Hollow vortices
Stefan Llewellyn Smith (UCSD)
Darren Crowdy (Imperial College)
Daniel Freilich (UCSD)
Support from NSF CMMI-0970113
Ellis Cumberbatch 80th birthday celebration
First meeting with Ellis
Mathematics in industry workshop, Cambridge, 1995. Almost twenty years ago
Design of moulded valve Craster, R. and Cumberbatch, E. and Day, R. and Fliert,
B. vd. and Harlen, O. and Lister, J. and Smith, S.L. and Malcolm, T. (1995) Design
of moulded valve. [Study Group Report]
http://www.maths-in-industry.org/miis/354/
Time-dependent contact problem with free boundaries dividing contact regions
from free regions.
Claremont workshop in 2009. Still working on resulting problem together (new
results right now).
No present but a credit card instead.
Vorticity
Introduced in the paper of Helmholtz (1858; translated by Tait 1867).
The “Scottish school”, including Kelvin and Thomson, sought a theory of “vortex
atoms” to explain the structure of matter.
Understanding elementary vortex structures has been a focus of much research.
Hard problem: simplify.
• Two-dimensional flows are a good approximation for flows that exhibit minimal
rate of variation in the third dimension or that are constrained by effects such
as stratification and rotation to move along near-horizontal surfaces. Will stay
with this for vortex motion.
• Use singular vorticity distributions, the simplest being point vortices. Will talk
about this. In many cases the scale of the vortices is much smaller than the
other scales in the system, so replacing the vortices by elementary structures
with no inherent scales is a natural modelling step. Want to keep this (reduction
approach), but need to keep physics in mind.
• Consider potential flow, neglecting other physical effects such as viscosity, stratification, compressibility, background rotation, or even other physical systems.
Another talk. . .
Examples
(a) Vortex visualized in aircraft wake. From https://www.nas.nasa.gov/Main/Features/2001/Fall/wake detection pic1.html. (b) Soap-film visualization
of wakes. From Zhang et al. (2000). (c) Experimental picture of a vortex dipole
in a stratified flow. From http://www.mun.ca/marcomm/gazette/2002-2003/jan9/resources/research2.jpg. (d) Vortex dipoles visualized in cloud layer.
From http://alg.umbc.edu/usaq/images/USA5.2006257.aqua.500m.crop.jpg
(e) Vorticity of a 2D flapping elliptical wing. From http://www.seas.ucla.edu/sofia/.
What is a hollow vortex? Fluid inside the vortex is stagnant in moving reference
frame. If the vortex translates at a constant speed, pressure inside is constant.
Boundary condition can be written as velocity of the fluid on the boundary is constant. If the vortex rotates (no known results), pressure field inside should be consistent with solid body rotation.
Related to theory of jets (Birkhoff & Zarantanello 1958). Classic method using two
conformal mappings between z -plane (physical), w-plane (potential) and ζ -plane.
Known hollow vortices:
• Pocklington (1894). Compressible extension by Pullin & Moore, Heister et
al. (1990) and Leppington (2006).
• Hill (1976): hollow vortex in strain. But wait. . .
• Baker, Saffman & Sheffield (1976): array of hollow vortices. Compressible
extension by Ardalon, Meiron & Pullin (1995).
• Crowdy and Roenby: hollow vortex surrounded by point vortices.
• Green and Crowdy: double vortex street.
Why? Always useful to find exact solutions to Euler equations. Also have advantage over vortex patches of involving a thermodynamic quantity, so useful as starting point for compressible vortices.
Hill’s Hollow Vortex
Not to be confused with Hill’s Spherical Vortex. First reference in Baker, Saffman
& Sheffield (1976); subsequently in papers by Baker and Pullin.
From: Library Help
Sent: 18 October 2009 14:50
To: Aref-Adib, Nadia
Subject: RE: Ask a question
Hello Nadia,
Someone is trying to find out about a PhD thesis done here at Imperi
in 1975 by Hill, F.M. I cannot find any record of this - searched ou
catalogue. British Library, Senate House library, Copac and Index to
Theses. This thesis was cited once in an article in J Fluid Mechanic
and in 1975 and there was a paper published by the thesis author in
the same journal in 1975 and this was later cited 20 times in other
articles from J Fluid Mechanics. I downloaded the paper in J Fluid
Mechanocs by Hill F.M. to check if there was any mention of the thes
there but did not find anything.
The only thing left now would be maybe to contact Alumni Office and
see if they have Mr Hill, F.M. on the database?
