150 Years of Vortex Dynamics (Mathematical analysis of the Euler equations : 150 years of vortex dynamics)

150

YearsofVortexDynamics HASSAN

AREF1

Center_forFluid Dynamics and Department_ofPhysics TechnicalUniversity_ofDenmarkKgs. LyngbyDK-2800, Denmark

and

Department_ofEngineeringScience andMechanics, College_ofEngineering,

VirginiaPolytechnic Institute&StateUniversity, Blacksburg,VA 24061, USA 1. Introduction

The subject of vortex dynamics

can

fairly be said tohave been initiated by the seminalpaper

[23] of HermannLudwigFerdinand Helmholtz

150 years ago.

Inthis

paper

Helmholtz established his three laws ofvortex

motion

in

roughly the form they

are

found today in textbooks

on

fluid mechanics. Onemotivation

seems

to havebeenhis interestin frictional phenomena, camied

over

fromhisinterest in energetics; another

was

his growing

awareness

ofthe

power

ofGreen’s theorem in hydrodynamics. In

a

speech [25] at

a

banquet

on

the occasion ofhis 70th birthday–

an

event

that brought together 260 friends and admirers at Kaiserhofon November 2, ₁₈₉₁ -Helmholtz gavethe followingaccount:

I have also beenin aposition to solve several problems in mathematical physics, someofwhich the greatmathematicianssincethetimeofEulerhadworkedoninvain –forexample, problemsconcerning

vortex motion and the discontinuity ofmotion influids, _{the problem of themotion}of soundwavesatthe openendsoforganpipes, andsoon. Butthe pridewhlchImight havefeltabout the final resultofthese $invesuga0ons$was considerablylessened bymyknowledge that I had only succeededin solving such problems, aftermanyerroneousattempts, by the gradual generalization of favorable examples and bya seriesoffortunateguesses. I wouldcomparemyself toamountainclimberwho,not knowingtheway, ascendsslowly and painfully and isoftencompelled to retracehis steps because he cangonofarther; who, sometimesbyrcasoning and sometimes by$acciden\zeta hiLs$uponsigns ofaffcsh path,which leads himalittlefarther:and who finally,when he has reached thesummit,discoverstohisannoyance aroyal roadonwhichhe might have riddenupifhehad been clever enough to findtherightstarting poimatthe

beginning.Inmypapers and memoirsI havenot,of course,giventhe readeranaccount ofmywanderings,

buthave onlydescribedthe beaten path along whichonemayreachthesummitwithout trouble.

Until the

appearance

ofHelmholtz’spaperthe integralsofthe hydrodynamicalequationshadbeen deternined almost exclusively on the assumption that the cartesian components of the velocity

of each fluid particle

are

partial first derivatives ofthe velocity potential. Helmholtz eliminated thislimitation,andtookintoaccounttheeffectsofffictionbetween different elements of the fluid

or

between the fluid and a solid boundary. At the time the effect offriction had not been fully understood mathematically. Helmholtz endeavored toidentifyaspectsofthemotionthatfrictional forces will produce in

a

fluid. Key

among

theseisthe spin-up of individual fluidparticles, which is measuredbythevectorfield known

as

the vorticity.

It is somewhat

rare

that

a

subject in

a

rather”mature” science such as fluid mechanics has

so

clear

a

starting date. Usually when this happens it isdue to

a

seminalpaperby a luminary of the

field,

a paper

that is far ahead of anything elseproduced by his contemporaries, and

a

paperthat is quickly embraced by thecommunity and sets thestagefordevelopments fordecades tocome. The early

papers

inthe

new

field ofvortexdynamics

were

scatteredamong

many

joumalsinmany countriesand

were

writtenin

a

multitudeoflanguages, primarily English,French, German,Italian andRussian. This diversity of publication

venue

and language, unfortunately, often makes the

150 YearsofVortex$Dynm\dot{u}cs$

literature ratherdifficulttoidentify and

access

forthe modemresearcher. An attemptto assemb]$e$

a

comprehensivc bibliography forthe firstcentury ofvortex dynamics

may

befound in [47]. For additional backgroundonHelmholtzand his workin hydrodynamics

see

[14].

2. Case studies

Some of the older papers collectedin the bibliography [47] have maintained themselvesinto modem researchwhileothers havebeen long forgotten. For example, thethesis ofGr\"obli [19,20] andthe later

paper

by Synge [69]

on

the solution of the three-vortex problem

were

revived about

30

years ago throughthe independent rediscoveries by Novikov [53] _{and Aref}[2]. For

a

review of the history of solution, neglect and re-discoveiy

see

[6]. While the three-vortex problem is

very

interesting of its

own

accord, the discovery of chaos in the four-voitex problem (cf. [3]) immediately propelledthis kindofproblem to the frontlines of”modern science”. Seealso

_{\S 2.2}

below.

Anotherexample ofthis kind

may

be found in the extensiveseries of works byDaRios([13]

andseverallaterpapers)

on

vortexfilament motionundertheso-called

localized

induction approx-imation. In spite ofhaving been done

as a

thesis under T. Levi-Civita,

one

the most illustrious mathematicians of his day, this work, somehow,

never

“took”. It

was

not until the $19ffl$_{’s when}

Arms&Hama[8] and Betchov[11] re-introduced thisidea-andBatchelorincludeditinhis well

knowntext [9]-thatit finally became

a

standardpartofthe subject. The beautifultransformation of Hasimoto [21], and the idea that vortexfilaments

can

support solitonwaves, also played

a

role inthis ”assimilation” into modem research. The histoiy ofDaRios’ work has been reviewed by Ricca[58,59].

2.1 Helmholtz’spaper

Helmholtzdiscovered

a

seriesoffundamentalpropositionsinhydrodynamics thathad entirely

es-caped his predecessors. He pointed out that already Euler hadmentioned

cases

of fluidmotion in which

no

velocity-potentialexists, forexample, the rotation of

a

fluid about

an

axis where

every

elementhasthe

same

angular velocity. Aminute sphereoffluidmay

move as a

whole in

a

definite direction, _{and change its shape, all while spinning about}

an

axis. This lastmotionisthe distinguish-ingcharacteristic ofvorticity. Helmholtz

was

the firsttoelucidate key propertiesof thoseportions of

a

fluid in which vorticity

occurs.

