VORTEX MOTION IN THIN FILMS by DAVID B.Sc. The U n i v e r s i t y A THESIS THE HALLY SUBMITTED IN REQUIREMENTS of Toronto, PARTIAL FOR THE 1976 F U L F I L M E N T OF DEGREE OF DOCTOR OF P H I L O S O P H Y i n THE FACULTY THE DEPARTMENT OF We a c c e p t to OF GRADUATE this the thesis required THE U N I V E R S I T Y 0 David PHYSICS as conforming standard OF B R I T I S H December, STUDIES COLUMBIA 1979 Hally, 1979 In presenting this an a d v a n c e d degree the shall I Library f u r t h e r agree for scholarly by h i s of at make that it written for may financial of University P of 'ri IC British 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1WS 6 is Apr. 23. ? 19 SO of British for for extensive be g r a n t e d It fulfilment of available by gain ^ Columbia shall the that not requirements Columbia, I agree r e f e r e n c e and copying t h e Head o f understood permission. Department Date freely permission purposes thesis in p a r t i a l the U n i v e r s i t y representatives. this The thesis of or that study. this thesis my D e p a r t m e n t copying for or publication be a l l o w e d w i t h o u t my ABSTRACT The been generalized depth on curved to lowest of fluid radii order is of vortex tion classical to theory include surfaces. in of A number small is of criteria on surfaces revolution the both result of von staggered effects radial of elliptical theory finite symmetry introduce to and small core in Karman, are core size core treated atmospheric rings 0 In the in detail. and are examined with the principal a generalized of the generate of are depth conserva- described. rings of vortices contradistinction streets) may be motion. are The superfluid in The from shown case of of the vortices are Applications to stable. Departures distributions vortex varying which (vortex examined. has in to configurations are of motion systems stability found. cyclones di s c u s s e d . used vortex vorticity in of motion fluids Existence and double symmetric thin comparison the vortex expansion surface. for wobbles is in simple particular, in equations proved In of vortices The the streamfunction laws. rectilinear a perturbation considered curvature of to an CONTENTS Page ABSTRACT i i LIST OF F I G U R E S LIST OF T A B L E S v vi ACKNOWLEDGEMENTS vii I. INTRODUCTION 1 II. COORDINATES 6 1. 2. 3. 4. III. IDEAL 1. 2. 3. IV. 3. 4. 5. 6. V. 15 FLUIDS 2. 3. IN T H I N FILMS The V o r t e x V e l o c i t y F i e l d The V e l o c i t y o f a V o r t e x i n a o f U n i f o r m Depth The V e l o c i t y o f a V o r t e x i n a o f V a r y i n g Depth The V o r t e x S t r e a m f u n c t i o n Conformal Transformations Constants of the Motion SIMPLE 1. 15 17 19 F i e l d E q u a t i o n s f o r an I d e a l F l u i d The K e l v i n C i r c u l a t i o n T h e o r e m The T h i n F i l m A p p r o x i m a t i o n THE MOTION OF V O R T I C E S 1. 2. 6 7 12 14 Harmonic Coordinates Surfaces of R e v o l u t i o n Thin Film Coordinates V e c t o r Components: Notation . . . . Fluid Fluid 23 23 30 36 43 48 50 54 VORTEX SYSTEMS The M o t i o n o f a S i n g l e V o r t e x : No E x t e r n a l V e l o c i t y F i e l d a) U n i f o r m D e p t h , C u r v e d S u r f a c e s b) P l a n e S u r f a c e , N o n - U n i f o r m D e p t h The M o t i o n o f a S i n g l e V o r t e x i n a Uni f o r m S t r e a m The M o t i o n o f a V o r t e x P a i r : Yi -Y2 = i i i 54 54 55 58 61 i v Contents 4. 5. VI. VIII. p The S t a b i l i t y o f a S i n g l e R i n g o f V o r t i c e s on a S u r f a c e o f R e v o l u t i o n The S t a b i l i t y o f V o r t e x S t r e e t s on S u r f a c e s of R e v o l u t i o n a) S t a g g e r e d V o r t e x S t r e e t s b) S y m m e t r i c V o r t e x S t r e e t s c ) E x a m p l e : The C y l i n d e r d) E x a m p l e : The S p h e r e VORTICES 1. 2. 3. 4. 5. VII. (Cont'd) WITH g 66 72 73 81 85 90 CORES 96 The P o s i t i o n a n d V e l o c i t y o f a V o r t e x C i r c u l a r Cores The V a l i d i t y o f t h e C i r c u l a r A p p r o x i m a t i o n E l l i p t i c a l Cores P e r t u r b a t i o n s of Plane S o l u t i o n s 96 98 102 104 113 APPLICATIONS FINITE a TO 1. Atmospheric 2. Superfluid CONCLUSION . REAL Cyclones Vortices ., . BIBLIOGRAPHY APPENDICES A. S y m p l e c t i c B. E v a l u a t i o n FLUIDS 118 118 122 124 126 Systems o f Sums 128 128 129 e LIST OF F I G U R E S Page I. The S u r f a c e of Revolution II. Vortex in a Fluid III. Streamlines a Core Fluid Near of Varying t h e Core of Varying IV. Fluid with Depression V. Fluid with Surface p = f(z) 8 Depth of a Vortex . . . 37 in Depth 40 near the Origin Curvature . . . . near the Origin VI. The P a t h VII. Staggered VIII. Surfaces 56 59 of a Vortex Vortex Street: f o r which Definite Pair 65 T h e Modes t h e Mode M=N . . . 78 M=N h a s Stability 79 IX. Staggered Vortex Street: T h e Modes M=%N . . 80 X. Symmetric Vortex Streets: The Modes M=N . . 84 XI. Symmetric Vortex Streets: T h e Modes M=%N . . 86 XII. y ( x ) and y"(x) XIII. A Vortex with XIV. The P a t h of a Vortex XV. Velocity Vortex and V o r t i c i t y o f a 89 + an E l l i p t i c a l with Core an E l l i p t i c a l 110 Core 111 Quasi-Steady 116 v LIST OF TABLES Page 1. Regions of S t a b i l i t y of S t r e e t s on a C y l i n d e r Staggered Vortex 2. Regions of S t a b i l i t y of S t r e e t s on a S p h e r e Symmetric Vortex Regions of S t a b i l i t y of S t r e e t s on a S p h e r e Staggered 3. 90 91 Vortex 94 vi ACKNOWLEDGEMENT I would like to thank this work and for initiating Prof. his F.A. advice Kaempffer and for criticism throughout. The financial assistance British Columbia (Macmillan Natural Sciences and Canada (Postgraduate of Graduate both University Fellowship) Engineering Research Fellowship) is vi i the and Council gratefully of the of acknowledged. 1 I. INTRODUCTION In beginning that in point are 1858 of Helmholtz the a fluid is the in of which gradient advected. a paper vorticity in the of went N isolated points, equations describing the to order system the positions describing the original associated of superposition: of the with Each field, is called vortex is not over some velocity vortex refer of core. the theorems tion fluid to of to region t h o s e " made by as the position and (1869) the C is: later T c the of = Although of vorticity the principle points the with the its is The velocity of is change velocity position field) of similar of a of and a position Helmholtz' , where circulation of distributed c i r c u l a t i o n ; the under the of vorticity but jv_ • ds_ that, equations point reformulated idea showed Helmholtz, rate reduced with core. the zero be field point, the with can velocity practice, a single is is differential obeys v o r t i c i t y , along confused Kelvin each lines fluid vorticity. which total be a contour velocity. field In the differential of each vortex vorticity fluid the showed at if partial the associated known introduced the points the a vortex. Kelvin and around to to Lord fields point its that marked He production non-linear, is, fluids. show and of the velocity confined small (not a are that velocity vorticity. to ordinary of which p o t e n t i a l , the then motion of equations is sum on two-dimensional, at a 2Nth momentum a scalar Helmholtz incompressible, except study published any circula- v_ is the assumptions advected 2 contour is constant enclose the same set strength, y around contour any , of Routh function the (a vortex Kirch offAs in esimal with vortex addition, angular moment Section IV.6. Since little much in work has been is his work Hell sparked day. the flows of time formal on interest applications bluff was are to It the Lin of in are to the to bodies of vortex of (e.g., there real which the centre moment of of In as in ( 1949) atmospheric infinit- p.530). quantized subject laws of the detail has motion. to ( 1943). conservation discussed (1943) by systems conservation Onsager vortices Lin under vortex application of and q u a n t i t y known is governing generalized (1931) (1967), conserved stream- equations much translations theory other. conservation (Batchelor circulation conservation For to of no Hamiltonian leads prediction support past the both The the vortex of Masotti there rotations another written renewed street). system, publication the would Other to (1921), circulation. p a r t i c u l a r , the fluid and although to and the system transformations. of on vortex the streamfunction there that C = r^, . for under C and proportional translations and if then introduced invariances under circulation, is first c i r c u l a t i o n , under the vortices Lagally coordinate Moreover, enclosing Hamiltonian invariance of of Hamiltonian motion) associated time. a vortex (1881) ( 1876), any in in been However, fluid that systems; super- circulation persists weather Karman to this systems vortex 3 Almost that and the vortices parallel flow in the briefly pal the is of thesis depth to an Lamb (1916), in his is gradient perturbation scheme III). Vortex are examined of shown to which casts relation energy be the is shown the and invariances under generated. In of circulation to in of that with is more text, motion comparison It long two-dimensional the the depth the the princi- purpose detail non-uniform coordinates II), the (one which the in and of to but in in useful in momentum the depth to the IV flow. motion and of field which are into laws particular, the is be equations of vortices are streamfunction and the form. of to angular invariance A kinetic transformations r e l a t e d to the additional corresponding conservation a small order symplectic streamfunction to and assumed the vortex production fluid lowest Systems the subjected the under infinitesimal to the derived conservation shown which a generalized vortex simple governing Section of prove equations are solutions by which velocity equations between infinitely assumption surface. of incompressible governed in assumed small. in of the systems a scalar) (Section assumption the such is component motion of small has well-known determining under fluids vertical of and fluid of for surface to ideal is reduced (Section of that date is defining sections to problem examine variation vortices the a method further on rectilinear: uniform motion the that work curvature After later are a curved depth generalize the plane. on radii this so outlines vortices fluid all are moment under 4 scale transformations. shown that conformal the The vortex Section variation motion of V, on of the some a single simple vortex surface Also, s t a b i 1 i t i e s o f some of vortices results Kelvin on are vortices vorticity the that cores The solution. tion an the is of must have in are of in it is under are depth examined. are derived Karman be of analyzed. configurations The by In to symmetrically stable Lord (1912). contradistinction rings past determined. first von and vortices rotating cores of are from each of while finite the placed similar circular vortex, travels systematic examined of the a vortex that length a precise of in the drift if can the that, order The of the is introduce for the core simplest be e x a m i n e d plane a vortex time form. cores but of It of times mean closely case of detail. core in distributions examined. a c i r c u l a r core. shown appreciable simply curvature systems pairs those effects vortex the having It the motion is of and may vortex evolution scheme of departures that vortex revolution double VI, separation, uniform, unstable. the small into elliptical for are surface rigidly that, streets) within approximates ation Karman, of and (1883) found Section enough vortex is rings that wobbles long it von In argued Thomson (vortex staggered of is transforms depressions generalizations particular, of and surfaces (1878), results bumps depth coordinates. effects localized the fluid streamfunction transformations In depth When t h e then is its as to its in a lowest remain vorticity perturborder circular distribu- 5 The attempt in to which just as every the the a linear ent model Coriolis e.g., of only order the other in in The are was earth of is or a cone. cyclones effects since important of atmospheric is of only surface by Almost variation the of of system curvature via curvathe not variations if not derived is an cyclones surface approximation (1963a)) largely treated exactly, latitudinal B-plane effects for a plane which the motivated model terrestrial Veronis parameter that of approximation parameter. (see, enough a first 1 1 - VI perturbation included Coriolis Sections curvature analytic are of provide 6-plane ture work are consistthe is large also important. Our forces tion have if one cyclone) in curvature approach been alone. compromise. model including are discussed Another A rotating fluid of revolution in depth of the theory is of of Its liquid Sections discussed. inherent Coriolis motivating is et.al. helium is this is assump- a terrestria1 of surface cyclone in must a simple affect vortex curvature also the VII.2 work known effects How applicable be does and extent to the following. support to motion the is to known (1963)). affect Section 11 - VI a drastic effects surface Coriolis VII.l. surface In of a large and liquid Hell (Osborne vortices? of The problem vortices. the of problems direction; (clearly a good model highlight Section bucket other A good model both in to the entirely expects attempt some from neglected really an is to a superparaboloid the variation distribution which superfluid the vortices 6 II. COORDINATES Throughout cular tractable. discussed II.l M be a t w o - d i m e n s i o n a l , nates such (x,y) g^-U.y) h(x,y) called harmonic and to with surface will always over with 2 dropped i f of these coordinates by In has Riemannian to choose coordi- the form: that D . a torus) is M . 2 part In no c h a n c e it completely to multiply More connected complex are defined by: ( 1 1 . 1 . 2) ty is: + r dty ) z of considered. y = r si n 2 be to a r e s t r i c t i o n or sphere. (r,cf>) are For s i m p l i c i t y amounts n o t be only that parametrized coordinates in. terms (D i s equivalent plane (x,y) we s h a l l flows). This will ; = h(x,y) of non-negative (1909)). general D topologically is possible Eisenhart (x,y) harmonic there oriented, d i f f e r e n t i ab l e the f l u i d = h* (r,cj>)(dr h*(r,<J>) the mathe- (II.1.1) sub-domain which element is the m e t r i c of t h e complex (e.g., ds parti- 2 x = r c o s (J> line t o make a .h (x,y) i j be a s s u m e d Polar The It a real-valued some surfaces sub-domains surfaces . coordinates. unambiguously those that = 6 is g.. (see, f o r example, concerned necessary that Coordinates metric function is be f o u n d length. manifold with where i t will Here' the p r o p e r t i e s at Harmonic Let the work choice! of c o o r d i n a t e s matics are this (II.1.3) 2 future the a s t e r i s k of confusion will between be h* and h . II.2 Surfaces The in the r <J) element = g (u)du 2 2 is = r(u) an and ds so line Revolution of a surface of revolution can be put form: ds where of angular require g|(u)dct 2 + 2 (r,d>) 2 = gj(u)du 2 + period be harmonic 2TT . Let polars. Then: 2 h (r)((r'(u)) du 2 of r d<j) ) = 2 (II.2.1) 2 coordinate that = h (r)(dr 2 .