Agnes
From: prvs=0596ea9827=baker@math.ohio-state.edu [prvs=0596ea9827=bak
Sent: Friday, December 11, 2009 7:50 PM
To: Crowdy, Darren G
Subject: Re: question
Hi Darren,
I enjoyed your talk very much and suspect there are aspects that cou
prove interesting in my current work. I hope to talk further with yo
I’m a little more organized in my thoughts.
I seem to recall the student was Mary Hill and worked for Derek Moor
think something happened to prevent her from completing her studies.
contacting Stephen Cowley in case he remembers more or corrects my
recollections.
Greg
Phil Hall knew who Mary Hill was and I have tracked her down. She is now a
community college instructor outside Chicago. I’ve asked her for more information
about what she did at Imperial, but no luck so far.
Problems statement
Turns out to be a very interesting problem. What is the shape of a hollow vortex in
a strain field with potential γz n at infinity?
hollow vortex
Free boundary value problem. Solve by constructing conformal map (classical
technique, cf. Birkhoff & Zarantanello). Map inside of unit circle |ζ| = 1 to outside of vortex, with z ∼ aζ −1 for large |z|.
Construction of conformal map
Consider a point vortex in a strain field. The potential is
w(z) = γz n +
Γ
log z,
2πi
which has n zeros in the fluid. Same topology here.
Define R(ζ) = dw/dz . Then (Constant velocity condition) |R(ζ)| = 1 on |ζ| = 1:
take
n
n
R(ζ) =
A
ζ n−1
ζ −α
ζ n − 1/αn
.
Condition at infinity gives
nγan−1
A=
.
|α|2n
The form of the complex potential is (Boundary is streamline)
W (ζ) = an
γ
iΓ
n
+
γζ
+
log ζ.
ζn
2π
Since dz/dζ cannot vanish in |ζ| < 1, zeros of R(ζ) in |ζ| < 1 must also be zeros
of dW0 /dζ . From
dW0
n
iΓ
= γan − n+1 + nζ n−1 +
,
dζ
ζ
2πζ
we must have
1
iΓ
nγan − n + αn +
= 0.
α
2π
Integrating gives
1
2iβ n−1 β 2ζ 2n−1
z(ζ) = a
−
ζ
+
,
ζ (n − 1)
(2n − 1)
where
4nπγan
µ=
,
Γ
αn +
2i
1
− n = 0,
µ α
αn = iβ,
β=−
µ
√
,
1 + 1 − µ2
and |µ| < 1 for physical solutions.
For n = 2, solutions exist up to |µ| = 0.76370794079042378256 . . ., at which point
the map is no longer univalent.
4
4
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−4
−4
−2
0
2
4
−4
−4
−2
0
2
4
Hollow vortex shapes for n = 2 with µ = 0.05, 0.245, 0.5 and µ = µ(2)
c (left) and
(3)
n = 3 with µ = 0.1, 0.4, 0.8 and µ = µc (right). Each vortex has area π .
Stability
The next thing to look at is stability. Can perturb the complex map and linearize or
adapt the method of BSS.
Results for γ = 0 vortex can be found analytically. In non-dimensional form
±
σm
p
= i(m ± |m|), m 6= 0,
There are modes sharing eigenvalues:
+
σ1− = σ−1
= 0,
σ1+ = σ4− = 2i,
−
+
σ−1
= σ−4
= −2i.
This suggests the possibility of resonance between modes with common eigenfrequencies.
For n = 2, the configuration is always unstable to a mode with growth rate ω = 2γ
for small γ : this corresponds to the instability associated with a point vortex situated
at the stagnation point of a linear straining flow.
In contrast, n = 3 and higher modes are linearly stable. Instability would be a
finite-area effect.
Formulation
BSS derive linearized equations to describe the stability of their basic state working
in the potential plane: W = φ + iψ is the independent variable. The perturbation
velocity potential Φ is a harmonic function in ψ < 0 decaying as ψ → −∞. In
these coordinates, the dynamic and kinematic boundary conditions are
1 ∂Φ ∂Φ
+
+
q02 ∂t
∂φ
∂Φ
1 ∂δ ∂δ
+
=
,
q02 ∂t ∂φ ∂ψ
2
∂ 1q
δ = 0.
