Tubes, Sheets and Singularities in Fluid Dynamics (Fluid Mechanics and Its Applications)

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Tubes, Sheets and Singularities in Fluid Dynamics

FLUID MECHANICS AND ITS APPLICATIONS Volume 71 Series Editor: R. MOREAU MADYLAM Ecole Nationale Supérieure d'Hydraulique de Grenoble Boîte Postale 95 38402 Saint Martin d'Hères Cedex, France

Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advancement. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particulary open to cross fertilisation with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages.

Tubes, Sheets and Singularities in Fluid Dynamics Proceedings of the NATO ARW held in Zakopane, Poland, 2–7 September 2001, Sponsored as an IUTAM Symposium by the International Union of Theoretical and Applied Mechanics Edited by

K. BAJER Warsaw University, Institute of Geophysics, Warsaw, Poland and

H.K. MOFFATT University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Cambridge, United Kingdom

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-48420-X 1-4020-0980-1

©2004 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

http://kluweronline.com http://ebooks.kluweronline.com

International Scientific Committee K. BAJER M. FARGE J. JIMENEZ S. KIDA R. KRASNY H.K. MOFFATT A. NORDLUND A.E. PERRY D. PULLIN

(co-chairman)

(co-chairman)

Warsaw Paris Madrid Nagoya Ann Arbor Cambridge Copenhagen Melbourne Pasadena

Local Organising Committee K. BAJER M. BRANICKI S. DROBNIAK A. MROWIEC M. NUREK-MALINOWSKA A. PALCZEWSKI A. STYCZEK

(chairman)

Sponsors

NATO International Union of Theoretical and Applied Mechanics (IUTAM) Office of Naval Research International Field Office (ONRIFO)

Warsaw University Institute of Geophysics (IGF UW)

PROFESSOR ANTHONY E. PERRY - 19.2.1937 TO 3.1.2001 Tony Perry published extensively in the field of fluid mechanics and aerodynamics in both experimental and theoretical areas. He pioneered many aspects of turbulence studies, three-dimensional flow separation, flow pattern topology, and vortex shedding processes. Tony was also internationally renowned for his work in the study of turbulent boundary layers using hot-wire anemometry. Throughout his career as a member of the academic staff at the University of Melbourne, he spent extended periods at places such as Caltech, Stanford, Princeton, NASA Ames, Harvard, DFVLR Gottingen, and Cambridge. In 1992 he was a Sherman Fairchild Distinguished Scholar at GALCIT Caltech, Pasadena. In 1996 he held the Clark B. Millikan Chair of Aeronautics for distinguished visitors at Caltech, and in 1999 a Rothschild Visiting Professorship at the Isaac Newton Institute for Mathematical Sciences at Cambridge University. In 1985 Anthony Perry was elected Fellow of the Australian Academy of Sciences and in 1998 he was elected Fellow of the American Physical Society. Tony was a great engineer and scientist but he will mostly be remembered for his lectures at many international conferences around the world. In his flamboyant style he would entertain his audience with his simple explanations of the topology of complex three-dimensional flow patterns. He was an inspiration to all the graduate students at the University of Melbourne. When there were signs that he would not be able to make it to Zakopane he told Keith Higgins, who had just started his PhD with me, that he had to select a topic which would be not only of scientific interest but also “entertaining”. IUTAM was kind enough to invite us to Zakopane and to invite Keith Higgins to “entertain in the style of Tony Perry”.

Associate Professor M.S. Chong Department of Mechanical and Manufacturing Engineering University of Melbourne.

Postal and Internet addresses of all authors and colour versions of some figures in this volume are available on the conference website:

www.igf.fuw.edu.pl/IUTAM

Contents

Preface

xv

Part I Vortex structure, stability and evolution Simulation of vortex sheet roll-up: chaos, azimuthal waves, ring merger Robert KRASNY, Keith LINDSAY & Monika NITSCHE

3

Merging of non-symmetric Burgers vortices Keith HIGGINS, M. S. CHONG & Andrew OOI

13

Effect of stretching on vortices with axial flow Maurice ROSSI, Ivan DELBENDE & Stéphane LE DIZÈS

19

Optimal two-dimensional perturbations in a stretched shear layer Stéphane LE DIZÈS

25

Advection–diffusion of a passive scalar in the flow of a decaying vortex Konrad BAJER, Andrew P. BASSOM & Andrew D. GILBERT

31

Linear stability of a vortex ring revisited Yasuhide FUKUMOTO & Yuji HATTORI

37

The Vortex–in–cell method for the study of three–dimensional vortex structures Henryk KUDELA &

49

L-transition from right-to left-handed helical vortices Valery OKULOV, Jens N. SØRENSEN & Lars K. VOIGT

55

Dynamical behaviour of counter-current axisymmetric shear flows Xilin XIE, Weiwei MA & Huiliang ZHOU

61

Part II

Singular vortex filaments

Complexity measures of tangled vortex filaments Carlo F. BARENGHI, David C. SAMUELS & Renzo L. RICCA xi

69

xii

TUBES, SHEETS AND SINGULARITIES IN FLUID DYNAMICS

Corotating five point vortices in a plane Tatsuyuki NAKAKI

75

On motion of a double helical vortex in a cylindrical tube Pavel A. KUIBIN

81

Intensive and weak mixing in the chaotic region of a velocity field Alexandre GOURJII

87

Evolution of the anisotropy of the quantum vortex tangle Tomasz LIPNIACKI

93

Motion of vortex lines in quantum mechanics

99

Part III Magnetic structure, topology and reconnection Magnetic dissipation: spatial and temporal structure Åke NORDLUND

107

Current sheets in the Sun’s corona Eric PRIEST

115

A model for magnetic reconnection H.K. MOFFATT & R.E. HUNT

125

Reconnection in magnetic and vorticity fields Gunnar HORNIG

133

Energy, helicity and crossing number relations for complex flows Renzo L. RICCA

139

Helicity conservation laws

145

A third-order topological invariant for three magnetic fields Christoph MAYER & Gunnar HORNIG

151

Asymptotic structure of fast dynamo eigenfunctions B. J. BAYLY

157

Part IV Vortex structure in turbulent flow Vortex tubes, spirals, and large-eddy simulation of turbulence D.I. PULLIN

171

Low-pressure vortex analysis in turbulence: life, structure, and dynamical role of vortices 181 Shigeo KIDA, Susumu GOTO & Takafumi MAKIHARA

Contents

xiii

Vortex bi-layers and the emergence of vortex projectiles in compressible 191 accelerated inhomogeneous flows (AIFs) Norman J. ZABUSKY & Shuang ZHANG Interaction of localised packets of vorticity with turbulence A. LEONARD

201

Extraction of coherent vortex tubes in a 3D turbulent mixing layer using orthogonal wavelets Kai SCHNEIDER & Marie FARGE

211

Vortex tubes in shear-stratified turbulent flows 217 Marie FARGE, Alexandre AZZALINI, Alex MAHALOV, Basil NICOLAENKO, Frank TSE, Giulio PELLEGRINO & Kai SCHNEIDER Coherent dynamics in wall turbulence Javier JIMÉNEZ,

229

Some characteristics of the coherent structures in turbulent boundary layers Davide POGGI

241

A singularity-free model of the local velocity gradient and acceleration gradient structure of turbulent flow Brian CANTWELL

247

Spiral small–scale structures in compressible turbulent flows Thomas GOMEZ, Hélène POLITANO, Annick POUQUET & Michèle LARCHEVÊQUE

261

Part V Finite-time singularity problems Discrete groups, symmetric flows and hydrodynamic blowup Richard B. PELZ

269

Diffusion of Lagrangian invariants in the Navier-Stokes equations Peter CONSTANTIN

285

Evidence for singularity formation in a class of stretched solutions of the 295 equations for ideal MHD J. D. GIBBON & K. OHKITANI Numerical evidence of breaking of vortex lines in an ideal fluid Evgeniy A. KUZNETSOV, Olga M. PODVIGINA & Vladislav A. ZHELIGOVSKY

305

Sufficient condition for finite-time singularity and tendency towards selfsimilarity in a high-symmetry flow 317 C. S. NG & A. BHATTACHARJEE Finite time singularities in a class of hydrodynamic models Victor P. RUBAN, Dmitry I. PODOLSKY & Jens J. RASMUSSEN

329

xiv


On stabilisation of solutions of singular quasi-linear parabolic equations with singular potentials 335 Andrey MURAVNIK Part VI Stokes flow and singular behaviour near boundaries Interactions between two close spheres in Stokes flow Nicolas LECOQ, René ANTHORE, François BOSTEL & François FEUILLEBOIS

343

Effective boundary conditions for creeping flow along a periodic rough surface B. CICHOCKI, P. SZYMCZAK & F. FEUILLEBOIS

349

Steady Stokes flow in a trihedral corner Vladimir S. MALYUGA & Alexandre M. GOMILKO

355

After-dinner speeches

361

List of Participants

365

Author Index

367

Topic Index

373

Preface

Vortex tubes and vortex sheets can be thought of as the fundamental building blocks of fluid flow at high Reynolds number, whether laminar or turbulent. It is therefore important to understand their structure, stability and evolution, and the various non-linear interactions that can occur among them. Similar problems in relation to magnetic flux tubes arise in magnetohydrodynamics (MHD) at high magnetic Reynolds numbers; and the analogies between MHD and vortex dynamics can be exploited in the development of insight in both fields. The dynamics of vortex tubes, sheets and more complex structures plays a central rôle in the description of turbulent shear flows, whose ‘coherent structures’ can be most naturally interpreted as vortical structures subject to both self-induced evolution and the complex interactions with a random environment and with boundaries. The interaction of skewed vortex tubes is a problem of acute current interest, because the intense stretching associated with the mutual interaction when such tubes are close to each other leads to rapid growth of vorticity. Whether this growth is or is not bounded within any finite time-interval is one of the famous open problems of fluid dynamics, and is the subject of much current analytical and numerical work. The papers collected in this volume range over the above topics, and constitute the Proceedings of a NATO ARW and IUTAM Symposium held in Zakopane, Poland, 2-7 September 2001. They are grouped in six parts as follows: Part I Part II Part III Part IV Part V Part VI

Vortex structure, stability and evolution Singular vortex filaments Magnetic structure, topology and reconnection Vortex structures in turbulent flow Finite-time singularity problems Stokes flow and singular behaviour near boundaries xv

xvi


The volume also contains two papers which were prepared for the Symposium, but whose authors (D. Poggi and B. Cantwell) were unfortunately unable to be present. The fresh mountain air of Zakopane stimulated one of us [HKM] to compose a limerick as an ‘alternative abstract’ for each of the lectures that were presented. At the suggestion of many participants these are included (with the permission of the relevant authors!) on the title page of each paper. Two authors have written ‘response limericks’, also included; another has responded by composing his entire abstract in limerick format! We thank IUTAM for its initial approval and sponsorship of this Symposium, NATO for its generous financial support as an Advanced Research Workoshop (ARW), and the Office of Naval Research (International Field Office in London) for additional generous support. We thank also Warsaw University for administrative support in the arrangements of the meeting. Konrad Bajer Keith Moffatt

12 April 2002

Note added in proof At a late stage in the production of this volume, we have learnt with great sadness of the sudden death of Richard Pelz on 24th September 2002. Richard’s contribution to this volume is contained on pp 269–283. His contributions to the finite-time singularity problem, some of which are described in this paper, have been at the cutting edge of research on this topic of central importance to fluid dynamics. He will be greatly missed throughout the extended worldwide family of fluid dynamics.

I

VORTEX STRUCTURE, STABILITY AND EVOLUTION

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Simulation of vortex sheet roll-up: chaos, azimuthal waves, ring merger Robert KRASNY1, Keith LINDSAY2 & Monika NITSCHE3 1

University of Michigan, Department of Mathematics, Ann Arbor, Michigan 48109-1109 USA [email protected]

2

National Center for Atmospheric Research, Climate and Global Dynamics Boulder, Colorado 80307-3000 USA

3

University of New Mexico, Department of Mathematics and Statistics Albuquerque, New Mexico 87131-1141 USA

Abstract This article reviews some recent simulations of vortex sheet roll-up using the vortex blob method. In planar and axisymmetric flow, the roll-up is initially smooth but irregular small-scale features develop later in time due to the onset of chaos. A numerically generated Poincaré section shows that the vortex sheet flow resembles a chaotic Hamiltonian system with resonance bands and a heteroclinic tangle. The chaos is induced by a self-sustained oscillation in the vortex core rather than external forcing. In three-dimensional flow, an adaptive treecode algorithm is applied to reduce the CPU time from to O(N log N), where N is the number of particles representing the sheet. Results are presented showing the growth of azimuthal waves on a vortex ring and the merger of two vortex rings. Vortex blob methods discrete, Applied to roll-up of a sheet, Will persuade any cynic That heteroclinic Tangles give insights quite neat.

1.

Introduction

Vortex sheets are commonly used in fluid dynamics to model thin shear layers in slightly viscous flow. This article reviews some recent simulations of vortex sheet roll-up in planar, axisymmetric, and threedimensional flow Krasny & Nitsche 2001; Lindsay & Krasny 2001. Vortex sheet simulations encounter difficulties due to Kelvin-Helmholtz in– stability and singularity formation Moore 1979 and the present work deals with these issues by applying the vortex blob method Chorin & Bernard 1973; Anderson 1985; Krasny 1987. This approach regularises

3 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 3–12. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

4

R. Krasny, K. Lindsay and M. Nitsche

the singular Biot-Savart kernel in the integral defining the sheet velocity. As a result, the instability is diminished and the computations can proceed past the singularity formation time into the roll–up regime. A comprehensive review of vortex blob methods is given by Cottet & Koumoutsakos (2000). The article is organised as follows. The onset of chaos in planar and axisymmetric flow is discussed in §2. A treecode algorithm for vortex sheet motion in three-dimensional flow is described in §3 and results are presented showing the growth of azimuthal waves on a vortex ring and the merger of two vortex rings. A summary is given in §4.

2.

The onset of chaos in vortex sheet flow

In planar flow, a vortex sheet is a material curve and it is represented on the discrete level by a set of particles with scalar weights for The particles are advected by the equations

where

is a regularised form of the 2-D Biot-Savart kernel. A similar approach is used for axisymmetric flow Nitsche & Krasny 1994. Figure 1 displays the roll-up of an initially flat vortex sheet in planar and axisymmetric flow, yielding respectively a vortex pair and a vortex ring Krasny & Nitsche 2001. At early times the roll–up is smooth, but at late times the sheet develops irregular small-scale features; a wake is shed behind the vortex ring arid gaps form between the spiral turns in the spiral core in both cases. Figure 2 shows a close-up at the final time. In the planar case, the irregular features are confined to a thin annular band around the core. In the axisymmetric case, the irregular features in the core are more dispersed and the sheet folds and stretches near the rear. It should be noted that the computations are well-resolved. As explained below, the irregular features are due to the onset of chaos. It is well-known that the motion of material points in planar incompressible flow is governed by a Hamiltonian system,

where the stream function plays the role of the Hamiltonian. A similar result holds for axisymmetric flow. Insights from dynamical

Simulation of vortex sheet roll-up

5

6


systems theory can then be used to shed light on the fluid dynamics Aref 1984; Ottino 1989. One example especially relevant for the present work is the oscillating vortex pair Rom-Kedar, Leonard & Wiggins 1990. The stream function in this model has the form

where is the steady flow defined by a pair of counter-rotating point-vortices, is a time-periodic perturbation strain field, and is the perturbation amplitude. The system is integrable for and chaotic for Figure 3 describes the dynamics of the model. In particular, the perturbed system has chaotic orbits associated with heteroclinic tangles and resonance bands Guckenheimer & Holmes 1983; Wiggins 1992. Returning to the vortex sheet flow, the first observation is that past the initial transient the flow enters a quasisteady state. In this regime it was found that the vortex core undergoes a small-amplitude oscillation which is close to time-periodic. In other words, the stream function of the vortex sheet flow is close to the form given in (4). Using the oscillation frequency, a Poincaré section of the vortex sheet flow was constructed and the result, shown in figure 4, has the generic features of a chaotic Hamiltonian system. The resonance bands and heteroclinic tangle in the Poincaré section are well–correlated with the irregular features in the shape of the vortex sheet (compare figures 2 and 4). Hence the vortex sheet flow resembles a chaotic Hamiltonian system, although the chaos is induced here by a self-sustained oscillation in the vortex core rather than external forcing. The oscillation resembles the periodic motion of a strained elliptic vortex Kida 1981.

3.

Azimuthal waves, vortex ring merger

In three-dimensional flow, a vortex sheet is a material surface and it is represented on the discrete level by a set of particles with vectorvalued weights for The particles are advected by the equations

where

is a regularized form of the 3-D Biot-Savart kernel Rosenhead 1930; Moore 1972. Evaluating the sum (5) for is an example of


7

8


an N-body problem. The simplest evaluation procedure is direct summation which computes particle-particle interactions per timestep. Since new particles are inserted to maintain resolution as the sheet rolls up, the CPU time quickly becomes prohibitive. The present simulations used a treecode algorithm to reduce the cost of evaluating the particle velocities Lindsay & Krasny 2001. The particles are divided into a nested set of clusters and the particleparticle interactions are replaced by a smaller number of particle-cluster interactions which can be efficiently computed using a multipole approx– imation. Treecode algorithms reduce the cost to O(N log N) Barnes & Hut 1986 or O(N) Greengard &; Rokhlin 1987. The algorithm used here follows an approach developed for two-dimensional vortex sheet motion and applies Taylor series in Cartesian coordinates to approximate the regularized Biot-Savart kernel Draghicescu & Draghicescu 1995. The algorithm implements several adaptive techniques including variable order approximation, nonuniform rectangular clusters, and a run-time choice between Taylor approximation and direct summation. The results presented below used up to particles. Figure 5 shows the growth of azimuthal waves on a vortex ring. The initial condition is a circular-disk vortex sheet with a transverse az– imuthal perturbation of wavenumber Such waves have been observed in experiments Lim & Nickels 1995; Shariff & Leonard 1992. Figure 6 presents a simulation of vortex ring merger. Experiments show that two vortex rings moving side by side in the same direction draw close to each other and merge into a single ring Schatzle 1987. This is a popular test case for numerical methods Cottet & Koumoutsakos 2000 and there is much interest in this flow as an example of vortex reconnection Kida & Takaoka 1994. In the present simulation, the rings are formed by the roll-up of two initially flat circular-disk vortex sheets. In figure 6, the material sheet surfaces are plotted in column and the associated vorticity isosurfaces are plotted in columns The vorticity was computed by differentiating the regularized Biot-Savart velocity integral. Two isosurfaces are plotted, (light gray) and (dark gray) of the initial maximum vorticity amplitude. The material surfaces representing the sheets approach closely but they do not actually touch. On the other hand, the vorticity isosurfaces apparently cancel and reconnect as the rings approach each other. Figure 7 shows a closeup of the material surfaces in the ring merger simulation at the final time. In the region where the two rings approach closely the core radius is small. The irregular small-scale features appear– ing in this region are artifacts of the graphics software; the simulation is well-resolved. The core radius becomes larger away from this region.


9

10



4.

11

Summary

Simulations of vortex sheet roll-up using the vortex blob method were presented. In planar and axisymmetric flow, the sheet develops irregular small-scale features due to the onset of chaos. A Poincaré section of the vortex sheet flow displays resonance bands and a heteroclinic tangle, the generic features of a chaotic Hamiltonian system. The chaos is induced by a self-sustained oscillation in the vortex core rather than external forcing. In three-dimensional flow, a treecode algorithm is applied to simulate the growth of azimuthal waves on a vortex ring and the merger of two vortex rings. In the latter case, the vorticity isosurfaces cancel and reconnect even though the material sheet surfaces do not touch. The simulation is nominally inviscid and apparently it is the regularized Biot-Savart integration that allows the vorticity to cancel. There is evidence that vortex blob simulations provide a good approximation for true viscous flow in certain cases Nitsche &; Krasny 1994; Tryggvason, Dahm & Sbeih 1991, but it will be necessary to perform comparisons with experiments and Navier-Stokes simulations to determine the physical validity of the present findings.

Acknowledgments This work was supported by the U. S. National Science Foundation through grants DMS-9506452, DMS-9996254, and DMS-0107187.

12


References ANDERSON, C. R. 1985 A vortex method for flows with slight density variations. Comput. Phys. 61, 417–444. AREF, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21. BARNES, J. & HUT, P. 1986 A hierarchical O(N log N) force-calculation algorithm. Nature 324, 446–449. CHORIN, A. J. & BERNARD, P. S. 1973 Discretization of a vortex sheet, with an example of roll-up, J. Comput. Phys. 13, 423–429. COTTET, G.-H. & KOUMOUTSAKOS, P. D. 2000 Vortex Methods: Theory and Practice. Cambridge University Press. DRAGHICESCU, C. & DRAGHICESCU, M. 1995 A fast algorithm for vortex blob interactions, J. Comput. Phys. 116, 69–78. GREENGARD, L. & ROKHLIN, V. 1987 A fast algorithm for particle simulations, J. Comput. Phys. 73, 325–348. GUCKENHEIMER, J. & HOLMES, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag. KIDA, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Jpn. 50, 3517–3520. KIDA, S. & TAKAOKA, M. 1994 Vortex reconnection, Annu. Rev. Fluid Mech. 26, 169–189. KRASNY, R. 1987 Computation of vortex sheet roll-up in the Trefftz plane. J. Fluid Mech. 184, 123–155. KRASNY, R. & NITSCHE, M. 2001 The onset of chaos in vortex sheet flow. J. Fluid Mech. 454, 47–69. LIM, T. T. & NICKELS, T. B. 1995 Vortex rings. In Fluid Vortices (ed. S. I. Green), pp. 95–153. Kluwer. LINDSAY, K. & KRASNY, R. 2001 A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow. J. Comput. Phys. 172, 879–907. MOORE, D. W. 1972 Finite amplitude waves on aircraft trailing vortices, Aero. Quart. 23, 307–314. MOORE, D. W. 1979 The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. Roy. Soc. Land. A 365, 105–119. NITSCHE, M. & KRASNY, R. 1994 A numerical study of vortex ring formation at the edge of a circular tube, J. Fluid Mech. 276, 139–161. OTTINO, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press. ROM-KEDAR, V., LEONARD, A. & WIGGINS, S. 1990 An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 314, 347–394. ROSENHEAD, L. 1930 The spread of vorticity in the wake behind a cylinder, Proc. Roy. Soc. Land. A127, 590–612. SCHATZLE, P. R. 1987 An experimental study of fusion of vortex rings, Ph.D. Thesis, California Institute of Technology. SHARIFF, K. & LEONARD, A. 1992 Vortex rings. Ann. Rev. Fluid Mech. 24, 235–297. TRYGGVASON, G., DAHM, W. J. A. & SBEIH, K. 1991 Fine structure of vortex sheet rollup by viscous and inviscid simulations, ASME J. Fluid Engin. 113, 31–36. WIGGINS, S. 1992 Chaotic Transport in Dynamical Systems. Springer-Verlag.

Merging of non-symmetric Burgers vortices Keith HIGGINS, M. S. CHONG & Andrew OOI Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia [email protected]

Abstract The merging of two Burgers vortices in an irrotational background straining flow at high Reynolds number is studied. Both axisymmetric and biaxial strain fields are considered. The merging events produce fine scale spiral vortex structures. For biaxial strain, a cat’s-eye streamline pattern emerges and vorticity is transported away by the background strain. For intermediate strain ratios, the onset of vortex merging is delayed and the resulting vorticity contours are distorted compared to the axisymmetric case. The merging is suppressed for sufficiently large strain ratios. With a strain that’s non-axisymmetric, The dance of two whorls is quite hectic; They pulse and converge And eventually merge; The whole process is quite apoplectic!

1.

Introduction

The discovery of coherent vortical structures in numerical simulations of turbulence has renewed interest in the properties of the Burgers vortex, a well-known equilibrium solution of the Navier-Stokes equations. The Burgers vortex has been used in the development of many theories of turbulent fine scales, where these are modelled using spatial ensembles of stretched vortices (Pullin & Saffman 1998 and references therein). The Burgers vortex results when a unidirectional vorticity field is embedded in an axisymmetric irrotational background straining flow, where and are the strain rates and field can be characterised by a strain ratio,

This strain

13 K. Bajer and H.K. Moffatt (eds.). Tubes, Sheets and Singularities in Fluid Dynamics, 13–18. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

14

K. Higgins, M. S. Chong and A. Ooi

For the case of axisymmetric strain, and Burgers (1948) obtained the equilibrium vorticity distribution in cylindrical coordinates

Here the circulation is constant and is the kine– matic viscosity. This represents a balance between viscous diffusion and vorticity intensification due to the stretching strain field. A Reynolds number based on the circulation is defined as

The study of the dynamics of non-symmetric stretched vortices was initiated by Robinson & Saffman (1984). They calculated numerically steady solutions for a single vortex in a non-symmetric axial strain field for small , showing how and affected the ellipticity and orientation of the vortex. Moffatt, Kida & Ohkitani (1994) developed a large asymptotic theory of a single stretched vortex in a non-symmetric strain field. Prochazka & Pullin (1998) extended this work to develop quasi-steady non-symmetric numerical solutions for

Merging of non-symmetric Burgers vortices

15

large and large Buntine & Pullin (1989) computed the time dependent merging of two Burgers vortices in an axisymmetric strain field for They showed that the merging produced spiral vorticity structures and eventual relaxation to an axisymmetric Burgers vortex. The aim of the present work is to examine the merging of two Burgers vortices separated by a distance at Figure 1 shows the two background strain scenarios considered. For the streamlines are almost circular in the far-field. However, in the case of biaxial strain where stagnation points appear in the flow and a cat’s-eye type streamline pattern emerges.

2.

Governing equations and numerical method For uni-directional vorticity, the vorticity transport equation is

where Here the full velocity field is written as the sum of the velocities induced by the vortex and the background straining flow

Length and time scales are chosen to be and The governing equations are non-dimensionalised accordingly. The hybrid spectral finite-difference method developed by Buntine & Pullin (1989) is employed to solve (5) numerically on an infinite radial domain. The prescribed initial condition is a superposition of two Burgers vortices, which can be characterised by their total circulation, The boundary condition ensures that all quantities vanish at

3.

Results Moffatt et al. (1994) introduced the function

which represents the excess of vortex induced dissipation over the background values due to the strain field. Here is the second invariant of the characteristic equation of the dimensionless rate-of-strain tensor (Chong, Perry &; Cantwell 1990). Figure 2 shows radial profiles of and D for and the corresponding vorticity contours for strain ratios (a) and (b) at Re = 5000.

16


The vorticity contours of figure 2 (a) show the emergence of spiral arm structures. These tighten under the action of differential rotation due to wind-up by the merging vortex cores and radial contraction due to the background straining flow. Portions of the arms are eventually wound into the resulting axisymmetric core, or are dissipated by viscosity. Turning now to the radial D profiles, we see that the initial local minimum in D is at the same radial position as the local maximum in The time sequence shows that the positions of the local minima in the D profile follow the positions of the spiral arms. At large times, a single axisymmetric Burgers vortex emerges. Here the local maximum in the D profile occurs at finite radius and in agreement with the asymptotic results of Moffatt et al. (1994). Figure 2(b) shows that the merging event is modified in a biaxial strain field. The vorticity contours and radial vorticity profile at show the vortex cores connected by a thin vortex sheet, produced from an earlier interaction of the cores. The cores are subsequently pulled apart by the positive axis strain. They reach an almost stationary position where the background strain field balances the convective ten– dency to merge. However, at time convective effects dominate and merging proceeds. Vorticity in the emerging spiral arms is transported away by the background strain. Fine spiral structure develops by although the tendency of the core to wind-up the arms is somewhat balanced by unwinding due to the positive strain. Hence the spiral structure is less persistent than in the case. At the vortices have merged and the resulting vorticity distribution begins to relax to a non-symmetric shape. Since the simulation was stopped at this time, it is not known if a steady solution would be obtained in this case. At large times the radial D profile has a local maximum at finite radius, although This is again in agreement with the asymptotic results of Moffatt et al. (1994) for Figure 3 shows the evolution of the vortex core separation, as defined in the inset. All curves intercept the axis at For initially decreases, and then oscillates as the cores begin to merge. For the and axes are aligned with the axes of maximum stretching and contracting strain rates respectively. Here the cores are initially pushed together, as they are aligned with the axis. As the cores orbit each other and become aligned with the axis, increases. Convective effects eventually dominate and decreases. Oscillations in ensue and the merging is completed at a later time compared to the axisymmetric case. However, for (these vorticity contours were not shown in figure 2), the background strain dominates and the core separation increases indefinitely.

Merging of non-symmetric Burgers vortices

17


18

4.

Conclusion

The merging of Burgers vortices in both axisymmetric and biaxial strain fields parametrised by a strain ratio has been studied at Fine vortex spiral arm structure is produced in both cases, provided that merging proceeds. For sufficiently large merging is suppressed. Qualitative comparisons of the function D with the asymptotic results of Moffatt et al. (1994) show good agreement. Support from NATO, IUTAM and The Australian Research Council is gratefully acknowledged.

References BUNTINE, J. D. & PULLIN, D. I. 1989 Merger and cancellation of strained vortices. J. Fluid Mech. 205, 263–295. BURGERS, J. M. 1948 A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199. CHONG, M. S., PERRY, A. E. & CANTWELL, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765–777. MOFFATT, H. K., KIDA, S. & OHKITANI, K. 1994 Stretched vortices - the sinews of turbulence; large-Reynolds-number asymptotics. J. Fluid Mech. 259, 241–264. PROCHAZKA, A. & PULLIN, D. I. 1998 Structure and stability of non-symmetric burgers vortices. J. Fluid Mech. 363, 199–228. PULLIN, D. I. & SAFFMAN, P. G. 1998 Vortex dynamics in turbulence Annu. Rev. Fluid Mech. 30 31–51. ROBINSON, A. C. & SAFFMAN, P. G. 1984 Stability and structure of stretched vortices Stud. Appl. Maths 163–181.

Effect of stretching on vortices with axial flow Maurice ROSSI1, Ivan DELBENDE 2 & Stéphane LE DIZÈS3 1

LMM, Université Paris VI, 4 place Jussieu, F-75252 Paris Cedex 05, France [email protected]

2

LIMSI, CNRS–UPR 3251, BP 133, F-91403 Orsay Cedex,France [email protected]

3

IRPHE, CNRS–UMR 6594, 49 rue F. Joliot-Curie, B.P. 146, F-13384 Marseille Cedex 13, France [email protected]

Abstract It is shown that a weak time-dependent stretching might rapidly destabilise a vortex, thus providing a mechanism for vortex bursts observed in turbulent flows. This study addresses the three-dimensional stability of a stretched viscous Batchelor vortex. In a fashion quite similar to Lundgren’s transformation, the strain field is almost eliminated from the linear equations that govern three-dimensional perturbations. Such transformed equations, which are reminiscent of those for the swirling jet instability, are then numerically solved in the simple case of a compression phase followed by a stretching phase. Simulations qualitatively demonstrate how strain and azimuthal vorticity cooperate to destabilise the vortex. When the strain on a tube is unsteady, For excitable modes please be ready! By scaling we claim: In a suitable frame, A vortex bursts into an eddy.

1.

Introduction

In turbulent flows, vorticity filaments have been individuated in real experiments (Cadot, Douady & Couder 1995) or numerical simulations (Vincent & Meneguzzi 1991). Their evolution clearly depends on the varying background stretching field generated by surrounding vortices. In this context, numerical studies have been performed where the stretching field acting on a straight vortex is non-uniform, time-periodic or generated by an array of vortex rings (see for instance Verzicco, Jiménez & 19 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 19–24. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

20

M. Rossi, I. Delbende and S. Le Dizès

Orlandi 1995; Verzicco & Jiménez 1999; Abid et al. 2002). Moreover new stretched vortex solutions have recently been found by Gibbon, Fokas & Doering (1999). These basic flows, which satisfy the Navier–Stokes equations, describe a vortex possessing both azimuthal and axial velocities, subjected to a time-dependent strain field oriented along with its axis. Such solutions are different from the previously studied cases and more pertinent to the description of real turbulent flows since both basic axial and azimuthal vorticity are present and the vorticity field is no longer aligned with a principal direction of strain. In this work, which addresses the dynamics of infinitesimal three-dimensional perturbations superimposed on such an unsteady stretched vortex, it is shown that another main consequence follows: the interaction between axial and azimuthal basic vorticity components gives rise, as in an unstretched swirling jet (see Lessen, Singh & Paillet 1974; Mayer & Powell 1992). to a powerful instability mechanism. Indeed, the cooperation between the classical swirling jet instability and stretching might destabilise the stretched basic vortex flow.

2.

The basic vortex flow.

A new set of exact Navier–Stokes solutions which include a global stretching effect, has recently been obtained by Gibbon et al. (1999). These solutions differ from the earlier solutions by Lundgren (1982) by two main features: stretching may be non-uniform and axial velocities are present. Here we put the emphasis on the second feature and we select an axisymmetric velocity field which reads, in polar coordinates as:

This flow is a combination of an unsteady linear strain field, an axisymmetric vortex (2) of constant circulation and an unsteady axial jet (3) of initial centerline velocity The core size which is identical for axial and azimuthal velocities, depends on kinematic viscosity and strain rate via

Effect of stretching on vortices with axial flow

where the modified time

21

is defined through the dimensionless quantity

and clearly coincides with real time for zero rate of strain At each time these expressions characterise an instantaneous Batchelor vortex of radius and swirl number (the typical ratio between characteristic azimuthal and axial velocities) Other flow profiles can be used but the present choice is natural since the stability properties of the unstretched Batchelor vortex are rather well-documented. Solutions (2)–(3) are clearly more general than the classical Burgers vortex as vorticity is not necessarily aligned along with the vortex axis or with a principal direction of strain. More importantly, it is easily checked that axial vorticity is enhanced (resp. reduced ) by stretching (resp. compression) while azimuthal vorticity is reduced (resp. enhanced). This dynamical exchange between vorticity components makes these solutions better candidates for vortex filament models. Note that quantity might a priori follow any time dependence. In a turbulent environment, this variation models the action of large structures on the vortex, i.e. it is linked to the dynamics of distant vortices. Such an explicit relation is not considered here. However, we expect quantity to fluctuate with a characteristic time larger than the vortex turnover time.

3.

Three-dimensional stability equations.

Using time and space rescalings, Gibbon et al. (1999) have shown that the stretched solution and the classical unstretched selfsimilar Batchelor vortex are connected. In the present section, we extend this property to the three-dimensional stability of this basic flow, i.e. we provide a connection between equations governing infinitesimal threedimensional perturbations of the stretched solution to those governing infinitesimal disturbances of its unstretched diffusing Batchelor vortex. Infinitesimal fluctuations of pressure and velocity (here written in polar coordinates) superimposed upon velocity field are governed by a linear system inhomogeneous with respect to time and spatial coordinates and Because of these explicit dependences, such a stability problem cannot be simplified by standard Fourier transform techniques. In order to handle this difficulty, three transformations

22


are now applied on the stability equations. The first two were previously used by Lundgren to relate the stretched and unstretched basic vortex solution. The change of variable for when written in terms of a time-dependent wavenumber, is commonly used in the context of inertial waves or rapid distortion theory. Both changes do neither modify the partial derivatives with respect to nor affect the elementary volume but they do eliminate terms relative to stretching or the explicit Finally, the additional transformations

are introduced to preserve the ratio between perturbation and basic state or else to remove the amplification factor affecting perturbation amplitudes due to external strain. After completion, the transformed linear system is quite close to the classical 3D stability of the vinstretched basic flow state: it differs only through the presence of terms associated with For details, the reader may consult a forthcoming article which will appear in JFM. The 2D stability of a stretched vortex (2)–(3) is directly related to the 2D stability problem for its unstretched and diffusing companion via these various transformations. For the general 3D stability problem, a quasi-static approximation may be assumed in which the linear stability characteristics are directly connected to those of an unstretched Batchelor vortex as follows. Near a time viscous diffusion may be neglected in the basic state: a char– acteristic length scale for radial variable is then given by the core size and a characteristic time scale by In turbu– lence studies, stretching due to large scales is small with respect to the velocity inside the vortex: we may hence assume As a result, although it might be of order one, is slowly evolving in time since in dimensionless form and can thus be considered frozen to its value Let us now employ as scale for velocity components and as a scale for veloc– ity component as a scale for pressure and as a scale for The dimensionless quasi-static equations become identical to equations governing the linear perturbations of an unstretched Batchelor vortex with swirl number and Reynolds num– ber Within this approximation, we infer that the strain however small may destabilise the vortex by bringing the unsteady swirl number in a region in which the companion unstretched Batchelor vortex is affected by a swirling jet instability. This results in a strong and rapid evolution of perturbations which may disrupt the previously coherent stretched vortex structure.

Effect of stretching on vortices with axial flow

4.

23

Numerical simulations.

Instead of the above mentioned quasi-static approximation, stabilisation of the stretched vortex is considered anew using numerical simulations of the three-dimensional equations. The simplest fluctuation containing a single stretching and compression phase is simulated: for the vortex is compressed with a constant negative strain rate and, for the vortex is stretched with the opposite strain rate Due to lack of space, only the main results are given here (for details, see the forthcoming article to appear in JFM) and only the case is shown. In that instance, the instantaneous swirl number, initially set to decreases during the compression down to the value and thereafter increases back towards its initial value: it thus lies initially outside the classical swirling jet instability bandwith, then penetrates it and finally leaves it. Moreover, the Reynolds number is fixed to Re = 667 and all simulations are initiated with perturbations in the form of a random field obtained by projecting a spatial white noise on divergence-free fields. The numerical simulations confirm the predicted stabilising effect of stretching and they agree with the theoretical framework presented within the quasi-static approximation: indeed, parameter is a good indicator to understand the observed time evolution. In particular, the fluctuation growth phase lies within the instability domain. The transverse flow structure is represented for different times in figure 1 where

24


isocontours of the axial component of the perturbation vorticity are plotted at a fixed streamwise location. The physical domain shown is a transverse cut of the vortex centred around its core. One observes first a filtering of small scales and the gradual appearance of a vortical structure. The observed symmetries indicate the presence of azimuthal wavenumbers and which are reminiscent of the most unstable azimuthal modes of the viscous unstretched Batchelor vortex at During the subsequent stretching phase, this structure is finally rolled-up. Compression and stretching effects appear on the average vortex core size. Moreover it can be shown that the growth rate achieves a maximum for axial wavenumbers around 2. In the nonlinear régime, this evolution might lead to the formation of several interwoven filaments. In turbulent flows, vortex filaments are subjected to fluctuating strain caused by large scale structures. The above simulations emulate such fluctuations and provide the following scenario: an initially stable vortex is brought inside the swirling jet instability domain by a long enough compression. If it remains in that region, perturbations may reach high amplitudes, causing the vortex to disrupt into filaments. This two-step mechanism tentatively explains the abrupt vortex destruction as observed in turbulent flows.

References ABID, M., ANDREOTTI, B., DOUADY, S. & NORE, C. 2002 Oscillating structures in a stretched–compressed vortex. J. Fluid Mech. 450, 207–233. CADOT, O., DOUADY, S. & COUDER, Y. 1995 Characterization of the low pressure filaments in three-dimensional turbulent shear flow. Phys. Fluids 7 (3), 630–646. DELBENDE, I., ROSSI. M. & LE DIZÈS, S. 2002 Stretching effects on the threedimensional stability of vortices with axial flow J. Fluid Mech. 454, 419–442. GIBBON, J.D., FOKAS, A.S. & DOERING, C.R. 1999 Dynamically stretched vortices as solution of the 3D Navier–Stokes equations. Physica D 132, 497–510. LESSEN, M., SINGH, P.J. & PAILLET, P. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753–763. LUNDGREN, T.S. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids , 25(12), 2193–2203. MAYER, E.W. & POWELL, K.G. 1992 Viscous and inviscid instabilities of a trailing line vortex, J. Fluid Mech. 245, 91–114. VERZICCO, R. & JIMÉNEZ, J. 1999 On the survival of strong vortex filaments in ‘model’ turbulence. J. Fluid Mech. 394, 261–279. VERZICCO, R., JIMÉNEZ, J. & ORLANDI, P. 1995 On steady columnar vortices under local compression. J. Fluid Mech. 299, 367–388. VINCENT, A. & MENEGUZZI, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20.

Optimal two-dimensional perturbations in a stretched shear layer Stéphane LE DIZÈS Intitut de Recherche sur les Phénomènes Hors Équilibre, 49, rue F. Joliot Curie, BP 146, F-13384 Marseille cedex 13, FRANCE [email protected]

Abstract The evolution of 2D linear perturbations in a uniform shear layer stretched along the streamwise direction is considered in this work. The shear layer is assumed to have an error function profile. The width and strength of the shear layer evolve in time due to the combined effect of viscous diffusion and stretching. The timedependent basic flow is therefore characterised by two parameters: the stretching rate and the Reynolds number . Using a direct-adjoint technique, perturbations which maximise the energy gain during a time interval are computed for various and . The results are compared with those obtained using a normal mode decomposition of the perturbations (WKBJ approach). Transient growths are shown to be weak in a stretched shear layer by opposition to what is observed in boundary layer flows. A sheet that is stretched, in collusion With limited viscous diffusion, Upon a flat table, Is grossly unstable; But frankly, it’s all a delusion.

1.

Introduction

A vortex sheet during its roll-up or the braid region between two adjacent vortices in a shear flow are typical examples of shear layers stretched along the streamwise direction. The goal of this article is to determine under which conditions these stretched layers are unstable with respect to the Kelvin-Helmholtz instability. Shear layers in a non-viscous flow have been known to be unstable with respect to two-dimensional perturbations since the works of Kelvin, Helmholtz and Rayleigh. Betchov & Szewczyk (1963) analysed the effect of viscosity and demonstrated that the instability extended down to zero Reynolds number. Using a WKBJ ansatz, they also took into account the weakly diffusing character of the shear layer and computed


26

S. Le Dizès

the maximum energy gain that a plane wave perturbation could reach during a time interval As viscosity, stretching is expected to also affect the stability properties of the shear layer. It is known to induce time-dependency by elongating scales along the stretching direction, and contracting the other ones (see for instance Delbende et al., this volume). My goal is here to understand the combined effect of stretching and viscosity by analysing the stability properties of a uniformly stretched viscous shear layer. As the flow is time-dependent, the stability analysis is performed by searching the optimal perturbations, that is those which maximise the energy gain on a prescribed time interval (see Trefethen et al. 1993) . This method allows to consider strongly time-dependent flows associated with slow Reynolds number and large stretching rate. It also captures transient growths associated with the non-normal nature of the perturbation equations. Here, the optimisation procedure is implemented using the direct-adjoint technique of Andersson, Berggren & Henningson (1999). For weakly time-dependent flows, the results are compared with the WKBJ approach of Betchov & Szewczyk in order to quantify the transient growth contribution in the maximum energy gain.

2.

Perturbations on a stretched shear layer

We consider a shear layer which is uniformly stretched along the streamwise direction and which possesses a velocity field of the form :

where erf is the error function. The stretching rate positive constant such that the velocity magnitude of the shear layer vary according to

is assumed to be a and the width

with Here, both initial width and initial velocity magnitude are normalised. Thus the basic flow evolution only depends on two parameters: the strain rate and the (initial) Reynolds number . This Navier-Stokes solution is different from the so-called Burgers layer which is stretched along the vorticity direction (see Beronov, 1997).

Optimal two-dimensional perturbations in a stretched shear layer

27

Here, the width of the shear layer goes to a non-zero constant for large time but the vorticity magnitude decreases exponentially to zero. The goal of the paper is to analyse the dynamics of linear perturbations on this time-dependent flow. The equations satisfied by 2D linear disturbances are obtained by linearising the Navier-Stokes equations around the basic flow (1). The stretching terms can be eliminated by transforming the spatial variables according to with given by (3). This leads to equations homogeneous with respect to the modified streamwise variable This permits to do a modeby-mode analysis where perturbation vorticity and streamfunction are given, for each wavenumber by

Using these expressions, the governing equations for to reduce to:

3.

and

are found

Optimal perturbations

As the perturbation equations (5a,b) contain explicit time-dependency in both S and the behaviour of the perturbations is expected to vary in time. The relevant quantity to characterise a perturbation is thus no longer its growth rate but instead its energy gain during a given time interval. Here, we shall consider an energy density defined for each wavenumber by

such that the energy gain on is given by When and the time-dependency is weak: the perturbation can be searched under the form of a local normal mode (WKBJ approach). Thus, at each instant, the WKBJ perturbation has a local growth rate Using equations (5a,b), one can show that is given by

where is the dispersion relation of a normalised shear layer with an error function profile. From (7), a WKBJ estimate for the

28

energy gain

S. Le Dizès

is provided by

The optimal gain is obtained by maximising the gain over all the perturbations without any assumption on the form of the perturbation. It can be computed by a optimisation procedure based on direct integrations of equation (5a) and of its adjoint as shown in Andersson et al. (1999). The optimisation procedure is not restricted to configurations where and Moreover, it captures transient growths associated with interactions of non-orthogonal modes.

Optimal two-dimensional perturbations in a stretched shear layer

29

In order to identify these effects, it is useful first to consider the case and for which the shear layer is stationary. Indeed, in that case, the WKBJ approach is exact and equivalent to a classical normal mode analysis. Differences between WKBJ gain and optimal gain can therefore only be due to algebraic growth generated by mode interactions. The results are displayed on figure 1(a) where the mean growth rate of the optimal perturbation is compared to the normal mode growth rate for various As expected, it is for the smallest that the departure from the normal mode analysis is the strongest. Note in particular that for the largest mean growth rate is obtained for a stable wavenumber. Nevertheless, the gain difference between WKBJ and optimal approaches remains small for all as seen on figure 1(b). In this figure are also analysed viscous effects. It shows that viscosity does not modify the difference between maximum WKBJ gain and maximum optimal gain. This difference is approximately constant and limited to a factor 5. The first conclusion of this work is therefore that transient growth associated with mode inter-

30

S. Le Dizès

actions remains limited in a shear layer. This flow is therefore different from boundary layer flows where the energy gain induced by transient growths can reach several thousands (Trefethen et al. 1993). Transient growths are also weak in presence of stretching (not shown here). This implies that the effect of stretching can be understood from the WKB J approach, and more precisely from the variation with respect to and of the local growth rate defined in (7). Characteristics of the most dangerous optimal perturbations are displayed on figures 2(a,b) for various Reynolds numbers and stretching rate. Figure 2(a) shows that the maximum gain diminishes with both viscosity and stretching for a fixed The stabilising effect of viscosity on the gain is directly related to the damping character of viscosity on the local growth rate. The same argument applies for stretching as the local Reynolds num[see expression (7)], decreases as ber, which is defined by stretching increases. With the same argument, the variation of the optimal wavenumber with respect to and shown on figure 2(b) can be understood. Indeed, as the local wavenumber decreases when or increases, the initial wavenumber must also increase with and in order to have the local wavenumber close to the most unstable wavenumber during a longer period of time. The spatial structures of the most dangerous optimal perturbation at and are displayed on figures 3 for a case dominated by viscosity (shown on the left) and and a case dominated by stretching (shown on the right). On these plots, both spatial variables are nondimensionalised by the initial width of the shear layer. This permits to compare the wavelengths of each case, and to see the spreading or contraction of the shear layer. It is interesting to note that the most apparent difference between both cases is the initial wavelengths.

Acknowledgments O. N. E. R. A. is gratefully acknowledged for its financial support.

References ANDERSSON, P., BERGGREN, M. & HENNINGSON, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11 (1), 134–150. BERONOV, K. N. 1997 Vorticity layers in unbounded viscous flow with uniform rates of strain. Fluid Dynamics Research 21, 285–302. BETCHOV, R. & SZEWCZYK, A. 1963 Stability of a shear layer between parallel streams. Phys. Fluids 6 (10), 1391–1396. ROSSI, M., DELBENDE, I. & LE DIZÈS, S. 2001 Effect of stretching on vortices with axial flow. This volume. TREFETHEN, L. N., TREFETHEN, A. E., REDDY, S. C. & DRISCOLL, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.

Advection–diffusion of a passive scalar in the flow of a decaying vortex Konrad BAJER1, Andrew P. BASSOM2 & Andrew D. GILBERT2 1

Warsaw University, Institute of Geophysics ul. Pasteura 7, 02-093 Warszawa, Poland [email protected]

2

University of Exeter, School of Mathematical Sciences Exeter, EX4 4QE, U.K. Here’s a blob that’s subjected to swirl; It’s a problem for somebody virile! But right at the core Where it turns more and more, That’s where I get in a whirl.

1.

Introduction

The shear due to the presence of a vortex is known to have strong effects on the advection–diffusion of a passive scalar. If there is an initial gradient of the scalar concentration with the spatial scale large compared with the size of the vortex, then this gradient will decay on a timescale much faster than normal diffusion, making the scalar concentration almost uniform in the vicinity of the vortex (Rhines & Young 1983). The process is governed by the advection–diffusion equation,

which is a linear equation for the scalar field The application of the linear advection–diffusion equation extends beyond the description of tracers carried by the fluid. In flows with a symmetry the same equation governs other important physical quantities. In the evolution of a weak magnetic field in two-dimensional MHD, for which both the velocity and the field vectors lie in the same plane, the evolution of the field is equivalent to the advection–diffusion of a flux function (Bajer 1998). The annihilation of the gradient of the flux function near a vortex implies the cancellation of the field — the phenomenon known in MHD as ‘flux expulsion’ (see Bajer, Bassom & Gilbert 2001 and the references therein). Similarly, the streamwise ve31 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 31–36. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

32

K. Bajer, A.P. Bassom and A.D. Gilbert

locity component in a boundary layer is carried by streamwise vortices, the streamwise momentum playing the role of a passive scalar (Pearson & Abernathy 1984, Moore 1985).

2.

Similarity solution

We consider the concentration of a passive scalar under the process of advection–diffusiori in the flow due to single axisymmetric Gaussian vortex where, in polar co-ordinates,

and the corresponding azimuthal velocity is equal to

We solve the advection–diffusion equation (1). The equation is linear and the flow is axisymmetric, so the Fourier modes of

evolve independently, satisfying

There is no natural length scale in this problem so we try for a similarity solution,

and with

given by (2) we obtain

Here is the Péclet number and is the Schmidt number. Therefore, the problem admits a similarity solution valid for all values of Pe and Sc. We will now analyse the regime when and but the vortex Reynolds number is fixed. We focus on the situation when the gradient of is uniform at the initial instant i.e., the mode but the following analysis can be easily adapted to other values of

Advection–diffusion of a passive scalar in the flow of a decaying vortex

Rescaling the similarity variable

33

we obtain

where is the Fourier mode with This equation has a regular singular point at and an irregular singular point at infinity, and is in the form suitable for the WKB analysis with as the perturbation parameter. We take

The regular solution takes the form,

where is now a similarity variable associated with the vortex rather than the scalar and At the second order we obtain

We plot with taken up to the second order in the WKB expansion, with Sc = 100 (fig. 1) and Sc = 10 (fig. 2), for different values of Re. The solution (9) is the ‘outer’ solution valid, in principle, in the region satisfying the correct boundary condition,

Obtaining the details of the solution at the centre of the vortex requires delicate matching of two regions, and the details of which will be given elsewhere. However when it transpires that the true value of the solution at is exponentially small (Moore 1985), so the WKB solution (9) vanishing at is then a good approximation almost everywhere.

3.

Conclusions

We have shown the existence of a similarity solution of the advection– diffusion equation representing the evolution of the initially uniform gradient of a passive scalar in the flow field of a diffusing Gaussian vortex.

34


Advection–diffusion of a passive scalar in the flow of a decaying vortex

35

36


As a special case, Sc=1 gives the solution of the Navier-Stokes equation obtained by Abernathy & Pearson (1984) and Moore (1985) where the diffusing quantity is the background vorticity perpendicular to the axis of the central vortex. The solution yields a quantitative description of the process of spiral wind-up (Bassom & Gilbert 1998, 1999), giving accelerated diffusion due to the reduction of radial scale and the enhancement of the concentration gradients. Such a situation occurs when a name sheet or another chemical reaction front is wrapped around a diffusing vortex. Marble (1985) studied the case when the initial fuel concentration is given by a step function. The similarity solution (9) and the analogous solutions for higher Fourier modes can be added to approximate the evolution of any initial concentration profile. Both small and large Sc regimes are of practical interest.

Acknowledgments We thank Andrew Soward and Dale Pullin for helpful comments and discussions. This work was supported by the British/Polish Joint Research Collaboration Programme of the British Council and and the State Committee for Scientific Research (KBN). K.B. also acknowledges the support from the KBN under the grant number 2 P03B 13517.

References BAJER, K. 1998 Flux expulsion by a point vortex. Eur. J. Mech. B/Fluids 17, 653–664. BAJER, K., BASSOM, A. P. & GILBERT, A. D. 2001 Accelerated diffusion in the centre of a vortex. J. Fluid Mech. 437, 395–411. BASSOM, A. P. & GILBERT A. D. 1998 The spiral wind-up and dissipation of vorticity in an inviscid planar vortex. J. Fluid Mech. abf 371, 109–140. BASSOM, A. P. & GILBERT A. D. 1999 The spiral wind-up and dissipation of vorticity and a passive scalar in a strained planar vortex. J. Fluid Mech. 398, 245–270. KAWAHARA, G., KIDA, S., TANAKA, M. & YANASE, S. 1997 Wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube in a simple shear flow. J. Fluid Mech. 353, 115–162. MARBLE, F. E. 1985 Growth of a diffusion flame in the field of a vortex. In Recent Advances in the Aerospace Sciences. Ed. C. Casci. Plenum 1985. MOORE, D. W. 1985 The interaction of a diffusing vortex and an aligned shear flow. Proc. R. Soc. Land. A 399, 367–375. PEARSON, C. F. & ABERNATHY, F. H. 1984 Evolution of the flow field associated with a streamwise diffusing vortex. J. Fluid Mech. 146, 271–283. RHINES, P. B. & YOUNG, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145.

Linear stability of a vortex ring revisited Yasuhide FUKUMOTO1 & Yuji HATTORI2 1

Graduate School of Mathematics, Kyushu University 33, Fukuoka 812–8581, Japan [email protected]

2

Faculty of Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan

Abstract We revisit the stability of an elliptically strained vortex and a thin axisymmetric vortex ring, embedded in an inviscid incompressible fluid, to three-dimensional disturbances of infinitesimal amplitude. The results of Tsai & Widnall (1976) for an elliptically strained vortex are simplified by providing an explicit expression for the disturbance flow field. A direct relation is established with the elliptical instability. For Kelvin’s vortex ring, the primary perturbation to the Rankine vortex is a dipole field. We show that the dipole field causes a parametric resonance instability between axisymmetric and bending waves at intersection points of the dispersion curves. It is found that the dipole effect predominates over the straining effect for a very thin core. The mechanism is attributable to stretching of the disturbance vortex lines in the toroidal direction. When water is pushed through a hole, The ring vortex plays a key role; When the core is quite thin, Res’nant waves are packed in; To grasp them, why, that is the goal!

1.

