On the topology of

vortex lines and tubes

Oscar Velasco Fuentes

CICESE, México

Velasco Fuentes, O.U. 2007: On the topology of vortex lines and tubes. Journal of Fluid Mechanics 584, 2007, 147 – 156.

http://www.cicese.mx/~ovelasco

Contents

• An epoch-making memoir

• A widespread misconception

• The not so well known shapes

• Conclusions

An epoch-making memoir

Background

• 1687 Isaac Newton– Philosophiae Naturalis Principia Mathematica.

• 1755 Leonhard Euler– Principes Généraux du Mouvement des Fluides

• 1822 Claude Luis Navier– Sur les Lois du Mouvement des Fluides, en ayant Égard à l’Adhésion

des Molecules

• 1845 George Gabriel Stokes– On the theory of the internal friction of fluids in motion, and of the

equilibrium and motion of elastic solids

• 1858 Hermann Helmholtz– Über Integrale der hydrodynamischen Gleichungen, welche den

Wirbelbewegungen entsprechen

Über Integrale der hydrodynamischen Gleichungen,

welche den Wirbelbewegungen entsprechen

• Motivation: understanding the motion that friction brings into the fluid

– How can this be done with the Euler equation? By studying motion that does not have a velocity potential.

• Section 1: Decomposition of the motion of a volume element of a continuous media

– Translation

– Compressions or expansions in three orthogonal directions

– Rotation about an instantaneous axis

• Section 2: Evolution of the vorticity

– Fluid particles originally free of vorticity remain free of vorticity

• Definitions

– Vortex line: a line that in all its points has the direction of the instantaneous vorticity

– Vortex filament: a portion of fluid limited by the vortex lines that pass through the perimeter of an infinitesimal element of surface.

– Fluid particles which at any time form a vortex line, however they move, continually form a vortex line.

– The product of the cross-section and the vorticity of a vortex filament does not change in time.

• Section 2: Evolution of the vorticity– The product of the cross-section and the vorticity of a

vortex filament is constant on the whole length of the

filament.

• Corollary: Vortex filaments must close on themselves or

extend to the boundary of the fluid.

• Helmholtz’s argument: if the vortex filament ended

somewhere within the fluid then it would be possible to

construct a closed surface through which the flux of

vorticity would not be zero.

(Chorin & Marsden, 1993: the argument is incomplete and does not constitute a proof.)

• Section 3: The inverse problem

– Helmholtz decomposition theorem:

determination of a vector from its curl and its

divergence

– Electromagnetic analogy: the velocity

corresponding to a given vorticity distribution

w, is the same as the magnetic force produced

by the electric-current distribution 2w.

• Section 4: Vortex sheets

• Section 5: Rectilinear vortices

– Conservation of vorticity in 2D flows

– Invariance of the vorticity centroid

– Equations of motion of a system of N

point vortices (solution for N=1,2)

– Image of a vortex

• Section 6: Vortex rings

– In a purely azimuthal vorticity field the sum

of the projected areas of all ring elements,

multiplied by their vorticity, is constant.

– Leap-frog interaction

– Head-on collision

– Collision with a wall

“German approbation, British

enthusiasm, and French suspicion”

• P.G. Tait read and translated

it in the fall of 1858

• W. Thomson (Kelvin) had

read it by the end of 1858.

• W. Thomson:

– On vortex atoms (1867)

– On vortex motion (1869)

(Darrigol’s characterization of reactions to Helmholtz’s memoir)

Thomson’s description of vortex tubes

• “In an unbounded infinite fluid,– a vortex tube must be either finite and endless

– or infinitely long in each direction.

• “In an infinite fluid with a boundary (…),– a vortex tube may have two ends, each in the boundary surface;

– or it may be infinitely long in each direction,

– or it may be finite and endless.

• “In a finite fluid mass,– a vortex tube may be endless,

– or may have one end, but, if so, must have another, both in the boundary surface.”

Lamb’s corollary about vortex lines

“A vortex-line cannot begin or end at any point in the

interior of the fluid. Any vortex lines which exist must either

form closed curves, or else traverse the fluid, beginning and

ending on its boundaries” (1879, p. 149).

“If the fluid is infinite the tube must be infinite, or else it must return into itself”

J.C. Maxwell (Atom, Encyclopaedia Britannica, 1875)

A widespread misconception

Classic textbooks

• “Any vortex lines which exist must either form closed curves, or else traverse the fluid, beginning and ending on its boundaries.”

– Lamb (1932, p. 203)

• “A vortex tube can never end within the fluid. It either reaches the boundary or it must be closed.”

– Sommerfeld (1950, p.136)

• “A vortex-tube cannot end in the interior of the fluid.”

– Batchelor (1967, p.93)

Modern textbooks

• “Vortex lines and tubes (...) must always form closed

curves or they must have their ends on the bounding

surface of the flow”

– Cottet & Koumoutsakos (2004, p. 8)

• “A vortex tube cannot end within the fluid. It must

either end at a solid boundary or form a closed loop.”

– Kundu & Cohen (2002, p. 134)

Nature

“This peculiar shape is an elegant example

of Helmholtz's laws, which govern the

structure of vortex filaments. Vortex

filaments cannot end abruptly within a

fluid, and so must join end to end, as in

a smoke ring, or attach to a wall or surface

discontinuity.”