∂ψ 2 q02 ψ=0
Now work in the ζ -plane. Equations on boundary become
1 ∂δ
1 ∂δ
1 ∂Φ
+
=
,
q02 ∂t φθ ∂θ ψρ ∂ρ
1 ∂Φ
1 ∂Φ
+
+
q02 ∂t φθ ∂θ
1 ∂ 1 q2
ψρ ∂ρ 2 q02
δ = 0,
ρ=ρ0
Finally
σΦ + Q
∂Φ
= Gδ,
∂θ
σδ + Q
∂δ
∂Φ
= −Q ,
∂θ
∂ρ
where σ = 2πλa2 /q0 Γ is the non-dimensional growth rate, and where Q and G are
known functions.
Numerical solution
Since Φ is harmonic, the functions Φ and δ can be written in the fluid region as
Φ=
∞
X
Φneinθ ρ|n|,
δ=
n=−∞
∞
X
δneinθ .
n=−∞
Obtain matrix equations
∞
X
−i
Qn−mmΦm +
m=−∞
−
∞
X
m=−∞
Qn−m|m|Φm − i
∞
X
Gn−mδm = σΦn,
m=−∞
∞
X
Qn−mmδm = σδn
m=−∞
(Gn and Qn obtained using FFT).
Truncate and solve for the vector r = [Φ−N , · · · , Φ0 , · · · , ΦN , δ−N , · · · , δ0 , · · · , δN ]T :
generalized eigenvalue problem.
10
10
8
8
8
6
6
6
4
2
0
0
Re σ
10
Im σ
Im σ
Results
4
2
0.2
µ
0
0
0.4
4
2
0.1
µ
0.2
0
0
0.3
0.05
µ
0.1
0.15
0.1
0.15
0.5
0.3
0.3
0.4
0.25
0.15
0.3
Re σ
0.2
Re σ
Re σ
0.25
0.2
0.1
0.15
0.1
0.1
0.05
0
0
0.2
0.2
µ
0.4
0
0
0.05
0.1
µ
0.2
0.3
0
0
0.05
µ
Imaginary and real parts of σ for the vortex in strain with n = 2, 3, 4.
BSS
470
G. R. Baker, P. G. Saffman and J . 8. Shefield
+=O
u=o
B
X
FIGURE
1. The physical plane for a regular array of vortices with
fore-and-aft symmetry.
Row of hollow vortices. BSS found solution using Schwarz–Christoffel mapping
2. The its
physical
plane
and computed
stability.
We consider an infinite linear array of identical vortices lying on the x axis
Extensive
literature
onnL,
arrays
vortex
patches
Saffman,
with
centres at
n = 0,ofkpoint
1, _+ 2,vortices
... . Eachand
vortex
is hollow
or has(Kamm,
a stagnant
Schatzman,
Jimenez).
core. Kida,
I n steady
flow, constant pressure inside the cores requires that the fluid
speed has a constant value, qo say, on the boundary of each vortex. The circulation r about each vortex is related to qo by
One-parameter family of solutions parametrized by the dimensionless ratio R =
U∞/q0. Shape of any hollow vortex is given parametrically by
L
X = (1 + R2) sin−1
2π
2R sin λ
,
1 + R2
L
Y = (1 − R2) sinh−1
π
2R cos λ
1 − R2
where L is a length scale and 0 ≤ λ < 2π is a parameter. Small R corresponds to
an array of point vortices or a single vortex, while large R gives a vortex sheet.
0.1
0.2
0.3
0.05
0.2
0.1
0.1
0
0
0
−0.1
−0.05
−0.1
−0.2
−0.3
−0.1
−0.1
−0.05
0
0.05
0.1
−0.2
−0.2
−0.1
0
0.1
0.2
−0.2
0
0.2
Comparison of BSS hollow vortex row (solid), with L = Γ = 1 and the n = 2
isolated hollow vortex solution (dashed) with appropriate parameters.
Stability
10
Im σ
8
6
4
2
0
0
0.2
0.4
0.2
0.4
β
0.6
0.8
1
0.6
0.8
1
1.5
Re σ
1
0.5
0
0
β
Upper panel: imaginary part of λ; lower panel: real part. Even modes are solid,
with the first instability arising around β = 0.43. Odd modes are dashed and include
the resonant mode.
Most unstable solution has period (in potential plane) of row: not subharmonic.