His investigation

was

restricted to

a

frictionless, incompress-iblefluid. He proved that in such

an

idealsubstance vortexmotioncouldneitherbe producedfrom irrotational flow

nor

be destroyed entirely by

any

natural forces thathave

a

potential. If vorticity exists within

a

group

of fluid particles, they

are

incapable oftransmitting ittoparticles thathave none. Theycannotbe entirely deprived of theirvorticitythemselves(althoughthe vorticity ofany individual particlemaychangeinthree-dimensionalflow; in two-dimensional flow the vorticity of

each particle is a constantofthe motion). For an ideal fluid the laws ofvortexmotion establish

a

curious andindissoluble fellowship between fluid particles and theirstateofrotation. In the Introductiontohis

paper

Helmholtz states:

Hence it appearedto metobe of importancetoinvestigate thespecies of$mo\dot{u}on$ for which thereisno

velocity-potential.

The following investigation shows that when there isa velocity-potentialthe elements of the fluid

have norotation, but thatthere is at least aportion of the fluid elements inrotationwhen there is no

$vel\propto ity-\mu ten\dot{0}d$.

Byvortex-lines(Vvirbellinien)1 denotel\’inesdrawnthroughthe fluidsoas ateveiy pointtocoincide

withtheinstantaneous axisofrotationofthecorrespondingfluid element.

150YcarsofVortex Dynamics

everypoint of theboundaryofaninfinitely smallclosedcurve.

Theinvestigationshowsthat,ifallthe forces whichactonthe fluidhaveapotential,–

1.Noelement ofthe fluid whichwasnotoriginally inrotation is made to rotate.

2. The elements whichatanytimebelongtoonevortex-line,however theymaybetranslated,remain ononevonex-line.

3. The productof thesection andtheangularvelocityofaninfinitelythinvortex-filamentisconstant

throughout itswhole length, andretains thesamevalueduringalldisplacements of the filament. Hence

vortex-filamentsmusteitherbeclosed curves,ormusthavetheirendsin the boundingsurface ofthefluid. According to Tmesdell [75, p.58] the

name

vorticity was introduced by Lamb [35] for the vector, $\omega$, whoseCartesian components, $(\xi, \eta, \zeta)$,

are

given in terms ofthe components $(u, v, w)$

ofthe(Eulerian)velocityvector $u$by

$\xi=\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}$, $\eta=\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}$, $\zeta=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$. (1)

In modemvectornotation

$\omega=\nabla\cross u$

.

(2)

Helnholtz’sresultin

\S 1

ofhispaperthat

an

arbitraryinstantaneous state ofcontinuous motion of

a

defomtable medium is at each point the superposition of

a

uniform velocity oftranslation,

a

motion of extension, a shearingmotion,andarigidrotation, precipitated

an

extendeddebate with the French academician Bertrand.

The mird lawcontains twostatements,$viz$that “vortex-filaments must either be closed curves”,

or

thatthey‘musthave theirendsinthebounding surface ofthe fluid”. Thefirststatementexcludes thepossibility of vortex lines that wander aperiodically and

never

close,

as one

finds, forexample, in

a

chaotic,

_{three-dimensional}

_flow2.

_{The second} is, in principle, correct only for vortex lines, although

an

example of

a

thin vortexfilament that ends at

a

point in the interiorofthe fluid has,

so

far

as we

are

aware,

never

been given. Thevorticity distribution insuch asffucture would be near-singular. See[18] for

a

modem perspective

on

thisproblem.

In

\S 3

ofhispaper [23] _{Helmholtz addresses the inverse problemoffinding the components of}

the velocity $u,$ $v,$ $w$ ffom the components ofvonicity $\xi,$ $\eta,$ $\zeta$ (up to

a

potential flow that

covers

theboundaiyconditions). He independently obtains the representationsofStokesfortheclassical problem of vector analysis of determining

a

vector field of known divergence $($‘hydrodynamic

integrals of the first class” in his terminology) and curl (“hydrodynamic integrals of the second

class“). _{Determination ofthe velocity field for incompressible fluid leads to the Biot-Savartlaw} ofelectromagnetism, which in the present

case

reads that each rotating element of fluid induces in

every

other element

a

velocity with direction perpendicular to the plane through the second element that contains the axis of the first element. The magnitude of this induced velocity

is

directly proportionaltothe volume of the firstelement,its angular velocity, and the sine of the angle betweenthe line that joins thetwoelements andthe axis ofrotation,and is inversely proportional

tothe

square

ofthe distance between thetwo elements.

Helmholtzalsoestablishesanalogiesbetweenthe induced velocity and the forces

on

magnetized

particles. Mostof these relationswould todaycomeundertheheading of potentialtheory.

In

\S 4

ofhis

paper

[23]_{Helmholtz derives}

_an

_{elegantexpression fortheconserved kinetic}

_energy,

“vis viva”inhisterminology,ofinfinite fluid with

a

compactdistribution ofvorticity within it. 2Thebestlcnown examplesmaybethc$ABC/lows$studicdby severulauthorseversincc their inuoductionin1965-66byArnokiand$H\ell non$;

see[4]foralxiefdesnipoonin the context of.chaotic advection“. Thereare manyotherinstanccswhere vortexlines donotclose.Iodeed,closed

150 Yeaisof VortexDynarnics

Figure 1: MotionoftwoparaUcl rcctilnearvortices (orpoint$vom\infty s$). Top: (a) circulations of thesamesign;$Cb$)

circulations ofopposite sign.Bottom: Motion ofavortexpair.From[27].

In \S 5, entitled “Straight parallel vortex-filaments”, Helmholtz studies certain simple

cases

in which therotationofthe elements

occurs

only in

a

set of parallel rectilinearvortex-filaments. In particular, he considers several infinitely thin, parallel vortex-filaments each of which carries

a

finite, limiting value, $m$, ofthe product of the cross-sectional

area

and the angular velocity. This

isthe

now

celebratedconcept of

a

point

vonex.

Helmholtz considers simple

cases

ofthe dynamics of such vortices. He establishes the law ofconservation of the center

of

vomcity of

an

assembly

of point vortices. The discussion is phrased in terms of the ”center of gravity” of the vortices

(consideringtheir values of $m$

as

the analog of”masses”): “Thecentre ofgravity ofthe

vortex-filaments remains stationaryduring theirmotionsabout

one

another, unless the

sum

of the masses bezero,in which

case

there isno centreofgravity:’ Withoutfurther explanation Helmholtznotes

the following two

consequences:

1. Ifthere be asingle rectilinear vortex-filamentofindefinitely small section in afluidindefinitein all

directionsperpendicularto it,the motionofanelement of the fluid at finite dlstancefromit depends only

onthe product$(\zeta dadb=m)$_{ofthe velocity}ofrotationandthesection, notonthe formofthat$\sec\dot{u}on$.