Dickinson (2003, v. 424, p. 621)

Hu, Chan and Bush (2003, v. 424)

Annual Review of Fluid Mechanics

• “Vortex lines cannot end

in the fluid.”– Widnall (1975, v. 7, p.142)

• “Vortex lines cannot

begin or end in the

fluid.”– Sarpkaya (1996, v. 28, p. 94 )

Journal of Fluid Mechanics

• It is “a consequence of the

kinematic theorem that

vortex lines cannot end

inside a fluid.”– Saffman (1990, v. 212)

• “According to the

Helmholtz theorem, a vortex

filament cannot begin or end

within a fluid.”– Zhang et al. (1999, v. 384, p. 207)

Physics of Fluids

• “A vortex tube cannot end in the

fluid, hence it must be connected

to a solid boundary or form a

closed loop.”– Webster & Longmire (1998, v. 10, p. 405)

• “Vortex lines are material curves

and must not end in the fluid.”– Nolan, D.S. (2001, v. 384, p. 207 )

About other vector fields

• “In this case (div u=0) a stream-tube cannot end in the interior of the fluid; it must either be closed, or end on the boundary of the fluid, or extend 'to infinity'.”

– Batchelor (1967, p.75)

• “Lines of B do not begin or end.”– Feynman et al. (1964,13-4)

• “The solenoidal characteristic of magnetic flux density distribution (…) results in magnetic flux lines becoming closed lines.”

– Cingoski et al. (IEEE Trans. Magn. 1996, v.32)

• “Q is solenoidal, div Q = 0; thus the flux tubes of Q cannot end in the fluid.”

– Fetter (Phys. Rev. 1967, v. 163, p. 391)

The not so well known shapes

Vortex lines

)(xds

xd

A vortex line is a curve that in all its points has the

direction of the instantaneous vorticity of the fluid.

If satisfies a Lipschitz condition the solution is

unique.

In other words: only one vortex line can pass through

each point and thus vortex lines cannot intersect.

A vortex line can begin or end at points where the

vorticity is zero.

Saffman (1995) denies it, Kellog (1953)

says it is a matter of convention.

Consider

Vortex lines appear to intersect on

the z axis.

This would violate the theorem of

uniqueness of solutions.

)0,,(

),0,0(

yx

xyu

Separatrix vortex lines

In the space (x,y,s) the two lines are in fact five

solutions:

• The equilibrium solution

• Two solutions that asymptotically

approach the equilibrium point as

(they end within the fluid)

• Two solutions that asymptotically

approach the equilibrium point as

(they begin within the fluid).

s

s

In general, a vortex line has

infinite length and passes

infinitely often infinitely close to

itself (Hadamard 1903, Moffat

1969).

This is a consequence of the

recurrence theorem of

Poincaré (1890) which can be

paraphrased as follows:

If a flow has only bounded vortex lines, then for any volume, however

small, there exist vortex lines that intersect the volume an infinite

number of times.

Dense vortex lines

Dense vortex lines in spherical and ring vortices with swirl

rrzr

1,,

1

)(

)(

ds

d

ds

d

Vortex lines lie on doughnut-shaped

surfaces where sayconst.,

rational

irrational

Vortex lines are given by

Vortex tubes

A vortex tube is a surface formed by all the vortex lines that

pass through a reducible closed contour

Since the chosen contour is arbitrary, an infinite number of

vortex tubes pass through each point.

Vortex tubes

that contain dense vortex lines

)(

)(

ds

d

ds

d

Self-intersecting, open vortex tube

A flow that is everywhere at rest, except in the disk -2<r<2,

-1<z<1, where the velocity and the vorticity are given by

A vortex tube

formed by separatrix vortex lines

12)(

)1()1(2)1()(

ˆ)()(

)('ˆ)(')(

ˆ)()(

24

35

zzzg

rrrrf

with

kzgr

rfrfrzgrf

zgrfu

Vortex tube that begins and ends

on surfaces within the fluid

A vortex tube

formed by separatrix vortex lines

A flow is everywhere at rest, except in the bulb

–g(z)<r<g(z), -1<z<1, where the velocity and the vorticity

are given by

)()(2)(),(

12)(

ˆ)(),(),(

ˆ)(),(

ˆ)(),(

35

24

zgrzgrzgrzrf

zzzg

with

kzgr

zrf

r

zrfr

z

zgzrf

zgzrfu

Vortex tube that begins and ends

on points within the fluid

Vortex tubes

that contain separatrix vortex lines

A flow that is everywhere at rest, except within the box

-inf < x inf, -1<y<1, -1<z<1, where the velocity and the

vorticity are given by

12)(

2)(

ˆ)()('ˆ)(')(

ˆ)()(

24

35

zzzg

yyyyf

with

kzgyfjzgyf

izgyfu

Vortex tubes with a slit

(of infinitesimal or finite width)

A bifurcation of a vortex tube?

kzjyizyx ˆ2ˆ)( 22

Conclusions

• Vortex lines

– Closed: finite but endless path.

– Open: extends to the boundary of the fluid (be it a solid

wall, a free surface or infinity).

– Separatrix: begins or ends within the fluid.

– Dense: it has infinite length but it is confined within a finite

region.

• Vortex tubes

– Since the contour used to generate a tube is arbitrary, this

may contain any combination of vortex-lines

Tubes formed by closed or open lines

Tubes containing separatrix lines

Tubes formed by dense lines