Theory of Concentrated Vortices

S.V. Alekseenko · P.A. Kuibin · V.L. Okulov

Theory of ConcentratedVorticesAn Introduction

With 233 Figures and 12 Tables

Professor S.V. AlekseenkoProfessor P.A. KuibinProfessor V.L. Okulov

Russian Academy of SciencesSiberian BranchInstitute of ThermophysicsLavrentyev Avenue 1630090 NovosibirskRussia

Translated from the first Russian Edition “Bведенuе в meopuю кoнценmpupoвaнныxвuxpeй” (Hoвocuбupcк, Iнcmumym menлoфuзuкu CO PAH, 2003).

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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 cg group velocity d diameter E Euler constant er, eθ, ez triple of unit orthogonal vectors in a cylindrical coordinate

system F force f frequency, function g mass force H helicity H Bernoulli constant, Hamiltonian I vortex momentum Im, Km modified Bessel functions i, j, k triple of unit orthogonal vectors k wave number, parameter L cut-off length L1, L2, L3 Lame coefficients 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)

Ri Richardson number 2g d dW

dr dr

−⎛ ⎞ρ ⎛ ⎞=⎜ ⎟⎜ ⎟⎜ ⎟ρ ⎝ ⎠⎝ ⎠

Ro Rossby number 2Wk⎛ ⎞=⎜ ⎟Ω⎝ ⎠

XII Nomenclature

S area, swirl parameter s arc length, distance T kinetic energy, tension force, parameter t time t, n, b triple of unit orthogonal vectors (tangent, normal and

bi-normal) U, V, u, v velocity vectors u, v, w, U, V, W velocity vector components in Cartesian cylindrical coordi-

nate systems V volume W complex potential x, y, z Cartesian coordinate systems z complex variable (= x + iy, = z1 + iz2)

Greece symbols

α, β angles, parameters Γ circulation, vortex intensity δ( ) Dirac's delta-function ε vortex core radius, small parameter ζ complex variable (= ξ + iη, = ζ1+ iζ2) θ phase, angle κ curvature ν kinematic viscosity, parameter ρ radius, density τ torsion, relative pitch ϕ potential, angle χ variable (= θ – z/l) χ vortex sheet intensity ψ, Ψ stream function ω vorticity, frequency ω vorticity vector Ω frequency, solid angle Ω angular velocity Bold type signifies complex or vector character.

Introduction

The class of concentrated vortices is distinguished among a great diversity of vortex flows and attracts attention from the point of fundamental re-search and practical applications. There is no an accurate definition for a concentrated vortex (actually, there is no concept of vortex in general). A concentrated vortex can be rigorously defined for an ideal fluid: this is a space-localized zone with non-zero vorticity surrounded by potential flow. Certainly, this definition does not cover the observed multitude of vortex phenomena. In this book, we would rely more on intuitive comprehension, taking concentrated vortices to be the vortex motion for which the vorticity is bound to spatial zones with localization occurring for at least one di-mension. The most illustrative examples of concentrated vortices are the following idealized objects: vortex sheet (localization in one dimension), infinitely thin vortex filament and its 2D analog – point vortex (localiza-tion in two dimensions), infinitely thin vortex ring of a finite diameter – closed vortex filament, vorton (localization in three dimensions). The more complex objects – a columnar vortex of Rankine vortex type (constant vor-ticity in a core of finite radius), fat vortex ring, Hill's vortex, and Hicks vortex – all of them have a non-zero volume with non-zero vorticity.

In more complex cases vorticity is non-zero over the entire space; none-theless a vortex core is easily distinguishable by much higher vorticity than in the rest of space. This is typical for viscous flow when vorticity diffu-sion takes place, and the Burgers vortex is a classic example.

A great variety of vortex flows can be realized in nature and technology. Many of them can be interpreted as concentrated vortices depending on the extent of their similarity to the above mentioned idealized objects. Un-doubtedly, one of the most common types of concentrated vortices is a co-lumnar vortex, or filament-type vortex. This is supported by Table I.1, pre-senting a short list of similar phenomena, along with some illustrations (Figs. I.1–I.4, Color Fig. I.11). This book focuses mainly on the mentioned types of vortex motion. Alternatively, these vortices are also termed the elongate concentrated vortices. In addition, vortex rings are considered as well (Fig. I.5), since in many aspects they are similar to elongate vortices,

1 All figures in color are available in Color plates.

2 Introduction

as well as point vortices. The analysis of the dynamics of point vortices would help us to explain some features of elongate vortices, especially when they interact with each other or with a solid surface.

In the literature there are no works devoted specifically to elongate con-centrated vortices. These problems are highlighted most comprehensively in the book “Vortex Dynamics” (Saffman 1992).