Theelements ofthe fluidrevolveabout itwithtangential velocity$= \frac{m}{\pi r}$,where$r$is$A\epsilon$distance fmm the centreofgravityofthe filament.

ne

positionofthecentre ofgravity, the angular velocity, theareaofthe

$s\propto uon$,andtherefore,of course, the magnimde $m$remainunaltered,evenif the form oftheindefinitely

small$\sec 0on$mayalter.

2. lfthere be two rectilinear vortex-fllaments of indeflnitely small section inan unlimitedfluid, each

willcause the otherto move ina direction perpendiculartothe line joining hem. Thus the length of

150YeaisofVortex Dynamics

Figure 2:Self-inducedforward motion ofavortex ring. From[27].

distancesfromit. If therotationbeinthesamedirectionforboth(thatis, ofthesamesign)theircentre ofgravity lies betweenthem. Ifin opposite directions(thatis,ofdifferentsigns), thecentreofgravity

lies in the linejoiningthemproduced. Andif,inaddition,the productofthevelocityand thesectionbe

thesame forboth, sothatthe centre ofgravityisataninfinitedistance, they travel forwards withequal

$ve1\infty ity$,andin parallel directions perpendiculartothe linejoiningthem.

See Fig. 1 for later illustrations ofthese motions. Inadditiontointroducing thisnotionof

a

vortex

pair”Helmholtzdescribes the motion of

a

$sing$]$e$ vortex-filament

near an

infiniteplane to whichit

isparallel. Hestatesthat the boundarycondition will be fulfilledifinstead oftheplane thereis

an

infinite

mass

of fluid with another vortex-filament

as

the image (with respect to the plane) ofthe first, _{and concludes: “From this it follows that the vortex-filament}

_moves

_{parallel to theplane in} thedirection in which the elementsofthefluid betweenit and the planemove, and withone-fourth ofthe velocity which the elements at the foot of

a

perpendicular from the filament

on

the plane have:’

In \S 6, entitled“Circularvortex-filaments”,_{Helmholtz smdies the} axisymmetric motion of

sev-eralcircular vortex-filaments whose planes

are

parallel to the xy-plane, andwhosecenters

are on

the z-axis. Here he considers the problem in full detail and amives at the conclusion that “in

a

circularvortex-filament of

very

smallsectionin

an

indefinitely extendedfluid,thecentreofgravity of the section has, from the commencement,

an

approximately constant and

very

great velocity paralleltotheaxisof the vortex-ring, and this is directed towards the sideto which the fluidflows through the ring.” (See_{Fig.2 for}

_a

laterillustration.)

Whentwosuchringsofinfinitesimal cross-sectionhave

a

common

axisand the

same

direction ofrotation, _{theytravel in the}

_same

_{direction. As they approach, the firstring widens and travels}

more

slowly, thesecond contracts and travels faster. Finally, if theirvelocities

are

not toodifferent,

150 YearsofVortex Dynsnics

Figure3: mustrationof leapfrogging” bytwovortexpairs.Theinduced velocitiesarcindicatedbyarrows.From[27]

baseduponGr(Sbli’s_calcuhuons[19, 20].

then repeated indefinitely (in principle- in reality the finite

cores

of the rings and the effects of viscosity$wiU$only allowone or_twocyclesof this_motion). If two_vortexringshaveequal radiiand

opposite angularvelocities, theywillapproacheach otherandwiden

one

another;andwhen they

are very

near

to

one

another,_{their velocity of approach becomes smaller and}smaller,andtheirrate of widening fasterand faster. Just

as

in the

case

ofthe straightvortexfilament

near

the plane wall,

this motion is similarto the motion ofa singlevortex ring mnning

up

against aplane wall. The imageof theringinthe wallis another similarringwith the opposite

sense

ofcirculation.

Lanchester

saw

thistype ofmotion involvingseveral

vomces

tobe relevant to vortex formation

behind

a

wing of finite span. Hewrote [36,p. 122]:

Groups offllamentsorringsbehave inasimilarmannertopairs: thus agroupofringsmayplay “leap-frog“ collectivelysolongas$Ae$totalnumber ofrings doesnotexceedacertainmaximum;congregations of vortex filaments likewise by their mutual interactionmove as a partofaconcentrated system, like

waltzers in aball-room; whenthe number exceeds a certain maximum the wholesystemconsists ofa

number oflessergroups.

Onlyin

rare

cases

does

a

singlepaperput foiward

so

many profoundideas and

open

so many

avenuesfor further investigation. Almost fifty years later, in 1906, Lord Kelvin, _{who had himself}

conducted

many

greatsmdies developingvortexdynamics further, wrote in the preface to

a

book aboutHelmholtz [33] that“his admuirabletheory ofvortexringsis

one

ofthemostbeautifulofall thebeautiful piecesof mathematical workhitherto done in the dynamics of incompressible fluids.’‘ Surprisingly Helmholtz

never

continuedhis investigations of the topic established in his ground-breaking

paper

[23]. Instead hewrote another remarkable

paper

[24]

on

_{discontinuous motion} of

an

inviscid fluid in which he used thenotion of

a

vortex sheetfrom[23].

2.2Point

vomces

Avast

area

of research started by Helmholtz’spaperis the smdy of themotionof straight, parallel,

infinitely thin vortex filaments (rectilinear vortices) in incompressible inviscid fluid or,

equiva-lently, the two-dimensional problem of point vortices

on a

plane. Through pioneering work of

hydrody-150 YearsofVortexDynanuCs

namics provided bysuch vortex elementsbecame the foundationfor

an

entire familyof numerical

methodsfor flow simulation todaycollective]y known

as

vortexmethods.

