Theory ofConcentrated Vortices · · S.V. Alekseenko P.A. Kuibin V.L. Okulov Theory of Concentrated Vortices An Introduction With 233 Figuresand 12 Tables ProfessorS.V.Alekseenko ProfessorP.A.Kuibin ProfessorV.L.Okulov RussianAcademyofSciences SiberianBranch InstituteofThermophysics LavrentyevAvenue1 630090Novosibirsk Russia Translated from the first Russian Edition “Bведенuе в meopuю кoнценmpupoвaнныx вuxpeй”(Hoвocuбupcк,Iнcmumymmenлoфuзuкu COPAH,2003). LibraryofCongressControlNumber:2007930219 ISBN978-3-540-73375-1 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright. 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Typesetting:bytheauthor Production:IntegraSoftwaresServicesPvt.Ltd.,India Coverdesign:ErichKirchner,Heidelberg Printedonacid-freepaper SPIN:11371441 543210 Preface Vortex motion is one of the basic states of a flowing continuum. Interest- ingly, in many cases vorticity is space-localized, generating concentrated vortices. Vortex filaments having extremely diverse dynamics are the most characteristic examples of such vortices. Notable examples, in particular, include such phenomena as self-inducted motion, various instabilities, wave generation, and vortex breakdown. These effects are typically mani- fested as a spiral (or helical) configuration of a vortex axis. Many publications in the field of hydrodynamics are focused on vortex motion and vortex effects. Only a few books are devoted entirely to vor- tices, and even fewer to concentrated vortices. This work aims to highlight the key problems of vortex formation and behavior. The experimental ob- servations of the authors, the impressive visualizations of concentrated vortices (including helical and spiral) and pictures of vortex breakdown primarily motivated the authors to begin this work. Later, the approach based on the helical symmetry of swirl flows was developed, allowing the authors to deduce simplified mathematical models and to describe many vortex phenomena. The major portion of this book consists of theoretical studies of vortex dynamics. The final chapter presents detailed results of experimentally observed concentrated vortices that provide the basis for analysis and stimulate development of vortex theory. The mathematical description of the dynamics of concentrated vortices is hindered by the requirement to consider three-dimensional and nonlinear effects, singularity, and various instabilities. For each particular problem, very different coordinate frames and equation systems must be used. Therefore the authors decided to open the work with a description of the basic laws of vortex motion and list in detail the flow equations of incom- pressible fluids1 in various reference systems (Chapter 1), even though this material may also be found in other books on fluid flow. Special attention is paid to flows with helical symmetry2, because the condition of helical symmetry makes it possible to simplify appreciably the formulation of problems and their solution, representing at the same time the properties of real flows, as shown in Chapter 7. When possible, all mathematical trans- 1 More detailed description of special models of compressible fluids flow can be found, for instance, in the work by Ovsyannikov (1981). 2 Additional information can be found in the book by Vasyliev (1958). VI Preface formations and analytical calculations, both in the first and subsequent chapters, are fully presented for the reader’s information and convenience. Chapter 2 can be considered as the key section, since it describes an in- finitely thin vortex filament - the fundamental object of the vortex motion theory. The Biot-Savart Law, that is the fundamental law of vortex fila- ment dynamics, as well as the self-induction mechanism of the filament motion are also presented in the second Chapter. Chapter 3 deals with principle models of vortex structures, which are of interest in themselves but also serve as a basis for considering more com- plex problems in the following chapters. Chapter 4 is devoted to stability analyses and waves on columnar vortices. The analyses have been carried out mainly using linear approximations. This allowed the authors to obtain exact solutions for different types of basic vor- tices and various modes, such as axially symmetric and bending modes. Chapter 5, titled “Vortex Filament Dynamics”, presents approximate methods of description because strongly nonlinear perturbations of vortex filament are addressed and analyzed therein. The principal approximate approaches used are the cut-off method and force balance method. A num- ber of examples are presented, including Hasimoto soliton. An introduction to vortex methods of flow calculation is presented in Chapter 6. Various mechanisms of vortex interactions are described and discussed. The possibilities of using vortex methods are shown for model- ing the nonlinear stage of instability development in shear flow, such as a classical shear layer, a starting vortex and a wake behind a plane. A model for the initiation of vortex precession in a cylindrical tube is proposed. In Chapter 7, which is based predominantly on the works of the authors and their colleagues, experimental results on observations of concentrated vortices obtained using laboratory equipment are shown. The major aim of this section is to show the existence of helical symmetry in real swirl flows and to illustrate theoretical fundamentals by means of experimental exam- ples of elongate concentrated vortices. The authors hope that this book will serve as an introduction to the the- ory of concentrated vortices and will be helpful for experts interested in vortex dynamics. Some of the authors results presented in this book were supported by The Russian Foundation for Basic Research (RFBR) under the grants 94- 02-05812, 96-01-01667, 97-05-65254, 00-05-65463, 01-01-00899; Grant of The President of The Russian Federation for the Support of Young Pro- fessors - 96-15-96815, grant 95-1149 by RFBR-INTAS, grant 00-00232 by INTAS, and grant 00-15-96810 by The Council for the Support of Leading Research Schools. All of these grants are gratefully acknowledged. The authors also would like to express their sincere gratitude to Mrs. E. Trifonova and Mrs. V. Bykovskaya, who kindly undertook the hard work of the manuscript preparation. Contents Introduction................................................................................................1 1 Equations and laws of vortex