Theory of Concentrated Vortices

Theory of Concentrated Vortices S.V. Alekseenko · P.A. Kuibin · V.L. Okulov Theory of Concentrated Vortices An Introd...

Author: S.V. Alekseenko | P.A. Kuibin | V.L. Okulov

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Theory of Concentrated Vortices

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

Theory of Concentrated Vortices An Introduction

With 233 Figures and 12 Tables

Professor S.V. Alekseenko Professor P.A. Kuibin Professor V.L. Okulov Russian Academy of Sciences Siberian Branch Institute of Thermophysics Lavrentyev Avenue 1 630090 Novosibirsk Russia

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

Library of Congress Control Number: 2007930219 ISBN 978-3-540-73375-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author Production: Integra Softwares Services Pvt. Ltd., India Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

SPIN: 11371441

543210

Preface

Vortex motion is one of the basic states of a flowing continuum. Interestingly, 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 manifested 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 vortices, and even fewer to concentrated vortices. This work aims to highlight the key problems of vortex formation and behavior. The experimental observations 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 incompressible 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 trans1

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 infinitely thin vortex filament - the fundamental object of the vortex motion theory. The Biot-Savart Law, that is the fundamental law of vortex filament 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 complex 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 vortices 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 number 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 modeling 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 examples of elongate concentrated vortices. The authors hope that this book will serve as an introduction to the theory 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 9402-05812, 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 Professors - 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 A a c cg d E er, eθ, ez

H I Im, Km i, j, k k L L1, L2, L3 l M m p Q R R r, θ, z Re

amplitude vector potential radius phase velocity group velocity diameter Euler constant triple of unit orthogonal vectors in a cylindrical coordinate system force frequency, function mass force helicity Bernoulli constant, Hamiltonian vortex momentum modified Bessel functions triple of unit orthogonal vectors wave number, parameter cut-off length Lame coefficients helix pitch vortex angular momentum azimuthal wave number pressure flow rate radius radius-vector cylindrical coordinate system Reynolds number ( = Wd/ν)

Ri

Richardson number ⎜ =

F f g

H

Ro

⎛

g dρ ⎛ dW ⎞ ⎜ ⎟ ρ dr ⎝ dr ⎠

⎜ ⎝ ⎛ Wk ⎞ Rossby number ⎜ = ⎟ ⎝ 2Ω ⎠

−2

⎞ ⎟ ⎟ ⎠

XII

Nomenclature

S s T t t, n, b

area, swirl parameter arc length, distance kinetic energy, tension force, parameter time 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 coordinate systems V volume W complex potential x, y, z Cartesian coordinate systems z complex variable (= x + iy, = z1 + iz2) Greece symbols α, β Γ δ( ) ε θ κ ν ρ τ ϕ χ ψ, Ψ

angles, parameters circulation, vortex intensity Dirac's delta-function vortex core radius, small parameter complex variable (= ξ + iη, = ζ1+ iζ2) phase, angle curvature kinematic viscosity, parameter radius, density torsion, relative pitch potential, angle variable (= θ – z/l) vortex sheet intensity stream function vorticity, frequency vorticity vector frequency, solid angle 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 research 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 dimension. 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 (localization 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 vorticity 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; nonetheless a vortex core is easily distinguishable by much higher vorticity than in the rest of space. This is typical for viscous flow when vorticity diffusion 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. Undoubtedly, one of the most common types of concentrated vortices is a columnar vortex, or filament-type vortex. This is supported by Table I.1, presenting 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 concentrated 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 Picture Description number in Fig. I.1 1 a Whirlpool in liquid flowing out a container through bottom orifice 2 b Tornado 3 c Vortices in flow over a delta wing under a high attack angle – Longitudinal vortices in a turbulent 4 boundary layer 5 d Longitudinal vortices in a boundary layer behind a body on a plane 6 e Set of vortex ropes formed behind the asymmetric jet injected into cross flow 7 f Set of vortex ropes in a rotating liquid layer heated from below 8 g Vortex ropes in the uprising vapor flow above a rotating liquid – Vortex filaments in the turbulence 9 model for superfluid helium 10 – Vortex filaments – bridges between vortex ropes in the wake behind a multi-blade propeller 11 – Vortex rings – closed vortex filaments 12 h Vortex filaments in flow over a dimple

References (Van Dyke 1982) (Snow 1984) (Payne et al. 1988) (Kim et al. 1971) (Tani et al. 1962) (Wu et al. 1988)

(Boubnov and Golitsyn 1986) (Vladimirov 1977b) (Donelly 1988) (Larin and Mavritskii 1971) (Widnall 1975) (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 (Alekseenko 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 disturbances). The disturbed states are characterized by a wide spectrum of different 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 selfinduced motion. This mechanism is also responsible for the motion of vortex rings and propagation of nonlinear wave packets known as a vortex soliton (or Hasimoto soliton) described (in first approximation) by the cubic Schrödinger equation. Existence of solid surfaces, other vortices, 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 restructuring of flow and it is significant for transfer processes and performance of industrial heat- and mass-transfer apparatuses of a vortex type (Alekseenko and Okulov 1996). The problem of vortex breakdown description has been one of main incentives for the study of concentrated vortex stability. In the book presented, the authors do not consider the theory of vortex breakdown because 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 laminarturbulent 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

b

40 mm

a

5

d

c

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 rotating 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 hydrogen 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 notices 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 ( ) 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 Turbulence 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 involves 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 result in catastrophic consequences. The complex vortex structures were discovered 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 largescale phenomena like oceanic vortices or atmospheric cyclones (anticyclones) also belong to the category of concentrated vortices. However, their scale is comparative (or larger) to the layer thickness of the atmosphere/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 vortices 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 developing mathematical description and experimentation. That is why the theory of concentrated vortices is based mainly upon approximate mathematical models. 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 experimental research, there is quite a limited number of works presenting tentative results on the stability and dynamics of concentrated vortices, acceptable for comprehensive validation of theoretical models.

8

Introduction

a b

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 comprehensive knowledge of concentrated vortex dynamics. This book presents the authors’ collection and systemization of knowledge, which should enable the reader to understand the important features of concentrated vortices. At the same time we have not here considered the myriad of examples and applications, which could be suitable for another book (or even book series).

1 Equations and laws of vortex motion

1.1 Vorticity. Circulation The term vorticity (or vorticity vector) is the important concept in fluid dynamics. It is a vector value, and in the Cartesian coordinate system (x, y, z) it is determined via projections (u, v, w) of the velocity vector u as

(r , t) ≡ ∇ × u = εijk

∂uk ⎛ ∂w ∂v ∂u ∂w ∂v ∂u ⎞ =⎜ − , − − ⎟, , ∂x j ⎝ ∂y ∂z ∂z ∂x ∂x ∂y ⎠

(1.1)

where εijk is a unit 3rd rank tensor of Levi – Civita. The motion of a continuous medium with zero vorticity (∇ × u = 0) is called irrotational. It follows from definition (1.1) and formulae of vector analysis that ∇

= ∇(∇ × u) = 0,

(1.2)

i.e., the vorticity vector is solenoidal. To understand the physical meaning of vorticity, let us analyze the relative motion of a medium in the vicinity of some point M(r) at a given instant of time. Let the fluid velocity at this point be u(r). Then, at some nearby point M'(r + δr) velocity alteration can be determined by series expansion to the values of the first infinitesimal order

δ ui =

∂ ui δ xj . ∂x j

Derivative ∂ui /∂xj is the 2nd rank tensor, which can be defined as the sum of the symmetric and antisymmetric tensors ∂ui ∂ui ( s ) ∂ui ( a ) = + . ∂x j ∂x j ∂x j Here, the symmetric part is written as ∂ui ( s ) 1 ⎛ ∂ui ∂u j = ⎜ + 2 ⎜⎝ ∂x j ∂xi ∂x j

⎞ ⎟ ≡ eij . ⎟ ⎠

(1.3)

10

1 Equations and laws of vortex motion

Value eij is called the rate of strain tensor, and the corresponding part of velocity (eij δxj) describes the pure straining motion. The antisymmetric tensor takes the form ∂ui ( a ) 1 ⎛ ∂ui ∂u j = ⎜ − 2 ⎜⎝ ∂x j ∂xi ∂x j

⎞ ⎟, ⎟ ⎠

and it can be expressed via vorticity components (1.1) ∂ui ( a ) 1 = − εijk ωk . 2 ∂x j The corresponding component of velocity δu is written as 1 1 1 − εijk ωk δx j or εijk ω j δxk , or 2 2 2

× δr ,

and it describes the rotational motion of a small element as a solid body with angular velocity /2. Actually, in the case of rigid body rotation with angular velocity δu =

× δr.

Hence, =

1 2

.

(1.4)

Therefore, any motion of a continuous medium can be separated into pure straining, rotational and, naturally, translational motions. The physical meaning of vorticity lies in the fact, that it equals double the local angular velocity of a medium. Correspondingly, the relationship for velocity at a point M'(r + δr) will take the following form 1 ui (r + δr ) = ui (r ) + eij δx j + εijk ω j δxk , 2

(1.5)

where derivatives in eij and ωj are taken at point M(r). Together with vorticity , circulation Γ is another important concept in the dynamics of fluids with vorticity. This value is a scalar, and is defined as a curvilinear integral of fluid velocity u along closed circuit s

1.1 Vorticity. Circulation

Γ=

∫ u ds .

11

(1.6)

s

If the curve is reducible (i.e., it can be contracted to a point inside a continuos medium), Stokes theorem can be applied, and a linear integral can be transformed into a surface one (Fig. 1.1)

Γ=

∫ u ⋅ ds = ∫ rot u ⋅ ndS = ∫ s

⋅ ndS.

(1.7)

S

Here S is an arbitrary surface bounded by circuit s; n is a unit normal vector. Besides this, a relationship between velocity curl and vorticity (1.1) is considered. The path-tracing rule is shown in the figure. Relationship (1.7) means that a vorticity flux through an arbitrary open surface equals the velocity circulation along a closed curve. This statement is true for simply connected areas of the flow, where any closed circuit is reducible. It is convenient to describe the kinematics of vortex flows using the concepts of vortex filaments and vortex tubes. They are introduced similarly to the concept of a streamline (a line whose tangent coincides with the direction of the velocity vector at any point) and stream tube (a portion of fluid bounded by a surface consisting of streamlines). In accordance with this, a vortex line is a line in a fluid, whose tangent is parallel to a local vorticity vector at any point, and a vortex tube is a set of streamlines passing through each point of some closed surface in a fluid. Vortex lines passing through its boundary form the lateral surface of a vortex tube. It follows from the definition of a vortex tube that the vorticity vector is parallel to the lateral surface of the vortex tube, i.e. · n = 0.

Fig. 1.1. On the derivation of the integral relationship (1.7) between circulation and vorticity

Fig. 1.2. Vorticity flux through the vortex tube

12

1 Equations and laws of vortex motion

Let us consider the part of a vortex tube bounded by two arbitrary open surfaces S1, S2 and a lateral surface St (Fig. 1.2). Thus, using the formula of Gauss – Ostrogradskii, the surface integral of the vorticity flux can be transformed into the volume integral in the following manner:

∫

∫

(1.8)

⋅ ndS = ∇ dV = 0 .

S

V

Here; the right side is reduced to zero due to the solenoidal character of the vorticity field (1.2). Expanding the left side of (1.8), we get

∫

⋅ ndS =

S

∫

⋅ ndS −

S1

∫

⋅ ndS +

S2

∫

⋅ ndS = 0 .

St

Since the vorticity flux through the lateral surface of the vortex tube is zero, the last relationship means that the vorticity flux through any transverse cross-section of the vortex tube is constant at a given instant of time. Thus, the vorticity flux can be considered as a characteristic of the vortex tube called the strength or the intensity of the vortex tube. Alternatively, if Eq. (1.7) is applied to the vortex tube, it can be concluded that the intensity of a vortex tube is equal to velocity circulation around any closed circuit on the tube surface, which encloses the tube once (Stokes theorem). It follows from the above consideration that vortex tubes cannot end in a continuous medium. Actually, if a vortex tube cross-section becomes equal to zero, according to the law condition, the angular velocity of fluid particles would increase to infinity. Obviously, to correspond to this conclusion, vortex tubes may be closed or end in infinity or end in solid or free surfaces. Let us introduce an additional important relationship between circulations of velocity and acceleration over a closed fluid circuit. With this purpose, at first, we notate a substantial derivative of velocity circulation over an open fluid circuit MP

d dt Considering that

P

∫

P

u ⋅ δr =

M

∫

M

du ⋅ δr + dt

P

d

∫ u ⋅ dt (δr ).

M

d (δr ) = δu , we get dt P

∫

M

⎛ u2 ⎞ 1 2 δ ⎜ ⎟ = uP2 − uM . ⎜ 2 ⎟ 2 ⎠ M ⎝ P

u ⋅ δu =

∫

(

)

1.2 Dynamics of vortical fluid

13

Then, reducing points M and P into a single point and obtaining closed circuit s, we get the following expression

d Γ(u) = Γ(u) , dt

(1.9)

where u = du dt , i.e., a substantial derivative of velocity circulation over a closed fluid circuit equals the circulation of acceleration over the same circuit. This is Kelvin’s circulation theorem. The above properties of vortical fluid motion represent merely kinematic theorems, not connected with specific features of fluids or peculiarities of their motion models. Proving of these theorems was only based on the general properties of medium continuity. Therefore, conclusions formulated in this section perfectly describe the reality. Other problems of vortical fluid motion concern dynamics and would significantly depend on the chosen flow model.

1.2 Dynamics of vortical fluid 1.2.1 Equations of ideal fluid motion

Hereafter, if not expressly stipulated, an ideal fluid will be considered. The theory of vortices in an ideal fluid due to its simplicity provides the solution to many specific problems with a practical value. First of all, this statement is related to the important problem of motions in ideal fluid caused by areas, whose vorticity differs from zero. The motion of ideal fluid is described by Euler equations ∇p du ∂u ≡ + (u ⋅ ∇)u = − +g dt ∂t ρ

(1.10)

and the equation of mass conservation dρ ∂ρ ∂ρ ≡ + u∇ρ = −ρ∇u or + div ( ρu ) = 0. dt ∂t ∂t

(1.11)

Equations are written for the general case of a compressible medium. Here g is the external force per a mass unit; p is the pressure; ρ is the density. Let us transform (1.10), taking into account vector identity (u∇)u = ∇(u 2/2) + curl u × u. As a result, we will get the Gromeka – Lamb equation

14

1 Equations and laws of vortex motion

⎛ u2 ⎞ ∇p ∂u . + ∇ ⎜ ⎟ + curl u × u = g − ⎜ ⎟ 2 ∂t ρ ⎝ ⎠

(1.12)

Let us consider a case for practice, when the motion is barotropic, and forces are potential, i.e. g = −∇Π, where Π is a potential of mass force, and ρ = ρ(p) and corresponding function exists

P ( p) = ∫

dp . ρ( p)

Then, the Gromeka – Lamb equation will take the form

∂u + ∇Η + curl u × u = 0 , ∂t

(1.13)

where H u2/2 + P + Π. Finally, applying operation curl to (1.13) and introducing vorticity, we get Helmholtz equation

d = ( ⋅ ∇ )u − dt

⋅ ∇u ,

(1.14)

which in the case of incompressible fluid is written as d ∂ = ( ⋅ ∇)u or = curl(u × ) . dt ∂t

(1.15)

The main advantage of Helmholtz equation in comparison with Euler equation (1.10) is an absence of pressure in (1.14). It is useful to notate equations of motion in a non-inertial frame of reference. Let us consider the moving frame (x', y', z') with velocity of coordinate origin v0 and angular velocity of its rotation 0 together with the immobile coordinate system (x, y, z). Let us denote the vector of relative velocity of a fluid particle as v. Considering the value of translational velocity v0 + 0 × r, the vector of the absolute velocity is written in the form of u = v + v0 + 0 × r. According to Coriolis theorem (Kochin et al. 1964), the absolute acceleration of a fluid particle is composed of relative dv/dt and translational dv0 /dt + d 0 /dt × r + 0 × ( 0 × r) accelerations, and Coriolis acceleration 2( 0 × v). Taking into account that (1.12) includes the absolute acceleration; the Gromeka – Lamb equation for the relative motion takes the form

1.2 Dynamics of vortical fluid

∂′v v2 + ∇ − v × curl v + 2( ∂t 2 dv d 0 ∇p =− +g− 0 − ×r − ρ dt dt

0

× v) =

0

×(

0

15

× r ),

where a prime of time derivative means differentiation in the moving frame of reference. Sometimes it is convenient to consider the absolute motion of fluid using the moving frame of reference

⎛ u2 ∂′u + ∇⎜ − u ⋅ (v0 + ⎜ 2 ∂t ⎝

0

⎞ × r ) ⎟ − ( u ⋅ (v0 + ⎟ ⎠

0

× r ) ) × curl u = −

∇p +g ρ

1.2.2 Theorems of motion for an ideal vortical fluid

Using different forms of motion equations and considering the case of barotropic motion in the field of potential forces, several consequences important for vortex dynamics can be deduced. Originally, they were formulated by Helmholtz (Helmholtz 1858). Initially, let us turn to Kelvin theorem (1.9) and notate its right side with consideration of (1.10): ⎛ ∇p ⎞ dΓ du ds = ⎜ − = + g ⎟ ds = − ∇(P + Π ) ds ≡ 0 . dt dt ⎝ ρ ⎠ Here the integral of ∇(P + Π) over the closed circuit equals zero, if functions P and Π are single valued. This is obvious for P = P (p). Hence, for the barotropic motion in the field of conservative forces with a single valued potential, circulation around any closed circuit is constant, i.e.,

∫

∫

∫

dΓ = 0. dt

(1.16)

Applying Stokes theorem (1.7), it can be concluded that vorticity flux through a fluid surface rounded by a fluid closed curve also stays constant d dt

∫

⋅ n dS = 0 .

(1.17)

S

It follows from (1.16), (1.17) that vortex tubes and vortex lines move together with fluid and the intensity of a vortex tube does not change with time. Allow us to show this using the reasoning of Batchelor (1967). We

16

1 Equations and laws of vortex motion

will consider a fluid tube (Fig. 1.3), which identically coincides with an arbitrary vortex tube at some instant of time t0. Let us mark out an arbitrary closed fluid circuit sc on the vortex tube surface, which encloses the tube once. In accordance with equation (1.16), circulation around this fluid circuit will stay constant in time. Now, let us again mark out an arbitrary small closed fluid circuit ss on the vortex tube surface, which does not enclose it. Apparently, the vorticity flux through the surface bounded by this circuit equals zero and according to (1.17) stays zero through time. Conversely, the intensity of the vortex tube will be maintained through time due to the invariance of circulation over the closed circuits enclosing the tube. This reasoning proves the above statement for the case of a vortex tube. Similar conclusions can be obtained for a vortex line, if a transverse cross-section of a vortex tube is reduced to a point and ultimately it is passed into a vortex line. The important relationship between the vorticity and the length of a fluid element follows from the conservation law of vortex tube intensity. Using t0, the length of a small fluid element (vortex tube element) is δr(t0), and the area of cross-section is δS(t0) (Fig. 1.4). Due to the small size of the element, directions of vectors δr and coincide. In the following instant of time t the tube will change its size and position as it is shown in the picture. Then, according to the law of vorticity flux conservation (1.17) we have | (t)|δS(t) = | (t0)|δS(t0), and according to the law of mass conservation ρ δS δs|t = ρ δS δs|t0 . Taking into account the direction of vectors, we get the following relationship:

Fig. 1.3. The scheme of an arbitrary vortex tube

Fig. 1.4. On the concept of vortex line stretching

1.2 Dynamics of vortical fluid

(

17

ρ)t δs(t ) = ρt δs(t0 )

(1.18)

(t ) δs(t ) . = (t0 ) δs(t0 )

(1.19)

0

or for incompressible fluid

Expression (1.19) has obvious physical meaning. At a change in the size of a small fluid element, the vector of local vorticity changes as does the length of a small element. Introducing the term stretching of vortex lines, as in the stretching of a small fluid element along the vortex line, we can assert that vortex line stretching leads to vorticity intensification. This effect is extremely important for the explanation of many phenomena connected with swirl flows and for the analysis of turbulence. However, in the last case, due to geometrically complex deformation of fluid elements, the integral value is introduced ∫1/2 ω2dV to estimate vorticity behavior. In the case of irrotational motion, Lagrange theorem is valid: if some volume of fluid moves irrotationally, it stays irrotational even after that. At that point, it is assumed that the fluid is ideal, the motion is barotropic, and body forces are conservative (Loitsyanskii 1966). The proof follows from Helmholtz equation (1.14) and analyticity condition of function (t) when using Lagrangian coordinates. Let at instant t0 (t0) = 0. Thus, according to (1.14), the substantial derivative d /dt at t = t0 is also equal to zero. It can be shown due to repeated differentiation of (1.14) that all following time derivatives also turn to zero at t = t0. This means that under the condition of analyticity of (t), vorticity remains zero at any instant of time t > t0. If it is required that the potential of body forces is single valued, Lagrange theorem can be easily proved using equation (1.17) without the condition of (t) analyticity. Ultimately, we can get from (1.17) that the vorticity flux through any open surface S is constant in time

∫

⋅ n dS = const .

S

If at some instant of time vorticity is equal to zero at any point of a control volume; const = 0. As the surface S and direction n are chosen arbitrarily, vorticity is also zero everywhere at any instant of time, which proves Lagrange theorem.

18

1 Equations and laws of vortex motion

1.2.3 Bernoulli theorem

One of the most important laws of conservation in fluid dynamics is Bernoulli’s theorem (Loitsyanskii 1966), which reads: ‘At steady barotropic motion of an ideal fluid in the field of potential body forces, parameter H (Bernoulli’s trinomial, see (1.13)) keeps its constant value along a streamline.’ H = u2/2 + P + Π = const (along a streamline) This law is easily derived from the Gromeka – Lamb equation (1.13). Actually, from the condition of steadiness, the first term in (1.13) becomes zero. Then, we take the scalar product of (1.13) with u. Obviously, u · (curl u × u) 0. Then u · ∇H = 0 or u(u/u · ∇H) = 0, whence taking into account definition of directional derivative, it follows that dH/ds = 0, which proves Bernoulli theorem. Here d/ds indicates the derivative taken along the streamline or fluid trajectory, which is equivalent in the case of steady motion. Taking the scalar product of the Gromeka – Lamb equation with the vorticity vector , we similarly get: /ω · ∇H = dH/ds = 0. Now d/ds indicates the derivative along the vortex line. Therefore, Bernoulli theorem is true even for a vortex line H = u2/2 + P + Π = const (along a vortex line).

(1.20)

Generally, Bernoulli constant H has different values for different streamlines and vortex lines. However, if everywhere in the space × u = 0,

(1.21)

it follows from (1.13) that in the whole space occupied by a continuous medium Bernoulli constant retains its value. Condition (1.21) is satisfied in two cases: 1) = 0 – irrotational motion; 2) || u − so-called helical motion or helical flow (Vasiliev 1958), and according to other publications (Dritschel 1991) − Beltrami flow. In the last case, vortex lines coincide with streamlines. Numerous examples of these flows are presented, for instance, by Vasiliev (1958). In the simplest case of incompressible fluid in the field of gravitation forces, Bernoulli constant takes the form H = u2/2 + p/ρ + gz = const,

(1.22)

where coordinate z is taken along the upward vertical axis. Bernoulli equation can be written even for unsteady flow, if it is irrotational, i.e.,

1.2 Dynamics of vortical fluid

19

∇ × u = 0. Then there is the scalar function ϕ(r, t), called the velocity potential and determined from equation u = ∇ϕ.

(1.23)

The velocity potential is determined unambiguously only for the single connected area. Substituting (1.23) into the first term of the Gromeka – Lamb equation (1.13), we have

∂ϕ u 2 + + P + Π = F (t ). ∂t 2

(1.24)

Here F(t) is an arbitrary function of time, which is similar for the whole flow field and can be determined from the boundary conditions. Expression (1.24) is also called the Couchy – Lagrange Integral. 1.2.4 Equations of viscous fluid motion

As mentioned above, the model of incompressible density-uniform ideal fluid is mainly used for analysis of vortex motion. However, some cases cannot neglect consideration of viscosity effects. Viscous liquids are described by Navier – Stokes equations ∇p ∂u + (u ⋅ ∇) u = g − + ν∆u, ∂t ρ

(1.25)

where ν is the coefficient of kinematic viscosity. Equation of continuity (1.11) at ρ = const takes the simple form div u = 0.

(1.26)

Let us derive a generalization of the Gromeka – Lamb equations for the case of viscous fluid. It follows from equation (1.26), identity ∆ u = grad div u − curl curl u

(1.27)

and vorticity definition (1.1), that ∆ u = − curl . Then, for description of the motion of viscous incompressible fluid in the potential field of mass forces, the vector equation of Gromeka – Lamb is written ∂u + ∇H + ∂t

× u = −ν rot

,

(1.28)

20

1 Equations and laws of vortex motion

an additional term ν∆ will appear on the right side of Helmholtz equation (1.15) due to the application of procedure curl to ν∆ u in Navier – Stokes equation (1.25). As a result, we have d = ( ⋅ ∇ ) u + ν∆ dt

or

∂ + (u ⋅ ∇) ∂t

= ( ⋅ ∇ ) u + ν∆ .

(1.29)

It follows from the last equation that at a given point of the flow of incompressible fluid, vorticity changes due to convection (the second term on the left side), deformation and rotation of a fluid element (the first term on the right) and diffusion caused by viscosity (the second term on the right).

1.3 Equations of fluid motion in orthogonal coordinates For convenience of application let us notate continuity equation (1.26), equations of motion (1.10) or (1.25), Gromeka – Lamb equations (1.12) or (1.28) and Helmholtz equations (1.14) or (1.29) in arbitrary orthogonal coordinates as well as in the most frequently used forms: Cartesian, cylindrical and spherical coordinate systems. Note that transition to equation for ideal fluid for any notation of equations might be accomplished by assuming ν = 0. 1.3.1 Arbitrary orthogonal system of curvilinear coordinates

Let us consider an arbitrary orthogonal system of curvilinear coordinates q1 , q2 , q3 , constrained with Cartesian coordinates by the following correlations qk = qk(x, y, z),

k = 1,2,3.

Equations qk(x, y, z) = const,

k = 1,2,3

determine the family of coordinate surfaces and three sets of curves, represented by the equations ⎧qk ( x, y, z) = const , k, n = 1, 2,3; k ≠ n, ⎨ ⎩qn ( x, y, z) = const are called coordinate lines. If unit vectors e1 , e2 , e3 , directed along the coordinate lines, are orthogonal at all spatial points, then the curvilinear co-

1.3 Equations of fluid motion in orthogonal coordinates

21

ordinates can be identified as orthogonal. In the case of non-zero Jacobian coordinate mapping, the latter can be solved using the Cartesian system, and then the position vector of an arbitrary point could be determined in the following form r = r(q1 , q2 , q3). Infinitesimal transition along the coordinate lines will be determined accordingly by increments drk =

∂r dqk , k = 1, 2, 3. ∂qk

Taking into account the orthogonality of a curvilinear system of coordinates, the second power of the length of a linear element can be written in the following form 2

3 ⎛ ∂r ⎞ = ds = dr ⋅ dr = ⎜ dq L2k dqk , ⎟ k ∂ q k ⎠ k =1 ⎝ k =1 3

∑

2

where Lk ( q1 , q2 , q3 ) =

( ∂x

∑

∂qk ) + ( ∂y ∂qk ) + ( ∂z ∂qk ) 2

2

2

are the Lame

coefficients. Let us illustrate the main processes of vector calculus in curvilinear orthogonal coordinates. Let ϕ be the scalar function, b the vector function, then

grad ϕ =

3

k =1

div b =

∆ϕ =

k

ek ,

k

1 ⎡ ∂ (L2 L3b1 ) ∂ (L1L3b2 ) ∂ ( L1L2b3 ) ⎤ + + ⎢ ⎥, ∂q2 ∂q3 ⎦ L1L2 L3 ⎣ ∂q1 rot b =

+

∂ϕ

∑ L ∂q

1 ⎡ ∂ (L3b3 ) ∂ (L2b2 ) ⎤ − ⎢ ⎥ e1 + ∂q3 ⎦ L2 L3 ⎣ ∂q2

1 ⎡ ∂L1b1 ∂L3b3 ⎤ 1 ⎡ ∂ (L2b2 ) ∂ (L1b1 ) ⎤ − − ⎢ ⎥ e2 + ⎢ ⎥ e3 , L3L1 ⎣ ∂q3 L1L2 ⎣ ∂q1 ∂q1 ⎦ ∂q2 ⎦

1 L1L2 L3

⎡ ∂ ⎛ L2 L3 ∂ϕ ⎞ ∂ ⎛ L1 L3 ∂ϕ ⎞ ∂ ⎛ L1 L2 ∂ϕ ⎞ ⎤ ⎢ ⎜ ⎟⎥ . ⎜ ⎟+ ⎜ ⎟+ ⎣⎢ ∂q1 ⎝ L1 ∂q1 ⎠ ∂q2 ⎝ L2 ∂q2 ⎠ ∂q3 ⎝ L3 ∂q3 ⎠ ⎦⎥

22

1 Equations and laws of vortex motion

Applying the second formula to (1.26) we obtain the equation of continuity in the following form ∂ (L2 L3 u1 ) ∂ (L1L3 u2 ) ∂ (L1L2 u3 ) + + =0. ∂q1 ∂q2 ∂q3

(1.30)

To notate Euler equations in an arbitrary orthogonal coordinate system we have to apply representation of the gradient in the direction from vector a · ∇b, and to notate the Navier – Stokes equations we need in additional representation for a Laplace operator acting on vector function. The Laplace operator components can be calculated by replacing the scalar function in a given representation of the Laplace operator by b = b1e1 + b2e2 + b3e3 vector and calculating proper derivatives including those of e1 , e2 , e3. However, the obtained result is too complex and usually in order to obtain components of ∆u the form (1.27) may be used as well as the above cited formulae of vector analysis. Let us notate the equations of fluid motion for ideal fluid in the form of Gromeka – Lamb in the directions e1 , e2 , e3 of an orthogonal curvilinear system of coordinates

∂u1 1 ∂ ⎛ u12 + u22 + u32 ⎞ + ⎜ ⎟⎟ − ( u2 ω3 − u3 ω2 ) = ∂t L1 ∂q1 ⎜⎝ 2 ⎠ ⎤ 1 ∂p ν ⎡ ∂ ∂ ( L3 ω3 ) − (L2 ω2 ) ⎥ , = g1 − − ⎢ ρL1 ∂q1 L2 L3 ⎣ ∂q2 ∂q3 ⎦ 2 2 2 ∂u2 1 ∂ ⎛ u1 + u2 + u3 + ⎜ L2 ∂q2 ⎜⎝ ∂t 2

= g2 −

1 ∂p ν − ρL2 ∂q2 L3 L1

⎡ ∂ ⎤ ∂ (L1ω1 ) − (L3 ω3 ) ⎥ , ⎢ ∂q1 ⎣ ∂q3 ⎦

∂u3 1 ∂ ⎛ u12 + u22 + u32 + ⎜ 2 ∂t L3 ∂q3 ⎜⎝ = g3 −

ν 1 ∂p − ρL3 ∂q3 L1L2

⎞ ⎟⎟ − ( u3 ω1 − u1 ω3 ) = ⎠

⎞ ⎟⎟ − ( u1 ω2 − u2 ω1 ) = ⎠

⎡ ∂ ⎤ ∂ (L2 ω2 ) − (L1ω1 ) ⎥ . ⎢ ∂q2 ⎣ ∂q1 ⎦

Substituting components of the vorticity vector = curl u into the above equations and performing some algebra according to the above cited formulae of vector analysis we obtain the set of equations of fluid motion in an arbitrary orthogonal system of coordinates

1.3 Equations of fluid motion in orthogonal coordinates

∂un 1 + ∂t Ln

3

u ⎡ ∂

∂L ⎤

23

∂p

∑ Lkk ⎣⎢ ∂qk (Lnun ) − uk ∂qnk ⎦⎥ = gk − ρLk ∂qk + 1

k =1

(1.31) ⎤ ⎫⎪ Ln ∂ ⎧⎪ L1L2 L3 ⎡ ∂ ∂ +ν (Ln un ) − (Lkuk ) ⎥ ⎬, n = 1, 2,3. ⎨ 2 2 ⎢ ∂qn L1L2 L3 k=1 ∂qk ⎪⎩ Ln Lk ⎣ ∂qk ⎦ ⎪⎭ 3

∑

1.3.2 Cartesian coordinate system

The equations of motion for an incompressible fluid (1.30), (1.31) take the simplest form in the absence of mass forces (g = 0) in a Cartesian coordinate system (x, y, z), where Lame coefficients are L1 = L2 = L3 = 1 ⎛ ∂ 2u ∂ 2u ∂ 2u ⎞ 1 ∂p ∂u ∂u ∂u ∂u +u +v +w =− + ν⎜ 2 + 2 + 2 ⎟, ⎜ ∂t ∂x ∂y ∂z ρ ∂x ∂y ∂z ⎠⎟ ⎝ ∂x ⎛ ∂ 2v ∂ 2v ∂ 2v ⎞ 1 ∂p ∂v ∂v ∂v ∂v +u +v +w =− + ν ⎜ 2 + 2 + 2 ⎟, ⎜ ∂x ∂t ∂x ∂y ∂z ρ ∂y ∂y ∂z ⎟⎠ ⎝ ⎛ ∂ 2w ∂ 2w ∂ 2w ⎞ 1 ∂p ∂w ∂w ∂w ∂w +u +v +w =− + ν⎜ 2 + 2 + 2 ⎟, ⎜ ∂x ∂t ∂x ∂y ∂z ρ ∂z ∂y ∂z ⎟⎠ ⎝

(1.32)

∂u ∂v ∂w + + = 0. ∂x ∂y ∂z Let us notate the first three equations (1.32) in Gromeka – Lamb form ∂ωy ⎞ ⎛ ∂ω ∂u ∂ ⎛ u 2 + v2 + w2 p ⎞ + + ⎟ = v ωz − w ωy − ν ⎜ z − ⎜⎜ ⎟, ∂t ∂x ⎝ ρ ⎠⎟ ∂z ⎠ 2 ⎝ ∂y ∂ω ∂v ∂ ⎛ u 2 + v2 + w2 p ⎞ ⎛ ∂ω + + ⎟ = w ωx − u ωz − ν ⎜ x − z ⎜⎜ ⎟ 2 ∂t ∂y ⎝ ρ⎠ ∂x ⎝ ∂z

⎞ ⎟, ⎠

⎛ ∂ωy ∂ωx ∂w ∂ ⎛ u 2 + v2 + w2 p ⎞ + ⎜ + ⎟ = u ωy − v ωx − ν ⎜ − ∂t ∂z ⎝⎜ ρ ⎠⎟ ∂y 2 ⎝ ∂x

(1.33)

⎞ ⎟. ⎠

The generalized Helmholtz equation (1.29) for viscous fluids in Cartesian coordinates can be written as

24

1 Equations and laws of vortex motion

∂ωx ∂ω ∂ω ∂ω ∂u ∂u + u x + v x + w x = ωx + ωy + ωz ∂t ∂x ∂y ∂z ∂x ∂y ∂ωy ∂ωy ∂ωy ∂ωy ∂v ∂v +u +v +w = ωx + ωy + ωz ∂t ∂x ∂y ∂z ∂x ∂y

∂u + ν∆ωx , ∂z ∂v + ν∆ωy , (1.34) ∂z

∂ωz ∂ω ∂ω ∂ω ∂w ∂w ∂w + u z + v z + w z = ωx + ωy + ωz + ν∆ωz , ∂t ∂x ∂y ∂z ∂x ∂y ∂z where vorticity vector components are determined by (1.1), while D = ∂ 2/∂x2 + ∂ 2/∂y2 + ∂ 2/∂z2. 1.3.3 Cylindrical coordinate system

In the cylindrical coordinate system (r, θ, z), the Lame coefficients have the following form: L1 = 1, L2 = r, L3 = 1. After substituting them into (1.30) and (1.31) for incompressible fluid we obtain

∂ur ∂u ∂u ∂u u 2 + ur r + uθ r + uz r − θ = r ∂θ r ∂t ∂r ∂z = gr −

u 1 ∂p 2 ∂u ⎞ ⎛ + ν ⎜ ∆ ur − 2r − 2 θ ⎟ , ρ ∂r r r ∂θ ⎠ ⎝

∂uθ ∂u ∂u ∂u uu + ur θ + uθ θ + uz θ + r θ = ∂t ∂r ∂z r ∂θ r = gθ −

1 ∂p 2 ∂u u ⎞ ⎛ + ν ⎜ ∆ uθ + 2 r − 2θ ⎟ , ρ r ∂θ r ∂θ r ⎠ ⎝

(1.35)

∂uz ∂u ∂u ∂u 1 ∂p + ur z + uθ z + uz z = gz − + ν∆uz , r ∂θ ∂t ∂r ∂z ρ ∂z ∂ (rur ) ∂uθ ∂ (ruz ) + + =0, ∂r ∂θ ∂z where ur, uθ, uz are respectively the radial, peripheral and axial components of the velocity vector, while operator ∆=

∂2 ∂r 2

+

1 ∂ 1 ∂2 ∂2 + 2 2 + 2 r ∂r r ∂θ ∂z

is equally defined for all equations of this Section.

1.3 Equations of fluid motion in orthogonal coordinates

25

Let us notate the Gromeka – Lamb equation in cylindrical coordinates only for fluid flowing in a potential field of mass forces ⎞ ∂ω ⎞ ∂ur ∂ ⎛ ur2 + uθ2 + uz2 p ⎛ ∂ω + ⎜ + + Π ⎟ = uθ ωz − uz ωθ − ν ⎜ r − 0 ⎟ , ⎟ 2 ∂t ∂r ⎜⎝ ρ ⎝ r ∂θ ∂z ⎠ ⎠ ⎞ ∂uθ ∂ω ∂ ⎛ ur2 + uθ2 + uz2 p ⎛ ∂ω + + + Π ⎟ = uz ωr − ur ωz − ν ⎜ r − z ⎜⎜ ⎟ 2 ∂t r ∂θ ⎝ ρ ∂r ⎝ ∂z ⎠

⎞ ⎟, ⎠

(1.36)

⎞ ∂uz ∂ ⎛ ur2 + uθ2 + uz2 p + ⎜ + +Π⎟ = ⎟ ∂t ∂z ⎜⎝ ρ 2 ⎠ ⎛ ∂ ( r ωθ ) ∂ωr ⎞ = ur ωθ − uθ ωr + ν ⎜ − ⎟. r ∂θ ⎠ ⎝ r ∂r Then generalized Helmholtz equation (1.29) accordingly takes the form ∂ωr ∂ω ∂ω ∂ω + ur r + uθ r + uz r = r ∂θ ∂t ∂r ∂z = ωr

∂ur ∂u ∂u ω 2 ∂ωθ ⎞ ⎛ + ωθ r + ωz r + ν ⎜ ∆ωr − 2r − 2 ⎟, ∂r ∂z r ∂θ r r ∂θ ⎠ ⎝ ∂ωθ ∂ω ∂ω ∂ω uω + ur θ + uθ θ + uz θ − r θ = ∂t ∂r ∂z r ∂θ r

∂u ∂u ∂u uω 2 ∂ωr ωθ ⎞ ⎛ = ωr θ + ωθ θ + ωz θ − θ r + ν ⎜ ∆ωθ + 2 − ⎟, ∂r ∂z r ∂θ r r ∂θ r 2 ⎠ ⎝

(1.37)

∂ωz ∂ωz ∂ωz ∂ωz + ur + uθ + uz = ∂t ∂r ∂z r ∂θ ∂u ∂u ∂u = ωr z + ωθ z + ωz z + ν∆ωz , ∂r ∂z r ∂θ where vorticity vector components are determined by the following formulae ωr =

∂ (ruθ ) ∂ur ∂uz ∂uθ ∂u ∂u − − , ωθ = r − z , ωz = . ∂z ∂r r ∂θ ∂z r ∂r r ∂θ

(1.38)

26

1 Equations and laws of vortex motion

1.3.4 Spherical coordinate system

Let us consider another widespread notation for viscous fluid flow equation in a spherical coordinate system (r, θ – latitude, φ – longitude), in which the Lame coefficients are L1 = 1; L2 = r; L3 = r sin θ. After substituting them in (1.30) and (1.31) for spherical components of velocity vector ur, uθ and uφ we obtain the following set of equations uφ ∂ur uθ2 + uφ2 ∂ur ∂ur ∂ur + ur + uθ + − = r ∂θ r sin θ ∂φ r ∂t ∂r = gr −

⎛ 2u 1 ∂p 2 ∂ ( uθ sin θ ) 2 ∂uφ ⎞ + ν ⎜ ∆ ur − 2r − 2 − 2 ⎟, ρ ∂r ∂θ r r sin θ r sin θ ∂φ ⎠ ⎝

uφ ∂uθ ur uθ uφ2 cos θ ∂uθ ∂uθ ∂uθ + ur + uθ + + − = r ∂θ r sin θ ∂φ r r sin θ ∂t ∂r = gθ −

∂uφ ∂t = gφ −

+ ur

⎛ u 1 ∂p 2 ∂u 2cos θ ∂uφ ⎞ + ν ⎜ ∆uθ + 2 r − 2 θ 2 − 2 2 ⎟ , (1.39) ρ r ∂θ r ∂θ r sin θ r sin θ ∂φ ⎠ ⎝

∂uφ ∂r

+ uθ

∂uφ r ∂θ

+

uφ

∂uφ

r sin θ ∂φ

+

ur uφ

+

r

uθuφ cos θ r sin θ

=

uφ ⎞ ⎛ 1 1 ∂p 2 2cos θ ∂uθ + ν ⎜ ∆uφ + 2 + 2 2 − 2 2 ⎟, ρ r sin θ ∂φ r sin θ r sin θ ∂φ r sin θ ⎠ ⎝

(

∂ r 2 sin θ ur ∂r where ∆ =

) + ∂ (r sin θ u ) + ∂ru θ

∂θ

φ

∂φ

= 0,

1 ∂ ⎛ 2 ∂ ⎞ 1 1 ∂ ⎛ ∂ ⎞ ∂2 . r sin + θ + ⎜ ⎟ ⎜ ⎟ ∂θ ⎠ r 2 sin 2 θ ∂φ2 r 2 ∂r ⎝ ∂r ⎠ r 2 sin θ ∂θ ⎝

In a spherical coordinate system Gromeka – Lamb equations (1.28) for fluid flow in a potential field of mass forces can be written as

1.3 Equations of fluid motion in orthogonal coordinates 2 2 2 ⎞ ∂ur ∂ ⎛ ur + uθ + uφ p + ⎜ + + Π⎟ = ⎟ 2 ∂t ∂r ⎜ ρ ⎝ ⎠ ν ⎛ ∂ ωφ sin θ ∂ωθ ⎞ ⎜ ⎟, = uθ ωφ − uφ ωθ − − ∂θ ∂φ ⎟ r sin θ ⎜ ⎝ ⎠ 2 2 2 ⎞ ∂uθ ∂ ⎛ ur + uθ + uφ p ⎜ + + +Π⎟ = ⎟ 2 ∂t r∂θ ⎜ ρ ⎝ ⎠ ν ⎛ ∂ωr ∂ rωφ sin θ ⎞ ⎜ ⎟, = uφ ωr − ur ωφ − − ⎟ ∂r r sin θ ⎜ ∂φ ⎝ ⎠ 2 2 2 ⎞ ∂uφ 1 ∂ ⎛ ur + uθ + uφ p ⎜ + + + Π⎟ = ⎟ 2 ∂t r sin θ ∂θ ⎜ ρ ⎝ ⎠ ν ⎛ ∂ ( rωθ ) ∂ωr ⎞ = ur ωθ − uθ ωr − ⎜ − ⎟, ∂θ ⎠ r ⎝ ∂r

(

27

)

(

)

(1.40)

and generalized Helmholtz equation (1.29) becomes uφ ∂ωr ∂ωr ∂ωr ∂ω ∂u ∂u + ur + uθ r + = ωr r + ωθ r + ∂t ∂r ∂r r ∂θ r sin θ ∂φ r ∂θ +

⎡ ωφ ∂ ur 2ω 2 ⎛ ∂ ( ωθ sin θ ) ∂ωφ ⎞ ⎤ + ν ⎢ ∆ωr − 2r − 2 + ⎜ ⎟⎥ , ∂θ ∂φ ⎠ ⎥⎦ r sin θ ∂φ r r sin θ ⎝ ⎢⎣

uφ ∂ωθ ur ωθ − ωr uθ ∂ωθ ∂ω ∂ω ∂u + ur θ + uθ θ + − = ωr θ + ∂t ∂r ∂r r ∂θ r sin θ ∂φ r + ωθ

ωφ ∂uθ ⎛ ∂uθ ω 2 ∂ω 2cos θ ∂ωφ ⎞ + + ν ⎜ ∆ωθ + 2 r − 2 θ − 2 2 ⎟ , (1.41) r ∂θ r sin θ ∂φ r ∂θ r sin θ r sin θ ∂φ ⎠ ⎝

∂ωφ ∂t = ωr

+ ur ∂uφ ∂r

∂ωφ ∂r + ωθ

+ uθ ∂uφ r ∂θ

∂ωφ r ∂θ +

+

uφ

∂ωφ

r sin θ ∂φ

ωφ ∂uφ r sin θ ∂φ

−

−

ωr uφ r

ur ωφ r −

−

uθ ωφ cos θ r sin θ

ωθuφ cos θ r sin θ

ωφ ⎞ ⎛ 2 ∂ωr 2cos θ ∂ωθ +ν ⎜ ∆ωφ + 2 + 2 2 − 2 2 ⎟, r sin θ ∂φ r sin θ ∂φ r sin θ ⎠ ⎝

+

=

28

1 Equations and laws of vortex motion

where vorticity is determined by the following correlations ωr =

ωθ =

(

)

∂u ⎞ 1 ⎜⎛ ∂ sin θ uφ − θ ⎟, ∂θ ∂φ ⎟ r sin θ ⎜ ⎝ ⎠

(

1 ⎜⎛ ∂ur ∂ r sin θ uφ − ∂r r sin θ ⎜ ∂φ ⎝ 1 ⎛ ∂ ( ruθ ) ∂ur ωφ = ⎜⎜ − r ⎝ ∂r ∂θ

) ⎟⎞ , ⎟ ⎠

(1.42)

⎞ ⎟⎟ . ⎠

1.4 Special cases of vortex motion 1.4.1 Helical flows (Beltrami flows)

The fundamentals of helical flows were first stated in 1881 by Gromeka in his obscure theses ‘Some Cases of Incompressible Fluid Flow’, and independently in the more known work of the Italian mathematician Beltrami in 1889 (due to this work, helical flows are also called as Beltrami flows). The most detailed description of this class of flows can be found in the book by Vasiliev (1958), which underpins the present Section. As already noted in Section 1.2.3, helical flow is a special case of the steady flow of an ideal fluid with vorticity when vortex lines coincide with streamlines (kinematic condition). The equivalent energetic condition concludes that mechanical energy is constant within the whole volume of flowing fluid, i.e. Bernoulli theorem is valid for the entire flow. For the general case of steady vortex flow of inviscid fluid, the particles moving along the different streamlines posses unequal amounts of energy, i.e. the Bernoulli constant assumes various magnitudes for the different streamlines. At the same time the amount of energy along each streamline remains the same, i.e. the Bernoulli constant remains equal. If all the streamlines originate from an area where the fluid is at rest or flows forward uniformly, then in all remaining areas, due to conservation of the Bernoulli constant along the streamlines, the energy of all particles will be identical, i.e. the flow will be either potential or helical. An example of such a flow may result in the formation of a helical flow in the discharge flowing from a vessel filled with liquid previously at rest. Another prime example is the formation of circulating flows in initially steady flows behind channel

1.4 Special cases of vortex motion

29

turns and tube bends. Yet when applying models of helical flow even in such evident examples we must remember that these results are valid only for steady flows of ideal fluid and stay in force only for those flows, where fluid viscosity and flow unsteadiness have a negligible role. The kinematic condition of helical motion can be expressed in the following way: u × curl u = 0

or curl u = λu,

(1.43)

at that, in general λ may be an arbitrary function of coordinates λ = λ (q1 , q2 , q3). If λ = const, then the helical flow is called uniform, while in the opposite case it is non-uniform. Taking the scalar product of (1.43) with u and accounting correlations curl

1 1 u 1 ⎛ ⎞ = curl u + grad × u and u ⋅ ⎜ grad × u ⎟ = 0, u u u u ⎝ ⎠

we find λ = e · curl e, where e = u / u is the unit vector, collinear to the velocity vector. An interesting property of the helical flow of compressible inviscid fluid results from the equation (1.43) if we apply div operation to both parts of the equation. Then we obtain λ div u + u grad λ = 0,

since

div curl u = 0.

Substituting the relationship for div u from the equation of continuity (1.11), we find u ⋅ grad

λ = 0. ρ

This means that along the streamline the ratio λ / ρ = const. For the special case of incompressible fluid, we obtain from the latter the following result: the streamlines are located on λ = const surfaces. Gromeka found that for a uniform helical flow with a solenoidal velocity field (div u = 0) velocity vector u should satisfy the vector equation ∆ u + λ2 u = 0. Indeed, this relationship follows from (1.43) after applying a rotor operation to the right and left sides of the equation. Moreover, from the fact that in this case div u = 0, it does not follow that the homogenous helical flow may exist only in incompressible fluid. From the equation of continuity

30

1 Equations and laws of vortex motion

(1.11) it follows that for a compressible gas a steady homogenous helical flow is possible, if u · grad ρ = 0, i.e. where the velocity vector is orthogonal to the density gradient – the streamlines run on the surfaces of equal density. The equations of non-uniform helical flow (1.43) in curvilinear orthogonal coordinates (see representation for rotor in Section 1.3.1) may be written as: 1 ⎛ ∂L3 u3 ∂L2 u2 − ⎜ L2 L3 ⎝ ∂q2 ∂q3

⎞ ⎟ = λu1 , ⎠

1 ⎛ ∂L1u1 ∂L3 u3 ⎞ − ⎜ ⎟ = λu2 , ∂q1 ⎠ L3 L1 ⎝ ∂q3

(1.44)

1 ⎛ ∂L2 u2 ∂L1u1 ⎞ − ⎜ ⎟ = λu3 . ∂q2 ⎠ L1L2 ⎝ ∂q1

The equations (1.44) together with the equation of continuity (1.30) for the case of stabilized helical motion of compressible fluid provide us with a set of four equations to determine four variables u1, u2, u3 and λ. Thus, the velocity field for helical flows by analogy with potential flows is completely determined by kinematic equations, while dynamic laws in the form of the Bernoulli integral (1.20) are employed for restoring the pressure field. 1.4.2 Two-dimensional flows

In the general case for each instant of time, the fluid flow velocity field is determined by three functions – u1, u2, u3, which are components of the velocity vector in some curvilinear coordinate system q1, q2, q3 u1 = f1(q1, q2, q3); u2 = f2(q1, q2, q3); u3 = f3(q1, q2, q3). The flow, which may be referred to a coordinate system, where all three velocity components – u1, u2, u3 – for each instant of time are functions of just two coordinates q1, and q2, and do not depend upon the third – q3, i.e. u1 = f1(q1, q2),

u2 = f2(q1, q2),

u3 = f3(q1, q2)

1.4 Special cases of vortex motion

31

let us call two-dimensional1. Geometrically this means that on all coordinate surfaces q3 = const velocity field is identical. In other words, ∂u1 ∂u2 ∂u3 = = =0. ∂q3 ∂q3 ∂q3 This implies that the Jacobian of the velocity components over the coordinates also equals zero

∂ ( u1 , u2 , u3 ) ∂ ( q1 , q2 , q3 )

=

∂u1 ∂q1

∂u1 ∂q2

∂u1 ∂q3

∂u2 ∂q1

∂u2 ∂q2

∂u2 = 0. ∂q3

∂u3 ∂q1

∂u3 ∂q2

∂u3 ∂q3

Therefore the relationship, which exists between functions u1, u2, u3 should not depend on coordinates q1, q2, q3, i.e. for two-dimensional flow the following functional correlation takes place F(u1, u2, u3) = 0.

(1.45)

This correlation allows a reduction in the number of variables in the equations, reducing at the same time the number of equations in the set. Plane-parallel flow

The simplest example of two-dimensional flow is plane or plane-parallel fluid flow when in some inertial reference frame all the particles are located on the same normal in some immovable plane (xy, for instance), posses similar magnitudes of pressure and density and move equally in parallel to this plane (Fig. 1.5), i.e. correlation (1.45) is given by w u3 = 0. 1

In the literature such flows are sometimes called ‘two-parametric’. For instance, the classification of flows by Vasiliev (1958) is based both on a number of parameters required for the description of velocity field, and the number of nonzero velocity components. In contrast to ‘two-parametric’, the author calls ‘twodimensional flows’, those in which one of the velocity components in some curvilinear coordinate system is equal to zero. It is not difficult to envisage a threeparametric, two-dimensional system: the particles are moving in parallel on some plane, but their velocities on a parallel to this plane are not equal. Nevertheless the stated classification is not generally accepted and we will use the more traditional term of ‘two-dimensional flows’.

32

1 Equations and laws of vortex motion

Fig. 1.5. The example of plane-parallel flow

In this case all the additives in equation set (1.32) containing z derivatives vanish. The third equation assumes its’ identity, while the fourth, the equation of continuity, takes the form div u ≡

∂u ∂v + =0, ∂x ∂y

(1.46)

where u and v are functions of coordinates x, y and time t, considered further as a parameter. Let us introduce function ψ(x, y) related to the velocity projections u=

∂ψ ∂ψ , v=− , ∂y ∂x

(1.47)

which identically follows the equation (1.46). Function ψ(x, y) has a simple hydrodynamic sense. Indeed, let us notate the streamline equation dx dy = u v

or udy − vdx ≡

∂ψ ∂ψ dx + dy = dψ = 0 . ∂x ∂y

It follows from the last equality that ψ function retains constant magnitudes along the streamlines; in other words the set of level lines ψ(x, y) = const represents the totality of the streamlines. In this context, function ψ(x, y) is called stream function. If we draw a curve between the points M and N in the xy plane, then fluid flow rate Q through this curve will be determined by the difference between the magnitudes of streamline function at points M and N. Indeed, if un is a normal curve velocity projection, then

1.4 Special cases of vortex motion N

N

∫

Q = ρ un ds = ρ M

∫

M

33

N

( −vdx + udy ) = ρ dψ = ρ ( ψ N − ψ M ) .

∫

M

Let us find the vorticity vector for plane fluid flow. Since in this case w = 0, yet u and v depend only upon x, y and time t, the vorticity vector will be determined only by its’ z projection, i.e. ωz = where ∆ =

∂2

+

∂v ∂u ∂2ψ ∂2ψ − = − 2 − 2 = −∆ψ , ∂x ∂y ∂x ∂y

(1.48)

∂2

is the Laplace operator. ∂x 2 ∂y 2 In the case of steady-state flow, using (1.46), we may notate the Gromeka – Lamb equation (1.13) in the form of two scalar equalities −ωz

∂ψ ∂H ∂ψ ∂H , − ωz = = . ∂x ∂x ∂y ∂y

Equating mixed derivatives of H function, we derive the form ∂ ( ωz , ψ ) ∂ωz ∂ψ ∂ωz ∂ψ − = 0 or Jacobian =0, ∂x ∂y ∂y ∂x ∂ ( x, y ) i.e. ωz should be dependant on ψ only. In applications the form of the above correlation is often used as given in this function ωz (x, y) = ωz {ψ(x, y)} which should satisfy Helmholtz equation (1.34), for which a flat case takes the form ∂ωz ∂ω ∂ω + u z + v z = ν∆ωz . ∂t ∂x ∂y

(1.49)

If relationship ωz = ωz (ψ) is known, then the investigation reduces to the solution of a boundary problem for ψ, which obeys the equation (1.48). Substituting (1.48) into (1.49) Landau and Lifshitz (1987) derived a full equation for ideal fluid, for which the stream function of a plane flow should satisfy ∂∆ψ ∂ψ ∂∆ψ ∂ψ ∂∆ψ − + =0. ∂t ∂x ∂y ∂y ∂x

34

1 Equations and laws of vortex motion

Nonswirling (longitudinal) axisymmetric flow

Another simple example of two-dimensional fluid flow is nonswirling (longitudinal) axisymmetric flow, when all the particles move equally along each of the planes, intersecting over the given line (axis), and have equal magnitudes of pressure and density (Fig 1.6). In a cylindrical coordinate system, for instance, those planes are rz, and the axis is Oz, while function (1.45) takes form of uθ u2 = 0. Equations (1.35) in this case lose all terms with θ-derivatives, the second equation reduces, while the fourth, the equation of continuity, takes the form ∂ (rur ) ∂ (ruz ) + =0. ∂r ∂z

(1.50)

The differential equation of streamlines for axisymmetric flow can be written as dr dz = ur uz

or

ur dz − uz dr = 0 .

(1.51)

Equation (1.50) shows, that r serves as an integrating factor for equation (1.51). After multiplying by r, the left side of this equation takes the form of the full differential of some function ψ dψ = rur dz − ruz dr , thus ur = −

1 ∂ψ 1 ∂ψ . ; uz = r ∂z r ∂r

(1.52)

Function ψ is called Stokes Stream Function. This function remains equal for each streamline and hence will stay constant on a surface (stream tube), which is obtained by rotating the given streamline around the axis of vvvvv

Fig. 1.6. The example of axisymmetric flow

1.4 Special cases of vortex motion

35

symmetry. If we draw an arbitrary surface between two concentric circles located on different stream tubes passing through the points M and N, then fluid flow rate Q, flowing through this surface, will be equal to the difference between the magnitudes of stream function on the stream tubes multiplied by 2πρ. Indeed, 2π

N

0

M

Q = ρ u ⋅ ndS = ρ ( ur nr + uz nz ) dS = ρ dθ

∫

S

∫ ∫ ( ur rdz − uzrdr ) =

∫

S N

= 2πρ dψ = 2πρ ( ψ N − ψ M ) .

∫

M

Let us calculate the vorticity components in the considered axisymmetric flow using stream function. Accounting that the velocity components do not depend on z and uθ = 0, and substituting their representation through the stream function (1.52) in (1.38), we get axial and radial vorticity components equal to zero. Then the equation to determine stream function by given distribution of circumference vorticity component ωθ takes the form 1 ∂ 2 ψ ∂ ⎛ 1 ∂ψ ⎞ + ⎜ ⎟ = −ωθ , r ∂z 2 ∂r ⎝ r ∂r ⎠

(1.53)

where ωθ being a function of coordinates and time should satisfy Helmholtz equation (1.37) ∂ωθ ∂ω ∂ω u ω + ur θ + uz θ − r θ = 0 or ∂t ∂r ∂z r

d ⎛ ωθ ⎞ ⎜ ⎟=0, dt ⎝ r ⎠

wherefrom it follows that in a steady-state case value ωθ /r is an arbitrary function of stream function ψ. Swirl axisymmetric flow

Another important example of two-dimensional flow is swirl axisymmetric flow of ideal fluid (Batchelor 1967). This flow is considered in many research works and referred to differently. For instance, it is described in the works by Vasiliev (1958) and Goldshtik (1981) as the circulating and vortex-type flow. The essential difference of such a flow as compared to the above considered nonswirling axisymmetric flow is the rotation occurrence, i.e. all velocity components may take non-zero magnitudes, including uθ u2. Though in this case the assumption concerning flow axisymmetry is still valid (all flow characteristics depend only on two cylindrical

36

1 Equations and laws of vortex motion

coordinates [r, z] and do not depend on θ; stream tubes represent surfaces of revolution), nevertheless the trajectories of the particles are no longer flat, but spatial curves (Fig. 1.7). For swirl axisymmetric flows, the equation of continuity retains the same form as (1.50). This allows us to introduce, according to Eqs. (1.52) the analog of stream function ψ, otherwise – stream function of the meridian section (Goldshtik 1981), which converts the equation of continuity to identity. For a steady flow taking the scalar product of equation (1.13) by the velocity vector and taking into account that u⋅( × u) = 0, and flow characteristics do not depend on θ, we obtain ur

∂H ∂H + uz = 0 or ∂r ∂z

∂ψ ∂H ∂ψ ∂H − = 0. ∂z ∂r ∂r ∂z

(1.54)

Equation (1.54) means, that there exists a relationship between functions H and ψ, which does not depend on coordinates r and z, i.e. H=

ur2 + uθ2 + uz2 p + + Π = H (ψ ) . 2 ρ

(1.55)

To derive an equation, describing steady axisymmetric flow of ideal fluid in a potential field of mass forces, let us introduce the function Γ = ruθ. In this case the second equation of the system (1.35) can be rewritten as ur

∂ (ruθ ) ∂ (ruθ ) 1 ∂ψ ∂Γ 1 ∂ψ ∂Γ + uz = − =0. r ∂z ∂r r ∂r ∂z ∂r ∂z

(1.56)

From the obtained relationship it also follows, that function Γ is only a ψ-dependant function, i.e. Γ = Γ(ψ). Let us further consider the first equation of the system (1.36). With regard to all the assumptions concerning flow and after substituting components of the vorticity vector (1.38), the equation takes the form

Fig. 1.7. Schematic diagram of axisymmetric swirl flow in a channel

1.4 Special cases of vortex motion

37

∂u ⎞ ∂H uθ ∂ (ruθ ) ⎛ ∂u = − uz ⎜ r − z ⎟ r ∂r ∂r ∂r ⎠ ⎝ ∂z or, taking into account (1.52), we obtain dH ∂ψ Γ ⎛ dΓ ∂ψ ⎞ 1 ∂ψ ⎛ 1 ∂ 2 ψ ∂ ⎛ 1 ∂ψ ⎞ ⎞ = ⎜ + ⎜ ⎜ ⎟+ ⎟⎟ . dψ ∂r r 2 ⎝ dψ ∂r ⎠ r ∂r ⎜⎝ r ∂z 2 ∂r ⎝ r ∂r ⎠ ⎟⎠ Hence r

∂ ⎛ 1 ∂ψ ⎞ ∂ 2 ψ dΓ 2 dH −Γ = −r ωθ . ⎜ ⎟+ 2 =r ∂r ⎝ r ∂r ⎠ ∂z dψ dψ

(1.57)

Functions H and Γ, appearing in equation (1.57), should be given. Sometimes it is possible to determine them by employing boundary conditions (see, for example, Goldshtik (1981)). Thus, the problem of vortex axisymmetric flow description is reduced to the investigation of an equation for meridian stream function, where the right-hand side of the equation needs additional consideration, similar to the equations for determination of stream function in plane (1.48) and longitudinal axisymmetric (1.53) flows. Note that Helmholtz equations (1.37) for components ωr and ωz in the concerned case apply identically. For ωθ we have d ⎛ ωθ dt ⎜⎝ r

⎞ 1 ∂ 2 ⎟ = 2 ∂z uθ . ⎠ r

The latter equation can be integrated (see Saffman (1992)) and we can obtain (1.57). Components ωr and ωz are linked with the respective velocity components by simple relationships ωr = ur dΓ dψ,

ωz = uz dΓ dψ .

Hence, we see that the projections of velocity and vorticity vectors to the meridian plane are parallel (antiparallel) to each other. 1.4.3 One-dimensional flows

Similarly, as for two-dimensional flows let us call one-dimensional flows those which can be related to such a reference frame where velocity components are functions of just one coordinate q1 u1 = f1 (q1 ); u2 = f2 (q1 ); u3 = f3 (q1 ).

(1.58)

38

1 Equations and laws of vortex motion

It is obvious that all considered examples of two-dimensional flow under the assumption of (1.58) could be simplified. Among the possibilities of transition from two-dimensional flow to one-dimensional let us consider just two important examples with uniform distribution of vorticity. 1. Let the velocity of a plane-parallel steady-state flow in an inertial reference frame have just one u component along the axis x (v = 0) and let this component be a function of just one y coordinate. Then, in view of Eq. (1.48), we obtain −

∂u ∂2ψ = − 2 = ωz ≡ ω = const . ∂y ∂y

Hence u = – ωy + u0, where u0 is velocity at the x axis. Consequently, the considered flow has its velocity, which changes linearly along the y axis. Stream function of such flow is given by ψ=−

ω 2 y + u0 y + ψ 0 . 2

This flow is shown in Fig. 1.8 and is called shear flow. In the particular case (ω = 0) all liquid particles move with equal constant velocities u0. Such flow is called uniform. 2. Let us consider steady-state flow with constant vorticity, which in a cylindrical coordinate system has only one circumferential component uθ, dependent on radial coordinate r. After notating (1.48) in cylindrical coordinates and taking into account the accepted assumptions we obtain −

1 ∂ (ruθ ) 1 ∂ ⎛ ∂ψ ⎞ = ⎜r ⎟ = −ωz ≡ −ω = const . r ∂r r ∂r ⎝ ∂r ⎠

Integrating this equation we find ruθ =

ω 2 ω r + C; ψ = − r 2 − C log r + ψ 0 . 2 4

Fig. 1.8. Scheme of shear flow

Fig. 1.9. Scheme of quasi-rigid-body fluid rotation

1.5 Flows with helical symmetry

39

For flows with restricted velocities constant C should be equal to zero. Streamlines of such flows are of concentric circumferences (Fig. 1.9), and the circumferential velocity varies linearly along the radius. Such a flow is usually called quasi-rigid-body rotation.

1.5 Flows with helical symmetry 1.5.1 Derivation of equations

Let us consider ideal incompressible fluid flow in the cylindrical reference system (r, θ, z) with the axis Oz, directed along the flow axis. Suppose that the flow has helical symmetry. This means that in the planes orthogonal to the z axis the flow pattern will remain the same at the translational motion along the z axis, accompanied by a simultaneous rotation on some angle θ. A possible scheme of such flow is presented in Fig 1.10. That is to say that the flow characteristics will maintain their magnitudes along the helical lines, which are described by equations r = const

z − θ ⋅ l = const .

(1.59)

Value h = 2πl corresponds to the pitch of helical symmetry, moreover the magnitude of l assumes positive values in the case of right-handed helical symmetry, and negative values in the case of left-handed symmetry. The vector tangent of the helical lines (1.59) will be as follows B = B 2 ⎡⎣ ez + ( r l ) eθ ⎤⎦ , where B 2 = (1 + r 2 / l 2 ) −1.

(1.60)

For such normalizing, the vector is called the Beltrami vector (Dritschel 1991). By means of vector B the condition of helical symmetry can be presented as B ⋅ ∇f = 0 ,

(1.61)

where f is an arbitrary scalar function characterizing the flow (velocity component, density and pressure). Vector B is orthogonal to unit radial vector er of the cylindrical coordinate system. Their vector product determines the third orthogonal vector = B × er = B2 ⎡⎣eθ − ( r l ) ez ⎤⎦ .

(1.62)

Let us introduce the values combined with the velocity projection onto the orthogonal directions er , B, ,

40

1 Equations and laws of vortex motion

Fig. 1.10. Scheme of a flow with helical symmetry

ur , uB =

B ⋅u B

2

⋅u r r = uz + uθ , uχ = 2 = uθ − uz . l l B

(1.63)

Further, following Alekseenko et al. (1999) for flows with helical symmetry, satisfying (1.61), let us rewrite the equation of continuity and Euler equations, expressed with the new variables (r, χ = θ – z/l) ∂ (rur ) ∂uχ + = 0, ∂r ∂χ 2

∂ur ∂u ∂u 1 ∂p B4 ⎛ r ⎞ + ur r + uχ r − , ⎜ uχ + uB ⎟ = − r ∂χ r ⎝ l ∂t ∂r ρ ∂r ⎠ B2 ⎛ r B−2 ∂p ⎞ ur ⎜ 2 uB + 2 − B−2 uχ ⎟ = − + ur + uχ − , r ∂χ r ∂t ∂r ρ r ∂χ ⎝ l ⎠

∂uχ

∂uχ

∂uχ

(

(1.64)

)

∂uB ∂u ∂u + ur B + uχ B = 0 . r ∂χ ∂t ∂r Below we consider two possible cases of vortex flows, which are described by the equations (1.64). 1.5.2 Flow with helical vorticity

Allow the ideal incompressible fluid flow to possess helical symmetry, i.e. all flow characteristics depend only on two spatial variables – r, χ and satisfy the equation set (1.64). Suppose that in the whole flow area the velocity projections of tangential direction to the lines (1.59) are proportional to the cosines of the inclination angle of the helical lines relative to the Oz axis. In this case functional relation (1.45) takes the simple form

1.5 Flows with helical symmetry

41

uB = const. This results in identical completion of the latter equation of the system (1.64). Besides, for the considered class of equations according to the first equation (1.64) we may introduce the analog of stream function ψ, which is determined by the relationships ur =

1 ∂ψ , r ∂χ

uχ = −

∂ψ . ∂r

(1.65)

Similar to previously considered uniform flow, where velocity remained constant across the whole area, this class of flows can be conventionally called ‘flows with helical symmetry’, which is characterized by “uniform” motion along the helical lines. According to (1.63) for circumferential and axial components in a cylindrical reference frame the condition of constant velocity uB takes the form r r u − u0 r uB = uz + uθ = u0 ≡ const, or uz = u0 − uθ , or z = − , (1.66) l l uθ l where constant u0 determines the value of the axial component of velocity on a flow axis. Solving (1.63) relatively uθ and uz, and taking into account (1.65) and (1.66) we obtain uθ =

∂ψ ⎞ ⎛r ⎞ 2⎛r ⎜ uB + uχ ⎟ = B ⎜ u0 − ⎟, ∂r ⎠ r +l ⎝ l ⎠ ⎝l l2

2

2

r ⎞ r ∂ψ ⎞ ⎛ ⎛ uz = 2 2 ⎜ uB − uχ ⎟ = B2 ⎜ u0 + ⎟. l ⎠ l ∂r ⎠ r +l ⎝ ⎝ l2

(1.67)

Analyzing (1.66) and (1.67) we may conclude that the considered class of flows conditionally could be referred to as a uniform type. Indeed, the flow velocity components ur, uθ, uz may take arbitrary values, under the stipulation that Eq. (1.66), associated only with axial and circumferential velocity components, holds true. Only in extreme cases, when l ∞, and helical lines (1.59) become straight, the axial velocity component will be constant in the whole area of the flow, i.e uz = u0 and the flow actually becomes uniform in z direction. A pioneer to introduce the class of flows satisfying condition (1.66), was Okulov (1993, 1995). However partial solution within the considered class was employed for combustion analysis (Borissov et al. 1993) and energy separation (Borissov et al. 1994) in swirl flows inside tubes. The most comprehensive analysis showing the effectiveness of use of the considered type of flows, describing real swirl flows, is given by Alekseenko et al. (1999).

42

1 Equations and laws of vortex motion

Let us consider the peculiarities of vorticity distribution in such flows. Then the vorticity vector components in cylindrical coordinates can be written as ωr = ωθ = − ωz = −

1 ∂uB ≡ 0, r ∂χ

r ⎡ 1 ∂ 2 ψ 1 ∂ ⎛ 2 ∂ψ ⎞ B4 ⎤ 1 ∂ur ∂uz 2 rB u − =− ⎢ 2 2 + − ⎥, 0 ⎜ ⎟ ∂r ∂r ⎠ l ∂χ l ⎢⎣ r ∂χ r ∂r ⎝ l ⎦⎥

(1.68)

⎡ 1 ∂ 2 ψ 1 ∂ ⎛ 2 ∂ψ ⎞ B4 ⎤ 1 ∂ur 1 ∂ (ruθ ) + =− ⎢ 2 + − rB u 2 ⎥. 0 ⎜ ⎟ 2 ∂r ⎠ r ∂χ r ∂r r ∂r ⎝ l ⎥⎦ ⎣⎢ r ∂χ 2

4 2

It is justified here, that ∂B /∂r = −2rB /l . It follows from (1.68), that the radial component of the vorticity field is equal to zero. In addition based on (1.68) we may obtain the relationship between the axial and circumferential components of the vorticity vector ωr = 0;

ωθ / ωz = r / l

or

ωθ = r ωz / l.

(1.69)

Condition (1.69) means, that for the given class of flows the vorticity vector is always directed along the tangent in helical lines. The converse proposition is also true (Okulov, 1993, 1995): if the vorticity field is collinear to the helical lines, then the condition (1.66) holds true. Indeed, if the flows with helical symmetry are conditioned by requirements (1.69), then after integrating the first equation of (1.69) and in view of definition of ωr we obtain uB = f(r). According to the second condition (1.69) let us consider the difference ωθ – r ωz / l = 0 and take into account (1.67) and (1.68). As a result we obtain f ' (r) = 0, i.e. uB = const. Thus, for the given class, the fulfillment of condition (1.66) imposed on a velocity field is equivalent to the requirement of collinearity of the vorticity field with tangents in helical lines (1.69). In that case the considered class of flows can be called flows with vorticity fields featuring vectors directed along helical lines, or simply flows with helical vorticity. The third equation in (1.68) can be presented as an equation determining stream function ψ according to the given distribution of the axial vorticity component ωz

∆ψ = 2u0

B2 ωz − 2, l B

(1.70)

1.5 Flows with helical symmetry

43

⎛ 2r 2 ⎞ 1 ∂ l 2 + r 2 ∂ 2 + − + 22 1 . ⎜ ⎟ r l ∂χ 2 ∂r 2 ⎝⎜ r 2 + l 2 ⎠⎟ r ∂r The vorticity vector for this class of flows can be directed according to the Eqs. (1.68) and (1.69) only along the tangent of the helical lines (1.59), and it can be completely determined by assigning just one component, ωz, for instance. At that, the value ωz, as a function of coordinates and time, should satisfy Helmholtz equation (1.37), which in helical variables r and χ reduces to one scalar equation where ∆ =

∂2

∂ωz ∂ωz ∂ωz ∂u r ⎛ ⎞ ∂u + ur + uχ = ωr z + ⎜ ωθ − ωz ⎟ z . ∂t ∂r r ∂χ ∂r ⎝ l ⎠ r ∂χ

(1.71)

Starting from (1.69) we deduce, that the right-hand part of the equation (1.71) equals zero. This means that the axial vorticity component does not vary along the trajectory of the fluid particle, and in steady state conditions ωz is an arbitrary function depending only on stream function ψ and not depending explicitly on spatial coordinates. Moreover, because of Eq. (1.69) the latter claim is valid also for the ratio ωθ /r. Note another interesting property of flows with helical symmetry with “uniform” motion along helical lines (1.66), which can be obtained, if considering flow in the coordinate system, uniformly moving with u0 velocity along the z axis. In this case, based on (1.66) – (1.68) the velocity and vorticity fields will take the form ⎛ 1 ∂ψ ∂ψ r ∂ψ ⎞ u=⎜ , − B2 , B2 ⎟ ∂r l ∂r ⎠ ⎝ r ∂χ r ⎛ ⎞ and = ⎜ 0, − B2 ∆ψ, B2 ∆ψ ⎟ . l ⎝ ⎠

(1.72)

It is easy to check that their scalar product is equal to zero, i.e. in some inertial reference frame the velocity and vorticity fields are orthogonal in contrast to the helical flows considered below. 1.5.3 Helical flows with helical symmetry of the velocity field

In the framework of the ideal fluid flow model with helical symmetry, let us consider a swirl flow in which velocity and vorticity fields are colinear. The velocity and vorticity fields of such flows in view of their solenoidal character may be represented by means of Beltrami vector B in the form of decompositions (Landman 1990; Dritschel 1991)

44

1 Equations and laws of vortex motion

u = φB + ∇ψ × B ,

= ζB + ∇ξ × B ,

(1.73)

where φ, ψ, ζ and ξ are some scalar functions of r and χ. Indeed, if taking into account relationships for div B = 0 and curl B = – 2 B2 B / l, then it is easy to prove that in view of the decompositions (1.73) the equation of continuity div u = 0 and equation div = 0 automatically hold true. Using definitions of vorticity = curl u and gradient, expressed in variables r and χ ∇ = er

∂ 1 ∂ , + eχ rB ∂χ ∂r

(1.74)

and based on the decompositions (1.73), we obtain ξ = – φ and ∆* ψ = 2

where

∆* ≡

B4 φ − B2 ξ, l

(1.75)

1 ∂ ⎛ 2 ∂ ⎞ 1 ∂ . ⎜ rB ⎟+ ∂ r ⎠ r 2 ∂χ 2 r ∂r ⎝

After introducing of the decompositions (1.73) into kinematic determination of uniform helical flows (1.43), we obtain φ = λψ,

ζ = − λφ = − λ2 ψ.

(1.76)

Therefore uniform helical flows with helical symmetry of flow field are completely determined by means of just one scalar function ψ, which satisfies the uniform equation ⎛ B4 ∆* ψ − ⎜ λ 2 B2 + 2λ ⎜ l ⎝

⎞ ⎟⎟ ψ = 0 . ⎠

(1.77)

Based on the first decompositions of (1.73) and (1.76), the velocity components in a cylindrical coordinate system in terms of ψ function, may be determined by means of the following formulae ur =

1 ∂ψ r ∂ψ ⎞ ∂ψ ⎞ ⎛r 2⎛ , uθ = B2 ⎜ λψ − ⎟ , uz = B ⎜ λψ + ⎟. l r l ∂r ⎠ ∂ r ∂χ ⎝ ⎠ ⎝

(1.78)

Thus, for uniform helical flows with helical symmetry, the problem of determination of velocity field can be fully reduced to the solution of a boundary problem for one scalar uniform linear equation (1.77). Then, based on the determined velocity field, the pressure may be recalculated by means of the Bernoulli integral.

1.6 Velocity field at specified distribution of sources and vortices

45

1.6 Velocity field at specified distribution of sources and vortices If vorticity field is known, for instance, from the solution of the Helmholtz equation, then the reverse problem; concerned with the construction of corresponding velocity field u, arises. The additional condition, imposed on u, would be the mass conservation equation (1.11). Nevertheless (1.11) includes another function as well, which is the density function ρ. In order to exclude this function, let us consider incompressible fluid satisfying ∇u = 0. Also, to demonstrate the generality of the mathematical operations for the two initial equations, let us introduce the density of volume sources ε(r, t), which will be included in the right-hand side of the equation of continuity for incompressible fluid. Then we obtain ∇u = ε,

∇×u = ,

(1.79)

where ε and are known functions of coordinates and time. Thus, the mathematical problem concludes in the determination of the vector field based on given distributions of divergence and the rotor of the sought vector. This problem is solvable, though the solution would be unique only under certain conditions. Let us consider the case of infinite space first, and then let us demonstrate the methods of accounting for solid boundaries following Sedov (1997) and Batchelor (1967). Let ε = ε(r) and = (r) be given within the whole infinite space and at the infinity (|r| → ∞) let ε → 0, → 0, u → 0. Then we can prove that the formulated problem has a unique solution, which can be expressed in the following form u(r, t) = uI(r, t) + uV(r, t).

(1.80)

Here uI is the solution of the problem (1.79) for the case ε ≠ 0, = 0 (irrotational or potential flow), while uV is the solution for ε = 0, ≠ 0, i.e. the initial problem splits into two new problems. Let us start with the first, when the density of the sources ε is known, and the flow is irrotational ∇u = ε,

∇ × u = 0.

(1.81)

Due to the second condition we may assume that u = ∇ϕ. Consequently we obtain the Poisson equation from the first condition ∇2ϕ = ε.

(1.82)

46

1 Equations and laws of vortex motion

Here the potential ϕ is a scalar function of coordinates. The solution of the Poisson equation is well known and takes the form ϕ(r ) = −

1 4π

ε(r ′)

∫ r − r ′ dV(r ′) .

(1.83)

The integral should be taken over the whole volume occupied by the fluid, and can exist only if ε is imposed by some limitations. The proper relationship for velocity uI takes the form uI (r ) = ∇ϕ =

1 (r − r ′) ε (r ′) dV (r ′) . 4π r − r ′ 3

∫

(1.84)

Now let us consider the second problem, when the vorticity field is given but the sources are absent ∇u = 0,

∇×u = .

(1.85)

Based on the first equation we may assign u = ∇ × A,

(1.86)

where A is the vector potential. Obviously the velocity field will not change if we substitute A ⇒ A + ∇f, where f is an arbitrary scalar function, i.e. potential A is not uniquely determined. Selecting f in such a way that ∇A = 0, and using (1.86) and the second expression in (1.85), we have ∇ × (∇ × A) = ∇(∇A) – ∇2A = ,

wherefrom for A we obtain the Poisson vector equation ∇2A = – .

(1.87)

Similarly to the first problem we may notate the expression for vector potential and respective velocity uV

A(r ) =

1 4π

(r ′)

∫ r − r′ dV(r′) ,

uV (r ) =

1 4π

∫

(r ′) × (r − r ′) r − r′

3

dV (r ′) .

(1.88)

Just as in the first problem for ε, certain restrictions are imposed on vorticity distribution . In particular, the existence of discontinuity surfaces of vector are permitted, but normal components ωn should be continuous on these surfaces. Another requirement is that the vorticity should be nonzero only in the finite part of the space limited by surface S, where ωn = 0.

1.6 Velocity field at specified distribution of sources and vortices

47

The resulting expression for the velocity vector in an infinite space including sources and vortices takes the form u = ∇ϕ + ∇ × A =

1 ε ⋅ ∆r 1 dV + 3 4 π ∆r 4π

∫

∫

× ∆r ∆r

3

dV ,

(1.89)

where ∆r = r – r'. Now let the space filled with fluid posses solid boundaries Σ. Then for the solution of problem (1.79) one should specify the boundary conditions on Σ. Due to the linearity of the equation, this means that the sought solution (1.80) should be rewritten in the following form u = uI + uV + u0 ,

(1.90)

where the additional velocity field u0 satisfies equations (1.79) with a zero-equal right side ∇u0 = 0,

∇ × u0 = 0.

(1.91)

Here uI, uV as before are determined by the Eqs. (1.84) and (1.88), which account for the distribution of ε and in a mass of fluid, while the boundary conditions at the solid boundary Σ are included by means of solution u0. As is obvious from (1.91), the vector field u0 is both solenoidal and irrotational. Condition = 0 gives the existence of velocity potential ϕ, so that u0 = ∇ϕ. As was already noted, the velocity potential is uniquely determined only for a simply connected domain. Substituting the expression for velocity into the first equation (1.91), we obtain the Laplace equation for the scalar function ϕ ∇2ϕ = 0.

(1.92)

Thus it is necessary to also solve the boundary problem to determine the harmonic function ϕ(r). The boundary conditions at Σ may be very different, usually however the normal velocity component un is given. Finally, to use Eqs. (1.84) and (1.88) it is required to extend functions ε and , given in a space filled with fluid, through Σ boundary into the whole space. Again, there may be various possibilities of such extension. This is concerned with the specific character of a particular problem as well as a certain degree of arbitrariness (see (Batchelor 1967, Sedov 1997,

48

1 Equations and laws of vortex motion

Saffman 1992)). Though we must take into account that the normal vorticity component ωn should be continuous on Σ 2. Note in conclusion that if the space is infinite and there are no sources, then the Poisson equation (1.82) transforms into the Laplace equation with solution ϕ = 0 for the case of infinite space. Then the resulting formula (1.89) will take the form u=

1 4π

∫

× ∆r ∆r

3

dV .

(1.93)

Based on analogy with the formulae of electromagnetic field theory, it can be said that vorticity induces velocity field. In the specific case of plain parallel flow the vector potential is A = (0, 0, ψ) and the vorticity vector is (r′) = (0, 0, ω). Stream function may be presented as a solution of the Poisson equation (1.87)

ψ (r ) = −

1 ω ( r ′ ) log r − r ′ dS′ . 2π

∫

(1.94)

After differentiating (1.94) we obtain the velocity field u=−

1 y − y′ 1 x − x′ ω(r ′) dS′ . dS′ and v = ω(r ′) 2 2 2π 2π r − r′ r − r′

∫

∫

(1.95)

For non-swirling axisymmetric flow with cylindrical variables (r, θ, z) the vector potential is A = (0, ψ/r, 0), while the vorticity vector is (r′) = (0, ω, 0). Introducing 12

2 s = ⎡( z − z′) + r 2 + r ′2 − 2rr ′ cos ( θ − θ′)⎤ ⎣ ⎦

and dV (r') = r'dz'dr'dθ, from (1.88) we obtain r ψ= 4π

∫∫

r ′ω ( r ′, z′ ) dr ′dz′

2π

∫ 0

cos θ dθ . s

The integral with respect to θ can be rewritten in the following form (Batchelor 1967) The latter condition can be removed, because we can always construct the Σ' surface, where · n = 0, beyond Σ boundary (concerning the extension of see Section 1.7.1.) 2

1.7 Vortex forces and invariants of vortex motion 2π

∫ 0

49

π⎡ 1 2⎤ −1 2 cos θdθ 1 2⎛ ⎛2 ⎞⎛ 2 2 θ⎞ 2 2 θ⎞ = − ⎜ 1 − k cos ⎟ ⎥ dθ = ⎢⎜ − k ⎟⎜1 − k cos ⎟ s k⎝ 2⎠ 2 ⎠ ⎥⎦ rr ′ 0 ⎢⎣⎝ k ⎠⎝ ⎤ 2 ⎡⎛ 2 2 ⎞ = ⎜ − k ⎟ K ( k ) − E ( k )⎥ , ⎢ k rr ′ ⎣⎝ k ⎠ ⎦

∫

12

2 2 where k2 = 4rr ′ ⎡( z − z′ ) + ( r + r ′ ) ⎤ ⎣ ⎦

K (k) =

π2

∫(

1 − k2 cos 2 θ

)

−1 2

, while

dθ and E ( k ) =

0

π2

∫ (1 − k

2

cos 2 θ

)

12

dθ

0

are complete elliptic integrals of the first and second kinds. Denoting ⎡⎛ 2 ⎤ 2 ⎞ f ( k ) = ⎢⎜ − k ⎟ K ( k ) − E ( k ) ⎥ , k ⎠ ⎣⎝ k ⎦

we get ψ=

1 2π

∫∫ f ( k )( rr ′)

12

ω ( r ′, z ′ ) dr ′dz ′ .

(1.96)

Accordingly, velocity components take the form ∂k 1 12 f ′ ( k ) ( rr ′ ) ω ( r ′, z′ ) dr ′dz′ , 2πr ∂z ⎡ f (k) ∂k ⎤ 1 12 uz = + f ′ ( k ) ⎥ ( rr ′ ) ω ( r ′, z′ ) dr ′dz′ . ⎢ 2πr ⎣ 2r ∂r ⎦ ur = −

∫∫

∫∫

(1.97)

1.7 Vortex forces and invariants of vortex motion 1.7.1 Vortex forces

The method of force balance is one of the most important approaches to the description of the dynamics of vortex structures. To understand and interpret correctly the forces in ideal fluid with vortices, firstly, we consider the forces acting on a solid body in uniform motion in ideal fluid with velocity U. Let us establish a coordinate system moving together with a body, and the coordinate origin is inside this body (see Fig. 1.11). In this system, the flow over a body has velocity at infinity U∞ = – U.

50

1 Equations and laws of vortex motion

Fig. 1.11. On the definition of forces in the ideal fluid

Let us apply the theorem of momentum to a fluid volume between the body boundary A and circumference s of a large radius R (in 2-D statement). Normal vectors n are directed outward from the body surface A and inward from the circumference s. Let us introduce the velocity vector V in the moving frame of reference V = u – U,

(1.98)

where u is the velocity of fluid in the absolute coordinate system. In fact, u is the perturbation of fluid velocity caused by a solid body. In this connection, we should note that in the case of 2-D circulation flow around the body, velocity perturbation at large distances, R decreases ∞ Actually, let us use for example the soas 1/R, i.e., |u| = O(1/R) at R lution to the problem on circulation flow around a cylinder (circle) with the radius of a (see Loitsyanskii (1966)): u =

U a2 Γ − ∞2 → at 2πiz z

R → ∞,

where z = x + iy. It is obvious that for large distances, the cylinder radius is not included in the solution, and, it can be assumed that the asymptotic solution does not depend on the body shape and is determined only by the value of the circulation. Considering the absence of tangential stresses in the ideal fluid, the momentum balance is written as

1.7 Vortex forces and invariants of vortex motion

51

∫ pn ds + ∫ pn ds + ∫ ρV (V ⋅ n ) ds = 0. A

s

s

Here the first integral is the pressure force from the body on the fluid. Correspondingly, from the fluid the body experiences the force

∫

F = − pn ds . A

The second integral is the force of pressure from the ambient fluid on a control volume, and the third integral is a momentum flux through the surface s. Therefore, it follows from the momentum balance that the force of fluid impact on a solid body is completely determined by distribution of pressure and velocity on cylindrical surface s distanced from the body F = ρ V (V ⋅ n ) ds + pn ds .

∫

∫

s

s

We should remember that here ds is the element of length, therefore, the force is determined per a length unit of the cylinder generatrix by Bernoulli’s equation p = const – ρV2/2,

where the constant is the same for the whole space because the flow is irrotational. Then, taking into account (1.98) and neglecting the small terms, which are quadratic by u, we have F = −ρU (V ⋅ n ) ds − u (U ⋅ n ) ds +

∫

∫

s

s

⎛ U + ⎜ const − ρ ⎜ 2 ⎝

2

⎞ ⎟⎟ n ds + ρ ( u ⋅ U ) n ds . ⎠s s

∫

∫

∫

Here the first term on the right side vanishes because (V ⋅ n ) ds is the fluid flow rate through the control surface s, which equals zero in the absence of sources and sinks. The third term also vanishes because for the circumference s one has the identity:

∫ n ds = 0. s

Two other terms can be combined and expressed via the double vector product

52

1 Equations and laws of vortex motion

∫ ⎡⎣u (U ⋅ n ) − ( u ⋅ U ) n ⎤⎦ ds = ∫ U × ( u × n ) ds. The substitution u u – U = V can be made in the last integral and since the additional integral is equal to zero, i.e.

∫ U × (U × n ) ds = U × (U × ∫ n ds) = 0 , s

s

we get

∫

F = −ρU × V × n ds .

(1.99)

s

The value U is taken outside the integral because this is the constant. Then, we introduce a unit vector t tangent to the circumference s, and a unit vector b, orthogonal to n and t, directed normally to the sketch plane in the reader’s direction as well as the z axis (see Fig. 1.11). Then, n = b × t. By this expression, we notate the integral in (1.99), again using the formulae of vector algebra

∫ V × n ds = ∫ V × (b × t) ds = ∫ (V ⋅ t ) b ds − t ∫ (V ⋅ b) ds = b Γ . This considers that V·b = 0 for a plane flow and, according to definition, ∫V·t ds = Γ is circulation. Sometimes the circulation vector is introduced = Γ b.

(1.100)

Finally, we have the following formula for the force F, acting on a solid body at its translational motion with velocity U in ideal fluid: F = – ρ U × b Γ.

(1.101)

In the case of uniform flow around an immobile body we get F = ρ U∞ × b Γ.

(1.102)

where U∞ = – U is the flow velocity at infinity. These relationships are the Kutta – Joukowski formula. According to Eqs. (1.101), (1.102), force F does not depend on the body shape. It is oriented normally to direction of the body or flow motion as it is shown in Fig. 1.11, and determined only by circulation around the body. Correspondingly, the module of this force is F = ρ U∞ |Γ|.

1.7 Vortex forces and invariants of vortex motion

53

To determine the direction of the equivalent force action, the following rule may be used: the direction of the resultant force affecting the body is obtained by rotation of the body velocity vector by the angle of 90° relative to the fluid towards the velocity circulation. Since the force component towards the body motion is zero, Eq. (1.101) also illustrates the known D’Alembert paradox on friction absence upon the motion of a body in ideal fluid. Now we consider the problem of the force balance in the flow with areas of concentrated vorticity. We rewrite Gromeka – Lamb equation (1.12), rearranging the convective terms to the right side ρ

⎛ u2 ⎞ ∂u = ρg − ∇p − ρ∇ ⎜ ⎟ + ρu × . ⎜ 2 ⎟ ∂t ⎝ ⎠

(1.103)

The value ρ u × is called the vortex force (Prandtl 1919, Saffman 1992). Integrating the equation (1.103) at ρ = const over the fixed volume V, we have ∂ ρu dV = − pT n dS + (ρg + ρu × ) dV. ∂t

∫

∫

∫

(1.104)

1 2 ρu is the total pressure. 2 The left side is the rate of fluid momentum alteration inside the volume V. At steady motion, this value vanishes. In this case, equation (1.104) provides the condition of the balance between pressure forces affecting the surface bounded by the volume V, external forces and vortex force. We should note that if there are no external forces beyond V and the Bernoulli constant at the volume surface is the same (this occurs when the surface is the streamline surface), the first integral on the right side of (1.104) vanishes, and hence, the external force becomes balanced by the vortex force. In other words, the value ρ u × in a steady flow can not be expressed by the gradient of some function, and hence, it cannot be compensated by the pressure forces because some external non-conservative mass force should be applied to the fluid for balance maintenance. The idea of kinematic substitution of the body moving relative to the fluid by vorticity distribution providing the required streamlining conditions on the body surface is directly connected with the concept of the vortex force. Joukowski called these systems bound vortices. This imaginary distribution of vorticity can be considered as a continuation of the free vorticity field into the body. Simultaneously, an imaginary velocity field appears inside the body. It is necessary to note that the field of bound vorticity does not satisfy Helm-

Here pT = p +

54

1 Equations and laws of vortex motion

holtz equation. Despite the infinitely many distributions of bound vorticity, the velocity field induced by this vorticity beyond the body should be the same. We should also note that the distribution of bound vorticity depends on the presence of other bodies and free vorticity. Among the many methods of vorticity field continuation into a body, we wish to mention the following: a) it is assumed that within the continuation area curl = 0 (Batchelor 1967); in this case = grad f, ∆f = 0; b) the properties of symmetry are used (for the area limited by a plane, sphere or cylindrical surface); c) the technique of conformal mapping of the flow area to the area with symmetrical properties is applied for the 2-D problems (see Chapter 6); d) if on the surface · n = 0, inside the body it is assumed to be = 0; the surface is thereby substituted by the vortex sheet (Saffman 1992). If the distribution of bound vorticity is determined and the total pressure on the body surface is constant, the force FB, affecting the body, is FB = ρ

∫ u×

dV .

VB

(1.105)

Using the vector identity and converting the volumetric integral to the surface one, we get for incompressible fluid

∫ u×

VB

dV =

⎡ 1

∫ ⎢⎣∇ 2 u

VB

2

⎤ ⎛1 ⎞ − (u∇)u ⎥ dV = ⎜ u 2 ⋅ n − u(u ⋅ n) ⎟ dS .(1.106) ⎦ ⎝2 ⎠ S

∫

B

The right side of equation (1.106) is determined by the velocity on the body boundary, and it does not depend on the method of imaginary system determination. Hence, the total force does not depend on the partial distribution of bound vorticity. As an example, we again consider the irrotational flow around the cylinder of radius a with circulation Γ > 0. The cylinder axis is directed along axis z, and its motion is 2-D and occurs in plane xy (Fig. 1.12). The flow potential in polar coordinates r, θ takes the form (e.g., see Loitsyansky (1966)) ϕ = − U∞ (r + a 2 r ) cos θ + Γ θ 2π .

Now we remove the cylinder and substitute it for the distribution of bound vorticity. Vortex lines are parallel to axis z, i.e., = (0,0, ω). One of the infinite possible distributions of vorticity is:

1.7 Vortex forces and invariants of vortex motion

55

Fig. 1.12. On the description of the irrotational flow of ideal fluid around the cylinder with circulation Γ

ω=

Γ πa

2

+

8 U∞ a2

r sin θ, r < a .

(1.107)

The corresponding velocity field inside the cylinder is solenoidal. It is described by the formulae u = − U∞ (1 + 2sin 2 θ)

r2 a

v = 2 U∞ sin θ cos θ

2

+ U∞ −

r2 a

2

+

Γ

2πa 2

Γ 2πa 2

r sin θ ,

r cos θ .

(1.108)

(1.109)

Expanding (1.105) FB = ρ

∫

u×

r ≤R

∫

∫

dV = ρ ⎡ − j uω dV + i vω dV ⎤ ⎣ ⎦

and substituting Eqs. (1.107) – (1.109), we find FB = ρ U∞ Γ j = ρU∞ × k Γ .

By doing so, we again come to the Kutta – Joukowski formula (1.102), where b k. The idea of the vortex force can also be applied to the determination of velocity of the straight vortex filament subject to the external force F. In-

56

1 Equations and laws of vortex motion

deed, in the coordinate system moving together with the vortex, the vortex force is ρ (u – uV) × , where u is the flow velocity, uV is the vortex velocity. The equilibrium condition requires that F + (u – uV) × = 0. If the vortex filament is oriented along basis vector k, we obtain uV = u + k × F/Γ.

1.7.2 Vortex momentum and vortex angular momentum

One of the advantages of the fluid motion description via vorticity distribution is in the existence of invariants of vortex motion, which are determined by the initial vorticity distribution and do not change with time. Therefore, some properties of the flow can be predicted without studying the flow details. The first two invariants include the so-called vortex momentum and the vortex angular momentum. We consider the flow of unbounded fluid resting at infinity without inner boundaries and in the presence of a confined area of vorticity3. The fluid density is assumed to be constant. The real momentum of the fluid ρ∫udV is determined conditionally because the value of the integral may depend on the shape of the surface, where the volume of integration tends to infinity. Thus, we proceed as follows: we calculate the resultant momentum, which should be applied to a limited fluid volume to provide the given motion from resting. This quantity is called the fluid impulse of the flow field (Batchelor 1967, p. 518). For the subsequent analysis we introduce a certain necessary integral identity. On the basis of vector identity ∇(b · a) = a × curl b + b × curl a + (a∇)b + (b∇)a

and assuming that b

r, we get

r × curl a = − a + ∇(r · a) − (r∇)a.

We notate the last term on the right side for each component of vector a (r∇)ai = ∇(r ai) − ai ∇r = ∇(r ai) − 3ai . Then, we integrate the vector identity over the volume, considering the last relationship, and convert two volumetric integrals containing gradients into the surface ones. Finally, using the formula of double vector product, we will obtain the integral identity 3

Results will not change if vorticity decreases exponentially at infinity.

1.7 Vortex forces and invariants of vortex motion

∫ r × curl a dV = 2∫ a dV + ∫ r × (n × a) dS.

57

(1.110)

This identity is true only for the 3-D space because at transformations it is taken into account that ∇r = 3. In the plane case, this value is 2 and, correspondingly, in (1.110) coefficient 1 will be used instead of 2 on the right side. Applying identity (1.110) to the velocity vector, we have 1 ρ u dV = ρ r × 2

∫

1 dV − ρ r × (n × u) dS. 2

∫

∫

(1.111)

A part of momentum, namely the value

1 I = ρ r× 2

∫

dV ,

(1.112)

which was introduced by Lamb (1932), is called the vortex impulse (or vortex momentum). The second term is finite, if surface S tends to infinity, but the value of the integral depends upon the surface shape. To verify this, we take into account that at infinity the flow field is potential, and without sources and sinks the main term of asymptotic expansion of potential is the dipole ϕ = – D·r/r3. The intensity of the dipole is proportional to the vortex momentum D = I/4π (Saffman 1992). Now we find velocity is asymptotic at infinity u ≡ ∇ϕ ∼

⎡3 ⎤ ( I ⋅ r )r − I ⎥ . ⎢ 2 4π r ⎣ r ⎦ 1

3 3

Substituting the result obtained into the surface integral in (1.111), we find that in spherical geometry, the integral equals – I/3, i.e., if the surface is a sphere, the “real” momentum is

∫

ρ u dV = (2 3) I. V

The same result will be obtained if V is a cube. If the volume has anisotropy, the result will differ. For instance, for the body in the form of a cylinder, whose height equals its’ diameter and the axis is directed along the axis of Ox, the integral is equal to ⎛ 4−2 2 2 2 ⎞ Ix , − Iy , − Iz . ⎜⎜ − 4 4 4 ⎝ ⎠

58

1 Equations and laws of vortex motion

We will show that the amount of vortex momentum, required to generate motion from resting, does not depend on time, even in the case of unsteady flow. Indeed, with consideration of Helmholtz equation (1.15) we have from (1.112): dI 1 ⎛ = ρ ⎜u× dt 2 ⎝

∫

+r×

d dt

1 ⎞ ⎟ dV = ρ 2 ⎠

∫ (u ×

+ r × ( ∇)u ) dV .