+ 2 2 + r (u)d<J) ) 2 2 (II.2.2) g (u)dd, 2 2 that: r'(u) r ( u) = 9u(u) g (u) (II.2.3) (J) when ce : U r(u) = h(r(u)) It will prove p i ( r ) exp 9u(s) 9cpt s J = h ' ( r) r = (II.2.4) -^y convenient \ ds -rhr"+ to (II .2.5) (II .2.6) define , . 1 Then : r (u) g ( u j r ' ( u) 2 p(r(u)) = (J) • gj,(u) = 3 guT u T _d_ rg<b( du [ r ( u) + 1 (II.2.7) Al s o : rp 1 - (r) r ( u) r ' (u) gj>(u)i [g (u) du u g<f>(u) _d_ f-gj>(u) gu(u) du [ g ( u ) J (II.2.8) u The Gaussian curvature i k * of a surface 3 1k n r in 2 9x' L + r h 9 x is defined m _ i m kn r r n by m i k mn_ (11. 2 .9 ) r with .km 9 2 ln In harmonic 8 3g mn 9jm 9x Sx" n 3 8xi a surface of h r 2 The h = h(r) only interested 3 consider Figure line 3X ^ p, apply determine in r, f o r any m a n i f o l d applications manifolds which h(r) of a function revolution these z and d> one i s p(r) about an by: in surfaces axis. in IR and 3 p = f(z) to parametrize is: generally for coordinates defined coordinates such c a n be i m b e d d e d and a n d be c y l i n d r i c a l One c a n u s e element (II.2.12) 2 in those of and rh (r) For p h y s i c a l cj) 2 h = h(r) definitions . (II.2.11) V £vrch P'(r) r. dh_ h dr the surface I). (II.2.10) n 1 by t h e r e v o l u t i o n Let The d We t h e r e f o r e obtained n revolution above that "IR . i coordinates: -1 For 9 8xm 1 the (see surface 9 Fig.I: The Surface of Revolution p = f(z) 10 ds whence, = (1 + ( f ( z ) ) ) d z 2 2 using + f (z)d<j) 2 2 (II.2.13) 2 (II.2.4-6) r = exp rz (1 + (f'(s)) r ds fjil 2 2 (II.2.14) Zo r h ( r) = f ( z ) p(r) = Notice that then 2 (II.2.16) (f'U))*)* -1 £ p ( z ) _< 1 . I f the slope of f(z) is tanO . Similarly, ds f ( z ) (1 + p(z) = sine z = b(p) . (II.2.15) one can d e f i n e a s u r f a c e The l i n e e l e m e n t is: = (1 + ( b ' ( p ) ) ) d p 2 2 of r e v o l u t i o n by: + p d<j> 2 (II .2.17) : and (1 + ( b ( s ) ) s 1 r = exp 2 )" ds 2 (II.2.18) Po rh(r) = p p( r ) I f the slope The 1ater. (II.2.19) (1 + of (II .2.20) 2 b(p) function is tane then -jp(&wh(r)) p ( r ) = cose will also be o f . importance N o t i ce t h a t , : jL(*«h(r)) since: 1 (b'(p)) )^ p(r) < 1 . = 1) < o (II.2.21) 11 Special a) Cases Plane For the plane b(p) = const, whence: r p = 1 ; = 0 - ; h(r) rp (r) 1 ; (II.2.22) K = 0 b) Cylinder For r the c y l i n d e r = exp(z/R) p(r) c) ; =0 = R = const, h(r).= R/r ; rp'(r) whence: = Rexp(-z/R) = 0 ; ; K = 0 (II.2.23) Sphere For 0 f(z) is the sphere g (0) the c o l a t i t u d e . r = tan%9 = R Q o ; 9.(9) (p ; h(r) = ( i + yU) Mr ) K = = R^T JT+YTJ .where Then: 2cos %0 = = c o s 8 ; r p ' ( r ; 2 2 P(r) = RsinB ) = 4r (1 + r ) 2 = - s i n 2 e (II.2.24) 12 II.3 Coordinates for a Thin Consider, now a fluid having similar Mi M and fluid is 2 is topological much z. requiring surfaces on z so each that of x,y that two their radii on such g the surfaces, The of is z z line the Mi distance M 3 g z z depth the nearly are 2 be x,y surfaces z=0,z=l. (x,y) element are are of The are harmonic. constant x,y defined orthogonal in of z z 9z the to + thin 3 g dy yy + 2 J the independent s g zz dz z coordinate of II.3.1 2 can be chosen z. n. fluid, the '• 1 1 << by k*(x,y) is 3.2 then: 1 /g k*(x,y) 0 The thinness comparison y 2 between curvature: 3g The and form: 3 is M that z=constant constant 2g dxdy xy fluid and them t o The has + 2 3 Mi Mi surfaces of z. then = g dx xx 2 the lines constant Since that of the coordinates ds so than s i m p l i c i t y we c h o o s e coordinates these by characteristics. coordinates a coordinate For bounded "thin". Choose Choose smaller Film 1 dz zz /g 3 of the fluid with the distance appreciably. ? II.3.3 1 ZZ requires over that which the a depth g ,g „ , g xx' xy yy a a is small vary J in 13 zz g 3 grad 3 3 'XX 'zz grad g xx << g << 'xy The z component Therefore g v 9 v and g xy n r a d k* It varies 3h 3x" will << (II.3.4) 3z independent are of z. harmonic: (II.3.5) h (x,y) 2 components 3 8x of grad y h(x,y) + « 2 be a s s u m e d only over 1 I Hi*, h 3y approximately 9 8y the f l u i d k * 3h_ F " dy are then is (II.3.6) thin become: ( r i . 3.7) 1 that the depth distances much of the larger fluid than the i.e., h 3x << Equations whe r e b y : ~ y y that 1 also appreciably depth: 9 x ~ h(x,y) the requirements W k*(x,y) vanishes horizontal hor is: are approximately ' ) 1 (II.3.3) Mi t h e ( x , y ) c o o r d i n a t e s x << yy on J g yy zz ( ' y 2 g Xjf nearly a The and ~ grad 1 9z , g . . ,g__, v since xx xy /g—« XX Moreover, 5 o f grad g r a d. 'zz 1 II.3.2.8 << suggest 1 (II.3.8) a perturbation scheme 14 where the g x x (x,y,z) = h^x.y) g x y (x,y,z) - g y y (x,y,z) = h (x,y) + A g g z z (x,y,z) = Xk (x,y) + l9[ a small parameter which A is vertical The k*(x,y) 1 X g ^ U . y . z ) 2 Z) z scales of the f l u i d , = Xk(x,y) 1 y y 2 ) + 2 and h o r i z o n t a l depth + A g ^ to ( x ,y , z ) + ... (II.3.8a) ... (II.3.8b) > (x ,y , z ) + (x ,y , z ) + measures of the lowest ... (II.3.8c) ... the r a t i o (II.3.8d) of system. order, is (II.3.9) 15 II.4 Vector In in three will always be d e n o t e d (e.g., letters subscripts by lower case components then will be the l i n e ds = hfdx v x - h l case components letters by F o r t h e most with upper V^) , a n d p h y s i c a l v^). element + hfdy 2 case components part physical = ^ , v is: + h dz 2 2 (II.4.1) 2 3 components V* Contravariant used. If the vector (e.g., c a n be components, by u p p e r 1 (e.g., a vector components. V ) , covariant letters system contravariant and p h y s i c a l superscripts with coordinate ways: components, components Notation any o r t h o g o n a l represented covariant Components: are r e l a t e d y - h V* 2 - ^ by: , v - z h V 3 z = ^ (II.4.2) Covariant and the symbol differential derivatives V operator 8a for any v e c t o r V 2 = will a^ . JlL- + will be d e n o t e d be r e s e r v e d defined so f o r the by semi-colons two-dimensional that: 8a Also: (TT 44) 16 III. IDEAL III-l Field An stress The FLUIDS Equations ideal f o r an I d e a l fluid is c a n be d e r i v e d equation defined Fluid t o be one f o r w h i c h t h e as t h e g r a d i e n t describing momentum of a s c a l a r conservation is function then n (in c o v a r i ant form) : 9 V i . „k + dt where V\ velocity V i;k = " V and V fields n ; i (III.1.1) are the c o v a r i a n t 1 respectively and c o n t r a v a r i a n t and s e m i - c o l o n s denote covariant d e r i v a t i ves . The f f where + (PV ).,1 is p In replace is equation o f mass density (III.1.2) of the of vortices (III.1.1) is: = 0 t h e mass studies conservation i t is fluid. most convenient by t h e v o r t i c i t y e q u a t i o n . to The v o r t i c i t y defined by: = £ w' 1 i n k v . g" ' i k (in.1.3) 2 n ink where e 1 by 2 3 is e = 1 , and taking with e i n the antisymetric g = d e t g. . . the c o v a r i a n t k and using tensor density An e q u a t i o n derivative (III.1.2) of for having W (III.1.1) to e l i m i n a t e 1 is obtained contracting V 1 . : »i 17 3t (III.1.2), {' P Note a r e now r e g a r d e d as t h e of motion. (III.1.4) may be w r i t t e n i n terms of derivatives: k 3 3x since (III.1.4) ;k and ( I I I . 1 . 4 ) equations that i p IP J (III.1.3) fundamental ordinary J W + V' the terms F ,w\ ^ P ' W " k P in the connections 3V 1 3x k ( I I I . 1 .5) r .. c a n c e l Jk 1 18 III.2 The K e l v i n In earlier 1869, Lord work. definition Circulation Kelvin H i s most reformulated important of the c i r c u l a t i o n Circulation Let much contribution and h i s p r o o f of Helmholtz was t h e of the Kelvin Theorem. C be a c l o s e d C(t) = {x (s,t) contour x (s,t) , k = 1,2,3 of s and such all t t in the fluid: : 0 < s < 1} where . Theorem a r e smooth that: The c i r c u l a t i o n (III.2.1) x (0,t) = x (l,t) k around real-valued , k - k C is functions defined 1,2,3 by: (III.2.2) V (x (s,t),t)^-(s,t)ds 1 o If C is k advected 9 x 3t then i t c a n be p a r a m e t r i z e d such that: (s,t) = V (x (x,t) ,t) k (III.2.3) i There fore : dr.. n ^ C _d_ q dt dt J . . k v (x (s,t))^-(s,t)ds 1 k ^ ( x \ t ) V + (x ,t)| (s t) 1 k ; n f ) ,2..k '1 ,o -V (x ,t)V,. (x ,t) k ;n n i i r i ds + n. (x\t) k 3x •(s,t) 3s r + + V (x ,t)V (x ,t) i k ; n n i V (x ,t)fL(x ,t) i k i for ds 3S •(s,t) 19 = nU^i.tht) - Thus, This - nix the Kelvin ( o , t ) ,t) (V V )(x (0,t),t) k i k the c i r c u l a t i o n around is 1 + (v v )(x (i,t),t) k = 0 any a d v e c t e d c o n t o u r Circulation Theorem. i k (III.2.4) is conserved. 20 III.3 The T h i n In introduced powers this parameter approximations The f l u i d i s " t h i n " c) is o equations in A of Sec.II.3. so t h a t a r e as follows: the approximation of equations = p + Q i s nearly p ( 1 ) constant: (x,y,z,t) + ... (III.3.1) constant, We d e f i n e : v =V h x x X where V , v =v h , v y y V ,V ,V J To l o w e s t so to this order, physical velocity It velocity ( I I I . 3.2) 2 i s assumed little v order x of the contravariant i n X, t h e m e t r i c , v , v z is diagonal a r e t h e components ofthe field. i s small varies ='v k* a r e t h e components field. that, z z velocity and of the f i e l d t o be i m p o s e d of the f l u i d p(x,y,z,t) p of approximations are may be u s e d . The d e n s i t y where a number the expansion of the small (11.3.8) b) Approximation section allowing The a) Film that the v e r t i c a l i n comparison with v (x,y,z,t) x = v with component ofthe the horizontal velocity height. 0 ) x (x,y,t) + X v ^ > ( x ,y , t ) 1 + ... (III.3.3a) v (x,y,z,t) y = v 0 ) y (x,y,t) + Av ! ) y (x,y,t) + ... (III.3.3b) v (x,y,z,t) z = Xv ! ) z (x,y,t) + A v 2 2 ) z (x ,y , z , t ) + . . . (III.3.3c) Similarly, vorticity components are defined: 21 w = W h ; X x (III.1.3) w = W h y and ( I I I . 3 . 3 ) v ( 1 then X w v 1 •i i-+:-Xw[ f - 9 ( 1 w = W k* z Z (III.3.4) imply: ) w 1 ; y ^ ( x - y . z . t ) + (III.3^5a) + ••• ) (x,y z t) 1 ) 5 3(hv[ w 0 ) ) l + l 3(hv<°') 3x ... (II1.3.5b) + ^l^U.y.z.t) + 3x (III.3.5c) Written e x p l i c i t l y in terms of w ,w ,w x y z becomes w _3_ X 3 t P.h I i w f X + ir- — w h 3x Ph I J 3 3x fw ] y 3 t Ph ' V x h v x h w _ x _3_ ph 3x JL 3t w. Pk which W ~ f + ph 4- 3y- i ~ VI 3 y 3x ph J _L h 3x Ph JL order X h I w » + _?. _L pk 3z r w ~ V X h X h J (III.3.6a) > w. 3y w z _L h 3z Ph J L w. 3y in (III.3.6b) pk ' 3 z + JL JL h 3y ph w ph Ph \ V v ..+ _z _L k 3z X (III.1.5) ' w v w + JL JL y • + -L _ L y k 3y Ph h 3y Ph v ph to the lowest ^ 3 h ay + w ^ _3_ ph 3x w N . v , ph A becomes: 3z (III.3.6c) 22 _3_ w. 3t To lowest w. h order 3(hkv 3x h the equation 0 ) x ) 9(bkv 3x The w ^ equation of this completely v A u The equations this order 0 order. for w of x the production In form: in that i t is approximated necessary assumed that to a r b i t r a r y x so analyticity of f l u i d through at the boundaries; v °> = v that y ° > = 0 a t z the only conditions is will * w z be d i s r e g a r d e d to lowest remain to solution the t r i v i a l conditions fwhich is, require (( ) with v ^ = may be function, are that " " l , b u t ^ 3z compatible for i n the term .There conditions solution be o f t h e b y an a n a l y t i c 0 = at the any f u n c t i o n may be a b s o r b e d no s l i p A l l physical order. valid function any b o u n d a r y . = 0 a n d z be o m i t t e d . (Since accuracy The assumed b o u n d a r y slip a n c y w a l l boundaries analytic. order flux v A. x # f o r some from no x v x ', v y a n d w_ o f o r d e r ( 1 1 1 . 3 . 3 a , b ,c) departures A). > v equations in may t h e r e f o r e y to i s , changes be a s s u m e d t o be t a k e n order is according y changes f ( x , y ) + 0 ( A ) = const, simplicity X approximation. will boundaries, ( and w : t h a t x H e n c e f o r t h , the s u p e r s c r i p t s quantities o f w ° > and of approximation, of w and w is (III.3.8) However, order induce (III.1.2) conservation = 0 independent w x ,* w y o f o r d e r (III.3.7) ) expressing ( 1 1 1 . 3 . 5a , b) , t o t h i s are Q ) y = 0 3z o f mass 3y are also y w ^ 3z no s l i p 0. of there however, that = 0 boundary Notice that is 23 since v z ^ » 0 = a n c ' surfaces z = const., with flux zero The reduced to boundaries independent t through problem of n upper e the lowest these of z than . w x lower order boundaries solution is are consistent surfaces. finding a two-dimensional (other and the fluid problem and w y motion since ) are, all to has now fields lowest been and order, 24 IV. VORTICES In in are which f i l m of equation of the 3(hkv ) 9x 3y general mixed real partial = defined except at = 0 z (x',y') r that 1 1 if , 2TTY : second flow. to be The the ; by the k(x,y)? (III.3.8) that is, 9y 2TTY<S is: 1.1) y (IV.1.2) j having continuous ip(x,y) field incompressible the is of a the vortex velocity region of (IV.1.1) field flow, D , and: x ^ 6(x-x')6(y-y') (IV.1.3) a contour (x-x ' ) 6 (y-y 1 3(hv ) 3(hv ) 9x 8y y ' satisfying 9x is u velocity with then: = function order. throughout 9(hv ) C 9 x ^(x,y) 3(hv^ 2 (x ,y )eG the of everywhere w_ = h r-T Notice hk derivatives at w described of (IV, function for having a system 0 9y valued is depth of is solution: streamfunction (x',y') fluid thesis y hk some the motion conservation = _L i i ( x , y ) for of of Field mass 3(hkv ) has problem ideal and Velocity x main equations h(x,y) Vortex The the the a thin function The FILMS section what N vortices IV.1 THIN this addressed: metric IN ) dxdy x dxdy interior G such that Thus, Y is by the Kelvin c a l l e d the (I V . 1 . 3 ) one C i r c u l a t i o n Theorem vortex strength. x , y ;x unit of strength 1 ,y ' ) at is (x the region 1 of 1 flow is for is 1 a vortex of a Green's function operator. {(x,y)eD> where the boundary is: boundary y = M U 3-D i=0 3.DH3.D = conditions for is 3-D often = ^ Theorem ^ 1 a i given, D assumed, o that ¥ ( x ,y ; x ^ , y . ) const, although = implies . (IV.1.6) on are: 3^ D, i =0 , . . . , M ; i = l , . . . ,M i l»---»M r <j> , i / k K (unspecified) r r s t r e a m f u n c t i on differential 1 1 It into (IV.1.5) 1 a self-adjoint elliptic 3D = The (IV.1.2) = -2TTS ( x - x ) 6 ( y - y ' ) x,y;x ,y ) D constant. Substituting ,y ). The of a has : £ v v ( x , y ; x ' , y ' )] where y is the • (IV. this Notice boundary is not that necessary, the conditions Kelvin are 1.7) that Circulation constant in t i me. The (IV.1.5) existence and the boundary ^(x , y ; x ' , y ) 1 is well known demonstrate and = (Courant the ^ and existence uniqueness of ¥ satisfying conditions: = const, Hilbert and on 3.D,i=0 , . . . , M (1962)). uniqueness up We u s e to an (IV.1.8) this to additive m constant of ¥ Set ¥ M -tuples, of the space satisfying = 0 and 0 ( Y ^,. . . , f ^ ) boundary of of map f: the A-*B having 1 prove constant Y under the to arises necessary to to possible the B be and is Y unique an show a unique and 3D to the r does that (r_. g_ equal D Clearly up to f an n is (the to is it linear. is only freedom fix Yo ,...,I\ n d the additive (IV.1.7) not There M ) of ). the Thus (0,...,0) = U pre-image. Suppose (IV.1.5) vector on d •U has values unique = l inverse one the Y F.. . conditions when the of (IV.1.7). finding setting this has A corresponding condition for f space Let ) (IV.1.7). vector conditions boundary that conditions M , and 0 exists constant only 3 D 9^D prove Q g boundary boundary around that necessary is (IV.1.8). , d e f i n e d by 3.j D, i - 1 , . . . , M, Y = 0 on circulation the corresponding condition values y ( x ,y ;x ' ,y ) it consider M a natural additive boundary M- tirpl:e.s, ( r ^ , . . . , r i possible To the ¥ ( x ,y ;x (IV.1.7) k 9 1 ,y ) and 1 with n Y * (x ,y ;x 1 ,y ) 1 satisfy r\ = 0,i = l,...,M. 3.U dxdy D n lyjy-y*) Then: » v ( d x d y (IV.1.9) where left denotes a d e r i v a t i v e normal t o the b o u n d a r y . dn. s i d e v a n i s h e s : , s i n c e from ( I V . 