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


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


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


⎛ 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


∂′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


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


(

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


1.2.3 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.,


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


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


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


∂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


∂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.


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


1.3.4 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


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


(1.11) 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)


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


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


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


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


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


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


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


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


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


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)


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


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


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


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


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


∫ ⎡⎣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|.


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


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:


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


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.


∫ 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


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

∫

∫

∫


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


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-


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 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


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


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).


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


(∫

∫

)

∫

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 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 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 ⎣ ⎦ ⎣ ⎦ ⎠ ⎝


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 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 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⎫

∞

∑


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


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)


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)


−

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


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 ⎠ ⎝ ⎠ ⎦⎥ ⎣


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)


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


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


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


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 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 ⎟ ⎥ ⎝ ⎠ ⎝ ⎠⎦


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


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-


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


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-


(

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


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


Fig. 6.1. For caption see facing page


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


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.


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


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


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


Fig. 6.5. Interaction of three vortices with various circulations: − Γ1 = Γ3 = 0.1Γ2; – Γ1 = Γ3 = 10Γ2. Calculation by Veretentsev and Rudyak (1986*)


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


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:


∑ ΓαΓβ = 0 ;

339

D = 0.

α 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


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


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


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


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


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


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-


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


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


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


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)


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


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)


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


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*)


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


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)


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


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))


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


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


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)


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


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


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


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


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:


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


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)


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


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)


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


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

.


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


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


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


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


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


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


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


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


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.

386


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.

388


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


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


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


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


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


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.


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

400


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),


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


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.


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


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 .


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


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)


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


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


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


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


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


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


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


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


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


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


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


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


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).


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


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


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)


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


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


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


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.


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.

434


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


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


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

438


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.


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


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-

442


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

444


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


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

446


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)

448


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


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

450

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*)


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

452


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


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*)

454


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


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).

456


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).


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*)

458


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


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

460


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


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

462


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


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


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


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


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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Tung C, Ting L (1967) Motion and decay of a vortex ring. Phys Fluids 10:901–910 Uberoi MS, Chow CY, Narain JP (1972) Stability of coaxial rotating jet and vortex of different densities. Phys Fluids 15:1718–1727 Valentine DT, Janke CC (1994) Flows induced in cylinder with both end walls rotating. Phys Fluids 6:2702–2710 Van Dyke M (1964) Perturbation methods in fluid mechanics. Academic Press, New York Van Dyke M (1982) An album on fluid motion. Parabolic Press, California, Stanford Vasiliev OF (1958) The Foundations of Mechanics of Helical and Circulating Flows (in Russian). Gosenergoizdat, Moscow-Leningrad Veretentsev AN (1988) Study of vorticity dynamics and instability development in shear flows (in Russian). Thesis … cand of phys-math, Inst of Theor and Appl Mech, Novosibirsk Figs. 6.10–6.12 are reprinted with permission from A.N. Veretentsev Veretentsev AN, Rudyak VYa (1986) Vorticity dynamics in 2D flows of inviscid fluid (in Russian). (Preprint, AN SSSR, Sib Otd, Inst Theor and Appl Mech, Novosibirsk, No 4) Copyright 1986. Figs. 6.1–6.6, 6.9 are reprinted with permission from Institute of Theoretical and Applied Mechanics SB RAS Veretentsev AN, Rudyak VYa (1987) On processes of origination and evolution of vortex structures in shear layers (in Russian). Izvestiya RAN, Mechanics of Liquid and Gas (1):31–37 Veretentsev AN, Kuibin PA, Merkulov VI, Rudyak VYa (1986) On derivation of equations of motion of discrete vortex particles for axisymmetric flows (in Russian). Izv SO AN SSSR, Ser Tekhn Nauk, 10(2):45–50 Veretentsev AN, Geshev PI, Kuibin PA, Rudyak VYa (1989) On development of the method of vortex particles in application to description of separated flows (in Russian). Zhurn Vychisl Matematiki i Mat Fiziki 29:878–887 Villat H (1930) Leçons sur la théorie des tourbillons. Gauthier-Villars, Paris, Vladimirov VA (1977a) On vortex impulse for flows of incompressible fluid (in Russian). Zhurn Prikl Mekhaniki i Tekhn Fiziki (6):72–77 Vladimirov VA (1977b) Formation of vortex ropes from uprising flows over evaporating liquid (in Russian). Doklady Akademii Nauk USSR 236:316–318 Vladimirov VA (1985) On similarity of effects of density stratification and rotation (in Russian). Zhurn Prikl Mekhaniki i Tekhn Fiziki (3):58–68 Vladimirov VA, Lugovtsov BA, Tarasov VA (1980) Turbulence suppression in cores of concentrated vortices (in Russian). Zhurn Prikl Mekhaniki i Tekhn Fiziki (5):69–76 Vladimirov VS (1976) Generalized functions in mathematical physics (in Russian). Nauka, Moscow Vlasov EV, Ginevskii AS (1986) Coherent structures in turbulent jets and wakes (in Russian). Itogi Nauki i Tekhniki, Ser Mekhanika Zhidkosti i Gaza 20:3–84 Vlasov EV, Ginevskii AS, Karavosov RK, Frankfurt MO (1982) Near-wall pressure pulsations behind 2D interceptors (in Russian). Trudy TsAGI 2137:3–22 Vogel HU (1968) Experimentelle ergebnisse über die laminare strömung in einem zylindrischen gehäuse mit darin rotierender scheibe. Bericht 6, Max Plank Inst

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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*)

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