Introduction

Vortex rings are invariably susceptible to wavy distortions, leading sometimes to violent wiggles and eventually to disruption. We revisit the linear stability problem of a thin vortex ring. It is widely accepted that the Moore-Saffman-Tsai- Widnall instability is responsible for genesis of unstable waves. Remember that this is an instability for a straight vortex tube subjected to a straining field in a plane perpendicular to the tube axis (Moore & Saffman 1975, Tsai & Widnall 1976). When viewed locally, a thin vortex ring looks like a straight tube. For simplicity, we restrict our attention to the Rankine vortex, a circular core of uniform vorticity. The Rankine vortex supports a family of neutrally stable three-dimensional waves of infinitesimal amplitude, being


38

Y. Fukumoto and Y. Hattori

well known as the Kelvin waves. The vortex ring induces, on itself, not only a local uniform flow but also a local straining field akin to a pure shear with principal axes oriented at 45° from the symmetric axis in the meridional plane (Widnall, Bliss & Tsai 1974, Widnall & Tsai 1977). A pure shear with the principal axes perpendicular to the vorticity deforms the core into an ellipse. This is a quadrupole field proportional to and in terms of polar coordinates in the meridional plane. The mechanism for the Moore-Saffman-Tsai-Widnall instability is a parametric resonance between the left- and right-handed bending waves via this quadrupole field. As shown by Bayly (1986) and Waleffe (1990), this instability has much in common with the elliptical instability, implying the ubiquity of the former; the influence of neighbouring vortices is, in the leading-order approximation, incorporated as a linear shear flow. The asymptotic solution of the Navier-Stokes equations for a thin vortex ring in powers of a small parameter the ratio of the core to the ring radii, starts with a circular-cylindrical vortex tube, at Then a dipole field proportional to follows at The quadrupole field comes as a correction at (Fukumoto & Moffatt 2000). Despite its dominance, the dipole field has not attracted as much attention as it deserves. This paper addresses a possible instability that the dipole field can trigger. According to Krein’s theory of parametric resonance in Hamiltonian systems (MacKay 1986), a single Kelvin wave cannot be fed by perturbations breaking the circular symmetry. An instability becomes permissible only for a superposition of at least two modes with the same wavenumber and the same frequency. Subjected to the dipole field, two Kelvin waves with angular dependence and can cooperatively be amplified at the intersection points of dispersion curves if the condition is met. As a first step, we investigate a parametric resonance that may occur between axisymmetric and bending waves. In §2, we give a concise description of the problem setting for linear stability analysis. In §3, the Kelvin waves are recalled. Before proceeding to a vortex ring, in §4, we have a look at the paper by Tsai & Widnall (1976), hereafter being referred to as TW76, for the stability of a strained vortex. A simplification is achieved by implementing the indefinite integrals left in TW76. In §5, we revisit the linear stability of a vortex ring. The growth rate of the dipole instability is compared with that of the Moore-Saffman-Tsai-Widnall instability. An interpretation is given of the mechanism for the new instability.

Linear stability of a vortex ring revisited

2.

39

Setting of linear stability problem

We write down the steady asymptotic solutions of the Euler equations for the circular vortex with uniform vorticity subject to some perturbations as the basic flow fields. We begin with an elliptic vortex. The perturbation is a plane pure shear. We expand the MooreSaffman elliptic vortex (Moore & Saffman 1971) in powers of strength of the shear. The notation follows that of TW76. Let us introduce cylindrical coordinates with the along the centreline of the circular core. The radial coordinate is normalised by the core radius and the velocity by the maximum azimuthal velocity Here is the circulation of the vortex. Let the and of velocity field be U and V, and the pressure be P inside the core The flow field in the exterior domain is assumed to be irrotational and we denote its velocity potential by The first-order truncation of the Moore-Saffman vortex is written as follows:

The leading-order flow is the Rankine vortex which is written, in dimensionless form, as

The first-order perturbation is a pure shear flow given by

a quadrupole field. The boundary shape of the core cross-section is an ellipse with the major axis along Next we turn to Kelvin’s vortex ring, a thin axisymmetric vortex ring with vorticity proportional to the distance from the axis of symmetry, which propagates steadily in an inviscid incompressible fluid. It is assumed that the ratio of the core radius to the ring radius R is very small: Introduce local cylindrical coordinates in the meridional plane, fixed to the ring, with the origin maintained at the centre of the circular core and with the angle measured from the direction parallel to the axis of symmetry. The detail of the asymptotic solution is found, for instance, in Widnall & Tsai (1977), hereafter being referred to as WT77. The leading-order

40


flow is the Rankine vortex (2). At the effect of curvature is called into play, and the perturbation flow field takes the following form:

The streamline pattern in the exterior region resembles that of the flow past a circular cylinder, being characteristic of the dipole field. It is noteworthy that a straining field appears at Roughly speaking, We inquire into evolution of three-dimensional disturbances of infinitesimal amplitude superposed on these steady flows. Following the prescription of Moore & Saffman (1975) and TW76, we seek the solution for disturbance velocity in a power series of a small parameter to first order: The wavenumber

and the frequency

are also expanded as

In the case of a vortex ring, is used in place of along the ring centreline is used in place of

3.

and the arclength

Kelvin waves

To leading order, the stability problem is reduced to oscillations of the circular core of constant vorticity. The dispersion relation for the Kelvin waves is described, for example, in Saffman’s textbook (1992). Suppose that an infinitesimal-amplitude wave of the form is imposed on the Rankine vortex. For later use, we write down the dispersion relation and the eigenfurictions only for the axisymmetric and the bending modes. We denote the or the toroidal component in the case of the vortex ring, of disturbance velocity by Define The dispersion relation for

reads


where we have simply written

where tions at For

and

where

and

4.

41

The corresponding flow field is

are constants constrained by the connecting condibut otherwise arbitrary. the dispersion relation and the eigenfunctions are

are constants related by the boundary conditions.

Linear instability of an elliptic vortex

The pure shear (3) at deforms the circular core into an elliptical shape and thus breaks the This quadrupole field excites a parametric resonance between two Kelvin waves with azimuthal wavenumber difference 2. Among them, the resonance between two bending modes is of particular significance. We shall make an attempt at simplifying the formulae of TW76. We recall that, in the core the Euler equations are reduced to a second-order ordinary differential equations for the disturbance pressure We find that this is explicitly solvable as

where are constants associated with the homogeneous part. The above expression simplifies TW76 by performing the indefinite integrals that were left in (3.15a, b). In accordance, the radial velocity in the

42


core is simplified. The velocity potential for the disturbance outside the core is the same as obtained by TW76:

where are constants for the homogeneous solution. Notice that the factor in front of diverges at Hence (13) is invalidated in this case and a separate treatment is necessary. Upon substitution from this solution, the kinematical and dynamical boundary conditions at are converted into a system of linear algebraic equations for and in the form:

where the matrix and

is, with use of the simplified notation

As is common, the matrices at are identical with those at In order for (15) to have non-trivial solutions for must belong to the spaces of the images of the corresponding matrices. This condition postulates that

Combining (17) for and – 1, we are left with a coupled system of homogeneous linear algebraic equations for and The flow is unstable when The non-real is possible only when (17) for simultaneously possess a nontrivial solution as demonstrated by Moore & Saffman (1975) and TW76. This requirement gives rise to the desired relation that holds between and For a collision of eigenvalues leading to instability, the maximum of growth rate occurs at The width in the wavenumber range of instability is determined by non-reality condition of The numerical computation of and is feasible with a high accuracy. All of the collisions of eigenvalues lead to instability, which does not agree with the result of TW76. We show in Table 1 the numerical results for a few crossing points of two dispersion curves (9) that


43

occur close to the axis of The bottom row is the values for a crossing point on this axis to be explained subsequently. Including other crossing points, at the crossing points with is of the same order with each other and is significantly, say a hundred times, smaller than the case of We thus confirm that the non-rotating waves are far more dominant than the rotating ones.

The case of The collision at calls for an individual treatment. The velocitypotential (14) for the exterior disturbance flow remains intact. The disturbance pressure in the core gives way to, to

and accordingly the expression of undergoes alteration. Repeating the same procedure, we arrive at compact formulae for the maximum growth rate and the unstable band width as

These formulae are valid only at values of satisfying (9) restricted to The most unstable mode is attained at the smallest root and the numerical result is shown in Table 1. The asymptotic behaviour of (19) and (20) at large values of is manipulated with ease as

44


The leading-order term of coincides with the growth rate of the elliptical instability as deduced by Bayly (1986) and Waleffe (1990). This term was reached by Eloy & Le Dizès (2001) by elaborating the limiting behaviour of the terms arising in Moore-Saffman’s approach.

5.

Linear instability of a vortex ring

For Kelvin’s vortex ring, it is the the dipole field (5) that breaks, to the of the Rankine vortex. The governing equations for the disturbance velocity and the disturbance pressure at inside the core and for the potential for the disturbance flow outside the core are written out in WT77. To have an idea, we give the of the Euler equations:

The boundary conditions require that, at

In view of (22) and (23), along with (5), a resonance instability is possible between the disturbance fields of the modes and satisfying at intersection points of the dispersion curves. Incidentally, this type of resonance is driven also by the Coriolis force (Mahalov 1993). It is illustrative to carry through a calculation for the case of and The leading-order disturbance velocity and the disturbance-velocity potential as well consist of a superposition of the axisymmetric and right-handed bending waves:

Substituting (5) and (24) into (22) and the remaining components of the Euler equations, we obtain equations for the axisymmetric and bending waves at Remarkably, these equations are again solvable. For the disturbance pressure and the velocity-potential disturbance are


For

the disturbance pressure

45

is

and the velocity-potential disturbance takes the same form as (26). Imposition of the boundary conditions (23) yields systems of inhomogeneous linear algebraic equations for and in the case of and for and in the case of The matrices at are identical with those at The solvability condition brings in a system of homogeneous linear algebraic equations for and The requirement that they possess a nontrivial solution for constitutes an eigenvalue problem for given Figure 1 displays curves of the dispersion relations, of the Kelvin waves, (11) for the axisymmetric wave and (9) for the bending wave Curves for are drawn with solid lines, whereas those for are drawn with dashed lines. The axisymmetric wave has two types of infinite branches symmetrically with respect to the horizontal axis either increasing or decreasing with Among the curves of the bending wave, the primary mode, being drawn with a thick solid line, is isolated. An infinite number of the remaining curves are called the Bessel modes. The branches increasing with correspond to waves rotating faster than the basic circulatory flow, while the opposite is true for the decreasing branches. The maximum of growth rate, if the instability occurs, is attained at We compute the value of with at many of the

46


intersection points. The primary branch, of has turned out to be totally irrelevant to the instability, and hence is ignored. The correction of the frequency takes non-real values only at the intersection points between the increasing branches of and the decreasing branches of Among all the intersection points looked at so far, the maximum growth rate is obtained at the intersection point with the smallest

between the first branches of and the Bessel mode maximum growth rate and the band width in

The are

Relatively large growth rate is attained at the intersections of the same branches of and the The longer waves may have some relevance to the oscillatory deformations of elliptical and triangular shapes touched upon by Shariff, Verzicco & Orlandi (1994). We need to be cautious about the smallness of the value of In the higher-order asymptotic solution for the vortex ring, the secondorder correction is a straining field (WT77, Fukumoto & Moffatt 2000). The growth rate of the resonance between and modes and the growth rate of the Moore-Saffman-TsaiWidnall instability (§4) are highly competitive. Comparison with the result of WT77 concludes that the present mechanism predominates over the other when the vortex ring is very thin:


47

Eloy & Le Dizès (2001) showed that the instability of the Rankine vortex subject to shear flow can be explained by parallelisation of the direction of stretching due to the shear and the disturbance vorticity. For the present case of a vortex ring, the strain tensor due to the first-order flow (5) is

Its eigenvalues and corresponding eigenvectors read

The direction of maximum stretching is the unit vector along the ring centreline, for and otherwise. Thus it would be interesting to investigate correlation between these eigenvectors and the unstable modes. Figure 2a shows the pdf’s of angles between vorticity and eigenvectors for the unstable case Here the pdf’s are weighted by the magnitude of vorticity. Note that uniform distribution corresponds to in three dimensions. It is seen that the vorticity is almost parallel to

48


Since the eigenvalue which corresponds to is positive for and negative for the corresponding pdf is decomposed into positive part and negative part in figure 2b. It shows that the positive part exceeds the negative part which implies that the alignment with gives positive stretching on average. The present instability is possibly due to the stretching in the toroidal direction. The solvability of problem rests on the assumption of uniform vorticity to leading order. This is not the case of a general vorticity distribution. To be worse, a critical layer would make its appearance in a thin region encircling the core as surmised by Kop’ev & Chernyshev (2000). These and other factors bearing with a practical situation wait for a further investigation.

References BAYLY, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163. ELOY, C. & LE DIZÈS, S. 2001 Stability of the Rankine vortex in a multipolar strain field Phys. Fluids 13, 660–676. FUKUMOTO, Y. & MOFFATT, H. K. 2000 Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity. J. Fluid Mech. 417, 1–45. KOP’EV, V. F. & CHERNYSHEV, S. A. 2000 Vortex ring oscillations, the development of turbulence in vortex rings and generation of sound. Phys. Uspekhi 43, 663–690. MACKAY, R. S. 1986 Stability of equilibria of Hamiltonian system. In Nonlinear Phenomena and Chaos (ed. S. Sarkar), pp. 254–270. Adam Hilger, Bristol. MAHALOV, A. 1993 The instability of rotating fluid columns subjected to a weak external Coriolis force. Phys. Fluids A 5, 891–900. MOORE, D. W. & SAFFMAN, P. G. 1971 Structure of a line vortex in an imposed strain. In Aircraft Wake Turbulence and its Detection (ed. J. H. Olsen, A. Goldburg & M. Rogers), pp. 339–354. Plenum. MOORE, D. W. & SAFFMAN, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. Roy. Soc. Land. A 346, 413–425. SAFFMAN, P. G. 1992 Vortex Dynamics, Chaps. 3, 10. Cambridge University Press. SHARIFF, K., VERZICCO, R. & ORLANDI, P. 1994 A numerical study of threedimensional vortex ring instabilities: viscous corrections and early nonlinear stage. J. Fluid Mech. 279, 351–375. TSAI, C.-Y. & WIDNALL, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73, 721–733. WALEFFE, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 776–780. WIDNALL, S. E., BLISS, D. B. & TSAI, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 35–47. WIDNALL, S. E. & TSAI, C.-Y. 1977 The instability of the thin vortex ring of constant vorticity. Phil. Trans. R. Soc. Land. A 287, 273–305.

The vortex-in-cell method for the study of three-dimensional vortex structures Henryk KUDELA, Pawel REGUCKI Wroclaw University of Technology, 27, 50-370 Poland [email protected], [email protected] Abstract The vortex particle method for numerical simulation of the 3D vortex structure evolution was used. Validation of the method was tested for the study of a single vortex ring by comparing the computed translation velocity with the theoretical formula and for the leap-frogging phenomenon for two rings with the same circulation. Our paper is clear as a bell On the method of vortex-in-cell; For vortical strings And leap-frogging rings The method works plausibly well.

1.

Introduction

Vorticity plays a fundamental role in all real fluid dynamic phenomena and for this reason the vortex method in the study of the fluid dynamics cannot be overestimated. In computation the “vortex particle” permits direct tracing for the evolution of the vorticity. Now it seems that the 2D vortex particle method is well grounded in that many numerical and theoretical results have been obtained (Ould Salihi & at al 2000, Kudela 1999). On the other hand, the 3D vortex method must still be developed. Generally the vortex method can be divided on the direct, free grid method based on the Biot-Savart law (Leonard 1985, Knio & Ghoniem 1990, Winckelmans & Leonard 1993) and vortex-in-cell methods where a grid is used for the velocity calculation but particles are used to track the vorticity (Christiansen 1973, Zawadzki & Aref 1991, Cottet 2000). The vortex-in-cell method is much faster then the free grid vortex method. Despite the fact that vorticity is divergence free, we introduced to the computation a vector particle that carries the “mass” of the vorticity. We are going to build a 3D program for the simulation of the viscous fluid flow using the vortex-in-cell method. Components that must be


50

H. Kudela and P. Regucki

included in such a program should be: a Euler inviscid solution of the flow by the vortex particles and a numerical procedure that takes into account the viscosity of the fluid (Cottet 2000). Here we present our primary results that relate to the inviscid Euler equations.

2.

Equation of motion and description of the computational algorithm

The equations that described the evolution of the vorticity in the inviscid and incompressible three-dimensional space are:

where is the vorticity vector and is the velocity. Now in agreement with the spirit of the vortex-in-cell method, the distribution of the vorticity is replaced by a discrete distribution of Dirac delta measures:

where

means vector particle at position When the domain of the flow is covered by the numerical mesh with equidistant spacing then the of the vector particle is:

Equation (2) assures the existence of vector potential the vorticity distribution to the velocity field:

which relates

where the components of vector potential are obtained by the solution of the Poisson equations (it was assumed that

The numerical calculation goes as follows:

The vortex-in-cell method for the study of 3D vortex structures

51

1) To solve equations (6) on the numerical mesh the strength of particles must be redistributed on the mesh nodes

where for we used the B-spline of the first order (it is equivalent to the volume-weighted scheme and influenced only 8 nodes). After the redistribution, equations (6) were solved by the fast Poisson solver. We used the periodic boundary conditions. 2) Using (5), the velocities at the grid nodes were computed by the central difference. The particles were advanced in time employing the Runge-Kutta scheme:

The velocity was computed from the grid nodes velocities by interpolation. We used the second order interpolator from the Fortran IMSL library. 3) Due to the vorticity stretching effect, in the new position the strength of the particles was updated:

The derivatives of the velocity were interpolated from the grid nodes on the position of the particles. For the solution of (9) we used the 4order Adams-Bashforth scheme. This completes one time step and the calculation returns to step 1.

3.

Examples of the numerical results

At first we used our program to test the motion of a single ring with uniform vorticity inside the core. The translation velocity is given by the the formula (Lim & Nickels 1995, Saffman 1993):

where of the core, - radius of the ring, For our calculation we took domain 10 × 10 × 10 and grid step The time step was used as The ring was divided into 100 slices, and in each slice the vorticity was redistributed between 100 particles where the summation is made for the particles that are in volume

52


a unit vector normal to the slice). For these parameters the velocity obtained from formula (10) is ~ 0.187 and it is nearly the same as the translation velocity for the ring in Figure 2. In Figure 3 we tried to simulate the motion of two rings with the same circulation that created the “vortex leap-frogging” phenomenon (Lim & Nickels 1995). We found that the evolution of two vortex rings strongly depends on their initial positions and parameters. In Figure 4 the initial position of the first ring was closer to the larger one and its diameter was smaller than in Figure 3. Qualitative changes in evolution between Figures 3 and 4 are clearly visible. The smaller ring passes through the larger one and starts to roll-up around the larger one producing the “tail”. Qualitatively the pictures resemble the experimental one published in Shariff & Leonard (1992).

4.

Closing remarks

It seems that the particle vortex method for 3D flow gives reasonable results and may be very useful for the simulation of the viscous 3D flow.

The vortex-in-cell method for the study of 3D vortex structures

53

54


During our calculations we monitored the kinetic energy helicity For the single ring For the case of “vortex game” (Figure 3) H changed from 0 to for the case from Figure 4 final helicity was Energy for the single ring (Figure 2) changed less then 1% but for the case from Figure 4 energy changed nearly about 40 %. We suspect that using a smoother redistribution of the “mass vorticity” (Cottet 2000) on the grid nodes will improve the results.

References CHRISTIANSEN, J. P. 1973 Vortex method for flow simulation. J. Comput. Phys. 13, 363–379. COTTET, G. -H. 2000 3D vortex methods: achievements and challenges. In Vortex Method, Selected Papers of the First International Conference on Vortex Methods, Kobe Japan 1999 (ed. K. Kamemoto & M. Tsutahara, Word Scientific (2000)), pp. 123–134. GHARAKHANI, A. & GHONIEM, A. F. 1997 Three–dimensional vortex simulation of time dependent incompressible internal viscous flows. J. Comput. Phys. 134, 75– 95. KNIO, O. M. & GHONIEM, A. F. 1990 Numerical study of a three–dimensional vortex method. J. Comput. Phys. 86, 75–106. KUDELA, H. 1999 Application of the vortex–in–cell method for the simulation of two–dimensional viscous flow. Task Quarterly 3, 343–360. LEONARD, A. 1985 Computing three–dimensional incompressible flows with vortex elements. Annu. Rev. Fluid Mech. 17, 523–559. LIM, T. I. & NICKELS, T. B. 1995 Vortex rings. In Fluid Vortices (ed. S. I. Green, Kluwer Academic Publishers), pp. 95–153. MEIBURG, E. 1995 Three–dimensional vortex dynamics simulations. In Fluid Vortices (ed. S. I. Green, Kluwer Academic Publishers), pp. 651–685. OULD SALIHI, M. L. & COTTET, G. -H. & HAMRAOUI, M. EL. 2000 Blending finite–difference and vortex methods for incompressible flow computations. SIAM J. Comput. 22 (5), 1655–1674. SAFFMAN, P. G. 1981 Vortex interactions and coherent structures in turbulence. In Transition and Turbulence (ed. R. E. Meyer, Academic Press), pp. 149–166. SAFFMAN, P. G. 1992 Vortex dynamics, (ed. Cambridge University Press) SHARIFF, K. & LEONARD, A. 1992 Vortex rings. Annu. Rev. Fluid Mechanics 24, 235–279. WINCKELMANS, G. S. & LEONARD, A. 1993 Contributions to vortex particle methods for computation of three–dimensional incompressible unsteady flows. J. Comput. Phys. 109, 247–273. ZAWADZKI, I. & AREF, H. 1991 Mixing during vortex ring collision. Phys. Fluids A 3 (5), 1405–1410.

L-transition from right- to left-handed helical vortices Valery OKULOV1’2, Jens N. SØRENSEN1 & Lars K. VOIGT1 1

Department of Mechanical Engineering, Technical University of Denmark Nils Koppels Allé, Bldg. 403, DK-2800 Lyngby, Denmark [email protected]

2

Institute of Thermophysics, SB RAS, Novosibirsk, Russia [email protected]

Abstract This study is devoted to analysing changes in the helical symmetry of axial vortex structures. The aim is to provide an improved understanding of the appearance of recirculating bubbles in swirl flows. The bubble generation is usually referred to as vortex breakdown, see e.g. Leibovich 1978 or Escudier 1988. The study concerns a viscous, incompressible, axisymmetric flow in a closed cylinder with co-rotation of the end-covers in the regime where the appearance of the first bubble takes place, see Brøns et al. 1999. For the investigated flow regime only one type of change in helical symmetry, denoted L-transition, was observed. The change of helical symmetry provides a possible explanation for the appearance of bubble structures in the vortex breakdown problem.

1.

Introduction

Recent progress in the understanding of vortex breakdown (Okulov 1996) shows that the phenomenon may be considered as a spontaneous transition from right- to left-handed helical vortices that both may exist under the same integral flow parameters. The difference between vortex structures with right- and left-handed helical symmetry (Alekseenko et al. 1999) is governed by the sign of the helical pitch of the vortex lines (a positive pitch denotes a right-handed helical vortex and a negative pitch a left-handed one). Although the theory (Okulov 1996) connects the flow before and after breakdown, it does not explain how the helical symmetry of the flow changes. As shown in figure 1, a continuous transition from a right-handed helical vortex to a left-handed one may take place in two different ways. Either the pitch goes through infinity or it goes through zero. The first (regular) case we refer to as L (linear)-transition (figure 1a) and the second (singular) one we refer to as R (ring)-transition (figure 16). If the vortex ring appearing in the R-transition was added to the swirl flow, without changing input and output conditions, it would 55 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 55–60. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

56

V. L. Okulov, J. N. Sørensen and L. K. Voigt

give a simple description of the appearance of a bubble as an induced axial reversal flow (see Saffman 1992). Unfortunately, only L-transition have up to now been observed experimentally (see e.g. Alekseenko et al. 1999). Employing a numerical code, as in the present study, allows one to obtain more detailed information about the bubble structures. Thus, the main objective of this study is to analyse locally the change in helical symmetry of the axial structures in the zone where vortex breakdown occurs and to define the type of transition correctly.

2.

Formulation of the problem and results

The fluid motion in a cylinder generated by rotating both end-covers (figure 1c) has proved to be useful for the study of the secondary reverse flow. Topological classification by Brøns et al. 1999 have demonstrated the richness of the flow patterns. Thus, sixteen different topologies have been detected as a function of the governing flow parameters. On the other hand, the basic topology change, from a flow without bubbles to a flow with one axial bubble, is still not fully understood. Thus, in the present work, we restrict our consideration to this basic case only. We consider a viscous, incompressible, axisymmetric flow in a cylinder of height 2H and radius R (figure 1c). The bottom and top covers rotate with constant angular velocities, and respectively. The three dimensionless numbers characterising the problem are and where is the kinematic viscosity of the fluid. The flow is described using cylindrical coordinates with corresponding velocity and vorticity depending only on In the we introduce an analogue to

L-transition from right- to left-handed helical vortices

the stream function

57

and a local circulation

Hence, the velocity and vorticity fields can be described by the scalar function and the tangential components of velocity and vorticity Formulated in these variables, the flow is governed by two transport equations arid a Poisson equation. Boundary conditions are established from the no-slip assumption of the velocity. For the numerical simulation we use the finite difference code developed at LIMSI/CNRS (see e.g. Daube 1991 or Sørensen & Loc 1989). The transport equations are discretised by a second-order central difference scheme and the Poisson equation is discretised to fourth-order accuracy using three-point expressions for the derivatives. Previous comparisons of results from the solver with experiments for the configuration with a fixed cover showed that the simulations with a grid size of 100 × 100h for and for predict correctly the onset of the reverse flow within experimental accuracy. In the present study we have investigated the particular case where the end covers either co- or counter-rotate with identical and constant angular velocity At counter-rotation, the axial and tangential velocities are equal to zero at the middle plane. The pumping in this case leads to the formation of a purely progressive motion along the cylindrical axis towards the disks (figure 2a). In this case, contours of the stream-surfaces are in the meridional intersection picture shown as two mirror images, consisting of closed and deformed tori. In figures 2 and 3, the intensity levels of stream surfaces and vortex tubes in the meridional intersection are nonuniformly spaced, with different numbers n of the levels determined by where B is or Profiles of the vortex-surfaces are two mirrored sets from deformed semitori, which encloses each other and break into disks (figure 2b). Such a semi-torus of the vortex tube is similar to a vortex semi-ring resting on a disk surfaces. Please note that the vortex rings only can induce a purely progressive motion along the axis toward the disks, and are not capable of generating recirculating bubbles. In the opposite case of co-rotation of the disks, another picture of the flow is obtained. Here the tangential velocities are non-zero in the middle cross-section, and the fluid corotates in the upper and lower half parts of the cylinder. Contours of the stream-surfaces in meridional intersection (figure 2c) shows the appearance of symmetrical bubbles, i.e. recirculation zones with weak reverse flow along the cylindrical axis. In this case the vorticity distribution is easy to describe (see figure 2d). Vortex tubes akin to enclosed cylin-

58


ders pass through the whole flow domain in the central part of the flow, whereas the structures of semi-torus shape are located near the cylindrical wall. In the following we neglect the influence of the induction of the torus vortices, and analyse only the influence from the vortex tubes on the bubble generation. The iso-curves of and from formula (1), are the intersection of the physical axisymmetric stream-surfaces and vortex-surfaces with any meridional plane. As the tangential velocity is non-zero, the real physical streamlines are three-dimensional spirals, as e.g. demonstrated in the computations by Sotiropoulos & Ventikos 2001. For the same reason, even though the flow is axisymmetric, the vortex lines are three-dimensional, with the near-axial vortex tubes consisting of helical-like vortex lines. To give a measure of their helical properties, we introduce the helical pitch defined as a the pitch of a helix filament touched by a vortex line in a given point. Thus, the dimensionless twist of a helix-like vortex lines is given as or in a more unified, form where is the angle of the inclination of the vorticity vector from vertical. The distinction between real vortex tubes and pure cylindrical forms are neglected, and the thin Ekman-type boundary layer zones on the rotating disks are not considered. In figure 3, two specific flow regimes are depicted. The first is a flow at a Reynolds number of 150, where no bubbles are present, and the other one is at a Reynolds number of 500, where one recirculating bubble appears in the middle of the flow domain. At the lower Reynolds number (figure 3a-c) the flow contains a right-helical vortex (l or K are positive) in the upper half part of the cylinder and a left-handed vortex (l or K are negative) in the lower part (see figure 3b). Thus, pure jet-like profiles of the axial velocity to both rotating disks are formed as a superposition of the flow

L-transition from right- to left-handed helical vortices

59

induced the central vortex structure and the peripheral deformed vortex semi-rings. By increasing Re the helical pitch changes sign (figure 3e). This change in helical symmetry induces an axial dent in the initial jetlike profile of the axial velocity. At increasing Re, the dent continuously enlarges together with a growth of the vortex-line screw is increasing) , corresponding to a reduction of the distance between the turns of the screw is decreasing). At a certain Reynolds number the influence of the dent makes the axial flow component to become zero and reversal flow appears on the cylindrical axis (figure 3f.) It should be mentioned that this behaviour was observed for all analysed aspect ratios. Furthermore, flow reversal took place approximately under the same value of the twist parameter or On this basis, as a generalisation, we conclude that a necessary (but not sufficient) criterion for the creation of a recirculating bubble in swirl flows are that a mirror change of the central vortex structure takes place. Furthermore, the cre-

60


ation of a recirculating bubble will first take place when the maximum pitch assumes a value

3.

Conclusion

In this study, numerical solutions of the axisymmetric Navier-Stokes equations were used to examine changes in the helical symmetry of vortex lines. As a specific case, we considered axial vortex structures in a steady swirl flow in a cylinder with co-rotating end-covers. The hypothesis that the phenomenon of vortex breakdown is associated with a transition from one helical vortex structure to a mirror-symmetrical structure has been confirmed. Only L-type transition was observed for the considered flow regime. Finally, the computations demonstrate that the creation of a recirculating bubble first takes place when the maximum pitch assumes a value

Acknowledgments This research has been supported in part by the INTAS (grant no. 0000232), RFBR (grant no. 01-01-00899), and the Otto Mønsteds Foundation.

References ALEKSEENKO, S.V., KUIBIN, P.A., OKULOV, V.L., & SHTORK, S.I. 1999 Helical vortices in swirl flow. J. Fluid Mech. 382, 195–243. BRØNS, M., VOIGT, L. K. & SØRENSEN, J. N. 1999 Streamline topology of steady axisymmetric vortex breakdown in a cylinder with co- and counter-rotating endcovers. J. Fluid Mech. 401, 275–292. DAUBE, O. 1991 Numerical simulation of axisymmetric vortex breakdown in a closed cylinder. Lectures in Applied Mathematics 28, 131–152. ESCUDIER, M. 1988 Vortex breakdown: observations and explanations. Prog. Aerosp. Sci., 25, 189–229. LEIBOVICH, S. 1978 The structure of vortex breakdown. Ann. Rev. of Fluid Mech., 10, 221–246. OKULOV, V.L. 1996 Transition from the right spiral symmetry to the left spiral symmetry during vortex destruction. Tech. Phys. Lett., 22(19), 47–54. SAFFMAN, P.G. 1992 Vortex dynamics. Cambridge Univ. Press. SOTIROPOULOS, F. & VENTIKOS, Y. 2001 The three-dimensional structure of confined swirling flows with vortex breakdown. J. Fluid Mech., 426, 155–175. SØRENSEN J.N. & LOC, T.P. 1989 High-order axisymmetric Navier-Stokes code: Description and evaluation of boundary conditions. Int. J. Numer. Math. in Fluids., vol. 9, 1517–1537.

Dynamical behaviour of counter-current axisymmetric shear flows Xilin XIE 1 , Weiwei MA 2 & Huiliang ZHOU 1 1

Department of Mechanics & Engineering Science, 220 Handan Road, Shanghai 200433, P.R.China

[email protected] 2

Donghua University, College of Basic Science 1882 Yanan Xi Road, Shanghai 200051, P.R. China

Abstract The dynamical behaviour of coherent structures in counter-current ax– isymmetric shear flow has been experimentally studied. Two kinds of vortices, i.e., axisymmetric and helical structures, were discovered with respect to different regimes in the forward velocity versus velocity ratio diagram. On axisymmetric structures, two global self-excited oscillation modes referred to as the shear layer and jet column modes could be set in the flow system and the effect of the velocity ratio on the dynamical behaviour of this kind of structures could be concluded as the principle of relative movement. On helical structures, it is a kind of streamwise vortex structures resulting from the jet column bifurcations whose temporal asymptotic behaviour could be described as a 2-torus. We measured the helical flow In a tube at some speeds high and low; We readily reckoned Two meters per second As a critical speed you should know.

1.

Introduction

The study of the dynamical behaviour of coherent structures in the free shear flow play an essential important role in the modern research of the generation mechanism of turbulence. The coherent structures in shear flow could be classified into two groups: spanwise (including axisymmetric) and helical structures. The dynamical behaviour of the wakes of a 2D circular cylinder have been experimentally studied in detail (Williamson 1996), a kind of amplitude modulation type of velocity fluctuations were widely discovered, e.g. near the borders between cells of different frequencies and oblique resonance regions, all of these cases corresponding to the interaction of

61 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 61–66 © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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different spanwise vortices. On round (or called axisymmetric) jets, two kinds of global self-excited oscillation modes termed as the shear layer and jet column modes could occur in actual flow systems. The former mode corresponds to the energetic vortices being quite regularly spaced in the streamwise direction and suppressing the vortex pairing resulting in the lengthening of the potential core (Strykowski & Niccum 1991). Oppositely; the latter peculiar to the jet column strengthens the vortex pairing in the shear layer resulting in the shortening of the jet potential core (Strykowski & Wilcoxon 1993). Danalia, Dusek & Anselmet (1997) discovered a kind of streamwise vortices, termed as the helical structure in our paper, in the case of the Reynolds number below a critical value

Dynamical behaviour of counter-current axisymmetric shear flows

63

in their numerical study of coherent structures in a round, unforced, spatially evolving, homogeneous jet.

64


Nowadays, the standard approach of dynamical system is being more and more widely used to study the dynamical behaviour of coherent structures. This study comprises two aspects: first in physical space and secondly in phase space. The former study includes not only topological structures of coherent structures and their spatial evolution, but also parametric evolution, that is the evolution of the form of coherent structures due to the varying of control parameters. The latter study puts emphasis on the statistic character of attractors.

2.

Experimental Results

The dynamical behaviour of counter-current axisymmetric shear flow have been studied in the forward velocity versus the velocity ratio R

Dynamical behaviour of counter-current axisymmetric shear flows

65

diagram as shown in Fig.l. The detailed description of the experimental facilities and instruments can be found in the paper (Ma, Xie & Zhou, 2001). The well-known axisymmetric structure (vortex ring) plays an important role in the case of in the range from to and a kind of unfamiliar vortex termed as helical structure is prevailing in the case about The main results are separated into two parts corresponding to axisymmetric and helical structures. (I) Dynamical behaviour of Axisymmetric Structures Based on the dynamical behaviour of axisymmetric structures as shown in Fig.2 and 3, the critical forward velocity is defined, subsequently the subcritical velocity regime and the supercritical regime In the subcritical velocity regime, the shear layer exhibits a self-excited oscillation in a certain range of the velocity ratio with a fixed forward velocity as shown in Fig.1 and Fig.4(a). In the supercritical regime, the effect of the velocity ratio could be described as the principle of relative movement, that is making measurements at a fixed spatial position when increasing the velocity ratio from an initial value is equivalent to moving the probe downstream to detect the spatial evolution of the flow system as the velocity ratio is fixed onto this initial value. The scenario of the spatial evolution of axisymmetric structures, based on this principle, could be deduced as follows: vortex rolling up, pairing and agglomeration gaining in intensity with structures unchanged (jet column self-excited oscillation) arranging in an ordered pattern (shear layer self-excited oscillation) after its intensity saturated ’ordered’ tearing ’weak’ turbulence when the forward velocity is less than as shown in Fig.4(b); the flow system evolves directly into ’weak’ turbulence rather than undergoing the ’ordered’ tearing process when the forward velocity is greater than or equal to Correspondingly, the spatial evolution of the temporal asymptotic behaviour of the dynamical system related to the axisymmetric vortices in the supercritical regime, as shown by Fig.5, could be described as follows: limit cycle (Hopf Bifurcation corresponding to vortex rolling up) limit cycle (subharmonic bifurcation corresponding to the vortex agglomeration) limit cycle (inverse superharmonic bifurcation corresponding to self-excited oscillations) limit cycle (superharmonic bifurcation or frequency-doubling bifurcation corresponding to ’ordered’ tearing) chaos (’weak’ turbulence) when the forward velocity is less than the ’ordered tearing’ does not exist when the forward velocity is greater than or equal to (II) Dynamical behaviour of Helical Structures The helical structure is a kind of streamwise vortex. The time trace of the velocity fluctuation is characteristic of amplitude modulation, and

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two base frequencies are contained in the flow system that coexist simultaneously. The spatial evolution of the helical structure could be described as follows: the jet column is separated into two parts at a certain spatial location and those entangle each other to form the helical structure, then the two parts reconnect further downstream resulting in the flow system evolving into turbulence in a catastrophic form. The spatial evolution of the mean streamwise velocity and streamwise turbulence intensity on the jet centre line corresponding to different velocity ratio are shown in Fig.6 and are quite different than those corresponds to axisymmetric structures. Correspondingly, the evolution of the temporal asymptotic behaviour of the dynamical system is as follows: the dynamical system directly evolves into a 2-torus through supercritical Hopf bifurcation corresponding to the jet column separation and undergoes transition from quasi-periodic attractor to strange attractor in a catastrophic form corresponding to the reconnection of the two separated jet columns. On parameter evolution, the evolution of the temporal asymptotic behaviour of the dynamical system as the velocity ratio increases from 1 to 3 could be described as follows: 2-torus limit cycle (inverse Hopf bifurcation) strange attractor (subharmonic bifurcation), as shown in Fig.7. The 3D phase trajectory corresponding to helical structure is revealed as a 2-torus as shown in Fig.8, and the correlation dimension is about 2.

Acknowledgments The work was supported by the National Natural Science Foundation of China under the Grant Number 10072018.

References DANALIA, I., DUSEK J. & ANSELMET, F. 1997 Coherent Structures in a Round, Spatially Evolving, Unforced, Homogeneous Jet at Low Reynolds Numbers. Phys. Fluids. 9(11), 3323–3342. MA, W.W. , XIE, X.L. & ZHOU, H.L. 2001 An experimental study on the coherent structures and chaotic phenomena in the axisymmetric counter shear flows. ACTA MECHANICA SINICA. 17(3), 214–224. STRYKOWSKI, P.J. & NICCUM, D.L. 1991 The stability of countercurrent mixing layers in circular jets. J. Fluid Mech. 227, 309–343. STRYKOWSKI, P.J. & WILCOXON, R.K. 1991 Mixing enhancement due to global oscillations in jets with annular counterfkrw. AIAA Journal 31(3), 564–570. WILLIAMSON, C.H.K. 1996 Vortex Dynamics in the Cylinder Wake. Annu. Rev. Fluid. Much. 28, 345–407.

II

SINGULAR VORTEX FILAMENTS


Complexity measures of tangled vortex filaments Carlo F. BARENGHI1, David C. SAMUELS1 & Renzo L. RICCA2 1

Mathematics Department, University of Newcastle, Newcastle NE1 7RU, United Kingdom. [email protected]

2

Mathematics Department, University College London, London, WC1 6BT, United Kingdom.

Abstract We introduce and test measures of geometric and topological complexity to quantify morphological aspects of a tangle of vortex filaments. The tangle is produced by standard numerical simulation of superfluid turbulence in Helium II. Complexity measures such as linking number, writhing number, average crossing number and helicity are computed, and their relation to the energy of the fluid is investigated. We found a complexity measure – It really is quite a treasure. – For a vortex entangled, By methods new-fangled; I'll explain if you have enough leisure.

1.

Introduction

Complex systems of filaments occur frequently in nature. Examples range from vortex structures to magnetic flux tubes to polymers, proteins arid DNA. We would like to relate the morphological complexity of such systems with physical properties, such as energy. The aim of this paper is twofold. First we introduce candidate measures of geometric and topological complexity; secondly, we choose superfluid turbulence as a convenient benchmark, compute these measures and compare them to energy.

2.

Vortex dynamics and superfluid turbulence

Superfluid turbulence (Barenghi 2001) consists of a disordered, apparently random tangle of vortex filaments. This state of turbulence is particularly simple if compared to traditional hydrodynamics turbulence. Firstly, the superfluid is inviscid. Secondly, all vortex filaments


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C.F. Barenghi, D.C. Samuels and R.L. Ricca

have the same circulation (the quantum of circulation). Thirdly, the vortex core radius is so small and the filaments are so long that the classical theory of thin-core vortex filaments applies well. Unlike what happens in classical turbulence, in which eddies can be of any size and strength, superfluid vorticity is always geometrically well defined (it is the location where both the real and the imaginary parts of the quantum mechanical wave function vanish). All these features make superfluid turbulence a convenient benchmark to study issues of complexity.

A superfluid vortex line can be described as a closed curve (Schwarz 1988) where is arc length and is time. The line interacts with the thermal excitations present in Helium II, which can be modelled as a viscous fluid of velocity field the interaction depends on a temperature dependent friction coefficient The instantaneous velocity of a point of the superfluid vortex line is given by

where is the tangent unit vector and the self-induced velocity is given by the classical Biot-Savart integral

where varies along the line and the integral extends to the collection of vortex lines which form the turbulent tangle Equations (1) and (2) are used to determine numerically the time evolution of an initial system of vortex lines in the presence of a given The computer code also performs vortex reconnections when two vortex lines become very close to each other. In superfluid turbulence

Complexity measures of tangled vortex filaments

71

reconnections do not involve dissipation and arise from the underlying quantum mechanics in a way which is well described by the Gross Pitaevskii equation for a Bose - Einstein condensate (Koplik and Levine 1993, Leadbeater et al 2001). In most experiments (Maurer and Tabeling 1998, Stalp et al 1999) the normal fluid is turbulent. To represent we perform calculations using two different models. The first is a steady ABC flow (Barenghi et al 1997) given by where and and A, B, C and are parameters. The second is a more realistic kinematic simulation of turbulence (Kivotides, Barenghi and Samuels 2001) for which

where J is the number of modes used, is a random unit vector, and the directions and orientations of and are chosen randomly but so that the energy of the mode has the dependence of Kolmogorov turbulence. A typical calculation starts with few seeding vortex rings as initial condition (our results do not depend on the initial condition). The initial vortex lines interact with each other and with the background normal fluid, which feeds energy into them. Soon the initial lines become distorted, grow, reconnect, and a vortex tangle is created (see Figure 1).

3.

Complexity measures

To analyse the tangle’s complexity we project it othogonally onto a given plane. The vortex loops are naturally oriented by the direction of the vorticity, so, using standard convention, we assign an algebraic value to each apparent point of self-intersection of the projected tangle. An algebraic measure of the complexity of two loops and is the average crossing number (Freedman and He 1991, Moffatt and Ricca 1992)

where and ing over all directions

and the angular brackets denote averagof projection. The generalisation to the entire

72


collection of filaments is

For computational simplicity we take only the projections onto the three principal planes and obtain the estimated average crossing number

where indicates that the solid angle average is replaced by the algebraic mean over the three principal planes of projection. Another interesting quantity is the writhing number which measures the average total coiling of a loop. For a single filament one can show that (Moffatt and Ricca 1992)

where the average is extended to the number of apparent, signed self-intersections of The generalisation to the tangle is Again, it is computationally convenient to approximate by the estimated writhing number given by

The linking number between two closed loops and a measure of the topological linking and can be denned as

provides

Unlike previous measures, the linking number is a topological invariant because it does not change under continuous deformations of the vortex strands performed by a sequence of Reidemeister moves (Adams 1994), so it is independent of the projection used. The total linking of a system of vortex lines can be defined by

where we deliberately exclude contributions from self-linking (due to writhe and twist of each vortex filament).

Complexity measures of tangled vortex filaments

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The final measure of complexity which we consider is the kinetic helicity (Moffatt 1969) defined by

where

is the vorticity and the integral is taken over the tangle volume . Since vorticity is confined only to vortex lines, is a delta function of strength in the direction along each filament of and we have

4.

Results

We compute complexity measures as function of time as the vortex tangle grows, and compare one another and against the tangle’s total length L and kinetic energy E. Typical results are shown in Figure 2

74


in the case of an ABC flow (Barenghi, Ricca and Samuels 2001) but essentially similar results are obtained when the Kolmogorov turbulence model is used. The initial rate of growth of L is in agreement with linear stability calculations. As the vortex structure unfolds, the dynamics develops structural complexity and entanglement. It is apparent that the length is a good measure of the energy of the system, as in the trivial case of a single straight vortex line. It is also apparent that, after the initial transient, the growth rate of all complexity measures is essentially the same, and is approximately twice the rate of growth of the length. This means that vortex tangle grows by curling up and folding upon itself more than by spreading and diffusing in space.

Acknowledgments R. L. R. acknowledges support from EPSRC (grant GR/K99015) and PPARC (GR/L63143).

References ADAMS, C. C. 1994 The knot book, W. H. Freeman and Co., New York, p. 14. BARENGHI, C. F., BAUER, G., SAMUELS, D. C. AND DONNELLY, R. J. 1997 Superfluid vortex lines in a model of turbulent flow. Phys. Fluids 9 2631–2643. BARENGHI, C. F. 2001 Introduction to superfluid vortices and turbulence. In Quantized vortex dynamics and superfluid turbulence (ed. C. F. Barenghi, R. J. Donnelly and W. F. Vinen), pp. 3–13. Springer Verlag. BARENGHI, C. F., RICCA, R. L. AND SAMUELS, D. C. 2001 How tangled is a tangle ? Physica D 157 197–206. FREEDMAN, M. H. AND HE, Z. -X. 1991 Divergence-free field. Energy and asymptotic crossing number. Ann. Math. 134 189–229. KIVOTIDES, D., BARENGHI, C. F. AND SAMUELS, D. C. 2001 Fractal nature of superfluid turbulence. Phys. Rev. Letters 87 155301. KOPLIK, J. AND LEVINE, H. 1993 Vortex reconnections in superfluid helium. 1375– 1378 (1993). LEADBEATER, M., WINIECKI, T., SAMUELS, D. C., BARENGHI, C. F. AND ADAMS, C. S. 2001 Sound emission due to superfluid vortex reconnections. Phys. Rev. Letters 86, 1410–1413. MAURER, J. AND TABELING, P. 1998 Local investigation of superfluid turbulence. Europhysics Letters 43 29–34. MOFFATT, H. K. 1969 The degree of knottedness of tangles vortex lines. J. Fluid Mechanics, 35 117–129. MOFFATT, H. K. AND RICCA, R. L. 1992 Helicity and the Calugareanu invariant. Proc. Roy. Soc. London A 439 411–429. 4 SCHWARZ, K. W. 1988 Three dimensional vortex dynamics in superfluid He. Phys. Rev. B 38, 2398–2417. STALP, S. R., SKRBEK, L. AND DONNELLY, R. J. 1999 Decay of grid turbulence in a finite channel. Phys. Rev. Letters 82, 4831–4834.

Corotating five point vortices in a plane Tatsuyuki NAKAKI Faculty of Mathematics, Kyushu University 6–10–1 Hakozaki, Fukuoka 812–8581 JAPAN [email protected]

Abstract Let us consider the motion of assembly of point vortices in the twodimensional incompressible ideal fluid. The motion is described by an ordinary differential equation, and we focus our attention on the relative equilibria which imply the corotation of vortices. In this paper, we analyse the stability of the equilibria when five point vortices satisfy some condition on the strength and initial configuration. When parameters, which appear in the above condition, belong to some range, we show that the equilibria are unstable and that the solution, which locates near the equilibria at the initial time, exhibits the relaxation oscillation. The stable equilibria appear in a narrow parameter range. For some equilibria, the stability is shown by a computer-assisted proof. On the problem of vortices five, For long I’ve continued to strive; Sometimes they’re stable, And then I am able To show how they jiggle and jive.

1.

Introduction

We consider the motion of assembly of point vortices in the twodimensional Euler fluid. When several vortices are in the fluid, every vortex drifts away with the flow due to the other vortices. Such a phenomenon is described by the following ordinary differential equation when the fluid occupies the whole plane:

where and are the complex position at time and the strength of vortex, respectively. The complex conjugate of is denoted by and The equation (1) is analysed for long times and many results are already known. We briefly summarise some of them. When two point vortices are in the fluid, the motion of vortices is easily analysed as is 75 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 75–80. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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T. Nakaki

shown in textbooks on fluid mechanics. A qualitative analysis of three point vortices are done by Aref 1979. When the number of vortices is greater than three, (1) is not solved yet in general cases, however, some special cases are treated. For example, Morikawa & Swerison 1971 and Cabrel & Schmidt 1999 deal with the problem of vortices at the vertices and centre of a regular polygon. They show the stability of corotation of the vortices. Other results are seen in O’neil 1987, Aref 1995, Castilla et al. 1993, Nakaki 2001, for examples. In this paper, we treat the case where vortices are at the vertices and centre of a diamond: Let and be real parameters. At the initial time we impose

which means that vortices are at the vertices and is at the centre of a diamond. The shape of the diamond is described by the parameter We also assume that

where is determined so that the five point vortices corotate around the origin, that is, there exists a real constant such that is constant in Here the constant shows the angular velocity of the corotation. In this case we call that five point vortices are in a relative equilibrium. The purpose of the paper is to discuss the stability of the relative equilibria. In the following section, the simple case where and is considered. The equilibria are unstable and we show that vortices, which locate near the equilibrium at the initial time, exhibit the relaxation oscillation. In Section 3, we shall focus our attention to the case where Our numerical simulation suggests that the corotation is stable only on a narrow parameter range.

2.

Square-shaped configuration

In this section, we treat the simple case where and the vortices with unit strength are at the vertices of a square and the fifth vortex is at the centre of the square. In this case it is easily found that five vortices are in the relative equilibria for all Nakaki 1999 already show the following theorem: Theorem 1.

which satisfies

Then there is a solution

of (1)

Corotating five point vortices in a plane

77

where

In the above theorem, the assumption implies that the angular velocity is zero, and that and are equilibria of (1). Theorem 1 implies the existence of a heteroclinic orbit. By applying Theorem 1 repeatedly, we find that there exist heteroclinic connections, which induce the oscillation of the five vortices as follows: When the initial configuration of five vortices are closed to an equilibrium (without loss of generality, we can assume that the equilibrium is , the solution of (1) travels near the heteroclinic orbit and approaches to the equilibrium . After approaching the solution goes away from due to the other heteroclinic orbit. In this way an oscillation of vortices occurs. Moreover the motion of the solution is slow when the solution is near equilibria and is fast when it is far from equilibria. As a result, we observe that the vortices do not move for a while, however, they begin to move suddenly and approach to the another equilibrium. After staying near the equilibrium, the vortices also resume moving (see Figure 1). Let us call this motion the relaxation oscillation.

Remark 1. When we can similarly show the existence of the heteroclinic orbit, and the relaxation oscillation also occurs. We omit the details here. If the relative equilibrium is stable against perturbations which satisfy the symmetric condition

According to our numerical simulations, the relaxation oscillation does not occur anymore.

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T. Nakaki

Diamond-shaped configuration

In this section we treat the case where 0 < b < 1. At first let the vortices at the vertices have same strength. By the standard lingered analysis we can easy find Theorem 2. There exist constants satisfy 1 If lingered sense;

and

which

the relative equilibrium is unstable in the

the relative equilibrium is neutrally stable; that is, 2 If all lingered eigenvalues are purely imaginary numbers. Numerically we find that · · · and · · · . In the latter case of Theorem 2, the stability of the equilibria is not become clear by checking the linear terms. To verify the stability the numerical simulations are done as shown in Figure 2, where we show the snapshots

of numerical simulations with b = 0. 57 (upper-left in the each window), b = 0. 585 (upper-right) and b = 0. 6 (lower-right). In view of Theorem 2, the configurations of vortices with b = 0. 57 and b = 0. 6 are unstable. We also observe that the configuration with b = 0. 585 seems to be stable from numerical points of view. So we can say that, when the configuration can be stable only in the narrow parameter range.

Corotating five point vortices in a plane

79

Next we consider the general case where The parameter range where the relative equilibrium is neutrally stable is shown in Figure 3. To prove the stability we examine the energy of

vortex system, that is, the Hamiltonian of (1) defined by

Let ,..., be the relative equilibrium. If attains the strictly local minimum of H, we can conclude that the relative equilibrium is stable in the Lyapunov sense. Here we remark that the Hamiltonian H is also the first integral of the vortex system in the rotating coordinate because We analyse the stability when the perturbation given to the equilibrium does not change the value of the angular velocity Without loss of generality, we remove the rotation of the configuration. To this end, after Cabrel & Schmidt 1999, we assume that the angular velocity is fixed to the initial value and is purely imaginary:

Here we use the fact that the angular velocity is determined by the moment of inertia and the circulations Under the condition (8), we try to show that the energy attains the strictly local minimum at the equilibrium by the following steps: 1. Compute the Hessian of H, namely

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2. Show that P is positive definite, that is, P > 0. To show P > 0, it suffices to prove that det > 0 for where and is the size of P. The elements of P is too complex to analyse by the standard methods. Hence we use the interval computations on workstations to estimate the value of det if the lower limit of the interval value of det is positive, we can rigorously conclude that det In other words, we apply a computer-assisted proof. The profil/bias library (see Knüppel 1994) is used to make interval computations. From now we show our results on the stability of equilibria. Let 4/3. The equilibria are neutrally stable if The interval computations to verify P > 0 are made for b = 0. 7, 0. 7 1 , . . . , 0. 94. Then we find that P > 0 when b = 0. 88, 0. 8 9 , . . . , 0. 93, which yields Theorem 3. When and b = 0. 88, 0. 8 9 , . . . , 0. 93, the relative equilibria are stable against the perturbations which satisfy (8). For b = 0. 7, 0. 7 1 , . . . , 0. 87 and b = 0. 94 we fail to show P > 0. For general values of our results are summarised in Figure 3. The parameters, for which we can prove P > 0, are in region B of Figure 3. For a lot of parameters in region B (almost any parameters in B), the relative equilibria are stable. Until now we do not succeed in proving the stability for all parameters in B because we use the computer-assisted proof for given discrete parameters We expect that the neutrally stable relative equilibria are stable when the circulations have the same sign, which is one of our future works.

References MORIKAWA, G. K. & SWENSON, E. V. 1971 Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14, 1058–1073. AREF, H. 1979 Motion of three vortices. Phys. Fluids 22, 393–400. O’NEIL, K. A. 1987 Stationary configurations of point vortices. Trans. Amer. Math. Soc. 302, 383–425. CASTILLA, M. S. A. C, MOAURO, V., NEGRINI, N. & OLIA, W. M. 1993 The four positive vortices problem: regions of chaotic behavior and the nonintegrability. Ann. Inst. H. Poincaré Phys. Théor. 59, 99–115. KNÜPPEL, O. 1994 PROFIL/BIAS — a fast interval library. Computing 53, 277–287. AREF, H. 1995 On the equilibrium and stability of a row of point vortices. J. Fluid Mech. 290, 167–181. CABRAL, H. E. & SCHMIDT, D. S. 1999 Stability of relative equilibria in the problem of N + 1 vortices. SIAM J. Math. Anal. 31, 231–250. NAKAKI, T. 1999 Behavior of point vortices in a plane and existence of heteroclinic orbits. Dynam. Contin. Discrete Impuls. Systems 5, 159–169. NAKAKI, T. 2001 The analysis to some point vortex problems using computers. Nonlinear Analysis 47, 3849–3857.

On motion of a double helical vortex in a cylindrical tube Pavel A. KUIBIN Institute of Thermophysics Lavrentiev ave., 1, 630090 Novosibirsk, Russia [email protected]

Abstract New approach is developed for the velocity estimation of a double helical vortex motion in boundless space as well as in a cylindrical tube. The system under consideration is two thin helical vortices winding around common axis and propagating in surrounding irrotational fluid. The problem on the determination of the velocity of the vortices motion is solved with help of formulae describing the velocity field induced by infinitely thin helical vortex inside a cylindrical tube (Okulov 1995) and using the technique of the singularities separation from this solution (Kuibin & Okulov 1998). As a result an analytical formula for the vortices propagation velocity is found with account for the self and mutual induced velocities, influence of the solid boundary and translational motion. The formula contains infinite series from the modified Bessel functions. Nonetheless it is shown that impact of these series is small enough and can be neglected to give a more simple formula. Vortices coiled up in doubles Bring great analytical troubles; But with series of Kapteyn We’re able to obtain Their speed, in the absence of bubbles!

