Joris Vankerschaver- The Motion of Solid Bodies in Perfect Fluids: A Geometric Outlook

8/3/2019 Joris Vankerschaver- The Motion of Solid Bodies in Perfect Fluids: A Geometric Outlook

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The Motion of Solid Bodies in Perfect Fluids:A Geometric OutlookInternational Young Researchers Workshop onGeometry, Mechanics and Control, Barcelona

Joris [email protected]

California Institute of Technology

Ghent University

December 16-18

Joris Vankerschaver (Caltech) Rigid bodies and fluids December 16-18 1 / 43
http://find/http://goback/


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Introduction & Motivation

Motivation

There are tons of situations where one would like to know the motion

of a rigid body in a fluid.

Unfortunately, the Euler equations are for all practical purposesintractable:

ut

+ u u = p,

plus boundary conditions.

Luckily, we can identify large-scale structures in the flow or make

simplifying assumptions and use those as elementary building blocksto construct a faithful approximation.

Potential flow, point vortices, . . .

Joint work with Eva Kanso (University of Southern California) and Jerrold E.

Marsden (Caltech).Joris Vankerschaver (Caltech) Rigid bodies and fluids December 16-18 2 / 43


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Introduction & Motivation Why geometry?

Geometric methods in fluid mechanics

Secret motivation behind this talk: the use of geometry in fluidmechanics is just too elegant to ignore.

Example: Euler equations (Euler, 1740): geodesic flow on the groupof volume-preserving diffeomorphisms (Arnold, 1966).



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Case in point: circulation

Circulation around a rigid body: measures how many times fluid

goes round the body.

Kutta-Joukowski theorem:

An airfoil in an airstream with velocity Vei and cir-culation experiences a lift force F = Vei.

F

V

(Partially) Explanation of why aircraft fly.

Underlying geometry: The KJ force is thegy-

roscopic force generated by the curvature ofthe Neumann connection.Buzzwords for now, will be explained later!


I d i & M i i Wh ?


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Introduction & MotivationWhy geometry?

Point vortices interacting with a rigid body

Elements of fluid dynamicsPotential flowVorticity

Reduction by stagesToy example: charged magnetic particlesParticle relabeling symmetryThe Neumann connection

Euclidian symmetry

Example: circulation

Conclusions & open questions


P i t ti i t ti ith i id b d


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Our model

We will study the dynamics of a circular, rigid body interacting with

N point vortices with strength i;

Fluid: inviscid and incompressible. Point vortex flow: superposition of elementary quanta of rotation in

the flow. Flow field of a single point vortex:




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History of solid-vortex interactions

Motion of solids in potential flow: Kirchhoff (1877); dynamics ofpoint vortices in a bounded domain: Lin (1941).

By contrast: motion of solids interacting with vortices: Shashikanth,Marsden, Burdick, Kelly (2002), Borisov, Mamaev, Ramodanov

(2003). Both groups came up with an (ad-hoc) Hamiltonian description on

se(2) R2N:

SMBK BMR

Poisson bracket canonical modifiedHamiltonian interaction-type kinetic energy.




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Different descriptions of the dynamics

BMR SMBK




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Our aim in this lecture

We will put the pioneering work of SMBK and BMR on a firm geometricalfooting.

By symplectic reduction we will obtain BMR and SMBK, and the linkbetween both;

The interaction between solids and point vortices is due to thecurvature of a certain hydrodynamical connection.

In the process, some interesting facts arise:

The system is fundamentally similar to the geometric description of a

charged particle in a magnetic field; The Kutta-Joukowski force on an airfoil is a gyroscopic force due to

curvature.


Elements of fluid dynamics Potential flow


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Elements of fluid dynamics Potential flow

Potential Flow

Ifu

(x, t) is the fluid velocity, then the vorticity is defined as

= u.

Basic assumption: = 0 (potential flow). Hence, u = . In otherwords, u is perpendicular to the level sets of a global function .

This assumption is somewhat artificial. Richard Feynman referred topotential flow as dry water. Nevertheless very good in some cases.


