150 Years of Vortex Dynamics

A new calculus for two dimensional vortex dynamics

Darren Crowdy

Dept of Mathematics

Imperial College London

d.crowdy@imperial.ac.uk

. p.1

In remembrance

Philip Geoffrey Saffman, FRS (19312008)

. p.2

In remembrance

Philip Saffman, FRS Derek Moore, FRS(19312008)

. p.3

Uniform ow past a cylinder

The first photograph in Van Dykes Album of Fluid MotionComplex potential

w(z) = U

(z +

1

z

)Simple. But what about flow past multiple objects?

. p.4

The biplane problem

Weierstrass -function and -function

. p.5

The Flettner rotor ship

Flettner rotors are rotating cylinders which exploit theMagnus effect for propulsionThis mechanism was explored in the 1920s and 1930s

. p.6

Triply connected analogues

The triplane 3-rotor Flettner yacht

These are examples where flows past three obstacles are relevant

. p.7

Quadruply connected rotor vessels

(source: BBC website)

Futuristic cloudseeder yachts wind-powered, unmanned vessels

(from Enercon press release)

Enercons E-ship uses sailing rotors to cut fuel costs by 30%. p.8

Civil engineering

Civil engineers are interested in forces on multiple objects (e.g. bridgesupports) in laminar flowsT. Yamamoto, Hydrodynamic forces on multiple circular cylinders, J. HydraulicsDivision, ASCE, 102, (1976)

. p.9

Topology of laminar mixing

Mixing in an octuply connected flow domain

. p.10

Biolocomotion

Recent interest in biolocomotion has led to resurgence in flowmodelling techniques originally pioneered in aeronautics

. p.11

Oceanic eddies

70 60 50 40

0

10

20

North equatorial

Current

Brazil

counter current

North

Lesser

Grenada Passage

SouthAmerica

North Brazil Current ring

Antilles

Geophysical fluid dynamicists want to model motion of oceanic eddiesin complicated island topographies

. p.12

Other engineering challenges

. p.13

Other engineering challenges

The World

. p.14

History of the two-obstacle problem

W.M. Hicks, On the motion of two cylinders in a fluid, Q. J. Pure Appl. Math., (1879)A. G. Greenhill, Functional images in Cartesians, Q. J. Pure Appl. Math., (1882)M. Lagally, The frictionless flow in the region around 2 circles, ZAMM, (1929).C. Ferrari, Sulla trasformazione conforme di due cerchi in due profili alari,Mem. Real. Accad. Sci. Torino, (1930)T. Yamamoto, Hydrodynamic forces on multiple circular cylinders,J. Hydr. Div, ASCE, (1976).E.R. Johnson & N. Robb McDonald, The motion of a vortex near two circular cylinders,Proc. Roy. Soc. A, (2004)Burton, D.A., Gratus, J. & Tucker, R.W., Hydrodynamic forces on two moving discs,Theor. Appl. Mech., (2004)

No prior analytical results for more than two aerofoilsStandard fluids literature contains almost nothing on multi-obstacleflowsThis talk seeks to fill this gap with an analytical treatment

. p.15

Riemann mapping theorem

Any simply connected domain Dz (bounded or unbounded) in theplane can be conformally mapped to the unit disc (and vice versa)

Let the unit disc in a complex -plane be denoted D

Let the conformal mapping from D to Dz be z()

If the domain is unbounded then a point D maps to infinity and,locally

z() =a

+ analytic

There are three degrees of freedom in the mapping theoremThis means, for example, that we can pick arbitrarily

. p.16

A point vortex outside a cylinder

Consider a single point vortex outside a unit-radius cylinderConformal map from the interior to the exterior of unit disc:

z() =1

We have chosen = 0 to map to z =.Let the unit circle || = 1 be denoted C0

3 2 1 0 1 2 33

2

1

0

1

2

3z-plane fluid region

3 2 1 0 1 2 33

2

1

0

1

2

3

z()=1/

plane

point vortex . p.17

A point vortex outside a cylinder

Complex potential for isolated point vortex at = :

w() = i2pi

log( )

But, we need it to be real on || = 1 (so it is a streamline)A function, built from w(), that is real:

w() + w()

But this is not analytic. On || = 1, = 1/, so consider

w() + w(1/)

= i2pi

log

(

1/

)= i

2pilog

(

||( 1/)

) i

2pilog + c

. p.18

Circulation around the cylinder?