The problem of $N$ interacting point vortices

on

the unbounded xy-plane, with _vortex $\alpha=$ $1,$

$\ldots,$$N$having strength

$\Gamma_{\alpha}$ (which is constant accordingto Helmholtz’s theorems) and position

$(x_{\alpha},y_{\alpha})$,consists in solving the folowingsystem of$2N$first-order,nonlinear, ordinarydifferential

equations

$\frac{dx_{\alpha}}{dt}=-\frac{1}{2\tau 1}\sum_{\beta=1}^{N}/\Gamma_{\beta}\frac{y_{\alpha}-y_{\beta}}{l_{\alpha\beta}^{2}}$, $\frac{dy_{\alpha}}{dt}=\frac{1}{2\pi}\sum_{\beta=1}^{N}’\Gamma_{\beta}\frac{x_{\alpha}-x_{\beta}}{l_{\alpha\beta}^{2}}$ , (3)

where $\alpha=1,2,$_$\ldots,$$N,$ $l_{\alpha\beta}=\sqrt{(x_{\alpha}-x_{\beta})^{2}+(y_{\alpha}-y_{\beta})^{2}}$is the distancebetween voitices $\alpha$ and $\beta$, and the prime

on

the summation indicatesomission of the singularterm_{$\beta=\alpha.$} TypicaUy,

an

initial value problemis addressed with the initial positionsofthevortices and their strengths given

so as

tocapture

or

model

some

flow situationofinterest.

The system (3)

can

alsobewritten

as

$N$ODEsfor$N$complex coordinates$z_{\alpha}=x_{\alpha}+iy_{\alpha}$

$\frac{dz_{\alpha}^{*}}{dt}=\frac{1}{2\pi i}\sum_{\beta=1}^{N}’\frac{\Gamma_{\beta}}{z_{\alpha}-z_{\beta}}$, $\alpha=1,2,$

$\ldots,$$N$, (4)

wherethe asterisk denotescomplexconjugation.

Inhislecmres[32,Lecmre20] _{Kirchhoff demonstrated that thesystem}(3)

_can

becastin

Hamil-ton’scanonical$fomi^{3}$

:

$\Gamma_{\alpha}\frac{dx_{\alpha}}{dt}=\frac{\partial H}{\partial y_{\alpha}}$, $\Gamma_{\alpha}\frac{dy_{\alpha}}{dt}=-\frac{\partial H}{\partial x_{\alpha}}$, $\alpha=1,2,$

$\ldots,$$N$, (5)

wheietheHamiltonian,

$H=- \frac{1}{4\pi}\sum_{\alpha,\beta=1}^{N}/\Gamma_{\alpha}\Gamma_{\beta}\log l_{\alpha\beta}$, (6)

is conserved during the motion ofthe point vonices. (Here and in what follows $\log$ denotes the namrallogarithm.)

In additionto$H$theHamiltonian _system(5) hasthree independem first integrals:

$Q= \sum_{\alpha=1}^{N}\Gamma_{\alpha}x_{\alpha}$ , $P= \sum_{\alpha=1}^{N}\Gamma_{\alpha}y_{\alpha}$, $I= \sum_{\alpha=1}^{N}\Gamma_{\alpha}(x_{\alpha}^{2}+y_{\alpha}^{2})$

.

(7)

Regardless of the values of thevortex strengths,theintegrals$H,$$I$, and$P^{2}+Q^{2}$

are

ininvolution,

that is, _{the Poisson bracket between any two of them is} zero;

see

the review paper [3]

or

the monograph$[50]^{4}$

.

AccordingtoLiouville’s theoreminanalytical dynamics the Hamiltoniansystem

(5) for$N=3$ is then integrable regardless of the values of the vortex strengths. Aterse general statementtothis effectwasincluded byPoincar\’e in hislecmres [56,

_{\S 77].}

Anextensive analytical smdy of integrability andof several special

cases

ofthree-vortexmotion had already been performed by Gr\"obli in his noteworthy 1877 G\"ottingen disseitation [19] (later 3Acomplete$cor\ddagger espnden\infty$followsby$set\dot{0}ng$the $gener|hzd$coordinates“$q_{\alpha}=x_{\alpha}$andthe $genei\cdot hrr4moment\iota’ p_{a}=\Gamma_{a}y_{\Phi}$. This

results inthe$oem\alpha\cdot ble$insight that the.phasespace –in thesenseofHamiltoniandynamics–forapoint vortex_{system is,}in essence,its

configurationspace, afactlaterexploited byOnsager inaseminalpaper[54]onthc statisticalmechanics ofasystem of pointvortices.

150 Yearsof VortexDynamics

also published

as an

extensive

paper

[20]$)$ thatmust rightly beconsidered

a

classic of the vortex

dynamics literature. An account ofthe life, _scientific achievements and tragic deathof the Swiss scientistandmathematician WalterGr\"obli (1852-1903)maybe foundin [6].

Thesolution of the three-vortex problem andthedissertation itself

were

mentionedinfootnotes by Kirchhoffin thethird(1883)editionof his]$ecmres$ [$32,$Lecmre20,

\S 3]

andinthefundamental

treatise by Lamb[35,

_{\S 155]}

(althoughina

way

that doesnotfullyrevealthe comprehensive

namre

of Grobli’s investigations). Based

on

these

cursory

citations it is not difficult to understand that almost

a

cenmry

later Batchelor would write in his important text [9] that the details ofmotion ofthree point vortices “arenotevident”. A lengthy excerpt (in Englishtranslation)from Gr\"obli’s dissertation is given in [6].

The Hamiltonian (6) depends only

on

the mumal distances $l_{\alpha\beta}$ between the vortices which

suggeststhat

one can

write equations of motion that involve only these distances. Suchequations

were

obtainedbyGr\"obli and later byLaura [37] whoalso expounded

on

the canonical formalism. They

are

$\frac{dl_{\alpha\beta}^{2}}{dt}=\frac{2}{\pi}\sum_{\lambda=1}^{N}//\Gamma_{\lambda}\epsilon_{\alpha\beta\lambda}A_{\alpha\beta\lambda}(\frac{1}{l_{\beta\lambda}^{2}}-\frac{1}{l_{\lambda\alpha}^{2}}I,$ $\alpha,$ $\beta=1,2,$

$\ldots,$$N$, (8)

where the twoprimeson thesummation sign

now

mean

that $\lambda\neq\alpha,$$\beta$

.

The quantity$\epsilon_{\alpha\beta\lambda}=+1$ if

vortices$\alpha,$$\beta$and$\lambda$ appearcounterclockwise inthe plane, and

$\epsilon_{\alpha\beta\lambda}=-1$if they

appear

clockwise.

Finally, $A_{\alpha\beta\lambda}$ is the

area

of the vortextriangle$\alpha\beta\lambda$whichcan,in

mm,

be expressedinterms ofthe

threevortexseparations (thesides ofthevortexmangle)byHero’s formula. Interestingly,Eqs.(8)

were

re-discovered independently at least twice: by Synge [69] in

1949

and by Novikov [53] in

1975.