1 . 2 ) and ( I V . 1 . 7 ) : 3 (Y-Y*) h.D 1 k 3 n ( ¥ _ y * ) d s = c o n s t . xT„ a = 0, n i i=1 The M U (I.V.I.10) 27 a n d when i the of right = 0 0 (IV.1.9) on vanishes 9 D 0 by Iy(Y-Y*) • V(Y-Y*)dxdy whence V ( Y-Y*) Thus, image and (IV.1.5) = 0 and, r r therefore and there (IV.1.7) term on Hence: (• I V. 1 . 1 1 ) on (0,...,0) = is first (IV.1.15). V = Y* exists which The = 0 since ( 3D»---» 3[)) . 0 has a Y( x ,y ;x unique 9 D 1 up t o , Y = Y* a unique ,y ' ) . pre- satisfying an additive constant. Y ( x , y ; x ' ,y ) a l s o has 1 V(x,y;x' ,y') and can the reciprocity property: =• Y ( x ' , y ' ; x , y ) be w r i t t e n i n the (IV-.1.12) form: V ( x ,y ;x ' ,y ) = - A ( x ,y ; x ' ,y ' )lnr + B ( x ,y x',y ) 1 1 ( I V. 1 . 1 3 ) with r = D if k is [(x-x ) 1 analytic circulation 2TT + 2 around f = 27T (y-y ) ] 1 (see, the I 91 k 9r 2 for and w i t h A and B analytic example, Sommerfeld contour r = e is: 2 ^ A ( x ' , y ' ; x ' ,,y' ) k~(7 Ty T as small rd6 2 r _ T in (1949)). The e + 0 whe nee A ( x ' ,y 1 ; x \ y ' ) = k(x' ,y' ) Substituting as r (IV.1.13) into (IV.1.14) (IV.1.5) one finds that 0 _9_ 9r , + 1 9A k 97 - „ 0 (IV.1.15) 2Q when ce : VA(x' ,y' ;x' , y ' ) The with I total positions = %vk(x',y' ) streamfunction (* ,y ) n (IV.1.16) f o r a system and r e s p e c t i v e n of N strengths vortices y ,n = l,...,N n s: = <r(x,y) n n where ty*{x,y) flows (e.g., N z Y Y ( x , y ; x ,y ) + n n n = is a uniform and H i l b e r t (1962)) The s p e c i a l (e.g., Let be a c l o s e d vortex lies tacitly of D to i n f i n i t y surface C ty*(x,y) must case a sphere) on presents and w i l l of vortices i s , however, contour C . assumed t h a t on s u c h Denote no r e a l n o t be what is by C e x t the e x t e r i o r o:h(v o f what i s , of course, dx + v d y ) = c is such C. . l nt unbounded interest. that no and i t s t h e i n t e r i o r and arbitrary). Then: K dxdy J considered a surface, by bounded. problems of p a r t i c u l a r its interior (the choice D is on a c l o s e d J exterior satisfy: (IV.1.18) f a r i t has been extension further. stream). imposed o So (Courant (IV.1.17) t h e s t r e a m f u n c t i on due t o o t h e r V," The t*{x,y) 1 Cint = s Y n z „ne C~ i n t ( I V . 1 . 19) 29 Integrating -o h ( v C around dx + v dy) J C in = Z n y the other d i r e c t i o n one y c finds: (IV.1.20) ext whence E y n=l = 0 Thus, The velocity must at be (IV.1.21) n there cannot and 1 a single vortex f i e l d due t o t h e s t r e a m f u n c t i o n i n t e r p r e t e d as (x'y ) be one of that due negative on such a surface. ¥ satisfying (IV.1.5) to a vortex of unit strength unit strength at infinity. (Note t h a t on a c l o s e d s u r f a c e t h e p o i n t i n f i n i t y i s j u s t l i k e a n y o t h e r point: e . g . , on t h e s p h e r e i t i s t h e s o u t h p o l e . ) position vortex of at put streamfunctions infinity If (we N such the satisfying the disappears. depth of the fluid is constant, simplicity) and x ,y ; x ' , y of the Laplacian. ¥ can then b e - w r i t t e n : Y(x,y;xiy') = -Inr Sommerfeld 1 ) k is for e.g., super- (IV.1.21) k= l function (see, Upon t h e constant becomes the + B(x,y;x',y') (1949)), whence, Green's (IV.1.22) upon comparison with (IV.1.13) A(x,y;x' ,y') Moreover, are no boundaries, = 1 (IV.1.23) since: V £nr = 2 2TT6 (x-x ' ) 6 (y-y ' ) , i f there then: B ( x , y ; x ' ,y') = 0 ; ^ ( x , y ; x ' , y ' ) = -h&n [ ( x - x ) + ( y - y ' ) ] 2 2 (IV.1.24) When the flow induced constant fluid by there depth is constant B to be due is no such to one the simple can therefore boundaries. decomposition If of regard k is the the not flow. 30 IV.2 The V e l o c i t y o f a V o r t e x i n a F l u i d o f U n i f o r m Depth The m o t i o n of the v o r t i c e s i s governed by e q u a t i o n (III.3.7): _3_ 3t which h implies Zl 3x that remains n W of that (IV.2.1) i s , each zl of a vortex system: i . e . , the s t r e a m f u n c t i o n sati sfies: but t h e x N = -2i-2 V^J F and n c a r r i e d along y Y 5(x-x )fi(y-y ) n n (IV.2.2) n are time-dependent, n moving t h e v e l o c i t y o f each expand the v e l o c i t y f i e l d near i t s s i n g u l a r i t y . (uniform It depth will fluid) prove is treated convenient variable z = x + iy variable z of Section are then z and symbols (there should II.3). z-= x - i y . in arguments. x - Equation 1 v y 2i h ( z ,z) 3z The c a s e first. be no c o n f u s i o n The i n d e p e n d e n t with the variables For s i m p l i c i t y the present (IV.1.2) ^ v o r t e x we t o i n t r o d u c e t h e complex are r e t a i n e d f o r a l l f u n c t i o n s v as i f by t h e f l o w . In o r d e r t o d e t e r m i n e k=l fluid c o n c e n t r a t e d and t h e f l o w r e t a i n s t h e characteristics always = 0 as i t moves a r o u n d i n Thus, the v o r t i c i t y c o n c e n t r a t e d at the p o i n t s fluid. (x^,y ) rW. is advected: -rH e l e m e n t -mai n t a i ns i t s v a l u e the z 3 h dy + is (z,I) despite their change then: (I V . 2 . 3 ) 31 Which, upon substitution (IV.1.17) of N r (IV.1.22) and -i.y . 3B(z,z;z. z-z,. T k + If |z - z is n ,z. ) 3z 2 i ^ ( z , z ) \ (IV.2.4) small : 8h n - Yn 8z •y. TV - becomes y z-z + + n i|>*(z,z) 97 2 Z Y k^n , k + 8z Y B(z,z;z ,z ) n i'(z,z;z K -1 dh (z-z ) V n' _ 1 z k ) } k n n + ° d - n z z z= z (IV.2.5) where h = h(z n The n ,z ) n' velocity field: concentric about z a to vortex. velocity The flows the velocity arise By velocity radially vortex the . n either. field Prefering field from z It should n not projecting the flow onto introduced The velocity giving field: i v seem contain does - z no hence might v_ are v and because terms i v. y x = y h ( v cannot ,z n V«_v = 0 plane to v it cannot 3h a that such as ) {z-z velocity a radial = They -v/V£nh source terms. to vortex this. but "fictitious" the - 1 impart 1 ^ ) impart terms is )(z-z ) n' n' i"Y ^ ) paradoxical satisfy rise v direction radial the hjz V x " i y v = hf f ^ 3h" 1 n ^ 32 + 2 Z Y Y(z,z;z k?n + uniform and t h e r e f o r e carries 4>*{z,z) +; £ ,z ) • z=z z= z„ k K is i Y. B(z,z;z ,z ) n induce vortex other terms any motion is vanish the vortex as in the vortex z at + 2 at z z therefore: ° that s n . n with 3 they cannot The v e l o c i t y of the J + ifi*(z,z) Z k^n alonq n 1 9h + n r it. All n Yk^Cz.zjz. K k ,z. ) . k Y B(z,z;z ,z ) n n n (IV.2.6) r z=z, z = z But u = h x x n n , u = h y y n^n ( I V. 2 . 7 ) Therefore : i l _ L nr, - £nh(z,z) h 9z n 2 + or, r e v e r t i n g + ip*(z,z) r Z k^n Y B(z,z;z ,z ) n n z.) • z= z •n y.V(z,z;z K + k (IV.2.8) k z = z n r to (x,y) coordinates: 3n. h (x ,y ) 2 n n -1 3y h .(x ,y ) 2 n = n y=yn x x n ^ n 9x x =xn y yn = (IV.2.9) 33 with Y fi = n ^-Jinh(x,y) ; + These Z k^n are + T ^ * ( X ,y) y B ( x ,y ; x + n ,y n ) p Y . V ( x ,y ;x. ,y. ) K K the (IV.2.10) K equations of motion for the vortex system. If there are no boundaries then from (IV.2.8) and (IV.1.24) if = X h n Example: on the method n order vortex to (From to This which in and is its , and 8 z r " u 1 (IV.2.9-10) determine also be with vortex strength of of in south The the harmonic pole). and z harmonic coordinates: n - ' • ii p -Yh'(r') 2h(r') at -y Since of i " by an give of a the vortex alternative (x',y') at the there ? , ) -y(p(r')-l) 2r' the corresponds are no ij;* = 0 depth, vortex on infinity. coordinates - hTT^7rT Tr' '- y velocity y uniform velocity indeed first. strength boundaries . the do determined agreement the no k = polar we can infinity flows, u a , i.e., % " '" in (IV.2.11) n that counterpart (II.2.4), external or, check sphere. 0 = ^ B = 0 h velocity, Consider sphere n z The V e l o c i t y o f t h e V o r t e x on a S p h e r e w i t h No B o u n d a r i e s a n d U n i f o r m D e p t h In correct * , , k/n V is , therefore: < - ' ' IV 2 12 (IV.2.13) 34 Using (II.2.4) : u r This = % = 0 c a n be d e r i v e d ; u„ p = £ Rt ^ a n (IV.2.14) ^ alternatively, for this case only, is zero, as f o 11ows: Since consider Y the v e l o c i t y t o be t h a t vorticity upon The t h e sum o f a l l v o r t e x such that superposing velocity there is o f each owing the vortex ordinary has vortex pole 1 Rsi n 0 w •3( v evaluated and t h e to one g e t s ay . zero completely Then vorticity symmetric of the sphere; motion. then strength hence, The v e l o c i t y satisfies field (using sin-9) 3v„-, 3 i<J> 89 •ay (IV.2.15) solution si n 0 If is of coordinates): 0 which fields vortex at the south polar and equal t o t h e symmetry no s ^ l f - i n d u c e d vortex incompressible constant the v e l o c i t y field i t s core of a single i t is i s everywhere about of field strengths v^ is t o be b o u n d e d by r e q u i r i n g • 2 iry Li m f 2TT cos 6 at B const. 0=0 , 6=1 (IV.2.16) a is that: VxRsin0d<j> = L i m - 2 i r a Y R ( 1 - c o s e ) 2 0+TT •4fTaYR 0+7T : (IV.2.17) 35 Therefore : 1 a = 2R (IV.2.18) 2 and (IV.2.19) v The given by velocity (IV.2.14) of the vortex since the vortex at the south (Of course, the v o r t e x that is at (IV.2.19) the is pole.) only i t is pole. at carried at is 1 along in The two methods at the south correct (e ,^') pole is the flow are also the i n s t a n t therefore in moving this of agreement. so vortex 36 IV.3 The V e l o c i t y o f a V o r t e x i n a F l u i d of Varying Depth The is more owing velocity complicated t o t h e more V(x , y ; x ' , y ) . 1 meet the upper that the boundaries curved must fluid of no l o n g e r core will reason curve uniform is the vortex in order (See F i g u r e so II); the v e l o c i t y fast.- T h u s , one of a vortex i n some way on t h e in a structure core. velocity as in Section field II.2, by e x a m i n i n g in the neighbourhood of z^. ( I V . 1 . 2 ) , ( I V . 1 . 1 5 ) , a n d (I V . 1 . 1 7 ) : 2i y h ( z , z ) k ( z , z ) 9z -y 2i ( h k ) ( z n , z A ( z + Y " n n l 'n z ' z A ( z ; n , z n ; z n , z n ) ) , z n : z T h k ) ( z n , z , z n ) 9 v( h k ) v ( z ' = , z n 9z ) 9 A ( z , z ; z ,z ) n n 9z \L Ln z=z f z-z. n ^ z- z n 3A(z Y, • ,z ;z 9z n to an i n f i n i t e s i mal l y , s m a l l infinitely depend in perpendicularly surfaces. propagate depth depth that slightly surfaces ( 1358) t h a t must of for this b u t must that of .varying of the s i n g u l a r i t y are streaming depth We p r o c e e d physical in a fluid bounding at the outset of varying its straight and lower in a f l u i d nature by H e l m h o l t z vortex expect that The p h y s i c a l is was s h o w n than complex core It of a vortex ,z ) n z=z z=z n x n & n l " n z z ) the. From Fig. II Vortex Core in a F l u i d of Varying Depth 38 + Y Z ) n + 3(hk),, 3z W " n' n (hk)(z ,z ) A ( Z n i *(z,z) K + From one K n and = h(z n 3 Y B ( z ,z ;z ,z ) 3F n n n' k n n 0(|z-z Un|z-z |) (IV.1.14) h . E Y | ^ ( z , z ;z. ,z. ) z=z k^n z=z + r , ( v (IV.1.16) ,z n (IV.3.1) n n' ) ; k and k(z E n writing ,z ) n n v (IV.3.2) gets : iv k ' n n -y 2i h k n n y I T ^ rnr' + ,Y v 'n n. Y 3z n n n ,z n ) + h iii* v ( r n 'z- z } n 3z n n z- n J z k 3h _n_ n h 3z n 3z "2 B(z,z;z , 9k Y _JT_ , i E Y vp(z,z;z + n 3k Y 3k ! n z ,z ) .zjl- n k^n z=z n Z-rZfj + As no before, velocity to the 0(|z-zj£n|z-zj) the terms vortex. of the term in carry vortex with them. source of section. the It n difficulties is remaining &n|z-z | divergent The as are term outlined z->z z- z an d z- z The exception the in (IV.3.3) terms at but with uniform in can z - z„ and the has beginning a the therefore is in\z-z^\ impart the of the definite n direction implies (along that the the curves vortex of moves constant with k infinite ). It therefore velocity along 39 k = const. small but surface and If, however, finite of the the -iy r—n 3k r— V n carried along v.. x is assumed c i r c u l a r core core eq u a l s : it in iv.. ' y velocity Ine radius due to . The n Z Y.Y(z,z;z k^n + n the volume advected of each (h h where n n n X k £ is and S e c t i o n = % the III.3. increases k the streamlines of not concentric (See the vortex Section in VI a more that this when (X the appears of 0 small As calculation + this the and, core the uniform is therefore h ^ 3z n (IV.3.4) k = since must be z n n the fluid is constant. incompress- Hence: (IV.3.5) perturbation parameter 0(k|V£nh|)) of the move rigorous manner. t e r m may be a in direction finite Suffice incorporated is n shrinking, term w i l l of Section approximately In the of it by a be the centre -Vk . cores say, However, circular, deferred sized to of II.3 constant. decreases. III). velocity effects , and core flow, while Figure to is e X] n n k n radius the at a ty*{z,z) ,z.) Jz=z k z ible, + n k is has flow: n w term boundary Y B(z,z;z ,z ) Now, , then this ^-r-(-2~ -^-(Ane^-l) h k \ 2 3z + vortex n uniform n the n 3 2 the of that for are of The until are the "renormalizing" treated present, : Fi g.111 Streamlines in a Fluid near of the Varying Core of Depth a Vortex 41 i.e., by replacing a n = a n by •3 e (IV.3.6) n whe r e 'cn * n n Y E is c n the k i n e t i c velocity and k of (IV.3.7) p energy the vortex r e n o r m a l i z i ng a 3 u x " 1 u y " within is, nth substituting core. (IV.3.5) Thus, into the (IV.3.4) : n 2i . I • 'n h k [ 4 n n 9z n f In a* -2 h k n n n K 2 Y B(z,z;z ,z ) 9z + the n + n 3z ^*(z,z) S Y * ( z , z ; z , ,z, ) k?n l An z=z (IV.3.8) k K k k J = n whence, using (IV.2.7) and s i m p l i f y i n g the equations of motion a re: X n " , 1 h k 2 n 9ft* n 9y ; y -1 h k n d 90* n 9x 2 n (IV.3.9) with: Y = -^ ft* h (x,y)k(x,y) 2 .k(x,y)JU + ijj*(x,y) + that constant) if (IV.3.10) one p u t s n reduces k=l y n B (x ,y ;x ,y ) ' n n' v , J s Y ^(x,y;x ,y ) k^n n n Notice + . n to J (IV.3.10) n n (IV.2.10) (within an additive The that the circular that velocity core is cores for will in cease the approximations, the vortex c i r c u l a r and (IV.3.9-10) estimates of periods are remains general to given be be been which Section derived circular. distorted valid. over in has Order by of However, advection magnitude (IV.3.9-10) VI.3. assuming are good so 43 IV.4 The Vortex It vortex is Streamfunction now s h o w n t h a t one c a n d e r i v e a generalized streamfunction. From (IV.1.14) A(x,y;x' ,y') and ( I V . 1 . 1 6 ) : = k(x\y') + &=f± 3 9k(x' ,y') + iXzlll 2 k ( + 3y ^ ' Q ( y 2 ) v , ) ' k ( x , y ) k ( x ' ,y') + 0 ( r ) , = h r 2 h = [(x-x ) 1 + 2 (y-y ) ] 1 ( I V. 4 . 