Another possible solution:

i2pi

log

(

||( 1/)

) i

2pilog

where is any real numberConsider the two terms separately:

i2pi

log

(

||( 1/)

) circulation 1 around cylinder, vortex at

and

i2pi

log circulation around cylinder, vortex at = 0 (z = )

Note: we are free to choose the round-obstacle circulationPick 1 = 0 if want zero circulation around cylinder

. p.19

The function G0(, )

It seems pedantic, but introduce the notation

(, ) ( )

Also introduce notation G0(, ):

G0(, ) = i

2pilog

(

||( 1/)

)= i

2pilog

((, )

||(, 1/)

)

Recall, this is complex potential for:

a point vortex, circulation +1, at it has circulation 1 around obstacle whose boundary is

image of C0 (hence subscript) it has constant imaginary part on C0

. p.20

What about three cylinders?

Now consider fluid region exterior of three circular cylinders

sss

z()=s/

d

q

Fluid region

q

D

obstacles

circular region

z() =s

with q =

s2

d2 s2 , =sd

d2 s2Geometry of D depends on geometry of given domain

. p.21

Generalized Riemann Theorem

D

Any multiply (M + 1)-connnected domain can be conformally mappedto from a circular domain D consisting of the unit disc with M smallercircular discs excisedThe radii of the discs will be {qj|j = 1, ..., M}The centres of the discs will be {j|j = 1, ..., M}Let unit circle be C0; all other circular boundaries {Cj|j = 1, .., M}

. p.22

Back to G0(, )

Lets go back to G0(, ) for the single cylinder example:

G0(, ) = i

2pilog

((, )

||(, 1/)

)

Recall, this is complex potential for:

a point vortex, circulation +1, at it has circulation 1 around obstacle whose boundary is

image of C0 (hence subscript) it has constant imaginary part on C0

What is analogous complex potential for the three cylinder example?

. p.23

Higher connected generalization?

Remarkable fact

G0(, ) i

2pilog

((, )

||(, 1/)

)

is the required complex potential!It has exactly the same functional form!!

It has

a point vortex, circulation +1, at a circulation 1 around object whose boundary is

image of C0 (hence subscript) circulation 0 around all other objects it has constant imaginary part on Cj (j = 0, 1, ....M )

. p.24

A fact from function theory

What can we possibly mean by this?

Fact: there exists a special transcendental function of two variables(., .) it is just a function of the data {qj, j|j = 1, .., M} such that:

(1) (, ) has a simple zero at = (2) G0(, ) has constant imaginary part on all the boundary circles

of D (so that all the obstacle boundaries are streamlines)

The function (, ) is called the Schottky-Klein prime function

It plays a fundamental role in complex function theory that extendsfar beyond the realm of fluid dynamics.

Consider it just a computable special function (cf: sin(x), Jk(x)) . p.25

Adding circulation around the otherobstacles

What if we want non-zero circulations around the other M obstacles?Then we need M additional complex potentials:

Gj(, ) = i

2pilog

((, )

||(, j(1/))

), j = 1, .., M

The complex potential Gj(, ) has

a point vortex, circulation +1, at a circulation 1 around cylinder whose boundary is

image of Cj (hence the subscript) circulation 0 around all other cylinders it has constant imaginary part on Ck (k = 0, 1, .., M )

. p.26

The functions {j()|j = 1, .., M}The functions {j()|j = 1, .., M} are simple functions of the data{qj, j|j = 1, .., M}:

j() j +q2j

1 j, j = 1, .., M

These functions are fully determined by {qj, j|j = 1, .., M}

Example: Suppose D is the annulus < || < 1 then there is justone Mbius map (M = 1): 1 = 0, q1 =

1() = 2

. p.27

Uniform ow past multiple objects

What about adding background flows?