Pocklington’ s hollow vortex
HENRY CABOURN POCKLINGTON was an unusual man, a solitary person but
not a lonely one. According to his own lights he lived a full and satisfying life, but
it was one almost completely filled with mathematics, physics and astronomy. [. . . ]
He rarely spoke, not even to his own brothers and sisters. He shunned people,
which makes it all the more surprising that he chose to become a schoolmaster
and was content to remain one all his working days. [. . . ]
Pocklington’s insistence that his vocation was to be a schoolmaster makes him,
perhaps, unique among the Fellows of the Royal Society of the twentieth century.
Other posts were offered to him but he rejected them. . . He refused to take up
residence in St John’s College, Cambridge, when he was elected to a Fellowship,
because he dreaded the social contacts of the high table. But he never appeared
to be unhappy, on the contrary he gave the impression of being almost invariably
calmly content. [. . . ]
Henry Cabourn Pocklington possessed an original, acute and understanding mind
in matters relating to physics and mathematics. [. . . ] A reading of his work fills
one with regret that he himself did not investigate more deeply the many problems
which interested him and on which he wrote with such economy of explanation.
HENRY CABOURN POCKLINGTON 1870–1952 (Obituary Notices of Fellows of
the Royal Society, Vol. 8, 555–565, by L. Rosenhead)
Pocklington’s solution
Pocklington uses a Schwartz–Christoffel mapping. Approach requires mastery of
elliptic functions.
Results are parameterized by 0 < k < 1. Write K = K(k), E = E(K); then
s
a =
K −E
,
2(K − E) − k 2K
φ0 = λ[E + (k 2a2 − 1)K],
√
2
V = U (2a − 1 − 2a a2 − 1),
where φ0 is half circulation of vortex, V is the speed of the flow at infinity, U is
the speed of the fluid on the vortex boundary and λ is an arbitrary quantity with
dimensions of circulation.
Shape given by
λ
x = (2a2 − 1)Z(u),
U
i
√
λh 2
π
2
y=
(2a − 1)
+ 2a a − 1 dn u ,
U
2K
where 0 ≤ u ≤ 4K parameterizes the boundary and Z(u) is the Jacobi elliptic
function (not implemented in Matlab but an AGM algorithm exists).
New approach
Construct conformal mapping z(ζ) from annulus ρ < |ζ| < 1 to the fluid region
outside the two hollow vortices.
Introduce special function
P (ζ, ρ) = (1 − ζ)
∞
Y
(1 − ρ2k ζ)(1 − ρ2k ζ −1).
k=1
P (ζ, ρ) has a simple zero at ζ = 1. We also need
ζP 0(ζ, ρ)
,
L(ζ, ρ) = ζK 0(ζ, ρ).
K(ζ, ρ) =
P (ζ, ρ)
√
Conditions on W (ζ): simple pole at ζ = ρ and ImW = const on |ζ| = ρ, 1.
Simple result:
iΓ
Ua −1
W (ζ) =
exp(−iχ)K(ζβ , ρ) − exp(iχ)K(ζβ, ρ) −
log ζ.
β
2π
Same approach as before: compute z(ζ) from an integral and obtain one-parameter
family.
Real advantage: simple formulation of stability problem.
6
2
1.5
4
1
2
0.5
0
0
−0.5
−2
−1
−4
−1.5
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−6
−4
−3
−2
−1
0
1
2
3
4
Pocklington vortices (normalized to have centroid at (0, ±1)). Limiting cases are
point vortex pair and vortex sheet along boundary. Family ranges from point vortex
pairs to vortex sheets along axis.
Double
street
Staggered streets
Green and Crowdy have extended Pocklington approach to cope
with periodicconstantly
triply
New applications
connected region and obtained double street solution. Just show pictures here.
U=0.4
1
Unstaggered streets
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
1.5
U=0.6
0.4
0.2
0
−0.2
−0.4
−1.5
−1
−0.5
0
0.5
1
1.5
. – p.32
Recent paper: Shao et al. (2007) Kármán vortexShao
street
patterning
in the
et al,assisted
“Kármán vortex
street assisted
pattern
Figure shows hollow vortices of different areas
growth of silicon nanowires, Chem. Commun., 8. silicon nanowires”, Chem. Commun., 8, (2007).
This is a periodic array of “Pocklington vortex pairs”; shapes are
reminiscent of the single vortex pair
. – p.31
Conclusions and further work
• Have obtained solution for hollow vortex in strain field. Presumably this is what
Hill had found. Extended to higher n.
• Stability of hollow vortex in strain shows critical difference between n = 2 and
n = 3 with resonant instability and higher modes. Related to MSTW instability
(finite strain); there is a reference to Hill’s work in the original Moore & Saffman
paper.