1) 2 whence : 3A ( x , y ' ;x ,y) 3x M (x,y;x' ,y' ) 3x |y=y and a similar both obey equation in the r e c i p r o c i t y 3B_ (x ,y ;x ,y ' ) 3x [y=y -r—. = c n be w r i t t e n 1 3ft Y h k 3y ' n n n n . 'n 2 J since 3B ( x ' ,y ' ;x ,y) 3y x=x |y=y can t h e r e f o r e Thus, Y 1 and Inr property: 1 (IV.3.9) (IV.4.2) 1 = in the (IV.4.3) x=x |y=y form: -1 3ft Y h k 3x n n n n (IV.4.4) 2 1 whe r e ft = \l y + 2 x k * k (x o n W k ^ V V V V ,y ) *j ^ n ^ n ' + ' h ( x ,y )k(x ,y ) ., n n n n y n >n>n x y 2 &n 17 J + 2Y ^ * ( X ,y ) n n n' (IV.4.5) } 44 is given may by be C.C. put in x = o where x^ 2-form, is of G Denote cores the symplectic As is vortex shown in streamfunction Appendix A, (IV.4.6) is the vortex exterior The {(x,y) boundary core : [(x-x ) by 3G may Gn + 2 n a symplectic derivative. streamfunction fluid. is a be of the ( y - y j The related 2 ] * region nth to the vortex < e > of is cores is: fluid outside the is: assumed t h a t are the kinetic e energy 's are sufficiently of the fluid small that the in D* is: (v +v )h kdxdy 2 D* y 2 L (IV.4.8) disjoint. The 2 2 x p( ? k I rati 2 + 3x since: kinetic (IV.4.7) H N D \ U 9G n n= l It (IV.4.4-5) form Vft V = its (1943). the -1 the n Lin of a 2 N - d i mens i o n a 1 v e c t o r , and The e n e r g3yj a generalization k 3x 'dip' 2" M 3y dxdy Ik 9yJ dxdy ( I V. 4 . 9) 45 Applying Green's where is to (IV.4.9): (I V . 4 . 1 0 ) 2 L Theorem 3D 1 the d i r e c t i o n a l derivative normal 3D* to Si nce : N * and k-^k^k^' = since (using <? y ; k' k x y = ds 3D one - = E. n ds 3D 1 k 3n d s (IV.4.12) 3n N n=l first ty* term n y r k ^ f 3^ ds k the energy by E^* 3 D * may be i n 3n (IV.4.13) 3D* be d e n o t e d M 3D* k -ds k=l boundary 3D* = y. n= l represents and w i l l The y N 8 ¥ • p " ^ d s k + p X so n D' ^ 3D^ by , y )) n has: •* The x - r ( I V . 4 . 11) ^*( 'y) + ~ ^( x , y ; x n V l)rr 1 ) r N U 3.D u U 3G. lj =l U=l 1 J due t o t h e f l o w induced . decomposed. i (IV.4.14) that 3D Since const., ty* * is k 3n ds = i=0 constant the f i r s t N z on k 3Di 3D n ds 3n z j-1 3D.n k is 3^„ ,4. 3G- 1 !Inds and t e r m on t h e r i g h t +• 3n constant. ii _Ji k 3n a d s s (IV.4.15) " YDn r Since = ! e 46is very small, ¥" * Therefore K ty* ^ n ds 3G,-: x , y ) on f] n n n K * k 3n k 3ri •ds = 3G„ . n n 2TT^*(x ,y ) j (IV.4.16) j S i mi 1 a r l y : 3D * M 1 , Y 3. n N Y { 3Y n =1 (IV.4.17) and t h e f i r s t % term i s c o n s t a n t - Y(Xj,y..;x ,y ) n on n .n ^ k I T -37T -3Gj 36^ as b e f o r e . If n^j , then and: y If n=j n 3G and n but * d s k^j (IV.4.18) ^ V ^ V V one can p r o c e e d as i n ( I V . 4 . 1 2 ) k , whence (IV.4.18) holds unless n=k=j to . transpose S i n c e on : k £ n r + B ( x , y ; x , y ) n n n n n J Y , J (IV.4.19) 3Y 3G, ' 2 7 T 1 3 17 3 f - n ( f k A n r + B ( >y; x x n >y ) n ende r-.e, = -2rr(k £ne n Therefore, using n - B( x ,y ;x ,y ) ) n R n (I V. 4.1 3,15 ,17 ,18 ,20) one f i n d s : N f E* = E, * + TTp £ { 2 y ^ * ( x 1 - (IV.4,20) n k £n n £ n ,y ) + y B ( x 2 ,y ;x ,y ) N | + T T P ^ k ^ Y Y ^ ( x , y ; x , y ) + const. (IV.4.21) n n n k k 47 The kinetic energy of a l l the f l u i d is therefore: N E = + 2TTP-U E k.= l N n=lL which, upon 2 N V(x ,y ;x ,y ) n n k k r YnB ( x n 2 + 2 Y k^n x ,y 17 n Y ^"(x ,y ) n n n + n substitution ;x of ,y 17 n 7 ) + y -^-ln 2 'n 2 a n const. (IV.4.22) (IV.3.7) and (IV.3.6) for E c n be come s : E = E ^ * + 2-n-pft + Thus, of Q is the flow which is const. proportional to the energy due t o t h e v o r t i c e s . of that part 4,8 IV.5 Conformal Transformations Since, function of Laplacian, when the a single it is fluid vortex natural depth is to the ask a conformal transformation since these the first considered by Lin tion (1943) the who vortex surfaces under N denote We now of the that constant function vortex and under the in the changes , question more conformal transforms of z+z This later, stream- motion coordinates, invariant. (1881) was generality, transforma- as: dz In dz n (IV.5.1) n transformed show of the the that depth quantities (IV.5.1) remains provided the K = 2iry (Lin's valid surface is on and curved invariant transformation. Let z = f(z) coordinates. harmonic showed - h Z Y n=l tildes W = -27rfi) . Routh streamfunction Si = where by Laplacian constant, Green's how under leave is be Notice that coordinates ds 2 a conformal (x,y) transformation such that x + iy of complex = z are since: = h (z,z)(dx 2 2 - h (z,z~)dzdf ~ TTzlT'TiT" 2 + dy ) = 2 h (z,7)dzdz 2 _ h (z T)(dx + f'(z)f'(z) 2 2 dy ) 2 , (IV.5.2) ? ) Th us : h(z,z) The dz dz = streamfunction n for $(2,f ; 2 \ Y ) r = (z,z) a vortex (IV.5.3) in ¥ ( z , z ; z ' ,z ) 1 the transformed system is: (IV.5.4) 49 and ifi*(2,z) since (IV.5.5) r(2,z) = the Laplacian is invariant. + B(z,l;z' ,f') -ln\z-T'\ From (IV.1.22) + = -ln\z-z'\ B(z,z;z ,z') 1 (IV.5.6) o r: B ( z , z ; z ' ,z"') = B ( z , z ; z ' , 7 ' ) + "An z - z z-z (IV.5.7) The r e f o r e : B ( z ' , z ' ;z' , ! ) 1 B ( z , z ; z ' , z ' ) + In = Lim z+z B(z' ,z' ;z' ,z') In the the transformed transformed ° = k y = Q (IV.5.8) the motion ^ 1 J/nV<VVV'V + dz dz In is derived from steamfunction: 2 ^ n h 2 ( z ,z N verifying system + z-z' z-z - (IV.5.1) h Z k=l ) + 2Y Y*(Z + * j [ y J B ( 2 , f ; z~ , t„ ) ,Z z i n n n ) n' n); dz In dz (IV.5.8) .50 IV.7 Constants As laws J.M. the and vortex the an Motion symplectic system, streamfunction ( 1969)). depth In streamfunction vortex k must and also induce all positions. N + -^X is a constant In particular, of ft Consider the is the = x.. + n metric invariant. be function are transformation be ) the e.g., invariant then the Symmetries of laws.' a real valued function |_G,fl] (IV. of # -n y V - a 9G 9x n n ^ n ^ n G ,y n (see, which Then: r *r Z conservation boundaries conservation G'( x i ,y i , . . . , x d_G dt are invariant particular, if function f l u i d therefore there infinitesimal transformations infinitesimal coordinate Let the the with vortex Souriau under the any associated leave h in of 9ft 3y n 9G J motion itself 9 y if 9G 9y n h k ' n n n J n and = nonly if [G,ftJ = 7.1) 0.. conserved. infinitesimal Y 9ft 9 x ' nn transformation: = y n J e Y h k 2 'n n n 9G 9x n (IV.7.2) for some small transformation. »ni e . G is c a l l e d the generator Then: ..,5 »n ) n n fi(xi,yi x y ) n > n of the 51 N Z „ ,Yh k n=1 n n = for some -[G,ft] 9 x n n 3 y 3G 3ft 3 y 3 n + ou ) 2 V + 0(e .) (IV.7.3) 2 e small 3G dtt 1 1 . Examples: a) If hEh(x) x = const, The , kEk(x) then generator ft of which is plane and is of b) hsh(r) r If £ ^n = x n are x n e - J is: (IV.7.4) 2 n conserved. constant , ksk(r) then = curves . n = y 'n n n When t h e this yields surface the of flow conservation circulation. = const, ^n boundaries under: transformation h (x)k(x )dx n therefore centre only ( Z y k the invariant this N G = is , and + £ y ft is n ' , and all boundaries invariant n n = y n " £ X under n The r e f o r e / are curves is of a 52 y which can be " ' ( I V - - ' 7 5 rewritten 2Y h'(r )k<r )r ) n n n (IV.7.6) ^ = 0 ; n Thus N 2 Z y n=l G = is conserved. yields the 2 For )k(r n / )r n n v dr flow in the conservation of moment Vortex another h (r n systems conserved in the quantity n plane known with of plane as (I V . 7 . 7 ) \ / - k = const, this circulation. with the k = const, angular exhibit moment of ci rcu1ati on: N ^Y (x y -x y ) n One v : can n n to Suppose and Physically h h(ax,ay) this means transformations preserved under must ft(axi in order that appropriately original = curved k surfaces are a h(x,y) , k(-ax.ay) that fluid is . the y the n motion system. follows: functions of = a k(x,y) . v invariant then order under scale boundaries are the stream- vortex also as: ,ay!,... ,ax ,ay ) scaled If transformation, transform the as homogeneous ( x ,y)->-(ax , a y ) the (IV.7.8) = const. n generalize i.e., function n = a fi(x, ,y u n in time) the is x ,...,x ,y ) transformed similar Differentiating by n n system + b(a) (with to the motion a and then in the setting 53 N 9ft E n= l Using 'n 9x (IV.4.4) + y 9ft lift + nay„ and t h e f a c t const that ft is itself conserved, one . h a s : f^n n n[Vn- n n^ n n The therefore k x y conservation of = c o n s angular r e l a t e d to the invariance transformations. That vortices pointed has been this is (IV.7.10) t moment of circulation of the f l u i d the case o u t by Chapman for under rectilinear (1978). is scale 54 V. SIMPLE VORTEX The examined behaviour in order differences varying V.l in depth to the conserved some gain of its the path Y - 2 insight into on vortices systems the a now qualitative a curved on is surface with plane. Vortex: Field motion is 2ft vortex vortices a Single Velocity first simple some rectilinear The M o t i o n o f No E x t e r n a l is of motion and Consider ft SYSTEMS of given a single vortex. Since by: 'h (x,y)k(x,y) _ k(x,yUn 2 - = B(x,y;x,y) '+ = const. (V.l.l) The vortex are two a) path may special Curved cases Surface, If the one depth B=0 supposes and the explicitly, of + is vortex may cross itself. There Depth uniform that moves one can put k=l and: = const. 2 velocity never interest. £nh (x,y) further its but Uniform B(x,y,x,y) If close there along are (V.l.2) no a curve boundaries h=const. then More is: v = -yzxVh(x,y). (V. 1.3) h (x,y) 2 The in motion the ledge is in plane, of determine which h(x,y) v marked . in contrast remains the to the stationary. vicinity of the motion of Notice core is a that vortex a know- sufficient to 55 b) Plane Surface, If the Non-Uniform surface B(x,y;x,y) Now, however, necessarily solving + even in are It the fluid the origin is The vortex Let the is x(x , y ) Y ^ One For can of only put _ h=l so (V.l.4) boundaries, B k in very few B does general uniform Figure , Q depth r > e assumed to be + the to but treat for the a small k = k ( r) , at = (x,y) the x flow case in which depression r < near (-a,0) (V.1.5) with a >> e be: (x,y) (V.1.6) s t r e a m f u n c t i on if k= k everywhere. Q Then sati sfies : e the is (V.l.7) region s u f f i c i e n t l y small in which -k'(r) Y nmrff form: by IV). ^(x ,y) = Y ( x , y ) If since: not analytic VY, across that: constant obtain however, for is may fk(x,y)) absence streamfunction 0 l one non-constant 0 where i possible, (see' = k planar known. has k is the vanish. (IV.1.5). solutions k Depth Y (x,y) 0 k ~ ~ -yk £n[(x +a) 0 V*F varies. 2 -k. k'(r) 0 k °(r)a + y ]^. 2 is 0 nearly constant Thus: Y c o s * x therefore ^ has A - ^ the vortex core Fig.IV: Fluid with Depression at the Origin 57 X = Since Vx ff(r)coscj> d = 0 2 some c o n s t a n t X . s at ( x . 0 r > e depending Thus, , y ) 0 for i f , f(r) ^ only as d~ at (x-x ) + induced d , where d is The d i r e c t i o n to the l i n e the vortex joining the depth the vortex is there along induced Notice It is fluids those largely of vortex and the depression is curves that, of constant depth depth. is of varying is property a r e more is perpendicular at the zxVk(x,y) k. vortex position velocity: (V.1.10) This the previous of flow of the varies depression. of the vortex unlike variation of the v e l o c i t y of constant due t o t h i s varying the distance a component the region (V.1.9) by t h e d e p t h of the f l u i d by t h e b e n d i n g k throughout and b 2 and t h e + 1 directec e 0 the depression. of At the (x,y) (y-y ) to When >> by of the vortex for large 3 is r then: 0 velocity on k ( r ) . the vortex B ( x , y ; x , y ) * The for velocity core. case, necessary that component to determine vortex difficult a knowledge systems to analyze in than of v_ . 5 8 V.2 The M o t i o n of a Single Consider constant is radially is the motion thickness curved near the o r i g i n ; by (so that The polar one chooses harmonic p V). is in a f l u i d of planar at i n f i n i t y but i t is One may t h e n smooth 1 1 . 2 ) r is given assumed suppose where at the o r i g i n coordinate so 0 is (see Section b ( p ) the surface that Stream vortex for simplicity (Figure z = in a Uniform of a single on a s u r f a c e symmetric defined Vortex by and the surface b ' ( 0 ) = 0 as b+0 p+<*> and i f ( I I . 2 . 1 8 ) that: /l+]b' (s) - Po ds = 1 ( V . 2 . 1 ) Po then as p - > - ° ° , r - > p , k - > - l . The as p ^ oo . external If field is one c h o o s e s such that i t is its direction a uniform t o be t h e stream x-direction then : ty*(r,<$>) = Uy = Ursine}) The path of the vortex JUh(r) where y = y 0 + U is Clearly, stream - ( y ^ the path y = y do n o t d e p e n d all. general, the o r i g i n 0 of the vortex is path depend the time upon far upstream. of the vortex f a r down- at i n f i n i t y , the paths on t h e c u r v a t u r e though., will the y + psincj) infinity pass therefore: = 0 y o ) t o o , and s i n c e In is ( V . 2 . 2 ) near taken the o r i g i n f o r the vortex the d e t a i l s at at to o f the c u r v a t u r e . 59 F i g . V: Fluid with Surface Curvature Near the Origin 60 The equations • _ yyh ' ( r) h 3 U - >_0 z there x = 0 , y motion of U : 2h (r)r x Assuming of is = r v ( r ) the = vortex are YXh'(r) (V.2.4) 2h (r)r y 3 a stationary point at (V.2.5) 0 whe r e : h'(r ) h( r ) 0 -2U y = 0 (Note that, point is from stable (II.2.21), if has Q, h'(r)/h(r) < 0 a maximum o r ). minimum The singular there: i.e., i f: 3 ft(o,r ) 2 0 9 ft(o , r ) T (p(ro)-l) 2 2 0 9 ft(o,r) d Q(o,r) 2 2 dydx 0 2rT dxdy dx- Y Crop-'(ro)-/(tJ(.irb)-l)3 g 27T dr (V.2.7) is positive or negative definite. Since p(r) < 1 this occurs when: r o P (<"o) 1 Since the (V.2.8) b(p) is - Gaussian predicts 0 + 1 < 0 curvature of s t a b i l i t y only decreasing sufficiently p( r ) close this to the occurs (V.2.8) the surface if K(r ) only origin. 0 if is: > 0 . the K = ~Pr/ ) rh ( r ) r , If stationary point is 61 V.3 The Motion, of Consider but opposite than of the in are (using of of by -y 2 vortices a distance depth. with much The equal smaller equations notation) : ix i b i + h? 2 y\'= of constant complex h?(zi-z ) Pair: a pair separated a fluid ii Zi Vortex motion strengths |V£nh| motion a Close z 3z iy hf hi(zj-z ) 2 2 3h : 3z 2 ( V.3.1) Therefore, defining Z = 1 Z = h{z\ - " „ 1 4d h ( Z + d ) . n + z ) 2 , d = %(zi - z ) 2 1 h ( Z + d) 1 h (Z-d) 2 z 3 1 h 3(Z-d7 IX 4d 1 ' 1 h (Z+d) " 2 3h(Z-d) az 2 1 hMZ-d) (The argument Expanding and Z = Note Z of dropping 2dhIz) 2 has terms TT a been of 3h(Z+d) 3z dropped order - i Y, h3(Z) 3h(Z) 3z for 3h(Z+d) 3 z. (V.3.2a) 3h(Z+d) 3z (V.3.2b) simplicity) ^ d d (V.3.3) that: d d so h 1 • Y h 3 ( Z + d7 h (Z-d) : that: 2 2dh (Z) 2 hTzT 3h(Z) 3z -2Z 3h(Z) hTzT 3z ( V. 3 . 4 ) 62 + d _ d\; = J d Z " [HTZT 3h(Z) , . I 3z hTD" 3h(Z)l 3z -2-—£nh(Z) (V.3.5) = E = const. (V.3.6) whence: h (Z)dd 2 This is the conservation of energy to this level of approxima- t i on. From v The pair 2 (V.3.3) and = h (Z)ZZ = ^ 2 moves One by with can = const. constant eliminate differentiating ' (V.3.6) : d Suppose from the equations of motion (V.3.3): 2 L speed. d - i>7 _ -Y • ~ 4 E •" 2Eh T Z T V (V.3.7) 3 hEh(r) 3h(Z) 3z Z _ Z put Z = and 7 2 3£nh(_Zj_ ^ 3z ' 2 , . Z Re 1 $ . Then (V.3.8) becomes: R* R$ R$ + 2R$ y /4E is | whence : - R $ )^£nh(R) 2 (V.3.9a) 2 = -2RR^£nh(R) 2 another • • 2 = h (R)(R 2 There = - (R 2 + 2 constant (V.3.9b) R $ ) 2 of (V.3.10) 2 the motion since: • + X = - 2 R ^ h ( R ) (V.3.11) 63 R $ h ( R) E J = 2 Using (V.3.12) D2 to eliminate J + R h"rr7 the surface and (V.3.12) is (V.3.12) in (V.3.10) I Y _ 2 2 K If const, 2 gives 2 ( V. 3 . 