For $N$ vortices

one

has $\frac{1}{2}N(N-1)$ quantities $l_{\alpha\beta}$ and, thus, $\}N(N-1)$ equationsofthe

form(8). However,_only$2N-3$ ofthese

are

independent. It

can

be shown that

$\frac{1}{2}\sum_{\alpha,\beta=1}^{N}\Gamma_{\alpha}\Gamma_{\beta}l_{\alpha\beta}^{2}=(\sum_{\alpha=1}^{N}\Gamma_{\alpha})I-P^{2}-Q^{2}$

.

(9)

Theequations(8),then,have twogeneralfirstintegrals,$viz$theHamiltonian(6)and thequantity

on

the left hand side of(9). Usingthesetwointegrals thethreeODEs for$l_{12},$ $l_{23}$and$l_{31}$maybereduced

to

a

single ODE that

can

be solved by quadramre, and thiswas, in essence, the solution method outlined byGr\"obliinhisdissertation [19, 20]. The

case

$N=3$ thus appears

as

acritical

one

since

for

more

vortices additional “scales of motion”appearwithoutany obvious integrals toconstrain them. One may, therefore, expect the problem to become non-integrable. Indeed, _{this is what}

happens and theconnectiontotherecentinterest in the

emergence

ofchaos innonlineardynamics is established. The

appearance

ofchaos inpoint vortex dynamics

as one

goes

from threetofour vortices is analogous tothe appearanceof chaos in the gravitational N-body problem ofcelestial mechanics

as one goes

from two to three bodies. For the

case

of point

masses

the

appearance

of chaos

or

the absence ofintegrability became part of the legacy ofPoincar\’e. For inexplicable

reasons

the analogous discussion for pointvortices hadto waitfor

more

than

a

cenmry

afterthe

solution of the three-vortex problem. Both Gr\"obli [19, 20] _{and later Laura} [37] _outlined how to

determinethe ”absolutemotion” ofthe voricesprovided thesolution for the ”relative motion”

as

givenbyequations (8)

was

alreadyknown.

The

namre

of the motion oftwo vortices had already been outlined by Helmholtz [23]. The motion of three vortices–both the relative and the absolute motion–with various intensities and initial conditions

was

extensively analyzed by Gr\"obli [19, 20]. The relative motion ofthree

150Yeaisof Vortex Dynamics

arbitrary voitices, _based upon Eqs.(8),

was

studied and classified by Synge [69] by introducing triangularcoordinates in

a

”phase space” of the three distances between thevortices. Gr\"oblihad actually found such arepresentation for the

case

ofthreeidenticalvortices, and thisconsmiction

was

foundindependently

a

cenmry

laterbyNovikov[53]. _{Synge’s comprehensive analysis}

was

re-discovered independently in[2]. Thus, todaythe three-vortex problemmaybeconsideredtohave arathercompletesolution. Gr\"obli [19,20] alsodiscoveredanunusualcasewherethethree vortices

converge on a

pointin

a

_finite

time. Except forSynge’s smdy[69],which

was

itselfoverlooked,this intriguing

case

of

vonex

collapse alsowent unnoticed for

a

century. It is admittedlyasomewhat special

case

requiring boththatthe harmonic

mean

of thethreevortex strengths be

zero

and that the integral ofmotion(9) vanish.

The integrable pmblem offour vortices arranged

as

two coaxial pairs has been addressed in many papers. Gr\"obli [19,20] investigated thecase of’ieapfrogging” when all vortices have the

same

absolute strength, and obtained

an

analytical representation forthe vortex trajectories, cf.

Fig.3. His analysis

was

repeated independently byLove [40] andHicks [26].

The

case

ofuniform rotation of

a

regular polygon of $N$vortices

was

addressed in the Adams

Prize essay of J. J. Thomson [71]. He proved that the regular N-gon is stable to infinitesimal permrbations for $N=2,3,4,5,6$but becomes unstablefor $N>7$

.

(For$N=7$ the polygon is marginallystableto linear order and one must go tothe next orderto decide the stabilityissue.) This smdy

was

extended by Havelock[22] andothers, and the problem continuestobeaddressed inthe hteramreinvarious forms. See therecentreview[5] andalsothe

extension

to”triple rings”

byAref&vanBuren [7].

Helmholtzwasalso the first to address problemsof pointvorticesinteracting withrigid

bound-aries [23]. As

we

have seen, he considered the

case

of

a

point vortex in the spacebounded by a planewall. Using

an

“image” vortex ofoppositestrength situated at the reflection of the original

vortexin theplane boundaryhereducedthe problemtothatofthe motionof

a

vortexpair

on

the unboundedplane. Thisuseof the”methodofimages” has since been widely employedin various problems of themotion of

a

singlepointvortexinvarious bounded$\beta omains$

.

Aparticular

case

of

an

equilibriumofavortexpair behind

a

cylinder inauniform potential flow isknown

as

the $F6ppl$

problem”after the senuinal

paper

[17].

$\Pi e$ general

case

ofthe motion ofpoint vortices in an arbitraiy domain

was

$smAed$byRouth

[63] using the theory ofconformal mappings.

.

The velocity ofa point vortex in the transformed

planeis not equal to the velocityobtained by simple $substim\dot{u}on$ ofthe conformal mapping into

theexpressionfor the velocityinthe original plane-

one

requires alsotheinfluence of theimages which is capmred by the so-called Routhcorrection”. Acompletemathematical theory

was

devel-oped byLin[39] whoshowed that the problem isalwaysHanuiltonian withaHamiltonian function that is

a

hybrid of Kirchhoff’s Hanuiltonian (6) forthe unbounded plane and the Hamiltonian that Routhfoundformotion ofasinglevortexin abounded domain[63].

W. Thomson [72]

was

the first to show that

a

vortex pair in steady motion

on

the unbounded plane is accompanied by

an

”atmosphere“, i.e.,

a

fixed,$c$]osedvolume (area)of fluid particles that

move

forward with the vortexpair. The bounding

curve

of this ”atmosphere” is todaysometimes called the ‘Kelvin oval”. Figure 4 reproduces the original drawing from [72] where

we

findthis description:

$\Pi e$diagram_represents precisely theconvexoutline referred_to,andthe lines ofmotionof theinterior

fluid camiedalong by thevortex,forthecaseofadouble vortex consisting oftwoinfinitelylong, parallel, sffaightvorices ofequal$rota0ms$in opposite directions.Thecurveshavebeendrawnby Mr. D.

