the stroke time. Our paper [1] establishes the fact that (2) and (3) agree with experiment and that, most astonishing, once the ring emerges from the gun, it does not appear to grow further in either ring radius or core radius. One would have thought that the stroke time T in (3) would have been increased by the time of flight t. We have found this result totally counter-intuitive, and have worked hard to disprove it. A start in this direction is explained at the end of this paper. The ring radius R is given by equating the volume of the inner ellipsoid in Fig. 1 to the volume of water displaced by the piston and leads to the expression  1/3 R = 3R02 L/4γ

(4)

where γ is the eccentricity of the ellipsoid shown in Fig. 1. γ is independent of velocity (as given by inviscid theory), and we obtain it simply by photography. We find γ  0.60. Similar arguments give an expression for the bubble size (see Fig. 1): R/Rb = (γ (1 + k)/3π)1/3

(5)

where  is the quantity in square brackets in (1). We then use two different expressions for conservation of momentum, which seem reasonable to use in a viscous fluid, since momentum is conserved in dissipative systems in classical mechanics. This allows us to derive expressions for the circulation in terms of the average piston velocity Vp = L/T .  = R02 L Vp /R 2 = R02 L 2 /R 2 T

(6)

which we believe is more accurate than the slug model, but gives comparable results in many cases. Note this means the circulation can be given in terms of the stroke time T , stroke length L and ring radius R (which itself comes from L in (4). Thus an experimenter can set the circulation in terms of gun characteristics only. With (4) in hand it is not hard to derive an expression for the velocity of the ring, which turns out to be proportional to the piston velocity Vp = L/T   γ Vp  8R V = ln − 0.558 =  (7) 4π R a 3π Equation (7) predicts V /Vp = γ /(3π)  0.30

(8)

which agreed well with our results using an earlier gun design discussed in [1]. Imagine our surprise when we discovered our new gun [1] gives V /Vp  0.63. The origin of this problem appears to be Kelvin waves on the cores of the rings at formation. Kelvin waves slow vortex rings in the manner described in section 4.4 of [1].

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Gharib et al. [4] have shown that for a “formation time”L/2R0 < 4 only a single isolated ring is formed whereas for L/2R0 > 4 the flow field consists of a vortex ring with a trailing jet. The former flow is the one taken for this study. Therefore we adopt the restriction L/2R0 < 4

(9)

These considerations have allowed us to construct a vortex ring scattering chamber. We have two identical vortex guns mounted on the circumference of a calibrated circle. The formulae we have developed allow us to know exactly what the properties of the projectile vortices are. If we set the stroke length L and stroke time T on each gun (they do not have to be identical) then the formulae given above allow us to know the core parameter (3), the ring radius (4), the bubble size (5) and the circulation (6). The velocity is given by (7). After the scattering event, we can obtain the circulation from (1) by measuring the velocity and radius of the product rings (and core size if needed) photographically. To the best of our knowledge this project offers a most sophisticated experimental apparatus for research on vortex dynamics. The characterization of rings by the formulae given above allows rapid exploration of a wide variety of initial collision conditions photographically. The initial conditions can be set and reproduced to machine shop accuracy. We are looking forward to productive use of this apparatus, and the kind of insight one can gather from vortex ring collision experiments. During this meeting we became aware of an important paper studying vortex rings by direct numerical simulation (DNS) from Coleman’s group at Southampton [5]. There is clearly much to be learned from a comparison of results in [1,5]. The comparison at the moment is hindered by the ratio ε = a/R. Our experiments typically have ε ≈ 4 × 10−2 , whereas the DNS have ε  0.2 → 0.4 because of the need to resolve details in the core. The first insight is a comparison of slowing vortex rings. We observe vorticity in the core decreasing in time and characterized by a drag coefficient. In one case we examined we had ε = 0.04, /ν = 3,400, their nearest case had ε = 0.20, /ν = 7,500. They observed a velocity decay to about 61% of the initial velocity after some 14 s, we drop to about 62% after a similar elapsed time. A second insight is gained by noting that they find a time t* needs to elapse before the core adjusts to its vorticity distribution. We find photographically that the ring needs to propagate at least one gun diameter before it adjusts its vorticity. This fact gives us our first clue as to why the core size does not appear to diffuse. Subtracting the time it takes for the ring to propagate a distance of the order of the gun diameter goes a long way toward understanding the apparent slow growth of the core (see Fig. 11 of [1]). A third insight is that the rings in Fig. 5(b) of [5] do not change much in radius, consistent with the results in Table 2 of our paper [1]. Finally, it is not true that one can make rings of all characteristics with one gun design. Our apparatus has a maximum stroke of 11 cm, but we have wide control over the stroke time with 100 ms being the most rapid. One can see that if we seek a/R = 0.2, we can do this with L = 10 cm, T = 7.5 s, but the Reynolds number /ν  290 far lower than the example B3 in [5] with /ν = 7,500. I am indebted to Hassan Aref for arranging this wonderful occasion to recall 150 years of vortex dynamics. References 1. Sullivan, I., Niemela, J.J., Hershberger, R., Bolster, D., Donnelly, R.J.: Dynamics of thin vortex rings. J. Fluid Mech. 609, 319–347 (2008) 2. Rayfield, G.W., Reif, F.: Quantized vortex rings in superfluid helium. Phys. Rev. 136, A1194–A1208 (1964) 3. Saffman, P.G.: The velocity of viscous vortex rings. Stud. Appl. Math. 49, 625–639 (1970) 4. Gharib, M., Rambod, E., Shariff, K.: A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121 (1988) 5. Archer, P.J., Thomas, T.G., Coleman, G.N.: Direct numerical simulation of vortex ring evolution from the laminar to the early turbulent regime. J. Fluid Mech. 598, 201–226 (2008)