Let us use one of conclusions from Gauss – Ostrogradskii theorem (divergence theorem)

∫ ⎡⎣b(∇a) + (a∇)b⎤⎦dV = ∫ b(a ⋅ n) dS,

(1.113)

which at substitution of r × b instead of b takes the form

∫ ⎡⎣r × (a∇)b + a × b + (r × b)∇a⎤⎦dV = ∫ (r × b) ⋅ (a ⋅ n) dS. Assuming a =

(1.114)

, b = u, for the derivative of the momentum we get

dI = ρ u× dt

∫

1 dV + ρ (r × u) ⋅ ( ⋅ n) dS. 2

∫

The surface integral becomes zero, since disappears at infinity. The first item is the integral of force and it can be reduced to the surface integral of velocity (see (1.106)). Owing to the fact that the velocity at infinity decreases as r –3, we see that this integral also becomes zero. Therefore, dI/dt = 0, which was to be proved. Another quantity required for the description of rotational motion of fluid from rest is the resultant angular momentum. Lamb (1932, § 152) has defined the momentum related to vorticity (or vortex angular momentum) as:

1 M = − ρ r 2 dV . 2

∫

(1.115)

Let us compare the quantity M with the “real” angular momentum of fluid ρ ∫ r × u dV. From the theorem about the curl (G. Korn and T. Korn 1968) we get

1 1 ρ r × u dV = − ρ r 2 dV − ρ r 2 (u × n) dS. 2 2

∫

∫

∫

1.7 Vortex forces and invariants of vortex motion

59

The surface integral in general diverges, nevertheless, if S is a sphere containing all the vorticity, then r 2 = const and if ( ·n) = 0 on the surface, the integral becomes zero, as it follows from the chain of transformations

∫ n × u dS = ∫

dV =

∫ [ r∇

+ ( ∇)r ] dV = r ( ⋅ n) dS = 0 .

∫

It is considered here that ∇ = 0. The definition of angular momentum provided by Batchelor (1967) is somewhat different

1 M = ρ r × (r × ) dV . 3

∫

From identity (1.113), assuming a =

(1.116)

, b = r 2·r, we find

1 1 2 1 2 r × (r × ) dV + r dV = r r ( ⋅ n) dS . 3 2 6

∫

∫

∫

Thus, it follows that definitions (1.115) and (1.116) are equivalent, if ( ·n) = 0 at the boundary. This is also true for the unbounded fluid because disappears at infinity. We will show that the angular momentum M is invariant. Let us notate the time derivative of Eq. (1.115) with consideration of (1.15) ⎡ ⎡ ⎤ dM r2 d ⎤ r2 = −ρ ⎢ (r ⋅ u) + ⎥ dV = − ρ ⎢(r ⋅ u) + ( ∇)u ⎥ dV. dt 2 dt ⎥⎦ 2 ⎢⎣ ⎣⎢ ⎦⎥

∫

∫

Applying Eq. (1.113) at a =

∫ ⎡⎣r

2

, b = r 2 u, we discover

∫

( ∇) u + 2u(r ⋅ ) ⎤ dV = r 2 u( ⋅ n) dS. ⎦

As a result, for the derivative of M we have

1 dM = ρ r × (u × ) dV − ρ r 2 u( ⋅ n) dS . 2 dt

∫

∫

It is obvious that the first term here is the integral of the vortex force moment; the surface integral tends to zero due to the disappearance of at infinity. Applying relationships u × = ∇(u 2/2) − (u∇)u, ∇ × (u 2 r) = ∇u 2 × r and Eq. (1.114) at a = b = u, we get

⎡ ⎤ ⎛ u2 ⎞ ⎡ u2 ⎤ dM = −ρ ⎢∇ × ⎜ r + r × (u∇)u ⎥ dV = ρ r × ⎢n − u(u ⋅ n) ⎥ dS . ⎟ ⎜ ⎟ dt ⎝ 2 ⎠ ⎣⎢ 2 ⎦⎥ ⎣⎢ ⎦⎥

∫

∫

60

1 Equations and laws of vortex motion

The integrand has asymptotic O(r –5) at infinity, and this provides zero value of the surface integral, i.e., dM/dt = 0. When examining the momentum of vortex motion of fluid in a bounded region, Vladimirov (1977 ) interprets the surface integral in relationship (1.111) as the bound impulse caused by vorticity distribution over a bounding surface with density u × n, i.e., the vortex momentum is the sum of the momentum intrinsic to a given fluid volume and attached momentum. The bound angular momentum is interpreted in the same way. The surface integral in the expression for angular momentum of the vortex fluid motion in a bounded region is

1 M= ρ 3

{∫ r × (r ×

}

∫

) dV + r × (r × [n × u]) dS .

The generalization of definition of vortex momentum for the flow of non-uniform density incompressible fluid is also presented in cited work

I=

1 r × curl(ρu) dV . 2

∫

Deficiency of absolute convergence of integral ∫ ρ udV can be eliminated, if we consider the motion of a slightly compressible medium (Saffman 1992). The motion generated by concentrated pulse force was considered as the example. It was demonstrated that the momentum of fluid confined in a sphere of radius ct, expanding with the sound velocity c, is equal to 2I0/3. Here index 0 means that a density of undisturbed flow is taken for ρ. The remainder, I0/3, is concentrated at a spherical front. It is interesting to note that invariants I and M stay valid even for a viscous unbounded fluid. Indeed, the action of viscosity is equivalent to supplementary mass force ν∇2u, added to the right side of equation (1.103). Correspondingly, the balance of total forces is supplemented with integral

∫

∫

ν ∇ 2 u dV = −ν ∇ ×

∫

dV = −ν n ×

dS,

and the integral

∫

∫

∫

∫

ν r × ∇ 2 u dV = −ν r × (∇ × ) dV = −ν r × (n × ) dS − 2ν n × u dS.

is added to the balance of force moments. Identity r × (∇ × ) = ∇(r · ) − (r ∇)

−

and Eq. (1.113) were applied at a = r, b = , to reduce the last integral to the surface one. The surface integrals vanish, since u = O(r –3), and dis-

1.7 Vortex forces and invariants of vortex motion

61

appears at infinity. Hence, viscous forces do not influence the rate of impulse and angular impulse change. Saffman (1992) presents the development laws for impulse

IV =

1 r× 2

∫

dVV

(1.117)

and angular momentum MV = −

1 2 r dVV 2

∫

(1.118)

of isolated vortices in the presence of external forces g and external velocity ue, which can be induced by other vortices or imaginary vorticity. The above relationships take the form dIV = gdVV + ue × dt

∫

∫

dVV ,

dMV = r × gdVV + r × (ue × ) dVV , dt

∫

∫

where integration is performed over volume VV, occupied by the vortex. Note that even in the bounded flows, the laws of momentum and momentum conservation can be partially satisfied. For instance, in a vortex flow in a region limited by a plane wall or two parallel walls, without mass forces the momentum component parallel to the wall is the invariant (Betz 1932). It is obvious that the flow in a sphere under conditions u · n = 0, · n = 0 at the surface has invariant M. Considered invariants I and M correspond to invariants of Euler equations relative to a spatial translation and rotation. Two additional invariants – kinetic energy and helicity, are connected with the invariance properties of Euler equations relative to time and plane reflection invariance. 1.7.3 Kinetic energy

The kinetic energy of fluid bounded by the volume V, is T=

1

∫ 2 ρu dV. 2

Introducing the vector potential A, at a constant fluid density, we have

62

1 Equations and laws of vortex motion

1 1 T = ρ u(∇ × A) dV = ρ A ⋅ 2 2

∫

∫

1 dV + ρ n ⋅ ( A × u) dS . 2

∫

In the infinite fluid the surface integral disappears, and considering (1.88), we have 1 T = ρ A⋅ 2

∫

dV =

1 ρ 8π

∫∫

(r ) ⋅ (r ′) dVdV ′. r − r′

(1.119)

Another expression for kinetic energy is presented by Lamb (1932)

∫

T = ρ u(r × ) dV . Equivalence of definitions can be checked by integration of identity 1 2 ⎡1 ⎤ u − u ⋅ (r × ) = div ⎢ u 2r − u ⋅ (r ⋅ u) ⎥ . 2 ⎣2 ⎦

The integral of the right side vanishes since it is reduced to the surface integral with the asymptotic of integrand O(r –5). Kinetic energy is invariant, if the incompressible fluid is unbounded or bounded by resting walls, external forces are conservative with the singlevalued potential Π and viscosity equals zero. Indeed, it follows from Euler equation (1.10) that

1 d du ρ u 2 dV = ρ u dV = dt 2 dt

∫

∫

∫ [ −u∇p + u ⋅ g] dV = −∫ (Π + p)(u ⋅ n) dS .

The surface integral vanishes, if u · n = 0 at the surface or the surface tends to infinity. 1.7.4 Helicity

The fourth invariant is helicity. It was defined by Moffatt (1969)

H = ∫ u ⋅ dV .

(1.120)

Helicity characterizes the linkage degree of vortex lines in a flow. We examine two wound vortex filaments C1 and C2 (Fig. 1.13) with intensities 1 and 2 as the simplest example. Assume that both lines are not knotted, i.e., they are continuously shrunk into a point. According to Stokes theorem, the circulation around the first contour is

1.7 Vortex forces and invariants of vortex motion

a

63

b

Fig. 1.13. The right ( ) and left (b) single winding of vortex filaments

K1 =

∫ u ds = ∫

C1

⋅ n dS .

S1

Since the vorticity flux through S1 is induced only by the second filament C2, then if the filaments are not linked ⎧0, K1 = ⎨ ⎩ ±Γ 2 , if the filaments are single-linked Sign “+” here corresponds to “right” winding, i.e., the circulation direction of the induced velocity around contour C1 coincides with the direction of the vorticity vector on the filament C1; sign “–” corresponds to “left” winding. In general, filament C2 may wind around C1 any whole number of times. In this case K1 = α12 · Γ2 where α12 (= α21) is an integer, which can be either positive or negative (‘the parameter of mutual winding of curves’ C1 and C2). Now we consider the value Γ1 K1, written as the integral over the volume V1, occupied by the vortex filament C1. Since ds is parallel to vector on the filament, Γ1 ds can be substituted by dV, i.e., Γ1K1 =

∫ Γ1u ds = ∫ u ⋅

C2

dV .

V1

While calculating the sum, we obtain

H = ∑ αij ΓiΓ j = 2α12Γ1Γ 2 = ∫ u ⋅ dV , i, j

V

64

1 Equations and laws of vortex motion

where V is the volume occupied by both filaments, or alternatively, the whole volume occupied by the fluid (since vorticity concentrates only on filaments). The value H = ±2n Γ1 Γ2, where n is the number of windings of one filament around another, and the sign indicates a right or left winding direction. The simplest description of helicity density h = u · is peculiar to the flow with helical symmetry (see section 1.5.1). In this case h = u0 · ωz. In real swirl flows, the value ωz changes slightly, whereas, u0 varies across a wide range and can change its sign, i.e., in these flows the helicity density is determined by the velocity at the flow axis. Helicity does not change with time at the motion of inviscid incompressible fluid under the action of a conservative force. The rate of helicity change is dH ⎛ du = ⎜ dt ⎝ dt

∫

+u

d dt

⎡1 ⎤ ⎞ ⎟ dV = ⎢ ( −∇p − ∇Π ) + u( ∇ )u ⎥ dV = ⎠ ⎣ρ ⎦

∫

⎛ − p − Π u2 ⎞ = ⎜ + ⎟( ⋅ n) dS . ⎜ ρ 2 ⎟⎠ ⎝

∫

Due to this, it is clear that in the infinite fluid with a bounded region of vorticity dH /dt = 0. In the bounded region helicity is the invariant, if the condition · n = 0 is satisfied at the region boundary. Conservation of helicity means that the “knottedness” structure of the vortex field is maintained (Moffatt 1969). The helicity invariant is related to the more common topological characteristic: the Hopf invariant (see Moffatt (1984)). Helicity is of special importance in magnetic hydrodynamics, particularly, in the dynamo theory (Moffatt and Tsinober 1992). 1.7.5 Invariants of two-dimensional flows

When studying the invariants of a two-dimensional fluid flow, it is necessary to consider some important differences. Firstly, the velocity at infinity may have the order of r –1. Secondly, vorticity is the maintained scalar value = (0, 0, ω) (there is no effect of vorticity amplification due to the stretching of vortex tubes). Finally, the velocity field u, v in variables x, y is determined by one scalar function namely the stream function (the third velocity component in the direction of axis z is usually neglected because it affects neither the velocity components in the flow plane, nor the pressure distribution).

1.7 Vortex forces and invariants of vortex motion

65

Due to a slow decrease in the velocity at infinity, it is impossible to apply directly the total momentum, angular momentum and kinetic energy of fluid4. However, there are some related quantities, which are invariant. Moreover, the additional invariant appears

∫

Γ = ω dS .

It is equal to the velocity circulation over a closed contour covering the whole region of the vortical fluid. The evidence of invariance Γ follows from the conservation of vorticity and the area of a fluid element (see also Section 1.2). For the following analysis we need the asymptotic of the stream function at infinity. Substituting expansion log |r − r′| = log r − r ⋅ r′/r2 + O(1/r2) into (1.94), we find ψ∼−

1 ⎡ Γ log r + 2π 2π ⎣⎢

( ∫ ωr × k dS ) × r ⎤⎦⎥ ⋅ rk + O ⎛⎜⎝ r1 ⎞⎟⎠. 2

2

(1.121)

Here k is a unit vector, normal to the flow plane. We should note that the second term has the order of O(1/r). To eliminate the problem of integral divergence, Batchelor (1967) suggests considering the characteristics of auxiliary flow with the stream function

ψ′ = ψ +

Γ log r . 2π

This flow is the difference between the given motion and the steady motion with the same total vorticity concentrated at the coordinate origin. Alternatively, this transformation can be called the reduction to zero total vorticity. The velocity of auxiliary flow decreases as r –2 at r ∞. Nevertheless, the integrals in determination of total momentum and angular momentum of fluid are still not completely converging. The magnitudes of corresponding values of the vortex momentum and the vortex angular momentum in the flow with a bounded region of the vortical fluid are determined correctly (the quantities are reduced to a unit length in zdirection) 4

The fourth invariant – helicity, has no sense for the plain and axisymmetrical flows without swirling because · u = 0.

66

1 Equations and laws of vortex motion

(∫

∫

)

∫

I = ρ ωr × k dS = ρ yωdS, −ρ xωdS, 0 ,

(1.122)

1 1 ⎛ ⎞ M = − ρ r 2 ωk dS = ⎜ 0, 0, − ρ ( x 2 + y 2 )ω dS ⎟ . 2 2 ⎝ ⎠

(1.123)

∫

∫

There is no multiplier 1/2 in determination of momentum because the vortex lines are not closed. If the vortex lines are closed through the end planes of a unit thickness layer, and this vorticity is taken into account, the expression will coincide with (1.112) (Saffman 1992). To establish a connection of I with the ‘genuine’ momentum, we will use a plane analog of identity (1.110)

∫ r × rot a dS = ∫ a dS − ∫ r × (n × a) ds , wherefrom at a = u we get

∫ u dS = ∫ r ×

dS +

∫ r × (n × u) ds .

Let us estimate the contour integral for a circle of large radius R. Asymptotics of the velocity field of an auxiliary flow (or the flow at Γ = 0) will be determined by the stream function (1.121) taking into account (1.122) u∼

1

⎡ − Ir 2 + 2( I ⋅ r )r ⎤ . ⎦ 2πρr 4 ⎣

Substitution into the contour integral at the limit provides lim ρ

R→∞

∫

u dS =

r
1 I. 2

Therefore, in a 2-D flow a half of the applied momentum converts into the momentum of unbounded fluid, and the second half goes to infinity. The vortex angular momentum of the auxiliary flow (or the flow at Γ = 0) coincides with the real angular momentum. Indeed, 1 1 M = − ρ r 2 ∇ × u dS = ρ r × u dS − ρ ∇ × (r 2 u) dS. 2 2

∫

∫

∫

The first integral on the right side is the real angular momentum, and the second integral is reduced to the contour one, which provides 1/2 ρR2Γ = 0 at integration over the circle of large radius R.

1.7 Vortex forces and invariants of vortex motion

67

Now we consider kinetic energy of the plane motion of fluid. According to Batchelor (1967), we notate ∂ψ ⎞ 1 1 ⎛ ∂ψ −v T = ρ (u 2 + v2 ) dS = ρ ⎜ u ⎟ dS = ∂x ⎠ 2 2 ⎝ ∂y ⎡ ∂ (uψ ) ∂ (vψ) ⎤ 1 1 1 = ρ ⎢ψω + − dS = ρ ψω dS − ρ ψu ds. ⎥ 2 2 2 ∂y ∂x ⎦ ⎣

∫

∫

∫

∫

∫

The first of these two integrals converges at the tending of S to infinity. Behavior of the second integral at high values of r will be estimated using asymptotics of the stream function (1.121). Choosing a circumference of radius R as a boundary of the integration domain, we find

∫ i.e., at R

ψu ds ~ −

1 log R 2π

(∫

ω dS

)

2

=−

Γ2 log R , 2π

∞ Γ2 1 T − ρ ψω dS − ρ log R → 0. 2 4π

∫

The value 1 ρ T0 = ρ ψω dS = − ρ 2 8π

∫

∫∫ ω( r ) ω ( r ′) ln r − r ′ dSdS′

(1.124)

equals the part of fluid kinetic energy, which depends on the distribution of total fluid vorticity. To notate the double integral, we use the presentation of the stream function (1.94). Since ψ is a component of the vector potential, it seems natural to use definition (1.119). Really, at Γ = 0 definition (1.124) can be obtained from (1.119). For the finite value of total circulation in the unbounded fluid, kinetic energy is infinite, and instead of T the value T0 should be used. To ensure that T0 is invariant we consider the rate of the stream function change with the help of (1.94). Since in the plane flow dω/dt = 0, ⎡ ⎤ 2( x − x ′) 2( y − y′) 1 dψ =− ω( x ′, y ′) ⎢u +v ⎥ dS 2 2 2 2 4π dt ( x − x ′) + ( y − y ′) ⎦⎥ ⎣⎢ ( x − x ′) + ( y − y′)

∫

and taking into account the formula of velocity determination (1.95), we obtain dψ/dt = 0, and hence, dT0 /dt = 0.

68

1 Equations and laws of vortex motion

Thus, in the plane flow of fluid with vorticity, quantities Γ, Ix, Iy, M and T0 are constant. When Γ ≠ 0, we can introduce other invariants related to I and M, and namely, the coordinates of vorticity centroid Xc =

∫ xω dS , Y = ∫ yω dS c ∫ ω dS ∫ ω dS

and the dispersion of vorticity distribution relative to the vorticity centroid

∫ [(x − Xc ) + (y − Yc )] ω dS = ∫ (x + y ) dS − X 2 − Y 2 , = c c ∫ ω dS ∫ ω dS 2

D

2

2

(1.125)

1 i.e., I = ρΓ(Yc, −Xc, 0) = ρΓRc × k, M = − ρ Γ( D2 + Xc2 + Yc2 ) k . 2 For the axisymmetrical flow with circular vortex lines in cylindrical coordinates (r, θ, z) = (0, ω, 0), A = (0, ψ/r, 0). In this case, the vortex momentum I = πρ

∫∫ ω r drdz 2

(1.126)

is directed along the axis of symmetry; and the kinetic energy is written as T = πρ

∫∫ ψω drdz.

(1.127)

It will be shown in Section 3.2.1 that knowing values I and T for the vortex ring, we can determine the velocity of its motion. Helicity and angular momentum in the considered flow are identically equal to zero. The spherical Hicks vortex (see Section 3.2) can be presented as the example of a vortex with non-zero helicity. The flow in the Hicks vortex is axisymmetrical with swirl, and it can be described by the stream function satisfying equation (1.57). The streamlines and vortex lines are situated on the stream surfaces, forming a family of nested tori. The flow possesses non-zero helicity, momentum, angular momentum and kinetic energy, which are constant in time.

2 Vortex filaments

2.1 Geometry of vortex filaments The key object in the vortical flow theory is the vortex filament; its widest definition is a vortex tube surrounded by fluid with zero vorticity. It is clear that this strict definition is applicable to the ideal fluid only. In real fluids the diffusion of vorticity occurs, however, the concept of the vortex filament is useful and contansive for a medium with a low viscosity. If we tend the filament cross-section to zero and maintain a constant value for circulation Γ, we obtain the vorticity distribution, different from zero only along a certain spatial curve. This vorticity distribution is called an infinitely thin vortex filament, or a line vortex (do not confuse this with a vortex line). In some literature, the term ‘vortex filament’ just means an infinitely thin vortex filament. We will use the term “vortex filament” also for interpretation of experimental results on swirl flow study, where vortex structures have elongated spatial shapes with vorticity concentrating along the axis. The examples are: tornado, waterspout, vortex behind the turbine runner, etc. The peculiarity of the listed structures is their three-dimensional geometry. Therefore we should consider the main methods of description and the basic (canonic) types of spatial curves with special emphasis on a helical line. Generally, the spatial curve might be given in parametric form (Bronshtein and Semendyaev 1985) r = r(α)

or

x = x(α), y = y(α), z = z(α).

(2.1)

Here α is a real parameter. The positive direction on this curve corresponds to the growth of α. The arc length s along the curve between points M0 and M, which are given by values of parameters α0 and α, is determined by formulae α

s=

∫

α0

2

ds,

⎛ dr ⎞ 2 2 2 ds = ⎜ ⎟ dα = x + y + z dα, d α ⎝ ⎠

70

2 Vortex filaments

where the dot represents a derivative with respect to α. If we fix point M0, the length s along the curve can be considered as a new parameter, i.e.

r = r(s)

or

x = x(s), y = y(s), z = z(s).

(2.2)

Besides Eqs. (2.1) and (2.2), other methods for description of a spatial curve are possible: 1 – by the introduction of a right-hand triple of orthogonal unit vectors (t, n, b), called a moving trihedron for point M of a spatial curve and representing three vectors directed along tangent, principal normal, and bi-normal; 2 – via two shape parameters – curvature and torsion – for this spatial curve at each point. Let us consider those parameters in detail. The unit vector of the tangent is t = dr(s)/ds. This vector coincides with the tangent at point M and is directed to the same side as the curve (Fig. 2.1). The unit vector n, co-directional to vector d2r (s)/ds2 is called a principal normal vector at point M (hereafter: normal vector). The binormal vector b is defined as b = t × n. One can prove that with the accuracy of the second infinitesimal order terms, the spatial curve element lies in the plane formed by vectors t, n. This plane is called the osculating plane. Note that the vector n is directed inside the arc on this plane. The curvature κ at point M is defined by relationships

d2 r ( s)

or κ = [ x ′′2 + y ′′2 + z ′′2 ]1/ 2 ≥ 0, ds 2 where the prime represents the derivative with respect to length s along the curve. The reciprocal value is a curvature radius R = 1/ κ. Note that for the plain case the curvature is a sign-alternated quantity. Torsion τ of a spatial curve is defined by the formula

κn =

τ=

(t × t′) ⋅ t ′′ t′

2

x ′ y′ z′ 1 = 2 x′′ y′′ z′′ . x′′ + y′′2 + z′′2 x′′′ y′′′ z′′′

(2.3)

Fig. 2.1. Right-hand triple of orthogonal unit vectors (t, n, b), used for the description of a spatial curve

2.1 Geometry of vortex filaments

71

The value τ = 0 implies that the curve is flat. Unlike the curvature parameter, the torsion has a sign that defines the direction of the curve twist. If τ > 0, the curve resembles a right-handed screw, while τ < 0 corresponds to a left-handed screw. If the curve is given in the form (2.1), then

κ2 =

1 R2

=

( x 2 + y 2 + z 2 )( x 2 + y 2 + z 2 ) − ( x x + y y + z z) 2 ( x 2 + y 2 + z 2 )3

τ=

x y z x y z . ( x 2 + y 2 + z 2 )3 x y z R2

,

(2.4)

(2.5)

The curvature and torsion are related to the triple of unit vectors (t, n, b) through Frenet – Serret equations

dt = κn , ds

dn = τb − κt , ds

db = −τn . ds

(2.6)

One can see from the above formula that the curve torsion may be interpreted as the angular velocity of the bi-normal rotation around the instant position of the tangent. Let us take a helical curve as an example; it is described in a parametric way

x = a cos α, y = a sin α, z = l α, a > 0.

(2.7)

Here a is the cylinder radius with the helical curve wound on, parameter α has the meaning of angle, counted from a certain point M0 (Fig. 2.2). The value 2πl is the helix pitch, and l > 0 corresponds to the right-handed screw depicted in the figure, and l < 0 - to the left-handed one. Taking the expression for the arc length s, we obtain α

∫

s = ( x 2 + y 2 + z 2 )1/ 2 dα = α a 2 + l 2 .

(2.8)

0

Correspondingly,

(

)

(

)

x = a cos s / a 2 + l 2 , y = a sin s / a 2 + l 2 , z = ls / a 2 + l 2 . Then the curvature and torsion are

κ = 1 R = x ′′2 + y ′′2 + z ′′2 = a (a 2 + l 2 ) = const ,

(2.9)

72

2 Vortex filaments

Fig. 2.2. Helical curve

−a sin α a cos α l l τ= −a cos α −a sin α 0 = 2 2 = const. (2.10) 3 ⎡(−a sin α)2 + (a cos α)2 + l2 ⎤ a sin α −a cos α 0 a + l ⎣ ⎦ R2

It is also easy to calculate the triple of unit vectors

l⎤ ⎡ − sin α, cos α, ⎥ , ⎢ a⎦ a2 + l2 ⎣ r ′′ = [ − cos α, − sin α, 0] , n= r ′′

t = r′ =

b =t×n =

a

(2.11)

a⎤ ⎡ sin α, − cos α, ⎥ . ⎢ l⎦ a2 + l2 ⎣ l

And the formula for the slope angle of the helical curve β is the following: tan β =

l τ = . a κ

(2.12)

Note that a circle is an example of a curve with zero torsion (τ = 0) and curvature is inversely proportional to the circle radius a

κ = 1/a,

(2.13)

while a straight line is an example of a curve with zero torsion and zero curvature.

2.2 Biot – Savart law

73

2.2 Biot – Savart law Let us derive a velocity distribution induced by an infinitely thin closed vortex filament with intensity Γ in infinite space with zero vorticity. We consider a closed vortex filament with a small, but finite cross-section δS (Fig. 2.3). Then vector is parallel to the filament element ds. Here we use Eq. (1.93), transformed into the following form, assuming dV = δS ds: 1 1 × ∆r × ∆r δSds = lim δSds . u= 3 3 0 S δ → 4π 4π ∆r ∆r

∫

∫

V

s

Here V is the vortex filament volume. According to the Stokes theorem ω · δS = Γ. With the limits δS 0, ω ∞ and condition Γ = const, we obtain the formula

u(r ) = −

Γ 4π

∫ s

∆r × ds (r ′) ∆r

3

.

(2.14)

This formula for velocity induced by an infinitely thin vortex filament is called the Biot – Savart law, since it coincides (formally) with the formula for magnetic field created by a closed conductor with a constant current. Formula (2.14) can be rewritten in the differential form

du = −

Γ ∆r × ds . 4π ∆r 3

(2.15)

Then du is an infinitely small velocity induced by element ds of the vortex filament at point M(r). There exists another, more simple method for the derivation of the Biot – Savart formula (2.14). We consider an infinitely thin vortex filament (also closed). Let n, b, t be the unit vectors of normal, bi-normal, and tangent, correspondingly, and xn, xb, xt are the coordinates along these directions. Obviously, in this case the vorticity can be presented in the form = Γδ(xn) δ(xb)t,

(2.16)

where δ is Dirac's delta-function. Substituting (2.16) in (1.93) and integrating over xn, xb, we obtain the desired formula Γ Γ ∆r × ds t × ∆r δ( xn )δ( xb ) dxn dxb dxt = − u= . 3 3 4π 4π ∆r ∆r

∫

Here we take dV = dxn dxb dxt and t dxt

∫

ds.

74

2 Vortex filaments

Fig. 2.3. On the formulation of the Biot – Savart law

The vorticity in the form (2.16) is written for the coordinate system bound to a curve. In the absolute coordinate system the following definition is more convenient (Leonard, 1985): ∂r = Γ δ ⎡⎣r − r ( s′ ) ⎤⎦ ds′. ∂s′ The Biot – Savart formula may take another form, if we transform the contour integral in (2.14) into the integral over arbitrary surface S bound to the contour s (looped infinitely thin vortex filament). Using one of the corollaries of Stokes theorem (G. Korn and T. Korn 1968) to the vector Γ r − r′ B≡ , 4π r − r ′ 3

∫

we obtain

∫ (dS × ∇) × B = ∫ ds × B ≡ u .

(2.17)

S

Here dS = ndS, dS is the surface element, n is the unit normal vector to this element. According to the Biot – Savart law, the right-hand side is the expression for induced velocity that has equivalent definition through the surface integral. We rewrite the vector product using the following vector identity: (dS × ∇) × B = ∇(B · dS) − (∇B)dS.

(2.18)

For convenience we perform the following transformations in the coordinate form. Let us expand ∇B, using the denotation

2.2 Biot – Savart law

75

a ≡ ∆r = ( x − x ′) 2 + ( y − y ′) 2 + ( z − z ′) 2 ,

∇B ≡

=

∂ ⎛ r − r ′ ⎞ ∂ ⎛ x − x′ ⎞ ∂ ⎛ y − y′ ⎞ ∂ ⎛ z − z′ ⎞ = + + = ∂r ′ ⎜⎝ a3 ⎟⎠ ∂x ′ ⎜⎝ a3 ⎟⎠ ∂y ′ ⎜⎝ a3 ⎟⎠ ∂z ′ ⎜⎝ a3 ⎟⎠

−a3 + 3( x − x ′) 2 a5

+

−a3 + 3( y − y ′) 2 a5

+

−a3 + 3( z − z ′) 2 a5

= 0,

i.e., the second right-hand term in (2.18) vanishes. Obviously, ∂/∂r' (B · dS) = − ∂/∂r (B · dS), because B = B(r − r'), and operator ∂/∂r may be taken outside the integral in (2.17), retaining the symbol ∇. At last, we obtain from (2.17)

u=−

Γ ∇Ω , 4π

(2.19)

where we bring in the notation

Ω(r ) =

r − r′

∫ r − r ′ 3 dS .

(2.20)

Since the flow beyond the vortex filament is potential, then by definition u = ∇ϕ we see that the potential ϕ is ϕ=−

Γ Ω. 4π

(2.21)

Note that the velocity potential is ambiguously determined, because the area beyond the vortex filament is not a simply connected domain. Quantity Ω is a solid angle subtended by area S viewed from point M(r); and the area is bound by the looped vortex filament (Fig. 2.4). Indeed, we have from (2.20) that dΩ = dSa/a2, where a = |r − r'| is the distance from point M to element dS, and dSa is the projection of the surdsssis

Fig. 2.4. Illustration of Biot – Savart law interpretation

76

2 Vortex filaments

face element on the plane normal to the vector (r − r'). The relationship obtained is a definition for the elementary solid angle. Finally, the velocity induced by a vortex filament can be determined through a vector potential: u = curl A. The vector potential of a vortex filament can be deduced from Eq. (1.88) by substitution of Eq. (2.16) for vorticity

A=

Γ 4π

∫

ds . ∆r

(2.22)

2.3 Rectilinear infinitely thin vortex filament 2.3.1 Vortex filament in ideal fluid A rectilinear infinitely thin vortex filament is a simple object, which can be easily described by the Biot – Savart law. It can be interpreted as a circle with an infinitely large curvature radius. Let axis z in a cylindrical coordinate system be directed along the vortex filament, as shown in Fig. 2.5. Obviously, we have only the tangential component of induced velocity u = u(r), the formula for this component can be obtained from (2.14)

u= u =

Γ 4π

∞

sin α

∫ r2 + z

−∞

π

dz = 2

Γ Γ . sin α dα = 4πr 2πr

∫

(2.23)

0

Hence, the circulation over a circle with radius r equals Γ. For r 0 the velocity u ∞, which is a consequence of the ideal fluid approach.

Fig. 2.5. Rectilinear vortex filament

2.3 Rectilinear infinitely thin vortex filament

77

Let us notate the formulae for stream function ψ and potential ϕ, which can be defined with the accuracy of a constant. In polar coordinates (r, θ) the definition gives us

uθ ≡ u = −

∂ψ 1 ∂ϕ = , ∂r r ∂θ

and accounting for (2.23)

ϕ=

Γ θ + const, 2π

ψ=−

Γ log r + const. 2π

(2.24)

Since these relationships are independent of coordinate z, it is enough to consider motion in a 2-D plane. Therefore, instead of a vortex filament, we refer to a point vortex. Then we can introduce a complex potential W(z)

W (z) = ϕ + iψ =

Γ Γ (log r + iθ) + const = log z + const . 2πi 2πi

Here the complex coordinate z = x + iy = reiθ, x, y are the Cartesian coordinates. If the vortex filament does not coincide with the z axis, but passes through a point on plane xy with a complex coordinate z0 = x0 + iy0, the complex potential is written as

W=

Γ log(z − z0 ) + const . 2πi

The benefits of using a complex potential is that due to additivity one can easily find the potential for a system of isolated or continuously distributed vortex filaments; after that the velocity field is simply recovered by formula

ux − iuy =

dW . dz

Here ux, uy are the components of velocity in a Cartesian coordinate system. We take as an example N rectilinear vortex filaments oriented in parallel to the z axis with complex coordinates z0k on plane xy. Then

W=

N

⎡Γ

⎤

∑ ⎢⎣ 2πki log(z − z0k ) + Ck ⎥⎦ , k =1

Γk dW N = , ux − iuy = dz 2πi( z − z0k ) k=1

∑

(2.25)

78

2 Vortex filaments

where Ck are the arbitrary constants that must be selected for better convergence at N ∞. The fluid velocity at the point coinciding with the vortex position z0p is determined by the following rule: in sum (2.25) we eliminate the term including z0p which is responsible for singularity. Consider the particular case of the last problem: there are two identical linear vortices with opposite signs and the distance between the vortices tends to zero. Then, taking the coordinates z0 + δz/2 and z0 − δz/2 for positive and negative vortices correspondingly, we obtain the formula W=

⎡ ⎛ Γ δz ⎞ δz ⎞ ⎤ Γ ⎛ δz ⎞ ⎛ lim ⎢log ⎜ z − z0 − ⎟ − log ⎜ z − z0 + ⎟ ⎥ = lim ⎜− ⎟. 2πi δz →0 ⎣ ⎝ 2 ⎠ 2 ⎠ ⎦ δz →0 2πi ⎝ z − z0 ⎠ ⎝

Assuming that lim Γ δz = m , we have δz →0

W=

iM . 2 π( z − z0 )

Here M = meiθm, where θm is derived from the initial vector orientation δz = | δz0|eiθm. To simplify the analysis we take z0 = 0, θm = 0, i.e., both vortices pass through the x axis on plane xy at both sides of the origin. Then

W=

im . 2 πz

(2.26)

This formula describes the so-called linear vortex dipole, or simply the vortex dipole with moment m. It is easy to demonstrate that the streamlines and equipotential curves are the circles passing the coordinate origins. Besides, the circle centers lie on the x and y axes, correspondingly. Note that for a usual dipole consisting of a source and a sink, the complex potential takes the form: W = m/2πz. One can see from comparison with (2.26) that the distinction between the vortical and usual dipoles is that streamlines replace equipotential curves and vice versa. We have described a rectilinear vortex filament in infinite space (or a point vortex on an infinite plane). In some particular cases, the analytical solution for space with rigid boundaries can be deduced using the image vortex method. In particular, for a point vortex in an area restricted with a real axis, the image vortex has a circulation equal in magnitude and opposite by the sign (Fig. 2.6). The complex potential of the system and induced velocity field are

2.3 Rectilinear infinitely thin vortex filament

W=

z − z0 Γ Γ ⎛ 1 1 ⎞ , ux − iuy = − log ⎜ ⎟, 2πi ⎝ z − z0 z − z0 ⎠ z − z0 2πi

79

(2.27)

where z0 is a complex coordinate of vortex, and the bar represents a complex conjugation. Here the vortex moves parallel to the wall with a speed equal to that induced by the image vortex at point z0, U = Γ/4πy0. Here y0 is the distance from the vortex to the wall. Another example is the motion of a point vortex inside or outside a rounded region with the radius a (Fig. 2.7). In this case the image vortex circulation is also equal in magnitude and opposite by sign to the actual one. The image vortex is placed on a radial half-line passing through the basic vortex at the distance of a2/r0 from the circle center. The complex potential and velocity for this system are described in the following way: W=

⎞ z − z0 Γ ⎛ 1 1 Γ − , ux − iuy = log ⎜⎜ ⎟⎟ . 2 2 2 πi ⎝ z − z0 z − a z0 ⎠ 2πi z − a z0

(2.28)

The vortex moves along a circular trajectory concentric to the area boundary with the velocity U=

r0 Γ , 2 2π a − r02

i.e., it rotates with the frequency f=

Fig. 2.6. Point vortex near a rigid plane

1 Γ . 2 2 4 π a − r02

Fig. 2.7. Point vortex inside a circular area

(2.29)

80

2 Vortex filaments

Here r0 = |z0| is the distance from the circle center to the vortex. The analytical solution for a vortex in a bounded area can be written for a more general case when the conformal image of the flow zone into a circle (or its outside) or to a half-plane is known. Introducing a complex variable in the flow zone and prescribing the conformity z( ) into a halfplane Im(z) > 0, we obtain

W=

z( ) − z( 0 ) Γ log . 2πi z( ) − z ( 0 )

Note that any external potential flow can be superimposed on the flow induced by the vortex and its image. 2.3.2 Vortex filament diffusion Now we analyze the contribution of viscosity. One of the consequences of the viscosity effect is that vortex lines do not coincide with the trajectories of liquid particles. This is clear from the conditions underlying the Eqs. (1.16), (1.17). Let us consider in detail the viscosity effect with the example of rectilinear vortex filament diffusion. At time instant t = 0 we have an infinitely thin vortex filament (of circulation Γ) coinciding with the z axis. Obviously, the solution with time remains axisymmetric, so the set of equations (1.37) simplifies to a single equation

⎛ ∂ 2 ω 1 ∂ω ⎞ ∂ω = ν⎜ 2 + ⎟, ⎜ ∂r r ∂r ⎟⎠ ∂t ⎝ where ω ωz is the axial component of vorticity. This equation coincides with the heat transfer equation for a problem of heat spreading from a linear source in a uniform medium. This solution is well known

ω=

c −r 2 e 4πνt

4 νt

.

(2.30)

Constant c follows from the initial condition, which is given through the Stokes theorem

Γ = Γ (r, t )

r

t =0

∫

= 2π ωrdr 0

Using the solution (2.30), we obtain

. t =0

2.3 Rectilinear infinitely thin vortex filament r

2 2π e −r Γ=c 4πνt

∫

4 νt

ξ

(

∫

rdr t =0 = c e −ξ d ξ t =0 = c 1 − e −r

0

0

2

4 νt

)

t =0

81

= c.

Thus

ω (r , t) =

Γ −r 2 e 4πνt

4 νt

.

(2.31)

Then the tangential velocity u is as follows: r

u(r , t) =

(

2 Γ 1 ωr dr = 1 − e −r 2πr r

∫ 0

4 νt

).

(2.32)

Distributions of velocity u are plotted in Fig. 2.8 for different instants of time. At t = 0 we have the velocity distribution induced by the infinitely thin vortex filament: u = Γ/2πl. For t > 0, a local maximum appears on these profiles and it drifts with time to infinity, with simultaneous decrease in the amplitude of the maximum. For r << 4νt the velocity u = Γr/8πνt, i.e., the liquid in the vortex core rotates as a solid body with the angular velocity Γ/8πνt. Thus the vorticity diffuses into the entire space filled with the fluid. This flow example was named the Lamb – Oseen vortex (Lamb, 1932). All the above analysis proves that taking into account the viscosity in the bulk of a fluid leads to vorticity diffusion, but in no way it is responsible for its generation.

Fig. 2.8. Radial distributions of tangential velocity at different time instants for the Lamb – Oseen vortex

82

2 Vortex filaments

2.4 Self-induced motion of a vortex filament Any bending of the vortex filament creates a situation where the given piece of filament is in the field of velocity induced by other sections of this filament. The resulting motion is known as self-induced vortex filament motion. According to Batchelor (1967) we can estimate the behavior of a curved thin vortex filament using the Biot – Savart law (2.14)

u (r ) = −

Γ 4π

∫

( r − r ′) × ds ( r ′ ). r − r′

3

Obviously, the integral diverges at r r', i.e., at the points of the vortex filament. Therefore, we estimate the induced velocity in the immediate vicinity of the given point on the vortex filament. Let this point be the origin of the coordinate system, and a small piece of filament is approximated by an arc with radius R = 1/κ, where κ is the curvature (Fig. 2.9). Let us introduce a triple of orthogonal unit vectors (t, n, b) allowing for b = t × n. Here n and b are the unit vectors of normal and bi-normal, and t is a tangent unit vector. Vector r' corresponds to the point on the filament, while r gives us the point where the induced velocity is calculated. We assume that the calculation region for velocity estimation is limited by conditions |r| = r

R and

s

R (or θ

1),

(2.33)

where s is the distance along the vortex filament from the origin O. The length scale is the curvature radius R. Thus

Fig. 2.9. Illustration on the calculation of self-induced motion of a vortex filament

2.4 Self-induced motion of a vortex filament

r = yn + zb,

83

s = Rθ,

r' = x' t + y' n = tR sin θ + nR(1 − cos θ) ≈ tRθ + n(R θ2/2) = st + (s2κ/2)n, dr' = δs ≈ (t + sκn)ds. Then we specify the numerator and denominator in the Biot – Savart integral

(

)

(r − r ′) × δs ≈ yn + zb − st − ( s 2 κ 2)n × ( t + sκn ) ds = = ⎡ zn − zsκt + ( y + s 2 κ 2) b ⎤ ds, ⎣ ⎦ 2

r − r ′ = yn + zb − st − ( s 2 κ 2)n = r 2 + s 2 (1 − yκ) + κ 2 s 4 4 . 2

Let us divide the Biot – Savart integral into two parts. After substitution of ready expressions into the first part of Biot – Savart integral, we integrate it at the limits – L ≤ s ≤ L, where L satisfies condition (2.33). The results is

Γ − 4π

∫

(r − r ′) × ds r − r′

3

Γ =− 4π

L

∫

−L

− zsκt + zn − ( y + κs 2 2)b ds + In , (2.34) ⎡r 2 + s 2 (1 − yκ) + κ2 s 4 4⎤ 3 2 ⎣ ⎦

where In is the remaining part of the Biot – Savart integral. So far we have used only this condition: s, L R. We can calculate the asymptotic expression for this integral, for the limit

r

0, L

∞.

0, L/r

To accomplish this, we have to use a new variable m = s/r

⎛ y κm 2 ⎞ z z − κ + − t n m ⎜⎜ 2 + ⎟b M 2 ⎟⎠ r r2 r Γ ⎝ − dm ≈ 32 4π 2 2 4 − M ⎡1 + m (1 − y κ + κ s 4) ⎤ ⎣ ⎦

∫

≈

r →0

Γ 4π

M

∫

(by − nz)(1 + 1.5m 2 yκ (1 + m 2 )) r 2 + bκ m 2 2 (1 + m 2 )3 2

−M

dm =

M M ⎞ 3κy dm m2 Γ (by − nz) ⎛ ⎜ ⎟+ = + dm 2 32 ⎜ ⎟ 4π 2 (1 + m 2 )5 / 2 r2 −M ⎝ − M (1 + m ) ⎠

∫

∫

84

2 Vortex filaments

+

κΓ b 8π

M

m 2 dm

∫

(1 + m 2 )3 2 −M

,

where M = L/r. We have neglected the first term in the numerator of the initial integral and have discarded the extra terms in round brackets in the denominator, except for yκ, which is taken into account as a correction for the numerator. This gives us M

dm

∫

−M M

∫

−M

m 2 dm (1 + m 2 )3 2

M

∫

=

1 + m2

≈ 2, −M

(

)

M

⎡ ⎤ 2L m , = ⎢− + ln m + 1 + m 2 ⎥ ≈ −2 + 2log 2 r ⎥⎦ − M ⎣⎢ 1 + m

m 2 dm

−M

(1 + m 2 )3 2

M

m

(1 + m 2 )5 2

⎡ ⎤ 1 1 4 2 dm ≈ 2 − = . − ⎢ ⎥ 2 3/ 2 2 5/ 2 3 3 (1 + m ) ⎦⎥ ⎢ (1 + m ) −M ⎣ M

=

∫

The result is the asymptotic formula for induced velocity in the vicinity of a curved vortex filament u≈

Γ 2πr

where

2

( y b − zn ) +

In0 = In

Γy 4πr

2

L→0 r →0

( y b − zn ) κ +

Γ L→0 4π

= lim

∫

s >L

2L ⎞ 0 Γκb ⎛ − 1⎟ + In , ⎜ log 4π ⎝ r ⎠

r ′ × ds (r ′) r′

3

(2.35)

.

Here the first term represents rotational fluid motion around the considered element of a filament and it does not produce any shift. The module of this term equals Γ/2πr and it coincides with Eq. (2.23) for the velocity induced by a rectilinear vortex filament. The second term is a correction to circulatory motion due to the filament curvature. The third term is responsible for motion by the bi-normal vector and it becomes infinite on the vortex filament. This means that the curved filament moves along the bi-normal vector with infinite velocity and may undergo deformations. Integral In0 , taken over the rest of the filament represents the non-local contribution to selfinduced motion of the filament. For finite values of L, this integral is limited. For L 0 the contribution from In0 in the origin vicinity is

2.4 Self-induced motion of a vortex filament

85

⎡ Γ ( st + ns 2 κ 2) × ds ⎤ Γ ⎡ ds ⎤ Γκ ⎥ −2 ⎢ =− ⎢ = −b log L. ⎥ 3 4π ⎢⎣ s ⎥⎦ 4π ⎢⎣ 4π ⎥⎦ s = s L s=L

∫

∫

Here coefficient 2 appears because the contributions from the vicinity of points s = ±L are the same, and the minus sign is because (+L) and (−L) are the lower and upper limits of integration, correspondingly. Returning to (2.35), we see that integral In0 liquidates the logarithmic singularity in the second term, occurring at L 0. Thus, in the asymptotic expression 0, i.e., at the point of for induced velocity (2.35), only ln r diverges at r the vortex filament. The general meaning of this is that the self-induced motion of a curved vortex filament at asymptotic limit is caused exclusively by local effect, and more specifically – by local curvature of filament axis κ. The divergence is related to the idealized concept of the vortex filament as an object with an infinitely thin core. Nevertheless, these formulae can be used for deducing stable forms of curved vortex filament (this is demonstrated below). With additional suppositions about the vortex core, these formulae are suitable for description of vortex filament dynamics. We rewrite (2.35) retaining only the term that makes a local contribution to self-induced motion, u=

2L ⎞ Γ ⎛ κb ⎜ log −1 . r 4π ⎝ ⎠

(2.36)

Note that in this approximate formula, the limit of integration L remains uncertain. This problem is discussed in the following sections. Now let us use Eq. (2.36) for the derivation of the equation for self-induced motion of a vortex filament. To avoid the divergence problem in (2.36), we make the assumption that the vortex filament is thin, but still has a finite radius ε (or its effective value – for a complex core structure) and that self-induced velocity is derived from (2.36) at r = ε. We denote the radius-vector of a point on the vortex filament as X(s, t), where s is the distance along the filament. By definition, the velocity of this point is

u = ∂X/∂t.

(2.37)

We introduce a new time coordinate t

Γ ⎛ 2L ⎞ − 1⎟ → t , ⎜ log ε 4π ⎝ ⎠

(2.38)

assuming L/ε = const. By making equal (2.37) and (2.36) at r = ε and taking into account (2.38), we obtain the equation for filament motion

86

2 Vortex filaments

∂X (t, s) = κb . ∂t

(2.39)

This approach is known as the local induction approximation and is usually attributed to Hama (1962) and Arms and Hama (1965), although it had been used previously in the work by Da Rios (1906) (as acknowledged by Ricca (1996)). We transform equation (2.39), accounting for these relationships:

b = t × n,

∂X ≡ X ′ = t, ∂s

X" = κn.

Thus κn = κ(t × n) = X' × X". The result is the so-called local induction equation (or LIE):

∂X (t, s) = X ′ × X ′′ . ∂t

(2.40)

The local induction approximation is one of the main simplified approaches to the dynamics of a vortex filament. We emphasize that within this approximation the induced velocity is directed by the bi-normal to the filament, but the effects of filament stretching or contracting remain beyond this approach, even though these effects can be significant in vortex filament dynamics (Klein and Majda 1991). Several important conclusions and LIE solutions, as well as other approaches will be described and discussed in the successive chapters.

2.5 Infinitely thin vortex ring Considering a vortex filament of circular shape (Fig. 2.10), we can determine the vorticity component normal to the plane (r, z), through Dirac's delta-function (see (2.16)) ω = Γδ(r − r0∗ )δ( z − z0∗ ).

(2.41)

Here (r0, z0) is the point where the filament passes through the plane (r, z) and Γ is its circulation. Substituting ω into (1.96), we deduce the stream function

ψ=

Γ (rr0 )1/ 2 2π

2 ⎡⎛ 2 ⎤ ⎞ ⎢⎜ k − k ⎟ K (k) − k E(k) ⎥ , ⎠ ⎣⎝ ⎦

(2.42)

2.5 Infinitely thin vortex ring

87

where k2 = 4rr0/[(z – z0)2 + (r + r0)2], K and E are full elliptic integrals of the first and second kind. Lamb (1932, §161), offered another form for stream function; he introduced the minimum and maximum distance from point (r, z) to the ring s12 = ( z − z0 ) 2 + (r − r0 ) 2 ; s22 = ( z − z0 ) 2 + (r + r0 ) 2 and used the new variable λ = (s2 – s1)/(s2 + s1). The result is the following

ψ=

Γ ( s1 + s2 ) [ K(λ) − E(λ)]. 2π

(2.43)

This expression can be derived through Landen's transformation (G. Korn and T. Korn 1968). Since a vortex filament with circulation Γ is equivalent to a distribution of dipoles over the circle area with uniform intensity Γ, it is possible to find the potential and stream function as the integral of the Bessel function (Lamb 1932, §161). ∞

ϕ=

Γ r0 e −λ ( z − z0 ) J0 (λr ) J1 (λr0 )dλ 2

∫ 0

∞

Γ ψ = rr0 e −λ ( z − z0 ) J1 (λr ) J1 (λr0 )dλ 2

z − z0 > 0 .

(2.44)

∫ 0

This form of stream function does not possess any special advantages in comparison with (2.42) and (2.43).

Fig. 2.10. Infinitely thin vortex ring

88

2 Vortex filaments

A round vortex filament is a simple object with non-zero curvature. As discussed above, this filament shifts by the binormal, i.e., in direction of the z axis, with infinite velocity. Due to its symmetry, the circular filament is not liable to deformation. The kinetic energy of the liquid motion, which is induced by the infinitely thin closed vortex filament, equals infinity. Note that the vortex impulse (despite a singularity in the velocity field) is a finite quantity. Indeed, substituting r = (x – r0 cos ϕ, y – r0 sin ϕ, z – z0), = ω · i0 = ω(−sin ϕ, cos ϕ, 0), dV = dSr0dϕ in (1.112) and taking into account (2.41), we find the non-zero component of the impulse

Iz = πρΓr02 = const .

(2.45)

The asymptotic expansion of a velocity field in the vicinity of a vortex filament with arbitrary geometry was considered in Section 2.4. For the specific case of a circular vortex filament, the stream function and velocity asymptotic can be found through the asymptotic properties of elliptic integrals (Abramowitz and Stegun 1964). Assuming s = s1 r0, we obtain from (2.42):

ψ=

uz =

Γr0 2π

⎧⎪ 8r ⎛ s2 8r0 s cos θ ⎛ s ⎞ ⎫⎪ ⎞ 0 O − + − + log 2 log 1 ⎜⎜ 2 log ⎟⎟ ⎬ , ⎨ ⎜ ⎟ s s r0 ⎠ ⎪⎭ 2r0 ⎝ ⎠ ⎪⎩ ⎝ r0

(2.46)

⎛ Γs 8r Γ Γ ⎛ 1 ∂ψ s⎞ ⎞ cos θ + log 0 − sin 2 θ ⎟ + O ⎜ 2 log ⎟ , =− ⎜ ⎜ r ∂r s r0 ⎠⎟ 2πs 4πr0 ⎝ ⎠ ⎝ r0

(2.47)

⎛ Γs 1 ∂ψ s⎞ Γ Γ sin θ − sin θ cos θ + O ⎜ 2 log ⎟ . = ⎜r 4πr0 r ∂z 2πs r0 ⎟⎠ ⎝ 0

(2.48)

ur =

Here s cos θ = r – r0, s sin θ = z – z0 (Fig. 2.11). As in the general case (see (2.35)), the first terms in the formulae for velocity correspond to a circulatory flow around a filament. In the formula for axial velocity (2.47), the second term has a logarithmic singularity that is responsible for translational motion of the vortex ring. Comparing Eqs. (2.47), (2.48) with (2.35), we obtain the formula for quantity L:

4πr0 0 ⎞ ⎛ L = 4r0 exp ⎜ 1 − In ⎟ . Γ ⎝ ⎠ Taking into account viscosity, the diffusion of a circular vortex filament within short time periods is described by Eq. (2.31) (Tung and Ting 1967).

2.5 Infinitely thin vortex ring

89

Fig. 2.11. Cylindrical and local polar co-ordinate systems in the vicinity of a circular vortex filament

With the growth of effective size (4νt)1/2, the influence of curvature should be taken into consideration. Assuming that the fluid convection relative to the moving vortex has no significant effect on vorticity distribution, Kaltaev (1982) derived the approximate Helmholtz equation

⎛ ∂ 2ω ∂ ω ∂ 2ω ⎞ ∂ω dz0 ∂ω + = ν⎜ 2 + − ⎟ ⎜ ∂r ∂t dt ∂r ∂r r ∂z 2 ⎟⎠ ⎝ with the exact solution

ω =

⎛ 4ν t ⎞ 2π−1/ 2 ⎜ 2 ⎟ ⎜ r ⎟ ⎝ 0 ⎠

−3 / 2

⎛ r 2 + r02 + ( z − z0 )2 ⎞ ⎛ rr ⎞ I1 ⎜ 0 ⎟ exp ⎜ − ⎟⎟ . ⎜ 4νt ⎝ 2ν t ⎠ ⎝ ⎠

(2.49)

Here z0(t) is the motion law for the vortex ring. A similar solution was deduced in the paper by Berezovskii and Kaplanskii (1988), where the problem was solved using the “quasi-self-similar” approach at the limit of small Reynolds numbers. For high Re numbers, the same authors (Berezovskii and Kaplanskii 1992) obtained the first approximation of the solution applying the technique of two-scale expansion (Van Dyke 1964) and using the small parameter ε = (2νt)1/2/r0 1. In this case ⎛ ρ2 + 2νt0 ⎞ ⎛ t0 ⎞ ρ ω ≅ ω0 (t ) exp ⎜ − I0 ⎜ ⎟ ⎟, ⎜ 4νt ⎟⎠ ⎜⎝ (2νt0 )1/ 2 t ⎟⎠ ⎝

(2.50)

90

2 Vortex filaments

where ρ2 = (r – r0)2 + (z – z0)2, t0 is a certain fixed point in time. A power function ω0(t) ~ tk is taken for the time dependence of the vorticity scale. The power law was applied also for the length scale – the vortex radius r0(t) ~ tp. The exponent value can be determined from the condition of the existence for a self-similar solution t d r0 (t ) + k + 1 = 0 r0 (t) dt

and the momentum conservation law (2.45). The circulation is obtained through the direct integration of vorticity (2.50) ∞ ∞

Γ=

∫ ∫ ω drdz = πω0 r0

2

.

(2.51)

−∞ 0

For p and k we obtain the equation system

p + k + 1 = 0,

k + 4p = 0,

from whence p = 1/3, k = −4/3. Thus, the vortex radius increases by the law r0(t) = R0(t/t0)1/3, where R0 is the vortex radius at t = t0. The law for the vorticity scale change is found from equation (2.51)

ω0 =

And the circulation is Γ =

p π2 r04

=

p ⎛ t0 ⎞ ⎜ ⎟ πR02 ⎝ t ⎠

⎛ t0 ⎞ ⎜ ⎟ π2 R04 ⎝ t ⎠ p

4/3

.

2/3

.

A similar model is elaborated in a paper by Lugovtsov (1970) for the turbulent diffusion of vorticity. In contrast to the laminar mode, when the viscous scale of the length is LL ~ t1/2, here the law for the length scale is LT ~ t1/4, and this gives us the scale for vorticity ωT ~ t−3/4. In conclusion of this Section, we give the examples of particular cases of bounded flows when an analytical formula can be found for stream function of a circular vortex filament. Consider a circular filament, which is placed parallel to a plane. Obviously, the condition of zero flux through the plane is fulfilled if we introduce another vortex filament, which is a mirror image of the first, relative to the plane. If we denote the expression in square brackets in (2.42) as f(k), we obtain

2.6 Infinitely thin helical vortex filament

ψ=

Γ(rr0 )1/ 2 ( f (k) − f (k1 )), 2π

91

(2.52)

where k12 = 4rr0 /[( z + z0 ) 2 + (r + r0 ) 2 ]. If the axis of the circular vortex filament passes through the center of a sphere of radius a (coordinate origin), the “image” vortex has the circulation −Γ (r02 + z02 )1/ 2 a , radius r1 = r0 a 2 (r02 + z02 ) , and axial coordinate

z1 = z0 a 2 (r02 + z02 ) . Therefore, the stream function is described by Eq. (2.52) with module k1, satisfying the condition 2 2 k12 = 4rr1 ⎡( z − z1 ) + ( r + r1 ) ⎤ . ⎣ ⎦

Finally, Brasseur (1986) found the additional stream function for the circular vortex filament, which is co-axial to a cylinder with radius a ∞

K1 (λa) Γ ψ1 = − rr0 I1 (λr0 ) I1 (λr )cos λ( z − z0 ) dλ . π I1 (λa)

∫ 0

To deduce the complete stream function, it is enough to add to ψ1 one of the forms for stream function (2.42) – (2.44).

2.6 Infinitely thin helical vortex filament 2.6.1 Helical vortex filament in infinite space Consider a flow of incompressible ideal fluid in infinite space induced by an infinitely thin helical vortex filament with pitch 2πl, radius a, and circulation Γ (Fig. 2.2). This elementary vortex structure is a fundamental object in the vortex flow theory (equally with the rectilinear filament and the vortex ring described in previous sections). The formula for the velocity induced by a helical filament can be deduced from the Biot – Savart law in the form of vector potential (2.22). Following Hardin (1982) and taking the geometry of a helical filament in the form x' = a cos φ, y' = a sin φ, z' = l φ, we notate the integral

I ( α, r ) =

∞

∫

−∞

exp ( iαφ ) dφ

r

.

(2.53)

92

2 Vortex filaments

Here r = (x – x', y – y', z – z'). Then the vector potential takes the form

A=

Γ {−a Im [ I(1, r)], aRe [ I(1, r)], lI(0, r )} , 4π

(2.54)

and the components of the velocity vector induced by the vortex filament are written, according to Eq. (1.86), as u=

∂Az ∂Ay , − ∂y ∂z

v=

∂Ax ∂Az , − ∂z ∂x

w=

∂Ay ∂x

−

∂Ax . ∂y

Let us rewrite the integral (2.53) for the cylindrical coordinate system (x = r cos θ, y = r sin θ)

I (α) =

∞

∫

−∞

(

exp ( iαφ) dφ r02 + ( z − lφ )

)

2 1/ 2

,

(2.55)

where r02 = r 2 + a 2 − 2ar cos ( θ − φ ) . Then we use the relationships known from the theory of cylindrical functions (Watson 1944): ∞

1 = exp ( − s σ ) J0 ( σr ) dσ , R

∫ 0

where R = r + s , and J0(⋅) is the zero-order Bessel function of the first kind that can be given in the form of series 2

2

J0 ( σr0 ) =

2

∑ δm Jm ( σr ) Jm ( σa ) cos m ( θ − φ) ,

⎧1, m = 0 where δm = ⎨ , ⎩2 , m ≠ 0

and the integral (2.55) is transformed into

I ( α) =

∞

∞

∞

0

0

−∞

∑ δm ∫ dσJm ( σr ) Jm ( σa ) ∫ dφ exp ( iαφ − z − lφ σ ) cos m ( θ − φ).

The direct integration of the inner integral over θ gives us ∞

∫ dφ exp ( iαφ − z − lφ σ ) cos m ( θ − φ) =

−∞

=

σ exp ( iα z l ) ⎛ exp ⎡⎣im ( θ − z l ) ⎤⎦ exp ⎡⎣ −im ( θ − z l )⎤⎦ ⎞ ⎜ ⎟. + ⎜ σ2 + ⎡ ( m − α ) l ⎤ 2 σ 2 + ⎡ ( m + α ) l ⎤ 2 ⎟ l ⎣ ⎦ ⎣ ⎦ ⎠ ⎝

2.6 Infinitely thin helical vortex filament

93

Besides, values for those integrals can be found in Watson (1944): for m±α≠0

σJm ( σr ) Jm ( σa )

∞

∫

dσ

0

σ2 + ⎡⎣( m ± α ) l ⎤⎦

2

⎧ ⎛ m±α ⎞ ⎛ m±α ⎞ a ⎟ Km ⎜ r⎟ ,a
and for m ± α = 0 ∞

∫ dσ

J1 ( σr ) J1 ( σa )

0

σ

⎧1 a ⎪⎪ 2 r , a < r =⎨ , ⎪1 r , r < a ⎪⎩ 2 a

where Im(⋅) and Km(⋅) are modified Bessel functions of m order. The final presentation of integral (2.55) for α = 0 and α = 1 is as follows:

I ( 0) =

∞

J ( σa ) J0 ( σr ) 4 ∞ ⎧ Im ( mr l ) Km ( ma l ) ⎫ 2 z⎞ ⎛ + dσ 0 ⎨ ⎬ cos m ⎜ θ − ⎟ ; σ l l m=1 ⎩ Im ( ma l ) Km ( mr l ) ⎭ l⎠ ⎝ 0

I (1) = +

∑

∫

2 ∞ mz ⎤ ⎞ ⎧⎪ Im−1 ( mr l ) Km−1 ( ma l ) ⎫⎪ ⎛ ⎡ exp ⎜ −i ⎢( m − 1) θ − ⎬+ ⎟⎨ l m=1 l ⎥⎦ ⎠ ⎪⎩ Im−1 ( ma l ) Km−1 ( mr l ) ⎭⎪ ⎝ ⎣

∑

⎧r a ⎫ ⎛ ⎡ mz ⎤ ⎞ ⎧⎪Im+1 ( mr l ) Km+1 ( ma l ) ⎫⎪ 1 2 ∞ exp ⎜ i ⎢( m + 1) θ − ⎬ + exp ( iθ ) ⎨ ⎬ . ⎟ ⎨I ⎥ l m=1 l ⎦ ⎠ ⎪⎩ m+1 ( ma l ) Km+1 ( mr l ) ⎪⎭ l ⎝ ⎣ ⎩a r ⎭

∑

Hereafter, the upper line in braces corresponds to the case r < a, and the lower line is for r > a. Substituting values I(0) and I(1) into the definition for vector potential (2.54) and performing several algebraic transformations using the recurrence equations for cylindrical functions of order m, m – 1, m + 1, and taking into account that

∂ ∂r

∞

∫ 0

J0 ( σa ) J0 ( σr ) σ

⎧ −1 ∞ ⎪ ,a
∫

the final formulae for velocity components induced by a helical filament take the form

94

2 Vortex filaments

ur =

Γa πl

2

∞

⎪⎧ I ′ ( mr l ) K ′ ( ma l ) ⎪⎫

⎛

z⎞

∑ m ⎨⎩⎪Imm′ ( ma l ) Kmm′ ( mr l )⎬⎭⎪ sin m ⎜⎝ θ − l ⎟⎠ ;

m =1

uθ =

Γ ⎧0 ⎫ Γa ∞ ⎪⎧ Im ( mr l ) Km′ ( ma l ) ⎪⎫ z⎞ ⎛ m⎨ ⎨ ⎬+ ⎬ cos m ⎜ θ − ⎟ ; 2πr ⎩1 ⎭ πrl m=1 ⎩⎪ Im′ ( ma l ) Km ( mr l ) ⎭⎪ l⎠ ⎝

uz =

Γ ⎧1 ⎫ Γa ⎨ ⎬− 2πl ⎩0 ⎭ πl 2

∑ ∞

⎪⎧ I

( mr l ) K ′ ( ma l )⎪⎫

⎛

(2.56)

z⎞

∑ m ⎨⎩⎪Imm′ ( ma l ) Kmm ( mr l )⎬⎭⎪ cos m ⎜⎝ θ − l ⎟⎠.

m =1

If we introduce

S ( a, b ) =

∞

∑ Im ( mb ) Km ( ma ),

m =1

then ∂S(a, b)/∂b is the well studied (Watson 1944) Kapteyn series and it converges if a > b ≥ 0. The series describing velocity field (2.56) is quite similar in format to this series, and for r < a and θ – z/l = 0 the series for uθ and uz coincides with the Kapteyn series; this means that it converges for all values = θ – z/l and r < a. The recent results on the analytical study of the Kapteyn series are available in Boersma and Yakubovich (1998). Hardin (1982) established through numerical techniques the convergence of a series incorporating the definition of ur, and also for uθ and uz at r > a. Note that if a 0, i.e., the helical filament transforms into a rectilinear one, the extreme values for solution (2.56) ur = uz = 0, uθ = Γ/2πr are in complete agreement with the known solution for a rectilinear filament (2.23). Formulae (2.56) give the velocity field induced by a right-handed vortex filament. To describe the effect of a left-handed filament, we should substitute the argument in trigonometric functions for θ + z/l and to change the signs for both components in the formula for uz. Note that solution (2.56) depends only on two helical variables r and = θ – z/l. This means that it belongs to the class of flows with helical symmetry described in 1.5.1, and a direct check proves that uB ≡ uz + ruθ /l = Γ/2πl ≡ const. The velocity component uχ = uθ – ruz /l orthogonal to it and to ur is as follows

⎧ ⎛ mr ⎞ ⎛ ma ⎞ ⎫ Im ⎜ Km′ ⎜ ⎟ ⎟⎪ ⎪ Γ ⎧⎪−r l ⎪ Γa ⎛ l r ⎞ ⎪ ⎝ l ⎠ ⎝ l ⎠⎪ uχ = ⎨ ⎬+ ⎬ cos mχ. ⎜ + ⎟ m⎨ 2π ⎪⎩ 1 r ⎪⎭ πl 2 ⎝ r l ⎠ m =1 ⎪ ⎛ ma ⎞ mr ⎞ ⎪ ⎛ I′ K ⎪⎩ m ⎜⎝ l ⎟⎠ m ⎜⎝ l ⎟⎠ ⎪⎭ 2⎫

∞

∑

2.6 Infinitely thin helical vortex filament

95

The velocity components ur and uχ obey the discontinuity equation in the form of the first equation in (1.64); this makes possible the introduction of the stream function (1.65), i.e., uχ = – ∂ψ/∂r and

⎧ ⎛ mr ⎞ ⎛ ma ⎞ ⎫ ⎪ Im′ ⎜ l ⎟ Km′ ⎜ l ⎟ ⎪ 1 ∂ψ Γa ⎪ ⎝ ⎠ ⎝ ⎠⎪ ur = m⎨ = 2 ⎬ sin mχ . r ∂χ πl m =1 ⎪ ⎛ ma ⎞ mr ⎞ ⎪ ⎛ I′ K′ ⎪⎩ m ⎜⎝ l ⎟⎠ m ⎜⎝ l ⎟⎠ ⎪⎭ ∞

∑

After integrating this we obtain

⎧ ⎛ mr ⎞ ⎛ ma ⎞ ⎫ ⎧ f1 ( r ) ⎫ ⎪ Im′ ⎜ l ⎟ Km′ ⎜ l ⎟ ⎪ Γar ⎪ ⎪ ⎪ ⎝ ⎠ ⎝ ⎠⎪ ψ=− 2 ⎨ ⎬ cos mχ + ⎨ ⎬, πl m=1 ⎪ ′ ⎛ ma ⎞ ′ ⎛ mr ⎞ ⎪ ⎪f ( r )⎪ I K ⎩2 ⎭ ⎪⎩ m ⎝⎜ l ⎠⎟ m ⎝⎜ l ⎠⎟ ⎪⎭ ∞

∑

where f1(r) and f2(r) are the arbitrary functions of r, defined, correspondingly, in areas r < a and r > a. Finding of f1(r) and f2(r) requires differentiation of ψ with respect to r. The use of recurrent ratios for derivatives of modified Bessel functions of m-th order and comparison with uχ yields

⎧ ⎡ ⎛ mr ⎞ mr ⎛ mr ⎞ ⎤ ⎛ ma ⎞ ⎫ Im′′ ⎜ ⎪ ⎢ Im′ ⎜ ⎟+ ⎟ ⎥ Km′ ⎜ ⎟⎪ ⎝ l ⎠⎦ ⎝ l ⎠⎪ ∂ψ Γa ⎪⎣ ⎝ l ⎠ l ⎪⎧ f1′( r ) ⎪⎫ = 2 ⎨ ⎬ cos mχ + ⎨ ⎬= ∂r πl m =1 ⎪ ⎛ ma ⎞ ⎡ mr ⎞ mr mr ⎞ ⎤ ⎪ ⎪⎩ f2′ ( r ) ⎭⎪ ⎛ ⎛ + I′ K′ K′′ ⎪ m ⎜⎝ l ⎟⎠ ⎢⎣ m ⎜⎝ l ⎟⎠ l m ⎜⎝ l ⎟⎠ ⎥⎦ ⎪ ⎩ ⎭ 2 Γ ⎧⎪−r l ⎫⎪ ⎪⎧ f1′( r ) ⎪⎫ = −uχ + ⎨ ⎬+⎨ ⎬ ≡ −uχ . 2π ⎩⎪ 1 r ⎭⎪ ⎩⎪ f2′ ( r ) ⎭⎪ ∞

∑

Integration gives f1(r) and f2(r), so the final form for stream function is as follows:

⎧ ⎛ mr ⎞ ⎛ ma ⎞ ⎫ Im′ ⎜ Km′ ⎜ ⎟ ⎟⎪ ⎪ Γ ⎪⎧ (r − a ) l ⎪⎫ Γar ⎪ ⎝ l ⎠ ⎝ l ⎠⎪ ψ= − ⎨ ⎬ ⎨ ⎬ cos mχ , 4π ⎩⎪− log(r 2 a 2 ) ⎭⎪ πl 2 m =1 ⎪ ⎛ ma ⎞ mr ⎞ ⎪ ⎛ I′ K′ ⎪⎩ m ⎜⎝ l ⎟⎠ m ⎜⎝ l ⎟⎠ ⎪⎭ 2

2

2

∞

∑

(2.57)

which is determined with a constant accuracy. One can see that the stream 0 reduces to a simple formula ψ = − Γ/2π log r and this function at a coincides with the solution for a rectilinear filament (2.24).

96

2 Vortex filaments

The isolines of stream function (2.57) at different values of helical pitch h = 2πl a are depicted in Fig. 2.12. For a higher pitch (Fig. 2.12 ), the isolines of ψ in the horizontal cross-section z = const are almost concentric circles; this corresponds to a flow induced by a rectilinear vortex filament (see Section 2.3.1). At the same time the isolines plotted in a vertical plane θ = const testify that the flow is spatial. Some isolines in the vicinity of the vortex filament crossing the plane are closed isolines. As the pitch decreases, the flow structure changes. For example, for h = 4 (Fig. 2.12c) we see the second family of loops in the horizontal section; this corresponds1 to the development of a "through" channel inside the helix. For a denser helix (Fig. 2.12d), we have a resting flow outside the helix and an almost uniform axial flow in the center (the dashed curves in Fig. 2.12 mark the cylindrical surface with radius 2a). At the limit h → 0 the coils of helical vortex merge and create a cylindrical vortex sheet (see Section 3.1). Solution (2.56), (2.57) for a helical filament in infinite space was obtained by Hardin (1982). Previous analysis has demonstrated that although the solution was obtained without additional suppositions (only by direct transformation of the Biot – Savart integral), it still belongs to the class of flows with helical symmetry with “uniform” motion along the helical lines described in Section 1.5.1. This conclusion is drawn also from the vorticity distribution along the helix. It is worth noting that for the case of a helical filament we cannot obtain a solution in the space limited by a tube using a simple image method (see Section 2.3.1) because for the helical vortex filament (unlike for rectilinear filaments) the image principle does not work. That is why in the next section we offer another approach for deducing the velocity field induced by a helical vortex filament in a tube. 2.6.2 Helical vortex filament in a cylindrical tube In this Section we follow Okulov's approach (Okulov 1993, 1995) and try to determine the velocity field for ideal fluid that is induced by a helical vortex filament inside an infinite cylindrical tube of radius R (the helix axis coincides with the cylinder axis) (Fig. 2.13). Since the vorticity is concentrated in the vortex filament (described by Eqs. (2.7)) and the vorticity vector is directed along the filament, the problem solution belongs to the class of flows (see Section 1.5.1) with helical symmetry and “uniform” motion along the helical lines or with a helical distribution of vorticity. In this case the stream function (1.65) inside the tube must satisfy the equation (1.70), and zero-flux condition has to be fulfilled at the tube wall 1

The correspondence is not strict because the lines ψ = const are not streamlines.

a

b

c

d

e

A

Fig. 2.12. Isolines of stream function for a flow induced by a helical vortex filament for horizontal (A, z = const) and meridian (B, θ = 0, π) cross-sections for different relative pitches of helix h = 2πl/a. a – 16; b – 8; c – 4; d – 2; e – 1

2.6 Infinitely thin helical vortex filament

B

97

98

2 Vortex filaments

Fig. 2.13. Helical vortex filament in a cylindrical tube

ur

r =R

= r −1 ∂ψ ∂χ r = R = 0.

(2.58)

Let us make more specific the right side of equation (1.70), i.e., determine the ωz. The vorticity is concentrated along the helical line L wound with pitch 2πl on the cylinder with radius a. The intensity of an infinitely thin vortex filament is described by circulation Γ, which is constant for the problem under consideration and related to vorticity through the Stokes formula

∫

⋅ ndS = Γ = const .

∆S

(2.59)

Here ∆S is the surface element where the filament passes, and n is the unit vector normal to this surface. Since in (1.70) we have only the zcomponent of vorticity, it is reasonable to take as ∆S a surface element with its normal parallel to the z-axis (Fig. 2.13). Then we have

∫

⋅ ndS =

∆S

∫ ωzdF = ∫ ωzrdϕdr.

∆S

∆S

After transition to variables (r, χ), we obtain the equation

∫ ωzr drdχ = Γ .

∆S

(2.60)

2.6 Infinitely thin helical vortex filament

99

Since the vorticity is different from zero only on the filament axis with coordinates (r = a, χ = χ0), the sub-integral function can be expressed through Dirac's delta-function

ωz = Γδ(r − a)δ(χ − χ0)/r,

(2.61)

and equation (1.70) takes the form

∂ 2ψ ∂r 2 =

+

l 2 − r 2 1 ∂ψ l 2 + r 2 ∂ 2 ψ + 2 2 = l 2 + r 2 r ∂r l r ∂χ 2

2lu0 r 2 + l2

− Γδ(r − a)δ(χ − χ0 )

r 2 + l2 r l2

(2.62)

.

To simplify calculations, we will search for a complex-valued function (ψ = Re ), that satisfies equation (2.62) and boundary conditions (2.58). Due to periodicity on χ, function can be written as

(r, χ) = r

∞

∑

m =−∞

exp ⎣⎡im ( χ − χ0 ) ⎦⎤ pm ( r , a ) .

(2.63)

We substitute the form (2.63) into (2.62) and multiply both sides by exp(−inχ), (n = 0, ±1, …) and integrate with respect to χ from 0 up to 2π. Taking into account the orthogonal property of functions {exp(imχ)}, we obtain for m ≠ 0

⎛ 2l 2 pm′′ + ⎜ 1 + 2 2 ⎜ ⎝ l +r

(

⎛ m2 l 2 + r 2 ⎞1 l2 − r 2 ⎜ − ⎟⎟ pm′ + ⎜ 2 l2r 2 ⎜ l + r2 r2 ⎠r ⎝ Γ l2 + r 2 = δ(r − a) 2 2 , 2π l r

(

)

) ⎟⎞ p ⎟⎟ ⎠

m

= (2.64)

and for m = 0 ⎛ 2l 2 p0′′ + ⎜ 1 + 2 ⎜ l + r2 ⎝

−

(l

2lu0 2

+ r2

⎞ 1 l2 − r 2 ′ + p p0 = ⎟⎟ 0 2 2 2 r + l r r ⎠

(

)

l + r2 Γ + δ (r − a ) . r2 r 2π 2

)

Firstly, we find the solution for equation (2.65)

(2.65)

100

2 Vortex filaments

(

c2l 2 + lu0 c1 c2r p0 ( r , a ) = + + r r 2

)

log r −

2 2 2 Γ ⎧⎪a / l + log a , ⎨ 4πr ⎪⎩ r 2 / l 2 + log r 2

where c1, c2 are the integration constants. The solution of equation (2.64) takes the form r

pm =

1 pm

∫

(

1

)

2

W pm , pm

0 1

R

2

Hpm dr

2 pm

+

∫W r

1

Hpm dσ

(p

1 m

2

, pm

)

,

(2.66)

2

here pm , pm are linear independent solutions for the corresponding (2.64) uniform equation, H is the right side of (2.64) and W is a Wronskian. To find the solutions of a homogeneous equation derived from (2.64), let us consider the Bessel equation l 2r 2 l +r 2

2

′′ + qm

l 2r 2 l +r 2

2

1 ′ − m 2 qm = 0, qm r

with linear-independent solutions in the form of modified Bessel functions Im(mr/l) and Km(mr/l). Let us differentiate this Bessel equation and divide the result by l2r2/(l2 + r2). The result is the homogeneous equation on derivative from the function q. This means that the solutions of equation (2.64) are the derivatives of modified Bessel functions or their linear combinations. To satisfy the boundary conditions (2.58), and meet the requirement of a regular solution at r = 0, we choose the following functions as the solutions of the homogenous equations: pm = Im′ ( mr l ) , pm = Km′ ( mr l ) − αm Im′ ( mr l ). 1

2

Then after calculations using Eq. (2.66) we find

pm = −

Γ ⎪⎧ Im′ ( mr l ) ( Km′ ( ma l ) − αm Im′ ( ma l ) ) , r ≤ a ⎫⎪ ⎨ ⎬. 2πl 2 ⎩⎪ Im′ ( ma l ) ( Km′ ( mr l ) − αm Im′ ( mr l ) ) , r > a ⎪⎭

By substituting pm into (2.63), taking the real part of the result, and considering that the relationship Z−m = − Zm is true for cylindrical functions, we obtain the formula for stream functions

ψ = c1 −

(

)

c2r 2 Γ + l 2 c2 + lu0 log r − 2 4πl 2

⎧⎪a 2 + l 2 log a 2 ⎫⎪ ⎨ 2 2 ⎬− 2 ⎩⎪r + l log r ⎭⎪

(2.67)

2.6 Infinitely thin helical vortex filament

−

Γar

101

⎪⎧Im′ ( mr l ) ( Km′ ( ma l ) − αm Im′ ( ma l ) ) ⎪⎫ ⎬ cos ⎡⎣m ( χ − χ0 ) ⎤⎦ . ′ ( mr l ) ) ⎭⎪ m m Im ⎪ m m=1 ⎩ ∞

∑ ⎨I′ ( ma l ) ( K′ ( mr l ) − α πl 2

Hereafter, the upper line in braces corresponds to the situation r < a, and the lower line to r ≥ a. In the solution (2.67) we have to specify the integration constants. The requirement for regularity of the solution at r 0 (a ≠ 0) yields the connection between constants u0 and c2: lc2 + u0 = 0. The constant c1 may be taken as zero, because the stream function is determined to within a constant. Thus, the solution is written in the form

ψ=

2 2 2 u0 r 2 Γ ⎪⎧a l + log a ⎪⎫ − ⎨ 2 2 ⎬− 2l 4π ⎪⎩ r l + log r 2 ⎭⎪ ⎧ ⎛ mr ⎞ ⎛ ma ⎞ ⎫ I′ ⎟ Zm′ ⎜ ⎟⎪ ∞ ⎪ m⎜ Γar ⎪ ⎝ l ⎠ ⎝ l ⎠⎪ − 2 ⎨ ⎬ cos m ( χ − χ0 ) , πl m=1 ⎪ ⎛ ma ⎞ ⎛ mr ⎞ ⎪ I′ Z′ ⎪⎩ m ⎜⎝ l ⎟⎠ m ⎜⎝ l ⎟⎠ ⎪⎭

(2.68)

∑

where Zm(x) = Km(x) − αmIm(x); αm = Km′ ( mR l ) Im′ ( mR l ) are taken to satisfy the zero-flux condition (2.58). Substituting (2.68) into (1.65) and using (1.67), we notate the components of the velocity vector

⎧ ⎛ mr ⎞ ⎛ ma ⎞ ⎫ ⎪ Im′ ⎜ l ⎟ Zm′ ⎜ l ⎟ ⎪ Γa ⎪ ⎝ ⎠ ⎝ ⎠⎪ ur = 2 m⎨ ⎬ sin m(χ − χ0 ), πl m=1 ⎪ ′ ⎛ ma ⎞ ′ ⎛ mr ⎞ ⎪ I Z ⎪⎩ m ⎜⎝ l ⎟⎠ m ⎜⎝ l ⎟⎠ ⎪⎭ ⎧ ⎛ mr ⎞ ⎛ ma ⎞ ⎫ Im ⎜ Zm′ ⎜ ⎟ ⎟⎪ ⎪ ∞ Γ ⎧ 0 ⎫ Γa ⎪ ⎝ l ⎠ ⎝ l ⎠⎪ uϕ = m + ⎨ ⎬ ⎨ ⎬ cos m(χ − χ0 ), 2πr ⎩1 ⎭ πrl m=1 ⎪ ⎛ ma ⎞ mr ⎞ ⎪ ⎛ I′ Z ⎪⎩ m ⎜⎝ l ⎟⎠ m ⎜⎝ l ⎟⎠ ⎪⎭ ∞

∑

∑

⎧ ⎛ mr ⎞ ⎛ ma ⎞ ⎫ Im ⎜ Zm′ ⎜ ⎟ ⎟⎪ ⎪ Γ ⎧ 0 ⎫ Γa ⎪ ⎝ l ⎠ ⎝ l ⎠⎪ uz = u0 − m⎨ ⎨ ⎬− ⎬ cos m(χ − χ0 ). 2πl ⎩1 ⎭ πl 2 m=1 ⎪ ⎛ ma ⎞ mr ⎞ ⎪ ⎛ I′ Z ⎪⎩ m ⎜⎝ l ⎟⎠ m ⎜⎝ l ⎟⎠ ⎪⎭ ∞

∑

(2.69)

102

2 Vortex filaments

Note that at R ∞ we have a helical filament in infinite space. Then αm 0 and solution (2.69) for the velocity components ur and uϕ coincides with the results of Hardin (1982). The value uz is different only by a constant u0, which corresponds to a uniform flow along the z axis. Both Hardin's solution (2.56), (2.57), and Eqs. (2.68), (2.69) are too complicated. Besides, the series diverges at the points of the vortex filament. The correct computation of the velocity field and stream functions requires isolation of singularities in the explicit form. Following Okulov (1993, 1995) and Kuibin and Okulov (1998), we can use the uniform decompositions of the modified Bessel functions at high values of argument (Abramowitz and Stegun 1964)

ξ t ⎡ ξ ⎤ emη ⎢1 + 1 + 22 + ...⎥ , 2πm ⎣ m m ⎦ πt −mη ⎡ ξ1 ξ 2 ⎤ 1 − + 2 − ...⎥ , Km (mx ) = e ⎢ 2m ⎣ m m ⎦ mη e ⎡ ζ1 ζ 2 1 ⎤ + 2 + ...⎥ , Im′ (mx ) = 1+ ⎢ 2πmt x ⎣ m m ⎦ Im (mx ) =

π e −mη ⎡ ζ1 ζ 2 ⎤ Km′ (mx ) = − 1− + − ...⎥ , 2mt x ⎢⎣ m m 2 ⎦ 1 1 1−t , t = (1 + x 2 ) −1/ 2 , η = + log 1+ t t 2 ξ1 = (3t − 5t 3 ) / 24, ξ2 = (81t 2 − 462t 4 + 385t 6 ) /1152,

(2.70)

ζ1 = (−9t + 7t3 ) / 24, ζ 2 = (−135t 2 + 594t 4 − 455t 6 ) /1152. If we substitute into the expressions for stream functions (2.68) the Bessel functions with their asymptotics (2.70), we find that the contribution into the singularity (of the logarithmic kind) is made only by the first terms of the series. Indeed, the replacement m m Car l 2 ⎡⎛ x ⎞ mx ⎞ my ⎞ ⎛ xy ⎞ ⎤ ⎛ ⎛ ′ ⎢ → − − Im′ ⎜ Z ⎜ ⎟ ⎟ m⎜ ⎟ ⎜ 2⎟ ⎥ 2mra ⎢⎝ y ⎠ ⎝ l ⎠ ⎝ l ⎠ ⎝ R ⎠ ⎥⎦ ⎣

makes possible the summation of the series produced from (2.68), and the presentation of the stream function in the form of a sum of singular and regular components

2.6 Infinitely thin helical vortex filament

ψ=

Sψ = −Car log

Γ Sψ + Rψ , 4π

(

)

103

(2.71)

a 2 + r 2 − 2ra cos ( χ − χ0 )

a*2 + r 2 − 2ra* cos ( χ − χ0 )

,

(2.72)

⎧⎪a 2 l 2 + log a 2 − Car log a 2 a∗2 ⎫⎪ Rψ = β 2 − ⎨ ⎬− 2 2 2 2 ∗2 l ⎩⎪ r l + log r − Car log r a ⎪⎭ 4ra ∞ ⎧⎪ Bm ( r , a ) ⎫⎪ − 2 ⎨ ⎬ cos m ( χ − χ0 ), l m =1 ⎩⎪ Bm ( a, r ) ⎭⎪ r2

(2.73)

∑

2 ⎛ xy ⎞ ⎛ mx ⎞ ′ ⎛ my ⎞ Car l ⎡⎢⎛ x ⎞ + Bm ( x, y ) = Im′ ⎜ Z ⎜ ⎟ −⎜ 2 ⎟ ⎟ m⎜ ⎟ l l mra y 2 ⎢⎣⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝R ⎠ m

m⎤

⎥ , β = 2πlu0 Γ . ⎥⎦

Here tilde stands for transformation

x = x 2 exp [Cx − 1] (Cx + 1) , Cx = 1 + x 2 l 2 (at l → ∞ x → x ; symbol x corresponds to a, r or R). The multiplier

Car = CaCr . The asterisk marks the “radius” of the “image vortex”

a* = R2 a . The singular component Sψ contains basic information about the flow pattern and dependency on the vortex parameters; it is useful for qualitative analysis of the flow. Nevertheless, the existence of a singularity in the primary presentation of stream functions via the series (2.68) forbids the operation of differentiation to Eq. (2.71) that would give us the formula for the velocity field. Therefore we have to separate the velocity field singularity directly in the series (2.69). Unlike in the case of stream functions, now we have to consider the second terms in series (2.70) of modified Bessel functions. Since the formula contains only functions ξ1(x) and ζ1(x), we can skip the subscript “1”, but we introduce the subscripts a, r, and R in Eqs. (2.70) when substituting a/l, r/l or R/l instead of x, correspondingly. The formulae for velocity (2.69) now takes the form:

104

2 Vortex filaments

ur =

Γa πl

2

( Sr + Rr ) ,

uϕ =

Γ ⎧ 0 ⎫ Γa Sχ + Rχ , ⎨ ⎬+ 2πr ⎩1 ⎭ πrl

(

)

Γ ⎧ 0 ⎫ Γa uz = u0 − Sχ + Rχ . ⎨ ⎬− 2πl ⎩1 ⎭ πl 2

(

)

(2.74)

The singular components Sr and Sχ are also written through elementary functions of distorted radial distances

⎡ ar sin(χ − χ0 ) a*r sin(χ − χ0 ) − + ⎢ 2 2 *2 2 * ⎢⎣ a + r − 2ar cos(χ − χ0 ) a + r − 2a r cos(χ − χ0 ) ⎧⎪ ( ζ r − ζ a ) atan ( r sin(χ − χ0 ) ( a − r cos(χ − χ 0 ) ) ) ⎫⎪ +⎨ ⎬− ⎪⎩( ζ a − ζ r ) atan ( a sin(χ − χ0 ) ( r − a cos(χ − χ 0 ) ) ) ⎪⎭ ⎛ r sin(χ − χ0 ) ⎞ ⎤ − ( ζ r + ζ a − 2ζ R ) atan ⎜ * , ⎜ a − r cos(χ − χ ) ⎟⎟ ⎥⎥ 0 ⎠⎦ ⎝ (2.75) Ca r l ⎡⎧ 0 ⎫ r 2 − ar cos(χ − χ0 ) r 2 − a*r cos(χ − χ0 ) − Sχ = + ⎢⎨ ⎬ + 2a ⎣⎢⎩−1⎭ a2 + r 2 − 2ar cos(χ − χ0 ) a*2 + r 2 − 2a*r cos(χ − χ0 )

Sr = −

Car l 2 2ar

⎛ ⎧⎪log a 2 ⎫⎪ ⎞ 1 2 2 ⎜ + ( ξr − ζ a ) log a + r − 2ar cos(χ − χ0 ) − ⎨ ⎬⎟ − 2 ⎟ ⎜ 2 r log ⎪⎩ ⎭⎪ ⎠ ⎝

(

)

a*2 + r 2 − 2a*r cos(χ − χ0 ) ⎤ 1 − ( ξr + ζ a − 2ζ R ) log ⎥. 2 a*2 ⎥⎦ Here Ca / r = Ca Cr , and the series residues are

Rr =

∞ ⎧⎪ Bmr ( r , a ) ⎫⎪ ⎧⎪ BmI ⎫⎪ m⎨ r m sin χ − χ = R m ( ) , ⎬ ⎨ Z ⎬ cos m ( χ − χ0 ) , (2.76) 0 χ B a r , ( ) ⎪⎩ Bm ⎪⎭ ⎪ 1 = m =1 ⎪ m m ⎩ ⎭

∞

∑

∑

where

⎛ mx ⎞ ⎛ my ⎞ Bmr ( x , y ) = Im′ ⎜ ⎟ Zm′ ⎜ ⎟+ ⎝ l ⎠ ⎝ l ⎠ m m Car l 2 ⎡⎛ x ⎞ ⎛ ζ x − ζ y ⎞ ⎛ xy ⎞ ⎛ ζ x + ζ y − 2ζ R ⎞ ⎤ ⎢⎜ ⎟ ⎜ 1 + + ⎟ − ⎜ 2 ⎟ ⎜1 + ⎟⎥ , m ⎠ ⎝R ⎠ ⎝ m 2mra ⎢⎝ y ⎠ ⎝ ⎠ ⎦⎥ ⎣

2.6 Infinitely thin helical vortex filament

105

⎛ mr ⎞ ⎛ ma ⎞ BmI = Im ⎜ ⎟ Zm′ ⎜ ⎟+ ⎝ l ⎠ ⎝ l ⎠ m m Ca / r l ⎡⎛ r ⎞ ⎛ ξr − ζ a ⎞ ⎛ ra ⎞ ⎛ ξr + ζ a − 2ζ R ⎞ ⎤ + + 1⎟ − ⎜ 2 ⎟ ⎜ 1 + ⎢⎜ ⎟ ⎜ ⎟⎥ , m 2ma ⎢⎣⎝ a ⎠ ⎝ m ⎠ ⎝R ⎠ ⎝ ⎠ ⎥⎦

⎛ ma ⎞ ⎛ mr ⎞ BmZ = Im′ ⎜ ⎟ Zm ⎜ ⎟+ ⎝ l ⎠ ⎝ l ⎠ +

m m Ca / r l ⎡⎛ a ⎞ ⎛ ξr − ζ a ⎞ ⎛ ra ⎞ ⎛ ξ + ζ a − 2ζ R ⎞ ⎤ − 1⎟ − ⎜ 2 ⎟ ⎜ 1 + r ⎢⎜ ⎟ ⎜ ⎟⎥ . m 2ma ⎢⎣⎝ r ⎠ ⎝ m ⎠ ⎝R ⎠ ⎝ ⎠ ⎥⎦

∞ the soluAnalysis of the forms (2.71) – (2.76) demonstrates that for l tion coincides with solution (2.28) for a rectilinear vortex in a cylinder (or for a point vortex in a circle). Numerical tests confirm that for a wide range of filament parameters, the series residues (2.73) and (2.76) are rather small in comparison with the main singular parts (2.72) and (2.75) presenting the stream functions (2.71) and the velocity field (2.74). Thus, we can neglect the residues (see quantitative estimates in (Okulov and Fukumoto 2004, Fukumoto and Okulov 2005)) in calculations during qualitative analysis. Since the key parts of (2.72) and (2.75) are expressed in simple form (through elementary functions), this gives us an opportunity to find illustrations for flow patterns in a wide range of vortex parameters. Plotting of stream function isolines helps in the understanding of several features of the flow. In particular (like in the case of a plain flow), a higher density of lines corresponds to a higher velocity (see. (1.47)). But the direction of the velocity vector does not coincide with the tangent to the isolines ψ = const. Therefore, the “stream lines” are plotted for the transverse (z = const) and longitudinal ([θ = 0] + [θ = π]) cross-sections of a tube for better understanding. This requires the integration of equation dr/ur = rdθ/uθ , in the plane z = const, and integration of dr/ur = dz/uz in the plane θ = const. Thus, the projections of velocity vector on these planes are directed by tangents to the constructed “stream lines”. Actually, these lines are the cross-sections of the stream tubes. Hereafter, we take the quantities normalized by the cylinder radius R. The variable parameters are the vortex pitch h = 2πl, the radius of helical filament a, and dimensionless velocity at the axis u0 = 2πlu0 / Γ (starting from this place, we skip the symbol of tilde for dimensionless velocity).

106

2 Vortex filaments a

b

c

A

B

Fig. 2.14. Isolines of stream functions in the horizontal cross-section z = const for a helical vortex filament in a cylindrical tube (A) and in infinite space (B). (a) h = 1, a = 0.5; (b) h = 8, a = 0.5; (c) h = 2, a = 0.9

Firstly, we have to consider the influence of the cylinder walls on the flow. For a small pitch h = 1 and moderate radius a = 0.5 (Fig. 2.14a) the isolines for stream functions in a tube are almost the same as the isolines in an unbounded flow. This is because the introduced transformation decreases the relative radius of the vortex. Indeed, for h = 1, a = 0.5, R = 1 we obtain a = 2.314 , R = 57.92 , so that a / R = 0.04 . Obviously, the influence of the walls in this situation is small. For a large pitch h = 8 (Fig. 2.14b), the flow patterns for a channel and infinite space are very different. In this case a = 0.519 , R = 1.155 , a / R = 0.45. Certainly, with the increase in the vortex radius, the wall effect becomes greater (Fig. 2.14, h = 2, a = 0.9), although the transformation has its effect also: a / R = 0.718 . Obviously, for a smaller helical pitch, the wall effect will be smaller. If at a = 0.9 we take h = 0.195, then a / R = 0.04 . The laws of flow pattern change, due to variation of the vortex parameters, are more obvious through the viewing of “stream lines”. Note that the value of u0 has no impact on the flow pattern in the tube crosssection. As it is obvious from Eqs. (2.69) for uz, the contribution u0 can be interpreted as a transition to the coordinate system moving with velocity –u0 along the z axis. It is clear that the quantity u0 is connected to the liquid flow rate through the tube and is important in the description of a swirling flow in a bounded zone. For example, at u0 = 0, h = 1 (Fig. 2.15)

2.6 Infinitely thin helical vortex filament

107

the axial motion in the vicinity of the tube axis is very weak because the axial motion mostly occurs through liquid flow at the periphery (near the channel walls). For u0 = 1, h = 1 and vice versa, the flow almost stops near the tube walls and the liquid flows through the central part of the channel. In the intermediate variant (Fig. 2.15, u0 = 0.5, h = 1) the fluids inside and outside the helical filament move in opposite directions. The same figure demonstrates the influence of vortex pitch on flow structure. For the case of a dense vortex (Fig. 2.15, h = 1) the near-axis flow is almost separated from the wall flow. The helical vortex filament and its vicinity play the role of a cylindrical shear layer. With a higher pitch (h = 2), the vortex becomes less dense and some portion of fluid in helical motion flows from the periphery towards the central part and vice versa, flow towards the periphery takes place. The higher the pitch, the larger the proportion of liquid which passes through both parts of the flow (periphery and near-axis). The flow patterns at different values of helical filament radius with a fixed pitch (h = 2) are plotted in Fig. 2.16. When the axial velocity is zero (u0 = 0) and the radius of the helix is small (a = 0.1), the flow is similar to the case of a rectilinear vortex filament and an almost uniform axial flow occupies most of tube cross-section. With an increase in radius (a = 0.5), a part of the axial flow concentrates in the vicinity of the vortex filament. The last variant: when the vortex is closer to the wall, and the axial velocity is zero (a = 0.9, u0 = 0), almost all motion is concentrated in a thin vortex helical tube near the filament. But when the axial velocity is non-zero (u0 = 0.5), the flow pattern changes: at moderate radius (a = 0.5; 0.7) the greater part of the flow passes through a helical stream tube of large crosssection, and for a large radius (a = 0.9) the fluid moves in the axial direction in the tube center except for the peripheral area, where we have a flow in a helical stream tube of moderate cross-section. The qualitative change in the flow structure at low radius of the helix occurs if the axial velocity corresponds to the case of a steady unbounded stream away from the vortex. As one can see from Fig. 2.16 (u0 = 1), even at the smallest radii considered (a = 0.1), a helical vortex stream tube develops and it spans from the wall to the significant part of the pipe cross-section and even overlaps the center. At a = 0.3 the efficient diameter of the stream tube is about half the channel diameter and it decreases drastically with the increase of a. More complete information about the flow structure is available by plotting the actual streamlines. This requires integration of the system of equations dr/ur = rdθ/uθ = dz/uz. Examples of streamlines are illustrated in the Color Fig. 1. For a curved vortex with zero velocity at the axis (Color Fig. 1a), streamlines near the vortex filament are slightly deformed spirals.