1 3 ) 4Eh (rT z Z = b(p) (see Section III.2) then may be r e w r i t t e n : p<S> = J (V.3.14) 2 i i _ J motion 2 (V.3.15) a+r (p)D P " 4E (V.3.14) (V.3.11) b , 2 and (V.3.15) are i d e n t i c a l to the equations of a p a r t i c l e moving under the influence of of the c e n t r a l potential : For with of each the vortex d£ d<J) a pair P is given the is fixed l also and i s c a is never n exactly one c i r c u l a r be l e s s - t h a n a. orbit The .path by: (V.3.17) TT+XbTpTpT^ that the distance p = a . the envelope of closest The c i r c u l a r o r b i t o f a l l the p o s s i b l e approach is b^-const, as qualitatively similar p -> °° to that the path shown orbits of the vortex in Figure to therefore for E and J . If is (V.3.16) 2 2 2 one f i n d s unstable + • from which origin (P)] (l [b'(p)] ) 9* fp y _ 4J E 2 = V 2 J and E there 2 J /E~ = ~ ~ = p J Xl 4E V(p) VI. pair The 64 impact parameter l - £ V is: Y - a ( V . 3 . 1 8 ) 65 Fig.VI: The Path of a Vortex Pair 66 V.4 The S t a b i l i t y o f a S i n g l e R i n g on a S u r f a c e o f Revolution Historically vortices deal which of vortices results concluded seven or in more that vortices at (Havelock or insufficient of this notably, polygonal the by vpirtices Thomson, (1933) is was who stable while erred in Thomson's in 1977 the method of a that configurations streets with (1977)) polygons. stability only equal Mertz stable. rigid vortex vertices (1883) showed t h a t It great regular are the a examined however, determine other the of J . J . Thomson such to first of of von a circular and infinite has also Karman (1912), container vortex lattices (1966)). configurations for positions received vertices Chapman Experimental obtained has at configuration of configurations (1878) vortices. stability Vortices Kelvin fewer Morton of bodies unstable. (1931), (Tkachenko rigid Lord six are studied, stability completed heptagon The been as placed case. fact regular showed were that heptagonal was move attention. strength His the of of confirmation within vortices the a circular in Hell, vortices are by of the stability boundary has a technique actually of some recently been in the which photographed (Yarmchuk, et.al.(1979)). We c o n s i d e r , sidered originally ring vortices of Let initially at: N on now, by an Lord extension Kelvin a surface vortices of of equal and of the question Thomson: revolution strength is a consingle stable? y be placed 6.7 2 TT i n on a surface fluid of whose revolution depth is (V.4.1) described constant (k=l) by h '= h(r) and has no and in a boundary N (B = 0) . 1 y j= 0 t h e s u r f a c e c a n n o t be n=1 of motion a r e , from ( I V . 2 . 1 1 ) : Since closed. n The equations •n " The of iYZ ' - T n h (r ) lk*nV n of initial 2 n symmetry the E 1 the z h (r ) n' 1 n v (V.4.2) 2r„h(rJ n n v configuration suggests a solution form: z = n "2TTJ n r(t)exp Substituting into r( t) = r 0 OJi t + N (V.4.2) one (V.4.3) finds 1-1 _Y_ ;wi r 2 h M r ) 0 k ^ r h'(r V 0 l-exp(2irik/N) 0 2h(r ) 0 ( V. 4. 4) The sum is k so evaluated ^ B giving (V.4.5) 2 that where, %[N-P(r )] for convenience, r h (r )/y 2 2 0 ring revolution stability (V.4.6) 0 The of Appendix l - e x p ( 2 7 T i k/N) CO be: in of unit of time has been taken to the axis . of with the the vortices angular rotates velocity configuration rigidly u> consider 1 . To small about examine the deviations from 68 the motion : (t) = n Substituting e [r exp(27Tin/N) + e ( 0 into (V.4.2) and tjje '" .* 1 n expanding (V.4.7) 1 to first order in the ' s (e k -e. ) n k' J (l-exp(2TT(k-n)/N))2 + P( r + Q(r ,u )e exp(4Trin/N) 0 1 )e -a) o n i ' n iexp(-4xrin/N) n (V.4.8) where: P(r,w) = % r p ' ( r ) + ( p ( r) - 1 ) (u-Jg) Q(r,o)) The solutions e = %rp'(r) to of the form 1 b exp[2Tri (l-M)n/N-i (V.4.9) Q(r .Wi))a + 0 S + + A t] M and (V.4.11) M equating coefficients of e 1 ^ s e p a r a t e l y . y i e l ds : M ( l M into M U are (V.4.10) M Substituting e (V.4.8) p(r)oj = a e x p [ ( 2 T r i (l+M)n/N)+i A t ] n + and + (V.4.9) P M + M ( o> i)) M r w a (S _ 2 + M (" M A + P(r ,a) ))b 1 M = 0 + Q ( ^ ^ ) ) b M = 0 0 1 (V.4.12a) (V.4:12b) with : c L N-l y ^ = " l-exp(2TriLk/N) (l-exp(2~Trik/N)) 2 - Js(N-L) ( 2 - L ) , L = l , . . . ,N ,( V . 4 . 1 3 ) (see Appendix (V. 4.12) if B). and There only if: are non-trivial solutions of 69 " Q (r , X ) 2 M 0 W l + (S + P(r ,o) ))(S _ 1+ M 0 1 1 + P(r , M 0 U l )) = 0 (V.4.14) The M t h mode Q 2 is stable K>^) " i f ( l S is + p + M real; K ' S > H s i.e.,i f : i - M ' P < r + A ) ) 0 >; 0 (V.4.15) Using (V.4.6), (V;4.9), (V.4.10) and ( V . 4 . 1 3 ) , (V.4.15) becomes: ^ g ( o ) r ° + p(r )(N-p(r )) 0 0 - ^i!iL > 0 , M=1,...,N-1 (V.4.16) In a linear s t a b i l i t y analysis symplectic system solutions: system along first its orbit, is have a stable mode; of Here been stability P . neglected displacements corresponding the second is modes in (V.4.16). ft forbidden are the They of the to small o f f the hyper-surface these of a = const. by t h e M=N modes cannot a f f e c t the of the c o n f i g u r a t i o n . (V.4.16) M=J5(N±1) ft solution two z e r o - f r e q u e n c y to small the other o f the system conservation which expects one c o r r e s p o n d i n g displacements The one a l w a y s o f any p e r i o d i c h a s a m i n i m u m when N odd. Hence, ' + p(r )[N-p(r )] g ( o ) r ° M=%N , N e v e n f o r s t a b i l i t y of the ring 0 0 - x > 0 N or of e v e vortices: n (V.4.17) >: -1 Notice that, i n c o n t r a d i s t i n c t i o n to the system the s t a b i l i t y i s enhanced at r is 0 . By m a k i n g possible N odd by n e g a t i v e the curvature to accomodate at in Section V.2, curvature of the surface r and negative 0 large a r b i t r a r i l y large numbers of i t vortices in a stable ring. Examples. a) Plane For For the plane: p(r) =l , p ' ( r ) = 0 (Section II.2). stabi1i ty: -N 2 + 8N - 8 > 0 N even > -1 whence there criterion is is inconclusive perturbation case stability theory on w h i c h N odd i f N < 7 . If a n d one must to determine Thomson (V.4.18) N=7 the s t a b i l i t y use h i g h e r the s t a b i l i t y . order This is the erred. b) ~ C y l i n d e r A cylinder the cylinder there is extends p(r)=0 around -(l+aHn((x-x' + a£n(x a perfectly circulation around r=r' valid around as ( V . 4 . 2 ) we h a v e c r i t e r i on i s r of However, 0 since a n d as r °° , ¥( x , y ;x , y ) 1 r=r' as r' ) (y-y') ) 1 °° a n d r=r' as is arbitrarily r 2 + 2 % + y) 2 2 streamfunction r'+O then : as . words : Y U . y ^ ' ,y' ) = is both in the choice on t h e c i r c u l a t i o n In o t h e r , rp'(r)=0 to i n f i n i t y an a r b i t r a r i n e s s depending r ' -»• 0 . has: 1 °° -2Tra . chosen (V.4.19) h f o r a unit is vortex. 2 TT ( 1 + a ) In w r i t i n g a = 0 . while the The that equations The s t a b i l i t y 71 N whence < 0 2 all rings The is a of the velocity vortices: , N even of velocity form: field v r vortices field = 0 cannot (V.4.20) , is N 2 < -1 are by the for odd non-zero v^ = c o n s t . affect valid N unstable. induced , , The stability all a . a in addition of a ring (V.4.19) of of such 72 V.5 The S t a b i l i t y Surfaces of If flow to is the also the in addition invariant axis of rotate of Vortex Streets Revolution to under rotational then about (1912) Let p = f(z) Then, these the r . z of II.2) the vortices rotation. In can view of of von streets. revolution f of perpendicular configurations vortex with of surface be described an., e v e n by: function. D * e x p f Q (V.5.1) 0 = 0 r(-z) Moreover, to a plane rings of the (II.2.14) Z Choosing axis called surface (Section from are double the qualitative similarities Karman symmetry r e f l e c t i o n in rotation rigidly on and since f(s) = f(-s): = yr^j (V.5.2) ( I I. 2 . 1 5 ) implies: h(i) r h (V.5.3) (r) is = —f- the invariant upon r=l % (z=0) (V.5.3) condition r e f l e c t i o n in We e x a m i n e strength y There two are st r e e t s . which at the r (z ) 0 0 distinct implies the plane s t a b i l i t y of and cases: N of that the surface containing a ring of strength staggered and N -y the is curve vortices at symmetric of -^-(-z ). 0 vortex 73 a) Staggered The initially Vortex Streets vortices of a staggered vortex street are situated at: r = r n , <j> 0 = 2-iTn/N , n=l,...,N , strength y (V.5.4a) r = m » * (2m+l)TTi/N = n , m=l , . . . ,N , s t r e n g t h -y (V.5.4b) In the the absence equations of of boundaries motion n h 2 ( r n ) in a fluid N 1X_ 'z_-z of uniform depth become: N z and I I + ,k=l n " k Lk^n z z £ i + m=l "m""n n 2r h ( r ) n n Y Z n h , ( r } v (V.5.5a) 1 h (r "m The of 2 symmetry the N of hi ) the , =i .k^m N z'm- z , k. k initial 1 n=l m i Yz h ( r ) ' m n' 2r h ( r ) n n (V.5.5b) v _LX_ zm „ - z .n configuration suggests solutions form: r = n r(t) = <j>m Substituting , * 2 ir i n / N = n (2m+l)Tri/N into (V.5.5) = ; o) and N-l r(t) r Q 2 - ^£ + N E m = 1 + o* t 2 , r m = ^ , + oo t (V.5.6) 2 making use of (V.5.3) gives 1 i-exp(2Trik/N) 1 l-r- exp((2m+l)TTi/N) " 2 r h'(r ) 2h(r ) 0 0 0 (V.5.7) The first second sum sum was is encountered (Appendix B): in the last section (V.4.5). The 74 N Z m=l 1 l - x e x p ( (2m+l)TTi/N) ( V.5.8) 1+x 1 Thus 03 The unit L(r? of To : time = + has again order the taken in the the e 's v k t n of 6 's and P(r ,oj )e 0 2 motion and (V.5.10b) expanding v 1 +. Q ( r 0 to : -e, ) k n set: (V.5.10a) 1 a ) m : kfn + 2 configuration m ( l - e x p ( 2 i r i (k-riJ/N)y 1 2 i u)„ t 6 (t)]e 2 + )'/y r h (r , a ) n equations (e be: i w„ t e (t).]e 2 + [r7exp( ( 2 m + l ) T r i / N ) into to s t a b i l i t y . o f the 0 Substituting ( V. 5 . 9) n been [r exp(27Tin/N) "m first P(r ) N determine n + r; , I (1 - ro-?exp(('.2(nwi)+l.)'Mi / N ) ) u )e exp(4Trin/N) J 2 nr n n i e x p ( - 4fri n/N) ( V. 5 . 11 a) N m \l x m- k (l-exp(2Tri(k-n)/N))' ( 6 6 (6 m-e n)' l-r§TxpTj2Tn-niH)Tri/N))2- ) v + E (V.5.lib) where P and P(r) so Q are = -p(i) d e f i n e d by ; rp'(r) (V.4.9,10). Using = ip'(i) (V.5.3): (V.5.12) that: P(p-u) = P(r) + 2u + p ( r ) ; Q(r,u)) = Q(£.-o>) (V.5.13) 75 The solutions e to (V.5.11) are of = a e . x p [ 2 T r i ( l + M)n/N + M n + the form: i X t] M b exp[2Tri(l-M)n/N - M iX^i] ( V . 5 . 14a) = c e x p [ - ' ( 2 m + l ) ( l + M)Tri/N 6 n + M i A t] M + • d e x p [ (2m+l.)(l-M)iri/N - M i X t] M (V.5.14b) Substituting of e l A r , and t into " '^ 1 e M (V.5.11) and e q u a t i n g separately t yields coefficients (using (V.5.9) and (V.5.13)) : U + Q(r M *M A l + T T (" M + ,u ))a + 0 A 2 l-M ( r r 2 ) o ) a M r*J M Q( ." ))V + M o V ( r Ab + M ( + 0 " M A n ( r ) d 2 J l +M T ( r QK'^^V + V A r 2 u ( X M + ^ r o ' W 2 ) o M = 0 M' ) £ = « M M d = (V.5.15b) 0 Ad =0 ) (V.5.15a) (V.5.15c) (V.5.15d) 0 where A = S 1 + M + P(r ,u) ) 0 2 + T (r ) = Q(r ,u> ) 2 N 0 —IT (r 2 M 1 + N 0 T i (x) = - x " T u ; x i 2 L - N _ L + 2 fx"M U ; - ,N-L Nx - ((L-l)x M L M - 7 ( 1 N 0 ^M(N-M) (V.5.16) ) 2 exp((2k + l ) U i / N ) _ e p((2k + l)ui/N)) X 2 X (N-L+l)) |\J (1 + x'V ^ r - , 5 L N 1 ,.. ., (V.5.17) 76 (see Appendix trivial B). S solutions of is L defined (V.5.15) in (IV.4.13). only if a = ±r There are d , , an d; b .= 2 non±r 2 c Then : U + Q(r ,u> ) M 0 Aa + (-X M whence for ± r 2 o(Ti M M ± 2 r T ; M AF, = 0 ( r )) b 2 2 1 + M (V.5.18) M M = 0 (V.5.19) solutions: (r ) T _ (r ))A - 2 + + 2 1 0 (Q(r0,u>2) - r T _ (r ))a + Q(r ,a3 ) M non-trivial M A ± 2 ± 1 2 M M A + 2 r T _ (r ))(Q(r ,o3 ) ± 2 2 1 M 0 2 r T 2 (r -)) 2 1 + | y | = 0 (V.5.20) The Mth modes o" r ( T l +M ± which can ( r ^oT be (2C are stable o " } T is l - ( o ) ) r 2 M (r ))(Q(r ,a)) 2 1+ M 0 simplified ± if D)(4q 2 real, > 4(A ± 2 i.e., - r ' T ^ f r if: (Q(r ,u>) 0 ) ) 2 (V.5.21) to: - 2C ± D) 2 = %x(l > 0 (V.5.22) whe r e : C E (Q - D -= r ( T A)/N 2 (r ) V 2 1 + M + _ xcosh((l-x)y) q If T (2C l M + ( r o) = Q(r ,co )/N 2 0 x = M/N ± D)(4q = T l-M 2 ( r (r M - - 2 x) - %sech (%y) (V.5.23) 2 ))/N 2 (l-x)cosh(xy) M/ C O A\ (V.5.25) 2 2 , y = Nfcnr , - 2C ± D) o) • 2 = 0 the solution is stable unless: (V.5.26) 77 r -+ r 0 1 Q on l y y The stability and M -> N - M > 0 and The Nth mode (Fig.VII; L lower signs). frequency the modes. It is is invariant under t h e r e f o r e s u f f i c i e n t to consider 1 . U denotes The to the perturbations upper t w o L modes U modes . L^lLA Ut (V.5.22) corresponds The .. , Q(r . h <x£ types the criterion are . signs are the stable if: r + r 0 in four V.5.22, the expected zero Bi^.mrUp L of +p(ri) ' < 0 0 (V.5.27) If > 0 p(r ) 0 stable. If unstable. are in p(r ) 0 Other If they N is tend to shown the and in 0 < 0 and cases even be the 0 must then IX. be e x a m i n e d the to modes go They criterion Villa) < 0 (Figure K(r ) first Figure stability > 0 (Figure K(r ) which VI11b) M = ^N all is mode then it are important The four similar. the this is is separately. unstable. are then same possibilities This for as is upper reflected and lower s i gns : (h If q is - This the if = 2arccosh is of surface of case (as von of the > 0 sech ^y) 2 N-*-«>) then (V.5.28) stability flow as sphere, is around: /2 (V.5.29) analogous Karman k + k when region infinite of 2 occurs a small stability those the to criterion the - 2 small restricted y h sech %y)(4q is to the staggered von Karman vortex a c y l i n d e r , our N+°°) . (V.5.28) condition streets (in for fact, results must However, as be s e e n is incompatible with often will approach for 78 2rr Fig.VII: Staggered Vortex S t r e e t : The Modes M=N. Dots i n i t i a l positions, crosses perturbed positions denote 79 a) Staggered vortex streets streets unstable. stable. Symmetric vortex p(r )<0, K(r )<0 o b.) Staggered vortex streets vortex streets stable. Fig.VIII: o unstable. S u r f a c e f o r which the Definite Stability Mode Symmetric M=N has 80 u St- u 0 Fig.XI: — Staggered Vortex S t r e e t : The Modes M=JgN. dots denote i n i t i a l p o s i t i o n s , the crosses perturbed positions. The 81 (V.5.27) leading to overall O t h e r mo.des a r e be examined b) Symmetric The initially considerably more for surfaces separately Vortex instability. specific complicated of and must flow. Streets vortices of a symmetric vortex street are located at: r = n r ; Q <|> = 2irin/N ; n=l,...,N , strength y (V.5.30) cf> m m ; 27Tim/N = -m=l,...,N , s t r e n ggtthh -y -y (V.5.31) Trying a s o l u t i o n , to z = n (V.5.5) r(t)exp(2TTin/N of + the form u t) ; 3 z m = r" 1 ( t ) e x p ( 2 T H m/N + u>. (V.5.32) one finds: r(t) = r 0 ; N(r w N r" ) N + (V.5.33) 3 A o-r ) r where use has k = (see mined been of B). The 1 and: (V.5.34) _ N': stability x of the configuration is deter- p u t t i ng: z = n r exp(2Trin/N 0 - i r exp(2uin/N 0 m substituting e 's (V.4.5) (l-xexp(27rik/N)) 1 Appendix by made 0 and into 6 's e = n and (V.5.5), trying + E„(t))e + 6 ; ( t ) ) ei m to solutions + (V.5.35a) u,3 t of i X t] M ,t l w expanding a e x p [ 2 T r i ( l + M)n/N n i to (V.5.35b) first the order in the form: + • b ^ x p [ 2 i r i (1 -M) n/N> i \t] (V.5.36a) 82 6 = c e x p [ 2 T r i (1 + M)m/N + i X t ] +: d ^ e x p [ 2 i r i ( 1 - M ) m / N - i X t ] f l m M M (V.5.36n) The solutions (X satisfy: + M Q(r ,o) ))a Ba :+(-A M R l +M R l-M ( ( r o ) M 0 ( + M a ^ M ( + Q(r ,aj3))b -:+ ^R M o ^ M r + BF --+ ^ 3 0 + ~ M A % B : ( 0 X + M Q r + (r )5 M + B d M o ' W = 0 3 ) ) d M = M M (V.5.37a) = 0 2 1 3 ( ^ 2 Q(r .u ))c -- + + r r (V.5.37b) (V.5.37c) = 0 (V.5.37d) 0 where S = B T _ M - R M ( ^ 2 ) P + K > 3 ) W " R R L = (*) U J x~ R fx"M N-L+.2 ' R Nx " [(N-M+l) N = U ( 1 _ Equations so that the (V.5.21): (V.5.37) stability N X 3 (V.5.38) 2 exp(27TJLk/N) (l-exp(2irik/n))2 (M-l)x ] , M= 1, . . . ,N N (V.5.39) ) 2 are exactly criterion is the analogous equation to (V.5.15) analogous to i.e., r MR ( r S ) ± can be R^U )) 2 + 1 + M 0 which + M , w ) - JsM(N-M) 0 ^W^KJ = y ~ klx 2 X Q(r = r R 2 3 > 4(B (r ))(Q(^ ,a) ) 2 1 + M 0 simplified (2E ± F ) ( 4 q 2 - 3 2 ± - (Q(r ,a> ) r ^ ^ r 0 2 3 ) ) ) (V.5.40) to: 2E ± F) > 0 , M=1,...,N (V.5.41) with: E = Jgx(l-x) + %csch (%y) 2 (V.5.42) 83 (l-x)cosh(xy) + xcosh((l-x)y) 2sinh (%y) F Q(r ,o) )/N 0 one may The y four signs the modes are signs N£nr that modes lower upper = suppose denotes L (V.5.44) 2 3 M/N x Again (V.5.43) 2 the y >_ 0 with in (V.5.45) 2 M=N V.5.41, , h±x<_\ . are in shown U upper expected zero Figure signs). As X . (L before, f r e q u e n c y modes. The give: > 0 (V.5.46) If i s p(r ) < 0 0 true, and K(r ) p(r ) > 0 0 Other (see Figure VI11b) then (V.5.46) si nee: + If < 0 0 cases p ( r ) > N - l > 0 and must K(r ) > 0 0 be then Q(r ,w ) 0 < 0 3 determined separately for .. ' each r surface of interest. We now criterion may Let show be that simplified g(x,y) - for (sinh(xy) y >_ 0 and + the stability +1 - (l-x)cosh(xy) xcosh(l-x)y - sinh(xy) s i nh ( ( 1 - x J y ) ) % j< x that further. 