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Author Index Andersen, A., 345 Aref, H., 1 Auguste, F., 321 Bisgaard, A.V., 315 Bohr, T., 345 Brandt, L.K., 179 Brøns, M., 315 Carton, X., 111, 133, 141, 147 Chavanis, P.-H., 233 Cichocki, T.K., 179 Crowdy, D., 25 Donnelly, R.J., 17, 449 Fabre, D., 321, 365 Farge, M., 225 Flierl, G.R., 141, 147 Fukumoto, Y., 351, 379 Gryanik, V., 133 Hattori, Y., 379 Hussain, F., 281 Iga, K., 299 Juul Rasmussen, J., 269 Kanso, E., 217 Kelly, S.D., 61 Kevlahan, N., 257 Kida, S., 275 Kimura, R., 339 Kimura, Y., 405 Kizner, Z., 81, 117, 127 Kolomenskiy, D., 185, 225 Koshel, K., 75 Krueger, P.S., 307 Kunnen, R., 331 Kuznetsov, E.A., 269 Le Dizès, S., 365 Leonard, A., 385 Llewellyn Smith, S.G., 211 Lopes Filho, M.d.C., 67 Luzzatto-Fegiz, P., 197 Magnaudet, J., 321 McDonald, N.R., 173 Meleshko, V.V., 419 Meunier, T., 141, 147

Michelin, S., 211 Moffatt, H.K., 9, 225 Moulin, F.Y., 339 Nakazawa, N., 275 Naulin, V., 269 Nelson, R.B., 173 Newton, P.K., 153 Nielsen, A.H., 269 Niino, H., 339 Noguchi, T., 339 Nomura, K.K., 179 Nussenzveig Lopes, H.J., 67 O’Neil, K.A., 55 Obi, S., 191 Okulov, V.L., 411 Orlandi, P., 263 Perrot, X., 111, 141, 147 Pirozzoli, S., 263 Pradeep, D.S., 281 Reznik, G., 81, 117 Ryzhov, E., 75 Sakajo, T., 167 Santbergen, R., 127 Schneider, K., 185, 225 Schnipper, T., 345 Sheel, T.K., 191 Sokolovskiy, M., 133 Sokolovskiy, M.A., 141, 147 Sørensen, J.N., 411 Spiegel, E.A., 93 Stepanov, D., 75 Stremler, M.A., 41 Trieling, R., 127, 331 Umeki, M., 399 van Heijst, G., 127 van Heijst, G.J., 331 Velasco Fuentes, O., 205 Verron, J., 133 Williamson, C.H.K., 197 Xiong, H., 61 Yukimoto, S., 339 Zakharov, V.E., 393

453