150 Yearsof VomDynamics

Figure 4:The “atnosphere” traveling withavortexpair.Fmm[72].

is

$\frac{y^{2}}{a^{2}}=\frac{2x}{a}\frac{N+1}{N-1}-(1+\frac{x^{2}}{a^{2}})$, where $\log N=\frac{x+b}{a}$.

The motionofthe surrounding fluid must be precisely thesame as it would be if the spacewithin this

surfacewereoccupiedbyasmooth solid.

Each passive fluidparticle maybe considered ‘a pointvortex of

zero

strength”, and the

equa-tionsofmotionfor all particles advected by the translatingvortexsystem

are

integrable.The defor-mation of

a

line offluid connectingtwo vortices within themovingbody

was

studied analytically byRiecke [60];

see

[48]foradditional illustrations.

2.3 Vonexatoms

In the $1860’ sW^{\cdot}m$

am

Thomson, later Lord Kelvin, became

very

interested in vortex dynamics

since he

was

convinced that atoms

were

to be modeled

as

vortex configurations in the aether. Taitmade

a

complete English translation ofHelmholtz’spaper [23] forhis

own use.

He also de-vised

some

extremely cleverexperimentstoillustratethe vortextheory using smoke vortex rings in air. Following completion of their famous Treatise on Natural Philosophy, referred to sim-ply

as

”Thomson and Tait”, and the successful laying of the Atlantic cable in 1866 (for whch Thomson

was

knighted andbecame Sir WilhiamThomson),ThomsonvisitedTaitin Edinburgh in nid-January 1867 and saw the smoke rings with his

own

eyes. Tait’s translation of Helmholtz’s

paper

was

published that

same

year in Philosophical Magazine. One must imagine that Kelvin encouraged his ffiend andcolleague topreparethis translationfor publication.

Thomson’s prodigious talent produced several firstrate studiesofvortex dynamics which,

al-though ultimatelywrong-headed intermsofatomic physics, havehad

a

lastinginfluence

on

fluid dynamics. Theidea ofcirculation, forexample, is fromthis period. The circulation is defined

as

thecontourintegral oftheprojectionof the flow velocity

on

the tangenttothecontour,

$\Gamma=\oint_{C}V\cdot ds$

.

(10)

Heshowedthat for

any

material contourmovingaccording toEuler’sequationforincompressible

flow,the circulation is

an

integral ofthemotion,

a

result known today

as

Kelvin $s$circulation

150 YearsofVonexDynamics

Figure5:Thit’sdrawings,reproducedin[74],that capture Thomson’sideasonhow atomsariseasvortex strucmresin

the aether.

W. Thomson(LordKelvin). This profound insight has continuedtoexert

an

influence

on

the entire field of fluid mechanics, including

in

such

areas

as

the assessment of the

accuracy

of numerical methods and in mrbulence modeling. Circulation is

a

distinctly topological entity, independent

of the shapeof the vortex andmeasurable by integration along any circuitthat loops around the

vortex. In thissense, thenotion of circulationmaybetaken as oneoftheearliestintroductions of

topological considerations into fluid mechanics. Tait’s seminal work

on

theclassification of knots

on

closed

curves

is a spin-offofhis interest in vortex atoms. It has stood the testoftime andis todayrecognized

as an

important contributiontotopology,knot theoryandgraphtheory. Maxwell

was an

importantcatalystfor Tait’s work

on

knots, sincehe had also become interested in

topo-logical ideas. Today the intersection of fluid mechanics and topology, in its multiple forms, has matured into

a

subfield often referred to

as

topological fluid dynamics. The

permanence

of circu-lation in

an

ideal fluid

was one

of thecomerstones ofvortex atom theory. Likeatoms, vortices in theaethercouldneitherbecreated

nor

destroyed.

Thomson’s fascination with the floating magnetexperiments by Mayer, e.g., [45, 46], _{and his} role inthe re-publication of these works in joumals such

as

Nature andPhilosophicalMagazine, werealso outgrowthsofhisconvictionthatvortices andatoms

are

intimately related. SeeSnelders’

article [66] for

a

comprehensive historical reviewof this topic. The famous quotefrom Thomson that ”Helmholtz’s [vortex] rings

are

the only tme atoms” summarizes the theme ofthis research thrust. Figure

5

depicts thekindof things heenvisioned.

Althoughitultimatelyfaded,the vortexatomideamaintaineditself for

many years

and through Kelvin’s boundless

energy

and great influence spread widely in the scientific community. The extensiveworkbyJ. J.Thomson,discovereroftheelectron,onvortexdynamicsin

was

stimulated by vortex atom theory. Even in his great

paper

of 1897 entitled ”Cathode Rays”, in which the discovery of the electron isannounced,

_we

findtheseremarks: “If

we

regardthesystemof magnets

as

a

model of

an

atom, the number ofmagnets being proportional to the atonicweight,

... we

should havesomethuing quite analogousto the periodiclaw...”, where by “periodic law” he

means

the periodic table of theelemerts. The referencetothe floatingmagnetsistoMayer’s experiments mentioned above. We see what

a

profound role these demonstration experiments played in the tluinking of these greatscientists. We shouldnotforgetthat atthe time analogexpeniments

were

one of the only

ways

ofexploring solutions to nonlinear equations that did not easily yield to

analytical methods. Computers and numerical solutions

were

stil

many years

inthe

mmre.

150 YearsofVortexDynamics

atomwritten for the 1878edition of Encyclopedia Britannica. He provided

a

detailed description ofpropertiesofvorticesin idealfluid andstrongly supported theidea of vortexatoms. Apparently,

he

was

reminded ofhis

own

earlier articles in whichhis celebrated electromagnetic theory

was

initially formulated based

upon a

mechanical modelthatalsomade referencetoHelmholtz’s

paper

[23].

2.5 Vonex ringsInspiteof the greatpopularityof Tait’s[70,pp. 291-294]smoke boxfor generating

vortexrings in air,the firstobservationof vortexrings probably corresponds with the introduction of smokingtobacco! Northmp [51,p.211] writes:

Itisnotimprobable thal the first observer ofvortexmotionswasSirWalterRaleigh; if popularbadition maybe credited regarding his useoftobacco, andprobably few smokeissince his dayhave failedto observe the curiouslypersistentfoms ofwhiteringsoftobacco smoke whichtheydelighttomake. But

sometwohundredeightyyearswentby,after the romanticdays of RaleighandSir FrancisDrake,who

madetobacco popular in England, beforeascientificexplanationofsmokeringswasattempted.