1.

Introduction

The phenomenon of double helical vortex is known for a long time. First of all such vortex is being generated in the wake of two-blade propeller or turbine. Double helix represents one of the possible vortex states after vortex breakdown (Faler & Leibovich 1977). A perfect double helix rotates and moves translationary without change of its form. The problem on determination of the vortex velocity and frequency of velocity pulsations induced remains actual one. In a bounded space double helix can be immobile. Such vortex system was observed in a vortex chamber of square cross-section with two-slopes bottom (Alekseenko et al. 1999).


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There are known few approaches to the determination of a double helix velocity. Takaki & Hussain (1984) gave a dispersion equation using the cut-off technique where is the wavenumber and is the phase velocity of the helical form propagation. The parameter A depends on the helix radius, and radius of its core, Here L and are the cut-off lengths which are in common case unknown functions of the helix parameters. Wood (1998) had analysed the case of small pitch, and showed that velocity has an asymptote for pair of vortices shifted circumferentially by radians and with equal circulations The problem solution becomes possible due to the progress in solution of an analogous problem on determination of the propagation velocity for a single helical vortex. Ricca (1994) had analysed different approaches of a helical vortex description. During numerical calculations of the velocity field induced by the helical vortex filament (with help of Hardin’s (1982) approach) in its vicinity, the author separated from the total velocity the part responsible for the drift velocity of the helix. Comparison of this result with the quadrature formula of the helix self-induced velocity based on the approach by Moore & Saffman (1972) showed that these quantities behave similarly in a wide range of the helix pitch variation and differ approximately by 0. 25. The solution for the velocity field given by Hardin (1982) represents infinite series which contain singularities of the polar and logarithmic type. Consequently difficulties arise at numerical calculations of the velocity field in the neighbourhood of the filament. Moreover the less is the helix pitch the slower is the series convergence. To avoid these difficulties a technique of the explicit separation of the singularities from the series was developed by Kuibin & Okulov (1998). They found in limit cases of small and large pitch that regular remainder of the velocity expansion in vicinity of a vortex filament differs exactly by 1/4 from the remainder presenting in the formula for the self-induced rotation of a vortex with finite circular core over which the vorticity is distributed uniformly. Recently Boersma & Wood (1999) proved that this difference equals exactly 1/4 for arbitrary vortex pitch. Now it is possible to find the self-induced velocity of a helical vortex through the constant appearing in the expansion of the binomial velocity in vicinity of an infinitely thin helical vortex filament. For the last constant depending of cause on the helix geometry an analytical formula was derived (Kuibin & Okulov 1998) and some simplification was proposed. The present work develops the method of singularities separation on the case of a double helix, both when the vortex propagates in boundless space and inside a cylindrical tube.

On motion of a double helical vortex in a cylindrical tube

2.

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Statement of the problem

Let’s consider a symmetric system of two identical helical vortices with radius pitch ) and circulation propagating in surrounding potential flow inside a cylindrical tube of radius R. The centrelines of the helices in cylindrical variables correspond to the equations and The vorticity is proposed to have a uniform distribution over vortex cores of small radius One additional parameter we will need is the value of velocity at the symmetry axis, The geometry of the vorticity distribution determines the helical symmetry of the velocity field which in turn yields the following relations (Alekseenko et al. 1999)

Thus analysis of a single velocity component, say will be enough for the problem solution. In accordance with the technique of the singularities separation (Kuibin & Okulov 1998) the circumferential velocity induced by a single helical vortex can be written as a sum of singular, and regular, parts

The members marked by asterisk relate to the velocity induced by the interaction of helical vortex filament with the cylindrical wall. For the singular parts we have

The quantities with a tilde may be interpreted as distorted radial distances with Coefficients before the logarithmic terms contain the functions The variables with indices a or R are constructed by the same rule. The quantity corresponds in some sense to a reflected or imaginary helical vortex outside the cylinder. The parameter combines the angular and axial coordinates,

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The regular parts read

where and are the modified Bessel functions (the prime denotes the derivative). Here and further, the upper and lower lines in the brackets correspond to the vortex interior and its exterior respectively. When we have a double helical vortex the velocity induced by the second vortex is being evaluated by the formulae (2)-(6) with the replacement by For the case of a single helical vortex with a finite uniform core the binormal velocity of its motion was found earlier (Kuibin & Okulov 1998). As the binormal velocity relates to the axial and circumferential components by the relation then in view of condition (1) we can write the rotational component of the helix motion

New parameter

introduced here means the dimensionless torsion i. e. ratio of the torsion to the curvature; in the same time equals the dimensionless helix pitch The term H appears from the regular part and - from the parts responsible for the interaction with the tube wall: The evaluation has showed that contribution of into does not exceed 1. 5% and may be neglected. In view of these notes H and read

On motion of a double helical vortex in a cylindrical tube

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Now the problem on determination of the double helical vortex motion is reduced to the evaluation of the mutual influence of the helices and in the case of vortices inside a cylinder - to additional evaluation of the velocity induced by the interaction of the second helix with the wall.

3.

Solution and analysis

We again consider a presentation for the motion velocity of (2) type. From the singular member taken at we find the contribution [– ln 2] entering into the square brackets and { –1/2} entering into the figure brackets in (7). The term arising from the regular part is similar to (8) and represents series with the members of altering sign (multiplied by and in combination with (8) it gives the series for the double helix

In the terms responsible for the interaction with the wall the singular part plays again the dominant role to give together with (9)

The impact from the regular part is negligible. Thus for the rotational motion of the double helical vortex we have

The analysis of the series yields asymptotes at large and at small is the Riemann zeta function). Numerical estimation of at moderate showed that it is two orders less then the first member in the figure brackets and also can be neglected. The axial component of the double helix motion in accordance with condition (1) can be found from a simple relation The dependencies of the velocity components are presented in figure 1. At the small the dimensionless rotational velocity tends to 1 and the axial velocity (at behaves as In At the large pitch the rotational velocity tends to asymptote + 1/2 and the axial velocity decreases as Naturally, at limit we obtain the motion of two thin straight vortices placed symmetrically relative the cylinder axis. Thus, a simple analytical formula is derived for the description of motion of a couple of thin helical vortices inside cylindrical tube.

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Acknowledgments This research has been supported in part by the RFBR Grant 00-0565463, Council for Scientific Schools Support, Grant 00-15-96810 and INTAS Grant 00-00232.

References ALEKSEENKO, S. V., KUIBIN, P. A., OKULOV, V. L. & SHTORK, S. I. 1999 Helical vortices in swirl flow. J. Fluid Mech. 382, 195–243. BOERSMA, J. & WOOD, D.H. 1999 On the self-induced motion of a helical vortex. J. Fluid Mech. 384, 263–280. FALER, J.H. & LEIBOVICH, S. 1977 Disrupted states of vortex flow and vortex breakdown. Phys. Fluids 20, 1385–1400. HARDIN, J. C. 1982 The velocity field induced by a helical vortex filament. Phys. Fluids 25, 1949–1952. KUIBIN, P.A. & OKULOV, V.L. 1998 Self-induced motion and asymptotic expansion of the velocity field in the vicinity of a helical vortex filament. Phys. Fluids 10, 607–614. MOORE, D.W. & SAFFMAN, P.G. 1972 The motion of a vortex filament with axial flow. Trans. R. Soc. London Ser. A 272, 403–429. OKULOV, V.L. 1995 The velocity field induced by vortex filaments with cylindric and conic supporting surface. Russian J. Eng. Thermophys. 5, 63–75. RICCA, R.L. 1994 The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241–259. TAKAKI, R. & HUSSAIN, A.K.M.F. 1984 Dynamics of entangled vortex filaments. Phys. Fluids 27, 761–763. WOOD, D. H. 1998 The velocity of the tip vortex of a horizontal-axis wind turbine at high tip speed ratio. Renewable Energy 13, 51–54.

Intensive and weak mixing in the chaotic region of a velocity field Alexandre GOURJII Institute of Hydromechanics, National Academy of Science of Ukraine 8/4 Zhelyabov str., Kiev 03680, Ukraine [email protected]

Abstract The local stirring properties of a passive fluid domain with arbitrary borders in known velocity field are discussed. Construction of maps for local stretching values in fixed moments allows to analyse informatively an evolution of regions, in which an intensive stirring takes place. The stirring process is explored in a sample of an advection problem of a passive impurity in the velocity field induced by a system of three point vortices moved periodically. It is shown that the regions of a chaotic motion of fluid particles and of an intensive stirring do not coincide. Chaotic region has a zone of weak stirring, in which contours are transported from one intensive stretching zone to another without any deformation. As three vortices follow their paths There’s a chance for some elegant maths; There’s absolute chaos From Kiev to Laos, You may see this in rivers and baths.

1.

Introduction

Mixing is a complex natural phenomenon that includes various mechanisms, the two most important being stretching due to velocity field and diffusion due to Brownian motion [Ottino, 1989]. In some cases diffusive effects can be neglected because of physical characteristics of fluid or time scale of the phenomenon, and the problem can be reduced to the analysis of deformation process of fluid domain in the velocity field. The velocity field is assumed to be given a priori. The problem on deformation of appointed regions, usually called the advection problem in literature [Ottino, 1989; Aref, 1990], is limited to the analysis of the trajectories of Lagrangian fluid particles, which form borders of the region under investigation, in Eulerian velocity field. Every fluid particle can be treated as a passive fluid particle, and governing equations of the problem are the system of differential equations of the 87 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 87–92. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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first order (Cauchy problem) with corresponding initial conditions. Every such particle moves along its own trajectory, and one can predict particle positions at any moment. Their ordered connection results to forming domain borders for a given moment. It is commonly accepted [Ottino, 1989; Aref, 1990; Meleshko & van Heijst, 1994, and references]: if the length of contours is increased exponentially in time, we have a chaotic system. If the length of contours is increased linearly, then the considered process is regular. Even for simple laminar flows, some hydrodynamical systems can be non-integrable and display a chaotic behaviour. Aref, 1990 proposed the terms chaotic advection or Lagrangian turbulence for this phenomenon, and it is the subject of intensive studying of many investigators.

2.

Formulation

Our objective directs to an investigation of the local properties of stretching for various segments and contours. Consider an arbitrary fluid particle in a two-dimensional unsteady velocity field U and another particle placed nearby for We determine in such way to introduce the velocity field U in the linearised form. Then the evolutionary equation can be presented in the form

where

Here our primary interest is the change of length of the vector in time, > 0. The components and define a temporal scale during which the particle A remains in the neighbourhood of O. The solution of equations (1) depends on the solution of the characteristic equation of the system, complex values and

where is a value, which has the largest real part; functions A, B, C, D, E and F are determined by gradients of the velocity field evaluated They depend on the sign of the function = at the point + (ad – cb).

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89

A smooth contour can be presented by a set of passive particles (markers). Their local coordinates at the initial moment can be given as where

is the parameter of the contour. The relative length of the contour is determined by the integral

=

Here the dot denotes differentiation of the functions with respect to It is important to note that the exponential part in eq. (4) depends on the velocity field. It is not controlled by the type of contour, which surrounds the point O. The other part of the equation is determined by an integral, which describes local stretching properties of given contours for An analysis above can be carried out for another base point of the velocity field for the same fixed moment The stretching peculiarities can be presented using topological maps (local stretching maps), where it is convenient to plot the value for chaotic advection regimes. The set of maps for various moments permits to analyse an evolution of regions, which have an intensive or weak stretching, in time.

3.

Advection problem analysis

Let us apply the method based on local stretching maps to the flow generated by a system of point vortices. The motion of N point vortices of intensity at position is described [Aref, 1983] by

Here the dot denoted the derivative with respect to time, the asterisk means the complex conjugate and the prime denotes omission of the singular term The motion of the system of point vortices is defined by a Hamiltonian dynamical system

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The integrability of the system depends on the number of vortices and, for the case of three and more vortices, the generated velocity field leads to a chaotic advection and, as a result, to an intensive stirring. The equation of motion of a passive Lagrangian fluid particle in the velocity field induced by the system of point vortices may be obtained by considering a marker at as a point vortex of zero intensity [ Meleshko et al., 1992]. The equation of passive fluid particle motion can be also transformed into Hamiltonian form. Therefore, the dynamical problem of a time-dependent Hamiltonian system with one degree of freedom is equivalent to the kinematic problem of passive fluid particle. Consider the periodic motion of three point vortices with the following non-dimensional initial conditions: The additional information about details of this interaction of such system of point vortices have been presented in Meleshko & Gourjii, 1994. The evolution of the vortex system is shown in fig. 1. During an interaction, vortices 1 and 2 form a vortex pair moving periodically with local period At the same time three vortices rotate around centre of vorticity with global period here First we can identify global chaotic regions of motion of fluid particle [Ottino, 1989]. Figure 2 shows the Poincaré section for fluid particles moved in the velocity field under the vortex interaction. There are both regular and chaotic regions of motion: the fuzzy collection of points corresponds to a chaotic motion, while the ordered set of points detects the regions of regular motion.

Intensive and weak mixing in the chaotic region of a velocity field

91

The map of exponential coefficient is plotted in fig. 3 using topological level method. The value of stretching of contours due to exponential multiplier has a dominant role in the region, where point vortices place. Chaotic region has also zones with weak stirring. These zones are located on the sides and on the top of the chaotic domain plotted in Poincaré section (fig. 2). To analyse an advection problem, one needs also to simulate this process for passive fluid particles in given velocity field induced by three point vortices. Calculations were performed for passive domain initially distributed in a circle of radius 0. 5, whose centre is placed in the chaotic domain of motion with coordinates (fig. 4). We apply the piece-spline interpolation method [Gourjii et al., 1996], which lets us to form the borders of a closed domain using an ordered set of markers. The moment is shown in fig. 5a. The system has a mutual position that is similar to the initial moment. The shape of the passive domain is practically unchanged. The passive domain remains in the region of the chaotic advection shown by dashed lines. The intensive regime begins only in this moment. Significant deformation takes place. Now the studied region has strong stretching, and is deformed almost into a line during the next half period of the interaction. Fig. 5b shows borders of the domain at The contour is subjected to a significant deformation: its length is increased exponentially. It confirms that a chaotic advection regime had risen. The construction of local stretching maps for various contours for the fixed moment gives a considerable utility in a preliminary analysis of the advection process of passive fluid contours in velocity field to be given.

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These maps, like snapshots of stretching values (or normalised stretching values) for various contours, helps not only to find the existence of regions of intensive stirring, but also, using the set of maps, to detect their drift in time.

Acknowledgments The author gratefully acknowledges financial support from the Organising Committee of the Symposium.

References AREF H. 1983 Integrable, chaotic and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345–390. OTTINO J. M. 1989 The Kinematics of Mixing: Stretching, Chaos and Transport. Cambridge. Cambridge University Press. 683. AREF H. 1990 Chaotic advection of fluid particle. Phil. Trans. R. Soc. London 333, 273–288. MELESHKO V. V., KONSTANTINOV M. YU., GURZHI A. A. AND KONOVALYUK T. P. 1992 Advection of a vortex pair atmosphere in a velocity field of point vortices. Phys. Fluids A4, 2779–2797. MELESHKO V. V. & VAN HEIJST G. J. F. 1994 Interacting two-dimensional vortex structures: point vortices, contour kinematics and stirring properties. Chaos, Solitons & Fractals 4, 977–1010. MELESHKO V. V. & GOURJII A. A. 1994 Stirring of an inviscid fluid by interacting point vortices. In Modelling of Oceanic Vortices (ed. G. J. F. van Heijst), Proceeding Colloquium Royal Netherlands Academy of Arts and Science, Amsterdam, North-Holland. 271-281. GOURJII A. A., MELESHKO V. V., VAN HEIJST G. J. F. 1996 Method of piece spline interpolation in an advection problem of passive impurity in a given velocity field, Dop. AN Ukrainy, N. 8. 48–54.

Evolution of the anisotropy of the quantum vortex tangle Tomasz LIPNIACKI Institute of Fundamental Technological Research, 21 Swietokrzyska St., 00-049 Warsaw, Poland [email protected]

Abstract The evolution of a quantised vortex tangle in superfluid depends on its line-length density, but it is also related to various geometrical measures of the vortex lines forming the tangle. In this paper microscopic dynamics of the vortex tangle is studied analytically to derive an evolution equation for an average binormal to the vortex lines, which is an important measure controlling the growth of the vortex tangle. This equation supplements the Vinen equation for the line-length density. The resulting system (in the contrary to the both Vinen equation alone and alternative Vinen equation) is applicable to an analysis of transients in which the counter-flow changes its direction as well as to the processes with counter-flow changing periodically with various frequencies. Research on fields anisotropic Is surely an interesting topic; The unit binormal Shows features abnormal, As seen if you’re not too myopic!

1.

Introduction

The variety of the dynamic phenomena exhibited by the superfluid He (He II) involves the appearance and motion of quantised vortices. We recall that at low velocities He II flows in a frictionless, presumably laminar manner and can be described within the ideal fluid model. When the characteristic velocities becomes sufficiently large, the superfluid laminar flow develops into a superfluid turbulent flow in which quantum vortices form a chaotic tangle. The quantum tangle is a complex system which behaviour can be analysed via various geometrical and topological measures. The topological complexity measures like linking number, writhing number etc. are on the current investigation, see Barenghi, Samuels & Ricca in this volume. Here we focus on geo4


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metrical properties of a tangle which is generated by counter-flow (the relative velocity of the two components) In such a case only the superfluid flow is turbulent while the normal flow may remain laminar, what significantly simplifies the analysis. Our description of dynamics of quantum vortices will base on the Localised Induction Approximation (LIA), supplemented by the assumption that vortex lines reconnect when they get close enough. Let us recall: If a curve traced out by a vortex filament is specified in a parametric form, an instantaneous velocity of a given point of the filament (in the superfluid reference frame) reads:

where the over-dot and the prime denote instantaneous derivatives with respect to time and arc length respectively, while non-dimensional friction coefficients, arid is a parameter of order of quantum circulation. If the motion of a vortex filament fulfils Eq. (l) its line-length satisfies the following equation (Schwarz 1988):

The main parameter characterising the vortex tangle is its line-length density L, i. e. the total length of vortices per unit volume The evolution equation for L follows from Eq. (2). Define (after Schwarz 1988) the following characteristic measures of a vortex tangle

Note, that vector I measures the anisotropy of the binorrnal to the vortex lines. For example, I = 0 for an isotropic tangle and if all binomials to the vortex lines forming the tangle are parallel. The parameter introduced by Schwarz is defined as Using the above measures one can rewrite Eq. (2) in the following form:

Evolution of the anisotropy…

95

The equation above corresponds to the Vinen equation

In Eq. (7) the phenomenological coefficients and has been interpreted as functions of the microscopic measures of the tangle. Moreover let us note that in Eq. (7) the generation term is proportional, not absolute value of counter-flow velocity like in Eq(8), but to its scalar product with anisotropy vector I. The anisotropy of the binomial is induced by the counter-flow, and for steady state turbulence the anisotropy vector I is parallel to However, when the counter-flow changes its direction abruptly to the opposite, then, before the polarisation of the vortex tangle will adjust to the new direction of counter-flow, the generation term in Eq. (7) will be negative. As a result, the line-length density must drop below its equilibrium value before the new equilibrium is reached. Note, that according to the classical Vinen equation (8) such transients are invisible since the generation term, which is proportional to the absolute value of counter-flow, remains constant. This example suggests that the Vinen equation should be supplemented by additional equation for time dependence of the anisotropy vector I, but, possibly, also by equations for other geometrical measures of the tangle like and

2.

Time evolution of anisotropy of the binormal to vortex lines in a tangle

In this section we propose an evolution equation for the anisotropy vector I. We assume that the time derivative dI/dt is composed of two terms:

Here, the generation term corresponds to the growth of the anisotropy due to the action of the counter-flow via mutual friction while the decay term describes the relaxation of the tangle due to reconnections. In order to devise the generation term let us recall the dynamics of a single vortex ring under the LIA (1) (eg. Lipniacki 2000 ). Let b be the unit binormal to a given vortex ring (normal to the plane containing the ring). Then

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Now, let us represent the tangle as a sum of vortex rings (or parts of the rings) having the same value of curvature: Such a representation is equivalent to neglecting the influence of higher derivatives (like ... ) on vortex dynamics, in the calculation of the generation term. Let f(b) denote the angular distribution of the binormal to the rings. Then we have

where denotes integration over unit sphere. To calculate the left hand side of Eq. (11) let us assume that f(b) is the most probable distribution of binomials to vortex rings which results in a given I i. e.

After standard calculations (see Lipniacki 2001a for details) one gets

where

is given by the implicit formula below:

Inserting Eq. (13) into Eq. (ll) one gets

The above expression can be easily calculate only in two limits; when the anisotropy is zero, or on the other extreme, when the i. e. when all rings have parallel binomials. The results obtained in these two cases suggest the following approximation of Eq. (15)

Evolution of the anisotropy…

97

We expect that the main contribution to the decay of the anisotropy of the binormal is due to the relaxation of the tangle caused by reconnections of vortices. Thus, to deduce this term we cannot anymore treat the tangle as a composition of vortex rings. Reconnections produce kinks and add the curvature to the vortex tangle. However, since any two kinks produced by a reconnection have opposite orientation, the reconnections do not alter the average binormal in the tangle. In the result the tangle anisotropy decreases at any reconnection event. Let denote time in which the reconnections produce the total curvature C, equal to that of lines forming the vortex tangle,

Time determine the characteristic time in which the tangle ”forgets” about is initial anisotropy. It may be calculated as follows. The reconnection frequency per unit volume, can be estimated, neglecting the influence of the counter-flow on vortex velocities, as (see Lipniacki 2001b)

Each reconnection introduces a curvature of the order of can be expressed as

Hence, time

Assuming an exponential decay of the anisotropy vector (in the case when the counter-flow is ”switched off”) we postulate the decay term in the following form:

where, according to Eq. (19), Finally, we get the evolution equation for the anisotropy vector I

which supplements the equation for line-length density, L.

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The above system is analysed in Lipniacki 2001a. Here we note only that the evolution of line-length density L predicted by Eqs. (21, 22) significantly differs from that predicted by Vinen equation alone or alternative Vinen equation, especially when the counter-flow velocity changed its direction during the transient. The above system is applicable for analysis of processes with counter-flow changing periodically. When counter-flow changes its direction periodically then there exists the critical frequency above which the line-length density L drops to zero. The prediction of that critical frequency (which is dependent to the amplitude of counter-flow signal) can be use to validate evolution system (21, 22) and so the proposed approach experimentally.

Acknowledgments The author is grateful to Carlo Barenghi, David Samuels for a stimulating discussion. This work was supported by the KBN Grant No 7T07A 01817.

References BARENGHI, C.F., SAMUELS D.C. & D.C. RICCA, R.L. Complexity measures of tangled vortex filaments This volume. LIPNIACKI, T. 2000 Evolution of quantum vortices following reconnection, Eur. J. Mech., B/Fluids 19, 361–378. LIPNIACKI, T. 2001 Evolution of the line-length density and anisotropy of quantum tangle Phys. Rev. B, 64, 214516-1-9. LIPNIACKI, T. 2001 From vortex reconnection to quantum turbulence. In Quantized Vortex Dynamics and Superfluid Turbulence (ed. C. Barenghi et al), Lecture Notes in Physics V. 571, Springer-Verlag. SCHWARZ, K.W.1988, Three-dimensional vortex dynamics in superfluid Homogenous superfluid turbulence, Phys. Rev. B, 38, 2398–2417.

Motion of vortex lines in quantum mechanics 1, 1

2

1,

&

3

Center for Theoretical Physics, Al. Lotników 32/46, 02-668 Warsaw, Poland [email protected]

2

Institute of Physics, Al. Lotników 32/46, 02–668 Warsaw, Poland

3

S. I. Witkiewicz High School, ul. 51, 01-737 Warsaw, Poland

Abstract In quantum theory, vortex lines arise in the hydrodynamic interpretation of the wave equation. In this interpretation, which is originally due to Madelung, the flow of the probability density for a single particle is described in terms of the hydrodynamic variables. For the sake of simplicity, the standard time-dependent Schrödinger equation, and the related vortex lines embedded in the probability fluid of the quantum particle, are considered here. A vortex line in this case is simply the curve defined by equating the wave function to zero. The linearity of the Schrödinger equation enables us to obtain a large family of exact time-dependent analytic solutions for the wave functions with vortex lines. Moreover, the method is general enough to allow for various initial configurations of the vortex lines. Although the equation of motion of the quantum mechanical probability fluid is different in its literal form from the equations describing the real physical fluid, we believe that the evolution of the vorticity in the quantum and in the real fluid share the same qualitative features that can be described in terms of the topology of the vortex lines configurations. The general phenomena such as the switch-over, creation and annihilation of vortices can be observed in the quantum mechanical fluid.

I’ll talk about quantum switch-over As felt going from Calais to Dover; The vortex lines shift Reconnect and then lift . . . I’m sorry my lecture’s now over!

1.

Introduction

The principle of probability conservation in non-relativistic quantum mechanics gives rise to a hydrodynamic interpretation of the quantum


100

wave equations. If we consider the Schrodinger equation for a quantum particle as an example, the probability density is and there is a simple formula for the probability flow current. Therefore, it is valid to interpret the probability density as the density of some “quantum fluid¨, and the corresponding conserved current as the flow of this “quantum fluid”. In et. al. (2000) it was argued that the zeros of a generic solution of the linear Schrödinger equation correspond to vortex line singularities in the quantum flow. In section 2 the objects of the study arc defined. In section 3 it is shown that the property of vorticity quantisation follows directly from the definition of the hydrodynamic variables. A very simple example of a straight vortex is given in section 4. A general method of generating more complex vortex solutions of linear wave equations is presented in section 5.

2.

Basic definitions and current conservation in quantum mechanics

The probability density the conserved current velocity field are connected with the wave function the formulae

and the through

where is the mass of the particle and is the electromagnetic vector potential. The continuity equation reads

Since the denominator in equation (4) may have zeros, the velocity field may have singularities like in case of a vortex line in a real fluid. In general, the zeros of a complex function of three real variables are one-dimensional varieties, i. e. lines. In the present context


they are vortex lines embedded in the solution equation

3.

101

of the Schrödinger

Quantisation of vortex strength

Since in the standard quantum mechanics the wave function is always a single valued function of the coordinates, the strength of a vortex line as measured by the circulation along any closed contour C encircling the vortex line and not intersecting other vortex lines,

must be quantised:

Indeed, the phase is well denned (up to a constant and modulo though), and the circulation is the total change of the phase along the contour C. If the velocity must tend to infinity as one approaches the vortex line, so as to satisfy the quantisation condition (8). Hence, nonzero vortex strength implies a singularity of the velocity field: quantum vortices are line singularities.

4.

An example: a straight vortex line

To give a simple example of a vortex line let us consider the wave function

The vortex line equation defines a straight vortex line along the (intersection of the planes and In this case the vortex strength

5.

A general method of generating vortex solutions

Let us choose as the initial condition for the Schrödinger equation an element of the continuous family of wave functions

102


103

104

parameterised be the components of the wave vector k. It is assumed that a solution of the time-dependent Schrodinger equation is known that corresponds to the initial condition By differentiating with respect to the components of k (any number of times), we obtain new wave functions. Each differentiation brings down a component of the position vector and in this manner we may generate an arbitrary complex polynomial that multiplies the initial wave function. Carrying out the differentiations, adding the results with appropriate complex coefficients, and setting k = 0 at the end, we arrive at the expression for the initial wave function of the form where and are real polynomials in the three variables and Conversely, given a function and two real polynomials arid we define two differential operators and by substituting in place of the variables in the polynomials and The two operators, acting on the family at yield the polynomial factors and again, providing a solution of the Schrödinger equation for the given initial wave function It is an obvious observation that is the polynomial part of the initial wave function, while the remaining part, denoted enters the solution via the initial condition

6.

Examples and conclusions

In section 5, a general method was presented to generate vortex solutions of the wave equations. Using this general method, example solutions were obtained for the free particle case, i. e. A(r, t) = 0, V(r) = 0. The pictures show the particularly peculiar phases of motion of two (fig. 1) and three (fig. 2–5) vortex lines, see the references for details. Our conclusion is that interesting phenomena occurring in the quantum mechanical probability fluid — in contrast with the “physical” fluids — can be easily described and visualised.

References MADELUNG, O. 1926, Z. Phys 40, 342. &

C. 2000, Motion of vortex

lines in quantum mechanics. Phys. Rev. A 61, 032110. &

Vortex lines in motion. Ada Phys. Pol. 100 — Supplement, 29.

C. 2001,

III

MAGNETIC STRUCTURE, TOPOLOGY AND RECONNECTION


Magnetic dissipation: spatial and temporal structure Åke NORDLUND Niels Bohr Institute and Theoretical Astrophysics Center, Juliane Maries Vej 30, DK-2100 Copenhagen Denmark [email protected]

Abstract A magnetically dominated plasma driven by motions on boundaries at which magnetic field lines are anchored is forced to dissipate the work being done upon it, no matter how small the electrical resistivity. Numerical experiments have clarified the mechanisms through which balance between the boundary work and the dissipation in the interior is obtained. Dissipation is achieved through the formation of a hierarchy of electrical current sheets, which appear as a result of the topological interlocking of individual strands of magnetic field. The probability distribution function of the local winding of magnetic field lines is nearly Gaussian, with a width of the order unity. The dissipation is highly irregular in space as well as in time, but the average level of dissipation is well described by a scaling law that is independent of the electrical resistivity. If the boundary driving is suspended for a period of time the magnetic dissipation rapidly drops to insignificant levels, leaving the magnetic field in a nearly force-free, yet spatially complex state, with significant amounts of free magnetic energy but no dissipating current sheets. Renewed boundary driving leads to a quick return to dissipation levels compatible with the rate of boundary work, with dissipation starting much more rapidly than when starting from idealised initial conditions with a uniform magnetic field. Application of these concepts to modelling of the solar corona leads to scaling predictions in agreement with scaling laws obtained empirically; the dissipation scales with the inverse square of the loop length, and is proportional to the surface magnetic flux. The ultimate source of the coronal heating is the photospheric velocity field, which causes braiding and reconnection of magnetic field lines in the corona. Realistic, three-dimensional numerical models predict emission measures, coronal structures, and heating rates compatible with observations. When driv’n by extreme agitation, I am subject to fierce dissipation; In each current sheet There’s created much heat, And my field thus achieves saturation.


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1.

Å. Nordlund

Introduction

Magnetic fields are ubiquitous in astrophysical objects; circumstances where there is no magnetic field present are exceptions. Indeed, much of the non-thermal activity that is observed in astrophysical systems is probably related to the presence of magnetic fields. Gravity is, indirectly, a major reason for the ubiquitous magnetic activity, because it tends to separate matter into dense and tenuous regions. Magnetic fields that connect such regions are subjected to stress in the dense regions, and are forced to dissipate in the tenuous regions. There, the magnetic field energy density can be many times higher than the thermal and kinetic energy density of the gas, and minor readjustments of the magnetic field may correspond to significant heating and acceleration of the gas. Understanding the principles that control the dissipation of magnetic energy when the plasma beta where is gas pressure and is magnetic pressure) is low and the magnetic Reynolds number where U is velocity, L size, arid magnetic diffusivity) is very high is a major challenge, and numerous research papers, review articles and books have been published on this subject over the years (e. g., Parker 1972, 1983, 1988, 1994; Sturrock & Uchida 1981; van Ballegooijen 1986; et al. 1989; Heyvaerts & Priest 1992; Longcope & Sudan 1994; Galsgaard and Nordlund 1996ab; Nordlund & Galsgaard 1997; Gomez, Dmitruk &; Milano 2000; to mention just a few). The solar corona is an ideal ‘test site’ for theories and models of magnetic dissipation, since it provides rich opportunities for observing both the spatial and temporal structure of a dissipating low beta plasma. The Sun is indeed a ‘Rosetta stone’ in the context of magnetic dissipation— once we understand how magnetic dissipation occurs under such well observed conditions we may be much more confident when extrapolating to more distant and less well observed circumstances. Numerical experiments have emerged as a complementary and rich source of inspiration in the quest to understand magnetic dissipation. Below I briefly summarise conclusions from two types of experiments; generic experiments where a low beta plasma is driven from two opposing boundaries, and realistic experiments that attempt to model solar coronal conditions as closely as possible.

2.

Boundary driven magnetic dissipation

Generic experiments demonstrate that magnetic dissipation does not occur in simple, monolithic current sheets, even if the boundary motions are large scale and slow (cf. Galsgaard and Nordlund 1996a for details

Magnetic dissipation: spatial and temporal structure

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about such experiments). Rather, a hierarchy of current sheets form, with smaller scale current sheets protruding from larger scale ones (cf. Fig. 1). On the average, the collective dissipation in the hierarchy of current sheets balances the boundary work, as it must, to satisfy energy conservation. The average level of work and dissipation at which the equilibrium is obtained does not depend noticeably on the resistivity (or, equivalently, the numerical resolution), as long as the magnetic Reynolds number is not too small. The work W, and hence the dissipation times the distance L between the boundaries, is proportional to the energy density of the magnetic field at the boundary, the average angle of inclination of the magnetic field at the boundary, and the boundary velocity

The crucial angle factor tan scales as tan where is the autocorrelation time of the boundary motions, is the “stroke length” of the boundary motions, and L is the distance between the driving boundaries. The average dissipation thus scales as

(Fig. 2a). Incidentally, it turns out that the scaling of the photospheric motions that drive the solar corona is such that which implies that similar contributions to the driving are obtained from an extended range of scales.

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A sequence of uncorrelated strokes at the two opposing boundaries would, if the connectivity between the boundaries was conserved, lead to further increase of the tilt angles. The magnetic field lines would become increasingly tangled, as each new stroke would cause the end points of field lines connected to neighbouring points at one boundary to move further apart at the other boundary. But the development of current sheets prevents connectivity from being conserved, and prevents the build-up of angles much larger than In the hierarchy of current sheets the smaller current sheets provide a dissipation path for the larger ones, down to the smallest current sheets at a few times the resistive scale. An increase of resolution (magnetic Reynolds number) allows even smaller current sheets to form, but does not change the large scale angles noticeably, and hence does not influence the average level of dissipation. In terms of the winding of one field line around another, the statistical steady state is one where the number of windings from one boundary to the other has an approximately Gaussian distribution, with a Gaussian width of order unity (Fig. 2b). It is indeed a well known result that a twist much larger than unity leads to violent instability (see Galsgaard & Nordlund 1996b and references cited therein).

3.

Suspended / resumed boundary driving

If the external work is suspended dissipation quickly drops, as the current sheets die out. Distributed (smooth) currents remain, but these dissipate much more slowly. The system is in an approximately forcefree state, similar to the one just before current sheets first turned on.


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When the external work is resumed current sheets return promptly, and dissipation quickly returns to the previous average level (Fig. 3). Note that, even though exactly the same velocity field is applied for exactly the same time, there are much sharper current concentrations (current sheets) in the right-most inset of Fig. 3, illustrating that the quiescent but spatially complex magnetic field in the suspended state is ”ripe” for quickly producing current sheets. The system thus reaches balance between driving and dissipation much more quickly from the suspended driving (~ force free) state than from the initial (potential) state.

4.

Coronal heating experiments

Realistic numerical models of coronal heating and activity are now within reach (Gudiksen & Nordlund 2002), even though the resolution

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constraints are quite severe. In order to apply realistic and well calibrated boundary conditions it is necessary to bridge the distance from the base of the corona to the photospheric surface, where both the magnetic field and the velocity field are known sufficiently well to specify the boundary driving. This limits the vertical resolution to better than, or of the order of, photospheric and chromospheric pressure and density scale heights. The horizontal resolution needs to be similar, to resolve the granular scales where the horizontal velocity amplitudes are largest. The size of a model, on the other hand, needs to be of the order of the size of an active region, or larger. The very high Alfvén velocity in the corona above active regions limits (via the Courant condition) the time step to of the order of 10–30 ms, at the given spatial resolution. Time intervals of at least several granulation turn over times (~ 5 min) need to be covered. The resulting overall constraints are demanding, but not excessively so, given the power of current, massively parallel supercomputers. With a relevant cooling function approximation (Kahn 1976) and Spitzer conductivity, and by computing synthetic emission measures that correspond closely to those observed from, e. g., the TRACE satellite (Aschwanden, Schrijver & Alexander 2001), one may compare the results directly with solar conditions. With initial conditions taken from a high resolution MDI magnetogram (Fig. 4a), and a random velocity field with scaling properties consistent with a 5 km/s rms solar photospheric velocity field, one obtains average heating rates over an active region and emission measure images that are similar to observed ones (cf. Fig. 4b).


5.

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Discussion and concluding remarks

The results of the numerical experiments, and the properties of the scaling law derived from them, provide evidence that we are finally approaching a basic understanding of magnetic dissipation. First of all, the basic form of the scaling law (2) follows from first principles, and agrees with previously proposed scaling laws (Parker 1983, van Ballegooijen 1986). In addition, it adds a prediction for the crucial inclination factor, for which Parker (1983) and van Ballegooijen (1986) had to make arbitrary assumptions. Secondly, the scaling law is robust in that the crucial inclination factor is bracketed from above and below. It is bracketed from below because in general current sheets do not develop until the local winding number is of the order of unity. It is bracketed from above, because a local winding number much larger than unity inevitably leads to instabilities that rapidly dissipate the surplus magnetic energy. Because of the spontaneous formation of a hierarchy of current sheets the scaling law is also robust against changes of the magnetic Reynolds number. If anything, an increased Reynolds number could in principle lead to an increase of the magnetic dissipation because, as pointed out by Parker (1988), if one assumes that a reduction of the magnetic diffusivity initially leads to a reduction of the magnetic dissipation the consequence is only that the boundary work for a while exceeds the dissipation, which leads to an increase of the average inclination and hence to a further increase of the boundary work. When the magnetic dissipation eventually comes into balance with the boundary work again it happens at a higher level than before. But in practice, an increase of the magnetic Reynolds number just leads to an extension of the hierarchy of electrical current sheets to smaller scales, which makes it possible to dissipate at the same rate even without increasing the average angle of inclination at the boundaries. Any claim to the contrary must be accompanied by a demonstration that it is possible to sustain a distribution of winding number that is substantially wider than unity at high The scaling law (2) is furthermore consistent with scaling laws derived from observations, in that it predicts magnetic dissipation to scale as in agreement with Porter & Klimchuk (1995). Also, since the solar photosphere has a very intermittent distribution of magnetic field strength B, where B is either very weak or of the order of 1 kG, the average heating is predicted to scale roughly as the magnetic surface filling factor. This again agrees with the observed scaling of coronal heating (Fisher et al. 1998).

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Acknowledgments This work was supported in part by the Danish National Research Foundation through its establishment of the Theoretical Astrophysics Center.

References ASCHWANDEN, M. J., SCHRIJVER, C. J. & ALEXANDER, D. 2001 Modeling of Coronal EUV Loops Observed with TRACE. I. Hydrostatic Solutions with Nonuniform Heating. Astrophysical Journal 550, 1036–1050. FISHER, G. H., LONGCOPE, D. W., METCALF, T. R. & PEVTSOV, A. A. 1998 Coronal Heating in Active Regions as a Function of Global Magnetic Variables. Astrophysical Journal 508, 885–898. GALSGAARD, K. & NORDLUND, Å. 1996a The Heating and Activity of the Solar Corona: I. Boundary Shearing of an Initially Homogeneous Magnetic Field. Journal of Geophysical Research 101, 13445–13460. GALSGAARD, K. & NORDLUND, Å. 1996b The Heating and Activity of the Solar Corona: II. Kink Instability in a Flux Tube. Journal of Geophysical Research 102, 219–230. GOMEZ, D. O., DMITRUK, P. A. & MILANO, L. J. 2000 Recent theoretical results on coronal heating. Solar Physics 195, 299–318. GUDIKSEN, B. & NORDLUND, Å. 2002 Bulk heating and slender magnetic loops in the solar corona. Astrophys. J. Letters, 572, L113–L116. HEYVAERTS, J. & PRIEST, E. R. 1992 A self-consistent turbulent model for solar coronal heating ApJ 390, 297–308. LONGCOPE, D. W. & SUDAN, R. N. 1994 Evolution and statistics of current sheets in coronal magnetic loops ApJ 437, 491–504. Z., SCHNACK, D. D., & VAN HOVEN, G. 1989 Creation of current filaments in the solar corona ApJ 338, 1148–1157. NORDLUND, Å. & GALSGAARD, K. 1997 Topologically Forced Reconnection. Lecture Notes in Physicsh 489, 179–200. PARKER, E. N. 1972 Topological Dissipation and the Small-Scale Fields in Turbulent Gases. Astrophysical Journal 174, 499–510. PARKER, E. N. 1983 Magnetic Neutral Sheets in Evolving Fields. II - Formation of the solar corona. Astrophysical Journal 264, 642–647. PARKER, E. N. 1988 Nanoflares and the solar X-ray corona. Astrophysical Journal 330, 474–479. PARKER, E. N. 1994 Spontaneous current sheets in magnetic fields: with applications to stellar x-rays. New York : Oxford University Press, 1994. PORTER, L. J. & KLIMCHUK, J. A. 1995 Soft X-Ray Loops and Coronal Heating. Astrophysical Journal 454, 499–511. STURROCK, P. A. & UCHIDA, Y. 1981 Coronal heating by stochastic magnetic pumping ApJ 246, 331 VAN BALLEGOOIJEN, A. A. 1986 Cascade of magnetic energy as a mechanism of coronal heating ApJ 311, 1001–1014.

Current sheets in the Sun’s corona Eric PRIEST School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland [email protected]

Abstract Current sheets play an important role in the Sun’s atmosphere, especially in coronal heating events and solar flares. They may form in response to motions of the magnetic footpoints in the solar surface or following a loss of equilibrium. In two dimensions, X-type null points may collapse to current sheets, as described by nonlinear self-similar solutions and by complex-variable theory. Magnetic diffusion resolves the sheets and allows fast reconnection to take place. There are many ways in which such reconnection can occur, depending on the boundary conditions, including one family of almost-uniform regimes and another family of non-uniform regimes. In three dimensions, nulls may also collapse to give a growing current along the spine or the fan of the null. (These are, respectively, isolated field lines or surfaces of field lines that approach or recede from the null point. ) Dissipation can then occur by either spine reconnection or fan reconnection, or also by separator reconnection when the current concentrates along, respectively, the spine, the fan or a separator (which is a field line that links one null point to another). Coronal heating may be produced by reconnection in the following ways: by converging photospheric motions at X-ray bright points; by binary reconnection when pairs of magnetic sources interact; by separator reconnection in complex fields due to tertiary flux interactions; by braiding of field lines; and by coronal tectonics heating at separatrix surfaces between intense flux tubes. Solar flares may occur when a magnetic catastrophe causes the slow eruption of a flux tube, which in turn drives the formation of a current sheet under the flux tube. As the sheet dissipates and reconnects, the overlying field lines holding down the flux tube are released so that rapid eruption and energy release can take place. The magnetic field of the Sun Engenders a whole lot of fun; There’s nothing to beat Collapse to a sheet, And that’s how the heating’s begun.

1.

Introduction

The Sun’s atmosphere has three layers. The surface, called the photosphere, is only 6000K in temperature and it possesses sunspots, dark areas of very strong magnetic field. Images of the line-of-site compo-

115 K. Bajer and H. K. Moffatt (eds. ), Tubes, Sheets and Singularities in Fluid Dynamics, 115–124. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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nent of the magnetic field reveal bipolar flux at sunspot pairs and also the presence of concentrated magnetic flux over the whole solar surface at the boundaries of convection cells (supergranules). Above the photosphere lies the warmer and rarer chromosphere and above that the temperature rises dramatically to over a million degrees in the corona. A key problem in astrophysics is then to understand how the corona is heated to such high temperatures. The energy that is required to heat the corona is about 5000 in an active region (a region lying above a group of sunspots where the magnetic field is stronger than normal). The energy flux through the surface of the Sun is plentiful. Since the Poynting flux is of order which is typically W for observed velocities (v) of 0. 1 km and field strengths (B) of 100G. But the key question is: how is this energy flux converted to heat? The induction equation

describes how the magnetic field changes in time. In the corona the ratio of the first to the second term on the right is of order 108 and so the diffusion term is completely negligible and the magnetic field is frozen to the plasma. The exception is in singularities where the electric current and therefore the gradient of the magnetic field is very large. So, how does the Sun create such singularities? In the following sections we describe: the formation and dissipation of 2D current sheets (Sections 2 and 3); the corresponding formation and dissipation in three dimensions (Section 4); and their importance for coronal heating (Section 5) and solar flares (Section 6). For more details of these subjects and for an extensive set of references, see Priest & Forbes 2000.

2.

2.1

Formation of current sheets in two dimensions X-points tend to collapse (locally)

If the field lines are free or there is an inflow of energy, an X-point tends to collapse (Figure 2). Physically, if you perturb a flux function from to then the force tends to continue the collapse. There are linear solutions with magnetic diffusion, which exhibit a decaying oscillation. Also, nonlinear self-similar solutions to

Current sheets in the Sun’s corona

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the ideal equations have been studied of the form

which produce a collapse in a finite time, but these are only local.

2.2

Sheet formation modelled with a complex variable

The formation of a sheet has been modelled using complex-variable theory For example, consider a potential field X-point (Figure la) having a field Move the footpoints of a finite region or add a finite energy to this minimum-energy configuration. Then, if the topology is preserved, to what configuration does this field evolve? Green suggested an evolution to a field of the form (Figure 1b) having a current sheet stretching from to Later, however, Somov & Syrovatsky 1976 suggested an alternative evolution to which contains singularities at the ends of the sheet (Figure 1c).

2.3

Self-consistent formation

More recently, self-consistent solutions satisfying the ideal MHD equations were discovered. Bajer 1990 gave numerical examples using the method of Moffatt 1985 to solve the ideal induction equation (which preserves the topology) and an equation of motion with an artificial damping term that ensures that the motions decay away in time. Furthermore, an analytical example of time-dependent sheet formation (Priest et al 1995) was obtained by expanding the MHD equations in powers of the Alfvén Mach number

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so that, to lowest order, the field evolves through a series of quasi-potential states with outside current sheets, in such a way that the velocity satisfies the ideal induction equation together with the condition that the acceleration be perpendicular to the field One example has singular end-points to the sheet.

2.4 Extensions to the basic theory The basic theory has been extended in many ways. First of all, the form of has been written down for other configurations. Also, a general method has been proposed for finding f(z) when the topology is preserved (Titov 1992), and techniques have been developed for modelling three-dimensional sheets. The formation of sheets in forcefree and magnetostatic fields have been treated (Bajer 1990; Bungey & Priest 1995). Finally, it has been realised that shearing motions of photospheric footpoints can form current sheets along separatrices.

3.

Dissipation of current sheets in two dimensions

When an X-point collapses to a sheet, dissipation allows the field lines to break and reconnect (Figure 1). The magnetic energy is converted to heat, kinetic energy and fast particle energy. This process is the source of heating and of many dynamic processes on the Sun, so how does it occur in both 2D and 3D? In 2D reconnection takes place at an X-type null point, where a localised breakdown of the frozen-field condition produces a change of field-line connectivity. The theory in two dimensions is well developed and includes: slow Sweet-Parker reconnection, fast Petschek reconnection and many other fast regimes that depend on the boundary conditions, such as Almost-Uniform and Nonuniform reconnection.

3.1

Classical reconnection Sweet and Parker suggested a model consisting of a simple current sheet with a uniform inflow (Figure 2a). The sheet has length 2L and width 21, while the inflow velocity and field are and and the outflow values are and Mass conservation into and out of the sheet implies while a steady balance between inwards advection of flux and its diffusion implies Furthermore, acceleration along the sheet by the force accelerates the plasma to an outflow speed equal to the Alfvén speed Eliminating L and between these three relations gives an expression for the reconnection rate which may be written in dimensionless form as


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where the magnetic Reynolds number is With say, the rate is only too slow to explain a solar flare. Later, Petschek proposed a much more rapid mechanism, in which the sheet bifurcates into two pairs of slow-mode shocks at which most of the energy conversion takes place (Figure 2b). The reconnection speed may be at any rate up to a maximum of which is typically 0. 01 - 0. 1. Here the subscript e refers to “external” values in the inflow region far from the current sheet. More recently, a family of regimes of Almost-Uniform reconnection was discovered in which the magnetic field lines are weakly curved in the inflow region (Figure 4b-e). The different regimes are produced by different boundary conditions on the inflow, producing either converging or diverging flows. Petschek’s mechanism is just one special case. These models were then generalised even further by allowing the inflowing field lines to be strongly curved (Figure 2f). The resulting NonUniform Reconnection regimes again depend on the boundary conditions and agree with numerical experiments provided the same boundary conditions are adopted (and the magnetic diffusivity is nonuniform).

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Three-dimensional reconnection Null-points - structure and collapse

In three dimensions the simplest null point has a field which possesses two families of field lines that pass through the null (Figure 3). An isolated “spine” field line approaches (or recedes from the null), here along the while a “fan” surface of field lines emanates from it (or approaches it), here in the (Priest & Titov 1996). Most generally, in the linear vicinity of a null the field can be written as

in terms of four parameters, namely (p, q) and the components of the current along the spine and normal to it (Parnell et al 1996). As in two dimensions, nulls can collapse to give a growing current along the spine or in the fan (Bulanov & Sakai 1997; Parnell et al 1996). There are linear and nonlinear treatments, which show the development of a singularity in a finite time, but the analysis is only local. When the fan current grows, the angle between the fan and spine decreases, whereas when the spine current grows the field lines in the fan rotate and close up (Figure 4).

4.2

Reconnection at a 3D null

At 3D null points there are three types of reconnection (Figures 5 and 6), namely spine reconnection (when the current focuses along the spine), fan reconnection (when it is concentrated in the fan) and separator reconnection, when the current grows along the separator joining one null to another (Priest & Titov 1996; Longcope 1996; Hornig &


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Schindler 1996). Figure 5 shows the result of a kinematic analysis, while Figure 6 gives an image from a 3D resistive MHD experiment of Galsgaard & Nordlund 1997a in which they explored the effect of imposing a shear on two boundaries of a box initially containing eight null points in equilibrium.

5.

Can reconnect ion heat the solar corona?

The solar corona is highly complex. It consists of three types of structure, which may well be heated differently. Coronal holes are magnetically open into interplanetary space, so the fast solar wind escapes from them, whereas coronal loops have both ends attached to the solar surface. There are also tiny points of emission called X-ray bright points, several hundred of which are present at any time. Reconnection may be heating in different ways. Simple reconnection may be driven at null points by photospheric motions to create an X-ray bright point according to the Converging Flux model. In this model new flux emerges in a supergranulation cell and migrates to the cell boundary, where it reconnects with pre-existfing flux on the boundary. A second way is by “binary reconnection” (Priest & Schrijver 1999) when pairs of oppositely directed flux sources in the solar surface move relative to one another and drive reconnection in the corona. Interactions between three or more sources in complex fields may instead involve separator reconnection (Priest & Titov 1996; Longcope 1996). Finally, braiding of field lines may also produce current sheets and heating. The relative motion of two photospheric sources in an overlying field has been studied numerically by Galsgaard et al 2000. Initially, the sources are unconnected, but as they move past one another reconnection takes place at a separator (Figure 7) and creates a twisted flux tube joining the two sources. Braiding was first proposed by Parker 1979 as

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a heating model and has been extensively studied (see Nordlund and Bhattacharjee). In particular, an experiment by Galsgaard & Nordlund 1997b shows the growth of current sheets and an impulsive heating.

6.

How do solar flares and coronal mass ejections occur?

The overall picture for a solar flare is that during the pre-flare phase a large flux tube with an overlying arcade of field lines becomes sheared


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and rises slowly. At some critical point, reconnection of the overlying arcade occurs at points below the rising flux tube. This disconnects some of the overlying field lines from the solar surface and so allows a rapid eruption to take place. As the eruption and reconnection continue, an arcade of rising hot loops is created with bright ribbons at its feet. A model in two dimensions for the cause of the eruption has been developed (Priest & Forbes 2000), consisting of a flux tube with an overlying arcade having sources that are a distance apart. As decreases and the sources slowly approach one another, the height of the flux tube slowly decreases and the configuration passes through a series of equilibria - until, that is, a critical point is reached beyond which there is no neighbouring equilibrium. If there is no reconnection, the flux tube rises towards a new equilibrium that contains a current sheet linking the tube to the solar surface. However, in numerical experiments, reconnection is driven at the current sheet and allows a rapid eruption of the tube. More recently, a three-dimensional version of this model has been developed.

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7.

Conclusion

On the Sun it is likely that the occurrence of current sheets is responsible both for coronal heating and for solar flares. In two dimensions, the formation and dissipation of current sheets is well understood, but in three dimensions the theory is only just beginning to be developed. However, there are many fascinating observations and numerical experiments to guide and inspire us in the future.

Acknowledgments I am most grateful to Konrad Bajer for organising an excellent meeting with such care and attention.

References BAJER, K. 1990 PhD Thesis, Cambridge University. BULANOV, S.V. AND SAKAI, J. 1997 Magnetic collapse in incompressible plasma flows. J. Phys. Soc. Jap. 66, 3477–3483. BUNGEY, T.N. AND PRIEST, E.R. 1995 Current sheet configurations in potential and force-free fields. Astron. Astrophys. 293, 215–224. GALSGAARD, K. AND NORDLUND, A. 1997a Heating and activity of the solar corona III. J. Geophys. Res. 102, 231–248. GALSGAARD, K. AND NORDLUND, A. 1997b Heating and activity of the solar corona I. J. Geophys. Res. 101, 13445–13460. GALSGAARD, K., PARNELL, C.E. AND BLAIZOT, J. 2000 Elementary heating events - magnetic interactions between two sources. Astron. Astrophys. 362, 383–394. HORNIG, G. AND SCHINDLER, K. 1996 Magnetic topology and the problem of its invariant definition. Phys. Plasmas 3, 781–791. LONGCOPE, D.W. 1996 Topology and current ribbons. Solar Phys. 169, 91–121. MOFFATT, H.K. 1985 Magnetostatic equilibria and Euler flows of complex topology. J. Fluid Mech. 159, 359–378. PARKER, E.N. 1979 Cosmical Magnetic Fields, Clarendon Press, Oxford. PARNELL, C.P., SMITH, J., NEUKIRCH, T. AND PRIEST, E.R. 1996 Structure of 3D magnetic neutral points. Phys. Plasmas 3, 759–770. PRIEST, E.R. AND FORBES, T.G. 2000 Magnetic Reconnection Camb. Univ. Press. PRIEST, E.R. AND SCHRIJVER, C.J. 1999 Aspects of 3D reconnection. Solar Phys. 190, 1–24. PRIEST, E.R. AND TITOV, V.S. 1996 Magnetic reconnection at 3D null points. Phil. Trans. Roy. Soc. Lond. 354, 2951–2992. PRIEST, E.R., TITOV, V.S. AND RICKARD, G.J. 1995 The formation of magnetic singularities by nonlinear time-dependent collapse of an X-type magnetic field. Phil. Trans. Roy. Soc. Lond. A. 351, 1–37. SOMOV, B.V. AND SYROVATSKY, S.I. 1976 Hydrodynamic plasma flow in a strong magnetic field. Proc. Lebedev Phys. Inst. 74, 13–72. TITOV, V.S. 1992 Calculating 2D potential fields with current sheets. Solar Phys. 139, 401–404.