• Limit of point vortex row pairing instability, as for hollow vortex in strain. Subharmonic modes (Floquet theory) not relevant: most unstable modes are periodic.
Similar to vortex patch single row.
• New method for obtaining Pocklington’s hollow vortex. Have investigated stability.
• Double street found. Stability?
• What other hollow vortices exist? Can one find rotating solutions?
Stagnant vortices
Stagnant vortices have fluid in their interior. Dynamics for equal densities can be
described by Birkhoff–Rott equation. Otherwise need different approach (such as
Orszag, Baker and Meiron 1981).
Stability will be different: Kelvin–Helmholtz instability is now possible as the interior
can support pressure disturbances. Should be able to stabilize using surface tension, but basic state changes and can no longer use such a simple conformal map
approach to find it.
Sadovskii vortices
Vortex patches have a vortex jump rather than a vortex sheet on the boundary.
Extensive literature in 2D and axisymmetric situations: contour dynamics.
Vortex patches and hollow vortices end-members of a continuous family of vortices
with both jump and delta-function in vorticity on the boundary. Sadovskii vortices.
Can one found continuous families of vortices going from patches to hollow? Do
the families end at some critical ration ωa/q ? In particular steadily rotating patches
are well-known (V states): what happens here?
Daniel Freilich current examining Sadovskii vortex in strain by simplifying approach
of Saffman & Tanveer (1984) using boundary correspondence method.
Compressible vortical flows
The incompressible Euler equation is a special case of the more general compressible equations.
Point vortices in a compressible flow have an obvious problem: close to the center of the vortex, the velocity increases without bound and becomes supersonic.
This problem can be avoided by considering vortices with internal structure such
as hollow vortices with constant pressure. See Barsony-Nagy, Er-El & Yungster
(1987).
The complex velocity potential
F (z, z) = φ(x, y) + iψ(x, y)
is used. It now depends on z as well as z since the flow is no longer incompressible,
We expand in M according to
F (z, z) = F0(z) + M 2F1(z, z) + o(M 2).
This is the Imai–Lamla version of the Rayleigh–Janzen expansion (Jacob 1959).
The second term in the expansion is
1 dF0
F1(z, z) =
4 dz
Z z
z1
dF0
dz
2
1
dz + G(z).
4
BNEEY outline how to obtain the function G(z). A number of considerations lead to
a standard problem in complex variable theory, one of these being that the force on
the vortex (obtained by the appropriate generalization of Blasius theorem) vanish.
The case of multiple point vortices is similar, with the force being required to vanish
at each vortex.
Question: what is the unsteady version of this?
BNEEY consider an interior structure, using MAE based on a result due to Taylor.
An appropriate inner coordinate is R ≡ r/(kM ) where k ≡ Γ/(2πU L). Then the
inner solution velocity potential can be expanded as
Φ = θ + ∆(M )Φ1 + · · · ,
where ∆(M ) is a gauge function and Φ1 satisfies
γ + 1 ∂ 2Φ1
γ − 1 ∂ 2Φ1 1
γ − 3 ∂Φ1
1
1−
+
1−
+
1−
= 0.
2R2
∂R2 R
2R2
∂R R2
2R2
∂θ2
A solution is obtained for Φ1 in terms of hypergeometric functions. However, BNEEY
state “It should be noted that, in our approach based on potential theory, a region
of vacuum is formed close to the centre of the vortex. A more realistic model that
avoids the region of vacuum can be obtained by introducing a rotational vortex
core. . . ”
It would be interesting to investigate other interior models.
Moving vortex singularities
Point vortices and applications
Equation of motion of point vortices goes back to Kirchhoff (1876). Aref (2007)
calls them a “classical applied mathematical playground”.
• Chaotic advection (Aref 2002).
• Integrable systems (Shashikanth et al. 2002; Borisov, Mamaev & Ramodanov
2007; Vankerschaver, Kanso & Marsden 2009).
• Control of fluid flows (Cortelezzi 1996; Protas 2008).
• Biological locomotion and models of vortex shedding and wakes (Cortelezzi
& Leonard 1993; Shukla & Eldredge 1997; Kanso & Oskouei 2008; Michelin,
Glover & Llewellyn Smith 2008; Michelin & Llewellyn Smith 2009a,b,c,d).
• GFD, e.g. hetons (Hogg & Stommel 1985) and vortices on the beta-plane (Reznik
1992).
Similar problems arise in superfluid mechanics (e.g. Pismen 1999).