0.5

1

2

4

8

108

u0 h

2 Vortex filaments

A

0

B 0.5

1.0

Fig. 2.15. Dependence of flow structure induced by a helical vortex in a cylindrical channel on helical pitch at different values of parameter u0 and fixed radius of the helical filament a = 0.5. (A) is the horizontal and (B) is the vertical cross-section

u0 a

0.1

0.3

0.5

0.7

0.9

A

0

0.5

1.0

109

Fig. 2.16. Dependence of flow structure induced by a helical vortex in a cylindrical channel on helix radius at different values of parameter u0 and a fixed helical pitch h = 2. (A) is the horizontal and (B) is the vertical cross-section

2.6 Infinitely thin helical vortex filament

B

110

2 Vortex filaments

As for the periphery, the streamlines are hardly curved, i.e., the axial component of velocity is much higher than the circumferential and radial components. For the case u0 = 1 and a small radius of the vortex filament (Color Fig.1 b,c), the streamlines are dense spirals either near or away from the filament, and the axial velocity at the periphery is much smaller than the circumferential one. For a vortex of large radius (a = 0.7) with zero axial velocity (Color Fig. 1c), the streamlines (except those closest to the filament) have complex structures. Otherwise, at u0 = 1 (Color Fig. 1d) the streamlines in the channel center are curved slightly. But the most drastic difference in the flow pattern is visible at low pitch of the helix (h = 0.5). At a = 0.5 and u0 = 0 (Color Fig. 1e), the wall flow is directed downwards and slightly swirling. Inside the helix, the flow is slow. An interesting streamline is observed: it passes near the tube axis and is directed by the diameter, then it winds around the vortex filament and again passes by the diameter (shifted by a half-period). For the opposite case, at u0 = 1, a vertical uprising flow occurs in the tube center, and the periphery is described by a spiral flow (Color Fig. 1f).

3 Models of vortex structures

3.1 Vortex sheet The infinitely thin vortex filament described in Sections 2.3 – 2.6 reflects the simplest marginal vorticity distribution when the whole vorticity is concentrated along some spatial curve coinciding with the vortex line. In another important extreme case, the vorticity is concentrated within an infinitely thin layer along some three-dimensional surface, which is called a vortex sheet. A vortex sheet can be conceived as being a thin layer with the thickness ε and vorticity ω, for which there exists a finite limit χ = lim εω ε→0 ω→∞

when ε 0, ω ∞. According to Stokes theorem, circulation Γ along the small rectangular contour with the thickness ε and length s is Γ = ωεs. Here the orientation of the contour is selected in such a way that vector is orthogonal to the plane of the contour. At the upper limit Γ = lim ωε s = χ s , ε→0 ω→∞

(3.1)

i.e. the quantity χ is the circulation per unit of vortex sheet length, and it can be considered as a measure of sheet intensity. Since there is a given direction connected with the vorticity vector, let us also introduce vector = χt, where t is the unit vector tangentional to the vortex line. Let us represent the vortex sheet element, as shown in Fig. 3.1. Here the vorticity vector (and t, accordingly) is orthogonal to the drawing plane and is directed towards the reader, while n is the unit vector to the sheet element. Then, if outside the vortex sheet the flow is irrotational, the vorticity can be expressed nominally by delta-function as follows

112

3 Models of vortex structures

Fig. 3.1. The vortex sheet element

= δ(xn),

(3.2)

where xn is the coordinate along the normal n. To determine the velocity induced by the vortex sheet let us substitute expression (3.2) into the Eq. (1.93)

u(r ) = −

1 ∆r × 1 ∆r × δ( xn ) dV = − dSdxn = 3 3 4π 4π r r ∆ ∆ V V

∫

∫

1 ∆r × dS(r ′) . =− 4π ∆r 3 S

∫

(3.3)

Here dS is the vortex surface element, r' is the coordinate of the point on the vortex sheet (Fig. 3.1; generally, vectors r, r', ∆r do not lie in the drawing plane). In the particular case of a single plane vortex sheet with the constant intensity χ, the integral in (3.3) can be easily solved. First of all let us expand the vector ∆r along the basis vectors (t, n, b)

∆r = (∆r · t)t + (∆r · n)n + (∆r · b)b. Then the first term on the right-hand side gives zero contribution to (3.3), because t || . The same applies to the second term, because ∆r · b is an odd function. As a result we obtain u(r ) =

4π

×

∫

(∆r ⋅ n)n ∆r

3

dS =

× n dS⊥ = 2 4π ∆r

∫

×n . 2

(3.4)

Here dS⊥ is the projection of the elementary area to the plane, orthogonal to ∆r. Therefore the last integral is the solid angle, at which the plane could be observed from point M(r), and equal to 2π. It follows from (3.4),

3.1 Vortex sheet

113

that the velocity vector in area 1 is parallel to the plane of the sheet and orthogonal to the vector (Fig. 3.1) and its scalar is the same everywhere

u1 = χ/2. In area 2 the velocity is directed to the opposite side and its scalar equals u1

u2 = −χ/2. Hence, on the vortex sheet with intensity nent undergoes discontinuity equal to [u] =

the tangential velocity compo-

× n.

(3.5)

And on the contrary, the existence of tangential discontinuity means the subsistence of the vortex sheet with the intensity, which can be found by = n × [u].

(3.6)

Relationships (3.5) and (3.6) are valid for any vortex sheet if considered locally. Another important case concerns the cylindrical sheet, when vector is orthogonal to the cylinder generating line. Let the cylinder have a circumferential cross-section (Fig. 3.2a, b). Taking into account that the element of length along the vortex line written in vector form is ds = tds, = tχ and dz(r') is the element of the generating line, we obtain from (3.3) the following

u=−

∞

∞

1 ∆r × χ ∆r × ds(r′) χ = − dS dz ( r′) = − 3 4π ∆r 3 4π 4 π ∆r −∞ −∞

∫

∫∫

⎡∞ χ ⎢ dz =− ⎢ 4π 2 2 ⎢⎣−∞ R + z

∫ ∫

(

∫∫

R× ds(r′)

(R

2

)

2 3/ 2

+z

dz =

⎤ χ R× ds(r′) ⎥ ′) = − × ( . d R s r ⎥ 3/ 2 2 2π R ⎥⎦

)

∫

Here it is taken into account, that ∆r = z + R, where R and z are projections of vector ∆r to the cross-sectional plane and the cylinder generating line accordingly, and that z component being odd does not contribute to the integral. The integral in brackets equals 2/R2. Let b be the unit vector parallel to the cylinder generating line and satisfying the b = t × n condition (Fig. 3.2a, b). Then, rewriting the expression under the integral via angles α and ϕ, we obtain for the point inside the cylinder

114

3 Models of vortex structures b

c

d

Fig. 3.2. Cylindrical vortex sheet of circular cross-section. Vortex lines: a, b – directed along the circumference, c – directed along the generating line, d – helical and I – selected vortex line

χb u= 2π

∫

ds sin α χb = R 2π

2π

∫ dϕ = χb = χ × n . 0

For the point outside the cylinder the integral apparently equals zero, i.e. u = 0. Thus, we have obtained a similar expression for the velocity jump as in (3.5) valid for a plane sheet, but now with different values of absolute velocities on both sides of the vortex sheet. This result has complete analogy with the magnetic field inside the infinitely long direct current solenoid. Note that similar conclusions are valid for a cylindrical vortex sheet with an arbitrary configuration of cross-section. Let us consider again the cylindrical sheet of circumferential crosssection with the radius a and intensity χ. But now the vortex lines are directed along the generating line (Fig. 3.2c). For this problem, velocity field can be obtained in a more simple way, using axial symmetry. This means that the induced velocity should only have the tangential component u, which is constant on the circumference of radius r. Then, according to the Stokes theorem, velocity circulation along the circumference L outside the cylindrical sheet, determined as Γ = 2πru, equals vorticity flux, which according to (3.1) is χ 2πa, wherefrom

u =

χa r

or

u =

Γ . 2π r

(3.7)

3.1 Vortex sheet

115

Apparently, the latter relationship exactly coincides with the velocity distribution (2.23) for an infinitely thin vortex line with the intensity Γ. Inside the cylindrical sheet the circulation along any concentric circumference in the cross-section of the cylinder obviously equals zero. Therefore, in the internal part u = 0. Note that the considered vortex sheet may be obtained by infinite compression of a helical vortex filament. Indeed, in the limit l ∞ in Eqs. (2.56), we find, that ur = uz = 0, while uθ agrees with (3.7) in an external area, whereas it equals zero in the internal area. In the third case of a cylindrical sheet (Fig. 3.2d), when vortex lines are helical, both axial and circumferential velocity components are induced. Such a sheet of circumferential cross-section with radius a, intensity χ and pitch of the helical vortex lines 2πl may be imagined as a superposition of the above considered vortex sheets: the first, generated by rectilinear vortex lines with intensity χ l

a 2 + l 2 , and the second, generated by vortex

lines in the form of circumferences with the intensity χ a

a 2 + l 2 . Thus,

χal Γ ⎧ = , uz = 0 ; r > a, ⎪uθ = r a 2 + l 2 2πr ⎪ ⎨ χa Γ ⎪u = 0, u = = ; r < a. z 2 2 ⎪ θ π l 2 a +l ⎩

(3.8)

Here Γ = 2π χ l a 2 + l 2 . We observe that the velocity field (3.8) obeys the general law of flows with helical symmetry (see Section 1.5) uz = u0 – r/l uθ, if we select for u0 the velocity inside the cylinder Γ/2πl. In conclusion of this Section, let us return to the solitary plane vortex sheet and calculate the velocity field based on the concept; that the vortex sheet is a continuous distribution of rectilinear vortex lines. As this occurs it is helpful to use complex variables. Let N be filaments (or point vortices) of an equal intensity n which are uniformly located along the x axis (Fig. 3.3). Passing to continuous distribution, let us rewrite the formula (2.25) in the integral form

∂w χ 1 dΓ = = ux − iuy = ∂z 2πi z − z0 2πi

∫

∞

∫

−∞

∞

dz0 χ =− log ( z0 − z ) . 2πi z − z0 −∞

Here dΓ = χdz0, z0 ≡ x, z = x + iy. Let us represent an argument of the logarithm in polar coordinates: z0 − z = reiϕ. If z0 ± ∞, then ϕ = 0, −π at

116

3 Models of vortex structures

Fig. 3.3. Modeling of the vortex sheet by continuous distribution of vortex filaments

y > 0 and ϕ = 0, +π at y < 0. Therefore, ux − iuy = ux = ∓ χ 2 at y

> <

0,

which agrees with the above obtained result (see Eq. 3.5). Formation of the vortex sheet occurs in many actual cases, such as flow over supporting areas, delta-shaped wings, and high-drag bodies. Due to self-induction impact, the vortex sheet is usually unstable and rolls into a spiral. This is how large-scale vortices appear, for instance, two tip vortices behind a plane, or a Karman vortex street behind a cylinder. The importance of representing the vortex sheet as a distribution of vortex filaments is that using a finite set of discrete vortices we may sufficiently simulate various flows, including the rolling-up process of the vortex sheet (see Section 6).

3.2 Spatially localized vortices 3.2.1 Vortex ring

The vortex ring is one of the few vortex objects in nature, which is commonly observed (see Fig. I.5). We can form a vortex ring, if we breathe out a puff of smoke through rounded lips. Often, we can see vortex rings coming from a car or tractor tailpipe. Dropping one fluid (colored) onto another illustrates another example of vortex ring formation. The first attempts of vortex ring mathematical description were undertaken at the end of 19th Century. Since the velocity field induced by a thin vortex almost coincides with the velocity field of an infinitely thin vortex, determination of the self-induced motion velocity of a vortex ring was the main problem. For a ring with circulation , having radius r0 , and core radius ε r0 (with uniform vorticity distribution), Kelvin (1867) suggested the formula for the velocity of translation motion

3.2 Spatially localized vortices

U=

8r 1 ⎞ Γ ⎛ log 0 − ⎟ . ⎜ ε 4⎠ 4πr0 ⎝

117

(3.9)

Hicks (1885) determined the velocity of a hollow (or filled with immobile fluid) vortex ring U=

8r 1 ⎞ Γ ⎛ log 0 − ⎟ . ⎜ ε 2⎠ 4πr0 ⎝

(3.10)

Proof of the validity of formula (3.9) for the round cross-section of a small area can be found in Lamb (1932, §§ 161-163). On the basis of Lamb’s ideas, Saffman (1970) developed the approach, which allows consideration of viscosity effect, swirling, unsteadiness and compressibility. Following the paper of Saffman, we will consider a thin vortex ring moving in a fluid at rest. The velocity field is shown in the form of the sum

u = U + u,

(3.11)

where U is the velocity of the vortex ring, which should be determined. Substituting expression (3.11) into the formula for kinetic energy T by Lamb’s definition (see Section 1.7.3) and taking into account the definition of impulse I (1.112), we obtain

∫

T = 2U ⋅ I + u ⋅ (r × ) dV .

(3.12)

We will use the local polar coordinates (s, θ) with the origin in the center of the cross-section (Fig. 3.4), which in the first approximation is assumed to be round. Fraenkel (1970) proved that vorticity in this case can be expressed as

ω(r, z) = ω0(s)[1 + O(ε/r0)].

(3.13)

The vortex ring impulse can be calculated using Eq. (1.126). It is directed parallel to the ring axis and amounts to (at a unit density) (3.14) ∫∫ ω (r , z)r drdz = πr0 Γ [1 + O(ε r0 )], ε Γ = ∫∫ ω (r , z)drdz = 2π ∫ sω0 ( s)ds . To determine the kinetic en0

I=π

where

2

2

ergy of the vortex ring using Eq. (1.127), we require the expression for the stream function, which can be obtained via the integration of Green function G over the vortex core

118

3 Models of vortex structures

Fig. 3.4. The coordinate system for a vortex ring

Ψ=

∫∫ ω (r , z)Gdrdz.

(3.15)

Function G coincides with the stream function of an infinitely thin vortex ring (2.42) with the unit circulation. Neglecting the small terms in expansion (2.46) and substituting it into (3.15) together with (3.13), we find

Ψ=

r0 2π

ε 2π

∫ ∫ s′ω0 (s′) ⎢⎣log (8r0 ) − 2 − 2 log ( s ⎡

1

0 0

2

)

⎤ + s′2 − 2ss′ cos ( θ − θ′ ) ⎥ ds′dθ′ . ⎦

For integration over θ' we will use the formula (Gradshtein and Ryzhik 1965) 2π

∫ 0

⎧4π log s′, s′ > s, log ⎡ s 2 + s′2 − 2 ss′ cos θ ⎤ dθ = ⎨ ⎣ ⎦ ⎩4π log s, s > s′.

From this formula we have for the points within the vortex cross-section ε

s

∫

∫

0

0

ε

∫

Ψ = r0 (log8r0 − 2) ω0 ( s′) s′ds′ − r0 log s ω0 ( s′) s′ds′ − r0 ω0 ( s′) log s′s′ds′ . s

Introducing function s

∫

Γ0 ( s) = 2π ω0 ( s ′) s ′ds ′

(3.16)

0

(where Γ = Γ0(ε)) and integrating by parts, reduces the expression for the stream function to the following:

3.2 Spatially localized vortices

119

ε

Ψ=

r0 Γ ⎛ 8r ⎞ r Γ0 ( s′) ds′. log 0 − 2 ⎟ + 0 ⎜ 2π ⎝ ε ⎠ 2π s s′

∫

Now substituting Ψ into Eq. (1.127), we deduce the main terms of kinetic energy expansion in the small parameter ε/r0

T=π

∫∫

ε

ωϕ Ψ drdz ≈

2 8r0 r0 Γ 2 ⎛ ⎞ r0 Γ0 ( s) log 2 ds . − + ⎟ 2 2 ⎜⎝ s ε ⎠ 0

∫

(3.17)

For a vortex ring with swirling, i.e., if the velocity component uϕ is nonzero, the following term will be added to the kinetic energy π

∫∫ ruϕ drdz, 2

and the last item in equation (3.12) will take the form

∫ u ⋅ (r ×

∫∫ rωϕ (ruz − zur )drdz + + 2π ∫∫ ruϕ ( zωr − r ωz )drdz.

) dV = 2π

(3.18)

Taking into consideration that

ωr = − ∂uϕ ∂z , ωz =

1 ∂ (ruϕ ) , r ∂r

we find that the second term on the right side of (3.18)

(

)

(

)

∂ 2 2 ⎫ ⎧∂ 2π ruϕ ( zωr − rωz ) drdz = π ruϕ2drdz − π ⎨ rzuϕ2 + r uϕ ⎬ drdz. ∂r ⎩ ∂z ⎭

∫∫

∫∫

∫∫

The last integral disappears here, if uϕ vanishes at infinity. As a result, equation (3.12) is written in the same manner both for a vortex with and without swirling

π

∫∫ ωϕ Ψdrdz = 2UI +2π∫∫ r ωϕ (ruz − zur )drdz.

(3.19)

The left side of Eq. (3.19) and the impulse are determined by the main terms with accuracy O(ε/r0) according to the Eqs. (3.14) and (3.17). They depend on ω0(s), vortex radius r0 and intensity Γ. We should note that these two terms have the order of O(r0Γ2). A problem is caused by the last term. Since ωϕ ∼ Γ/ε2, uz ∼ Γ/ε, r ∼ r0, drdz ∼ ε2, the order of this term is r02 Γ2/ε, and the left side of (3.19) is not considered in the estimation of U.

120

3 Models of vortex structures

However, due to more accurate estimation of ωϕ and uz, we see that this is not correct. To prove this, let us use Lamb’s transformation (1932, § 162). We assume that the motion is steady. Components of velocity uz, ur and vorticity ωϕ satisfy the steady motion equations in the coordinate frame moving together with the vortex ∂u 1 ∂ (rur ) + z = 0, r ∂r ∂z

ur

∂ωϕ ∂r

+ uz

∂ωϕ ∂z

=

ur ωϕ

1 ∂uϕ + . r ∂z 2

r

Hence,

r 2 uz ωϕ − rzur ωϕ = ruϕ2 − 3rzur ωϕ −

(

)

(

)

∂ 2 ∂ 2 uϕ rz − r 2 zuz ωϕ + r zur ωϕ . ∂z ∂r

Integrating this relationship over the vortex cross-section, we get

2π

∫∫ r ωϕ (ruz − zur )drdz = −6π∫∫ rzur ωϕ drdz + 2π∫∫ ruϕ drdz . 2

This formula is accurate for the steady flow. Now the terms on the right side have the order O(r0Γ2) and can be estimated using the approximation (3.13), i.e., ωϕ = ω0(s), ur = Γ0(s) cos θ/2πs, uϕ = uϕ0(s), drdz = s ds dθ. This gives

2π

∫∫

ε

3 r ωϕ (ruz − zur )drdz ≈ − r0 Γ 2 + 4π2 r0 suϕ20 ds . 4

∫ 0

Substituting the calculated integral into (3.19), we have ε

ε

2 8r 1 2 ⎛ 3 ⎞ r Γ0 ( s) Γ r0 ⎜ log 0 − 2 ⎟ + 0 ds = 2Uπr02 Γ − r0 Γ 2 + 4π2r0 suϕ20 ds . ε 2 4 ⎝ ⎠ 20 s 0

∫

∫

Finally, the expression for the velocity of the vortex ring motion can be written as:

U=

⎛ π2 ε 2 uϕ20 ⎞ π2 ε 2 uθ2 8r 1 Γ ⎜ ⎟. log 0 − + 2 4 − 4πr0 ⎜ ε 2 Γ2 Γ2 ⎟ ⎠ ⎝

(3.20)

Here uθ = Γ0(s)/2πs, and the lines above the quantities represent an averaging over the vortex cross-section. The estimate of the kinetic energy with the same accuracy gives

3.2 Spatially localized vortices

⎛ π2 ε 2 uϕ20 ⎞ 8r π2 ε 2 uθ2 1 ⎜. 2 T = r0 Γ 2 ⎜ log 0 − 2 + 2 + ⎜ ⎜ 2 ε Γ2 Γ2 ⎝ ⎠

121

(3.21)

The effect of swirling, i.e., the motion along the vortex axis, on the velocity of the vortex ring has also been studied by Widnall et al. (1971) with the help of the matched asymptotic expansions method. We should note that swirling, ultimately decreases the vortex velocity to zero and even causes motion in the opposite direction. When there is no swirling, formula (3.12) can be written through the integral characteristics

T = 2UI −

3 I2 , 4 π2 r03

i.e., the velocity of the vortex ring is expressed through energy, impulse and a typical length scale U=

T 3 I . + 2 I 8 π2 r03

Without swirling, formulae similar to (3.20) and (3.21) were derived by Fraenkel (1970) with the help of expansion in the small parameter ε/r0 1 in the solution of integral equation on the stream function

1 Ψ (r , z) = − Ur 2 + 2

∫∫ r ′F(Ψ(r ′, z′))G(r , r ′, z − z′) dr ′dz′.

(3.22)

Here rF(Ψ) = ωϕ; G is the Green function of the Laplace operator for infinite space under the condition of axial symmetry; it coincides with the stream function (2.42) at = 1. Using an expansion of type (2.46), Fraenkel proved the existence of a solution for equation (3.22) and determined the motion velocity of the thin vortex ring

U=

1 ⎤ ⎪⎧ ⎛ ε2 Γ ⎡ 8r0 1 ε ⎞ ⎪⎫ ⎢log − + 4 V 2 ( s) sds ⎥ ⎨1 + O ⎜ 2 log 2 ⎟ ⎬ , ⎜ 4πr0 ⎢ r0 ⎟⎠ ⎭⎪ ε 2 ⎝ r0 0 ⎣ ⎦⎥ ⎩⎪

∫

s

where

1 Ω( s ′) s ′ds ′, V ( s) = − s

∫ 0

πε 2 ω0 ( s) . Ω( s ) = Γ

122

3 Models of vortex structures

It is clear that this result coincides with the formula (3.20). For the vortex without any swirling and with uniform vorticity distribution we have from (3.16) that Γ(s) = Γ s2/ε2, and correspondingly uθ = Γ s/2π ε2 and uθ2 = Γ 2 8π2 ε 2 , and relationship (3.9) follows from there. For a hollow vortex,

there is no inner motion in the core, and hence, u θ2 = 0 , uϕ20 = 0 and (3.20) provides the Hicks formula (3.10). If in the core there is no clear boundary of vorticity at s = ε, Eqs. (3.20) and (3.21) become insufficiently definite. For the case of smooth vorticity distribution rapidly decreasing with distance from the axis, Saffman (1992) suggests combining the items

− log ε +

1 Γ2

ε

∫ 0

Γ 02 ( s ) ds . s

This sum does not depend on ε, if ε is sufficiently high, therefore, Γ0(ε) = Γ. Particularly, for the viscous vortex ring for short periods t << r02 4ν , the core structure can be considered as the local 2-D, then from (2.31) we have

(

Γ0 ( s) = Γ 1 − e− s

2

4 νt

).

As a result, U=

Γ 4πr0

8r0 1 ⎧ ⎫ − (1 − γ + log 2) ⎬. ⎨log 4ν t 2 ⎩ ⎭

(3.23)

The motion of a viscous vortex was studied by Tung and Ting (1967) using the method of the matched asymptotic expansions, but due to an arithmetical mistake the constant equals 0.688 instead of (1 − γ + log 2)/2 ≈ 0.588. Strictly speaking, under the condition of viscosity the motion is not steady. Nevertheless, according to Saffman (1970), if the velocity of the vorticity centroid transportation X=

1 (r × ) ⋅ I rdV , 2 I2

∫

U=

dX , dt

is considered as the vortex ring velocity, the Lamb transformation is true and Eq. (3.23) represents the velocity of the vortex centroid motion. Applying the solution for vorticity diffusion in the vortex ring of the type (2.49), valid in a wider time range, Kaplanski and Rudi (1999) derived the formula for the vortex ring velocity in the form of integral

3.2 Spatially localized vortices

U=

Γ0 4π

∞

{π [1 − erf(µ)] (1 − 6µ 2ν t ∫ 0

2

123

⎛ µr ⎞ ) + 6 π µ exp(−µ 2 ) J12 ⎜ 0 ⎟ dµ, ⎝ 2ν t ⎠

}

coinciding with (3.23) at low values of t and providing the right asymptotic for the decaying vortex ring (Rott and Cantwell 1993 , b)

U AS =

7 πr02 Γ 0 . 15 (8πνt )3 / 2

On the basis of the approach by Saffman (1970), Moore (1985) examined the motion of a uniform vortex ring in a compressed medium and derived a correction for the velocity formula

U=

8r0 1 5M2 ⎞ Γ ⎛ − − ⎜⎜ log ⎟. 4πr0 ⎝ ε 4 12 ⎟⎠

Here M = Γ/2πac∞ is the Mach number, c∞ is the sound velocity at infinity. This formula considers only the lowest degree of expansion in M. As for “fat” vortex rings, their geometry and motion velocity are determined by numerical calculations only. When solving equation (3.22) with function F(Ψ) = const, Norbury (1973) obtained a one-parametric family of steady vortex rings. The dimensionless average core radius α = S πr02 was chosen as the parameter, where S is the area of the vortex crosssection, and r0 = (rmin + rmax)/2. The range of parameter α is from α 1 (corresponding to a thin vortex ring) to α = 2 (corresponding to a Hill spherical vortex ring with the known accurate analytical solution) were studied. The shape of the vortex ring cross-section for different values of α is shown in Fig. 3.5. With an increase in α, the motion velocity and the kinetic energy of the vortex both decrease (Fig. 3.6), and the impulse increases. Dependencies for the thin vortex core, obtaining from (3.18), (3.19) and (3.14) by means of normalizing with the same length and velocity scales, as in the paper by Norbury (1973) 1⎛ 8 1⎞ 8 7⎞ π2 ⎛ U = ⎜ log − ⎟ , T = ⎜ log − , 4⎝ α 4⎠ α 4⎠ 2 ⎝

I = π2 .

are presented in Fig. 3.6 for comparison. Expansions of U, T and I with consideration of small values of the second order by α are given in the paper by Fraenkel (1972).

124

3 Models of vortex structures

Fig. 3.5. The shape of the vortex ring cross-section vs. the dimensionless average core radius α (Norbury 1973*)

Fig. 3.6. Dependency of velocity U, kinetic energy T and impulse I of the vortex ring on dimensionless core radius α. Dashed lines correspond to the “thin” vortex core (Norbury 1973*)

3.2.2 Hill’s spherical vortex Now we consider the extreme case α = 2 (Hill 1894). The area of vortical fluid is assumed to be confined to the interior of a sphere of radius a, translating with the constant velocity U. The vortex lines are circles with a common axis passing through the center of the sphere, and the streamlines lie in meridian planes. Outside the sphere the motion is potential. We use the cylindrical variables (r, ϕ, z) with the coordinate origin in the center of the sphere and consider the distribution of vorticity ω = (0, ωϕ, 0)

3.2 Spatially localized vortices

⎧⎪ Ar , r 2 + z 2 < a 2 , ωϕ = ⎨ 2 2 2 ⎪⎩0, r + z > a ,

125

(3.24)

satisfying Helmholtz equation (1.15). The velocity field relative to a moving vortex satisfies the continuity equation and we can introduce the stream function ur = −

1 ∂Ψ , r ∂z

uz =

1 ∂Ψ . r ∂r

We should note that the vector potential is A = (0, Ψ r , 0) . The equation for the stream function can be obtained from the kinematic relationship = curl u

r

∂ ⎛ 1 ∂Ψ ⎞ ∂ 2 Ψ = −r ωθ . ⎜ ⎟+ ∂r ⎝ r ∂r ⎠ ∂z 2

(3.25)

The boundary conditions on the sphere surface follow from the velocity continuity. In the same time ωϕ r = F ( Ψ ) , whence it follows that

Ψ = const on the surface, bounding the region of non-zero vorticity. Far from the sphere, the velocity uz relative to the sphere is –U. It is more convenient to solve the equation (3.25) using spherical variables (ρ, ϕ, χ) [ρ sin χ = r, ρ cos χ = z] ∂2Ψ ∂ρ2

+

1 ⎛ ∂ 2 Ψ cos χ ∂Ψ ⎞ ⎪⎧− Aρ2 sin 2 χ, ρ < a, − ⎜ ⎟=⎨ ρ2 ⎝⎜ ∂χ 2 sin χ ∂χ ⎠⎟ ⎪⎩ 0, ρ > a

(3.26)

and employing the variable separation method. The general solution to the uniform equation is ⎛ d⎞ Ψ c = ⎜ cρ2 + ⎟ sin 2 χ . ρ⎠ ⎝

Obviously, inside the sphere we have d = 0, and outside it c = 0. The particular solution to the non-uniform equation is

Ψp = −

A 4 2 ρ sin χ . 10

Satisfying the conditions on the sphere surface, we determine

126

3 Models of vortex structures

⎧1 2 2 2 2 ⎪⎪10 Aρ sin χ(a − ρ ), ρ < a , Ψ=⎨ ⎪− 1 Aa 2ρ2 sin 2 χ(1 − a3 ρ3 ), ρ > a , ⎪⎩ 15

(3.27)

or, returning to cylindrical variables,

⎧1 2 2 2 2 2 2 2 ⎪⎪10 Ar (a − r − z ), r + z < a , Ψ=⎨ ⎪− 1 Aa 2r 2 ⎡1 − a3 (r 2 + z 2 )3 / 2 ⎤ , r 2 + z 2 > a 2 . ⎣ ⎦ ⎪⎩ 15

(3.28)

The constant A is derived from the condition of flow uniformity far from the sphere 2 − a 2 A = −U . 15 The streamline pattern in the Hill vortex is shown in Fig. 3.7. When differentiating the stream functions (3.28), we get the velocity components

3 rz ur = U 2 , 2 a

3 ⎛ 2r 2 + z 2 ⎞ 2 2 2 uz = U ⎜1 − ⎟⎟ , r + z < a , 2 2 ⎜⎝ a ⎠

⎡⎛ a2 uz = U ⎢⎜ 2 ⎢⎜⎝ r + z 2 ⎣

⎞ ⎟⎟ ⎠

5/ 2

⎛ a2 ⎜⎜ 2 2 ⎝r + z

5/ 2

⎞ ⎟⎟ , ⎠ ⎤ 2z 2 − r 2 ⎥ , r 2 + z2 > a 2 . 1 − ⎥ 2a 2 ⎦

3 rz ur = U 2 2 a

(3.29)

(3.30)

In the frame of reference related to the vortex, the motion is steady, and this provides easy determination of the pressure distribution. Indeed, it is found from (1.57) and (3.24) that inside the sphere the Bernoulli constant is H = –AΨ, and from the Bernoulli equation without mass forces and at the unit density we find that

P = P∞ − AΨ − (ur2 + uz2 ) 2 with velocity components (3.29). Outside the sphere, the Bernoulli constant is a global constant and correspondingly

3.2 Spatially localized vortices

127

1

0 -1

0

1

Fig. 3.7. Streamlines in the Hill vortex

1

0 -1

0

1

Fig. 3.8. Pressure isolines in the Hill vortex

P = P∞ − (ur2 + uz2 ) 2 with values ur and uz, determined by functions (3.31). The distribution of pressure in the vortex is shown in Fig. 3.8. The following invariants can be easily calculated for the spherical Hill vortex: circulation Γ = 5aU, impulse I = Iz = 2πa3U, angular momentum M = Mϕ = –5π/16a4U, and kinetic energy T = 10/7 π a3U2. We should note that at the calculation of invariants it is necessary to use the absolute reference frame

1 ur = ur , uz = uz + U , Ψ = Ψ + Ur 2 . 2

(3.31)

3.2.3 Hicks spherical vortex The Hicks spherical vortex (Hicks 1899) is a more spatially complex vortex structure with three non-zero components of velocity and vorticity. For the general case of axisymmetrical swirling flow, the fields of velocity and vorticity in cylindrical coordinates (r, ϕ, z) are determined by two functions (see section 1.4.2)

128

3 Models of vortex structures

1 ∂Ψ ⎞ ⎛ 1 ∂Ψ , uϕ , u = ⎜− ⎟, r ∂r ⎠ ⎝ r ∂z ⎛ ∂uϕ ⎞ 1 1 ∂ = ⎜− , − D2 Ψ , ruϕ ⎟ , r r ∂r ⎝ ∂z ⎠

(3.32)

(3.33)

∂ 1 ∂Ψ ∂ 2 Ψ + 2 . ∂r r ∂r ∂z Let us assume that a region of swirling fluid propagates without a shape change, i.e., for some moving reference frame, where ∂Ψ/∂t ≡ 0, the motion is stationary. Then, in accordance with (1.57)

where D2 Ψ = r

ωϕ r

=−

dH (Ψ ) Γ dΓ(Ψ ) . + 2 dΨ dΨ r

(3.34)

Here Γ = ruϕ and H = 1 2 (ur2 + uϕ2 + uz2 ) + P ρ + Π (Bernoulli function) are the arbitrary functions of Ψ . We again use the variables with tilde, i.e., the values of velocity and stream function in the reference frame translating with some constant velocity U along the axis Oz. As this occurs, relationships (3.31) and uϕ = uϕ remain true. We will skip tilde above ur, uϕ. Thus, for the stream function, we have the following differential equation

D2 Ψ = r 2

dH dΓ −Γ . dΨ dΨ

(3.35)

For the Hill spherical vortex considered above H = AΨ , Γ = 0 inside the sphere, and H = const, Γ = 0 beyond the sphere. Moffatt (1969) (see also Hicks (1899)) showed that equation (3.35) can be solved, if it is assumed that 2 2 2 ⎪⎧ H = H0 − AΨ , Γ = αΨ , r + z < a , ⎨ 2 2 2 ⎪⎩ H = H0 , Γ = 0, r + z > a .

(3.36)

Here H0 , A, α are constants. In addition, equation (3.35) becomes linear 2 2 2 2 2 ⎪⎧− Ar − α Ψ , r + z < a , D2 Ψ = ⎨ 2 2 2 ⎪⎩0, r + z < a ,

(3.37)

3.2 Spatially localized vortices

129

and the simple relationship between the velocity and vorticity fields is implemented

ωr = αur,

ωϕ = αuϕ + Ar,

ωz = αuz.

(3.38)

The boundary conditions for Ψ coincide with the conditions in the problem of the Hill vortex. The external solution at r2 + z2 > a2 will coincide with that described above. We introduce the quantity

Ψ1 = Ψ + Ar 2 α 2

(3.39)

and reduce equation (3.37) to a uniform one

D2Ψ1 = – α2Ψ1. As in the case of deducing a solution to the Hill vortex, it is reasonable to employ the spherical variables (ρ, ϕ, χ) [ρ sin χ = r, ρ cos χ = z] and solve the problem using the variable separation method

∂ 2 Ψ1 ∂ρ2

+

1 ⎛ ∂ 2 Ψ1 cos χ ∂Ψ1 ⎞ 2 − ⎜ ⎟⎟ + α Ψ1 = 0 . 2 ⎜ 2 χ ∂χ sin ρ ⎝ ∂χ ⎠

To provide matching with the external flow, it is necessary to choose the following dependency on χ: Ψ1 ∼ sin2 χ, i.e., Ψ1 = R(ρ) sin2 χ. For R(ρ) we have the equation ⎛ 2 ⎞ R′′ + ⎜ α 2 − 2 ⎟ R = 0 , ⎜ ρ ⎟⎠ ⎝ whose solution can be expressed via the spherical Bessel function of the first order1 (Abramowitz and Stegun 1964)

R = bαρ j1 (αρ),

j1 ( x ) =

sin x x

2

−

cos x , b = const . x

Therefore, considering Eq. (3.39), we get

Ψ = ⎡bαρ j1 (αρ) − A ρ2 α 2 ⎤ sin 2 χ , ρ < a . ⎣ ⎦

(3.40)

Constants b and A are determined from the condition of matching the stream function (3.40) with function (see (3.27), (3.28))

1

Moffatt (1969) uses the Bessel function of the fractional order

J3/ 2 ( x) = 2 x / π j1 ( x) instead of j1.

130

3 Models of vortex structures

Ψ=−

U 2 2 ⎛ a3 ρ sin χ ⎜1 − 3 ⎜ ρ 2 ⎝

⎞ ⎟⎟ , ρ > a , ⎠

and their derivatives with respect to ρ at ρ = a. As a result, A=

α3 j1 (αa ) b, a

2 U = α 2 j2 (αa) b . 3

(3.41)

Assuming that parameters α and b are given, and varying them, we discover the family of vortices. The velocity of the vortex motion is determined by Eqs. (3.41). Two subfamilies of the determined solutions should be noted specially. The first of them is given by the values of α, when j1(αa) = 0. In this case A = 0 and according to (3.38), ω = αu, i.e., the Beltrami flow is obtained. The first root of function j1(x) gives the value α = 4.493/a. Another distinguished subfamily appears at j2(αa) = 0, and this corresponds to the value of U = 0, i.e., to immobile vortices. The minimum value of parameter α for such vortices equals 5.763/a. Finally, passing to the limit α → 0 from Eqs. (3.40), (3.41), we get the stream function for the Hill vortex (3.27). When the stream function (3.40) is known, the velocity components are determined by differentiation uz, uz

ur = bα2 sin χ cos χ j2(αρ),

(3.42)

uz = 2bα [ j1 (αρ) ρ − j1 (αa) a ] − bα 2 α 2 (αρ) sin 2 χ .

(3.43)

The velocity component uϕ inside the vortex is obtained from definition Γ (Γ = ruϕ) and Eq. (3.36)

uϕ = bα2 ρ [j1(αρ)/ρ – j1(αa)/a] sin χ.

(3.44)

A simple connection between the velocity curl and the velocity itself (3.38) allows easy determination of the flow vector potential. Indeed, since rot u = αu + Ar iϕ , it is natural to seek the vector potential in the following form 1 A= u+F, α and for the determination of vector F we get the equation curl F = −

Ar iϕ . α

(3.45)

3.2 Spatially localized vortices

a

Fig. 3.9. Hicks vortex. surfaces (αa = 3)

131

b

– behavior of the vortex line and streamline; b – stream

The second equation arises from the condition that vector A must be solenoidal div F = 0.

(3.46)

Obviously, the problem of (3.45), (3.46) is similar to that of the inner flow in the Hill spherical vortex (3.26), and in correspondence with (3.29) this gives

Ar =

1 A 2 1 A 2 ur − ρ (1 + sin 2 χ) . ρ sin χ cos χ , Az = uz + α α 5α 5α

Finally, the component Aϕ is proportional to uϕ:

Aϕ = uϕ α . Beyond the sphere ρ = a, the vector potential has a single non-zero component Aϕ = Ψ / r . Now, we consider the structure of the vortex field inside the sphere. It is clear from Eqs. (3.38) that the vortex lines, like the streamlines lie on the surfaces Ψ = const (Fig. 3.9 ). In turn, the latter are torus-like nested surfaces (Fig. 3.9b). At αa < 5.763, the stream function has one extreme at the point with coordinates χ = π/2, ρ = ρ0, satisfying the condition uz = 0 . At αa > 5.763, the second extreme appears at point χ = π/2, ρ = ρ1. The picture of isoline Ψ = const becomes two-celled (Fig. 3.10). An increase in the number of cells occurs after the transition of the value αa through the hhhhhhhhhhhhhhhhh

132

3 Models of vortex structures

Fig. 3.10. The two-cell structure of the stream surfaces in the Hill vortex at α = 7

Fig. 3.11. Radial positions of stream function extreme points ρ0 and ρ1 in the Hill vortex vs. parameter α

next root of function j2(x). Dependencies ρ0(α) and ρ1(α) are shown in Fig. 3.11. At α → 0 ρ0 → a/ 2 , and this corresponds to the extreme of the stream function in the Hill vortex. In another extreme case α → ∞ ρ0 → s0/α, where s0 = 2.744 is the first root of equation (s2 – 1) sin s + s cos s = 0.

(3.47)

Respectively, ρ1 → s1/α, etc. According to Moffatt (1969), the period of a vortex line increases from zero at Ψ = 0 (at the boundary of a sphere ρ = a) to infinity at Ψ = Ψ max at the vortex “axis” (χ = π/2, ρ = ρ0). However, according to the detailed analysis (Kuibin 2003), the vortex line period increases logarithmically at Ψ → 0 , and at Ψ → Ψ max, the period becomes finite and equals

(

2 παr0

2 α2r02 − 2 + Ar04 Ψ max

)

⎛ A r02 ⎞ . ⎜⎜ 1 + 2⎟ ⎟ ⎝ Ψ max α ⎠

Considering the simple dependencies of velocity components on coordinates, we can express all invariants via the elementary functions. The impulse (see (1.126)) of the Hicks vortex is directed along axis z, and equals

Iz = π

∫∫ r

2

ωϕ drdz = 2πa3U =

4π abf0 (αa), 3

f0 ( x) = x 2 j2 ( x) ,

and coincides with the impulse of the Hill vortex at the same velocity U. The angular momentum (see (1.115)) has two non-zero components

Mϕ = −π

∫∫ r (r

2

+ z 2 )ωϕ drdz = −2π2 ba 2 f1 (αa),

3.2 Spatially localized vortices

133

b

Fig. 3.12. Diagrams of functions f0(x), f1(x), f2(x) ( ) and f3(x), f4(x) (b), included in relationships for invariants of the Hicks vortex

f1 ( x ) = ⎡(8 + 4x 2 − x 4 ) j0 ( x ) − (8x − 4 x3 ) j1 ( x ) − 8⎤ x 2 , ⎣ ⎦ Mz = −π

∫∫ r (r

2

+ z 2 )ωz drdz = −

8π 2 ba f2 (αa), 15

f2 ( x) = x 2 j1 ( x) − 5xj2 ( x). Kinetic energy can be calculated using the formula (1.119)

T = 16π

b2 f3 (αa), a

⎡ j 2 ( x) j1 ( x ) j2 ( x) j12 ( x) ⎤ − + f3 ( x ) = x 4 ⎢ 2 ⎥. 2x 10 ⎥⎦ ⎣⎢ 9

The last invariant of the Hicks vortex is helicity

H = 2π ∫∫ ru ⋅ drdz =

8π b 2 f4 (αa), 3 a2

f4 ( x) = x 4 ⎡ xj12 ( x) − xj0 ( x) j2 ( x) − 2 j1 ( x) j2 ( x) ⎤ . ⎣ ⎦ Diagrams of functions f0, f1, f2 are shown in Fig. 3.12 , while f3, f4 are presented in Fig. 3.12b. If we know two invariants, for instance; impulse and energy, we can determine (at a given radius a) the velocity of the vortex motion and the value of parameter α. Applying the third invariant, Mz or helicity H, we can determine all three values, which assign the vortex parameters: a, b, α (velocity U is connected with parameter b by Eqs. (3.41)).

134

3 Models of vortex structures

3.3 Columnar vortices in ideal fluid 3.3.1 Rankine vortex The model of a cylinder vortex with a finite round core of radius a, and a constant vorticity ω inside it (Fig. 3.13), is more realistic than the model of an infinitely thin vortex filament. Outside the core, the flow is assumed to be irrotational. As in the case of a vortex sheet, this vortex can be approximated by the continuous distribution of rectilinear vortex filaments in the core. Then, according to Stokes theorem, the contribution of the core cross-section element dS to circulation dΓ, equals

dΓ = ω dS. The circulation around any circuit once and enclosing the whole vortex core is Γ = ωπa2 = const.

(3.48)

Applying formula (2.25) for the velocity field induced by N vortices and turning to the continuous distribution, we have

ux − iuy =

dΓ

∫ 2πi(z − z0 ) =

ω dS(z0 ) . 2πi z − z0

∫

S

This integral is convenient for calculation, but it is simpler to get the result by considering the axial symmetry of the problem, i.e., the presence of only the circumferential velocity component u = u(r). From Stokes theorem, for a circle with radius r > 0, we have 2πru = ω, and further taking into account (3.48) we find the expression for velocity in the region of irrotational (potential) flow b

Fig. 3.13. Profiles of velocity ( ) and pressure (b) in the Rankine vortex

3.3 Columnar vortices in ideal fluid

u=

Γ a2 ω = , r > a. 2r 2πr

135

(3.49)

As in the case of a cylindrical sheet, this distribution coincides with the velocity field induced by an infinitely thin vortex filament with intensity Γ at distance r > a. Inside the core, in the same way we obtain

u=

2πru = πr2ω or

Γr 2πa 2

, r < a.

(3.50)

Linearity of the profile indicates the solid-body rotation of fluid in the vortex core with angular velocity Ω equal to

Ω = Γ/2πa2.

(3.51)

The resultant velocity distribution is shown in Fig. 3.13 . Apparently, there is a break in the velocity profile at the boundary of the core r = a, caused by a vorticity jump. Nevertheless, this model called the Rankine vortex is the most popular. It reflects the main features of concentrated vortices. The radial distribution of static pressure is characterized by a drastic decrease in the vortex core. To calculate the pressure profile, we use the Euler equation, which in polar coordinates and accounting for axial symmetry takes the form

ρ

u 2 dp = , r dr

(3.52)

then r

p = p∞ + ρ

∫

∞

u2 dr , r

(3.53)

where ρ is the fluid density, p∞ is the pressure at infinity. Substituting velocity profiles (3.49), (3.50) into (3.53), we find the distribution of pressure p = p∞ − ρ

p = p∞ − ρ

ω2 a 4 8r 2

, r > a,

ω2 a 2 ω2 2 r , r < a, +ρ 4 8

(3.54)

(3.55)

136

3 Models of vortex structures

the diagram of which is shown in Fig. 3.13b. The minimum pressure is obtained at the vortex axis

pmin = p∞ − ρω2 a 2 / 4 = p∞ − ρ

Γ2 4π 2 a 2

.

(3.56)

At the core boundary, p – p∞ = (pmin – p∞)/2. For an infinitely thin vortex filament of intensity Γ, the pressure at the axis tends to (– ∞), as it is obvious from (3.56) at a → 0. Such a drastic decrease in pressure explains the formation of funnels on the free surface of the fluid upon intensive rotation and objects being drawing into the near-axial zone of tornadoes. 3.3.2 Gauss vortex

The smooth profiles of vorticity and azimuthal velocity were obtained by Hopfinger and van Heijst (1993) upon numerical examination of monopolar vortex instability. In the dimensionless form they are written as q ⎡ ⎛ r ⎞q ⎤ ⎡ ⎛ r q ⎞⎤ ωR ⎡ 1 ⎛ r ⎞ ⎤ u 1 r exp ⎢ − ⎜ ω≡ = ⎢1 − q ⎜ ⎟ ⎥ exp ⎢ − ⎜ ⎟ ⎥ , u ≡ = ⎟⎥ . U ⎣⎢ 2 ⎝ R ⎠ ⎦⎥ U 2R ⎢⎣ ⎝ R ⎠ ⎦⎥ ⎢⎣ ⎝⎜ R ⎠⎟ ⎦⎥ Here R, U are the scales of length and velocity, q is a shape factor. In particular, assuming that q = 2 and introducing the dimensionless coordinate r = 2r / R, we obtain ⎛ r2 ⎞ ⎛ 1 ⎞ ω = ⎜1 − r 2 ⎟ exp ⎜ − ⎟ . ⎜ 2 ⎟ ⎝ 2 ⎠ ⎝ ⎠

(3.57)

This vortex is called the Gauss vortex because the dimensionless stream function Ψ is described by the Gaussian distribution

Ψ=

1 r2 exp(− ). 2 2

(3.58)

Due to their simplicity, formulae (3.57), (3.58) are often used for the generalization of experimental data. 3.3.3 One-dimensional helical flow Let us consider an example of a steady one-dimensional helical flow, where the streamlines coincide with the vortex lines (Section 1.4.1), in the framework of the model of swirling axisymmetrical flow. With this pur-

3.3 Columnar vortices in ideal fluid

137

pose we notate (1.44) in cylindrical coordinates. We assume that the velocity components depend on the radial coordinate only and λ ≡ const ≠ 0. Then, it follows from the first equation (1.44) that ur ≡ 0, i.e., we obtain the flow with the streamlines located on the co-axial cylinders. Due to the combination of the second and third equations in (1.44), we get

uθ ∂ (r uθ ) ∂u 1 ∂ uz + uz z = 0, at that =λ. uθ ∂r ∂r ∂r r

(3.59)

We should note that the first equation in (3.59) can be also obtained from (1.36) after substitution of (1.38) and considering the above assumptions and the fact that for steady helical flows the Bernoulli constant is the same within the whole flow area. Equations (3.59) determine nothing but the sustained helical flow of inviscid fluid in a round tube at uniform energy distribution. Equations (3.59) can be integrated at various initial assumptions. However, according to Vasiliev (1958), we restrict oneself to a case of simple distribution for the circumferential velocity component uθ = αr,

α = const,

which is equivalent to the velocity distribution inside the Rankine vortex core (3.50). Then, after integration of (3.59) we get for the axial velocity component uz = u02 − 2αr 2 , where u0 is the velocity at the flow axis. Therefore, in contrast to the Rankine vortex, where the axial velocity may only be constant, this class of flows allows non-uniform radial distribution of the axial velocity component. 3.3.4 One-dimensional (columnar) helical vortices Elementary helical vortex structures (an infinitely thin vortex filament and a vortex sheet consisting of helical vortex filaments) were described formerly. However, real vortices have a finite sized core. We commence the consideration of this class of helical vortices with the simplest case of axisymmetrical or columnar vortices. In contrast to the Rankine vortex, consisting of uniformly distributed rectilinear vortex filaments (or point vortices in a circle), we consider the axisymmetrical helical vortex made by superposition of helical vortex filaments (Fig. 3.14) (Kuibin and Okulov 1996). If the distribution of filament intensities within a cylindrical hhhhhhhhhhhhhhhh

138

3 Models of vortex structures

Fig. 3.14. Schemes of the axisymmetrical helical vortex

region is known, the problem of the velocity field is reduced to the integration of statement (2.56) or (2.69). However, the problem can be solved without the application of results from Section 2.6. Analysis of equations (Section 1.5) determining the flow demonstrates that the condition of axial symmetry allows rigorous development of the general model of axisymmetrical helical vortices, since ∂/∂χ ≡ 0, ur ≡ 0. Hence, the Helmholtz equation (1.71) is true for any distribution of a vertical vorticity component by r. From (1.68) we have

∂ ( ruθ ) = rωz . ∂r

(3.60)

Integrating (3.60) with respect to r, we get r

1 ωz (r ′)r ′dr ′. uθ = r

∫

(3.61)

0

Since the axial velocity component stays connected with uθ by relationship (1.66), we have, correspondingly r

uz = u0 −

1 ωz ( r ′ ) r ′dr ′. l

∫

(3.62)

0

The condition of axial symmetry, significantly simplifies the problem of pressure calculation from the Euler equation. Actually, one of two equations of system (1.64), which includes the pressure, is satisfied identically, and the second equation is reduced to

3.3 Columnar vortices in ideal fluid

139

1 2 1 ∂p uθ = , r ρ ∂r then, r

∫

p = p0 + ρ uθ2 0

dr ′ , r′

(3.63)

where p0 is the pressure at the tube axis. It is obvious from (3.63) that the axial velocity component does not affect the pressure in the swirling axisymmetrical flow, induced by the distribution of helical vortex filaments. Later, this important feature will be used for evaluation of vortex parameters through distribution of the bottom pressure (see Chapter 7). For more convenient description, we introduce the dimensionless function r

2π Φ (r ) = ωz (r ′)r ′dr ′. Γ

∫

(3.64)

0

At that, in accordance with (3.61) and (3.62), the velocity components are expressed via function Φ(r)

uθ =

Γ Γ Φ ( r ) , uz = u0 − Φ (r ). 2πr 2πl

(3.65)

We also introduce a dimensionless vorticity and pressure drop ωz =

8π2 ε 2 p − p0 πε 2 , ωz , ∆p = Γ ρ Γ2

(3.66)

where ε is the typical size of the vortex. Let us consider three important examples of the simplest vorticity distributions (Table 3.1): uniform distribution within the core of radius ε (model I); fractional-power distribution (model II) and Gaussian distribution (model III). Like the distributions of pressure in the above examples, the profiles of circumferential velocity coincide with those for the vortices of Rankine (model I), Scully (1975) (model II) and Lamb (see (Hopfinger and van Heijst 1993), model III), correspondingly. Simultaneously, the vorticity vector has a non-zero circumferential component and, thus, we get the axial velocity profile, which is non-uniform over the radius. We should note that the difference uz – u0 is inversely proportional to the pitch of helical symmetry.

140

3 Models of vortex structures

Table 3.1. Models of axisymmetrical vortices Model

ωz

I

∆p

ε

ε

⎧1, r < ε ⎨ ⎩0, r ≥ ε Φ (r )

II

⎧⎪r 2 ε 2 , r < ε ⎨ r≥ε ⎪⎩ 1,

⎧⎪ r 2 ε 2 , r<ε ⎨ 2 2 ⎪⎩2 − ε r , r ≥ ε

ε

⎛ r2 ⎜⎜1 + 2 ⎝ ε r2 r +ε 2

III

2

r2 r 2 + ε2

⎞ ⎟⎟ ⎠

−2

⎛ r2 exp ⎜⎜ − 2 ⎝ ε

⎞ ⎟⎟ ⎠

⎛ r2 1 − exp ⎜⎜ − 2 ⎝ ε

⎞ ⎟⎟ ⎠

⎛ r2 ε2 ⎡ 2 log 2 − 2 ⎢1 − exp ⎜⎜ − 2 r ⎢⎣ ⎝ ε

2

⎞⎤ ⎟⎟ ⎥ + ⎠ ⎥⎦ 2 2 ⎛ r ⎞ ⎛ r ⎞ +2 Ei ⎜⎜ − 2 ⎟⎟ − 2 Ei ⎜⎜ −2 2 ⎟⎟ ⎝ ε ⎠ ⎝ ε ⎠

We consider the uniform vorticity distribution over the annular crosssection b1 ≤ r ≤ b2 as a generalization of model I

r < b, ⎧ 0, ⎪ 2 2 ⎧0, r < b1 , r > b2 Γ ⎪ r − b1 , Φ = r , b1 ≤ r ≤ b2 , (3.67) ωz = ( ) ⎨ ⎨ 2 2 π(b22 − b12 ) ⎩1, b1 ≤ r ≤ b2 ⎪ b2 − b1 ⎪ 1, r > b2 . ⎩ Such “hollow” vortices may be formed by near-wall swirling of the flow, and they will be used for construction of more complex models of vortices (see Section 7.3.2). The pressure field for the helical vortex with ringshaped vorticity distribution takes the form

3.3 Columnar vortices in ideal fluid

0, r < b1 , ⎧ ⎪ 4 4 ⎛ r − b1 1 r ⎞ 2 ⎪ 4 log b − ⎜ ⎟⎟ , b1 ≤ r ≤ b2 , 1 2 ⎪ 2 2 2 ⎜ ρΓ ⎪ b r b b − 1 ⎠ 2 1 ⎝ p = p0 − 2 ⎨ 8π ⎪ 4b12 b ⎪ − 1 + 2 − log 2 , r > b2 . 2 2 2 2 ⎪ b1 2 2 r b2 − b1 − b b 2 1 ⎪⎩

(

)

(

141

(3.68)

)

At b1 = 0 and b2 ≡ ε, the model described by Eqs. (3.67), (3.68), coincides with model I. Profile families of tangential and axial velocity components calculated by Eqs. (3.65) are shown in Fig. 3.15 together with function Φ(r) from (3.67). In contrast to the Rankine vortex, this model admits a great variety of axial velocity distributions depending on the values and signs of the helical pitch l and the velocity at axis u0. In the extreme case, l → 0, Γ → 0, Γ/l = const, (b2 − b1) → 0 the ring vortex degenerates into the vortex sheet localized on the cylindrical surface r = b2 = b1, with a uniform axial flow inside (see Section 3.1). In contrast to models (3.67), (3.68) and model I, in real flows the distributions of vorticity and velocity demonstrate a smooth behavior. The profile of tangential velocity in model II corresponds to a special case of generalized empirical profile used by Shtym (1985) for the vortex chambers. The pressure in this model does not depend on l and it is linearly connected with the axial velocity

p = p0 + ρ

Γ 4πε 2

( u0 − uz ) .

Apparently, model II is more convenient for the description of turbulent vortices (Murakhtina and Okulov 2000). At the same time, it is more natural to assume a Gaussian vorticity distribution for the laminar vortices. It is known that in a viscous fluid the vorticity concentrated in a rectilinear vortex filament (see Section 2.3.2) diffuses in time as follows ω (r, t ) =

⎛ r2 ⎞ Γ exp ⎜ − ⎟⎟ . ⎜ 4πνt ⎝ 4ν t ⎠

(3.69)

For a steady axisymmetrical swirling flow in a cylindrical channel, the vortex size will increase the downward flow, and it can be assumed for some region of the channel that this vortex has a core of a constant size ε (see Table 3.1, model III). The velocity field corresponding to this model is written as

142

3 Models of vortex structures

Fig. 3.15. The family of profiles of tangential uθ and axial uz velocity components of the columnar vortex. Helical pitch l and velocity at axis u0 are the parameters of the flow

3.3 Columnar vortices in ideal fluid

uθ =

(

)

Γ ⎡ 1 − exp −r 2 / ε 2 ⎤ , ⎦ 2πr ⎣

uz = u0 −

(

)

Γ ⎡ 1 − exp −r 2 / ε 2 ⎤ ⎦ 2πl ⎣

143

(3.70)

or uθ =

⎛ ⎛ Γ ⎡ Γ ⎡ r 2 ⎞⎤ r 2 ⎞⎤ ⎢1 − exp ⎜⎜ −1.256 2 ⎟⎟ ⎥ . ⎢1 − exp ⎜⎜ −1.256 2 ⎟⎟ ⎥ , uz = u0 − 2πl ⎢⎣ 2πr ⎢⎣ rm ⎠ ⎥⎦ rm ⎠ ⎥⎦ ⎝ ⎝

Here, the radius of the maximum tangential velocity is introduced instead of scale ε rm = 1.256 ε . The pressure distribution in model III is expressed via the integral exponent

Ei ( − x ) =

−x

∫ exp ( −x )

x dx.

−∞

At a large argument (small ε), function Ei(− x) decreases as exp(− x)/x. As a result, the relationship for the pressure at a distance r > ε simplifies significantly. In the paper by Bühler (1988) the formula for pressure was derived in a different form (Ei was expanded into a series). Models I, II, and III are compared in Fig. 3.16 by two different methods. In all cases, the parameters are as follows: u0 = 0 and l = –1. The values of ε were chosen due to the demand, that radius rm – maximum of tangential velocity profile is the same for all three models (ε = 1 – for models I, II and ε = 1/1.12 – for model III). The first comparison in Fig. 3.16 is made for the vortices with the same circulation Γ = 2π. As a result, we can conclude that the solutions corresponding to models I and III are the closest. However, model I is simpler and more illustrative because the size of the vortex core can be easily determined by the break in the velocity profile. Therefore, model I will be used for the following analysis. A significant distinction (especially for the pressures) of models I and III from model II is explained by the fact that the vorticity in the last vortex is less concentrated. In our opinion, if it is assigned that for all vortices pressure differences ∆p at the vortex axis and its periphery (r → ∞) are the same, the comparison will be more correct. But vortex circulations will differ:

Γ1 = 2π, Γ 2 = 2π 2, Γ3 = 2π

1.256 log2 .

In this case, profiles of vorticity and velocity for all three solutions become more agreeable with each other, especially the profiles of pressure (Fig. 3.16b). Comparison of all calculated pressure profiles at the maximumpoint of tangential velocity rm provides the important information. For vortices I and II, this maximum point coincides with point r0.5, where the pres-

144

3 Models of vortex structures b

ωz

ωz

2

2

1

1

0 uθ

0 uθ

0.5

0.5

0.0 uz

0.0 uz

1.0

1.0

0.5

0.5

0.0 ∆ p/ρ

0.0 ∆ p/ρ

0.5

0.5

0.0

0

1

2

3

r

0.0

0

1

2

3

r

Fig. . Comparison of radial distributions of the axial vorticity component ωz, tangential uθ and axial uz velocity components and pressure ∆p/ρ for different models ( – I, – II, – III) of a columnar vortex (see Table 3.1). Comparison conditions: – equal circulations ; b – the difference between the values of static pressure in the vortex center and at its periphery are the same

sure is half of the total pressure difference at the vortex axis and the periphery (half of the total drop). This fact follows directly from the analysis of corresponding formulae for the pressure. While analyzing the pressure distribution, we should note that even here points rm and r0.5 almost coincide. This means that the vortex radius can be determined using the point, where the pressure equals half of the total pressure drop.

3.3 Columnar vortices in ideal fluid

145

3.3.5 Q-vortex To describe a flow before and after vortex breakdown, empirical formulae are usually used (Leibovich 1978, 1984)

uθ =

(

(

))

(

K 1 − exp −αr 2 , r

uz = W1 + W2 exp −αr

2

) =W +W 1

2

(

(

− W2 1 − exp −αr

2

)),

(3.71)

where K, W1, W2, α are constants determined empirically. Distribution of tangential velocity in (3.71) exactly corresponds to the Burgers vortex (Burgers 1940), the velocity field of which satisfies the Navier-Stokes equations (see Section 3.4.1). However, for the axial velocity component in (3.71) and, respectively, in the experiment, there are significant differences from Burgers solution. In Burgers vortex, the axial velocity is the function of coordinate z only, whereas in (3.71) uz does not depend on z, but changes considerably along the radial coordinate. The velocity field in the form of (3.71) was first studied against instability in the paper by Lessen et al. (1974), and it was derived from the self-similar solution of Batchelor (1964) for a swirling wake with neglect for some small terms. We should note that profiles (3.71) describe the experiment with great accuracy, and they are widely used for the processing of data on swirling flows (Faler and Leibovich 1977; Escudier 1988; Alekseenko and Shtork 1992). Considering this fact, the above formulae are widely applied for theoretical analysis of swirling flow stability. Indeed, for these purposes a simpler one-parametric model is used

uθ =

(

( ))

( )

q 1 − exp −r 2 , uz = exp −r 2 , r

(3.72)

which is called the Q-vortex. However, it is important that in the structure, the solution (3.70) obtained previously exactly coincides with the empirical formulae (3.71). The direct comparison of the exact solution of (3.70) and the empirical formulae (3.71) gives

Γ = 2πK,

l = K/W2 ,

u0 = W1 + W2

and ε = 1

α,

and this both specifies the physical sense of empirical constants and identifies a large class of swirling flows, described by (3.71), as flows induced by the columnar helical vortices with Gaussian distribution of vorticity in the core. Correspondingly, for Q - vortex

146

3 Models of vortex structures

Γ = 2πq,

l = q,

u0 = 1,

which significantly constricts the considered class of flows. Therefore, profiles (3.71), (3.72) are the solution to the equations of ideal fluid motion, which satisfactorily describe the experimental data. 3.3.6 Helical vortex with a finite-sized core For theoretical description of rotating helical vortices, the solution derived in Section 2.6 is of little use because the self-induced velocity of the infinitely thin filament equals infinity. In a real fluid, the vortex core always has a finite size. According to this, we elaborate the model of the vortex with a core in the shape of a helical rope of a round cross-section ε in the plane normal to the vortex axis (Fig. 3.17). Let us consider the simplest distribution of vorticity, satisfying Helmholtz equation (1.71): ωz = const inside the core. The velocity field induced by this vortex can be easily presented via solution of (2.56) or (2.69), if the core is considered a superposition of infinitely thin vortex filaments, spread uniformly over the core. Note that the module of vorticity ω is not a constant, since vortex filaments in such a vortex are twisted, and the angle of the vorticity vector inclination changes over the vortex cross-section. Indeed, according to geometrical plotting

ωz = ω cos α = ωl / l 2 + r ′2 . Here r′ is the radial coordinate of the vortex filament; α is the angle between ω and axis z (Fig. 3.17). It is clear that at ωz = const and l = const the value of ω increases with distance r′.

Fig. 3.17. The model of a helical vortex with a finite-sized core. The local coordinate system (σ, ϕ) is shown on the right

3.3 Columnar vortices in ideal fluid

147

Now we consider the solution to (2.69) for an infinitely thin helical vortex filament with circulation in the form of u = F, where F is the function of filament geometry only. Turning to continuous distribution of vortex filaments, for a thin vortex tube we can write the following differential relationship

duε = Fd , where F is the same function as for an infinitely thin vortex filament, d is the circulation of a distinguished vortex tube with a cross-section dSn, index ε represents the solution for a vortex with the core radius ε. It is more convenient to calculate the solution using the horizontal cross-section (z = 0) of the vortex tube with the area Sz. From the Stokes equation we have

d

= ω · n dS = ωz dSz.

Thus, it follows that: = ωz Sz. We should stress that here is the circulation for a helical vortex with a finite-sized core. In the same time ωz dSz = ω dSn , and taking into account the connection between ωz and ω, we obtain

dSz = dSn l 2 + r ′2 l

or

Sz =

∫

Sn

l 2 + r ′2 dSn . l

Using the last formula, we can determine the area of the helical vortex cross-section in the plane z = 0. Considering all these relationships, we obtain the equation for the velocity induced by the helical vortex with a core of finite-sized radius

∫

∫

uε (r , θ, z) = FdΓ = ωz FdSz =

=

1 lSz

Γ FdSz = Sz

∫

(3.73)

ε 2π

∫∫

l 2 + r ′2 F (r , θ, z; r ′, θ′, z′)σ dσ dϕ .

0 0

It is also considered here that dSn = σ dσ dϕ. Integration is performed over a circumference of radius ε with the center at point r′ = a, θ′ = θ0, z′ = 0. The local polar coordinates σ, ϕ with the center at point (a, θ0, 0) are connected with coordinates of the basic cylindrical system r′, θ′, z′ by relationships

148

3 Models of vortex structures

⎧r ′ cos(θ′ − θ0 ) = a + σ cos ϕ , ⎪ ⎨r ′ sin(θ′ − θ0 ) = σ sin ϕ cos α , ⎪ z′ = −σ sin ϕ sin α , ⎩ where α is the angle between the vortex axis and axis Oz (tan α = a/l). This model differs from the conventional one described in the literature, which deals with the theoretical description of helical vortices, where uniform distribution of the vorticity module over the core cross-section is assumed (Widnall et al. 1971; Moore and Saffman 1972). However, this distribution of vorticity does not satisfy Helmholtz equation. Actually, if we substitute the following relationship ωz = ωl

r 2 + l2 ,

into the left side of (1.71), where ω = const, we obtain

dωz l l ∂ω ∂ ⎛ = + ur ω ⎜ ⎜ 2 2 2 dt ∂r ⎝ r + l 2 r + l ∂t

⎞ ⎟⎟ , ⎠

r − a ≤ ε.

(3.74)

In general, the right side of (3.74) is not zero. Helmholtz equation is accurately satisfied only in some particular cases, for instance, for the considered axisymmetrical helical vortex, when ur = 0 and ω does not depend on time. For an arbitrary helical vortex, the model (3.73) is the approximate one. The smallness of the right side of (3.74) is the required condition for application of this model. Particularly, this is possible for weakly curved vortices (a l). In approximation of thin vortex filaments, component ωz changes insignificantly over the core cross-section, and it can be considered almost constant, then the right side of (3.74) is also small.

Fig. 3.18. On the determination of function F(r) in (3.76). The tube and core cross-sections in a horizontal plane ( ) and the plane normal to the vortex axis (b)

3.4 Viscous models of vortices

149

The simpler form of the approximate model (3.73) is obtained for characteristics, averaged over the angular coordinate θ, uε =

1 2π

2π

∫ uε dθ.

(3.75)

0

Let us substitute (3.75) into (3.73) and change the order of integration. Considering that the integrals from series entering into solution (2.69) for u, are equal to zero, we can reduce the problem of average velocity determination to the calculation of a single function F(r)

urε ≡ 0,

Γ Γ F (r ), uzε = u0 − F (r ) , 2πr 2πl 1 ⎧0, r < r ′⎫ F (r ) = ⎨ ⎬ dS . Sz ⎩1, r ≥ r ′ ⎭ uϕε =

∫

(3.76)

Sz

The integral in (3.76) equals the intersection area of the circle of radius r with cross-section Sz. The ratio of areas in (3.76) does not change, if both figures are projected on the plane normal to the helical axis of the vortex (Fig. 3.18). Therefore, we find that F(r) = S(0)/πε2, where S(0) is the intersection area of the circle σ = ε with the ellipse prescribed by the formula .

(a + σ cos θ)2/l2 + (σ sin θ)2/(a2 + l2) = r2/l2. Equation (3.73) represents a formal notation of the velocity induced by the helical vortex with a core of round cross-section. Fukumoto and Okulov (2005) elaborated a more rigorous approach for searching the solution for the velocity field induced by a helical vortex with an arbitrary vorticity distribution in a core of finite but small size.

3.4 Viscous models of vortices 3.4.1 Burgers vortex Taking into account viscosity enables us to smooth out peculiarities which appear in the vicinity of the vortex core in the models of both an infinitely thin vortex filament and a Rankine vortex. Such a solution for steady-state conditions was considered in Section 2.3.2 as an example of the viscous diffusion of vorticity. To compare time dependent profiles of vorticity (2.31) and velocity (2.32) with the experimental data we must introduce a

150

3 Models of vortex structures

certain scale a = 2 νt , which is a linear measure of the vortex at a point in time t. Then we obtain the following correlations Γ exp ( − r 2 a 2 ) , π a2

(3.77)

Γ ⎡1 − exp ( − r 2 a 2 )⎤ , ⎦ 2πr ⎣

(3.78)

ω=

v=

describing the so called Lamb vortex (see Saffman and Baker (1979); Hopfinger and van Heijst (1993)). By their structure the formulae (3.77) and (3.78) coincide with the vorticity distribution and azimuthal velocity for a three-dimensional steady-state Burgers vortex, which was pioneered by Burgers (1940, 1948) and Rott (1958) for the description of turbulent swirl flows. Burgers vortex pertains to the class of axisymmetric exact solutions of the Navier-Stokes equations expressed in form

ur = u(r), uθ = v(r), uz = w(r) = zf(r),

(3.79)

where u, v, w are the radial, azimuthal and axial velocity components, respectively. Donaldson and Sullivan (1960) described a family of solutions similar to (3.79). It was assumed that the flow was formed within an infinitely long rotating porous tube with the proper boundary conditions at the wall. A solution to the Navier-Stokes equations in form of (3.79) was also found by Sullivan (1959), but within boundless space. Assigning velocity circulation and an additional constant at the infinity, he obtained the solution in the form of a two-cell vortex (see Section 3.4.2). In particular, Burgers vortex follows on from this solution. To find the solution for Burgers vortex let us consider the special case of (3.79)

u = u(r), v = v(r), w = αz, α = const,

(3.80)

and rewrite the Navier-Stokes equations and the equation of continuity (1.35) in cylindrical coordinates taking into account (3.80)

u

∂u v2 ∂ ⎛1 ∂ 1 ∂p ⎞ − =− +ν ⎜ (ru) ⎟ , ∂r r ρ ∂r ∂r ⎝ r ∂r ⎠

(3.81)

∂v ∂ ⎛1 ∂ ⎞ uv =ν (rv) ⎟ − , ⎜ ∂r ∂r ⎝ r ∂r ⎠ r

(3.82)

u

3.4 Viscous models of vortices

w

∂w 1 ∂p , =− ∂z ρ ∂z

151

(3.83)

∂w 1 ∂ (ru) + = 0. r ∂r ∂z

(3.84)

The relationship for radial velocity can be obtained simply from the equation of continuity (3.84)

u = − αr/2.

(3.85)

Then (3.82) yields the equation for the azimuthal component

−

α ⎡1 ⎤′ (vr )′ = ν ⎢ (rv)′⎥ , 2 ⎣r ⎦

where prime means r derivative. Integration under the boundary condition v = 0 at r → ∞ gives v = c1[1 − exp(−αr2/4ν)]/r. Integration constant c1 can be found by assigning circulation Γ at the infinity:

Γ = 2πrv|r → ∞ = c1 2π. As a result we derive the following expression for v:

v=

⎛ αr 2 ⎞ ⎤ Γ ⎡ ⎢1 − exp ⎜ − ⎟⎥, 2πr ⎣ ⎝ 4ν ⎠ ⎦

Γ = Γ r =∞ .

(3.86)

The velocity profile has just one local maximum vm, determined by the condition v′ = 0 or 1 + 2ϕ = eϕ, where ϕ = αr2/4ν. The solution of this transcendental equation is ϕ = 1.2565… , from which

rm = 2.242

vm = 0.16

ν . α

Γ α 0.36 Γ . = π ν π rm

(3.87)

(3.88)

Quantity rm is usually interpreted as the effective vortex radius. Using extreme values, the profile of azimuthal velocity can be expressed in the convenient dimensionless form

152

3 Models of vortex structures

Fig. 3.19. Azimuthal velocity profiles in Rankine and Burgers vortex models

⎛ 1.39 ⎡ v r 2 ⎞⎤ = ⎢1 − exp ⎜ −1.26 2 ⎟ ⎥ . vm (r / rm ) ⎣ rm ⎠ ⎦ ⎝

(3.89)

Let us consider asymptotic expressions (3.86). At r → 0

v=

Γ αr 2 Γα Γ r (1 − 1 + , r = 1.26 + ...) ≈ 2πr 4ν 8πν 2π rm2

i.e., near the axis we have rigid-body rotation. In addition, if we assign α2 = rm2 /1.26, the velocity profile will concur with distribution (3.50) for a Rankine vortex. At r → ∞ we have asymptote v = Γ/2πr which conforms both to a Rankine vortex and a singular vortex of the same intensity Γ. Comparison of azimuthal velocity for Burgers and Rankine models under the conditions Γ = Γ|r = ∞ = idem and a = rm is presented in Fig. 3.19. Now let us determine the pressure from the remaining two equations. Equation (3.83) takes the form of α2z = −(1/ρ)∂p/∂z, from which p = −ρα2z2/2 + ρc(r). Substituting this expression in (3.81) we obtain the desired equation for c(r): c′( r) = −α2r/4 +v2/r. Integrating and substituting c(r) into the expression for p, we derive the calculation formula r 2 ⎛ r2 ⎞ v 1 p = p0 − ρα 2 ⎜ z 2 + ⎟ + ρ ∫ dr , r 2 4⎠ ⎝ 0

(3.90)

involving an integral which can be computed using numerical methods. The existence of a steady solution for viscous fluid is accounted for by the circumstance that viscous vorticity diffusion is compensated by the radial transfer of vorticity due to axial straining of the vortex (because w = αz).

3.4 Viscous models of vortices

153

3.4.2 Sullivan vortex Sullivan (1959) obtained an exact solution to the Navier-Stokes equation in the form of a steady two-cell vortex. This solution was discovered in the form of (3.79) and written as

⎛ αr 2 ⎞ ⎤ α r 6ν ⎡ + ⎢1 − exp ⎜ − ⎟⎥ , 2 r ⎣ ⎝ 4ν ⎠ ⎦ ⎤ Γ ⎡ ⎛ αr 2 ⎞ v= ⎢H ⎜ ⎟ H (∞ ) ⎥ , 2πr ⎣ ⎝ 4ν ⎠ ⎦ 2 ⎡ ⎛ αr ⎞ ⎤ w = αz ⎢1 − 3exp ⎜ − ⎟⎥ , ⎝ 4ν ⎠ ⎦ ⎣

u=−

α 2r 2 ν2 1 ⎧⎪ + 36 2 p = p0 − ρ ⎨α 2 z 2 + 2 ⎪ 4 r ⎩

⎡ ⎛ αr 2 ⎞ ⎤ − 1 exp ⎢ ⎜− ⎟⎥ ⎝ 4ν ⎠ ⎦ ⎣

(3.91) 2

r 2 ⎫⎪ v + ρ ⎬ ∫ dr , r 0 ⎪⎭

x t ⎪⎧ ⎪⎫ H ( x ) = ∫ exp ⎨−t + 3∫ [1 − exp(− s )]s −1ds ⎬ dt. 0 0 ⎩⎪ ⎭⎪

Here Γ is the circulation at the infinity; α is the constant; p0 is the pressure at point r = 0; z = 0; ν is the kinematic viscosity. As an example, various velocity profiles are illustrated in Fig. 3.20. When r → ∞ the Sullivan vortex exactly coincides with the Burgers vortex. Bellamy-Knights (1970) generalized the Sullivan solution for a timedependent situation. As a result he obtained a solution, describing the timedependent breakdown of a two-cell vortex. Earlier, Rott (1958) studied an unsteady one-cell vortex solution.

Fig. 3.20. Velocity distribution in the Sullivan vortex. α = 0.001, ν = 0.01, z = 1

4 Stability and waves on columnar vortices

4.1 Types of perturbations Numerous observations confirm that many kinds of swirling flows (either bounded or free) exhibit instability. This instability leads to the formation of secondary vortical motions, linear or nonlinear waves, and may cause the vortex breakdown. However, different kinds of perturbations (for example, neutral (inertial) waves) can be observed even for stable flows. This Chapter focuses on columnar vortices only. The key aim is to identify the criteria for vortex instability and describe the waves on vortices. The characteristic feature of these disturbances is a 3-D structure, which usually exhibit as spiral or helical forms (Fig. 4.1). Before we begin the theoretical analysis of the instability problem, let us consider the main types of axisymmetrical and non-axisymmetrical perturbations of a columnar vortex (for other types of disturbances, like those on vortex rings, see the review by Kop’ev and Chernyshev (2000)). For clarity, we will consider a vortex with a distinguished core of radius r = R that has a discontinuity in the velocity profile (or velocity derivative) on its boundary. For the most general presentation, the core boundary affected by a linear monochromatic disturbance is described by the formula r = R + aRe ei(kz + mϕ − ωt), where amplitude a

R;

(4.1)

Re is the real part; k is the axial wave number;

m is an integer (the azimuthal wave number); ϕ is the azimuthal angle (Fig. 4.2); ω is the frequency. For analysis of disturbed core shape, we may consider the case when k is real and t = 0. Then we obtain r = R + a cos (kz + mϕ).

(4.2)

The main types of perturbations given by Eq. (4.2) are shown in Fig. 4.2. The case of m = 0 corresponds to the axisymmetrical mode. The wavelength is 2π/k. The cross-sections z = const are concentric circles with a radius from (R − a) to (R + a).

156

4 Stability and waves on columnar vortices a

b

Fig. 4.1. Spiral perturbations in a swirling flow: a – spiral breakdown of a vortex after a diaphragm in a vortex chamber (Alekseenko and Shtork 1992*); b – spiral disturbance of a flow axis in a container with a rotating bottom (Hourigan et al. 1995*)

Fig. 4.2. Types of perturbations for a core of columnar vortex: 1 – core boundary; 2 – lines of fixed phase kz + mϕ = const

4.2 Intsability of a vortex sheet

157

For m ≠ 0 we obtain non-axisymmetrical modes. The modes with m = ±1 are usually called bending modes. The core cross-section z = const is a circle of radius R, shifted by distance a along the radius r at the angle ϕ = −kz/m. The mode m = +1 takes the form of a left-handed helix, and m = −1 of a right-handed one. The helix pitch equals 2π/k, the same as the wavelength for an axisymmetrical disturbance with m = 0. Fig. 4.2 shows the isolines {r = const, kz + mϕ = const} in the form of helical lines with the pitch of 2π/k|m|. We also plotted the isolines of constant velocity – isotaches with a more complex shape. For |m| = 2 the circular shape of the core cross-section transforms into an ellipse. For all the cases with |m| ≥ 2, the vortex axis (due to symmetry in disturbance) remains undisturbed. If the amplitude of disturbance is not small, we cannot use the simple canonic form. For the limiting case of an infinitely thin vortex filament, the disturbed state can be described by one of the canonic curves – a helical line r = const,

kz + mϕ = const,

as described in detail in Section 2.1.

4.2 Intsability of a vortex sheet It follows from Eq. (3.6) that a tangential discontinuity produces a vortex sheet. Several basic vortex models have a tangential discontinuity. Before we begin to study the problem of columnar vortex stability, we have to pay attention to a vortex sheet; the stability of this object can be of special interest (Batchelor 1967; Saffman 1992). Helmholtz was the first to demonstrate vortex sheet instability in response to small disturbances (Helmholtz 1868). We will consider the simplest case of a plain vortex layer formed by two uniform fluid flows of the same density moving in opposite directions with the velocity of 0.5U (Fig. 4.3). In the general case of 3-D disturbances, the deviation of this vortex layer from its undisturbed state is y = η(x, z, t). Since vorticity is concentrated only in a vortex sheet, the flow in the rest of the space remains potential and it is described by the Laplace equation with potentials

158

4 Stability and waves on columnar vortices

Fig. 4.3. Diagram for vortex sheet disturbance. The vorticity vector is directed normally towards the reader

1 − Ux + ϕ1 , y > 0, 2 1 Ux + ϕ2 , y < 0 , 2 where ϕ1, ϕ2 are disturbances of the potential. For a vortex sheet, which is the interface between two regions of fluid and represents a material surface, we can notate the kinematic conditions and pressure continuity (if there is no surface tension). The kinematic condition from the side of area 1 is written as

∂ϕ1 ∂y =

= y =η

dη ∂η ∂η dx ∂η dz = + + = dt ∂t ∂x dt ∂z dt

∂ϕ ⎞ ∂η ∂ϕ1 ∂η ∂η ⎛ 1 . + − U+ 1⎟+ ⎜ ∂t ∂x ⎝ 2 ∂x ⎠ ∂z ∂z

Assuming a small disturbance, we obtain in the first approximation

∂ϕ1 ∂y

=

∂η 1 ∂η − U . ∂t 2 ∂x

(4.3)

=

∂η 1 ∂η + U . ∂t 2 ∂x

(4.4)

y =0

Similar, for area 2

∂ϕ2 ∂y

y =0

The dynamic condition is the equality of pressure p1

y =η

= p2

y =η .

(4.5)

4.2 Intsability of a vortex sheet

159

The formula for pressure is obtained from the Cauchy – Lagrange equation (1.24) ⎛ ∂ϕ 1 2 ⎞ + u ⎟ = const , p + ρ⎜ ⎝ ∂t 2 ⎠ where u is the velocity magnitude. We substitute this into (4.5) and keep the terms of the first infinitesimal order, and then obtain ∂ϕ ⎞ 1 ⎛ ∂ϕ ⎛ ∂ϕ2 ∂ϕ1 ⎞ + U⎜ 1 + 2 ⎟ = const . ⎜ ∂t − ∂t ⎟ ∂x ⎠ y =0 ⎝ ⎠ y =0 2 ⎝ ∂x

(4.6)

To determine the stability of a vortex sheet, we present variables η, ϕ1, ϕ2 in the form

⎡η ⎤ ⎡a ⎤ ⎢ϕ ⎥ = ⎢ Ψ ( y) ⎥ ei(kx x +kz z −ωt ), ⎢ 1⎥ ⎢ 1 ⎥ ⎢⎣ϕ2 ⎥⎦ ⎢⎣ Ψ 2 ( y) ⎥⎦

(4.7)

where kx, kz are the components of the wave vector; k = kx2 + kz2 is the wave vector module; ω is a complex frequency; a is a constant. We substitute those expressions into the Laplace equation for potential and obtain Ψ1′′ − k2 Ψ = 0, Ψ ′′2 − k2 Ψ = 0.

Then, assuming a limited solution at infinity, the potential is

Ψ1 = b1e−ky, Ψ2 = b2e−ky, b1, b2 = const.

(4.8)

Substituting (4.7), (4.8) into the conjugation conditions (4.3), (4.4), and into Eq. (4.6) yields the dispersion equations

1 −b1k = −aωi − Uiakx , 2 1 b2 k = −aωi + Uiakx , 2

1 −ib2 ω + ib1ω + U ( b2 ikx + b1ikx ) = const = 0. 2 The dispersion relationship follows from these equations 1 ω = ±i Ukx = iβ 2

160

4 Stability and waves on columnar vortices

or

1 β = ± Ukx , 2

(4.9)

where β is an increment. One can see that the real part of ω, i.e., the real frequency, is equal to zero. This means that the disturbance is not a running wave. Existence of a positive root for the increment testifies that the vortex sheet is unstable for any periodic disturbance with kx ≠ 0. And the smaller the wavelength, the higher the growth rate for disturbances. A similar result was obtained on the basis of the Birkhoff–Rott equations by Saffman (1992). The above considered type of instability is classified as a Kelvin – Helmholtz instability.

4.3 Waves in fluids with solid-body rotation As shown in the previous chapters, there exists a zone of solid-body rotation near the axis of a columnar vortex. Hence, it is desirable to consider this special case, all the more so since it is related to the motion and stability of a fluid in a rotating container (Greenspan 1968). At first, let us consider the propagation of plane waves and then the case of axisymmetrical waves. Further, we will describe the phenomenon of the Taylor column. Kelvin (1880) was the first to analyze the wavy flow in a liquid with solid-body rotation (for a space between two disturbed cylindrical surfaces). In this Section, except for one particular case, we are going to consider infinite space. 4.3.1 Plane waves The pioneering studies of plane waves were carried out by Long (1951), Fultz (1959), Chandrasekhar (1961), S. Nigam and P. Nigam (1962). Consider the following problem statement. Let the incompressible fluid rotate with the angular velocity Ω around the z axis. Besides, the fluid translates along axis z with a constant velocity W (Whitham 1974). It is important to take into account W, since the existence of axial flow is typical for columnar vortices. Let us choose the coordinate system rotating with the fluid. Then we have to introduce additional forces – centrifugal and Coriolis forces. The centrifugal force 1/2∇[ × r]2 brings a radial change in the

4.3 Waves in fluids with solid-body rotation

161

static pressure, and this contribution can be united in the Euler equation with the static pressure by introducing an effective pressure

p−

ρ [ 2

× r ] − p0 → p. 2

Here, for the effective pressure we keep the symbol p. After subtraction of the hydrostatic pressure p0, the value p represents the pressure disturbance. The Coriolis force is equal to 2[u × ], and this force is responsible for wave spreading in a fluid with solid-body rotation. Thus, the Euler equation (1.10) takes the following form in the rotating coordinate system: ∂u + (u∇)u + 2 [ ∂t

1 × u ] = − ∇p . ρ

(4.10)

The mass conservation equation remains the same

∇u = 0.

(4.11)

In the case of small disturbances the linearized Euler equation is as follows: du + 2[ dt

where

1 × u ] = − ∇p , ρ

(4.12)

d ∂ ∂ = +W , ∂z dt ∂t

and the value of disturbed velocity u retains the same symbol. To deduce the dispersion equations, we have to formulate a relationship for pressure disturbance. Let us rewrite Eq. (4.12) through the velocity components

du 1 ∂p ⎫ − 2Ωv = − , dt ρ ∂x ⎪⎪ dv 1 ∂p ⎪ ,⎬ + 2Ωu = − dt ρ ∂y ⎪ ⎪ dw 1 ∂p . =− ⎪ dt ρ ∂z ⎪⎭

(4.13)

By differentiation of these equations with respect to corresponding variables x, y, z and composing a sum in view of (4.11), we obtain ⎛ ∂v ∂u ⎞ 1 2 ∇ p = 2Ω ⎜ − ⎟. ρ ⎝ ∂x ∂y ⎠

162

4 Stability and waves on columnar vortices

Taking the total time derivative and considering (4.13), we obtain ∂w 1 d 2 . ∇ p = 4Ω 2 ρ dt ∂z

Then we differentiate this equation, again with respect to t, and obtain the ultimate equation for p 2

2 ∂ ⎞ 2 ⎛∂ 2 ∂ p = 0. ⎜ + W ⎟ ∇ p + 4Ω ∂z ⎠ ⎝ ∂t ∂z 2

(4.14)

For the particular case of W = 0, Eq. (4.14) for periodic disturbances p = Φe–iωt takes the form

∂2 p ∂x 2

+

⎛ 4Ω 2 + − 1 ⎜ ∂y 2 ⎜⎝ ω2 ∂2 p

⎞ ∂2 p ⎟⎟ 2 = 0 . ⎠ ∂z

(4.15)

For ω > 2Ω the coefficient at ∂2p/∂z2 is positive. This means that this equation is elliptical and can be reduced to the Laplace equation. In this case the influence of a point source disturbance spreads through the entire volume. If ω < 2Ω, the equation becomes hyperbolic and disturbance is propagated in a limited space − within a cone of which the axis coincides with the z axis, and the half-angle equals arcsin(ω/2Ω). Now we return to Eq. (4.14) and, taking p ∼ ei(kr − ωt), we obtain the following dispersion relationship

( ω − Wkz )2 k2 − 4Ω2 kz2 = 0.

(4.16)

Here ω and k are the real frequency and the wave number; kz is a zcomponent of the wave vector. Since k2 ≡ kx2 + ky2 + kz2 > kz2 , it follows from (4.16) that the waves exist already under condition (ω − Wkz)2 < 4Ω2. For W = 0 this is in agreement with the condition for hyperbolicity of Eq. (4.15). The solution of the dispersion equation has two roots

ω = Wkz ±

2Ωkz , k

(4.17)

corresponding to two wave modes. Existence of a real root only means that the image part of the frequency (increment) equals zero, i.e., the waves in a solid-body-rotating liquid are neutrally stable. They are also called inertial waves.

4.3 Waves in fluids with solid-body rotation

163

The phase velocity of the wave is equal to c≡

ω kz = k k

2Ω ⎞ 2Ω ⎞ ⎛ ⎛ ⎜W ± ⎟ = cos θ ⎜ W ± ⎟, k ⎠ k ⎠ ⎝ ⎝

where θ is the angle between the vector k and axis z. One can see that at W ≠ 0 the dispersion takes place even for a fixed value of wave number k. The group velocity vector is defined as cg = ∂ω/∂k, and its components are the following

cgx = ∓ cgy = ∓

2Ωkx kz k3 2Ωky kz k3

cgz = W ± 2Ω

, ,

kx 2 + ky 2 k3

.

The corresponding module of group velocity is equal to

cg =

2 2 2Ω ⎛ Wk ⎞ W k sin 2 θ ⎜1 ± . + ⎟ k Ω ⎠ ⎝ 4Ω 2

If W = 0, c = 2 Ω cos θ/k, cg = 2 Ω sin θ/k, cg ⋅ k = 0. The latter relationship means that the wave vector and the group velocity vector are mutually perpendicular (Fig. 4.4). For a fluid with translational motion (W ≠ 0), the angle γ between the wave vector and the group velocity vector is not equal to 90°; it depends on the wave propagation direction θ and parameter Ro = Wk/2Ω, which can be interpreted as the Rossby wave number. As usual, at Ro 1 the dominating force is the Coriolis force, and at Ro 1 the inertia forces become dominating. The dependencies for γ vs. θ at different values of Ro are shown in Fig. 4.5, and diagrams for wave propagation are depicted in Fig. 4.6. For Ro = 0 cg ⊥ k (see Fig. 4.4), and at Ro → ∞ the first and last modes coincide, and also cg becomes parallel to axis z, and |cg| = W, meaning the drift of the disturbance by translational flow. Consider the velocity field. First we exclude the pressure from Eq. (4.12), and apply the operation rot. The result is

d ∂u ( ∇ × u ) = 2Ω . dt ∂z

(4.18)

164

4 Stability and waves on columnar vortices

Fig. 4.4. Diagram of plain inertial wave propagation in a fluid with solid-body rotation at W = 0

Fig. 4.5. Dependency of angle γ between the wave vector and the group velocity vector on the direction of wave propagation and parameter Ro = Wk/2Ω: I, II are the first and second modes

Then we present the velocity disturbance in the form of plain wave u ∼ Nei(kr − ωt).

(4.19)

Then from (4.18) we have the equation (ω − Wkz)[k × u] = 2iΩkzu,

(4.20)

and from continuity equation (4.11) we obtain the condition of perpendicularity k ⋅ N = 0 or k ⊥ u. Taking into account the dispersion relationship (4.17), Eq. (4.20) takes the form

Fig. 4.6. Diagram of wave propagation as a function of angle θ at Ro = 2

4.3 Waves in fluids with solid-body rotation

165

[e × u] = ±iu, where the unit vector is e = k/k. In general, the complex amplitude of the wave is presented as N = A + iB. It follows from the last equality that [e × B] = ± A, i.e., vectors A and B are mutually perpendicular, they are equal by module and lie in the plane normal to vector k. Let us introduce a local coordinate system {X, Y, Z}, in which axis Z coincides with vector k, and vectors A and B lie on axes X and Y. By taking the real part from (4.19), we obtain the velocity components

ux = A cos (ωt − kr), uy = A sin (ωt − kr). One can see that for every fixed spatial point the vector of velocity perturbation u rotates in time with constant amplitude, i.e., the wave has a circular polarization. 4.3.2 Axisymmetrical waves Analysis of axisymmetrical waves in a fluid with solid-body rotation is best performed in a cylindrical frame of reference {r, θ, z} with axis z along the vector Ω. Let us represent the velocity components {u, v, w} as

u ∼ f(r)ei (kz − ωt)

etc.

(4.21)

Making the transition to cylindrical coordinates, we obtain from Eqs. (4.11), (4.12) taking into consideration the axial symmetry 1 ∂p ⎫ , ρ ∂r ⎪ ⎪ −iωv + 2Ωu = 0, ⎪ ⎪ ⎬ ik −iωw = − p, ⎪ ρ ⎪ ⎪ 1 ∂ ( ru ) + ikw = 0. ⎪ r ∂r ⎭ −iωu − 2Ωv = −

(4.22)

The equation for mass conservation allows us to introduce the stream function Ψ through relationships u=−

1 ∂Ψ 1 ∂Ψ , w= . r ∂z r ∂r

(4.23)

166

4 Stability and waves on columnar vortices

By excluding v, w, p from (4.22), it is easy to obtain the equation for u. For variables defined in (4.21) it takes the form

r2

d2 f dr 2

+r

⎞ ⎤ df ⎡ 2 2 ⎛ 4Ω 2 + ⎢r k ⎜ 2 − 1⎟ − 1⎥ f = 0. ⎜ ⎟ dr ⎢⎣ ⎝ ω ⎠ ⎥⎦

The solution of this equation (it becomes zero at the flow rotation axis r = 0) is as follows:

⎛ ⎞ 4Ω 2 ⎟, f = f0 J1 ⎜ kr − 1 2 ⎜ ⎟ ω ⎝ ⎠ where J1 is Bessel function of the first kind of order 1; f0 is a constant. Note that the solution exists only for the condition ω < 2Ω. Since the Bessel function J1(x) has an infinite series of zeros x1, x2, …, the flow is split into isolated areas, limited with co-axial cylinders of radius ri according to the equality

kri

4Ω 2 ω2

− 1 = xi .

(4.24)

To determine the flow pattern, we have to find the formula for stream function using (4.21) and (4.23), Ψ=−

⎛ ⎞ f0 4Ω 2 ⎟ sin ( kz − ωt ) . 1 rJ1 ⎜ kr − 2 ⎜ ⎟ k ω ⎝ ⎠

(4.25)

The examples of instant streamlines calculated by (4.25) are plotted in Fig. 4.7. One can see how the flow splits into isolated cells, with the size along axis z equal to π/k. The perpendicular plane flow picture is described by Eq. (4.24). This pattern corresponds to a harmonic wave running along axis z with the phase velocity c = ω/k. In the coordinate system moving along z with velocity c we observe a standing wave. In contrast to the plane waves, the peculiarity of this solution is that the wave frequency ω is independent of the wave number k, i.e., the axisymmetrical waves in infinite fluid with solid-body rotation are dispersionless. However, dispersion arises immediately when the flow has restricting walls. Consider the flow in a cylindrical tube with radius R. The requirement is that a whole number of cells fit the radius R, i.e., we impose condition R = xi, where xi is the i-th zero of function J1(x). As a result, we obtain from (4.24)

4.3 Waves in fluids with solid-body rotation

167

Fig. 4.7. Instant picture of streamlines for propagation of axisymmetrical waves in a fluid with solid-body rotation (calculation by (4.25) for f0/k = 1.06)

4Ω 2

(4.26) − 1 = xi , ω2 and the relationship between ω and k is established for given values of R, Ω and xi. Correspondingly, we get the expression for the group velocity

kR

cg ≡

xi2 dω ω = , dk k xi2 + k2 R2

which is obviously smaller than the phase velocity c = ω/k. If the flow is restricted with both cylindrical and plane boundaries normal to the rotation axis, a traveling wave does not exist. In this case the solution must be in the form of a standing wave with condition L = nπ/k, where L is the cylinder length; n is the integer. Substituting k from this condition into (4.26), we obtain the formula for eigen frequency of small oscillations of rotating fluid in a cylindrical vessel with radius R and length L

⎛ x 2 L2 ⎞ ω = 2Ω ⎜ 1 + 2 i 2 2 ⎟ ⎜ n π R ⎟⎠ ⎝

−1/ 2

.

4.3.3 Taylor column In the experiment performed by Taylor (1923), it was discovered that the slow pushing of a ball along the axis of solid-body rotation causes a mo-

168

4 Stability and waves on columnar vortices

tion of the entire fluid column where this ball is inscribed (see photo in Fig. 4.8). This phenomenon can be interpreted using the language of wavy phenomena. But firstly we have to analyze the outcome of the motion equations in a rotating coordinate system for slow steady motion. The slowness means that inertia effects are small, i.e., the Rossby number Ro = W/aΩ 1. Here W is the characteristic velocity; a is a characteristic length scale. Then we obtain from Eq. (4.10) 2Ωv =

1 ∂p 1 ∂p ∂p = 0. , 2Ωu = − , ρ ∂x ρ ∂y ∂z

(4.27)

Here we use the Cartesian coordinate system with axis z along the rotation axis. We see that pressure p and velocity components u, v are independent of coordinate z. We exclude pressure and obtain ∂u ∂v + = 0. ∂x ∂y

(4.28)

This equality and the mass conservation equation (4.11) give us

∂w = 0. ∂z

(4.29)

hhhh

Fig. 4.8. Visualization of Taylor column for the floating of a silicon droplet (radius of 2 cm) in a container with large diameter; the container rotates at the rate of 56 r.p.m (Bush et al. 1995*)

Fig. 4.9. Diagram of flow over a disc in a solid-body rotating fluid (Taylor column)

4.3 Waves in fluids with solid-body rotation

169

Equations (4.28) and (4.29) mean that any slow motion in a fast-rotating fluid (as a whole) is a superposition of two independent motions – a twodimensional flow in the plane normal to axis z and axial flow (independent of coordinate z). This statement is the content of the Proudman theorem (Proudman 1916). Equation (4.29) yields another surprising result. Consider a disc of radius a placed perpendicular to the axis of fluid rotation. The incident flow is uniform with axial velocity W. Since the normal velocity on the disc surface is w = 0, it remains zero throughout the cylinder with the radius of a according to (4.29) (Fig. 4.9). Beyond this cylinder we have w = W. The flow pattern offers the explanation to the phenomenon of “Taylor’s column”. Naturally, at the surface of discontinuity of axial velocity, the approximate equations (4.27) are not applicable. From the wave theory standpoint, existence of the Taylor column is explained by the fact that disturbance from a solid body propagates along axis z, i.e., the group velocity has only z -components (cgz). Then we use formulae from Section 4.1 and find the component of group velocity

cgx = cgy = 0, kz = 0, cgz = W ± 2Ω/k. Besides, one of the roots for the group velocity has to be less than zero for the perturbation to develop upstream; this means that

cgz = W − 2Ω/k < 0. If we assume that the main contribution is made by the wave with length λ ≡ 2π/k ∼ a, where a is the typical size of the body (a radius if it is a sphere), this gives us the final condition for existence of the Taylor column (written through the Rossby number): Ro ≡

W 1 < ≈ 0,3. aΩ π

Firstly, this criterion is in agreement with the condition of slow motion (Ro 1); secondly, this is very close to the experimental conditions under which Taylor (1923) observed the development of this column.