A model for magnetic reconnection H.K. MOFFATT & R.E. HUNT Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 90E, UK [email protected]

If you’re not too dreadfully weary, You may ask: what’s the point of this theory? Just please recollect That B-lines reconnect; I hope that will answer your query!

1.

Introduction

Magnetic reconnection is a diffusive process whereby the topology of a magnetic field imbedded in a highly conducting fluid may change with time. The process is frequently represented by the simple diagram shown in figure 1 in which the oppositely directed lines of force AB, CD reconnect to the configuration AC, BD. A huge literature emanating from the early papers of Sweet (1958), Parker (1957) and Petschek (1964) has evolved; for a recent thorough exposition including discussion of the important application of the theory in solar, magnetospheric and other contexts, see Priest & Forbes (2000). Despite the prolonged research on this topic, there seems to be as yet no simple model which demonstrates unambiguously the necessarily time-dependent process indicated in figure 1 which takes account of the curvature of the field lines before and after reconnection. It is the purpose of this short paper to provide such a model. We restrict attention here to two-dimensional evolution in an incompressible fluid; generalisation to the more challenging problem of three-dimensional skewed flux tubes may however also be possible.

2.

The initial configuration

It is natural to represent the curves AB, CD in figure 1 by the hyperbola where k and Y are constants. We first construct a magnetic field 125 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 125–138. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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which is confined to two oppositely directed flux sheets centred on this hyperbola. With this is achieved by the choice where is the transverse scale of the flux sheet. We shall adopt nondimensional variables using as the unit of length; so from here on we may take We shall assume further that Y > 1 (so that the two flux sheets are ‘separated’ at t = 0, well separated if and we shall assume that the curvature parameter is < 1. The field components associated with (2) are

With sheet is

the flux (per unit distance in the

in the upper

and the flux in the lower sheet is similarly In order to induce rapid reconnection, we subject this field to the irrotational straining flow with (Figure 2). We may adopt as the unit of time, so that henceforth The Lorentz force with will obviously perturb this flow; but we may adopt (6) as an initial condition for the velocity field, and as an outer boundary condition as for all The initial streamfunction associated with (6) (with now is

Magnetic Reconnection

3.

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The evolution equations Consider first the magnetic induction equation

where is the (dimensionless) magnetic diffusivity parameter. For the two-dimensional situation, we take with

Note that is in the applied electric field in the be ‘uncurled’ to give.

Equivalently, with

and There is no In these circumstances, (8) may

and

The Navier-Stokes equation including the Lorentz force is, in standard

With vorticity

the vorticity equation is


128

or equivalently

Equations (11) and (14), coupled with the initial conditions A = at and the outer boundary conditions

provide a complete specification of the problem. Note the symmetries, A even in and odd in and which are respected by the evolution equations.

4.

Boundary-layer approximation

As the straining flow sweeps the two sheets towards each other, it is obvious that the field gradient in the will increase, while that in the will diminish. In these circumstances, a boundarylayer approach is legitimate. We expand A and in the form

where

is odd in

and g and

are even in

Note that then,

In each case, the leading-order term will be dominated by the part involving the i.e.

With this boundary-layer approximation, we have


129

and so, to leading order, (11) yields

and (14) yields

where a measure of the strength of the initial field. This equation integrates with respect to (as expected within the boundarylayer approximation) to give

where the constant of integration (+1) is fixed by the boundary conditions We have thus reduced the problem to three nonlinear coupled evolution equations (24), (25), (27). Clearly the initial condition on is just To obtain initial conditions on

and h, we expand (2) in the form

so that, from (4. 2), these initial conditions are

5.

Weak field limit

If then the field behaves, at least in the initial stages, as a passive vector field. We then have and (24), (25) become

If diffusion is negligible

then the field is ‘frozen in’,

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and so

We then have

Thus, although the magnetic field strength increases exponentially (like the relevant part of the Lorentz force in (27) (that part not compensated by the pressure field) remains this is because the curvature of the field lines near decreases in proportion to There may be a weak cumulative effect of the Lorentz force over a long time; but the perturbation velocity field driven by this Lorentz force certainly does not increase exponentially, and remains relative to the imposed strain field. When this frozen field development lasts only for so long as the diffusion terms in (33), (34) remain negligible. From (36), becomes comparable with when

at which stage the (dimensionless) field strength is of order From this point on, diffusive reconnection proceeds in a layer whose thickness is and presumably does not decrease much below this.

6.

Saddle point reconnection Being even in

the functions

admit expansions

where, from (31), (32),

Of course, for

we have

Near the origin, the B-lines A = cst. are hyperbolic and given by


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The separatrices through O are the lines

Note that at , although the field is exponentially weak near O, these separatrices are well-defined as Under the frozen-field evolution (36), (37), they move with the fluid to position

i. e. the angle between the separatrices decreases exponentially. At time , the flux crossing the positive is at time , it is

The difference, is the flux that has reconnected and is associated with the field in the region (see figure 3). Clearly, this reconnection process occurs in an -neighbour-

hood of the saddle point of the field B where and cannot occur in the absence of such a saddle point. Figure 4 shows the computed evolution of based on equations (24), (25), (27), for the following parameter values:

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As expected, remains constant at near -1 for small t, while diffusion is still negligible; it then rises quite rapidly and tends to +1, the reconnection process being 99% complete by Contrary to received opinion, there is no indication here that the Lorentz force has a dominating effect on this process. The time-scale of the process is O(1), i.e. when we return to dimensional variables; this is because, however small the value of the transverse scale decreases to before the onset of diffusion, and on dimensional grounds the reconnection time-scale is necessarily Of course this conclusion may need modification if the magnetic field parameter M is greatly increased. Numerical experiments over the space of the parameters and are in progress.

References PARKER, E.N. 1957 Sweet’s mechanism for merging magnetic fields in conducting fluids. J. Geophys. Res. 62, 509–520. PETSCHEK, H.E. 1964 Magnetic field annihilation. In Physics of Solar Flares (Ed. W. N. Hess), NASA SP-50, Washington, DC, 425–439. PRIEST, E.R. & FORBES, T. 2000 Magnetic Reconnection Camb. Univ. Press. SWEET, P.A. 1958 The neutral point theory of solar flares. In Electromagnetic Phenomena in Cosmical Physics, IAU Symp. 6, (Ed. B. Lehnert) Camb. Univ. Press, 123–134.

Reconnection in magnetic and vorticity fields Gunnar HORNIG Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, 44780 Bochum, Germany [email protected]

Abstract Reconnection is an important process of structure formation in fluid dynamics. It occurs in form of vortex reconnection in hydrodynamics as well as in form of magnetic reconnection in plasmas. Two basic types of reconnection are known.

The planar two-dimensional reconnection and the more generic, but more complicated three-dimensional case. These two types differ for example with respect to their production of helicity in reconnection. Simple analytic examples of vortex reconnection are given and compared with corresponding solutions of magnetic reconnection. It is shown that, while for magnetic reconnection two-dimensional stationary solutions exist, vortex reconnection always requires a time-dependent velocity field. This explains why vortex reconnection in spite of all similarities often has a more complicated geometry than magnetic reconnection . The process of line reconnection Results from diffusion-advection; With a flick of the wrist I can generate twist, And this for your greater detection!

1.

Introduction

Reconnection is an important process both in magnetohydrodynamics (MHD) and hydrodynamics (HD). It describes a change in the topology of field lines of a divergence-free field, i. e. the vorticity field in HD or the magnetic field in MHD. The equation which determines the evolution of the divergence-free field is in the case of vorticity the curl of the NavierStokes equation and for the magnetic field the curl of Ohm’s law. We can summarize both in


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where is either the vorticity field or the magnetic field and denotes the resistive term in Ohm’s law or the right hand side of the NavierStokes equation, i. e. The process of reconnection is most pronounced in fluids or plasmas which have respectively a high Reynolds number or magnetic Reynolds number. Here the effect of is negligible almost everywhere in the fluid so that is frozen-in with respect to the flow v. For a vanishing the flux of and the topology of field lines are conserved due to Kelvin’s theorem (Alfvén’s theorem in MHD). However, the approximation of small compared to can break down easily near null points of Figure 1 shows an example of a planar two-dimensional reconnection process, that is a process where can be written as The images show the evolution of a set of G-field lines which initially form a single flux tube. The field lines are integrated from two crosssections (black) which are moving with the fluid. The non-ideal term dominates the evolution only near the X-point of the field (z-axis) and thereby enables reconnection.

For the planar case eqn. (1) can be rewritten as an ideal evolution,

where the velocity u is to be considered as the transport velocity of the lines. For the case of a non-vanishing at an X-point of we get a stagnation flow with a singularity at the X-point. In the generic case, i. e. if does not vanish in higher order at the null, the singularity scales as where is the distance from the null point. This results in a transport of the field lines onto the x-point in a finite time (Hornig 2001b). Correspondingly the newly connected field lines are transported in a finite time downstream away from the null. The result is in accordance with numerical calculations by De Waele (1994), which show a quadratic dependence of the reconnection time on the distance between the flux tubes. The particular structure of the singularity

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is the mathematical reason for the ‘cut and reconnect’ process. It can be used for a definition of reconnection in planar configurations (Hornig 2001b). Simple examples of this type of reconnection for both MHD and HD were given in Hornig (2001a).

2.

Three-dimensional reconnection

In the investigation of magnetic reconnection it was soon realized that the planar, two-dimensional concept of reconnection is not easily transferred to three-dimensional systems. Already adding a small constant component to parallel to the former invariant direction does not significantly change the dynamics of the process but inhibits the determination of a transport velocity The latter exists if and only if which is no longer satisfied if there is a parallel component of It was shown in Hornig (2001b) that both the planar and the three-dimensional reconnection can be described in a general framework of covariant flux conservation. Within this description there exist corresponding transport 4-velocities, which in the case of three-dimensional reconnection cannot be mapped to 3-vector flows The dynamics of flux tubes in the three-dimensional reconnection is different form the planar case in that there does not exist a well-defined place where the field lines are cut, instead the flux tubes undergoing reconnection split as soon as they enter the region where is relevant for the dynamics (see Figure 2). Thereupon they flip around each other before they merge to two newly connected flux tubes. In MHD this is sometimes called magnetic flipping. The splitting of the flux tubes reflects the fact that no transporting flow neither smooth nor singular for this process exists. This process differs from the planar case in another aspect. It does not conserve the total helicity. The total helicity measures an averaged linkage of field lines in a certain volume

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V and is defined as

where is the vector potential for This quantity is well defined if the volume V is bounded by a surface which is tangential to lines. The dynamics of this quantity can be derived from the identity where is the electric potential in MHD and in HD we can use for all non-relativistic flows. Integrating over a volume with a boundary on which the helicity current vanishes (e. g. where and yields

The term on the right hand side is to be considered as the source term of helicity. For MHD the source term vanishes for the planar reconnection since For HD it vanishes only if we assume divergencefree, isentropic flows since in this case not only has to vanish, but also Note that this holds also for threedimensional fields or which show a planar type of reconnection. It is not required that the field is planar outside the reconnection region. For the three-dimensional case the helicity production will be in general non-vanishing. This effect causes a twist of the two flux tubes in the final state of Figure 2, provided the initial flux tube was untwisted.

3.

Reconnection solutions in HD

In the following it is shown how two- and three-dimensional reconnection solutions are constructed for the Navier-Stokes equation. We assume a constant viscosity , a constant density and a divergencefree velocity so that the continuity equation is trivially satisfied and it remains to solve First we consider two-dimensional solutions of the form and refer to the direction of the z-axis). This implies that both components, and are divergence-free and eqn. (6) splits into two equations:

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A solution of the system can be found by first solving eqn. (7) and substituting this into eqn. (8). The latter is the same type of equation that we have to solve for two-dimensional resistive MHD. There we express the magnetic field with the help of a flux function so that the resistive Ohm’s law reads

For a prescribed velocity that is for the so called kinematic problem, the solution of this equation are well studied both analytically and numerically, see for example Moffatt 1978. In particular we know that for reconnection to occur the velocity field has to have a stagnation flow, that is it has to have a hyperbolic null point. In addition the electric field has to be non-vanishing at the stagnation point. The rate of reconnected flux is given by an integral along the reconnection line, which is the line obtained by extending the X-point of in the invariant direction,

This implies that in vortex reconnection even if the vorticity field is planar the velocity field has to have a non-vanishing component in the third direction since Although the non-vanishing reconnection rate requires a time-dependent and three-dimensional velocity field for vortex reconnection, we can solve the system (7, 8) by starting with a time-independent solution of (7) with parameters

For this solution is a simple linear stagnation flow. Inserting it into Eq. (8) and using corresponding initial conditions we get a time-dependent solution which resembles the planar reconnection process shown in Figure 1. Note however, that the vorticity field is still invariant in one direction. The figure shows only a set of vorticity field lines forming a flux tube. This is different from the case where the vorticity is concentrated in tubes, as for instance in the numerical simulations by Kida (1991). By using in (11) we superimpose the stagnation flow with a field of constant vorticity in z-direction. This yields a simple extension from a planar two-dimensional vorticity field to a field which now has a component in the third direction. Reconnection in such a field evolves like in the example shown in Figure 2. To be more precise the figure shows a situation where i. e. the component, is comparatively small. For

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higher values of the process becomes increasingly asymmetric. Finally for the hyperbolic structure of turns into an elliptic structure and inhibits reconnection. This feature is not present in magnetic reconnection, it is due to the coupling of and by In MHD a similar relation between and does not exist. This restriction also limits the helicity production per reconnected flux in HD.

4.

Summary

A selection of fundamental features of reconnection in magnetic fields and vorticity fields were given and the differences for planar and threedimensional reconnection were discussed. It turns out that while for magnetic reconnection stationary two-dimensional reconnection solutions exist, vortex reconnection requires always a three-dimensional and timedependent velocity field. There exist, however, solutions where the vorticity field is planar and depends only on two coordinates. This shows that the complicated three-dimensional dynamics shown in many numerical simulations of vortex reconnection is mainly due to the global interaction of vortex structures and not a requirement of the local reconnection process.

Acknowledgments The author gratefully acknowledges financial support from the Volkswagen Foundation.

References DE WAELE, A.T.A.M. & AARTS, R.G.K.M. 1994 Route to vortex reconnection, Phys. Rev. Lett. 72, 482-485. HORNIG, G. 2001a The Geometry of Magnetic and Vortex Reconnection. In Quantized Vortex Dynamics and Superfluid Turbulence (ed. C. F. Barenghi, R. J. Donnelly & W. F. Vinen), Lecture Notes in Physics 571, Springer. HORNIG, G. 2001b The geometry of reconnection. In Geometry and Topology of Fluid Flows (ed. R. Ricca) Kluwer. KIDA, S., TAKAOKA, M. & HUSSAIN, F. 1991 Collision of two vortex rings, J. Fluid Mech. 230, 583-646. MOFFATT, H.K. 1978 Magnetic field generation in electric conducting fluids, Cambridge University Press.

Energy, helicity and crossing number relations for complex flows Renzo L. RICCA Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom [email protected]

Abstract Algebraic and topological measures based on crossing number relations provide bounds on energy and helicity of ideal fluid flows and can be used to quantify morphological complexity of tangles of magnetic and vortex tubes. In the case of volume-preserving flows we discuss new results useful to determine lower bounds on magnetic energy in terms of topological crossing number and average spacing of the physical system. New relationships between average crossing number, energy and helicity are derived also for homogeneous vortex tangles. These results find interesting applications in the study of possible connections between energy and complexity of structured flows. Topological arguments show That the energy’s bounded below; But what’s so engrossing’s The number of crossings, From which my new insights will flow.

1.

Magnetic and vortex knots as standard embeddings

Consider an incompressible and perfectly conducting fluid in an unbounded domain of that is simply connected, with fluid velocity smooth function of the position vector and time and such that in and at infinity. Consider the class of magnetic fields {B} that are solenoidal and frozen in that is

Under ideal conditions these fields have their prescribed topology conserved during time evolution. We restrict our attention to fields that are localised in space and that are indeed confined to tubular neighbourhoods of knots and links. By construction the field is standardly embedded into nested tori centred on smooth loops that are knot139 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 139–144. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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ted and linked in (Ricca 1998). For simplicity we take field lines that are closed on themselves within each tube and that have same flux We therefore identify an magnetic link with the standard embedding of a disjoint union of magnetic solitori in

Similarly, by replacing the magnetic field B with the vorticity field chosen so as to satisfy (1) and (2), we obtain vortex knots and links Let us denote by the total volume of the magnetic link and consider the evolution of under the action of the group of volumeand flux–preserving diffeomorphisms

The magnetic energy and the helicity of

are defined by

where A is the vector potential associated with An important issue in topological fluid mechanics is to relate geometric and topological properties to the energy and helicity of the fluid system (for an introductory review see Ricca & Berger 1996).

2.

Helicity, linking and average crossing numbers

It is well known that helicity admits topological interpretation in terms of linking numbers. We have: Theorem (Moffatt 1969; Berger & Field 1984; Moffatt & Ricca 1992): Let be a collection of magnetic links (knots). Then

where denotes the linking number of respect to the framing induced by the embedding of the in denotes the (Gauss) linking number of with

with and

The Gauss linking number between and is a fundamental topological invariant of link types and it is obviously conserved under frozen field evolution. The linking number admits interpretation in terms of minimal number of crossings of the link type: by

Energy, helicity and crossing number relations

141

assigning to each crossing present in a given projected diagram of the oriented link (omitting self-crossings and following a standard sign convention), it can be expressed as sum over signed crossings, according to the formula

where is the classical Gauss integrand form and denotes over- and under–crossings in plane projections (omitting self–crossings; see Figure 1). The White linking number is also a topological invariant associated with the field embedded in and it can be decomposed in two geometric quantities the writhing number (which measures the average space coiling of in ) and the total twist present in Linking numbers are closely related to the average crossing number which is an algebraic measure of the link complexity in space (Freedman & He 1991). For any pair and of components this quantity is simply given by the average total number of apparent crossings present in the link and is defined by

where # denotes all possible crossings (including self-crossings) of the curves and < • > average over all possible planes. Now, since

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and holds true for any given pair of components and the inequality can be extended to the whole system, hence by using (6) and (8), we have

3.

Topological bounds on magnetic energy

We consider the magnetic relaxation of the magnetic link subject to a volume- and flux-preserving diffeomorphism, as discussed by Moffatt (1992). For simplicity we assume that all link components have same flux and that are all zero–framed, that means for each An interesting result that relates magnetic energy and complexity of the physical system is given by the following: Theorem (Freedman & He 1991): Let link (or knot), then we have

Note that

where

be an essential magnetic

denotes the topological crossing

number of the knot or link type. Moreover Moffatt has shown (1992) that the energy is bounded from below, according to the inequality

where depends on the geometry of supp(B), with a spectrum of ground states given (1990) by

where is related to the topology of the system. The problem whether these infima can actually reach their minimum value remains open. Both and were left undetermined. By using the result of Freedman & He, combined with the inequality (10), we can show (for a detailed discussion, see Ricca 2002) that

Equation (14–i) and (12) state that for given linking complexity (with notable exceptions: see Figure 2b), the smaller the volume, the higher the energy level. This is in agreement with the intuitive idea that for a

Energy, helicity and crossing number relations

143

knotted rope the tighter the knot, i. e. the smaller the physical space it occupies, the higher is its potential energy. Equation (14–ii) identifies with the topological crossing number of the knot/link type. This result, combined with eq. (13), provides an interesting and powerful relationship between ground state energy and topology and shows the intimate relationship between the two. By direct inspection of the link tabulation, it is however immediately evident that there may be countably many topologically distinct links with equal number of components and same topological crossing number leaving partially open the fundamental question of identifying uniquely ground state energies with topology. An example is given by Figure 2, where three distinct link types with and are shown: assuming for these links same volume and flux, then all three links have same

4.

Vortex tangles and complexity

Finally, let us consider a tangle of (zero-framed) vortex filaments in a steady state. As above, all filaments are assumed to have same strength Kinetic energy T and helicity are defined as usual by

Since

by applying Hölder inequality to the second term, we have

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where is the total enstrophy of the tangle. Work on possible new relationships between complexity, topology and energy of vortex tangles is currently under way and new results on complexity measures can be found in the paper by Barenghi, Ricca & Samuels (2001) (see also the paper in this volume).

Acknowledgments The present author wishes to acknowledge financial support from EPSRC (grant GR/K99015) and PPARC (GR/L63143).

References BARENGHI C.F., RICCA, R.L. & SAMUELS D.C. 2001 How tangled is a tangle? Physica D 157, 197–206. BERGER, M.. & FIELD, G.B. 1984 The topological properties of magnetic helicity. J. Fluid Mech. 147, 133–148. FREEDMAN, M.H. & HE, Z.-X. 1991 Divergence-free fields: energy and asymptotic, crossing number. Ann. Math. 134, 189–229. MOFFATT, H.K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129. MOFFATT, H.K. 1990 The energy spectrum of knots and links. Nature 347, 367–369. MOFFATT, H.K. 1992 Relaxation under topological constraints. In Topological Aspects of the Dynamics of Fluids and Plasmas (ed. H. K. Moffatt et al.), pp. 3–28. Kluwer, Dordrecht, The Netherlands. MOFFATT, H.K. & RICCA, R.L. 1992 Helicity and the invariant. Proc. R. Soc. Lond. A 439, 411–429. RICCA, R.L. 1998 Applications of knot theory in fluid mechanics. In Knot Theory (ed. V. F. R. Jones et al.), pp. 321–346. Banach Center Publ. 42, Polish Academy of Sciences, Warsaw. RICCA, R.L. 2002 Energy, helicity and topological bounds. In preparation. RICCA, R.L. & BERGER, M.A. 1996 Topological ideas and fluid mechanics. Phys. Today 49 (12), 24–30.


Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, and Institute of Fundamental Technological Research, ul. 21, 00-049 Warsaw, Poland [email protected]

Abstract Using the language of differential forms on a space-time, one can write the equation of an ideal fluid in a form similar to the Maxwell equations. Vorticity current plays then the role of the source term and the Euler equations can be interpreted as the generalisation, to the whole space-time, of the well-known fact that the number of vortex lines passing through any two-dimensional surface spanned on a closed contour can be expressed by a circulation associated with this contour. A similar procedure can be used for the ideal MHD. It appears that by using this formulation various helicity conservation theorems may be derived in the natural and straightforward manner . My passion for diff’rential forms Defies all traditional norms; It may make you queasy, But it’s really quite easy; It ought to be taught in the dorms.

1.

Introduction

It appears that many equations of mathematical physics when formulated in the language of differential forms take especially simple and elegant form (Flanders 1963, Arnold, Khesin 1998). This, as we will see, concerns also equations of fluid dynamics 1990, 1991), including MHD. The simplest example of a differential form (apart from 0-forms which are just functions) is a 1-form i. e. a field of covectors. Such a form can be evaluated on a vector-field X to obtain (with Einstein summation convention observed). Having such forms one defines their exterior product – a simple which when evaluated on vector-fields gives


146

So, this is always zero when ( is the dimension of the space) or when 1-forms are linearly dependent. Any is a linear combination (with coefficients depending on ) of such simple forms. For being respectively and we have

so

if is an odd number (or Forms can be also differentiated

).

The operation takes into derivative. We have the useful relation

it is called the exterior

For any given vector-field V we can define also, what is sometimes called the interior derivative of the by

Thus the interior derivative takes into purely algebraic operation. Assuming that is a

and it is a we have

In principle we will be working here with differential forms defined on the four-dimensional space-time. However, for clarity and better exposition of the correspondence with the commonly used vector notation we will often decompose any 1-form into temporal and spatial part, where the spatial part is For convenience we assume that Greek indices take values 0, 1, 2, 3 whereas the Latin ones – values 1, 2, 3. Similarly, we can decompose the exterior derivative where This decomposition makes sense as long as we use the Galilean system of coordinates. In the description that we propose the vortices (vortex lines) are treated as a sort of particles responsible for the induced fields. So, in this language both ”fields” and ”particles” are treated as equally real. This corresponds to the situation which one encounters in superfluids, where vortices are even better visible than the fluid velocity.


2.

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The Euler equations

To begin with, we define the action 1-form

where are the covariant components of fluid velocity and specific enthalpy, which in the case of incompressible fluid is and in the more general case of a barotropic fluid

is the

Here denotes the pressure. The ”density” of vortex lines can be measured by a vorticity current 2form J, which in a system of coordinates co-moving with vortex lines is given by a purely spatial 2-form

The dot above the equality sign reminds here that the equality takes place in a specific system of coordinates. Using Galilean transformation

with being the vortex line velocity we may pass to the laboratory system to obtain The contravariant velocity of vortex lines in general can be different from the fluid velocity, as it occurs, for example, in the case of superfluid helium In the case of the ordinary fluid Let S be a 2-dimensional surface in the space-time, spanned on a contour The equality

by the Stokes law leads to which appears to be equivalent in the vector notation to

where in the last two equations is a vorticity vector-field which corresponds to the vorticity 2-form in 3-dimensional space by Obviously, the continuity equation can be also written by

148

using the fluid current 3-form where again in the system of coordinates moving locally with the fluid

and where

is the fluid density. In the laboratory system again

where is contravariant fluid velocity. The continuity equation div reads then as Thus finally the full set of flow equations is given by Let us note that in the four-dimensional space-time a vortex line is represented by a two-dimensional surface (the history of a vortex line). In general, two two-dimensional surfaces (e. g. planes) in a four-dimensional space have just one point in common. Therefore it makes sense to speak of vortex lines passing through a two-dimensional surface in the spacetime. As so taking the exterior derivative of Eq. (10), one obtains

which in the vector notation is equivalent to the equations for vorticity

Here the dot denotes the time derivative.

3.

MHD equations

In addition to the previously introduced forms 2-form of the magnetic field.

and J we define the

Let be a contravariant vector-field of the electric current divided by the fluid density then the equation

is equivalent in the vector notation to the following system


149

Similarly, let us define the 1-form of the electromagnetic potential

and the ”magnetic current”

Then the equation is equivalent to the following system of vectorial equations

Taking the exterior derivative of (18) we have

which again is equivalent to

Summarising the equations of the ideal MHD can be written as

and

4.

is their consequence.


Theorem 1 The following 3-forms in the space-time define conserved currents

where in

we assume

(the case of ordinary fluid dynamics).

To prove it, we have to demonstrate that

150

Indeed, in the appropriate co-moving system of coordinates this is equivalent respectively to 1. 2. 3. & 4. Again and are vanishing since these are 4-forms living in 3-space (there is no dt). Since we have also

which proves that indeed For example, in the vector notation the equation expressed as

where defines helicity and the flux of helicity. In particular, if

can be

is is vanishing at infinity, we have

As noticed by Moffatt (1969) helicity expresses some topological properties of vortex (respectively magnetic) lines. This work was supported by KBN grant 7T07A01817.

References ARNOLD, V.I., KHESIN, B.A. 1998 Topological Methods in Hydrodynamics. SpringerVerlag, New York–Berlin–Heidelberg. FLANDERS, H. 1963 Differential Forms with applications to physical sciences. Academic Press, New York–London. MOFFATT, H.K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, part 1, 117-129. 1988 Properly posed boundary condition and the existence theorem in superfluid In Trends in Applications of Mathematics to Mechanics, Proc. 7th Symposium (ed. J. F. Besseling & W. Eckhaus), pp. 224-237. Springer Verlag. 1990 Helicity theorem and vortex lines in superfluid Int. J. Theoret. Physics 29, 1277-1284. 1991 Differential forms and fluid dynamics. Arch. Mech. 43, 653661.

A third-order topological invariant for three magnetic fields Christoph MAYER & Gunnar HORNIG* Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, 44780 Bochum, Germany [email protected], [email protected]

If you notice a third-order link You can colour it green, brown and pink; Invariants follow, As hills follow hollow; So a field triply-linked cannot shrink.

Abstract The topology of divergence-free fields is important in many parts of physics, e. g. in magnetohydrodynamics, plasma physics, hydrodynamics, superfluids etc. With the focus on applications in magnetohydrodynamics, our principal aim is the characterisation of magnetic fields by the means of invariants. In this report an introduction to the problem of finding higher order invariants is given. Then a third-order link integral of three magnetic fields is presented, which can be shown to be a topological invariant and therefore an invariant in ideal magnetohydrodynamics. This integral generalises the known third-order link invariant derived from the Massey triple product, which could only be applied to isolated flux tubes. As an example three magnetic fields not confined to flux tubes are given that possess a third-order linkage.

1.

Introduction

In recent years topological considerations have become increasingly important in the study of physical problems. In plasma physics for example the topology of magnetic fields plays an important role when dealing with problems such as the stability and time evolution of a plasma or when estimating its energy content. The structure of magnetic fields is studied with the aid of topological measures of complexity, i. e. measures which are invariant under diffeo-

*The

authors gratefully acknowledge financial support from the Volkswagen Foundation.


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morphic isotopies of the magnetic field. The simplest of such measures is magnetic helicity. For a magnetic field B with vector potential A it is defined as the volume integral

which is gauge invariant if Magnetic helicity measures the total mutual linkage of magnetic flux (see Moffatt 1969). In ideal magnetohydrodynamics, i. e. in the regime of large magnetic Reynolds numbers, the magnetic field is frozen into the plasma flow and magnetic helicity is a conserved quantity. Since it is quadratic in magnetic flux it is often referred to as a second-order topological invariant. As examples the Hopf link and the double twisted torus are shown in the following figure, which both have the same magnetic helicity of

It has been known for a long time that magnetic field configurations exist which are non-trivially linked, but for which the magnetic helicity vanishes. Helicity e. g. fails to detect the interlocking of three flux tubes in the form of the Borromean rings (Figure 1c). This raises the question whether higher order invariants exist which detect these linkages.

2.

Higher order invariants

In knot theory different invariants are known which distinguish for example the Borromean rings from unlinked rings. To use these invariants e. g. to measure the complexity of magnetic fields, they have to be expressed in terms of physical quantities, i. e. in terms of the magnetic field B. For most knot invariants a translation into a physical setting is unknown and still no higher order invariant with > 2 has been found which is applicable to an arbitrary magnetic field We will see that matters simplify, if the field splits into isolated flux tubes each with magnetic field In this case is to be considered as an invariant which measures the -th order cross-linkage of distinct magnetic flux tubes.

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153

It was recognised by Monastyrsky & Retakh (1986) and independently by Berger (1990) and Evans & Berger (1992) that the link invariants based on so-called Massey higher products (see Massey 1958, 1969) can be written as invariants applicable to magnetic flux tubes. Massey higher products in fact yield a hierarchy of higher order invariants, which are expressible as cross-linkage integrals of the above mentioned type. Unfortunately, their usage is restricted to magnetic fields confined to flux tubes which must not possess a linkage lower than the linkage which is measured. The situation is summarised in the following table Total-linkage

Cross-linkage

cross-helicity for arbitrary fields fields restricted to flux tubes with no 2nd order linkage fields restricted to flux tubes with no 2nd or 3rd order linkage where and are constructed respectively from the Massey triple and quadruple product.

3.

Generalisation of the third-order invariant

In this section we generalise the third-order invariant known from the Massey triple product to an invariant of three magnetic fields not confined to flux tubes. Theorem: Let and be three magnetic fields with potentials and let In an ideal dynamics, i.e. with in all three components, the integral

is a gauge invariant, conserved quantity, if the three potentials obey for all and the boundary condition for n being the normal vector to the boundary of the integration volume V.

154


A complete proof of this theorem is given in Hornig & Mayer (2001) and only the main ideas shall be presented here: For an arbitrary gauge field we have the following general identity

where the term on the left is known as the second Chern class and the three-form on the right as the Chern-Simons three-form. For the gauge group U(1), which yields electrodynamics on identity (2), together with an ideal dynamics, i. e. with leads to the continuity equation for helicity density

This proves the well known result that the integral over helicity density, for the boundary condition is a conserved quantity. Using instead of the group U(1) the gauge group SU(2), the potential A naturally has three components A summation over repeated indices is understood. Here the Pauli matrices were chosen as a basis for the Lie algebra su(2). Interpreted as three independent potentials of electromagnetic fields identity (2) together with leads (i) to the helicity conservation of and (ii) to a new continuity equation, which with and cyclic permutations of it, reads

Analogous to the case of helicity this continuity equation yields a conservation law for the density Together with the gauge invariance (see Hornig & Mayer 2001) this proves the above theorem. Let us note that the presented derivation reveals a close relationship between magnetic helicity and the generalised third-order invariant. In fact, both conservation laws result from the same identity (2). In the special case of three fields confined to flux tubes, condition (i) of the theorem implies that the tubes must be mutually unlinked. It can easily be confirmed that in this case integral (1) coincides with expressions derived from the Massey triple product given by Monastyrsky & Retakh (1986), Berger (1990) and Ruzmaikin & Akhmetiev (1994). Finally we want to remark, that the third-order invariant can be interpreted as the cross-helicity

of the pair of divergence-free fields

and

for each

A third-order topological invariant for three magnetic fields

4.

155

Example magnetic field

We know that the requirement of the above theorem can always be satisfied for fields confined to mutually unlinked flux tubes. However, it is not easily satisfied for three arbitrary fields. In the following Yang-Mills theory is used to construct a natural example of three magnetic fields not confined to flux tubes, to which the invariant can be applied. In an SU(2) gauge symmetry the field strength has components for Taking the exterior derivative we observe that is sufficient for all to vanish. These potential components meet the above requirement and therefore shall be interpreted as three potentials of independent electromagnetic fields. Since is trivially fulfilled for i. e. for a pure gauge Yang-Mills field, we are now going to construct such a field on a time slice of Using the mapping

we perform a gauge transformation from the classical vacuum

This yields the potential of a vacuum Yang-Mills field (see Hornig & Mayer 2001) with a winding number of the vacuum of 1 (see ItzyksonZuber 1980). The three independent magnetic fields, calculated from are well defined and decay sufficiently with

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for large radii. As is shown in Hornig & Mayer (2001) the non-vanishing winding number of the Yang-Mills vacuum directly implies a non-trivial third-order linkage. Explicitly calculating integral (1) we find

The constructed magnetic fields are in fact highly symmetric. Each field is invariant with respect to rotations about the and all three fields are similar in that they can be obtained from one field by rotations which map the axis of symmetry on one another. As seen in Figure (2) all field lines are closed and have an elliptical shape. Even though all cross-helicities vanish, individual field lines among different fields are in general linked. To be more precise, every pair of such field lines is either linked exactly once or intersects twice.

5.

Conclusions

An integral expression is given which generalises the third-order invariant known from the Massey triple product. It is interesting that in the presented derivation helicity and are the result of the same identity. An example shows that the new invariant is a real generalisation. Still an open question is whether a third-order invariant for the “total-linkage” of a single arbitrary magnetic field exists and whether it can be constructed from the “cross-linkage” similar to the case of helicity. There might e. g. exist a subdivision of a single magnetic field into three components, such that Unfortunately, the antisymmetry of the “cross-linkage” seems to be one of the key problems for a further generalisation analogous to helicity.

References BERGER, M.A. 1990 J. Phys. A: Math. Gen. 23, 2787. EVANS, N.W., BERGER, M.A. 1992 in Moffatt et al: Topological aspects of fluids and plasmas, Nato ASI Series E, Vol 218 (Dordrecht, Kluwer Academic Publ. ), p 237. HORNIG, G., MAYER, C. 2001 J. Phys. A: Math. Gen. (accepted), arXiv: physics/0203048. ITZYKSON, C. & ZUBER, J.-B. 1980 Quantum Field Theory. McGraw-Hill, New York. MASSEY, W.S. 1958 Some higher order cohomology operations. Symp. Intern. Topologia Algebraica, Mexico, UNESCO. MASSEY, W.S. 1969 “Higher order linking numbers”. In Conf. on Algebraic Topology, Univ. Illinois at Chicago Circle, Chicago, Ill., June 1968, ed. Victor Gugenheim. Reprinted in: J. of Knot Theory and Its Ram., 7, No. 3, (1998), 393-414. MOFFATT, H.K. 1969 Journal of Fluid Mechanics 35, 117. MONASTYRSKY, M.I. & RETAKH, V.S. 1986 Commun. Math. Phys. 103, 445. RUZMAIKIN, A. & AKHMETIEV P. 1994 Phys. Plasmas 1, 331-336.

Asymptotic structure of fast dynamo eigenfunctions B. J. BAYLY Mathematics Dept., University of Arizona Tucson AZ 85721, USA [email protected]

Abstract The eigenfunctions of the kinematic dynamo problem exhibit complicated spatial structure when the magnetic diffusivity is small. When the base flow is spatially periodic, we may study this structure by examining the Fourier components of the eigenfunction at large wavevectors. In this regime we may seek a WKB form in terms of slowly-varying functions of wavevector. The resulting hierarchy of equations may be systematically analysed for both zero and small nonzero diffusivities. Eigenfunctions of dynamos fast Have fine structure of gradient vast; I ask you to gaze At equations of phase, And singular things that don’t last. —HKM 2001

1.

A Professor in old Zakopane Wrote limericks clever and fane, Then read them aloud To an exuberant crowd Whose response was to throw lots of mane. –BJB 2001

Introduction

This symposium is devoted to understanding the origin and role of the near-singular events and structures observed in real flows and numerical simulations. The direct approach, pursued by most participants at this meeting, is to start from nonlinear evolution equations (exact or approximated or modelled) and ask under what circumstances the solutions become infinite at some points in space and time. An alternative, less direct, approach is to start from the linearised equations for perturbations to known solutions, and formulate hydrodynamic or hydromagnetic stability problems. The resulting eigenproblems are again nonlinear, because the unknown eigenvalue multiplies the unknown eigenfunction, but in a different way from the original evolution equations. These eigenproblems may also have singular solutions, but the singularities will be different from the singularities of the evolution equations. Despite the indirectness of this formulation, it is likely that

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some of the near-singular structures observed in real and simulated flows are closely related to the singular eigenfunctions of stability problems. The techniques of this paper are applicable to a wide variety of hydromagnetic stability problems. For simplicity, however, we will focus on the kinematic dynamo problem, in which a flow of electrically-conducting fluid is specified and the behaviour of a weak magnetic perturbation is analysed in the linear regime. For positive values of the magnetic diffusivity this problem is relatively well-understood as an elliptic eigenvalue problem. But in the limit of zero magnetic diffusivity the eigenfunctions develop more and more complex spatial structure, and there are open questions about what (if anything) they converge to. Understanding the behaviour of the eigenvalues and eigenfunctions in the non-diffusive limit is the fast dynamo problem. The fast dynamo problem was identified almost thirty years ago ( Vainshtein & Zeldovich 1972) and has been considerably studied since. We draw the reader’s attention to the monograph of Childress & Gilbert (1995) for a comprehensible introduction and comprehensive discussion of what is now a sizable body of scientific literature. The bibliography of the present note will of necessity only mention a small selection of the work on this subject.

2.

Kinematic dynamo basics

The usual formulation of the kinematic dynamo problem is as follows. The velocity field of an electrically-conducting fluid is prescribed. It is typically smooth, with “Jacobian” strain-rate derivative matrix

and solenoidal: Then the induction equation for the magnetic field can be written in advection-stretching form:

where represents the magnetic diffusivity. We will suppose that all physical quantities are non-dimensionalised with respect to sensible scales, so that is the inverse of the magnetic Reynolds number. The velocity field is typically bounded in space, and steady in time. This allows the possibility that solutions to (2) might grow exponentially in time with a well-defined spatial form:

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with Indeed, the classical kinematic dynamo problem is to determine for what flows there are such growing solutions, and what the growing solutions look like. The fast dynamo problem further asks whether the growth rate remains bounded above zero as the diffusivity and to what (if anything) do the eigenfunctions converge. In virtually all studies appears to approach a well-defined limit as even though develops extremely complicated structure on ever-smaller scales. In this work I shall assume that the eigenvalue converges, and examine the detailed structure of the magnetic field eigenfunctions.

3.

Steady spatially-periodic velocity field

We shall assume that the given velocity field is steady and spatially periodic, and write it explicitly as

where is a subset of In many examples is actually finite, as with ABC flows (Arnold & Korkina 1983, Galloway & Frisch 1986, Lau & Finn 1993). Otherwise it is reasonable to assume rapidly as so that is smooth. Eigensolutions of the induction equation (2) then may be sought in the form (choosing the Bloch wavevector equal to zero)

This form is not necessarily localised in wavevector space. We anticipate that if the diffusivity is positive then at some point will decay as increases, but if is small then this decay may only begin at large values of Furthermore the limit of as with m held fixed might not decay with increasing at all. The noninterchangeability of these limits is a subtle but important aspect of the fast dynamo problem. The induction equation can then be separated into its own Fourier components:

This form is frequently computationally convenient, especially when is a small finite set). Unfortunately it is conceptually rather opaque, as most of us have little experience with multivariable difference equations with non-constant coefficients.

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Slowly varying Fourier amplitudes

The existence of structure on many length scales in indicates that the significant Fourier amplitudes are ”spread” over a wide region of wavevector space. More precisely, the size of the region occupied by significant Fourier amplitudes is large compared with the typical size of the wavevectors in Over most of this region the separation between and is relatively small and seems relatively unimportant, just as the differences between Cambridge and Oxford appear small and unimportant when viewed from outside Britain. In the fast dynamo problem this impression is correct. We may incorporate it in a WKB (or GLJWKB, as enthusiasts for historical accuracy at the workshop suggested) ansatz

where and are slowly varying functions of This ansatz also assumes that m may be treated as a continuous variable, and that Taylor series may be used to approximate the slowly varying functions when evaluated at The vector field is given unit magnitude everywhere, and merely plays the role of indicating the direction of There are orientability issues concerned with the choice of sign for but these will not be pursued in this paper. The important information lies in which has both real and imaginary parts to reflect both the phase and amplitude of respectively. The exponential ansatz (7) will also allow us to seek S as the sum of a leading-order part plus corrections. Important aspects of such as power laws or diffusive regularisation will emerge transparently from additive corrections to It should be noted that itself is not slowly- varying by the above definition. Our ansatz does not prevent from varying by an O(1) amount from one lattice point in to its neighbours. It is subtle, even fortuitous, that we are able to assume slowly-varying and to draw inferences regarding the values of the non-slowly-varying function at discrete wavevectors. Substituting (7) into the induction equation, and multiplying through by yields


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Now we use the assumption of slow variation to write

where indicates higher-order, i.e. asymptotically smaller, terms. We can use (9) to approximate the exponentials in (8) by

bearing in mind that we can use the linear approximation to the exponential for o(1) but not O(1) arguments. Before we attempt to substitute the approximations (9), (10) into the induction equation (8), note that many of the terms will have similar structures. Using F, G as dummy symbols, these common structures are of the form

Anything like that does not depend on factors out of the sum, and the remaining summations simply amount to reconstituting the original physical-space function and its second derivative, respectively:

These derivatives are of course matrix-valued, but we are omitting indices in the hope that the reduced clutter enhances clarity. Returning to the quantities appearing in the induction equation, we observe that the original physical-space forms of the velocity and velocitygradient fields reappear when the slowly-varying ansatz (7) for the Fourier amplitudes of the magnetic field eigensolution is substituted into the induction equation (6) in wavevector space. This algebraic felicity reflects a general geometric principle: that the main contributions to the Fourier amplitude at a wavevector of large magnitude tend to come from a localised region of physical space in which the gradients are aligned with

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the direction of (the principle of stationary phase). These formulae also tell us the location of the spatial region giving the dominant contribution to it is the argument of the reconstituted physical-space formulae in (12). Because it is so important we give it its own symbol: At this stage it might be objected that we seem to have inconsistent interpretations of S. When S was first introduced we said it should have both real and imaginary parts to accommodate phase and amplitude variations of But if we identify with a physical point in space, this would seem to prohibit any non-constant imaginary part of S. The resolution is that the principle of stationary phase is intentionally an approximate statement that describes only the leading order of a short-wave asymptotic theory. We will find typically that at leading order and are real, and imaginary components enter only as corrections. To be sure, there are circumstances in which even the leading order parts of S and X are complex, but we shall not pursue them in this paper. As mentioned before, the unit vector field indicates the direction of but does not play a major role in determining the asymptotic behaviour. For simplicity we shall modify the whole problem to a scalar form (Bayly 1993). We drop and replace the stretching matrix by the scalar “stretching function” The result is an eikonal equation for the phase function alone:

For those concerned about throwing the baby out with the bathwater, the scalar dynamo model allows complex-valued fields, so that vector processes such as cancellation of oppositely-directed fields may still be faithfully represented. Numerical investigations of complex-valued scalar dynamo models show behaviour completely analogous to the original vector problem.

5.

Diffusionless asymptotics

The situation of diffusivity exactly zero is of course not the same as the limit as diffusivity goes to zero. This can be seen directly in (6), in which the diffusivity multiplies while all other terms involve lower


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powers of m. No matter how small is, as long as it is positive the diffusive term dominates for sufficiently large On the other hand, if is very small then there will be a sizable finite range of m for which the non-diffusive terms dominate. For the high-wavenumber part of this range we may still use the slowly-varying ansatz and focus on the nondiffusive terms only. What we are doing, of course, is working out an intermediate asymptotics for m in an “overlap” region between the very high at which diffusivity dominates and the low at which is O(1). In a global theory we would work out asymptotics for these other regimes also and fix undetermined quantities by matching. In this work, however, we shall restrict attention to this intermediate regime. Even apart from the diffusive term, some terms in the equation will dominate others. The solution will then consist of a big part that approximately satisfies the dominant balance, plus a little part that collectively accounts for all the corrections. Our ansatz suggests that will be with gradient of size O(1). The correction is expected to be O(l), or perhaps with gradient Without the diffusive term, the leading order in (14) is and at this order we have Recalling the stationary-phase interpretation that is a physical point at which the spatial structures are aligned perpendicular to m, equation (15) indicates that to leading order the spatial structures at are aligned parallel to the velocity field. This is not at all surprising; indeed it a check on the sensibility of the asymptotics. Equation (15) is an Eulerian statement about the alignment of structures at a fixed point in space. There is a natural Lagrangian version of this statement that is perhaps slightly stronger: if the spatial structures are aligned parallel to the velocity field at a certain point at a certain time, and then allowed to follow the material trajectories of the flow, they will continue to remain so aligned. More precisely, if we introduce a time-like parameter along a Lagrangian trajectory, then the position and wavevector evolve along the trajectory according to

Then, if (15) is true at any point of the trajectory, it will be true at every point. As mentioned in a previous section, the leading order functions and are purely real, consistent with the stationary phase interpretation.

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The next order appearing in the diffusionless (14) is O(1):

This can be interpreted in the Lagrangian representation (16) as

Recalling the definition (7) of S, we recognise this as simply the logarithm of the magnitude of Again, this bears out our previous remark that the corrections to the purely real leading-order behaviour may be complex, with the imaginary part containing amplitude information. The exact nature of solutions to (18) depends of course on the Lagrangian trajectories, i. e. fluid particle paths, of the flow. It is a common situation for a substantial portion of the flow to exhibit “Lagrangian chaos”, in which details of individual trajectories are chaotic and unpredictable but long-time averages along trajectories take well-defined values. We would therefore expect the right-hand side of (18) to have a well-defined trajectory average The logarithmic amplitude would then be expected to go roughly like as a function of the Lagrangian parameter The amplitude itself goes roughly like

At the same time, if the trajectory is in the chaotic region, we expect the magnitude of the wavevector to increase exponentially at a rate given by the Lyapunov exponent along the trajectory,

Putting (19) and (20) together implies a power law type of behaviour:

The exponent in this law depends in an intricate way on several quantities. Some of them have simple geometric significance, such as the Lyapunov exponent or the trajectory average of the stretching function But the eigenvalue is not so simple; it appears to be determined by the interaction of large-scale structures, i. e. at wavevectors of size O(1). And the third term on the right-hand side of (18) seems to defy simple interpretation, let alone its trajectory average. Whatever the power is, the fact that it is a power law at all indicates that the zero-diffusivity limit of the eigenfunction is not smooth.


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If the power is negative the spatial eigenfunction may be continuous or even differentiable a certain number of times. It appears in most situations that the power is actually positive. This implies that no pointwise convergence is to be expected, or even convergence to a distribution like a The most likely candidate so far advanced for zerodiffusivity limit of the eigenfunction is a sign-singular measure (Du, Tel, &Ott 1994).

6.

Diffusive asymptotics

Because the diffusivity multiplies the highest power of in the induction equation (whichever form we consider) the point at which it enters the asymptotics depends on the regime. For example, when or greater, diffusion is as strong as advection, and both are stronger than stretching. At the opposite extreme, when diffusion is smaller than advection and stretching, both of which are O(1). As mentioned before, we are interested in small scales, but for which diffusive effects are still relatively small. More precisely, we will assume so that the leading order remains non-diffusive and diffusive effects enter as corrections. There still remains the question of the relative magnitudes of and the formally O(1) terms in the diffusionless expansion. In practice there can be a variety of subtle balances and fractional powers (e. g. or in the results, depending on details of the velocity and stretching fields. The reality is that now we have a problem with two small quantities in which we want to expand the solution, one of which is a constant and the other a variable. The complete theory must involve consideration of all the possible balances and appropriate matches amongst them, and is beyond the scope of this note. We will therefore proceed informally from here on. Algebraically, the crucial difference between the diffusive and diffusionless eikonal equations is that the diffusive equation is quadratic in albeit with the small prefactor Therefore the correction to the phase function must also contain a term quadratic in and we expect its magnitude to go to zero as although as yet its precise limiting behaviour is not specified. We use the same notation as in the diffusionless expansion, with diffusive corrections

The eikonal equation uses the derivatives of the phase functions

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and similar expressions for equation yields an order

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Substituting into the eikonal component

where is the same quantity that appeared at the leading order of the diffusionless expansion. Details of the solution to (24) depend of course on the flow-field and the stretching function But we can observe that is now a mere inhomogeneous term. i. e. a regular perturbation, no longer a singular perturbation. Whatever the details bring, we can anticipate that solutions at small finite will connect continuously with the solutions when exactly. For example, suppose we are in the situation described at the end of the diffusionless asymptotics: that we are in a region of chaotic flow. Then, following the Lagrangian trajectory with time-like parameter we have upon which (24) becomes

therefore

In other words, the overall behaviour of the diffusive correction to the phase function is a negative imaginary (hence regularising) part quadratic in Keith Moffatt remarked that, according to this formula, diffusive effects become important when i. e. at physical lengthscales on the order of in agreement with an estimate obtained by quite different methods(Moffatt & Proctor 1985). We may pursue higher-order terms in the diffusive asymptotics, though the terms rapidly increase in number and decrease in usable information. As is typical of many such expansions, successive corrections satisfy linear equations with inhomogeneities defined in terms of lower-order terms already found. Provided resonance does not occur, successive diffusive corrections are expected to have the same dependence on as with decreasing powers of


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Resonance does not appear to occur at but it does at O(1). Indeed, it would be surprising if it did not, because at O(1) will appear any modification of the eigenvalue due to the presence of small finite diffusivity. Now, the complete determination of the diffusive correction must involve “boundary conditions” on the diffusive corrections to which can only be defined by matching to other asymptotic regimes. Nonetheless, the order-of-magnitude dependence on is governed by the structure of the asymptotic expansion, and we therefore expect the diffusive correction to the eigenvalue to be of the same magnitude in as The claim of the foregoing paragraph may appear to be a detail of high-order asymptotics, of marginal interest at best. But the last paragraph argues that whatever determines the eigenvalue, the difference between the diffusive and non-diffusive eigenvalues should vanish at a well-defined rate as the diffusivity goes to zero. This result is almost, but not quite, an affirmative answer to the fast dynamo problem. In order to establish a positive fast dynamo result, very little more has to be done. What is needed is an accurate numerical calculation at a sufficiently small but finite diffusivity, yielding an eigenvalue with real part greater than zero by an O(1) amount. What “sufficiently” means is that the real part of the calculated eigenvalue should be greater than the magnitude of the diffusive correction to the eigenvalue. This would establish that for any smaller diffusivity, the real part of the corresponding eigenvalue would also be positive, and that the limit of the eigenvalue as diffusivity goes to zero would also exist and have positive real part. Of course, such a conclusion would still lack mathematical rigour by the nature of both the numerics and the asymptotics. But it would be a substantial step in our understanding of the fast dynamo problem. It has been almost 30 years since Vainshtein and Zel’dovich first formulated it; we really ought to be asymptotically approaching a solution by now.

7.

Summary

Many important fluid mechanical phenomena involve structures developing on the smallest spatial scales available. These smallest scales are usually determined by a diffusivity of some kind, and in the limit of zero diffusivity the small scale structures often appear to approach some kind of singularity. The kinematic dynamo is a well-studied problem of this type, and this paper has presented a general approach to investigating its small-scale structure. Since we only specified classes of problems rather than specific examples we were not able to provide explicit results,

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but we have identified the nature of the asymptotic limit and described some features we expect to be generic or at least common. We only considered the kinematic dynamo problem in this paper, and indeed obtained most of our conclusions from a scalar modification. However it does not appear that the overall results depend crucially on the special structure of either the scalar model or the kinematic dynamo problem. In particular, we expect this kind of behaviour to manifest itself in a wide variety of hydrodynamic and hydromagnetic stability problems. The technique of using a WKB-type analysis in wavevector space instead of physical space was the key to making progress in this problem, and it will be of interest to see in what other problems this technique proves to be useful.

Acknowledgments I first worked on fast dynamos with Steve Childress over fifteen years ago. I would like to acknowledge his continuing influence and inspiration as I have pursued this ever-more intriguing problem.

References ARNOLD, V. I. & KORKINA, E. I. 1983 The growth of a magnetic field in a threedimensional steady incompressible flow. Vest. Mask. Un. Ta. Ser. 1, Matem. Mekh. no. 3, 43-46. BAYLY, B. J. 1993 Scalar dynamo models. Geo. Astro. Fluid Dyn. 73, 61-74. CHILDRESS, S. & GILBERT, A. D. 1995 Stretch, Twist, Fold: The Fast Dynamo, Springer. DU, Y., TEL, T., & OTT, E. 1994 Characterization of sign-singular measures. Physica D 76, 168-180. GALLOWAY, D. J. & FRISCH, U. 1986 Dynamo action in a family of flows with chaotic streamlines. Geo. Astro. Fluid Dyn. 36, 53-83. LAU, Y. -T. & FINN, J. M. 1993 Fast dynamos with finite resistivity in steady flows with stagnation points. Phys. Fluids B 5, 365-375. MOFFATT, H. K. & PROCTOR, M. R. E. 1985 Topological constraints associated with fast dynamo action. J. Fluid Mech. 154, 493-507. VAINSHTEIN, S. I. & ZELDOVICH, YA. B. 1972 Origin of magnetic fields in astrophysics. Usp. Fiz. Nauk 106, 431-457. [English translation: Sov. Phys. Usp. 15, 159-172. ]

IV

VORTEX STRUCTURE IN TURBULENT FLOW


Vortex tubes, spirals, and large-eddy simulation of turbulence D.I. PULLIN Graduate Aeronautical Laboratory, California Institute of Technology, Pasadena CA 91125

Abstract Progress in the quantitative modelling of turbulence using vortex-based models of the fine scales is reviewed. Recent work on the calculation of the spectrum of a passive scalar convecting and diffusing within a stretched-spiral vortex is briefly described. This is followed by a discussion of the application of ideas from the study of the vortex structure of the small scales of turbulence to the development of subgrid models for the large-eddy simulation (LES) of turbulent flows at large Reynolds numbers. Examples are given including the LES of rotating and non-rotating plane channel flow and of the mixing of a passive scalar by forced isotropic turbulence. A scalar transported by flow; In a spiral of type that we know, Itself becomes spiral, Possibly chiral, And its spectrum is found without woe. —HKM 2001

1.