2 = x ( l - x ) [sinhy Now: so = 2x(l-x)sinh %y ; Then: 2 E ± F ^ 0 <_ 1 since - s i n h ( (1-x) )y] = y(.sinh(xy) sinhx is - s i nh ( ( l - x ) y ) ) increasing. 84 U 0 — U 0 — 0 — 0 — Fig.X. Symmetric Vortex S t r e e t : The Modes M=N. Dots denote i n i t i a l p o s i t i o n s , crosses perturbed p o s i t i o n s . 85 Therefore: whence hsinhhy < sinh(xy) |£ > 0 . dy + sinh((l-x)y) < sinhy Therefore: — g(x,y) > g(x,0) = l-(l-x)-x = 0 The r e f o r e : 2E The ± F > 2E - stability 4q The M = HN y z > 0 y i f y > 0 c r i t e r i o n may t h e r e f o r e - 3 l\^ l F = 2E - F > 0 critical (N e v e n ; ( V . 5 . 4 7) be s i m p l i f i e d t o : , M=1,...,N-1 modes (V.5.48) are in general see Figure XI) f o r which t h e L modes with the s t a b i l i t y c r i t e r i on i s : > (1 + c o s h ^ y ) Q q For sufficiently c) large N The N^ 0 0 , q •+ 0 so t h a t 3 a symmetric vortex street 4 g -^ ) j for will always The Cylinder stability of vortex streets on t h e c y l i n d e r is examined. For (V.5.49) streets the c y l i n d e r then are immediately 4C 2 For modes. p(r) = 0 implies , p'('r) that so that q a l l symmetric 2 = q 3 = 0 vortex unstable. Staggered tion As 5 unstable. Example: now > yg- . y lv,3 2 q 3 ( 16tanh Uy) 2 stability become = 16 s i n h % y 1 3 1 2 - D 2 vortex are stable i f : < 0 M = 1 T h e L modes o f moment streets (V.5.50) a l l four have modes already of c i r c u l a t i o n now y i e l d been and zero explained. angular frequency The..conserva- 86 U U Fig. XI: Symmetric Vortex S t r e e t : denote i n i t i a l p o s i t i o n s , positions. The the Modes M % N . The d o t s c r o s s e s pe r t u r b e d ; 87 moment of c i r c u l a t i o n Z y n n £ n r n S-Y 4> n = c conservation These modes and must N is i.e., From (from - \ by t h e M = N . , be n e g l e c t e d U modes. a n d do n o t a f f e c t t h e is M=%N(x = Jg) . D is then zero, (V.5.22)). sech (J- y) 2 2 i 2 unstable 0 (V.5.53) unless: /2 (V.5.54) and ( V . 5 . 2 6 ) , the separation R is = the separation of the rings of = c within each ring of the c y l i n d e r . only is: Therefore: arccosh/2" with c (V.5.56) 2TTR/N the s t a b i l i t y The / ^ (V.5.55) of the v o r t i c e s the radius agreement determined N are v i o l a t e d Rarccosh/2 N = d -T— that (V.5.52) is: d in (V.5.51.) o n e may p u t = 2arccosh , where - laws even | (II.2.23) v o r t i ces while _ t h e mode y t configuration. for stability • 2 s therefore of the If n = const. These stability o f o r the c y l i n d e r a r e : ,,. (V.5.57) c von K a r m a n . of this by h i g h e r stabilities When mode order is M=%, T^ + M (x) = T^_ (x) m indeterminate. perturbation of the other modes It c a n be theory. f o r N even and f o r o d d a r e now d e t e r m i n e d . -^[(D-2C)cosh isy] 2 = x ( \ ~ x ] [s i nh ( (1 - x ) y ) -2cosh(%y)sinh(Jgy)] so - sinh(xy) < 0 88 since sinhx is increasing (D-2C)cosh %y = x 2 i f x>0 > 0 2 . when y = o as y + °° -*- -°° ( x ^ 1) Thus, i f XT*1, Denote these D-2C h a s e x a c t l y zeros: one z e r o x(1-x) 2 2 = 3y + (l-e ) 8 2 ( l - e where 2 ) These y"(x) when 2 inh(Jgy) > 0 y -»• °° one z e r o f o r each x (x^l). y (x). + y = arccosh/2 , C=0 a n d D_>0 2arccosh/7T< y ( x ) and + y"(x) sinh(xy) y=0 as oo are denoted < 2cosh(Jsy)s - Moreover: D + 2C h a s e x a c t l y When sinh((l-x)y) 2sinh(%ye)cosh(%y)+2cosh(Jay)sinh(i2y) D + 2C = - ( 1 - x ) Therefore x. s i n h { h y ( l - e ) ) - s i nh{hy(1 + e ) ) + 2 c o s h ( % y ) s i n h ( % y ) e = 2x-l . -> f o r each y"(x). 3(D+2C)cosh (%y) _ But: < y ; therefore D + 2C _> 0 when: < y (x) %<x<l + (V.5.58) Define y" The vortex y" = max y " ( x ) x street is < y + Figure < y XII and y~ c o r r e s p o n d , stable y + = min y ( x ) x (V.5.59) + i f : (v.5.60) is a graph o f y ~ ( x ) and y ( x ) . to the values + of x closest t o h. Clearly y + For N odd 90 this is: Thus, around given y in if Table N = 1: 3 x = h a small first . region of few N t h e s e stability regions are Regions of S t a b i l i t y of Staggered V o r t e x S t r e e t s on a C y l i n d e r < 2,079 4 A = 1.6 2 78 < y < 1.9 344 A = 0 . 0 30 7 , 1 . 6 6 34 < y < 1.8806 A = 0.. 01-55 , 1.6842 < A = N = 9 is is the 7 the actual < y < y 1.8525 width of these stability is only regions 0.0,879 0.0094 measured in radi i : + A y If N is one even higher somewhat of more (II.2.24) , Q(r ,o) ) 0 perturbation if theory y=arccosh/2, to check Sphere analysis From order possible ^N. The The is -y 2N needs M'= Example: sphere For , cylinder d) . there 1.5 5 2.2 A mode odd N even: , N = the For 1. N = 5 but N is = arccosh/2 Table A=0 + k). x = %(1 3 = -h the stability of c o m p l i c a t e d than (V.5.33) -H and tanh (y/2N) 2 vortex streets on cylinder. the on the (V.5.45): + -JgNtanh(y/-2N) c o t h ( J g y ) (V.5.61) Since: t a n h ( a x ) c o t h x : <^ 1 if a < 1 : 91 Moreover: E =-Jsx(l-x) + %cosech (%y) > Jgx(l-x) 2 ; F>0 (V.5.63) Therefore: 4q If N is even 4q If N is for 2E - 3 2E there 4q - 3 - there odd Therefore The - 3 2E - F < is a mode - x=% , 2 a mode F < -(N -8N+3)/N 2 symmetric vortex stability criterion (V.5.48) (V.5.64) whence 2 < 0 , if , whence < 0 , if x=%(l+^) all N = 2 , . . . ,6 x(l-x) F < -(N -8N+4)/N is - + ^ 2 streets has with been N>7 N>7 N>7 are examined unstable numerically . Table 2: Regions of S t a b i l i t y of S t r e e t s on a S p h e r e N = 2 , stable if y > 4.2451 , 9 < 38.1727° N = 3 , stable if y > 5.3020 , 9 < 44.9072° N = 4 , stable if y > 7.5957 , 9 < 42.3078° N = 5 , stable if y > -10 ..4306 , 0 < 38.8225° N = 6 , stable if' y > 1 7.7602 , 9 < 25.6478° N _> 7 , unstable From : ( 1 1 . 2 . 2 4 ) , Q(r ,oo ) 0 2 = -k (V.5.9) and -%tanh (y/2N) 2 Symmetric Vortex (V.5.26): + %N t a n h ( y / 2 N ) t a n h (%y) (V.5.65) The r e f o r e : 92 § - sech (y/2N) ^ 2 [ t a n h ( y ) _l t + ^tanh(y/2N)sech (Jgy) f o r each N there Denote it by y Let y(N) > y(N) a = y/2N . is The (see . exactly vortex n n ( > 0 2 Thus, a y / 2 N ) ] for y > 0 one y such street is that (V.5.66) Q = 0 unstable if: (V.5.27)). Then N = y/2a and: 1 + tanh a - —tanhatanh(%y) 2 a tanha 4a M_ N o ( l +tanh a)a (V.5.67) a 2 tanha w ytanh^y > > < 1 : i.e., if ^ , 1 y y(N) > 1.543 Let f(a) = a - x u • £ Therefore < 1.543 . f(0) = 0 a(l+tanh a) tanha . (l+a)tanha a ytanh(%y) Let y*(N) . = + . a t a n , h < a < 0 if 1 l 0 1 + 2 a • T T , h u a > 0 and: . s 1 f : , then Q > 0 . largest y*(N)tanh(%y*(N)) Then: a < (l+a)tanha > 1 + 2a = 1 + be t h e if (V.5.68) Therefore: 2 . , n Q < 0 Therefore: f ' ( a ) =• ( 1 - t a n h a ) ( 1 - ( 1 + a ) (1 + t a n h a ) ) But n value = 1 + of y such that (V.5.69) Then : 1.543 < y(N) Differentiating < y*(N) (V.5.69) by (V.5.70) N: 93 tanh(Jgy*(N)) Since y*(N) y*(N) < One can always check < y(N) N > 4 , that y*(4) A staggered vortex street on and > 1.6 is M=%N Let N _> 4 now s h o w n or = 0.648 that M=^(N±1) f(x,y) < 1.6 . Thus and: for unstable i f modes numerically < 1.6 It -y*(N) N ^ Q for all 1.543 _ > t a n h (JgXl. 5 43) 2 **|M 9y*(N) 9N : + % y * ( N ) s e e n (*gy*(N) ) < 1.6 ± 2 > 1.543 tanh(Jgy*(N)) whence + %y*(N)sech (%y*(N) ) - y if are N > 4 y the (V.5.71) sphere is therefore . < 1.6 and N >^ 6; then the unstable. = cosh(y(l-x)) + (1-f)cosh(yx) . Then: A ) 3f 1 = ^ycoshUy) [ l - x y t a n h ( x y ) ] + y ( s i n h ( x y ) - With f(x,y) y < 1.6 sinh((l-x)y))> (V.5.72) > f(h,y) = 0 holds i f 0 for h < x D = %xf(x,y)csch (h>y) x < 3/4 > 0 2 i f i f xy < 3/4 . < 1.1997 . (V.5.72) Thus: Therefore: h<x < 3/4 , y < 1.6 (V.5.73) and: f£ = % c s c h 2 ( J a x ) ^ | ^ - > 0 i f h < x < 3/4 , y < 1.6 (V.5.74) Moreover, q from 2 (V.5.65) 1 - 7m and (V.5.66): ( V . 5 . 75) 94 The modes M=J§N or M=%(N + 1) have: x < whence C = Jsx(l-x) - hsech (hy) < h - ^ - z - %sech (%y) 2 (V.5.76) and, using 4q ( V . 5 . 73 ,75 , 7 6 ) : " 2 2C + D > ^ + ;sech (J5X1.6) 2 3 4N2 + 0.0297 (V.5.77) if y < 1.6 4q Moreover, y = 1.6 for y . The r e f o r e : -2C + D > 0 2 one c a n c h e c k and Using fore on has unstable the sphere been , t h e modes (V.5.22), The , N > 6 < 1.6 then: (V.5.78) i f 2C + D < 0 . • Since — y < 1 . 6 M=%N (V.5.78) i f with y < 1.6 or , J s < x < 0 . 5 8 M=%(N + 1) and ( V . 5 . 7 9 ) . N ^> 6 Hence are determined numerically 3: have these modes a l l staggered ( V . 5 . 79) x < 0.58. are t h e r e vortex streets unstable. s t a b i l i t y of staggered Table < 0 dX 3 , x < j : < 1.6 N _> 6 y numerically that x = 0.58 2 C + D < 0 For i f vortex (Table streets for N <_ 5 3). Regions of S t a b i l i t y o f Staggered V o r t e x S t r e e t s on a S p h e r e N = 2 stable i f 0 < y < 1. 7 6 2 7 , 65 .•5.302°<e < 9'0° N = 3 s t a b l e i f 1 . 5 5 2 2 < y < 1.6 306 , 7 4 . 6 1 7 1 ° < 8 < N > 3 unstable 75.3404° 95 These cylinder vortex (or results von streets streets. differ Karman exhibit q u a l i t a t i v e l y from vortex greater streets) stability where than those the on the staggered symmetric vortex 96 VI. VORTICES WITH FINITE CORES Until now, it been i n f i n i t e s i m a 1ly finite ular, core the small is is shown not motion The that the the the Y = the are then all section vortices the of flow a wobble is is In Velocity extends of the of a Vortex over a finite vortex region G into one can w (x,y,t)xh (x,y)dxdy (VI.1.1a) w_(x,y,t)yh (x,y)dxdy (VI.1.1b) w_(x,y,t)h (x,y)dxdy (VI. 2 2TTY the by: 2 2TTY and introduced 2 1 of partic- curved z 2TTY have effects considered. surface symmetric core 1 this vorticity if and position X = In that vortex. Position When define of radially of assumed cores. distributions it VI.1. has 1.2) Since: _1_ h w. 9(hv )' 9(hv ) x y 3(hv ) 3(hv ) y 2 Try (VI.1.3) 3y 2 x dxdy 3x = o j h(v 9 G x dx + v dy) y = r db a r ( VI.1.4) so that taining depend 2TTY the on is core. still By the circulation Kelvin's Circulation time. The velocity of the around core is: a contour Theorem y condoes not 97 U X = 3w (x,y,t) z X - xh (x,y)dxdy 1 2 2 Try J 2TTY 3t w .(x,y,t)xh(x,y) [(v -v '9G x G x )dy - (v -v g G y )dxJ (VI.1.5a) 9w U y = Y = 1 2iryJ 2 Try 9G fx,y,t) -yh 3t (x ,y) dxdy 2 w (x,y,t)yh(-x,y)[(v > x v Q x )dy - (v -v Q )J.dx (VI.1.5b) where on v_g 9G is the the v e l o c i t y of boundary terms the core vanish/ boundary. Using Since (III.3.7) w = 0 and (III.3.8): x IT = fW -1 2TTY 1 • V V -1 z xhkdxdy k r w. 2TTY 1 2nyj V • (vxhk)dxdy w v hdxdy (VI.1.6a) w (VI.1.6b) z x Similarly: 2TTY In terms of the G v hdxdy z y s t r e a m f u n c t i on 1 2 h k - i r h *ik ^ ITY 2TTY 2 G ty d e f i n e d by (IV.1.2) |^dxdy ( VI.1.7a) ffdxdy (VI.1.7b) 98 VI. 2 Circular A derived Cores "circular" vortex is defined by a s t r e a m f u n c t i on o f t h e iMx,y) = -yA(x,y;X 0 t o be one w h i c h is form: ,Y )f(r) + y B ( x ,y ; X 0 ,Y„) + 0 ty*{*,y) (VI.2.1) where r = [(x-X) f ( r) f has continuous). one and w = 0 r (VI.2.1), and 1 5 > (VI.2.2) (so A that and the v e l o c i t y B are those field is defined by satisfies: ty* =o for r (VI.2.3) > e and t h e c o r e (VI.1.2) and ( V I . 1 . 3 ) boundary is: to evaluate r = e (VI.1.1) finds: X = X where of , The f u n c t i o n s k Using 2 derivatives „ . f1V * * Thus, (y-Y) ] = Inr continuous (IV.1.13) + 2 is v + 0(v) 0 a small the core radius ; Y = Y parameter + 0 of 0(v) order and t h e d i s t a n c e (VI.2.4) e|V£nh| over which , the ratio h and k vary appreciably. We w i s h vanish as e -> 0 f(r) There f o r e : dxdy to ^ determine with y O(Ane) w ^ 0 ^ 0(e ) 2 , h e one 2 : finds. U and U x up t o t e r m s y which constant. f'(r) v U ^ ^ 0(f) ^ , T^- , O(-^r) , so f"(r) and that % 0(A") • since terms of 99 order e° e~ in must VIJJ be r e t a i n e d and t e r m s of k in V* order V 0 2 , e f(r) in + terms h . of order Us i ng (IV. 4.1) 0(£ne) (VI.2.5) k (x,y) 2 9k, 2 dy 9k dy T [ r > k r K o T + ^ 0 2 Tx" 2 9y (VI.2.6) 1 1 h k '2 2 h^k 3 ' 3/ 0 2 J- 2x 3_h_o 9x uh - 3x_ 2k 0 9j<o_ 9x 0 _2y_ 3h_o_ • J3J/_ 9k_p_' h 3y " 2 k 9y 0 0 + 0(e ) (VI.2.7) 2 For zero convenience denote (X,Y) has evaluation been at made the o r i g i n the o r i g i n . and subscripts Substituting into (VI.1.7a) : U' f 2TT 1 1 k 2 h 2TTY o J " 0 0 9k l 0 3y 'k with x = rcoscj) U' , y 1 k FT" ( Z 2 K + 0 0 "0 + yk 2 0 2x_ 9 h ~h 9x 0 0 0 yk r J x j 3/ = rsincf> 9y J ' 2 . 9k 2y 3x ~ h 9h 9y zXfrriALo2 dy " IX r 0 0 d frdf(r)] dr dr l!io. + Z iko.1 9x 3x " 2k 0 T + ' { YIP^ "9y Y r ) + ' 9i 9y d rdf dr h 0 9h iL 9y + n 9y n -1 2k 0 9k, dr 3y JJ r 2 (VI.2.8) J 2 ( ) dr f rd<|>dr +• 0 ( e £ n e ) Therefore: _d f r d f ( r ) l dr dr 0 9y + 9y 0(e£n e) 2 100 koh* Y9Bo 3^-" 3y 3y I _1_ 3 h 3y 0 1 2ko + [ho Xjkj 2 3B ko " o L 3y K 0 3 ^ + 2 Y3k, 3y df dr 'o J + 2 0 (e£ y3k 3y 0 + ° 0 y3k 2 3y 0 3y 4 k e ) 2 2 dr • 2 y Y_k_o_ 3 £ n ( h k ) _ + y 3 (ef'(e)) ef(e)f'(e) 2 fdf(r) l dr dr ay J lh 8k o 3y J • In e (VI.2.9) 0(e£n e) 2 D e f i ne : re fdf] dr r 0 The kinetic-energy V£ c dr of ( V I . 2 . .10) the 2 o 'o 'o 'o of the Section core £ v h krdcj)dr 2 2 fdrt k„Y IV.3, is the _ r£ 2 2 in in C2TT r e r TT As fluid a r dr d<J)dr = h k £ 2 0 2 0 constant. J„ '0 [ Y 2 1 T '0. = core . 2 H + 3x k 7TpY k 2 is is 0 3 the if/ 2" d<j>dr by, (VI.2. 