Suppose, as , we have

z() =a

+ analytic

for some constant a, and we want the complex potential for uniformflow of speed U at angle to the x-axis

The required complex potential is

2piUai

(ei

G0

ei G0

) =

It is just a function of G0(, )

. p.28

By the way...

If we let (, ) = ( ) so that

G0(, ) = i

2pilog

(

||( 1/)

)

then the formula

2piUai

(ei

G0

ei G0

) =

withz() =

1

flow past circular cylinder ( = 0)

reduces to

U

(1

+

)= U

(z +

1

z

)(choosing = 0)

. p.29

Straining ows around multiple objects

What about higher-order flows?

Suppose, as , we have

z() =a

+ analytic

for some constant a, and we want the complex potential for strainingflow tending to eiz2 as z

The required complex potential is

2pia2i

(ei

2G0

2 ei

2G0

2

) =

Again, it is just a function of G0(, )

. p.30

What if the objects move?Complex potential when jth obstacle moves with complex velocityUj :

WU() =1

2pi

C0

[Re[iU0z()]

] [d log

((, )

(1, 1)

)]

M

j=1

1

2pi

Cj

[Re[iUjz()] + dj

] [d log

((, )

(j(1), 1)

)]

U (U0, U1, ..., UM)

The constants {dj|j = 1, ..., M} solve a linear systemThis time, expression is an integral depending on (., .)

(Useful for modelling biological organisms, vortex control problems). p.31

Strategy for problem solving

Step 1: Analyse the geometry and determineD, the data {qj , j|j = 1, ..., M} and map z()(use numerical conformal mapping if necessary)

Step 2: Construct the Mbius maps {j()|j = 1, ..., M}and compute (., .) (it all depends on this function!)

Step 3: Do calculus with the functions{Gj(, )|j = 0, 1, .., M} to solve the fluid problem

Lets do some examples!. p.32

Three point vortices near three circularislands

1

2

3

12 0

d

s

D is the unit -disc with two smaller discs excised, each of radius qand centred at .The point = 0 maps to infinity.Assume point vortices all have circulation Assume circulation j around island Dj

D

qq

. p.33

Three point vortices near three circularislands

1

2

3

12 0

d

s

w1() =3

k=1

G0(, k) point vortices

3G0(, 0) round-obstacle circulations zero

2

j=0

jGj(, 0) add in round-obstacle circulations. p.34

What is the lift on a biplane?

U

ds

Take D as the annulus < || < 1:

z() = i

d2 s2(

+

),

=1 (1 (s/d)2)1/21 + (1 (s/d)2)1/2 ,

We have z() =a

+

+ analytic, a = 2i

(d2 s2)

D

. p.35

What is the lift on a biplane?

U

ds

w2() = 2piUai

(G0

G0

) =

uniform flow ( = 0)

1

j=0

Gj(,) add round obstacle circulations

(now use Blasius integral formula for Fx iFy). p.36

Generalized F oppl ows with two cylinders

U 1

2

2

1

ds

w3() = 2piUai

(G0

G0

) =

uniform flow ( = 0)

+ G0(, 1) G0(, 2)+ G0(, 1) G0(, 2) point vortices

(now search finite dimensional parameter space for equilibria). p.37

Cylinder with wake approaching a wall

d

U=i

Take D as < || < 1

z() =i(1 2)

2

( +

), d =

(1 )22

.

|| = 1 maps to the boundary of the cylinder|| = maps to wall.