PVE
Use complex notation. Complex potential for a point vortex at zn (t) = xn (t) + iyn (t)
with circulation Γn is
Γn
log (z − zn).
2πi
Complex velocity: un (z) − ivn (z) = dFn /dz .
Fn =
PVE:
żn = w̃n,
where the tilde indicates the desingularized (total) complex velocity at zn :
Γn 1
w̃n = lim w −
.
z→zn
2πi z − zn
PVE has nothing to do with a sum of other point vortex velocity fields. It is the
statement that the translational velocity of the point vortex is obtained by removing
the leading-order singularity due to the point vortex, when computing its velocity.
Why?
History (for bibliography see Meleshko & Aref 20071)
The pioneers: derivation (1858–1912) Helmholtz (1858, translated by Tait 1867)
Über Wirbelbewegungen: “If there be two rectilinear vortex-filaments of indefinitely
small section in an unlimited fluid, each will cause the other to move in a direction
perpendicular to the line joining them. Thus the length of this joining line will not
be altered. They will thus turn about their common centre of gravity at constant
distances from it.”
Kirchhoff (1876) Vorlesungen über Mathematische Physik. Zwanzigste Vorlesung,
§ 3: “
u=
∂W
,
∂y
v=−
∂W
,
∂x
1
W = − m log ρ, ”
π
Routh (1881) Some Applications of Conjugate Functions: “the current function of
P is obtained from that of Π by subtracting m2 log µ”. Leads to point vortices as
singular structures.
Thomson (1883) A Treatise on the Motion of Vortex Rings: vortices far enough
away from each other remain circular to leading order. Boundaries of vortices are
deformed, but these deformations are neutral modes leads them and ignored if
other line vortices are far enough away.
1
Both tragically died in 2011.
The classics: formalization (1912–1954) A number of textbooks still in print
today originally date from the period 1912–1954. The complex variable formulation
of irrotational flow is mature at this point in time, but the justification of the PVE has
not changed since Helmholtz.
The following books all state that a single vortex is at rest and that point vortices
move due to the velocity field of other point vortices.
Villat’s 1930 Leçons sur la Théorie des Tourbillons, Lamb’s 1932 Hydrodynamics (the first edition dates from 1878), Ewald, Pöschl and Prandtl’s The physics of
solids and fluids, with recent developments, Rouse’s 1938 book Fluid mechanics
for hydraulic engineers (a hydraulics textbook which might be expected to have
a practical bent), Sommerfeld’s Lectures on Physics (1950, vol. 2, IV.21.2) and
Milne-Thomson’s Theoretical Hydrodynamics (first edition in 1938, making it a successor to Lamb, and final edition in 1968).
E.g. Lamb § 155: “Since this centre remains at rest, the filament as a whole will be
stationary. [. . . ] The motion of each filament as a whole is entirely due to the other,
and is therefore always perpendicular to AB.”
The golden age: expansion (1952–1984) Research into supersonic flow past
delta wings led to BME (Brown–Michael equation), but the treatment of the PVE in
the textbooks and monographs of the time such as Batchelor (1967) shows no real
change from before.
There is one exception: Friedrichs’ 1966 Special Topics in Fluid Dynamics. In it,
he computes the force exerted by the fluid on a vortex filament (point vortex) and
argues that if the vortex is free (as opposed to bound), this force must vanish. The
idea of the force acting on a vortex filament was presumably inspired by the BME
work mentioned above and will recur in later books.
Russian literature. Kochin, Kibel’ and Roze’s book Theoretical Hydromechanics, a
1964 English translation of the 1955 Russian original and Sedov’s 1971 (1968 in
Russian) A course in continuum mechanics. Vol 3: Fluids, gases and the generation of thrust both consider point vortices. They use the traditional verbal argument:
a single vortex does not move so its self-induced velocity is ignored even if more
vortices are present.
The moderns: Marchiori and Pulvirenti (1984) Vortex Methods in Two-Dimensional Fluid Dynamics: Proof that system of small vortex patches converges to vortex
dynamics.
Ting and Klein (1991) Viscous Vortical Flows (updated in 2007 with Knio): MAE
calculation for a Rankine vortex in a uniform stream.
Saffman (1992) Vorticity Dynamics: 2.3: “it is appropriate to give a direct argument
based on momentum conservation. [. . . ] For an alternative argument based on
vortex force, see 3.1.”