Big eddies have plenty to spare; But the little ones hardly do care, For at scales small, Dissipation is all, If energy cascade is fair. —DIP 2001

Introduction

The modelling of turbulence using ensembles of superposed local solutions to the Navier-Stokes and Euler equations appears to provide a viable approach to the calculation of some properties of turbulent small scales. Although Hill spherical vortices and Burgers vortices have been used as the basic elements for vortex-based modelling of turbulence (He et al. 1999; Kambe & Hatakeyama 2000; Pullin & Saffman 1998), the most interesting structure proposed to date has perhaps been the stretched-spiral vortex (Lundgren 1982). Its nearly axisymmetric vorticity may be viewed as tube-like on scales which are large compared with its cross-sectional dimension, and sheet-like on smaller scales. Experimental and numerical studies of turbulent flow have revealed the presence of vortex tubes (Porter et al. 1998) and spiral structures (S. Kida, this volume). Indeed, it may be argued that spirals are unavoidable once tubes are present. The formation mechanism for tubes/spirals,

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if they exist at intermediate and small turbulent scales, remains an open question. In the present paper we discuss a recent application of the stretchedspiral vortex to the description of turbulent mixing of small scales. This is followed by a brief review of recent work on subgrid-scale modelling for LES based on the idea of stretched vortices as subgrid elements.

2.

Small-scale model of passive-scalar mixing in turbulence

Pullin & Lundgren (2001), henceforth referred to as PL, considered passive-scalar transport and the dynamics of axial velocity inside spiral vortices. They argued that the evolution of the axial velocity and of a passive scalar inside stretched-vortices both obey a generic convectiondiffusion equation of the form

where are stretched polar co-ordinates in the cross-sectional plane of the vortex, is a stretched time, are either the axial velocity and viscosity or the scalar intensity and scalar diffusivity, D) respectively and is the known stream function associated with the azimuthal/radial velocity field of the vortex in the presence of a strain field which stretches vortex lines. PL solved (1) by a two-time analysis. We discuss two main results.

2.1

Spectrum of the axial velocity

At large wavenumbers, PL find that the leading-order term in the spectrum of the axial velocity is produced by the stirring of the initial distribution of axial velocity by the axisymmetric component of the azimuthal velocity. This gives the spectrum

where is the wavenumber, N is the rate of creation of vortex length per unit time per unit volume, are the Fourier components of the initial axial velocity, is the radial derivative of the angular velocity associated with the axial vorticity and is the stretching strain.

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At large this spectrum is subdominant compared to the component for the azimuthal velocity itself (Lundgren 1982). This is because axial vortex stretching attenuates axial components of velocity while amplifying the axial vorticity. The result suggests that axial motion is of secondary importance in stretched-vortex models of turbulent fine scales.

2.2

Spectrum of a passive scalar

When in (1) the solution gives the evolution of a passive scalar mixed by the stretched vortex. PL use this to show that the scalar spectrum is the sum of two power laws with coefficients that are integrals involving the Fourier components of both the initial scalar field and of the axial vorticity. The coefficients can be evaluated explicitly when simple forms for these fields are assumed. If, further, the stretching strain is scaled as where is the energy dissipation, the resulting scalar spectrum is

where is the Kolmogorov length, is the energy dissipation, is the scalar dissipation, and is the Schmidt number. The first term in (3) agrees with the spectrum (Batchelor 1959) which is expected in the viscous convective range when It is produced by the winding of the scalar by the axisymmetric part of the axial vorticity. The second term is generated by the convectiondiffusion of the scalar by the non-axisymmetric spiral velocity. It is of the classical Obukov-Corrsin form (see Tennekes & Lumley 1974) in the inertial-convective range. At large the scalar dissipation is dominated by the Batchelor term. Although the model does not assume isotropy, when this assumption is made, the one-dimensional scalar spectrum can be obtained from (3). This is shown in Figure 1 for values of (heat in water) and (salt in water) compared to the data of Gibson & Schwartz (1963). The scalar spectrum (3) is obtained from an approximate solution of (1) using a velocity field which is itself a long-time asymptotic solution of the Navier-Stokes equations. These solutions contain many dimensional variables. Hence, the power-law scalings in the leading order terms of (3) are obtained from dynamicalbased, and not from dimensional-based reasoning.

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Large-eddy simulation of turbulence

The continuity and Navier-Stokes equations, filtered on a scale much larger than the smallest viscous scale are (Leonard 1974),

where

is the velocity, the filtered (resolved) fields are defined by and represents the effect of subgrid dynamics. These equations are not closed and require a scheme or algorithm for computing from This is the SGS model. When this is given the numerical solution of (4) is referred to as ‘large-eddy simulation’ (LES). The term itself implies that all turbulence properties of interest are contained within This may not be true. There is now a substantial LES literature. See Lesieur & Métais (1996) for a review. Stretched vortex models of the small scales can be adapted to the evaluation of if it is supposed that the subgrid motions are those produced by nearly straight subgrid vortices which are, on average, undergoing stretching by the resolved scale straining field. With this ansatz, can


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be written as (Misra & Pullin 1997)

where is the subgrid energy spectrum, K is the subgrid energy and are the direction cosines of the subgrid vortex axis. A simple model for can be obtained by adapting the stretched-spiral form of (Lundgren 1982)

where is the axial strain along the subgrid vortex axis provided by the local resolved flow. The prefactor can be estimated using second-order velocity structure function information contained in the resolved flow (Métais & Lesieur 1992; Voelkl 2000; Voelkl et al. 2000). Several simple models of have been implemented where vortices align with eigenvectors of and/or the resolved vorticity (Misra & Pullin 1997).

4.

Large-eddy simulation of channel flow

Large-eddy simulations using the stretched-vortex SGS model have been performed for decaying turbulence, plane-channel flow and channel flow under spanwise rotation (Voelkl 2000; Voelkl et al. 2000). The turbulent flow in an open channel under spanwise rotation can be simulated using channel-fixed axes which rotate with constant angular velocity about the spanwise or . The configuration is illustrated in Figure 2.

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The non-dimensional numbers which characterise this problem are the Reynolds number and the rotation number where is the friction velocity and where and refer to the friction velocity on the suction and pressure sides respectively. Figure 3 shows the mean velocity profile for and (Voelkl et al. 2000), compared with the experiments


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of Wei & Wilmath (1989). The effect of the SGS model, compared to a run in which the model is turned off, is substantial. Voelkl (2000) conducted LES of rotating channel flow at and and to match the direct numerical simulations (DNS) of Kristoffersen & Andersson (1993). All cases were run using de-aliasing and with an effective resolution of 48 × 64 × 43 in a domain of size Constant flowrate was enforced (matching the flowrate of the DNS in each case) to drive the flow. The mean velocity profiles are compared to the DNS results in Figure 4. The turbulence intensity on the suction side is increasingly suppressed with increasing rotation number to the extent that, at the highest rotation number investigated, the maximum of turbulence kinetic energy in the buffer layer (not shown) has almost completely disappeared on the suction side. The effects of rotation can also be illustrated by comparing flow visualisations of rotating and nonrotating channel flow (Figures 3 and 4). The isosurfaces of constant wall-normal component of the vorticity, indicate that the turbulent structures are suppressed on the suction side of the channel. For the rotating flow, the vorticity structures extend from the top wall far into the centre region of the channel.

5.

Large-eddy simulation of passive-scalar mixing The filtered equation for a scalar c is

where is the subgrid flux of c by the turbulent velocity field A model for can be developed based on solutions to equation (1) (Pullin 2000). Briefly, the SGS scalar flux is approximated in vortex-fixed co-ordinates as a volume-time integral over a cylindrical volume of radius where and over a mixing time T, using the product of the scalar field obtained from (1) and the vortex velocity. To simplify the resulting expression, an average is taken over the spin angle of the non-axisymmetric part of the vortex velocity field, assumed uniformly distributed on the unit circle. Because the scalar flux produced by the vortex is linear in the Fourier components of the nonaxisymmetric velocity field, only the flux produced by the axisymmetric velocity survives. When a further average is taken over the expectation of the vortex circulation assumed equally likely to be positive/negative, and a transformation is made from vortex fixed to laboratory axes

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reduces to an approximate form

where is the vortex energy. The dependence on the vortex radius and radial velocity profile is now contained entirely in Equation(8) takes the form of a tensor eddy-diffusivity model of the subgrid scalar flux. The mixing time T is approximated as a typical eddy turnover time for the subgrid motions, where is a dimensionless constant. An estimate (this value is used presently) can be obtained by considering the special case of local isotropy, but the present model is generally anisotropic. To include (8) in the stretched-vortex SGS model, the identity is made, where K is the subgrid energy. Both K and are available in individual cells from the SGS stress model described earlier. The combined model was tested (Pullin 2000) using the LES of passive scalar mixing by a forced turbulent flow field in the presence of a mean scalar gradient in the “1” direction. The turbulence is spatially periodic in a box. The filtered, forced Navier-Stokes equations together with (7), and incorporating the stretched-vortex SGS model and (8), were solved numerically at resolution using a Fourier-Galerkin pseudo-spectral method 2 with 3/2 dealiasing After the turbulence became statistically stationary, turbulence statistics were taken over 20 – 30 large-eddy turnover


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times and over several independent realisations. Taylor Reynolds numbers in the range were considered. Some results of the LES are displayed in Figure 5 which shows the variation of the scalar variance versus compared to the fully resolved DNS results of Overholt & Pope (1996). The general trend of the DNS and the present LES is similar but for the LES results are consistently below the DNS. This persists to low where both DNS and LES are fully resolved and may result from different implementation in the forcing at low wavenumbers. For the scalar variance becomes essentially independent of at near the constant value of This is consistent with the idea of a mixing transition at near (Dimotakis 2000).

6.

Concluding Remarks

The stretched-spiral vortex model appears to reproduce many of the Kolmogorov-like spectral features of the inertial range at large wavenumbers including a –5/3 velocity spectrum, a –7/3 pressure spectrum and both the –5/3 and –1 spectra for a passive scalar. This model also appears to be well suited as a basic subgrid structure for the construction of useful subgrid scale models for the large-eddy simulation of turbulent flows. Nevertheless, some features of this approach to SGS modeling re– main unsatisfactory. In particular, the present models for alignment of the subgrid structure axes with respect to the resolved vorticity vector and the eigenvectors of the resolved scale rate-of-strain tensor clearly require improvement. A major challenge ahead is to attempt to extend this structure-based modelling approach to the near-wall region of the turbulent boundary layer in order to avoid the necessity for near-DNS resolution of the near-wall structures.

Acknowledgements This work was supported in part by the National Science Foundation under Grant CTS-9634222. Useful discussions with P. Dimotakis, T.S. Lundgren and T. Voelkl are gratefully acknowledged.

References BATCHELOR, G.K. 1959 Small-scale variation of converted quantities like temperature in turbulent fluid, part I general discussion and the case of small conductivity. J. Fluid Mech. 5, 113–133. DIMOTAKIS, P.E. 2000 The mixing transition in turbulent flows. J. Fluid Mech. , 409, 69-98.

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GIBSON C.H. & SCHWARZ, W.H. 1963 The universal equilibrium spectra of turbulent velocity and scalar fields. J. Fluid Mech. 16, 365-384. HE, G.W., DOOLEN, G.D. & CHEN, S.Y. 1998 Calculations of longitudinal and transverse velocity structure functions using a vortex model of isotropic turbulence. Phys. Fluids 11 3743-3748. KAMBE, T. & HATAKEYAMA N. 2000 Statistical laws and vortex structures in fully developed turbulence. Fluid Dyn. Res. 27, 247-267 . KRISTOFFERSEN, R. & ANDERSSON, H.I. 1993 Direct simulations of low-Reynoldsnumber turbulent flow in a rotating channel. J. Fluid Mech. 256, 163-197. LEONARD, A. 1974 Energy cascade in large-eddy simulations of turbulent fluid flows. In Advances in Geophysics (ed. F.N. Frankiel, R.E. Munn), pp. 237-248. Academic (New York). LESIEUR, M. & MÉTAIS, O. 1996 New trends in large-eddy simulations of turbulence. Ann. Rev. Fluid Mech. 28 45–82. LUNDGREN, T.S. 1982 Strained spiral vortex model for turbulent fine structure. Phys Fluids 25, 2193-2203. MÉTAIS, O. & LESIEUR, M. 1992 Spectral large-eddy simulation of isotropic and stably stratified turbulence. J. Fluid Mech., 239 157-194. MISRA, A. & PULLIN, D.I. 1997 A vortex-based subgrid stress model for large-eddy simulation. Phys. Fluids , 9, 2443–2454. OVERHOLT, M.R. & POPE, S.B. 1996 Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence. Phys. Fluids 8 3128-3148. PORTER, D.H., WOODWARD P.R. & POUQUET A. 1996 Inertial range structures in decaying compressible turbulent flows. Phys. Fluids 10 237-245 PULLIN, D.I. 2000 A vortex-based model for the subgrid flux of a passive scalar. Phys. Fluids , 12 2311-2319. PULLIN, D.I. & LUNDGREN, T.S. 2001 Axial motion and scalar transport in stretched spiral vortices. Phys. Fluids , 13 2553-2563. PULLIN, D.I. & SAFFMAN, P.G. 1998 Vortex dynamics in turbulence. Ann. Rev. Fluid Mech. , 30, 31–51. TENNEKES, H. & LUMLEY, J.L. 1974 A first course in turbulence, The MIT Press VOELKL, T. 2000 A physical-space version of the stretched-vortex subgrid-stress model for large-eddy simulation of incompressible flow. PhD Thesis, California Institute of Technology VOELKL, T., PULLIN D.I. fe CHAN, D.C. 2000 A physical-space version of the stretched-vortex subgrid-stress model for large-eddy simulation. Phys. Fluids, bf 12, 1810-1825. WEI, T. & WILLMARTH, W.W. 1989 Reynolds-number effects on the structure of a turbulent channel flow J. Fluid Mech. 204, 57–95.

Low-pressure vortex analysis in turbulence: life, structure, and dynamical role of vortices Shigeo KIDA, Susumu GOTO, Takafumi MAKIHARA Theory and Computer Simulation Center, National Institute for Fusion Science, Oroshi-cho 322-6, Toki 509-5292, JAPAN [email protected]

Abstract The low-pressure vortex analysis is performed for the study of dynamical properties of tubular vortices in turbulence. An automatic tracking scheme of arbitrarily chosen vortices is developed which makes it easier to examine the history of individual vortices. The low-pressure vortices have typically two distinct regions of high vorticity, that is, the tubular central core and surrounding spiral arms. The vorticity in these two regions is perpendicular to each other. It is observed that both the length of fluid lines and the area of fluid surfaces increase, in average, exponentially in time with growth rates of 0.17 and 0.30 respectively. The main contribution to these stretching comes from the velocity induced by vortices. Vortex core pressure is low, And it’s usually curved like a bow; Computational sketching Shows non-uniform stretching And spirals wrapped up by the flow.

1.

Introduction

Turbulence is full of vortical motions of various types. Among others, the tubular swirling vortices are commonly observed in many kinds of turbulent flows. They play central roles in turbulence dynamics, such as the enhancement of mixing, diffusion, resistance, etc. The dynamical properties of vortical motions in isotropic turbulence is the main theme of the present paper. Stationary isotropic turbulence is investigated by the low-pressure vortex analysis which we have recently introduced (Miura & Kida l997). The fluctuations of physical quantities are statistically invariant in stationary turbulence, but individual vortices, if identified, have their own lives of finite time. It is anticipated that they are born through a


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kind of Kelvin-Helmholtz instability (Porter et al. 1994), interact with other vortices and the background shear flow, and then they are broken down into scattered vorticity, which will be sources of new vortices, and so on. Although plausible, this scenario is hard to be observed because a lot of other vortices coexist and tend to hide such a typical scenario behind them. In this paper we develop an automatic tracking scheme of arbitrarily chosen vortices for the study of the dynamical properties of vortices.

2.

Low-pressure vortex

The low-pressure vortex is a rotating low-pressure region. Its axis is defined by a minimal line of the pressure field which is parallel to the eigenvector of the pressure Hessian of the smallest eigenvalue. The pressure takes a minimum value in a plane perpendicular to the minimal line. The core of the vortex is defined as a radially concave region around the axis, so that the core boundary is given by the inflection surface of the pressure field. Here, is the radial coordinate in a plane perpendicular to the axis. The axis and core boundary of the low-pressure vortex are constructed as follows. We start to prepare the data of the pressure, its gradient, and its Hessian at all the grid points. In terms of these data the pressure field is expressed, around each grid point, as a Taylor series up to the second order. This Taylor series is transformed into the normal form by the use of the eigenvalues and eigenvectors of the pressure Hessian. If the two largest eigenvalues are positive, the pressure (approximated by the Taylor series) takes a minimum in any plane parallel to the two associated eigenvectors. The point closest to the master grid point on this minimum-pressure line is taken as a node of a vortex axis if it is located within a half grid distance and if there is a swirling motion around the minimum line. This Taylor series analysis is repeated for all the grid points. These node points are then connected appropriately to construct vortex axes, and the core boundary is determined as the inflection curve, on planes perpendicular to the vortex axis, of the pressure field. Each vortex axis is composed of many node points. All vortex axes are labelled by integers. To each node are attached the axis number, a serial number inherent in the axis, and the spatial coordinates of the node as well as of the master grid point. There is one-to-one correspondence between all the nodes and their master grid points. See Kida (2000) and references therein for detail of the procedure.

Low-pressure vortex

3.

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Tracking of individual vortices

Since many vortex axes are moving about in a turbulent flow, it is hard to trace, by bare eyes, the movement of each vortex. It is therefore helpful if an arbitrarily chosen vortex axis can be tracked automatically. The tracking of individual vortex axis is not an easy task because they are not conserved objects. They may split, merge, disappear, or even newly be created. A complete identification scheme therefore is very difficult. Nevertheless, we develop here an automatic tracking scheme of vortex axes, which turns out to work pretty well. The present scheme of identification of vortex axes at two consecutive times is based upon the idea of ‘majority decision’ (Miura 2001). Those vortices which take many votes from the master grid points of all the nodes on a chosen vortex axis at a previous time and from their nearest neighbours are identified as descendants of the concerned vortex. The numerical procedure of automatic tracking scheme is as follows. [1] Prepare two data sets of vortex axes at times and where is taken small enough that the vortices may move at most by the grid width. [2] Choose a vortex axis arbitrarily at time [3] Pick up all the master grid points associated with this vortex (solid circles in figure 1). [4] Register the above master grid points and their nearest neighbours (open circles) as voters. [5] Count the number of votes of all the vortex axes at time which were given by these registered grid points. [6] Descendants of the original vortex are determined by the following two criteria. First, the vortex axes which get only a single vote or have node points more than double of the original one are discarded. Then, those that have the largest votes or get more than votes are chosen as descendants. This method is applied to a forced turbulence of Reynolds number at resolution and the results are shown in figure 2. The

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core of an arbitrarily chosen vortex is drawn with a black tube in a small cubic region of side 128. The number of time step is 30 between consecutive panels. All the other vortex axes are drawn with grey lines. By this method we can trace easily any vortex to see its temporal evolution. This particular vortex is being stretched and split into three at the last panel. We checked that the stretching rate along the vortex axis is mostly positive. Simultaneous tracking of many vortices is straightforward.

4.

Structure of Tubular Vortices

Since the discovery of tubular vortical regions in isotropic turbulence in early 1980s (e.g. Siggia 1981), they have attracted many researchers as a fundamental structure of turbulence. The high-energy-dissipation and high-vorticity regions are found to be located closely to each other but not overlap completely (see Hosokawa & Yamamoto 1989). The idea of concentrated vorticity has been applied to the numerical simulation

Low-pressure vortex

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of turbulent flows as the vortex stick method in which the motion of vortices in an irrotational flow is calculated by the Bio-Savart law (e.g. Chorin 1982). The actual flow, however, is not so simple because vorticity spreads over the whole space through various dynamical processes such as instability, tearing, splitting, stretching, and viscous diffusion though tubular vortices of strong vorticity surely exist. As an example of complex vortical structure, we plot, in figure 3, a cross-section of a low-pressure vortex on a square which is perpendicular to the vortex. The vortex axis is located at the centre. Fluids rotate counter-clockwise. Figure (a) shows the core boundary, the diameter of which is about It is deformed from a circular shape and elongated in the direction of the vortex arms [see figures (b, d)]. The structure of vorticity is shown in figures (b), (c), and (d) for the magnitude, the axial component, and the cross-axial component, respectively. Two distinct regions of high vorticity are observed at the round centre and two arms wrapping it. The axial component contributes to the former, and the cross-axial component to the latter. A prototype of this double spiral arm structure was first derived by Moore (1985) around a straight diffusing vortex tube in a simple shear flow, and was extended to an oblique orientation to examine the combined effects of wrapping, tilting, and stretching around a tubular vortex of initiallyuniform background vorticity (Kawahara et al. 1997). The spiral vortex arm was observed in a decaying isotropic turbulence at (Kida & Miura 2000) and in a forced turbulence at (Kida 2000). Note that this structure is different from Lundgren’s (1982) spiral vortex in which vorticity in the spiral is parallel to that in the core. The contours of the Laplacian of pressure are drawn in figure (e), where is the strain-rate tensor. It takes large positive values in the core where the vorticity dominates the strain The contours are similar to the core boundary, and has been frequently used for visualisation of vortex tubes (see Tanaka & Kida 1993). Figure (f) shows that the energy dissipation takes place more actively at the periphery of vortex core mostly in the vortex arms.

5.

Reynolds Number Dependence

It is one of the most interesting questions whether the tubular vortices keep their dynamical importance at larger Reynolds numbers. The Reynolds number dependence of the characteristic quantities of vortices may offer some hints to answer this question. We compare two sets of stationary turbulence at different Reynolds numbers (see tables 1 and 2). The upper and lower rows give the values at resolutions and

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187

respectively. The turbulence is forced at large scales by fixing the magnitude of the Fourier components of velocity at lower wavenumbers The ratio of the cut-off wavenumber to the Kolmogorov wavenumber is approximately the same. This value is large enough to guarantee the numerical accuracy. The viscosity is different by 4 times between the two runs, whereas the energy and the energy dissipation rate are almost the same. The Taylor length and the Kolmogorov length obey the Kolmogorov scaling law as where is the order of the integral length. The mean values in table 2 are calculated by the arithmetic average of 23 snapshots over 5 eddy turnover times and 3 snapshots over 0.6 eddy turnover times for smaller and larger Reynolds numbers, respectively. Here, D is the core diameter, the circulation, the mean azimuthal velocity at the core boundary, the Kolmogorov velocity, the RMS velocity, the relative volume occupied by vortices to the flow volume, and are the relative contributions of the enstrophy and the energy dissipation rate from the vortex core to the totals, respectively. We note that the core diameter is about 10 times the Kolmogorov length and the azimuthal velocity at the core boundary is about 3 times the Kolmogorov velocity with a weak dependence on The azimuthal velocity does not seem to scale with The enstrophy inside the vortex cores is about 60% of the total, and the energy dissipation is about 30% of the total, both of which hardly change with the Reynolds number. The vortex Reynolds number is about 100 and increases slowly with The relative volume occupied by the vortex decreases also slowly with the Reynolds number, but this is within the statistical

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fluctuations. These behaviours encourage us to expect that the tubular vortex (defined by the low-pressure vortex) keeps the dynamical role at larger Reynolds numbers.

6.

Role of Vortices

One of the most prominent features of turbulence is the mixing ability. Floating materials in turbulence are stirred much more strongly than in laminar flows. The effective mixing is fundamental in many practical situations such as dilution of contaminations, enhancement of combustion and chemical reaction, etc. It is certain that the vortical motion may play key roles in mixing, but we really do not know where and how the fluid elements are deformed and mixed. The mixing of materials is after all caused by the molecular diffusion, the rate of which is proportional to the spatial inhomogeneity enhanced by turbulence. The measure of mixing is not easy to be defined, but the stretching rates of fluid (or material) lines and surfaces seem to be good candidates for it. A direct and naive way to see how fluid elements are mixed is to follow the deformation of fluid blobs in turbulence. As a first step for the study of deformation we have conducted direct numerical simulations of fluid lines (Kida & Goto 2002). It has been known that the length of fluid lines in turbulence increases exponentially in time with a stretching rate of order of the inverse of the Kolmogorov time (Batchelor 1952). The stretching rate is accurately estimated as by our fluid-line simulation. On the other hand, the tradi– tional method using many fluid-line-elements give which under– estimates the true value by about 30% (Girimaji & Pope 1990, Huang 1996). The shape of deformation is an important factor to measure the mixing rate. The motion of a fluid surface is simulated using a net of triangles. In figure 4, we plot the temporal evolution of a fluid surface starting with a flat plane together with the axes of low-pressure vortices near the surface. The time elapses from (a) to (d). By comparing them carefully, we find that the fluid surface is being wrapped along these axes (see a big wrinkle at bottom-left, for example). An axis crossing perpendicularly the surface at the bottom-right twists it counter-clockwise. The surface area increases exponentially with growth rate of which is equal to the sum of the true stretching rates of fluid line and the one obtained by the fluid-line-elements simulation. Concerning the most effective stretching regions we should examine not only the periphery of tubular vortices but also such regions as streams and convergence zones (Hunt et al. 1988).

Low-pressure vortex

7.

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Concluding Remarks

The vortical structure of turbulence is investigated by the method of low-pressure vortex analysis. The double spiral vortex arms wrapped around tubular vortex cores are captured for the first time by this method. The energy dissipation is taking place more actively in these arms. For the purpose of focusing on the evolution of a particular vortex among so many vortices in turbulent flows we have developed an automatic tracking scheme. This enables us to examine the whole life of individual vortices, from their generation to break-down, which is a key to understand the turbulence dynamics.

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Acknowledgments This work has been supported by a Grant-in-Aid for Scientific Re– search on Priority Areas (B) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

References BATCHELOR, G. K. (1952). Material-Element Deformation in Isotropic Turbulence. Proc. Roy. Soc. London A 213, 349-366. CHORIN, A. J. (1982). The Evolution of a Turbulent Vortex. Commun. Math. Phys. 83, 517-535. KIDA, S. & GOTO, S. (2002) Line statistics: Stretching rate of passive lines in turbulence. Phys. Fluids, 14, 352–361. GIRIMAJI, S. S. & POPE, S. B. (1990). Material-Element Deformation in Isotropic Turbulence. J. Fluid Mech. 220, 427–458. HOSOKAWA, I. & YAMAMOTO, K. (1989). Fine Structure of a Directly Simulated Isotropic Turbulence. , J. Phys. Soc. Japan, 58, 20–23. HUANG, M.J. (1996) Correlations of Vorticity and Material Line Elements with Strain in Decaying Turbulence. Phys. Fluids, 8, 2203–2214. HUNT, J.C.R., WRAY, A.A. & MOIN, P. (1988) Eddies, Streams, and Convergence Zones in Turbulent Flows. CTR Proc. of Summer Program 1988, pp. 193–208. KAWAHARA, G., KIDA, S., TANAKA, M. & YANASE, S. 1997 Wrap, tilt and stretch of vorticity lines around a strong straight vortex tube in a simple shear flow. J. Fluid Mech. 353, 115–162. KIDA, S. (2000). Computational Analysis of Turbulence — Description by LowPressure Vortex. Proc. IV International Congress on Industrial and Applied Mathematics, Edinburgh 5-9 July 1999, Eds. J.M. Ball & J.C.R. Hunt. pp. 117–128. KIDA, S. & MIURA, H. (2000). Double Spirals around a Tubular Vortex in Turbulence. J. Phys. Soc. Japan, 69, 3466–3467. LUNDGREN, T.S. (1982). Strained spiral vortex model for turbulent fine structure. Phys. Fluids, 25, 2193–2203. MIURA, H. (2001). Private communication. MIURA, H. & KIDA, S. (1997) Identification of Tubular Vortices in Complex Flows. J. Phys. Soc. Japan, 66, 1331–1334. MOORE, D.W. (1985). The Interaction of a Diffusing Line Vortex and Aligned Shear Flow. Proc. Roy. Soc. London A 399, 367–375. PORTER, D.H., POUQUET, A. & WOODWARD, P.R. (1994) Kolmogorov-like spectra in decaying three-dimensional supersonic waves. Phys. Fluids, 6, 2133–2142. SIGGlA, E.D. (1981) Numerical Study of Small Scale Intermittency in Three-Dimensional Turbulence. J. Fluid Mech. 107, 375–406. TANAKA, M. & KIDA, S. (1993). Characterization of vortex tubes and sheets. Phys. Fluids A 4, 2079–2082.

Vortex bi-layers and the emergence of vortex projectiles in compressible accelerated inhomogeneous flows (AIFs) Norman J. ZABUSKY & Shuang ZHANG Laboratory of Visiometrics and Modelling, Dept. of Mechanical and Aerospace Engineering, Rutgers University P. 0. Box 909 Piscataway, New Jersey 08855 [email protected]

Abstract Vortex bi-layers and “Vortex Projectiles” (VPs) are the essential coherent structures which emerge in the shock accelerated inhomogeneous (RichtmyerMeshkov) flows, in particular the light planar curtain configuration. In our visiometric mode of working, we identify and quantify several vortex processes which emerge in 2D simulations during four time epochs. In particular: large positive and negative secondary circulations that arise from incompressible baroclinic processes;

upstream and downstream moving VPs; and an intermediate stratified decaying turbulent sub-domain containing VPs. I use methods visiometric, And I tell you, this is my pet trick; If in my pot-boiler Compressible Euler Gives a vortex, then I will project it.

1.

Introduction and Overview

Interest in accelerated inhomogeneous flows (AIFs) has recently increased because they can explain hydrodynamic phenomena in supernova astrophysics, supersonic combustion and laser fusion. These flows, often called Rayleigh-Taylor or Richtmyer-Meshkov flows are reported at the biannual International Workshop on Compressible Turbulence Mixing. The eighth “MIX” meeting was held at Pasadena in December 2001. (http://www.llnl.gov/IWPCTM/). Reviews are in Zabusky (1999) and Brouillette (2002). Figure 1 shows several classical configurations where an incoming shock from the left ( M for Mach number) strikes an interface ( near contact discontinuity) between media of different density. These have been investigated experimentally, theoretically and computationally. From 191 K. Bajer and H.K. Moffatt (eds.). Tubes, Sheets and Singularities in Fluid Dynamics, 191–200. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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left to right, we see: (a) the classical small-amplitude sine wave interface (Kotelnikov et al. 2001); (b) inclined planar (Samtaney &; Zabusky 1994); (c) Dropped tank with re-acceleration (Kotelnikov et al. 2000); (d) inclined and sinusoidal curtains ( or two close interfaces); (e) cylinder or sphere (Zabusky & Zeng 1998); (f) ellipse (Ray et al. 2000). The references are to work of NJZ and colleagues. Arid the name in italics are pioneering experimentalists for the configuration above

Here our computational simulations of the compressible Euler equations focus on the emergence and interaction of nearby opposite-signed vortex sheets or bi-layers. This configuration, like that of the von Karman wake is unstable and usually results in the formation of compact regions of vorticity which have a dominant translational velocity and we call this general class of states “Vortex Projectiles” (VPs). The quintessential examples of a vortex projectiles are the point vortex dipole in 2D with velocity proportional to the (circulation/separation distance) and the thin core vortex ring in 3D with uniform velocity proportional to the (circulation/radius). For a general VP these singular distributions of vorticity may be relaxed into distributions or a turbulent cluster, which on average are “dipolar”. Laminar analytical examples include the Lamb-Chaplygin vortex pair in 2D and the Hill’s vortex ring in 3D.

Structures in compressible accelerated inhomogeneous flows

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We illustrate the generic process that forms these objects in 2D planar flows by investigating the planar curtain (d) with the PPM algorithm.

where the mass conservation equation is

For the gases we take an ideal equation of state.

2.

Geometry, parameters and numerical methods

The geometry and main parameter space for this sawtooth-shaped “light” (slow/fast/slow) gas layer or “curtain” in Fig. 2 are identical to our previous study (Yang et al. 1990), where the curtain is inclined at 30° to the vertical with a horizontal thickness 0.427 times the width of the shock tube. The density ratio of the first slow/fast is 0.14 (Atwood which models a helium layer in air. (However, for convenience we choose the ratio of specific heats to be the same in both media).

We study incident shock waves of M = 1.5 and 2.0, and we observe and note many new phenomena. We use inflow and outflow at the left and right x-boundaries, and reflecting boundary conditions for the horizontal y-boundaries. This yields a re-acceleration configuration at later

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time, when localised acoustic wave fronts are reflected from the horizontal boundaries and continue perturbing the stratified isolated vortex structures. Laboratory experiments with the same physical parameters and a tube length of cm have been performed by Sturtevant (1985,1987) and Bonazza et al. (1985). Our 2D compressible Euler equations were solved with the PPM algorithm (Collela & Woodward 1984 and Blondin & Hawley 2001) in a Cartesian frame. The frame velocity is where k is a coeffi– cient related to M and a is the ambient sound speed. This optimised our computation, for it kept the upstream and downstream moving vortex projectiles in a relatively small domain.

3.

Phenomena and time epochs

In Yang et al. (1990), we showed that an incident planar shock wave passed through both interfaces and was reflected in a variety of complicated ways. However, the dominant phenomenon in the early time epoch was the deposition of narrow vortex layers due to baroclinic processes, as seen again in Fig. 3a (density above arid vorticity below) for The layers were on the contact “discontinuities” and were of opposite-sign: (+, red) on the upstream and (-, blue) on the downstream interfaces. As the deposition is occurring, the vortex layers begin evolving and their lower tips translate. The first major vortex phenomenon is a collision and “lift-off” of the bottoms of the opposite-signed layers, which leads to a sudden topology change or “breakthrough” from upstream to downstream. This is also seen in Fig.3a, as well as a new large intensification of density (localised red region on axis). The shock strikes the upstream interface at Note, time is the normalised time, where is the time required for a planar shock (without inhomogeneity) to traverse a distance equal to the horizontal extent of the curtain The corresponding evo– lution of circulations, positive, negative and total is shown in Fig. 4 as: and respectively. In the first panel Fig. 4a at (the beginning of epoch II. or eII, hereafter), the incident shock has been transmitted through the curtain (as seen in density by the blue-green transition at right and an upstream reflected wave (orange-green) is seen at right. As the complex collision domain (which we designate as VP1) tra– verses the shock tube (Fig. 4b, it evolves into a “mushroom” structure, which collides with the upper horizontal boundary (the end of the eIIa). Also seen is an alternating signed vortex wake of at least four vortices, designated as VP4 and VP5. In density we see a low-


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density (blue) “bubble” formed at the upper boundary and an intrusion (due to the upward moving V P1) at “mushroom” 1. (Note, the collision of oppositely moving coaxial point vortex dipoles and subsequent transverse motion is discussed in Lamb (1932) and the generalisation to colliding finite-area vortex configurations is presented in Overman & Zabusky (1983) via contour dynamics. V P1’s collision with the upper wall separates eII into phases, eIIa and eIIb. At (see Fig. 4c, during phase eIIb), we see that, after collision with the upper boundary, V P1 has separated into upstream and downstream projectiles (V P2 and V P3, respectively) with a complex turbulent state between them. The separation is easily explained as the consequence of each signed region interacting and binding with its mirror image about the upper horizontal boundary. Four of the alternatingsigned wake vortices have bound into V Ps (V P4 and V P5), which are apparent in density as “mushroom” structures. During eII, the circulation increased by nearly 50% after the incident shock front was far downstream. This important secondary baroclinic process is due to: 1) refracted waves up-and- downstream that are reflected from the horizontal boundaries; and 2) complex vortex interactions among VPs and their boundary images. The circulation reached a maximum at the end of eII. During eIII (see Fig. 3d, some isolated vortices continue to merge into new V Ps like V P6 (although still not obvious in density) while V P4 dissociates when it hits the upper boundary (and simultaneously enhances the formation of V P6). In eIII, we note that the baroclinic generation process is comparable to the compressible dilatation process. Finally, at (see Fig. 3e, in eIV), the dominant V P2 and V P3 have evolved into nearly circular domains which are translating upstream and downstream, respectively at nearly constant velocity. Both contain approximately 30% of the total absolute circulation are the positive (+) and negative (–) vorticity in the domain). Thus, 70% of the circulation is in a “turbulent” state, which also contains V Ps (eg. V P5 and V P6, which are translating upward and downward, respectively). At this time, the half V P’s displayed (i.e., the mirror image is not shown), are dominantly domain of one sign but are surrounded by oppositesigned thin vortex layers (which we quantify in Fig. 5). The circulation is nearly constant in eIV because of the competition among the baroclinic sources re-accelerating the dominant V Ps, the compressible dilation which is more significant as shown in Fig. 4, and numerical dissipation.

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197

Fig. 4 shows the normalised positive, negative and total circulations vs time for M = 1.5 and 2.0 runs. We normalised the M = 1.5 curve by multiplying it by the factor max which collapses the data of the two runs. We chose the time epochs in accord with phenomena of the M = 2.0 run. For example, V P1 hits the upper boundary at (the time between eIIa and eIIb). This is close to when V P1 hits the upper boundary in the M = 1.5 simulation. In eIII, the signed circulations both decay until they reach a near- constant value and in eIV their magnitudes begin to increase slightly. Fig. 5 shows the variation of total enstrophy (which emphasises larger magnitude vortex domains) and two enstrophy generating terms of the integrated enstrophy-variation equation. The baroclinic source term

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dominates in eIIa and eIIb and maintains a near-constant enstrophy over numerical dissipation. In eIII the dilatation term exceeds the baroclinic term , but both are too small to prevent enstrophy decay. The enstrophy decays in eIII and eIV as (note, the rate of change of enstrophy rate decays as More work is required to establish the meaning of these exponents when compared with evolutions of very-high Reynolds number Navier-Stokes systems.

4.

Discussion, conclusion and future directions

We believe the inclined planar curtain is fundamental to understanding turbulent mixing inherent in accelerated inhomogeneous flows and in particular the re-acceleration problem. This follows because of the quick appearance and close interaction of oppositely- signed vortex layers and the emergence of vortex projectiles (V Ps). Furthermore, this configuration is easily obtained in the laboratory, as indicated by the accomplishments of the Los Alamos group (Rightely et al. 2000).


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We have continued our pioneering numerical study of the interaction of a planar shock with a planar inclined light fluid curtain (a 2D slow/fast/slow environment) to much longer times. We introduce four time epochs to focus on dominant physical processes. At the end of the second epoch, we observe the separation of a traversing complex vortex projectile (V P1) into upstream and downstream projectiles with an intermediate stratified (lower density) turbulent domain. We have quantified some of the characteristic properties of the flow including the upstream and downstream V Ps and the evolution of circulation and functions related to enstrophy evolution. A brief report of our results was published recently by Zabusky & Zhang (2002), including colour images of density and vorticity. Our current research in planar curtains is focused on examining the secondary vorticity generation and the emergence of vortex projectiles and turbulent sub-domains at late time. We will extend the parameter space to include: initially finite thickness interfacial layers (the issue of well-posedness); configurations with less symmetry; higher Mach numbers; heavier-than-ambient (f/s/f) curtains; and three spatial dimensions

Acknowledgments We acknowledge recent collaborations with Sandeep Gupta and Gaozhu Peng. This work has been supported by the Department of Energy (Grant DE-FG02-98ER25364) under Dr. Daniel Hitchcock and by Rut– gers University. We also acknowledge the use of PPM for the Euler equations obtained from the superior code work of the VirginiaHydrodynamics Group .

References BLONDIN, J. M. & HAWLEY, J. 2001 Virginia Hydrodynamics Code. http://yonka.physics.ncsu.edu/pub/VH-1/index.html.

BONAZZA, R., BROUILLETTE, M., GOLDSTEIN, D., HAAS, J.-F., WINCKELMANS, G. AND STURTEVANT, B. 1985 Bull. Am. Phys. Soc. 30, 1742. BROUILLETTE, M. 2002 The Richtmyer-Meshkov Instability. Ann. Rev. Fluid Mech. 34, 445–468. COLELLA, P. & WOODWARD, P. R. 1984 The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations. J. Comp. Phys. 54 (1), 174–201. KOTELNIKOV, A., RAY, J. AND ZABUSKY, N. J. 2000 Vortex morphologies on reaccelerated interfaces: Visualization, quantification and modeling of one-and-two mode compressible and incompressible environments. Phys. Fluids, 12, 3245–3264. KOTELNIKOV, A., GULAK, Y. AND ZABUSKY, N. J. 2000 Nonlinear evolution and vortex localization: Different phases of a single mode Richtmyer-Meshkov unstable interface. Accepted, subject to modifications J. Fluid Mech. (March 2002).

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L A M B , H. 1895 Hydrodynamics, edition, Cambridge University Press, 1932. OVERMAN II, E. A. & ZABUSKY, N. J. 1982 Coaxial, Scattering of Euler Equation Translating V-states via Contour Dynamics. J. Fluid Mech., 125, 187. RAY, J., SAMTANEY, R. AND ZABUSKY, N. J. 2000 Shock interactions with heavy gaseous elliptic cylinders: Two leeward-side shock competition modes and a heuristic model for interfacial circulation deposition at early times. Phys. Fluids, 12 707–716. RIGHTLEY, P. M., PRESTRIDGE, K., ZOLDI, C. A., BENJAMIN, R. F. AND VOROBIEFF, P. 2000 Velocity field measurements of a shock-accelerated gas cylinder. The 53rd Annual Meeting of the American Physical Society’s Division of Fluid Dynamics (Meeting ID: DFD00), Session MF.001, Nov. 19-21, 2000. Washington, D. C. SAMTANEY, R. & ZABUSKY, N. J. 1994 Circulation Deposition on Shock Accelerated Planar and Curved Density-Stratified Interfaces: Models and Scaling Laws. J. Fluid Mech. 269, 45–78. STURTEVANT, B. 1985 Caltech unpublished reports. STURTEVANT, B. 1987 In Shock Tubes and Waves. Edited by H. Gronig (VCH, Berlin, 1987), p. 89. YANG, X., ZABUSKY, N. J., AND CHERN, I.-L. 1990 ‘Breakthrough’ via DipolarVortex/Jet Formation in Shock-Accelerated Density-Stratified Layers. Phys. Fluids, A2 (6), 892–895. ZABUSKY, N. J. & ZENG, S.-M. 1998 Shock cavity implosion morphologies and vortical projectile generation in axisymmetric shock-spherical F/S bubble interactions. J. Fluid Mech. 362, 327–346. ZABUSKY, N. J. & ZHANG, S. 2002 Shock - planar curtain interactions in 2D: Emergence of vortex double layers, vortex projectiles and decaying stratified turbulence Phys. Fluids, 14(1), 419–422. ZABUSKY, N. J. 1999 Vortex paradigm for accelerated inhomogeneous flows: Visiometrics for the Rayleigh-Taylor and Richtmyer-Meshkov environments. Ann. Rev. Fluid Mech. 31, 495–535.

Interaction of localised packets of vorticity with turbulence A. LEONARD Graduate Aeronautical Laboratories, California Institute of Technology, Mail Code 301-46, Pasadena, CA 91125-4600, USA [email protected]

Abstract The evolution of initially weak structures of vorticity as they evolve in an incompressible turbulent flow is investigated. Such objects are candidates for important structures in the inertial range and in the dissipation range of scales. As these structures are strained by the flow, fine-scales of vorticity are produced along the direction of maximum compression with a consequent flow of energy to the high wavenumbers. It is shown that, under certain circumstances, the self-energy spectrum of such a structure may be time-averaged, producing a fractional power law. The exponent of the power law depends on the ratio of the first two Lyapunov exponents of the strain tensor. I follow a vortical blob Through stretching and turning and throb; I can go very far

With pancake or cigar; You’ll see it’s a very fine job!

1.

Introduction

In this paper we continue our investigation of the evolution of localised structures of initially weak vorticity as they evolve in an incompressible turbulent flow. The first results were presented in Leonard (2000), hereafter known as L2000. Initially, these structures evolve passively by the induced velocity field of the large-scale vorticity field. This field is threedimensional and time-dependent so that these objects are subjected to straining apropos of lagrangian chaos, characterised by a distribution of finite-time Lyapunov exponents. Because of compression along at least one direction, fine scales of vorticity are produced. Therefore energy is shifted to higher wave numbers and, as we will see, backscatter of energy also occurs. One question might be - are such structures candidates for inertial range turbulence? Thus we will be concerned with their instantaneous and time-averaged spectral properties. It is shown that the time-averaged energy spectrum 201 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 201–210. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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is a power law with exponent depending on the ratio of area growth rate to line element growth rate or, equivalently, the ratio of the first two Lyapunov exponents. In addition we are able to connect our results with Lundgren’s strained spiral vortex model (Lundgren 1982), consisting of asymptotic solutions to the two-dimensional viscous equations for axial vorticity with an imposed axisymmetric strain. In Lundgren’s case, appropriate averaging over such structures yields Kolmogorov’s spectrum. Gilbert (1993) has shown that certain generalisations of Lundgren’s scenario can lead to the same result. An example of early work that considered the effect of strain on weak turbulence is that of Pearson (1959). The present work also has some similarities to the problem of determining the evolution of a weak magnetic field being advected by a velocity field that is either known or has known statistics. This is termed the kinematic dynamo problem. For example, Bayly (2001) studies the time eigenvalue problem for the magnetic field with a known velocity field that is steady in time and periodic in space. Chertkov et al. (1999) assume the statistics of a turbulent velocity field and study the state of a magnetic dynamo that remains smaller than the viscous scale of the turbulence with the magnetic diffusivity that is much smaller than the fluid viscosity.

2.

Kinematics of passive vorticity

Let the total velocity and vorticity into active and passive components as follows:

fields be decomposed

where and are, respectively, the velocity and vorticity fields of the active, large-scale turbulence and and the fields of the initially weak, passive turbulence. The exact momentum equations are

It is imagined that the passive vorticity is formed as a result of pairing, reconnection, or breakdown of large-scale active vorticity or from vortic– ity that is shed from a parent structure. The passive vorticity is assumed to be initially relatively weak so that self-induced motions are negligible. Thus, we neglect the term in (4). The approach here is somewhat

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different from rapid distortion theory (RDT). In RDT, perturbations to by are included as a source term for (Cambon & Scott 1999). Here they are not. We assume to be expressed as a superposition of discrete elements or blobs,

where for each initial blob we take to be a spherical vortex ring with a gaussian profile,

where is a measure of the initial blob diameter, the magnitude of is a velocity scale for the blob, and is the location of the blob. The evolution of is given by

In addition, the blob packet is assumed to remain small relative to spatial variations in beyond linear so that the velocity field observed by each blob may be approximated by

where

is the velocity gradient tensor

and is the component of Therefore, the vorticity field of each blob will satisfy the following linearised vorticity transport equation

where is a function of time only. As shown in L2000, (10) can be solved exactly for each blob to give

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i.e., a compact object, ellipsoidal in shape. Here F is the deformation tensor satisfying

The tensor M takes care of viscous effects by

and the impulse vector satisfies

with the solution given in terms of the deformation tensor as

The blob index has been suppressed on the tensors U, F, and M.

3.

Properties of the deformed blob

For a velocity field that produces chaotic advection we expect exponential behaviour in time for the eigenvalues of F. In any case, because is positive definite and symmetric it can be diagonalised by a rotation matrix

defining the finite-time Lyapunov exponents,

ordered so that Incompressibility

yields the relation

In general, infinitesimal line elements increase in length as while are area elements increase by For convenience, we transform each blob to the coordinate system


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We find that

Viscous effects are assumed negligible in the

and

directions so that

If viscous effects become important, they will first manifest themselves in the direction, the direction of maximum compression. Then is given approximately as

where we have assumed that the rotation matrix remains constant and equal to during the time interval that the integrand is exponentially large, i.e., In (21) the quantity is given by where of F

is the rotation matrix associated with the polar decomposition

and V is the positive-definite and symmetric matrix related to F by

4.

Spectral analysis of a single blob

To investigate the spectral properties of a single packet or blob we compute the Fourier transform of given by (21) to find

The self-energy spectrum for blob over spherical angles in space

requires the following integration

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For convenience we chose

as follows:

so that

where

is defined from (23) as

In addition, using (24) we can write

where

of

has magnitude

as

and is defined by

Using simple asymptotics we find that, depending on the magnitude is given approximately as follows: small

intermediate

large

The small result for is seen to be proportional to and the square of the impulse as expected. For intermediate values is proportional to This is the result one obtains for two vortex tubes that are nearly parallel to each other, separated by a small distance and with equal and opposite circulation.


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For large the sheet-like cross section of the vortex tubes becomes apparent, hence the behaviour above and observed in the numerical simulations by Reyl et al. (1998). But also the fact that there are two opposing sheets is manifested by the constant, component proportional to These power-law contributions extend to large values of up to ing that becomes

where falls off exponentially. Assumin (34) and using

Then this above-mentioned high wavenumber cutoff is given by

Integration of (34-36) over shows that the total self energy, is increasing as with the increase being split roughly as one-quarter in the intermediate range and threequarters in the large range, independent of viscous effects and independent of for If the above results are valid up to the time

5.

Time-averaged spectra

It is clear that the time-average of the self-spectrum of a single blob does not exist given the exponential behaviour in time (see (34-36)). The reason is that this behaviour is only valid for early times and for an iso– lated blob. Even an isolated blob will elongate in the direction to the point when the strain will be nonuniform in that direction. Ultimately, the blob will extend over a random, non-intersecting space curve with variable width and curvature in the direction. The more general initial condition (5) with blobs instead of the single one considered in the previous section would require a detailed analysis mainly because the energy spectrum would contain important cross terms. However, we suggest that the following features will be present in the general case. As compression takes place in the direction, multiple sheets of vorticity of alternating sign will be observed, instead of the single or double sheets of the single blob case. These

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sheets will extend in the and directions and be stacked upon each other in the direction. The continual thinning of these sheets will produce a corresponding flow of energy to high wavenumbers.Viscous effects will prevail when for a particular sheet that sheet thickness approaches We propose a model based on the above scenario.Assume that the spectrum for an isolated blob given by (36) for high wavenumbers is valid up until another structure “collides” with the blob. The direction along which other objects may approach rapidly is the direction. Assume that another such structure is at an initial distance L away from the blob in the direction.At a later time this distance is reduced to Thus, at a fixed the spectrum given by (36) will continue up till a time given by Thus, using (36), the time-integrated spectrum will be

where is defined above and is the time at which the high wavenumber contribution first appears at wavenumber and is given by If we use (37) for (t) and assume that and are independent of time then the time integral in (39) may be computed readily to give i.e., power-law behaviour with an exponent depending on possibilities are listed in Table 1.

Various

The entries “Pancakes” and “Cigars” just correspond to the two possible limits for The second entry is based on the stretched vortex


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model used by Lundgren (1982). That is, if a rectilinear vortex tube and the surrounding fluid are being stretched longitudinally with constant Lyapunov exponent then a material element in the surrounding fluid also suffers an elongation rate of in the plane normal to the tube axis. Interestingly, this ratio yields a –5/3 exponent for as found by Lundgren(1982).Indeed Gilbert (1993) shows that the –5/3 exponent may be obtained more generally using a simple cascade argument in which The entry is from an earlier study by Girimaji & Pope (1990) at relatively low Reynolds numbers, using direct numerical simulation (DNS). Their result for the average values of the exponents was essentially independent of Taylor Scale Reynolds number for the cases 63, and 90. A recent DNS study of line and surface element stretching in homogeneous turbulence by Kida et al.(2002) may be found in these proceedings. Galluccio & Vulpiani (1994) have computed Lyapunov exponents for a flow map derived from ABC flow, a simple, steady three-dimensional flow known to produce chaotic advection. In this case they find that the expected value of It is argued that this result is related to the fact that the dynamics induced by ABC flow can be transformed into a system that is invariant under time reversal. Hence the term “reversible dynamics” used in Table 1.

6.

Summary

We have studied the evolution of weak structures of vorticity as they evolve in a turbulent flow. Straining along the direction of maximum compression produces fine scales and thus high wavenumber contributions to the energy spectrum. Viscous effects come into play by limiting the maximum wavenumber that can be achieved. For the general initial condition we would expect multiple thin layers of vorticity of alternating sign in the direction of maximum compression We model this process by assuming an initial structure in the form of a simple vortex ring that evolves into an ellipsoidal blob and eventually collides with another structure. This allows us to take a meaningful time average of the self-energy spectrum of the vortex blob. We find that the time-averaged spectrum to have power-law behaviour with an exponent depending on the ratio of the first two Lyapunov exponents, If this ratio is 1/2 then the spectrum is Kolmogorov’s law. This result provides an interesting connection to the work of Lundgren (1982), who found the same result for the time-averaged spectrum of weak, two-dimensional distributions of vorticity in the velocity field of an axisymmetric vortex tube that is being stretched at a constant rate. A similar connection was also

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found by Gilbert (1993) who considered generalisations to Lundgren’s stretched vortex scenario. Further work needs to be done to see if the scenario and related processes described in this paper hold up under further scrutiny and are indeed connected to inertial-range turbulence. For example, timeindependent values of Lyapunov exponents should be replaced by ensemble and time averaging over a time-dependent joint probability distribution for and Consideration of more general initial vortex structures leading to more complex geometrical shapes as they evolve would be useful.

References BAYLY, B. J. 2001 Asymptotic structure of fast dynamo eigenfunctions. Proceedings IUTAM Zakopane CAMBON, C. & SCOTT, J. F. 1999 Linear and nonlinear models of anisotropic turbulence. Ann. Rev. of Fluid Mech. 31, 1–53. CHERTOV, M., FALKOVICH, G., KOLOKOLOV, I., & VERGASSOLA, M. 1999 Smallscale turbulent dynamo. Phys. Rev. Lett. 83, 4065–68. GALLUCCIO, S. & VULPIANI, A. 1994 Stretching of material lines and surfaces in systems with Lagrangian chaos. Physica A 212:, (1-2) 75–98. GILBERT, A. D. 1993 A cascade interpretation of Lundgren’s stretched spiral vortex model for turbulent fine structure. Phys. Fluids A 5, 2831–2834. GIRIMAJI, S. S. & POPE, S. B. 1990 Material-element deformation in isotropic tur– bulence. J. Fluid Mech. 220, 427–458. KIDA, S.,GOTO, S. & MAKIHARA, T. 2002 Low-pressure vortex analysis in turbulence: Life, structure, and dynamical role of vortices. Proceedings IUTAM Zakopane LEONARD, A. 2000 Evolution of localized packets of vorticity and scalar in turbulence. In Turbulence Structure and Vortex Dynamics (ed. J. Hunt & J. Vassilicos), pp. 127–139. Cambridge University Press. LUNDGREN, T. S. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25, 2193–2203. PEARSON, J. R. A. 1959 The effect of uniform distortion on weak homogeneous turbulence. J. Fluid Mech. 5, 274–288. REYL, C. & ANTONSEN JR., T. M. 1998 Vorticity generation by instabilities of chaotic fluid flows. Physica D 111, 202–226.

Extraction of coherent vortex tubes in a 3D turbulent mixing layer using orthogonal wavelets Kai SCHNEIDER1 & Marie FARGE2 1

Centre de Mathématiques et d’Informatique, Université de Provence, 39, rue F. Joliot–Curie, 13453 Marseille Cedex 13, FRANCE [email protected]

2

LMD-CNRS, Ecole Normale Supérieure, Paris 24, rue Lhomond, 75231 Paris Cedex 9, FRANCE

Abstract We present a new technique to extract coherent vortex tubes out of turbulent flows. The method is based on an orthogonal vector-valued wavelet decomposition of the vorticity field using the fast wavelet transform. A nonlinear thresholding of the wavelet coefficients is applied, where the threshold depends on the Reynolds number and on the total enstrophy of the flow, only. The coherent vortex tubes are reconstructed from the strong wavelet coefficients while the remaining weak coefficients correspond to an incoherent background flow. As example we present an application of this method to a turbulent mixing layer computed by high resolution direct numerical simulation. We find that only few wavelet coefficients are necessary to represent the coherent vortex tubes of the flow. The incoherent background flow reconstructed from the remaining weak coefficients is structureless and exhibits an energy equipartition. Of wavelets I’m an adherent; Though some people think me quite errant, Like the structures extracted By techniques compacted, My lecture’s entirely coherent.