. p.38

Cylinder with wake approaching a wall

d

U=i

w4() = G0(, ) G0(,) point vortices+ WU() flow due to moving cylinder

where U = (i, 0)

. p.39

Model of school of swimming sh

Point vortices

Moving objects

U

U0U1U2

Conformal mapping non-trivial in this case. It happens to be

z() =

[a

=0

+b

=0

]G0(, ) + c

Near = 0 (so = 0):

z =a

+ analytic

D

qq

. p.40

Model of school of swimming sh

Point vortices

Moving objects

U

U0U1U2

w5() =6

k=1

kG0(, k) point vortices

(

6k=1

k

)G0(, 0) round-obstacle circulations zero

+ 2piUai

(G0

G0

) =0

uniform flow ( = 0)

+ WU() flow due to moving bodies U = (U0, U1, U2). p.41

How to compute (., .)?

Option 1: There is a classical infinite product formula for it:

(, ) = ( )k

(k() )(k() )(k() )(k() )

Example: In the doubly connected case, D to be < || < 1There is just a single Mbius map given by 1() = 2The infinite product is then

(, ) P (/, )

where

P (, ) (1 )

k=1

(1 2k)(1 2k1).

. p.42

Innite sum representations

P (, ) is analytic in < || < 1, so also has Laurent series

P (, ) = A

n=(1)nn(n1)n,

where A is a constant. This converges faster than product!Crowdy & Marshall have extended this idea to produce a fastnumerical algorithm for higher connectivityIt is based on Fourier-Laurent representations (not infinite products)

MATLAB M-files will be freely available soon atwww.ma.ic.ac.uk/ dgcrowdy/SKPrime.

Crowdy & Marshall, Computing the Schottky-Klein prime function on the Schottkydouble of planar domains, Comput. Methods Func. Th., 7, (2007)

. p.43

Relation to Lagally?

It can be shown that

P (, ) = iCe/2

1/41(i/2, )

where = log and 1 is first Jacobi theta function

The Jacobi theta function can be related to the Weierstrass and function

Recall: Lagally (1929) used the latter functions in his solution tothe biplane problem

Our calculus simplies and extends this two-obstacle result

. p.44

Streamlines for uniform ow

5 4 3 2 1 0 1 2 3 4 51.5

1

0.5

0

0.5

1

1.5

6 4 2 0 2 4 61.5

1

0.5

0

0.5

1

1.5

1 0 1 2 3 4 5 6 7

3

2

1

0

1

2

Very easy to plot using analytical formulae for complex potentialConformal maps from circular domains D are just Mbius mapsThis answers the question prompted by Van Dykes first photograph!

. p.45

Two aerofoils in unstaggered stack

2 0 24

3

2

1

0

1

2

3

4

2 0 24

3

2

1

0

1

2

3

4

2 0 24

3

2

1

0

1

2

3

4

Two aerofoils with gradually increasing circulation

. p.46

Kirchhoff-Routh theory

In 1941, C.C. Lin wrote two papers in which he established thatN -vortex motion in multiply connected domains is HamiltonianHe relied on the existence of a special Greens functionThis special Greens function is precisely G0(, )!He also showed the following transformation property of

Hamiltonians:

H(z)({zk}) = H()({k}) +N

k=1

2k4pi

log

dzdk

where zk = z(k)This fact completes the theory!A general analytical framework now exists for N -vortex motionCrowdy & Marshall, Analytical formulae for the Kirchhoff-Routh path function in multiplyconnected domains, Proc. Roy. Soc. A, 461, (2005)

. p.47

Lifes little ironies

My ofce door at MIT. p.48

Modelling geophysical ows

Simmons & Nof, The squeezing of eddies through gaps, J. Phys.Ocean., (2002).

. p.49

Vortex motion through gaps in walls

0

0

.

5

1

1

.

5

2

2

.

5

3

3

.

5

4

4

.

5

5

2

.

5

2

1

.

5

1

0

.

50

0

.

51

1

.

52

2

.