170

4 Stability and waves on columnar vortices

4.4 Linear instability of Rankine vortex with an axial flow 4.4.1 Dispersion relationships As was demonstrated in the previous Section, a columnar vortex with linear distribution of azimuthal velocity is stable to any small perturbations; only neutral (inertial) waves can propagate in this kind of vortex. The model of the Rankine vortex is a combination of solid-body rotation and potential flow outside the core; in many cases this is a good approximation for real columnar vortices and vortex filaments. However, it was Kelvin (1880) who demonstrated that the Rankine vortex is also stable for small disturbances (both for axisymmetrical and spiral modes). The instability develops if the axial flow in the core is imposed upon the Rankine vortex (Chandrasekhar 1961; Krishnamoorthy 1966). In the process of stability analysis, a rectangular profile of axial velocity is usually employed (i.e., a constant for the core and zero outside the core). Put differently, this is a swirled jet surrounded by a potential flow. The detailed study of temporal instability was performed by several authors (Moore and Saffman 1972; Uberoi et al. 1972; Lessen et al. 1973; Drazin and Reid 1981; Saffman 1992). The spatial instability was studied by Wu et al. (1992); spatial-temporary instability with focus on the vortex breakdown problem was analyzed in a paper by Loiseleux et al. (1998). We will follow these two publications in our linear analysis of the Rankine vortex stability with an axial flow. We consider the flow of inviscid incompressible fluid with density ρ, which is described by the Euler equation (see (1.35) at ν = 0) in the cylindrical frame of reference {r, θ, z} with corresponding velocity components {u, v, w}. The undisturbed velocity profiles are written as U = 0, V ( r ) = Ωr , W ( r ) = W∞ + ∆W , r < R, U = 0, V ( r ) = ΩR2 / r , W ( r ) = W∞ , r > R,

(4.30)

where R is the core radius; Ω is the angular velocity of rotation in the core; W∞ is the external axial velocity; ∆W is the jump of axial velocity on the surface r = R, which is a vortex sheet. Taking into account the axial velocity is important for analysis of spatial instability and also for finding the condition of transition between absolute and convective instabilities, which was linked to the situation of vortex breakdown by Loiseleux et al. (1998). From Eq. (3.56) we obtain the pressure distribution

p = p∞ − ρΩ2(R2 − r2/2), r < R, p = p∞ − ρΩ2R4r−2/2, r > R,

4.4 Linear instability of Rankine vortex with an axial flow

171

where p∞ is the pressure at r → ∞. This problem statement affords us the introduction of two dimensionless parameters: the dimensionless velocity a and the swirl parameter S a=

ΩR W∞ , S= . ∆W ∆W

Note that the swirl parameter is related to the Rossby number (Ro), through a simple ratio: S = 1/Ro. The typical profiles of axial velocity as a function of a, and the profile for tangential velocity are plotted in Fig. 4.10. By varying parameter a, we can obtain different combinations in the form of a jet or a wake in the core with external flow. When we denote the perturbations for velocity and pressure as u, v, w, p and linearize the Euler equations and mass conservation law, we obtain the following equations

∂u V ∂u ∂u 2Vv 1 ∂p + +W − =− , ∂t r ∂θ ∂z ρ ∂r r ∂v ∂V V ∂v ∂v uV 1 ∂p +u + +W + =− , r ∂t ∂r r ∂θ ∂z ρr ∂θ ∂w V ∂w ∂w 1 ∂p + +W =− , ∂t r ∂θ ∂z ρ ∂z ∂u ∂w ∂v +r + = 0. u+r ∂r ∂z ∂θ

(4.31)

Aiming for the derivation of dispersion relationships, we can present perturbations in the form

{u, v, w, p} = {uˆ ( r ) , vˆ ( r ) , wˆ ( r ) , pˆ ( r )} ei( kz+mθ−ωt ) ,

(4.32)

where the wave number k and the frequency ω can be complex or real depending on the problem statement; m is the integer azimuthal wave number. Since the velocity distributions are different, we have to consider separately the areas r < R and r > R, and also we have to write the matching condition on the interface of the axial velocity discontinuity, and this must yield the dispersion equation. Vortex core, r < R

We obtain from (4.31), (4.30), and (4.32)

172

4 Stability and waves on columnar vortices

pˆ ′ , ρ im ˆ − 2uˆ Ω = ivg pˆ , ρr kpˆ ˆ = , wg ρ uˆ imvˆ ˆ+ = 0. uˆ ′ + + ikw r r

(4.33)

g = ω – mΩ – (W∞ + ∆W)k,

(4.34)

ˆ + 2Ωvˆ = iug

Here

and prime means the derivative with respect to r. The physical meaning of (4.34) is the frequency with a correction for Doppler effect in the frame of reference moving with the velocity of undisturbed flow. After eliminating ˆ , we have the ordinary differential equation from (4.33) variables uˆ , vˆ , w for the eigenfunction

(

)

r 2 pˆ ′′ + rpˆ ′ + β2r 2 − m 2 pˆ = 0 ,

where

⎛ 4Ω 2 ⎞ β2 = k 2 ⎜ 2 − 1⎟ . ⎜ g ⎟ ⎝ ⎠

Fig. 4.10. Profiles of tangential (a) and axial (b–e) components of velocity as functions of parameter a: (b): a ≥ 0, jet with co-current flow; (c): –0.5 ≤ a ≤ 0, jet with counterflow; (d): –1 ≤ a ≤ −0.5, wake with counterflow; (e): a ≤ –1, wake with co-current flow

4.4 Linear instability of Rankine vortex with an axial flow

173

The general solution for this equation is a linear combination of the Bessel function of the first kind Jm(βr) and second kind Ym(βr) of the order |m|. Due to solution finiteness at r = 0, we can retain only the first term, i.e.

pˆ ( r ) = AJm ( βr ) , A = const.

(4.35)

We also require a formula for uˆ , which can be obtained from (4.33) and (4.35)

uˆ ( r ) =

2ΩmJm ( βr ) ⎤ iAk2 ⎡ ′ (βr ) − J β ⎢ ⎥, m rg β2 ρg ⎣ ⎦

(4.36)

where the prime is the derivative with respect to the argument. The sign of mode m is taken into account through the ratio J–m = (–1)mJm. External area, r > R

Since the flow is potential, it is convenient to use the potential ϕ defined from condition u = ∇ϕ. Assuming ϕ = ϕˆ ( r ) e (

i kz + mθ−ωt )

and substituting this expression into the Laplace equation ∆ϕ = 0, one obtains the equation for ϕˆ ( r )

(

)

r 2 ϕˆ ′′ + r ϕˆ ′ − m 2 + k2r 2 ϕˆ = 0 . The general solution for this equation is expressed through the modified Bessel functions of the first Im(ekr) and second Km(ekr) kind of order |m|, where e = sign (kr). Again, the requirement for a finite solution allows us to acquire only one term

ϕˆ ( r ) = BKm ( ekr ) , B = const, and the wave number enters into the argument with the sign of e, for the solution must be finite at infinity. Correspondingly, for uˆ and pˆ we obtain

′ ( ekr ) , uˆ ( r ) = BekKm ⎡ ⎤ Ω R2 m + W∞k⎥ . pˆ ( r ) = −iρBKm ( ekr ) ⎢ −ω + 2 r ⎣ ⎦

(4.37)

Due to the property K–m = Km Eqs. (4.37) are the same either for positive or negative modes m.

174

4 Stability and waves on columnar vortices

Vortex sheet, r = R

The kinematic and dynamic conditions should be given for this surface. The kinematic condition means that the normal component of velocity u at the disturbed boundary

η = R + αei(kz + mθ − ωt) is equal to u≡

dη ∂η ∂η v ∂η . = +w + dt ∂t ∂z r ∂θ

Substituting the formulae for v and w (see the analysis above) we obtain

uˆ− = −iαg, uˆ+ = −iα ( g + ∆W k ) , where the subscripts “–” and “+” refer to the values of uˆ ( R) on the left and right from r = R. The elimination of amplitude α gives us the kinematic condition

uˆ− uˆ+ . = g g + ∆W k The dynamic condition means the equality of normal stresses, that means

pˆ − = pˆ + . Using Eqs. (4.35) – (4.37), one obtains a set of two algebraic equations A

2ΩmJm ⎤ ekKm′ ik2 ⎡ ′ β − = , J B m ⎢ ⎥ Rg ⎦ g + ∆Wk β2ρg2 ⎣

AJm = BiρKm ( ω − Ωm − W∞ k ) . The condition of solvability of this system with respect to A, B is the zero determinant, so the following dispersion equation is true:

⎡ βRJm′ ( β R )

( g + ∆W k )2 ⎢

⎣⎢ Jm ( βR )

−

g2β2 R2 Km′ ( ekR ) 2Ωm ⎤ . = − ⎥ g ⎦⎥ ekRKm ( ekR )

(4.38)

For convenience, let us rewrite it in a dimensionless form using the parameters R and ∆W as the characteristic scales

4.4 Linear instability of Rankine vortex with an axial flow

⎡ Jm′ ( β )

( g + k )2 ⎢β

⎣⎢ Jm ( β )

−

2Sm ⎤ g2 β2 Km′ ( ek ) = 0, ⎥+ g ⎦⎥ ek Km ( ek )

g = ω − mS − (1 + a)k,

where

β = k 4S 2 / g2 − 1.

175

(4.39) (4.40) (4.41)

Note that in the general case the arguments of the Bessel function are complex. Hollow vortex

In several publications (e.g., Uberoi et al. (1972)), a more general case is considered, when the medium densities inside the core and in the external region are different. The most interesting case is the so-called “hollow vortex”. It has zero density in the core and rotation of fluid (with density ρ) by the law of potential flow in the external region with zero axial velocity. This kind of flow may occur after the formation of a cavitation zone. For the first time the waves on a hollow vortex were studied theoretically by Kelvin (1880), Pocklington (1895) (hollow vortex ring with a small crosssection). The results of the last paper were later used by Maxworthy et al. (1985) for the analysis of experimental data. To obtain the dispersion equation for a hollow vortex, we must have the solutions for eigenfunctions uˆ ( r ) and pˆ ( r ) in the area of potential flow (r > R) and for condition on the interface (r = R). Let W∞ = 0. Expressions (4.37) remain valid for the region r > R. For the undisturbed interface

r = η = R + αei(kz + mθ − ωt) we have the unchanging kinematic condition for uˆ+

uˆ + = −iα ( g + ∆Wk ) = −iα ( ω − mΩ ) .

(4.42)

However, the dynamic condition takes a slightly different form because we start from the equality of total normal stresses (not perturbations) p0 = p∞ − ρ

Ω 2 R4 2r

2

ˆ i(kz +mθ−ωt ) , r = η. + pe

176

4 Stability and waves on columnar vortices

Here p0 = const is the pressure in the hollow core; p∞ is the pressure of fluid at r → ∞; the second right-side term is caused by centrifugal forces, and the third term is the pressure perturbation. For the undisturbed state, p0 = p∞ – ρΩ2R2/2. Linearization of the condition brings the equality

pˆ + = −ρΩ 2 Rα.

(4.43)

Substituting into (4.42), (4.43) the Eqs. (4.37) at r = η ≈ R and excluding the amplitudes α, B, we obtain the dispersion relationship for a hollow vortex

K′ ( ekR ) ω . = m ± −ekR m Km ( ekR ) Ω

(4.44)

In analysis of temporal instability, we assume the wave number k to be real. Since the ratio under the square root is always positive, the roots for ω are always real. Hence, the flow in a hollow vortex is stable relative to small perturbations, either axisymmetrical or spiral modes. In the extreme cases of small and high wave numbers, we obtain from (4.44) the following formulae:

k R→0:

ω = ±Ω ( − ln k R ) ω = Ω ⎡m ± ⎣

k R >> 1:

−1/ 2

→ 0,

m ⎤, ⎦

m = 0, m≠0,

ω1,2 = 2; 0,

m = 1,

ω1,2 = 0; −2,

m = −1 ,

ω = Ω ⎡m ± ⎣

k R⎤ , ⎦

ω = ±Ω k R ,

m = 0.

The negative sign for the frequency means a change of sign ahead of time t in exponent exp[i(kz + mθ – ωt)]. 4.4.2 Linear analysis of temporal instability In the analysis of temporal instability we use k = kr, = ωr + i ωi. There is no loss of generality in taking the velocity of external flow as equal to zero: a ≡ W∞/∆W = 0. Indeed, a appears only in the parameter g (see

4.4 Linear instability of Rankine vortex with an axial flow

177

(4.40)). For the real wave number the recalculation for the case a ≠ 0 is available by simple substitution: ωr → ωr − ak. The dispersion equation (4.39) can be solved only by numerical methods, so firstly we have to obtain asymptotic results at the limits of small and large (by module) values of Bessel function arguments. Short waves, k

Assume |ω|

1; |β|

1

1. Then |g| ≈ |ω – k|

1, β ≈ ik, Jm′ ( ik ) / Jm ( ik ) ≈ i,

Km′ ( k ) / Km ( k ) ≈ −1. Taking these relations, one obtains from (4.39) the dispersion formula = k/2 ± i k/2

(4.45)

and in dimensionless form = k∆W/2 ± i k∆W/2. Existence of a positive imaginary root for the complex frequency means flow instability. Note that the imaginary part is exactly the same as expression (4.9) for the increment in the problem of vortex sheet, i.e., this is an obvious case of Kelvin – Helmholtz instability, caused by a tangential jump in the axial velocity ∆W. There is no effect of the swirl parameter S. It follows from (4.45) that admission |ω| 1 is valid. Short waves, k

1; |β|

k

In this case the simplified connection between ω and k is derived directly from the definition for β. Those conditions are satisfied if the expression under the square root in (4.41) vanishes. With account for (4.40), this gives us the desired relationship

ω = (m ± 2)S + k

(4.46)

and in a dimensionless form

ω = ∆Wk + (m ± 2)Ω. One can see that these waves are neutrally stable. For m = 0 the latter formula coincides with Eq. (4.17) describing the inertial waves in infinite fluid with solid-body rotation.

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4 Stability and waves on columnar vortices

Long waves, k

1

Here we have to distinguish three cases: |β| 1, |β| ≈ 1, |β| 1. The asymptotic formulae derived by Loiseleux et al. (1998) are summarized in Table 4.1. 1) (Loiseleux et al. 1998*)

Table 4.1. Asymptotic formulae for long waves (k β

m=0

|β|

ω0±

1

m≠0 ωm = k + (m + 2µ) S ˆ m = (m − µ), m ≠ 1 ω

= k ± 2S

ˆm =µ ω

|β| ≈ 1

⎛ 2S ω0,n = ⎜ 1 + ⎜ β 0,n ⎝

⎞ ⎟⎟ k + ⎠

k2 ⎛ 1 ⎞ ⎜ − NS ⎟ , m = 1 2 ⎝S ⎠

⎛ 2S ωm,n = ⎜ 1 ± ⎜ β m,n ⎝

⎞ ⎟⎟ k + mS ⎠

2

2 Sk ⎛ 1 1 ⎞ 3 + ⎜⎜ ⎟ k log k β0,n ⎝ β0,n 2 S ⎟⎠ k2 log k ω= k±i 2S +

|β|

1

ω = k + mS − µ

k2 k3 ±i 2S 2S m

Here µ = m/|m|; N = ln(2/|k|) – E + 1/4, E = 0.5772 is the Euler constant; bm,n is the n-th zero of the Bessel function Jm(β). The frequencies of neutral waves are marked by indices obvious from the equation form. Symbol “∧” means the second root for the case |β| 1, m ≠ 0. For |β| 1 and m ≠ 0 the second root of the dispersion equation vanishes at m = ±1, so for the case |m| = 1 we have a more accurate formula. The main conclusion from the asymptotic results obtained is that for every azimuthal number m there exists a single unstable wave (the lower line in the Table) and an infinite number of neutral inertial waves. We can write the imaginary part of the complex frequency in the dimensional form

ωi = ±

( ∆W )2 R2 2Ω m

– for |β|

1,

m ≠ 0,

one can see that it is determined by a jump in the axial velocity ∆W, i.e., this is the same type of Kelvin – Helmholtz instability. Now let us analyze the results of numerical calculation for the dispersion equation (4.39) performed by Loiseleux et al. (1998).

4.4 Linear instability of Rankine vortex with an axial flow

179

Axisymmetrical mode, m = 0

Dispersion curves for the unstable mode are plotted in Fig. 4.11 in relation to the swirl number S. Here ωr and ωi are the real and imaginary parts of a complex frequency. It follows from the graph that the flow swirl has a stabilizing effect, and in any case the flow remains unstable. Figure 4.12 shows the comparison of exact and approximate solutions for the unstable mode. For low k the agreement between solutions is quite good. Spiral modes, m ≠ 0

Dispersion curves for neutral inertial waves are shown in Fig. 4.13. Note that for the mode ω0± the numerical solution and approximation coincide. Typical examples of dispersion curves for two unstable modes m = ±1 are shown in Fig. 4.14 for two positive values of swirl parameter S. Note that both the magnitude of the swirl parameter and its sign are essential for the spiral mode. These dispersion curves have several features. Firstly, for S ≠ 0 the symmetry between positive and negative spiral modes is broken. Secondly, for S > 0 the increment for the negative mode is always higher, i.e., it has higher instability. That is why we have to analyze the negative spiral mode with m = −1. It is worth noting that the increment depends not only on the sign of m but also on the magnitude. For a small wave number k 1, the most unstable mode is m = 0, and for the case k 1 the highest instability is inherent in the mode with m → −∞.

Fig. 4.11. Dispersion curves for unstable mode at m = 0 (Loiseleux et al. 1998*)

180

4 Stability and waves on columnar vortices

Fig. 4.12. Dispersion curves for unstable mode at m = 0, S = 0.7: 1 − numerical calculation; 2 − asymptotic formula (Loiseleux et al. 1998*)

Fig. 4.13. Dispersion curves for neutrally stable inertial modes at m = 0, S = 0.7: 1 − numerical calculation; 2 − asymptotic formulae (Loiseleux et al. 1998*)

The third feature of a dispersion curve is that its shape changes after the swirl parameter passes through a certain critical value S = Scr. The effect of S on dispersion curves is illustrated in detail in Fig. 4.15 for the case m = −1. It follows from the graph that the impact of the swirl parameter on stability is different for various ranges of S. If S > Scr = 0.46, the growth of a swirl brings flow stabilization (see. Fig. 4.15d). If S < Scr, at high k > ks the growth in swirl parameter destabilizes the flow (see Fig. 4.15b). However, for the range of small wave numbers, less than the critical level ks = ks(S), we observe a complex behavior of dispersion curves, which is illustrated by Fig. 4.16.

4.4 Linear instability of Rankine vortex with an axial flow

181

Fig. 4.14. Dependency of dispersion curves for unstable spiral modes with m = +1 (1) and m = –1 (2) on the swirl parameter S: a, b – S = 0.4; c, d – S = 0.7 (Loiseleux et al. 1998*)

Curve II in Fig 4.16a corresponds to the case S = 0. Appearance of a very small swirl produces a sequence of instability zones divided by small bands of neutral stability. For k > ks all wave numbers are unstable. If S Scr, the instability zones expand and merge at S = Scr. With further growth of the swirl parameter, the dependency ωi (k) becomes smoother (see Fig. 4.17a) and one can see that it is described adequately through a longwave approximation for small k. For understanding the mechanisms of formation of separate instability zones, we have to consider the dispersion curves for the real frequency ωr. It is more convenient to deal with the value Re g/k2, i.e., the frequency shifted due to the Doppler effect (see Fig. 4.16b, 4.17b). The unstable ˆ −1 , ωn−1 mode (1), its long-wave approximation (I), neutral modes ω−1,n , ω

ˆ −1 in the long-wave approximation (3) are plot(2), neutral modes ω−1,n , ω ted here. In addition to the known modes listed in Table 4.1, one more neutrally-stable wave with frequency ωn−1 appears.

182

4 Stability and waves on columnar vortices

Fig. 4.15. Effect of swirl number S on dispersion curves at m = –1: a, c – frequency; b, d – increment (Loiseleux et al. 1998*)

The character of dispersion curves testifies to resonance interactions between the unstable modes and neutrally stable waves; this is the reason for the formation of instability zones (Loiseleux et al. 1998). This mechanism is illustrated by a scheme in Fig. 4.18. Here q is the ordinal number of the instability zone; q = 1 – the first zone to the left from k = ks; kq,1 and kq,2 are the wave numbers for the left and right boundaries of q-th zone. For 1 the point k = kq,1 the merging of two neutral modes ω−1,q and ωq+ −1 produces the generation of the Kelvin – Helmholtz unstable mode. And at k = kq,2 the resonance interaction of the unstable mode with the neutral wave ω−1,q splits the unstable mode into two neutral waves ω−1,q and ωq−1. The resonance condition assumes that ωm,n (k) = ωr (k), where the rightside is the real frequency of the unstable mode. Substituting this equality into approximate formulae in Table 4.1, we obtain the discrete set of resonance wave numbers

km,n ≈ 4S2/βm,n,

(4.47)

4.4 Linear instability of Rankine vortex with an axial flow

183

Fig. 4.16. Dispersion curves at low wave numbers; S = 0.35 < Scr, a = 0, m = –1: 1 – unstable mode of Kelvin – Helmholtz; 2 – neutral modes; 3 – long-wave approximation for ωˆ −1(k) and ω−1,n(k), 4 – resonance wave numbers. I – long-wave approximation for unstable mode, II – case S = 0 (Loiseleux et al. 1998*)

184

4 Stability and waves on columnar vortices

Fig. 4.17. Dispersion curves at low wave numbers, S = 0.55 > Scr, a = 0, m = –1 (Loiseleux et al. 1998*). The symbols are the same as in Fig. 4.16

4.4 Linear instability of Rankine vortex with an axial flow

185

Fig. 4.18. Diagram for resonance interactions producing the q-th zone of instability (Loiseleux et al. 1998*)

where βm,n are the zeros of Bessel function Jm(β). These values of wave number are depicted in Fig. 4.16a by crosses; they coincide with the right boundaries of the instability zones. Thus, a discrete character of the instability zones for the range of small wave numbers is caused by a discrete set of Bessel functions in dispersion relationships. 4.4.3 Linear analysis of spatial instability In analysis of spatial instability we assume that k = kr + iki,

ω = ωr .

The flow is unstable if ki < 0. Substituting these expressions into the dispersion equation (4.39), we obtain the dispersion relationships: kr = kr(ω; S, m, a), ki = ki(ω; S, m, a). Let us consider the case where there is no external axial flow (a = 0). The examples of computation performed by Wu et al. (1992) are shown in Figs. 4.19 and 4.20. Figure 4.19 shows the effect of azimuthal wave number m on curve ki(ω). We should emphasize that in the spatial problem statement the Rankine vortex with axial flow is always unstable to small disturbances (both for axisymmetrical and spiral modes). Besides, the growth in frequency produces a growth in instability, i.e., (−ki) increases. A similar conclusion was obtained for relationships ωi(k) in the analysis of temporal instability (see Figs. 4.11 and 4.15).

186

4 Stability and waves on columnar vortices

For small values of the swirl parameter S, the dispersion curves for different m almost coincide (Fig. 4.19a). But the growth in S brings a higher difference between the increments of various modes (Fig. 4.19b). Among the seven tested modes, the highest instability is inherent in the mode with m = −3. It is natural to expect that more “negative modes” will be more unstable. But for the range of small frequencies the curve behavior is more complicated, and additional analysis is required here. Figure 4.20 shows the effect of the swirl number S on the dispersion curves. For positive modes (m = 2, see Fig. 4.20a) the swirl growth enhances the flow stability, and for negative modes (see Fig. 4.20c) the stability decreases. For the axisymmetrical mode (m = 0) there is no effect of swirling (see Fig. 4.20b).

4.5 Kelvin waves Kelvin, in his classical work (Kelvin 1880) titled “Vibrations of a columnar vortex”, presented an analysis of wave motion of the columnar vortex for three main cases: ) rigid-body rotating vortex, b) hollow vortex, and c) Rankine vortex without axial flow. Usually Kelvin waves are regarded as core perturbations of the Rankine vortex. In this Section we will consider such a kind of wave motion. Remember that we have considered cases a) and b) in Sections 4.3 and 4.4.1 respectively. Kelvin (1880) derived exact dispersion equations for arbitrary infinitesimal harmonic perturbations of the Rankine vortex core. He revealed gggg

Fig. 4.19. Dependency of spatial increment ki on frequency ω for different azimuthal modes m: S = 0.1 (a), 0.4 (b) (Wu et al. 1992*)

4.5 Kelvin waves

187

Fig. 4.20. Dependency of spatial increment ki on frequency ω for different values of swirl parameter S: m = 2 (a), 0 (b), 3 (c) (Wu et al. 1992*)

that these perturbations were neutral and presented quantitative results for axisymmetric and bending modes in the long-wave approximation. Saffman (1992) analyzed an additional case for wavelength, comparable or less than the vortex core size. By means of numerical simulation Arendt et al. (1997) displayed how perturbations, initially localized on the vortex tube with constant vorticity, evolve into Kelvin wave packets.

4.5.1 Dispersion equations

The problem statement is exactly the same as in Section 4.4.1, with exception of the circumstance, that in the undisturbed velocity profile (4.30) there is no axial flow, W∞ ≡ ∆W ≡ 0, i.e. the regular Rankine vortex is

188

4 Stability and waves on columnar vortices

considered. Then, omitting derivation and keeping in mind the former symbols, we may obtain the dispersion equation directly from (4.38) Jm′ (βR) Km′ (kR) 2 Ωm . − 2 2 =− β R Jm (β R) gβ R kR Km (kR)

(4.48)

g = ω – mΩ,

(4.49)

β2 = k2[4Ω2/g2 − 1].

(4.50)

Here

The solutions are found in the form of traveling neutral waves, i.e. in form of exponent exp[i(kz + mθ – ωt)], where frequency ω and wave number k are real quantities. By virtue of symmetry, we may assume keeping a general approach, that k > 0 and m > 0, then e = sign(kz) ≡ 1 ( . (4.38)). 4.5.2 Axisymmetric mode, m = 0 Let us rewrite Eqs. (4.48) and (4.50) for m = 0. Then from (4.49) we have g = ω. Introducing denotation q ≡ βR, we can write the equations in the following form

J0′ (q) K0′ (kR) =− , q J0 (q) kR K0 (kR) ω=±

2ΩkR k R +q 2

2

2

.

(4.51)

(4.52)

Solving Eq. (4.51) relative to q, we find the sought dispersion relationship ω = ω(k) from (4.52). First, let us consider long waves, kR 1. Expanding Bessel function K0 and its derivative into a series in the small argument, for the second member of Eq. (4.51) we obtain −

K0′ (kR) 1 kR ≈ ⎯⎯⎯⎯ → −∞ , kR→0 kR K0 (kR) kR(log 2 − E − log kR)

where E = 0.5772 is the Euler constant. This condition means that the solution of Eq. (4.51) reduces to determination of the root of the Bessel function J0 (q)

4.5 Kelvin waves

J0 (q) = 0.

189

(4.53)

This function is presented graphically in Fig. 4.21. Apparently, q01 < q02 < q03 …, and the magnitude of the least root equals q01 = 2.4049 ≈ 2.4. In a long-wave approximation for the frequency ω from (4.52) we obtain the positive root

ω = 2ΩkR/q0,

(4.54)

where q0 is the root of Eq. (4.53). Phase velocity is accordingly

c = ω/k = 2ΩR/q0.

(4.55)

For the least root q0 = q01 we obtain the maximum phase velocity c = 2Ω/1.2, resembling closely the value of circumferential velocity at the boundary of the vortex core. Apparently, at the long-wave limit the axisymmetric waves are dispersionless. If the wavelength is the same or less then vortex core size, i.e. kR Q 1, then the solution for Eqs. (4.51), (4.52) can be obtained only using numerical methods. Note, that at a fixed value of wave number k Eq. (4.51) has an unlimited number of roots qn(k). Each root (radial mode) under the number n agrees with its own dispersion curve. The examples of numerical calculation for frequency ω, phase c and group velocities cg based on Eqs. (4.51) and (4.52) are represented in a dimensionless form in Fig. 4.22. Here the number of the curve agrees with the serial number of the root. The limiting value of frequency for all the roots is ω = 2Ω, which follows from (4.52) at kR → ∞ and q = const. The behavior of ω at small k is described by the Eq. (4.54). For phase velocity c = ω/k we may apply (4.55) at a long-wave limit, while in the case of short waves all curves tend to the same asymptotic dependence: c = 2ΩR/k. Eventually the group velocity for long waves concurs with the phase velocity (dispersionless waves), whereas in a short-wave area it tends to zero. As it follows from the calculations, for a given root the group velocity is less (or equal) to the phase velocity.

Fig. 4.21. Diagram of Bessel function J0 (q)

190

4 Stability and waves on columnar vortices

Fig. 4.22. Dependencies of dimensionless values of frequency ( ), phase velocity (b), and group velocity (c), on dimensionless wave number at m=0

4.5.3 Bending mode, m = 1 Long waves, kR

1

At m = 1 Eqs. (4.48) – (4.50) take the form

g = ω – Ω, β = k [4/(ω/Ω − 1)2 − 1],

(4.56)

J1′ (q) K1′ (kR) 2 , − 2 =− q J1 (q) q (ω Ω − 1) kR K1 (kR)

(4.57)

2

2

where q = βR. For the long-wave approximation (kR 1) the second member of Eq. (4.57) can be simplified using asymptotic formulae for Bessel functions

−

K1′ (kR) 1 1 − k2 R2 (log kR) 2 − k2 R2 (2 γ + 1 − log 4) 4 ≈ 2 2 ≈ kR K1 (kR) k R 1 + k2 R2 (log kR) 2 + k2 R2 (2γ − 1 − log 4) 4 ≈

1

⎡1 + k2 R2 (log 2 − γ − log kR) ⎤ . ⎦ k R ⎣ 2

2

4.5 Kelvin waves

191

In order to obtain an analytical solution for (4.57), let us also assume that ω Ω. Then it follows from (4.56) that β ≈ k and q ≡ βR 1, i.e. we consider the least roots of Eq. (4.57). Upon such assumptions we may employ expansion in a small parameter for both terms in the left-hand part of (4.57)

J1′ (q) 1 1 − 3q2 8 1 1 ≈ 2 ≈ − ≈ q J1 (q) q 1 − q2 4 q2 4 ≈

1 k R ⎡ 4 (ω Ω − 1) − 1⎤ ⎣ ⎦ 2

2 q (1 − ω Ω) 2

2

=

2

−

1 1 ⎛ 8 ω⎞ 1 ≈ 2 2 ⎜1 − ⎟− , 4 3k R ⎝ 3 Ω ⎠ 4

1 k R ⎡ 4 (ω Ω − 1) − 1⎤ (1 − ω Ω) ⎣ ⎦ 2

2

2

≈

⎛ 5 ω⎞ ⎜1 − ⎟. 3k R ⎝ 3 Ω ⎠ 2

2

2

Here we have taken into account Eq. (4.56) and relationship q = βR. As a result we obtain an approximate equation

2 ⎛ 5 ω⎞ 1 ⎡ 2 ⎛ 8 ω⎞ 1 ⎞⎤ 2 2⎛ − γ ⎟⎥ , ⎜1 − ⎟ − + 2 2 ⎜1 − ⎟ = 2 2 ⎢1 + k R ⎜ log kR 3k R ⎝ 3 Ω ⎠ 4 3k R ⎝ 3 Ω ⎠ k R ⎣ ⎝ ⎠⎦ 1

2

2

wherefrom the ultimate dispersion relationship can be derived (Kelvin 1880)

ω 1 1 1 ⎛ ⎞ = − k2 R2 ⎜ log + + 0.1159 ⎟ . Ω kR 2 4 ⎝ ⎠

(4.58)

For the long-wave limit this formula takes an even more simple form:

ω/Ω = − 0.433k2R2. The given mode causes a drift of the vortex axis, which for this reason takes the form of a left-handed helix (with the large step and small amplitude due to the accepted assumptions). The negative value of the frequency means that the perturbation moves around the cylindrical surface against the flow rotation. Therefore such a mode is called a retrograde mode. If m = − 1, then the axis of vortex perturbation becomes a right-handed helix. But the above conclusions remain true. Indeed, the second member of (4.57) does not change because K−m = Km. The first term in the lefthand side member also remains unchanged, since both the numerator and the denominator of the Bessel function Jm change sign. Other terms include m in form of mΩ product. Changing m from (+1) to (–1) is equivalent to alteration of the fluid rotation direction. Thus, the perturbation is

192

4 Stability and waves on columnar vortices

still propagating against the fluid rotation direction. A diagram for both cases is presented in Fig. 4.23. Arbitrary value of kR

In this case we have the equation set (4.56), (4.57), which can be solved using numerical methods. For each fixed magnitude of kR , Eq. (4.57) has an unlimited number of roots. Each root is related to its own dispersion curve, consisting of one or two branches. The examples of the numerical calculation of dispersion curves for the first four roots are presented in Fig. 4.24. Here, the number of the curve corresponds to the number of the root (or the number of the radial mode). Modes, situated in the area, where ω < 0, are retrograde modes. Modes with ω > 0 are co-grade modes, because the perturbations are propagating towards fluid rotation. As the graphs reveal, the roots 1, 2, 3 further consist of two branches, proper to “retrograde” and “co-grade” modes. The root “0” represents just the retrograde mode, which barely has asymptote ω → 0 at kR → 0. In a short-wave area (kR → ∞) the frequencies of all retrograde modes tend to the common limit ω = − Ω, while those of co-grade modes tend to the value of ω = 3 Ω. The ability of the frequency ω (as well as the phase velocity) to vanish at certain values of kR is an interesting peculiarity of the retrograde modes. It follows from (4.56), that if ω = 0, then βR = 3kR. Therefore, zero frequency occurs for those values of kR , which satisfy the following equation

K1′ (kR) 1 J1′ ( 3kR) 2 . + 2 2 =− kR K1 (kR) 3kR J1 ( 3kR) 3k R The first three roots are: (kR)0 = 2.5; 4.4; 6.2. The existence of an extreme point for the phase velocity c at some value of dimensionless wave number kR = (kR)* (see Fig. 4.24b) is another peculiarity of the retrograde modes. It follows from the condition of extreme existence

dc/dk = d(ω/k)/dk = (cg − c)/k = 0 that this extreme occurs at a parity point between the group and phase velocities. Comparing the phase and group velocities we may distinguish three characteristic areas of wave numbers (Fig. 4.25a) for the roots with numbers equal to or greater than 1. At kR < (kR)0 phase and group velocities have different signs. At (kR)0 < kR < (kR)* |cg| > c, while at kR > (kR)* |cg| < c. For the “co-grade” mode phase velocity always exceeds group velocity (Fig. 4.25b) and either of them is positive.

4.5 Kelvin waves

193

Fig. 4.23. Diagram of the vortex axis at the perturbations in the form of the bending modes m = ±1

Fig. 4.24. Dispersion curves for bending mode m = 1: – frequency; b – phase velocity; c – group velocity

194

4 Stability and waves on columnar vortices

Fig. 4.25. Comparison between phase c and group cg velocities for retrograde ( ) and co-grade (b) modes. Root number is 1

Remember, that the phase and group velocities are defined by the axial wave number and therefore imply phase velocities and perturbation energy propagation along the vortex axis. 4.5.4 Evolution of initially localized perturbations. Mechanisms of wave propagation In this Section let us consider the evolution of initially localized perturbations of various types on the Rankine vortex. Arendt et al. (1997) studied such a problem and revealed that any type of initial perturbation evolves into Kelvin wave packets. This problem is important for two reasons. First of all, it enables us to follow the experiments, where the generation of such perturbations is possible; secondly, one can explain the mechanisms of Kelvin wave propagation. The latter is of general importance even for the vortices of another structure rather than the Rankine vortex. Due to the awkward algebra we do not present the method of calculation here, which can be found in the original paper. Note that in the cited paper the Laplace transformations have been used to obtain analytical solutions. Also some of the symbols used here may differ from those applied in the original paper. As before, let us represent the elementary harmonic perturbation by means of exponent exp[i(kz + mθ – ωn t)]. Also note that the dispersion equation (4.48) is still valid. However since we use ω for vorticity, then let us designate ωn, as an angular velocity, where index n is the radial number (or the number of the root of Eq. (4.48) at a fixed m). Arendt et al. (1997) solved the problem of evolution of the initial perturbations based on linearized Euler equations and the matching conditions on a vortex sheet (of the same type as presented in Section 4.4.1) applying Laplace transformations. Let us consider the following cases for m = 0, –1, –2 with the corresponding initial conditions.

4.5 Kelvin waves

195

Case m = 0

The initial perturbation is given in form of the local increase for the axial component of the vorticity ωz in the vortex core at a length in the order of R (radius of the vortex core)

ωz (0) =

2Ωε R 2π

e− z

2

/ 2 R2

.

Such a notation agrees with the Gauss distribution of vorticity perturbation along the z axis. Here 2Ω is a non-perturbed value of vorticity; the ratio ε/R 1 means infinitesimal perturbation of vorticity as compared with 2Ω. Dimension of the perturbed area R along z is selected based on experimental observations. Besides, we assume absence of the azimuthal component of the vorticity at t = 0

ωθ(0) = 0. Based on primary equations we reveal that the initial perturbation of the tube radius δR is described by the following law

δR R

=− t =0

ε 2 R 2π

ez

2

/ 2 R2

,

i.e. local increase in vorticity leads to local thinning of the vortex tube. Further, Figs. 4.26–4.28 present calculation results for the distribution of ωz, ωθ, ∂ωz/∂t and δR/R evolution. Quantity ∂ωz/∂t is of great importance since it represents the variation in the axial vorticity, caused by stretching/scrunching of the vortex lines (see also Section 2.2). Figures 4.26, 4.27 are presented as level profiles of the proper quantities. Here, the areas corresponding to positive values are shaded. Time t is normalized by 2π/Ω. We derive the main conclusions as follows. First of all, let us draw attention to the circumstance that ωz = ∂uθ/∂r. If ωz varies along z axis at r = const, then this relationship means the existence of differential rotation, causing helical twisting of the vortex lines. This is distinctly seen in Fig. 4.26 for the azimuthal component of the vorticity ωθ. The areas with negative values of ωθ appear at t = 0.5, while for long time periods positive and negative areas alternate and propagate along both sides of the initial position of perturbation. In turn, ωθ induces flow within the plane (z, r), since fffggffffff

196

4 Stability and waves on columnar vortices

Fig. 4.26. Distribution of the vorticity components for a wave packet with m = 0: level profiles are presented with intervals of 0.1 (Arendt et al. 1997*)

4.5 Kelvin waves

197

Fig. 4.27. Distribution of ∂ωz/∂t for a wave packet with m = 0: level profiles are presented with intervals of 0.05 (Arendt et al. 1997*)

Fig. 4.28. Perturbation of the vortex tube radius for a wave packet with m = 0 (Arendt et al. 1997*)

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4 Stability and waves on columnar vortices

ωθ = ∂ur/∂z − ∂uz/∂r. The appearance of axial flow, non-uniform along z is very important. As is obvious from Fig. 4.26 (∂ωz/∂t ≠ 0) such a flow results in stretching/scrunching of the vortex lines, i.e. it causes variation of ωz. Appropriate behavior of the vortex tube radius is shown in Fig. 4.28. The above described process enables us to consider the propagation of axisymmetric perturbations on a vortex tube as a phenomenon of twisting and untwisting vortex lines. This is obvious for Kelvin waves also. Figure 4.29 exhibits simultaneous pictures of ωθ, ∂ωz/∂t and ωz distributions for Kelvin waves with m = 0, n = 0 and k = 1/R (see Arendt et al. (1997) n = 1). Interdependence between the initial perturbation evolution and the Kelvin waves makes it possible to explain the complex structure of flow, shown in Figs. 4.26 – 4.28. The reason concludes, that for m = 0 there is a set of modes with the various radial number n (number of the root of Eq. (4.51)), which have different radial structures (the higher n is, the more complicated the structure) and various frequencies. In the same time Kelvin waves possess dispersion (see Fig. 4.22). In the long-wave area, the less n is, the higher the group velocity. For a given n, long waves move faster in comparison with short waves. Therefore the complex pattern for the later time points shown in Figs. 4.26 – 4.28 is stipulated by dispersion of perturbations. In particular, Fig. 4.28 reveals that at t = 4 the wave which is faster, runs forward, while slow waves are segregated behind.

∂ωz/∂t

Fig. 4.29. Distributions of ωz, ωθ, ∂ωz/∂t for Kelvin waves with m = 0, n = 0, k = 1/R: level profiles are presented with intervals of 0.33 (Arendt et al. 1997*)

4.5 Kelvin waves

199

Case m = – 1

The only modes which cause a drift of the vortex core, are those with m = ±1. It is convenient to assign the initial perturbation of the vorticity in Cartesian coordinates

2Ωε ∂ − z2 / 2 R2 e , R 2π ∂z ωy (0) = ωz (0) = 0.

ωx (0) =

This perturbation is corresponded by drift of the vortex core center towards the x-axis, which can be calculated using formula ∆=

ε R 2π

e− z

2

/ 2 R2

,

i.e. we have deformation of the vortex tube in the (x, z) plane. Coordinates x and r are coupled by the usual relationship x = r cos θ. All remaining symbols and assumptions are the same as described previously in the present Section. The results of calculations, obtained by Arendt et al. (1997), are presented in Fig. 4.30 ,b for early and late stages of evolution, correspondingly. The amplitude of perturbation is shown in an augmented scale. At the initial stage (t = 0.5; 1.0) the strained element of the tube bends towards the (−y) direction and starts rotating around the tube. For the later points in time (t > 2.0) two spiral perturbations can be distinguished: a right-handed perturbation moving in z-axis direction, and a left-handed one, propagating in the opposite direction. Thus, behavior of the deformed vortex can be easily explained within approximation of a thin vortex filament. Indeed, any bending of the vortex filament leads to self-induced motion of the bent part along its bi-normal. The bi-normal is determined using the formula t × n, where t is the unit tangential vector, directed along the vorticity vector, while n is the unit normal vector directed along the vector of radius of the curvature. For the point in time t = 0, t × n = – ey, where ey is the unit vector directed along the y -axis. Similarly, for large time intervals the propagation of dedicated Kelvin waves is driven by self-induced motion of bent parts of the vortex. Besides, as in the case m = 0, at large distances perturbations represent packets of Kelvin waves, diverging due to dispersion. In order to comprehend the structure of a packet, we have to take into account, that the

200

4 Stability and waves on columnar vortices

mode n = 0 has a dominating impact at m = −1. Then it is obvious from Fig. 4.24c that the waves with kR ≈ 0.5 are the fastest. Short waves have lower velocity and can be observed in the trail of the main perturbation (see Fig. 4.30 , t = 7.0). There are longer waves as well, which are less discernable.

a

t=0

t = 1.0

t = 2.0

b

t = 4.0

t = 7.0

Fig. 4.30. Evolution of a vortex tube surface under the impact of a wave packet with m = –1 for the earlier stage at –7R ≤ z ≤ 7R ( ) and later stage at –20R ≤ z ≤ 20R ( ). On the left, the z axis is directed to the left, while on the right it is directed to observer and angled down slightly (Arendt et al. 1997*)

4.5 Kelvin waves

201

Case m = – 2

Modes with |m| > 1 cause non-axisymmetric deformation without a shifting of the vortex tube axis (see Section 4.1). Let us assign for a given mode the initial deformation in the form of a vortex core flattening

ωx (0) =

2Ωε R 2π

x

∂ − z 2 / 2 R2 , e ∂z

ωz (0) = 0,

δR(θ) =

ωy (0) = −

2Ωε R 2π

y

∂ − z 2 / 2 R2 , e ∂z

2 2 ε cos(2θ)e − z / 2 R . 2π

a

t=0

t = 1.0

t = 2.0

b

t = 4.0

t = 12.0

Fig. 4.31. Evolution of a vortex tube surface under the impact of a wave packet with m = –2 for the earlier stage at –7R ≤ z ≤ 7R (a) and later stage at –20R ≤ z ≤ 20R (b). On the left, the z axis is directed to the left, while on the right it is directed to observer and angled down slightly (Arendt et al. 1997*)

202

4 Stability and waves on columnar vortices

Such a notation represents widening of the core towards the x axis and thinning along the y axis in the vicinity of point z = 0. Figure 4.31 ,b presents calculation results, obtained by Arendt et al. (1997) for the earlier and later stages of the evolution, accordingly. It is obvious from the figures, that we have splitting of the initially deformed part into two bent vortex tubes starting to rotate around the z axis. Again, this is caused by self-induced motion. Since the intensity of each tube is twice as small as compared with a non-perturbed vortex, all the processes occur slowly in comparison with the case m = –1, when the whole tube is bent. For large time periods, packets of two twisted helical perturbations are distinguished. Similarly as for m = –1, right-handed perturbations move along the positive direction of the z axis, while left-handed perturbations move in the opposite direction. Again, the mechanism of propagation is related to the self-induced motion. All the principle motion regularities agree with the case m = –1, including the influence of dispersion. Only the maximum of the group velocity falls in the wave numbers range equal to kR ≈ 1.5.

4.6 Instability of Q-vortex. Instability criteria 4.6.1 Instability criteria The analysis of linear stability for one of the simplest models of inviscid vortex flows, namely the Rankine vortex, was presented in Section 4.4. However there are many other models and types of vortex flow. Thus the problem of instability criteria, which can predict flow stability based on given velocity fields, is of great importance (Leibovich 1984; Escudier 1988). Such a criterion was first obtained by Raleigh (1880, 1916), and it is named after him. Raleigh criterion states, that an inviscid axisymmetric vortex flow is stable if Φ≡

1 d (Vr ) 2 > 0 . r 3 dr

(4.59)

Here, function Φ = Φ(r); V is the tangential velocity, while other velocity components are absent. Later, based on analysis of axisymmetric perturbations, Synge (1933) revealed that (4.59) is yet sufficient condition, i.e. Raleigh criterion indicates necessary and sufficient condition of stability for merely vortex flow relative to the axisymmetric perturbations.

4.6 Instability of Q-vortex. Instability criteria

203

If we assume the conservation of circulation for a liquid particle at small radial displacements being equal to Γ = 2πrV = 2πK, then Raleigh criterion can be clearly interpreted physically both in terms of energy and forces (Goldshtik 1981). Let us consider the second option. At the equilibrium condition the centrifugal force is balanced by the pressure gradient: ρV2/r ≡ ρK 2/r 3 = dP/dr. Let the fluid particle with the coordinate r1 be displaced at a distance equal to r, keeping the value K1 constant. If the velocity distribution is such that ρ

K12 r3

<

dP K2 =ρ 3 , dr r

then the restoring force will apply to the particle. Hence the condition (4.59) follows. Instability criterion relative to helical perturbations in the flow rotating in an annular narrow channel was derived by Ludwieg (1960) though this is a particular result. Howard and Gupta (1962) also considered vortex flow in an annular channel, but for the case of non-zero axial velocity component W(r). They obtained a generalization of Raleigh criterion, revealing that the flow is stable relative to the axisymmetric perturbations, on condition that 1 d (Vr ) 2 3 dr 1 =r ≥ . J≡ 2 2 4 ⎛ dW ⎞ ⎛ dW ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ dr ⎠ ⎝ dr ⎠ Φ

(4.60)

This is sufficient condition. The analogy between criterion (4.60) and the stability criterion for stratified fluid was drawn in the cited paper by means of Richardson number Ri: flow is stable if Ri ≡ gρ'/ρW' 2 ≥ 1/4. Here, g is the acceleration due to gravity; ρ = ρ(r) is the density, prime is the r coordinate derivative. We will demonstrate below, that assigning Φ ≡ r–3(Vr)' = gρ'/ρ, the stability equation for vortex flow with the axial component for the case of axisymmetric perturbations (see Eq. (4.64)) will concur with that for stratified fluid in a radial field. Then indeed, the quantity J in inequality (4.60) will have an implication on the local Richardson number. Leibovich and Stewartson (1983) obtained the most common criterion for incompressible fluid. They determined that the sufficient condition of instability for an unbounded inviscid column vortex with axial flow can be written as follows

204

4 Stability and waves on columnar vortices 2 d ⎛V ⎞ ⎡ d ⎛V ⎞ d ⎛ dW ⎞ ⎤ V ⎜ ⎟ ⎢ ⎜ ⎟ (Vr ) + ⎜ ⎟ ⎥ <0. dr ⎝ r ⎠ ⎢⎣ dr ⎝ r ⎠ dr ⎝ dr ⎠ ⎥⎦

(4.61)

Moreover, they considered both axisymmetric and spiral modes. Note, that in case of three-dimensional perturbations we may refer to the analogy between stratified and vortex flows (Vladimirov 1985). The above considered instability criteria (as well as some others) are predominantly of sufficient character and prevent the need to obtain the necessary information and perform rigorous assessments. Therefore, the stability analysis of a particular flow pattern remains a basic method of study. Below we will consider the instability of Q-vortex, which is closer to real column vortices as compared with the Rankine vortex, and at the same time is in agreement with the latter at r → ∞ and r → 0. 4.6.2 Instability of Q-vortex. Inviscid analysis Let us notate the non-perturbed velocity field of Q-vortex (see Section 3.3.5) in dimensionless form

U (r ) = 0, V (r ) =

q⎡ 1 − exp(−r 2 ) ⎤ , W (r ) = a + exp(−r 2 ) . ⎦ r⎣

(4.62)

As for the Rankine vortex, maximum axial velocity difference ∆W is selected here as a velocity scale. Moreover, unlike (3.72) the constant component of axial velocity a = W0/ ∆W is taken into account. The examples of velocity profiles for the case q = 1, a = 0 are presented in Fig. 4.32. At the point of the tangential velocity maximum we have Vmax = 0.639, rmax = 1.122. Parameter q = Vmax/0.639 is apparently proportional to the swirl parameter S, introduced in Section 4.4.1 for the Rankine vortex. Therefore let us call q a swirl parameter as well. Let us first derive the equations for perturbations in a general form (Howard and Gupta 1962), assuming that non-perturbed flow has two nonzero velocity components V(r) and W(r). As usual, let us present the perturbations in the following form

{u, v, w, p} = {uˆ (r ), vˆ (r ), wˆ (r ), pˆ (r )} exp [i(kz + mθ − ωt)] . Substituting these expressions into Euler equations and mass conservation equation (4.31), and linearizing them, we obtain

4.6 Instability of Q-vortex. Instability criteria

205

Fig. 4.32. Velocity profiles for the Q-vortex model at q = 1, a = 0

ˆ + iug

2V vˆ = pˆ ′, r

ˆ + iuV ˆ ∗= vg

m pˆ , r

ˆ ′ + wg ˆ = kpˆ , iuW ˆ− iuˆ ∗ − kw where

mvˆ = 0, r

g = g(r) = ω – mV/r – kW.

Here prime means (d/dr) operator, while asterisk (∗) means (d/dr + 1/r) operator. In this case Doppler-shifted frequency g is a function of the radial coordinate in contrast to Eq. (4.35) applicable for the Rankine vortex. Note also that at the derivation of these equations we take the non-zero term u ∂W/∂r in the third equation of the set (4.31), unlike the case of the Rankine model. ˆ , we have two intermediate equations (GoldFurther, excluding vˆ and w shtik 1981) ⎡ 2VV∗ ⎤ 2Vm ˆ ⎢1 − pˆ ′ = iug pˆ , ⎥+ rg 2 ⎦ r 2 g ⎣

206

4 Stability and waves on columnar vortices

⎡ 1 kW ′ mV∗ ⎤ pˆ iuˆ ′ = −iuˆ ⎢ + + ⎥+ , g rg ⎦⎥ gs ⎣⎢ r where s = r 2/(m 2 + k2r 2). Eventually substituting pˆ (r ) from the second equation to the first, we obtain an ordinary differential equation of the second order for the eigenfunction uˆ (r ) ′⎫ ⎧ ⎡ s ⎛ 2mV 2Vsk ⎪ ⎞⎤ ⎪ (uˆ ∗ s)′g2 − uˆ ⎨ g2 − 2 (V ∗kr − mW ′) − rg ⎢ ⎜ 2 − g′ ⎟ ⎥ ⎬ = 0. (4.63) r ⎠⎦ ⎪ ⎣r ⎝ r ⎪⎩ ⎭

Howard and Gupta (1962) were the first to derive the above equation. Boundary conditions can be written as follows (see Batchelor and Gill 1962)

uˆ (0) = 0,

m ≠ 1,

uˆ ′(0) = 0, m = 1, uˆ → 0, r → ∞ . Note that Eq. (4.63) is invariant to the following transformation: q → −q, m → −m, i.e. changing the sign of the azimuthal mode will not influence the result if the swirl direction changes simultaneously. This is obvious in a case where there is no axial movement, but is not trivial for W ≠ 0. For axisymmetric perturbations (m = 0) Eq. (4.63) is simplified essentially after the replacement uˆ = gF (r ) ⎡ (c − W ) 2 F ∗ ⎤ ′ − k 2 (c − W ) 2 F + Φ F = 0 , ⎣ ⎦

(4.64)

where c = ω/k is the phase velocity of perturbations, while function Φ is determined by (4.59). Howard and Gupta (1962) used this equation to derive the stability criterion (4.50). In the general case of 3-D perturbations, Eq. (4.63) should be solved numerically. Basic results were obtained by Lessen et al. (1974), Duck and Foster (1980), Leibovich and Stewartson (1983), Mayer and Powell (1992), Olendraru et al. (1996). Lessen et al. reduced Eq. (4.63) to canonic form by means of the transformation F = guˆ r

f2 (r )F ′′ + f1 (r )F ′ + f0 (r )F = 0

4.6 Instability of Q-vortex. Instability criteria

207

with the boundary conditions F(0) = F(∞) = 0. The problem of eigenvalues was solved numerically by the Runge-Kutta method in temporal statement, i.e. the complex phase velocity c was determined as eigenvalues at a given k, m, q. At the same time it was assumed that the value a in (4.62) was equal to zero. This means the absence of axial flow at a distance from the vortex core. The calculations have been performed for the first six nonaxisymmetric negative modes and the first positive mode. We have emphasized the negative modes as they are more unstable compared with the positive ones. Axisymmetric perturbations were not considered since they were assumed to be unstable based on the general analysis provided by Howard and Gupta (1962). Later, Mayer and Powell (1992) displayed stability of the Q-vortex with respect to m = 0 mode, employing direct numerical calculations. Typical examples of calculation are shown in Figures 4.33–4.35. For positive azimuthal mode (m = +1, Fig. 4.33) phase velocity cr increases monotonously with an increase in the wave number, while dependence ci(k) at a fixed q has a local maximum. One can see that swirling stabilizes flow (relative to perturbations with m = +1). Full stabilization occurs at some critical value q, equal to 0.0739 (according to the corrected calculations by Mayer and Powell (1992)).

Fig. 4.33. Dependence of dispersion curves for Q-vortex on vortex parameter q at m = +1 (Lessen et al. 1974*)

208

4 Stability and waves on columnar vortices

For the negative spiral modes, the increment of the amplitude ωi is much higher than that for the positive modes with the same value of |m|, and it has local maximum ωimax as well (see Fig. 4.34). At a fixed magnitude of q the value ωimax increases with an increase of |m|. The influence of the swirling on stability is non-monotone (see Fig. 4.35b), and the limiting value of ωimax for each azimuthal mode is reached at certain values of the parameter q. Based on the asymptotic theory of Leibovich and Stewartson (1983) it is shown that ωimax monotonously approaches the limiting value of 0.459 at m → −∞. Later Mayer and Powell (1992) proved these conclusions by numerical calculations (see Table. 4.2). Another important conclusion of the cited works concerns the Q-vortex, which is stable with respect to any linear perturbations at a quite strong swirl, exceeding the level of q ≈ 1.5. Thus Q-vortex essentially differs from the Rankine vortex with axial flow, which is unstable for any values of swirl (see Fig. 4.15b,d). Phase velocity of the negative modes rises with increasing k (Fig. 4.35 ) as in the case of positive values of m. The most comprehensive analysis of the inviscid instability of the Qvortex within the whole range of m was carried out by Mayer and Powell (1992). The more exact method for numerical analysis based on a spectral collocation method with basis Chebyshev functions has been used. Thus a set of two equations for uˆ and vˆ was employed, including the linear eigenvalue of g instead of using Eq. (4.63) for uˆ by Howard and Gupta. The data obtained in preceding study were corrected and the resulting patterns have been mapped in {k, q} coordinates. The example of the regime map for m = −1 is presented in Fig. 4.36. Here the isolines for ωi = const > 0 are presented with an interval of 0.01. The outer circuit is a neutral curve confining the outside area where the Table 4.2. Position and increment of perturbations of maximum growth for different values of the azimuthal wave number m (numerical calculation by Mayer and Powell (1992*))

m –1 –2 –5 –10 –100 –∞ – ∞** **

q 0.458 0.693 0.833 0.861 0.872 0.871 0.870

k/|m| 0.811 0.591 0.539 0.531 0.530 0.531 0.532

ωimax 0.24244 0.31382 0.39217 0.42152 0.45006 0.45876 0.45900

Asymptotic theory by Leibovich and Stewartson (1983)

4.6 Instability of Q-vortex. Instability criteria

209

flow is stable relative to the given mode. It is revealed that the instability area is the greatest for the azimuthal wave number m = −1. Moreover, the instability area covers the range of the negative values of the swirl parameter q < 0 only for m = −1. Recalling invariance regarding the transformation of q → −q, m → −m, the existence of an instability zone at q < 0 for ggg

Fig. 4.34. Dependence of increment ωi on wave number k for negative spiral modes at q = 0.8 (Lessen et al. 1974*)

Fig. 4.35. Dependence of the real ( ) and imaginary (b) parts of the complex phase velocity on wave number at various magnitudes of q parameter for m = –1 (Lessen et al. 1974*)

210

4 Stability and waves on columnar vortices

Fig. 4.36. Isolines ωi = const > 0 with an interval of 0.01 for the most unstable mode at m = –1 (Mayer and Powell 1992*). The area of instability is within the outline ωi = 0

m = −1 means, that the positive mode m = +1 at q > 0 will be unstable as well. This agrees with the calculations by Lessen et al. (1974) (see Fig. 4.33). The limiting negative value of q equals –0.0739 and is achieved at k = 0.63. Conversely, the limiting positive value of q for the instability zone is q ≈ 1.5 (at k ≈ 0.54). In the absence of swirl (q = 0) unstable wave numbers are within the range of 0 < k < 1.18. The upper boundary of the instability is located at the point of α ≈ 2, q ≈ 0.42. The maximum value of the perturbation increment is ωimax = 0.2424 and is reached at q = 0.458, k = 0.811. This data differs from the appropriate result obtained by Lessen et al. (1974) (ωimax = 0.147, q = 0.32, k = 0.3). This can be explained by calculation errors. Among the peculiarities of the result map shown in Fig. 4.36, we have to note the dual-domain structure of the instability area with the rapid change of flow parameters at q ≈ 0.7. Mayer and Powell also revealed the existence of several unstable modes for each azimuthal wave number, in particular 10 for m = −1. Figure 4.36 represents data for the first mode as the most unstable. Each consequent mode has a narrower range of instability and smaller values of increment. With the increase of |m| the instability range in coordinates {q, k/|m|} converges as compared to the case of the most unstable mode at m = −1, though maximum values of the increment increase up to the limiting value at m = −∞. Some values on maximum increment are presented in Table 4.2. Here the wave number is presented in a reduced form (k/|m|), because in this form it tends to the finite limit for m → −∞.

4.6 Instability of Q-vortex. Instability criteria

211

4.6.3 Instability of Q-vortex. Viscous analysis It is obvious that for real viscous flows the results obtained in the previous Section and demonstrating the growth of instability with the increase of azimuthal and axial wave numbers are invalid. Therefore the analysis of the stability of the Q-vortex without taking into account the viscous effects cannot be considered as a full study. Lessen and Paillet (1974) were the pioneers, who investigated the influence of the viscosity on inviscid modes. They noted that the viscosity plays a stabilizing role and there is a critical Reynolds number, limiting the range of stable flow. Akhmetov and Shkadov (1987), and Khorrami (1991) revealed the existence of specific viscous unstable modes. Mayer and Powell (1992), Delbende et al. (1998), Akhmetov and Shkadov (1999), and Olendraru and Seller (2002) conducted studies within a broader range of parameters and corrected results obtained in previous works. The initial linearized equations for velocity and pressure perturbation amplitudes, following from the Navier-Stokes equations and the equation of continuity, can be written in the following form (Mayer and Powell 1992)

⎤ 1 ⎛ m2 + 1 2⎛ uˆ ′∗ ⎡ im ⎞ 2⎞ + ⎢ig − ⎜⎜ 2 + k ⎟⎟ ⎥ uˆ + ⎜ V − ⎟ vˆ = pˆ ′ , Re ⎢⎣ Re ⎝ r r⎝ r Re ⎠ ⎠ ⎦⎥ ⎞⎤ impˆ vˆ ′∗ ⎡ 1 ⎛ m2 + 1 2mi ⎞ ⎛ uˆ = , + ⎢ig − + k2 ⎟ ⎥ vˆ + ⎜ V ∗ − 2 ⎜⎜ ⎟ 2 ⎟ Re ⎢⎣ Re ⎝ r r r Re ⎠ ⎝ ⎠ ⎦⎥ ⎤ ˆ ′∗ ⎡ 1 ⎛ m2 w 2⎞ ˆ − W ′uˆ = ikpˆ , + ⎢ig − ⎜⎜ 2 + k ⎟⎟ ⎥ w Re ⎢⎣ Re ⎝ r ⎠ ⎥⎦ iuˆ ∗ − kwˆ −

mvˆ = 0. r

Here Reynolds number Re is defined according to the maximum difference of axial velocity and the vortex core radius. All other symbols are as per Section 4.6.2. Figure 4.37 represents the examples of the numerical calculations provided by Delbende et al. (1998) for inviscid modes subject to viscosity. Apparently, viscosity results in a decrease of increment. The higher |m| is, the more profound the decrease is. At Re = 667 and q = 0.8 all modes with |m| > 12 are stable. Another important inference concludes that in the existence of the critical Reynolds number Rec confining the region where all inviscid modes decay. Some results obtained by Mayer and Powell (1992) hhhhhhhhhhhhhhhhh

212

4 Stability and waves on columnar vortices

Fig. 4.37. Influence of the viscosity on inviscid unstable modes (Delbende et al. 1998*): q = 0.8; Re = 667

are presented in Table 4.3. The minimum critical Reynolds number limiting the region of stable flow is equal to 13.905 and corresponds to the mode with m = −1. With increase of |m| Reynolds number Rec rapidly increases, while Rec/m2 ratio tends to a finite limit. Purely viscous instability was discovered both for axisymmetric and spiral modes (Khorrami 1991). The main calculation results for the unstable viscous axisymmetric mode are presented in Table. 4.4 and Fig. 4.38. The critical Reynolds number is equal to 322.42. The area of instability rapidly grows with an increase of Re number. It follows from the table, that the values of ωimax are less than those for inviscid modes by a factor of tens (see Table. 4.2). However, by virtue of symmetry the instability will also occur for q < 0, i.e. in the region where inviscid modes decay. The importance of taking into account viscous mode confines right in latter statement. Data for the unstable viscous spiral mode with m = −1 are presented in Table 4.5 and Fig. 4.39. Maximum increments here are essentially greater Table 4.3. Critical parameters determining the initiation of the unstable inviscid modes allowing for viscosity (Mayer and Powell 1992*)

m –1 –2 –3 –5 –10 –100 –10000

q 0.337 0.466 0.515 0.545 0.557 0.550 0.542

k/|m| 0.415 0.475 0.479 0.473 0.464 0.456 0.456

Rec 13.905 26.26 46.91 105.29 356.32 2.9922⋅104 2.8376⋅108

Rec/m2 13.905 6.5650 5.2122 4.2116 3.5632 2.9922 2.8376

4.6 Instability of Q-vortex. Instability criteria

213

when compared with the axisymmetric viscous mode, while the critical Reynolds number is much lower (Rec = 17.527) and quite close to Reynolds number Rec for inviscid mode (Rec = 13.905, see Table 4.3). It is a principle, that the instability region is situated in the range of negative q, where inviscid spiral modes are stable. Table 4.4. Dependence of the position and value of the maximum increment as well as the range of unstable values of q and k on Re for axisymmetric viscous mode (Mayer and Powell 1992*) Re 322.45 350 103 104 106

q 1.08 1.08 1.05 0.929 0.400

k 0.468 0.471 0.444 0.280 0.180

qωi max 2.55⋅10–7 1.82⋅10–4 9.03⋅10–4 2.23⋅10–4 3.07⋅10–6

qunst – 1.01...1.15 0.83...1.26 0.54...1.29 0.12...1.30

k unst – 0.33...0.63 0.08...1.06 0...1.29 0...1.41

Table 4.5. Dependence of the position and value of the maximum increment as well as the range of unstable values of q and k on Re for spiral viscous mode m = –1 (Mayer and Powell 1992) Re 17.6 20 100 1000

q –0.477 –0.510 –0.475 –0.240

k 0.338 0.369 0.418 0.210

ωi max 8.17⋅10–5 2.55⋅10–3 9.61⋅10–3 1.66⋅10–3

qunst k unst – – 0.26...0.7 0.14...0.55 0...1.08 0...1.13 0...1.08 0...1.14

Fig. 4.38. Dependence of neutral curves for axisymmetric mode on Reynolds number (Mayer and Powell, 1992*)

214

4 Stability and waves on columnar vortices

Fig. 4.39. Isolines ωi = const for the spiral viscous mode m = –1 (Mayer and Powell 1992*)

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex) The Q-vortex, the stability of which was analyzed in the previous Section, is a very convenient model of real columnar vortices with smoothed velocity profiles. Nevertheless, this is a particular case. In the more general form, the velocity field of a columnar vortex can be introduced as the 5parametrical family

⎫ ⎪ Γ ⎡ ⎪ V (r ) = 1 − exp(−αr 2 ) ⎤ , ⎬ ⎣ ⎦ 2πr ⎪ W (r ) = W∞ + W0 exp(−β r 2 ), ⎭⎪ U = 0,

(4.65)

where Γ, α, W∞, W0, β are constants. In corresponding dimensionless form with Γ/2π = q, α = β = 1, we have the Q-vortex (see (4.62)). This Section deals with the waves on columnar vortices with the velocity fields described by (4.65). At that, parameters can be different, particularly when they can be chosen on the basis of experimental data. In contrast to Section 4.6, which deals with linear stability of the Q-vortex, the main purpose of this Section is the analysis of steady nonlinear waves. Linear waves are analyzed as a result of linearization of initial nonlinear equations. The Howard-Gupta equation (4.63) can be used for the complete analysis of linear perturbations on a columnar vortex with an arbitrary velocity profile. Since the equations and approaches to the description of axisymmetrical and bending waves of finite amplitude are

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)

215

significantly different, they will be considered separately. We should also note that in the literature, the wave phenomena on vortices are only studied in connection with an attempt to describe vortex breakdown. We do not consider this problem here. 4.7.1 Axisymmetrical nonlinear standing waves The main motive to study the standing waves in the swirling flow is connected with the wave model of vortex breakdown (Benjamin 1962; Leibovich 1984). In the same time the case of axisymmetrical standing waves is distinguished by the fact that the Euler equations are reduced to one partial derivative equation of the elliptical type for stream function ψ, related to radial u and axial w velocity components by relationships u=−

1 ∂ψ , r ∂z

w=

1 ∂ψ . r ∂r

(4.66)

The flow is assumed to be steady and incompressible, then, the equation of mass conservation identically satisfies Eqs. (4.66). Vorticity components (ωr, ωθ, ωz) in cylindrical coordinates (r, θ, z) in view of axial symmetry take the form ωr = −

∂v , ∂z

ωθ = −

∂u ∂w , − ∂z ∂r

ωz =

1 ∂ (rv). r ∂r

(4.67)

Using (4.66), the azimuthal velocity component can be expressed via the stream function 1 ⎛ ∂ 2 ψ ∂ 2 ψ 1 ∂ψ ⎞ ωθ = − ⎜ 2 + 2 − ⎟. r ⎝⎜ ∂z r ∂r ⎠⎟ ∂r

(4.68)

Then, instead of Euler equations, it is more convenient to use GromekaLamb equations (1.13), which are written for the steady axisymmetrical flow of incompressible fluid without mass forces as (see Section 1.3.3) ∂H , ∂r wωr − uωz = 0, ∂H uωθ − vωr = . ∂z

vωz − wωθ =

Here the Bernoulli constant is

(4.69)

216

4 Stability and waves on columnar vortices

H (r , z ) =

1 2 p u + . ρ 2

Taking into account the definitions of vorticity components (4.67), the second of the above equations is written as u

∂ ⎞ ∂ ∂ ⎛ (rv) + w (rv) = 0 or ⎜ u ⋅ ⎟ (rv) = 0, ∂r ∂z ⎝ ∂r ⎠

or in the most common form

d (rv) = 0, dt

(4.70)

where d/dt is a substantial derivative. This format indicates constant circulation over a fluid circuit in the form of a circumference with radius r in the plane normal to the axis of symmetry. We introduce the notation

rv = F(ψ).

(4.71)

It is clear that this is circulation around the axis of revolution without coefficient 2π. Since the trajectories of liquid particles coincide with the streamlines for the steady flow, it follows from the condition (4.70) that value F is determined by the value of stream function ψ. The same relates to Bernoulli constant H, i.e.,

(

)

p 1 2 u + v 2 + w 2 + = H (ψ ) . ρ 2 Using definitions (4.66) and (4.71), vorticity components in the axial plane can be written as

ωr = u

dF , dψ

ωz = w

dF . dψ

Since ωr/ωz = u/w, the components of velocity and vorticity vectors are parallel in the axial plane. From the first (or third) equation of system (4.69) we can find the expression for ωθ depending on H and F ωθ =

1 ∂H v dH F dF . + ωr = −r + u ∂z u dψ r dψ

Comparing this expression with (4.68), we obtain the equation for stream function ψ (see also (1.57))

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)

∂2ψ ∂z

2

+

∂2ψ ∂r

2

−

1 ∂ψ dH dF . = r2 −F r ∂r dψ dψ

217

(4.72)

Introducing a new independent variable y = r 2, we get a more convenient form of equation

∂2ψ ∂z

2

+ 4y

∂2ψ ∂y

2

=y

dH dF . −F dψ dψ

The equation of (4.72) type was used for the analysis of vortex flows in many studies (e.g., see Long (1953), Fraenkel (1956), Squire (1956), Benjamin (1962), Pritchard (1970), Batchelor (1967), Leibovich (1986), Leibovich and Kribus (1990)). As noted in the most recent of these, the equation (4.72) was first derived by Bragg and Hawthorne (1950). Therefore, it is suggested to refer to it as the equation of Bragg-Hawthorne (or BHE). To solve equation (4.72), first it is necessary to determine functions H(ψ) and F(ψ). This procedure is easy for the simplest case, when for some value of z = z0, there is no the radial velocity component, and the radial distributions of the remaining two velocity components are known (Benjamin 1962), i.e., u(r, z0) = 0, v(r, z0) = V(r), w(r, z0) = W(r) ≠ 0.

(4.73)

Let conditions (4.73) be assigned. Then, from Euler equations at z = z0 we have 1 ∂p V 2 (r ) . = r ρ ∂r

Then, we find the stream function from (4.66) at z = z0 r

∫

ψ(r , z0 ) = rW (r )dr ≡ Ψ (r ) , 0

and the inverse dependence is determined

r = R(ψ). At z = z0 W is the function of ψ, therefore W (ψ ) =

1 dψ 1 dΨ (r , z0 ) = ( R( ψ ) ) . r dr R(ψ ) dR

(4.74)

218

4 Stability and waves on columnar vortices

From (4.74) and definitions of H and F we obtain the required expressions

p( ψ ) 1 = p(0, z0 ) + ρ ρ

R( ψ )

∫ 0

V 2 (r ) dr , r

F(ψ) ≡ R(ψ)V(R(ψ)), H (ψ ) =

p 1 2 (V + W 2 ) + , ρ 2

dH dH dR 1 dW F dF . = = + 2 dψ dR dψ R dR R dψ

Particularly, the problem statement described above was used by Leibovich and Kribus (1990) in search of solutions to (4.72) in the form of the standing periodic and single waves. Here are the main conclusions of the mentioned work. For more convenient analysis it is assumed that the functional form of azimuthal velocity V(r) at z = z0 is fixed, but the level changes, i.e., in the dimensionless form we can present F as

F(ψ) = λf(ψ), where f(ψ) is the dimensionless fixed function, and λ is the regulating constant. For corresponding normalization we have

λ = V0/W0.

(4.75)

Here V0 and W0 are the typical values of the azimuthal and axial velocity components. Hence, quantity λ indicates the level of flow swirling. We consider the flow in a round tube. The initial flow, which is the solution to BHE, is assumed to be the columnar

ψ(r, z0) ≡ Ψ(r). We search for other solutions periodical in z, with the wavelength of L. The static bifurcational analysis is applied for their search. Since calculations use numerical methods, the initial flow should be detailed. A particular case of the family (4.65) was chosen

W(r) = 1, V(r) = λ{1 – exp(–αr2)},

(4.76)

which is known as the Burgers – Rott vortex. Formulae are written in the dimensionless form, and parameter λ corresponds to definition (4.75). However, even before specification of the initial flow field, some general conclusions can be made. Thus, it is shown that at L → ∞ perturbation

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)

219

of the stream function Φ ≡ ψ − Ψ (with some accuracy) is written as Φ = A(Z)φ0(r), where φ0(r) is the eigenfunction of the linearized problem for the minimum proper value, and A(Z) satisfies the equation d2 A dZ2

+ αA2 + BA = 0.

(4.77)

Here Z is the “slow” axial variable; α, β are parameters. This equation was first derived by Benjamin (1967). It assumes the solution is in the form of a single wave

⎡⎛ 1 ⎞ A = a sech 2 ⎢⎜ aα ⎟ ⎠ ⎣⎢⎝ 6

1/ 2

⎤ Z⎥ ⎦⎥

(4.78)

under the condition aα > 0. The constant a is related to the parameters of Eq. (4.77). It is necessary to note that the solution to (4.78) correlates with the solution to the Korteweg-de Vries equation derived by Leibovich (1970) for weakly- nonlinear waves of a swirling flow in a tube (see also Section 4.7.2). For the case of a certain velocity field (4.76), numerical calculations of Leibovich and Kribus (1990) allowed for the following conclusions. On the basis of initial columnar vortex (4.76), four types of new solution can be developed by the method of bifurcation. Type I consists of the columnar flows and one of them is of particular interest. This flow is called the “principal conjugate” and branches off the initial flow at some critical swirl level. It is shown that this flow is supercritical, if the initial flow is sub-critical, and vice versa. The idea of conjugate flows as well as supercritical and sub-critical states for the swirl flows, were introduced by Benjamin (1962) in his theory of vortex breakdown. We should remember that under the sub-critical state, infinitely small perturbations can propagate both up and down the flow, and under the supercritical state, they can only propagate down the flow. Solutions of type II are the single standing waves, which exist at supercritical values of parameters, and at large distances upwards and downwards, the flow coincides with the initial one. These solutions are significantly nonlinear and admit the stagnation point and reverse flows similar to those existing at vortex breakdown. Simultaneously, it is interesting that weakly-nonlinear solutions (see (4.77), (4.78), and Leibovich (1970)) are a good approximation for this type of solution. Periodic standing waves available at sub-critical parameters relate to type III. For instance, if we increase the wavelength at a constant level of

220

4 Stability and waves on columnar vortices

swirling, depressions take the form of single waves, and tops become smoother and equivalent to the regions of the columnar flow, coinciding with the “principle conjugate” one from solutions of type I. At the limit we have solutions of type IV: a single wave, which is supported by the “principle conjugate” flow, but not the initial flow as in solutions of type II. It is necessary to note that the analysis of the steady finite-amplitude waves, as described above, cannot be considered as complete, since the question on stability of nonlinear wave modes against 3-D perturbations arises. Similar investigations were conducted by Kribus and Leibovich (1994) with consideration of viscosity effects. The main conclusion indicates that the finite-amplitude periodic and single waves on the columnar vortex are stable to axisymmetrical perturbations, but they are unstable to bending modes, if the wave amplitude exceeds the critical value. 4.7.2 Axisymmetrical weakly-nonlinear traveling waves Now, let us consider the general case of axisymmetrical waves on the columnar vortex, bounded by a cylindrical surface (the swirling flow in a tube). As in the previous Section 4.7.1, expressions for the stream function (4.66) and vorticity components (4.67), (4.68) in cylindrical coordinates are as before. The notation for circulation F also remains

F(r, z, t) = rv(r, z, t),

(4.79)

however, here it is the function of two coordinates and time. For derivation of governing equations, we again use the equation of ideal fluid motion in the form of Gromeka-Lamb, but the unsteady ones (see 1.36). From the second equation of system (1.36), with consideration of definitions (4.66), (4.67), condition ∂H/∂θ = 0 and new variable, y = r2, we obtain ∂F ∂ψ ∂F ∂ψ ∂F −2 +2 = 0. ∂t ∂z ∂y ∂y ∂z

(4.80)

Then we exclude Bernoulli function H from the first and third equations of system (1.36) by means of cross differentiation ∂ ⎛ ∂u ∂w ⎞ ∂ ∂ − ⎜ ⎟ = (vωz − wωθ ) − (uωθ − vωr ) . ∂t ⎝ ∂z ∂r ⎠ ∂z ∂r

Substituting vorticity and velocity components through expressions (4.66) – (4.68), (4.79) and again using variable y, we obtain the second governing equation

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)

D2

∂ψ 2 ∂F ∂ψ 2 ∂ψ ∂ψ ∂ ⎡ 1 2 ⎤ + F +2 − 2y D D ψ ⎥ = 0. ∂t y ∂z ∂y ∂z ∂z ∂y ⎢⎣ y ⎦

221

(4.81)

Here operator D2 is

D2 ≡ 4 y

∂2 ∂y 2

+

∂2 ∂z 2

.

Therefore, the system of accurate nonlinear equations (4.80), (4.81) is derived for the stream function ψ and circulation F. Weakly-nonlinear long waves, described by approximated equations are the subject of investigation in this Section, therefore, for the following analysis it is necessary to employ dimensionless values. For this purpose, according to Benney (1966), Leibovich (1970), we introduce two length scales: transverse b and longitudinal L (e.g., the tube radius and wavelength, correspondingly). The typical value of azimuthal velocity V0 (for example, the maximum) is taken as the velocity scale. Then, the transition to dimensionless quantities is performed in the following manner: y = b 2 y,

z = Lz , t = (L / V0 ) t ,

ψ = b 2V0 ψ ,

D 2 = b 2 D2 = 4 y

F = bV0 F , ∂2 ∂y 2

+ k2

∂2 ∂z 2

.

Retaining the previous notations for use in this Section, we will assume Eqs. (4.80), (4.81) to be dimensionless. It is obvious that the single parameter of this problem is k = b/L, which is included into the definition of operator D2 and means the dimensionless wave number. In the long-wave approximation k 1. Undisturbed solutions to Eqs. (4.80), (4.81) are y

F = F(y),

1 ψ= W ( y)dy , 2

∫ 0

where F(y) and W(y) are the arbitrary functions, corresponding to the undisturbed columnar vortex. The following aim is the search for solutions in the form of axisymmetrical long waves of small amplitude. Thus, we assume that

222

4 Stability and waves on columnar vortices y

1 ψ = W ( y)dy + εϕ( y, z, t), 2

∫ 0

F = F(y) + εf(y, z, t), where ε 1. According to Benney (1966), Leibovich (1970), we make expansions of ϕ and f to the first order by ε and k2 in the form of

ϕ = φ0 ( y) A( z, t) + εφ1 ( y)

1 2 A + k2 φ2 ( y) Azz + ..., 2

f = γ 0 ( y) A( z, t ) + εγ1 ( y)

1 2 A + k 2 γ 2 ( y) Azz + ... . 2

Substituting ψ and F into (4.80) and (4.81), we obtain in the first approximation by ε and k2 respectively

ft − 2F ′ ϕz + Wfz + 2ε(ϕy fz − ϕz fy ) = 0, ϕtyy +

F 2y 2 +

fz + W ϕzyy − W ′′ϕz + ε 2y2

k2 k2 ϕtzz + W ϕzzz + 4y 4y

ffz + 2εϕy ϕzyy − 2εϕz ϕyyy = 0.

Here the subscripts mean the corresponding partial derivatives and the prime means the derivative with respect to y. Now, substituting expansions for ϕ and f into these equations, we have the following system of equations: γ 0 At + εγ1 AAt + k2 γ 2 Atzz − 2F ′ ⎡φ0 Az + εφ1 AAz + k2φ2 Azzz ⎤ + ⎣ ⎦ +W ⎡ γ 0 Az + εγ1 AAz + k2 γ 2 Azzz ⎤ + 2ε [ φ0′ γ 0 − φ0 γ 0′ ] AAz = 0 , ⎣ ⎦ φ′′0 At + εφ1′′AAt + k2 φ′′2 Atzz +

F ⎡ γ A + εγ1 AAz + k2 γ 2 Azzz ⎤ + 2 ⎣ 0 z ⎦ 2y

+W ⎡ φ0′′ Az + εφ1′′AAz + k2 φ2′′ Azzz ⎤ − W ′′ ⎡φ0 Az + εφ1 AAz + k2 φ2 Azzz ⎤ + ⎣ ⎦ ⎣ ⎦ +

k 2 φ0 k2W φ0 εγ 2 Atzz + Azzz + 02 AAz + 2ε(φ0′ φ0′′ − φ0 φ0′′′) AAz . 4y 4y 4y

This system of equations is solvable relative to functions φi, γi, (i = 0,1,2) in the mentioned approximation, if A satisfies the equation

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)

At = −c0Az + εc1AAz + k2c2Azzz,

223

(4.82)

where coefficients c0, c1, c2 should be determined. Initially, let us substitute At into the first equation. Then, in the first approximation, equating to zero the expressions at ε and k2 and terms of zero infinitesimal order, we find γi as functions of φi γ0 =

γ1 =

2F ′ φ0 , W − c0

⎧ ⎡ F ′ ⎤′ ⎫⎪ c1φ0 F ′ 2 ⎪ + 2φ02 ⎢ ⎨φ1F ′ − ⎥ ⎬, W − c0 ⎪ W − c0 W − c0 ⎦ ⎪ ⎣ ⎩ ⎭ γ2 =

⎤ c2 2F ′ ⎡ φ0 ⎥ . ⎢φ2 − W − c0 ⎣ W − c0 ⎦

Substituting γi and At in the second equation and performing the above procedure, we derive the ordinary differential equations for the functions φi ⎫ ⎪ Lφ0 = 0, ⎪ ⎪ Lφ1 = 2 ⎡ c1φ0 S − φ02Q⎤ , ⎬ ⎣ ⎦ ⎪ φ0 Lφ2 = 2c2φ0 S − . ⎪ ⎪⎭ 4y

(4.83 a, b, c)

Here operator L is L= q( y ) =

where

d2 dy 2

+ q( y ) ,

FF ′

y 2 (W − c0 ) 2

−

W ′′ . W − c0

Correspondingly S= Q=

q W ′′ 1 , + W − c0 2 (W − c0 ) 2

2 (W − c0 )

3/ 2

( yW ′) ′′ d ⎡ . y W − c0 q ⎦⎤ + ⎣ dy (W − c0 )2

Using the kinematic conditions u(0) = u(1) = 0, we determine the boundary conditions

224

4 Stability and waves on columnar vortices

φi(0) = φi(1) = 0

(i = 0,1,2).

(4.84)

Now, let us determine the coefficients c0, c1, c2. At the given W(y), F(y) c0 is the eigenvalue for the problem (4.83 ), (4.84). Values of c1 and c2 are chosen to make the right-sides of Eqs. (4.83b,c) orthogonal to function φ0. Otherwise, non-uniform problems (4.83b), (4.83c) and (4.84) have no solutions. Thus, we obtain

⎫ ⎪ ⎪⎪ 0 ⎬ 1 ⎪ 1 −1 2 c2 = y φ0 dy / ∆, ⎪ 8 ⎪⎭ 0 1

∫

c1 = φ30 Qdy / ∆,

(4.85)

∫

1

where

∫

∆ = φ02 Sdy . 0

Since expressions for Q and S contain denominators (W – c0), at W = c0 integrals may diverge. However, the theorem of Chandrasekhar (Chandrasekhar 1961) states that there are at least two values of c0, one of them is lower than min W(y), and another is higher than max W(y) in the range (0,1), and integral divergence can be avoided. Another method of deriving the solution, applied by Leibovich (1970) is possible. The values of c0 and F(y) are fixed. Then, if we notate W(y) = µm(y), where m(y) is the fixed function, µ can be considered as the eigenvalue (see also Chandrasekhar (1961)). Knowing coefficients ci, we can easily write a soliton solution to the Korteweg-de Vries equation (4.82). It is more convenient to do this in coordinates x ≡ z - c0t, τ ≡ εt (slow time). Then, Eq. (4.82) takes the form Aτ = c1 AAx +

k2 c2 Axxx . ε

(4.86)

Remembering that ε = O(k2), and assuming that ε = k2 without loss of generality, we can write the soliton solution in the known form ⎡ 1 ⎛ c a ⎞1/ 2 ⎛ 1 ⎞⎤ A = a sech ⎢ ⎜ 1 ⎟ ⎜ x + ac1τ ⎟ ⎥ , 3 ⎢ 2 ⎝ 3c2 ⎠ ⎝ ⎠ ⎥⎦ ⎣ 2

where a = const. In case of the radially-unbounded space, the above procedure becomes invalid due to the appearance of singularity. Therefore, another approach

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)

225

is used (Leibovich 1970). It is assumed that vorticity is concentrated in the vortex core, and far from the core the flow is potential. Perturbations are assumed to be axisymmetrical and long. Solutions for the inner and outer areas are derived separately using the method of asymptotic matching and corresponding boundary conditions. As a result, we obtain the integraldifferential equation ∞

A ( ξ, τ ) dξ c3 ∂3 , Aτ = c1 AAx + 3 1/ 2 2log (1 k ) ∂x 2 2 −∞ ⎡( x − ξ ) + k ⎤ ⎣ ⎦

∫

(4.87)

which describes propagation of weakly-nonlinear axisymmetrical waves along the core of a columnar vortex in an unbounded space. Here, quantities A, x, τ have the same sense as in Eq. (4.86). As before, in parameter k = b/L, the longitudinal scale L is identified with the wavelength, and the typical size of the vortex core is taken as the transverse scale b, instead of the tube size. Coefficients c1 and c3 are written as (compare with (4.85)) ∞

1 2 c3 = φ∞ ∆∞ , 4

∫

c1 = φ30Qdy ∆ ∞ , 0

∞

where

∆∞ =

∫ φ0 Sdy, and φ 2

0

→ φ∞ at y → ∞.

−∞

Accurate within derivation error, Eq. (4.87) can be rewritten as (Leibovich and Randall 1972) Aτ = c1 AAx + c3

−1 ∞ 1⎞ ∂ 3 ⎧⎪ ∂A ⎫⎪ ⎛ 2log log ( 2 x − ξ ) sign ( x − ξ ) A dξ ⎬ . + ⎜ ⎟ 3⎨ k⎠ ∂ξ ⎪ ∂x ⎪⎩ ⎝ −∞ ⎭

∫

This form is convenient because at the limit of infinitely long waves (k → 0) the second term in braces becomes zero, and thus, we again come to the Korteweg-de Vries equation (4.86). 4.7.3 Bending waves The bending waves are characterized by the fact that they cause a shift of the vortex axis relative to the undisturbed position. Azimuthal wave number m = ±1 corresponds to the bending modes (see Section 4.1). The linear bending waves were considered above for the particular cases at analysis of Rankine vortex and Q-vortex stability. In the current Section we con-

226

4 Stability and waves on columnar vortices

sider the linear neutral steady and weakly-nonlinear bending waves on the columnar vortices of the Q-type. The linear analysis is carried out using Howard – Gupta equation (HGE) as for the Q-vortex. In contrast to the case of Q-vortex, new solutions for neutral perturbations, corresponding to so-called “slow” and “fast” waves, are presented. Weakly- nonlinear bending waves are described by the nonlinear equation of Schrödinger, and at the limit of infinitely small wave numbers, they are described by the nonlinear integral-differential equation. It is shown that soliton solutions to the Schrödinger equation exist only within certain ranges of the carrier wave numbers. The results presented are based on the works by Leibovich and Ma (1983), Leibovich (1986), Leibovich et al. (1986), they are presented here without detailed computations due to their awkwardness. We will assume that the undisturbed velocity field of the columnar vortex U = (0, V(r), W(r)) is assigned to the family (4.65). For the specific examples, the flow parameters take the following values: Γ = 2π, α = 1, W∞ = W0 = 0 (flow A); Γ = 1.39 2π, α = 1.28, β = 0.54, W∞ = 0, W0 = 0.4 (flow B). For the last example we used the empirical velocity profile from Maxworthy et al. (1985). Here, in Section 4.7.3 all quantities are reduced to the dimensionless form via the typical scale of tangential velocity V0 and the typical radial scale a0. In the general form, the velocity field in cylindrical coordinates is written as v = U(r) + εu(r, θ, t; ε),

(4.88)

where ε is the small amplitude parameter. Perturbation for the velocity vector is expanded into a series in ε

u = u1 + εu2 + ε2u3 + …

(4.89)

Similar expansions are applied to the pressure. Linear neutrally stable bending waves

For the linear analysis we use Eq. (4.88) only. The solution is found in the form of a traveling wave u = A uˆ 0 (r ) exp[i(kz + mθ − ωt)] + c.c., ⎪⎫ ⎬ P = A Pˆ0 (r ) exp[i(kz + mθ − ωt)] + c.c., ⎪⎭

(4.90)

where c.c. is the complex-conjugate value, m = ±1; k is the real wave number; ω is the real frequency. For fixed k and m under corresponding boundary conditions, the linear problem represents that of the eigenvalues ω = ω(k, m). Substituting (4.88) – (4.90) into the Euler equations and mass

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)

227

conservation equation, and linearizing them, we obtain the Howard – Gupta equation (4.63), as derived above. It can be shown that for the neutrally stable bending modes, the condition of symmetry ω(−k, −m) = − ω(k, m) is satisfied. The Howard – Gupta equation is true for arbitrary wavelengths, and in the general case, it requires numerical methods of solution. Numerical calculations of Leibovich et al. (1986), performed for partial basic velocity profiles, revealed several branches of dispersion relationship for neutral perturbations. The first branch has an asymptote: frequency ω → 0, phase velocity c → 0 and group velocity cg = dω/dk → 0 at k → 0. Waves, relating to this branch were called the “slow” waves. The asymptote of other branches is ω → mΩ(0), c → ∞ at k → 0, where Ω(0) = V(r)/r at r = 0. The corresponding waves are known as “fast” waves. Now, we will analyze different types of waves separately. First, let us examine the slow waves. According to the experiment, the long waves are of greatest importance. Therefore, it is convenient to have the asymptotic expression for ω at k 1, which is taken from the mentioned papers without derivation ω=−

⎤ 2 mΓ 2 ⎡ k ⎢ log + K − E ⎥ . 4π ⎣⎢ k ⎦⎥

(4.91)

Here, the Euler constant is E = 0.5772…, and constant K for the profile family (4.65) at W∞ = 0 is

K=

4π2W02 ⎤ 1⎡ ⎛1 ⎞ log E α + − ⎢ ⎜ ⎥. ⎟ 2 ⎢⎣ ⎝ 2 ⎠ βΓ 2 ⎥⎦

We should note that previously the dispersion relationship in the form of (4.91) was approximately derived by Moore and Saffman (1972) on the basis of the Biot-Savart law and the cut-off method with reference to the helical vortex filament (see Sections 5.1, 5.2). The dispersion curves numerically calculated by Leibovich et al. (1986) for mode m = −1, are shown in Fig. 4.40 ,b for the basic velocity profiles A and B. The asymptotic formula (4.91) correlates well with numerical calculation in the range of ⎢k⎪ < 0.2. For vortex B, dispersion curves are not symmetrical relative to axis k = 0. It can be recalculated for m = 1 on the basis of condition ω(−k, −m) = − ω(k, m). According to comparison of diagrams and b, the velocity profile of the basic flow significantly affects the wave characteristics. The infinite number of modes, characterized by number M, taking integer values M = 0, 1, 2, 3..., was found for the fast waves. The first five

228

4 Stability and waves on columnar vortices

modes are shown in Fig. 4.41. The feature of the “fast” modes is that their eigenfunctions damp fast with distance from the vortex axis. It can be assumed that they differ from zero in the area of radial distances, which are significantly less than the size of the vortex core. Taking into account this condition, Leibovich et al. (1986) derived the asymptotic formula for the dispersion relationship in the case of long waves (k 1), with an arbitrary value of m ω = mΩ0 + kW0 + kΩ0 ( mW2 − 2kΩ0 ) ( mΩ 2 M ) .

(4.92)

Fig. 4.40. Dependency of frequency ω, phase c and group cg velocity on wave number k for the “slow” waves at m = –1. Numerical calculation by Leibovich et al. (1986*)

Fig. 4.41. First five “fast” modes for vortex A at m = –1. Numerical calculation by Leibovich et al. (1986*)

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)

229

This formula is true for kΩ0(2kΩ0 – mW2) > 0. Here

Ω0 = Ω(0), Ω2 = Ω″(0), W0 = W(0), W2 = W″(0),

(

1 ⎡1 M = ⎢ m + m2 + 8 2 ⎣2

)

1/ 2

(

1 ⎤ ⎡1 + M ⎥ ⎢ m + m2 + 8 2 ⎦⎣2

)

1/ 2

⎤ + M + 1⎥ , ⎦

M = 0, 1, 2, 3..., the prime means the derivative with respect to r. The values of parameters Ω0, Ω2, W0, W2 are shown in Table 4.6 for vortices of type A and B, and for the Q-vortex at two values of parameter q. For bending mode m = −1 and main branch M = 0 Eq. (4.92) takes the form ⎛ ΩW ⎞ Ω2 ω = −Ω0 + ⎜ W0 + 0 2 ⎟ k + 0 k2 , kΩ0 ( 2 KΩ0 + W2 ) > 0. 6Ω 2 ⎠ 3Ω 2 ⎝

(4.93)

Expressions for the phase and group velocities follow from this formula

c=

Ω ⎛ Ω W ⎞ Ω2 ω = − 0 + ⎜ W0 + 0 2 ⎟ + 0 k2 , k k ⎝ 6Ω 2 ⎠ 3Ω 2

(4.94)

ΩW dω 2 Ω02 k. = W0 + 0 2 + dk 6Ω 2 3 Ω2

(4.95)

cg =

Comparison with numerical calculations indicates that the asymptotic formulae are acceptable for a wide range of wave numbers. Therefore, wave characteristics of “fast” waves calculated using (4.93)–(4.95) are shown in Fig. 4.42 in the range of moderate k = 0 ÷ 0.5. It is seen that at k → 0, the phase velocity truly tends to infinity. In general, the behavior of dispersion curves differs significantly from the case of the “slow” waves (see Fig. 4.40) both in shape and numerical values. Particularly, at low k |cg| |c|, and frequency does not depend on k. Table 4.6. The values of parameters from (4.92), for vortices of type A, B, and Q-vortex Parameter A

B

Ω0 Ω2 W0 W2

1.78 –2.28 0.40 –0.432

1 –1 0 0

Q-vortex q = 1.0 q = 0.4 1 0.4 –1 –0.4 1 1 –2 –2

230

4 Stability and waves on columnar vortices

Fig. 4.42. Dependency of frequency ω, phase c and group cg velocities on wave number k for the “fast” waves at m = –1, M = 0 on the vortex of type A. Calculation by asymptotic Eqs. (4.93) – (4.95)

The asymptotic formulae are also applicable to the Q-vortex. If we compare the calculation of Lessen et al. (1974) in Fig. 4.35 for the phase velocity of unstable perturbations on the Q-vortex with Eq. (4.94), it becomes obvious that the results are identical. Thus, we can conclude that the perturbations described by Lessen et al. are related to the “fast” waves (the main mode) by the classification of Leibovich et al. (1986). However, the question about the imaginary part remains unsolved because the first work deals with unstable perturbations (ci ≠ 0), and the second work considers neutral perturbations (ci = 0). As mentioned above, the long waves are of particular interest, since they are observed in the experiment. Therefore, asymptotic Eqs. (4.91), (4.92) of the analytical form are convenient for the analysis of kinematics and geometry of the bending waves on the vortex. The condition of phase constancy has the form

kz + mθ – ωt = const, |m| = 1. It describes the behavior of the helical structure of the fixed radius r in space and time. In the case of a “slow” wave with consideration of (4.91) we have

kz + mθ + m

⎤ Γ 2⎡ ⎛ 2 ⎞ k ⎢log ⎜ ⎟ + K − E ⎥ t = const, 4π ⎣⎢ ⎝⎜ k ⎠⎟ ⎦⎥

m = 1.

Thus, it follows that at fixed z and any r, the helical structure (the surface of a constant phase) rotates with the angular velocity

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)

231

⎡ ⎛2⎞ ⎤ dθ Γ = − k2 ⎢log ⎜ ⎟ + K − E ⎥ , 4π ⎢⎣ ⎜⎝ k ⎟⎠ dt ⎥⎦ independently of the sign of m. For clear definition, we assume that the direction of the axial vorticity component of the basic flow coincides with the direction of axis z. Then, Ω0 > 0, Γ > 0. Hence, the helix rotates in the direction opposite to the fluid rotation in the basic vortex. Then, we obtain

dθ k =− , dz m ⎤ dz Γ ⎡ ⎛2⎞ = c(k) = −mk ⎢log ⎜ ⎟ + K − E ⎥ . 4π ⎢⎣ ⎜⎝ k ⎟⎠ dt θ=const ⎥⎦ According to these relationships, if mk < 0, the wave propagates towards axis z, and dθ/dz > 0. The last condition means that the helix is righthanded. In other words, the right-handed structure propagates along with the axial vorticity of the basic vortex, and vice versa, the left-handed structure moves in the opposite direction. This rule is schematically shown in Fig. 4.43 . Using (4.92) at k < 0 for the “fast” waves, the condition of phase constancy is written as

kz + mθ – mΩ0t = const.