Most textbooks approach the PVE in the traditional way: Lighthill’s 1986 An informal introduction to theoretical fluid mechanics, Chorin and Marsden’s 1993 A
Mathematical Introduction to Fluid Mechanics, Chorin’s 1994 Vorticity and Turbulence and Newton’s 2001 book The N -Vortex Problem – Analytical Techniques.
Faber (1995) Fluid Dynamics for Physicists: vortex lines are viewed as physical
entities that exert forces on each other, which forces ultimately make the vortices
move.
Majda and Bertozzi’s 2002 book Vorticity and Incompressible Flow is standard:
“Ignoring the fact that the velocity of a point vortex is infinite at its center, [. . . ] we
find that a point vortex induces no motion at its center”, but does refer to the MAE
approach of Ting. So do Wu, Ma and Zhou in Vorticity and Vortex Dynamics (2006)
and Alekseenko, Kuibin and Okulov’s Theory of Concentrated Vortices (2007).
General vortex singularities
Fridman & Polubarinova (1928) First derivation using a Laurent series approach.
Bogomolov (1976) Some applications.
Saffman and Meiron (1986) Investigation of weak solutions. See also Winckelmans
and Leonard (1988) and Greengard and Thomann (1988).
Chefranov (1987, 1989, 1991) Desingularization of energy.
Newton (2005) Dipole system; ad hoc equations.
Borisov & Mamaev (2006) Source/sink systems.
Yanovsky (2009) Derivation of general equation.
Four main approaches have been proposed for deriving GVE (see also Flucher &
Gustafsson 2007):
Desingularization/renormalization. Singularities are bad, so we remove them.
Generalized conservation of momentum.
MAE approach. Fails for dipole: infinite self-advection speed. Maybe OK for singularities with zero impulse.
Weak functions and generalized functions Integrate the vorticity equation against a
smooth test function, or else just balance generalized functions.
Alternatively, use viscosity and obtain moment equations (Nagem and collaborators).
Conservation of momentum
Saffman’s conservation argument treats a moving contour C with velocity uc . NII:
Z
d
dt
Z
Z
ρu dS = −
S
pn dl −
C
ρu [(u − uc) .n] dl.
C
Write in complex notation and shrink the contour around the vortex. Obtain
Ṁ = iρΓn(żn − w̃n).
Now
Z Z
Z
M=
ρw dS = ρ
S
0
0
2π
Γne−iθ
+ O(1) r dr dθ = O(2),
2πir
so Ṁ = 0.
This gives PVE. NII is satisfied in an integral sense around the vortex. Turns out
that angular momentum is conserved. The critical point is the result Ṁ = 0.
Can generalize for dipoles and get
Ḋ = −w̃0nD,
żn = w̃n
as in Yanovsky (2009). Approach problematic for higher singularities: integrals
depend on arbitrary regularization.
Hollow vortices
Understanding less singular structures can lead to insight into core dynamics of
point vortices. Vortex patches are well known (contour dynamics, etc. . . ). Look at
a different family of exact solutions to the Euler equations.
HENRY CABOURN POCKLINGTON 1870–1952 (Obituary Notices of Fellows of
the Royal Society, Vol. 8, 555–565, by L. Rosenhead) HENRY CABOURN POCKLINGTON was an unusual man, a solitary person but not a lonely one. According
to his own lights he lived a full and satisfying life, but it was one almost completely
filled with mathematics, physics and astronomy. Perhaps his only real confidants
were his father, Henry, and his sister, Ida; otherwise he seems to have had no
friends nor to have had any desire for acquaintance to ripen into friendship. He
rarely spoke, not even to his own brothers and sisters. He shunned people, which
makes it all the more surprising that he chose to become a schoolmaster and was
content to remain one all his working days. To all intents and purposes he walked
through life with unhurried pace, interested neither in events nor people. He was
certainly unaffected by them nor had he any influence on them; his sister, Miss E.
Ida L. Pocklington, says that his attitude and character remained unchanged from
boyhood to death.
With children, however, he seems to have had great sympathy and patience; he
spent much time with them trying to elucidate their difficulties in mathematics and
physics. But he had to be assured of the genuineness of their interest. It was for
this reason that even though he chose to remain a schoolmaster, he made no mark
in his profession; he found it impossible to impose his will upon classes of boys as
a whole and transmit his interests to them, but he was kind and patient with those
in whom he could detect genuine scientific curiosity. His pupils used to plague him
unmercifully, cruelly, sometimes even wittily, but he hardly ever appeared to no-
tice. Only when something particularly foolish had been perpetrated in the school
physics laboratory was he known to fly into a rage – and his rages, even though
they were of short duration, were terrifying to small boys.