motion....................................................9 1.1 Vorticity. Circulation........................................................................9 1.2 Dynamics of vortical fluid..............................................................13 1.2.1 Equations of ideal fluid motion...............................................13 1.2.2 Theorems of motion for an ideal vortical fluid........................15 1.2.3 Bernoulli theorem....................................................................18 1.2.4 Equations of viscous fluid motion...........................................19 1.3 Equations of fluid motion in orthogonal coordinates.....................20 1.3.1 Arbitrary orthogonal system of curvilinear coordinates..........20 1.3.2 Cartesian coordinate system....................................................23 1.3.3 Cylindrical coordinate system.................................................24 1.3.4 Spherical coordinate system....................................................26 1.4 Special cases of vortex motion.......................................................28 1.4.1 Helical flows (Beltrami flows)................................................28 1.4.2 Two-dimensional flows...........................................................30 1.4.3 One-dimensional flows............................................................37 1.5 Flows with helical symmetry..........................................................39 1.5.1 Derivation of equations...........................................................39 1.5.2 Flow with helical vorticity.......................................................40 1.5.3 Helical flows with helical symmetry of the velocity field.......43 1.6 Velocity field at specified distribution of sources and vortices......45 1.7 Vortex forces and invariants of vortex motion...............................49 1.7.1 Vortex forces...........................................................................49 1.7.2 Vortex momentum and vortex angular momentum.................56 1.7.3 Kinetic energy.........................................................................61 1.7.4 Helicity....................................................................................62 1.7.5 Invariants of two-dimensional flows.......................................64 2 Vortex filaments....................................................................................69 2.1 Geometry of vortex filaments.........................................................69 2.2 Biot – Savart law............................................................................73 2.3 Rectilinear infinitely thin vortex filament......................................76 VIII Contents 2.3.1 Vortex filament in ideal fluid..................................................76 2.3.2 Vortex filament diffusion........................................................80 2.4 Self-induced motion of a vortex filament.......................................82 2.5 Infinitely thin vortex ring...............................................................86 2.6 Infinitely thin helical vortex filament.............................................91 2.6.1 Helical vortex filament in infinite space..................................91 2.6.2 Helical vortex filament in a cylindrical tube...........................96 3 Models of vortex structures...............................................................111 3.1 Vortex sheet..................................................................................111 3.2 Spatially localized vortices...........................................................116 3.2.1 Vortex ring.............................................................................116 3.2.2 Hill’s spherical vortex...........................................................124 3.2.3 Hicks spherical vortex...........................................................127 3.3 Columnar vortices in ideal fluid...................................................134 3.3.1 Rankine vortex.......................................................................134 3.3.2 Gauss vortex..........................................................................136 3.3.3 One-dimensional helical flow................................................136 3.3.4 One-dimensional (columnar) helical vortices........................137 3.3.5 Q-vortex.................................................................................145 3.3.6 Helical vortex with a finite-sized core...................................146 3.4 Viscous models of vortices...........................................................149 3.4.1 Burgers vortex.......................................................................149 3.4.2 Sullivan vortex.......................................................................153 4 Stability and waves on columnar vortices........................................155 4.1 Types of perturbations..................................................................155 4.2 Intsability of a vortex sheet...........................................................157 4.3 Waves in fluids with solid-body rotation......................................160 4.3.1 Plane waves...........................................................................160 4.3.2 Axisymmetrical waves..........................................................165 4.3.3 Taylor column.......................................................................167 4.4 Linear instability of Rankine vortex with an axial flow...............170 4.4.1 Dispersion relationships........................................................170 4.4.2 Linear analysis of temporal instability..................................176 4.4.3 Linear analysis of spatial instability......................................185 4.5 Kelvin waves................................................................................186 4.5.1 Dispersion equations.............................................................187 4.5.2 Axisymmetric mode, m = 0...................................................188 4.5.3 Bending mode, m = 1............................................................190 4.5.4 Evolution of initially localized perturbations. Mechanisms of wave propagation..................................................194 Contents IX 4.6 Instability of Q-vortex. Instability criteria...................................202 4.6.1 Instability criteria...................................................................202 4.6.2 Instability of Q-vortex. Inviscid