Fig. 4.43. The scheme of bending wave motion on the columnar vortex: – “slow” mode; b – “fast” mode (the motion depends on the sign of W″(0)). In the vortex crosssection: 1 – direction of liquid particle rotation; 2 – direction of phase front rotation

232

4 Stability and waves on columnar vortices

Therefore, the angular velocity of helix rotation is

dθ = Ω0 , dt i.e., at z = const, the helix rotates as the liquid particles near the axis of the basic vortex. Respectively dθ k =− , dz m

dz m = Ω0 , dt θ=const k

the condition from (4.92) should be also satisfied: –kmΩ0W2 > 0. If W2 = 0, the left-handed helix propagates towards axis z, and the righthanded one – in the opposite direction. If W2 ≠ 0, it is necessary to consider two cases: I – W2 < 0, II – W2 > 0. At W0 > 0 (I) corresponds to the jet velocity profile, and case II corresponds to a profile of the wake type. At W0 < 0, the situation is inverse. In the case of I, the above restriction allows the waves with km > 0 only, i.e., this is the left-handed structure moving towards axis z. In the case of II, the waves with km > 0 are acceptable, and this indicates the right-handed helix, propagating in the negative direction of axis z. These conclusions are also schematically shown for W2 ≠ 0 in Fig. 4.43b. Weakly-nonlinear bending waves. Solitons

The model equations for nonlinear waves are derived in the following manner (Leibovich et al. 1986). In expansion (4.89), for perturbation of the velocity vector we assume that u1 identically coincides with the solution of (4.90) for the linearized problem at the fixed values of k = k0 and m = m0. Then, we substitute expansions for the velocity (4.89) and pressure into equations of Euler and mass conservation, and search for the terms of a higher infinitesimal order u1 and u2. However, expansion (4.89) is not well defined at εt = O(1), until the secular terms are unsuppressed. This can be done, assuming that amplitude A in (4.90) is a function, slowly changing in space and time. This presentation means modulation of the initial signal with the wave number k0, and it is accompanied by the introduction of “slow” variables

Z = ε(z − cgt), τ = ε2t. Partial solutions to the non-uniform differential equations for u2 and u3 exist, if the right-sides are orthogonal to the conjugate linear problem. The

4.7 Linear and nonlinear waves in columnar vortices (like Q-vortex)

233

expression for the group velocity cg is determined from the above requirement, and the equation for A is derived. This is the nonlinear cubic Schrödinger equation i

∂A ∂2A 2 + µˆ + νˆ A A = 0 . 2 ∂τ ∂Z

(4.96)

Here, coefficients µˆ and νˆ are determined via the integral expressions. The group velocity cg and coefficient µˆ can be alternatively found from the linear dispersion relationships cg = ω′(k0), µˆ =1 2 ω′′(k0 ) . As is known, Schrödinger equation (4.96) allows soliton solutions under ˆ ˆ > 0 . Otherwise, the initial perturbation, which has the the condition of µν shape of the envelope of the initial periodic signal with the carrier wave number k0, diverges in time due to the dispersion effect. The single-soliton solution (4.96) is written as

A( Z, τ) = A0 ε

where

Φ=

iΦ

⎧⎪ ⎛ νˆ ⎞1/ 2 ⎫⎪ sech ⎨ A0 ⎜ ⎟ ( Z − Cτ) + ∆ ⎬, 2µˆ ⎩⎪ ⎝ ⎠ ⎭⎪

⎛ A2 νˆ C 2 ⎞ C Z+⎜ 0 − ⎟ τ + Φ 0. ⎜ 2 2µˆ 4µˆ ⎟⎠ ⎝

Here, parameters A0, C, ∆ and Φ0 are determined by the initial conditions. Let us write the corresponding solution to u in the initial physical variables

u(r , θ, t ) = U (r ) + A0 εuˆ 0 (r )exp[i(kz + mθ − ωt )] × × sech[εA0 (νˆ / 2µˆ )1/ 2 ( z − Ct )] + c.c.,

(4.97)

where

⎫ ⎪ εC ⎪⎪ k = k0 + , 2µˆ ⎬ 2 2 εCcg ⎡c A νˆ ⎤ ⎪ ω = ω0 + + ε2 ⎢ − 0 ⎥ . ⎪ 2µˆ 2 ⎦ ⎪ ⎣ 4µˆ ⎭

C = cg + εC,

It is assumed that ∆ = φ0 = 0, and uˆ 0 (r ) is the solution to the linear problem (4.90). As seen nonlinearity provides the corrections of the group velocity, wave number and frequency of carrier waves.

234

4 Stability and waves on columnar vortices

ˆ ˆ > 0 , the soliton solution to (4.97) exists only within Due to restriction µν certain ranges of the carrier waves, depending on the basic velocity profile, and this is the feature of the soliton solution. Thus, Leibovich and Ma (1983) showed that soliton solutions on the main “fast” branch at |m| = 1 and for the vortex of type A exist within the “frame’’ of wave numbers 0.68 P k P 1. Leibovich et al. (1986) determined the soliton “frames” for the “slow” branch, assuming that it better corresponded to the experiment, than the “fast” branch. Their results can be summarized as follows: the ranges of wave numbers, which accept soliton solutions on the “slow” branch at m = −1, are |k| < 0.45; |k| > 0.96 for flow A,

−0.3 < k < 0.8; k > 1.2; k < −1 for flow B. In conclusion of the current Section, we should note that at the limit of infinitely long “slow” waves, dispersion relationship (4.91) has no longer analytic property. Correspondingly, Schrödinger equation (4.96) becomes invalid. A similar problem arises upon consideration of axisymmetrical waves on the vortex in radially- unbounded space (see Section 4.7.2). Using a certain procedure, Leibovich (1970) derived the integral-differential equation (4.87). Using a similar approach, Leibovich et al. (1986) also derived the integral-differential equation, which is valid for infinitely long weakly-nonlinear bending waves on the “slow” branch.

5 Dynamics of vortex filaments

In the previous Chapter we presented an analysis of stability and waves on columnar vortices with different structure. The main condition was that the perturbation amplitude must be much smaller than the size of the vortex core. For the given undisturbed velocity field, the problem of linear stability or linear waves allows the existence of exact solutions (analytical or numerical). The basic equations are the Euler or Navier – Stokes equations. However, in the case of concentrated vortices (vortex filaments), the size of the core is much smaller than any other linear scales – filament curvature radius, wavelength or perturbation amplitude, size of external bodies and channels. Since the nonlinearity is strong, the direct calculation of vortex filament dynamics seems to be a causa mortis. For this problem statement we have to apply approximate methods valid for a thin vortex filament. One of the key methods is the cut-off method (Batchelor 1967; Rosenhead 1930; Crow 1970) related to the use of the Biot – Savart law, and also the method employing the balance of forces (Widnall et al. 1971; Widnall and Bliss 1971; Moore and Saffman 1972). This Chapter deals with these approximate methods and their modifications; we also consider the self-induced motion of vortex filaments with different spatial shape and make the analysis of vortex filament stability for several specific cases.

5.1 Cut-off method In Chapter 2, the velocity field induced by an infinitely thin vortex filament was described through the Biot – Savart law (2.14). It was proven that the curvature of the vortex axis creates self-induced filament motion caused entirely by the local effect (local curvature of the filament). The asymptotic formula (2.36) for the velocity field was deduced. The formula is valid for an infinitely thin vortex filament at the distance r from the axis, which is much smaller than the curvature radius ρ at the given point. However, the direct application of asymptotic formula (2.36) to the calculation of self-induced velocity of the filament is impossible because of logarith-

236

5 Dynamics of vortex filaments

mic divergence at r → 0, i.e., at the vortex axis. One of the approaches to avoid divergence is the assumption that r ≡ a, where a is a certain effective radius of vortex filament (in reality it is different from zero). Then we obtain the formula for the velocity of self-induced motion u u=

2L ⎞ Γ ⎛ κb ⎜ log −1 . a 4π ⎝ ⎠

(5.1)

At the limit of an infinitely thin filament a → 0 we can retain the main term and obtain u∼−

Γ κb log a . 4π

(5.2)

Using the new time coordinate through transfiguration (2.38) or t

Γ ( − log a ) → t , 4π

we obtain the local induction equation (2.39) or (2.40). However, Eq. (5.1) (and certainly Eq. (5.2)) containing a dimensional value under the logarithm, does not fit for the quantitative estimation of vortex velocity. The reason for this is the uncertainty at the limit of integration L and efficient radius of vortex a. In the framework of local induction approximation, this problem is solved on the basis of the “cut-off method”. The essence of this method is that on integration of the Biot – Savart formula over the filament contour we exclude the filament section with length L at both sides from the point considered |s| < L, and the cut-off length L is assumed to be proportional to the vortex radius a L = δa.

(5.3)

Besides, the proportionality coefficient δ = O(1). Then the Biot – Savart integral takes the form u( R) = −

Γ 4π

∫

[δ]

( R − R′) × ds ( R′) R − R′

3

,

(5.4)

where r ≡ R is the coordinate of the considered point on the filament; R′ is the coordinate of an arbitrary point on the filament; [δ] means that interval 2δa is excluded.

5.1 Cut-off method

237

According to the cut-off method, the unknown values δ and a are found through comparison of calculations by Eq. (5.4) with the known solutions gained by other techniques. The form (5.4) was used by Thomson (1883), but the numerical value for the cut-off length was first calculated by Crow (1970) for the simplest case of a constant vorticity in the core. Before deriving the expressions for δ and a, let us make asymptotic estimates for Eq. (5.4). This is an easy task if we use the results from Section 2.4. Since the vicinity of the given point of the filament is responsible for the main contribution into the integral, we can write (5.4) in the form

⎛ Γb L (− κs 2 2)ds ⎞ Γ L ⎟= u ≈ 2⎜ − . κb log 2 3 / 2 ⎜ 4π ⎟ 4π a δ ( ) s δa ⎝ ⎠

∫

(5.5)

Here the upper limit of integration is the same as in Section 2.4 (see (2.34)); the lower limit is the cut-off length δa ≡ L; coefficient 2 appears due to integration over one area instead of two (taking into account the symmetry of the integrand). This integrand was derived in (2.34) and is rewritten here for the case r = y = z = 0, i.e., for the vortex axis. If in (5.5) for the limit a → 0 we keep only the main term, it is transformed into u ∼ – (Γ/4π) κ b log a, which is exactly (5.2). One can see also that Eq. (5.5), although it was derived with the cut-off length excluded from the integration limits, it still coincides with (5.1) with accuracy to a numerical coefficient. For several problems, authors directly employ the asymptotic formulae in the forms (5.1), (5.2), (5.5) in slightly modified view. Parameters L, δ, a may be indefinite (as in case of the LIE method) or be taken from an experiment (see Maxworthy et al. (1985)). We are however, interested in more accurate calculations using (5.4). In this case the result can be compared with ready known solutions, so we could find parameters δ and a. Besides Eq. (5.4), let us use another form of cut-off method developed by Rosenhead (1930): u=−

Γ 4π

∫⎡

( R − R′) × ds ( R′)

2 2⎤ ′2 ⎣( R − R ) + µ a ⎦

3/ 2

.

(5.6)

Here µ is a numerical coefficient; a is the core radius. It was demonstrated (Moore 1972; Saffman 1992), that this formula is more preferable for numerical integration and it is equivalent to (5.4) under a certain connection between µ and δ. The formula for cut-off length was first obtained by Crow (1970) for the simplest case when the vorticity is constant and concentrated in the core

238

5 Dynamics of vortex filaments

with radius a. The author also compared the cut-off method results with known solutions of Lamb (1932) for the velocity of a thin vortex ring and Kelvin (1880) for the frequency of long helical waves on a columnar vortex. Let us consider initially, a vortex ring. According to Lamb, the velocity of its motion is (see 3.2.1) U=

Γ ⎡ 8ρ 1 ⎤ log − ⎥ , a ρ << 1, 4πρ ⎢⎣ a 4⎦

(5.7)

where ρ is the vortex ring radius. Now we calculate the self-induced velocity of the ring by means of the cut-off formula (5.4) using the diagram in Fig. 5.1,

U ≡u=

Γ 4π

2 π−δa ρ

∫

δa ρ

⎡ ⎛ δa ⎞ ⎤ ⎢1 − cos ⎜ ⎟ ⎥ ρ sin(θ 2) Γ ⎝ 2ρ ⎠ ⎥ ⎢ d θ = log ⎢ 8πρ ⎛ δa ⎞ ⎥ a ρ→0 4ρ2 sin 2 (θ 2) ⎢1 + cos ⎜ ⎟ ⎥ ⎝ 2ρ ⎠ ⎦⎥ ⎣⎢

≈

Γ

(5.8)

4ρ

log . ≈ δa a ρ→0 4πρ Here ρ = |R| = |R′|. Equating Eqs. (5.7) and (5.8), we find the coefficient δ and cut-off length L log 2δ = 1/4, δ = 0.642, L ≡ δa = 0.642a.

(5.9)

Fig. 5.1. Diagram for the calculation of self-induced motion of the vortex ring with the cut-off method

5.1 Cut-off method

239

Now we apply the cut-off method to a helical vortex filament, twisted (for better comparison) to the left (see Saffman (1992)). Then we use the formula in Section 2.1 for a helical line, by rewriting it with current symbols and with consideration for left-handed helical symmetry. Then we obtain from (2.7) the formula for the radius-vector of a point on a moving helix (Fig. 5.2)

⎡ ⎤ k R = R ⎢i cos θ + j sin θ − (θ − ωt) ⎥ . γ ⎣ ⎦

(5.10)

Here θ is the parameter with the meaning of polar angle; ω is the frequency; 1/γ = h/2πR is the dimensionless helix pitch; h = 2πl is the helix pitch; R is the radius. We can direct the tangential unit vector t as shown in the Figure, and direct the vorticity vector along t. Then from Eqs. (2.8), (2.9), (2.11) we obtain the formulae for an element of arc length ds, curvature radius ρ and vector t:

⎫ ⎪ ⎪ ds = tds, ⎪ ⎬ ρ = R(1 + γ 2 ) / γ 2 , ⎪ 1 sin θ, − cos θ, 1 γ ].⎪ t= [ ⎪ 1 + 1 γ2 ⎭ ds = − R 1 + 1 γ 2 dθ,

(5.11)

The sign “–” in the formula for ds is used to provide an increase in the arc length s along z axis.

Fig. 5.2. Diagram of a left-handed vortex filament

240

5 Dynamics of vortex filaments

Application condition for the cut-off method is a sider the case γ 1,

ρ. Further, we con-

for future comparison with Kelvin’s solution for long waves on a columnar vortex. Then the condition a ρ, in view of (5.11), takes the form

R/γ2.

a

(5.12)

The Kelvin waves are considered in linear approximation, i.e., for R a, but this condition does not contradict (5.12), since for very long waves we have γ → 0. Without loss of generality, we can calculate the self-induced velocity u at t = 0 and θ = 0. Then R = R i, R – R′ = R[i(1 – cosθ) – jsinθ + kθ/γ] and (5.4) is written in the form

u=−

Γγ 2 4πR

∫

[ δ]

j (θ sin θ − 1 + cos θ) − k γ (cos θ − 1) ⎡ θ2 + 2 γ 2 (1 − cos θ) ⎤ ⎣ ⎦

3/ 2

dθ.

Integration is taken at infinite limits with respect to θ, except the range

θ < θc =

δa

δaγ . R 1 + 1 γ 2 γ→0 R ≈

For γ 1 the integrand can be simplified, so that this integral, with regard to symmetry, can be written as ⎧ ⎫ ∞ ∞ (θ sin θ − 1 + cos θ) (1 − cos θ) ⎪⎪ Γγ 2 ⎪⎪ u=− dθ + k γ dθ ⎬ . ⎨j 2πR ⎪ aδγ θ3 θ3 ⎪ aδγ R ⎩⎪ R ⎭⎪

∫

∫

(5.13)

The integrals in braces can be written through the integral cosine Ci(θc), which in the case of a small argument θc = aδγ/R 1 can be simplified to ∞

Ci(θc ) = −

∫

θc

cos θ dθ ≈ E + log θc , θ

where E = 0.5772 is the Euler constant. Finally, we derive

5.1 Cut-off method

u=−

Γγ 2 ⎧ ⎛ 1 ⎞ ⎛3 ⎞⎫ ⎨ j ⎜ − E − log θc ⎟ + kγ ⎜ − E − log θc ⎟ ⎬ . 4πR ⎩ ⎝ 2 ⎠ ⎝2 ⎠⎭

241

(5.14)

Obviously, the helix rotation frequency ω is determined by the selfinduced velocity u. To find ω, we use the partial time derivative from (5.10): ∂R/∂t = Rωk/γ and we take into account that vector [∂R/∂t – u] is parallel to vector t. The last fact follows from Fig. 5.3, where the shift of filament element for time δt is shown. Note that vector u is nonperpendicular to the filament. This follows from analysis of (5.14) and the expression for t in (5.11) at θ = 0. The vector ∂R/∂t is directed along z axis by its definition. Since vector (–j + k/γ) at θ = 0 is parallel to t, we then have

∂R/∂t – u = A(–j + k/γ), where A is a coefficient. Excluding A through scalar multiplication of the equality by j and k, we obtain the formula for the frequency

ωR = (u ⋅ k)γ + (u ⋅ j). Substituting u from (5.14) and neglecting the small terms of second order in γ, we obtain the formula for the helix rotation frequency ω=

Γk 2 4π

⎡ 1 ⎤ ⎢ − 2 + E + log ( ak ) + log δ ⎥ , ⎣ ⎦

(5.15)

where k ≡ γ/R = 2π/h is the wave number. Note that due to a high pitch (γ 1) the vortex axis is almost parallel to z axis. Obviously, ω < 0 for ak 1. This means (see Eq. 5.10) that the helix rotates to the left. The Kelvin formula (4.58) for long helical waves on a columnar vortex with uniform distribution of core vorticity takes the form (at R ≡ a, Γ = 2πΩR2)

ω =

Γk 2 ⎡− log ( ak ) + 0.366 ⎤⎦ . 4π ⎣

Comparison with (5.15) gives us

δ = 0.642, or log 2δ = 1/4, which coincides with calculation (5.9) based upon vortex ring motion. Let us consider one more specific case – a sinusoidal vortex filament, which rotates as a whole with angular velocity ω around axis z (Fig. 5.4).

242

5 Dynamics of vortex filaments

Fig. 5.3. Diagram for the shift of a vortex filament during time δt

Fig. 5.4. Diagram of plain sinusoidal vortex filament rotating as a whole around axis z

There is no motion along axis z. The sinusoidal line is a superposition of two helical lines with different helical symmetry (see Saffman (1992), p. 234) and it is described in parametric form through relationships

x = R cos ϕ cos (θ − ωt), y = R cos ϕ sin (θ − ωt), z = Rϕ/γ. Here R is the sinusoid amplitude, ϕ is the sinusoid angle, γ = 2πR/h is the dimensionless wave number. Let us reproduce the calculation for selfinduced motion using the cut-off method, which is made simpler than for the helical filament. As usual we assume a high pitch, i.e., γ 1. Without loss of generality, we calculate the velocity for self-induced motion of the filament point with coordinates θ = 0, ϕ = 0 at t = 0. Then we obtain R = R i, R′ = R(i cos ϕ + kϕ/γ). Using formulae from Section 2.1, we obtain

ds = R sin 2 ϕ + 1 γ 2 dϕ ≈ Rdϕ γ , ρ = R/γ2 cos ϕ ,

t = − i γ sin ϕ + k. Substituting the expressions in (5.4) and taking into account that γ obtain the integral

1, we

5.2 Self-induced motion of helical vortex filament with an arbitrary pitch

u=−

Γγ 2 j 4πR

∫

( ϕ sin ϕ − 1 + cos ϕ) dϕ

2 2 2 [δ] ⎡⎣ϕ + γ (1 − cos ϕ) ⎤⎦

3/ 2

243

( ϕ sin ϕ − 1 + cos ϕ) dϕ. Γγ 2 j 2πR aδγ ϕ3 ∞

≈−

∫

R

Here, as in (5.13), integration is carried out for infinite limits except for the cut-off length 2L. Comparison with (5.13) shows that this integral coincides with the first integral in (5.13). Taking into account that the second term in (5.13) vanishes on calculation of ω for a long-wave approximation, we draw the conclusion that rotation frequencies for sinusoidal and helical vortex filaments with large pitch coincide with each other in a first approximation. At the end of this Section we present the formula for δ in the case when the core has an axial component of velocity and vorticity distribution is non-uniform. This formula can be easily obtained through comparison of (5.8) with the appropriate analytical solution for the vortex ring (3.20) a

a

1 4π 2 8π2 log 2δ = − 2 v2rdr + 2 w2rdr , 2 Γ 0 Γ 0

∫

∫

(5.16)

where v and w are the tangential and axial components of velocity in the vortex core, correspondingly. Without axial flow and at constant vorticity in the core we obtain from (5.16) that ln 2δ = 1/4, which coincides with (5.9). Analysis performed by Moore and Saffman (1972) demonstrated that the cut-off method with cut-off parameters given by ratio (5.16) is valid with the accuracy of O(a/ρ).

5.2 Self-induced motion of helical vortex filament with an arbitrary pitch Consider a thin vortex filament of helical shape in infinite space with the assumption ε ρ, but without restrictions on the helix pitch (Fig. 5.5). The vorticity is uniform, concentrated in a round core with radius ε and directed along to the tangent to the vortex filament centerline. There is no axial flow in the core. The calculation is performed using the Biot – Savart equation (2.14), avoiding singularity in the Biot – Savart integral by the method of Moore and Saffman (1972), which is described more completely by Ricca (1994). The essence of the method is a special form for the writing of self-induced velocity u (see also the explanatory diagram in Fig. 5.6)

244

5 Dynamics of vortex filaments

u( R) =

( R − R′ ) Γ ⎢⎡ ( R − R′) ds − t × ds t× 3 3 4π ⎢ ′ ′ − R R − R R ⎣

∫

⎤ ⎥ + u ( R). ⎥ ⎦

(5.17)

Here R is the point on the filament where we calculate the self-induced velocity; subscript refers to the auxiliary vortex ring. The first integral is the Biot – Savart integral for a helical filament; the second integral is the same for the vortex ring touching the filament at point R; it has the same curvature radius as the filament at R, but the opposite direction of vorticity. Obviously, the difference of these two integrals eliminates the singularity. The last term is the known expression for velocity of the same vortex ring, but with the opposite sign of vorticity. Therefore, addition or subtraction of a vortex ring in (5.17) does not change the result for a helical filament, but this helps us to eliminate the singularity (a formal proof follows). The applicability condition remains ε ρ, where ρ is the filament curvature radius (Moore and Saffman 1972). This procedure can be considered as a modification of the cut-off method, since a part of the filament is substituted by a part of vortex ring. In this framework a similar approach was developed in the paper by Widnall et al. (1971).

Fig. 5.5. Diagram for a helical vortex filament

Fig.5.6. Geometric interpretation of Moore and Saffman (1972) method: the self-induced velocity of helical vortex H (a) is split into two components through subtracting (b) and adding (c) of the touching vortex ring L of the equivalent curvature (Ricca 1994)

5.2 Self-induced motion of helical vortex filament with an arbitrary pitch

245

For clear definition, we consider a right-handed helix. Let us notate the key relationships and definitions based upon formulae from Section 2.1, using the current symbols: τ = h/2πa = l/a is dimensionless pitch of helix (coincides with the torsionto-curvature ratio in the helix); ρ = (a2 + l2)/a = a(1 + τ2) is curvature radius; ds = a 2 + l 2 dθ = a 1 + τ2 dθ is element of arc length;

t = (1 + τ2)−1/2 [−i sin θ + j cos θ + τk]; n = [−i cos θ − j sin θ]; b = τ(1 + τ2)−1/2 [i sin θ − j cos θ + k/τ]; R′ = a[i cos θ + j sin θ + kτθ]; R = a[i]; R − R′ = a[i (1 − cos θ) − j sin θ − kτθ]. From this point forward, we are interested only in the bi-normal component of induced velocity, i.e., u(R)⋅b(R), where b(R) = b|θ = 0 = τ(1 + τ2)−1/2 [ − j + k/τ].

Using the set of relationships above, the bi-normal component in the integrand of the first integral in (5.17) takes the form

(

)

−2 ⎡ ( R − R′ ) ⎤ 1 + τ−2 θ sin θ + τ − 1 (1 − cos θ ) ⎥ ds = dθ . (5.18) b ( R) ⋅ ⎢t × 3 3/ 2 ρ ⎢⎣ R − R′ ⎦⎥ ⎡ 2τ−2 (1 − cos θ ) + θ2 ⎤ ⎣ ⎦

For θ → 0 the integrand is written as

1 dθ 2ρ θ

(5.19)

and, indeed, it reflects the existence of logarithmic singularity at the point θ = 0. Now we have to notate the integrand for the second integral in (5.17), i.e., for a vortex ring. This procedure has already been done in 5.1 (see Eq. 5.8). Since the induced velocity for the ring is directed by the bi-normal, then it follows from (5.8) for the new notations ⎡ (R − R b ( R ) ⋅ ⎢t × ⎢ R−R ⎣

) ⎤⎥ ds

3

⎥ ⎦

=

dθ 4ρ sin ( θ

2)

=

dθ 23 / 2 ρ 1 − cos θ

. (5.20)

246

5 Dynamics of vortex filaments

To compare this with the first integral in (5.17), we have to know the relationship between θ and θ . This relationship is given by the condition of equality for arc lengths s ≡ s in the interval [−πρ, πρ] θ = θ / 1 + τ2 .

(5.21)

For θ → 0 we obtain the limiting condition dθ /2ρθ = dθ/2ρθ, which is exactly (5.19). In this way, for the point R′ = R the logarithmic singularities from two integrals vanish. For convenience, we can rewrite the bi-normal component of velocity ub from (5.17) in the form

ub = u ( R ) ⋅ b ( R ) =

Γ ( I1 + I2 ) + u , 4πρ

(5.22)

where the velocity of the vortex ring according to (5.7) is u = u ⋅b =

Γ ⎡ 8ρ 1 ⎤ log − ⎥ , 4πρ ⎢⎣ ε 4⎦

and integrals I1 and I2 taking into account (5.18), (5.20), (5.21) are rewritten in the following equivalent form: πρ πρ ⎡ ⎡ ( R − R′ ) ⎤ (R − R ⎢t × ⎥ b I1 = 2ρ b ( R ) ⋅ ⎢t × ds 2 − ρ ⋅ 3 ⎢ R − R′ ⎦⎥ R−R 0 0 ⎣⎢ ⎣

∫

∫

π 1+τ2

=2

⎧ ⎪ −2 ⎨ 1+ τ ⎪ ⎩

∫ ( 0

(

− 1+ τ

)

2 −1/ 2

)

1/ 2

2−3 / 2

(

) ⎤⎥ ds

3

)

⎥ ⎦

⎡θ sin θ + τ−2 − 1 (1 − cos θ ) ⎤ ⎣ ⎦− 3/ 2 − 2 2 ⎡ 2τ (1 − cos θ ) + θ ⎤ ⎣ ⎦

⎡ ⎛ θ ⎢1 − cos ⎜ ⎜ 2 ⎢⎣ ⎝ 1+ τ

⎞⎤ ⎟⎥ ⎟⎥ ⎠⎦

=

(5.23)

−1/ 2 ⎫

⎪ ⎬ dθ, ⎪ ⎭

∞ ⎡ ( R − R′ ) ⎤ ⎥ ds = I2 = 2ρ b ( R ) ⋅ ⎢t × 3 ⎢ ⎥⎦ ′ R R − πρ ⎣ ⎡ θ sin θ + τ−2 − 1 (1 − cos θ ) ⎤ ∞ ⎦ dθ. −2 1/ 2 ⎣ 1+ τ =2 3 / 2 ⎡ 2τ−2 (1 − cos θ ) + θ2 ⎤ π 1+τ2 ⎣ ⎦

∫

∫ (

)

(

)

(5.24)

5.2 Self-induced motion of helical vortex filament with an arbitrary pitch

247

In this format, both integrals become finite. The only dimensionless parameter is the dimensionless pitch of helix τ, which is equal to the dimensionless torsion ˆ = τ = τρ

l a + l2 2

ρ

and this form was used in similar analysis by Ricca (1994). The results of numerical simulation by Ricca are plotted in Fig. 5.7. Note that at high values of the dimensionless helix pitch τ > 0.5 I2 ≈ 0 and it can be excluded from calculations of ub. For further comparison, we have to present solution (5.22) in dimensionless form with the velocity scale (Γ/4πρ) uˆb = I1 + I2 + uˆ = log ( ρ / ε ) + CMS,

CMS = C(γ) = I1 + I2 + log 8 – 1/4.

where

(5.25) (5.26)

Subscript MS represents the approach developed by Moore and Saffman (1972). Let us consider the case of a helix with a high pitch, when τ 1. Then ∞

I2 ≈ 2

∞

dθ 1 θ sin θ − 1 + cos θ dθ ≈ = → 0. 3 2 πτ τ→∞ θ θ πτ πτ

∫

∫

(5.27)

Fig. 5.7. Dependency of integrals I1 (5.23) and I2 (5.24) vs. dimensionless torsion τ

248

5 Dynamics of vortex filaments

The first integral is rewritten in the form

I1 = I11 + I12, θ0 ⎡

⎤ θ sin θ − 1 + cos θ 1 ⎥ dθ , − I11 ≈ 2 ⎢ 3 1/ 2 ⎥ 3/ 2 ⎢ θ 2 τ (1 − cos ( θ / τ ) ) ⎦ 0 ⎣

∫

πτ

πτ

2γ ⎡ θ sin θ − 1 + cos θ ⎤ (1) ( 2) . − I12 I12 ≈ 2 ⎢ dθ − dθ = I12 ⎥ 3 1/ 2 3 / 2 θ ⎣ ⎦ (1 − cos ( θ / τ ) ) θ0 θ0 2

∫

∫

At θ0 → 0 θ0

1 ⎞ ⎛ 1 − ⎟ dθ = 0 . I11 = 2 ⎜ ⎝ 2θ 2θ ⎠ 0

∫

(1) is taken equal to ∞. Then Taking into account (5.27), the upper limit in I12

this integral is expressed through the integral cosine Ci(θ0), as considered in Section 5.1: () I12 =2 1

∞

∫

θ sin θ − 1 + cos θ θ

3

θ0

dθ ≈

1 − E − log θ0 . 2

( 2) is easily calculated through the substitution of variables Integral I12

( ) I12 = 2

πτ

2dθ

∫ 23 / 2 τ (1 − cos ( θ / τ ) )1/ 2

θ0

=

1

π

dα

= 2 θ ∫/ τ (1 − cos α )1/ 2 0

π/ 2

= θ0

dϕ = log 4 + log τ − log θ0 . sin ϕ / 2τ

∫

Inputting all this into (5.25), (5.26), we obtain

uˆb = log ρ / ε − log τ − E + log 2 + 1/ 4; CMS = − log τ – E + log 2 + 1/4.

(5.28)

According to the previous Section, the angular velocity ω in the long-wave approximation (high pitch of helix) is merely the induced velocity divided by the helix radius a:

5.2 Self-induced motion of helical vortex filament with an arbitrary pitch

ω=

ub Γ Γ ⎡ ρ 1⎤ uˆb = = log − log τ − E + log 2 + ⎥ = ⎢ 2 2 a 4πρa ε 4⎦ 4πa τ ⎣ 2 Γk = [ − log kε − E + log 2 + 1/ 4]. 4π

249

(5.29)

Here, we take into account that wave number k = 1/aτ, and ρ = aτ2 for τ 1. The obtained results coincide with the Kelvin formula (4.58) for long helical waves on a columnar vortex with uniform distribution of vorticity in the core with radius ε. We remember that in the previous Section a similar formula was obtained using the cut-off method, and the cut-off parameter was chosen through comparison with the Kelvin formula (4.58). For a moderate and low values of pitch, calculation of the bi-normal component of velocity using Eqs. (5.23) – (5.26) becomes more complex. In particular, this is due to a highly-oscillating integrand in Eq. (5.24). In view of this, Ricca (1994) made an attempt to estimate the self-induced velocity of a helical vortex filament through the velocity induced in its vicinity through the Kapteyn series (Hardin 1982), described earlier in Section 2.6.1. Indeed, expansion of a bi-normal component in the vicinity of curvilinear vortex filament (2.35) may be presented in a form similar to (5.25) a uˆb( ) = −2cos χ

ρ ρ + log + CH , σ σ

(5.30)

where (σ, χ) are the local polar coordinates in the filament vicinity, and subscript H represents Hardin’s approach. With the notation δ = σ/ρ, we can consider the velocity in the vicinity of point {rext = a[1 + δ(1 + τ2)], θ = 0}, and considering that

(

uˆb = ( uˆ z − τuˆθ ) 1 + τ2

)

12

(5.31)

,

we obtain for the zone outside the helix

Cext = −

(

2τ 1 + τ 2

(

)

12

1+ δ 1+ τ

2

)

−

(

4 (1 + δ ) 1 + τ2

(

)

32

)

τ ⎡1 + δ 1 + τ ⎤ ⎣ ⎦ 2

2

S ( rext ,0 ) +

2 + log δ, δ

(5.32)

where S stands for Kapteyn series that is included into formulae for velocities uθ and uz (2.56). Similarly, for the inner part of the helix {rint = a[1 – δ(1 + τ2)], θ = 0} we obtain

250

5 Dynamics of vortex filaments

Cint = −

(

2 1 + τ2

)

12

−

τ

(

4 (1 − δ ) 1 + τ2

(

)

32

S ( rint ,0 ) −

)

τ ⎡1 − δ 1 + τ ⎤ ⎣ ⎦ 2

2

2 + log δ. δ

(5.33)

Ricca (1994) calculated the value

CH = (Cext + Cint)/2

(5.34)

as a function of dimensionless pitch and compared it with calculations using Eq. (5.26). Since the Kapteyn series has a strong polar-like singularity, the velocity value was determined at a finite distance from the filament. The smaller the helix pitch, the greater the distance required. For example, for τ = 0.5 the distance from the filament was σ = 0.125a, and this brings a significant error into CH. In his calculations Ricca (1994) had shown that values CH and CMS differed by approximately 0.25 for the considered range of τ. Asymptotic analysis of the mutual correspondence of values CH and CMS at low and high τ is presented in a paper by Kuibin and Okulov (1998). To eliminate the problems of singularity in the solution, the authors used the technique of direct separation of singularities from a series of the Kapteyn type described in 2.6.2. Following this approach, we write the bi-normal component of velocity according to (5.31) and (2.74)

(1 + τ ) =2

2 12

uˆb

τ

⎡ ⎛ ⎞⎤ 2 a ⎞ ⎛ ⎧0 ⎫ 2 ⎢1 − ⎜ 1 + τ ⎟ ⎜ ⎨ ⎬ + {Sχ + Rχ } ⎟ ⎥ . r ⎠ ⎝ ⎩1 ⎭ τ ⎠ ⎦⎥ ⎣⎢ ⎝

(5.35)

Equating the right-sides of Eqs. (5.30) and (5.35) and assuming that χ = 0 we tend σ → 0. Then, taking into account the explicit expression for Sχ (2.75)1 we obtain

CKO =

(

2 1 + τ2 τ

)

12

(1 + τ ) −

2 32

2 32

+ 1 + τ2

τ

H=

(

4 1 + τ2 τ2

(1 + τ ) − log

)

τ

− H, (5.36)

32

Rχ

r =a χ= 0

.

Equations (5.35) and (5.36) are derived for a boundless case, i.e. u0 = Γ/2πl and in formulae (2.75), (2.76) we neglect the effects of a cylinder wall. 1

5.2 Self-induced motion of helical vortex filament with an arbitrary pitch

251

The subscript KO stands for approach developed by Kuibin and Okulov (1998). Actually, CKO is the lim CH . The value of H depends on the single δ→ 0

parameter τ and in view of (2.76) it is written in the form of the series

(

⎧ 2 ⎪2 1+ τ H= ⎨ τ m =1 ⎪ ⎩ ∞

∑

)

32

⎫ m⎞ ⎛m⎞ ⎤ 1 ⎪ ⎡ 2m ⎛ ⎢ τ Km′ ⎜ τ ⎟ Im ⎜ τ ⎟ + 1⎥ + m ⎬ . ⎝ ⎠ ⎝ ⎠ ⎦ ⎣ ⎪ ⎭

(5.37)

Note that series (5.37) converges rapidly. However, its convergence can be accelerated. Indeed, if we apply uniform expansion (2.70), we obtain

ξ + ζ ξ + ζ 2 − ξ1ζ1 2m ⎛m⎞ ⎛m⎞ − Km′ ⎜ ⎟ Im ⎜ ⎟ + 1 ∼ − 1 1 − 2 τ m m2 ⎝τ⎠ ⎝τ⎠ −

ξ3 − ζ 3 + ξ1ζ 2 − ξ 2 ζ1 m

3

=−

τ

(

2 1+ τ

⎡ 1 ⎢ 3τ2 30τ4 a3 = ⎢ − 8 1 + τ2 1 + τ2 ⎢⎣

(

)

2 32

⎤ 35τ6 ⎥ . + 3⎥ 1 + τ2 ⎥ ⎦

) ( 2

⎛ 1 a3 ⎞ ⎜m + 3 ⎟, m ⎠ ⎝

)

One can see that the terms of series (5.37) for high m diminish as m–3. Denoting the terms of the series as Hm, we rewrite H in the following form:

H=

∞

⎛

a ⎞

∑ ⎜⎝ Hm + m33 ⎟⎠ − a3 ζ ( 3) ,

(5.38)

m =1

where ζ is Riman’s zeta function (Abramowitz and Stegun 1964); (3) = 1.20205... . Calculation of H using Eq. (5.37) at τ = 5 with accuracy 7⋅10–5 is achieved with 74 terms of series, and the two terms are enough by Eq. (5.38). The terms of series (5.38) have the order m–5, and the convergence acceleration can be strengthened. The character of relation H(τ) is shown in Fig. 5.8. Let us consider the asymptotic behavior of H at high τ. Using the modified Bessel function at a low value of argument for this expansion (Abramowitz and Stegun 1964), we obtain

H1 τ

1

( )

5 = − log2τ + + Ε + O τ−2 , Hm τ 4

1

=

(

−1

)

m m −1 2

( )

+ O τ−2 , m = 2,3,....

252

5 Dynamics of vortex filaments

Fig. 5.8. Function H (5.37) vs. dimensionless torsion τ

Summation of series (5.37) and substitution into (5.36) gives us CKO |τ

1 = log

( )

2 1 + − E + O τ −2 . τ 2

(5.39)

From comparison with (5.28) one can see that at τ → ∞ CKO = CMS + 1/4. In Figure 5.9, the asymptotic (5.39) is compared with the plot calculated by Eq. (5.36) for 5 terms of series (5.38), and also with calculations from Ricca (1994) for the range 5 ≤ τ ≤ 12 and 29 ≤ τ ≤ 37. Let us show that even for another limit τ 1 the constants are different by 1/4. Indeed, at a high value of argument (Abramowitz and Stegun 1964) 12

⎛m⎞ ⎛ τ ⎞ Im ⎜ ⎟ = ⎜ ⎟ ⎝ τ ⎠ ⎝ 2πm ⎠

12

⎛ m ⎞ ⎛ πτ ⎞ Km′ ⎜ ⎟ = ⎜ ⎟ ⎝ τ ⎠ ⎝ 2m ⎠

⎡ 4m 2 − 1 τ ⎤ + ...⎥ , em τ ⎢1 − m 8 ⎣⎢ ⎦⎥

⎡ 4m 2 + 3 τ ⎤ + ...⎥ . e −m τ ⎢1 + 8 m ⎢⎣ ⎥⎦

As a result, the terms of series (5.37) have the asymptotic form Hm |τ

1= −

( )

3 τ2 + O τ4 . 8 m3

After series summation and substitution into (5.36) we obtain

CKO |τ

1=

τ 3ζ ( 3) − 4 2 1 + log τ + 1 − + τ + O τ3 . τ 2 8

( )

(5.40)

5.2 Self-induced motion of helical vortex filament with an arbitrary pitch

253

Fig. 5.9. Dependency of CKO (5.36) on dimensionless torsion τ (1) and comparison with calculations by Ricca (1994) (2) and asymptotic function (5.39) for high τ (3). The numbers in parentheses represent relative distance to the filament σ/ρ

Fig. 5.10. CKO (5.36) vs. τ (1) and comparison with calculation by Ricca (1994) (2) and asymptotic form for small τ (3)

In Fig. 5.10 the curve calculated by Eqs. (5.36), (5.37) is compared with the asymptotic form (5.40) and data from the work by Ricca (1994). Taking into account just two terms of the series (5.38) yields complete coincidence of the calculation with curve CH(0.1), produced by Ricca for the range 0.5 ≤ τ ≤ 1.4 with use of the Kapteyn series (Hardin 1982).

254

5 Dynamics of vortex filaments

Fig. 5.11. τ-dependencies of quantities I20 , I22

Now we have to find the asymptotic form for integrals (5.23), (5.24) at low τ. The quantity I1, as it was shown by Ricca (1994), tends to zero at → 0 (see Fig. 5.7). The value I2 at small τ is determined by the integral ∞

I21 = 2

1 − cos θ

∫⎡

2 2 π 2 (1 − cos θ ) + τ θ ⎤ ⎣ ⎦

32

dθ.

The dependence of the residue I20 = I2 − I21 on τ is plotted in Fig. 5.11. The latter integral I21 we transform into the following form: τ ⎞ ⎛1 I21 = I22 + ⎜ + log + 1⎟ , 8 ⎠ ⎝τ ∞⎧

I22

⎫ 1 − cos θ 21 2 − θ + P τ2θ2 ⎪ ⎪ =2 ⎨ + dθ, 3/ 2 ⎬ 2 2 32 ⎪ θ ⎡ 2 + τ2θ2 ⎤ π ⎪ ⎡ 2 (1 − cos θ ) + τ θ ⎤ ⎦ ⎣ ⎦ ⎩⎣ ⎭

∫

P=

( )−

log 8π2 12

8

π 12

2

(5.41)

.

The behavior of I22 at small τ is also shown in Fig. 5.11. Thus, I20 → 0 and I22 → 0 at τ → 0. Therefore, the value I2 at τ 1 is given by the expression in brackets in (5.41). Substituting the integrals into (5.26), we obtain

5.2 Self-induced motion of helical vortex filament with an arbitrary pitch

1 3 + log τ + . τ 4

1=

CMS |τ

255

Comparison with (5.40) shows that at a low pitch we again have CKO = CMS + 1/4. The overall analysis of the relationship between CMS and the limit of CH (at δ → 0), i.e., CKO, is given by Boersma and Wood (1999). After several transformations with integrals I1 and I2, authors presented the quantity CMS in the form:

(

1 CMS = − + log 2 + 2τ2 − 2τ 1 + τ2 4

)

(

) (

1 − log 1 + τ2 + 1 + τ2 2

12

)

32

W ( τ ) , (5.42)

∞⎧

⎫ 2 (1 − cos θ ) χ (1 − θ ) ⎪ 1 ⎪ W ( τ) = ⎨ − ⎬ dθ, 32 2 2 32 θ ⎪ ⎡1 + τ 2 ⎤ 0 ⎪ ⎡ 2 (1 − cos θ ) + τ θ ⎤ ⎦ ⎣ ⎦ ⎩⎣ ⎭

∫

(5.43)

where χ(x) is the step-like Heaviside function (χ(x) = 0, x < 0; χ(x) = 1, x ≥ 0). On another side, the authors applied the integral form of the Kapteyn series (Boersma and Yakubovich 1998) ∞

∑ Km ( ma ) Im ( mb ) =

m =1 ∞⎡ = ⎢ t 2 + a 2 + b 2 − 2ab cos t 0⎢ ⎣

∫(

)

−1 2

π

∫(

1 s 2 + a 2 + b 2 − 2ab cos s π0

−

)

−1 2

⎤ ds ⎥ dt, ⎥⎦

and demonstrated that series S(rext, 0) (see (5.32)) in the vicinity of a vortex filament has the asymptotic form

S ( rext ,0 ) =

+

τ2

(

(

(1 + τ ) log

(

2 12

τ2 2 1 + τ2

)

32

2 1 + τ2

τ2 1 + δ 4 1 + τ2

)

32

2

−

)

32

log δ +

τ2 W ( τ ) + o (1) , 4

where o(1) means the value that tends to zero at δ → 0. The result is

(

CKO = log 2 + 2τ2 − 2τ 1 + τ2

)

12

(

) (

1 − log 1 + τ 2 + 1 + τ 2 2

)

32

W ( τ ).

256

5 Dynamics of vortex filaments

Comparison with (5.42) shows that at any value of pitch CKO = CMS + 1/4. The paper by Boersma and Wood (1999) presents the complete analysis and a table with the values of function W(τ). For small τ W ( τ ) |τ

1=

(

1 + 1 + τ2 τ

)

−3 2

τ log + 1 + 2

7⎤ 75 43 ⎤ ⎡3 ⎡135 ζ ( 5 ) − ζ ( 3) + ⎥ τ4 + + ⎢ ζ ( 3) − ⎥ τ2 + ⎢ 2⎦ 16 8⎦ ⎣8 ⎣128

( )

9065 1225 337 ⎤ 6 ⎡ 7875 +⎢ ζ (7) − ζ ( 5) + ζ ( 3) − τ + O τ8 . ⎥ 256 64 48 ⎦ ⎣ 1024

For another limit – high τ W ( τ ) |τ

( )

3 ⎛ 25 4 ⎛3 ⎞ ⎞ − E ⎟ τ −3 − ⎜ − log 2 − E ⎟ τ−5 + O τ −7 . 2 ⎝ 12 3 ⎝2 ⎠ ⎠

1= ⎜

Therefore, calculation of self-induced velocity is available through Eqs. (5.42), (5.43) with function W(τ) given through the integral, or through Eqs. (5.36), (5.37) with function H(τ), given through the modified Bessel functions. Recently Okulov (2004) derived formulae for W(τ) and C(τ) in a whole range of τ variation. To obtain fast calculations without high accuracy, Kuibin and Okulov (1998) offered the approximation of function CKO(τ) with a simple relationship app CKO = log

τ 1 + τ2

+

1 + 1.455τ + 1.723τ2 + 0.711τ3 + 0.616τ4 τ + 0.486τ2 + 1.176τ3 + τ 4

,

(5.44)

that has the same asymptotic forms (5.39) and (5.40) as function (5.36). The maximum deviation from the accurate dependency is 0.007 for τ ∈ (0, ∞). Since we have the exact relationship between CKO and CMS, the Eqs. (5.36), (5.37), or (5.42), (5.43), or (5.44) according to (5.25) can be used for deriving the self-induced velocity of a helical vortex with uniform distribution of vorticity in the core. For generalization of results for a vortex with inner structure, we can write, by analogy with a vortex ring (see (3.20)) ⎡1 ρ π2 ε 2 u 2 π 2 ε 2 w2 ⎤ ⎥, uˆb = log + CKO − ⎢ − 2 4 + ε Γ2 Γ 2 ⎥⎦ ⎢⎣ 2

(5.45)

5.3 Hasimoto soliton

257

where u and w are the tangential and axial components of velocity in the core, and the bar means the procedure of averaging over the vortex crosssection.

5.3 Hasimoto soliton In (2.4) we derived the equation on the basis of local induction approximation

∂X/∂t = κb

(5.46)

expressed in relative (but not dimensionless) coordinates. For example, time t is normalized according to (2.38) and its dimensionality is [m2]. It was demonstrated by Hasimoto (1972) that (5.46) can be reduced to a nonlinear Schrödinger equation (NSE), and one of the possible solutions is a spiral soliton, which was later called the Hasimoto soliton. In this Section we derive the NSE and then analyze the single-soliton solution using the publications of Hasimoto (1972) and Lamb (1980). First of all let us rewrite the last two Frenet – Serret equations (2.6) in a complex form (n + ib)′ + iτ(n + ib) = – κt,

(5.47)

where the stroke means the derivative with respect to the arc length s. We introduce two new complex functions

⎛ s ⎞ N = (n + ib)exp ⎜ i τds ⎟, ⎜ ⎟ ⎝ 0 ⎠

∫

(5.48)

⎛ s ⎞ = κ exp ⎜ i τds ⎟ . ⎜ ⎟ ⎝ 0 ⎠

∫

(5.49)

* Obviously, | |2 = = κ2, where the asterix means a complexconjugate value. Then, we take (2.6) and (5.47) and obtain the relationships for s-derivatives from N and t

N′ = – t ′ = ( *N +

t, N*)/2.

(5.50)

For the time derivate of vector t, taking into account definition and Eq. (5.46), we obtain

tt = (X)t = (Xt)′ = (κb)′ = κ′b – τnκ = i( ′N* –

*

′N)/2.

(5.51)

258

5 Dynamics of vortex filaments

It is more difficult to derive Nt. In the general case, any vector can be written as a linear combination of unit vectors (t, n, b) or any other triple of independent vectors (t, N, N∗)2, which are linear combinations of unit vectors, as was proposed by Hasimoto (1972)

Nt = N + N∗ + t,

(5.52)

where coefficients , , must be defined. This requires consideration of the following relationships:

N⋅N = N⋅t = 0, N⋅N∗ = 2. Then, after multiplication of (5.52) by N and then by N∗, we obtain 2 = N⋅Nt = (N⋅N)t /2 = 0,

2 = N∗⋅Nt,

2( +

∗

) = (N⋅N∗)t = 0.

Thus, = 0, and coefficient is imaginary, so we assume is a real function. The condition (Nt)t = 0 gives us = t⋅Nt = −N⋅tt = −iN ( ′N* –

= iF, where F

′N)/2 = −i ′.

*

Then (5.52) takes the form

Nt = iFN–i ′t.

(5.53)

To find the unknown function F, we take the derivative with respect to s for (5.53) and the derivative with respect to t for (5.50). Making equal the right-sides of the equations obtained (separately for t and N; the terms with N∗ vanish), we obtain two equations t *

F′ = (

′+

− i ″ − iF *

= 0,

′)/2 or F = [| |2 + A(t)]/2,

where A(t) is an integration constant. Substituting F into the first equation, we obtain one equation for

i

t

+

″ + [| | + A(t)] /2 = 0. 2

(5.54)

After one more transformation

=

1 2

⎡ it ⎤ A(t)dt ⎥ , exp ⎢ − ⎢⎣ 2 0 ⎥⎦

∫

and finally we obtain Schrödinger’s cubic equation (SCE) 2

Note that N is not a usual vector: it represents a pair of complex numbers.

5.3 Hasimoto soliton

i

2

t

+ ″ + 2| |

= 0.

259

(5.55)

This equation is well studied in the literature (e.g., Lamb (1980)). It was shown that the equation has a multi-soliton solution. Now we consider the most important case of single-soliton solution. We assume that there exists a solitary wave spreading with a constant velocity cg in the positive direction. Then we introduce a running coordinate ξ = s – cgt, and derive the solution in the form

⎡ s ⎤ = κ(ξ) exp ⎢i τ(ξ)ds ⎥ ⎢⎣ 0 ⎥⎦

∫

with condition κ = 0 at s → ∞. Substituting this expression into (5.54) and separating the real and imaginary parts, we obtain two ordinary differential equations for torsion τ and curvature κ – cgκ [τ(ξ) – τ(cgt)] = κ″ – κτ2 + (κ2 + A)κ/2, – cgκ′ + 2κ′τ + κτ′ = 0. Note that the unsteady equations for curvature and torsion (intrinsic equations) were derived by Da Rios (1906) and Betchov (1965) directly from LIE. It follows from the second equation that (cg – 2τ)κ2 = B. The condition κ|s→∞ = 0 gives a constant of integration B = 0. Therefore

τ = cg /2 = const.

(5.56)

Then the first equation takes the form

κ″ – κτ2 + (κ2 + A)κ/2 = 0.

(5.57)

Since this is a differential equation of second order, we have to, besides the condition κ|s→∞ = 0, assign one more condition, for example, the maximum value of curvature κmax. Then this equation can be solved at the value of constant A = 2(τ2 – ν2), where ν = κmax/2, and its solution is written through a hyperbolic secant

κ = 2ν sech νξ.

(5.58)

The solution of the problem on the spreading of a solitary wave along the vortex filament was obtained in variables of torsion – curvature. The recovery of a spatial form of the curve from the given values of torsion and curvature is a standard problem in the theory of curves and it can be solved using the Riccati equation. However, the calculation procedure is rather cumbersome and we show here only the final results.

260

5 Dynamics of vortex filaments

Let the x axis be directed along the undisturbed vortex filament. Then we present the unit vectors in the Cartesian coordinate system as

t = it1 + jt2 + kt3, n = in1 + jn2 + kn3, b = ib1 + jb2 + kb3 and we take into account that t = X′, and obtain

∫

x = t1ds,

∫

∫

y = t2 ds,

z = t3ds.

(5.59)

The components of tangential vector are found from the Frenet – Serret equation with the known values of torsion and curvature

ti′ = κni ,

ni′ = τbi − κti , bi′ = −τni ,

i = 1, 2, 3.

The equations are written in index form, and the following relationship is true: ti2 + ni2 + bi2 = 1. Let us introduce new functions i

= (ti + ini)/(1 − bi),

− 1/

i

=

∗ i

= (ti − ini)/(1 − bi),

which are easily expressed through components of unit vectors. Differentiating with respect to s and taking into account the Frenet – Serret equations, we obtain the Riccati equation both for i

′i + i and

i.

i i

1 + iτ(1 − 2

2 i)

= 0,

The general solution takes the form i

=

ci P + Q , ci R + S

where ci is the integration constant, and functions P, Q, R, S are the same for all components. Omitting a long procedure of calculation for these functions and several transformations, we write the final relationship for (5.59) in parametric form

5.3 Hasimoto soliton

261

x = s – (2µ/ν) tanh η, y + iz = r eiθ, tx = 1 – 2µ sech2 η,

ty + itz = – νr(tanh η − iT) eiθ,

nx = 2µ sech η sinh η, ny + inz = – [1 – 2µ(tanh η − iT) tanh η] 2

bx = 2µT sech η,

(5.60)

eiθ,

by + ibz = iµ(1 – T2 – 2iT tanh η] eiθ, 2µ κ sech η = 2 , ν ν + τ2

r=

where

θ = Tη + ν2(1 + T2)t = τs + (ν2 – τ2)/t, η = νξ = ν(s – 2τt), µ = 1/(1 + T2), T = τ/ν. Here in the formula for r we have taken into account Eq. (5.58). Let us analyze the soliton shape, for example, at the time moment t = 0. Then, using the relative variables

X = xν, Y = yν, Z = zν, L = sν, we obtain

X = L− Y= Z=

2 1 +Τ 2

2 1 +Τ 2 2 1 +Τ 2

tanh L,

sech L cosΤ L,

(5.61)

sech L sinΤ L.

Obviously, the only parameter that characterizes the soliton shape is the quantity T, which is a ratio of the torsion τ = const to the maximum curvature ν = κmax/2. The typical forms of soliton are shown in Fig. 5.12 in projections to planes xy, xz, yz and in isometrical projection at different values of parameter T. The dashed line in Fig. 5.12e depicts the envelope for radius r, which varies from r = 2µ/ν at x = 0 to zero at infinity. The maximum curvature is achieved at x = 0. One can see from the figures that the soliton is a spiral bounded by an envelope. However, its shape largely depends on the value of parameter T. For T ≥ 1 y and z are the single-valued functions of x. For example, the torsion τ > 0 due to condition (5.56) and assss

262

5 Dynamics of vortex filaments

Fig. 5.12. For caption see next page

5.3 Hasimoto soliton

Fig. 5.12. Dependency of vortex soliton shape on parameter

263

264

5 Dynamics of vortex filaments

assumption cg > 0 means that the spiral is right-handed and the parameter T is also positive. For T = 1, we have dX/dL = 0 at L = X = 0 and a sharpening occurs on the envelope, although there is no singularity on the vortex filament. For T < 1 “overlapping” of a curve occurs and a closed loop originates in the plane xy. If the torsion τ → 0, then for the limit T → 0 the spiral tends to a flat curve in plane xy, but without crossing (no joining of filament points). The typical length of soliton L is determined by the condition of reduction of transversal size by “e” times, i.e., at νs = 1. Then L = 2x ≈ 2s = 2/ν = 4/κmax. The wavelength λ is found from condition: ∆θ = 2π = τ∆s. Then

λ = x(s + 2π/τ) − x(s) = 2π/τ + 2µ[tanh(νs) − tanh(νs + 2π/T)]/ν,

(5.62)

i.e., the wavelength changes along the x-axis. The wavelength in the vicinity of x = 0 (in view of symmetry) is the following

λ=

2π 4µ ⎛π⎞ tanh ⎜ ⎟ . − τ ν ⎝Τ ⎠

For a high torsion T → ∞ or at a large distance s → ∞ (x → ∞) we obtain a simple ratio

λ = 2π/τ. The number of spiral coils n on a characteristic size of soliton L is n = L/λ. For the case of T → ∞

n = L/λ = T/π. Now we can analyze the motion of the Hasimoto soliton (also called a vortical soliton). The velocity of a material point on a vortex filament according to (5.46) is u = X = κb . Then, taking into account (5.58) and (5.60), we obtain

ux = 4µτ sech2 η = τ(ν2 + τ2)r2,

w = uy + iuz = [iν2(1 − T2) + 2ν2 T tanh η](y + iz), where w is the velocity vector in a complex plane yz. The last expression can be written more conveniently through the radial wr and azimuthal wθ components of velocity in a complex plane

w = wθ + wr,

5.3 Hasimoto soliton

where

265

wθ = iν2(1 − T2)(y + iz) = i(ν2 − τ2) (y + iz), wr = rν2 T tanh η (y + iz) = 2ντ tanh η (y + iz).

Note that multiplication by the imaginary unit i means the rotation of vector (y + iz) by 90° counterclockwise. However, wθ also comprises the coefficient (1 − T2), so the resulting direction of a spiral rotation depends on parameter T. For |Τ | < 1 the direction of spiral rotation is the same as the direction of the vorticity vector at infinity, i.e., at Τ = 2 and Τ = 5 the spiral rotates to the right if we look along the direction of axis x. Besides, it is swirled to the right and moves in the line of x growth with the velocity cg = 2τ at τ > 0. For |Τ | > 1 the direction of rotation is the opposite. For |Τ | = 1 there is no rotation. The quantity cg = 2τ, obviously represents a group velocity, since it characterizes the motion of a spiral structure as a whole, or equivalently, of its envelope. Indeed, let r = rmax = 2µ/ν. Then it follows from (5.60) that η = ν(s – 2τt) = 0 or s = 2τt. Correspondingly, the longitudinal coordinate x of filament point with r = rmax is x = s – 2µ/ν tanh η = cgt, i.e., the center of the package (soliton) moves with the same velocity cg = 2τ both in the absolute frame of reference and along the arc length s. Now we determine the phase velocity cp. By definition, this is the velocity of a point with a constant phase

θ = τs + (ν2 − τ2)t = τ(s – c0t) = const. For the coordinate system linked to the arc length s, the phase velocity cp = c0 = (τ2 – ν2)/τ = const. Let us find the relationship for cp in the absolute frame of reference. Since s = c0t + θ/τ, the coordinate x at this point is

x = c0t + θ / τ −

2µ tanh ⎡ν c0 − cg t + θν / τ ⎤ . ⎣ ⎦ ν

(

)

If we make the time differentiation and consider, for clear definition, a case when at time t = 0 the soliton center is in the point x = s = 0, we obtain

cp =

2µ(c0 − cg ) dx = c0 − . dt t =0 cosh 2 (θν / τ)

(5.63)

It follows from this formula that the phase velocity varies for different points of the wave packet. Let us write two limiting relationships for phase velocity – for the soliton center (θ = 0) and for infinity (θ → ±∞):

266

5 Dynamics of vortex filaments

Fig. 5.13. Diagrams for vortex soliton motion at different values of parameters and τ

cp0 = cp|θ = 0 = c0 − 2µ(c0 − cg) = (τ2 + ν2)/τ,

(5.64)

cp∞ = cp|θ → ±∞ = c0 = (τ2 − ν2)/τ.

(5.65)

Then we take into account a striking fact that according to differential equations of soliton motion (see (5.57) for curvature κ) nonlinearity is de-

5.4 Application of momentum balance to description of vortex filament dynamics

267

fined by the vortex filament curvature. In the formula for phase velocity this is governed by the term ν2 = (κmax/2)2. For weakly-nonlinear waves (ν → 0) the phase velocity is constant along the wave packet and equals the group velocity

cp0 = cp∞ = τ = cg/2

at ν → 0.

The important parameter is the ratio of group velocity to phase velocity, which is a function of parameter T

cg cp0

=

2τ2 τ +ν 2

2

=

2 1 + 1/Τ 2

.

At high torsion (T → ∞) cg = 2cp0, at T = 1 cg = cp0 (no rotation), and at T = 0 (plain wave) cg = 0. Note that due to renormalization of time t in Eq. (5.46), the dimensionality of velocity is [m–1], i.e., the same as for torsion and curvature. The overall picture of vortex soliton motion is shown in Fig. 5.13. The vorticity vector for the filament at infinity is up-directed vertically. Although originally the solution was sought for cg > 0, the case of cg < 0 is easy for description. Then τ < 0 (left-handed spiral) and T < 0. The proper examples are shown in the Figure.

5.4 Application of momentum balance to description of vortex filament dynamics 5.4.1 Forces acting on a vortex filament The approach to the description of thin vortex filament dynamics, which is based on momentum balance, was developed by Moore and Saffman (1972) (see also Saffman (1992)). The idea is to balance the forces applied to an element of vortex filament. There exists several forces applied to the filament, and Kutta – Joukowski force and filament strain are the main examples of them. The concept of forces acting on the filament can be introduced in different ways: the intuitive way, with stress upon physical reasoning, or through interpretation of some terms of motion equation, that is, derived through accurate procedures. That is why the force definitions can vary for different authors. The concept of the Kutta – Joukowski force for a vortex filament was first used in a paper by Widnall et al. (1971) for an example of a vortex ring. It was derived that the vortex ring velocity decreases in the case of axial flow and it equals

268

5 Dynamics of vortex filaments ∞

U = U0 −

2πrw2 dr , ρΓ 0

∫

(5.66)

where ρ is the ring radius; Γ is the circulation; r is the radial coordinate counted from the vortex filament axis; w = w(r) is the axial velocity; U0 is the ring velocity without axial flow. The expression in the numerator is the axial flux of momentum. Then the turning of the axial flow by the circle requires the force ∞

F=−

2πrw2dr , ρ 0

∫

which is directed towards the ring center and making allowance for (5.66), can be written as Γ (U – U0). By the notation form, this is the Kutta – Joukowski force (see Section 1.7.1), and the relative velocity is determined as a difference between the real velocity of ring motion U and the equilibrium velocity U0 calculated for zero axial flow. In other words, the centrifugal force arising due to the axial flow along the ring is compensated by the Kutta – Joukowski force, which appears when the ring velocity deviates from the equilibrium value. Thus, in this approach the Kutta – Joukowski force is introduced only for consideration of the axial flow along the vortex filament. In this interpretation, the Kutta – Joukowski lifting force was used in a paper by Widnall and Bliss (1971), where the concept of force balance was first formulated. The authors of this paper (see also Widnall (1975)) considered the motion of a vortex filament under the action of a plain sinusoidal or spiral perturbation, and the momentum balance was reduced to the equality of the Kutta – Joukowski force and inertia forces (with consideration for the attached mass). Moore and Saffman (1972) (see also Saffman 1992) give a different definition for the Kutta – Joukowski force. Let us consider again the example of a thin vortex ring. Then we can write the lifting force acting on the element of vortex filament ds in the form Fk = Γt × Uds (see Fig. 5.14). Here U = Ub is the ring velocity relative to the liquid (motionless at infinity) and the force is directed by bi-normal b = t × n. In this case, force Fk is in the plane of the ring and is directed outward. In leading order, it is balanced by the tension which is defined as FT = T0dsn/ρ. The concept of tension force T0 is introduced for a curvilinear section of filament and has a simple physical interpretation: on the inner side of a curved section the streamlines become denser and the pressure decreases and vice veeerr

5.4 Application of momentum balance to description of vortex filament dynamics

269

Fig. 5.14. Forces acting on an element of vortex ring (according to Saffman (1992))

versa, the pressure increases in the outside zone. This pressure difference creates the force directed to the curvature center. The balance of these two forces gives us the equation

Τ0 ρ

n + Γt × U = 0.

(5.67)

If we know the ring velocity, as for the case of a constant vorticity in the core, then in view of (5.7) we obtain from (5.67) the formula for tension

Τ0 =

8ρ 1 ⎞ Γ2 ⎛ ⎜ log − ⎟ . 4π ⎝ a 4⎠

(5.68)

The formula for a hollow vortex with the velocity given by Hicks (1885) is as follows:

Τ0 =

8ρ 1 ⎞ Γ2 ⎛ ⎜ log − ⎟ . 4π ⎝ a 2⎠

(5.69)

Note that the concepts of tension and force balance for vortex filaments have been used previously for the study of quantum vortices in superfluid helium (Hall 1958, 1970), although these attempts did not produce good quantitative results. One can see from the given examples that the definitions for the Kutta – Joukowski force according to Widnall and Bliss (1971) and Moore and Saffman (1972) are different not only in magnitude, but in direction. Obviously, the difference emerges from different definitions and calculations of the relative velocity. In particular, one can avoid the concept of tension by including the tension force into the Kutta – Joukowski force (see Lundgren and Ashurst (1989)), that corresponds to the definition by Widnall and Bliss.

270

5 Dynamics of vortex filaments

Herein we will use the approach developed by Moore and Saffman (1972), using the formal derivation of motion equations, and the concepts of different forces will be used primarily for interpretation of different terms in the motion equations. The main advantage of the method of force balance is that we do not have to know in detail the flow structure in the vortex core, as in other integral hydrodynamic approaches, e.g., the Karman – Polhausen approach. 5.4.2 Derivation of force-balance equations We begin with consideration of force-balance method from derivation of momentum balance equations for an element of a thin vortex filament (Fig. 5.15). Let the vortex filament be described in parametric form as

r = R(ξ, t), where parameter ξ determines the position along the vortex filament. Then the distance along the filament is s = s (ξ, t), and the unit tangential vector is given by formula

t=

∂R ∂ξ

∂s . ∂ξ

Let the point ξ = const move with the velocity of a liquid particle, i.e., ξ is a material variable. Then the vortex velocity is ∂R/∂t. Since the velocity of a spatial curve is defined unambiguously only for a normal component, the above introduced velocity of a vortex allows an arbitrary component proportional to t. The additional condition for the tangential component of velocity (see Eq. 5.75) gives unambiguously the parameter ξ. The vortex motion is driven by two factors – self-impact (self-induced motion) and external velocity field, e.g., from another vortex. The problem consists of the derivation of an equation for ∂R/∂t.

Fig. 5.15. On the application of the force-balance method for a vortex filament

5.4 Application of momentum balance to description of vortex filament dynamics

271

We can make the momentum balance for element ∆ of a vortex tube with length ds, limited by lateral cylindrical surface Σ and two end faces S1 and S2, which are normal to t. The balance equation takes the form (for a liquid with unit density)

∂ udV + pndA + ∂t

∫

∫

∆

Σ

∫ [ pn + u(urel ⋅ n)] dA = 0 .

S1 + S2

(5.70)

Here u is the local velocity of the liquid; urel is the liquid velocity relative to the control surface; n is the outside normal to the surface. The first term represents the temporal change of momentum inside the control volume, the second term is the pressure force on the lateral surface, and the third term is the pressure force and momentum flux through the end faces. The second term can be rewritten in the form

∫ (− p)n dA = ds FE ,

(5.71)

Σ

where FE is the force per unit of vortex length, caused by the action of the liquid outside the vortex core on surface Σ. Let us introduce the notion w = urel⋅t for the tangential component of the relative velocity. Then we obtain

∫ [ pn + u(u

⋅ n) ] dA = ds

S1 + S2

∂ ( pt + uw)dA, ∂s S ( s )

∫

where S(s) is the vortex cross-section area as a function of length s. Combining it with the first term from (5.70), we obtain

∂ ∂ udV + ds ( pt + uw)dA = −ds FI . ∂t ∆ ∂s S ( s )

∫

∫

(5.72)

Here FI means the force per vortex length unit acting on the core boundary from the liquid inside the vortex filament. Therefore, the momentum balance equation (5.70) is written as the equality of forces

FI + FE = 0,

(5.73)

which are determined by Eqs. (5.71) and (5.72). Note that the existence of relative velocity urel can be caused by a non-uniform profile of axial velocity and by difference between the velocity for the control end face and the liquid, even for a uniform profile.

272

5 Dynamics of vortex filaments

Then we assume that there is no external velocity field and the filament motion is caused only by self-induction. Let us introduce an artificial concept of self-induced velocity VI as

VI ( ξ ) =

⎧ R ( ξ ) − R ( ξ′ ) R(ξ) − R Γ ⎪ ′ ds − × t ⎨t ′ × 3 4π ⎪ R ( ξ ) − R ( ξ′ ) R(ξ) − R ⎩

∫

⎫ ⎪ ds ⎬. 3 ⎪⎭

(5.74)

Here subscript refers to a vortex ring inscribed into the filament at point ξ (see (5.17)), i.e., this is the velocity according to the Biot – Savart law, minus the vortex ring velocity. This is how to eliminate the singularity from the Biot – Savart integral, as demonstrated in Section 5.2. The meaning of definition (5.74) and quantity VI are discussed below. With the known velocity VI, we can impose the condition

VI(ξ) ⋅t = ∂R/∂t⋅t

(5.75)

with the purpose of unambiguous determination of parameter ξ. That is why we can introduce the relative velocity Q

Q(ξ) = VI − ∂R/∂t,

(5.76)

that characterizes the relative motion of liquid and filament. It follows from (5.75) that Q ⊥ t. Existence of the relative velocity creates the opportunity to artificially introduce the Kutta – Joukowski force and other forces. However, we will instead act in a formal way and use the Euler equations for description of the velocity field outside the vortex core. Let us pass a plane through a point on vortex filament R(ξ); the plane contains the vectors of normal n and bi-normal b (Fig. 5.16). We introduce a polar coordinate system (r, θ) in this plane, and also the auxiliary caarte

Fig. 5.16. On the definition of coordinate system

5.4 Application of momentum balance to description of vortex filament dynamics

273

Cartesian system x, y with unit vectors i, j. Then the position of arbitrary point M in this plane is given by the vector

r = R(s(ξ)) + rer,

(5.77)

where er is a unit radial vector. Obviously, for any point in the vicinity of the filament we can pass a plane and assign it the proper value of parameter ξ. Then we will use the local toroidal coordinate system {r, θ, s}, proposed by Widnall et al. (1971) for a plain vortex filament considered by Ting (1971) in a general case (see also Callegari and Ting (1978)). This system is orthogonal if we define angle θ as

θ = ϕ − θ0(s, t),

(5.78)

and – dθ0/ds = τ is a torsion. Here θ0 is the angle between i and n, and ϕ is the angle between n and er. If we use ϕ as an angular coordinate, which seems to be reasonable, then the system {r, ϕ, s} would not be orthogonal. In the orthogonal curvilinear coordinate system {r, θ, s}, the length element dr is written in the form

dr = Lrerdr + Lθeθdθ + Lsesds,

(5.79)

where Lr, Lθ, Ls are the Lame coefficients. To calculate these coefficients and to prove the condition (5.78), we have to use the Frenet – Serret equations (2.6)

dt = κn , ds

dn = τb − κt , ds

db = −τn , ds

and also use the definition for a tangential vector

t = ∂R/∂s.

(5.80)

Here κ ≡ 1/ρ is the local curvature of the vortex filament at point R(ξ). Using the geometrical constructions depicted in Fig. 5.16, we can write the formulae for unit vectors er and eθ in the plane nb

er = er (θ, s, t) = n cos ϕ + b sin ϕ, eθ = eθ (θ, s, t) = b cos ϕ − n sin ϕ, where ϕ = θ + θ0(s,t). For a certain time moment t, the differential dr in view of (5.77) and (5.80) are written as de ⎛ de ⎞ dr = dR + rder + er dr = tds + er dr + r ⎜ r dθ + r ds ⎟ . ds ⎝ dθ ⎠

(5.81)

274

5 Dynamics of vortex filaments

Using the relationships for er and eθ and the Frenet – Serret equations, we obtain der = −n sin ϕ + b cos ϕ = eθ , dθ

∂θ ∂θ der dn ∂b cos ϕ − n sin ϕ 0 + sin ϕ + b cos ϕ 0 = = ds ds ∂s ∂s ∂s ∂θ ⎞ ⎛ = eθ ⎜ τ + 0 ⎟ − κt cos ϕ . ∂s ⎠ ⎝ Substituting those relationships into (5.81), we obtain ∂θ ⎞ ⎛ dr = er dr + reθdθ + (1 − κr cos ϕ ) tds + r ⎜ τ + 0 ⎟ eθds . ∂s ⎠ ⎝

Comparing it with presentation (5.79), which is valid for orthogonal curvilinear coordinates, we see that orthogonality requires the condition ∂θ0/∂s = – τ. Naturally, for Lame coefficients we obtain

Lr = 1 ,

Lθ = r ,

r Ls = 1 − cos ϕ . ρ

The local system is well-defined if the distance from the vortex filament is limited by values of O(ρ). The flow outside the vortex core with circulation Γ is a potential flow and is described by the Laplace equation, which for the curvilinear coordinates {q1, q2, q3} is as follows (see Section 1.3.1)

∇2Φ =

1 ⎡ ∂ ⎛ L2L3 ∂Φ ⎞ ∂ ⎛ L1L3 ∂Φ ⎞ ∂ ⎛ L1L2 ∂Φ ⎞⎤ ⎢ ⎜ ⎜ ⎟⎥ = 0, (5.82) ⎟+ ⎜ ⎟+ LL 1 2L3 ⎢ ⎣ ∂q1 ⎝ L1 ∂q1 ⎠ ∂q2 ⎝ L2 ∂q2 ⎠ ∂q3 ⎝ L3 ∂q3 ⎠⎥⎦

where Φ is the velocity potential. In coordinates {r, θ, s} the Laplace equation takes the form ∂ 2Φ ∂r 2

+

1 ∂h ∂Φ 1 ∂Φ 1 ∂ 2 Φ 1 ∂h ∂Φ ∂ ⎛ 1 ∂Φ ⎞ + + 2 + 2 +r ⎜ ⎟ = 0, 2 ∂s ⎝ h ∂s ⎠ h ∂r ∂r r ∂r r ∂θ r h ∂θ ∂θ

where h ≡ Ls = 1 – r cos ϕ/ρ. According to the problem conditions, a/ρ 1. Therefore, the solution outside the core is to be found in the form of expansion

Φ = Φ0 + Φ1 + Φ2 + …,

5.4 Application of momentum balance to description of vortex filament dynamics

275

a = a0(s) + a1(s, θ) + a1(s, θ) + … . The index i = 0, 1, 2, ... means that the quantity has the order O(ai/ρi). The undisturbed potential Φ0, according to (2.24), is

Φ0 = Γθ/2π,

r > a0,

and the undisturbed core is round, i.e., a0 = a0(s). Substituting the expansion for Φ into the Laplace equation (5.82) and assuming that r ≈ a, s ≈ ρ, we obtain for Φ1

∂ 2 Φ1 ∂r 2

+

1 ∂Φ1 1 ∂ 2 Φ1 Γ sin ( θ + θ0 ) . + 2 =− 2 r ∂r r ∂θ 2πρr

(5.83)

Two boundary conditions must be given for Eq. (5.83): the internal condition at r = a0 and the external condition at r → ∞. The internal boundary condition is determined by the fact that the core boundary r = a(s, θ, t) is a material surface. In other words, we have to impose the kinematic condition on this boundary (see also the kinematic condition for a vortex sheet in 3.1)

ur = da/dt for

r = a(s, θ, t).

(5.84)

This means that for the coordinate system {r, θ, s} the substantial derivative from the core boundary is equal to the radial component of liquid velocity at this point. By definition

ur =

dr d ∂ = + urel ⋅ ∇, , dt dt ∂t

where the operator is

∇ = er

∂ eθ ∂ es ∂ + + , ∂r r ∂θ h ∂s

and urel is the liquid velocity in the system {r, θ, s} at the core boundary. If the local coordinate system rotates with the angular velocity and moves with the filament translatory velocity ∂R/∂t, then for this moving frame of reference we have

urel = ∇Φ − ∂R/∂t −

× (r0 − R).

(5.85)

The relation for the substantial derivative d/dt in the moving frame of reference can be derived in a more accurate way. Note that the total derivative d/dt in the absolute frame of reference is as follows

276

5 Dynamics of vortex filaments

∂ d = + u0 ⋅ ∇ 0 , dt0 ∂t0

where u0 is the absolute liquid velocity, and ∇0 is the Laplace operator in a fixed Cartesian coordinate system. For a moving coordinate system {r, θ, s}, the Laplace operator is substituted with ∇r θs, and the partial timederivative transforms into

⎛ ∂ ⎞ ∂ ⎛ ∂t ⎞ ∂ ⎛ ∂r ⎞ ∂ ⎛ ∂θ ⎞ ∂ ⎛ ∂s ⎞ ⎜ ⎟ = ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ , ⎝ ∂t0 ⎠r0 ∂t ⎝ ∂t0 ⎠r0 ∂r ⎝ ∂t0 ⎠r0 ∂θ ⎝ ∂t0 ⎠r0 ∂s ⎝ ∂t0 ⎠r0 where t0 = t. Since r0 = R(s,t) + rer (θ, s, t), then for the fixed r0 we obtain

∂e ⎛ ∂R dr0 = ⎜ +r r ∂t ⎝ ∂t

∂e ⎞ ⎛ ∂R +r r ⎟ dt + ⎜ ∂s ⎠ ⎝ ∂s

∂er ⎞ + er dr = 0. ⎟ ds + r ∂θ ⎠

It follows from geometric construction that

∂R/∂s = es ≡ t, ∂er /∂s = −κ cos ϕt, ∂er /∂θ = eθ . Taking those relationships and multiplying the above identity by unit vectors t, er, eθ, we obtain

∂s ∂e ⎛ ∂R = −⎜ +r r t ∂t ∂ ∂t ⎝

⎞ t ⎟⋅ , ⎠ h

∂r ∂R er , =− ∂t ∂t ∂θ 1 ⎛ ∂R ∂e ⎞ =− ⎜ + r r ⎟ ⋅ eθ . r ⎝ ∂t ∂t ∂t ⎠

Finally, substituting these relationships into the formula for (∂ ∂t0 )r0 and collecting the proper terms into the Laplace operator, we obtain

∂e ∂ ∂ ⎛ ∂R = −⎜ +r r ∂t0 ∂t ⎝ ∂t ∂t

⎞ ⎟ ∇rθs . ⎠

Here we take into account that er⋅(∂er/∂t) = 0. Then for the total time derivative in a moving frame of reference we have (see also Section 1.2.1) ∂ ⎛ ∂R ∂e d = + ⎜ u0 − −r r ∂t ∂t dt ∂t ⎝

⎞ ⎟ ∇ rθs . ⎠

5.4 Application of momentum balance to description of vortex filament dynamics

277

Since u0 = ∇Φ, and r(∂er/∂t) ≡ Ω × (r0 – R), the expression in brackets identically coincides with the relative velocity (5.85). Thus, disclosing the condition (5.84) we have

⎧∂ ⎡ ∂R − ⎨ + ⎢∇Φ − ∂t ⎩ ∂t ⎣

⎤ ⎫ × (r0 − R) ⎥ ⋅ ∇ ⎬ ( r − a ( θ, s, t ) ) = 0 ⎦ ⎭

∂Φ1 ∂R 1 ∂Φ 0 ∂a1 1 ∂Φ1 ∂a0 − ⋅ er − 2 − + ∂r ∂t r ∂θ ∂θ h 2 ∂s ∂s

or

+

∂R ⎛ eθ ∂a1 es ∂a0 ⎞ ⋅ + + r[ ∂t ⎝⎜ r ∂θ h ∂s ⎠⎟

⎡ e ∂a e ∂a ⎤ × er ] ⋅ ⎢ θ 1 + s 0 ⎥ = 0. ⎣ r ∂θ h ∂s ⎦

Here, in the expansion for Φ and a we retain only the terms of the first approximation that do not vanish. To estimate the time derivative, we have to take into account that the self-induced velocity by its order of magnitude is equal to |∂R/∂t| ∼ (Γ/ρ) log (ρ/a) = O(1/ρ). Then ∂/∂t = O(1/ρ2), since |δR| = O(ρ), and | | ∼ 1/t = O(1/ρ2). The estimates for the terms of this equation show that the first three terms have the magnitude of order 1/ρ, and the others – 1/ρ2 and higher. Then, keeping these three terms and formulating er through n and b, we obtain the boundary condition at r = a0(s,t) ∂Φ1 ∂R Γ ∂a1 . = ⋅ ⎡n cos (θ + θ0 ) + b sin (θ + θ0 )⎤⎦ + ∂r ∂t ⎣ 2πa02 ∂θ

(5.86)

The boundary condition for Φ1 at r → ∞ is derived by matching with the inner limit of external flow, which is described by the Biot – Savart law. Using the form (5.74) not for the filament, but for a certain point M(r0) near the filament, we obtain

u(r0 , ξ) =

r −R r − R′ Γ Γ ds′ = UI (r0 , ξ) + t′ × 0 t × 0 3 4π 4π r0 − R′ r0 − R

∫

∫

3

ds ,

where UI (r0 , ξ) r = R = VI ( ξ) . Decomposing the integral at the condition 0

r

ρ yields

u = VI (ξ) +

Γ 2πr

(Xb − Yn) + 2

⎛ Γr ⎞ Γ ⎡ XY Y2 ⎛ 8ρ ⎞⎤ ⎢− 2 n − 2 b + b log ⎜ ⎟⎥ + O ⎜ 2 ⎟ . 4πρ ⎢⎣ r r ⎝ r ⎠⎥⎦ ⎝ρ ⎠

278

5 Dynamics of vortex filaments

Here r = xi + yj = Xn + Yb, and r0 is a radius-vector in the absolute frame of reference. Assuming that r → ∞ and taking into account that u = ∇Φ0 + ∇Φ1 + …, and ∇Φ0 = eθΓ/2πr = Γ(Xb − Yn)/2πr2, we find the boundary condition

∇Φ1 = VI +

or

Y2 Γ ⎡ XY ⎛ 8ρ ⎞ ⎤ − − n b + b log ⎜ ⎟ ⎥ ⎢ 2 2 4πρ ⎣⎢ r r ⎝ r ⎠ ⎦⎥

Φ1 = VI ⋅ r +

Γ ⎛ 8ρ ⎞ Y log ⎜ ⎟ + VI ⋅ t ds. 4πρ ⎝ r ⎠

∫

(5.87)

One can verify that the solution of Eq. (5.83) with boundary conditions (5.86) and (5.87) is the following

Φ1 = VI ⋅ r −

a02 ⎛ ∂R ΓY ⎞ ⎛ 8ρ ⎞ log ⎜ ⎟ + − VI ⎟ ⋅ r + 2 ⎜ ∂t 4πρ r ⎝ ⎠ ⎝ r ⎠

ΓYa02 ⎛ 8ρ ⎞ + log − 1⎟ + VI ⋅ t ds + Φ1def , 2⎜ a0 4πρr ⎝ ⎠

(5.88)

∫

where the last term appears due to the contribution from a1, i.e., due to deformation of the core cross-section. Let us calculate the pressure on the surface of the vortex core using Bernoulli’s equation. Actually, this is a difficult task, because the calculation is conducted in a non-inertial frame of reference and we have to take into account the rotation and the translatory motion of the local frame of reference. However, if only the terms of order O(a/ρ) are kept in the Bernoulli equation, we obtain a rather simple formula 2

p+

2 urel 1⎛ ∂R ⎞ + Π ≈ p + ⎜ ∇Φ − ⎟ ≈ ∂t ⎠ 2 2⎝

⎛ Γ2 ⎞ 1 ∂R ≈ p + (∇Φ 0 ) 2 + ∇Φ 0 ⋅ ∇Φ1 − ⋅ ∇Φ 0 = O ⎜ 2 ⎟ . ⎜ρ ⎟ 2 ∂t ⎝ ⎠ Note that the Bernoulli constant is the same for the entire zone in the case of irrotational flow, and pressure p has a reference from the static pressure far away from the filament. Then we substitute the expression for Φ0 and Φ1, and also take into account that the core deformation, i.e., the term Φ1def does not make a contribution to the forces. Then we assume r = a( = a0) and obtain the relationship for the pressure at the core surface

5.4 Application of momentum balance to description of vortex filament dynamics

p=−

Γ2

Γ ⎛ ∂R ⎞ V − ⎟ ⋅ (Yn − Xb) − 2⎜ I ∂t ⎠ πa ⎝ 8π a ⎛ Γ2 ⎞ 8ρ 1 ⎞ Γ2 X ⎛ − 2 2 ⎜ log − ⎟ + O ⎜ 2 ⎟ . ⎜ρ ⎟ a 2⎠ 4π ρa ⎝ ⎝ ⎠ 2 2

279

+

(5.89)

The force per unit of filament length FE is derived by integration of the pressure over the curvilinear surface of the core

∂R ⎞ Γ2 ⎛ Γ 2 ∂a 2 Γ2 8ρ 1 ⎞ ⎛ FE = Γ ⎜ VI − t n t n. × + − − − log ⎟ ⎜ ⎟ ∂t ⎠ a 2⎠ 4πρ ⎝ 8πρ 8πa 2 ∂s ⎝

(5.90)

For this notation, the first term is the Kutta – Joukowski lifting force, and the second term represents the tension that was introduced earlier on the intuitive level. The last term is a result of the pressure reduction due to fluid rotation around the filament axis, and it is compensated by an analogous term in the inner force FI. The third term is a force that is directed along the vortex axis and caused by variation in the core cross-section along the filament. It will be demonstrated below that this contribution is negligible. 5.4.3 Hollow vortex Let us apply the method of force balance to the simplest case of a hollow vortex, which nevertheless remains incompressible. Since we are considering the case of thin vortices, we have to take into account the surface tension σ at the core interface. Then we have the formula for pressure pI inside the core (first approximation) deduced from (5.89)

pI = −

Γ2 8π a

2 2

+

σ , a

(5.91)

where σ/a was added due to capillary forces. Since the pressure inside the core is constant (based upon (5.91)), then a = a(t). The following relationship is valid

a2L = const,

(5.92)

where L is the length of the vortex filament. Now we use the condition of force balance (5.73), where FE is determined by Eq. (5.90), and FI is calculated through (5.72)

280

5 Dynamics of vortex filaments

FI = −

(

)

∂ ∂ ptdA + Fσ = − pI πa 2t + Fσ . ∂s S ( s ) ∂s

∫

Here we see an additional term Fσ, caused by the existence of surface tension σ. To find Fσ, we have to consider an element of vortex core with the length ds (Fig. 5.17), which is hollow. By definition, the circumference of length 2πa at the interface is exposed to the force (tension) with magnitude T = 2πaσ. The resultant force df, acting on element ds, is equal to df = Tt s + ds − Tt s =

∂ (Tt )ds. ∂s

The force per unit of vortex length is

Fσ = df ds =

∂ (2πaσt ) . ∂s

Then for FI in view of (5.91) and definition n = ρ dt/ds we obtain

FI =

⎛ Γ2 ⎞n ∂ ⎡ 2πaσ − πa 2 pI t ⎤ = ⎜ + πaσ ⎟ . ⎟ρ ⎦ ⎜ 8π ∂s ⎣ ⎝ ⎠

(

)

(5.93)

Substituting (5.90) and (5.93) into (5.73) and making vector multiplication by t, we obtain the motion equation for a hollow vortex filament

∂R b = VI + (T0 + πaσ) . ∂t ρΓ

(5.94)

In the derivation procedure we took into account the Eqs. (5.75) for velocities and (5.69) for tension T0 in a hollow vortex ring. As one can see, the surfff

Fig. 5.17. Sketch of a hollow vortex

5.4 Application of momentum balance to description of vortex filament dynamics

281

surface tension gives the correction term πaσ to the filament tension T0. Therefore the correction is a resultant force of capillary pressure and actual tension arising due to surface forces. Let us derive a similar equation using the cut-off method. We take Eq. (5.74) for VI, and perform integration using the cut-off method with 2δL excluded

R( ξ ) − R ∂R Γ t × = VI + ∂t 4π [ δ ] R( ξ ) − R

∫

3

ds = VI +

8ρ Γ ⎛ ⎞ ⎜ log − log 2δ ⎟ b. 4πρ ⎝ a ⎠

Here, the integral for the ring has been calculated in (5.8). For a hollow vortex ring, the cut-off method gives us ln 2δ = 1/2. Comparing it with (5.94), we see a complete coincidence at σ = 0. Consideration of σ gives us a modified formula for δ: log 2δ =

1 4π2 aσ . − 2 Γ2

With (5.92) and (5.94), we can calculate the vortex radius a = a(t). This requires an expression of the vortex filament length through ∂R/∂t. That is why we have to proceed with the following chain of transformations: L=

dL = dt

∂s

∫ ds = ∫ ∂ξ

∂ ⎛ ∂s ⎞ ∂ξ

dξ ,

∂⎛

∂s ⎞

∫ ∂t ⎜⎝ ∂ξ ⎟⎠ ∂s ds = ∫ ∂t ⎜⎝ log ∂ξ ⎟⎠ ds,

∂ ∂R ∂ ⎛ ∂R ⎞ ∂ξ ∂ ⎛ ∂R ⎞ ∂ξ = ⎜ ⎟ = ⎟ = ⎜ ∂s ∂t ∂ξ ⎝ ∂t ⎠ ∂s ∂t ⎝ ∂ξ ⎠ ∂s =

(5.95)

∂ ⎛ ∂s ⎞ ∂ξ ∂t ∂⎛ ∂s ⎞ + t ⎜ log ⎟ . ⎜t ⎟ = ∂t ⎝ ∂ξ ⎠ ∂s ∂t ∂t ⎝ ∂ξ ⎠

Then, we multiply the last equation in a scalar way by t and with identity t⋅∂t/∂t = 0, obtain the following:

∂⎛ ∂s ⎞ ∂ ∂R ∂ ⎛ ∂R ⎞ n ∂R = ⎜t ⋅ . ⎜ log ⎟ = t ⋅ ⎟− ∂t ⎝ ∂ξ ⎠ ∂s ∂t ∂s ⎝ ∂t ⎠ ρ ∂t

(5.96)

Substitution of this relationship into ∂L/∂t and consideration of one more identity

∫ ∂/∂s (t⋅∂R/∂t)ds = 0 gives us the required formula

282

5 Dynamics of vortex filaments

dL ∂R n =− ⋅ ds. dt ∂t ρ

∫

(5.97)

5.4.4 Vortex filament with an inner structure Let us consider a more complex case of vortex filament, one with an inner structure. If we want to calculate FI using Eq. (5.72), we have to find the force of pressure and momentum flux through an end face of a control cylindrical element in the vortex core (see Fig. 5.15). We start our consideration with the pressure. In leading order, we obtain for the zone inside the core ∂p uθ2 , = ∂r r

and for the boundary (from (5.89)) p = − Γ2/8π2a2,

r = a.

Then the pressure force on the end face is as follows (with the negative sign because we want to take into account the sign in (5.72)) a a ⎡ a ∂p ⎤ Γ 2 π 2 2 − 2πrp dr = −π ⎢ pr 2 − r 2 + a uθ , dr ⎥ = 0 r 8 2 ∂ π ⎢ ⎥ 0 0 ⎣ ⎦

∫

∫

where the bar means averaging over the core cross-section, and the initial integral is integrated by parts. Now we consider the momentum flux. The liquid velocity in the vortex core is

∂R/∂t + wt + uc, where ∂R/∂t is the velocity of a point on the vortex filament at ξ = const, w is the axial velocity in the local coordinate system moving with a fixed ξ, and uc is the velocity in the core cross-section. Then the momentum flux through this cross-section is determined as ∂R ⎛ ∂R ⎞ − w⎜ + wt + uc ⎟ dA = −πa 2w − πa 2 w2t − wuc dA. ∂t ⎝ ∂t ⎠

∫

∫

The last integral differs from zero only for a bended vortex filament. The direct calculation is a tedious procedure. Therefore let us apply the

5.4 Application of momentum balance to description of vortex filament dynamics

283

technique described by Moore and Saffman (1972). We assume, with the accuracy of O(1/ρ), that an element of vortex core is considered as a part of a vortex ring, which is fixed in a uniformly moving liquid. Let us use a cylindrical polar coordinate system, as shown in Fig. 5.18, with the velocity components (u1, u2, w). In the first approximation, the equation for the ring surface is x12 + ( x2 − ρ ) = a 2 . The desired integral is written in the form: wuc dA = w ( e1u1 + e2u2 ) dA, 2

∫

∫

where e1, e2 are the unit vectors. Then the integral components can be presented in a general form

⎡ ∂

∂u j

∂w ⎤

∫ wuidA = ∫ ⎣⎢⎢ ∂x j ( wuj xi ) − wxi ∂x j − xiuj ∂x j ⎦⎥⎥ dA.

(5.98)

Here i, j = 1,2, and summation is assumed for repeating indices. The derivatives in the integrand can be obtained from the equations of motion written in cylindrical coordinates

∂u j ∂x j

=−

u2 , x2

uj

uw ∂w =− 2 . ∂x j x2

The integral from the first term in (5.98) equals zero, since it is reduced to the velocity component normal to the vortex surface. It is zero because the vortex surface is the surface composed by the streamlines. Substituting the expressions for derivatives in (5.98), we obtain u

∫ wui dA = 2∫ x22 wxi dA .

Fig. 5.18. On the analysis of a vortex filament with an inner structure

284

5 Dynamics of vortex filaments

In leading order, we accept that

w ≈ w0(r), uθ ≡ v ≈ v0(r),

u2 = (x1/r)v0(r),

where r 2 = x12 + ( x2 − ρ ) ≤ a 2 , θ is the polar angle in the core cross2

section; so the final results with the accuracy of O(1/ρ2) are as follows

∫

∫

wu1dA = 2 w0v0

a

x12 r 2 cos 2 θ 2π dθdr ≈ 2 w0v0 dθdr = w0v0r 2 dr , x2 ρ ρ

∫

∫ 0

∫ wu2dA = 2∫ wu2dA = 0 . Therefore

∫

a

wuc =

2πb 2 r w0v0 dr = λΓw0 a 2b / ρ. ρ

∫ 0

Here we use identity e1 ≡ b, and coefficient λ is introduced for calculation convenience. For the case of a uniform vortex w0 = w = const, v0/r = const, λ = 1/4. Calculation of FI by (5.72) requires estimation of the temporal change in the momentum for the volume ∆ = ds πa2, which moves at ξ = const. Then from (5.72) we obtain

⎞ ∂s ⎤ 1 ∂ ⎛ 1 ∂ ⎡⎛ ∂R ⎞ ∂R ⎢⎜ wπa2t + πa 2 + uc dA ⎟ dξ ⎥ . ⎜ wt + uc + ⎟ dsdA = ⎟ ∂ξ ⎥ ∂t ⎠ ∂t ∆ ds ∂t ∆ ⎝ ds ∂t ⎢⎜ ⎠ ⎣⎝ ⎦

∫

∫

The calculations are made with the accuracy of O(1/ρ3). Then we can neglect the last two terms through the estimates

⎛ 1 ⎞ ∂R ⎛1⎞ ∂ = O⎜ 2 ⎟, = O⎜ ⎟ , ∂t ⎝ρ⎠ ⎝ ρ ⎠ ∂t

⎛1⎞

∫ ucdA = λΓa b / ρ = O ⎝⎜ ρ ⎠⎟ . 2

Factoring out dξ (as an independent variable) and expanding the first term on the right-side, we have ∂ ∂⎛ ∂s ⎞ wπa 2t + wπa 2t ⎜ log ⎟ . ∂t ∂t ⎝ ∂ξ ⎠

(

)

Further calculations and estimates for some terms would require the integral equation of mass conservation. Let us derive it for a cylindrical ele-

5.4 Application of momentum balance to description of vortex filament dynamics

285

ment of the vortex core with the length of ds = s|ξ+dξ – s|ξ, which moves at ξ = const. Then the mass balance is the following

(

)

∂ ⎡ s ξ+ dξ − s ξ πa 2 ⎤⎥ = wπa 2 − wπa 2 , ⎢ ξ ξ+ dξ ⎦ ∂t ⎣ wherefrom we have ∂ ⎡ 2 ∂s ⎤ ∂ =− a wa 2 . ⎢ ⎥ ∂t ⎣ ∂ξ ⎦ ∂ξ

(

)

(5.99)

In particular, this gives us the estimate

(

)

( )

∂ wa 2 = O ρ−2 . ∂s Substituting those expressions into (5.72), and taking into account (5.96) and making estimates for the terms, we find with the accuracy of O(1/ρ3) that

FI = −

∂ ∂⎛ ∂s ⎞ (wπa 2t ) − wπa 2t ⎜ log ⎟ + ∂t ∂t ⎝ ∂ξ ⎠

⎡ Γ2 ⎛ u2 ⎞⎤ n ∂ ⎡ ⎛ u2 ⎞⎤ + ⎢ + πa 2 ⎜ θ − w2 ⎟ ⎥ + ⎢ πa 2 ⎜ θ − w2 ⎟ ⎥ t − ⎜ 2 ⎟ ⎥ ρ ∂s ⎢ ⎜ 2 ⎟⎥ ⎢ 8π ⎝ ⎠⎦ ⎝ ⎠⎦ ⎣ ⎣ ∂t ∂⎛ ∂s ⎞ −πa 2w − πa 2wt ⎜ log ⎟ + ∂t ∂t ⎝ ∂ξ ⎠ +Γλτwa 2

(5.100)

n ∂ ⎛λ⎞ − Γwa 2b ⎜ ⎟ . ρ ∂s ⎝ ρ ⎠

Finally, combining (5.73), (5.90), (5.100) and retaining only the components perpendicular to t, we obtain the desired equation for the vortex filament motion

∂R ⎞ n ⎛ 2 ∂t 2 ∂ ⎛λ⎞ 2n Γ ⎜ VI − ⎜ ⎟ b − Γλτwa , (5.101) ⎟ × t + T = 2πa w + Γwa t t s ∂ ρ ∂ ∂ ρ ρ ⎝ ⎠ ⎝ ⎠ 1 T = T0 + πa 2 uθ2 − πa 2 w2 , 2 T0 coincides with the tension in a hollow vortex ring determined by Eq. (5.69), and τ is the torsion, which appears here due to the application of the

where

286

5 Dynamics of vortex filaments

Frenet – Serret formula. Note that the terms on the left-side of (5.101) have the order O(1/ρ), and for the right-side terms the order is O(1/ρ2). Since we want to check (5.101), let us consider two particular cases. For the first case, we take a vortex ring spreading in a resting medium. Then VI ≡ 0, and all terms in the right-side become zero. Vector multiplication by t yields the equation

∂R ⎛ 1 ⎞ b = ⎜ T0 + πa 2 uθ2 − πa 2 w2 ⎟ , ∂t ⎝ 2 ⎠ Γρ which was derived previously with other methods in the papers by Saffman (1970) and Widnall et al. (1971). The second example is a left-handed vortex filament with a high pitch; it rotates with angular velocity ω and has a uniform core. Then, inside the core λ = 1/4, Γ = 2πΩa2, uθ2 = Ω 2 a 2 / 2, w = W , w2 = W 2 , where Ω is the angular velocity of fluid rotation in the core; W is the constant axial velocity. Velocity VI is easily found through direct calculations using (5.74). A similar procedure for both integrals was carried out in Section 5.1, where the cut-off method was considered. If the cut-off parameter δ is the same for a helix and a ring, then it vanishes during calculation of the integrals. For a long-wave limit, we obtain from (5.8) and (5.14) the following relationship VI =

4ρ ⎤ 2 8ρ 1⎤ Γ ⎡1 δaγ Γ ⎡ − E − log − log ⎥ b = k2 R ⎢log − log − E + ⎥ b . ⎢ R a δa ⎦ 4πρ ⎣ 2 4π 2⎦ ⎣ ka

Here we take into account that for a high pitch helix the curvature radius is ρ = R/γ2, j ≈ –b (see Fig. 5.17), τ = −k, where γ = 2πR/λ, k = 2π/λ, λ and R are the helix pitch and radius. However, we cannot use the ready formulae from Sections 5.1 and 5.2 for the determination of ∂R/∂t and ∂t/∂t, since another form of curve presentation was given there. The value ∂R/∂t is found by the following reasoning. Since VI ∼ b, then we obtain from the helix geometry and (5.75) that ∂R/∂t ∼ b. Due to a high pitch, the filament is almost parallel to axis z (see Fig. 5.2). Therefore, ∂R/∂t = ωRb. The derivative ∂t/∂t is easily calculated from (5.95), (5.96), (2.6) and the last formula ∂t ∂ ⎛ ∂R ⎞ ∂b = ⎜ = −ωRτn = kRωn. ⎟ = ωR ∂t ∂s ⎝ ∂t ⎠ ∂s

5.4 Application of momentum balance to description of vortex filament dynamics

287

Substituting all the obtained expressions into (5.101), we find the dimensionless frequency of helix rotation

ω ⎛ k2 a 2 K k2W 2 Wk3a 2 ⎞ ⎛ Wk ⎞ =⎜ − − ⎟ ⎜1 + ⎟≈ 4Ω ⎟⎠ ⎝ Ω ⎠ Ω ⎜⎝ 2 2Ω 2 ≈

k 2 a 2 K k2W 2 Wk3a 2 1 W 3k3 1 a 2k3 KW − − + − , Ω 2 4Ω 2 Ω3 2 2Ω 2

where K = log(2/ka) – E + 1/4. Note that here we do not use the condition R a, accepted for Kelvin waves. 5.4.5 Consideration of the inner core structure For the general case of an arbitrary vortex filament the equation (5.101) is not closed because the changes in the velocity field inside the core in dependence on time and the coordinates have not been determined. This refining can be accomplished with two unused equations – mass conservation (5.99) and balance of the longitudinal component of impulse, which takes the form (projection of Eq. (5.73) on t)

∂ Γ2 ∂a2 ∂ ⎛ 1 2 2 ∂ ⎛ ∂s ⎞ ⎞ + ⎜ πa uθ − πa2 w2 ⎟ − 2πa2w ⎜ log ⎟. (5.102) (πa2w) = − 2 ∂t ∂t ⎝ ∂ξ ⎠ 8πa ∂s ∂s ⎝ 2 ⎠ It was demonstrated in the paper by Moore and Saffman (1972) that a variation in the core radius along the filament has the order O(a2/ρ2). The first approximation gives us

a = a(t), and the time dependence is determined from the condition of volume conservation in the filament core with length L

La2 = const.