0 constant Therefore, '3 since velocity the of 11) volume the core i s: ,x U' 1 k h 0 2 3 3y Y B(x,y;X,.Y) r + \k(x,y)ln similarly: ^*(x,y) h (x,y)k(x,y)l 2 a, and + + Jx=X y^ 0(££n £) 2 (VI.2.12a) 101 y _ - i k„h U YB(x,y;X,Y) 3x 2 + ty*(x,y) fh (x,y)k(x,y) 2 + in agreement been Y fk(x y)£n s included in propagate as a with vortex provided its core the will be core within the in it. effects The the the e is < k not than of effects finite a core of and due (VI. of other size 2.12b) vortices remains to of will have therefore infinitesimal the the size circular. advection therefore, distortion the of A v ^ induced it is the to of have core. (D = by V in vortex In the some This induced pj-|_V£nh| ^ is the approximation the surface necessary larger motion then the v_ that than general, vorticity idea of discussed is to e >> the V the : depth by be m o r e the The II.3 Section curvature k surface . the approximation (D, (D . o. the if induced must of Thus, v by by v(°^e|V£nh| ^ j f ^ °^ I VJlnh | . vortex suggested the order: neglected negligible, V with of necessary, considerably but a core circular velocity negligible, be the order: vortex must of is velocities of is beX (the distorted It 2 VI.3. Section curvature is 0(eiln e) (VI.2.12)). in ty* A vortex a* (IV.3.9) with + of motion be nearly (III.3.4a): radius fluid. surface of non- core the are If curvature horizontal i.e., 102 VI. 3 The core of Validity of the Circular Suppose that the a vortex is nearly Approximation vorticity distribution circular: that within is: w. where v (VI.3.1) is the v = v h = h W The z the 0 o (r) ) + + vh 0 parameter ( l ) vv ( 1 ) of velocity (VI.2.12). Within the r,(j)) (VI.3.3) (VI.3.4) fe-*0(v) of the core (VI.3.2) (r,<M = ^ ) U = ( small vortex is: 1 [ w vhdxdy 2 Try 'G " z hn , 2 Try v ( 0 )w(°)dxdy v + +v v ^w^M ) (v_ 2TTYJ 1 first core. even term The small relatively in other z is and advected periodic terms deviations large However, w (r) (VI.3.5) from changes w with v( )w 1 z )h + + w 0(v ) 2 v( )w 1 o + (o) (l) v ,h )dxdy ( 0 ) z 0 (VI.3.5) 0(v ) 2 obtained a circular core order can of a circular the w the z core ' on — r ^ — r where w ir ; n so that produce vortex. distribution perturbation around for magnitude velocity a circular of ( l ) 0 comparable the h h )dxdy 0 ) of carried period z ( o W d ) that localized is w v is in consider a small orbit are ( o + + The ( 1 ) f of vorticity . o .S i,n c, e^ a nearly r is o the 103 radius at orbits the w^^ ment is which core, also of localized initially. direction through vortex zero is the sweeps the almost w ^ z due (i.e., an to it the angle this is of of As velocity 2TT . velocity in V order: The w ^ z induced net the displace- time 2 (1) —>—r V —T-rn—^ w ' v by 2 w^ ) £ 0 J u z so that the time averaged the departure from The motion the the path of of a c i r c u l a r core vortex a circular Similarly, that the motion position motion of of of of order: vv/ The at most vortex. of net X ~ the vortex small is (to for is, vortex. The the only of due order to v^v} ^ . 0 wobble about v). more to velocity a periodic order frequencies complicated lowest of order, order amplitude of w^^ a super- y/e z upon the wobbles the is 2 = v which e systematic order: For vortex with of therefore expects a circular e is one wobbles (o) component vU v. this quite addition where c is is U is to negligible. the the c neglible. vortex velocity velocity of a is circular 104 VI.4 Elliptical The explicitly wobble if distribution This is (see, Cores the its in the core within motion is it. of a elliptical The depth vortex with of the g e n e r a 1 i z a t i on o f Ki, r c h h o f f ' s e.g., Lamb (1916) p.226) to can be a uniform fluid is vorticity constant. elliptical non-planar demonstrated vortex surfaces. Suppose : w z = w = Q = const. (x,y) £ G 0 (x,y)£ G (VI .4.1) with: G = {(x,y) If there are no ; (f) 2 (£] < boundaries V iMx,y) 2 or -w h (x,y) = 2 0 + = 0 i j 2 + external flows: = -w h [l + ax 0 0(v ),(x,y) 2 , (x,y) i ( € 0 + V I gy] G G (VI .4.3) whe r e : It may is convenient use to introduce complex conformal transformations. f^ffU.z) = -%w h 0 = 0 2 0 coordinates so that one Then: [ l + %(a-iB)z + Js(a+i3)z] , (z,z) £ G (z,I) £ G (VI.4.5) 105 whence: i>(z,z) = -%w h [zz+J (a-i3)z z+%(a+i3)zz ] 2 2 0 + 4>. ( z ) * e = ( z 2 s ) + V + (z,z) € G *.(z) z (z,i) $ > G (VI.4.6) the subscripts the interior e and i denoting and e x t e r i o r Consider the of the terms dz d? Moreover: Therefore if r, ell—d- = n (a-b) UTTbT t 0 1 field P *»S + for core. a+b the core £2 the mapping the v e l o c i t y 4> of ' potentials mapping: c = In complex if boundary U l > 1 is: since is conformal outside is to at z n=l vanish |?|=1 "| d| < 1 . the core. Thus, infinity: (VI.4.7) Pn?" Moreove r: I q z n n = I n=l The that coefficients the v e l o c i t y q c (VI.4.8) n n n= l p n and field q n are determined be c o n t i n u o u s by requiring on t h e b o u n d a r y . One f i n ds : n > 4 Pn (VI.4.9) wo ho d ^[a(2+d)-i3(2-d)] (VI.4.10) wo ho d 8 (VI.4.11) L 4 106 ,w h c (l 2 q i = WphocM Po d ) 2 0 4 2 + ( 1 + 2 = w h c (l-d )d 8 p 3 = w h c^(l-d )d 48 2 w is z + i g ( 1 _ i\/ " V a ( 1 + d ) 2 + i B ( d ) 2 -| (VI.4.14) 1 _ d ) 2 j uniformly future of the core is ( V I . 4. 16) over the core times therefore since it it is determined boundary. con t r a v a r i a n t l 2 d i s t r i b u t e d at a l l y " ) distributed by t h e shape o f t h e x d 2 The s t r u c t u r e V (VI.4.12) d ) : ] (VI.4.15) L uniformly v _ 2 0 Since The i e ( 1 0 2 solely _ d ) 2 p advected. + (VI.4.13) 0 2 remains 1 z w h c^(l-d )^ = , l-d ) 2 P l [ a ( velocity f i e l d on the b o u n d a r y is: lij 3?| 2 1 h (z,z) 2 i w o c (1 - d ) l-(a-iB)z 2 2 \ + (a+iB)z ^ (a(l+d) +i6(l-d) )] 2 T i wo ab Ca~+~bT - •aa+i b B + e " 1 + 2 T 9 +| 2i e | ? [ = 1 0(v ) 2 aa(d-3)-iBb(d+3) (VI.4.17) where e is The tional d e f i n e d by: first velocity: c = se two t e r m s 1 9 are constant implying a transla- 107 u x Consider, Only can the . ,,y l U = now, _ i wo ab 2(a + b ) C the term: component change V its = x (e ve 1 o c i t y 1 is to the ) surface of iVw -de is: i Woabe ( V I . 4 . 19) -1 field = -icoi of this of a uniform rotation with = -icoc(e velocity _ 1 9 +de field n 9 ) , |?|=1 perpendicular i e sin -de- 2w ab (a+b) to 29 i 0 the velocity third given The V (VI.4.19) to the if: (VI.4.22) 2 term by causes the ellipse to rotate with angular (VI.4.22). pe r p e n d i cu 1 a r • c o m p o n e n t w ab 4(a+b) x (VI.4.20) (VI.4.21) | 0 Thus, angular i s : 2codc = core is: identical w the a+b 1 B |e which = - — y component -de (a+b)le co J i 9 sin28 boundary V x 2woabd component core (VI.4.18) g] V -iV -de- 9 velocity v£ The l b This e The - perpendicular shape. Re a a 0 of the aa(d-3)(sin6-dsin39) fourth + term is: 6b(d+3)(cos 8-dcos30) (VI.4.23) The terms in sin9 and cos0 imply an additional contribution to 108 the trans 1ationa1 velocity of the vortex B(d+3)-ia(d-3) Since the field, perpendicular U -iU x The j = x ( a + ) D velocity from taken (VI.4.24) y a distortion TTW ab a constant U bcos9+U asin9 a+b contribution causes of is: y V" component (VI.4.23) of for the terms the ellipse. the in ellipse sin and However, to 1 39 rotate cos in the once 39 time the displacer 0 ment D ^ of e2a which ^ changes . through by The small The net The l U as the terms seen words, displacement D n„ e ^ t+ . ~ v 2 £ times of distorting of order w term w i l l h order: in appears phase the Thus the a frame of the of . of from gradient other is of a boundary the core to + iB thereremains 0 henceforth translation of the be core disregarded, is, from (VI.4.23) : _ - circulation 2Try net above time, in velocity x U (or, for the rest, cancels: elliptical lj _iijy to this at The nearly and is 2TT 2TT) . (VI.4.18) due During ellipse very nearly boundary ve the rotate fore the = w ab 4 ( a + b) 9 around Wo h 2 3(b-2a)+ia(a-2b) the dxdy core (VI.4.25) is 2 T r a b w 0 h 0 + 0(v2) (VI.4.26) Therefore : Y : Wp ab 2 (VI . 4 . 2 7 ) 109 Using the core is X for that y TlT+bT when circular one chooses and 9ho_ 3y o is the v e l o c i t y 3 i (a-2b) 3h_o_' 3x h3 n i n agreement XIII) X derived axes the x-axis U -iU with using of (VI.4.28) the result t o be a l o n g Vh an a n g l e 0 with of the vortex -iy 2h 1 + 3(a-b) 9h 3x 3 the e l l i p s e 0 rotates with angular whose instead, the major the x-axis is: 2i Ta+BT 1 If, so t h a t the v e l o c i t y y coordinates of the e l l i p s e . now m a k e s 0 Since h3 has been of the e l l i p s e (Figure "(b-2a) a=b t h i s are the p r i n c i p l e axis a cores. (VI.4.28) axes of then: U -iU Notice the definitions (VI.4.29) velocity co g i v e n by (VI.4.22): (^ Uu - i U ) ( t ) = -2Xh _3 Ilia 8x x Over many y periods considered nearly Y 2uh v Y 3y A changes constant. 9h 3x . 0 w t + very little 0 i n 2 a (VI.4.30) so t h a t Integrating s 3ho ( a - b ) ^ ^ ^ 2icot T a + bT 1 . 3(a-b) • , 2(a + b ) " 4^hJ 3 7 T i W This amplitude 3 0 h 1 + 3(a-b) then gives: .1 j (VI.4.31a) t . (VI.3.31b) C O S 2 c j t i s the equation of a trochoid (Figure of the o s c i l l a t i o n i n the motion is: . ^ 4a)h 3 n jjutejq.,. 3(.;-b') 3x (a + b) 16 h, i t may be j ^ 3x , 0 ( v e ) X I V ) . The 110 Fig.XIII: A Vortex with an Elliptical Core The The but P a t h s o f a V o r t e x w i t h an E l l i p t i c a l Core. e c c e n t r i c i t i e s of the e l l i p s e s are correct, t h e i r s i z e s a r e much r e d u c e d . 112 It is interesting (VI.4.23) the core constant at with a much obey the terms slower causing note distortion rate than other c i r c u l a r approximation time. that vorticity within lengths of core found which is to In the has next it cores for one puts disappear. is therefore and w i l l another vorticity d=0 in A circular distorted therefore correspondingly section a continuous if such greater circular distribution. 113 VI.5 Perturbations Several uniform By depth using exact flows the to uniform depth flows general vortex anid l e t flow. is LI c a n the be is are example, of such known Batchelor a solution for obtain for perturbation (1967) as a to but this expansions p.534). first non-planar solutions planar more in the the the s t r e a m f u n c t i on known of the s t r e a m f u n c t i on non-planar of the flow planar that: 0 by can the for ty so few that h of their. terms and k Taylor : (x - U t , t ) of r > e first expand = velocity + v^ translation 1 ; (x,t) of the +... (VI.5.1) zeroth order solution. expanded: also equation _3_ at ( 1 ) + v U 2 ( 2 ) +. .. (VI .5.2) expanded: h = h The be 2 One U = vll h can -v ^ ) + o -ty{x,t) is for as be approximated expansions. v_U one assumed = solutions . 0 ; 0 be v Solutions streamfunction ty(x,t) w^ ) may (see, problem if/ (x_,t) It vortex the parameter Let Planar streamfunction approximation small of ( 1 ) + ... governing 1 h2 Substituting + vh 0 V 2 ij, (VI.5.1) h ty 2 = h 0 is (from l i JL [3y and 3x » " + T x^JL expanding y y^fl- dy one —2 h + •^ y9y * ' (III.3.7) l i JL 3x + 3>T" 3x X_ V v has 2 + < and $ to V I - - > 5 3 (IV.1.2)): 0 (VI.5.4) lowest order: : 114 vV 2 9 ^ ° > _3_ 9y 9x hi which is uni form the requirement a.. 9X 0) (VI.5.5) ,(0) ijr be a s o l u t i o n that for planar, ; depth f1ow. To order v : _i _9_ 9t 9^ 9y h 2 , I 9 9x ( 0 ) u X ( l ) V _9_ _ 9x 3y 9^ 9x ( Q a' ) f ^ ,••(1). 9y ( 1 ) + _9_ uy(D 9y ax 2!jiiL»»«»- 0 t no (VI.5.6) This boundary is a linear In only general subject to the initially as r •> ; = 0 the solution is (VI.5.7) , t=0 c o m p l i c a t e d a n d c a n be numerically. We w i s h long t o be s o l v e d conditions: -> 0 found equation to look for solutions c i r c u l a r and r e t a i n periods of time. Thus, their we l o o k to (VI.5.4) circular form for solutions which are for relatively of (VI.5.6) having: ^ ° ) '= * < ° > ( r ) a n d ty ( (VI.5.6) {1) = 0 . becomes : r (0) rdty dr dr (0) 1r d Ar r d ty dr (0) dr (0) d ty dr 9h 9x 9h 0 ( •dr[r (0) rdty dr dr s l n' d I cTr r (0) A rd^ dr dr c o s ty = 0 n y ( l ) • ' j d . f l _1 - U x(l) (VI.5.8) 115 There is a solution '_d_ r a > dr dr [dr dr U with: a Putting f(r) y ( 1 ) ( if: W j dr J dr 0 ) 0 ) + a ,(0) — ( ( 0 (VI.5.9) -1 \f^f) = £ (0) d frdifr dr dr _d_n dr r (VI.5.10) and u.-= r , (VI.5.9)- 2 becomes ( V I . 5 . 11) The r boundary -* oo - f/ of s conditions 0 r as require r -> 0 . that: There f -> c o n s t . is as a one-parameter family solutions: x 4u _ 4r a +u a + r 2 _ 2 2 (VI.5.12) 2 The r e f o r e : (0) d± _ = (0) 4ar (a +r ) dr 2 2 i f ; ; ( 0 ) 2a£n(a + r ) 2 = 2 (VI.5.13) The strength of the vortex ,(0) _ Y = Lim r v is: •4a (VI.5.14) r->oo so that — (0) = a + r 2 ,y(D 2 -Y 2h 0 liLo. dx , i i x ( l ) _x_ = 2h u 0 l b . dy (VI.5.15) The vorticity w The (0) radius initial of is, _ to 2 a TT2 + lowest order: : (VI.5.16) Y r ) 2 the core vorticity is of order distribution a has. t h e . A vortex form whose (VI.5.16) will 116 1.0 .5 1.0 0 2.0 3.0 r a Fig.XV: The Velocity and Vorticity of a Quasi-Steady Vortex therefore vortex remain might In be circular called general it is not depth of fluid of not good a p p r o x i m a t i o n they a satisfy long periods of time. Such quasi-steady. analysis be the for possible to is 4> different equations: to perform varying since since outside V i|J 2 0 ; = 0 , a similar need the V* core TJV;^] 118 VII. APPLICATIONS TO VII. 1 Atmospheric As for an this atmospheric the curvature mid-latitudes the a Coriolis dominant also phere the the of in taking earth. is the the provide full the of the observation neglected e n t i r e l y , Coriolis may motion be this is assumption relaxed for the evidence section, force of that we In is for effects geostrophic have motivation model roughly unrealistic of the of a simple account However, the density equations to atmosphere. including that introduction, part attempt wind constant The an the f o r c e , which shown has in cyclone role possibility is was of FLUIDS Cyclones explained thesis REAL that plays the examined. that at It the atmos- may be somewhat. atmosphere wri tten : sV |= where the v_ + and V hydrostatic centrifugal gravity the v-vv curl and of VP = - = ^ - - 2 w x v - £ are pressure force are G J is obtains the three-dimensional and e f f e c t s ignored. the (VII.1.1) -|| + V - ( p v ) one both angular and using £ of is the vectors. viscosity the velocity = 0 vorticity (VII.1.1) of P and acceleration the equation of earth. is the due to Taking continuity: (VII.1.2) equation: 119 We s u p p o s e order, that as tangential the density earth. While a much III that to the surface is a function a very (VI I . 1 . 2 ) lowest is + W p at the v e l o c i t y then i s , to of the e a r t h . only than that satisfied this above is the density i f : V/v = 0 suppose the certainly is uniform. . However: = 0 order. lowest We a l s o of the height r e s t r i c t i v e assumption better approximation Equation to in Section (VII.1.4) One c a n t h e n show ^)'l ) 0 (see, f o r example, Veronis (1963b))that: f(_ 9 t + v- V which i n terms becomes + 1*. at with h equation by w ax The so w Thus, i t is force does. isolated vortex a region 2u is acts no l o n g e r is (VII.1.7) the density in the equations parameter vorticity as II.3 (VII .1.6) and ( I I I . 3 . 