Edwin Fitch Northmp(1866-1940)

was

a

professor of physicsatPrinceton and author of

a

science fiction book entitled “Zero to Eighty: Being my Lifetime Doings, Reflections, and Inventions;

also my Joumey around the Moon.” The book

was

published in Princeton in

1937

under the

pseudonym Akkad Pseudoman. It gives

a

fictional account, supported by valid scientific data, _of

a

MorrisCounty resident’s trip around the

moon.

It

appears

to havea sustained followingin the world ofscience fiction.

Bycurious coincidence the firstexperimental observations of the generation ofvortexringsin air

were

performedby Rogers [61] inthe

same

year

(1858)that Helmholtz publishedhisseninal

paper

[23]. William BartonRogers$(1804arrow 1882)$will bebetter known today

as

the founder ofMIT.

Indeed,he

was

heavily engagedin this enterprise ataboutthe timehis paper

on

vortexrings

was

written.

The extensive smdy by Northrup [51,52] should also be mentioned here. It contains

a

very

detailed description of

a

”vortexgun”, including all the parameters,togetherwithbeautifulphotos

ofinteracting vortex rings andvortexringsinteractngwith rigidobstacles,

e.g.,

with

a

small watch chain. The modem reader

may

be intriguedto

see

in these$neararrow cenmy$old

papers

an

essentially

contemporary elucidationoftheinteraction of twocircular vortexringstilted towards

one

anothcr

so

as

tointeract$aRer$havingpropagatedfor

some

dismce, cf. Fig.6.

Theoretical smdiesofthe motion ofa ciicularvortex ring ofclosed toroidal shapewith

core

radius$a$andradius$R$of thecenterline of thetoms,where$a\ll R$, in

an

ideal fluid ledto

a

formula

fortheself-induced translational velocity$V_{ring}$,directed normallytothe plane ofthering: $V_{ring}= \frac{\Gamma}{4\pi R}(\log\frac{8R}{a}-C)+O(a/R)$

Here $\Gamma$ is the (constant) intensity of the voitex ring, equal to the circulation along

any

closed

path around the vortex core, and $C$ is

a

constant. There

was some

disagreement inthe literamre

conceming thevalue of$C$

.

The value $C= \frac{1}{4}$

was

given (without$pr\infty f$)byW.Thomson [73] and

later by Hicks, Basset, Dyson and Gray. This corresponds to the

case

where the vonicityinside the

core

varies$d\dot{u}$ectly

as

the distance from the centerline of the ring. The value$C=1$

was

given

by Lewis [38], J. J. Thomson [71], Chree, Joukovsk\"u, and Lichtensteinforauniformdistribution of vorticitywithin the

core.

For

a

hollow vortex core,

or

if

one

assumes

the fluid inside the

core

is stagnant, the value $C= \frac{1}{2}$ results $[$?$]$. The reviewby Shariffand Leonard [65]

on

vortex ring

150 YeaisofVortexDynamics

FIG. $\iota 4$

.

$\epsilon$

$t$

$c$ _$d$

Figure6: Sketch of interaction of twoidenticalvortexrings launchedon acollisioncourse. From Northmp[52].

2.5 Vonexstreets

Most students of fluid mechanicsknow thatthe

common

staggered

array

of

vortices

that forms in the wakeofacylinder(or anybluffbody) is called theK\’amdn

vonex

street. Theconceptof the vortexstreet isamong the bestknown in all offluid mechanics,in the same ’ieague”asReynolds

number, Bemoulli’sequation and theconcept ofthe boundary layer. Thefornation and

smcmre

of vortex wakes downstream of bluff bodies had been smdied extensivelyin experiments going backtoLeonardodaVinci,butvon$K4rm4n$’stheorywasthefirst real analysis of the phenomenon. In his charnuing book [30] he explains that his interest

was

aroused by

an

early picmre ofsuch vortices in

a

fresco in

one

of the churches in Bologna, Italy, where St. Christopher is shown carrying the child Jesus

across

a

flowing stream. Altemating vortices

are

seen

behind the saint’s

foot;

_see

[49] _for a beautiful color picmre of this fresco at the Church of St Dominic, entitled Madonna$con$bambino$tra$ ISantiDomenico, PietroMamre $e$ Critoforo,painted by

an

unknown

artistofthe fourteenth$cenmi\gamma$

.

Altemating vortices in air

were

observedand imaged by the English scientistMallock [41,42] whileimpressivephotos of suchvortices inwater

were

obtained by the German scientistAhlbom [1]. The French scientistB\’enard[10] also observed the altemating formation ofdetached vortices

on

the two sides of

a

bluff obstacle in water and later in many viscous fluids and in colloidal solutions.

Analysis shows that only two such configurations will propagate inthe streamwise direction: Thevortices musteitherbe arrangedin

a

symmetric

or

in

a

staggered configuration. Numerically the intensities ofthe vortices, $\Gamma$, are _$aU$

equal, but the vortices

on

the two horizontal

rows

have opposite signs. Inremarkable theoretical investigations [28, 29]

von

K4rm\’an examined the

ques-tionof stability of such processions inunbounded, incompressible,inviscid,_{two-dimensional flow}

withembedded pointvortices. Hebecame interestedin this problem when he

was

appointed

as a

graduateassistantin G\"ottingen in Prandtl’s laboratory in 1911. Prandtl had

a

doctoral candidate,

K.Hiemenz, towhom hehadgiven the taskofconstructing

a

waterchannelinordertoobserve the separationoftheflowbehind

a

cylinder. Much to his surprise, Hiemenzfound thattheflow inhis channel oscillated violently,andhefailedto achievesymmetrical flow about

a

circularcylinder.

150Yess ofVortexDynanics

$.-rightarrow—arrow 0———–\oplus---arrow---arrow-\oplus---\sigma$

$\ovalbox{\tt\small REJECT}_{\backslash }^{\wedge’}$

-.

$——\oplus---\otimes----arrow---\Theta^{--arrow---\oplus--}$

.

$.——-\Theta---rightarrowrightarrow---\oplus--arrow----arrow-rightarrow-\Theta-rightarrow---arrow--rightarrow--\oplus---$

$\ovalbox{\tt\small REJECT}_{\backslash }’.\mathfrak{G}\cdot-\cdot----rightarrow--rightarrow--\Theta^{---\mathfrak{G}^{---\oplus---}}$

Figure7: Schematic ofasymmetric andastaggeoed vortex street downstream ofabluff body. From[30].