(5.103)

Similarly, the tangential velocity uθ in the first approximation is independent of s and can be found from the condition for circulation conservation:

uθ (r , t ) =

Γ ⎛r⎞ g ⎜ ⎟ , g(1) = 1. 2πr ⎝ a ⎠

(5.104)

Here, the dimensionless function is determined by the initial state of the vortex. Then we can write

288

5 Dynamics of vortex filaments

a 2 uθ2 =

Γ2 8π 2

µ,

(5.105)

1

µ=4

where

g 2 (η) dη. η 0

∫

For a core with uniform vorticity g(η) = η2, µ = 1. The analysis of axial velocity w(t, r, ξ) is more complex. In the general form it can be presented as w = W (t) + q(ξ, t ) +

Γ ⎛r⎞ ϕ⎜ ⎟ m ⎝a⎠

(5.106)

according to the following reasoning. With the uniform stretching of a tube, the velocity profile is kept, and this is reflected in the third term, where the length m is determined from the initial conditions. The first term is a contribution to the average velocity due to a change of a(t). The second term is responsible for variation of average velocity along the tube and it has the order O(1/ρ), since it is estimated by the velocity component V|| ≡ t ⋅ ∂R/∂t. Without loss of generality, we can apply an additional condition 1

∫ ηϕ(η)dη = 0, 0

which means a zero contribution from the third term into the total liquid flow rate through the tube cross-section. Then the values averaged over the tube cross-section are the following w = W + q,

(

)

a 2 w2 = a 2W 2 + νa 2 Γ 2 m 2 + 2a 2Wq + O 1 ρ2 ,

1

∫0

where ν = 2 ηϕ2 (η)dη = const, which is found from initial conditions. The equation for q is derived by substitution of (5.106) into the mass conservation equation (5.99) in view of Eqs. (5.96) and (5.103)

∂V n ∂R 1 dL ∂q ∂⎛ ∂s ⎞ 1 dL = − ⎜ log ⎟ + = − II + ⋅ + , ∂s ∂t ⎝ ∂ξ ⎠ L dt ∂s ρ ∂t L dt

(5.107)

where the vortex length L, in turn, is given by Eq. (5.97). Note that (5.107) allows the calculation of q with the accuracy of an arbitrary function of time. Therefore the unique determination of q requires an additional condition

5.4 Application of momentum balance to description of vortex filament dynamics

∫ qds = 0,

289

(5.108)

that will be explained below. Now we determine the axial component of velocity, W(t). To do this, we substitute (5.106) into (5.102) and taking into account (5.107) we obtain the equation

2a 2W dL d 2 + (a W ) = 0. L dt dt Whence

Wa2L2 = const,

or WL = const.

The latter relation with regard to (5.108) means conservation of the axial velocity along the axis of a closed vortex filament. Indeed,

∫ wds = ∫ Wds + ∫ qds = WL = const.

L

There is one more parameter which characterizes the inner structure of the vortex – parameter λ; it can be found from the definition

λ=

a

2π Γwa

1

2

⎡

Γ

⎤

∫ r uθdr = ∫ ⎢⎣1 + Wm ϕ(η)⎥⎦ ηg(η)dη + O(1 ρ). 2

0

0

Obviously, λ = λ(t). In this way, the parameters of the inner structure are determined by initial conditions and the behavior of filament length L, which is found from the vortex motion equations. The result is a closed system of equations for vortex filament dynamics. The filament motion equation (5.101) can be rewritten in the form

n n ∂R ⎞ ∂t ⎛ 2 2 Γ ⎜ VI − + ⎟ × t + T = 2πa Wq + 2πa W ∂t ⎠ ρ ρ ∂t ⎝ ∂ ⎛b⎞ + λΓWa 2 ⎜ ⎟ + O(1 ρ3 ), ∂s ⎝ ρ ⎠

(5.109)

where

T = T0 +

Γ 2µ πa 2 Γ 2 ν − πa 2W 2 − , 16π m2

T0 =

Γ2 ⎛ 8ρ 1 ⎞ ⎜ log − ⎟ . 4π ⎝ a 2⎠

290

5 Dynamics of vortex filaments

The first right-side term in Eq. (5.109) was transferred from the left-side of the expression for tension T. The vector multiplication by t gives the equation

Tb b ∂R ∂t ∂ ⎛b⎞ − VI − = −2πa 2t × − 2πa 2Wq − λΓWa 2t × ⎜ ⎟ . (5.110) ∂t Γρ ∂t ρ ∂s ⎝ ρ ⎠ In leading order, only the left-side remains

∂R Tb = VI + . ∂t Γρ

(5.111)

Solving of (5.110) requires estimation of ∂t/∂t and q, and this can be performed using (5.95), (5.107), and also the first approximation for ∂R/∂t. Then

t× s

q=

∫

Tb ⎞ ∂t ∂⎛ = t × ⎜ VI + ⎟, ∂t ∂s ⎝ Γρ ⎠ s

⎡ ∂ ⎛ ∂R ⎞ n ∂R ⎤ ∂VI s dL s dL ⎢ − ∂s ⎜ t ⋅ ∂t ⎟ + ρ ⋅ ∂t ⎥ ds + L dt = − t ⋅ ∂s ds + L dt . ⎝ ⎠ ⎣ ⎦

∫

In the integral for q we did not specify the lower limit, because according to (5.107), an arbitrary function of time must be added to q to satisfy condition (5.108). Thus, (5.110) together with auxiliary equations, creates a closed system of equations which is enough for the calculation of a deformed vortex filament motion. We must also note that if any external velocity field VE(ξ) exists, it can be simply added to the quantity VI. 5.4.6 Modified equations of vortex filament motion The method of force balance was applied by Lundgren and Ashurst (1989) for the derivation of modified equations. The main difference from the previous derivation is that the local stretching of the filament was taken into account, i.e., a = a(ξ, t). In particular, this allows an opportunity for the description of traveling waves with variable core cross-section. Omitting computations and transformations that resemble the path of the previous analysis, we can write the closed system of the equation of Lundgren and Ashurst (1989) for the description of evolution of the shape and radius of a vortex filament

5.5 The method of matched asymptotic expansions

291

∂R ⎧ = VB + u, ⎪ ∂t ⎪ u ⋅t ∂ ⎪∂ Γσ Γ 2t ∂ (log σ), ⎪ ∂t [ 2u − (u ⋅ t )t ] = σ ∂ξ [ u − (u ⋅ t )t − VB ] + 2 t × u + a0 8πa02 ∂ξ ⎪ ⎨ ∂R ∂s 1 ∂R ⎪ = , , σ= t= ⎪ σ ∂ξ ∂ξ ∂ξ ⎪ ∂σ ∂ ⎪ = t ⋅ (VB + u), σa 2 = a02 . ⎪ ∂t ∂ξ ⎩ Here R = R(ξ,t); a = a(ξ, t); a0 is the initial radius of the core, parameter σ is the velocity of the local stretching; ξ is a material variable (a point with fixed value of ξ moves with the velocity V = ∂R/∂ξ); u = V – VB is the liquid velocity in the core relative to the ambient medium; VB = VB(s, t) is the self-induced velocity at point R(s, t), determined by the formula of Rosenhead (5.6) with parameter µ2 = 0.22 (a uniform vorticity in the core). The last of these equations represents mass conservation for a local element in the core. Here, we emphasize again that the concept of relative velocity for a vortex is rather conventional: in this case it is defined relative to the velocity VB.

5.5 The method of matched asymptotic expansions The method of matched asymptotic expansions (MAE) was first used by Widnall et al. (1971) for the description of dynamics of a vortex filament with internal structure. Calculation of self-induced velocity was performed with the accuracy of O(Γ/ρ). Callegari and Ting (1978) extended this method for the case of vortex motion in a viscous medium. Fukumoto and Miyazaki (1991) made the computations with method with the accu2 racy of O(Γa/ρ ) and derived the filament motion equation obtained previously by Moore and Saffman (1972) with a less strict method of force balance (see Section 5.4). The main aim of that study was the determination of the effect of the internal vortex structure (mostly axial velocity) on motion and stability of the vortex filament. The method is valid for thin vortex filaments (a/ρ 1). The key point is that within approximation of a thin filament, we can find the outer (using the Biot – Savart law) and inner (on the basis of Euler or Navier – Stokes equations) solutions for velocity field. Then we match these solutions.

292

5 Dynamics of vortex filaments

Herein we outline the method following the paper by Fukumoto and Miyazaki (1991), which here we denote simply as FM. This method will be applied for derivation of the Moore – Saffman equation (MSE), which has no viscosity and has the accuracy of O(Γa/ρ2). Later within the framework of local induction approximation (LIA) we obtain from MSE the simplified equations of evolution: the generalized local induction equation, Hirota equation, nonlinenear Schrödinger equation, and modified Korteweg – de Vries equation. As an example, the accurate N-soliton solutions of those equations will be considered; some comments on other kinds of solutions are also given. 5.5.1 Derivation of the equation for vortex filament motion We begin the derivation procedure with the estimation of self-induced velocity u, and for this we take the Biot – Savart law (2.14) in the case of a vorticity vector directed along the filament axis u(r ) = −

Γ (r − R′) × t′ ds′. 3 4π r − R′

∫

(5.112)

Here R(s′, t) is the radius-vector of a point on the vortex filament axis. We use the local orthogonal cylindrical coordinate system (r, θ, s) introduced in Section 5.4.2 and depicted in Fig. 5.16. We also notate the required relationships from 5.4.2 retaining the previous symbols

r = R(s, t) + rer , dr = er dr + reθdθ + htds, h = 1 – κr cos ϕ, θ = ϕ – θ0(s, t),

∂θ0/∂s = −τ.

Let us expand R′ in the Taylor expansion relative to point R up to the cubic terms in (s′ – s), taking into account the Frenet – Serret equations. A similar procedure was performed (up to the terms of second order) in Section 2.4. The result is as follows 1 1 R′( s′, t ) = R( s, t ) + t ( s′ − s ) + κ n( s′ − s ) 2 + (−κ 2t + κ s n + κτb)( s′ − s )3 , 2 3!

where κs = dκ/ds. Substituting this formula into (5.112) and integrating into the limits (−L, L), yields for the case a r ρ (see Section 2.4)

5.5 The method of matched asymptotic expansions

u (r ) = +

293

Γ Γκ ⎡ 2L ⎤ Γκ eθ + log cos ϕ eθ + − 1⎥ b + ⎢ r 2πr 4π ⎣ 4π ⎦

3Γκ2r ⎧ 1 ⎫ ⎡ 2L 4 ⎤ 1 − ⎥ + cos 2ϕ eθ + eθ ⎬ + (5.113) ⎨(er sin 2ϕ + eθ cos 2ϕ ⎢log 16π ⎩ r 3 2 18 ⎣ ⎦ ⎭ ⎛ Γa 2 ⎞ +Q + O ⎜ 3 ⎟ . ⎜ R ⎟ ⎝ ⎠

Here Q represents the remaining non-local contribution into the Biot – Savart integral, and the first three terms on the right-side coincide exactly with the quadratic approximation (2.35). Equation (5.113) cannot be used directly for finding the vortex filament velocity because it is valid in the tube vicinity, but not in the tube itself, where the inner structure (i.e. the velocity field) has to be considered. The Euler equation for a vortex tube and perturbation method will be used for calculation of the filament velocity. Then the solution obtained must be matched with (5.113). Let us give a more accurate definition of the radius-vector R(s,t), which characterizes the position of the vortex axis: that is the center of axial vorticity distribution in each cross-section of the vortex. We have an additional requirement that the curve R moves as a material line, i.e., it consists of liquid particles. For this purpose, we introduce a material variable ξ, which is defined unambiguously in the additional condition for R, for example

R(ξ, t ) ⋅ t = Q(ξ, t) ⋅ t = Q . Here R = ∂R(ξ, t) ∂t gives the velocity of a point on the filament axis at ξ = const; Q is a non-local component of self-induced velocity from (5.113) at r → 0. Since the scalar multiplication by t makes all the terms in (5.113) zero (except Q), this equality means the equality of tangential components of self-induced velocity and the velocity of the vortex axis ∂R/∂t. The equations of motion are more convenient in a dimensionless form. For the given problem, we have two scales of length – the characteristic radius of core a0, and we accept the initial undisturbed value as this length, and the characteristic radius of curvature ρ0 for the curve R(s,t). Let us introduce the ratio of these lengths ε

ε = a0/ρ0 and then we search for a solution in the form of expansion in ε on condition

294

5 Dynamics of vortex filaments

ε

1.

With an assumption that the scale of changes along the vortex filament has the order of O(ρ0), we normalize the variables s, ξ, R, ρ to the value ρ0. For the transversal coordinate r, we apply the scale a0 for the case of inner expansion and the scale ρ0 is used for outer expansion. As the timescale, we choose the value 2πρ02 / Γ . This means that the vortex evolves as a whole in accordance with LIE and we consider the slow bending modes. Finally, the scale for velocity is the maximum tangential velocity (Γ/2πa0). The procedure is as follows. The external solution is found from the Biot – Savart law, when the vortex tube is considered as a spatial curve. Asymptotic behavior of the external solution in the intermediate area is governed by Eq. (5.113). We use the same equation for the matching of outer and inner solutions as a boundary condition for the inner solution at infinity. The inner solution, naturally, is searched for in local cylindrical coordinates (r, θ, s). Let us denote the liquid velocity in this coordinate system as V = (u, v, w). Then for the coordinate system in the remainder, the dimensionless velocity is given by expression

u(r , θ, s, t) = εR + V (r , θ, s, t ). Expansions for velocities and the pressure in parameter ε are written in the form

u = εu(1) (r, θ, s, t) + ε2u(2) (r, θ, s, t) + …, v = v(0) (r, t) + εv(1) (r, θ, s, t) + ε2v(2) (r, θ, s, t) + …, w = w(0) (r, t) + εw(1) (r, θ, s, t) + ε2w(2) (r, θ, s, t) + …, p = p(0) (r, t) + εp(1) (r, θ, s, t) + ε2p(2) (r, θ, s, t) + …, R = R(0) (ξ, t ) + εR(1) (ξ, t )t... .

(5.114)

The superscript (0) symbolizes the undisturbed values for a rectilinear vortex. The time dependence for them is conditioned by the vortex line stretching; this stretching is uniform relative to s. The boundary condition at the vortex axis is

u = v = 0 at r = 0. The rest of the boundary conditions appear after matching of the inner and outer solutions. For this purpose, we rewrite (5.113) in dimensionless inner variables. Then the matching conditions at r → ∞ take the form

5.5 The method of matched asymptotic expansions

v(0) ~ u (1) ~

v(1) ~

295

1 + 0(r −2 ), w(0) ~ 0(r −2 ), r

1 ⎡ ⎛ 2L ⎞ ⎤ (1) (0) κ ⎢ log ⎜ ⎟ − 1⎥ sin ϕ + ⎡⎣Q − R ⎤⎦ er , 2 ⎢⎣ ⎝ εr ⎠ ⎥⎦

1 ⎡ ⎛ 2L ⎞ ⎤ 1 (1) (0) κ ⎢ log ⎜ ⎟ − 1⎥ cos ϕ + κ cos ϕ + ⎡⎣Q − R ⎤⎦ eθ , 2 ⎣⎢ ⎝ εr ⎠ ⎦⎥ 2

⎡ ⎛ 2L ⎞ 4 ⎤ 3 (2) (1) u (2) ~ κ 2r ⎢ log ⎜ ⎟ − ⎥ sin 2ϕ + ⎡⎣Q − R ⎤⎦ er , 8 ⎢⎣ ⎝ εr ⎠ 3 ⎥⎦ ⎡ ⎛ 2L ⎞ 5 ⎤ κ 2r ⎡ (2) 3 − ϕ + + Q − R(1) ⎤ eθ , cos 2 v(2) ~ κ 2r ⎢log ⎜ ⎥ ⎟ ⎣ ⎦ ε 8 6 48 r ⎠ ⎣⎢ ⎝ ⎦⎥ where L = L / ρ0 . If we need to find the internal solution, we have to sustitute expansions (5.114) into the Euler equations written for a moving curvilinear coordinate system. Avoiding a long transformation chain (see the paper of Fukumoto and Miyazaki (1991) and Callegari and Ting (1978)), we present here the resulting expression for the velocity of vortex filament motion in dimensional coordinates

⎡ κΓ ⎧⎪ 4π2 r ⎛ 2L ⎞ 1 ⎛ r ⎞ ⎫⎪ (0) 2 − + lim R ( ξ, t ) = ⎢ log ⎜ ⎨ 2 r ⎡⎣v ⎤⎦ dr − log ⎜ ⎟ ⎬ − ⎟ ⎢ 4π ⎝ a ⎠ 2 r →∞ ⎩⎪ Γ 0 ⎝ a ⎠ ⎭⎪ ⎣ ∞ ∞ ⎤ ⎤ 8π 2 4π ⎡ 1 (0) 2 (0) (1) ˆ 0 dr ⎥ b − − 2 r [w ] + 2εw w ⎢ rw(0) dr ⎥ t × Rξ(0) − (5.115) Γ ⎢0 Γ 0 ⎥⎦ ⎥⎦ σ ⎣ ∞ ⎤ 2π ⎡ ∂ − ⎢ r 2v(0) w(0) dr ⎥ t × ( κb) + Q. Γ ⎢0 ⎥⎦ ∂l ⎣

∫

∫{

}

∫

∫

Here the rate of local stretching is σ ≡ |Rξ| = ∂s/∂ξ; the uniform part a = a(t) of core radius is determined by expansion a(l, t) = a(t) + ε2a(2)(s, t), and the requirement for volume conservation ˆ 0(1) is a symmetric part of quandictates relationship a(t)2s(t) = const; and w (1) (1) (1) (1) tity w : w = wˆ 0 (r , s, t ) + wˆ (r , s, t )cos ϕ . We emphasize that Eq. (5.115) was derived with the second order of accuracy and coincides with the corresponding formula of Moore and Saffman (1972). However, for the latter case the computation was performed through a less strict procedure, using the force balance method.

296

5 Dynamics of vortex filaments

Before the derivation of model equations from (5.115), we analyze the dispersion relationships to understand the contribution of the second-order terms and to see the influence of axial velocity in the vortex core. Considering the bending mode m = ±1 in a long-wave approximation, i.e., at |k| 1, where k is the azimuthal wave number, we obtain the analytical formula for frequency ω (see Fukumoto and Miyazaki (1991))

ω=−

∞ ⎡⎛ ∞ ⎞ mΓk2 2π ⎛ 2 (0) (0) ⎞ ⎤ ⎜ r v w dr ⎟ ⎥ , C0 + k3 ⎢⎜ rw(0) dr ⎟ C0 + ⎟ ⎟⎥ 4π Γ ⎜0 ⎢⎜⎝ 0 ⎠ ⎝ ⎠⎦ ⎣

∫

∫

where the integration constant C0 equals

⎛ 2 C0 = log ⎜ ⎜ ⎝a k

⎞ ⎟⎟ − γ + lim r →∞ ⎠

2 ∞ ⎡ 4π 2 r ⎛ r ⎞ ⎤ 8π r[v(0) ]2 dr − log ⎜ ⎟ ⎥ − 2 r[w(0) ]2 dr. ⎢ ⎝ a ⎠ ⎥⎦ Γ 0 ⎢⎣ Γ 0

∫

∫

This expression is valid for a columnar vortex with arbitrary, but axisymmetrical distributions of axial and azimuthal components of vorticity. A similar relationship was obtained by Moore and Saffman (1972) for a vortex filament with a high pitch.

Fig. 5.19. Asymptotic behavior of dispersion curves for m = –1. Columnar vortex with empirical velocity profile is taken from Maxworthy et al. (1985). 1, 2 – calculations for the first and second orders of accuracy in ε (Fukumoto and Miyazaki 1991*)

5.5 The method of matched asymptotic expansions

297

Fig. 5.20. Asymptotic values of frequency ω, phase velocity c and group velocity cg for the second order of approximation in ε (Fukumoto and Miyazaki 1991*). Conditions are described in the captions of Fig. 5.19

The examples of computations for velocity profile (4.65) with empirical constants W0/Vm = 0.4, β = 0.54, α = 1.28, Γ/(2πaVm) = 1.37 (Maxworthy et al. 1985), where Vm is the maximum tangential velocity, are plotted in Fig. 5.19, 5.20 for the mode m = –1. Figure 5.19 shows the comparison of calculated values of frequency for the first (1) and second (2) approximations. One can see that taking into consideration the second order corrections results in asymmetry of relationship ω(k) relative to the axis k = 0. Figure 5.20 is a plot of distributions for frequency ω, phase velocity c and group velocity cg as functions of wave number k in dimensionless form. Note that this data is in agreement with numerical computations for the Howard–Gupta equation, performed by Leibovich et al. (1986) for the same velocity profiles and valid for the entire range of wave numbers. 5.5.2 Local induction approximation In the framework of a local induction approximation (see Section 2.4), we neglect the contribution from the non-local induction Q and a change in cut-off parameter L/a along the vortex filament. We rewrite (5.115) in the form

298

5 Dynamics of vortex filaments

R = κAb − Bt × Rξ(0) σ − Ct ×

∂ ( κb), ∂s

where A, B, C are the corresponding coefficients. We transform the derivative relative to ξ in the following way: Rξ(0) σ = Rs(0) , and R(0) is replaced with its first approximation, that is κAb. Then we change the normalization for the time variable: tA → t. Then, using the Serret – Frenet equations and definitions for unit vectors, we obtain

R(ξ, t) = κb + β ( κ s n + κτb),

(5.116)

∞

β=

where

4π rw(0) dr + Γ

∫ 0

∞

+

2π 2 (0) (0) r v w dr Γ

∫

0 r

∞ ⎛π r ⎞ 2π 2L 1 ⎞ Γ ⎛ Γ (0)2 ⎜ ⎟ rv dr rw(0)2 dr log lim log − + − − ⎜ ⎟ a 2 ⎠ r →∞ ⎜ Γ a⎟ Γ 2π ⎝ 4π 0 ⎝ 0 ⎠

∫

.

∫

In the formula for A we neglected the small term with ε. Now we have to make the transition to variable s in the left-side of Eq. (5.116). Variables ξ and s are related through the dependency ξ = s − Wt, where W is a certain slip velocity that has to be found. We denote ∂R(s, t)/∂t as Rt, and ∂R(ξ, t)/∂t as R, and then perform the following chain of transformations:

R=

∂ ∂R ∂R R( s − Wt, t) = −W ( s, t ) = −Wt + Rt . + ∂t ∂s ∂t

Then instead of (5.116) we obtain

Rt = κb + β(κsn + κτb) + Wt. Differentiation of this equation in respect to the arc length s and use of the Serret – Frenet formula gives us the following

Rst = κst − κτn + β[κssn + κs(τb − κt) + (κsτ + κτs)b − κτ2n] + Wst + Wκn. Then we use the identity Rst ⋅ Rs = 0, which is a consequence of Rs ⋅ Rs = t ⋅ t = 1 after time differentiation. After scalar multiplication of the last equa-

5.5 The method of matched asymptotic expansions

299

tion by Rs, we obtain Ws = βκsκ or W = βκ2/2. The resulting equation for Rt takes the form ⎛1 ⎞ R = κb + β ⎜ κ 2t + κ s n + κτb ⎟ = ⎝2 ⎠ 3 ⎡ ⎤ = Rs × Rss + β ⎢ Rsss + Rss × ( Rs × Rss ) ⎥ . 2 ⎣ ⎦

(5.117)

This equation is a generalization of LIE and it takes into account the axial velocity with the second order of accuracy through parameter β. It was demonstrated in Section 5.3 that through the Hasimoto transformation (Hasimoto 1972) the LIE reduces to a nonlinear Schrödinger equation. Using the same procedure, the vector differential equation (5.117) can be reduced to the well-known equation of Hirota (1973) for a scalar function. For this purpose, let us introduce the complex functions and N (see Section 5.3)

⎛ s ⎞ = κ exp ⎜ i τds ⎟ , ⎜ ⎟ ⎝ 0 ⎠

∫

⎛ s ⎞ N = ( n + ib ) exp ⎜ i τds ⎟ , ⎜ ⎟ ⎝ 0 ⎠

∫

where τ is the torsion. Then the Serret – Frenet formulae

ts = κn, ns = – κt + τb, bs = – τn can be presented in the form

ts = (

*

N+

Ns = –

N*)/2,

(5.118)

t.

Here subscript s means a derivative with respect to the arc length. Then we differentiate (5.117) with respect to the arc length and taking into account the definition for tangential vector t obtain, that

⎡⎛ 1 ⎤ ⎞ tt = κ sb − κτn + β ⎢⎜ κ ss + κ3 − κτ2 ⎟ n + ( 2κ s τ + κτ s ) b ⎥ = 2 ⎠ ⎣⎝ ⎦ * * = −( N + N )/2, where

=−i

s

−β

2

ss

− β| |

/2.

(5.119)

(5.120)

It was shown in Section 5.3 for the derivative Nt that it can be presented in the form

300

5 Dynamics of vortex filaments

Nt = iFN + ft,

(5.121)

where functions F and f are to be determined, and F is a real function. We again use the relationships: N⋅N′ = N⋅t = 0, N⋅N∗ = 2. From the condition (N⋅t)t = 0 and Eqs. (5.119), (5.121) we obtain f = . Then we differentiate (5.118) with respect to t and (5.121) with respect to s. The rightsides of the obtained equations are equal, so we have two new equations for the determination of function F *

Fs = i( t

− iF

Substituting the expression for

F=

1 2

2

t

+

ss

+

+

s

(5.122)

)/2,

(5.123)

= 0.

in (5.122) and integrating, we find F

1 + iβ 2

(

∗ s

−

s

∗

).

and F in (5.123), we obtain the Hirota equa-

Substituting formulae for tion (Hirota 1973) i

*

–

2

2

⎡ − iβ ⎢ ⎣

sss

+

3 2

2⎤ s

⎥ =0. ⎦

(5.124)

This equation at β → ∞ reduces to the modified Korteweg – de Vries equation t

−β

sss

3 − β 2

2 s

= 0.

With zero axial flow in the vortex core (β → 0), the Hirota equation is reduced to the nonlinear Schrödinger equation NSE (see (5.55)). The NSE was used in Section 5.3 for the description of a single-soliton solution – the Hasimoto soliton. The Hirota equation relates to a class of integrable equations and it has an accurate solution in the soliton form. To understand the effect of axial flow in the vortex core, we demonstrate the FM results for a single vortex soliton and for a two-soliton solution. This data is important for the description of soliton interaction. 5.5.3 N-soliton solution The Hasimoto method described in Section 5.3 is unsuitable for finding a multi-soliton solution, i.e., at N ≥ 2. Therefore FM used the soliton surface

5.5 The method of matched asymptotic expansions

301

approach previously developed by Sym (1984) and applied by Levi et al. (1983) to the description of N solitons on a vortex filament without axial flow (using the NSE as the basis). According to this approach, the Nsoliton solution is generated by a consecutive application of the Bäcklund transformation, starting from the undisturbed state of the vortex filament, i.e., the solving is purely algebraic. Calculations are the same as for the case of LIE. Therefore, we can take the results, for instance, from the paper by Levi et al. (1983), with only one variation: one has to replace the dispersion relationships. According to FM, we assume ∼ exp[2i( s − t)] and obtain from (5.124)

( i) = 2

2 i

+ 4β i3 .

(5.125)

Here j = pj + iqj are discrete eigenvalues. Then the exact solution for the shape of a single soliton (j = 1) is written in Cartesian coordinates as

X + iY = A sech [2q1(s − cg t)] exp[2ip1(s − ct)],

(5.126)

Z = s − A tanh [2q1(s − cg t)],

(5.127)

where 1 = p1 + iq1. Here the soliton amplitude A, the phase velocity of the carrier wave c and group velocity cg (envelope velocity) are as follows:

A = q1 c=

(p

2 1

)

+ q12 ,

(

)

2 ⎡ 2 p1 − q12 + 2β p1 p12 − 3q12 ⎤ , ⎣ ⎦ p1

(

)

cg = 4 ⎡ p1 − β q12 − 3p12 ⎤ . ⎣ ⎦

With zero axial flow (β = 0) this solution coincides with the Hasimoto soliton (see Section 5.3). It follows from these formulae that the axial flow has no effect on the soliton shape, but only changes the values of the group and phase velocities. The solution for Hasimoto soliton in Section 5.3 was given through the maximum curvature κmax and torsion τ. For convenience of comparisons between eigenvalues of p1 and q1 we can reformulate them through the parameters κmax and τ, if we take their definitions (see Section 2.1)

p1 = τ/2, q1 = κmax /4. Note that (see also Section 5.3) velocities cg and c are constant along the arc length s. At transition from variable s to variable Z the shape of the soliton, in view of (5.127), is described by the expression

302

5 Dynamics of vortex filaments

X(Z, t) + iY(Z, t) = =A sech [2q1{Z − cg t + E(Z, t)}] exp[2ip1{Z − cg t + E(Z, t)}], where E is a solution to transcendental equation

E(Z, t) = A tanh [2q1(Z − cg t + E)]. Obviously, the phase and group velocities in the laboratory coordinate system are not constants anymore. Let us derive the formulae for group Cg and phase C velocities in the laboratory coordinate system. We begin with the condition of a constant phase dθ =

∂θ ∂θ dZ + dt = 0, ∂t ∂Z

so the velocity for a point with a constant phase is dZ ∂θ ∂θ . =− ∂t ∂Z dt Then, assuming θ = 2q1(Z – cgt + E) and θ = 2p1(Z – ct + E), we find the group velocity (velocity for the envelope) Cg and phase velocity (phase velocity of the carrier wave) C

Cg = cg ,

∂E ∂Z . C= ∂E 1+ ∂Z c + cg

In this derivation we have used

∂E ∂E . = −cg ∂t ∂Z Thus, the group velocity has the same constant value both along the arc length s and in the absolute coordinate system. Now the phase velocity C = C(Z, t) is not constant in the laboratory coordinate system. However, for a filament with small curvature (q1 1) dE/dZ 1 we can take

C ≈ c = const. Similarly, if we take the dispersion relationship (5.125), according to Levi et al. (1983), we obtain the two-soliton solution (N = 2)

5.5 The method of matched asymptotic expansions

X2 + iY2 =

Im

1 sech η1eiξ1 2

+

1

Im

2 2

2

303

1 × B

⎧ ⎡⎛ 1 ⎤ 1 2 2 ⎞ × ⎨ ⎢⎜ D + Re D ⎟ cosh η2 + D tanh η1 sinh η2 − i Im D sinh η2 ⎥ eiξ1 + 2 ⎠ ⎦ ⎩ ⎣⎝ 2 1 2 ⎡ ⎤ + ⎢(1 + Re D)cosh η1 + D sech η1 + i Im D sinh η1 ⎥ eiξ2 + 4 ⎣ ⎦ 1 2 ⎫ + D sech η1ei(2ξ1 −ξ2 ) ⎬ , 4 ⎭ Im 1 Im 2 1 Z2 = s − × tanh η1 − 2 2 B 1 2

⎧⎛ 1 1 2⎞ 2⎞ ⎛ × ⎨⎜ Re D + D ⎟ sinh η1 cosh η2 + ⎜ 1 + Re D + D ⎟ sinh η2 cosh η1 − 2 2 ⎠ ⎝ ⎠ ⎩⎝ 1 2 ⎫ − D [sech η1 sinh η2 − tanh η1 cos(ξ1 − ξ 2 )] + Im D sin(ξ 2 − ξ1 ) ⎬ . 2 ⎭ Here

j

= pj + iqj, j = 1,2, qj ≥ 0,

1 2⎞ ⎛ B = cosh η1 cos η2 + ⎜ Re D + D ⎟ [cos(ξ1 − ξ 2 ) + cosh(η1 + η2 )], 2 ⎝ ⎠ η j = 2(Im j ) s − 2 Im[ ( j )]t + η0j , ξ j = 2(Re

D=

j )s − 2

∗ ∗ 2( 1

Re[

( j )]t + ξ0j ,

− 1 ) [ 1∗ ( 1 −

∗ 2 )],

η0j , ξ0j are the real constants. The expressions for a two-soliton solution are rather cumbersome, but they are shown here to allow the reader to perform their own calculations. As an example, we show in Fig. 5.21 two cases of soliton interaction; their parameters are: 1,2 = ±0.5 + 0.25i. For the first case (Fig. 5.21a) there is no axial flow (β = 0), but for the second case (Fig. 5.21b) β = – 0.5. The initial states are identical. One can see that the main difference is in different propagation velocities for corresponding solitons depicted in figures a and b. Note also that the calculations fail to explain a high shift in phases after soliton interaction that was observed in the experiments of Maxworthy et al. (1985). The reason for this discrepancy is the neglect of local induction and variation of the cut-off parameter along the vortex filament during derivation of Eq. (5.117).

304

5 Dynamics of vortex filaments

Fig. 5.21. Oncoming interaction of two vortex solitons with parameters ζ1,2 = ±0.5 + 0.25i. β = 0 (no axial flow) (a); 0.5 (b) (Fukumoto and Miyazaki 1991*)

In the paper by Konno and Ichikawa (1992), a multi-soliton solution for a vortex filament with axial flow was found using the method of reverse scattering problem. The initial equation was a hybrid equation of Wadati et al. (1979), which is equivalent to Eq. (5.117). The calculation results are slightly different from the results presented by Fukumoto and Miyazaki (1991). In particular, the two-soliton solution is symmetric relative to the solitons interchange. Here we show two illustrative examples of simulation. Figure 5.22 depicts the comparison of the vortex solitons interaction for the cases of zero axial flow (a) and with axial flow (b). The soliton parameters and axial velocity w are shown in dimensional (still conditional) values because of time renormalization in LIE, but time variable t is dimensionless. The parameters of the solitons are chosen in such a way that at w = 0 they move together. The introduction of non-zero axial velocity immediately creates the situation where the solitons disperse in different directions. Interaction of vortex solitons with parameters taken from the experiment of Maxworthy et al. (1985) is shown in Fig. 5.23. In general, there is a good correlation between calculation and experiment. However, the phase shift after soliton interaction (as for the FM approach) is much smaller than in experiment.

5.5 The method of matched asymptotic expansions

305

Fig. 5.22. Temporal evolution of two vortex solitons with parameters ζ1 = −0.5 + 2i [cm–1] and ζ2 = –0.5 + 3i [cm–1]. w = 0 (no axial flow) (a); 0.2 cm (b) (Konno and Ichikawa 1992*)

Fig. 5.23. Oncoming interaction of two vortex solitons with parameters ζ1 = −0.45 + 0.10i [cm–1] an ζ2 = –0.78 + 0.36i [cm–1] on a vortex filament with w = –0.53 cm: a – expanded view, b – projection on a plane (Konno and Ichikawa 1992*)

306

5 Dynamics of vortex filaments

5.5.4 Comments In the previous sub-section we used LIE approximation to demonstrate the influence of axial flow in the vortex core on the dynamics of vortex solitons. In this Section we present the most important results from FM (without details of derivation). These results cover the problem of the axial velocity and dynamics and stability of helical vortices. We also discuss several non-local effects. Helical vortex

One of solutions for Eq. (5.117) is a helical vortex filament that moves as a body without a change in shape in the direction of axis z with translatory velocity WT and rotates around z with angular velocity Ω

R= Here

κ κ + τ2 2

( cos

)

ex + sin ey + Zez .

θ = (κ2 + τ2)1/2{s – [τ + β(τ2 − κ2/2)]t} + θ0, Z=

τ

(κ

2

+ τ2

)

1/ 2

⎡ κ2 ⎛ 3 ⎞ ⎤ s + ⎢ ⎜1 + βτ ⎟ t ⎥ + Z0 . τ ⎝ 2 ⎠ ⎦⎥ ⎣⎢

Expressions for WT and Ω take the forms

WT =

κ2

(κ

2

+τ

)

2 1/ 2

⎛ 3 ⎞ ⎜ 1 + βτ ⎟ , ⎝ 2 ⎠

Ω = − (κ2 + τ2)1/2[τ + β(τ2 − κ2/2)] In an experiment, the frequency Ω0 is usually measured in a fixed plane Z = const. We use this condition and assume θ = Ω0t + θ0, then we obtain

Ω0 = − (κ2 + τ2)1/2(1 + τβ)/τ. Comparing this with solution (6.70) for a helical filament without axial flow, we see that the effect of the axial flow is merely a change in the value of translation velocity and angular velocity. Linear stability of a helical vortex filament

The first step in the analysis of helical filament stability is substitution of into (5.124) and derivation of evolution equation in variables of curva-

5.5 The method of matched asymptotic expansions

307

ture κ and torsion τ, which are the parameters of a helical filament. Then we take the perturbations in κ and τ in the form δκ ∼ cos (ks − ωt), δτ ∼ sin (ks − ωt), and obtain the dispersion relationship 3 ω = 2kτ0 + β k(k 2 + 3τ02 − κ 02 ) ± 1 + 3βτ0 k(k2 − κ 02 )1/ 2 . 2

Here τ0 and κ0 are undisturbed values. For β = 0 we get the equation derived previously by Betchov (1965) for a helix without axial flow. It was demonstrated in this paper that the helix is unstable to perturbations with the wavelength higher than 2π/κ0. The new equation demonstrates the effect of the axial velocity on helix stability. If βτ0 > 0 or βτ0 < –2/3, the growth rate for the unstable mode increases. If –2/3 < βτ0 < 0, the growth rate decreases. Finally, for torsion τ0 = –1/(3β) the helical vortex filament is stable relative to any small perturbations. Thus, the axial flow may be a serious factor in helical filament stability. However, these conclusions are based upon the local approach, which fails to describe some phenomena, in particular, instability (Crow 1970) due to the interaction of two trailing vortices. Contribution of the entire filament to the stability of the helical vortex filament

Here we present the results from FM regarding the influence of the entire length of helical vortex filament on its stability. This study required use of the cut-off method (the same as in the paper by Widnall (1972)), but the accuracy was refined to the second order in the filament curvature. It was shown by Widnall (1972) that there are three types of instability – longwave, short-wave, and mutually-inductive. For the local approximation, we can find only the long-wave mode of instability. That is why it is natural to make a comparison with this mode. Calculations were conducted by numerical integration of the Biot – Savart law with use of the Rankine model; this model has a constant axial velocity W in the vortex core. The natural dimensionless parameter is the ratio W/V, where V is a maximum tangential velocity. The conclusions of this analysis are as follows (for long-wave instability). For a low ratio |W/V| the axial velocity W tends to stabilize the left(right)-handed vortices and destabilize the right(left)-handed vortices, if the given helicity is positive (negative). The predictions of the local model and cut-off model coincide on the qualitative level. The discrepancy becomes higher with an increase in |W/V|. Above a certain ratio |W/V| the growth rate for the long-wave mode decreases (both for left- and righthaaadddnnnn

308

5 Dynamics of vortex filaments

Fig. 5.24. Boundaries of long-wave instability for the Rankine vortex with dimensionless axial velocity W/V = – 0.4 (a) and 0.4 (b) . The digits at the curves stand for ratio ε/a (Fukumoto and Miyazaki 1991*)

handed vortices). If the helix pitch is not too low, there exist bands of |W/V| where the long-wave mode is forbidden. The boundaries of stability windows depend on the pitch, core radius and torsion of the helical vortex, as illustrated in Fig. 5.24. Here the torsion τ = l/(l2 + a2), 2πl is the helix pitch, a is the helix radius, ε is the radius of the vortex core. The main difference between these two models is that the suppression of the long-wave mode is achieved due to the first-order effects in curvature (for the cut-off model) and only by the second order effects for the local model.

6 Dynamics of two-dimensional vortex structures

6.1 The method of discrete vortex particles To calculate vortex flows we may use various methods. Recently the approaches based on the direct numerical solution of Navier – Stokes equations have been used for quite a broad class of problems. The calculation method of 2-D problems in terms of variables “stream function – vorticity” may be considered as representative of such approaches. Vortex methods based on the Lagrange approach in description of fluid flow are used successfully in the case of localized vorticity, especially for large Reynolds numbers, when influence of the viscosity on the vorticity dynamics is small. A fairly comprehensive review of vortex methods can be found in the papers by Leonard (1980, 1985). Let us consider here, solely the problem of simulation of the plain twodimensional flows. Note that for calculation of the plain flows with vorticity, uniformly distributed in confined domains, the contour dynamics method is often used (see the review by Pullin (1991)), since it is characterized by lower dimensionality (only the dynamics of domain boundary are calculated rather than for all elements simulating vorticity distribution). And in the case of arbitrary distribution of the vorticity the vortex methods can be useful. 6.1.1 Motion equations of vortex particles in infinite liquid The simplest form for the set of motion equations of discrete vortices can be written in the case where vorticity carriers are singular objects. These objects are comprised of infinitely thin rectilinear vortex filaments (or point vortices if only plain motion is considered). Since the point vortex has no self-induced velocity, then its motion velocity equals the sum of the velocities induced by other vortices. If the vortices with intensity Γα, α = 1, …, N have coordinates rα = (xα, yα) at a certain instant of time, then according to (2.25) we have

310

6 Dynamics of two-dimensional vortex structures

xα = −

yα − yβ xα − xβ 1 1 Γβ Γβ . , yα = 2 2 2π β≠α 2π β≠α rα − rβ rα − rβ

∑

∑

(6.1)

Here the dot over the symbol means the time derivative. Note that when point vortices approach each other their velocities increase infinitely. Therefore, using this method in practice, one or another regularization technique is employed. Belotserkovsky and Nisht (1978) suggested to introduce a certain truncation radius δ. When vortices come together at a distance of less than δ, the induced velocities are imposed by a certain restriction: the velocity inside a circle with the radius δ is determined by linear interpolation between the velocity magnitude on the circle and zero value of the velocity on the vortex axis. Rosenhead (1930), who was one of the founders of vortex methods, used in his work simple algebraic regularization, which concluded in the replacement of the denominators in the equations (6.1) with the value of |rα – rβ|2 + δ2. A more complete list of regularization techniques can be found in the overview by Leonard (1980). Thus in fact, the objects with finite dimensions, i.e. vortex particles are used instead of point vortices. The basic approaches to the development of vortex methods are described in the above considered overviews. Let us consider here the method of construction of discrete vortex particles model, which is based on the variation principle (Veretentsev and Rudyak 1986). The advantages of such models conclude in their conservatism: all the conservation laws intrinsic to the continual flow model are automatically transferred to the discrete flow model. The velocity field for a 2-D infinite flow of incompressible liquid reconstructed by means of vorticity field, can be written as (see (1.95)) 1 y − y′ ⎧ ω (r ′, t ) dS(r ′), ⎪u(r , t) = − 2π 2 r − r′ ⎪ ⎨ x − x′ ⎪v(r , t) = 1 ω (r ′, t ) dS(r ′). ⎪ 2π r − r ′ 2 ⎩

∫

∫

(6.2)

The integral is calculated over the whole flow area. Let us make the transition from the Eulerian variables r in (6.2) to the Lagrangian variables = (ξ, η) which represent the initial position of the material point = r|t=0. In the plain flow the vorticity and the liquid element area are constant, i.e. ω( , t) = ω( , 0) = ω0 ( ); dS = dS0 = dξdη. Taking into account that u( , t) = x ( , t), v( , t) = y ( , t), let us write motion equations for the material point as

6.1 The method of discrete vortex particles ⎧ y ( , t ) − y ( ′, t ) 1 ⎪x ( , t ) = − ω0 ( ′) dS0′ , 2π r , t − r ′, t 2 ⎪ ( ) ( ) ⎪ ⎨ x ( , t ) − x ( ′, t ) ⎪ 1 ω0 ( ′ ) dS0′ . ⎪y ( , t ) = 2π r , t − r ′, t 2 ⎪ ( ) ( ) ⎩

311

∫

(6.3)

∫

Motion equations (6.3) can be written in Hamiltonian form:

⎧ ⎪ω0 ( ⎪ ⎨ ⎪ω ⎪ 0( ⎩

) x ( ,t ) =

δH [ x, y ] δy ( , t )

) y ( ,t ) = −

,

δH [ x, y ] δx ( , t )

(6.4)

,

where variation derivatives are taken from the functional H [ x, y ] = −

1 4π

2 2 ∫∫ log {⎡⎣ x ( ,t ) − x ( ′,t )⎤⎦ + ⎡⎣ y ( , t ) − y ( ′, t )⎤⎦ } ×

×ω0 ( ) ω0 ( ′) dS0 dS0′ .

(6.5)

Quantity H, multiplied by fluid density ρl, concurs with the kinetic energy of the fluid vortex flow. It is determined by the vorticity distribution character and does not depend on time (Batchelor 1967). Moreover, Hamiltonian (6.5) is invariant with respect to translation and rotation of the (x, y) plane. These properties lead to the familiar laws of conservation of momentum and angular momentum (see Section1.6). Motion equations (6.4) can be derived from the Hamilton variation principle t2

∫

δ Ldt = 0,

δx( , t1) = δx( , t2) = 0, 0 ≤ t1 < t2 ≤ t.

(6.6)

t1

Hamilton variation principle is applied to Lagrange function L, determined as follows: L = x ( , t ) y ( , t ) ω0 ( ) dS0 − H [ x, y ] .

∫

As a checking procedure let us find the Lagrange function variation

(6.7)

312

6 Dynamics of two-dimensional vortex structures

δL = δH =

∫

∫ [δxy + xδy] ω0dS0 − δH,

⎡ δH ⎤ δH ⎢ δx δx ( , t ) + δy δy ( , t ) ⎥ dS0 . ⎣ ⎦

(6.8)

Taking into account the evident equality t2

t2

∫∫

δx y ω0 dS0 dt = −

t1

∫ ∫ y δx ω0dS0dt

t1

and substituting (6.8) into (6.6), we obtain t2

∫∫

t1

⎪⎧ ⎡ ⎨− ⎢ω0 ( ⎩⎪ ⎢⎣

⎡ + ⎢ω0 ( ⎢⎣

) y ( ,t ) +

) x ( ,t ) −

δH [ x, y ] ⎤ ⎥ δx ( , t ) + δx ( , t ) ⎦⎥

⎫⎪ δH [ x, y ] ⎤ ⎥ δy ( , t ) ⎬ dS0 dt = 0. δy ( , t ) ⎥⎦ ⎪⎭

Equations (6.4) follow due to the arbitrariness of the variations δx, δy. The formulated variation principle enables us to develop conservative discrete vortex methods for calculations of vortex flows. Upon discretization, the area of initial distribution of the vorticity is split into the cells Bα (α = 1,…, N), while ω is approximated by the assembly of vortex particles

ωN =

N

⎛ r − rα ⎞ ⎟ σα ⎠

Γ

∑ σα2 f ⎜⎝ α=1

α

(6.9)

with intensities

Γα =

∫ ω0 ( ) dS0 .

Bα

The particle centers are located at the points rα(t) = (xα(t), yα(t)), coinciding with the mass centers of the images of Lagrange cells Bα in the plane of Euler variables. Characteristic dimensions σα depend on the area of the proper Lagrangian cells Sα , so that σα → 0 at Sα → 0. Function f determines the configuration of vorticity spreading in the particles. It is normalized by unit, while the function f ( r − rα / σα ) σα2 weakly converges to

δ-function at σα → 0.

6.1 The method of discrete vortex particles

313

Substituting ωN into the Lagrange function, rewritten in Eulerian variables, and applying the variation principle, results in Hamiltonian equations for vortex particles

Γ α xα =

∂HN , ∂yα

Γ α yα = −

∂HN ∂xα

(6.10)

with Hamilton function HN = −

1 N Γ α Γβ 4π α,β=1 σα2 σβ2

∑

∫∫

⎛ r − rα (t) log r − r′ f ⎜ ⎜ σ2 α ⎝

⎞ ⎛ r′ − rβ (t) ⎟⎟ f ⎜ σβ2 ⎠ ⎜⎝

⎞ ⎟ dSdS′. (6.11) ⎟ ⎠

In some cases the motion equations can be conveniently written in polar coordinates rα = |rα|, ϕα = atan (yα/xα)

Γ αrα =

∂HN , rα ∂ϕα

Γ α ϕα = −

∂HN rα ∂rα

(6.12)

or in complex form (zα = xα + iyα) Γ α zα =

dHN . dzα

(6.13)

Here, the dash above the symbol means a complex conjugation. Veretentsev and Rudyak (1986) have shown that the dependence of the terms of sum in Hamiltonian HN (6.10) on rα and rβ are determined by the function of the scalar variable rαβ = |rα – rβ|

HN = −

1 4π

∑ Γ α Γβ ∫ αβ

Vαβ (rαβ ) rαβ

drαβ ,

(6.14)

rαβ

Vαβ (rαβ ) = 2π

∫ F(r , σα , σβ )rdr ,

(6.15)

0

F ( r , σα , σβ ) =

1 σα2 σβ2

∫

⎛ r + r′ f⎜ ⎜ ⎝ σβ

⎞ ⎛ r′ ⎞ ⎟⎟ f ⎜ ⎟ dS(r ′) . σ α ⎝ ⎠ ⎠

(6.16)

Then the equations of motion (6.10) can be rewritten using function Vαβ

314

6 Dynamics of two-dimensional vortex structures

yα − yβ ⎧ 1 Γβ V rα − rβ , ⎪ xα = − 2 αβ 2π β rα − rβ ⎪⎪ ⎨ xα − xβ ⎪y = 1 Γβ V rα − rβ . α 2 αβ ⎪ 2π β r r − α β ⎪⎩

(

∑

)

(

∑

)

(6.17)

Let us note the important properties of the Hamilton equation set for the motion of vortex particles (6.10). Hamiltonian HN does not depend on time explicitly and is invariant regarding translation and rotation of (x, y) plane. Hence, it follows that the interaction energy of vortex particles ρlHN , impulse of the system IN = (INx, INy), where

INx = ρl

∑ Γα yα ,

INy = −ρl

α

∑ Γα xα , α

(6.18)

and angular impulse

1 MN = − ρl 2

∑ Γα (xα2 + yα2 ) = − 2 ρl ∑ Γαrα2 1

α

α

(6.19)

are invariants and approximate respective invariants of the continual distribution of the vorticity (see Section 1.7.5). To verify this statement let us consider the rate of change of quantity INx :

d ρl dt

∂H

∑ Γα yα = ρl ∑ Γα yα = −ρl ∑ ∂xαN . α

α

α

(6.20)

Let us represent the Hamiltonian in the form of sum

HN =

∑ Hαβ . αβ

Due to (6.14) we have: ∂Hαβ/∂xα = –∂Hβα/∂xα . Taking into consideration that the self-induced velocity equals zero ∂Hαα/∂xα = 0, we are convinced that the sum in (6.20) equals zero. Verification of the second component of the momentum can be conducted similarly. It is convenient to consider the conservation law for angular momentum in polar coordinates

dMN 1 = − ρe dt 2

∂H

∑ Γαrαrα = − 2 ρe ∑ ∂ϕαN . 1

α

α

6.1 The method of discrete vortex particles

315

2

Note, that rα − rβ = rα2 + rβ2 − 2rα rβ cos (ϕα − ϕβ ). Taking into account the symmetry properties according to the indices, we find that the last sum equals zero. Selecting the proper shape function f and parameters σα , we may obtain from equations (6.17) all the known discrete vortex models for plain flows. In particular, if we tend all σα to zero, we find Vαβ = 1 and arrive at the equations of motion for the system of point vortices (6.1). Neglecting the size of the observed particle but conserving the sizes of all other particles, we obtain a generalized vortex model, in which velocity of the particle coincides with the velocity induced by the whole system at the point where the center of the particle is situated at a given instant of time. Note, that in this case the set of equations (6.17) ceases to be Hamiltonian. Nevertheless such models are prevailing. The corresponding function F, determined by the equation (6.16), is proportional to the shape function

F ( r , σβ ) =

1 f ( r σβ ) . σβ2

(6.21)

Usually the simple shape functions are employed in calculations (see the overview by Leonard (1980)). Function f (r ) =

1 θ (1 − r ) 2πr

corresponds to the model, where the velocity, induced by β particle, becomes zero, if the distance from α particle is less than σβ (Chorin and Bernard 1973). The above considered model, which was used by Rosenhead (1930), follows from the generalized vortex model with function F (6.21) and shape function

f (r ) =

1 1 . π (1 + r 2 ) 2

In the model proposed by Kuwahara and Takami (1973), the vorticity of each particle was varied in time according to the diffusion law for an isolated vortex (see Section 2.3.2). This agrees with function F (6.21), which includes the shape function f (r ) =

( )

1 exp −r 2 π

316

6 Dynamics of two-dimensional vortex structures

and the parameters σβ = (2νt)1/2, where ν is some artificial viscosity. Veretentsev and Rudyak (1986) used Gaussian shape function as well, but with constant dimensions of the particles. The function Vαβ for this case takes the form 2 ⎛ ⎞ rαβ ⎟. Vαβ (rαβ ) = 1 − exp ⎜ − 2 2 ⎜ σα + σβ ⎟ ⎝ ⎠

Selecting the dimensions of vortex particles, the authors equate vorticity in the center of α particle and that in the material particle, have the same Lagrange coordinate α

Γα πσα2

= ω0 (

α

).

6.1.2 Motion equations of vortex particles in limited simplyconnected domains To employ the method of discrete vortices in the flow around bodies and internal flow problems, two circumstances should be taken into account: firstly, the equations of motion need to satisfy the no-flow condition on a rigid surface; and secondly, in the case of flow separation, some simulation of vorticity generation must be provided. In a more complete formulation of the problem, the non-slip condition at the body surface must be taken into account. At that, the boundary-layer equations in the near-surface area need to be solved, and matching with the external inviscid flow must be performed (Belotserkovsky et al. 1993). The method of bound vortices (Belotserkovsky and Nisht 1978) is a convenient technique enabling us to allow for the no-flow condition on a surface of the body with arbitrary configuration. Since the surface of the body placed in inviscid fluid flow, is in fact the line of tangential discontinuity of the velocity, it is substituted by the attached vortex sheet, which in turn, is simulated by means of a point vortices assembly. At the same time, the no-flow condition is formulated only in a fixed number of checkpoints situated between the vortices. The problem of arranging bound vortices and checkpoints, as well as determining their number, was studied in most detail by Gorelov (1980, 1990). In contrast to the uniform distribution, which is usually used (see Belotserkovsky and Nisht (1978)). Gorelov proposed finding the location of the checkpoints based on the condition that the velocity induced at these points by bound vortices is equal to the velocity induced by the continuous vortex layer. This enables one to signifi-

6.1 The method of discrete vortex particles

317

cantly increase the accuracy of determination of descending vortices intensities or the time step of integration. The total accuracy of the calculations depends on the number of bound vortices. An increase in this number is restricted by computer resources, since we have to solve sets of linear equations with a great number of variables. The same reason causes some problems in employing the method of bound vortices in the problems on the motion of vorticity domains close to the elongate boundaries (near plane, in the channel etc.). At the same time, the methods of complex function theory can be applied in a broad class of problems, such as mapping the flow area on a more simple plane, for instance, on the upper part of a complex plane, and introducing a specularly reflected vortex system. Such an approach was used in the problems of fluid inflow into the channel (Sabelnikov and Smirnykh 1985), flow over obstacles (Vlasov et al. 1982; Evans and Bloor 1977), jet outflow (Shimizu 1986), etc. It is easy to account for vorticity generation in the problems of flow around a body accompanied by separation on a sharp edge. According to the Kelvin theorem (see Sections 1.1, 1.2.2) the velocity circulation along the contour covering the body, and the vortex wakes descending from this body, do not change in time. This condition gives the equation determining the vorticity, descending from the body to the flow. Exactly the same approach was used in the works by Belotserkovsky and Nisht (1978), Il’ichev and Postolovskii (1972), Molchanov (1975). If the problem contains infinite or half-infinite elements, such as plane, channel etc., some other considerations should be used. Usually such problems enable us to write the Kutta - Joukowski condition in explicit analytical form (Sabelnikov and Smirnykh 1985; Veretentsev et al. 1989; Evans and Bloor 1977), which allows us to determine the generated vorticity. Further, let us consider the derivation of the equations of motion of vortex particles in order to simulate plain flows in simply-connected domains with possible separation on sharp edges. Following Kuibin (1993), let us consider a plain flow of incompressible inviscid fluid in the domain D, having the boundary ∂D with a salient point. Locally the boundary in the vicinity of the salient point is represented in the form of a wedge with the opening angle β. Let us introduce Cartesian coordinates z1, z2 in domain D, selecting the point of origin on the edge of the wedge, as well as the appropriate complex variable z = z1 + iz2. The conformal image (z) of the domain D to the half plane = ζ1 + iζ2 (ζ2 > 0) is known. Thereby, boundary ∂D maps into the line ζ2 = 0. Without losing generality we may assume that (0) = 0. Separation is simulated by the descent of an infinitely thin vortex layer (vortex sheet) from the sharp edge. Let us imagine the

318

6 Dynamics of two-dimensional vortex structures

vorticity field ω in the form of the sum for external vorticity ωe generally occurring in the flow at the initial point in time, and vorticity, generated as a result of the separation, ωs. Knowing the vorticity field and Green function of the Laplace operator for the half plane (Vladimirov 1976), the stream function can be found using the familiar technique

ψ ( z, t ) = K ( z, z′ ) ω ( z′, t ) dS′ + ψ p ( z, t ) ,

∫

(6.22)

D

where ω = ωe + ωs; dS' = dz'1 dz'2; ψp is the stream function of the potential flow; K ( z, z ′ ) = −

( z ) − ( z′ ) ( z ) − ( z′ )

1 log 2π

.

Let us calculate the velocity field using stream function u1 =

∂ψ , ∂ z2

u2 = −

∂ψ , ∂ z1

or in a compact form

uj =

∂ψ

2

∑ ajk ∂ zk ,

(6.23)

k =1

where ajk is the antisymmetric matrix: a11 = a22 = 0; a12 = –a21 = 1. Substituting stream function (6.22) into (6.23), we obtain

u j = M j ( z, z′ ) ω ( z′, t ) dS′ + u jp ,

∫

(6.24)

D

Mj =

2

∑ ajk k =1

∂K ( z, z′ ) ∂ zk

,

j = 1, 2,

(6.25)

where u1p, u2p are the velocity components of the potential flow. In the ideal incompressible fluid the vorticity of a fluid particle remains constant, i.e. if we consider the initial distribution ωe of fluid particles to be given, then the problem of vorticity dynamics reduces to the tracing of particle migration. Concerning the value of ωs it changes in time and depends on the current distribution of vorticity in the flow area. Note, that during separation the fluid particle passing the edge of a wedge and obtaining a certain portion of vorticity, behaves just like the particle from the ex-

6.1 The method of discrete vortex particles

319

ternal field ωe, i.e. its intensity does not change. In order to determine the intensity, inherent in the particle upon its descent from the edge, let us use the hypotheses on velocity finiteness on a sharp edge (Kutta - Joukowski hypotheses) similarly as in works by Sabelnikov and Smirnykh (1985), Vlasov et al. (1982), Evans and Bloor (1977). It is as well to note that in the case of a wedge with the opening angle β > 0 we must also indicate the direction of the vortex descent. Usually it is supposed that the wake descends tangentially to the wedge from the windward side (Pullin 1978). Let us derive the mathematical condition of the velocity finiteness on the edge. With this aim in mind let us proceed to variables ζ1, ζ2 in (6.23)

uj =

2

⎛ ∂ζ ∂ψ

∂ζ ∂ψ ⎞

∑ ajk ⎜⎝ ∂zk1 ∂ζ1 + ∂zk2 ∂ζ 2 ⎟⎠ ,

j = 1, 2.

k =1

Since the determinant of Jacobi matrix J = |dz/d |2 at the edge becomes infinite, condition | uj | < ∞ requires that

∂ψ ( 0 ) = 0, ∂ζ1

∂ψ ( 0) = 0 . ∂ζ 2

The first equation is satisfied by virtue of the no-flow condition. Taking into account (6.21) the second leads to the equation determining the generated vorticity

∂ψ p 1 ζ2 ( z ) , t dS ω + z ( ) ( 0) = 0 . π D (z) 2 ∂ζ 2

∫

(6.26)

Applying Lagrange method for describing the motion of a fluid particle belonging to the separated layer, let us select in the capacity of the Lagrange variable the point in time τ, respective to the descent of a particle from the edge. Let us write the vorticity generated during the separation in the form of integral t

∫ (

)

ωs ( z, t ) = δ z − z 0 ( τ, t ) γ ( τ ) dτ ,

(6.27)

0

where z0(τ, t) indicates the location at the instant of time t of the elementary segment of the vortex sheet, descending from the edge at time point τ; γ(τ) is the intensity of the vortex sheet at the z0(τ, t) point, which corresponds to the velocity shear; δ is the delta-function.

320

6 Dynamics of two-dimensional vortex structures

In consequence of the incompressibility of a fluid dS = dS0 = da db. Then, in view of the definition of ωs, we derive the velocity of the points associated with the field ωe from (6.24),

u j = M j ( z ( a, b, t ) , z ( a′, b′, t ) ) ω ( a′, b′, t ) dS′0 +

∫

D t

+

0 ∫ ∫ M j ( z ( a, b,t ) , z′) δ ( z′ − z ( τ,t ) ) γ ( τ ) dS′dτ + ujp .

D0

The velocity of material particles related to the vortex sheet is described by a similar expression. Let us change in the second term of the sum the order of integration and let us integrate using primed variables. In order to shorten the mathematical notation, let us introduce the following indications: c1 = a + ib, c2 = τ are Lagrange variables; B1 = D, B2 = [0, t] are respective ranges of their variation; ω1(c1, t) = ωe(a, b, t), ω 2(c2, t) = γ(τ); dS10 = dS 0 , dS20 = dτ . Then the velocity in Lagrange variables can be written as: u j ( z ( cα , t ) , t ) =

∑ ∫ M j ( z ( cα ,t ) , z ( cβ ,t ) ) ωβ ( cβ ,t ) dSβ0 + 2

β=1 Bβ

+u jp ( z ( cα , t ) , t ) ,

(6.28)

α = 1, 2, j = 1, 2.

Hence, we obtain the analogue of equation (6.3), describing the fluid motion law z j ( cα , t ) =

∑ ∫ M j ( z ( cα ,t ) , z ( cβ ,t ) ) ω β( cβ ,t ) dSβ0 + 2

β=1 Bβ

+u jp ( z ( cα , t ) , t ) ,

(6.29)

α = 1, 2, j = 1, 2,

which along with the condition of velocity finiteness (6.26), written in Lagrange variables as

1 2 π β=1

∑∫

Bβ

( ( )) ω c dS ( ) ( z ( c ,t ))

ζ 2 z cβ , t

2

β

0 β

β

0 β

+

∂ψ p ∂ζ 2

( 0 ) = 0,

(6.30)

form a closed system of equations. Similarly, as for the infinite fluid, the equations (6.30) can be written in Hamiltonian form

6.1 The method of discrete vortex particles

⎧ 0 δH [ z1 , z2 ] , ⎪ ωα ( cα ) z1 ( cα , t ) = δz2 ( cα , t ) ⎪ ⎨ ⎪ω0 c z c , t = − δH [ z1 , z2 ] , α = 1, 2, ⎪ α( α) 2( α ) δz1 ( cα , t ) ⎩ H [ z1, z2 ] =

321

(6.31)

∑ ∑ ∫∫ K ( z ( cα ,t ) , z ( cβ ,t )) ωα0 ( cα ) ωβ0 ( cβ ) dSα0dSβ0 + β=1

1 2 2 α=1

2

Bα Bβ

+

2

∑ ∫ ψ p ( z ( cα ,t ) ,t ) ω0α ( cα ) dSα0 .

α=1 Bα

Equations (6.31) can be derived from the variation principle (6.6), applied to the Lagrangian

L=

2

∑ ∫ z1 ( cα ,t ) z2 ( cα ,t ) ωα0 ( cα ) dSα0 − H [ z1, z2 ] .

α=1 Bα

In the case where stream function of the external potential flow does not change in time (∂ψp/∂t = 0), and the flow is without separation, the Hamiltonian H does not depend on time explicitly, i.e. the energy conservation law of the vortex motion remains true ρlH (Batchelor 1967). If the flow area possesses any other symmetry with respect to translation (half plane or stripe for instance) or rotation (circle), then the Hamiltonian will remain invariant relative to them. Then for translation symmetry (for example along the Oz1 axis) the conservation law of respective momentum component will remain true 1 I1 = ρl z2 ω dS , 2

∫

while for the rotational symmetry the conservation law of angular momentum applies (Batchelor 1967) as 1 2 M = ρl z ω dS . 3

∫

Let us proceed to the discrete model by digitization of the vorticity field similarly as in Section 6.1.1. In contrast to the ωe field, for which all of the above considered approaches of discretization are suitable (see approximation (6.9))

322

6 Dynamics of two-dimensional vortex structures Ne

∑

0 ωeN =

k =1

Γ ek ⎛ z − zk f⎜ σk2 ⎝ σk

⎞ ⎟, ⎠

vorticity field ωs, generated as a result of separation, can be approximated by breaking the variation range of the Lagrange variable τ – time axis – on time frames by markers t0 = 0, t1, t2, ... . For representation of the vorticity constituent produced during the time interval [ tk–1, tk ] we obtain the following function ωs ( z , t ) =

tk

∑ ∫ δ ( z − z 0 ( τ,t ) ) θ (t − τ ) γ ( τ ) dτ , k =1 N

tk−1

where θ is the Heaviside function . Assuming the time increment to be rather small, we approximate ωs(z,t) as follows: ωsK =

K (t )

∑

k =1

Γ sk fk ( z − zsk (t ) , σ sk ) ,

Γ sk =

tk

∫ γ ( τ) dτ ,

tk−1

the number of fragmentations K (t ) =

∑ θ (t − τk ) ; Γsk is the total intenk

sity of the vortex layer part with the center situated at point zsk(t), descending from the edge during the time ∆tk = tk – tk–1. Here, function fk approximates the vorticity distribution of the current part of the layer. It should weakly converge to δ-function at σsk → 0. Parameter σsk depends on ∆tk so that σsk → 0 at ∆tk → 0. Now it is obvious, that at the discrete level the fields ωe and ωs are described by similar expressions. Further, the indices “e” and “s” are omitted. Thus, the structure of the motion equations for vortex particles will be the same as for infinite fluid (6.10). The Hamiltonian of a discrete system for Gauss function in the form of (6.22) can be written as follows (Veretensev et al. 1989)

HN = ⎡ N (t ) 1 ⎢ − Γk Γ n ⎢ log 8π k,n =1 ⎢⎣

∑

N (t )

∑ Γkψ p ( zk ,t ) − k =1

2

k

−

n

k

−

n

2 ⎞ ⎤ (6.32) ⎛ 2 ⎞ ⎛ k − n ⎟⎥ ⎜ k − n ⎟ + Ei − , − Ei ⎜ − 2 2 2 ⎟⎥ ⎜ ⎜ Dk + Dn2 ⎟ ⎜ Dk + Dn ⎟ ⎥ ⎝ ⎠ ⎝ ⎠⎦

6.1 The method of discrete vortex particles

323

= (zk), N(t) = N + K(t). Parameters Dk2 are the dimensions of the vortex particles in an auxiliary plane assigned by the following expression k

Dk2 =

⎛ z 2−α σα ⎞ ⎤ σk2 ⎡ ⎢1 − exp ⎜ − k ⎟⎥ , 2 ⎜ ⎟⎥ Jk ⎢ σ k ⎝ ⎠⎦ ⎣

α=

2π , 2π − β

where β is the opening angle of the edge of the wedge; σ is the parameter of solution regularization near the edge, which is proportional to ∆t. It was supposed for distinctness that at each time increment σ = σN. The quantity Jk = J(zk), where J is Jacobian of the conformal map J(z) = |dz/d |2. With no sharp edges we may suppose that Dk2 = σk2 Jk . Taking into account an explicit form of the Hamiltonian, the equations of motion of vortex particles take the form

zk =

1 2πizk′

2 ⎞⎤ ⎧ ⎡ ⎛ − ⎪ 1 ⎢1 − exp ⎜ − k n ⎟ ⎥ − Γn ⎨ 2 2 ⎟ ⎜ ⎥ ⎪⎩ k − n ⎢⎣ n =1 ⎝ Dk + Dn ⎠ ⎦ 2 ⎞⎤ ⎡ ⎛ 1 ⎢ ⎜ k − n ⎟⎥ − ⎢1 − exp ⎜ − D2 + D2 ⎟ ⎥ − k − n ⎢ ⎜ k n ⎟⎥ ⎝ ⎠⎦ ⎣

N (t )

∑

⎡ ⎛ ⎞⎤ 2⎞ ⎛ k − n ⎟⎥ Dk4 zk′′zk′ ⎢ ⎜ k− n ⎟ − exp − − ⎢exp ⎜ − 2 2 ⎟ 2 2 ⎟⎥ 2 ⎜ 2 2 ⎜ ⎜ Dk + Dn ⎟ ⎥ σk Dk + Dn ⎢⎣ ⎝ Dk + Dn ⎠ ⎝ ⎠⎦ 2

(

k = 1,2, ..., N(t),

zk′ =

dz ( d

k ),

zk′′ =

d2 z d

2

)

(6.33)

⎫ ⎪ ⎬ + up ( zk , t ) , ⎪ ⎭

( k).

Let us analyze the equation (6.33). The first component in braces conforms to the interaction of vortex particles with each other. The second one describes the interaction of the particles with the flow boundary. The last component in braces depends on variation of the dimensions of the vortex particle images in an auxiliary -plane. Eventually, the condition of velocity finitude at the edge, written in discrete form, gives us the equation determining the intensity of the next vortex particle originating from the edge

324

6 Dynamics of two-dimensional vortex structures

1 π

N (t )

∑ Γk k =1

Im

k 2

k

2 ⎞⎤ ⎡ ⎛ ∂ψ p k ⎢1 − exp ⎜ − ⎟⎥ + ( 0 ) = 0. 2 ⎜ Dk ⎟ ⎥ ∂ζ 2 ⎢ ⎝ ⎠⎦ ⎣

(6.34)

The equations (6.33) and (6.34) enable us to calculate the dynamics of plain vortex flows in simply-connected domains with rigid boundaries taking into account vorticity generation during the separation of flow over sharp edges. Besides, due to the Hamiltonian character of the motion equations of vortex particles (see (6.10)) in the case where ∂up/∂t = 0, the energy conservation law ρH = const remains true in the discrete model. If the motion occurs near the plain infinite wall or within the infinite channel, then the conservation law for momentum projection on a boundary line follows from the condition of Hamiltonian character of the system N (t )

I1N

∑

1 = ρ Γk Im ( zk ) = const , 2 k=1

while for the flow in a circle we have the conservation law for angular momentum N (t )

∑

1 2 MN = ρ Γk ⎡ zk + σk2 ⎤ = const . ⎣ ⎦ 3 k=1 6.1.3 Motion equations of the system of co-axial vortex rings

In contrast to plane flows, simulation by means of vortex methods of 2-D flows belonging to another class, such as axisymmetric flows, meets with additional difficulty, shown by the fact that an infinitely thin vortex ring has an infinite velocity of the self-induced motion. Various techniques have been proposed in order to overcome such a problem: according to Brutyan and Krapivskii (1984) the self-induced velocity is excluded from consideration and only the velocity induced by the system of surrounding vortices is considered. Instead of considering a single ring vortex Belotserkovsky and Nisht (1978) suggested taking two vortices and assigning them the velocity determined at the mid-point between the vortices. Instead of vortex filaments Belotserkovsky and Ginevsky (1995) considered vortex tubes with the radius of the cross-section, determining the rate of discreteness, at that the velocity inside these tubes varied linearly. Let us consider the approach of developing vortex methods proposed by Veretentsev et al. (1986). Here, motion of the system consisting of the elemen-

6.1 The method of discrete vortex particles

325

tary ring vortices of a finite cross-section is described by Hamilton equations. Within the framework of the above mentioned technique, let us rewrite in Lagrange variables = (σ0, z0) the equation (1.97), describing the axisymmetric motion of a nonswirling fluid flow. Let us take the initial coordinates of the material point (σ( , 0), z( , 0)) in the capacity of these variables (each elementary vortex ring is conformed by a corresponding point in the plane ϕ = 0):

σ ( ,t ) =

12 df ∂k′ −1 σ ( ,t ) σ ( ′,t ) ) ω0 ( ′) dS0′ , ( 2πσ ( ,t ) dk′ ∂z

∫

1 z ( ,t ) = 2πσ ( ,t )

∫

⎛ f 12 df ∂k′ ⎞ + ⎜⎜ ⎟⎟ ( σ ( , t ) σ ( ′,t ) ) ω0 ( ′) dS0′ . ⎝ 2σ ( , t ) dk′ ∂σ ⎠

(6.35)

When deriving the equations (6.35) we have to take into account the conservation laws for both vorticity (Batchelor 1967) ω( ,t)/σ( ,t) = = ω( ,0)/σ( ,0) = ω0( )/σ0 and volume σdσdz = σ0dσ0dz0 = σ0dS0. Similarly, as for the plane problem, the equations of motion can be written in Hamiltonian form

ω0 ( ) z( , t) =

δH [ σ, z ]

δH [ x, y ] , ω0 ( ) ⎡σ2 ( , t) ⎤ = − 2 ⎣ ⎦ δz( , t) δσ ( , t) i

(6.36)

with the Hamiltonian H [ σ, z ] =

1 2π

∫∫ f ( k ) ( σ ( ′,t ) σ (

′′, t ) ) ω0 ( ′ ) ω0 ( ′′) dS0′ dS0′′. 12

(6.37)

The variation principle, from which the equations (6.36) follow, is similar to the principle (6.6) with the Lagrangian L = z ( , t ) σ 2 ( , t ) ω0 (

∫

) dS0 − H [σ, z].

Hamiltonian (6.37) does not depend explicitly on time and is invariant with respect to z-shift. This leads to the well-known conservation laws for energy ρH and momentum (see Section 1.7.5) I = πρ σ 2 ( , t ) ω0 (

∫

) dS0 = πρ∫ σ2 ω ( r , t ) dS.

Here, the radius-vector r is introduced within the plane of variables (σ, z).

326

6 Dynamics of two-dimensional vortex structures

Breaking the area of fluid with vorticity in a number of cells Bα (α = 1, …, N) and placing the particles (thin vortex rings) into the cells, let us proceed to the discrete distribution of the vorticity ωN, approximating the original ω, ωN ( r , t ) =

N

∑ Γαχ (r , rα (t ) , εα (t ) ) . α=1

χ is some distribution function of vorticity within the vortex particle having the center rα = (σα , zα), which coincides with that of a proper Lagrange cell. The intensities of the vortex particles Γα =

∫ ω0 ( ) dS0

Bα

do not change in time. At the same time, the characteristic dimension of particle εα(t) is time dependent in contrast to the plane problem. It is natural to assume, that the value εα2 is proportional to the area of the Lagrange cell, which in turn, varies so that the volume of the toroidal area occupied by the cell, remains constant. Calculating the Lagrangian for the discrete model LN =

(p

α

(

= Γ α σα2 + εα2

))

N

∑ pα zα − HN

α=1

and applying variation principle (6.6), we obtain

Hamilton equations describing the motion of a system of ring vortices pα = −

∂HN ∂HN , zα = . ∂zα ∂pα

(6.38)

A certain form of equation of motion depends on selection of χ function. This function should be weakly convergent to δ-function as well as normalized to its unit. Such conditions are met in particular by function ⎡ σ 2 + σ 2 + ( z − z )2 ⎤ ⎛ 2σσ ⎞ 2σ α α ⎥ I0 ⎜ 2 α ⎟ , χ = 1 2 3 exp ⎢ − 2 π εα εα ⎢⎣ ⎥⎦ ⎜⎝ εα ⎟⎠

(6.39)

which can be considered an analog of the Gauss distribution of the vorticity in the plane problem. Parameters εα are proportional to the effective di-

6.1 The method of discrete vortex particles

(

ameters of the proper cells: εα = µ α Sα0 σα0 σα

)

12

327

. Let us determine co-

efficients µα based on a condition of the best approximation of the vorticity field at the initial point in time. Applying the Laplace method for the calculation of integrals, Veretentsev et al. (1986) derived a Hamiltonian of a discrete model for the distribution function (6.39) HN =

⎛ ⎞ 32 σ 2 1 N 2 2 Γ α ϕα ( σα ) ⎜ log 2 α − 4 + E ⎟ + ⎜ ⎟ 4π α=1 εα ⎝ ⎠

∑

2 ′ ⎡ ⎞⎤ rαβ 1 N 1 ⎛ ⎢ ⎜ ⎟⎥ . + Γ α Γβ ϕα ( σα ) ϕβ ( σβ ) f ( kαβ ) + Ei − 2 2π α ,β=1 2 ⎝⎜ εα + εβ2 ⎠⎟ ⎥ ⎢⎣ ⎦

(6.40)

∑

(

)

Here ϕα ( σ ) = 2σ3 2 π1 2 ε α−1 exp ⎡ −2σσα ε α−2 ⎤ I0 2σσα εα−2 ; E = 0.5772… is ⎣ ⎦ the Euler constant; prime in summation means that the terms at α = β are omitted; Ei is the integral exponential function. The Hamiltonian of a discrete model does not depend explicitly on time and is invariant with respect to z-shift. Therefore, the conservation laws for energy and momentum remain true for a discrete model as well. If conditions εα2 σα2 for discrete vortex rings are valid (let us call vortex rings ‘thin’ in this case), then the Hamiltonian (6.40) essentially reduces, because ϕα ( σα ) ≈ σ1α 2 . In a special case, when such vortex rings are situated at a rather large distance from each other, we may neglect the contribution of Ei function to the energy of the vortices interaction. As a result we obtain the model of thin vortex rings with the excluded core (Acton 1980; Brutyan and Krapivskii 1984). Note, that here the proposed model has certain advantages as compared with the other known vortex models. Firstly, the approximation of the model improves if the number of particles increases, at least in terms of integral characteristics of flow, such as the momentum of the vorticity field, for instance. Secondly, all the parameters of a discrete model at a given number of vortex particles are identically determined by the initial data of a specific hydrodynamic problem. Thirdly, the velocities of vortex rings and the energy of their interaction as well as selfinteraction remain finite in all situations including those when vortex rings approach each other. This enables us to employ the model studying flows with vorticity concentration, such as the generation of a vortex ring, development of Kelvin – Helmholtz instability in a shear layer, etc.

328

6 Dynamics of two-dimensional vortex structures

When developing the model, it was supposed that the flow area is infinite. Nevertheless, the model can be used in solving boundary problems as well. At that it is necessary, first of all, to introduce the system of vortices which provides the no-flow condition on a rigid boundary; and secondly, to model somehow the generation of the vortices at these points of a rigid surface, where separation of the boundary layer occurs.

6.2 Motion of the system of rectilinear vortices When a vortex system in infinite fluid has non-zero total circulation, then the momentum conservation condition (6.19) enables us to introduce additional invariants. Indeed, the coordinates of the vorticity center xc =

∑ Γα xα , ∑ Γα

yc =

∑ Γα yα ∑ Γα

and the dispersion of vorticity distribution D2 D

2

(6.41)

1

Γ α ⎡( xα − xc )2 + ( yα − yc )2 ⎤ ∑ ⎣ ⎦ = ∑ Γα

(6.42)

remain constant, and are similar to the quantities introduced in Section 1.7.5, in being quite useful for analyses of vortex system motion. Since the interactions of large-scale vortex structures play an important role in a transport phenomenon in shear flows, consideration of the main modes of interaction of isolated vortex structures and infinite vortex chains is of certain interest. There are quite a number of works devoted to the interaction of vortices (see Saffman and Baker (1979), Meleshko and Konstantinov (1993)). New mathematical methods enabling us to study vortex motion on surfaces, including those of spherical shape, are given in the book by Borisov and Mamaev (1999). To simplify matters, let us stay with the analysis of interaction of circular vortices with finite diameters, and uniform distribution of vorticity in the boundless fluid at rest. Let us consider a 2-D flow of inviscid incompressible fluid caused by a vortex system. To simulate evolution of such a flow employing the method 1

It

follows

⎛ ⎞ D = ⎜ ∑ Γ α Γβrαβ2 ⎟ ⎝ αβ ⎠ 2

bers α and β.

from

⎛ ⎞ 2 ⎜ ∑ Γα ⎟ ⎝ α ⎠

the

paper

by

Novikov

(1975)

that

quantity

2

, where rαβ is the distance between the vortices with num-

6.2 Motion of the system of rectilinear vortices

329

stated in Section 6.1.1, each vortex is broken into N cells of equal area. The vorticity centers of the respective cells are assigned as initial coordinates of vortex particles. To determine further motion of vortex rings we use the system (6.17) with the shape function f(r) (6.22) that gives

Vαβ (rαβ ) = 1 − exp ⎡ − rα − rβ ⎢⎣

2

( σα2 + σβ2 ) ⎤ . ⎥⎦

Parameters σα are accepted to be equal to the reduced diameters of the cells

σα2 = d2 N , where d is the vortex diameter. Equations (6.17) are integrated using the Runge – Kutta method of the 4th order. Computational accuracy is controlled by the validity of conservation laws for momentum IN (6.18), angular momentum MN (6.19), and kinetic energy ρHN , provided by the Hamiltonian (6.14). 6.2.1 Interaction of two identical vortices at various initial distances Calculation results on the interaction of two vortices with diameter d and circulations Γ1 = Γ2 at the various initial distances l = var are shown in Fig.6.1. Here T = (2πl)2/(Γ1 + Γ2) is the rotation period of the respective system of point vortices. At l = 3d the vortices interact in a manner peculiar to point vortices: they rotate around the vorticity center of the system without changing their shape and they retain constant angular velocity Ω = 2π/T. At l = 1.7d the flow pattern changes: vortices distinctly deform, while their rotation velocity relative to the vorticity center increases, however the aggregation process is not observed. This agrees with the data obtained using the contour dynamics technique (Roberts and Christiansen 1972; Zabusky et al. 1979). Upon further decrease of l the interaction process becomes more intense and is accompanied by the exchange of particles. Starting with the distance l = 1.66d the process changes qualitatively. Yet at the time point t = T/4 the vortices “catch on” each other. This leads to them merging with each other, which is accompanied by the origination of a unified vortex structure. At smaller distances the vortex pairing occurs more routinely and the originating vortex structure becomes more compact. Interaction of vortices with an elliptic shape occurs according to the same scenario. In addition, the “critical” distances, at which either exchange of particles or their complete merging starts, are closer to each other. Interaction between vortices, having initial shape, which is determimminnmmmmmmmmmmnnmmimmimi

330

6 Dynamics of two-dimensional vortex structures

Fig. 6.1. For caption see facing page

6.2 Motion of the system of rectilinear vortices

331

Fig. 6.1. Interaction of two vortices of the same circulation but with different initial distances between them: l/d = 3.00 (a); 1.70 (b); 1.66 (c); 1.10 (d). Calculation by Veretentsev and Rudyak (1986*)

332

6 Dynamics of two-dimensional vortex structures

mined based on the “matching” condition of the vortex and potential flows, is described in the work by Saffman (1992), where the author indicates, in particular, the critical value of parameter S/l2 = 0.3121 (S is the area of each vortex), at which the vortices become unstable against infinitesimal 2-D perturbations. Passing on to the reduced diameter of the vortex d = (4S/π)1/2, we obtain the critical distance l = 1.586d, which is somewhat lower than that for circular and elliptic vortices. 6.2.2 Interaction of two vortices of the same size but with different circulations When the initial distance is large, the interaction occurs similarly to the case of identical vortices. With decrease in l the vortex with the smaller circulation starts to deform, while the angular rotation velocity of the vortices with regard to the vorticity center proves to be greater, as compared to the system of two point vortices with the same circulations. If the initial distance between the vortices is less than the critical value, then the character of the interaction is determined by the correlation between their circulations. At Γ1 >> Γ2 the vortex with smaller circulation starts to wind up on the vortex with larger circulation and then eliminates (Fig. 6.2). Here the shape of the vortex with large circulation remains invariable. In due course the vortex system originates with the core, formed by the vortex of larger circulation, and surrounded by the cloud of particles, previously belonging to the vortex of smaller circulation. If the vortex circulations are of the same order, then the vortex with lower circulation starts to wind up on the more intense vortex, but at that the latter deforms and eliminates. 6.2.3 Interaction of two vortices of the same circulation but with different sizes Here the smaller vortex possesses higher initial energy and the character of the interaction of the vortices is similar to that described in Section 6.2.2. In general, at the interaction of vortices with various circulations and sizes, critical distance depends on the correlation of circulations and their dimensions. If the initial distance between the vortices is less than critical, then two mechanisms of origination of large vortices might occur. The first represents the vortex pairing, while the second one concludes in the entrainment of the vortex with smaller initial energy by the more powerful vortex.

6.2 Motion of the system of rectilinear vortices

333

Fig. 6.2. Interaction between two vortices of the same size but with different circulations: l/d = 1.70; Γ2 = 0.1Γ1. Calculation by Veretentsev and Rudyak (1986*)

6.2.4 Interaction of three vortices with circulations of the same sign At the interaction of three vortices there exist the additional mechanisms of origination of large vortices. Thus, the system of three identical vortices with circulations Γ, situated initially at the apical points of an equilateral triangle with the sides l, at the rather large values of l/d, is rotating as a whole with the constant angular velocity Ω = 3Γ/(2πl2) and period T = 2π/Ω accordingly (Fig. 6.3 ). As is known, similar configuration of point vortices is stable independently of l. A decrease in the initial distance between the vortices of finite sizes leads to their deformation and change in their angular rotation velocity. Starting from a certain value l, the vortices lose their stability and combine into a single large structure (Fig. 6.3b). Another mechanism of large vortices origination is observed in the case where three identical vortices are situated on a common straight line at a distance l from each other (Fig. 6.4 ). The corresponding system of point vortices rotates with the constant angular velocity Ω = 3Γ/(4πl2) (with the period T = 2π/Ω). At a large initial distance l the vortices interact as point vortices. When l is less than critical, then the middle vortex breaks through the outermost vortices. Two vortex structures originate as a result (Fig. 6.4b). Later, depending on the initial distance, two originating structures may either rotate around each other, or pair with each other. If the vortices in the linear chain of three vortices have various circulations, then the more intense vortex entraps both vortices of smaller intensity (Fig. 6.5 ). In the case where a less intense vortex is situated in the middle, then the more intense outermost vortices disrupt the middle vortex, forming a double-vortex configuration (Fig. 6.5b).

334

6 Dynamics of two-dimensional vortex structures

Fig. 6.3. Interaction of three vortices initially situated at the apical points of an equilateral triangle: l/d = 3.00 (a); 1.70 (b). Calculation by Veretentsev and Rudyak (1986*)

Further increase in the vortex number of the system leads to yet more complex processes of large vortex origination. Nevertheless, these processes represent consequent combinations of two main mechanisms, such as pairing, tripling, etc., as well as vortex growth, conditioned by takeover or break-up of neighboring less intense vortices. 6.2.5 Interaction of two vortices with circulations of contrary signs If two vortices have similar circulations but contrary signs and initially they are situated far from each other, then these vortices move further in a direction orthogonal to the line connecting their centers, without changing their shape and with a constant velocity V = Γ/2πl. In other words, they behave as a pair of point vortices (see Section 2.3.1). Reduction in the distance between the vortices results in their deformation (Fig. 6.6) and an increase in the progressive motion velocity. The shape of the vortices, remaining constant over the motion of the vortex pair, is studied numerically jhgygjgjj

6.2 Motion of the system of rectilinear vortices

335

Fig. 6.4. Interaction of three vortices initially situated in a straight line: l/d = 3.00 ( ); 1.70 (b). Calculation by Veretentsev and Rudyak (1986*)

336

6 Dynamics of two-dimensional vortex structures

Fig. 6.5. Interaction of three vortices with various circulations: − Γ1 = Γ3 = 0.1Γ2; – Γ1 = Γ3 = 10Γ2. Calculation by Veretentsev and Rudyak (1986*)

6.2 Motion of the system of rectilinear vortices

337

Fig. 6.6. Interaction of two vortices with equal circulations but contrary signs: l/d = 1.10. Calculation by Veretentsev and Rudyak (1986*)

in the work by Pierrehumbert (1980) depending on parameter α = dc/l, where dc is the angular diameter of the vortex (see Fig. 6.7). The problem was solved using the matching technique for vortex and potential flows. Parameter α varies from 0 to 2.16. The limiting shape of touching vortices in the cited work is defined incorrectly. Proper solution on the shape of wwwjhvvvvvvvvvvvvvvvvvvvvvjhhww

338

6 Dynamics of two-dimensional vortex structures

Fig. 6.7. The family of steady vortex pairs. Magnitudes of parameter α, starting from the outside: 1.97; 1.55; 1.22; 0.844; 0.639; 0.500; 0.390; 0.302; 0.225; 0.159; 0.100; 0.048. Calculation by Pierrehumbert (1980*)

Fig. 6.8. A system of two vortices with circulations of contrary signs and different values

touching vortices, or on the shape of the vortex associated with the plane, which is reciprocal, was found by Sadovskii (1971) and later by Saffman and Tanveer (1982). If circulations of the vortices differ from each other both by sign and absolute values, then at large initial distances, vortices move along the circular orbits around the vorticity center of the system situated on the line, which crosses the centers of the vortices behind the vortex with higher intensity (Fig. 6.8). If the value l is less than critical, a more intense vortex may entrain part of another vortex with smaller circulation. 6.2.6 Interaction of three vortices with circulations of contrary signs. Vortex collapse Principally, a new mechanism on the origination of large vortex structures, namely vortex collapse, might occur in the system consisting of three vortices. Collapse, as well as the opposite phenomenon – anti-collapse of the system of point vortices are considered in the works by Novikov and Sedov (1979), Aref (1979), where the terms of origination of indicated effect are formulated as follows:

6.2 Motion of the system of rectilinear vortices

∑ ΓαΓβ = 0 ;

339

D = 0.

α<β

Here D is the dispersion of vorticity distribution, introduced in the beginning of Section (6.41). The equality of the values ϕα = ω0 and

ρα ρα = t∗−1 for all vortices may serve as an additional condition, where (ρα, ϕα) are the polar coordinates of the vortices with respect to the vorticity center. Later in time the system evolves according to the following laws: t

ρα (t)

= ρα0

λ(t) ,

ϕα (t)

= ϕα0

∫

+ ω (t) dt , 0

where λ2(t) = 1 – t/t*; ω(t) = ω0/ λ2(t). Note that ϕα (t) − ϕβ (t) = ϕ0α − ϕβ0 , 0 rαβ (t) = rαβ λ (t) , i.e. the system is rotating with the angular velocity ω(t),

retaining similarity of the initial configuration with the scale factor λ(t). If t* < 0, then the anti-collapse (dispersal) of the vortices occurs. At t* > 0 the system shrinks to the point in time t = t*. The point of collapse concurs with the vorticity center of the system. Later in time the vortices skip the vorticity center initiating the anti-collapsing process. At that, the vortex configuration is reflected by the vorticity center, while the value t* changes its sign while keeping its modulus. For the system of three vortices

t∗ =

( )

2 S0 γ ⎡⎢( Γ2 − Γ1 ) r120 ⎣

ω0 =

(

2

) − Γ ) (r )

0 0 3π r120 r23 r13

2

+ ( Γ3

0 2 13

1

( )

0 2⎤ + ( Γ3 − Γ 2 ) r23 ⎥⎦

,

1 ⎡ Γ1 + Γ 2 Γ 2 + Γ3 Γ1 + Γ3 ⎤ + 0 2 + 0 2 ⎥. ⎢ 4π ⎣ (r120 )2 (r23 ) (r13 ) ⎦

Here γ is the orientation of the vortex triplet, equal to 1, when rounding them counter-clockwise according to their numbering, and equal to –1 otherwise; S0 is the area of the triangle originated by the vortices. The collapse of three point vortices is shown in Fig. 6.9 . Here Γ1 = Γ2 = −2Γ3. The initial configuration in that case is the right triangle with the legs 3 and 3 2 . A phenomenon which is similar to the collapse of the point vortices, can be observed for finite regions of vorticity as well. The dynamics of a triple-vortex system with circulations and initial coordinates of the centers, hghhhhhhhhhjjhhhjjjjjj

340

6 Dynamics of two-dimensional vortex structures

Fig. 6.9. Interaction of three vortices with circulations of different signs: – point vortices; b – vortices of a finite size; t* – time of collapse. Calculation by Veretentsev and Rudyak (1986*)

6.3 Modeling the dynamics of shear flows

341

similar to that of the previous case, is shown in Fig. 6.9b. When the vortices draw together to a distance less than critical, they lose their stability: the vortices with the same sign associate with each other, originating a two-vortex structure. In contrast to point vortices, where the collapse is unstable with respect to small perturbations, for vortices of finite size; collapse is a rather stable phenomena relative to perturbations of the initial coordinates and circulations. The presented examples give concepts on some mechanisms of large vortices origination within the isolated systems. It will be shown in the following Sections that these mechanisms can operate in various shear flows and separated flows.

6.3 Modeling the dynamics of shear flows The free shear flows: jets, wakes, mixing layers, shear layers, etc., are widely spread in nature as well as in engineering. Instability, which leads to the formation of large-scale vortex structures is one of the main features of shear flows. The classical experiments of Brown and Roshko (1974) on the investigation of mixing layer evolution stimulated the appearance of the large number of papers on the dynamics of large-scale vortex structures in the free shear flows. Reviews in this field (Vlasov and Ginevskii 1986; Rabinovich and Sushchik 1990; Cantwell 1981; Fiedler and Fernholtz 1990; Bridges et al. 1989; Liu 1989) demonstrate a wide range of methods of computational fluid dynamics applied for the shear flow studies. Since the turbulent flows at high Reynolds numbers are of particular interest, different models of turbulence are used for these studies. The most promising method with the minimum of assumptions is the method of direct numerical simulation based on the numerical solution to the unsteady NavierStokes or Euler equations with the following averaging by time, space or an ensemble of realizations as performed in the experiments. Different spectral, pseudo-spectral and finite-difference methods are applied for the direct numerical simulations. Nevertheless, the main mechanisms of large structure formation in the free shear flows can be studied using the simple but illustrative method of discrete vortex particles. 6.3.1 Mechanisms of formation for the large vortices in the shear layer Now, let us consider the 2-D motion of an inviscid incompressible fluid, whose undisturbed state is described by the velocity profile

342

6 Dynamics of two-dimensional vortex structures

u(y) = −u0th(2y/δ), where δ is the typical thickness of the shear layer. At the initial moment, the following perturbations are superimposed on the flow: the main one with the wavelength of λ and the subharmonic one with the wavelengths of nλ (n is an integer). The shear layer will be modeled by a set of vortex particles, whose initial coordinates are assigned by the formulae xα

t =0

= Xα ,

yα

t =0

= Yα +

⎛k

⎞

∑ A n sin ⎜⎝ n Xα + ϕn ⎟⎠ , n

where Xα, Yα are undisturbed coordinates of the vortex particles; An and ϕn are respectively the amplitudes and phases of introduced perturbations; k = 2π/λ is the wave number. The solution to the problem on the development of perturbations is reduced to the integration of the equation on vortex particle motion (6.17) with the function 2 Vαβ = 1 − exp ⎡⎣ −rαβ (σα2 + σβ2 ) ⎤⎦ .