7 ) allowing a n d on t h e s p h e r e that = 0 o are (III.3.6) no c h a n g e s Coriolis of Section continuity: ay Coriolis meter ay h ax z coordinates w +2co z z JL a(hkv ) produces the ll + A. a(hkv ) + co . film (VII.1.5) order. of equations z p p x These 2 of the thin to f i r s t JL + in z is as vortex of i s o l a t e d as where a constant possible to t o have cores. w vary of motion, known 2cocos0 with replaced z with but including the C o r i o l i s 9 is source flows para- the c o l a t i t u d e of v o r t i c i t y i n which the The w h o l e v o r t i c i t y then height idea breaks of a down. The 120 only k satisfactory co and effects are both of surface Thus this thesis Coriolis the that to neglecting all variation. the v o r t i c e s in a simple on a c u r v e d vorticity that throughout discussed in way t o a c c o u n t rotating the flow for surface and n o t will just core. been modelled way, one m i g h t for the C o r i o l i s Y large force numbers vorticity vortices. vortex IV. by r e q u i r i n g v(x r>— y n n ) a by a l a r g e It is that In the vortex have similar number possible to more k simply k(x ,y ) + n n (VII.1.8) W ( ^ n l ^ V V + n o • as: • no Y = c o n s t ' ' *nl = c o n s t - (VII.1.9) Unfortunately, Ey constraint such n = 0 a system does . Thus, there will pole) whose fixed at infinity (the south wanes as the flow progresses. very large negligible model (which the e f f e c t of this and the system f o r the flow. would involve If solving still the appear vortex the equations waxes of vortices will become provide investigation required t o be a strength t h e number single should A proper not respect a vortex and is satisfactory of this of motion system by of account strengths „ n " FTvV^ Y extended of point in Section 2 n " with to: may be r e s t a t e d Y flows an a t m o s p h e r i c discussed according - planar model vortices ,*, many by the which amounts and depth conclude A vortex t o be t h e a p p r o x i m a t i o n which be e x t e n d e d effects. However, vary constant one must have seems curvature cannot necessarily at resolution 121 computer initial f o r l a r g e numbers o f v o r t i c e s , l o n g times and conditions) i s beyond the scope of t h i s t h e s i s . varied 122 IV.2 Superfluid In Vortices recent almost exclusively Hell. While tions which completely that years in work relation the complete describe of of component Vortices vortex having are observed. The c o r e s ized, n o t by r e g i o n s of singular arity. of one o r two describe £ equations m = mass is zero constant not incomin of are is is units helium characterin truly at the and i s which singul- the order Angstroms. expects, n n nearly The v o r t i c i t y of the core the motion h velocity quantized is equa- accepted b u t by a r e g i o n of the f l u i d liquid is of the v o r t i c e s rapidly. of been are s t i l l it very constant, vorticity, but the d e n s i t y On The decreases The r a d i u s Hell circulation h/m = 1 0 " crrr/se.c (h = P l a n c k ' s density liquid cores, has differential of the f l u i d atom) the to the b e h a v i o u r (see Putterman(1974)) irr.otationa 1 and, o u t s i d e pressible. vortices set of p a r t i a l the flow understood the s u p e r f l u i d on i s o l a t e d = 6 n = therefore, of c o vortices n s t of motions o- c< dk 2i n n h k l2 az but that (IV. 3.4) should now: (VII.2.1) f o r a vortex in •h. n n + equation ' 1 2 + >*(z,z> that +1 n + system ..n h n E Y^(Z,Z;Z. k^n k K n are then: a Y B (z ,z ;z ,z ) n n n az + az ,z, ) (VII.2.2) K z=z„ There the is no v o r t e x fluid outside streamfunction the core is and the k i n e t i c not conserved. energy However, in of 123 both the vortex case of uniform streamfunctions flow exist. and These of flow are, a ,y 2 h(x 2 ,y ) + 2y ^ * ( X ,y J 1 r ' J )1 k(x ,y )(£n6 -l) n n n (VII.2.3) v N E iy B(x =i l f ,y 2 n n 2y ^x ,y )} + ; x . ,y. ) n , k= l , J ^ plane respectively: ' n n n' n " n n J N ft = % E E Y y.^x ,y ; x . ,y. ) + n = l k/n n n k k' - on N r E y if. *F(x ,y ; x , ,y. ) + % E i y B ( x k/n n'^n k'-'k' n=l^ N ft = h E n=l + Y depth n n n , n n ; x , ,y. ) k k' h=l (VII.2.4) tion of The conservation the k i n e t i c energy These problems date. in equations more of it difficult is methods the seem free to of corresponds the now be used geometries with surface solve necessary. of for to the fluid external to than k = k 0 is + ar a rotating ¥ ( x ,y : x ,y 1 2 been that 1 ) of since fluid. and conserva- to discuss have particular interest a cylindrical container of ft can general A problem shape of the core. vortex used to vortices this is in the Unfortunately, numerical 124 VIII. CONCLUSION In this vortex motion fluids of has small To a similar fluid thesis been but the the generalized varying author's nature is by "orthomorphic two The fields in shown in to purely it fluids be a small horizontal He has is has vortex motion. in only also describing If the the and to a plane or a sphere, explicitly in terms As did shown the more that (1943) there is The of the be Lamb may only be example The the vortex approximations ratio of equations there is of of the of velocity perturbation the of choice used expansion vertical are to examined. a function surface are h(x,y) be curvature fluid is uniform, flow is topologica11y similar of motion written equations there on may be are h(x,y). of case motion elliptic for might motion order effects the if plane." a suitable determine that general of fluid a systematic surface of equations functions Lin the by depth. depth boundaries Green's to lowest shown no the that, varying scales; the that, fields the work sphere. p a r a m e t e r A r e p r e s e n t ! * ng for depth, a shown consistent suited In on possible of onto published two-dtrriension.aT.flow. vortices author coordinates, projection j u s t i f i c a t i o n that discussed suggested velocity in surfaces. only vortex rectilinear vortices the depth, by by curved who obtained is on of include (1916) constant approximated to theory Lamb of without depth knowledge, is assumes conventional with can boundaries be w r i t t e n partial terms differential rectilinear vortices, a vortex in and streamfunction varying of the operators. the author for these has equations 125 of motion, fluid, under that and that, examined depth extend cores cause discussed. It cores is core much that it energy of the transforms surface of these simply the has been curvature and differences stability qualitative of rigidly differences also stability. was of shown depends is systems qualitative effects introduces the vortex marked of the wobble kinetic uniform, Studies VI asymmetries the showing showed t h a t Section If the is simple systems. symmetric vortex. of of considerations were radially depth V, can systems In a Section vortex to the behaviour in rotating r e l a t e d to transformations. variation between is if conformal The that it non-infinitesimal that only small on than the velocity their wobbles approximately smaller the into of strength the circular core vortex motion the radius but of amplitude, and may be neglected. The theory developed in Sections etical interest as a generalization vortex motion. It is applied seems and is to atmospheric impossible of surface quite that model of to reconcile these that important in such an role in vortices a cyclone. an in in atmosphere Of As the simple would course, cyclogenesis, be of vortices but provide if ask in effects the then greatly is of classical to shown a similar, would the however, motion. curvature possible phere natural, of 1 1 1 - VI how Section the of theortheory it the first force incidence suppressed. it force type. non-rotating, Coriolis be VII.l, Coriolis this a simple may of It atmosorder plays of cyclones 126 Vortices theory that of the longer Section core cases IV of flow with of calculating one remain uniform does makes the depth depth can In for the and of planar the most there is still V described and general, flow, For functions be a modification constant. exist. varying field. if helium streamfunction of streamfunction velocity superfluid radii a vortex special in B but flow to for a which the require there interesting the by is no the vortex problems practical describe problem the 127 BIBLIOGRAPHY B a t c h e l o r , G . K . ( 1 9 6 7 ) An I n t r o d u c t i o n t o F l u i d (Cambridge U n i v e r s i t y P r e s s , Cambridge, Chapman, D.M.F. (1977) C o l u m b i a) Chapman, D.M.F. Courant, ( 1978) M.Sc. J. Thesis Math. (University Phys. R. , a n d H i l b e r t , D. ( 1 9 6 2 ) P h y s i c s (John W i l e y & Sons, Sec.IV, p.290. Eisenhart, of Havelock, 19, (1931) Phil.Mag. of British p.1988. Methods o f M a t h e m a t i c a l New Y o r k ) V o l . 1 1 , C h . I V , L . P . ( 1 9 0 9 ) A T r e a t i s e on t h e C u r v e s and S u r f a c e s ( G i n n a n d T.H. Dynamics England) 1_1, D i f f e r e n t i a l Geometry Co., Boston), p.93. p.617. H e l m h o l t z , H. ( 1 8 5 8 ) C r e l l e ' s J o u r n a l , t r a n s l a t e d : S e r . 4 , No.226, Supp.33, p.485 (1867) Kelvin, Lord ( 1878), Nature Kelvin, Lord ( 1869), T r a n s . Roy . S o c . , E d i n . Karman, T.von (1912) 18., p.13 Phys.Zeits., K i r c h h o f f , G. ( 1 8 7 6 ) , V o r l e s u n g e n Mechanik ( L e i p z i g ) p.255 Koege, p. Lagally, Lamb, Lin, H. (1918) M. Acta (1921) Math. 41, Math.Seits., Phil.Mag. 2_5 13, p.49 uber mathematishe Physik, p.306 10., p.231 (1916) Hydrodynamics (Cambridge U n i v e r s i t y Cambridge, England) 4th ed. , A r t . 8 0 , Ch.IV, CC. ( 1 9 4 3 ) On t h e M o t i o n ( U n i v e r s i t y of Toronto o f V o r t i c e s i n Two Press, Toronto) Press, p.101. Dimensions M a s o t t i , A. (1931) Atti.Pontif.Accad.Sci.Nuovi.Lincei Mertz, ( 1978) Phys.Fluids. G.J. Morton, W.B. Onsager, L. Osborne, D.V. ( 1935) ( 1949) 2_1, P r o c . R o y . I r i s'h Nuovo et.al. Cim. (1963) Supp. 84, p.209. p.1092 Acad. 6_, Can.J.Phys. A 4_2, p.21 p.249 4J_, p . 8 2 0 P u t t e r m a n , S . J . (1974) S u p e r f l u i d Hydrodynamics P u b l i s h i n g Co. , .Amsterdam, 1974) p . l 9 f f (North Holland and p.267ff. 128 Routh, E.J., Proc.L.M.S., 1_2 p.83 Sommerfeld, A.(1949) P a r t i a l D i f f e r e n t i a l Equations ( A c a d e m i c P r e s s I n c . , New Y o r k (N . Y .)) p . 5 0 . Souriau J . M . ( 1969) S t r u c t u r e des P a r i s , 1969) Thomson, J . J . (1883) (Adams P r i z e Tkachenko, V. ( 1966) T r e a t i s e on Essay 1882, Sov.Phys. the Motion of Vortex Rings M a c M i l l a n , London) p.95. JETP G. , ( 1963a) J.Mar.Res. Zl_ Veronis, G. , ( 1 9 6 3 b ) J.Mar.Res. 21, E.J., et.al. (1979) Physics S y s t e m e s Dynami ques ( D u n o d , Veronis, Yarmchuk, in t 2_3, p.1049 p.110 p.199 Phys.Rev.Lett. 43, p.214 129 APPENDIX A: MATHEMATICAL Let with where and k N g . One may = e(-det e is the is extension M M be a t w o - d i m e n s i o n a l metric o* FORMALISM define a scalar o* , namely, the Riemannian a two-form manifold by: a* g)\ (Al) antisymmetric of smooth tensor function to on a two-form unique M . on two-form There the N with is a e = 1 2 1 natural 2N-dimensiona1 a (dx'x...xdx ,dy'x...xdy ) N density manifold satisfying: N E yo*(dx ,dy ) n=l = n (A2) n n where and the a y is ,n=l,...,N are constants. Note d i f f e r e n t i ab 1 e e v e r y w h e r e , a symplectic non-trivial structure on three-forms M . N cannot a (Note exist a ker therefore that on that Va* a=0 induces = 0 since two-dimensional mani f o l d . ) Suppose A natural flow that is ft is induced, some the scalar function equations of on motion M of . N which a re : HY -1 ^| = a A Vfi = grad n (A3) N where denotes x derivative. Notice 4 f = Vft so and t h a t ft i s (A3) matics of position M and V is the exterior that: = a(fl,n) = 0 conserved. becomes on In (IV.4.4). symplectic systems harmonic For see coordinates further Souriau reference (1969). o* = h ( x , y ) k ( x , y ) e 2 on the mathe- 130 APPENDIX B: E V A L U A T I O N OF All Section the V may V ~ sums be e v a l u a t e d N R, ( z ) special SUMS e x p ( 2TTJ necessary easily = calculations of Ln/N) (l-zexp(27rin/N7T " 1 the once: ' 2 n for Z c o m P l e > x L = 1 (Bl) is known. Suppose R. ( z ) The infinite reordering R, ( z ) The = of 00 E k z the 1/1 K - is absolutely n=l second sum vanishes unless E N( r N - L + l ) z r 1 r N _ L+k-1 = L N-fa N((N-L+l)z ' +(L-l)z N L (l-z ) N right regions hence, sides of by the of both complex analytic To = n = convergent (Bl) and plane (B3) exp( z ~ = rN , r an If I I- L ) (B4) are excluding (B4) is z=xe -rri/n (2n + l ) T r i L / N ) N + M (l x ) N analytic the l (l-exp((2n + l)Tri/N)) + Nth valid , - (L-l)x e 2 2 N ^ = " L - N v n 1 l, l - e x p ( 2-TTi L k / N ) (l-exp(2Trik/N)r = for ' - ) . . ^ m / <N n R ( / \ " L Z } R V all of all ( x e one: z. 7 7 1 ^ M ' (B5) 2 L in boots + c integer. ,_N-L + 1 (B4) put L N(-(N-L l)x - S allowing 2 T (x) N T, ( x ) 2 N continuation, evaluate (B2) e x p ( 2 T r i (L + k - l ) n / N ) = The Then: N k-1 = < 1 . sums: E i |z| ,k-l £ kz e x p ( 2 77 i ( k - 1 ) / N) k=l series L R. ( z ) that N E exp(2-rriLn/N) n=l = L the first / n ( Z } w ) 131 L i N ( l + ( N - l ) z m z-1 N - ( N - L + l ) z ( l - z i n N ( N ( N - l ) z m N " zll 1 N - - ( L - l ) z L - ( N - L + l ) ( N - L ) z N - 1 ( 1 - Z N z-1 2 ( 1 - Z +l) (N-L) - L ) N N N ~ L " 1 - ( L - l ) ( 2 N - L ) z 2 N " L " 1 ) ) (N(N-l)-(N-L + l)(N-L)z~ KN-L - 2 ) - 2 N Z Mm N L -(L-l);(2N-L)z N - L ) ) (N-L)(L-1)(2N-L) 2N Sg(N-L) ( 2 - L ) L ' H o p i t a l ' s [ij r^^wm' ( B 6 ) rule has s been used > • '»' - > N 1 twice. (B7)
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Vortex motion in thin films Hally, David 1980
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Title | Vortex motion in thin films |
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Hally, David |
Date Issued | 1980 |
Description | The classical theory of rectilinear vortex motion has been generalized to include vortices in thin fluids of varying depth on curved surfaces. The equations of motion are examined to lowest order in a perturbation expansion in which the depth of fluid is considered small in comparison with the principal radii of curvature of the surface. Existence of a generalized vortex streamfunction is proved and used to generate conservation laws. A number of simple vortex systems are described. In particular, criteria for the stability of rings of vortices on surfaces of revolution are found. In contradistinction to the result of von Karman, double rings (vortex streets) in both staggered and symmetric configurations may be stable. The effects of finite core size are examined. Departures from radial symmetry in core vorticity distributions are shown to introduce small wobbles in the vortex motion. The case of an elliptical core is treated in detail. Applications of the theory to atmospheric cyclones and superfluid vortices are discussed. |
Subject |
Thin films Vortex-motion |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-23 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085760 |
URI | http://hdl.handle.net/2429/22397 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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