Von$K4rm4n$ _{addressed themodel problem oftwoinfinite}

rows

_ofpoint vortices and derived

a

criterionforwhen such

a

configuration isnotunstableto

linearized

perturbations. He showed that the symmetnic configuration, cf. Fig.7, is always unstable and thatthe staggered configuration is also unstableunlessthespacingbetweensuccessive vortices in either

row

andthedistance between th$e$

rows

has

a

definiteratio.

If thespacing between successive vorticesin the

same row

is called $l$, andif the distance

be-tween thetwoparallel

rows

iscalled$h$,

von

Kkmdn’s criterion [29]is

$\cosh\frac{\pi h}{l}=\sqrt{2}$,

or

$\frac{h}{l}=0.283$

.

(11)

The velocity, $U$,ofhorizontal translation of theinfinite

rows

is foundtobe

$U= \frac{\Gamma}{2l}\tanh\frac{\pi h}{l}$

.

(12)

$7bis$ is todayvery_{well known. What is probablyless well known is that in the original}paper

[28]

von

$K4rm4n$_{foundthe criterion} (11) with $\sqrt{3}$$(or h/l=0.365)$ on the righthand side rather

than $($the $correct)\sqrt{2}$, which

was

confirmed subsequently in [31,64] (with reference to the then

newly created theory of

an

infinite system of linear differential equations due to O. Toeplitz in 1907) and by LordRayleigh [57]. The original drawings [31] of thestreamlines in

a

coordinate systemmoving steadily with the vortices

are

reproduced in Fig.8. (Whentheratio $h/l$is given by

Eq.(ll),the propagationspeed of the street,$U$_,inEq.(12)is $\Gamma/l\sqrt{8}.)$

The

erroneous

value was, for example, later used by Synge [68] in his re-denivation of the

$Karm4n$dragfomiula,althoughthe analysisiseasily corrected.

Von $K4 4n$’s analysis precipitated huge amounts of work, both experimental, analytical-and much later–numerical. On July 18, 1922,

a young

W. Heisenberg, then a smdent of A. SommerfeldattheInstimteforTheoretical PhysicsatUniversity ofM\"unchen,submitted

an

article

$[$?$]$ in which he tried to define

an

absolute size of the K\’arm\’an vortex street behind

a

flat plate

ofwidth $d$ placed perpendicularly to the oncoming flow of velocity $U_{\infty}$ farupstream. Based on

physical arguments heamived atAe numericalvalues $l/d=5.45$ and $h/d=1.54$

.

Thesevalues

fit

von

$K4rmdn$’s second value for the ratio ofwidth to intra-row spacing, $h/l=0.283$ and the ratioof the speed ofpropagationofthe

row

relativeto the flow speed at infinity, $U/U_{\infty}=0.229$

.

150 Years of Vortex Dynamics

Figure8: Sbeamlinesofavoilex$sn\cdot eet$in the co-moving$\hslash ame$basedon$m\epsilon$pointvortexmodel[31].

of

a

new

matrix” version of

quanmm

mechamics for whichhe

was

toreceive the Nobel prize in

1933

togetherwith E. Schrodinger and P. A. M.

Dirac.

Nevertheless, his doctoral dissertation,

completedin July 1923,

was on

hydrodynamics, in particularstabilitytheoryandmibulence, and

he would retum briefly to the topic of fully developed turbulence in the period following World WarII.

The

necessary

condition for absenceoflinear instability

was

generalizedtovortex streets

mov-ing obliquely to thedirection ofthe “free stream” by Dolaptschiew andMaue. Wluile the

paper

by Maue [43] _{will probably be} famuiliar, in part because this work

was

highlighted in the well-known]$ecmres$ of Sommerfeld [67], _{theextensive work of Dolaptschiew is less well known than} it oughttobe. Insofar

as

assimilationinto theliteramreinthe Westis concemed,thesimation

was

nothelped by severalof Dolaptschiew’spapers beingpublished in Bulgarian andRussian, albeit usuallywith

an

abstract

or

summary

in Gennan.

3. Conclusion

History,toparaphraseLeibnitz,is

a

useful thing, forits smdynotonly givestothe researchers of thepast theirjustdue butalso provides those of thepresentwith

a

guide forthe

orientation

of their

own

endeavors. While Helmholtz’s 1858

paper on

vortex dynamics andvorticityis of great importance andspawned the

new

subfieldof vortex dynamics,

one

must admit that in the greater scheme of things Helmholtz is today primarily remembered for other contributions to science. There

are

several individuals who would not today be immediately associated with the field of

vortex dynamics, since they did work in other fields– often well outside fluid mechanics–that becameof

even

greaterimportance. We

may

listDirichlet, Fiiedmann, Hankel,Heisenberg,Klein, Lin, Love,J. J.Thomson,Zermeloand probably

even

Lord Kelvin.

In the “case studies” in SectionII

we

havefocused

on

what

one may

callthe classical

applica-tions

of Helmholtz’s vortex theory. It is the test of

any

significant advance thatit elicits interest far beyond the boundaries anticipated by itscreator. Thus, the importanceof vortexdynamics

was

realized in meteorology and oceanography by such towering figures

as

VilhelmBjerknes whose seminal work[12]bearsthe title “On the dynanics of thecircularvortex: with applications tothe atnosphere and atmospheric vortex and

wave

motions.” At the offier end ofthe size-scale

spec-tmm

we

may

cite the application of classical vortex dynamics to superfluid Helium [16], where

150YeaisofVortex Dynamics thefamous footnote in [54] announcedthequantization ofcirculation inthis

case.

As

one surveys

the

now

vastliteramre in vortex dynamics

some

150 years

after Helmholtz’s

paper

one

is stmck bythe richnessofthesubjectmatter,andby how thevaniousaspectsenterdifferent applicationsin almost infinitely variedways. Kuchemann’s figurative characterization of vortices

as

“the sinews and muscles of fluid motions” [34] is no less apt today than it

was

when it

was

written 40years

ago.

Acknowledgments Theauthor acknowledgessupportof

a

Niels BohrVisiting Professorship atthe Danuish Technical University provided by the Danish National Research Foundation. This

paper

is largelybased on the joint

paper

with V. V. Meleshko [47]. I thank Slavafor sharing his many insightsinto the history and practice ofvortexdynamics.

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