Simultaneously, it is assumed that the flow is periodic by x with the period of l = mλ, where m is an integer, depending on the type of subharmonic perturbations superimposed on the flow. The pattern of perturbation development for the system “harmonics + subharmonics” with initial amplitudes A1 = 0.003, A2 = 0.001 for the shear layer with thickness δ = 0.3/k is shown in Fig. 6.10 . Calculations are performed for phase difference ∆ϕ = ϕ2 – ϕ1 = π/2. At the first stage, when perturbations are low, they do not interact with each other and according to the linear theory of instability, they increase exponentially without a change in the sinusoidal shape. In Fig. 6.10b, time range τ = tu0/λ < 0.5 corresponds to this stage. To determine the energy of the harmonics shown in the picture, the spectrum of longitudinal velocity pulsation was analyzed and averaged by the transverse coordinate:

Ek =

1 (uk′ ) 2 dy . 2 32θ0u0

∫

Here θ0 is the initial thickness of the impulse loss (for the considered case, θ0 = δ(1 – ln 2) ≈ 0.307δ). The numbers of lines correspond to the wave numbers: 1 – k = 2π/λ, 2 – 2k, 3 – k/2, 4 – 3k/2. When the velocity perturbation reaches several percent of u0, the nonlinear effects become obvious. The shear layer assumes a serrated shape, then curls, and the formation of primary vortex structures occurs (Fig. 6.10 , τ = 1.0). Simultaneously, interaction of the harmonics commmemmemememememememeememe

6.3 Modeling the dynamics of shear flows

343

Fig. 6.10. Development of disturbances with wavelengths λ and 2λ in a shear layer. Initial conditions: A1 = 0.003, A2 = 0.001, ∆ϕ = π/2. a) the shear layer shape at different time instants; b) time dependence of the developing disturbance amplitude. Calculation by Veretentsev (1988*)

344

6 Dynamics of two-dimensional vortex structures

mences, in particular, the rates of the main harmonics decrease, the higher (Fig. 6.10b, line 2) and combined (line 4) harmonics appear. The fast growth in the energy of subharmonics (line 3) and combined harmonics (4) occurs not only due to the energy of the main flow, but due to the energy of other spectrum components, in particular, harmonics with the wave number 2k (line 2), whose energy after τ = 1.0 starts to decrease. On average, the degree of its reduction has the same order as the energy acquired by the subharmonics and the combined harmonics. When the energy of subharmonic perturbation reaches ~1% of u0, secondary instability, presented as the alternating (up-down) transverse shift of vortex structures, starts to develop. Then, pair rotation of structures begins, and they pair (see Fig. 6.10 , τ = 4.0). At this stage the dominant role is played by the resonance interaction of subharmonic perturbation k/2 with combined perturbation k − k/2, formed in the previous stage due to weak nonlinear interaction of main harmonics k and subharmonics k/2. The evolution of the shear layer is significantly dependent on amplitudes A1, A2 and phase shift ∆ϕ. Simultaneously, the amplitude of the main harmonics determines the character of the shear layer evolution at the initial stage of instability development, and the subharmonics amplitude and phase difference ∆ϕ make this at the stage of secondary instability. The effect of the mentioned parameters on the behavior of momentum loss thickness is shown in Fig. 6.11. At fixed A2 = 4.17 ⋅ 10–3, ∆ϕ = π/2 (Fig. 6.11 ) and a relatively small amplitude of the main perturbation, there is a gentle region, corresponding to the linear stage of instability development. With a rise in A1, this region shortens, and the layer thickness increases more quickly due to more intensive development of initial instability. The reverse pattern is observed at the later stage: with a rise in A1/A2, the growth rate of momentum loss thickness decreases. This is explained by the fact that the development of initial instability stops before intensification of the subharmonic perturbation and commencement of the secondary instability stage. According to Figs. 6.11b, c, the layer thickness at the stage of initial instability development does not depend on subharmonic amplitude and phase shift. The higher the value of A2, the earlier the development of secondary instability. The rate of this process is maximum, when ∆ϕ = π/2. This is due to the fact that the area of subharmonic antinode coincides with the centers of the initial vortex structures. With a decrease/increase in the phase difference, development of secondary instability is protracted. The subsequent development of the shear layer is connected with intensification of subharmonics with higher wavelengths. A situation, where subharmonics with wave number k/4 (k is the wave number, at which initial instability develops within the shear layer) become excited and quickly hhhhhhhhhhhhhhhh

6.3 Modeling the dynamics of shear flows

345

Fig. 6.11. Time dependence of momentum loss thickness in the shear layer at variation of the main harmonics amplitude ( ): A2 = 4.17⋅10–3, ∆ϕ = π/2, A1 = 8.34⋅10–4 (1), 4.17⋅10–3 (2), 8.34⋅10–3 (3); subharmonics (b): A1 = 7.41⋅10–3, ∆ϕ = π/2, A2 = A1 (1), A1/2 (2), A1/4 (3) and the value of phase shift ∆ϕ (c): A1 = 7.41⋅10–3, A2 = A1/2, ∆ϕ = π/2 (1), π/4 (2), 0 (3). Calculation by Veretentsev (1988*)

346

6 Dynamics of two-dimensional vortex structures

Fig. 6.12. Development of perturbations in the shear layer with the wavelengths λ and 3λ. Calculation by Veretentsev (1988*)

grows in the flow, is most typical. In calculations with superposition of perturbations with non-zero amplitudes A1, A2, A4 on the flows, it was observed that after pairing of the initial structures, the growth in the shear layer thickness did not stop, and the pairing of secondary structures and the formation of vortex structures of the next generation continued. Therefore, development of the shear layer is possible due to pair merging of the forming vortex structures. Thus, subharmonics with wave numbers k/2n (n = 1, 2…) should dominate in the spectrum of initial perturbations. If these conditions are not satisfied, other formation mechanisms for the large vortex structures are possible, hence, another pattern of shear layer development can be observed. For instance, at superposition of perturbations with two wavelengths (λ and 3λ (Fig. 6.12)) on the main flow, development of initial instability occurs according to the scenario shown in Fig. 6.10. Simultaneously, the stages of secondary instability development differ significantly. According to Fig. 6.12 (τ = 3.5), and pairing of initial vortex structures oc-

6.3 Modeling the dynamics of shear flows

347

curs. Following evolution, this leads to the break-up of the middle vortices in every triplet and the formation of a chain of double-vortex structures. A similar process occurs upon quadruple merging of initial vortices, when main harmonics and subharmonics with the wavelength 4λ are excited in the flow. However, in this case, the quadruple-vortex structure becomes a triple-vortex one at first, and then a double-vortex structure. The interaction processes are even more complex, if vortex symmetry inside the secondary structures is broken for some reason. Upon vortex tripling, this situation may arise, if perturbations of low amplitude at the wavelength 2λ are additionally superimposed on the flow. Finally, we should note that in the case where several subharmonics are simultaneously excited in the flow, development of secondary instability becomes more complex, and it is caused by nonlinear interactions of subharmonics. At that, formation of secondary vortex structures may occur in different ways. The choice of the way is mainly determined by the initial conditions: amplitudes and subharmonic phases. Varying these conditions, we can control the development of the shear layer. 6.3.2 Instability of a starting vortex There are almost no ideal shear flows where the flows on different sides of the shear layer are directed towards each other, and velocities beyond the layer are equal. If we impose the flow with velocity u1 > u0 on the whole field, we will have a more realistic unidirectional flow with velocities u1 + u0 and u1 – u0 far from the shear layer. The character of instability development in this flow is the same as in the ideal shear layer with velocities ±u0. The closest real prototype of the shear layer is the mixing layer: the flow, developed behind the edge of a plate, when a liquid or gas flows along this plate with a different velocity on each side of the plate. The description of the flow and the character of instability development can be found in the reviews mentioned at the beginning of Section 6.3. The mechanisms of instability development at the nonlinear stage are very close to those described in Section 6.3.1. Here we will describe the objects which are more difficult to study: the shear flow formation, when the stream over a plate edge is not parallel to the plate or in case of the flow over a wedge. Unsteadiness, when a growing “starting” vortex is formed at the edge, is the specific feature of that flow. Let us consider a separated flow of inviscid incompressible fluid over the edge of a semi-infinite plate, starting from rest, as the idealized problem on the starting vortex. We assume that on the complex plane this plate

348

6 Dynamics of two-dimensional vortex structures

occupies the negative real semi-axis Re(z) ≤ 0, Im(z) = 0. The main term of the complex potential of separation-free flow over the plate has the form (Pullin 1978)

Wp (z, t ) = −ig (t ) z ,

(6.43)

where g(t) is the function determining the dependence of the complex potential on time. This potential leads to a velocity field with a singularity at point z = 0. Physically, the boundary layer formed on the windward side of the plate can not flow over the sharp edge without separation. The boundary layer detaches and a free shear (vortex) layer, spread in the outer potential flow, is formed. At high Reynolds numbers, this shear layer has low thickness, and it can be modeled by a vortex sheet. The equation of vortex sheet motion in the external potential field (6.43) takes the form

d z ( Γ, t ) = dt Γ (t ) ⎫⎪ (6.44) ⎤ 1 ⎧⎪ 1 T ⎡ 1 1 = − ⎨ g(t ) − ⎢ ⎥ dΓ′⎬ , 2 (Γ, t ) ⎪ 2πi 0 ⎣ (Γ, t) − (Γ′, t) (Γ, t) − (Γ′, t ) ⎦ ⎩ ⎭⎪

∫

where = i z . The condition of Kutta - Joukowski requires a finite velocity at the plate edge. Hence, the expression in braces should be equal to zero at z = 0

1 g(t ) + 2πi

ΓT (t )

∫ 0

⎡ 1 1 ⎤ − ⎢ ⎥ dΓ′ = 0 . (Γ′, t ) ⎦ ⎣ (Γ′, t )

(6.45)

We should note that the system of equations (6.44), (6.45) is similar to system (6.29), (6.30). It is shown in the papers by Nikol’skii (1957) and Pullin (1978) that this system has the self-similar solution, if g(t) = atm. The starting vortex formed is geometrically self-similar at different moments in time. Equation (6.44) has one integral, namely, the law of change of longitudinal component of the vortex impulse is true (Kuibin 1993)

dI d ≡ dt dt

ΓT ( t )

∫

z2 ( Γ, t ) dΓ =

0

Thus, for the self-similar problem we have

π 2 g (t ) . 4

6.3 Modeling the dynamics of shear flows

I=

π 2 t 2m +1 a . 4 2m + 1

349

(6.46)

Different self-similar laws can be formulated for the considered problem, for example, the law of vortex size increase, the law of total circulation growth, etc. The last law 4/3 (4m + 1)/3

ΓT(t) = Jma t

(6.47)

was considered by Pullin (1978). The exact value of coefficient Jm is unknown. Unfortunately, the calculation accuracy was not mentioned in the above paper, where values of Jm for some m were presented. When solving the considered problem (6.44), (6.45) with the method of discrete vortex particles, the equations of particle motion (6.33) can be simplified considerably. The conform reflection of the flow region (the plate with a slit along the negative real semi-axis) on semi-plane Im ( ) > 0 is assigned by formula z( ) = – 2, and the complex potential of the external (separation-free) flow (6.43) takes the form Wp( , t) = – g(t) . Thus, we can find the complex velocity of the flow: up = g(t ) /2 .

Considering that zk′ = –2 k, zk′′ = −2, the equations of vortex particle motion (6.33) take the form 2 ⎞⎤ ⎧ ⎡ ⎛ ⎪ 1 ⎢ k − n ⎜ ⎟⎥ − 1 − exp − 2 Γn ⎨ 2 ⎟ ⎜ − ⎢ D D + k n =1 ⎪⎩ k n ⎣ k n ⎠⎥ ⎝ ⎦ 2 ⎡ ⎛ ⎞⎤ 1 ⎢ ⎜ k − n ⎟⎥ − ⎢1 − exp ⎜ − D2 + D2 ⎟ ⎥ − k − n ⎢ ⎜ k n ⎟⎥ ⎝ ⎠⎦ ⎣

1 zk = − 4πi

N (t )

∑

2 ⎞⎤ ⎫ ⎡ ⎛ 2 ⎞ ⎛ k − n ⎟⎥ Dk4 4 k ⎪ g(t ) ⎢ ⎜ k − n ⎟ − exp − − ⎢ exp ⎜ − 2 ⎜ D2 + D2 ⎟ ⎥ σ 2 ( D2 + D2 ) ⎬ + 2 , ⎜ Dk + Dn2 ⎟ k ⎜ k n ⎟⎥ k k n ⎪ ⎢⎣ ⎝ ⎠ ⎝ ⎠⎦ ⎭

k = 1, 2,..., N (t),

Dk2

(6.48)

2 ⎞⎤ ⎛ σk2 ⎡ k σ ⎥ ⎢ ⎜ ⎟ . = 1 − exp − 2 2 ⎟ ⎜ σ 4 k ⎢⎣ k ⎠⎥ ⎝ ⎦

When particles leaving the edge, do not approach the plate within a distance of about 3 –5 times their diameter, exponential terms responsible for interaction with the wall, can be neglected. As a result, the simplified

350

6 Dynamics of two-dimensional vortex structures

equations of motion and the formula for the finiteness of velocity (6.34) can be written as: 1 zk = − 4πi −

⎧ ⎪ Γn ⎨ ⎪⎩ n =1 n ≠k

N (t ) k

∑

Γk 4πi k

k

⎡ 1 + ⎢ ⎣⎢ 2 k

1 π

1 −

n

2 ⎡ ⎛ − n ⎞⎤ ⎢1 − exp ⎜ − k ⎟⎥ − ⎢ ⎜ Dk2 + Dn2 ⎟ ⎥ ⎝ ⎠⎦ ⎣

⎫ ⎪ ⎬− − n⎪ ⎭ 1

k

(6.49)

⎤ g(t) , k = 1, 2, ..., N (t) , ⎥+ − k ⎦⎥ 2 k 1

k

N (t )

Im (

k =1

k

∑ Γk

k 2

)

= g(t) .

(6.50)

As in the initial equation (6.48), the equation (6.49) has a multiplier tending to infinity, when a particle approaches the edge; this creates some inconveniences for numerical application of the method. In the same time, substitution of function g(t) from (6.50) into the motion equation eliminates this peculiarity. Finally, noting that k(

1 k −

2 k − n Dk2 + Dn2

n)

=−

k(

1 k +

n)

zk − zn

≈ σk2

k + n 2 k

−

2 , zk − zk

2

2

+ σn2

k + n 2 n

2

,

and hence, 2 ⎞ ⎛ ⎛ z −z 2 ⎞ k − n ⎜ ⎟ exp − 2 ≈ exp ⎜ − k2 n2 ⎟ , 2 ⎜ ⎟ ⎜ ⎟ ⎝ Dk + Dn ⎠ ⎝ σk + σ n ⎠

we obtain the following equations: ⎡ ⎛ z − z 2 ⎞⎤ n ⎢1 − exp ⎜ − k ⎟⎥ − 2 2 ⎟ ⎜ ⎢ σ + σ k n ⎠⎥ n =1 ⎝ ⎣ ⎦ ⎤ 1 1 − ⎥ , k = 1, 2, ..., N (t) . k( k + n) k( k − n)⎥ ⎦

1 zk = 2πi ⎡ 1 − Γn ⎢ 4πi n =1 ⎢⎣ N (t )

∑

N (t )

∑

Γn zk − zn

(6.51)

6.3 Modeling the dynamics of shear flows

351

After analysis of the derived equations we can see that the second term completely describes the interaction of the flow with the plate surface. If we neglect it, we derive the equation of particle motion in an unbounded fluid. Upon software implementation of the method, the question about the choice of position for the next descending vortex arises. In schemes where the condition of no-flow on a solid surface is satisfied approximately (at a finite number of points), and the surface is substituted by a set of bound vortices (Belotserkovskii and Nisht 1978), it is usually assumed that free vortices descend by the tangent to the plate. The descending vortex is assumed to be at distance ∆l from the closest vortex, where ∆l is the distance between the bound vortices. However, in this case, the step of integration of motion equations by time is rigidly connected with the number of bound vortices. On one hand, this leads to an uneconomical computational algorithm, and on the other hand, this leads to significant errors in the problem solution because of the high curvature of the separation line in the initial stage. In the paper by Zobnin (1986), it is suggested to determine the parameters of descending vortices by the data of asymptotic analysis of the shape and intensity of the vortex sheet in the vicinity of the edge. Rott (1956) determined that near the edge 3/2

2(m + 1)/3

z(Γ, t) = (K1λ + iK2λ )t

,

where λ = 1 – Γ/ΓT(t). The values of constants K1 and K2 should be determined for every m. In the paper by Kuibin (1993), the coordinates of descending vortices are determined as follows. For every time step, a marker (a vortex particle with zero intensity) is placed on the edge. During time ∆t the marker moves to a new position (as do all the other particles; we should also note that the velocity at the edge is finite even for the discrete model). Then, a particle with an intensity determined by the equation of velocity finiteness at the edge substitutes the marker. Apparently, at integration of the motion equations by the Euler method of the first order, the vortex will be on the line, which continues the plate. The schemes of the higher order also provide the transverse shift. As for the first vortex, it should substitute the starting vortex, formed during time ∆t. Here, it is suggested to determine its coordinates using the following conditions: the law of vortex impulse change should be satisfied (6.46); the velocity, induced at the plate edge by the first separated vortex, should have only a longitudinal non-zero component (i.e., the vortex is above the edge). The system of motion equations (6.50), (6.51) was integrated using the method of Runge - Kutta of the second order. The accuracy of the self-

352

6 Dynamics of two-dimensional vortex structures

similar problem solution was estimated by satisfaction of laws of changing of vortex impulse and total circulation, notated for the discrete system as: IN ≡

N (t )

∑

Γ n Im ( zn ) =

n =1 N (t )

ΓTN ≡

πa 2 t 2m +1 , 4 2m + 1

∑ Γn = Jma4 / 3t(4m+1) / 3 .

(6.52)

n =1

The typical formation pattern of the spiral vortex structure on a sharp edge for the self-similar flow with index m = 1 is shown in Fig. 6.13. Here, the points mean the centers of vortex particles. It is obvious that the starting vortex is a smooth spiral structure. Simultaneously, numerous experiments (Van Dyke 1982; Pierce 1961; Freymuth et al. 1983) demonstrated that the process of vortex formation is accompanied by instability development and the formation of coherent structures over the vortex perimeter. Instability of the starting vortex was studied in a numerical experiment on the basis of the generalized method of discrete vortex particles in papers by Veretentsev et al. (1989); Kuibin et al. (1991). The separated flow over a semi-infinite plate was calculated at the introduction of controlled perturbations of a given frequency into the flow. In particular, the effects of edge and plate vibration, different external fields (including the acoustic one) background perturbations were studied. It was shown that these perturbations were transformed into vortex ones, which caused the development of instability in the starting vortex. The difference in the problem statement from the undisturbed starting vortex considered above, is the complex potential of the external separation-free flow Wp(z, t). Indeed, when determining the potential asymptotic of the disturbed flow in the edge vicinity, we obtain

Fig. 6.13. The shape of the starting vortex in the case of the self-similar flow with index m = 1. The number of time steps is N = 200. Calculation by Kuibin (1993)

6.3 Modeling the dynamics of shear flows

Wp(z, t) = − ig(t) z + h(t)q(z).

353

(6.53)

Here, function h(t) determines the character of the dependency between the intensity of perturbation source and time. If perturbations are harmonic, h(t) is presented as: h(t) = C1 sin(ω1t + ϕ1) + C2 sin(ω2t + ϕ2).

(6.54)

Function q(z) is connected with the type of perturbations. Thus, q(z) = z corresponds to the perturbations generated by longitudinal oscillations of the plate. For transverse oscillations, q(z) = i z . Background pulsation in the ascending flow may lead to perturbations of potential both with function q(z) = z and q(z) = i z or to their combination. As shown by Veretentsev (1988), the effect of the external acoustic field on the studied flow is reduced to the generation of vortex perturbations, whose intensity equals Af sin (2πft + ϕf), where f is the frequency of acoustic oscillations. A part of the potential corresponding to acoustic oscillations has the form Bf sin (2πft + χf) i z . Amplitudes Af and Bf are proportional to the amplitude of an ascending acoustic wave. Perturbations caused by vibration of the “edge”, more precisely, by low-amplitude oscillations of the plate of a finite length l, hinge-jointed to the semi-infinite plate, are described by the same law (Kuibin 1993). Since the undisturbed problem is self-similar, the scales of the main flow are determined by a single dimension coefficient a and current time t. In particular, the typical size of vortex L and the time of its evolution are connected by relationship L3/2 = aTm+1. When solving the problem with perturbations, it is natural to take the period of introduced perturbations τ as the time scale. Experimentally (Pierce 1961), the most expressive pattern of instability development was observed at the stage of uniformly accelerated motion, and this corresponds to index m = 1. This case will be considered further in this Section. According to numerical experiments, the character of instability development in the starting vortex for all the mentioned types of perturbations (with potentials of type (6.53)) and function q(z) = i z or q(z) = z) is qualitatively similar. The flow pattern at successive moments of time at m = 1, q(z) = i z , h(t) = sin 2πt/τ, T = 25 τ is shown in Fig. 6.14. The shape of the vortex at t = 25 τ, calculated without perturbations is presented in Fig. 6.13.

354

6 Dynamics of two-dimensional vortex structures

Fig. 6.14. Development of perturbations in the starting vortex: perturbation frequency f = 25/T, where T is the calculation period of vortex formation. The points correspond to the centers of vortex particles. The dashed line is the interpolation of the vortex sheet shape. Calculation by Kuibin (1993)

Fig. 6.15. Development of instability in the starting vortex, as observed experimentally (Pierce 1961*)

6.3 Modeling the dynamics of shear flows

355

It can be clearly seen that the amplitude of the developing vorticity perturbation increases quickly with distance from the plate edge. This enables the formation of small vortex structures in the spiral of the starting vortex. This pattern of instability development qualitatively correlates with those observed experimentally (Pierce 1961). The pictures of a shadow pattern of a separated flow over the profile, moving with a constant acceleration (taken from the above paper) are shown in Fig. 6.15. Apparently, perturbations in experiments are caused by vibration of the profile, related to imperfection of the operating mechanism. To obtain quantitative information on this process, it is necessary to analyze the spectral composition of the developing perturbations. The situation is complicated by the fact that the undisturbed flow is also unsteady, and the ordinary stability theory cannot be applied. However, we can use the self-similar properties of the starting vortex. With the self-similar variable (Pullin 1978) λ = 1 – Γ/ΓT(t),−2(geometry of the vortex sheet is dem + 1)/3 scribed by function (λ) = z(Γ, t) t and remains constant. The re(1 − 2m)/3 is also retained. Therefore, it is duced velocity field z (Γ, t) t convenient to study the spectral properties of the function, derived with consideration of geometrical similarity of the flow at different instants of time F (λ, t) = ⎡Un ⎣

(

( λ )t

2(m +1) / 3

,t

)

− Un0

(

(λ )t

2(m +1) / 3

⎛t ⎞ ,t ⎤ ⎜ ⎟ ⎦ ⎝ t0 ⎠

)

(1− 2m ) / 3

, (6.55)

where Un0 (z, t) is the velocity component of the undisturbed flow at moment t, calculated at point z of the spiral and directed normally to it; Un(z, t) is the corresponding velocity of the disturbed flow, projected to the same normal; t0 is some fixed moment of time. At low perturbation amplitudes and at a short time alteration in the vicinity t0, the function (6.55) becomes the periodic function of time, and this allows its expansion into the Fourier series and the analysis of spectral composition of developing perturbations. According to the calculations, development of initial instability occurs when the frequency of external perturbation f = 1/τ. The dependency of perturbation amplitude of the main mode on self-similar variable λ (in the flow shown in Fig. 6.14) is presented in Fig. 6.16 (line Af). The Fourieranalysis was conducted during the time range (23τ, 25τ). There is an initial region with monotonous perturbation growth, which can be called the linear region. Then, the amplitude of the main mode develops notmonotonously (nonlinear effects), however, up to the point λ = 0.17 it increases at an average rate. An important feature of the flow is the generation of subharmonic perturbations (line Af/2) with frequency f /2. Spectrograms indicate the existence of active interrelations of the resonance type

356

6 Dynamics of two-dimensional vortex structures

between the main and subharmonic modes with frequencies f /2 and 3f /2 (curves Af/2 and A3f/2 in Fig. 6.16). This is distinctly illustrated in the spectrograms, where the minimums of the main mode correspond to the maximums of the subharmonic mode (see Fig. 6.16, values λ = 0.35; 0.42; 0.47; 0.52, etc.). The amplitude of the subharmonic perturbation Af/2 within the range λ = 0 ÷ 0.38 is always lower than the main mode amplitude Af, and at λ > 0.5, it is higher. In particular, this leads to an increase in the size of the initial vortex structures with time. Finally, it is necessary to note that despite Af/2 > Af in the final stage, the absolute level of subharmonics is low, therefore, there is no secondary instability and vortex pairing as it occurs in the shear and mixing layers. With a rise in λ, or in terms of vortex coordinates with an approach to the core, perturbations at all frequencies decay. The last fact is connected with the generation of intensive rotation, which suppresses pulsation in the vortex core (Vladimirov et al. 1980).

Fig. 6.16. Dependencies between the amplitudes of developing perturbations in the starting vortex and the self-similar variable (Kuibin 1993)

Fig. 6.17. Development of starting vortex instability at excitation of frequencies f = 25/T and 12.5/T. Calculation by Kuibin (1993)

6.3 Modeling the dynamics of shear flows

357

Instant of time t0, in whose vicinity the Fourier-analysis of function (6.54) is performed, characterizes the ratio of perturbation wave length to the size of the starting vortex. Analysis of perturbation development and dynamics of their amplitudes at different ratios t0/τ (Kuibin 1993) has shown that the maximum perturbation amplitude is obtained at t0 = 24τ. This corresponds to the fact that six wavelengths are positioned on the right side of the external convolution of the starting vortex (see Fig. 6.14). In the experiment by Pierce (1961), the secondary vortex structures are especially obvious at the same ratio of scales. To study interaction of perturbations with different wavelengths, perturbation development was studied upon simultaneous excitement of the main and subharmonic modes with the same amplitudes. In contrast to the shear layer, there is no vortex pairing. This could be observed at T τ, when the length of perturbation wave is low in comparison with the curvature radius of the external convolution of the starting vortex. Qualitative changes also occur: alternate vortex structures become stronger, others become weaker (see Fig. 6.17). 6.3.3 Wake instability behind a thin plate

At first sight, a wake behind a thin plate is a particular case of a mixing layer, where velocities on both sides of the plate are the same. Nevertheless, we cannot transfer investigation results obtained for the mixing layer to the wake problem. First of all, this is due to the symmetry of the average velocity profile relative to the wake axis, hence, there are two instability modes with different parity. The existence of two instability modes in the flows with symmetrical velocity profiles was indicated in the papers by Michalke and Schade (1963), Tatsumi and Kakutani (1958). They also inform us that the antisymmetrical mode (with an odd profile of longitudinal velocity pulsation) is the most unstable in a wake. Therefore, in most papers on wake instability, especially experimental ones (e.g., see Sato and Kuriki (1961), Mattingly and Criminale (1972)), the second mode (symmetrical) is neglected. At the same time, perturbation, which is superposition of two modes is excited in the experiments. In the works of Wygnanski et al. (1986), Marasli et al. (1989), attempts were made to experimentally generate each of the modes separately. Conversely, it is clear that if both modes are independent at the linear stage, their interaction at the nonlinear stage will significantly affect the character of the flow development. It was shown above, that the main mechanism of instability development at the nonlinear stage in the mixing layer is the subharmonic one.

358

6 Dynamics of two-dimensional vortex structures

According to the general considerations, a similar mechanism seems to be possible even for a wake (Gertsenshtien and Shtemler 1977). However, in the paper by Kelly (1968), it was shown that despite the possibility of resonance interaction of neutral perturbations of symmetrical and antisymmetrical modes for the symmetrical profile, the coefficient of mode bond (calculated in the framework of the weakly- nonlinear theory of the Stuart-Watson type) is identically equal to zero. The negative result, obtained in the work of reference upon numerous strong assumptions, can not be considered as the ultimate result, besides, it does not answer the question how the nonlinear stage of instability development occurs in the wake. Development of perturbations in the flow of the wake type (more precisely, in the flow with the profile of longitudinal velocity u = 1 − 0.7exp(–0.9y2)) was studied numerically by Gertsenshtien et al. (1985). Several interesting results were obtained. In particular, the separation of long-wave components of the spectrum was determined. Moreover, upon nonlinear interaction of these perturbations, the greatest increase is observed for their difference component. The determination of secondary instability of finite-amplitude wave modes to transverse 3-D perturbations also seems important. From the point of development of control methods for the considered flow, it is necessary to study the mechanisms of the nonlinear stage of instability generation for each mode separately and at their interaction. The question on possible secondary instability related to resonance intensification of 2-D subharmonic perturbations has a principle meaning. Now, let us consider the plain laminar wake behind the plate, parallel to the flow of incompressible fluid on the basis of the work by Kuibin and Rudyak (1992). The flow velocity is represented as u = u0 + u′, v = v′, where u0(x,y) is the undisturbed velocity of the steady flow. This velocity is assumed to be known, and we approximate it by function u0 = U∞ – ∆u exp (–0.6931y2/b2) (Nishioka and Miyagi 1978), where U∞ is the velocity of the ascending flow. Velocity defect ∆u diminishes with distance x from the back edge of the plate by the law ∆u = U∞(4x/l + 1)–1/2, and the halfwidth of wake b increases as b = l[0.6931(4x/l + 1)/Re]1/2. Here l is the plate length, Re = U∞l/ν is the Reynolds number. Axis x is directed downward along the flow, and y is across it, the coordinate origin is at the back edge of the plate. Then, we consider that the source of perturbations is in the vicinity of the back edge of the plate. This allows us to assign perturbations to the vorticity ω′(0, y, t) in cross-section x = 0. In this statement, the problem can be solved by the method of discrete vortex particles, described above. In this case, the field of vorticity pulsation is modeled by a set of discrete vortex particles, whose motion equations take the form (see (6.51))

6.3 Modeling the dynamics of shear flows

359

N ⎛ z − z 2 ⎞⎤ 1 f Γn ⎡⎢ zk = 1 − exp ⎜ − k2 n2 ⎟ ⎥ − ⎜ ⎟⎥ 2πi n =1 zk − zn ⎢ ⎝ σk + σ n ⎠ ⎦ ⎣

∑

N

⎡ 1 f − Γn ⎢ 4πi n =1 ⎢⎣

∑

1 k( k +

n)

−

k = 1, 2,..., N (t),

⎤ ⎥ + u0 (zk ), n)⎥ ⎦

1 k( k − 2 k

(6.56)

= −zk ,

where zk = xk + iyk; xk, yk are the coordinates of the center of the k-th vortex particle; Nf = 2Nt is the total number of vortices descending from the plate; Nt is the number of the time step. Circulation of the vortex particles descending from the plate is determined as Γ n = ∆xn I0± , where ∆xn = ∆t u0(0, yn0) and ∞

∫

Iγ± = ± y γ ω′(0, y, t ) dy, γ = 0,1, 2, 0

∆t is duration of the time step; t = Nt ∆t, and signs “plus” and “minus” refer to the particles above and under the plate. The initial position of the vortex particle is assigned by coordinates xn0 = 0 and yn0 = I1± I0± . Particle dispersion is determined from the conditions of the best approximation of vorticity separated at the n-th step and is equal to σn2 = (∆xn ) 2 12 + I2± I0± − yn20 . It is convenient to use function ω′(0, y, t) to provide the circulation of vortex particles in the form of

(

)

(

)

Γn± = A1± sin 2π f1tn + ϕ1± + A2± sin 2π f2 tn + ϕ±2 .

(6.57)

Then, if An+ = An− and ϕn+ = ϕn− , the symmetrical mode is generated (it is called varicose due to the shape of the wake), and at An+ = An− , but

ϕn+ − ϕn− = π, it is the antisymmetrical (sinusoidal) mode, fi are the frequencies of the introduced perturbations. Therefore, the problem of instability development in a wake under the action of assigned external perturbation is reduced to the solution of the system of nonlinear differential equations (6.56). The system was integrated using the method of Runge – Kutta of the second order with a uniform time step ∆t U∞ /l = 0.0625.

360

6 Dynamics of two-dimensional vortex structures

Calculations correspond to the Reynolds number Re = 2.5⋅105 (Reθ = 443, θ = 0.5l π Re is the displacement thickness), they were conducted with Strouhal numbers Stk = fk θ/U∞ = 0.015/k. Dimensionless amplitudes of perturbations ak± = Ak± θU∞ varied from 0.281 to 2.81. Then, we skip the subscripts, and the dimensionless amplitudes of antisymmetrical and symmetrical modes will be indicated as ak and sk. Nonlinear interaction of perturbations of the same mode

When small perturbations of frequency f of some mode of the same parity are introduced into a flow, initial instability will be developed just at this frequency. Typical calculation results are shown in Fig. 6.18., the flow was calculated up to distances x′ = x/θ = 600. At excitation of the antisymmetrical mode, formation of a sinusoidal wake is observed (see Fig. 6.18 , a1 = 2.81). At the initial linear stage of perturbation development, the shape of the perturbation remains sinusoidal, and its amplitude increases. Then, nonlinear effects appear, the shape of the perturbation stops being sinusoidal and a vortex street like the Karman street is formed. If perturbation of the varicose mode is excited in the flow, its development leads to formation of the vortex street, symmetrical relative to the axial line of the wake (Fig. 6.18b). The typical feature of the nonlinear stage of instability development is the generation of combination frequencies of different parity. Thus, if frequency f of the antisymmetrical mode is excited in the flow, during flow evolution perturbations of frequencies 2f, 3f, etc. are generated, and the perturbation at frequency 2f is symmetrical, and at 3f, it is antisymmetrical. If the initial perturbation of frequency f was symmetrical, it generates symmetrical perturbations of combination frequencies. This is due to the fact that the nonlinear evolution of the longitudinal perturbation of velocity is regulated by the nonlinear quadratic term of transfer equation v′∂u′/∂y with parity opposite to u′ for the antisymmetrical mode. As predicted by the linear theory, the flow is less sensitive to the symmetrical mode. In particular, this explains the fact that the appearance of combination frequencies at excitation of the symmetrical mode is observed further down the flow than at excitation of the antisymmetrical mode. A decrease in the initial amplitude of the introduced perturbation, protracts the linear stage of perturbation development and does not qualitatively change the character of flow evolution. If the amplitude of one mode is significantly higher at simultaneous excitation of both modes at a given frequency, this relationship remains true far along the flow as was observed in the work of Gertsenshtein et al. (1985).

6.3 Modeling the dynamics of shear flows 361

Fig. 6.18. Development of instability in the wake behind a plate: a – sinusoidal (antisymmetrical) mode; b – varicose (symmetrical) mode. Calculation by Kuibin and Rudyak (1992)

362

6 Dynamics of two-dimensional vortex structures

If the amplitudes of induced perturbations are of the same order, the profile of the average flow velocity becomes smoother, i.e., at the axis, the flow velocity decreases and far from the axis it increases. Depending on the phase shift between perturbations of symmetrical and antisymmetrical modes, different degrees of asymmetry of the average velocity profile can develop. The nonlinear stage of instability development proves to be very complex. Perturbation of combination frequencies of both modes, their interaction with each other and main harmonics are observed there. This interaction may provide both suppression and intensification of perturbation depending on the phase relation. According to the commonly accepted theory (Maslowe 1981), there is no subharmonic resonance in the wake, whereas in the mixing layer, it is the standard channel of secondary instability development (Veretentsev and Rudyak 1987). Whether it is possible or not to reach subharmonic resonance at interaction of two perturbations of antisymmetrical mode (the main and subharmonic) can be easily deduced from a simple kinematic model, when the wake is modeled by two rows of vortices with vorticity of different signs (Karman street, see Fig. 6.19 ). As a result of initial instability of the initial main perturbation at frequency f (or with a wavelength λ), the Karman street is formed of vortices located in staggered rows. Secondary instability, which provides vortex pairing in every row, is obtained at the wavelength 2λ. Perturbation developing at this wavelength breaks the equilibrium of the vortices and forms their pairs. It is obvious that the most favorable phase shift between harmonic and subharmonic for this is ∆ϕ = ϕ1 − ϕ2 = 0, π/2, π, …, and unfavorable shift is π/4 + nπ, n = 1, 2, …. When the phase shift equals π/4, the centers of primary vortices are in the nodes of subharmonic perturbation, and secondary instability should not hghghg

Fig. 6.19. The kinematic scheme of interaction between the main and subharmonic perturbations in the wake behind a plate: – sinusoidal mode; b – varicose mode (Kuibin and Rudyak 1992)

6.3 Modeling the dynamics of shear flows

363

develop. Conversely, development of instability in the varicose mode leads to the formation of a symmetrical vortex street (Fig. 6.19b). Here, the phase shift, most favorable for development of secondary instability, equals ∆ϕ = 0. At that, vortex pairing, which leads to the known “leapfrog” action observed at the interaction of two vortex pairs occurs. On the contrary, the phase shift of subharmonic perturbation by π (∆ϕ = π) makes this vortex pairing impossible. Calculations prove the described qualitative pattern of interaction between the main harmonic and subharmonic. The character of interaction between introduced perturbations significantly depends on the phase shift between perturbation of the main frequency and subharmonic. In a particular case, when both introduced perturbations are antisymmetrical (a1 = a2), dependency between dimensionless, averaged by the y-coordinate energy of pulsation of longitudinal velocity at frequencies f1 = f and f1 = f /2

εk =

Ekd ( f1 , f2 ) Ekd ( fk )

mθ

, Ekd =

1 uf′2 2mθ −mθ k

∫

x′= d

dy,

(6.58)

calculated in the given cross-section x′ = d (it is selected that m = 10), is shown in Fig. 6.20 . Here Ekd ( fk ) is the energy of longitudinal velocity pulsation, obtained at perturbation of only one harmonic f, and Ekd ( f1, f2 ) – at perturbation of the harmonic and subharmonic. Analysis of the diagram demonstrates that despite active interaction between introduced perturbations, there is no resonance intensification of the subharmonic at any ∆ϕ. Moreover, subharmonic perturbations are suppressed. This is due to the fact that during nonlinear interaction between main perturbation a1 and subharmonic a2, subharmonic perturbation s2 is generated. As a result, initial ffff

Fig. 6.20. Kinetic energy of pulsation at interaction of the main (1) and subharmonic (2) perturbations of the same modes in the wake behind a plate: – antisymmetrical modes; b – symmetrical modes. Calculation by Kuibin and Rudyak (1992)

364

6 Dynamics of two-dimensional vortex structures

subharmonic perturbation and the one generated have opposite parities, and this cannot enhance the total perturbation, but actually suppresses it. When the perturbations s1 and s2 interact at the nonlinear stage, a subharmonic is generated together with a combination subharmonic. The most favorable phase shift of the main frequency and subharmonic is actually ∆ϕ = 0 (see Fig. 6.20b), and at ∆ϕ > π/4, subharmonics suppression is observed. Correlation of perturbation energies of the main frequency and subharmonics proves their active interaction, despite an insignificant increase in subharmonic energy even at the favorable phase shift ∆ϕ. This is due to the fact that the phase of generated subharmonics differs from the phase of the introduced subharmonic. In the simplest model of two connected oscillators, the phase shift between introduced and generated subharmonics is π/2. Therefore, the resulting subharmonic perturbation has a phase shift which is unfavorable for development of secondary instability relative to the phase of the main perturbation. A decrease in the amplitude of introduced perturbations does not qualitatively change the pattern of flow development; it only elongates duration of the linear stage. Resonance interaction between perturbations of different modes

Since at nonlinear interaction of two perturbations in a wake, combination frequencies of both modes are generated, it becomes clear that even if subharmonic resonance is possible, it can only be observed when the generated combination perturbation has the same parity as the initial one. From this point of view, it is necessary to consider the interactions of a1 with s2 and s1 with a2 (ak, sk are perturbations of antisymmetrical and symmetrical modes of frequency fk). It can be expected that resonance interactions in the first case will be significantly weaker because the symmetrical mode has a very low amplification factor. And if a1 = s2 at the initial moment of time, the amplitudes of generated subharmonic perturbations are very low even at distances x′ ∼ 200. Calculations prove these qualitative considerations. At interaction of a1 with s2, a weak amplification of subharmonics is found. The most favorable phase shifts are ∆ϕ = π/4 and π (Fig. 6.21 ). On the contrary, at interaction of s1 with a2, a drastic amplification of subharmonic perturbations of the clear resonance character is observed. Amplification of subharmonic perturbations occurs in a wide alteration range of ∆ϕ: 0 ≤ ∆ϕ ≤ 3π/4. Maximum amplification is reached at ∆ϕ = π/2 (see Fig. 6.21b). In Fig. 6.22, evolution of the amplitude maximum of longitudinal velocity perturbations

6.3 Modeling the dynamics of shear flows

365

Fig. 6.21. Kinetic energy of pulsation at interaction between the main (1) and subharmonic (2) perturbations of different modes in the wake behind a plate: – harmonic of the antisymmetrical mode and subharmonic of the symmetrical mode; b – harmonic of the symmetrical mode and subharmonic of the antisymmetrical mode. Calculation by Kuibin and Rudyak (1992)

A = u′fk2

1/ 2 max

is shown along the flow. The solid curve corresponds to subharmonic perturbations, and the dashed line corresponds to perturbations of the main frequency. Light and dark points indicate amplitude levels of single perturbations of subharmonic and main frequencies. The effect of subharmonic perturbation amplification at interaction between modes s1 and a2 is observed in a wide alteration range of initial perturbation amplitudes. This effect occurs even in plain-parallel approximation, when ∆u = const and at different values of velocity defect, which vary from 0.2 to 0.75. The method of perturbation generation for each mode suggested by Wygnanski et al. (1986), Marasli et al. (1989), allows experimental observation of subharmonic perturbation amplification. For this purpose, it is necessary to introduce perturbation of the main frequency of the symmetrical mode and antisymmetrical subharmonic perturbation, shifted by phase relative to the main one by π/4 ÷ π/2 into the flow.

366

6 Dynamics of two-dimensional vortex structures

Fig. 6.22. Amplitudes of perturbations developing in the wake behind a plate: 1 − at perturbation of symmetrical mode harmonic; 2 − at perturbation of antisymmetrical mode subharmonic; 3 − harmonic amplitude and 4 − subharmonic amplitude at excitation of harmonics of both perturbations with the phase shift π/2. Calculation by Kuibin and Rudyak (1992)

6.4 Motion of vortices in cylindrical tubes The necessity to study the vortex motion in cylindrical tubes is caused by numerous problems of the applied character arising from flow swirling in tubes. One important problem among them is the generation of lowfrequency regular pulsation of velocity and pressure. This occurs, for instance, in the suction tubes of water turbines (Murakami 1961; Fanelli 1989), upon combustion of a fuel mixture in a swirl burner (Knysh and Uryvskii 1981), upon operation of cyclone separators (Gupta et al. 1984) and other devices, whose operation principles provide an application for rotating flows. The physical mechanism of oscillation generation is fairly clear: it is shown experimentally that pulsation is caused by the precession of a vortex formed in the flow. The phenomenon of vortex precession, known as “precession of the vortex core” (PVC), will be described in detail in Chapter 7. The current Section deals with theoretical approaches and numerical simulation of the phenomenon. The theoretical model of PVC for a jet flow in a tube with a ring shear layer was developed by Knysh and Uryvskii (1981). They studied the

6.4 Motion of vortices in cylindrical tubes

367

process starting from initial instability of the shear layer, which leads to formation of discrete vortices. Then, as a result of secondary instability, vortices join into a vortex “cloud”, whose center is shifted relative to the tube axis, and the “cloud” itself makes a circular precession motion. When modeling secondary instability, the authors used a plain model of point vortices. However, as already mentioned above, instabilities that are atypical for the physical properties of the flow, are developed in the system of point vortices. Most works on velocity and pressure pulsation in swirling flows of the wake type are based on the assumption of the formed rectilinear vortex, whose axis does not coincide with the tube axis (Murakami 1961), or the helical vortex (Bondarenko and Zav’yalov 1979; Fanelli 1989). Further, we will consider possible mechanisms for the loss of axial position of the vortex, the problem of helical vortex motion in a cylindrical tube and the effect of three-dimensionality (a pitch of helical vortex lines) on the character of flow instability development in a tube. 6.4.1 Motion equations for vortex particles in a circular domain

General form of motion equations for vortex particles in a bounded domain with sharp edges are presented in Section 6.1.2. If the domain boundary is circumference |z| = 1, there are no sharp edges and it natural to assume that in (6.32) there is no additional potential flow: ψp = 0. Projection of a circle on the plane is assigned by formula = i(1 − z)/(1 + z). Consideration of integral exponents in (6.32) is important only as particles approach each other or the domain boundary. Therefore, as in derivation of (6.51), we should substitute arguments of integral exponents with −

zk − zn

2

σk2 + σn2

and −

zk − 1 zn

2

σk2 + σn2

correspondingly. Then, for the circular geometry problem, Hamiltonian (6.32) takes the following form: ⎡ z − zn 1 N Γk Γ n ⎢ log k HN = − 8π k,n=1 1 − zk zn ⎢ ⎣

∑

⎛ z −z 2 ⎞ ⎛ z −1 z n −Ei ⎜ − k2 n2 ⎟ + Ei ⎜ − k 2 2 ⎜ σk + σ n ⎟ ⎜ σk + σ n ⎝ ⎠ ⎝

2

− 2

⎞⎤ ⎟⎥ . ⎟⎥ ⎠⎦

(6.59)

368

6 Dynamics of two-dimensional vortex structures

Substituting the obtained expression into Hamiltonian equations (6.13), we derive the equation of particle motion in a circle ⎧ ⎡ ⎛ z − z 2 ⎞⎤ 1 N ⎪ 1 ⎢ zk = − Γn ⎨ 1 − exp ⎜ − k2 n2 ⎟ ⎥ − ⎜ σk + σ n ⎟ ⎥ 2πi n=1 ⎪ zk − zn ⎢ ⎝ ⎠⎦ ⎣ ⎩ ⎛ z − z 2 ⎞ ⎤ ⎫⎪ 1 ⎡ n ⎢1 − exp ⎜ − k ⎟ ⎥ ⎬ , k = 1,..., N. − ⎜ σk2 + σn2 ⎟ ⎥ ⎪ zk − zn ⎢ ⎝ ⎠⎦ ⎭ ⎣

∑

(6.60)

We should note that as in the system of point vortices (Geshev and Chernykh 1983), the system of vortex particles in a circle allows motion integrals, independent of time, i.e., invariants. Firstly, this is Hamiltonian HN (6.59), which corresponds to the kinetic energy of vortex motion of the fluid. Secondly, since the domain of fluid motion is a circle, due to Hamiltonian invariance (6.59) relative to rotations, there is the motion integral, connected with the law of angular momentum conservation MN =

N

∑ Γn zn

2

= const.

(6.61)

n =1

6.4.2 Precession of a rectilinear vortex in a tube

Let us consider the model flow with a rectilinear vortex at the tube axis. We assume that at the initial instant of time, there is additional vorticity in the flow, non-uniformly distributed over a circumferential coordinate. In the wake behind a water turbine, this vorticity may be generated due to separation of the flow by the rotor blades, and circumferential nonuniformity may be caused by asymmetry of the vortex chamber. According to Kuibin (1993), let us elaborate the mathematical model of the main vortex withdrawal from the center. For this purpose, we consider a swirling flow of inviscid incompressible fluid in a cylindrical tube of radius R with average velocity U along the tube axis (axis Ox). Further, all the values are presented in dimensionless form with ranging by R and U. We assume that at the initial instant of time, there is a vortex with uniform vorticity distribution (Rankine vortex) of diameter d0 with circulation Γ0 in the tube center. Circumferential non-uniformity of vorticity distribution is modeled by M vortices (in accordance with the number of rotor blades, when the wake behind a turbine is considered), whose axial lines lie on the cylindrical surface of radius ρ (Fig. 6.23). The sizes of disturbing vortices are, vorticity inside the vortices is uniform, and circulation is assigned by the sinusoidal law:

6.4 Motion of vortices in cylindrical tubes

369

Fig. 6.23. On the model of generation of rectilinear vortex precession

Γk = aΓ0 sin(2πk/M + χ),

k = 1, 2, …, M,

where a is the value characterizing the intensity of vorticity disturbance; χ is the phase shift. Each of the vortices is divided into nk equivalent cells. At the initial instant of time, we place vortex particles into vorticity centroids of corresponding cells. Intensities of particles attributed to one vortex are assumed to be the same Γkj = Γk nk . Parameters σj, characterizing the sizes of particles are determined via the cell areas (σ2j )k = dk2 4nk . The equations of motion (6.60) in the paper by Kuibin (1993) were integrated using the method of Runge – Kutta of the second order. Calculation accuracy was controlled by satisfaction of the laws of energy (H = const) and angular momentum conservation (6.61). The deviation during calculations did not exceed 1%. The pattern illustrating evolution of the vorticity field in a circle is shown in Fig. 6.24. Vortex particles marked by squares refer to the axial vortex. Particles with sign “+” model the vortices which cause perturbations with positive circulation, and sign “−” corresponds to negative ones. The following parameters are used for this calculation: a = 0.1; ρ = 0.6; M = 8; χ = 0; n0 = 16; nk = 4; d0 = dk = 0.2 (k = 1, …, M). It is clear that the axial vortex quickly leaves the center and starts moving along the trajectory close to the circular one. The motion trajectory of the main vortex center, leading out of the axial position, is shown in Fig. 6.25. According to the law of angular momentum conservation, disturbing vortices with negative circulation are forced back to the tube wall, and vortices with positive circulation migrate to the center. The motion velocity of the main vortex, which achieves a quasi-stationary orbit, is close to the velocity of one Rankine vortex in a circular domain (Murakami 1961)

370

6 Dynamics of two-dimensional vortex structures

Fig. 6.24. Calculation of vorticity field dynamics for the model of precessing rectilinear vortex: 1 – particles of the main vortex; 2 – particles of disturbing vortices with positive circulation; 3 – with negative circulation. Calculation by Kuibin (1993)

6.4 Motion of vortices in cylindrical tubes

371

Fig. 6.25. Trajectory of the center of a precessing vortex. Calculation by Kuibin (1993)

u0 =

where

Γ0 r , 2π 1 − r 2

(6.62)

2 ⎡ ⎤ ⎛ d02 d02 ⎞ 1 ⎢ 2 2 2⎥ 1 + r0 − − ⎜ 1 + r0 − r= ⎟ − 4r0 ⎥ . ⎜ 2r0 ⎢ 4 4 ⎟⎠ ⎝ ⎣⎢ ⎦⎥

Here, r0 is the radius of orbit of the Rankine vortex center, and in our case we take the mean radius of the main vortex orbit rm as the above value. The calculated values of velocity differ from the values calculated by Eq. (6.62) by no more than 2.2%. Let us consider the expression for the frequency of Rankine vortex precession Ω=

Γ0 1 . 2π 1 − r 2

(6.63)

At low values of rm the value r is also small, and hence, according to (6.63), dependency of frequency Ω on rm is weak. Therefore, it is appropriate to take rm, but not Ω, as the value, which characterizes the forming flow. Then, we study the dependency of rm on initial parameters of the problem. According to calculations with different assigned values of perturbation a, orbit radius rm grows linearly with a rise of a from 0 to 0.05 (Fig. 6.26; other parameters are the same as in the previous example). With the further increase in perturbation amplitude, the growth of rm slows and reaches its maximum at a = 0.16. A decrease in rm at high perturbations (a > 0.16) can be explained by the active interaction of disturbing vortices with possible merging of vortex particles of different-sign circulation, which ddddddddddddd

372

6 Dynamics of two-dimensional vortex structures

Fig. 6.26. Dependency of the average radius of the precessing vortex orbit on perturbation amplitude. Calculation by Kuibin (1993)

Fig. 6.27. Dependency of the average radius of the precessing vortex orbit on the initial radius of the disturbing vortex position. Calculation by Kuibin (1993)

decreases their influence on other particles. In calculations where a = 0.25, the flow did not reach the steady state: the trajectory of the main vortex motion became irregular. Dependency of rm on the initial position of disturbing vortices also has not-monotonous character (Fig. 6.27; parameters: a = 0.1; M = 8; χ = 0; n0 = 16; nk = 4; d0 = dk = 0.2). If the radius of circumference ρ (where the centers of vortices which model perturbations of the vorticity field were located at the initial moment) is small, these vortices mix quickly and slightly affect the main vortex. If they are located near the tube walls (ρ > 0.7), there cannot be a considerable shift of the main vortex from the center due to the law of angular momentum conservation: there is no allowance for the displacement of vortices with negative circulation. An increase in the number of disturbing vortices M leads to a growth of rm, and at high M, the radius of the main vortex orbit approaches asymptote rm → 0.32, which corresponds to continuous distribution of perturbatttt

Fig. 6.28. Dependency of the average radius of the precessing vortex orbit on the number of vortices modeling the disturbing vortex field. Calculation by Kuibin (1993)

6.4 Motion of vortices in cylindrical tubes

373

tion over the circumference coordinate (see Fig. 6.28; a = 0.1; ρ = 0.6; χ = 0; n0 = 16; nk = 4; d0 = dk = 0.2). The effect of other parameters (χ, dk) on rm is insignificant. With a rise of fragmentation number nk, calculation accuracy increases but value rm remains almost the same. In the numerical experiments described above, the period of vorticity perturbation over the angle was assumed to be equal to 2π (the azimuthal wave number is m = 1). Since real perturbations do not usually have sinusoidal distribution over the circumference coordinate, we consider similar perturbations with high wave numbers. In this case, circulation of vortices modeling vorticity perturbations, are determined by formula Γk = aΓ0 sin(2πkm/M + χ),

k = 1, 2, …, M,

It is obvious that for an even m, there will be zero effect on the central vortex because vortices with similar intensity and sign are of one diameter, but at opposite sides of the center. At odd m, the effect decreases with a rise of m. Thus, if for m = 1 (a = 0.1; ρ = 0.6; M = 8) rm = 0.25, for m = 3, we obtain rm = 0.09. Another interesting type of perturbation is observed when the total vorticity from each blade equals zero. We can model these perturbations using a set of vortices positioned as two concentric circumferences of different radii, for when vortices are of the same intensity but opposite sign, are at every radial beam. According to calculations, in this case, the axial vortex leaves the central position and begins precessing. In conclusion, we consider pressure pulsation on the tube wall. As was already discussed, since calculated frequency of the main vortex precession is close to the precession frequency of a single Rankine vortex in a circular domain, probable pulsation characteristics of the flow will be close to the case of a single vortex (at least for precession frequency). To be certain that this is true, we consider the velocity induced by a traveling vortex

Fig. 6.29. Pulsation of the induced velocity on the channel wall (Vz0) and in the model of the precessing vortex with orbit radius rm(V0) (see (6.64)). Calculation by Kuibin (1993)

374

6 Dynamics of two-dimensional vortex structures

at a circle boundary: at point z0 = 1. We will use Eq. (6.60), neglecting the exponential terms and assuming N = 1, z1 = rm exp(iΩt). The velocity at point z0 is tangential to the wall, and its value is V0 =

1 − rm2 Γ0 . 2π 1 + rm2 − 2rm cos Ωt

(6.64)

Comparison of dependency (6.64) with the velocities at point z0 during computations is shown in Fig. 6.29. The obtained result means that the pressure pulsation on the tube wall in calculations will be close to the pulsation in the case of one precessing vortex. 6.4.3 Motion of a helical vortex in a tube Above, we have considered the model for a flow with a precessing rectilinear vortex in a tube. In fact, the vortex, which loses the axial position, takes the shape of a helix. The helix pitch depends on the degree of flow swirling. To describe the motion of a helical vortex in a suction tube of a water turbine, Fanelli (1989) used analytical results from the work of Hashimoto (1971), where the field of velocity, induced by a helical vortex filament in a cylindrical tube, was determined in a form similar to (2.69). However, Hashimoto made some errors in deriving the velocity field and in the subsequent description of the tube wall effect on the velocity of helical vortex motion. Moreover, Hashimoto (1971), Fanelli (1989) did not consider the self-induced velocity of a helical vortex. The most complete and correct investigation of helical vortex motion in a tube with analysis of different factor effects is presented in the paper by Kuibin and Okulov (1998). As in the case for an unbounded flow (see Section 5.2), firstly, we should determine the bi-normal component of the vortex motion velocity. The effect of cylindrical tube walls on the vortex velocity can be determined using the value of velocity, induced by the “reflected vortex” at the point where the vortex center lies. In Eqs. (5.36), (5.44) this corresponds to consideration of the additional term H* in value C ∗

H =

(

4 1 + τ2 τ

2

)

32

(S

∗ χ

+ Rχ∗

)

r =a χ= 0

.

6.4 Motion of vortices in cylindrical tubes

375

Values Sχ∗ and Rχ∗ represent the terms of Eqs. (2.74), (2.75) with a* or R. According to the estimate of values in a wide range of parameters, contribution of Rχ* does not exceed 1.5 %, and it can be neglected. Therefore, H* =

⎡ 1 ⎢ 9η k= ⎢ 12 1 + η2 ⎢⎣

(

(

2 1 + τ2

)

32

τ

)

12

−

⎡ a2 R2 − a 2 ⎤ − log k ⎢ 2 ⎥, 2 R2 ⎦⎥ ⎣⎢ R − a

7η

3

(1 + η )

2 32

−

3τ

(6.65) ⎤ l τ ⎥ , η= . + 2 32⎥ R 1+ τ ⎥⎦ 3

(1 + τ )

2 12

(

)

Comparison of value C , calculated with (Fig. 6.30, curve 1) and without consideration of H* at different values of a/R, demonstrates a significant effect of the tube walls. With an increase in τ this effect becomes stronger and at high l/a and l/R we obtain asymptote

C → − 2l2/(R2 – a2).

(6.66)

Upon description of the vortex motion it is necessary to consider additionally the contribution of a potential flow – uniform translation along axis z with velocity u0. The contribution into C and correspondingly into the bi-normal velocity, using parameter β = u02πl/Γ, is written as: 2(β − 1)(1 + τ2)1/2/τ ,

(6.67)

and at high τ this leads to a parallel vertical shift of the curves in Fig. 6.30 by 2(β – 1), and at low τ this contribution becomes larger: 2(β – 1)/τ. Summarizing contributions of all factors influencing the motion of a helical vortex in a cylindrical tube, we can write the resulting formula for the bi-normal velocity l 1 + 1.455τ + 1.723τ2 + 0.711τ3 + 0.616τ4 − uˆb = log + ε τ + 0.486τ2 + 1.176τ3 + τ4

(

2 ⎡1 π2 ε 2 u 2 π 2 ε 2 w2 ⎤ 2 1 + τ ⎢ ⎥ − −2 +4 + τ Γ2 Γ 2 ⎥⎦ ⎢⎣ 2

−

(

2 1 + τ2 τ

)

32

)

12

(β − 1) −

⎛ a2 R2 − a 2 ⎞ log k − ⎜⎜ 2 ⎟. 2 R2 ⎠⎟ ⎝R −a

(6.68)

376

6 Dynamics of two-dimensional vortex structures

Fig. 6.30. Dependency of CKO on τ for a helical vortex in a cylindrical tube at different values of a/R (Alekseenko et al. 1999)

Dependency of uˆ b on τ for the fixed values of β = 0 and core radius (with uniform vorticity distribution) ε/R = 0.5 at different values of a/R is shown in Fig. 6.31. According to analysis of Eq. (6.68) and the diagram, we can conclude that stationary (immobile) helical vortex structures may exist, when the self-induced velocity of the helical vortex caused by its curvature and swirling is completely suppressed by velocity, as induced by the wall and axial velocity. In Fig. 6.31, points where curves cross with abscissa, i.e., uˆ b = 0, correspond to the stationary vortices. It follows from equation (6.68) that for any vortex we can select the value of β0, which corresponds to the immobile vortex. Dependency of β0 on τ at different values of a/R is shown in Fig. 6.32. We should note that at low τ for all values of a/R, β0 → 0.5. At high τ in accordance with (6.66), curves tend to asymptote τ2/(R2/a2 – 1). When studying the flow in a tube, distributions of characteristics are usually considered over the tube cross-section. For the 2-D flow, the key information is obtained from the distributions of radial ur and circumference uθ velocity components, and the vortex precession is characterized by its angular velocity. Let us find the angular velocity for a helical vortex. From the connection between the bi-normal velocity, the axial and circumference ones (5.31), as well as the condition of helical symmetry (1.62), we notate

6.4 Motion of vortices in cylindrical tubes

u 1 Γ Ω= θ = 2 a 4πa 1 + τ2

⎡ τ ⎢ ⎢ 2β − uˆb 1 + τ2 ⎢⎣

(

⎤ ⎥ . 12⎥ ⎥⎦

)

377

(6.69)

Fig. 6.31. The τ - dependency of the bi-normal velocity of the helical vortex in a cylindrical tube at different values of a/R. ε/R = 0.05, β = 0

Fig. 6.32. The τ - dependency of parameter β0, corresponding to immobile helical vortices at different values of a/R

Taking into account Eq. (6.68), we deduce that value β (or u0) does not affect the angular vortex velocity, and this corresponds to the formula for the circumference velocity component (2.69). However, we should note that if we consider the transition velocity of the vortex core trail in a certain cross-section of the tube, its value does not

378

6 Dynamics of two-dimensional vortex structures

coincide with Ω. This is due to the fact that the helical vortex has a nonzero velocity of its own motion along the tube axis. Indeed, since the vortex has a helical structure, its pure displacement along axis Z at Ω = 0 provides rotation of the vortex cross-section image in some certain plane, normal to the tube axis, i.e., during time ∆t the vortex trail moves by uθ∆t in the plane due to the circumference velocity and by uz(a/l)∆t due to the axial velocity and helical shape. As a result, we obtain that the angular velocity of the vortex trail in the tube cross-section is

Ω1 = Ω

τ2 1+ τ

2

−

u0 uˆb Γ =− 2 l 4πa τ 1 + τ2

(

)

12

.

(6.70)

At a high pitch of the helix, Ω and Ω1 coincide. At low τ the contribution of u0 may be predominant.

7 Experimental observation of concentrated vortices in vortex apparatus

This Chapter presents experimental results and its main aim is to demonstrate the existence and features of real elongate vortices. This is a way to obtain justification and proof for the key conclusions of the theoretical models developed in previous chapters. Most of the experimental results belong to the authors of this book and their colleagues.

7.1 Experiment methods 7.1.1 Experiment equipment As demonstrated in the Introduction, concentrated vortices may be observed under quite different conditions. As for experimentation, a vortex chamber with controllable geometric and regime parameters is a convenient experiment facility. Any kind of vortex chamber comprises a working volume and a swirling device. Often the flow swirling is achieved directly in the working volume. Among the diversity of swirling devices we focus on the most typical designs; some of these designs were used in the experiments mentioned in this Chapter. There are two classes of device – tangential (Fig. 7.1a–e, 7.2) and axial (Fig. 7.1f) vortex generators, combined schemes are also possible. For the tangential design, only the circumferential component of velocity is imparted to the flow at the vortex chamber inlet. The velocity vector is directed tangentially to the cylindrical surface of a channel or to a conditional circle with diameter d (Fig. 7.2). A widespread kind of tangential swirler is a cylindrical channel with a tangential inlet fitting (Fig. 7.1a). Variants for the inlet fitting are as follows: slot (Fig. 7.1b (Escudier 1988)), round or rectangular nozzle with several slits (Fig. 7.1c (Kutateladze et al. 1987)) or several nozzles (Fig. 7.2 (Alekseenko and Shtork 1992)). A more perfect design is a swirler with a spiral guide – snail (Fig. 7.1d), which is often used in the Ranque – Hilsch vortex tube (Shtym 1985; Piralishvili et al. 2000).

380

7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.1. Design of flow swirling devices

In a tangential vane swirler, the circumferential momentum to the flow is conveyed by a cascade of vanes with variable angle relative to the radial direction from the chamber center. The profiled insertion provides a flow turn from the radial to the axial direction. Fig. 7.1e shows a vane-type swirler with a profiled insert. Such a swirler was used in the experiments of Sarpkaya (1971), Faler and Leibovich (1977, 1978), Garg and Leibovich (1979), and others. A similar kind of swirler (without an insert) was used by Guarga et al. (1985). The vane swirler of axial type is a cascade of flat (Brüker and Althaus 1992) or curved blades (Khalatov 1989) oriented in an angle to the mainstream in the working section. The swirling of the flow in the working section can be also achieved with the help of a screw-type (Fig. 7.1f) or band swirler. A vane swirler of axial type is used often in the measurement equipment, for instance, in a vortex flow meter (Zalmanzon 1973). A diagram of the vortex chamber employed in the papers of Alekseenko and Shtork (1992), Shtork (1994), Alekseenko et al. (1999) is shown in pppppppp

7.1 Experiment methods

381

Fig. 7.2. Diagram for a rectangular hydraulic chamber of tangential type

Fig. 7.2. The results of these experimental researches are the basis of this Chapter. The working section is made of plexiglass; it is a vertical chamber of square cross-section and measures 188×188×600 mm. The direct-flow rectangular nozzles with an outlet size of 14×23 mm are arranged in three rows and united into corner assemblies. The nozzle axis is shifted relative to the chamber corner by 15 mm. The tilt angle γ for every nozzle is regulated in the horizontal plane within the limits 0 ÷ 35°. The swirling flow is controlled by directing the nozzle axes tangentially to a conditional circle with diameter d and with the center in the chamber axis. This corresponds to the tangential method of flow swirling. It is known (Shtym 1985; Escudier et al. 1980; Gupta et al. 1984), that the outlet conditions and those at the blanked-off end face are essential for the flow structure. In particular, a concentrated vortex in the form of a vortex filament is observed in a chamber with an outlet diaphragm. In the tangential model, the geometry is changed through the variation of nozzle orientation and the bottom shape, through diaphragm provision in the outlet and through a shift of the outlet. Although this experimental vortex chamber cannot be a classical object because of its square cross-section, nevertheless, it is a useful setup for studying various vortex structures. Furthermore, tests with cylindrical inserts prove that the flow patterns for concentrated vortices in the near-axis zone are almost the same for cylindrical and rectangular configurations.

382

7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.3. Diagram of the electro-diffusion method for measuring local velocity

Water is used as the working liquid. The flow visualization is made with small air bubbles. The static pressure distributions on the walls and inside the flow, as well as the velocity field, are measured during the experiments. The pressure intakes on the wall are apertures with diameter 0.4 mm, and we use standard pressure tubes of diameter 1.4 mm for measurements in the flow. The pressure is recorded using tensor-resistance transducers. The profiles for tangential and axial components of velocity are measured using three different techniques: standard pressure tubes of 1.4 mm diameter (far from the vortex axis); the method of stroboscopic visualization for particles; the electro-diffusion method. The essence of the stroboscopic visualization method is the registration of tracer particles while the flow is illuminated with short light pulses (Shtork 1994). The tracer particles are small air bubbles of 20–30 µm; the light source is a pulse-operated flash-lamp. The electronic stroboscope supplies a series of electrical pulses with a regulated interval and controllable number of pulses (from two to six) to a flash-lamp. The light pulse duration is 20 µs, and the accuracy for intervals between pulses is 0.5 µs. The velocity measurement principle is based upon the measuring of shift for tiny particles that follow the stream motion during a certain time period given by the stroboscope. The principle of the electro-diffusion method (Alekseenko and Markovich 1993) is the measurement of the redox reaction rate in an electrochemical cell, which consists of a probe – cathode , anode , the special electrolyte and a measurement circuit (Fig. 7.3). The cathode is a platinum wire of 50 µm thickness welded into a glass capillary. A negative potential is applied to the cathode (relative to the anode). The anode is a holder made of stainless steel with a surface area much larger than that of

7.1 Experiment methods

383

the cathode; so the total rate of electrochemical reaction is determined by the cathode. The electrolyte is 0.005% solution of K3Fe(CN)6 and K4Fe(CN)6 with an addition of Na2CO3 stock solution with the concentration of 0.1 M. This electrolyte does not participate in the electrochemical reaction; it is required only for the suppression of the migration current driven by the electric field. At a certain potential difference, the diffusion regime appears when the current reaches its maximum and it is governed by the hydrodynamic situation near the cathode. The dependency of diffusion current on the liquid velocity u in the vicinity of the probe element usually takes the form of I = A u + B, where A, B are the calibration constants. The probe is calibrated directly in the working section using a calibration nozzle. The wedge-shaped electro-diffusion probe used for measuring the 2-D velocity field is a pair of platinum electrodes welded into a glass capillary (see Fig. 7.3). The external diameter of the active element of the probe is 1 mm. The probe signals are supplied to the inlets of the electro-diffusion transducers (a development of the Institute of Thermophysics SB RAS) that comprise a DC amplifier and a DC source. The outlet signals are sent to the analog-digital converter for computer processing. Taking into account the calibration characteristics, one can obtain the total velocity from the sum of signals taken from the electrodes; the difference between the signals gives us the direction of the velocity vector. 7.1.2 Parameters of a swirling flow As for any viscous flow, the main regime parameter for a flow in a vortex chamber is the Reynolds number Re Re = Q/(Σ⋅ν), where Q is the liquid flow rate; Σ is the cross-sectional area of the chamber; ν is the kinematic viscosity. The degree of flow swirling in a vortex chamber is described by an additional parameter – swirl parameter S. There are different forms of this definition. Quite simple expressions are the ratio of the maximum tangential to the maximum axial velocity or the ratio of the average tangential to the average axial velocity. A widely recognized definition is given through the formula (Gupta et al. 1984) S = Fmm/Fm L, where the quantity

(7.1)

384

7 Experimental observation of concentrated vortices in vortex apparatus

Fmm =

∫ (ρvw + ρv′w′) rdΣ Σ

is the angular momentum flux in the axial direction and it takes into account the contribution of the z-θ component of the turbulent stress; the quantity

Fm =

∫ ( ρw

2

)

+ ρw′2 + ( p − p∞ ) dΣ

Σ

is the momentum flux in the axial direction: it takes into account the contribution of normal turbulent stress and pressure; w and v are the axial and tangential components of velocity; r is a radial coordinate; ρ is the liquid density; L is the typical size (radius for the case of a cylindrical chamber); p is the overpressure. Usually the pressure and turbulent pulsations are neglected and the swirl parameter is calculated by formula (7.1) using simplified definitions for the fluxes

∫

(7.2)

∫

(7.3)

Fmm = ρvwrdΣ , Σ

Fm = ρw2 dΣ . Σ

The exact computation of the swirl parameter by formulae (7.1) – (7.3) is practically impossible because the velocity fields are usually unknown a priori. However, it is possible to estimate the swirl parameters through the geometrical parameters of the chamber. Let us make this kind of estimate for the tangential chamber depicted in Fig. 7.2. The axial component of angular momentum flux may be approximately written as

Fmm =

n

∑ (GnVnd 2)i = GVnd 2 = G2d 2ρΣn . i=1

Here i is the nozzle ordinal number; N is the total number of nozzles; G is the mass flow rate; Gn is the mass flow rate through a nozzle; Vn is the average velocity at the nozzle outlet; Σn is the total cross-sectional area for all nozzles. We assume that all the nozzles are identical. The momentum flux is estimated as Fm = GV = G2/ρΣ, where V is the mean velocity in the chamber cross-section. It follows from (7.1) that

7.1 Experiment methods

S=

385

1 Σ d . 2 Σn L

Assuming that Σ = m2, L = m/2, Σn = nσ, where m is the width of the chamber, σ is the nozzle cross-sectional area, we obtain the ultimate estimate for the swirl parameter S=

md . nσ

(7.4)

We see that this is a purely geometrical ratio, so it is usually called a design swirl parameter. For a described design, the diameter of the conditional circle d(m) is related to the nozzle tilt angle γ (see Fig. 7.2) through the formula d = 0.246 sin γ. If all nozzles are open and they are directed at the same angle γ, then the swirl parameter is equal to S = 12 sin γ. Feikema et al. (1990) performed direct experimental comparison between the design swirl parameter (7.4) and the exact definition (7.1) for the case of an axisymmetrical channel. Despite a large quantitative difference, there is a good correlation between these two definitions. However, further numerous studies of swirling flows, including research by Alekseenko and Shtork (1992), Alekseenko et al. (1994), Okulov and Martemianov (2001), Martemianov and Okulov (2002, 2004) revealed that the Reynolds number and swirl parameter are not the unique characteristics of the flow regime. In particular, the outlet conditions and the conditions at the blanked-off end face of the chamber play a significant role. For example, for identical values of flow rate Q and swirl parameter S but different boundary conditions, quite different structures of flow may be observed in the vortex chamber depicted in Fig. 7.2. These are vortices: a) precessing (flat bottom; open outlet of the chamber); b) columnar (flat bottom; central outlet orifice); c) helical (inclined bottom or a shifted outlet orifice); d) two entangled vortices (bottom with two flat slopes; central outlet orifice). These structures will be described in detail in the following sections. Two important facts follow from analysis of the listed regimes. Firstly, number Re and swirl parameter S do not unambiguously denote the flow structure. The new parameters introduced in Section 7.2 can be additional characteristics. Secondly, the observed vortical structures have helical symmetry, that is, a spatial period along axis z exists. This fact is taken as a primary assumption in the theoretical models outlined in Sections 1.5, 2.6.2, 3.3.3, 3.3.4, 3.3.6.

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7 Experimental observation of concentrated vortices in vortex apparatus

7.2 Helical symmetry of vortex flows As mentioned in Section 7.1.2, the swirl parameter S at a given Reynolds number (or flow rate Q) does not serve as an unambiguous characteristic for the vortical flow structure in the vortex chamber. For a flow with helical symmetry, the alternative approach is the introduction of two new parameters. These parameters follow from the theoretical analysis given previously (Section 1.5) and they are the helix pitch h = 2πl (or simply l) and the velocity of uniform flow u0. These parameters enter into relationship (1.66) between axial and tangential components of velocity in twodimensional helical flows. In the particular situation of zero axial velocity u0, the helix pitch l is determined unambiguously through the integral swirl parameter S. Indeed, at u0 = 0 from (1.66) by multiplication with w and integration over the flow cross-section, we obtain in view of definition (7.1), the relationship: l = −Fmm /Fm ≡ −LS,

∫

∫

where Fmm = ρ v wr 2 drdϕ, Fm = ρ w2 rdrdϕ are the axial components of momentum flux and angular momentum flux, correspondingly; L is the efficient radius of the chamber. If the axial velocity at the flow symmetry axis is not zero (as often happens experimentally), then we obtain from (1.66) that l = −Fmm /(Fm − u0G) = −LS/(1 − u0G/Fm),

(7.5)

where G is the mass flow rate. Hence, for an arbitrary swirled flow with helical symmetry the helix pitch depends not only on swirl parameter S, but also on the velocity value at the axis u0 (it determines a uniform flow). Note that in real flows, the condition of helical symmetry is not satisfied for the entire length of axis z. Indeed, the flow structure undergoes substantial changes with distance from the swirling device – from vortex breakdown to a complete decay of the swirling. The assumption about helical symmetry in the developed model means an identical flow structure with period 2πl along axis z for the entire infinite span. Obviously, this approach does not work for the entire flow zone. However, several researchers have noted (Leibovich 1984; Escudier 1988; and others) that swirling flows have rather lengthy zones (up to several scale sizes) where the velocity profile is almost unchanged. It would be most reasonable to apply locally the hypothesis about helical symmetry (the two-dimensional model developed above). This requires that condition (1.66) is satisfied for a real vortex flow. In that case the parameters l and u0, constituting (1.66), are naturally taken as the new characteristics of swirl flows.

7.2 Helical symmetry of vortex flows

387

For validation of the hypothesis about local helical symmetry in swirling flows, we have to compare the profiles of axial velocity (measured for fixed cross-sections) with the values of w, calculated from the measured tangential velocity through formula (1.66). This calculation requires parameters l and u0. If the local characteristics of flow u0 for a given crosssection are found through direct experimental measurements, the parameter l can be found directly from (7.5) but it is not always possible to calculate l with the desired accuracy. This is because the computation of momentum flux and angular momentum flux requires detailed knowledge of the velocity field over the entire cross-section of the working area (tube). The problem of finding l can be simplified due to a linear relationship between the velocity components in (1.66). After averaging of (1.66) we obtain a simpler formula l = 〈rv〉/(u0 − 〈w〉),

(7.6)

where the brackets represent averaging. The procedure of averaging in (7.6) may be done for the entire tube cross-section or for a part of the cross-section, where helical symmetry is assumed. In particular, treatment of the experimental data of Alekseenko et al. (1994) excluded the zone near the tube walls, where viscous effects occur. Testing of averaging with weight functions demonstrated that for few measurement points on the radial coordinate, it is better to use the averaging over the radius with the weight function 1/r. In many experiments, the flow velocity at the axis was not determined or determined with a low accuracy. In such cases, the parameters u0 and l were found through minimization of the quadratic form 〈(lv − lu0 + rv)2〉. Substituting u0 (determined in (7.6)) into the form, we obtain the formula for l l = [〈rv〉〈w〉 − 〈rvw〉]/[〈w2〉 − 〈w〉2].

(7.7)

When we have determined l by (7.7), we find that u0 = 〈w〉 + 〈rv〉/l. The checking of local helical symmetry for real swirl flows was performed in the papers of Alekseenko et al. (1994, 1999) for different types of swirlers, flow regimes and diagnostic methods (described previously). The parameters of helical symmetry u0 and l were determined by one of the three methods described previously. Figure 7.4 shows the comparison of the experimental value of axial velocity with the quantity calculated from the measured tangential component by formula w = u0 − rv/l. The measured values of w are presented with bold points and calculated ones are marked with empty points.

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7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.4. Testing of local helical symmetry in swirl flows with different types of swirlers (Alekseenko et al. 1999*). 1 – measurements of axial velocity w; 2 – values of w calculated by formula w = u0 – rv/l. a – slot swirler (Escudier 1988), type A, cross-section iii: V – velocity at the inlet slot; b – rotating container with central suction (Maxworthy et al. 1985): Ω = 1.51 s–1, Q = 180 l/h, Vm – maximum tangential velocity, r0 – radius, where v = Vm; c, d – vortex chamber with a nozzle swirler (Shtork 1994) (see Fig. 7.2); c – diaphragmatic outlet: de = 70 mm, ze = 430 mm, Re = 2.8⋅104, S = 3, cross-section z = 323 mm; d – chamber without a diaphragm: Re = 3.2⋅104, S = 1, z = 385 mm; e – vane swirler (Garg and Leibovich 1979): Re = 11480, swirl parameter Ω = 0.79, z = 193 mm, R – chamber radius; a – c – steady flow; d, e – swirling flow with vortex core precession

7.2 Helical symmetry of vortex flows

389

Fig. 7.5. Testing of local helical symmetry through comparison of vortex shape with a helix (Alekseenko et al. 1999*): a – left-handed vortex; b – double helix; 1 – projection of vortex axis on a vertical plane; 2 – sinusoids with parameters: h = 355 mm (a), 238 mm (b)

Figure 7.4d,e presents the data for an unsteady swirl flow with distinct expression of the phenomenon of vortex core precession. For this case, the determination of parameters u0 and l by formulae (7.6) and (7.7) includes averaging over time. Data analysis confirms that helical symmetry is present almost for the entire flow zone, except the near-wall zone. In this area, the viscosity effects become significant: this is revealed by the formation of a boundary layer and the near-wall Görtler vortices. A small difference for the main flow zone remains within the measuring accuracy limits. Similar conclusions follow from data analysis obtained by Faler and Leibovich (1977), Guarga et al. (1985), Kutateladze et al. (1987). Another method for the testing of helical symmetry is to check the quality of a helical shape for vortex structure produced in swirl flows. Since there is no data available in the literature (see the Introduction), this kind of test was conducted for two flow regimes with explicit helical structures (see Section 7.5). The idea of checking, is that the projection of a helical curve on a plane must be a sinusoid. For finding projections of real helical structures, computer processing was employed for instant video imaging of an air filament that shows the vortex axis. Fig. 7.5 shows comparison of a sinusoid

390

7 Experimental observation of concentrated vortices in vortex apparatus

with the average position of the axis for single (a) and double (b) vortices. The coincidence is almost perfect. The sinusoid parameters give us the pitch of helical symmetry: h = 335 (a) and 238 mm (b). This analysis demonstrates the presence of helical symmetry in swirl flows for all types of swirlers and allowance for using the ideal fluid model for swirl flow description.

7.3 Concentrated vortex with a rectilinear axis 7.3.1 Generation of concentrated vortices

As mentioned before, the main aim of Chapter 7 is to demonstrate the existence of elongate vortices and describe their key properties. Naturally, the primary task of an experiment is the generation of concentrated vortices belonging to a canonical type. In the Introduction (Table I.1 and Fig. I.1) we gave miscellaneous examples of concentrated vortices. However, in most cases these vortices cannot be used as objects convenient for detailed experimental description. Among the reasons for this are the: spontaneous generation, temporal and spatial instability, and lack of control over the parameters. Vortex chambers and tubes with flow swirling as well as chambers with rotating lids are the most convenient devices for the generation and observation of concentrated vortices (see Figs. 7.1, 7.55). In vortex flow-through chambers the concentrated vortex originates due due to the presence of a diaphragm at the outlet (Shtym 1985; Escudier et al. 1980). Unless otherwise stated, we define concentrated vortices as elongate concentrated vortices of the filament type. We apply the term vortex filament to the class of experimentally observed vortices with a width (core diameter) much smaller than the chamber size or vortex length. Let us consider in detail the typical flow regimes, methods of control, and process of filament generation for the tangential hydraulic chamber depicted in Fig. 7.2 (Alekseenko and Shtork 1992; Alekseenko et al. 1999). The key parameters for the flow regime are the following: Reynolds number Re, design swirl parameter S, and other geometrical parameters, like the diameter de of the outlet orifice of the diaphragm, and its position ze relative to the chamber bottom. The number Re may vary through a change in the liquid flow rate. Usually the values of Re for vortex chambers are rather high, so most of the phenomena and quantitative characteristics (in dimensionless coordinates) are self-similar relative to number Re, that is to say, Reynolds number is not a governing parameter.

7.3 Concentrated vortex with a rectilinear axis

391

In the same time the swirl parameter S (it varies due to control of the nozzle number and their tilt angle γ or diameter of the conditional circle d; see Fig. 7.2) has a strong impact on the flow regime. Finally, control over the flow symmetry is maintained by a shift of the outlet orifice and through a change in the shape and inclination of the chamber bottom. For a completely open outlet, the swirling flow in a tangential chamber has a complex structure. The velocity maximum is near the lateral walls of the chamber, and recirculation flow occurs in the near-axis zone, and one (or a few) indistinct concentrated vortex is formed at the zone boundary; this vortex undergoes precession around the geometric axis of the chamber. This flow regime is described in Section 7.4. The diaphragm in the outlet stabilizes the flow and shifts the velocity maximum towards the chamber axis. Almost complete suppression of a vortex core precession occurs for a diaphragm with relative orifice de/m < 0.85 (for the given chamber). This flow restructuring means localization of vorticity at the chamber axis; it is stronger if the outlet orifice diameter is smaller (to a certain degree), and it generates a vortex filament type structure. Such a regime is described in this Section. If for the regime with a vortex filament the asymmetric boundary conditions are provided, for example, by means of shifting the outlet orifice, the filament becomes deformed and assumes a spiral shape (stationary in time and space). Depending on how we create this asymmetry, it is possible to observe left- and right-handed spirals, a double helix and a helix with alternating helical symmetry (see Section 7.5). If the swirl parameter is increased for a steady filament regime, the flow eventually loses its stability. This leads to formation of waves or perturbations in the form of a vortex breakdown. The same phenomena occur after introducing artificial perturbations into the flow (see Section 7.6). In this Section we consider in detail the case of vortex filament flow. The flow visualization in this regime and its diagram are presented in Fig. 7.6a,b. The flow visualization is achieved through small air bubbles with vertical illumination by a light “knife”. A drastic increase in the local circumferential velocity of liquid in the chamber center (see Fig. 7.13) is accompanied by a drastic decrease in pressure (see Fig. 7.14). This effect creates a cavitation zone at the vortex axis: this is a thin air-filled continuous filament with a constant thickness of up to 0.1 mm (Fig. 7.6a). This air filament is formed by tiny air bubbles (they are added to the liquid for flow visualization). The filament spreads from the chamber bottom to the diaphragm opening and beyond. As the gas content increases, the cavity thickness grows up to 5–10 mm (Fig. 7.7a), and the air filament becomes non-uniform – it becomes thinner at the bottom and thicker at the outlet, but the flow remains stable. This structure may be considered as a hollow vortex.

392

7 Experimental observation of concentrated vortices in vortex apparatus a

b

Fig. 7.6. Visualization (a) and diagram (b) of a flow with generation of a rectilinear concentrated vortex (Alekseenko et al. 1999*): de = 70 mm, ze = 560 mm, Re = 104, S = 2.9. The light curve in the photograph is the air filament for vortex axis visualization

When one decreases the air supply and liquid flow rate, the air filament becomes very thin; then due to capillary forces it breaks into separate small bubbles which are useful as a tracer for flow visualization (Fig. 7.7b). Thus, generation of an air cavity is an efficient way to visualize the axis of a concentrated vortex in a liquid. Figure 7.8a presents flow visualization in the chamber cross-section. One can see that the particle trajectories are almost circular; this testifies to the axial symmetry of the swirl flow in the near-axis zone even for a square cross-sectional chamber. Flow visualization in the lower zone provides important information (Fig. 7.8b). It follows from this picture that the particles move in spiral trajectories towards the center. This behavior is related to the formation of an end-wall boundary layer due to vortex interaction with the plane; this explains the localization of vorticity in the central part of the end-wall. Indeed, for the core of a two-dimensional vortex, the centrifugal forces are compensated by the radial pressure gradient

v2 ∂p ρ = . ∂r r

7.3 Concentrated vortex with a rectilinear axis

a

393

b

Fig. 7.7. Examples of visualization of concentrated vortices: a – vortex filament at the chamber bottom with a large supply of air for visualization, Re = 2.7⋅104, S = 3, de = 70 mm; b – magnified image of the vortex filament with a small air supply and a very low liquid flow rate a

b

Fig. 7.8. Flow visualization in horizontal cross-section of a tangential chamber with a diaphragmed outlet. S = 3, de = 70 mm: a – z = 90 mm, Re = 2.7⋅104; b – z < 5 mm (at the bottom), Re = 104

394

7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.9. Vortex breakdown at the bottom of tangential chamber (flat bottom under the lower array of nozzles; outlet diaphragm with a central orifice)

In a boundary layer, the radial gradient cannot be equalized by centrifugal effects due to viscous retardation of the liquid. This causes a radial motion of liquid directed towards the center. Due to conservation of mass and angular momentum, vorticity is localized; a vortex filament is generated with an axial flow-through along the axis. The vortex filament may lose its structure because of instability or due to the phenomenon of vortex breakdown. An example of vortex breakdown for a vortex localized at the bottom is shown in Fig. 7.9. The diaphragm in the outlet cross-section in the chamber allows us to sustain a vortex filament throughout the entire chamber, as demonstrated in Fig. 7.6. Thus, generation of concentrated vortices in a vortex chamber of the given type is possible under specific conditions at the chamber outlet and on the end wall (bottom). Obviously, a change in the bottom shape must cause a significant change in the flow structure inside the vortex chamber. This problem was considered in detail in several papers, mostly focused on applications (Goldshtik 1981; Kutateladze et al. 1987; Smulsky 1992). In this case, the problem of vortex-plane interaction has a dual interest: generation of a concentrated vortex and calculation of vortex filament characteristics through measurements in the near-bottom area. The latter problem is crucial for experimentation because often it is difficult to make direct measurement of filament parameters (especially for helical ones) inside the chamber. This aspect will be considered below. Although due to diaphargming of the chamber outlet it is possible to generate a concentrated vortex, the whole flow structure remains very complex. Figure 7.10 depicts visualization of a swirl flow for different vertical cross-sections. One can see from these images that the velocity vector exhibits non-monotonous alternation of its direction with a departure from fffffffff

7.3 Concentrated vortex with a rectilinear axis a

b

c

d

395

e

Fig. 7.10. Flow visualization in vertical crosssections of a tangential chamber at different distances r from the vortex chamber axis. Re = 1.6⋅104; S = 3; de = 70 mm; r = 12 (a), 30 (b), 54 (c), 84 mm (d); e – location of visualization cross-sections

the vortex axis. The vector inclination angle ϕ as a function of the distance r to the chamber center is shown in Fig. 7.11 for two cross-sections of the chamber. One can see that the distribution of angle ϕ varies drastically with the chamber height. However, these changes are determined mainly by restructuring of the axial velocity profiles (see Fig. 7.12). Experimental data on profiles of tangential and axial components of velocity is presented in Figs. 7.12, 7.13 (in dimensionless and dimensional coordinates, correspondingly). Figure 7.12 is a plot of velocity profiles for two cross-sections when the outlet orifice has a rather large diameter hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

396

7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.11. Inclination angle ϕ for velocity vector in a rectilinear vortex (relative to the horizon). S = 3, ze = 430 mm. 1 – z = 63 mm, de = 70 mm, Q = 5.17 l/s; 2 – correspondingly 323; 70; 5.17; 3 – 63; 100; 5.17; 4 – 323; 100; 5.17; 5 – 63; 100; 6.5; b – 323; 100; 6.5 a

b

Fig. 7.12. Dimensionless profiles of tangential ( ) and axial (b) velocity components in a vortex chamber (Alekseenko et al. 1999*). de = 100 mm, S = 3. 1 – formula (3.74); 2 – empirical formula by Escudier et al. (1982) (see formula (7.10)); 3 – z = 63 mm, Re = 2.7⋅104; 4 – correspondingly 323 mm, 2.8⋅104; 5 – 63 mm, 3.5⋅104; b – 323 mm, 3.5⋅104

7.3 Concentrated vortex with a rectilinear axis a

397

b

Fig. 7.13. Profiles of tangential ( ) and axial (b) velocity components in a vortex chamber with a small outlet orifice. de = 40 mm, ze = 430 mm, Re = 0.8⋅104, S = 3: 1 – experimental data; 2 – empirical formula (3.71)

de = 100 mm. These results were obtained with a pressure tube, since the radius of the vortex filament was rather large (rm ≈ 14 mm). Here, rm was determined from the position of maximum tangential velocity vm. It is important that the profile of tangential velocity does not change for the entire channel height, but the profile of the axial component undergoes significant changes (closer to the outlet orifice, the velocity maximum drifts towards the vortex axis). An illustrative feature of axial velocity distribution is the existence of a local minimum on the axis. However, whenever a concentrated vortex is generated, we can observe an intense flow along the vortex. As the outlet size decreases, the dip on the axial velocity profile fades, as one can see from the experimental data in Fig. 7.13 (for de = 40 mm). In this case the vortex radius is only 1.8 mm, and measurements were available only through the method of stroboscopic visualization. The velocity distribution is described perfectly by empirical relations (3.71) with the values of constants: K = 5, W1 = 0.3, W2 = 1.6, α = 0.3. Experimental data allows us to classify this rectilinear vortex as a vortex filament because the vortex core diameter (3–10 mm) is much smaller than the vortex length (430–560 mm) and chamber width (188 mm). To apply exact solutions (see Table 3.1) to the description of experimental profiles, we have to know all the key characteristics of the vortex: circulation Γ, core size ε, pitch of helical symmetry l and velocity at the chamber axis u0. They can be recalculated through the empirical constants K, W1, W2, α but we should be aware that finding these

398

7 Experimental observation of concentrated vortices in vortex apparatus

constants implies several assumptions. For example, Garg and Leibovich (1979) approximated the dependency (3.71) to experiments by the leastsquares method. Therefore, we prefer direct methods for the determination of model parameters. The calculation methods for l and u0 were described in Section 7.2, and below we review the methods for calculation of Γ and ε. For vortex models I and II, the parameter ε coincides with the radius of maximum tangential velocity rm, and for model III it is recalculated by formula ε = rm /1.12, i.e., this parameter is assumed to be known (if so the experiment gives us directly the radius rm). If we know the value of maximum velocity vm, circulation Γ for the corresponding models is determined by formulae Γ = 2πrm vm for I and II, Γ = 4πrm vm /0.715 for III. Therefore, the vortex models can be determined if in addition to parameters l and u0 (which are found by one of the methods formulated in Section 7.2) we know also the position and value of the maximum for the tangential component of velocity. Another approach to find and ε is through pressure profiles. Indeed, if we can determine the point r0.5, where the pressure variation equals half of the pressure drop between the values at the vortex axis and periphery then this point can be identified (see the theoretical models in 3.3.4) as point rm where the maximum of tangential velocity is observed. Alternatively, if we know the pressure drop ∆p0, we can find the value for circulation (see formulae for pressure in Section 3.3.4 and Table 3.1): Γ = 2πr0.5(η∆p0/ρ)1/2, where η = 1, 2 and 1/ln 2 for models I, II, and III, correspondingly. Generally, the characteristics of velocity field and pressure field in a vortex filament are very difficult to deduce because of the small size of the core, the three-dimensional nature of the flow in helical vortices and the unsteadiness (precession and turbulence). However, some parameters of vortex might be estimated from simple measurements, e.g., the bottom pressure (Kutateladze et al. 1987). Since the boundary layer thickness is usually small, it is natural to expect that distributions of static pressure on the chamber bottom and in the vortex core (near the bottom) must be identical. Surely, we have to exclude the possibility of vortex breakdown – this phenomenon assumes the existence of a significant pressure gradient along the axis (Escudier 1988). The example of vortex breakdown at the chamber bottom is shown in Fig. 7.9. If there is no vortex breakdown, we can restore the field of tangential velocity from pressure measurements and find other characteristics, in particular, the size of the vortex core.

7.3 Concentrated vortex with a rectilinear axis

399

Figure 7.14 compares radial distributions for static pressure in different cross-sections of a chamber and at the bottom. Obviously, a pressure tube is a poor instrument for measuring inside a swirling liquid flow near the vortex axis. However, analysis of the available experimental data in zones suitable for measuring, testifies that the pressure profile does not change with height and it is close to that at the bottom. A similar conclusion was drawn by Kutateladze et al. (1987) for the case of a vortex chamber with a diaphragm (with a different design). Hence, it is possible to perform analysis from bottom pressure data and to make estimates for the parameters of a vortex filament interacting with a flat bottom. Flow visualization in the bottom zone (Fig. 7.8b) testifies that the particles move in spiral trajectories towards the center. Therefore, in the bottom vicinity the radial component of velocity ur ≠ 0. Hence, the fact depicted in Fig. 7.14 is crucial. If according to (3.63) the pressure is independent of the axial component of velocity, the pressure must remain constant for different cross-sections (if the core size remains constant). However, the pressure profile is conservative even when the vortex interacts with the bottom (here, we observe a radial flow). Thus, the calculation of vortex filament parameters through measurements in the bottom zone is a justified procedure.

Fig. 7.14. Distribution of static pressure in different cross-sections of the chamber (Alekseenko et al. 1999*). ze = 430 mm, Re = 0.8⋅103, S = 3. de = 70 (a), 100 mm (b); points for z = 0 correspond to the pressure at the bottom

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7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.15. Generalized profiles of bottom pressure at different values of Re and S (Alekseenko et al. 1999*). de = 70 mm, ∆p0 = ⏐p0 – p∞⏐, where p0 and p∞ are the static pressures at the vortex axis and the periphery

In Fig. 7.15, we present data on bottom pressure at different Reynolds numbers Ren and swirl parameter S generalized in coordinates [∆p/∆p0, 2r/m]. The diameter of the outlet de is fixed and equal to 70 mm. Here ∆p0 = |p0 – p∞|, p0 is the pressure on the vortex axis; p∞ is the pressure at a distance (on the lateral wall of the channel). The Reynolds number Ren = Vndn/ν, Vn is the velocity at the nozzle outlet; dn is the hydraulic diameter of the nozzle. Figure 7.16a presents experimental data on the characteristic size rm (position of the maximum for tangential velocity) and r0.5 (position of half the pressure drop) as a function of outlet diameter de. The scale size rm is usually associated with the radius of the concentrated vortex. Here, the data coincide with the measurements of Escudier et al. (1980) for a vortex tube with a slot inlet. The most intriguing fact is the coincidence of values rm and r0.5. Hence, the vortex radius can be determined from the point where the pressure loss is half of the total pressure drop. The same conclusion follows from theoretical models. Attempts at theoretical calculation of rm were performed by Abramovich (1976) based on the principle of maximum flow rate and by Goldshtik (1981) based on the principle of kinetic energy flux maximum. Formulae were obtained for the relative vortex size 2rm/de written through a design parameter of the vortex chamber mc = 4Σn/(πded),

7.3 Concentrated vortex with a rectilinear axis

401

where Σn is the total area of the nozzle cross-sections; d is the diameter of the conditional circumference (see Fig. 7.2). Smulsky (1992) demonstrated that the simple empirical formula of Ovchinnikov and Nikolaev (1973) was more convenient:

2rm 0.35 = , de mc

(7.8)

which is valid at de/2Rc > 0.1 and mc > 0.1. Here, Rc is the radius of the vortex chamber. The measured values for maximum tangential velocity are plotted in Fig. 7.16b as a function of the outlet diameter de. Here V′ is the velocity component at the nozzle outlet, which is parallel to the tangential component in the chamber. A simple estimate for vm can be obtained on the basis of profile (3.70), with the assumption that it is valid up to the channel wall, i.e., r = Rc (Goldshtik 1981). In this case, the value for velocity on the wall V′ is calculated from the formula vm R = 0.715 c . V′ rm a

(7.9) b

Fig. 7.16. Characteristic scales for transversal sizes (a) and velocity (b) in a concentrated vortex (Alekseenko et al. 1999*): rm – coordinate for the maximum of tangential velocity vm, r0.5 – coordinate for the half-drop of pressure. 1 – data from Escudier et al. (1980), S = 1.8; 2 – Alekseenko et al. (1999), S = 3.1; 3 – measurements of 2r0.5/m (a) and calculation of velocity from measured pressure profiles using the Rankine model (b). Other symbols – experimental values of 2rm/m (a) and relative maximum of velocity (b); 4, 5 – calculations at S = 3.1 and 1.8, correspondingly, using Eqs. (7.8) (a) and (7.9) (b)

402

7 Experimental observation of concentrated vortices in vortex apparatus a

b

Fig. 7.17. Dependency of rarefaction on the vortex axis (relative to the periphery) on liquid flow rate Q, diameter of outlet de and nozzle tilt angle γ (Shtork 1994)

The vortex radius rm can be calculated, for instance, by the empirical formula (7.8) or by another method. For example, the calculated relationships in Fig.7.16 can be obtained on the basis of (7.9) using the model developed by Goldshtik (1981) for rm. For another approach, the maximum of tangential velocity can be found from rarefaction ∆p0 in the vortex center, e.g., by the Rankine model vm = ∆p0 ρ . This method for the calculation of vm gives a result which closely agrees with the direct measurements (see light circles in Fig. 7.16b). The rarefaction value is influenced by swirl parameter S (or the nozzle tilt angle γ), liquid flow rate Q and the diameter of the outlet de, as demonstrated in Fig. 7.17. The main conclusion of this analysis is that the key parameters (as used in the theoretical models of Section 3.3.4) can be found from a simple test – the study of vortex interaction with a plane. Moreover, the heuristic models such as (7.8) and (7.9) can be elaborated upon for the calculation of vortex parameters. As for the comparison of models with experiments, wide experience has been accumulated. The profiles (3.71) (we now know that this is equivalent to model III) were compared with the measurements of Leibovich (1984), Escudier (1988) and others. Since models I and III are similar, we can expect a close fit also for a model with uniform vorticity distribution in the core. As an example, we present in Fig. 7.18 the comparison of an annular vortex model with the experimental data of Kutateladze et al. (1987). This comparison indicates the possible existence of a vortex with annular distribution of vorticity.