1.

Introduction

Many turbulent flows exhibit organised structures (e.g., Jimenez & Wray 1993) evolving in an unorganised random background. A separation of the flow into these two components is a prerequisite for a sound physical modelling of turbulence. Since these coherent vortices are well localised and excited on a wide range of scales, we have proposed to use the wavelet representation of the vorticity field to analyse (Farge 1992), to extract (Farge, Schneider & Kevlahan 1999) and to compute them


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(Schneider & Farge 2000). Recently, we generalised the vortex extraction technique for three-dimensional flows using a vector-valued wavelet decomposition (Farge, Pellegrino & Schneider 2001). Orthogonal wavelet bases are well suited for these tasks, because they are made of self-similar functions well localised in both physical and spectral spaces (Daubechies 1992, Farge 1992) leading to an efficient representation of intermittent data, such as turbulent flow fields. The vortex extraction method is based on an orthogonal wavelet decomposition of the vorticity field, a subsequent thresholding of the wavelet coefficients and a reconstruction from those whose modulus is above the given threshold. Its use is motivated by mathematical theorems yielding optimal min-max estimators for denoising of intermittent data (Donoho 1993). It depends on the enstrophy of the flow and the Reynolds number only. In (Farge, Schneider & Kevlahan 1999) we showed that a few strong wavelet coefficients represent the organised part of the flow, i.e., the coherent vortices. The remaining many weak wavelet coefficients represent the background flow which is structureless and may be modelled by some stochastic process. After a short presentation of the wavelet based vortex extraction algorithm, we demonstrate its efficiency by applying it to a turbulent mixing layer computed my means of high resolution direct numerical simulation. Finally, we conclude and give some perspectives for modelling turbulent flows.

2.

Wavelet algorithm for vortex extraction

We consider the vorticity field at a given time t, being the velocity field, computed at resolution where N is the number of grid points and J the number of dyadic scales. We use a three-dimensional vector-valued Multi-Resolution Analysis (MRA) of i. e., a set of nested subspaces for representing the flow at different scales Considering the orthogonal complement spaces we obtain a wavelet representation. Therefore the vorticity vector is developed into an orthogonal wavelet series,

Extraction of coherent vortex tubes in a 3D mixing layer

with

213

and

where and are the one-dimensional scaling function and the corresponding wavelet, respectively. Due to orthogonality, the scaling coefficients are given by and the wavelet coefficients are given by where denotes the product. The extraction algorithm can be summarised as follows: given sampled on a grid the total enstrophy

for

N – 1, and

perform the three-dimensional wavelet decomposition (i.e., apply the Fast Wavelet Transform to each component of ) to obtain for compute the threshold coefficients to obtain

and threshold the

perform the three-dimensional wavelet reconstruction (i.e., apply the inverse Fast Wavelet Transform) to compute and from and respectively, use Biot-Savart’s relation to reconstruct the coherent and incoherent velocity fields from the coherent and incoherent vorticity fields, respectively. Note that the decomposition of is orthogonal and hence it follows that The complexity of the Fast Wavelet Transform (FWT) is of O(N), where N denotes the total number of grid points.

214

3.


Application to a turbulent mixing layer

Here we apply the above algorithm to a forced three dimensional turbulent mixing layer calculated by high resolution direct numerical simulation (Rogers & Moser 1994) evolved for 40 eddy turn over times. The modulus of vorticity for the total flow is shown in Fig. 1. We observe four transverse rollers, which are produced by the 2D Kelvin– Helmholtz instability, together with well pronounced longitudinal vortex tubes, called ribs, resulting from three-dimensional instability. In Fig. 2 (left) we plot the coherent vorticity, reconstructed from 3% of the total number N of wavelet coefficients. The incoherent vorticity (Fig. 2, right), reconstructed from 97% of the wavelet coefficients, contains 1% of the turbulent kinetic energy and 82% of the enstrophy. It is nearly homogeneous with very weak amplitude and contains no structure. Now we focus on the extraction of two vortex tubes, i.e., ribs in between two rollers. In Fig. 3 we plot the isosurfaces of the vorticity mod-

Extraction of coherent vortex tubes in a 3D mixing layer

215

ulus for the total flow, the coherent part (3% of the wavelet coefficients) and the incoherent part (97% of the wavelet coefficients). We observe that the coherent flow almost perfectly preserves the vortex tube present in the total flow, while the incoherent flow is structureless.

The corresponding longitudinal energy spectra in Fig. 4 show that the coherent part exhibits the same power-law behaviour as the total flow over the whole inertial range. In contrast, the incoherent flow has a flat energy spectrum, i.e., an equipartition of energy, which means in other words that it is decorrelated. The PDFs of vorticity show that the coherent part preserves the same strongly non Gaussian behaviour as the total flow, in particular its extreme values. For the incoherent flow we observe a strongly reduced variance of the vorticity PDF which shows an exponential behaviour.

4.

Perspectives

These results give the motivation to develop a new turbulence model, called CVS (Coherent Vortex Simulation), where the evolution of the coherent part of the flow (vortex tubes) is deterministically computed in an adaptive wavelet basis, while the the influence of the incoherent flow onto the coherent one is statistically modelled (Farge, Schneider & Kevlahan 1999; Schneider & Farge 2000; Farge & Schneider 2001).

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Acknowledgments We thank Michael Rogers for providing us with the DNS data and Giulio Pellegrino for developing the 3D wavelet code. We acknowledge financial support from the French-German program Procope (contract 01220ZE).

References DAUBECHIES, I. 1992 Ten Lectures on wavelets, SIAM, Philadelphia. DONOHO, D. 1993 Unconditionnal bases are optimal bases for data compression and statistical estimation. Appl. Comp. Harmonic Analysis, 1, 100–115. FARGE, M. 1992 Wavelet Transforms and their Applications to Turbulence. Ann. Rev. of Fluid Mech., 24, 395–457. FARGE, M., SCHNEIDER, K. & KEVLAHAN, N. 1999 Non–Gaussianity and Coherent Vortex Simulation for two–dimensional turbulence using an adaptive orthonormal wavelet basis. Phys. Fluids, 11(8), 2187–2201. FARGE, M., PELLEGRINO, G. & SCHNEIDER, K. 2001 Coherent Vortex Extraction in 3D Turbulent Flows using orthogonal wavelets. Phys. Rev. Lett., 87(5), 054501 FARGE, M. & SCHNEIDER, K. 2001 Coherent Vortex Simulation (CVS), a semideterministic turbulence model. Flow, Turbulence and Combustion, in press JIMENEZ J. & WRAY A. A. 1993, The structure of intense vorticity in isotropic turbulence, J. Fluid Mech., 255, 65–90. ROGERS, M. & MOSER, R. 1994 Direct simulation of a self-similar turbulent mixing layer. Phys. Fluids, 6(2), 903 – 923. SCHNEIDER, K. & FARGE, M. 2000 Numerical simulation of temporally growing mixing layer in an adaptive wavelet basis. C. R. Acad. Sci. Paris Série II b, 328, 263–269. SCHNEIDER, K., FARGE, M., PELLEGRINO, G. & ROGERS, M. 2000 CVS filtering of 3D turbulent mixing layers using orthogonal wavelets. Proceedings of the 2000 Summer Program, Center for Turbulence Research, Nasa Ames and Stanford University, 319-330.

Vortex tubes in shear-stratified turbulent flows Marie FARGE1, Alexandre AZZALINI1, Alex MAHALOV2, Basil NICOLAENKO2, Frank TSE2, Giulio PELLEGRINO3 & Kai SCHNEIDER 3 1

LMD-CNRS, École Normale Supérieure, Paris 24, rue Lhomond, 75231 Paris Cedex 5, FRANCE [email protected]

2

Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, USA 3 Centre de Mathématiques et d’Informatique, Université de Provence, 39, rue F. Joliot–Curie, 13453 Marseille Cedex 13, FRANCE Abstract Coherent vortex extraction using wavelets is applied to a shear-stratified turbulent flow computed by Direct Numerical Simulations (DNS) to compute the atmospheric jet stream at the tropopause. The basic state is characterised by a jet centred at the tropopause and stable density stratification profile with increased stratification above the tropopause. Quasi-equilibrium turbulent flow-fields are obtained after long-time integration of the governing equations written in primitive variables using adaptive spectral domain decomposition method. The coherent vortex tubes are extracted from the vorticity and potential vorticity fields, using a nonlinear filtering in wavelet space. It is finally checked that the coherent vortex tubes exhibit the same dynamics as the total flow and therefore drive the residual background flow. I use without any compunction A well-chosen wavelet function; With theorems nice, And methods concise, My results are immune to debunction.

1.

Shear-stratified turbulent flows

We study a generic situation encountered in geophysical turbulence where there is competition between shear and stable stratification. Stratification produces sheet-like structures (‘pancakes’) and waves which may inhibit turbulence caused by the shear, and therefore reduce the turbulent mixing. Shear where is the mean horizontal velocity profile) tends to destabilise the interfaces and gives rise to Kelvin-


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Helmholtz instabilities producing vortex tubes which drive strong mixing. We model jet streams encountered in the Earth atmosphere at the tropopause. This region is the transition between the troposphere and the stratosphere at an altitude of 10 to 15 km depending on the latitude. We consider a jet, centred at a vertical coordinate (Fig. 1), in a stably stratified fluid. The flow consists of a jet core surrounded by two shear layers: the layer below has a positive shear and a weak potential temperature gradient while the layer above has a negative shear and a stronger potential temperature gradient. The buoyancy (BruntVäisälä) frequency with the gravity, is increased by a factor two across the tropopause.

The gradient Richardson number which quantifies stratification and is based on the vertical gradient of the mean horizontal velocity increases towards the jet edges (near in figure (1) where the effect of stratification tends to reduce turbulence since the shear length-scale exceeds the buoyancy scale there (Tse, Mahalov, Nicolaenko & Fernando 2001). The flow in the vicinity of the edges of the jet is locally out of geostrophic equilibrium and thus produces nonlinear gravity waves which travel further away and break (near in figure (Fig. 1). These regions, far from the jet edges, have much weaker veloc-

Vortex tubes in shear-stratified turbulence

219

ity fluctuations than in the core, therefore turbulent mixing is reduced there, although potential energy and temperature fluctuations remain strong. This is in agreement with observations (Bedard, Canavero & Einaudi 1986). Indeed, critical levels where waves extract, energy from the mean flow (near in figure (Fig. 1) correspond to those regions of enhanced turbulence, where the gradient Richardson number is below hence generating Kelvin-Helmholtz instabilities (Kaltenbach, Gerz & Schumann 1994). The flow evolution has been computed using 3D Navier–Stokes equations coupled with the heat equation under the Boussinesq approximation. The numerical integration is made using spectral domain decomposition (Tse, Mahalov, Nicolaenko & Fernando 2001) and considering periodic boundary conditions in the streamwise and spanwise directions, but not in the vertical direction, where a non-uniform in spectral domain decomposition is implemented. This permits more realistic boundary conditions in the vertical and allows shear and stratification profiles to adjust during the flow evolution. For each horizontal wavenumber the vertical domain is broken down into 127 subdomains of variable sizes, which match their boundary conditions by a mortar method. The flow is first integrated during several eddy turnover times until it reaches a quasi-stationary state where viscous dissipation balances the external forcing. The external forcing corresponds to a non-uniform mean shear and a potential temperature profile (Fig. 1) which has been calculated using the meso-scale code MM5. After reaching the quasistationary regime, we then perform very long-time integrations to obtain the turbulent fields we study in this paper. The code has been parallelised using MPI and the computation has been performed on a parallel machine (ARL MSRC SGI Origin 3800) at Los Alamos National Laboratory. The spatial resolution is 256x256x512 and the CPU time to compute one time step is 15s, using a cluster of 32 SGI processors.

2.

Coherent vortex extraction using wavelets

We consider the vorticity field at a given time t, being the velocity field, computed at resolution where N is the number of grid points and the number of dyadic scales. We use a three-dimensional vector-valued Multi-Resolution Analysis (MRA) of a set of nested subspaces for representing the flow up to scale Considering the orthogonal complement spaces we obtain a wavelet representation which is based on two sets of functions, the scaling functions span-

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ning the approximation spaces and the wavelets spanning the complementary spaces Therefore the vorticity vector is developed into an orthogonal wavelet series,

with

and

where and are the one-dimensional scaling function and the corresponding wavelet, respectively. Due to orthogonality, the scaling coefficients are given by and the wavelet coefficients are given by where denotes the inner product. The extraction algorithm can be summarised as follows: given sampled on a grid the total enstrophy

for

and

perform the three-dimensional wavelet decomposition (i.e., apply the Fast Wavelet Transform to each component (which requires O(N) operations) of to obtain for 1,

compute the threshold coefficients to obtain

and threshold the

perform the three-dimensional wavelet reconstruction (i.e., apply the inverse Fast Wavelet Transform) to compute and from and respectively, use Biot-Savart’s relation since we neglect the potential part of the flow, to reconstruct the coherent and incoherent velocity fields from the coherent and incoherent vorticity fields, respectively.


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Note that the decomposition of is orthogonal and hence it follows that We have already applied this wavelet method to extract coherent vortices in three dimensional homogeneous and isotropic turbulent flows (Farge, Pellegrino & Schneider 2001).

3.

Application to the vorticity field

We will first consider the vorticity vector field (Fig. 2a) which has been computed at resolution N = 256 × 256 × 512, but for practical reason has been undersampled to We split the total flow into coherent (Fig. 2b) and incoherent flows (Fig. 2c). We find that only 2.5% of the modes are sufficient to represent the coherent flow, which contains 99% of the total energy and 89% of the total enstrophy. We observe that the coherent vorticity field presents the same vortex tubes as those observed in the total vorticity field when we plot the isosurfaces (Fig. 2a and 2b). In contrast, the incoherent vorticity field (Fig. 2c) does not exhibit any organised structures. If we consider the probability distribution function (PDF) of the vorticity (Fig. 3a), we find the same variability for the coherent vorticity as for the total one, while the variability of the incoherent vorticity is reduced by a factor 4. As to the PDF of vertical velocity (Fig. 3b), we find very different behaviours: the total and coherent vertical velocities are skewed, varying from –2.3 to 1.5; in contrast, the incoherent vertical velocity is not skewed and its behaviour is close to Gaussian (Fig. 3b), with a reduced variability between –0.1 and +0.1. If we now plot the energy spectrum (Fig. 3c), we find the same scaling for the total and coherent velocity fields up to the wavenumber with a maximum around reflecting large scale correlation. For higher wavenumbers, the spectral slope becomes much steeper (about and we observe that the coherent contribution is slightly shifted below the total spectrum. The incoherent field, which contributes no more than 1% to the total energy, is spread all over the wavenumbers, presenting a scaling. This shows some tendency towards decorrelation, although energy equipartition in three dimension corresponds to a scaling. We now propose to investigate other variables, looking for a sharper decorrelation.

4.

Application to the potential vorticity field

We decompose the flow into geostrophically balanced and unbalanced components. This is achieved through Craya basis decomposition (Craya

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1958). In the Boussinesq case we consider here (for which velocity remains divergent-free), the primitive variables, velocity and temperature T, are replaced by three new variables: the quasi-geostrophic (QG) potential vorticity the horizontal divergence and the thermal wind imbalance The first variable corresponds to the flow component which is in geostrophic balance, while the two other variables characterise the departure from geostrophic and hydrostatic balance. The QG potential vorticity is a scalar field which is transported by the geostrophically balanced horizontal components of the total velocity.


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In this paper we focus on the QG potential vorticity field (Fig. 5a), which has been computed at resolution and undersampled to Using the wavelet decomposition as described in (Farge, Schneider & Kevlahan 1999), the QG potential vorticity field is split into coherent and incoherent components which are orthogonal. We find that only 6.8% of the modes are sufficient to represent the coherent QG potential vorticity, which contains 99.6% of the total QG potential enstrophy. We observe that the coherent QG potential vorticity field presents the same tube-like structures as those observed in the total QG potential vorticity field when we plot the isosurfaces where (Fig. 5a and 5b). In contrast, the incoherent QG potential vorticity field (Fig. 5c) does not exhibit any organised structures.

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If we consider the probability distribution function (PDF) of the QG potential vorticity (Fig. 4a), we find the same variability for the QG potential coherent vorticity as for the total one, while the variability of the incoherent vorticity is reduced by a factor 25. We find the same spectral behaviour for the total and coherent QG potential enstrophy (Fig. 4b) up to wavenurnber the incoherent QG potential enstrophy presents a scaling, characteristic of equipartition between all wavevectors in three dimensions. We now check how accurately the nonlinear flow dynamics is preserved by the wavelet filtering. As a first test, we integrate during 3 eddy turnover times both the total and the coherent potential vorticities and compare the final states. On Fig. 6a and Fig. 6a we check that the two final vorticity fields are very similar and exhibit the same vortex tubes, in both shapes and locations. The fact that the initially filtered flow thus preserves the short term evolution tends to confirm the hypothesis that the coherent vortex tubes drive the incoherent background flow, which is slaved to them. This result, plus the previous observation that the incoherent potential vorticity is quasi-Gaussian and decorrelated, give us some confidence in being able to statistically model the influence of the incoherent contributions onto the coherent flow, the evolution of the latter, only, being deterministically computed. This is the principle of the CVS (Coherent Vortex Simulation) method, which has been already applied in the case of two dimensional flows (Farge & Schneider 2001), but not yet in the case of three dimensional flows.


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Conclusion

In this paper we have applied the wavelet filtering method to a stably stratified and sheared turbulent flow in three dimensions. Such a flow is highly inhomogeneous since it consists of a central jet subjected to different stratifications above and below the jet core. We have shown that the coherent structures are well extracted by retaining very few wavelet coefficients, which are actually sufficient to predict the flow evolution on few eddy turnover times. We have also verified that the residual flow is incoherent, with Gaussian PDFs and presents a tendency towards decorrelation. These results give us motivation to try to extend the CVS method to compute such shear-stratified turbulent flow. In this paper we have also explored the relevance of using the Craya decomposition to gain a better insight into the dynamics of sheared stratified flows. We have considered, as a first step, the QG potential vorticity which is weakly coupled to the two other Craya variables. We plan for future work to extend the wavelet filtering to study the departure from geostrophic balance, and filter the horizontal divergence and the thermal wind imbalance.

Acknowledgments This work is supported by the AFOSR Contract FG9620-99-1-0300, the DoD HPC Challenge Program, the NSF-CNRS-DAAD collaboration grant between ASU, LMD and ICT, the French-German program Procope (contract 01220ZE) and the European program TMR (contract FMRX-CT98-0184).

References BEDARD, A.J., CANAVERO, F. & EINAUDI, F. 1986 Atmospheric gravity waves and aircraft turbulence encounters. J. Atmos. Sci., 43, 2838-2844. CRAYA, A. 1958 Contribution à l’analyse de la turbulence associée à des vitesses moyennes. P.S.T. Ministère de l’Air (Paris), 345. KALTENBACH, H.J., GERZ, T. & SCHUMANN, U. 1994 Large-eddy simulation of homogeneous turbulence and diffusion in stably stratified shear flow, J. Fluid Mech. 280, 1–40. TSE, K.L., MAHALOV, A., NICOLAENKO, B. & FERNANDO, H.J.S. 2001 A spectral domain decomposition method and its application to simulations of shear-stratified turbulence. Lecture Notes in Physics, Springer-Verlag, 566, 353-378. FARGE, M., SCHNEIDER, K. & KEVLAHAN, N. 1999 Non–Gaussianity and Coherent Vortex Simulation for two–dimensional turbulence using an adaptive orthonormal wavelet basis. Phys. Fluids, 11(8), 2187–2201. FARGE, M., PELLEGRINO, G. & SCHNEIDER, K. 2001 Coherent Vortex Extraction in 3D Turbulent Flows using orthogonal wavelets. Phys. Rev. Lett., 87(5), 054501

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FARGE, M. & SCHNEIDER, K. 2001 Coherent Vortex Simulation (CVS), a semideterministic turbulence model. Flow, Turbulence and Combustion, 66, 393–426.

Coherent dynamics in wall turbulence Javier JIMÉNEZ 1,2 1

School of Aeronautics, Universidad Politécnica 28040 Madrid, SPAIN [email protected]

2 Centre for Turbulence Research, Stanford University

Stanford, CA 94305-3030, USA Abstract

It is of this paper the goal to study the turbulent wall where structures abound whose effect is profound even though their size is small. It will thus first be shown that the wall can survive on its own without any interaction, save perhaps in the form of a passive reaction, with the turbulent flow higher on. What makes this wall layer unique is a steady nonlinear streak which cycles chaotically while deforming locally, and which organises into objects of much larger sizes. The rest of this paper is concerned with it. In turbulence near to a wall, There are structures much longer than tall; With the breakdown of streaks, The vorticity peaks; And the speed can slow down to a crawl.

1.

Introduction

Walls are present in most flows and profoundly influence turbulence. That makes wall-bounded turbulence an interesting physical phenomenon, and one of prime technological importance. Wall-bounded flows are responsible for most of the drag of moving vehicles, as well as for a substantial part of aerodynamic noise, and it is also through walls that heat is transferred into the fluid and that aerodynamic loads are imposed on structures. We shall see that many issues in these flows are still open, and that they differ in many ways from the isotropic and homogeneous turbulent case.


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We shall focus in this paper on the structural organisation of the nearwall viscous and buffer layers, with emphasis on the flow of energy and on the large-scale organisation of the elementary structures.

2.

The energy balance and the near-wall region

As long as the pressure gradient is not too strong, we can assume to a first approximation that the flow near the wall is homogeneous in the streamwise and spanwise directions. In the wall-normal direction it is inhomogeneous and anisotropic. This is due to two effects. Firstly the impermeability condition prevents wall-normal velocity fluctuations from developing length scales much larger than inducing a natural scale stratification in which larger structures are only present away from the wall. Secondly the wall enforces a no-slip condition for the other two velocity components, so that viscosity cannot be neglected even at high Reynolds numbers. In a thin layer very close to the wall viscosity dominates, and the flow scales to a first approximation in wall units. If is the friction velocity, where is the kinematic viscosity and U the mean velocity profile, the dimensionless flow thickness characterises the global Reynolds number. To lowest order, the near-wall region is independent of although it may depend on the properties of the wall (smoothness, etc.) Away from the wall, where is of the order of the flow thickness the Reynolds number is too high for viscosity to be important. The length scale there is but the scale for the velocity fluctuations is still An overlap layer is needed to match the inner and outer length scales, whose ratio is In this layer the distance to the wall is too small for to be relevant, while the Reynolds number is too large for the viscosity to be important. The only remaining length scale is itself, and the classical logarithmic mean velocity profile is obtained from dimensional arguments,

The constant A has to be determined independently, and controls the overall friction coefficient. If we assume that holds up to we obtain in terms of A and of The viscous layer lies underneath the level in which applies, and acts as the boundary condition which fixes A, and the drag. The empirical limits for the logarithmic layer are and (Österlund et al. 2000) The energy argument used by Townsend (1976) provides a useful alternative derivation for the logarithmic velocity profile. Assume a flow which is statistically homogeneous in the streamwise direction, such as

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a turbulent pipe, and separate the velocity into its mean value and fluctuations. The balanced equation for the mean turbulent kinetic energy, is

The first term in the right-hand side is the production of turbulent energy by the interaction between the mean shear and the Reynolds stresses, the last one is the large-scale viscous diffusion of the kinetic energy, which can be neglected everywhere except in the viscous sublayer, and is the dissipation by the small scales. These three terms are balanced in the left-hand side by the divergence of a spatial energy flux where are the pressure fluctuations. If we assume that the energy production and dissipation are in equilibrium within the logarithmic layer, and take the latter to be the energy balance becomes,

which integrates directly to The left-half side of this equation is an exact expression for the energy production, while the right-hand side depends only on generic properties of the turbulent cascade. The assumption is that the cascade is implemented by approximately isotropic eddies whose velocities are and whose sizes are Note that this assumption is equivalent to the scaling hypotheses used to derive but that in this form it gives a more concrete interpretation of as the proportionality constant defining the integral scale of the eddies of the logarithmic region. A consequence of this analysis is that the energy flux should be constant across the logarithmic layer, and this is tested in figure 1(a), which contains data from three numerical channels at moderate Reynolds numbers. Positives fluxes flow away from the wall, and an increase of the flux with implies a local excess of energy production. The three curves collapse well near the wall, where energy production dominates. Part of the excess energy diffuses into the wall, but the rest flows through the logarithmic layer into the outer flow, where dissipation is dominant. In this sense, the wall provides part of the power needed to maintain turbulence in the outer region. There is a plateau in the logarithmic region, in agreement with together with perhaps a second energy production peak near the outer edge of the logarithmic layer. It is difficult to measure experimentally all the terms of the energy equation, and there are no data equivalent to those in figure 1(a) at

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higher Reynolds numbers. Some parts of have however been measured and can be used to check how representative are the numerical results. Figure 1(b) shows the turbulent transfer of streamwise fluctuations, which accounts for about half of the flux in That figure includes the numerical channels used in figure l(a), together with experimental boundary layers at much higher Reynolds numbers. Notwithstanding the uncertainties in the experimental data, both sets of measurements coincide where they overlap, giving some confidence in the conclusions from the numerics. The suggested conceptual picture is that of an energy-producing region near the wall which exports some of its extra turbulent energy across the logarithmic layer into the outer flow. This spatial energy transport is an energy cascade different from the usual Kolmogorov one. In the latter, the energy is transferred locally to the smaller scales, where it is eventually dissipated by viscosity. In the spatial cascade the energy is transferred to other locations before being eventually dissipated by cascading to smaller scales. It can be estimated that the turbulent energy transferred out of a layer bounded by the wall and some level within the logarithmic region is equivalent to the energy generated in its top 15%.

3.

The energy generation mechanism

Because of this energy-generating property, and because of its role as a boundary condition determining the wall drag, the region below


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100 has been the subject of intensive study. In this layer the wall-normal Reynolds numbers are low, and we may expect quasi-laminar structures that can be understood deterministically. In some sense this region corresponds to the Kolmogorov viscous range of isotropic turbulence, and it is easy to show that the Kolmogorov and wall-unit lengths are identical. The dominant structures in this region are streamwise velocity streaks and quasi-streamwise vortices. The former are long sinuous arrays of alternating streamwise jets superimposed on the mean shear, with an average spanwise separation (Smith & Metzler 1983). At the spanwise locations where the jets point forward, the wall shear is higher than the average, while the opposite is true for the ‘low velocity’ streaks where the jets point backwards. It was shown by Orlandi & Jiménez (1994) that the net effect is to increase the mean wall shear, and that this is the immediate cause of the excess turbulent wall drag. The quasi-streamwise vortices are slightly tilted away from the wall, and each one stays in the near-wall region only for (Jeong et al. 1997). Several vortices are associated with each streak, with a longitudinal spacing of the order of (Jiménez & Moin 1991). Some of them are connected to the trailing legs of the vortex hairpins of the outer part of the boundary layer, but most merge into disorganised vorticity after leaving the immediate wall neighbourhood (Robinson 1991). It has long been recognised that the vortices cause the streaks by advecting the mean velocity gradient (Blackwelder & Eckelmann 1979). This process is independent of the presence of the wall, as shown by the observation by Lee, Kim & Moin (1990) of streaks in uniformly sheared flows. There is less agreement on the mechanism by which the vortices are generated. It was proposed by Kim, Kline & Reynolds (1971) that streaks and vortices are part of a cycle, in which the latter are the result of an instability of the former. The argument was made more specific by Swearingen & Blackwelder (1987), who noted that the spanwise velocity gradients that separate the low- from the high-velocity streaks are subject to inflectional instabilities. This conceptual model has been elaborated and used to explain several properties of disturbed boundary layers by Jiménez (1994) and Hamilton, Kim & Waleffe (1995), among others, and is the essence of the quasi-linear models developed at MIT (Landahl & Mollo–Christensen 1992) and Cornell (Holmes, Lumley & Berkooz 1996). It can be summarised as that the quasi-streamwise vortices act on the mean shear to create the streaks, which become inviscidly unstable and eventually produce tilted streamwise vortices.

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A slightly different view is that the cycle is organised around a nonlinear travelling wave, which represents a permanently disturbed streak. This is not very different from the previous model, which essentially assumes that the undisturbed streak is a fixed point, and that the cycle is a homoclinic orbit running through it. Candidate nonlinear waves have been computed by Waleffe (1998) and others, while reduced models based on this approach have been formulated in Waleffe (1997). Two recent studies have strengthened the connection between these simple structures and real turbulent flows. The first one identifies an unstable travelling wave as a saddle point in a particular transition scenario in a turbulent channel (Toh & Itano 2001). Its stable manifold connects to a perturbed quasi-laminar state of the channel, reminiscent of the large-scale velocity ‘streaks’ associated by Andersson, Berggren & Henningson (1999) to bypass transition. The unstable manifold connects to full-blown turbulence. Both this wave and the one in Waleffe (1998) are computed for the minimal flow introduced by Jiménez & Moin (1991), which substitutes the full turbulent channel by a periodic array in and of identical computational boxes. Each box contains a single copy of the relevant structures, and the resulting reduction in the number of degrees of freedom is an important factor in simplifying the flow. The interesting observation is that this simplification modifies only slightly the skin friction or the near-wall turbulent intensities, which are in this way shown to be relatively independent of the chaotic character of turbulence. The second paper mentioned above takes this simplification one step further by using a numerical mask to damp the turbulent fluctuations away from the wall (Jiménez & Simens 2001). This technique was pioneered by Jiménez & Pinelli (1999) to show that the near-wall region was ‘autonomous’ and essentially independent of the outer flow. It was shown in that way that an autonomous turbulence-generating ‘engine’ resides in the region When the flow is masked in that region turbulence decays globally, but as long as that layer is untouched, it remains indefinitely turbulent. The study in Jiménez & Simens (2001) uses this masking technique on a minimal box. The result is that, when the flow is masked above turbulence reduces to a stable single travelling wave which is shown in figure 2. It consists of a wavy lowvelocity streak, associated to which there is a pair of staggered quasistreamwise counter-rotating vortices. Raising slightly the height of the mask results in a bifurcation cascade, starting with a limit cycle during which the wave grows deeper and shallower, while the two vortices grow and wane. Further raising of the mask leads to a two-frequency torus, mild chaos, and full-scale bursting turbulence.


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Figure 2 is qualitatively similar to the travelling waves obtained by Waleffe (1998) and Toh & Itano (2001), and to the structures deduced from larger channels by Jiménez & Moin (1991) and Jiménez & Pinelli (1999). Visual observation of the differences between movies of these and larger flows suggests the root of the complexity in the latter. As the streak goes through the cycle, it ejects some vorticity into the outer flow, essentially as a small hairpin vortex. In full-depth flows this vorticity evolves, becomes disorganised, and eventually comes back to modify the next cycle of the streak. In autonomous flows the vorticity is damped by the mask as soon as it is ejected, and this randomising mechanism is not present.

4.

The inactive motions near the wall

In the previous section we have shown that there is a relatively small self-sustaining structure in the near-wall region which can explain the local production of turbulent energy. It would however be wrong to conclude that these are the only turbulent scales active near the wall. Townsend (1976) noted that, although the impermeability condition implies that the vertical scale of the wall-normal velocity fluctuations should at most be no such requirement exists for the wall-parallel fluctuations, which can be larger. He named those hypothetical largescale structures ‘inactive’ because they could not, by themselves, create

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Reynolds stresses. We have mentioned that, even in the buffer layer, the low-velocity streaks have lengths of the order of 1000 wall units, suggesting that large-scale organisation exists even below One possibility is that its origin is the interaction with the outer flow, which imposes on the wall its own large scales. Another is that the wall organises itself, even at large scales. Such self-organisation is observed in many nonlinear systems, and it is of some interest to find whether the large scales in this case are autonomous or exogenous. Their expected behaviour would be different in each case. If they were forced from outside it might for example be possible to control them by acting away from the wall, while if the organisation is self-induced, such controls would probably be ineffective. Some guidance can be obtained from experimental spectra. If the origin of the large-scale organisation were the core flow, the structures would grow longer as the Reynolds number increases. This is indeed true, as can be seen in figure 3, which shows longitudinal pre-multiplied spectra of the streamwise velocity in four boundary layers at increasing Reynolds numbers. It is clear that the long-wavelength end of the plateau moves to the right as the Reynolds number increases. It has been shown that this end of the spectrum scales in the overlap layer with the boundary layer thickness (Österlund et al. 2000a). It is not obvious however that these very long structures represent individual


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streaks. The short-wavelength end of the plateau remains relatively constant at at all Reynolds numbers. A more detailed view of these large structures is obtained from numerical simulations. Figure 4 displays two-dimensional spectra from a numerical channel at (del Álamo 2001), and from an autonomous wall. Both are computed in periodic boxes which are large enough to minimise interference of the computational domain with the large structures. The first surprise is the almost perfect correspondence between the autonomous and the fully turbulent case, which strongly suggests that the organisation of wall into long streaks is not due to the outer flow. There are no turbulent fluctuations in the autonomous case above 80. The second surprise is that the autonomous streaks are actually longer than those of the full channel, which also runs counter to the idea that the outer flow is the source of large scales. It is clear from this figure that the longer scales are also wider, and that this part of the flow is far from being isotropic. This is specially true of the longitudinal velocity, and it is interesting that the ‘ridge’ in that spectrum follows fairly closely the power law

The origin of this relation, if confirmed by further experiments, is unknown. The spectra of the other two velocity components are closer to being isotropic, and correspond approximately to the dimensions of the elementary structures described in the previous section. The model suggested by the discussion in this section is that of a nearwall region in which the self-sustaining small-scale structures organise themselves into streaks a few thousand wall units long and about 100 wall units wide, essentially independently of the outer flow. There is an additional self-similar organisation of these streaks into longer and wider structures which, on the one hand, are present in autonomous walls without any outer flow, while on the other are limited in real boundary layers to lengths shorter than a fixed multiple of the outer flow thickness. These two observations are difficult to reconcile, although a possibility that immediately offers itself is that an autonomous wall layer would form structures of infinite length with an essentially flat spectrum. This is not as unlikely as it may seem, since the aspect ratio of the structures at the right hand side of figure 4(a) is already 300:1, and it is difficult to see how such a number might differ from infinity in terms of dynamics. The effect of the outer flow would then be to limit these structures to lengths shorter than a given one.


5.

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Summary

We have reviewed some of the open problems in the physics of the near-wall and logarithmic layers of wall flows. We have seen that the former is an active site for turbulence generation, which can run autonomously and which exports energy to the rest of the flow. Its dynamics are reasonably well described in terms of a regeneration cycle involving long streamwise velocity streaks and shorter quasistreamwise vortices. The cycle appears to be associated to a nonlinear structure, a wavy streak, which has been identified theoretically and observationally, and has been shown to be self-sustaining in severely truncated numerical experiments. These structures organise themselves self-similarly at larger scales, of the order of several thousand wall units long and wide. This organisation occurs autonomously in turbulent walls without an outer flow, but in natural boundary layers its maximum length is proportional to the boundary layer thickness. The spanwise width of the structures scales as a power law of their length, although the reason is not understood.

Acknowledgments This work was supported in part by the Spanish CICYT contract BFM2000-1468 and by ONR grant N0014-00-1-0146. M. P. Simens, whose work is discussed in section 3, was supported in part by the EC under the TMR grant CT98-0175. The autonomous spectra in figure 4 are due in large part to the work of Manuel Gareía–Villalba and Oscar Flores.

References ANDERSSON, P., BERGGREN, M. & HENNINGSON, D.S. 1999 Optimal disturbances and bypass transition in boundary layers, Phys. Fluids, 11, 134–150. BLACKWELDER, R. & ECKELMANN, H. 1979 Streamwise vortices associated with the bursting phenomenon. J. Fluid Mech. 94, 577–594. DEL ÁLAMO, J.C. 2001 The large-scale organization of turbulent channels, Ph. D. Thesis, U. Politécnica Madrid. FERNHOLZ, H.H., & FINLEY, P.J. 1996 The incompressible zero-pressure-gradient turbulent boundary layer: an assessment of the data. Prog. Aerospace Sci. 32, 245–311. HAMILTON, J.M., KIM, J. & WALEFFE, F. 1995 Regeneration mechanisms of nearwall turbulence structures. J. Fluid Mech. 287, 317–348. HITES, M.H. 1997 Scaling of high-Reynolds number turbulent boundary layers in the National Diagnostic Facility, Ph. D. Thesis, Illinois Inst. of Technology. HOLMES, P., LUMLEY, J.L. & BERKOOZ, G. 1996 Turbulence, coherent structures, dynamical systems and symmetry, Cambridge Univ. Press.

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JEONG, J., HUSSAIN, F.. SCHOPPA, W. & KIM, J. 1997 Coherent structures near the wall in a turbulent channel flow. J. Fluid Mech. 332, 185–214. JIMÉNEZ, J. 1994 On the structure and control of near-wall turbulence. Phys. Fluids 6, 944–953. JIMÉNEZ, J. & MOIN, P. 1991 The minimal flow unit in near wall turbulence. J. Fluid Mech. 225, 221–240. JIMÉNEZ, J. & PINELLI, A. 1999 The autonomous cycle of near wall turbulence, J. Fluid Mech. 389, 335–359. JIMÉNEZ, J. & SIMENS, M.P. 2001 Low-dimensional dynamics in a turbulent, wall, J. Fluid Mech., 435, 81–91. KIM, H.T., KLINE, S.J. & REYNOLDS, W.C. 1971 The production of turbulence near a smooth wall in a turbulent boundary layers. J. Fluid Mech. 50, 133-160. KIM, J., MOIN, P. & MOSER, R.D. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166. LANDAHL, M.T. & MOLLO–CHRISTENSEN, E. 1992 Turbulence and random processes in fluid mechanics, Cambridge Univ. Press. LEE, M.J., KIM, J. & MOIN, P. 1990 Structure of turbulence at high shear rate, J. Fluid Mech. 216, 561–583. MOSER, R.D., KIM, J. & MANSOUR, N.N. 1999 Direct numerical simulation of a turbulent channel flow up to Phys. Fluids 11, 943–945. ORLANDI, P. & JIMÉNEZ, J. 1994 On the generation of turbulent wall friction. Phys. Fluids 6, 634–641. öSTERLUND, J.M., JOHANSSON, A.V., NAGIB, H.M. & HITES, M.H. 2000 A note on the overlap region of turbulent boundary layers, Phys. Fluids 12, 1-4. ÖSTERLUND, J.M., JOHANSSON, A.V., NAGIB, H.M. & HITES, M.H. 2000 Spectral characteristics of the overlap region in turbulent boundary layers. Extended abstract to ICTAM 2000, Chicago, August 2000. ROBINSON, S.K. 1991 Coherent motions in the turbulent boundary layer. Ann. Rev. Fluid Mech. 23, 601–639. SMITH, C.R. & METZLER, S.P. 1983 The characteristics of low speed streaks in the near wall region of a turbulent boundary layer. J. Fluid Mech. 129, 27-54. SWEARINGEN, J.D. & BLACKWELDER, R.F. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255–290. TOH, S. & ITANO, T. 2001 On the regeneration mechanism of turbulence in the channel flow – role of the traveling-wave solution. In Proc. IUTAM Symp. on Geometry and Statistics of Turbulence (ed. T. Kambe, T. Nakano & T. Miyauchi), pp. 305–310. Kluwer. TOWNSEND, A.A. 1976 The structure of turbulent shear flow. Cambridge U. Press, second edition, pg. 135. WALEFFE, F. 1997 On a self-sustaining process in shear flows, Phys. Fluids 9, 883–900. WALEFFE, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Letters 81, 4140–4143.

Some characteristics of the coherent structures in turbulent boundary layers Davide POGGI Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino ITALY [email protected]

Abstract The present work describes the results of an experimental study about

the kinematics of the coherent structures in an open channel flow. The coherent events, detected in the near-wall region, are analysed by an original method based on conditional-sampling. Using simple kinematic characteristics occurring during the bursting events the coherent structures are grouped into several clusters obtaining a substantial improvement over the classical conditional average. In this way different kinds of coherent structures are distinguished, and probably ascribed to different phases of the same group of coherent structures. In particular, we show experimental evidence supporting the hypothesis about coherent packets of hairpin vortices formulated by Zhou et al. 1999 even at high Reynolds number. This analysis of the near-wall region is realised by means of a new experimental technique which allows an accurate measurement of the streamwise and wall-normal turbulent velocity components ( and respectively) in the flow field very close to the wall.

1.

Introduction

It is well known that the near-wall region of turbulent boundary layers shows a complex temporal-spatial organisation dominated by streaky structures (e.g. Robinson 1991, Jiménez & Pinelli 1999). Even if the scenario is composed of a large number of different elements (low-speed streaks, vortical structure, ejections, sweeps, streamwise vorticity), it is generally accepted that there exists an overall structure that comprises all of them. Although different scenarios have been proposed (Smith et al. 1991, Jeong et al. 1997, Zhou et al. 1999), it seems that in the near-wall region a fundamental role is played by the so-called “bursting cycle” which is responsible for most of the turbulence production. The core of the cycle consists of low-speed streaks that intermittently become unstable, break down and erupt into the outer region. Nevertheless, several studies have suggested that the previous scenario of the bursting cycle may have even more complex evolution (Smith and Walker, 1997;

241 K. Bajer and H. K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 241–246. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Zhou et al., 1999). In particular, according to Zhou et al. (1999) and Adrian et al. (2000), hairpin vortices very frequently occur in groups and the individuals within the groups propagate coherently forming a packet of hairpin vortices. From an experimental point of view, a significant body of quantitative information on coherent structures has been gathered by various conditional sampling methods, even though several studies (e.g. Yuan and Morhtarzadeh, 1994) have highlighted the great difficulty of the classical conditional methods to detect unambiguously the bursting events. In particular serious doubts have been raised on the ability of the classical methods to distinguish events of different nature, or different phases of the same event. In this work we stress the necessity of discerning the different events of the bursting cycle and of grouping these events into clusters with different kinematic characteristics. Thanks to this preliminary analysis, we show that a more accurate and reliable conditional analysis is possible showing a lower sensitivity to phase alignment. The resulting method is able to extract from the signal a principal and a secondary peak of velocity. We argue that the secondary peak and its different position from the primary one is connected to the passage of a train of vortices. The present method is capable of analysing the succession of primary and secondary vortices even at high Reynolds numbers where the numerical and PIV methods fail.

2.

Experimental facilities and test

The near-wall turbulent velocity measurements have been performed in the closed circuit schematically shown in Figure 1. The measurement technique has been thoroughly documented previously (Poggi et al. 2002); only the key aspects will be repeated here. The main part of the circuit, which uses water as a working fluid, is a rectangular channel, 9 m long, 20 cm high and 30 cm wide. The channel walls, made of glass, allow the passage of laser light on one side of the channel and the collection of the scattered light on the other side. The glass bottom of the channel is covered with a 2 cm plexiglass sheet, having a very narrow cut in correspondence of the lower vertical laser beam. Thanks to the bottom slot filled with water, all the four beams cross only the vertical glass wall, following the same optical path. This makes it possible to acquire reliably the vertical component of velocity in the zone closest to the wall (see Figure 1).

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The measurements were performed with a two-component Dantec LDA system used in forward scatter mode. The dimensions of the measurement volume are estimated to be 0.090, 0.090 and 0.80 in the stream-wise, wall-normal and span-wise directions, respectively. No measurement correction or preliminary analysis was needed and a high data rate was obtained even very close to the wall (the data rate was always above 900-1000 Hz also in the worst conditions). No artificial seeding of the channel water was employed, and care was taken that the variation of the fluid temperature during each series of measurements was less than 0.5°C. The flow field has been measured in flow with Reynolds number, (Re is defined as with U bulk mean velocity obtained by integration of the average velocity profile, and the water depth).

3.

Clustering and conditional sampling analysis

A particular conditional-sampling quadrant analysis with threshold H (Lu & Willmarth 1973) is used to detected the burst events. According to the quadrant method, the ejection events of the bursting process are associated with large instantaneous values of Reynolds stress in the second quadrant, while the sweep phases are associated with instantaneous Reynolds stress of large amplitude in the fourth quadrant. The threshold on the plane that is the parameter that defines the significant events, has been evaluated by modifying the method proposed by Nakagawa & Nezu 1981 to account for the dependence from the wall distance (Poggi et al. 2001). Although the quadrant method has a high

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probability of detecting the visual ejection and a low probability of false ejections, it may also be overly simplistic (Smith & Walker 1997). In fact, while it is possible to say that an ejection and a sweep take place with a large contribution to the second and the fourth quadrant respectively, experimental results show that many regions of the second and the fourth quadrant motion are not clearly connected with the bursting events. This suggests that some of the inconsistency between various conditional sampling methods (Yuan & Mokhtarzadeh 1994) could be associated with a possible confusion between the ejection events above described and the more generic events in the bursting cycles.

In order to overcome such difficulty, in this work we try to select the ejections and the sweeps using some kinematic characteristics of these events. According to the classical point of view, the ejections are characterised by a rapid deceleration of the streamwise velocity turbulent component at the leading edge followed by a dramatic positive velocity gradient at the trailing edge, while the normal velocity fluctuation shows a strong negative gradient after a gradual acceleration. From this description of the events, it is clear that the most important clues of bursting events are present in the amplitude of the velocity fluctuations ( and ) and in the velocity gradient and Therefore, in order to describe more accurately the bursting events, it is convenient to group events into specific clusters characterised by both the same sign of the velocity fluctuation and the same sign of the maximum absolute velocity gradient. In this way, the ensemble-average velocity can be computed by considering only the events belonging to each cluster. The clustered conditionally-averaged behaviours (clustered with the maximum absolute velocity gradient, and normalised with the friction

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velocity and aligned at the maximum gradient) of the streamwise turbulent velocity components during the detected bursting events are plotted, as continuous lines in figures 2, for Reynolds number and distance from the wall The typical patterns of are observed with the ejection events taking the form of a gradual deceleration followed by a rapid acceleration. When the events are grouped in suitable clusters, the disagreement between different phase alignments are very small, thus avoiding the low reliability of the phasing in the classical conditional sampling (Yuan & Mokhtarzadeh 1994). Figure 2 (a) shows the average of the streamwise velocity belonging to the same cluster and aligned with respect to the maximum positive velocity gradient (continuous line) and the maximum negative one (dashed line). In spite of the different phase alignment the average shape is very similar. However, the ensemble-average velocity made with the same phase alignment but of events correspondent to different clusters shows deeper differences (Figure 2(b)).

While the reliability of the clustering in the phase alignment is evident the physical explanation of this improvement is more complicated. In fact, it is difficult to distinguish if the method allows to group the events of different nature or the event belonging to different phase of the same event. In order to answer this question let us pay particular attention to the behaviour of the conditional-average. The presence of a secondary peak occurs later than the principal peak when the maximum positive velocity gradient is used to group the events and earlier than the peak when the negative one is used. The events that collaborate to form the secondary peak are correlated with the primary peak. We can argue that these peaks are probably connected with the passage of different

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vortices belonging to the same packet, and this could confirm the coherence between the primary and the secondary peaks. In particular, while figure 3(a) shows the detection of a primary hairpin vortex followed by a secondary vortex, figure 3(b) presents a probable downstream hairpin vortex follows by the primary one. This scenario, in agreement with that found by Zhou et al (1999) and Adrian et al. (2000), supports the hypothesis that the method is able to discern the events belonging to different phases of the same event. It is important to remark that the secondary peaks are absent in the traditional conditional average. This explains the unreliability of the single grouping used in the classical methods because of the noncorrelation between the events of different clusters.

References ADRIAN, R. J., MEINHART, C. D. AND TOMKINS, C. D. (2000). Vortex organisation in the outer region of the turbulent boundary layer. J. Fluid Mech., 422, 1-54. JEONG, J., HUSSAIN, F., SCHOPPA, W. AND KIM, J. 1997 Coherent structure near the wall in a turbulent channel flow. J Fluid Mech. 332, 185-214. JIMÉNEZ, J. & PINELLI, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 225, 213-241. Lu, S. S. & WILLMARTH, W. W. 1973 Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J Fluid Mech. 60, 481-511. NAKAGAWA, H. & NEZU, I. 1981 Structures of space-time correlations of bursting phenomena on open-channel flow. J Fluid Mech. 104, 1-22. POGGI, D., PORPORATO, A. AND RIDOLFI, L. An experimental contribution to near wall measurements by means of a special laser Doppler anemometry technique. Exp. Fluids. 32, 366-375. POGGI, D.; PORPORATO, A.; RIDOLFI, L. Experimental investigation of the near-wall flow field during bursting events, in Ed World Scientific Publishing, Singapore In press, FMTM2001, December 4-6, 2001, Tokyo, Japan. ROBINSON, S. K. 1991 Coherent motion in the turbulent boundary layer. Ann. Rev. Fluid Mech. 23, 601-622. SMITH, C. R. & WALKER, J. D. A. 1997 Sustaining mechanisms of turbulent boundary layers: The role of vortex development and interactions. Panton 13-47. ZHOU, J., ADRIAN, R. J., BALACHANDAR, S. AND KENDALL, T. M. 1999 Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech., 387, 353-396. YUAN, Y. M. & MOKHTARZADEH-DEHGHAN M. R. 1994 A comparison of conditionalsampling methods used to detect coherent structures in turbulent boundary layers. Phys. Fluids 6, 2038-2057.

A singularity-free model of the local velocity gradient and acceleration gradient structure of turbulent flow Brian CANTWELL Department of Aeronautics and Astronautics, Stanford University cantwell@ stanford.edu

Abstract Research on the fine scale structure of turbulence has led to a greatly improved understanding of the basic geometry of the local flow patterns associated with kinetic energy dissipation. One model of the local flow that has been considered previously is based on a simplification of the transport equation for the velocity gradient tensor called the Restricted Euler Equation. This equation is exactly solvable and, although the solution reproduces many of the geometrical features observed in direct numerical simulations of turbulence, the solution also exhibits a finite time singularity. It is well known that the velocity and acceleration gradients in free turbulent flows actually decrease continuously with time when measured by a Lagrangian observer. For example, an observer convecting with the flow on the dividing streamline of an ensemble averaged turbulent plane mixing layer would measure large scale gradients that decrease in proportion to 1/time and microscale gradients that decrease like (Cantwell 1981). The power law in time associated with this decay can generally be estimated using dimensional analysis together with classical balances relating turbulent kinetic energy production and dissipation. This paper will describe a procedure for removing the singularity in the Restricted Euler model while maintaining the convenience of an exact solution. The resulting system is useful for generating large ensembles for statistical modelling. The new model is matched to decay rates derived from dimensional analysis and accurately predicts many of the geometrical features of both the velocity and acceleration gradient tensors. Probability density functions for both gradient fields generated by the model are compared with results from direct numerical simulation.

1.

Introduction

Once the Reynolds number of a viscous flow is large enough to produce instability and once the amplitude of the instability is large enough to produce turbulence then further amplification ceases and the overall behaviour of the flow tends to be independent of the viscosity. Define – –

Integral velocity scale characterising the overall motion Integral length scale characterising the overall motion 247

K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 247–260. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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From a wide variety of experiments it is observed that the intensity of turbulence scales with the characteristic integral velocity of the flow, independent of If the viscosity is decreased keeping everything else the same the rms turbulent velocity fluctuations would not be expected to change. The spectrum of turbulent fluctuations broadens as the Reynolds number is increased; the range of scales increases, but stays about the same and the size of the largest scale eddies stays about the same; the scale of the flow, tends to be independent of

2.

One-parameter flows Consider the two-parameter dilation group of the Euler equations.

where and are arbitrary. Note that we have invoked Reynolds number invariance in writing down the group (1) where is stretched by the square of the factor used to stretch Furthermore all three coordinate directions are stretched by the same factor. If we act on the Reynolds equations for the ensemble-averaged flow using this group the result is

The point of this is that when we remove the viscous stress term from the Reynolds equations and assume that fluctuating velocities scale with the mean, the result is a system which is invariant under the two parameter dilation group of the Euler equations rather than just the one-parameter group of the full viscous equations obtained by setting in (1). One-parameter flows are (usually turbulent) shear flows in open domains governed by a single global parameter with units

Usually M is an integral invariant determined by the forces which create the flow. Following Cantwell (1981) we can use invariance under the group (1) to develop a general set of similarity rules for characterising the space-time evolution of ensemble-averaged, one-parameter flows. This is accomplished by solving the characteristic equations of (1). These are

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with integrals

where the time-dependent length and velocity scales are

The group parameter, parameter M.

is determined by the units of the governing

These rules can be used to determine the temporal or spatial evolution of the flow Reynolds number

From (8) we can see that if the Reynolds number increases with time and we would expect the range of scales in the flow to increase. If then the Reynolds number decreases with time and there is a tendency for the flow to re-laminarise. If then the Reynolds number is constant.

3.

Fine scale motions

Now let’s turn our attention to the fine scales and see what we can learn about the physics of energy dissipation. This means looking closely at fluctuating strain rates and since, in a turbulent flow, the strain is closely linked to the vorticity one is eventually led to a general study of the behaviour of the velocity gradient tensor. Considerations of the balance between production of turbulent kinetic energy and dissipation can be used to develop estimates for the microscale motions that are responsible for most of the dissipation of turbulent kinetic energy. The Taylor microscale, scales as

According to (9) there is always some eddying motion in the flow with a characteristic length that varies like and is independent of the governing parameter M. Note that the velocity gradients of the large scale motion vary as

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which is independent of M and In a sense the large scale gradients constitute a clock which can be used to date the evolution of the flow. We can define the Kolmogorov velocity and length scales as those motions that constitute the lower limit for instability; motions with a characteristic Reynolds number of order one. This leads to the classical estimates of the Kolmogorov velocity and length scales,

and

In a sense the Taylor and Kolmogorov microscales bracket the range of scales that can contribute significantly to kinetic energy dissipation in the flow. At scales larger than the Taylor microscale the turbulent motion is considered to be essentially inviscid. At the smallest scale are the Kolmogorov microscales with a local Reynolds number of order one. The fine scale gradients over the whole range of dissipating motions vary according to,

4.

The inertial subrange

We can derive one of Kolmogorov’s (1941) most famous results using purely dimensional reasoning and the similarity rules worked out earlier. Let’s accept Kolmogorov’s basic tenet and assume that a range of scales exists where the turbulent motion is independent of both and M and is governed only by the volumetric rate of kinetic energy dissipation. We can think of the inertial subrange as a kind of universal one-parameter flow governed by where

is the turbulent kinetic energy dissipation with units and exponent The temporal evolution of the characteristic scales of the inertial subrange should follow the similarity rules in (6),

The value implies very strong local forcing of the flow, typically much stronger than the forcing in most common situations. For example, to produce at the largest scale of a jet one would need to apply


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a force which increased in proportion to the fourth power of the time. We can use (15) to establish scaling laws for the turbulent kinetic energy spectrum. Assume a range of scales exists which is of order Ask: how is the kinetic energy distributed among various scales? Let be the wave-number The kinetic energy per unit wave-number at a given wave-number can be related to the time as follows.

Using (16), solving for the time in (15a) and substituting the result into (17) produces the classical result first postulated by Kolmogorov.

The behaviour has been more or less confirmed in a wide variety of high Reynolds number experiments and so the arguments of Kolmogorov and the postulated existence of the inertial subrange are generally accepted to be correct.

5.