analysis..............................204 4.6.3 Instability of Q-vortex. Viscous analysis..............................211 4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)....................................................................................214 4.7.1 Axisymmetrical nonlinear standing waves............................215 4.7.2 Axisymmetrical weakly-nonlinear traveling waves..............220 4.7.3 Bending waves.......................................................................225 5 Dynamics of vortex filaments.............................................................235 5.1 Cut-off method.............................................................................235 5.2 Self-induced motion of helical vortex filament with an arbitrary pitch..........................................................243 5.3 Hasimoto soliton...........................................................................257 5.4 Application of momentum balance to description of vortex filament dynamics............................................267 5.4.1 Forces acting on a vortex filament.........................................267 5.4.2 Derivation of force-balance equations...................................270 5.4.3 Hollow vortex........................................................................279 5.4.4 Vortex filament with an inner structure.................................282 5.4.5 Consideration of the inner core structure...............................287 5.4.6 Modified equations of vortex filament motion......................290 5.5 The method of matched asymptotic expansions...........................291 5.5.1 Derivation of the equation for vortex filament motion..........292 5.5.2 Local induction approximation..............................................297 5.5.3 N-soliton solution..................................................................300 5.5.4 Comments..............................................................................306 6 Dynamics of two-dimensional vortex structures..............................309 6.1 The method of discrete vortex particles........................................309 6.1.1 Motion equations of vortex particles in infinite liquid..........309 6.1.2 Motion equations of vortex particles in limited simply-connected domains.............................................316 6.1.3 Motion equations of the system of co-axial vortex rings.......324 6.2 Motion of the system of rectilinear vortices.................................328 6.2.1 Interaction of two identical vortices at various initial distances...............................................................329 6.2.2 Interaction of two vortices of the same size but with different circulations.........................................................332 X Contents 6.2.3 Interaction of two vortices of the same circulation but with different sizes...................................................................332 6.2.4 Interaction of three vortices with circulations of the same sign..........................................................333 6.2.5 Interaction of two vortices with circulations of contrary signs..........................................................334 6.2.6 Interaction of three vortices with circulations of contrary signs. Vortex collapse..............................338 6.3 Modeling the dynamics of shear flows.........................................341 6.3.1 Mechanisms of formation for the large vortices in the shear layer................................................341 6.3.2 Instability of a starting vortex................................................347 6.3.3 Wake instability behind a thin plate......................................357 6.4 Motion of vortices in cylindrical tubes.........................................366 6.4.1 Motion equations for vortex particles in a circular domain...367 6.4.2 Precession of a rectilinear vortex in a tube............................368 6.4.3 Motion of a helical vortex in a tube.......................................374 7 Experimental observation of concentrated vortices in vortex apparatus................................................................................................379 7.1 Experiment methods.....................................................................379 7.1.1 Experiment equipment...........................................................379 7.1.2 Parameters of a swirling flow................................................383 7.2 Helical symmetry of vortex flows................................................386 7.3 Concentrated vortex with a rectilinear axis..................................390 7.3.1 Generation of concentrated vortices......................................390 7.3.2 Vortex composition...............................................................403 7.4 Precession of a vortex core...........................................................409 7.5 Stationary helical vortices............................................................417 7.5.1 Single helical vortices............................................................417 7.5.2 Double helix..........................................................................422 7.6 Perturbations of a vortex core.......................................................426 7.6.1 Waves on concentrated vortices............................................426 7.6.2 Vortex breakdown in a channel.............................................431 7.6.3 Vortex breakdown in a container with a rotating lid.............445 References...............................................................................................467 Index.......................................................................................................485 Nomenclature A amplitude A vector potential a radius c phase velocity c group velocity g d diameter E Euler constant e , e , e triple of unit orthogonal vectors in a cylindrical coordinate r θ z system F force f frequency, function g mass force H helicity H Bernoulli constant, Hamiltonian I vortex momentum I , K modified Bessel functions m m i, j, k triple of unit orthogonal vectors k wave number, parameter L cut-off length L , L , L Lame coefficients 1 2 3 l helix pitch M vortex angular momentum m azimuthal wave number p pressure Q flow rate R radius R radius-vector r, θ, z cylindrical coordinate system Re Reynolds number ( = Wd/ν) ⎛ gdρ⎛dW⎞−2⎞ Ri Richardson number ⎜= ⎜ ⎟ ⎟ ⎜ ρ dr⎝ dr ⎠ ⎟ ⎝ ⎠ ⎛ Wk⎞ Ro Rossby number ⎜= ⎟ ⎝ 2Ω ⎠