7.3 Concentrated vortex with a rectilinear axis

403

Fig. 7.18. Comparison of experimental data from Kutateladze et al. (1987) (nearwall swirled jet in an open-ended chamber) with the model of a vortex with annudistribution of vorticity (Alekseenko et al. 1999* ). 1 – experiment; 2 – Eq. (3.68)

We see that the generalized model of an axisymmetrical helical vortex with an arbitrary distribution of the axial component of vorticity by a radial coordinate (in particular, three cases – I, II, III in Section 3.3.4) give us various velocity and pressure fields; in limiting configurations those fields agree with known theoretical models and they fit the experimental data. It is evident that vortex parameters for these models can be obtained through measurement at the bottom or lateral wall of the chamber, where the vortex interacts with the plane. 7.3.2 Vortex composition

A completely new vortex structure develops in the chamber as we increase the diaphragm orifice diameter. A dip appears in the profile of axial velocity, and the profile of tangential velocity at large radii deviates more from the empirical profiles (3.71) or as was established in Section 3.3.4 from the exact solution (3.70). Let us return to Fig. 7.12, where we plotted velocity profiles measured in two cross-sections for a case of a rather large outlet de = 100 mm. The velocity profiles mostly correspond to the LDA measurements performed by Escudier et al. (1980) in a vortex chamber with tangential fluid fed along the entire length of the chamber (Fig. 7.1b), and also with the data of Brüker and Althaus (1992), obtained using the PIV technique in a chamber with an axial swirler. The remarkable feature of axial velocity distribution is a local minimum at the flow axis; it was observed also that the tangential

404

7 Experimental observation of concentrated vortices in vortex apparatus

velocity deviates at high radius from the empirical relationships (3.71). For values r/rm > 2, the experimental points deviate from the empirical formulae (3.71) so greatly, that Escudier et al. (1982) had to modify the profile using a linear addition with the empirical coefficient ω1

v=

(

)

K⎡ 1 − exp −αr 2 ⎤ + ω1r . ⎣ ⎦ r

(7.10)

However, this modified formula fails to correspond to the Burgers solution that has a clear physical meaning. As for the axial velocity profile, the situation is even worse. It is impossible to describe this non-monotonous behavior through simple manipulations with formulae (3.71). Nevertheless, the problem of description of a complex velocity profile (such as depicted in Fig. 7.12) can be resolved in the framework of vortex models I and III. Primarily, the question arises about the existence of helical symmetry in a flow of that kind. We consider this question using the example of velocity profiles (see Fig. 7.12) measured away from the chamber bottom at a distance of 323 mm (where the effect of the bottom is negligible). Let us divide the flow into two zones. Zone 1 – from the axis up to r* (the point of maximum axial velocity) and zone 2 – the annular zone from r* up to the periphery (chamber wall). According to the technique for the finding of helical symmetry parameters, we have to determine them for each of the two flow zones: l = −46.7 mm, u0 = 0.12 m/s for zone 1 and l = 46.7 mm, u0 = 0.80 m/s for zone 2. One can see that the pitches of helical symmetry are of the same magnitude, but with opposite signs, hence, the circumferential component of vorticity is alternating. This result brings us to the conclusion that: such complex behavior of velocity components might be modeled through composition of several simple vortices. This composite vortex is related to disjointed annular zones (in our case these are zones 1 and 2). The example of composition of two vortices with rectilinear axes is shown in Fig. 7.19. Here, the cylindrical area with radius b1 = r* and uniform vorticity inside represents the first vortex (vorticity vector is directed at a certain angle to the z-axis – the left-handed vortex). The second vortex occupies the annular zone with radii b1 and b2, with a uniform vorticity inside, but the vector of vorticity has a different orientation (the right-handed vortex). The resulting velocity field (due to linear connection between the vorticity and velocity) is determined as a sum of contributions from two vorticity zones. We can consider different combinations for different zones (including geometry other than rings) and with different orientations of .

7.3 Concentrated vortex with a rectilinear axis

405

Fig. 7.19. Composition of two columnar vortices

To obtain a theoretical foundation for making a composition from two vortices, we have to analyze the solution (2.68) for the stream function corresponding to an elementary helical filament. When we exclude the term with u0 in (2.68), it is easy to see that the solution is invariant with respect to the sign of l. In particular, u0 is a constant of integration and it must be determined separately for every zone. Therefore, there is no problem with the term having u0/l in (2.68). Then we can make a vortex model by combination of zones with left-handed and right-handed helical filaments (except as their crossing). The annular zones introduced above do not contradict this requirement. Using the hypothesis on the composition of right- and left-handed vortices in zones 1 and 2, we can write the ultimate relationship for the checking of helical symmetry, that gives the connection between the tangential and axial velocities (the analog of relationship (1.66) for the checking of helical symmetry in simple flows), which takes the form ⎧ rv ⎫ ⎪w1 + l , r < r∗ ⎪ r∗v(r∗ ) rv − r∗v(r∗ ) ⎪ ⎪ − w=⎨ . ⎬ ≡ u0 + rv l l ⎪w − , r > r ⎪ ∗ ⎪ 2 l ⎪⎭ ⎩

(7.11)

Here, w1 and w2 are the integration constants. Analysis of (7.11) concludes that for the inner zone the constant w1 = u0, i.e., it coincides with the axial component of velocity on the chamber axis, and for the outer vortex w2 = u0 + 2r*v(r*)/|l|, which is determined

406

7 Experimental observation of concentrated vortices in vortex apparatus

by the matching of profiles on the boundary between the zones. Therefore satisfaction of condition (7.11) means the existence of generalized helical symmetry in a flow with a non-monotonous profile of axial velocity and we can deal with a composite of two helical vortices. The results of checking for the existence of a generalized helical symmetry for the aforementioned regimes are shown in Fig. 7.20. The results indicate that for the data of Brüker and Althaus (1992) and Shtork (1994) this generalized condition of helical symmetry is satisfied with a high degree of accuracy. Therefore, one can state that a combined helical vortex originates in these regimes. To make an approximation for the velocity field of a vortex with a rectilinear axis, the combination depicted in Fig. 7.19 is sufficient. The velocity field induced by each of these vorticity zones is calculated by model I with equal magnitude (but opposite sign) parameters l. The parameters of basic vortices are chosen to achieve the best concordance between calculation and experiment. One can see from Fig. 7.21 that this theory neatly describes the profile of the axial velocity w and slightly less well for v. This is related to the fact that the theoretical model at the point of the maximum for v always has a flexure because the vorticity jumps at this point from zero to a finite value. Returning to Fig. 7.12, we can focus on one feature: the profile of tangential velocity remains the same over the channel length, meanwhile, the profile of the axial component undergoes considerable variation: the tttttt a

b

Fig. 7.20. Checking for helical symmetry in a swirl flow with the assumed composition of two vortices (Alekseenko et al. 1999*). a – data of Brücker and Althaus (1992), l/R = 0.67, w1 = 0.72 cm/s, w2 = 1.53 cm/s, r*/R = 0.31; b – data of Shtork (1994), l = 46.7 mm, w1 = 0.12 m/s, w2 = 0.8 m/s, r* = 16 mm, de = 100 mm, ze = 430 mm, Re = 2.8⋅104, S = 3, z = 323 mm. 1 – measurements of axial velocity; 2 – axial velocity calculated from measurements of tangential velocity using the condition of helical symmetry (7.11)

7.3 Concentrated vortex with a rectilinear axis

407

Fig. 7.21. Comparison of experimental profiles with the exact solution (7.11) for a composite of two vortices (Alekseenko et al. 1999*). 1 – experiment by Brücker and Althaus (1992); 2 – experiment by Shtork (1994), the same data as in Fig. 7.20. Calculation using formula (7.11): 3 – right-handed vortex, 4 – left-handed vortex; 5 – composition of vortices

top of the axial velocity drifts to the vortex axis as the measurement crosssection approaches the outlet. The similarity for profiles of tangential velocity at different heights of a chamber was noted by Escudier et al. (1980). The flow created by a composite of two vortices also follows this regularity. If we take into account that for an axisymmetrical flow the pressure is determined only by the tangential velocity (3.63), then this fact creates an opportunity to identify the key characteristics of a composite vortex from bottom pressure measurements (as made for simple vortex filaments in Section 7.4.1). Figure 7.14 confirms this assumption. Indeed, for a single vortex (and for a composite of two vortices) the pressure profiles on the bottom and in the flow are identical. The results of research on flow regimes with composites of two vortices have to be completed with the description of measurement for bottom pressure: we wish to generalize these results for different diameters of the outlet. As was noted regarding the coordinates used in Section 7.4.1, the pressure profiles on the bottom cannot be generalized for different diameters of the outlet de. However, we can apply the method developed

hhhhhhhhhhhhhhhh

408

7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.22. Generalization of bottom pressure profiles for different outlet diameters (Alekseenko et al. 1999*). ze = 430 mm, Re = 2.8⋅104, S = 3. 1 – Eq. (7.12)

for jet flows. As a distance scale, we use the value of radial coordinate r0.5, where the pressure change equals half the total pressure drop ∆p0; then the profiles of bottom pressure at different de may be generalized in coordinates [∆p/∆p0, r/r0.5] with high accuracy (Fig. 7.22). These experimental points are described by the simple relationship

(

)

∆p = − 1 1 + r 2 .

(7.12)

Here ∆p = ∆p ∆p0 , r = r r0.5 . The same relationship is suitable for approximation of dimensionless profiles of bottom pressure in a vortex chamber with lateral injection (Smulsky 1992). In this way, the hypothesis on a two-vortex composite in a swirling flow made possible the description of complex velocity distributions measured in experiments. However, the significance of this composite approach is not only in search of approximation for empirical velocity fields, but also an opportunity to give a physical interpretation of a real object. In our example, the vorticity concentration in the core of a real vortex is caused by an intense left-handed vortex, and outside the core the vorticity is described a weaker right-handed vortex. Then the principle of composition will be applied for the explanation of a structure of more complex helical vortices.

7.4 Precession of a vortex core

409

7.4 Precession of a vortex core Let us describe a swirl flow regime in a vortex chamber (Fig. 7.2) where there is no diaphragm in the outlet. Unlike the case of a diaphragmed chamber with a small outlet, here the flow becomes unsteady due to loss of axial symmetry. In this situation we can distinguish an explicit precessing vortex core (Fig. 7.23). The loss of axial symmetry is not related to the asymmetry of the outlet of the working section. Further in this Section, we will consider the flow below the level z = z* (see Fig. 7.2). Observations imply that the configuration of the outlet part of the chamber at z > z* has no influence on the flow in the zone with z < z* because here the main factor is a drastic expansion of flow at z > z*. Figure 7.23 gives a sketch of a swirl flow structure in a vortex chamber with an open outlet. The peculiarity of the flow is the formation of a wide near-axis zone of recirculation flow (the boundary is depicted by the dashed line). The maximums for axial and tangential velocity are shifted to tttttt a

b

d

Fig. 7.23. Diagrams (a, c) and visualizations (b, d) of flow in a vortex chamber with an open outlet (Alekseenko and Shtork 1992*). b, d – z = 235 mm, Re = 4.3⋅104, S = 3 (see Fig. 7.2). The pictures were taken at different times. The precessing vortex core is explicitly visible

410

7 Experimental observation of concentrated vortices in vortex apparatus

the periphery (this follows from the time-averaged profiles). The analysis of time-averaged regimes of this type is considered in numerous publications (see Shtym (1985)) but we are interested in unsteady phenomena. It follows from visual observation that a concentrated elongate vortex arises at the boundary of zones for uprising and descending flows. This vortex rotates with the flow around the geometric axis of the chamber (see Fig. 7.23) and has a slightly pronounced helical structure. In addition to checking for helical symmetry in the averaged velocity profiles (Fig. 7.4d), this fact provides the justification to apply the theory of flow with helical symmetry to this kind of flow regime (this was accomplished in Section 3.3.6). The rotating concentrated vortex generated in an unstable swirl flow is usually named a precessing vortex core (PVC). However, all the available descriptions for a PVC use experimental conditions that differ from ours. Besides, these descriptions have a lack of identification for the spatial structure of a precessing vortex. For example, Kutateladze et al. (1987) observed the axis precession of a swirling air flow (accompanied by sound generation) in a cylindrical channel with a tangential inlet and an open outlet. Zalmanson (1973) described the phenomenon of precession in an expanding channel, which is applied to the measurement technique. Gupta et al. (1984) considered the precession in swirling devices and in burners; the impact of PVC on combustion was also analyzed. According to Gupta et al. (1984) and Sozou and Swithenbank (1969), a precessing vortex core is one possible state of a swirl flow occurring after vortex breakdown. A hydrodynamic regime with PVC typically has an extensive near-axis zone of recirculation with high velocities of reverse flow. The large-scale 3-D perturbations occur at the outlet of the swirling device. Combustion usually has a strong effect on PVC (either suppression or amplification). A flow with combustion of premixed species has higher amplitudes and frequencies of pulsation (Gupta et al. 1984). For the described vortex chamber, the PVC is most explicit when the nozzles tilt with angle γ = 5 – 10° (see Fig. 7.2). However, even under these conditions the core is not always continuous and stable. Flow visualization in the chamber cross-section demonstrates the occasional formation of two or more vortices instead of one; later they merge back into one core. As for the air cavity along the vortex axis, this cavity is occasionally broken into parts and these parts moves by circle independently and then combine into a single feature. The data on precession frequency as a function of liquid flow rate at different swirl parameters S (or nozzle tilt angle γ) is depicted in Fig. 7.24 a,b both in dimensional and dimensionless forms. The measurements were tttttttt

7.4 Precession of a vortex core a

411

b

Fig. 7.24. Precession frequency vs. flow rate in dimensional (a) and dimensionless (b) coordinates (Alekseenko et al. 1999*)

made with electro-diffusion transducers, strain gauge transducers and were supported by visual observations. Here, f is the vortex precession frequency; Sh = fm/wm is the Strouhal number; Rem = wm m/v is the Reynolds number; wm is the superficial flow rate through the chamber. One can see that frequency f increases linearly with flow rate Q. Such a dependency was established also for other types of tangential swirlers (Chanaud 1965; Cassidy and Falvey 1970). A linear relationship between f and Q for a wide range of flow rates was observed in many studies and was the basis for the design of vortex flowmeters (Zalmanson 1973). The Strouhal number is self-similar relative to the Reynolds number but it still depends on the swirl parameter S (Fig. 7.25). At S ≤ 4 the Strouhal number exhibits linear growth with an increase in S parameter. Many researchers do not identify a PVC with the motion of large-scale vortical structures (helical vortices) but they do see a correlation between the generation of PVC and the development of counter-flow along the axis of the swirl flow (Yazdabadi et al. 1994). However, we already demonstrated in Section 2.6.2 that the existence of a counter-flow is a good indicator of a helical vortex in the flow. The lack of information about flow spatial structure in most papers may be attributed to the complexity of structure recognition in a precessing flow or explained by the lack of appropriate experimental techniques for the study of three-dimensional swirl flows (see the review by Alekseenko and Okulov (1996)). Indeed, PVC can be easily observed in the plane normal to the flow axis (Fig. 7.23). However, recovery of spatial geometry of the flow from a plain picture is an extremely complicated task. hhhhh

412

7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.25. Dimensionless frequency of precession vs. swirl parameter for the range Re = (1.5 ÷ 4.7)⋅104 (according to Fig. 7.24b)

Fig. 7.26. Diagram for precession of the vortex filament

All quantitative measurements are usually made in the same plane. The common set of experimental data usually comprises the averaged velocity and the pressure distribution along the radial direction (see the review by Alekseenko and Okulov (1996)). Only Mollenkopf and Raabe (1970) and Yazdabadi et al. (1994) obtained the distribution for velocity components for the entire cross-section using the technique of conditional averaging. These results usually give us the precession frequency and allow us to establish the existence of a counter-flow but do not recover the spatial flow structure. Numerous tests have proved that the precession motion is typical not only for swirl flow regimes in open-ended chambers, but also for cases of vortex filaments in chambers with an orifice at one end. Therefore, the vortex filament is not absolutely steady but is able to undergo a slight (but explicit) precession movement under certain conditions (more specifically, when nozzle tilt angle γ > 15°). The diagram for vortex filament precession is shown in Fig. 7.26, and the precession frequency as a function of flow rate (nozzle outlet velocity) is plotted in Fig. 7.27 for two values of γ. As for PVC in a chamber with an open end, the dependency of frequency on the flow rate is linear. Also, the higher the swirl parameter, the higher the frequency. Besides, at γ = const, all experimental data for a vortex filament and PVC are on one straight line plot (see Fig. 7.28). These experimental results indicate that PVC can be interpreted as a helical vortex that rotates around its axis in the bounded space (chamber). A similar structure, i.e., a helical vortex with a core of finite width inside a gggggggggggggggggggg

7.4 Precession of a vortex core

Fig. 7.27. Dependency of precession frequency for a vortex filament on flow rate (liquid velocity at the nozzle outlet) and level of swirling (angle γ)

413

Fig. 7.28. Comparison of precession frequency for a vortex filament (1) and PVC for the regime with a completely open outlet (2)

cylindrical tube, was considered theoretically in Section 3.3.6. Let us compare the results of the theoretical calculations with experimental data on velocity profiles and precession frequency. The velocity profiles depicted in Fig. 7.29a are for a flow in a cylindrical chamber with a tangential swirler and a blanked-off left end face (see. Fig. 7.1c, Kutateladze et al. (1987)). The data for a tangential chamber with a square cross-section (Fig. 7.2) are plotted in Fig. 7.29b (Alekseenko et al. 1999). All the flow regimes exhibit a vortex structure with precesc c a

b

Fig. 7.29. Averaged velocity profiles in a swirl flow with a precessing vortex core (Alekseenko et al. 1999*). 1 – calculated using formula (3.76) for a rotating helical vortex; 2 – experiment; a – Kutateladze et al. (1987) (near-wall swirled jet in a round channel with a blanked-off end face, x = 175 mm); b – Alekseenko et al. (1999) (tangential chamber with an open outlet. Re = 3.2⋅104, S = 1, z = 63 mm, see Fig. 7.2)

414

7 Experimental observation of concentrated vortices in vortex apparatus

sion. The experimental velocity profiles are time-averaged. Therefore, the comparison is performed with the theoretical formulae (3.76), which were also derived for averaged velocity fields. The parameters of vortices can be found from measured profiles of axial and circumferential velocities. The parameters l and u0 are found through the checking of helical symmetry (see 7.2), and circulation and effective sizes a and ε are taken from the model (3.76) using the mean-squares method (the notations are given in Fig. 3.22). The derived parameters of the vortex structures have the following values: Figure

Γ, m2/s

l, m

a, m

ε, m

u0, m/s

R, m

7.29a

5.15

–0.076

0.011

0.025

–3.50

0.0375

(7.13)

7.29b

0.16

–0.078

0.028

0.057

–0.09

0.0940

(7.14)

Let us consider one more flow regime for the same chamber with a square cross-section and an open outlet (at Re = 4.3·104, swirl parameter S = 3), when the velocity distribution in the cross-section z = 323 mm corresponds to a precessing vortex with the parameters Γ, m2/s 0.24

l, m –0.063

ε, m

a, m 0.038

u0, m/s

0.044

–0.24

R, m 0.094

(7.15)

When we refer to vortex precession frequency, we usually mean the frequency for a bundle of vorticity spots passing a fixed point (sensor): this sensor may be fixed in the chamber volume or on the wall. Hence, we take frequency f for rotation of the vortex cross-section (vortex core) in a fixed plane. The formula for angular velocity of movement of a vortex wake in a tube cross-section (6.70) was derived in Section 6.4.3, we obtain from it that

f =−

uˆb Γ 2 2 8π a τ 1 + τ 2

(

)

12

,

(7.16)

where τ = l a , uˆb is the dimensionless bi-normal velocity for a helical vortex in the tube according to (6.68). The dependency of dimensionless frequency of precession f = 2πfR2/Γ on dimensionless parameter a/l for β = u02πl/Γ = 1, ε/R = 0.05 and different a/R is plotted in Fig. 7.30. The contribution from self-induced velocity may be perceptible only at low τ but the influence of velocity along the axis is much more significant. This contribution to the frequency is

7.4 Precession of a vortex core

415

Fig. 7.30. Dimensionless precession frequency f vs. reciprocal relative pitch a/l (1) and contribution to this frequency corresponding to the wall influence (2). β = 1, ε/R = 0.05

approximately ∼u0/l (or ∼β/l2). With an increase in τ (decrease in a/l), the graphs show that the precession frequency approaches the asymptote. Indeed, at high τ from (6.68), (7.16) we obtain

f

τ

−

= 1

⎡ a2 − ⎢ 4π2 a 2 ⎢⎣ R2 − a 2 Γ

R2 R2 − a 2 R2 − 2a 2 a⎞ 1 ⎛ ⎛ 1 ⎞⎤ O β − + + + τ + τ 2 2 log log 2 log ⎟ ⎜ ⎜ ⎟⎥ ε ⎠⎟ a2 R2 R2 2τ2 ⎝⎜ ⎝ τ4 ⎠⎥⎦

and the limit τ → ∞ gives us the known formula for a point vortex precession in a circular area (2.29). Now, we have to calculate the precession frequency for vortices with the parameters given above (see (7.13) – (7.15)). Table 7.1 presents dimensionless values for frequency measured in experiments fexp and theoretical value fth calculated by the formula (7.16). The contributions of different effects were also estimated: curvature fκ = −

torsion fτ = −

1 R2 1 4π a 2 τ 1 + τ2

(

1 R2 1 2 4π a τ 1 + τ2

(

)

12

)

12

(

)

⎡a ⎤ log ⎢ 1 + τ2 ⎥ , ε ⎣ ⎦

⎡ ⎛ τ ⎞ + ⎢ log ⎜ 2 ⎟ ⎣ ⎝1+ τ ⎠

(7.17)

(7.18)

416

7 Experimental observation of concentrated vortices in vortex apparatus

Table 7.1. Parameters of vortices with precession Formula τ = l/a ε/ρ

a/ε

f exp

f th

fκ

fR

fτ

fβ

(7.13) (7.14) (7.15)

0.44 0.49 0.86

0.17 0.21 0.14

0.17 0.20 0.17

–0.06 –0.16 –0.18

0.07 0.30 0.36

0.17 0.13 0.13

–0.01 –0.06 –0.14

+

6.9 2.8 1.7

0.05 0.23 0.31

1 + 1.455τ + 1.723τ2 + 0.711τ3 + 0.616τ4 τ + 0.486τ2 + 1.176τ3 + τ4 tube walls fR =

(

1 + τ2 1 − −2 τ 4

)

12

⎤ ⎥, ⎥ ⎥⎦

R2 − a 2 ⎞ 1 R2 1 + τ2 ⎛ a 2 − log k ⎜ ⎟, 2π a 2 τ2 ⎜⎝ R2 − a 2 R2 ⎟⎠

velocity at the axis fβ = −

1 R2 β . 2 π a 2 τ2

(7.19)

(7.20)

Here f th = fk + fτ + fR + fβ ; and the values with tilde are defined as

(

)

1 + x 2 l 2 + 1 (see Section 2.6.2); coeffix = x 2 exp ⎡ 1 + x 2 l 2 − 1⎤ ⎢⎣ ⎥⎦ cient k in (7.19) depends on τ and η = l/R (see Section 6.4.3) ⎡ 1 ⎢ 9η k= ⎢ 12 1 + η2 ⎣⎢

(

)

12

−

7η3

(1 + η )

2 32

−

3τ

(1 + τ )

2 12

⎤ ⎥. + 3 2 ⎥ 1 + τ2 ⎦⎥

(

τ3

)

The tabulated data manifests that, firstly, the theoretical formula gives a frequency close to the experimental one, secondly, the contributions of different effects must be taken into account for a correct estimate of vortex precession frequency. Besides, a situation is possible when all effects compensate each other and the vortex remains immovable. Strictly speaking, the formula for bi-normal velocity of a helical vortex in a tube (6.68) is valid either for a helical vortex with a thin core ε/ρ 1, or for a weakly curved columnar vortex with a/ε 1. For the sample vortex (7.13) we obtain ε = 0.05ρ, and for (7.14) – ε = 0.23ρ. For these two cases the vortex is rather thin and the accuracy of the frequency is high. For the third case (7.15), the vortex radius is larger (ε = 0.31ρ). The degree of deflection is also high (a = 0.86ε). Hence, the accuracy for estimation of

7.5 Stationary helical vortices

417

frequency is lower. Obviously, for a “thick” vortex we have to take into account the inner structure of the vortex core (but the procedure for the finding of vortex parameters implies a model with uniform vorticity within the core). Finally, we have to note that since the helix pitch is rather high, we can use instead the formula (7.18) for torsion impact, the formula with long-wave approximation (see (5.29)), and this formula gives us 1 R2 1 fτ = − 2 4π a τˆ 1 + τˆ 2

(

)

12

(

⎡ 1 + τˆ 2 ⎢ ⎢ 0.366 − log τˆ − 2 τˆ ⎢⎣

)

12

⎤ ⎥ ⎥. ⎥⎦

For the smallest value of ratio l/a = 1.7, this approximation has an error of 3.5%.

7.5 Stationary helical vortices Analyzing the rotation frequency of helical vortices in the previous Section we have indicated the possible situations, when vortex rotation is compensated by average motion of the fluid in the channel. In such cases we may expect appearance of steady helical structures, which indeed were observed in the experiments by Alekseenko and Shtork (1992), Alekseenko et al. (1999) and are described in the present Section. Helical vortices may appear either due to instability of the axisymmetric flow relative to the spiral modes, or in consequence of deformation of the rectilinear filament due to an induced distortion of boundary conditions. In the first case, helical vortices are unsteady and predominantly threedimensional (such as helical waves, spiral breakdown of the vortex, see Section 7.6). Here we will consider only the second method of helical vortex generation, which allows us to observe stationary (immovable) structures. As in previous experiments, the vortex chamber of square cross-section presented in Fig. 7.2 was employed. Through displacement of its outlet and a change of bottom shape, the following stationary helical structures were observed: right-handed vortex, left-handed vortex, vortex with exchangeable helical symmetry, and a double helix – two entangled helical vortices. 7.5.1 Single helical vortices

Let the flow in a vortex chamber be twisted to the right as before. Let also a diaphragm with an orifice of diameter de be mounted in the outlet crosssection. The central location of the orifice results in the origination of a stable rectilinear vortex (vortex filament) with a diameter dependent on the hhhhhhaaaahhhh

418

7 Experimental observation of concentrated vortices in vortex apparatus a

Fig. 7.31. Scheme ( ) and visualization (b) of a swirling flow with a left-handed vortex (Alekseenko et al. 1999*). de = 70 mm, ze = 560 mm, δ = 62 mm, Re = 2.4⋅104, S = 4.5. Helical shape of the vortex is generated due to displacement of the outlet by distance δ from the chamber axis

outlet dimension – the smaller the orifice, the smaller the vortex diameter (see Section 7.3.1). Displacement of the outlet by distance δ with respect to the chamber axis, results in a drastic change of flow structure (Fig. 7.31). The air filament, which visualizes the vortex axis, rolls up into helical form. As a whole, such a structure is immobile. At the same time, fluid particles move around the helical axis, thus accomplishing double helical motion. The velocity in the vicinity of the geometric axis of the channel is small and directed downwards, though we may observe an intense flow along the vortex filament towards the outlet. The direction of the vortex axis screw is opposite to the direction of the moving particles, i.e. a left-handed helical vortex is formed. Maximum deviation of the air filament from the channel axis increases with increasing displacement of the outlet. It reaches 43% of halfwidth of the chamber at δ = 65 mm. This testifies to the non-linear character of flow perturbation. Note particularly, that in such a complex turbulized flow (Re numbers calculated by the nozzle diameter reach up to 4⋅104) the very thin air filament remains unbroken. In particular, this fact may be explained by turbulence attenuation at the axis of intense vortex (Vladimirov et al. 1980). Formation of a helical axis is observed at all angles of nozzle turning greater than γ > 5°. Nevertheless, the most stable state of air filament is observed at some intermediate location of the nozzles γ = 10 – 20°. In other

7.5 Stationary helical vortices

419

cases, intermittent breakdown of the air filament, caused by small-scale perturbations occurs. The shape of the outlet does not practically influence the flow pattern (specifically when changing a round orifice for a slit of the same area). Depending on the displacement of the diaphragm orifice, the intersection point between the vortex axis and the chamber bottom shifts slightly by a distance not exceeding 10-20 mm. The number of helical coils depends on the distance ze between the bottom and the diaphragm (see Fig. 7.31). Thus at ze = 420 mm we may observe the formation of a half coil, while at ze = 560 mm – a full coil is d

d

Fig. 7.32. Spatial shape of the helical vortex axis (in three projections). de = 70 mm, ze = 560 mm, δ = 62 mm, Re = 2.4⋅104, S = 4.5

420

7 Experimental observation of concentrated vortices in vortex apparatus

formed. Generally speaking, the shape of the coil axis is not an ideal helix. If drawing attention to interaction between the vortex and the flat bottom, it becomes obvious, that the axis must be orthogonal to the plane of the bottom, while the plane must be inclined at a certain angle to the horizon in order to maintain the ideal shape. Therefore, in the near-bottom area the helical shape distorts. This is well shown in Fig. 7.32, revealing the 3-D position of the vortex axis (in the projections). Here, the same digits indicate spatially similar points. It is obvious, that the helix is projected vertically as a sinusoid, while in the horizontal plane, the projection represents the circumference. Apparently these conditions hold true approximately for the points 5-14. However, in the bottom vicinity (points 1-5) distortion of the helix, concluding in variation of the amplitude and a change in screwing direction occurs. Therefore, for the time being let us neglect this region from consideration. The formation of a helical shape upon deformation of the helical filament can be clearly explained by self-induced motion, while its immobility is stipulated by compensation of the helix motion by the average flow in a bounded space. However the question as to why the vortex axis is wound to the left and not to the right remains unanswered. The explanation follows from the analysis of an exact solution for velocity field (2.69). Indeed, the vortex filament wound to the left results in deceleration of the flow in the near-axis area of the chamber. Consequently, displacement of the orifice prevents the fluid from flowing out along the geometrical axis of the chamber (helix), and thus the left-handed winding is selected. Conversely, for a right-handed helix, according to the theory an intense flow along the chamber axis and its decay at the periphery might be observed. This enables us to make the assumption that in experiments a righthanded vortex may appear both in the case of a centrally located outlet (in order to let the fluid flow out) and, for instance, in the case of an inclined bottom (in order to initiate initial deformation of the vortex axis). Indeed, a right-handed structure is revealed under the mentioned conditions. This is clearly demonstrated in Fig. 7.33. Based on the theoretical model proposed in Section 7.3.2, we may state that within the initial combined vortex; the right-handed helical vortex was suppressed in the first case, while the left-handed vortex was inhibited in the second case. The performed analysis allows us to hypothesize that there should be more complex helix-like vortex structures with a transition from right-handed to left-handed symmetry. To prove the hypotheses we made some additional modifications of the experimental test section, which enabled us to vary the inclination angle of the chamber bottom across the experiments (Alekseenko et al. 1995). The diaphragm is placed at the outlet of the chamber, with the

7.5 Stationary helical vortices

421

b

Fig. 7.33. Scheme ( ) and visualization (b) of a swirling flow with a right-handed vortex (Alekseenko et al. 1999*). de = 70 mm, ze = 390 mm, β = 18°, Re = 1.3⋅104, S = 2.9. The helical vortex is generated due to the inclined bottom

b

c

Fig. 7.34. Immobile helical vortex with variable helical symmetry (Alekseenko et al. 1999*). – flow scheme: 1 – right-handed vortex; 2 – left-handed vortex; 3 – imagined cylindrical surface; 4 – flow swirl direction; 5 – inclined bottom; 6 – displaced outlet; b and c – flow visualization; the pictures are taken from two orthogonal directions. de = 70 mm, ze = 420 mm, δ = 62 mm, bottom inclination angle β = 20°, Re = 2.8⋅104, S = 3

422

7 Experimental observation of concentrated vortices in vortex apparatus

orifice 70 mm in diameter, which is shifted relative to the axis by a distance equal to 62 mm. At an inclination angle of the chamber bottom with respect to the horizon equal to 20°, the flow rate Q = 5.25 l/s, and swirl design parameter S = 3 we are able to obtain a well pronounced stationary (immobile) vortex structure with a change of helical symmetry from righthanded to left-handed. Photography of an air filament fixing vortex geometry for such a regime is presented in Fig. 7.34 b, c. Pictures of the same immobile vortex were taken from two orthogonal positions. Convergence of a number of coils in the pictures can be explained by the changing direction of screw of the vortex axis (see scheme in Fig. 7.34 ). The right-handed vortex originates in the lower part of the chamber, while the left-handed vortex is observed in the upper part. The transition zone in the central part of the chamber has a smooth character. Here, the helical symmetry breaks down similarly to conjugation of the left vortex with the horizontal plane (see Fig. 7.32). 7.5.2 Double helix

The above described experimental observations of helical structures relate to single vortex filaments. However, theory on helical vortices (see Section 2.6) assumes possible existence of an arbitrary number of interacting helical vortex filaments. We may state that the experimental observation of such phenomena might be extremely difficult. Indeed, there are only a limited number of citations on the existence of double helix structures, at the same time being unsteady and non-uniform. Chandrsuda t l. (1978) indicated spiral binding of two elongate vortices in the mixing layer (Fig. 7.35). Boubnov and Golitsyn (1986) observed unsteady spiral pairing of two vortex filaments in natural convection within a rotating vessel. Moreover, the final stage concluded in the merging of two vortices into a stronger one. A system of hairpin vortices interacting in the wake behind a body in a boundary layer (Acarlar and Smith 1987) and in jets (Perry and Lim 1978) apparently has a bi-spiral character. A double helix originates upon vortex breakdown as well (F 1er and Leibovich 1977), but here the authors did not demonstrate that the colored jets are the vortex axes (see Table. 7.2, type 5). The interaction of vortex filaments seems to play a principle role in hydrodynamics. It is supposed (Takaki and Hussain 1984) that spiral pairing is the elementary interaction in turbulence. Attempts at generating a stationary double helix in the tangentional chamber were undertaken after observation of single stable helical filament. These attempts succeeded due to the trial of a great number of variations (Alekseenko and Shtork 1992, 1994).

7.5 Stationary helical vortices

423

Fig. 7.35. Spiral binding of two elongate vortices in the mixing layer in the process of blowing a plain air jet into still air (Chandrsuda et al. 1978*)

It is revealed that a double helix appears in a vortex chamber provided both with a centrally located outflow orifice and two inclined flat slopes at the bottom of the chamber (Figs. 7.36, 7.37). A double helix represents two entangled helical vortex filaments of the same sign. Screwing of the filaments in the form of right-handed helical vortices corresponds to the direction of flow swirling. Based on a developed theoretical model, it becomes clear that the screwing of the vortex axes to the right is caused by the central location of the outflow orifice (which allows intensive flow along the chamber axis). As for a single right-handed vortex, the use of two inclined slopes at the bottom of the chamber is related to the necessity of forming initial deformation of the vortex axes. Visually observed kinematics of the flow field (by trajectories of air bubbles) fully agree with the theoretical calculations (see Fig. 7.38, N = 2). In contrast to the single-helix vortex flow, which is rather stable, a double-helix pattern should be considered as quasi-stationary for the following reasons. Firstly, the air filaments which visualize the axes of the vortices are somewhat fuzzy, they oscillate chaotically and break down occasionally. As in other experiments, the Reynolds numbers do not significantly influence the flow pattern. Secondly, one of the filaments always is dominant, since in order to separate a double helix it is necessary to provide fine adjustment of the nozzle directions, and sometimes to control the flow rates through the individual nozzles. Though in general, a double-helix vortex flow is distinctly observed. The inclination angle of the slopes at the bottom of the chamber exerts the main influence on the double helix parameters (such as the distance between filaments 2a, helix pitch h = 2πl, and the number of half-waves j).

424

7 Experimental observation of concentrated vortices in vortex apparatus b

Fig. 7.36. Scheme ( ) and visualization (b) of the flow with a double helix (Alekseenko et al. 1999*). de = 65 mm, ze = 420 mm, β = 50°, Re = 4⋅104, S = 3. A double-helix pattern is formed due to the dual-sloping bottom

Fig. 7.37. Flow visualization in a vertical plane for a regime with a double helix. Re = 4⋅104, S = 3, de = 65 mm, β = 50°

At β = 50° we have j = 3, h = 250 mm, 2a = 60 mm (see Fig. 7.36) while at β = 30° we have j = 6, h = 115 mm, 2a = 25 mm (Fig. 7.39 ). The latter case is not fully stable since a double helix with five half-waves sometimes originates (Fig. 7.39b). The theoretical model allows for an arbitrary number of helical vortices in a multi-vortex structure. The example of streamlines for four vortices is presented in Fig. 7.38 (N = 4). Though in practice, observation of structures consisting of more than two vortices is rather difficult due to instability upon interaction, which causes their breakdown and merging. Besides, the origination of several vortices instead of just one means that their strength decreases by several times. This circumstance prevented the obtaining of a system consisting of four well-pronounced stationary vortices (Alekseenko et al. 1999). Seemingly for such a system, a chamber of square cross-section could be favorable in terms of perfect geometry. In order to generate four vortices, four slopes are placed in every quarter of the chamber bottom. We clearly observe four vortices in the near-bottom area; while in the main flow, the vortices are almost always broken and visible only for short time periods. The peculiarity of four-vortex structure is that the vortices are weakly curved and weakly pronounced. At the same time, the intense flow along the channel axis (the axis of symmetry of the vortex structure) is clearly observed. This description agrees with the calculated flow pattern for a four-vortex structure.

7.5 Stationary helical vortices N =2

425

N =4

w0 = 0

w0 = Γ/h

w0 = 2 Γ/h

Fig. 7.38. Multi-vortex structures in a cylindrical tube (calculation) (Alekseenko et al. 1999*). N is the number of infinitely thin helical vortex filaments

426

7 Experimental observation of concentrated vortices in vortex apparatus b

Fig. 7.39. Formation of a double helix with a large number of coils. Re = 4.3⋅104, S = 3, de = 70 mm, β = 30°. , b – various points in time: j = 6 (a), 5 (b)

7.6 Perturbations of a vortex core 7.6.1 Waves on concentrated vortices

The non-perturbed vortex filaments presented in Figs. 7.6, 7.7 can exist only under certain conditions. In fact the perturbed state is a specific feature of elongate concentrated vortices. The waves with various modes as well as the vortex breakdown refer to perturbations of vortex filament. The helical structures described in Sections 7.3–7.5 represent perturbed states of vortex as well. Origination of perturbations may be stipulated by the instability of a swirling flow as well as external impacts including artificial excitation. The present Section aims to demonstrate at a qualitative level, i.e. in terms of flow visualization, the existence of various perturbation types of the vortex filaments. Thus, in Chapters 4-6 we underline the importance of theoretical consideration of stability and dynamics of concentrated vortices. Here the detailed quantitative description is omitted for the following reasons. The experimental data concerning waves on vortices published in the literature is limited. Therefore it is rather impossible to give an appropriate description of wavy phenomena. Conversely, broad empirical data on vortex breakdown has accumulated in the literature. Though, as mentioned in the Introduction, this subject is beyond the consideration of this book.

7.6 Perturbations of a vortex core

427

The experimental data presented in the current Section was obtained mainly in a tangential chamber (Fig. 7.2 (Alekseenko and Shtork 1992; Shtork 1994)), a weakly-diverging channel (see Fig. 7.40 (Faler and Leibovich 1977, 1978)), in a rotating vessel with local suction (see Fig. 7.40 b, c (Maxworthy et al. 1985; Khoo et al. 1997)), in a swirl jet (Panda and McLaughlin 1994; Billant et al. 1998), and in some other configurations (Brücker and Althaus 1992, 1995; Brücker 1993). Figure 7.41 represents the examples of disturbances on a vortex filament originating in a tangential chamber provided with an outlet diaphragm (Fig. 7.2). The specific feature of such a design is that the flow behind the diaphragm may have a strong influence on the generation of perturbations, since the perturbations are able to propagate along the vortex filament in any direction. The swirl jet is formed in the space behind the diaphragm, i.e. in fact the area in front of the diaphragm serves as a swirling device for the following area behind the diaphragm, and in many cases can be considered as infinite. Propagation of a swirl jet in free space at a sufficient level of swirling is accompanied by vortex breakdown, formation of reverse flow areas and flow turbulization. These phenomena may serve as a perturbation source of the vortex filament existing both in the forepart area of the diaphragm and beyond the outlet. Based on the above mentioned circumstance, we present separate data for three regions: before the diaphragm, its vicinity, and far from it. Let us return to Fig. 7.41. The traveling perturbations presented in the pictures originate spontaneously and propagate downwards from the outlet (i.e. upwards along the flow). Here, the following typical patterns of perturbations may be observed: – helical waves (in the form of a train); b – helical waves with variable wavelength; c – double helix; d – high frequency waves of a helical type; e, f – waves in the form of a traveling vortex breakdown. Artificial excitation must be employed in order to achieve more detailed determination of the perturbations structure. Such experiments were conducted by Maxworthy et al. (1985) and Alekseenko and Shtork (1992, 1997). A vertical cylindrical chamber provided with a thin suction tube from above and rotating around its own axis was employed in the first paper (Fig. 7.40b). The excitation was caused in the following ways: ) generation of fluid flow rate impulse in the suction tube, b) periodic movement of the tube, c) touching the vortex filament with a thin rod. As a result six types of waves were observed, namely: helical waves (1); plane standing waves (2); isolated kink waves (3); isolated kink waves with tail-end perturbation (4); helical waves with increasing wave length (5); axisymmetric waves (6). Let us briefly describe each wave type.

428

7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.40. Diagrams of experimental setups for studying vortex breakdown and the waves on vortices: – Faler, Leibovich (1977); b – Maxworthy et al. (1985); c – Khoo et al. (1997)

7.6 Perturbations of a vortex core

d

b

c

e

f

429

Fig. 7.41. The examples of vortex filament perturbations in a tangential chamber (Alekseenko and Shtork 1992*): – Re = 1.7⋅104, exposure 1/30 s; b – 4⋅104, 1/1500 s; c – 4.3⋅103, 1/30 s; d – 4.2⋅103, 1/30 s; e – 2.4⋅104, 1/15 s; f – 1.0⋅103, 1/30 s

430

7 Experimental observation of concentrated vortices in vortex apparatus

Helical waves with constant amplitude are the helices curled in a counter stream form (right-handed helix) and rotating and propagating counter to the current flow. Group velocity is greater than phase velocity. Plane standing waves result from interaction between helical waves traveling from above and those reflected from the bottom. These waves were first described by Kelvin (1880). Such waves rotate counter to the flow. Isolated kink waves look like a coil of spiral propagating counter to the flow with the group velocity exceeding phase velocity. These waves resemble Hasimoto soliton, though rigorous proof of this similarity has not been obtained. Similar solitary perturbations originated spontaneously on vortex filaments generated by a fluctuating grid in a rotating vessel (Hopfinger et al. 1982; Hopfinger and Browand 1982). Possible theoretical schemes of motion for Hasimoto soliton are presented in Fig. 5.13 (for a vortex filament without axial flow). A distinctly pronounced soliton-like wave was produced in a tangential chamber (see Fig. 7.2) by means of artificial excitation (Alekseenko and Shtork 1992). Evolution of such a wave and the scheme of its propagation are presented in Fig. 7.42. The perturbation resembles a right-handed spiral coil rotating and propagating counter to the flow. Unlike in the work of Maxworthy et al. (1985), here the group velocity cg is less than phase velocity c. This obviously follows from Fig. 7.42, if we trace the trajectories of both the wave packet center and the wave phase. Generally speaking, this fact does not agree with the theory of Hasimoto soliton (see Section 5.3). Indeed, from the last formula of Section 5.3 for weakly-nonlinear waves it follows that cg < c at the torsion value |T| < 1. However, this case corresponds bbbb b

c

d

e

f

Fig. 7.42. Evolution of the solitary spiral wave on a vortex filament in a tangential chamber: – e – freeze-frames over 125 ms; f – scheme of perturbation motion. Re = 1.6⋅103, S = 3.5

7.6 Perturbations of a vortex core

431

to tipping over spiral soliton (see Fig. 5.13, right lower scheme), which does not agree with the scheme in Fig. 7.42 e. Thus the problem concerning the nature of the observed spiral waves still requires full consideration. 7.6.2 Vortex breakdown in a channel

The observation of flow patterns in a vortex chamber (Fig. 7.2) reveals that traveling perturbations may take the form of a vortex breakdown both in the case of their spontaneous origination and upon artificial excitation. A striking example of traveling vortex breakdown is presented in Fig. 7.43 (Alekseenko and Shtork 1997). Here the perturbation propagates counter to the flow leaving behind a turbulized wake rather localized edgewise. The transverse dimension of the perturbation is quite small (about several diameters of bubbles visualizing the flow). Therefore, it is impossible to talk about the detailed structure of the perturbation. Most likely it is of the same type as bubble breakdown. Maxworthy et al. (1985) noted that only axisymmetric traveling perturbations could cause vortex breakdown, while Benjamin (1962) and Leibovich (1970, 1978) in their theoretical works proposed vortex breakdown models based exactly on axisymmetric nonlinear soliton-like waves of the Korteweg-de Vries type. Let us turn from traveling perturbations to those localized in space and representing mainly vortex breakdown. For perturbations of this type, either rapid widening of the vortex core or abrupt deviation of the vortex axis from the initial position may occur at some point of the elongate vortex. As this occurs a stagnation point appears in the interior of the vortex on its axis or nearby, generating a zone of reverse flow. Peckham and Atkinson (1957) were the first to describe vortex breakdown for the case of flow over a delta wing (see also Fig. 5 (Lambourne and Bryer 1961)). Two main types of breakdown, namely bubble and spiral ff

Fig. 7.43. Freeze-frames of forced traveling vortex breakdown. The time interval between the frames is 80 ms. Vortex flow is directed upward. The impact on the vortex core was localized at a distance of 50 mm from the upper edge of the frame. Q = 1.31 l/s, S = 3.5 (Alekseenko and Shtork 1997*)

432

7 Experimental observation of concentrated vortices in vortex apparatus

breakdown were revealed. The majority of the research was conducted later, in weakly-diverging tubes. It is worth noting the pioneering work by Sarpkaya (1971), where double helix breakdown was first discovered. Faler and Leibovich (1977, 1978) presented the most comprehensive classification of breakdown types in a weakly- diverging tube (see Fig. 7.40 ). According to these, a total of seven types of breakdown can be distinguished. Visualization and brief characteristics of these types are presented in Table 7.2. The eighth type of breakdown, namely conical breakdown was discovered for various conditions independently by Alekseenko and Shtork (1992), Sarpkaya (1995), Khoo et al. (1995) (see also Khoo et al. (1997) and Billant et al. (1998)). A similar classification of vortex breakdown for a free vortex filament was made by Khoo et al. (1997). The respective experimental test section is presented in Fig. 7.40c. Here, the vortex filament is generated in a rotating vessel provided with a thin suction tube from above (as in the experiment by Maxworthy et al. (1985)). Corresponding results are presented in Table 7.3, which is placed opposite Table 7.2 for convenient comparison of the data. It is assumed that the data presented in both Tables completes all major types of vortex breakdown. For more comprehensive understanding, regime maps are shown in Figs. 7.44 –7.47, while additional illustrations on vortex breakdown, obtained under other conditions, are presented in Figs. 7.48 –7.55. The regime map in Fig. 7.44 shows the results obtained by Faler and Leibovich (1977) in [Re, z/R0] coordinates for fixed values of Ω. Here the Reynolds number is Re = WD/ν; R0 = D/2; z is the average location of the breakdown point; Ω = Γ/WD is the swirl parameter; Γ is the circulation; W is the superficial velocity in a neck with diameter D (see Fig. 7.40 ). The data of Khoo et al. (1997) is presented in Fig. 7.45 in [Re*, S] coordinates for fixed values of fluid flow rate Q. Here Re* = Γ/2πν is the analog of the Reynolds number, which is flow rate independent; Γ = V0R0 is the circulation; S = R0Γ/2Q is the swirl parameter; V0 is the tangential velocity corresponding to the radius of the vessel R0 (see Fig. 7.40c). The data of Faler and Leibovich (1977) in Fig. 7.46 is recalculated in the coordinates proposed by Khoo et al. (1997). This enables us to make appropriate comparison between regime maps for unbounded and bounded flows. In Figure 7.47, the internal scale of the vortex core, such as an effective radius r1 is used instead of chamber size R0 . Such a substitution enables us to disengage from design features of the setup and proceed to a more universal presentation of the data.

7.6 Perturbations of a vortex core

433

Table 7.2. Vortex breakdown in a weakly-diverging channel (bounded flow) according to the data by Faler and Leibovich (1977*) (types 0-6, see test section in Fig. 7.40 ), and by Sarpkaya and Novak (1997) (type “c”). See additional explanations in the text below Type Designation c Conical

Image a

b 0

1

2 3

4

5

Axisymmetric closed (or bubble) vortex breakdown

Re = 2.56·103; Ω = 1.777

Axisymmetric open vortex breakdown Spiral breakdown

a b c d

6

Filament disruption

One or two tails rotating together with the flow. Simultaneous emptying and filling of the bubble occur through the tail-part. A colored trickle is occasionally observed inside the bubble. Fully opened tail-part.

The helix is screwed along the flow and rotates with the flow.

Modification of spiral breakdown Flattened buba ble breakdown b Double helix breakdown

Description Strongly turbulized wake with a conical envelope. In a short exposure time, two or three entangled helices screwed and rotating against the flow are observed. – ∆τ = 1/30 s; Re = 105; Ω = 0.61 b – ∆τ = 6·10–9 s; Re = 1.15·105; Ω = 0.31

No rotation – side view b – top view Appears from type 6. The structure formed is stable in shape and location. –t=0 b – t = 5 min c – t = 10 min d – t = 15 min The state is stable and fixed. The trickle is situated within the plane at slight deviations, and screwed along the flow at large deviations.

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7 Experimental observation of concentrated vortices in vortex apparatus

Table 7.3. Vortex breakdown in an unbounded flow according to the data of Khoo et al. (1997*) (see the scheme in Fig. 7.40c). See additional explanations in the text below. The images are horizontally tilted. Visualization is performed by means of various dyes Type Designation c Conical

Image

Description Represents a spiral with a conical envelope rotating with the flow. Re* = 3.5 103; S = 6.0

0

Closed bubble breakdown

Strong mixing inside the bubble. The inclined toroidal vortex ring responsible for filling and emptying is observed inside. Re* = 2.5 103; S = 2.5 High rotation velocity, strong mixing. Rapid expansion of the filament in the form of a cup is observed from the beginning, followed by origination of a densely wound helix, which generates vortex rings in the open tail-part. Re* = 3.0 103; S = 4.0 The spiral is screwed against and rotates with the flow. Re* = 103; S = 9.0 Re* = 750; S = 7.5

1

Open bubble breakdown

2

Spiral breakdown

3

Distorted spiral breakdown

4

Flattened bubble breakdown

Re* = 750; S = 7.5

5

Double helix breakdown a

– originates from type 6 due to formation of the second helix; the filament is screwed along the flow.

b 6

Filament disruption

b – branches of a double helix are screwed against and rotate with the flow. Re* = 103; S = 1.5

7.6 Perturbations of a vortex core 435

Fig. 7.44. Dependence of breakdown types and their average axial location z on Reynolds number and swirl parameter Ω (Faler and Leibovich 1977*). See notations of the types in Table 7.2

436

7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.45. Dependence of the vortex breakdown types in unbounded space on Reynolds number and swirl parameter S (Khoo et al. 1997*). The insertion shows the areas of dominant breakdown types. See notations of the types in Table 7.3. 1 represents the lines of constant flow rate; 2 is the less dominant type of breakdown; 3 is the equally dominant type of breakdown

7.6 Perturbations of a vortex core

437

Fig. 7.46. Regime map by Faler and Leibovich (1977) in coordinates proposed by Khoo et al. (1997*). The insert shows the areas of dominant breakdown types. See notations in Fig. 7.45

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7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.47. Integrated regime map of vortex breakdown (Khoo et al. 1997*). See notations in Fig. 7.45

Since our interest is focused mainly on perturbations on free vortex filaments, let us give a description of the breakdown in unbounded flow first (see Table 7.3), comparing it with other data. Parameter S, determined by Khoo et al. (1997), is related to Ω (Faler and, Leibovich 1977) by the approximate relationship: S = 1.91Ω. Let us note that among the general properties, the location of the breakdown point may shift along the flow, while the breakdown type may change spontaneously from one form to another.

7.6 Perturbations of a vortex core

439

Let us begin the analysis with the small value of swirl parameter S and Reynolds number Re*, i.e. from the lower part of Table 7.3. Remember that here Re* does not depend on fluid flow rate in contrast to the experiments by Faler and Leibovich (1977), and other authors. Let us designate further the work by Faler and Leibovich (1977) as FL, while the work of Khoo et al. (1997) as KYLH. Type 6 (filament disruption) agrees with the data by FL and is characterized by smooth deviation of the colored trickle (vortex axis) from the initial position. The flow is laminar, and the trickle is observed up to the drain tube, as a whole the trickle is stationary. In a weakly- diverging channel, the trickle at slight deviations still lies within the plane; while at large deviations (up to almost full contact with the wall) it is screwed along the flow. The increase in the swirling in a channel results in transition to types 4 and 5. Type 5 represents double helix breakdown. Though its manifestation might be different for bounded and unbounded areas. In a weaklydiverging channel a double helix originates out of type 6 as presented in Table 7.2. once the originated pattern is stable in its shape and position (does not rotate). Branches are screwed along the flow. In unbounded space, Khoo et al. found two diverse forms of double helix breakdown designated as 5 and 5b (see Table 7.3). Type 5 also originates out of type 6. In particular, another vortex filament appears around the initial helix filament originating due to separation of the flow in the boundary layer on flat bottom of the chamber. Filaments are screwed along the flow direction. Type 5b is observed at lower values of Re* (< 750). The spirals are screwed against though rotate together with the flow. Apparently, double helical types of breakdown are the most diverse in terms of their manifestation. In tangential chambers double helices obviously have different forms (Alekseenko and Shtork 1992). Figure 7.41c demonstrates double helical perturbation originating spontaneously on a vortex filament, while a stable double helix breakdown in the vicinity of a diaphragm outlet is shown in Fig.7.48c. Here the helices rotate along the flow. It is obvious, that these structures differ from the double helices presented in Tables 7.2 and 7.3. Just as for the previous types of breakdown, types 3 and 4 are observed in the range of small values of Re*. They are distinguished by unstable behavior. Type 4 relates to flattened bubble breakdown. This type of breakdown develops equally both in bounded and unbounded areas. It appears differently from two orthogonal directions. The pattern does not rotate; it originates in the channel either out of type 6, spontaneously or accompanied by increased swirling of the flow.

440

7 Experimental observation of concentrated vortices in vortex apparatus b

c

Fig. 7.48. Vortex breakdown in the vicinity of a diaphragm in a tangential chamber. Re = 1.4⋅105 is determined by orifice parameters, de = 70 mm; S = 6.86 is determined by characteristics of the chamber and diaphragm; – exposure 1/30 s; b – 1/60 s; c – photoflash. The short exposure demonstrates a double helix type of breakdown. In a long exposure the vortex is perceived as conical

Type 3 is the distorted spiral breakdown. Unlike type 2, it is characterized by a strong reverse flow. It agrees with type 3 for diverging channels. Type 2 is the spiral breakdown. Dominant in many cases, this type has the broadest domain of existence in terms of a wide range of Re numbers and swirl parameters. In unbounded space the helix is screwed opposite to the flow and rotates along the flow similar to spiral vortex breakdown in a flow over a delta wing. At the same time, in a diverging channel, the spiral is conversely screwed to the flow direction and yet rotates with the flow. Type 2 may occasionally change into types 0 and 1. A well-defined type of spiral breakdown is observed in a tangential chamber behind a diaphragm orifice, i.e. intrinsically in a swirl jet entering unbounded space (Alekseenko and Shtork 1992; Shtork 1994). Figure 7.49 reveals the detailed structure of the flow. Here the flow is swirled to the left, while the spiral is right-handed, i.e. it is screwed opposite to flow and rotates with the flow. Such a scheme agrees with the observations by Khoo et al. (1997). In the internal area a reverse flow forms. The exposure was equal to 1/30 or 1/60 second, so that velocity and motion direction could be determined using the traces of small marking bubbles. Analyzing bubble trajectories at a vortex axis it becomes obvious, in particular, that the vortex filament shifts perpendicular to the generatrix.

7.6 Perturbations of a vortex core b

c

441

d

Fig. 7.49. Spiral breakdown of a vortex in a tangentional chamber behind a diaphragm. Re = 4⋅104 is defined by outlet parameters, de = 110 mm; S = 5.52 is defined by chamber characteristics in front of the orifice; – visualization in general lighting, exposure 1/60 s; b – light sheet in a central cross-section, 1/30 s; c – light sheet in a short distance vertical cross-section, 1/30 s; d – flow pattern

For the bubble breakdown, we may distinguish two types, namely open bubble and closed bubble. Open bubble breakdown can be described as follows. Firstly, rapid expansion of the colored trickle in the form of a cup is observed, followed by origination of a tightly coiled helix, which in turn generates vortex rings in a fully opened tail-part (see Color Fig. 2). High rotation velocity and intense mixing characterize such a structure. It originates out of a spiral type due to the increase in swirl degree. In general, the structure corresponds to type 1 in a bounded channel (FL). The closed bubble breakdown of 0 type corresponds to type 0 in a bounded space (FL). The structure has the definite shape of a bubble, which can be visualized by means of dyeing. Usually, a single axisymmetric tail (occasionally two tails) extends behind the bubble and rotates in a cocurrent direction, screwing into the spiral at some distance from the bubble. The internal structure of a bubble is unsteady and three-dimensional. Simultaneous filling and emptying of the internal area occurs. Faler and Leibovich suggested two mechanisms of mass transfer. According to the first, there exists just one tail, which serves as the emptying channel, while filling occurs at a diametrically opposite point of the tail area. The second mechanism occurs very seldom at lower Re numbers. In this case, two diametrically opposite empty tails as well as two diametrically opposite filling points, orthogonal with respect to the tails, are observed. The bubble always drifts slowly along the channel axis near the midpoint, changing its size at the same time. It grows when drifting upstream and diminishes when drifting downstream. It is worth noting that the col-

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7 Experimental observation of concentrated vortices in vortex apparatus

ored filament, originating from the bubble nose, frequently appears inside the bubble. A more detailed structure of bubble breakdown was revealed in the experiments by KYLH and Brücker and Althaus (1992, 1995). The inclined toroidal vortex ring, precessing around the axis and responsible for the filling and emptying of the internal area of the bubble, was discovered (see visualization in Color Fig. 2b and reconstruction of the ring image according to the measurements, employing PIV technique, in Fig. 7.50). The generalized scheme of the bubble in the form of projection of streamlines to the meridian plane is presented in Fig 7.51. Obviously, there are two stagnation points. Filling of the bubble occurs near the front point S1, while the emptying takes place at the rear point S2. Figure 7.52 represents additionally the examples of both the bubble and intermittent type of vortex breakdown in a tangential chamber behind the orifice of a diaphragm. The breakdown of the vortex, known as conical was found after all the other types of breakdown under various conditions and by different researchers. The conical breakdown of Sarpkaya (1995) was represented as a turbulent conical wake at high Reynolds numbers in a weakly-diverging channel. Yet later, Sarpkaya and Novak (1997) discovered in an extremely short exposure (6 ns), that in fact conical breakdown constitutes the structure, consisting of two or more entangled helices with a conical envelope Fig. 7.50. Reconstruction of the 3-D shape of bubble breakdown (Brücker and Althaus 1992). The slope of the vortex ring is shown with respect to axis plane e

Fig. 7.51. The scheme of vortex breakdown of the bubble-type. The projections of streamlines to the meridian plane (Brücker and Althaus 1995)

7.6 Perturbations of a vortex core a

443

b

Fig. 7.52. Vortex breakdown of bubble type ( ) and intermittent type (b) behind the diaphragm orifice in the tangential chamber. Re = 3.7⋅104 is determined by orifice parameters, de = 110 mm; S = 4.9 is determined by parameters in fore-part of the orifice; exposure 1/30 s

(see Table 7.2, type “ ”). Khoo et al. (1997) (see Table 7.3 and Color Fig. 2) derived a similar conclusion though they had observed conical breakdown within a very narrow range of conditions and only for a single helix. In both cases, the conical breakdown was formed from the bubble breakdown. As also follows from experiments by Alekseenko and Shtork (1992), conical breakdown registered in the vicinity of the diaphragm outflow in a tangential chamber is nothing but a rotating double helix. Figures 7.48 ,b are obtained under similar conditions, though the first picture was taken with an exposure equal to several milliseconds (flashbulb), while the latter – at 1/30 of a second. A conical vortex breakdown, different from as described above, was found in the experiments on a swirling submerged jet conducted by Billant et al. (1998). Here, the main difference concludes in the magnitude of the opening angle of the cone, which was equal to 90 degrees (see Color Fig. 3). Such a large value is essentially greater as compared with that indicated in the other works, besides, the flow was laminar. Four types of breakdown, namely: bubble, conical, asymmetric bubble, and asymmetric conical were selected in the cited work. All four types can be referred to the vortex breakdown phenomenon, since only these types of breakdown include a stagnation point. The latter two types form out of the first two with increasing

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7 Experimental observation of concentrated vortices in vortex apparatus

Reynolds number and are characterized by the availability of a stagnation point precession around the jet axis. Besides, the hysteresis effect of the patterns was seen. It is as well to note that flows with conical symmetry prevail in vortex dynamics (Goldshtik 1981; Shtern and Hussain 1998). Properties such as ambiguity and hysteresis are peculiar only to them. Two more examples of conical vortex breakdown are described below. Figure 7.9 reveals conical breakdown immediately near the bottom of a tangential chamber. Note that after the vortex breakdown the vortex filament reconstructs immediately. Outwardly, a similar picture might also be observed for a tornado (see Color Fig. I.1.). Another example concerns the case of a tangential chamber provided with two diaphragms (Fig. 7.53). In the vicinity of the lower compartment, situated between the deck of nozzles, a conical breakdown of the vortex occurs; however, the vortex filament re-forms in the space between the two diaphragms. This filament is thrust almost perpendicular to the internal surface of the cone. In conclusion of this Section we should note that even at the qualitative level, there is no complete description of the types of concentrated vortex perturbations. As will be shown later, it is impossible to deduce appropriate theories about changes in the flow structure based only on the visual pattern of the flow.

Fig. 7.53. Flow structure in a tangential chamber with two diaphragms. D is the diaphragm with a shifted outflow orifice; Di is the intermediate diaphragm with a central orifice

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445

7.6.3 Vortex breakdown in a container with a rotating lid The typical feature of vortex breakdown in swirl jets and flows in channels is the variety of its manifestation and instability. From this point of view, vortex breakdown in a container with a rotating lid seems to be canonical and, respectively, more appealing for investigations. Such a system also has important practical applications, in particular, for laboratory modeling of a tornado (Maxworthy 1972; Maxworthy et al. 1985; Okulov et al. 2004), for investigation of vortex breakdown control (Husain et al. 2003) or when modeling the process of crystal growth using the method of Chokhralsky (Berdnikov 2000). The flow pattern in a container with a rotating cylindrical rod (the model of a crystal) is shown in Fig. 7.54. It is clearly seen that a recirculation zone of the bubble breakdown type is formed under the crystal surface, and a toroidal vortex is formed near the lateral walls. The typical scheme of the setup for the investigation of vortex breakdown (Spohn et al. 1998) is shown in Fig. 7.55. A swirling flow with a concentrated vortex is formed at the axis of a cylindrical container due to gggg

a

b

Fig. 7.54. The pattern of flow in a container with a rotating rod upon isothermal modeling of the process of crystal growth by the method of Chokhralsky (Berdnikov 2000*). Re = Ω2 R/ν = 2570. – flow is visualized by aluminum powder; b – the scheme of the flow

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7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.55. The scheme of the experimental setup of Spohn et al. (1998) for the investigation of vortex breakdown in a container with a rotating bottom: 1 – tube for supply of the fluorescent dye; 2 – wire ring in the lid for tracer generation by electrolytic method; 3 – rotating disc; 4 – wire ring on the cylinder wall for tracer generation

the rotation of a disc near the bottom with angular velocity Ω. In the upper part, there can be either a free surface of fluid or an immobile lid. In some experiments a rotating top lid was used, the lower one was immobile or simultaneous rotation of both lids was performed. Some experiments have used a container of non-circular cross-section (see for example, the papers by Okulov et al. (2003), Anikin et al. (2004)). The physical mechanism of fluid motion initiation in these systems has a dual origin. On one hand, the rotation of the disc is transferred to the fluid due to the friction forces. On the other hand, rarefaction at the axis of rotation provides axial motion of fluid towards the center of the rotating disc along the cylinder axis and a reverse flow near the cylinder walls (this is shown by the arrows in Fig. 7.55). However, this simple scheme is broken by the formation of recirculation zones connected with the bubble-type vortex breakdown (Figs. 7.56 –7.58). The problem of its generation is not yet solved. It is shown in the fundamental works of Vogel (1968) and Escudier (1984) that the character of vortex breakdown depends on the Reynolds number Re = ΩR2/ν and the ratio H/R, where H is the height and R is the radius of the cylindrical chamber. Figure 7.59 represents the regime map in the mentioned coordinates, constructed by Escudier (1984) on the basis of flow visualization within the diametrical cross-section of a light ggggggggggggggggggggggg

7.6 Perturbations of a vortex core b

447

c

Fig. 7.56. 1-bubble vortex breakdown in a container with an upper solid lid at H/R = 1.5 and different Re: 1139 (a), 1492 (b) and 1854 (c). Visualization is by the fluorescent dye supplied from above along the vortex axis (Escudier 1984) b

c

Fig. 7.57. 2-bubble vortex breakdown in a container with an upper solid lid at H/R = 2.5 and different Re: 1994 (a), 2126 (b) and 2494 (c) (Escudier 1984)

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7 Experimental observation of concentrated vortices in vortex apparatus b

c

Fig. 7.58. 3-bubble vortex breakdown in a container with an upper solid lid at H/R = 3.25 and different Re: 2686 (a), 2752 (b) and 2819 (c) (Escudier 1984)

sheet by means of dye injection through a central orifice in the immobile top lid. Besides, it is found that the visual pattern of the flow does not depend on which lid rotates: the upper (Vogel) or the lower (Escudier) ones with accuracy of up to the mirror refection relative to the horizontal axis, i.e., we can neglect gravitation effects in this problem due to their infinitesimal influence. Therefore, to make the presentations uniform, the flow images taken from different sources are oriented in the manner that their bottoms correspond to the rotating disc and tops – to the immobile one. The bubble breakdown with one bubble is the predominant type (see Fig. 7.56). Outwardly it resembles a closed bubble breakdown in a weaklydiverging tube but it is more stable in its shape and position. The zone of one-bubble breakdown in the form of an elongated tongue was determined by Vogel (1968) (see Fig. 7.59). Escudier (1984) repeated these results and determined additionally that with a rise of Re and H/R, two bubbles appear fffffffff

7.6 Perturbations of a vortex core

449

Fig. 7.59. The map of vortex breakdown regimes in a container with a rotating bottom and an immobile lid (1, 2) (Escudier 1984), supplemented with data of Sørensen (1992) and Stevens et al. (1999) for the boundary of transition from axisymmetric to non-axisymmetric unsteady flow regimes (3, 4), and the calculation of Gelfgat et al. (1996 ) of critical Re for the axisymmetric perturbation mode (5)

(see Fig. 7.57), whose domain of existence is inside the zone of one-bubble breakdown. Then, formation of three bubbles is possible in a very narrow range of parameters (see Fig. 7.58). If we fix ratio H/R and increase Re, then after exceeding some critical Reynolds number (the dashed line in Fig. 7.59), the flow becomes unsteady. Moreover, according to the experiments of Escudier (1984), unsteadiness within the ranges H/R < 3 and H/R > 3.1 has a different nature. In particular, for H/R < 3, the position of the stagnation point starts oscillating along the axis, and the flow retains axial symmetry. For H/R > 3.1, the first indicator of unsteady motion appears as precession of the lower area of the bubble vortex breakdown around the axis. This obviously means asymmetry of the flow. Sørensen (1992) expanded these results for a wider range of velocity of disc rotation (at H/R = 2) and visualized the axisymmetric flow regime with an oscillating breakdown zone in the whole domain of existence up to the boundary, where axial symmetry disappears, and further until formation of developed 3-D unsteady flow. To eliminate the external influence, which may occur due to dye injection into the flow area upon visualization, light-scattering particles with neutral floatation were introduced into the container in advance of fffffffffffff

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7 Experimental observation of concentrated vortices in vortex apparatus 2002

2301

2598

4014

2707

4996

2505

2896

5981

8001

3500

10016

Fig. 7.60. Visualization of the flow in a cylindrical container for different Re (Sørensen 1992*)

7.6 Perturbations of a vortex core

451

these experiments. Transitions from steady axisymmetric to unsteady axisymmetric and then to 3-D unsteady flow regimes are illustrated in Fig. 7.60. In the range of Re from 2000 to 2500, there is only one area of vortex breakdown with a complex bubble structure. At higher Reynolds numbers axisymmetric oscillations of the bubble arise. As was described by Escudier (1984), it follows from these observations that the flow becomes unsteady at Re = 2550. In the range of Reynolds numbers above 3000, the oscillating bubble area of vortex breakdown gradually fails, and at Re > 4000, the 3-D helical vortex structure starts to form. This structure rotates around the flow axis. Stevens et al. (1999) repeated the visualization of this regime transition using another ratio H/R = 2.5 and a different method of particle introduction (injection). The experiments proved the scenario of flow development at H/R < 3 described above. Both results extended the boundary of the area of asymmetry formation within the flow for H/R < 3 in the diagram of Fig. 7.59 (4). It is necessary to note that there exists many works on the numerical modeling of steady axisymmetric bubble breakdown in a closed container (Lugt and Haussling 1982; Lugt and Abboud 1987; Neitzel 1988; Sørensen and Loc 1989; Lopez 1990; Lopez and Perry 1992). Their main conclusion is that the numerical solution to the 2-D axisymmetric equations of NavierStokes perfectly describes not only the appearance of vortex breakdown but also its position, size and number of “bubbles” on the axis. It is also shown that the streamlines are closed inside the axisymmetric recirculation zone, and this is the main difference from breakdown in diverging tubes, where the bubble is asymmetric and open. However, the possibility of axisymmetric flow regimes was revised in experiments (Hourigan et l. 1995; Spohn et l. 1998). In particular, bubble openness and asymmetric flow pattern, visualized by a light sheet within the diametric cross-section of a container were distinguished for the typical axisymmetric regime at Re = 1850, H/R = 1.75 in the paper by Spohn et al. (1998). The electrolytic method of finely-dispersed particle generation was applied in these experiments for visualization. In contrast to injection and seeding, this allowed the dosed input of tracers into the flow. Particles were let out by the impact of 15 Volt rectangular pulses of 10 Hz frequency on a wire ring of 30 mm diameter, situated axisymmetrically on the surface of the upper lid (see Fig. 7.55). The typical example of the time evolution of tracers in completely steady flow, when the bubble is formed and steady, is shown in Fig. 7.61. At first (Fig. 7.61 , b), particles within a flat cross-section of the light sheet are located asymmetrically along the front surface of the bubble. During the next phase of particle spreading, the asymmetry becomes more dddd

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7 Experimental observation of concentrated vortices in vortex apparatus a

b

c

d

e

f

g

h

Fig. 7.61. Time evolution of tracers upon bubble vortex breakdown in a container with a solid upper lid. Re = 1850, H/R = 1.75 (Spohn et al. 1998*): dimensionless time t∗ = tΩ is measured from the beginning of the container bottom rotation: – t∗ = 1027; b – 1068; c – 1099; d – 1129; e – 1160; f – 1191; g – 1232; h – 1262

obvious in the form of different time-stable folds on the left and the right (Fig. 7.61c, d). The third stage demonstrates the open character of the bubble (Fig. 7.61e, f). Upper folds move inwards of the bubble, radially shrinking and simultaneously stretching along the axis. Lower folds move downward (outwards from the bubble). Finally, in the last stage (Fig. 7.61g, h), particles inside the bubble reach the upper bubble boundary, revealing the upper stagnation point. Oval zones stay free of particles, and this indicates

7.6 Perturbations of a vortex core

453

the existence of a closed toroidal ring area inside the bubble. It is interesting to trace the change in the flow structure with a rise of Re (Fig. 7.62). Exceeding the critical Reynolds number at H/R = 1.75 results in bubble disappearance as in Fig. 7.60. In contrast to visualization via particle seeding, in this case the tracers outline a helical surface, whose projection is shown in Fig. 7.62b. According to observation within horizontal crosssections, the particles are concentrated along the spirals, especially in the tail area of a bubble, i.e. the spatial distribution of particles is not axisymmetric. Therefore, the asymmetry and open character of bubble breakdown are obvious. This contradicts previous experiments and numerical calculations. This contradiction can be solved by consideration of the flow as a complex 3-D motion of fluid. The point is that in the steady swirl axisymmetric flows the streamlines have a 3-D helical shape (see Section 1.4.2 and examples from Chapter 3), only their combination into the stream surface has axial symmetry. Really, trajectories of tracers describe the complex 3-D helical motion, and this is illustrated by the calculation of Sotiropoulos and Ventikos (2001), based on the solution of the 3-D unsteady Navier-Stokes equations (Color Fig. 4). Therefore, to obtain a symmetric pattern in the plain longitudinal and transverse cross-sections of the light sheet, it is necessary to provide a dense and uniform distribution of particles along the whole stream surface. It was impossible to achieve this upon particle introduction controlled by generation with a given frequency, but the simpler visualization methods (injection and seeding) provide more uniform particle distribution over the whole stream surface. To solve this problem completely and determine the difference between the real flow in an area of axisymmetric regimes and its axisymmetric image, Okulov et al. (2004) compared calculated and experimental data. gggggg b

Fig. 7.62. Visualization of the flow in a container with a solid upper lid and a rotating bottom at different Re: 2250 ( ) and 2750 (b), H/R = 1.75. The supply of tracers is over the perimeter of the lateral wall (Spohn et al. 1998*)

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7 Experimental observation of concentrated vortices in vortex apparatus

Diametric cross-sections of stream tubes calculated by direct numerical simulation by means of 2-D axisymmetric code (Sørensen and Loc 1989) were compared with those calculated using data of PIV-measurements. Cross-sections of 25 stream tubes with the constant flow rate Qi marked with a non-uniform step are shown in Color Fig. 5 (as in Color Fig. 6 and Table 7.4). Here Qi is defined by relationship Qi = min(Q) + [max(Q) – min(Q)]×(i/30)3.

The comparison proves good correspondence not only for the main changes in the flow topology but also for the distributed characteristics. This fact, without doubt, may be interpreted in favor of the axisymmetric flow regime, despite the weak asymmetry of the real flow. Hence, the axisymmetric equations of Navier-Stokes can be successfully used for diagnostics of the real flow in the area below the boundary of transition to the asymmetric flow (4 in Fig. 7.59). Another important conclusion of this comparison is connected with the fact that PIV-measurements, as numerical calculations, register the closed type of bubble breakdown (Color Fig. 5). A similar conclusion follows from the works by Sørensen et al. (2001) and Pereira and Sousa (1999). To stabilize breakdown, experiments in the latter paper were conducted using a rotating bottom in the form of a convex cone. The flow structure was reconstructed by the streamlines using the measurements of the axial velocity component (Fig. 7.63). According to Fig. 7.63 , in this case bubble breakdown represents a closed recirculation zone of a toroidal shape with two stagnation points. This topology of the flow fixes two stagnation points, and the bubble surface between them should be impenetrable to tracers, at least, in two methods of visualization: injection and generation (upon seeding, some part of the particles can be within the area of bubble formation). In all cases, particles penetrate into the bubble (see Figs. 7.56 – 7.58, 7.60 and 7.61). It is necessary to note that the regime of perfect axisymmetric steady flow (free of any small pulsations and asymmetric distortions) cannot be obtained experimentally. In fact, in experimental setups, low asymmetry and flow unsteadiness always exist due to inevitable distortions caused by geometric, thermal, dynamic or other non-uniformities. The example of the influence of asymmetric mounting of the upper lid from the paper by Sotiropoulos et al. (2002) is shown in Fig. 7.64. It is necessary to note that this asymmetry connected with the experimental error is significantly less than that registered by Hourigan et l. (1995) and Spohn et l. (1998) and is caused by visualization of streamlines, but not stream tubes. However, it leads to the formation of weak unsteadiness. Really, lid inclination distorts the stream surfaces, and they make a rotary motion, which naturally leads gggggggggg

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455

Fig. 7.63. Streamlines in the flow regimes with vortex breakdown (from measurements of axial velocity component (Pereira and Sousa 1999*)). – one bubble, Re = 2200, H/R = 2; b – two bubbles, Re = 2570, H/R = 3

to weak pulsation. Therefore, a simple explanation can be given to tracer penetration into the closed area of the bubble. It relates to the divergence of fluid particle trajectories (and tracers, correspondingly) with instantaneous streamlines in the case of unsteady motion of fluid. This fact becomes more obvious, if we consider the expressed unsteady regimes as described in detail below. Even then, despite the fact that experimental imperfection provides negligibly small unsteadiness, the presence of special points (stagnation and corner) in the flow leads to significant divergence between the visual (Lagrange) and measured (instantaneous – Euler) flow patterns for conventionally steady regimes. It is necessary to note that this simple explanation does not contradict the more complex analysis of Lagrange characteristics of the flow (Sotiropoulos et al. 2002).

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7 Experimental observation of concentrated vortices in vortex apparatus a

b

Fig. 7.64. Visualization of steady bubble breakdown. Re = 1850, H/R = 1.75: a – ‘ideal’ container; b – container with an immobile lid, inclined approximately by 0.4° relative to the horizon (Sotiropoulos et al. 2002*)

Let us turn to more detailed consideration of unsteady flow regimes, which are traditionally distinguished in the diagram of Fig. 7.59. The question of how we can distinguish these regimes from the described background unsteadiness connected with the imperfection of experimental setups is naturally posed. With this purpose, Naumov et al. (2003) measured velocity pulsation with a Laser Doppler Anemometer (LDA) in a container with H/R = 2 for different Reynolds numbers at distances of R/2 from the cylinder axis and H/4 from the rotating lid. The dependency of velocity dispersion on the Reynolds number is shown in Fig. 7.65. This data does not contradict the regime diagram in Fig. 7.59: with a rise of Re to 2500, the steady flow regime with a similarly negligible level of pulsation is observed, then intense oscillations appear, and this is characterized by a linear growth in dispersion of the axial velocity component. The example of visualization of unsteady vortex breakdown (Sørensen 1992) at Re = 3000 is shown in Fig. 7.66 for different instants of a complete oscillation period. According to the picture, this regime may be classified as the conic type of vortex breakdown with a funnel slightly oscillating along the vortex axis. These oscillations significantly hamper any measurements of distributed flow characteristics. To overcome the arising difficulties, Naumov et al. (2003) applied the combined approach using two non-contact methods of velocity field measurement: Laser Doppler Anemometry (LDA) and Particle Image Velocimetry (PIV). Firstly, parameters of velocity pulsation at some fixed point were obtained by the LDA method. The temporal recording of the axial velocity component for H/R = 2 and Re = 3000 (Fig. 7.67 ) demonstrates the periodic character of oscillations with frequency f = 0.63 Hz (Fig. 7.67b).

7.6 Perturbations of a vortex core

457

Fig. 7.65. Relationship between dispersion of the axial velocity component, measured by LDA in a cylindrical container and Re (Naumov et al. 2003*)

t=0

t = T/4

t = T/2

t = 3T/4

Fig. 7.66. Visualization of the near-axis vortex structure for an unsteady regime with developed axisymmetrical oscillations at Re = 3004 (Sørensen 1992*)

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7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.67. Temporal recording of the axial velocity component, measured by LDA (a), and corresponding spectral density (b) at Re = 3004 (Naumov et al. 2003*)

The time period for the averaging of instantaneous velocity fields measured by PIV was determined using this information. The velocity fields were measured at four instants of a complete oscillation period in a step divisible by a quarter of period (T/4 = 0.39 s). For the instants of time chosen by this method, the velocity field was determined by means of statistic averaging of four PIV-images of the flow, obtained at corresponding instants with a delay of 0, T, 2T, 3T. As for the steady case, this divisibleperiodic averaging of instantaneous velocity fields allows a significant decrease in the random measurement error. In the same time it completely eliminates the shift error caused by non-stationary changes in the flow structure. In Color Fig. 6, the results obtained are compared with the 3-D unsteady calculation using the method described in detail by Shen et al., (2001). In this illustration, there are cross-sections of 25 instantaneous stream tubes of a constant flow rate with a non-uniform step as in Color Fig. 5. The size of the window is determined by coordinates [–3R/4; 3R/4] in

7.6 Perturbations of a vortex core

459

horizontal and [H/8; 7H/8] vertical directions. According to the presented stream tubes, we can see a shift of the bubble breakdown area of a vortex structure downward along the axis, and the oscillation amplitude exceeds significantly the oscillation amplitude of the visualized flow structure (Fig. 7.66). Besides, the PIV-images of the flow indicate the existence of a closed bubble, whereas it is not registered by visualization. At t = 0, the bubble is at the highest point of its trajectory (near the immobile bottom) and it grows reaching its maximum size at t = T/4. Then, it is carried by the main flow down to the rotating lid, simultaneously decreasing in size up to complete disappearance. At t = T/2, the bubble is at the lowest point of its trajectory, and it has yet to register distinctly. At t = 3T/4, we cannot see the bubble, but its transportation upward is indicated by local expansion of the stream tubes near the axis, clearly seen in the top part of the pictures. Reaching its highest point, the bubble appears again (at t = 0) and starts to grow. The cycle repeats. This comparison allows connection between the flow structure at axisymmetric unsteady vortex breakdown and intense axial oscillations of the closed bubble area, which disappear and generate again. In this case, there is a significant distinction with visual diagnostics of the flow caused by a large difference between particle trajectories and instantaneous streamlines: tracers here have no time to follow the bubble alterations. It is interesting to note that in the area of transition to the 3-D flow (Re = 4000), the bubble was not registered at all, as in the visualization. Now, let us turn to consideration of the regime map (see Fig. 7.59) in the area of unsteady flows. As was already mentioned before, unsteadiness in the ranges H/R < 3 and H/R > 3.1 has a different nature. The transition boundary in the first zone, determined by Escudier (1984), is the line ascending monotonously, and above this line, the bubble oscillates axisymmetrically (see Color Fig. 6), and at H/R > 3.1, it is a descending, almost horizontal line. Possible explanation for such behavior may be that the boundary consists of two curves, representing different regimes of transition to unsteadiness, and they cross at H/R ≈ 3. The first curve describes the boundary of axisymmetric mode growth, and the second is responsible for the growth of asymmetric perturbations. The hypothesis about the existence of different zones with predominantly axisymmetric or asymmetric perturbations was made by Sørensen et al. (1996) and supported by Gelfgat et al. (1996 ) for one of two methods of transition to the unsteady flow. Hence, the transition to the unsteady flow regime can be described as a competition between two types of unsteadiness with predominantly axisymmetric or asymmetric perturbations. According to this idea, we can suppose that both types may exist at all values of H/R, and

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7 Experimental observation of concentrated vortices in vortex apparatus

the regime with a minimum critical Reynolds number of the growth of axisymmetric or asymmetric modes determines the beginning of the transition to the corresponding type of unsteady flow. New experimental data described previously and results of numerical simulation prove this hypothesis and allow for the continuation of both boundaries. The line consisting of circles was obtained as a neutral curve for axisymmetric perturbation upon numerical analysis of the axisymmetric solution to the Navier-Stokes equations (Gelfgat et al. 1996 ), and the dashed line indicates the beginning of the transition to asymmetric unsteady flow in the visualizations of Escudier (1984), Sørensen (1992) and Stevens et al. (1999). At 1.8 < H/R < 3, the calculated critical parameter (see Fig. 7.59, 5) for the axisymmetric type of transition perfectly correlates with the experiment (see Fig. 7.59, 2) and data from other papers on numerical simulation (Lopez and Perry 1992; Daube and Sørensen 1989; Tsitverblit 1993). Close correspondence of the values for H/R between the measured boundary of transition and the calculated neutral curve once again proves the efficiency of axisymmetric modeling for total diagnostics of the flow in a cylindrical container up to transition into unsteady flow. However, for H/R > 3, comparison with experimental data (see Fig. 7.59) shows that the critical Reynolds numbers obtained by numerical analysis (Gelfgat et al. 1996 ), are higher than in the experiment. This means that asymmetric perturbations start growing earlier than the axisymmetric ones, and 3-D numerical analysis is required. However, there are now several examples of numerical solution to the 3-D Navier-Stokes equations for the problem under consideration (for instance, see Serre and Pulicani (2001), Sotiropoulos and Ventikos (2001), Marques and Lopez (2001)). In particular, calculations from the latter paper only for H/R = 3 demonstrate the predominant growth of the fourth azimuthal mode at an increase in Re = 2730, 2850 and 2900. We should also note that new analysis, performed by Gelfgat et al. (2001), did not solve this problem because it was conducted only for the azimuthal modes without consideration of the asymmetric helical modes, which are more consistent for the helical structure of the real flow (see Color Fig. 4). In conclusion, we will consider one more type of flows in containers with rotating bottom (see Fig. 7.55) and without upper lids. Experiments with the free fluid surface demonstrate significant differences both in the regime maps and the bubble shapes (Spohn et al. 1993, 1998). Actually, the bubble attached to the free surface (Fig. 7.68), becomes the main form of breakdown, and the areas with a detached bubble are very narrow in coordinates [Re, H/R] (Fig. 7.69). If we neglect the distortion of the free surface, the considered problem is equivalent to the flow in a closed cylinder of double height with face lids, rotating simultaneously with one frequency

7.6 Perturbations of a vortex core

Fig. 7.68. Visualization of the bubble attached to the free surface upon vortex breakdown in a container with a rotating bottom and a free fluid surface (Spohn et al. 1993*). Re = 1850, H/R = 1.0

461

Fig. 7.69. The map of vortex breakdown regimes in a container with a rotating bottom and a free fluid upper surface (Spohn et al. 1993)

and in the same direction (Brøns et al. 2001). For these problems (where both end faces rotate), only numerical investigations have been conducted (Valentine and Janke 1994; Gelfgat et al. 1996b; Brøns et al. 2001). Alteration of the vorticity field and related flow topology (formation of recirculation zones – bubbles) was analyzed in the papers of Okulov et al. (2001, 2002, 2005) for a particular case of the same angular velocities of lid and bottom rotation. In this case, symmetric flows appear in the upper and lower parts of the cylinder with the plane of symmetry in the middle horizontal cross-section (z = 0). The physical mechanism of fluid motion in this system was discussed above but in this case it is an applied reflection, symmetrical to the upper and lower parts of the flow (Fig. 7.70 ). In the case of counter-rotation of the discs, two families of mirrored vortex tubes are formed as nested deformed tori, closed on the surface of the rotating discs (Fig. 7.70b). Vorticity distribution in the form of a torus is well known in fluid mechanics as the vortex ring. It induces pure translation at the torus axis, which occurs in the given case from the cylinder center towards the rotating edge. Naturally, deformation of vortex tubes from clearly toroidal shapes and the helical character of composing vortex filaments distort the flow pattern in comparison with the flow field induced by the vortex ring (Sections 2.5 and 3.2.1), but it is not sufficient to change the translation character of the flow towards the end faces along the axis. Possibly, this is the reason why the recirculation zones are not found there. If the discs rotate in the same direction, the pattern of vorticity distribution

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7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.70. The scheme of flow ( ) and cross-section of the vortex tubes for Re = 2000 and H/R = 2 at counter- rotation of discs (b) and at their co-rotation (c)

changes considerably (Fig. 7.70c). A system of almost cylindrical vortex tubes, closed at the rotating discs, is formed in the middle part of the flow. Symmetrical toroidal families of vortex tubes are driven to the cylinder periphery (in the figure the boundary is marked by a dotted line). In this case, the velocity field is induced by three vortex structures. If the influence of two toroidal vortices provides the axial velocity component, directed towards the rotating end faces (similar to the example described above), the effect of the central vortex structure on the velocity behavior at the axis may be ambiguous. This is connected with the induction of different types of axial velocity profiles (convex and concave) at different signs of torsion of the vortex filaments, which compose the columnar-like vortex tubes (Okulov 1996; Murakhtina and Okulov 2000; Alekseenko et al. 1999). The structure of the vortex lines, located on the central vortex tube, is shown in Fig. 7.71 for two values of Reynolds number: Re = 150, when a bubble is not formed, and Re = 1300, when a bubble exists. The data presented clearly demonstrates a change in the torsion of the vortex lines for the regimes with different flow topology. A more complete illustration of this interrelation is shown in Table 7.4 for the case of co-rotation of the discs and H/R = 1. The quantity, reverse of local pitch 1/l (Sections 7.1.2 and 7.2) is used there as the measure of torsion intensity of the helical-like vortex lines. In the first column of the table, there are tubes, along which value 1/l was calculated. Its change is presented in the second column by

7.6 Perturbations of a vortex core

463

the same type of lines, which indicate the corresponding vortex tube. At low Reynolds numbers (Re = 150), a vortex structure with right-handed helical symmetry (l > 0) is formed in the upper part of the flow, and a vortex structure with left-handed helical symmetry (l < 0) in the lower part. They induce convex and concave profiles, correspondingly, or the profiles of axial velocity, convex towards rotating discs like the peripheral toroidal vortices (the third column in the Table). With increasing velocity of disc rotation (Re = 400), the helical symmetry of the vortices changes spontaneously: left-handed at the top and right-handed at the bottom. These vortex structures induce axial flow, already oppositely initiated (from rotating discs towards the center, See Fig. 7.71). In superposition with the flow, induced by toroidal vortices, we obtain the profile of axial velocity with a depression at the axis. The further growth of Re up to 500 increases the degree of vortex filament torsion (decreases the pitch), and this amplifies the counter-flow, induced by them, and leads to the formation of bubbles (the fourth column of the Table). One more change in helical symmetry was registered at Re = 1300. It occurs for the inner vortex tubes of the considered structure. On the contrary, for the outer vortex tubes of columnar structures, an increase in the degree of vortex line torsion (along the left-handed helix from above and along the right-handed helix from below) is observed. As a result, the zone of counter-flow moves from the cylinder axis towards this area and an annular recirculation zone is formed instead of a bubble. With a more considerable increase in Re up to 2400, the annular recirculation area shifts closer to the periphery and beyond the considered central vortex structure, where torsion of the vortex lines almost diminishes. Now, similar to the Rankine vortex (Section 3.3.1), it consists of almost straight vortex filaments and does not induce the axial velocity component, whose value is close to zero in the near-axis area. Therefore, for flows with vorticity inside a cylindrical cavity, there is a connection between formation of recirculation zones and spontaneous changes in helical symmetry of the near-axis vortex structure in the flow. In conclusion of this problem description, we should note the correlation between the calculated flow patterns in the lower part of the cylinder and visual flow patterns for the flow with a free surface (Spohn et al. 1993, 1998). Nonetheless, there is no complete correspondence because of natural distortion of the free surface shape during the experiment, especially at a rise of the flow swirl. Investigation results on vortex breakdown in a closed container as described above have shown that a comprehensive approach with application of different measurement methods and numerical simulation is required for the correct diagnostics of this complex phenomenon. Classification of vortex breakdown types (see Section 7.6.2) and investigation of unsteady

464

7 Experimental observation of concentrated vortices in vortex apparatus

Fig. 7.71. The shape of the 3-D vortex lines, composing the; central, close to cylindrical, vortex tubes and induced profiles of axial velocity along the cylinder with co-rotating top and bottom lids at Re = 150 (a) and Re = 1300 (b) with H/R = 1

7.6 Perturbations of a vortex core

465

Table 7.4. Flow structures in a container with a lid and bottom, rotating synchronously Re Cross-sections of vortex tubes

Degree of vortex Profiles of axial line torsion velocity

150

H

1/l -4

-2

0

2

4

-H

400

H

1/l

0 -4

2

-2

4

-H

500

H

1/l

0 -4

2

-2

4

-H

1300

H

1/l -4

2

-2

4

-H

2400

H

1/l -4

2

-2

-H

4

Cross-sections of stream tubes

466

7 Experimental observation of concentrated vortices in vortex apparatus

perturbations of concentrated vortex cores (see Section 7.6.1) only on the basis of visual study of the flow pattern are not sufficient for correct description of these complex phenomena. Besides, the problem of the development of a consistent theory on these unique phenomena is still unsolved.

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Index

A approximation, local induction, 86 B Biot – Savart law, 73 C circulation, 10, 12 coordinate system toroidal, 273 Couchy – Lagrange integral, 19 Lame coefficients, 21 curvature, 70 radius, 70 curve, helical, 71 cut-off method, 235 D density of volume sources, 45 dispersion relationships, 170, 187 dipole, vortex, 78 discrete vortex models, conservative, 310 dynamics of shear flows, 341 E energy kinetic, of fluid, 61 equation continuity, in orthogonal curvilinear system of coordinates, 21 Bragg-Hawthorne, 217

Korteweg-de Vries, 224, 225 Korteweg-de Vries, modified, 300 Helmholtz, 14 Hirota, 300 Howard and Gupta, 206 mass conservation, 13 Poisson, 45 Schrödinger, 233, 258 equations Euler, 13 Frenet – Serret, 71 Gromeka – Lamb, 13 in orthogonal curvilinear system of coordinates, 22 Navier – Stokes, 19 of fluid motion Hamiltonian form, 311 in a non-inertial frame of reference, 14 in orthogonal curvilinear system of coordinates, 22 in Cartesian coordinate system, 23 in cylindrical coordinate system, 24 in spherical coordinate system, 24 vortex particles motion, of, 309 F flow helical (Beltrami), 18, 28 one-dimensional, 136 non-uniform, 29 uniform, 29

486

Index

with helical symmetry, 43 laminar, 439 nonswirling (longitudinal) axisymmetric, 34 one-dimensional, 37 plane-parallel, 31 shear, 38 swirl axisymmetric, 35 two-dimensional, 31 uniform, 38 with helical symmetry, 39 with helical vorticity, 40 fluid ideal, 13 incompressible, 45 fluid circuit, 12 force Coriolis, 160 Kutta – Joukowski, 267 tension, 268 vortex, 53 I impulse fluid, 56 vortex, 57 instability Kelvin – Helmholtz, 160, 176, 179 spatial, 185 temporal, 176 instability criteria, 202 invariants of vortex motion, 49 two-dimensional, 49 H Hamiltonian of the vortex motion, 311 Hamiltonian equations for vortex particles, 313 Hasimoto soliton, 257 helical symmetry, 39 change, 422 helicity, 62

K Kutta - Joukowski hypotheses, 319 Kutta–Joukowski formula, 52 L Lagrangian of the vortex motion, 311 liquid ideal, 13 viscous, 19 local induction approximation, 297 M method bound vortices, of, 316 image vortex, 78 matched asymptotic expansions, of, 291 momentum balance, 267 discrete vortex particles, of, 309 mode axisymmetrical, 155, 179, 188 bending, 157, 190 viscous, 211 co-grade, 192 radial, 189, 194 retrograde, 191 spiral, 179 varicose, 360 momentum angular, 58 angular, vortex, 58 vortex, 57 motion, 9 irrotational, 9 pure straining, 10 rotational, 10 O orthogonal system of curvilinear coordinates, 20

Index P perturbations, subharmonic, 342 plane, osculating, 70 precessing vortex core, 410 Q Q-vortex, 145 stability of, 204, 211 R Raleigh criterion, 202 regularization of the velocity field of point vortices, 310 Reynolds number, 211 Richardson number, 203 ring vortex, 116 infinitely thin, 86 Rossby number, 163 rotation quasi-rigid-body, 39 solid-body, 160 S soliton, 232 solution soliton, 224, 234 multi-soliton, 300 two-soliton, 302 stability of vortex flows, 155 stagnation point, 431 stream function, 32 of meridian section, 36 Stokes, 34 stream tube, 11 streamline, 11 swirl parameter, 171, 383 design, 385 T Taylor column, 167 tensor, rate of strain, 10 theorem Bernoulli, 18

487

Kelvin, 13 Lagrange, 17 Proudman, 169 Stokes (intensity of vortex tube), 12 Stokes (transformation of integrals), 11 torsion, 70 trihedron, moving, 70 tube vortex, 11 vortex, intensity, 12 V variation principle for construction of discrete vortex models, 310 vector, 9 Beltrami, 39 bi-normal, 70 normal, 70 solenoidal, 9 tangent, 70 velocity group, 163 phase, 163 velocity potential, 19 vortex bound, 53 Burgers, 149 Burgers – Rott, 218 Gauss, 136 helical double helix, 422 left-handed, 418 one-dimensional, 137 right-handed, 420 with a finite-sized core, 146 Hicks, 127 Hill, 124 hollow, 175, 279 Lamb – Oseen, 81 point, 77, 309 precession, of, 368 Rankine, 134 stability, 170

488

Index

starting, 347 Sullivan, 153 vortex breakdown, 431 bubble, 431, 446 closed, 441 conical, 432, 442 distorted, 440 double helix, 432, 439 flattened, 439 open, 441 spiral, 431 traveling, 431 vortex collapse, 338 vortex filament, 69, 390, 391 diffusion, 80 free, 432 infinitely thin, 69 helical, 91 rectilinear, 76 motion, self-induced, 82 sinusoidal, 241 vortex line, 11 stretching, 17, 195 vortex particle, 310 shape function, of, 315 vortex ring, 238 vortex sheet, 111, 157, 174 cylindrical, 113, 114 plane, 112, 115 stability, 157 vortical soliton, 264

vortex street, 360 vorticity, 9 centroid, 68 generation, 317 distribution, dispersion, 68 flux, 12 intensification, 17 W wake, instability, of, 357 wave number, azimuthal, 155 waves axisymmetrical, 165 bending, 225 circular polarization of, 165 dispersionless, 164 dispersion of, 164 “fast”, 227 helical, with constant amplitude, 430 inertial, 162 Kelvin, 186 kink, isolated, 430 nonlinear, 214 plane, 160 standing, 430 “slow”, 227 standing, 215

a

d

e

b

f

c

g

Color Fig. I.1. The observed shape of tornadoes. a – e, taken from the Photo Library of the National Oceanic & Atmospheric Administration (NOAA), http://www.photolib.noaa.gov/nssl/; f – g, photos from the website http://www.tornadochaser.net/may72002a.html (reprinted with permission from Mr. Tim Baker)

Color Fig. I.2. Coherent structures in a submerged impinging jet (Markovich 2003*): a - flow scheme; b, c - visualization using hydrogen bubbles of the nondisturbed jet and the jet excited by external oscillations with Strouhal number Sh = 0.5; d - velocity vector field and e - vorticity field measured using the method of Particle Image Velocimetry, Reynolds number Re = 7600, the color corresponds to the values of velocity and vorticity respectively

a

b

c

d

e

f

Color Fig. 1. Spatial shape of the streamlines in the flow induced by a helical vortex filament in a cylinder. a: h = 2, a = 0.1, u0 = 0; b: 2, 0.1, 1; c: 2, 0.7, 0; d: 2, 0.7, 1; e: 0.5, 0.5, 0; f: 0.5, 0.5, 1

a

c

b

Color Fig. 2. Visualization in the cross-section of flows for the open bubble (a), closed bubble (b) and conical (c) breakdown types (Khoo et al. 1997*). (a): Re = 3000, S = 4.0; (b): 2500 and 2.5; (c): 3500 and 6.0

Color Fig. 3. Visualization of the conic breakdown type in a swirling submerged jet of water (Billant et al. 1998*). The nozzle diameter is 25 mm, Re = 606, S = 1.37 (determined as the ratio of tangential velocity at a half of the nozzle radius to axial velocity at the axis near the nozzle aperture)

a

b

Color Fig. 4. A three-dimensional pattern of tracer trajectories at bubble vortex breakdown for Re = 1492, H/R = 2 (numerical calculation of Sotiropoulos and Ventikos (2001*) under the experimental conditions of Spohn et al. (1998)). Particles were ejected axisymmetrically and uniformly at the radius of 0.004R from the axis with a localization of 1.52R upstream from the bubble. The blue particles enter the bubble, red particles move around it, and green particles demonstrate typical toroidal trajectories inside the bubble

a

b

Color Fig. 5. Comparison of stream tube cross-sections calculated by PIVmeasurements and calculated numerically for the steady flow modes at Re = 1560 ( ) and 2043 (b) in a sector with the coordinates x = [72; 216] and z = [30; 174] (Okulov et al. 2004*)

a

b

t=0

T/4

T/2

3T/4

Color Fig. 6. Comparison of stream tube cross-sections calculated by PIVmeasurements (a) and numerically (b) for the unsteady flow mode with strong axisymmetrical oscillations of the bubble-like area of vortex breakdown at Re = 3004. Top to bottom: successive phases during oscillation period (Naumov et al. 2003*)