It is pertinent to also note the books by Villat (1930), Joukowski (1937a, b), Lamb (1932), Kochin et al. (1964), Milne-Thomson (1938), Sedov (1997), Batchelor (1967), Lavrentyev and Shabat (1973), Loitsyanskii (1966), Goldshtik (1981), Lugt (1983), Gupta et al. (1984), Ting and Klein (1991), Meleshko and Konstantinov (1993), Lugt (1996), Kozlov (1998), and reviews by Hall (1966), Widnall (1975), Saffman and Baker (1979), Leibovich (1984), Spalart (1998), Andersson and Alekseenko (2002).

Table I.1. Examples of concentrated vortices

Phenomenon number

Picture in Fig. I.1

Description References

1 a Whirlpool in liquid flowing out a container through bottom orifice

(Van Dyke 1982)

2 b Tornado (Snow 1984) 3 c Vortices in flow over a delta wing

under a high attack angle (Payne et al. 1988)

4 – Longitudinal vortices in a turbulent boundary layer

(Kim et al. 1971)

5 d Longitudinal vortices in a boundary layer behind a body on a plane

(Tani et al. 1962)

6 e Set of vortex ropes formed behind the asymmetric jet injected into cross flow

(Wu et al. 1988)

7 f Set of vortex ropes in a rotating liq-uid layer heated from below

(Boubnov and Golitsyn 1986)

8 g Vortex ropes in the uprising vapor flow above a rotating liquid

(Vladimirov 1977b)

9 – Vortex filaments in the turbulence model for superfluid helium

(Donelly 1988)

10 – Vortex filaments – bridges between vortex ropes in the wake behind a multi-blade propeller

(Larin and Mavritskii 1971)

11 – Vortex rings – closed vortex filaments (Widnall 1975) 12 h Vortex filaments in flow over a

dimple (Kiknadze et al. 1986)

Introduction 3

Fig. I.1. Examples of generation of concentrated vortices (see comments in Table I.1). f – vortex core

The description of concentrated vortices summarized in a single book is deemed by importance of concerned problem rather than by deficiency of certain books covering the subject. The concept of vortex filament is one of the fundamental concepts of fluid dynamics. The vortex filament (point vortex) is a simple and convenient model for describing real vortices. Moreover, this is a basis for developing mathematical models for more complex vortex flows (e.g., the Method of Point Vortices (Belotserkovsky and Nisht 1978) and the Model for a Flow with Helical Symmetry (Alek-seenko et al. 1999)).

4 Introduction

In reality, the concentrated vortices of the vortex filament class almost never have a rectilinear axis due to arising from different instabilities and the capacity of the vortex cores to be a waveguide (i.e., to transfer distur-bances). The disturbed states are characterized by a wide spectrum of dif-ferent modes – axisymmetric, bending, etc. but the most typical forms of disturbances are those with a helical or spiral shape (Fig. I.2b). The key mechanism for the propagation of these disturbances is a self-induced motion. This mechanism is also responsible for the motion of vortex rings and propagation of nonlinear wave packets known as a vor-tex soliton (or Hasimoto soliton) described (in first approximation) by the cubic Schrödinger equation. Existence of solid surfaces, other vor-tices, type of basic flow field – have a strong impact on the behavior of a concentrated vortex.

It seems that the most striking and fascinating phenomenon is a vortex breakdown. This phenomenon is manifested in a sudden deviation of the concentrated vortex axis from the current direction or it manifests in a sudden expansion of the vortex core with formation of counter flow zones. The classic examples of vortex breakdown are presented in Fig I.4 for the case of flow over a delta wing and in Fig. I.6 - for a swirl flow in a slightly diverging pipe. There are many types of vortex breakdown, but the most frequent are the bubble type (Fig. I.6a, I.2d) and spiral type (Fig. I.6b, I.2c) breakdown. Vortex breakdown brings a radical restruc-turing of flow and it is significant for transfer processes and performance of industrial heat- and mass-transfer apparatuses of a vortex type (Alek-seenko and Okulov 1996).

The problem of vortex breakdown description has been one of main in-centives for the study of concentrated vortex stability. In the book pre-sented, the authors do not consider the theory of vortex breakdown be-cause of its incompleteness. The main focus is on systematic description of experimental data in Section 7.6. The outline of this problem is available in several reviews: Hall (1972), Leibovich (1978, 1984), Escudier (1988), Althaus and Weimer (1997).