5

70 60 50 40

0

10

20

North equatorial

Current

Brazil

counter current

North

Lesser

Grenada Passage

SouthAmerica

North Brazil Current ring

Antilles

Critical vortex trajectories for two offshore islandsCrowdy & Marshall, The motion of a point vortex through gaps inwalls J. Fluid Mech., 551, (2006)

. p.50

Critical vortex trajectories

0 1 2 3 4 5 6 7 8 92.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

Four offshore islands

0 1 2 3 4 5 6 7 8 9 10 112.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

Five offshore islandsNote: Even the conformal slit maps are obtained analytically! . p.51

Other applications of the calculus

The calculus has many other applications:

Contour dynamics: Facilitates numerical determination ofvortex patch dynamics(kernels in contour integrals expressed using (., .) )

Crowdy & Surana, Contour dynamics in complex domains, J. Fluid Mech., 593, (2007)

Surface of a sphere(need to endow spherical surface with complex analytic structureby means of stereographic projection)

Surana & Crowdy, Vortex dynamics in complex domains on a spherical surface,J. Comp. Phys., 227, (2008)

. p.52

Test of the methodComparison with free space code [Dritschel (1989)]:

2 0 2 4 6 8 104

3

2

1

0

1

2

3

4height=1.05

t=0 t=1 t=2 t=3 t=4t=5

t=6

2 0 2 4 6 8 101

0

1

2

3

4

t=0t=1 t=2 t=3 t=4

t=5t=6

height=1.05

. p.53

Patch motion through a gap in a wall

Compares well with Johnson & MacDonald, Phys. Fluids, (2004):

Crowdy/Surana Johnson/McDonald

. p.54

Patch motion near a spherical cap

. p.55

Patch motion near a barrier on a sphere

Simulation of vortex patch penetrating a barrier on a spherical surface

. p.56

References and resources

For

Published papers A PDF copy of this talk A preprint of the paper: A new calculus for two dimensional

vortex dynamics Downloadable MATLAB M-files for computing (., .) (soon)

Website: www.ma.ic.ac.uk/ dgcrowdy

d.crowdy@imperial.ac.uk

. p.57

150 Years of Vortex Dynamics~~~f In remembrance~~f In remembrance~~f Uniform flow past a cylinder~~~f The biplane problem~~~f The Flettner rotor ship~~~f Triply connected analogues~~~f Quadruply connected rotor vessels~~~f Civil engineering~~~f Topology of laminar mixing~~~Biolocomotion~~~f Oceanic eddies~~~f Other engineering challenges~~~f Other engineering challenges~~~f History of the two-obstacle problem~~~f Riemann mapping theoremf A point vortex outside a cylinder~~~f A point vortex outside a cylinder~~~f Circulation around the cylinder?~~~f The function $G_0(zeta ,alpha )$~~~f What about {underline {three}} cylinders?~~~f Generalized Riemann Theorem~~~f Back to $G_0(zeta ,alpha )$~~~f Higher connected generalization?f A fact from function theory~~~f Adding circulation around the other obstacles~~~f The functions $lbrace heta _j(zeta )|j=1,..,M brace $~~~f Uniform flow past multiple objects~~~f By the way...f Straining flows around multiple objectsWhat if the objects move?~~~f Strategy for problem solving~~~f Three point vortices near three circular islandsf Three point vortices near three circular islandsf What is the lift on a biplane?f What is the lift on a biplane?f Generalized F"oppl flows with two cylinders~~~f Cylinder with wake approaching a wall~~~f Cylinder with wake approaching a wall~~~f Model of school of swimming fish~~~f Model of school of swimming fish~~~f How to compute $omega (.,.)$?~~~f Infinite sum representations~~~f {Relation to Lagally?~~ }~~~~~~f {Streamlines for uniform flow }~~~~~~f {Two aerofoils in unstaggered stack}f {Kirchhoff-Routh theory~~~} f Life's little ironies~~~f { Modelling geophysical flows~~~~~~~} f { Vortex motion through gaps in walls~~~~} f Critical vortex trajectories~~~{f Other applications of the calculus~~~} {f Test of the method~~~} {f Patch motion through a gap in a wall~~~} {f Patch motion near a spherical cap~~~} {f Patch motion near a barrier on a sphere~~~} f References and resources~~~