Elements of fluid dynamics Vorticity


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y y

Vorticity in two dimensions

Vorticity measures rotation: a paddle-wheel at x will rotate withangular velocity (x, t). Hence: rotation = no potential flow.

Point vortices: singular distributions of vorticity: (x) = (x x0).

Point vortices keep their structure and are finite-dimensional flowstructures characterized by their strength and location x0.


Elements of fluid dynamics Vorticity


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y y

ViscosityBig whirls have little whirls that feed on their velocity, and little whirls have lesser

whirls and so on to viscosity. Lewis Fry Richardson (1920)

Viscosity term in the Navier-Stokes equations:

u

t+ u u = p 2u.

Here, we put = 0: no decay of vorticity. Conservation law in twodimensions:

t+ u = 0.

Vorticity is advected with the flow.

Kelvins theorem: ddt

= 0, where

=

C

u dl =

S

n dS


Reduction by stages


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y g

Rigid bodies interacting with point vortices

We will study the dynamics of a circular, rigid body interacting withN point vortices with strength i;

Fluid: 2D, inviscid;

General body shapes can be treated with conformal mappingtechniques.

x

(x0, y0)

X

yY


Reduction by stages


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Outline of the method Consider the motion of a fluid and a rigid body as geodesic motion on

Emb(F0,R2

) SE(2)

with respect to the kinetic energy metric (Arnold). Do symplectic reduction with respect to the particle relabeling group;

this imposes the condition that the vorticity is given by N point

vortices.Lamb (1896), 144

To some writers the matter has presented itself as a much simpler one. The

problems are brought at one stroke under the sway of the ordinary formula of

Dynamics by the imagined introduction of an infinite number of ignoredco-ordinates, which would specify the configuration of the various particles of the

fluid. The corresponding components of momentum are assumed all to vanish,

with the exception (in the case of a cyclic region) of those which are represented

by the circulations through the several apertures.


Reduction by stages Toy example: charged magnetic particles


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Toy example: a charged particle in a magnetic field

The motion of a charged particle of mass m and charge e in a magneticfield B = dA can be determined in two equivalent ways:

Through the minimal coupling Hamiltonian

H =1

2m p eA2

;

By using the kinetic energy H = p2 /2m and the magneticsymplectic form:

B = can + eB.

Surprisingly, the relation between both has an exact analogue in theinteraction of solids and vortices!




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Reconciling both points of view: the Kaluza-Klein approach

Kaluza-Klein: the trajectories of the charged particle are geodesics onQ = M U(1).

Let M = R3 with the Euclidian metric h and let A be a vectorpotential on M. Define a new metric g on Q by putting

gab =

h + AA AA 1

.

Kinetic energy Hamiltonian:

H =1

2mp pA2 +

p2

2.




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Symplectic reduction recovers the minimal coupling picture

Starting point: U(1) acts on Q and leaves the Hamiltonian invariant.

Momentum map:

J : TQ R, J(x, , p, p) = p.

Reduced spaces: for e R, we have J1(e) = {(x, , p, e)} and

J1(e)/U(1) = {(x, p)} = TM,

with the canonical symplectic form.

The Hamiltonian on Q drops to TM:

H =1

2mp eA2 .

Conclusion: symplectic reduction yields the minimal coupling picture.




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Shifting the symplectic form

What if we would like to work in the modified symplectic picture?

Define a shift map : TM TM

: (x, p) (x,p = p + eA)

Note that HA = Hkin.

is a symplectic map:

can = (dp dx)

= d(p + eA) dx

= can + eB

Conclusion: pulls back the minimal coupling picture to the modifiedsymplectic picture.




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The geometry behind all this: connections and curvature

Define the formsA = A + d and B = dA.

A is a U(1)-connection on Q.

The curvature B is equal to the magnetic field B.

Hamiltons equations for the modified symplectic form B:

iXB = dH iXcan = dH eiXB

The Lorentz force eiXB is a gyroscopic force, due to the curvature B.