Pocklington’s insistence that his vocation was to be a schoolmaster makes him,
perhaps, unique among the Fellows of the Royal Society of the twentieth century.
Other posts were offered to him but he rejected them; perhaps the one which interested him most was the offer of a Chair of Mathematics in the University of Cape
Town, but when correspondence had reached the stage at which a decision had to
be made, he rejected that offer also. He refused to take up residence in St John’s
College, Cambridge, when he was elected to a Fellowship, because he dreaded
the social contacts of the high table. But he never appeared to be unhappy, on the
contrary he gave the impression of being almost invariably calmly content. His confessed hobbies were Chinese and music, but his real hobbies were mathematics,
physics and astronomy. [. . . ]
Henry Cabourn Pocklington possessed an original, acute and understanding mind
in matters relating to physics and mathematics. His work showed versatility, power
and elegance. His writings, if they were addressed to anybody, were intended for
those likely to understand. A reading of his work fills one with regret that he himself
did not investigate more deeply the many problems which interested him and on
which he wrote with such economy of explanation.
BM
1952–1956 :
# Date
1 1952
2 6/1953
3 2/1954
4
5 3/1954
6 10/1954
7 4/5/1955
8 4/1955
9 5/1956
interest in delta wing vortices. Historical table:
Author
Quoted in
Notes
Legendre, JAS
2, 3, 5, 6, 7, 8 submitted 6/23/1952
Adams, JAS
3, 5, 6, 7, 8
submitted 3/23/1953
Edwards, JAS
5, 7, 8, 9
submitted 11/10/1953
Cheng, Tech. Rep 5, 8
can’t find it!
Cheng, JAS forum 8
submitted 12/1/1953 2D u/s
BM, JAS
7, 8, 9
presented 1/1954
BM, Tech. Rep
Cheng, JAS
9
submitted 6/11/1954
Rott, JFM
8
submitted 1/1/1956 2D u/s
BM follow-ups Smith, J. H. B. (1968) Delta wing vortex sheets. Uses BM vortex to
model the end of the sheet.
Graham (1980) BM for vortex shedding at a corner. Blasius argument in the appendix equivalent to BM.
Cortelezzi and Leonard (1993) BM for vortex shedding past a semi-infinite plate.
See also Cortelezzi (1995).
Howe (1996) Force-based approach yielding different equation. Interested in aeroacoustic applications.
Boundary layer shedding Clements (1973) Kirchhoff vortices used to model boundary layer shedding: continuous release of vortices.
Longuet-Higgins (1980) Clements idea for shedding from ripples.
Vortex sheets Pullin (1978) Numerical modelling of vortex sheet roll-up using Birkhoff–Rott equation. Uses BM vortex as last point in discretized vortex sheet to
represent infinite number of turns of spiral.
Krasny (1989) Vortex-dipole sheet model for a wake.
Jones (2003, 2005) Vortex sheet shedding numerically. Hard to see what is going
on at the end of the vortex sheets: BM vortex?
Shelley & Alben (2008) Vortex sheet and flexible body.
Summary: explosion of interest and arrival of computers. BM developed for steady
delta vortices; suggested for 2D unsteady case by Cheng (1954) and Rott (1956).
Subsequently used for vortex shedding; simplest canonical problem is probably
Cortelezzi and Leonard.
Generalized momentum argument
The momentum argument becomes problematic for higher singularities. Attempting to apply it to dipoles leads to the contradictory result w̃n = 0. Also need equation for the dipole strength.
This can be fixed by considering the rate of change of the momentum deficit with
respect to the translational velocity of the entity. This still leads to integrals that do
not exist in an ordinary sense. Multiply Euler equation by “test” function T (x, t) and
integrate. Then by taking T = (z − zn )m−1 and T = (z − zn )m successively,
Ḋ = −w̃0nD,
żn = w̃n
for m ≥ 2. The resulting system satisfies the Euler equation pointwise everywhere
outside the singularity and integrals of it around any contour enclosing the singularity. The result for the dipole is different from that of Fridman & Polubarinova (1938),
who do not have an evolution equation for Ḋ, and from that of Newton (2005).
Same result as Yanovsky (2009).
This approach is still problematic for higher singularities. Yanovsky (2009) argues
there are no equations for such singularities. Can interpret them as regularizations,
but, frustratingly, some indeterminacy remains.