The geometry of dissipating fine scale motion

Now let’s turn our attention to a physical picture of these small scale motions. We first develop the transport equation for the velocity gradient tensor by taking the gradient of the Navier-Stokes equations. When the Poisson equation for the pressure is subtracted the result is

where

is the trace-free part of the acceleration gradient tensor In the remaining sections and will be the main objects of interest. Note that we are working in Cartesian coordinates throughout this paper. But before we attempt to do anything with the system of equations (19) it is natural to ask what the Cartesian tensors and look like in a turbulent flow. For this we will turn to direct numerical simulation data. The procedure begins with a numerically-generated time-dependent flow simulation. At a given instant, the velocity gradient tensor, is evaluated at every grid point in the computational space. The first invariant,

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due to incompressibility. This leaves the second and third invariants as the two scalars that determine the local flow pattern at each point. The character of the eigenvalues is determined by the cubic discriminant,

Also of interest are the nonzero invariants of the strain and rotation tensors.

The pdfs of the invariants contain a great deal of information concerning the geometry of the fine scales. Points near the origin correspond to low gradient values associated with the large scale motions; points far away characterise the high gradient fine scales. There is a general tendency for the (Q, R) pdf to develop a roughly elliptical shape with major axis of the ellipse aligned with the upper left and lower right quadrants. In fact the shape is really more like an inclined teardrop with the point of the teardrop lying along the branch. The strongest energy dissipating motions in the flow have a saddle-saddle unstable node geometry. This implies that the eigenvalues of the rate-of-strain tensor are ordered according to Points of high dissipation are generally characterised by high levels of vorticity, although there is usually a fairly broad distribution about a 45° line in this space. At any point one can construct a locally orthogonal system of coordinates from the eigenvectors of the rate-of-strain tensor. When the vorticity vector is located relative to this system one finds, in a region of high dissipation a strong tendency for the vorticity to be aligned with the direction of the smaller of the two positive rate-of-strain eigenvectors. What about the acceleration gradient tensor, Cheng (1996) carried out a detailed study of this tensor using low-Reynolds number computations of homogeneous and isotropic turbulence as well as the wake computations of Sondergaard et al. (1996). Figure 1 shows the invariants of computed by Cheng (1996). It appears that looks a lot like but with a change in sign, When the data is conditioned on higher and higher rates of dissipation, the invariants of tend to gather closer and closer to the branch and for points within 75% of the maximum kinetic energy dissipation the data literally hugs the line. It appears that in regions of high dissipation, is a good approximation. Note that the figures are logarithmic contour plots. In regions where the data is sparse, the contour


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plotting algorithm will surround one or a few data points and the plots take on the appearance of scatter plots. In the higher Reynolds number wake simulations of Sondergaard shown below in Figure 2 the strain invariants of are seen to lie closer and closer to the origin as the data is conditioned on higher and higher rates of kinetic energy dissipation. In contrast the strain invariants of always lie far from the origin in regions of high dissipation. Although the observations described above

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are for two specific flows, a variety of flow cases have been studied, and the basic geometry of both tensors in a region of high dissipation tends to be the same for all flows; they are universal.

Now we will attempt to develop a universal model of the small scales that reproduces the geometry of and We begin with (19) and (20). The tensor describes the effect of viscous diffusion and anisotropic pressure forces on the evolution of the velocity gradient tensor. It is instructive to look at the solution of the homogeneous case In this case equation (19) becomes a set of quadratically coupled, nonlinear


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ordinary differential equations for the nine components of the velocity gradient tensor called the restricted Euler equation. The cubic discriminant, is conserved for all particles under this approximation. With this integral of the motion known, the time evolution of the two invariants can be determined in terms of elliptic functions. Once is known the complete system can be solved exactly for The discriminant defines an appropriate time scale for normalising all the variables in the problem, Let

For any initial condition, the solution evolves to,

where

satisfies the following matrix relationship

The restricted Euler system has the property that solutions become singular in finite time. To remedy this let’s return to (19). Let

Equation (19) becomes,

Here is the crucial step; assume that,

and divide (29) through by This transforms the full equation (51) to the restricted Euler but with the flexibility to define a new time. The equation becomes,

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Now we define the new time,

This enables us to remove the singularity and continue to exploit the exact solution of the restricted Euler equation. In this model, the geometry of the velocity gradient field is taken to be determined by the restricted Euler solution and is chosen to match the time evolution of the microscale velocity gradients of a specified flow. Given that, the question is: what behaviour of the acceleration gradient tensor is implied by (30) and is that behaviour consistent with simulations? The discriminant is used again to define a normalising time,

and let

The asymptotic behaviour of the solution is

According to our previous discussion of fine scales the physical velocity gradient tensor should behave according to (13), Let’s require that the asymptotic behaviour of the physical velocity gradient tensor is,

Now using,

we can write,

or using (32),


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Integrating (39) enables one to express For

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in terms of physical time

For

Now the sought after function

is for

and for

For any value of

The physical gradient tensors are related by

The quantity

plays the role of an initial Reynolds number. In terms of this parameter the model becomes

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This completes the model. To develop an ensemble one specifies and an ensemble of random initial values of subject to the continuity equation.

6.

Concluding remarks Note that for all

and for large

This leads to the properties of observed in the simulations described above. Figure 3 shows the development of from an initially random ensemble. Figure 4 illustrates the tendency for to lie along the branch and to have relatively small values of the invariants in regions of high dissipation. In summary the model embodied in (30), (42) and (43) culminating in (47) reproduces many of the geometrical features of the velocity gradient tensor and the trace free part of the acceleration gradient tensor observed in direct numerical simulations.

References CANTWELL, B. J. 1981 Organized motion in turbulent flow. Ann. Rev. Fluid Mech. 13, 457-515. CHENG, W. P. 1996 Study of the velocity gradient tensor in turbulent flow. Stanford University Joint Institute for Aeronautics and Acoustics Report TR 114. KOLMOGOROV, A. N. 1941 The local structure of turbulence in incompressible flow for very large Reynolds number. C. R. Acad. Sci. U.R.S.S. 30, 301. SONDERGAARD, R., CANTWELL, B. J. & MANSOUR, X. 1996. The effect of initial conditions on the structure and topology of temporally evolving wakes. Stanford University Joint Institute for Aeronautics and Acoustics Report TR 118. SORIA, J., SONDERGAARD, R., CANTWELL, B. J., CHONG, M. S. & PERRY, A. E. 1994 A study of the fine-scale motions of incompressible time-developing mixing layers. Phys. Fluids, 6 (2), Pt. 2, 871–883.

Spiral small-scale structures in compressible turbulent flows Thomas GOMEZ 1 , Hélène POLITANO 2 Annick POUQUET 3 & Michèle LARCHEVÊQUE 1 1

LMM, Université Pierre & Marie Curie, 8 rue du Capitame Scott, 75015 PARIS, FRANCE [email protected]

2

Observatoire de la Côte d’Azur, Nice, B.P. 4229, 06304 NICE, FRANCE 3 ASP/NCAR, P. O. Box 3000, Boulder, Colorado 80307–3000, USA Abstract We extend the spiral vortex solution of Lundgren 1982 to compressible turbulent flows following a perfect gas law. Lundgren’s model links the dynamical and spectral properties of incompressible flows, providing a Kolmogorov spectrum. A similar compressible spatio-temporal transformation is now derived, reducing the dynamics of three-dimensional (3D) vortices stretched by an axisymmetric incompressible strain into a 2D compressible vortex dynamics. It enables to write the 3D spectra of the incompressible and compressible square velocities and in terms of, respectively, the 2D spectra of the enstrophy and of the square velocity divergence, by use of a temporal integration (Gomez & al. 2001). New numerical results are presented now using gridpoints; initially, the Mach number is 0.32, with local values up to 0.9, the Reynolds number is 1,400, and A inertial behaviour is seen to result from the dynamical evolution for both the compressible and incompressible three-dimensional kinetic energy spectra.

I’ll try to put it in words: You may think I’m away with the birds! Turb’lence compressible Gives spectrum of decibel K to the minus five-thirds.

1.

Introduction

To study turbulence through the dynamics of the small scale structures which develop and of their spectral counterpart, Lundgren (op. cit.) introduced a model based on the intermittent fine scales of incompressible turbulent flows thought as consisting in a collection of uncorrelated stretched spiral vortices, randomly oriented in space and individu-


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ally subject to an axisymmetric irrotational straining field produced by larger scales. The basic small scale structures are assumed to be created by large scale processes, excluded in the dynamics of the model, like Kelvin-Helmholtz instabilities or vortex interaction mechanisms. These processes would produce a given quantity N of vortex length per unit time and unit volume, constant for a stationary turbulence. This model is actually the only one which provides analytically the famous spectrum (Kolmogorov 1941) from the Navier-Stokes equations, although this kind of approach had been introduced already in the Townsend model (Townsend 1951). The central point of these vortexbased models is the use of a spatial set of small scale structures randomly oriented, which are taken as local solutions of the Navier-Stokes equations. The model due to Townsend predicts a scaling law for the energy spectrum for the axisymmetric Burgers vortex and a law for the plane Burgers layer for small scales Obviously, the Kolmogorov exponent (–5/3) lies between these values. This suggests that, in order to obtain the law, the vorticity field might be composed of a mixture of both tube-like and sheet-like structures. It can thus be expected that any Navier-Stokes solution which includes the roll-up of fine vorticity gradients in a strain field may produce a scaling law as noted by Gilbert (Gilbert 1993).

2.

The compressible Lundgren transformation

Like in the Lundgren’s problem, we consider a problem in which the fluid is strained by an axisymmetric flow of the form

with and where is the uniform strain rate of the external field. We are looking for a solution in the vicinity of the in which there is only a component of vorticity, with all the variables independent of The compressible Navier-Stokes equations for the vorticity and the velocity divergence with a constant dynamic viscosity are written in (Gomez & al. 2001). In order to reduce the three-dimensional dynamics to a two-dimensional one, the change of variables in space and time is defined, similarly to the incompressible case, as:

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and, in the compressible case, the spatio-temporal transformation between the three- and two-dimensional fields take the form

where the subscript 2 denotes the two-dimensional flow, is the mass density, the internal energy per unit volume and the pressure. The variables are the stretched variables in space and time, corresponding to a purely two-dimensional evolution.

3.

The compressible spectrum

We consider a stationary and homogeneous three-dimensional compressible turbulence, and assume that the velocity divergence is concentrated in the vicinity of the vortex filaments which are themselves isolated and created at a constant rate with the same structure and the same strength. We can thus invoke the ergodic hypothesis as in Lundgren to transform a space integration into a time integration on the temporal evolution of a single structure. The compressible velocity spectrum is thus expressed as

where is a characteristic time of interaction between the structures of the velocity divergence and of the vorticity, the filament length and F the velocity divergence spectrum obtained by integration in spherical shells of radius in wavenumbers space (Gomez & al. 2001).

4.

Numerical experiments

We consider a medium of characteristic length of mean density and mean velocity used to normalise to unity density and velocity. The internal energy is normalised by and is related to the temperature as usual by with the non-dimensionalised constant parameter the initial Mach number is where the speed of sound is defined as with the adiabatic index and R the perfect-gas constant. The normalisation temperature

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on which is based is set to ensure that the temperature variable is of order unity. Furthermore, we take the eddy turn-over time as the dependent unit. In the code, the equations for conservation of mass, momentum and internal energy are written in the following nondimensionalised form:

where and D are respectively the stress tensor and the strain tensor. The two dimensionless parameters that arise are the Reynolds number and the Prandtl number where is the constant thermal conductivity, the kinematic viscosity and the thermal diffusivity. The perfect gas law is written as

4.1

Two-dimensional dynamical evolutions We now present numer-

ical results for flows with an initial ratio of compressible to incompressible energies This ratio is stabilised around 0.03 at The Mach number, initially equal to 0.32, decreases very slowly (by 5% at the final time However, the maximum of the local Mach number, initially equal to 0.9, decreases steadily to stabilise around 0.55 at a time Note that it takes a sound wave a time to cross the computational box. The intense fine compressible structures are weak shocks which interact strongly with the spiral branches of the vortex during a characteristic time comparable to One can observe these interactions on the isocontours of the square velocity (Fig. 1); note as well the persistence of the spiral structure of the solenoidal component of the velocity, even though the maximum initial Mach number is close to unity. This is due to the fact that the energetic structures of the compressible velocity are at scales smaller than the characteristic scale of the spiral arms. In fact, the most intense shocks are localised along the branches of the spiral vortex. Moreover, these compressible structures locally, in the vicinity of the spiral arms, are perpendicular to the arms, leading globally to what can be called an “ortho-spiral” structure; the divergence field is then carried along in the global rotation of the flow enticed by the strong vortex; hence, the complex structure for

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4.2 Three-dimensional spectral properties In this section, we compute numerically the three-dimensional spectrum of the compressible square velocity from the temporal evolution of the two-dimensional spectra of the square velocity divergence according to relation (9), as well as its incompressible counterpart from the two-dimensional enstrophy spectra( Lundgren 1982). The temporal integration is performed up to a time chosen according to the dynamical evolution of the two-dimensional flow. For the compressible part of the flow, we choose to take into account the interactions between the shocks and the vortex structure. In the model, after this time, the compressible structures, which have interacted with the spiral vortex, leave the influence domain of the local vortex; they travel away while weakening because of dissipation until another intense spiral vortex is encountered which reenergise them again. During this travel between intense vortices, the compressible fluctuations are assumed to have irrelevant contributions to the three-dimensional compressible kinetic spectrum.

5.

Conclusion

Our results display a inertial range simultaneously for compressible and incompressible velocity modes (Fig. 2). Moreover, we point out that kinetic energy may have a better Kolmogorov scaling range than the square velocities when density gradients become strong. Indeed such spectral scaling laws are observed only for velocity power spectra in

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three-dimensional numerical simulations (Porter & al. 1998), for which it should be interesting to perform kinetic energy spectra for comparison. These differences could be thus explain by the density spectrum which should change when r.m.s. Mach number is increased.

References PORTER, D., POUQUET, A. & WOODWARD, P. 1998 Phys. Fluids 10, 237–245. LUNDGREN, T. S. 1982 Phys. Fluids 25, 2193–2203. GOMEZ, TH., POLITANO, H., POUQUET, A. & LARCHEVÊQUE, M. 2001 Phys. Fluids 13, 2065–2075. TOWNSEND, A. A. 1951 Proc. R. Soc. Land. A 208, 534–542. KOLMOGOROV A. 1941 Dokl. Akad. Nauk. SSSR 30, 9–13. GILBERT A. 1993 Phys. Fluids A 5(11), 2831–2834.

V

FINITE-TIME SINGULARITY PROBLEMS


Discrete groups, symmetric flows and hydrodynamic blowup Richard B. PELZ Rutgers University, Department of Mechanical and Aerospace Engineering, 98 Brett Road, Piscataway, NJ 08854-8058, USA [email protected]

Abstract Discrete group theory is applied to certain well-known flows with discrete

symmetries. A flow is associated with particular discrete point or space group if flow components possess the same symmetries as the irreducible representations of the group. The flows examined are or have been considered candidates for finite-time blowup. The properties of the groups are related to the characteristics of the flows. Singularity? Yes, I conceive it, The problem is how to achieve it; With array octahedral Like a modern cathedral – When you see it, you better believe it!

1.

Introduction

This work is motivated by the observation that most of the incompressible flows that have been suggested as candidates for spontaneous generation of finite-time singularities possess some discrete symmetries. The theory of discrete point and space groups, which has been useful in categorising and predicting properties of molecules and crystals, is introduced here with the goal of doing the same for symmetric flows. A flow will be defined as “being associated with” a discrete group if the flow variables, the components of velocity and vorticity and pressure, can be related to the group’s irreducible representations, a term defined in the next section. The properties of discrete groups and associated flows, may be correlated with finite-time collapse in the flows. It may help answer the question of whether flow solutions have global regularity or blowup in finite time, which is an open question for solutions of the Navier Stokes and Euler for incompressible flows.


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The properties of a 3-D discrete point group, such as commutivity and degeneracy, and whether there is a centre of symmetry, have analogs in the associated symmetric flows in such as the structure of nulls and critical points. The properties of a space group whose unit cell is in the point group, may provide a link between periodic flows associated with the space group and the problem. To the author’s knowledge, the association of symmetric flows with discrete groups has not been performed. Making such a connection may represent at least lateral movement in fluid mechanics and in particular on the blowup problem. To begin the connection, the finite set of crystallographic groups will be considered here. There are 32 point groups, 7 lattices and 230 space groups. Since they contain up to six-fold rotations, they represent a subset of the discrete groups that can be realised in a flow. Mathematical books on discrete groups usually mention only briefly the 3-D crystallographic groups. Books on solid-state physics, crystallography, and physical chemistry, such as Kettle 85; Wherrett 86; Mirman 99 are more informative (Vanderbilt 01). The structure of paper is as follows. A brief discussion of groups and representations is given in the Section 2 followed by a short section on blowup solutions. Then, four examples of symmetric flows that have been considered for blowup are examined in Sections 4-7. These flows are: anti-parallel vortex tubes, orthogonal dipoles, the Taylor-Green vortex, and the high-symmetry, octahedral flows. The discrete group associated with each flow is established and the representations related to the flow variables. Present numerical evidence from some of these flows suggests that blowup does not occur; nevertheless, it is useful to establish their link to discrete groups. The last flow, related to the full octahedral group, is one in which evidence contrary to blowup has not been given (yet). Here the tidal / politely decomposition about the collapse point is introduced. The ramifications of the degeneracy are discussed. Chirality and the relation between the point group and the space group is presented.

Discrete groups, symmetric flows and hydrodynamic blowup

2.

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Representation theory of 3-D discrete groups

In this section a short discussion is given on the relevant concepts in discrete group theory. The five symmetry operations are: the identity E, the n-fold rotation about a given axis the reflection about a given plane in space the inversion about a point in space (about the origin, and the improper rotation which is a rotation about an axis composed with a reflection about the plane normal. One can form groups with sets of these operations; the order of the group being the number of operations. Using the closure property of groups, every composition (or multiplication) of operations in a group is also in the group. If the order of composition does not matter, it is a commutative or Abelian group. For each group, a multiplication table can be formed with element being the product of the and operation. Given an nth-order symmetry group, a matrix representation of the group, is a set of corresponding matrices that has the same multiplication table. then Each matrix representation, can be identified by its vector basis, the set of orthonormal vectors which transform the same by and The matrix representations of reduced form are the representations with the smallest block-diagonal form. All the reduced blocks of all the reduced forms form the set of irreducible representations (irreps). The sum of the squares of ranks of these irreps equals The trace of the blocks is called the character. Non-unity characters indicate degeneracy. The group, for example, is a fourth order group with two reflectional symmetries in planes and The elements are E, and is needed because A rectangular pyramid is invariant under the action of these group elements as is a water molecule. With the basis a matrix representation in reduced form is

With the basis

a matrix representation is

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Because all these matrices are diagonal, the irreducible representations can be separated into actions on vectors and alone. The totally symmetric irrep associated with is normally labelled The and irreps are labelled and respectively. Table 2 shows the character table for

A scalar field of which is an odd function of would transform according to the characters of irrep That is, a±l character indicates whether the field is even or odd under the symmetry operation. Such a field “is associated with” the irrep Likewise a field odd in is associated with A field even in and is associated with the fully symmetric irrep a field odd in both and with

3.

Blowup and Symmetry

Is it a coincidence that symmetric flows have been a focus of the search for a blowup solution? The increased resolution offered by such flows may simply allow a larger temporal range to be computed and. hence, more evidence of scaling. Symmetry is, however, a constraint, on the flow which may restrict the solution to follow unstable branches. It is possible that all non-symmetric flows are regular in time, whereas a symmetrically constrained one blows up. Phenomenologically, blowup in an incompressible flow can be visualised when the cross-sectional area of a vortex tube of nonzero circulation becomes zero in finite time. Another scenario is that the curvature of vortex lines becomes infinite. In each case both length and time scales go to zero in finite time. Blowup theorems and constraints for frequency and wavelength are given in Beale, Kato, & Majda 85; Ponce 85; Constantin & Fefferman 93; Constantin 94; Constantin 95; Constantin, Fefferman & Majda 96. All of the candidate flows examined below indicate that vorticity (an inverse time) blows up as it approaches a reflectional symmetry plane.


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Here, as in the rest of the paper, a reflectional symmetry plane will contain a stream surface, that is, the normal velocity is an odd function and the parallel velocities are even with respect to the symmetry plane. The vorticity then is only normal on the plane. One could have a reflectional symmetry with velocity and vorticity fields switched, but there is no axial strain rate on the vorticity in the plane in this situation. The functional form for the divergent vorticity about a symmetry plane is then dipolar; the blowup occurs as a collapse about a vorticity null on the symmetry element. A reflectional symmetry is also a logical way to constrain a vortex dipole to move parallel to the plane. The addition of another reflectional symmetry plane normal to the first which cuts across the dipole makes the dipole track be the line intersecting the two planes. It focuses attention on the motion of level curves of vorticity in the second plane. The dimension of the set of singular points is reduced if blowup is restricted to symmetry planes and rotation axes. Many of the papers cited above give theorems of blowup in the maximum norm, which can be realised even with a set of zero dimension. The Navier-Stokes result of Caffarelli, Kohn & Nirenberg 82, that the 1-D Hausdorff measure for the space-time set of singularities is zero, suggests a spatial point-wise singularity. In this paper, it will be assumed that the blowup set for both Euler and Navier-Stokes solution, if it exists, will be at one point in space. Analysis using discrete groups is particularly useful if the blowup point is at the centre of symmetry. An analogy can be made between a point collapse of vortical structure in a symmetric flow and a molecule invariant under the operations of the associated discrete group. In addition to this isolated collapse in there is a relationship between the unit cell of a lattice or space group and the point group. A collapse in a periodic domain with symmetries can then be formally linked to the isolated one in A spherical coordinate system with the centre of collapse at the origin, and decomposition into toroidal and poloidal components may be advantageous for analysis of point collapse of a symmetric flow. If there is a self-similar collapse, a renormalised variable involving and is likely. If the centre of symmetry is a vorticity null, the collapse would then involve the increasing confinement of the toroidal component of vorticity.

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Anti-parallel vortex collapse and the group

One of the symmetric flows which has received a lot of attention as a candidate flow for finite-time singularity in the Euler equations is the evolution of perturbed, initially oppositely directed vortex tubes. See Siggia 85; Melander & Hussain 88; Kerr & Hussain 89; Pumir & Siggia 90; Kerr 93; Waele & Aarts 94. In many of these studies, vortex tubes are perturbed so that the Crow instability occurs. In the nonlinear evolution, the tubes collapse towards each other. A cartoon of a typical initial arrangement is given in Figure 1.

Many researchers have used the inherent reflectional symmetry of the problem to reduce the computational domain (c.f. Pumir & Siggia 90; Kerr 93). Others who have not, have observed that when the tubes are of the same strength, the flow is attracted to one with reflectional symmetry (c.f. Siggia 85). It was shown with filament simulations that when the separation distance is much larger then the de-singularisation length (or core radius), this separation distance scales as Behaviour of


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filament solutions diverge from that of the Euler system when the vortex tube starts to distort as it approaches the symmetry plane. Researchers have imposed not only the reflection symmetry between the two tubes, but another reflectional symmetry with a plane perpendicular to the first and to the direction of the vorticity, as discussed in Section 3. This plane divides each vortex tube in two and along with the other symmetry makes the fundamental domain be one quarter of the original. Let the former plane be at and the latter plane be The is the intersection and also the direction of induced velocity at the pinch. These two reflectional symmetries generate as discussed in the example of Section 2. Now the last column of the character table, Table 2, may be understood; the entries of a row are the flow components associated with the irrep of that row. It is clear that the velocity component in the direction and the pressure are even with respect to all operations and hence they are associated with the fully symmetric irrep which has only ones as characters. The a odd function of both and is in The group is Abelian and non-degenerate. Since there is only a line of symmetry, the collapse point is not defined, which may be at the root of conflicting numerical evidence on blowup in this flow. Usually, periodic boundary conditions are imposed in at least the and directions, hence, the periodic problem is a 2-D cubic lattice.

5.

Orthogonal Dipoles and the Group

Recently, Moffatt 00 suggested the configuration of two orthogonally aligned vortex dipoles as a candidate for point collapse and blowup. In the frame moving with a 2-D dipole, there are fore and aft hyperbolic points. The manifold orientation of the front critical point produces a positive axial strain rate on the second dipole pair. The second produces a similar strain rate on the first. The strain rate will increase as the dipoles move closer to each other from their induced motion. Figure 2 shows a sketch of a possible orientation of the orthogonal dipoles. Equal-strength dipoles are placed along the diagonals of opposite faces of the cube. (Obviously, vortex lines must close or extend to infinity, and the tube will distort in time.) Let the origin be at the centre the cube and the Cartesian axes extend through the centres of the sides of the cube. The vortex system in Figure 2 is then invariant under 2-fold rotations about the three axes. These three along with the identity form the operations of the dihedral group, whose character

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table is given in Table 2. The flow components in the last column of the table are then associated with the four irreducible representations A, as shown. This point group is non-degenerate and Abelian with a centre of symmetry at the intersection of the three rotation axes, the origin, which is the conjectured collapse point.

Mutual interaction may halt a linear strain rate / vorticity coupling, as was shown in the simulation of Boratav 92. The idea of strain rate experienced by the fundamental coming from an image, strain rate expe-


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rienced by the image coming from the fundamental, and collapse of the image and fundamental towards some centre of symmetry are, however, key to a possible singularity scenario.

6.

Taylor-Green Vortex and the Group

The Taylor-Green vortex (Taylor & Green 37) is a symmetric flow that has long been suspected of developing spontaneous singularities. Due to the finding that the width of the analyticity strip has exponential behaviour, it is now generally believed that this flow remains smooth for all time. See Morf, Orszag & Frisch 80; Brachet, Meiron, Orszag, Nickel, Morf & Frisch 83; Brachet, Meneguzzi, Vincent, Politano, & Sulem 92. The symmetries of the Taylor Green vortex include the zero-planes as reflectional symmetry planes. These three operations are the generators of the non-degenerate, eighth-order, point group – The other five symmetry operations of are the identity E, two-fold symmetries about each axis and the inversion The Taylor Green vortex is a periodic flow. With periodicity and reflectional symmetry imposed at The centre of symmetry is then repeated in a cubic lattice. One more symmetry that TG possesses make the space group more interesting, however. There are three two-fold rotational symmetries about the lines These, as was described in the previous section, generate the point group at the point With the group at the origin and at a more complicated space group is created. The centres are at where is even, and centres are at where Table 3 shows the character table for the group. Shown are the flow components associated with the irreps. The subscripts and stand for gerade and ungerade which mean even or odd with respect to inversion. The radial velocity, which scales as about the origin, is associated with and the radial vorticity, which scales as is associated with The non-degeneracy of the group is linked with the non-degeneracy of the origin as a critical point of the flow. Expansion of velocity about the origin yield nonzero linear terms and manifolds that are the axes. Simulations do not indicate a collapse about either of the centres of symmetry, but they indicate that thin shear layers form along the reflection planes, which indicates two-dimensionalisation of the flow and hence depletion of nonlinearity.

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7.

Octahedral Flows and the Group

A number of studies have produced evidence that particular vortical flows associated with the full octahedral group, may blowup in finite time. See Boratav & Pelz 94; Boratav & Pelz 95a; Boratav & Pelz 95b; Pelz & Boratav 95; Pelz & Gulak 97; Pelz 97; Pelz, Gulak, Greene & Boratav 99; Greene & Pelz 00; Pelz 01; Pelz 02. The first few studies use a highly symmetric flow in originally developed by Kida 85 as an extension of the Taylor-Green vortex for direct simulations of turbulence. He used the TG symmetries but added a four-fold rotational symmetry at the point It was observed in these studies that three orthogonal vortex quadrupoles appeared centred about the origin and accelerated towards the origin, all the time shrinking in scale but not changing in form. This visual self-similar collapse could not be verified quantitatively owing to lack of resolution in the equally-spaced Fourier pseudo-spectral method. A high-precision, extended series of this flow, Pelz & Gulak 97, indicated a blowup of enstrophy at a time similar to the collapse time in the simulations. In Pelz 97 the problem was recast in keeping some of the symmetries, and a filament method was used to explore this problem. Reflections at the zero planes were retained as well as a permutation symmetry: which is a three-fold rotational symmetry about the diagonals. From these simulations it was clear that at least for this model system, a self-similar collapse occurs. Figure 3 shows a late time view of the surface created by the local cores around the twelve vortex filaments. The symmetries are enforced; only half of one filament is the fundamental. Figure 4 shows the behaviour of two length scales: the closest distance of the filament to the origin and the


279

local core radius there, versus time to the estimated blowup time. It is clear that these lengths have a square-root behaviour as suggested by Leray 34.

The three zero-plane reflections together with the three-fold rotations assumed in the problem are the generators of the full octahedral group, This group is the set of 48 rotations and reflections that leave invariant an octahedron. The octahedron is oriented with the centre at the origin and the six vertices on the Cartesian axes. The character table is given in Table 4. There are four-fold symmetries about the three axes and three-fold symmetries about the four diagonals two-fold symmetries about the six diagonals in the zero planes three zero-plane reflections six diagonal-plane reflections the identity, the inverse and 14 improper rotations which make up the 48 elements. This is a degenerate point group as indicated by the non-unitary characters. The degeneracy can be thought of as coming from the permu-

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tation symmetry mentioned above. Only one of the three Cartesian velocity or vorticity components is needed; the symmetry make the other two redundant. The Cartesian velocity and vorticity fields then


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belong to the triply-degenerate and irreps, respectively. The critical point at the origin is then degenerate, with the first non-zero terms at third order. The subscripts and in Table 4 need to be explained. The irreps with subscripts 1 and 2 differ in that 1 is even with respect to reflections about diagonal planes, for example, and 2 is odd. The flow components do not have this symmetry, but can decomposed as an even (sym) plus an odd (anti): The flow components T and S also need to be explained. Following Moffatt 78 the toroidal T / poloidal S decomposition of the vorticity is

where is the unit vector in the radial direction. The scalar fields S and T can be found from the vorticity by

where

is the shell Laplacian. The velocity is

where U is a potential which can be found from solving the following Poisson problem The toroidal vorticity, which is even with respect to zero plane reflections and the diagonal rotations, is associated with the fully symmetric irrep. The poloidal vorticity, which is odd with respect to zero plane reflections, is associated with the ungerade irrep, as shown in Table 4. The problem suggested by Kida can then be seen as a flow about the origin. Instead of the group at as was the case in TG, the four-fold rotational symmetries there are the generators of the octahedral group O. This group has as a symmetry about the diagonals. One of these diagonals intersects the origin and together with the reflections there, are the generators of the group at the origin. Hence, the space group has point groups at points and O point groups at There is a chirality to the collapse of the three orthogonal quadrupoles. The handedness is based on the sign of the poloidal vorticity in the first quadrant. If the origin is positive, the adjacent collapse at is negative. Thus the non-symorphic space group has a further chequerboard pattern in orientation.

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Summary

Discrete group theory has been introduced for flows which possess discrete symmetries. Apart from allowing a classification of these flows, the properties of groups can be related to the behaviour of the flow. A group with a centre of symmetry indicates that in an associated flow a critical point exists there around which a collapse may occur. The degree of degeneracy of the critical point is reflected in the degeneracy of the group. These properties may be useful in showing whether a flow solution has a finite-time singularity through point-wise collapse.

References BEALE, J. T., KATO, T., & MAJDA, A. 1985 Remarks on the breakdown of smooth solutions for 3-D Euler equations. Commun. Math. Phys. 94 61–66. BORATAV, O. N. 1992 Ph.D Thesis, Rutgers University. BORATAV, O. N. & PELZ, R. B. 1994 Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. The Physics of Fluids 6 no. 8, 2757–2784. BORATAV, O. N. & PELZ, R. B. 1995 On the Local Topology Evolution of a HighSymmetry Flow. The Physics of Fluids 7, no. 7, 1712–1731. BORATAV, O. N. & PELZ, R. B. 1995 Locally Isotropic Pressure Hessian in a HighSymmetry Flow. The Physics of Fluids 7, no. 5, 895–897. BRACHET, M. E., MEIRON, D. I., ORSZAG, S. A., NICKEL, B. G., MORF, R. H. & FRISCH, U. 1983 Small-scale Structure of the Taylor-Green Vortex. J. Fluid. Mech. 130 441–452. BRACHET, M. E., MENEGUZZI, M., VINCENT, A., POLITANO, H., & SULEM, P. L. 1992 Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows. Phys. Fluids A 4 12, 2845–2854. CAFFARELLI, L., KOHN, R. & NIRENBERG, L. 1982 Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35 771–831. CONSTANTIN, P. 1994 Geometric statistics in turbulence. SIAM Review 36, 73–98. CONSTANTIN, P. 1995 Nonlinear inviscid incompressible dynamics. Physica D 86, 212–219. CONSTANTIN, P. & FEFFERMAN, C. 1993 Direction of Vorticity and the Problem of Global Regularity for the Navier-Stokes Equations. Indiana Univ. Math. J. 42, 775. CONSTANTIN, P., FEFFERMAN, C. & MAJDA, A. J. 1996 Geometric constraints on potentially singular solutions for the 3-D Euler equations. Commun. Part. Diff. Eqns. 21, 559–571. GREENE, J. M. & PELZ, R. B. 2000 Stability of Postulated Self-Similar, Hydrodynamic Blowup Solutions. Physical Review E 62, 7982–7986. KERR, R. M. 1993 Evidence for a singularity in the three-dimensional, incompressible Euler equations. Phys Fluids A 5 1725–1746.


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KERR, R. M. & HUSSAIN, A.K.M.F. 1989 Simulation of vortex reconnection. Physica D 37 474. KETTLE, S. F. A. 1985 Symmetry and Structure Wiley. KIDA, S. 1985 Three-dimensional periodic flows with high-symmetry. J. Phys. Soc. Jpn. 54, 2132–2140. LERAY, J. 1934 Sur le mouvement d’un liquide visqueux emplissant l’space. Acta Math. 63, 193–248. MELANDER, M. & HUSSAIN, A. K. M. F. 1988 Cut and connect of two antiparallel vortex tubes. Center for Turbulence Research Report S88, 257-286. MIRMAN, R. 1999 Point Groups, Space Groups, Crystals, Molecules World Scientific. MOFFATT, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press. MOFFATT, H. K. 2000 The interaction of skewed vortex pairs: a model for blow-up of the Navier-Stokes equations. J. Fluid Mech. 409, 51–68. MORF, R. H., ORSZAG, S. A. & FRISCH, U. 1980 Spontaneous Singularity in ThreeDimensional Inviscid, Incompressible Flow. Phys. Rev. Letts. 44, no 9, 572–574. PELZ, R. B. 1997 Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Physical Review E, 55, 1617–1620. PELZ, R. B. 2001 Symmetry and the Hydrodynamic Blowup Problem. Journal of Fluid Mechanics, 444, 299–320. PELZ, R .B. 2002 (in press) A Candidate for Hydrodynamics Blowup: Octahedral, Vortical Flow. In Geometry and Topology of Fluid Flows, (ed. R. Ricca) NATOASI. PELZ, R. B. & BORATAV, O. N. 1995 On a Possible Euler Singularity During Transition in a High-Symmetry Flow. In Small-Scale Structures in Three-Dimensional Hydro and Magneto Hydrodynamic Turbulence, (ed. M. Meneguzzi, A. Pouquet & P.L. Sulem), pp. 25–32. Springer. PELZ, R. B. & GULAK, Y. 1997 Evidence for a Real-Time Singularity in Hydrodynamics from Time Series Analysis. Physical Review Letters , Vol. 79, No. 25, 4998–5001. PELZ, R. B., GULAK, Y., GREENE, J. M. & BORATAV, O. N. 1999 On the Finitetime Singularity Problem in Hydrodynamics. In Fundamental Problematic Issues in Turbulence, (ed. A. Gyr, W. Kinzelbach, A. Tsinober), pp. 33–40. Birkhäuser. PONCE, G. 1985 Remarks on a Paper by J.T. Beale, T. Kato and A. Majda. Commun. Math. Phys. 98, 349–353. PUMIR, A. & SIGGIA, E. D. 1990 Collapsing solutions to the 3-D Euler equations. Phys Fluids A 2 220–240. SIGGIA, E. D. 1985 Collapse and amplification of a vortex filament. Phys. Fluids 28 3, 794–805. TAYLOR, G. I. & GREEN, A. E. 1937 Mechanism of the Production of Small Eddies from Large Ones. Proc. Roy. Soc. A 158 499–521. VANDERBILT, D. 2001 private communication. WAELE, A. T. A. M. & AARTS, R. G. K. M. 1994 Route to Vortex Reconnection. Phys. Rev. Letts. 72 no. 4, 482–485. WHERRETT, B. S. 1986 Group Theory for Atoms, Molecules and Solids Prentice Hall International.


Diffusion of Lagrangian invariants in the Navier-Stokes equations Peter CONSTANTIN The University of Chicago, Department of Mathematics Chicago, Il 60637, USA. [email protected]

Abstract The incompressible Euler equations can be written as the active vector system where

is given by the Weber formula

in terms of the gradient of A and the passive field (P is the projector on the divergence-free part.) The initial data is so for short times this is a distortion of the identity map. After a short time one obtains a new and starts again from the identity map, using the new instead of in the Weber formula. The viscous Navier-Stokes equations admit the same representation, with a diffusive back-to-labels map A and a that is no longer passive. I’ll analyse Navier-Stokes, For dynamics of vodkas and cokes; [A] is an entity Near the identity; That’s how I’ll baffle you folks!

1.

Introduction

The classical method of characteristics for partial differential equations (John 1991) allows one to prove the existence of finite time singularities for hyperbolic conservation laws. The classical maximum principle allows one to prove that singularities are absent in certain nonlinear parabolic equations. The equations of incompressible fluids do not fit neatly in either of these classes of PDE. Ideal, frictionless fluids in Eulerian coordinates are a hyperbolic system with non-local additive forcing due to the gradient of pressure. The gradients of velocity are affected by non-local fluctuations in the pressure Hessian, and the classical blow-up proofs do not apply. Viscous, incompressible fluids in Eulerian coor285 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 285–294. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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dinates are a parabolic system with non-local additive forcing, due to the gradient of the pressure. Because of these non-local effects in the accelleration, the Navier-Stokes equations do not have a maximum principle for the velocity. Lagrangian coordinates for fluids correspond to the classical method of characteristics. We present an approach to the fluid equations in which the non-local aspects become multiplicative, rather than additive. Using this approach one can interpret the well-known criterion (Beale, Kato & Majda 1984) for finite-time singularities as a Lagrangian incompressible shock, or splitting of particles. Topological changes in incompressible fluids or plasmas could be explained in one of two possible ways: either by dynamical singularities in the ideal equations, or by viscous effects. In many cases, viscous effects are deemed to be too slow to explain the phenomena. Following ideal singularity formation and subsequent dynamics is beyond the reach of present analytical and numerical tools. In the event of a near singularity in the inviscid fluid, small scales are produced rapidly, and viscous effects can become rapidly essential. In the approach we discuss here (Constantin 2001 (a), Constantin 2001 (b)), the Navier-Stokes equations and their various approximations can be described in terms of near identity transformations. These are diffusive particle path transformations of physical space that start from the identity. The active velocity is obtained from the diffusive path transformation using the Weber formula (Serrin 1959) and a virtual velocity. The active vorticity is computed from the diffusive path transformation using the Cauchy formula (Serrin 1959) and a virtual vorticity. The path transformation and the virtual fields are computed in Eulerian coordinates. In the absence of kinematic viscosity, both the virtual velocity and the virtual vorticity are passively transported by the flow. In the presence of viscosity, these fields obey parabolic diffusion equations with diffusion coefficients that are proportional to the kinematic viscosity. When the viscosity is absent, these equations revert formally to passive transport equations. Apart from being proportional to the viscosity, the coefficients of these diffusion equations involve second derivatives of the near identity transformation and are related to the Christoffel coefficients. The diffusive path transformations are used for short time intervals, as long as the transformations do not stray too much from the identity. The duration of these intervals is determined by the requirement of invertibility of the gradient map. If and when the viscosity-induced change in the Jacobian reaches a pre-assigned level, one stops, and one restarts the calculation from the identity transformation, using as initial virtual field the previously computed active field.

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Lower bounds on the minimum time between two successive resettings are given in terms of the maximum enstrophy.

2.

Ideal Fluids

The classical Lagrangian description (Arnol’d 1984) of fluids uses the map as basic variable. This is the map that assigns to a label the position where a fluid particle arrives at time t if it starts at time from the position The velocity of the particle is The map is a volume preserving diffeomorphism, i.e., it is smooth, invertible and its spatial gradient has determinant equal to one. The inverse map is denoted The Eulerian velocity is The fact that the label a of the position does not change in time is reflected in the equation

The particle labels coincide with the particle position at the initial reference time, The velocity can be computed using the Weber formula (derivable from the Euler equations) Here P is the projector on divergence-free functions, posed gradient matrix and obeys

is the trans-

The solution of this equation is

where

is the initial velocity. In this manner, use of the Weber formula

allows one to compute from explicitly, using a non-local equation of state. The system (1, 4) is an active vector system, is “active” in contradistinction with “passive”; passive vectors solve a linear equation (1) with prescribed independently of A. Active scalars have been discussed in (Constantin 1994).) The vorticity of the fluid obtained by differentiating the Weber formula (2) can be written as

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where

is the sign of the permutation,

and

obeys

The solution of (6) is given by and thus (5) is the classical Cauchy formula. The relation between and is that is differentiation in Lagrangian coordinates. The well known criterion of (Beale, Kato & Majda 1984) states that no singularity occurs in the time interval [0, T] if and only if i.e. if the time integral of the maximum modululus vorticity is finite. Using this criterion and the Cauchy formula above it is easy to prove that no singularity occurs in the solution of the Euler equations if and only if

Indeed, because is bounded, the boundedness of the above integral implies the boundedness of the integral in the Beale-Kato-Majda criterion, so the condition is sufficient for regularity. It is also necesssary, because, if the flow is regular, then one can prove that the solution of (1) is smooth, and consequently, the integral above is finite. Filtering approximations of the Euler equations can be obtained by composing the Weber formula (2) with a filter

The operation is an approximation of the identity that commutes with translation, for instance

with and positive kernel J that is normalized smooth, and decays sufficiently fast at infinity. Two canonical examples of such J are the Poisson and the Gaussian kernels. The filtered active vector equation coupled with and (7) has global smooth solutions for If the solutions of the active vector equation (1, 4) are smooth on some time interval then they are the limit as of the corresponding solutions of the filtered active vector equations. These filtered active vector equations are related to generalized vortex methods (Chorin 1973): the curl of obeys the filtered vorticity equation.


3.

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Viscous Fluids

The considerations above can be generalized and applied to viscous fluids. The Weber formula (2) is retained as is. The inviscid material derivative is replaced in the viscous case by the operator of advection and diffusion

The inviscid active vector equation viscous

is replaced by the

More interesting is the change in the equation for the virtual velocity The inviscid is replaced by

where the coefficients C are computed from the transformation A via

These coefficients are related to the Christoffel coefficients in a straightforward manner. The system (10, 11) together with the Weber formula (2) is equivalent to the Navier-Stokes equation. This means two things. First, if a solution of the Navier-Stokes equations is given, and if one computes A and according to the linear equations (10, 11), then the Weber formula (2) holds. And secondly, if a solution of the nonlinear system (10,11,2) is found, then the velocity given by (2) solves the viscous Navier-Stokes equations. Remarkably, the vorticity satisfies the same Cauchy formula (5) as in the inviscid case. The virtual vorticity is related to the virtual velocity by the same relationship, namely where is given by the same formula as in the inviscid case. There is no need to invert the map A in order to perform this operation, but if one could invert, then the operation would be simply the curl of the virtual velocity in A coordinates. Just as for the virtual velocity, the equation of the virtual vorticity is modified in a non-trivial manner

When one sets then this equation reverts formally to the passive transport equation The diffusive map is no longer automatically invertible for all time, because the inviscid constraint

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P. Constantin

can be violated by diffusion. The Jacobian of the diffusive transformation A obeys

This equation has initital datum identically zero. The coefficients C have dimension of reciprocal of length, so the right hand side of (14) has dimensions of reciprocal time. Recalling the connection with the Christoffel symbol, we remark that regions of high particle path curvature have the potential to trigger rapid changes in the determinant. The viscous Cauchy formula can be interpreted as The time scale for topological change is roughly of the order of Even with small viscosities this time scale can be short if is large. These facts are verified in numerical experiments of vortex reconnection (Ohkitani & Constantin 2002). Filtering approximations of the Navier-Stokes equations can be obtained by applying the same procedure as in the inviscid case. One solves together with and with given by the same formula (7) as in the inviscid case. The coefficients C are computed from A by the same formulas (12) as in the Navier-Stokes case. These equations have global smooth solutions for positive As long as the solution of the Navier-Stokes equations are smooth, they are the limit as of the solutions of the filtered equations. The Cauchy formula holds for the filtered equations. More precisely, one considers the virtual vorticity defined as before by It obeys the same equation (13) as the viscous virtual vorticity, except that instead of one has to use One then considers the curl where The filtered viscous Cauchy formula states that is equal to the right hand side of (5):

The filtered Weber formula (7), the advecting velocity is obeys the filtered Cauchy formula

implies that the curl of It follows that


291

As in the inviscid case, obeys a filtered vorticity equation or generalized vortex method, with viscosity this time:

The equation obeyed by is the same as (14) except that is replaced by In the viscous case one has to reset the calculation whenever the matrix valued function is no longer invertible. This needs to be done even for the filtered equations. The resetting times are times at which the calculation starts afresh. More precisely, one fixes One chooses an initial datum for the vorticity and one computes for a short time This precaution is necessary only if the initial datum is not analytic. One sets then the initial active vorticity at One sets One solves the filtered system for for a time interval and one computes the new active on this time interval using the Cauchy formula (17). One then resets: at the map A is set to the identity and the virtual vorticity is reset to be the previously calculated active vorticity. Inductively, The active vorticity paths in function space are continuous, by construction One can obtain lower bounds for the minimal resetting time (Constantin 2002). These bounds depend on the filter length only via a bound on the maximum enstrophy. One considers the analytic norm

A function which has finite analytic norm is real analytic: it is the restriction of a complex analytic function defined in a strip The norm is smaller than the analytic norm. Consider a time interval [0, T] and define the total enstrophy as

Here is associated to a filtered equation with in the manner described above. If the quantity is bounded above, independently of then it is well known that the solution of the Navier-Stokes equation is the limit as of and that it is smooth on the time interval [0, T]. Denote by G the non-dimensional maximum norm of vorticity:

292

P. Constantin

Consider a small an aribitrary small starting time ciate to it the length

Here and below say

Asso-

denote certain explicit absolute constants. Take 0 < with Then the velocity obeys

with

For any resetting time and distance to to the identity matrix, in analytic norm obeys

Given a nondimensional (small) number

for t in the time interval

4.

the

one can guarantee that

if we take

Summary

The inviscid incompressible fluid equations have relabeling invariants, frozen in the fluid. For instance, the statement that the vortex lines are transported is embodied in the Cauchy formula that relates the Eulerian vorticity to the gradient of the inverse Lagrangian transformation A: the Cauchy invariant is conserved along particle trajectories. The inverse Lagrangian map is defined as long as the solution is smooth, and the Cauchy invariant at is related to its value at simply by a relabeling transformation. The relabeling invariants acquire a non-trivial dynamical character in the Navier-Stokes equation. The equations for the corresponding objects are parabolic. The use of diffusive path transformations leads to both the exact preservation of the classical Weber and Cauchy formulae, and to well-posed (allbeit complicated) diffusive equations for the virtual vorticity (the Cauchy invariant), in which the highest derivative term is a pure Laplacian diffusion term. Traditional non-diffusive particle transformations have been used in dynamo theory (Soward 1972, see Moffatt 1978, Chapter 8) to derive


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mean field equations for the magnetic induction equation. Under these transformations the induction equation is naturally modified in its highest derivative term, by coefficients involving up to third order derivatives of the Lagrangian map. If one applied traditional Lagrangian transfortmations in a non-linear setting, the corresponding terms may lead to high wave number instability. In the diffusive dynamics of virtual vorticity and particle paths presented here is a second order parabolic, well posed system. The diffusive coefficients in the evolution equations for the virtual fields, proportional to the viscosity, contain second order derivatives of A and are related to the Christoffel coefficients of the Riemannian connection obtained via the transformation A. These coefficients have units of reciprocal length and, together with the viscosity, determine a time scale for topological change. The time scale is short if the diffusive particle trajectories have high curvature. The invertibility of the diffusive relabeling map may be lost for time increments exceeding this time scale.

Acknowledgments Research partially supported by NSF DMS-0101022 and by by the ASCI Flash Center at the University of Chicago under DOE contract B341495.

References ARNOL’D, V. I. 1984 Mathematical methods of classical mechanics, Springer-Verlag, GTM 60 New York. BEALE, J. T., KATO, T., MAJDA, A. 1984 Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys. 94, 61-66. CHORIN, A. 1973 Numerical study of slightly viscous flow, J. Fluid. Mech. 57, 785 796. CONSTANTIN, P. 1994, Geometric statistics in turbulence, Siam Review, 36, 73-98. CONSTANTIN, P. 2001 (a) An Eulerian-Lagrangian approach for incompressible fluids: Local theory, J. Amer. Math. Soc. 14, 263-278. CONSTANTIN, P. 2001(b), An Eulerian-Lagrangian approach for the Navier-Stokes equations, Commun. Math. Phys. 216, 663-686. CONSTANTIN, P. 2002, Near Identity Transformations for the Navier-Stokes Equations, preprint xxx.lanl.gov/list/math.AP/0112128. JOHN, P. 1991 Partial Differntial Equations Applied Math. Sciences 1, SpringerVerlag, New York. MOFFATT, H.K. 1978 Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, Cambridge. OHKITANI, K., CONSTANTIN, P. 2002, in preparation. SERRIN, J. 1959 Mathematical Principles of Classical Fluid Mechanics, In Handbuch der Physik, 8 (ed. S. Flugge and C. Truesdell), 125-263.

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P. Constantin

SOWARD, A.M. 1972 A kinematic theory of large magnetic Reynolds number dynamos, Phil. Trans. Roy. Soc. A 272, 431-462.

Evidence for singularity formation in a class of stretched solutions of the equations for ideal MHD J. D. GIBBON1, K. OHKITANI2 1

Department of Mathematics, Imperial College of Science, Technology and Medicine, London SW7 2BZ, UK [email protected]

2

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.

Abstract A class of stretched solutions of the equations for incompressible, ideal 3D-MHD are studied using Elsasser variables This class takes the form where and The chosen domain is of a tubular form which is infinite in the with periodic cross-section. This follows a previous study by the authors on this same class of solutions for the 3D Euler equations. In both cases the systems are of infinite energy. Strong numerical evidence for a finite time singularity in the Euler case was subsequently confirmed by a rigorous analytical proof by Constantin. In the MHD case, pseudo-spectral computations of the 2D partial differential equations for and valid on the cross-sectional domain provide evidence for a finite time blow-up in both the fluid and magnetic variables although an analytical proof for the existence of this singularity remains elusive. We consider the problem of blow-up, The vortices rapidly grow up; They flourish like petals On flowers among nettles; You can purchase them down at the Co-op.

1.

Introduction

The question whether a singularity develops in the vorticity field of the three-dimensional incompressible Euler equations is an important one. The vorticity undoubtedly accumulates into local structures such as vortex tubes and sheets, but whether this process occurs sufficiently rapidly for to become singular in a finite time has been an issue which has yet to be settled conclusively. Numerical and analytical evidence suggests that the close interaction of vortex tubes at very strong angles


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J. D. Gibbon and K. Ohkitani

is a situation where a singularity is the most likely to occur (Kerr 1993; Constantin, Fefferman & Majda 1996). It should be stressed, however, that this problem is still unsolved. The same question remains open for the related but more complicated problem of ideal incompressible MHD (Kerr & Brandenburg 1999; Grauer & Marliani 1998; Caflisch, Klapper & Steele 1997). Singularity formation is an important issue in MHD as it is thought to be relevant to reconnection processes in solar phenomena (Priest & Forbes 2000). These flows are generally taken to be finite domain, finite energy flows. A different system has been proposed by Gibbon, Fokas, & Doering (1999) who examined a class of 3D Euler velocity fields of the form

They showed that the variables and and the third component of the vorticity obey a simple set of coupled two-dimensional partial differential equations in which plays no part. In terms of the above variables and the 2-dimensional velocity field these equations are

and

together with div (Gibbon, Fokas, & Doering 1999). As they stand, no boundary conditions have been applied to the above, leaving the second partial of the pressure, as an arbitrary function of time. Ohkitani & Gibbon (2000) studied equations (2) and (3) in a 3-dimensional tubular domain which is infinite in the with a finite cross-section with periodic boundary conditions applying across it. The divergence-free condition div is important as it reflects the fact that all 2-dimensional cross-sections are connected. Integration of this condition across implies that must satisfy the mean-zero condition Applying this condition to the equation for in (2) determines (t)

The 2D velocity vector Hodge decomposition

can be reconstructed from

and

using a

Singularity formation in solutions of the equations for ideal MHD

297

Ohkitani & Gibbon (2000) provided strong numerical evidence that the dominant variable develops a finite time singularity. Subsequently, by solving the problem exactly on characteristics, Constantin (2000) proved that this singularity in exists and that it must be two-sided; that is, from general initial data blows up simultaneously to and at different places in The positive growth in is later and steeper than the negative. Malham then showed that the support of negative regions of collapses to zero in a finite time while the L 1-norm remains non-zero (Malham 2000). The three-dimensional vortices that develop in the tube just before blow-up have a flower-like spatial structure, with petals of strong vorticity interleaved with hollow regions of weak vorticity. These vortices formally have infinite energy and are destroyed when suggesting that the Euler equations will not sustain a solution of the form expressed in (1) past the singularity time. The initial condition used in Ohkitani & Gibbon (2000) is a simple sinusoidal one

In order to give readers the general idea on the behaviour of and these fields starting from the above initial condition are shown in perspective plots in Figures 1 and 2, respectively. Strongly negative dips in are remarkable, which are associated with flat regions in See Ohkitani & Gibbon (2000) for the details of the properties of this solution.

2.

The ideal MHD problem

2.1 Equations in Elsasser variables It is our intention in this paper to give a description of similar behaviour1 in the equations for three-dimensional, incompressible ideal MHD coupling a magnetic field to an inviscid fluid

where is the fluid velocity and where is the hydrodynamic pressure (Priest & Forbes 2000). The above equations for 1

A fuller account can be found in Gibbon & Ohkitani (2001).


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ideal MHD (7) and (9) can be recast into a simpler form using Elsasser variables With this combination, equations (7) and (8) become

together with div can be used in (11)

The linear

with the two component vectors

and

given by

of

displayed in (1)


With the

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two-dimensional material derivatives defined by

the velocity fields

satisfy

where is the two-dimensional gradient. In the last section it was shown how the equation for for Euler satisfies a Riccati equation which includes an arbitrary function of time Here, also makes an appearance together with


300

These equations are generalisations, to ideal MHD, of the ideas in Gibbon, Fokas, & Doering (1999).