Concentrated vortices are related to coherent structures, which may be identified as vortices and have a key role in the processes of laminar-turbulent transition as well as in developed turbulent flow (Boiko et al. 2002; Kachanov 1994). Most important of those are the longitudinal vortices in a turbulent boundary layer (Kim et al. 1971), and horseshoe-shaped vortex structures (see review by Cantwell (1981)). The coherent structures in the form of vortex rings are clearly distinguishable in axisymmetric free shear flows. This kind of example for an impinging jet is shown in Color Fig. I.2, where experimental data obtained by Markovich (2003) is plotted in terms of velocity vector field as well as vorticity field. The concept of vortex rrggggggggggggggggggggggggggggggggggggggggggggggggggggggrrrggggggggrrrrrr

Introduction 5

a

40 m

m

b

c

d

Fig. I.2. Undisturbed (a) and disturbed (b – d) vortex filaments. a, b – tangential hydraulic chamber (Alekseenko and Shtork 1992*)2; c – swirl air jet, Re = 1.4⋅104, nozzle diameter 152 mm (Panda and McLaughlin 1994*); d – chamber with rotat-ing bottom of diameter 91.3 mm (Spohn et al. 1998*)

a b

Fig. I.3. Formation of vortex filaments in a boundary layer ahead of an obstacle (cylinder) with slot suction (Seal and Smith 1997*): a – flow visualization with hy-drogen bubbles; b – diagram, slot with sizes 64 × 2 mm is placed at 88.5 mm from the cylinder with diameter 89 mm

2 For references marked by the asterisk see the copyright and permission no-tices in the reference list.

6 Introduction

Fig. I.4. Bubble (above) and spiral (below) types of vortex breakdown in flow over a delta wing. Dye visualization in a water channel (Lambourne and Bryer 1961*). Flow velocity is 5.1 cm/s

a b c

Fig. I.5. Laminar (a) and turbulent (b, c) vortex rings. Smoke visualization (Akhmetov 2001)

a

b

Fig. I.6. Bubble (а) and spiral (b) breakdown of a vortex. Dye visualization of flow in a slightly diverging pipe (Sarpkaya 1971*)

Introduction 7

filaments (quantum vortices) was especially fruitful in developing the Tur-bulence Theory for Superfliud Helium (Donelly 1988; Nemirovskii and Tsubota 2000).

Concentrated vortices play an important (and often dominant) role in technical applications. For example, the design of the vortex flow meter in-volves measuring the liquid flow rate through the precession frequency of a concentrated vortex in swirl flow. The generation of precessing vortex ropes behind the hydroturbine wheel may induce high pressure pulsations and re-sult in catastrophic consequences. The complex vortex structures were dis-covered in the Ranque-Hilsh vortex tube (see Fig. I.7) (Arbuzov et al. 1997; Piralishwili et al. 2000). Non-stationary vortex structures are significant for combustion processes in vortex furnaces and vortex burners (Gupta at al. 1984; Alekseenko and Okulov 1996). The formation of vortex ropes and their breakdown in flow over a delta wing can influence the lift force and wing control. The key mechanism of heat transfer enhancement on a surface with dimples concludes in the formation of elongate concentrated vortices often called “tornado-like structures”.

Among the natural phenomena a tornado (see Color Fig. I.1) (Nalivkin 1969) and its small-scale analog – a whirlpool as well as a “dust devil” are the most revealing examples related to concentrated vortices. The large-scale phenomena like oceanic vortices or atmospheric cyclones (anti-cyclones) also belong to the category of concentrated vortices. However, their scale is comparative (or larger) to the layer thickness of the atmos-phere/ocean, so their description is a special subject.

The concentrated vortices are revealed even at astrophysical level. The hydrodynamic mechanism of forming galactic spiral structures is related to the generation of nonlinear local disturbances similar to the Rossby vor-tices and they are the source of spiral waves in a galaxy disc (Nezlin and Snezhkin 1990; Alekseenko and Cherep 1994).

All these features of real concentrated vortices demonstrate complexity and variety of their behavior; this creates great difficulties both in develop-ing mathematical description and experimentation. That is why the theory of concentrated vortices is based mainly upon approximate mathematical mod-els. The common approach to the description of dynamics of a deformed elongate vortex is based on the Biot-Savart Law in approximation of a thin vortex filament; although the low-disturbance states of a columnar vortex can be calculated with rather simple analytical or numerical methods on the basis of the exact equations of Euler or Navier-Stokes. As for experi-mental research, there is quite a limited number of works presenting tenta-tive results on the stability and dynamics of concentrated vortices, accept-able for comprehensive validation of theoretical models.

8 Introduction

a

Fig. I.7. Double spiral vortex in the Ranque-Hilsh vortex tube (Arbuzov et al. 1997*): a – flow diagram; b – Hilbert-visualization with exposition of 2.5⋅10-4 s (visualization of spatial gradient of optical density), chamber with rectangular cross-section of 34×34 mm

The above mentioned difficulties explain why so far there is no compre-hensive knowledge of concentrated vortex dynamics. This book presents the authors’ collection and systemization of knowledge, which should en-able the reader to understand the important features of concentrated vor-tices. At the same time we have not here considered the myriad of exam-ples and applications, which could be suitable for another book (or even book series).

b