2.2

Equations in a tubular domain

Our equations have not yet had any boundary conditions applied. Let us use the same tubular domain as in the Euler problem; that is, a tube infinite in the with a finite cross-section with periodic boundary conditions applying across it. The divergence-free conditions in 3D become div in 2D. In turn these mean that must satisfy the pair of mean-zero conditions Applying these mean-zero conditions to (17) determines (t)

This pair, together with the equations for

and

constitute a double set of equations that are almost the generalisation of those found for 3D Euler in Ohkitani & Gibbon (2000). What is missing from equations (19) and (20) is that there is no pair of independent equations for curl because of the extra complication that occurs when the curl is taken of In the Euler case in (5) it was possible to reconstruct from and without having to integrate the velocity equations directly. A direct numerical integration of (20) to find is unavoidable here. This means solving for the Laplacian of the pressure which can be found by taking the 2D divergence of (20)

where is the 2D Laplacian. Equation (19) has been used to derive (21), which is invariant under the exchange A simple pair of relationships for can be found through their definitions

In fact which simply means that the fluid and magnetic circulations are constant. None of this class of flows are finite in energy or helicity so they fall into a different category from the more


301

familiar class of 3D finite domain flows whose energy and helicity are also finite. The case closest to 3D Euler is that of a force-free state in the sense that Lorentz force reduces to a gradient. Here the magnetic field is taken to be Taking and we obtain two equations

2.3

A material treatment of the magnetic field

Consider the two-dimensional basic variables and in the decomposition The equation for the two-dimensional part of the magnetic field reads

which has an integral (Cauchy formula) of the form where is the initial magnetic field and is a Lagrangian particle label The determinant of the Jacobian matrix satisfies

where follows that

and the total derivative is simply

It

for Hence, if in a finite time, then at the same time. Therefore some components of the Jacobian matrix must also be singular at that time. From (26) this establishes the important result that must also become singular if there is negative blow-up in except for the pathological case where the product with accidentally cancels the singular contributions. We will show an example of numerical solutions, which corresponds to the initial condition

A – dealiased standard pseudo-spectral method was used to solve (16) and (21) simultaneously. The grid points used were 2562, 5122 and 10242. For time marching we used a fourth-order Runge-Kutta method, with

302


typical time step Numerical results using grid points are described below since no significant differences were observed at higher resolutions. Spatial averages of squared field variables are defined as follows

and

The time evolution of these norms are shown in Figure 3. All the four

quantities apparently blow up around It should be noted that the strength of singularity in and appears to be weaker than that of and In Figure 4 the time evolution of and are shown. It is remarkable that at these two fields are close to each other, as can be checked by calculating their mutual correlation. The subtle point is that while the fluid and magnetic variables blow up simultaneously, this should not be interpreted as and pointwise. This issue can be resolved if blows up at a slower rate than other fields, which is not inconsistent with numerical results.


3.

303

Concluding remarks

In this paper we have extended the stretched solutions of the type expressed in (1) for the three-dimensional Euler equations to the equations for ideal MHD on a tubular domain. Similar to Ohkitani & Gibbon (2000) it is found that the solutions for the MHD case apparently blow

304


up in finite time. The numerical results have passed all the criteria for an apparent blow up and some analytic constraints that control this have been obtained. However, unlike the Euler case, it is not possible at present to prove the blow up purely analytically. Also, because the the growth of magnetic field is weaker than the fluid velocity, it is difficult to observe clear power-law behaviour that shows finite-time blow up. Numerical simulations at higher spatial resolutions will be helpful to settle the matter, as well as new analytical techniques to handle the system of equations considered in this paper.

References CAFLISCH, R., KLAPPER, I. & STEELE, G. 1997 Remarks on Singularities, Dimension and Energy Dissipation for ideal Magneto-Hydrodynamics. Commun. Math. Phys. 184, 443–55. CONSTANTIN, P. 2000 The Euler Equations and Nonlocal Conservative Riccati Equations. Internat. Math. Res. Notices (IMRN) 9, 455–65. CONSTANTIN, P., FEFFERMAN, CH. & MAJDA, A. 1996 Geometric constraints on potentially singular solutions for the 3D Euler equations. Comm. Partial. Diff. Equns 21, 559–571. GIBBON, J. D., FOKAS, A. & DOERING, C. R. 1999 Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations. Physica D 132, 497–510. GIBBON, J. D. & OHKITANI, K. 2001 Singularity formation in a class of stretched solutions of the equations for ideal MHD. Nonlinearity 14, 1239–1264. GRAUER, R. & MARLIANI, C. 1998 Geometry of singular structures in magnetohydrodynamic flows. Phys. Plasmas 5, 2544–2552. KERR, R. 1993 Evidence for a singularity of the 3-dimensional, incompressible Euler equations. Phys. Fluids A 5, 1725–46. KERR, R. & BRANDENBURG, A. 1999 Evidence for a singularity in ideal magnetohydrodynamics: Implications for fast reconnection. Phys. Rev. Lett. 83, 1155–58. MALHAM, S. J. A. 2000 Collapse of a class of three-dimensional Euler vortices. Proc. Royal Soc. Lond. 456, 2823–33. OHKITANI, K. & GIBBON, J. D. 2000 Numerical study of singularity formation in a class of Euler and Navier-Stokes flows. Phys. Fluids 12, 3181–94. PRIEST, E.& FORBES, T. 2000 Magnetic Reconnection (Cambridge, Cambridge University Press)

Numerical evidence of breaking of vortex lines in an ideal fluid Evgeniy A. KUZNETSOV1,3, Olga M. PODVIGINA2,3 & Vladislav A. ZHELIGOVSKY2,3 1

L.D.Landau Institute for Theoretical Physics, 2 Kosygin str., 117334 Moscow, Russian Federation [email protected]

2

International Institute of Earthquake Prediction Theory and Mathematical Geophysics, 79 bldg. 2 Warshavskoe ave., 113556 Moscow, Russian Federation; Laboratory of general aerodynamics, Institute of Mechanics, Lomonosov Moscow State University, 1, Michurinsky ave., 119899 Moscow, Russian Federation

3

Observatoire de la Côte d’Azur, CNRS UMR 6529, BP 4229, 06304 Nice Cedex 4, France

Abstract Emergence of singularity of vorticity at a single point, not related to any symmetry of the initial distribution, has been demonstrated numerically for the first time. Behaviour of the maximum of vorticity near the point of collapse closely follows the dependence where is the time of collapse. This agrees with the interpretation of collapse in an ideal incompressible fluid as of the process of vortex lines breaking.

1.

Introduction

The problem of collapse in hydrodynamics, i.e. of a process of singularity formation in a finite time, is essential for understanding of the physical nature of developed turbulence. Despite a progress in construction of the statistical theory of Kolmogorov spectra within both diagram and functional approaches (see, e.g., Monin & Yaglom 1992; L’vov 1991 and references therein), so far the question whether the Kolmogorov spectrum is a solution to the statistical equations of hydrodynamics remains open. Another important problem, as yet unsolved, is the one of intermittency. In statistical sense intermittency can be interpreted as a consequence of a strongly non-Gaussian distribution of turbulent velocity, resulting in a deviation of exponents for higher correlation functions from their Kolmogorov values (Frisch 1995). Non-Gaussian behaviour


E. A. Kuznetsov, O. M. Podvigina and V. A. Zheligovsky

306

implies that odd correlation functions do not vanish; this indicates the presence of strong correlations between velocity fluctuations, suggesting existence of coherent structures in turbulence. Analysis of both numerical and experimental data reveals (see Frisch 1995 and references therein) that in the regime of fully developed turbulence the distribution of vorticity is strongly inhomogeneous in space – it is concentrated in relatively small regions. What is the reason of this? Can such a high concentration be explained by formation of singularity of vorticity in a finite time? How can one derive from this hypothesis the Kolmogorov spectrum? This question is not rhetoric: it is well known that any singularity results in a power-law kind of spectrum in the short-scale region. Thus, the problem of collapse is of ultimate importance in hydrodynamics. The most popular object in the studies of collapse in hydrodynamics is a system of two anti-parallel vortex tubes, inside which vorticity is continuously distributed (Kerr 1993), or in a more general setup – flows with a higher spatial symmetry (Boratav & Pelz 1994; Pelz 1997). It is well known, that two anti-parallel vortex filaments undergo the so-called Crow instability (Crow 1970) leading to stretching of vortex filaments in the direction normal to the plane of the initial distribution of vortices and to reduction of their mutual distance. It was demonstrated in numerical experiments (Kerr 1993) that point singularities are formed in cores of each vortex tubes at the nonlinear stage of this instability, and near the point of collapse increases like being the time of collapse (see also Grauer, Marliani & Germaschewski 1998).

2.

Basic equations

In this paper we present results1 of a numerical experiment, which can be interpreted as emergence of singularity of vorticity at a single point in a three-dimensional ideal hydrodynamic system, where initial data lacks any symmetry. The representation of the Euler equation for vorticity in terms of vortex lines is employed, which was introduced in Kuznetsov & Ruban 1998:

Here the mapping represents transition to a new curvilinear system of coordinates associated with vortex lines, so that is a tangent 1

Preliminary results were communicated in Zheligovsky, Kuznetsov & Podvigina 2001.

Numerical evidence of breaking of vortex lines

vector to a given vortex line, mapping (2). Dynamics of the vector

where is the flow velocity at a point projection to the vortex line at this point:

307

is the Jacobian of the satisfies

and

is the transverse

Equations (1)-(3) are closed by the relation between vorticity and velocity:

The system of equations (1)-(5) can be regarded as a result of partial integration of the Euler equation

The vector field incorporated in (1), is the Cauchy invariant manifesting frozenness of vorticity into the fluid. If is the initial distribution of vorticity. The Jacobian J can take arbitrary values because the description under consideration is a mixed, Lagrangian-Eulerian one (Kuznetsov & Ruban 1998; Kuznetsov & Ruban 2000). In particular, J can vanish at some point, which by virtue of (1) generically implies singularity of vorticity. It was demonstrated by Kuznetsov & Ruban 2000 that collapses of this type are possible in the three-dimensional integrable hydrodynamics (Kuznetsov & Ruban 1998), where in the Euler equation (6) a modified relation between vorticity and velocity (both generalised) is assumed:

Emergence of singularity of vorticity at a point, where means that a vortex line touches at this point another vortex line. This is the process of breaking of vortex lines. Being analogous to breaking in a gas of dust particles (dynamics of a gas with a zero pressure), this process is completely determined by the mapping (2).

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3.


Breaking of vortex lines

Let us assume now that a collapse in the Euler hydrodynamics occurs due to breaking of vortex lines. Denote by a solution to the equation and let where the minimum is achieved at Near the point of the minimum the Jacobian can be expanded (cf. Kuznetsov & Ruban 2000): where i s a positive definite matrix and The Taylor expansion (8) is obtained under the assumption that J is smooth, which is conceivable up to the moment of singularity formation. At the numerator in (1), i.e. the vector does not vanish: the condition is satisfied when the three vectors lie in a plane, but generically none of them equals zero (that were a degeneracy) so that near the point of singularity

Furthermore, implies that an eigenvalue of the Jacobi matrix (say, vanishes, and generically the other two eigenvalues and are non-zero. Therefore, there exist one “soft” direction associated with and two “hard” directions associated with and It follows from (8), that in the auxiliary the self-similarity is uniform in all directions. However, in the physical space the scales are different. Following Kuznetsov & Ruban 2000, we show how an anisotropic self-similarity emerges in the flow. The analysis for the Euler equation coincides with that for the integrable hydrodynamics (7). Decompose the Jacobi matrix in the bases of eigenvectors of the direct and conjugate spectral problems:

The two sets of eigenvectors are mutually orthogonal: In a vicinity of the point of collapse the eigenvectors can be regarded as approximately constant. Decompose the vectors and in (10) in the respective bases and denote their components by and

309


The vector

can be represented in terms of

as follows:

As a result, (10) can be expressed as

where

and is assumed to be small. Consequently, size reduction along the directions and is the same as in the auxiliary a-space, i.e., but in the soft direction, the spatial scale is Therefore, in terms of the new self-similar variables and integration of the system yields for and a linear dependence on and for – a cubic one:

Together with (9), relations (13) and (14) implicitly define the dependence of on r and The presence of two different self-similarities shows, that the spatial vorticity distribution becomes strongly flattened in the first direction, and a pancake-like structure is formed for Due to (1) and the degeneracy of the mapping the vorticity lies in the plane of the pancake. Near the singularity the behaviour of is defined by the following self-similar asymptotics:

In essence, in the above analysis one is concerned with the behaviour of the mapping near a fold, and thus breaking of vortex lines can be naturally explained within the classical catastrophe theory (Arnold 1981; Arnold 1989).

310

4.


Super-weak collapse

According to the collapse classification of Zakharov & Kuznetsov 1986, breaking of vortex lines is not a weak collapse but a super-weak one, because already the contribution of the singularity to the enstrophy characterising the energy dissipation rate due to viscosity is small, the contribution to the total energy is However, the integral is divergent as Thus, the breaking solution cannot be continued beyond in the Sobolev space with the norm According to the theorem proved by Beale, Kato & Majda 1984, this suffices for

to hold. The condition (16) is necessary and sufficient for a collapse in the Euler equation, and it is satisfied for (15). Another restriction follows from the theorem by Constantin, Feferman & Majda 1996, stating that there is no collapse for any if

where the supremum is over a region near the maximum of vorticity Occurrence of collapse implies divergence of the integral (17) for Consequently, sup has to increase at least like It is evident that, due to solenoidality of either the derivative in the direction along the vector in the pancake-like region should have no singularity at the scales of the order of or larger, or the singularity should be weaker than However, this does not rule out large gradients of in a region separated in the soft direction from the pancake-like region, for instance, with the behaviour with This conjecture is plausible, since transition from the a-space to the physical one involves a significant contraction in the soft direction of the region near the point of breaking: a sphere of radius is mapped into the pancake-like region. Thus, a sphere in the r-space of radius containing the pancake includes a large preirnage of the region outside the sphere in the a-space of radius (the shape of the preimage is governed by higher order terms in the expansion (8) ). Hence in the process of breaking of vortex lines three scales can appear: and an intermediate scale with (whose presence makes sure that there are no contradictions with the theorem of Constantin, Feferman & Majda 1996).


5.

311

Numerical results

To verify the hypothesis that formation of singularity in solutions to the Euler equation can be due to vortex line breaking, we performed a numerical experiment for the system of equations (1-5). Two features of this system are notable. First, in contrast with the original Euler equation, possessing an infinite number of integrals of motion – Cauchy invariants, the system (1-5) is partially integrated and therefore contains the Cauchy invariants explicitly. Hence, while the invariants are guaranteed to be conserved when (1-5) is solved numerically, it is necessary to test how accurately they are conserved in the course of direct numerical integration of the Euler equation (6). Second, in the system (1-5) integration in time (in (3) ) is separated from integration over space (in (5) ), i.e. from inversion of the operator curl. The system (1-5) is considered under the periodicity boundary conditions and inversion of the operator curl can be performed by the standard spectral techniques with the use of the Fast Fourier Transform. The main difficulty in numerical integration of the system stems from the necessity of transition (both direct and inverse) between the variables r and a at each time step. It was circumvented by the use of two independent grids in the r-space: a moving one (the R-grid), the motion of whose points is governed by (3), and a steady regular one (the r-grid), which coincides with the a-grid. The numerical algorithm consists of the following steps: (i) by integrating (3) in time, find new positions of the R-grid points; (ii) compute new values (1) of on the R-grid by finite differences; (iii) by linear interpolation from the values of vorticity at nearby points ofthe R-grid, determine on the r-grid (to do this, for each point of the regular grid it is necessary to find a tetrahedron, containing the point, whose vertices are the nearest points of the R-grid); (iv) solve the problem (5) to determine the flow velocity v on the r-grid; (v) by linear interpolation, determine v on the R-grid. Computations are performed with the resolution of 1283 grid points. In order to check numerical stability of the algorithm test runs are made for several initial conditions, which are ABC flows. Any ABC flow is an eigenfunction of the curl and hence it is a steady solution to the force-free Euler equation (6). They are found to remain steady in computations with the time step up to with the relative error of the solution being within the threshold, and the Jacobian J being reproduced with the accuracy. An initial vorticity which we consider is a solenoidal field comprised of random-amplitude Fourier harmonics with an exponentially decaying spectrum; the decay is by 6 orders of magnitude from the first to the last

312


spherical shell in the Fourier space, the cut-off being at wavenumber 8. It satisfies everywhere in the box of periodicity (this enables one to perform the projection (4); it is checked that this condition is never violated during the run). This field does not possess any symmetry. In the course of numerical integration we monitor energy conservation: kinetic energy of the flow remains constant with the accuracy better than 1%. For such an initial condition we observe formation of a peak of at a single point. At this point the Jacobian J and are minimal over space for all times close to the time of collapse, and the minimal values decrease in time to a high precision linearly (Fig. 1). (In this run the time step is for and afterwards.) In this run the maximum of vorticity increased almost 20 times before integration was terminated. The final width of the peak of is 2-3 times the length of the interval of spatial discretisation (Fig. 2 shows a strong localisation of at the end of the run). Figure 3 illustrates concentration of vorticity lines and formation of a fold near the point of singularity. Formation of similar peaks of vorticity accompanied by


313

a decrease of J to zero is also observed in several runs for other initial conditions of vorticity from the same class. At different times the global (over space) minima of J and of are achieved at four different points. Behaviour of the Jacobian in time at one of these points (short-dashed line on Fig. 4) suggests that the second singularity can also be developing; it is not traced down to the time of its collapse, because this is prevented by formation of the first singularity. The peak of vorticity turns out to be narrow from the moment of its birth. In order to verify that it is not spurious (i.e. it emerges not due to a numerical instability of our algorithm) we have reproduced its formation in computations by a modified algorithm, with different interpolation techniques employed for linear interpolation at step (iv). These techniques introduce some smoothing intended to inhibit formation of a spurious singularity. However, in the new run all numerical data has been reproduced with the relative precision

314


To check that the process can be identified as breaking of vortex lines we compute the time dependence of the Hessian of the Jacobian at the point of the minimum of J. At the final stage of the saturated asymptotic linear behaviour of the minimum we did not find any essential temporal variation of its eigenvalues. This agrees qualitatively with the expansion (8). Figure 3 illustrates final positions of vortex lines near the point of collapse. Some anisotropy is observed in the spatial distribution of near the maximum of vorticity. How-


315

ever, due to an apparent lack of spatial resolution we cannot claim that two essentially different scales emerge. The following questions also remain open: Why is the time of occurrence of collapse small compared to the turnover time? Why is the peak of vorticity quite narrow from the moment of its appearance? The obtained results can be interpreted as the first evidence of the vortex line breaking; the collapse, which is observed numerically, is not related to any symmetry of the initial vorticity distribution and in particular the collapse occurs at a single point.

Acknowledgments The authors are grateful to the Observatory of Nice, where this work was initiated and the paper was completed. Visits of E.K. to the Observatory of Nice were supported by the Landau-CNRS agreement, and those of O.P. and V.Z. – by the French Ministry of Education. Participation of E.K. in the project was also financed by RFBR (grant no. 0001-00929), by the Program of Support of the Leading Scientific Schools of Russia (grant no. 00-15-96007) and by INTAS (grant no. 00-00292).

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References ARNOLD, V.I. 1981 Theory of Catastrophe. Znanie, Moscow (in Russian) [English transl.: Theory of Catastrophe 1986, 2nd rev. ed. Springer]. ARNOLD, V.I. 1989 Mathematical Methods of Classical Mechanics. 2nd ed., SpringerVerlag, New York. BEALE, J.T., KATO, T. & MAJDA, A.J. 1984 Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94, 61–66. BORATAV, O.N. & PELZ, R.B. 1994 Direct numerical simulation of transition to turbulence from high-symmetry initial condition. Phys. Fluids 6, 2757–2784. CONSTANTIN, P., FEFERMAN, CH. & MAJDA, A.J. 1996 Geometric constrains on potentially singular solutions for the 3D Euler equations. Commun. Partial Diff. Eqs. 21, 559–571. CROW, S.C. 1970 Stability Theory for a pair of trailing vortices. Amer. Inst. Aeronaut. Astronaut. J. 8, 2172–2179. FRISCH, U. 1995 Turbulence. The legacy of A.N.Kolmogorov. Cambridge Univ. Press. GRAUER, R., MARLIANI, C., & GERMASCHEWSKI, K. 1998 Adaptive mesh refinement for singular solutions of the incompressible Euler equations. Phys. Rev. Lett. 80, 4177–4180. KERR, R.M. 1993 Evidence for a singularity of the 3-dimensional, incompressible Euler equations Phys. Fluids A 5, 1725–1746. KOLMOGOROV, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Doklady AN SSSR 30, 9–13 (in Russian) [reprinted in 1991 Proc. R. Soc. Lond. A 434, 9–13]. KUZNETSOV, E.A. & RUBAN, V.P. 1998 Hamiltonian dynamics of vortex lines for systems of the hydrodynamic type, JETP Letters 67, 1076–1081. KUZNETSOV, E.A. & RUBAN, V.P. 2000 Collapse of vortex lines in hydrodynamics. JETP 91, 776–785. L’VOV, V.S. 1991 Scale invariant theory of fully developed hydrodynamic turbulence – Hamiltonian approach. Phys. Rep. 207, 1–47. MONIN, A.S. & YAGLOM, A.M. 1992 Statistical hydro-mechanics. 2nd ed., vol.2, Gidrometeoizdat, St.Petersburg (in Russian) [English transl.: 1975 Statistical Fluid Mechanics. Vol. 2, ed. J.Lumley, MIT Press, Cambridge, MA]. PELZ, R.B. 1997 Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E, 55, 1617–1626. ZAKHAROV, V.E. & KUZNETSOV, E.A. 1986 Quasiclassical theory of three-dimensional wave collapse. Sov. Phys. JETP 64, 773–780. ZHELIGOVSKY, V.A., KUZNETSOV, E.A. & PODVIGINA, O.M. 2001 Numerical modeling of collapse in ideal incompressible hydrodynamics. Pis’ma v ZhETF (JET Letters) 74, 402–406.

Sufficient condition for finite-time singularity and tendency towards selfsimilarity in a high-symmetry flow C. S. NG & A. BHATTACHARJEE Center for Magnetic Reconnection Studies, Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA [email protected]

Abstract A highly symmetric Euler flow, first proposed by Kida (1985), and recently simulated by Boratav and Pelz (1994) is considered. It is found that the fourth order spatial derivative of the pressure at the origin is most probably positive. It is demonstrated that if grows fast enough, there must be a finite-time singularity (FTS). For a random energy spectrum a FTS can occur if the spectral index Furthermore, a positive has the dynamical consequence of reducing the third derivative of the velocity at the origin. Since the expectation value of is zero for a random distribution of energy, an ever decreasing means that the Kida flow has an intrinsic tendency to deviate from a random state. By assuming that reaches the minimum value for a given spectral profile, the velocity and pressure are found to have locally self-similar forms similar in shape to what are found in numerical simulations. Such a quasi self-similar solution relaxes the requirement for FTS to A special self-similar solution that satisfies Kelvin’s circulation theorem and exhibits a FTS is found for On the singular blow-up of Euler, I am an inveterate toiler; I look near the null, You may think this is dull, But there's no need to be such a spoiler!

1.

Introduction Consider the three-dimensional (3D) Euler equation

with the divergence-free condition The self-consistent pressure must satisfy An important question is whether the solution of (1) can become singular in finite time for a smooth initial condition with finite energy. Some useful and rigorous constraints on 317 K. Bajer and H.K. Moffatt (eds.), Tubes, Sheets and Singularities in Fluid Dynamics, 317–328. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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the nature of a possible finite-time singularity (FTS) have been established (Beale, Kato & Majda 1984, Constantin 1994), but the singularity has not been explicitly demonstrated in a mathematically rigorous way. Recently, an analytical model has been developed for a highly symmetric initial condition that exhibits a FTS (Bhattacharjee & Wang 1992, Bhattacharjee, Ng & Wang 1995), with some assumptions which, though physically plausible, have not yet been proved. These analytical results suggest that if an initial state with symmetries that are preserved by the Euler equation is considered, then the problem of demonstrating a FTS becomes technically less difficult. Another motivation for choosing highly-symmetric initial conditions is that they fix a geometry in place for all times, and the hope is that such studies will help identify geometrical sites where a FTS will tend to develop in less constrained flows. A recent numerical experiment has been performed (Boratav & Pelz 1994, 1995, hereafter, BP) on a highly symmetric initial flow, first proposed by Kida (1985). The symmetries of the Kida flow are discussed in Section 2. Due to the high symmetry, BP are able to do the simulation with high spatial resolution (5123). Within the limits of the resolution, BP report that the maximum vorticity scales as as viscosity becomes small. Pelz (1997) has also proposed a vortex filament model in an effort to explain the simulation results. This model, although based on strong assumptions, predicts a FTS which evolves in a locally self-similar way near the origin. More recently, Pelz and Gulak (1997) have performed additional numerical simulations on the Kida flow. They calculate the Taylor series in time, sum the series using the Padé approximation, and recover the FTS with a value of close to that found in the simulations of BP. Ng and Bhattacharjee (1996) have proposed a sufficient condition for finite-time singularity of the Kida flow, based on analytical and numerical evidence. One way to satisfy this condition is to require that the fourth order spatial derivative be positive at the origin and within a finite range X for all time. If the range X actually tends to zero as the singularity develops, a FTS can still occur if grows rapidly enough. In Section 3, it is shown that if grows at least as fast as there must be a FTS. It follows that, for a random isotropic 1-D energy spectrum the condition for a FTS with growing can be satisfied with the spectral index This seems only marginally satisfied in BP’s simulations where it is observed that In Section 4, we present statistical calculations, which suggest that the is very possibly positive at the origin for a random energy spec-

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trum. This is a remarkable result since without the Kida symmetries, the expectation value of should be zero everywhere. As discussed below, this result has significant dynamical implications. The positivity of has the intrinsic dynamical effect of favoring the development of a more coherent flow. In particular, the third order spatial derivative of the velocity along the at the origin tends to decrease by the action of a positive In Section 5, by assuming that approaches the minimum allowed value for a given spectral profile, a coherent state is found which is locally self-similar around the origin. Furthermore, the velocity and pressure self-similar profile for this state is found to be similar to those observed in BP’s simulations. The development of such a quasi self-similar structure is evidence supporting the realization of a FTS. Specifically, the requirement for the spectral index is now relaxed to which appears to be consistent with simulation data. A special self-similar solution that satisfies Kelvin’s circulation theorem and exhibits a FTS is found for The timedependence of this solution is consistent with the filament model of Pelz (1997).

2.

The Symmetries of the Kida flow

The symmetries of the Kida flow have been discussed in detail by Kida (1985). Here we build these symmetries into the representations for and The components of the velocity field can be written as where can be expressed in Fourier series,

Here are natural numbers which represent the three components of a wave vector In order to satisfy the symmetries and the condition • the following conditions must hold:

where the last summation (denoted by C) is over all permutations of any three natural numbers i.e., By (2) and (3), it can be seen that for close to the origin, In particular, the initial state considered by both Kida (1985) and BP is with all other terms set to zero. With


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represented by (2), it can be shown that the pressure

where defined by

By (3) and (4), we know that easy to see that the time evolution equation,

is of the form

with

It is

preserves the Kida symmetries, where an overdot denotes time derivative. Equation (6) and the mode-generation scheme provide a dynamical mechanism for the excitation of modes in the small scales. From (1) and (3), we find where denotes the second spatial derivative of at the origin. This means that close to the origin and thus (0) (hereinafter denoted simply by is the first non-zero pressure derivative at the origin. Thus, the quantity has a special dynamical significance.

3.

Sufficient condition for finite time singularity

Let us now consider the flow along the line By (2), we obtain and Note that the vorticity is also identically zero along this line. This leaves open the possibility that the vortex singularity can happen infinitesimally close to this line, since the non-zero vorticity outside this line can collapse infinitesimally close to it. (This is similar to a type of FTS discussed by Bhattacharjee, Ng &; Wang 1995). From (1), we obtain

where now the overdot denotes total time derivative along a fluid element moving in a trajectory which is close to zero, with Defining we obtain

Note also that for the initial flow, at for all From (8), we see that if for all time, then can become negative


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infinite in finite time due to the presence of the term. Although this is a rigorous statement, in practice, it is difficult to calculate by following a fluid element. Therefore, we seek alternative formulations of this sufficient condition. One possibility that can be shown to lead to a FTS is to assume that there exists a range in which for all time before a possible singularity appears. (Since we will later see that is most probably positive for a general Kida flow, we exclude the possibility that becomes negative in finite time.) For the simplest case, we assume that where C is a finite positive constant for all time (including Since it follows from the assumption above and by simple integration from that the quantities and are also positive within the range Then by (7) and the fact that there exists a fluid element with the Lagrangian coordinate within this range (0, C) that is always accelerating towards the origin However, since the condition is always maintained by the symmetry of the Kida flow, the fluid element cannot pass through the origin. The system resolves this contradiction by having the velocity derivative blow up in finite time, since the fluid element with finite and increasing velocity is forced to go infinitesimally close to the point with zero velocity This behavior is reflected in Eq. (8) according to which tends to negative infinity in finite time due to the presence of the term. If the time-dependence of is determined dominantly by the term (or if the term is of the same order as the term), then as Thus, under the assumptions discussed above, we see that the condition at the origin becomes a sufficient condition for a finite-time singularity (Ng & Bhattacharjee 1996). This is not a point condition because it is derived under the assumption that there exists a range Note that if we allow . with a fixed value of it is possible that may shrink so rapidly that it leaves behind any trajectory before a FTS can develop by the above mechanism. In the discussion above, we have not considered the relationship between and X. In reality, they are related since X is a measure of length scale and is found by taking spatial derivatives which should involve the inverse of the length scale. Let us assume the relation where is a positive constant. By this we have also effectively assumed that is a smooth function within with characteristic length of the order of X. It can then be shown that:


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There must be a FTS for the case with for any smooth function with continuous derivative and satisfying (A) To show Statement (A), let us first consider the case when the speed is a monotonically decreasing function for with some constant By the above assumptions, we can find a finite constant such that the Taylor approximation is good for for all time. Then we have, in this range,

Consider a fluid element at at time such that for all Because of the initial condition and the fact that this fluid element is always being accelerated toward the origin, it must have a finite negative speed. Without affecting the argument, we can treat this speed as very small compared with the speed It will then take a time

for the element to be accelerated up to the same speed as The position will still be very close to this fluid element during this time period if Since the magnitude of is monotonically decreasing for this case and the fluid element is accelerating, will never pass through the fluid element at any later time. Therefore the fluid element will always stay in a region in which it is always accelerated towards the origin. Since it cannot pass through by symmetry, a FTS must appear to resolve the contradiction. By Eqs. (10) and (11), if the inequality

can be satisfied in finite time with obeying the conditions stipulated in Statement (A), then there must be a FTS. It is easy to show that this is always possible when Since if this is not so, then we must have

for all where equation

is a finite constant. However, the solution to the

with an always positive

is


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which must become zero in finite time and thus violate the required condition for in Statement (A). (For the case with the above argument does not go through and therefore a FTS does not necessarily appear.) It is possible to generalize the proof to the case with the speed not monotonically decreasing. However, we will omit such a discussion here since this would take up too much space. We remark that the total time of such non-monotonic periods must be finite otherwise either a FTS will appear or X goes to zero in finite time. From the above discussion, we see that the value of is very important to the problem of FTS, especially when it is positive and grows fast enough as the characteristic length scale of the system shrinks.

4.

Positivity of

We summarize here the results obtained by Ng and Bhattacharjee (1996) that support a positive First, we need to introduce an independent set of modes of the Kida flow such that any flow satisfying the symmetries of the Kida flow can be expressed in the form

where are real constants. The choice of the set of independent modes is not unique, but we will choose to work with the simplest possible choice that each consists only the Fourier modes of a given set of three all odd or all even natural number that satisfy (2) and (3). All modes are normalized to have the same energy. The quantity is a quadratic function of so we can write For expressed in (16), we obtain

where We now state a theorem: For all the independent modes defined above, The proof of this theorem, although straightforward, is quite complicated and needs the help of computer programs such as Mathematica to do the algebra, and thus is omitted here. From (17) and by the theorem stated above, we see that the contributions from the self terms is always positive. Though this strengthens the case for positive we need to consider the cross terms which cannot be neglected in principle, and they can be positive or negative depend on the sign of Let us first consider an example with two modes

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so that Note that the cross term between an odd mode and an even mode is always zero. In order to have positive, it is sufficient to have This condition is not always true for any two modes. However, it is also proved using Mathematica that

where are the wave number of the modes Although (18) is an relation between two modes, it can have an important effect on for a state with many modes. To see this possibility in a qualitative way, let us assume that the second term in the right hand side of (17) includes positive and negative terms in a rather random manner so that its magnitude is most probably within a range defined by the standard deviation

where the second relation is obtained by assuming that in the average sense because (18) holds for most terms in the double sum for large N. This means that the magnitude of the second term (which can be positive or negative) in the right hand side of (17) is most probably of the same order of magnitude of the first term, which is always positive. So the total of the two is most probably positive. Although this is just a qualitative argument, it has been confirmed in statistical calculations. This involves a Monte Carlo calculation of as defined in (17). The energy is distributed randomly in the Fourier space according to a given energy spectrum of the form filling up to N modes. In most of our calculations, we choose which is the spectrum observed by BP near However, the qualitative trends observed are not sensitive to variations in In Fig. 1, we plot the numerical values of the percentage of cases with negative values of as a function of N. A straight line fit of Fig. 1 gives The expectation value of for a random flow can be calculated directly without implementing the Monte Carlo method. In Fig. 2, values of as a function of for different spectral indices are plotted. Combined with another result that where is the expectation value of the range of positive in the we have By Statement (A) of the previous section, we know that for the random model if we should have a FTS.


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For a general periodic flow, the expectation value of the pressure derivative should be zero for at all spatial point x. The positivity of at the origin of the Kida flow is therefore quite a special case, which has important dynamical implications examined in the next section.

5.

Quasi self-similarity

So far, in establishing the positivity of only random statistics are used. In reality, the flow may evolve in a much more coherent way. In fact, BP’s simulations show that the flow pattern near the origin evolves in a quasi self-similar way, which suggests a highly coherent state. In this section, we will examine the possible dynamical consequence of a positive and how it can increase the probability of FTS formation. Let us define to be the value of at the origin. By (7) or (8), we obtain

This is the first and simplest dynamical equation involving We immediately see that this equation tends not to favor a random state since we have already establish that over all random states. For an initial state with it is therefore reasonable to expect that will be negative for all time. Thus the true dynamics of the system tends to follow a path that is different from the random path because in the latter (This is because is a linear function of the amplitudes.) Based on the above considerations, we make a drastic assumption:

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During the dynamical evolution, after a certain initial transient time, the energy is distributed among the Fourier modes such that is very close to its minimal value (maximum in magnitude) for a given energy spectral profile. Note that this assumption has not constrained the shape of the spectral profile. To simplify the discussion, we only consider spectral profiles characterized by two parameters: a spectral index so that the isotropic energy spectrum is given by up to a sharp cut off wave number K. Although a sharp cut off is physically unrealistic, this can be used to approximate the limit of a fast decay of spectral energy for high wave numbers. Note also that if in finite time, then we must have a FTS. Furthermore, if this happens, has to be small enough such that the maximum vorticity and the stress tensor elements tend to infinity (Beale, Kato & Majda 1984). For a given spectral profile, it is straightforward to find the minimum state. Note that because of the Kida symmetry (3), only the even modes contribute to

It is not hard to see that the flow will be locally self-similar around since a minimum state implies definite correlation among the Fourier amplitudes. Fig. 3 plots of the minimum state as a function of (in unit of ) for 26 different N from 107 to 132330, rescaled to the coordinate of the plot of to show local selfsimilarity. We use in these plots, but the profiles for other v are in similar shape. Also plotted on Fig. 3 are the rescaled simulation data of BP at and (*) Similarly, Fig. 4 plots the profiles of the minimum states, for N from 107 to 1624, rescaled to the coordinate of the plot of with simulation data over plotted. The agreement between the simulation data and the self-similar profile of the minimum state appears to be reasonably good. For a minimum state, it is straightforward to estimate that and By Statement (A) in Section 3, the condition for FTS is relaxed to which appears to be quite easily satisfied, and is, in fact, consistent with the numerical results of BP. Moreover, the flow should develop a quasi self-similar form for a constant Since this form is so restrictive, we can actually determine the value of by making use of global dynamical constraints (since it may not be possible to find self-similar solution satisfying the equation of motion for any Let us consider the constraint imposed by the Kelvin’s circulation theorem:


327

where C represent any close loop transported by the fluid. Substituting the self-similar form into (21), we obtain,

with Note that the loop C is being transported by the field according to the relation where we have used for with some positive constant c. Any quasi self-similar solution with a profile has to satisfy (22), besides satisfying (1) of course. This is a strong restriction on the flow. One special solution of (22) is simply obtained by choosing This automatically makes the coefficient in (22) constant. To ensure that the loop integral remains constant also, we can choose a profile function that all the vorticity concentrate into vortex filaments. Therefore the integral for any loop that does not contain one or more filaments is zero, and for any loop that does the integral has a definite value. Moreover, the values of these integrals will not change in time. For such a quasi self-similar flow with we obtain,

All the scaling relations (23) agree with the results of the filament model of Pelz (1997). Note that the velocity tends to blow up in finite time.


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6.

Summary

We have shown that the Kida flow has a distinctive property that at the origin is most probably positive, which provides an intrinsic tendency for the flow to develop a more organized state by decreasing By simply assuming that it actually goes to the state with minimum we have found a quasi self-similar flow profile that agrees reasonably well with the simulation data. For a random energy spectrum it is found that the condition for the development of a FTS is to have the energy spectral index smaller than 6. By means of the Kelvin’s circulation theorem, a simple local self-similar solution is found for with vortex filaments.

Acknowledgments The authors would like to thank R. B. Pelz and O. N. Boratav for providing their simulation data. This research is supported by the NSF Grant No. ATM-9801709 and the DOE Cooperative Agreement No. DEFC02-01ER54651.

References BEALE, J. T., KATO, T. & MAJDA, A. 1984 Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66. BHATTACHARJEE, A. & WANG, X. 1992 Finite-time vortex singularity in a model of 3-dimensional Euler flows. Phys. Rev. Lett. 69, 2196–2199. BHATTACHARJEE, A., NG, C. S. & WANG, X. 1995 Finite-time vortex singularity and Kolmogorov spectrum in a symmetrical 3-dimensional model. Phys. Rev. E. 52, 5110–5123. BORATAV, O. N. & PELZ, R. B. 1994 Direct numerical-simulation of transition to turbulence from a high-symmetry initial condition. Phys. Fluids 6, 2757–2784. BORATAV, O. N. & PELZ, R. B. 1995 On the local topology evolution of a highsymmetry flow. Phys. Fluids 7, 1712–1731. CONSTANTIN, P. 1994 Geometric statistics in turbulence. SIAM Review 36 73–98. KIDA, S. 1985 3-dimensional periodic flows with high-symmetry. J. Phys. Soc. Jpn. 54, 2132–2136. NG, C. S. & BHATTACHARJEE, A. 1996 Sufficient condition for a finite-time singularity in a high-symmetry Euler flow: Analysis and statistics. Phys. Rev. E 54, 1530–1534. PELZ, R. B. 1997 Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E 55, 1617–1626. PELZ, R. B. & GULAK, Y. 1997 Evidence for a real-time singularity in hydrodynamics from time series analysis. Phys. Rev. Lett. 79, 4998–5001.

Finite time singularities in a class of hydro dynamic models Victor P. RUBAN1,2, Dmitry I. PODOLSKY1 & Jens J. RASMUSSEN2 1

L. D. Landau Institute for Theoretical Physics, 2 Kosygin St., 117334, Moscow, Russia [email protected]

2

Optics and Fluid Dynamics Department, OFD-129, Risø National Laboratory, DK-4000 Roskilde, Denmark

Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form in 3D Fourier representation, where is a constant, Unlike the case (the usual Eulerian hydrodynamics), a finite value of results in a finite energy for a singular, frozen-in vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularisation procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of anti-parallel vortex filaments and an analog of the Crow instability is found at small wave-numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like where is the singularity time. Abstract

We consider the fate of string links, That can tangle with all sorts of kinks; I won’t try to disguise What we regularise; It's more subtle than anyone thinks!

1.

Introduction

This talk is based on the recent article by Ruban, Podolsky, & Rasmussen (2001). Here we take the point of view that infinite curvature of frozen-in vortex lines is in some sense a more fundamental characteristics of hydrodynamic singularity than infinite value of the vorticity maximum. To illustrate this statement, we consider a class of models of an incompressible inviscid fluid, different from Eulerian hydrodynamics, such that finite energy solutions with infinitely thin frozen-in vortex filaments of finite strengths are possible. Thus, we deal with a situation


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V. P. Ruban, D. I. Podolsky and J. J. Rasmussen

when the vorticity maximum is infinite from the very beginning, but nevertheless, this fact itself does not imply a singular behaviour in the dynamics of vortex strings, while their shape is smooth and the distance between them is finite. However, the interaction between filaments may result in formation of a finite time singularity for the curvature of vortex strings. It is the main purpose of present work to study this phenomenon analytically. In general, our approach is based on the Hamiltonian formalism for frozen-in vortex lines, that is a natural language for speaking about inviscid incompressible flows (Kuznetsov & Ruban 2000). It is a well known fact that absence of solutions with singular vortex filaments in Eulerian hydrodynamics is manifested, in particular, as a logarithmic divergence of the corresponding formal expression for the energy functional of an infinitely thin vortex filament having a finite circulation. More important is that the self-induced velocity of a curved string is also infinite. This is the reason, why we cannot work in the framework of usual hydrodynamics with such one-dimensional objects, that are very attractive for theoretical treatment. The situation becomes more favourable, when we consider a class of regularised models, with the divergence of the energy functional eliminated. However, dynamical properties of a desingularised system depend on the manner of regularisation. For instance, it is possible to replace in the energy functional the singular Green’s function of the Laplace operator by some analytical function which has no singular points near the real axis in the complex plane. In that case we may not expect any finite time singularity formation, because the corresponding velocity field created by the vortex string appears to be too smooth with any shape of the curve, and this fact prevents drawing together some pieces of the string. In this paper we consider another kind of regularisation of the Hamiltonian functional, when the Green’s function is still singular, but this singularity is integrable in the double contour integral

with a small but finite positive constant If is not small, we actually have models that are rather different from Eulerian hydrodynamics. Nevertheless, such models still have many common features with usual hydrodynamics, which are important for singularity formation in the process of the interaction between vortex filaments — a similar hydrodynamic type structure of the Hamiltonian and a power-like behaviour of the Green’s function, with negative exponent. Therefore we believe that it is useful to investigate these models, especially the formation of a finite time singularity in the vortex line curvature.

Finite time singularities in a class of hydrodynamic models

2.

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Hamiltonian dynamics of vortex filaments

To clarify the meaning of the suggested models (1) and to explain the employed theoretical method, let us recall some basic relations for frozen-in vorticity dynamics in a perfect fluid, starting from the Lagrangian formalism (Ruban 1999; Ruban 2001). Let a Lagrangian functional specify the dynamics of some incompressible medium of unit density, with the solenoidal velocity field Especially we are interested here in systems with quadratic Lagrangians, which in 3D Fourier representation take the form:

where M is some given positive function of the absolute value of the wave vector k. This expression should be understood as a kinetic energy on the group of volume preserving mappings. It is clear that the usual Eulerian hydrodynamics corresponds to the simplest case In the general case, the systems (2) may be viewed as models for some inviscid unusual fluids. Due to the presence of the Noether type symmetry with respect to relabelling of Lagrangian labels of fluid points, all such systems have an infinite number of integrals of motion, which can be expressed as conservation of the circulations of the canonical momentum field along any closed contour advected by flow, thus the generalised theorem of Kelvin is valid: These integrals of motion correspond to the frozen-in property of the canonical vorticity field curl In particular, the flows with frozen-in singular vortex filaments are possible. The dynamics of the shape of such infinitely thin vortex filament is determined in a self-consistent manner by the variational principle with the Lagrangian

where the vector function must have unit divergence: (Ruban 2001), and where the Green’s function in the Hamiltonian is equal to the following integral

332


The suggested Hamiltonians (1) correspond to the function The choice of the longitudinal parameter is not unique, but this does not affect the dynamics of the vortex string which is an invariant geometric object. Sometimes it is convenient to use parameterisation of the vortex line shape by the Cartesian coordinate: Then the function gives immediately that and are canonically conjugated quantities. Now, for some fixed value of the parameter let us consider the symmetrical dynamics of a pair of oppositely rotating vortex filaments, with and with the symmetry plane Due to this symmetry, it is sufficient to consider only one of the filaments. The exact expression for the Hamiltonian of this system is the following:

where is the mean distance between the two filaments. The first term in this equation describes the non-local self-interaction of the filament, while the second one corresponds to the interaction with the second filament.

3.

Singularity in long-scale nonlinear dynamics

The system with the Hamiltonian (5) possesses the exact stationary solution which describes the uniform motion of straight anti-parallel filaments. But examination of the quadratic on X and Y part of the Hamiltonian (5) has shown that this solution is unstable respectively to long-scale perturbations. The instability is an analog of the well known Crow instability. A general nonlinear analysis of the nonlocal system (5) is difficult. Therefore we require some simplified model which would approximate the nonlinear dynamics, at least in the most interesting long-scale unstable regime. Now we suggest such an approximate model and find a class of solutions describing the formation of a finite time singularity. The idea how to derive the required long-scale approximation is the following. In general, the exact expression for the Hamiltonian of a pair of singular filaments, after integration by parts, can be represented as the integral over a surface drawn between the filaments:

Finite time singularities in a class of hydrodynamic models

333

In long-scale limit we obtain from here the local nonlinear Hamiltonian

where the coordinate is measured from the symmetry plane. This Hamiltonian approximates the exact nonlocal Hamiltonian of a symmetrical pair of vortex filaments in the case when the ratio of a typical value of Y to a typical longitudinal scale L is much smaller than In particular, this means that the slope of the curve with respect to the symmetry plane should be small, and also Y should be small in comparison with the radius of the line curvature. With the Cartesian parameterisation of the curve, the corresponding approximate local nonlinear equations of motion have the following form (after appropriate time rescaling)

These equations admit the self-similar substitution

with arbitrary constants X*, t*, and with the exponent After substituting Eqs.(8) into Eqs.(7), we obtain a pair of ordinary differential equations for the functions and :

where

With using cylindrical coordinates, the general solution of this system can be represented in the following parametric form:


334

where the parameter runs between the limits C and are arbitrary integration constants. The constant C determines the asymptotic slope of the curve at large distances from the origin: when while the constant reflects the symmetry of the system with respect to rotations in plane. The condition for applicability of the local approximation (6) is satisfied if It is interesting to note that the total angle between two asymptotic directions in plane does not depend on the parameter C in the long-scale local approximation used above: At small values of this angle approaches

4.

Discussion

Thus, we observed that in the local systems (6) with finite time singularity formation is possible in the self-similar regime. Inasmuch as the condition for applicability of the approximate Hamiltonian (6) is satisfied in a range of the parameter C related to the self-similar solutions (10-12), we conclude that in the nonlocal systems (1) self-similar collapse of two symmetrical singular vortex filaments can also take place. The principal question is whether this is also possible for filaments having finite width. If yes, then such solutions are analogous to the assumed self-similar solutions of the Euler equation. Though the exponent differs from 1/2, the difference is small if is small. However, an important difference exists between infinitely thin filaments and filaments with finite width: inside the latter, longitudinal flows take place, caused by a twist of the vortex lines constituting the filament. Those flows keep the width homogeneous along the filament if a local stretching is not sufficiently fast. This mechanism acts against singularity formation and, probably, in many cases it can prevent a singularity at all. Thus, a more or less consistent analysis of the general situation should take into account, besides the dynamics of a mean shape of the filament, at least the dynamics of the width and the conjugated dynamics of the twist.

References K UZNETSOV , E.A. & R U B A N , V.P. 2000 Hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems. Phys. Rev. E 61, 831. RUBAN, V.P. 1999 Motion of magnetic flux lines in magnetohydrodynamics. JETP 89, 299. R U B A N , V.P., P ODOLSKY , D.I., & R A S M U S S E N , J.J. 2001 Finite time singularities in a class of hydrodynamic models. Phys. Rev. E 63, 056306. R UBAN , V.P. 2001 Slow inviscid flows of a compressible fluid in spatially inhomogeneous systems. Phys. Rev. E 64, 036305.

On stabilisation of solutions of singular quasi-linear parabolic equations with singular potentials Andrey MURAVNIK* Moscow Aviation Institute, Dept. of Differential Equations Russia 125993, Moscow A-80, GSP-3, Volokolamskoe shosse 4 [email protected]

Abstract We consider singular quasi-linear parabolic equations containing Bessel operator and a singular potential. We find a class of (non-classical) well-posed boundaryvalue problems for those equations and a necessary and sufficient condition of the stabilisation of their solutions.

1.

Introduction

It is well-known that for singular differential equations problems with weighted boundary-value conditions are well-posed. For example, a classical boundary-value condition for equations of principal type containing Bessel operator is boundedness (in the neighbourhood of the hyperplane of singularity) of the solution multiplied to the power function (see 6 and references therein); the power of the weight is entirely defined by the parameter at the singularity of Bessel operator. If, however, a parabolic equation contains low-order terms (even regular ones), then the situation is changed - a condition linking the weighted solution and its normal derivative on the same hyperplane (hereafter called special hyperplane) becomes well-posed (while their weights are still defined by the same parameter). In this paper we consider the case where the coefficient at the zero-order term (hereafter called potential) has a singularity (at the same hyperplane) while the equation contains so-called non-linearity of BurgersKardar-Parisi-Zhang type (see e. g. 3, 4) with a singular non-linear coefficient. It turns out in that case the weighted boundary-value con*The

author was supported by INTAS, grant 00-136


A. B. Muravnik

336

dition depends also on the potential and the coefficient at non-linearity, while solutions of a constant sign form the class of uniqueness. Further, for those solutions we prove the necessary and sufficient condition of stabilisation (in terms of non-linear weighted means of the initial-value function).

Preliminaries

2.

The following notations will be used:

According to 6 we define (for real ) the following function space: Now we have to quote (for the completeness) the results obtained in 5 for the linear case. The following problem is considered:

where The following assertions are valid: Theorem 1 There exists a classical solution of problem (1)-(3), belonging to and that solution is unique.

Theorem 2 Let

Then for any if and only if

Singular quasi-linear parabolic equations with singular potentials

337

where

Finally, we consider the following problem:

Since is continuous and bounded on and then (see 1) there exists a unique classical bounded solution of (4)-(6); we will denote that as The following relation is known from 5:

3.

Unique solvability We consider equation

together with initial-value condition (2) and boundary-value condition

Here

is non-negative and belongs to

to equation belonging to to check that under the above assumptions about least one such root). Let

denote Then is bounded too Apart from problem (8)-(2)-(9) we consider equation

belongs is any root of the (it is easy there exists at is bounded

338

A. B. Muravnik

together with initial-value condition (5) and boundary-value condition

Since then, by the virtue of Theorem 1, there exists a unique classical solution of (10)-(5)-(11), belonging to let us denote that solution by v(x,t). (7) and positivity of the solution of problem (4)-(6) in case of non-negative initial-value function (see 7) imply the positivity of by cause

in

and define is positive. Then

bounded

Let us denote as - it is well-defined beis positive too. Moreover, is

is bounded (since it is equal to

Further,

therefore

direct substitution to (10) yields:

hence, by the virtue of positivity of a + 1 and

i. e. is a classical solution of equation (8) and belongs to the space Further, by the virtue of continuity of and smoothness of

for any

hence

satisfies initial-value condition (2) as well.

Finally,

because is positive and satisfies condition (11). So constructed above function satisfies problem (8)-(2)-(9). Thus the existence of classical positive solution of problem (8)-(2)-(9) belonging to is proved.

Singular quasi-linear parabolic equations with singular potentials

339

In order to prove its uniqueness we suppose to the contrary that and are classical positive solutions of problem (8)-(2)-(9), belonging to and define and as and respectively (they are well-defined because and are positive). Then, as we have shown above, and belong to and are classical solutions of problem (10)-(5)-(11) with the same initial-value function from Therefore by the virtue of Theorem 1 Thus the following assertion is proved: Theorem 3 There exists a unique classical positive solution of the problem (8)-(2)-(9) belonging to

4.

Stabilisation Let

and let from the space be the classical positive solution of the problem (8)-(2)-(9). Then let denote (as in the previous section) the classical solution of (10)-(5)-(11) belonging where the initial-value function is equal to finally, let A denote

Then by the virtue of Theorem 2 we have (for any non-negative l) the following assertion: for any if and only if

Since

it follows that for any if and only if

However

from

340

Now we can denote

A. B. Muravnik

by B, and this yields the following neces-

sary and sufficient condition of the stabilisation of solution for problem (8)-(2)-(9): Theorem 4 Let

Then for any if and only if

where

Remark. Summability of the potential with respect to t could be replaced by a weaker assumption of the convergence of

the last

assumption, however, is substantial. Otherwise the nature of the problem is in general changed on an essential way: even in case of heat equation a weighted stabilisation with an exponential weight with respect to t takes place. In our case all the weights (including the stabilisation condition) depend on spatial variables and are generated by the singularity of the equation while dissipation generates no weights here. The author is grateful to A. L. Skubachevskii for useful considerations.

References KIPRIJANOV, I. A., KATRAHOV, V. V. & LJAPIN, V. M. 1976 Boundary value problems in domains of general form for singular parabolic systems of equations. Sov. Math. Dokl. 17, No. 5, 1461–1464. POHOŽAEV, S. I. 1982 Equations of the type Math. USSR Sb. 41, No. 2, 269–280. KARDAR, M., PARISI, G. & ZHANG, Y.-C. 1986 Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892. MEDINA, E., HWA, T., KARDAR, M. & ZHANG, Y.-C. 1989 Burgers equation with correlated noise: Renormalization group analysis and applications to directed polymers and interface growth. Phys. Rev. Lett. A39, 3053–3075. MURAVNIK, A. B. 1994/1995 On asymptotic behaviour of solutions of singular parabolic equations with singular potential. Reports of Mittag-Leffler Inst. 11. KIPRIJANOV, I. A. 1997 Singular elliptic boundary problems. Nauka [in Russian]. MURAVNIK, A. B. On stabilization of solutions of singular quasi-linear parabolic problems. Math. Notes (to appear).

VI

STOKES FLOW AND SINGULAR BEHAVIOUR NEAR BOUNDARIES


Interactions between two close spheres in Stokes flow 1

, Nicolas LECOQ2, René ANTHORE2, François BOSTEL & François FEUILLEBOIS 3 2

1

Institute of Fundamental Technological Research, Polish Academy of Sciences, 21, 00-049 Warsaw, Poland [email protected]

2

Université de Rouen, UMR 6634 CNRS Place E. Blondel, 76821 Mont Saint Aignan Cedex, France

3

PMMH, École Supérieure de Physique et de Chimie Industrielles 10 rue Vauquelin, 75231 Paris Cedex 05, France

Abstract We investigate if two close spheres in a fluid flow at low-Reynolds-number can touch each other and interact mechanically. We outline how this problem relates to microhydrodynamics of suspensions. We measure the translational and rotational motion of a sphere, which settles in a silicon oil onto another, fixed sphere of the same size. We use simultaneously a video system and a laser interferometer coupled with encoders. We calculate the motion, assuming that the particles come into contact and that the mechanical interactions superpose with the gravitational and hydrodynamic forces. The experiment confirms the model and determines its parameters.

When spheres are tending to impact, The question is: do they make contact? When stick turns to slip, The forces can dip; And this is the tough nut that I've cracked.

1.

Introduction

Theoretical description of structure and macroscopic properties of suspensions is based on the calculation of hydrodynamic interactions between particles immersed in a fluid (Felderhof 1990). Such interactions increase indefinitely with the decrease of the gap between the particle surfaces (Jeffrey & Onishi 1984). Therefore methods of numerical computations and values of the suspension transport coefficients are sensitive to the nature of other than hydrodynamic small-distance interactions, e.g. the existence of mechanical contact (Wilson & Davis 2000).


344

M.L.

N. Lecoq, R. Anthore, F. Bostel, F. Feuillebois

2.

Hydrodynamic interactions

Consider a group of N spherical rigid particles translating and rotating in a fluid flow characterised by the low Reynolds number, Re

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