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Relation with the fluid-solid problem

Magnetic terms Minimal coupling

Symplectic form curvature canonical

Hamiltonian canonical connection form.

For the solid-fluid system, G will be the group Diffvol ofvolume-preserving diffeomorphisms. The magnetic picture willcorrespond to Borisov et al. and the minimal coupling picture toShashikanth et al.

The analogue of the Lorentz force will be the Kutta-Joukowski forceon a rigid body with circulation.




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Dictionary

Charged particle Fluid-solid problem

U(1) Diffvolcharge e vorticity

vector potential A connection form Amagnetic field B magnetic two-form

Lorentz force Kutta-Joukowski force

Minimal coupling BMR equationsModified symplectic picture SMBK equations


Reduction by stages Particle relabeling symmetry


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The fluid-solid configuration spaceWe label each fluid particle by its initial location, and track its motion as afunction of time. If the initial label space is denoted by F0, then the

configuration space is hence

Emb(F0,R2) SE(2),

with the following conditions:

for Emb(F0,R2), we have d2x = d2xF0 (volume preservation); The image of together with the rigid body fills up R2 (no

cavitation/interpenetration, slip boundary condition).

(, g)

x

yy

x




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The tangent space to Emb(F0,R2) SE(2)

Consider an element (, g) Emb(F0,R2) SE(2). How can we describe

the tangent space T(,g)(Emb(F0,R2) SE(2))?

Take a curve t (t, gt) such that 0 = and g0 = g.

The derivative 0 is a map from F0 to TR2, defined as

0(x) =t(x)

t

t=0

T(x)R2.

In other words, 0

is a vector field along .




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The kinetic energy

Let (, g; , g) be an element of T(Emb(F0,R2) SE(2)) and defineu = 1 and (,V) = g1g.

T =

2

F

u2 d2x

fluid

+I

22 +

m

2V2

body.

The system describes a geodesic on Emb(F0,R2) SE(2) with

respect to the kinetic-energy metric.

This is valid for arbitrary distributions of vorticity. To bring in point vortices: symplectic reduction.




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Particle relabeling symmetry

The physics is indifferent to the way in which we label the individual

fluid particles.

The group Diffvol of volume-preserving diffeomorphisms acts on theright on Emb(F0,R

2) SE(2):

(, g) = ( , g),

leaving the kinetic energy invariant.

The projection : Emb(F0,R2) SE(2) SE(2) is a principal fiber

bundle with structure group Diffvol.




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Interlude: bundles and connections

Let G be a group acting on a manifold Q so that Q/G is again amanifold. Then : Q Q/G is termed a principal fiber bundle withstructure group G.

A connection on Q is one of the following:

1. A G-equivariant one-form A : TQ g such that

A(vq g) = Adg1 A(vq) and A(Q) =

for all g.2. A G-invariant distribution H such that TQ = H V.


Reduction by stages The Neumann connection


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What happens if we move the rigid body?

For each infinitesimal motion (g, g) TSE(2) of the rigid body,

consider the vector field , where solves

2 = 0 and

n= (V + X) n on F

and (,V

) = g

1

g. gives the response of the fluid to the rigid body motion (g, g).

Example: unit speed motion in x-direction.

= R

X

X2 + Y2




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Making this rigourous: the Neumann connection

For all (, g), define a horizontal lift mapping

h(,g) : (g, g) ,

Define H(, g) =Im

h(,g); this is a principal fiber bundle connectioncalled the Neumann connection.

The connection form A is given by the Helmholtz-Hodgedecomposition:

A(, g, , g) = uV Xvol,

where u = + uV.




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The curvature of the Neumann connection

Question: move the rigid body around in a closed loop. Will the fluid

particles also return to their original location?

Failure of fluid particles to return to their original locations:geometric phase due to curvature of the Neumann connection.

Classically known as Darwin drift (1953).




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Interlude: geometric phases

Use the connection to lift closed path (t) inQ/G to a path H(t) in Q.

Failure ofH(t) to close: geometric phase.

Ambrose-Singer: infinitesimal geometricphases are generated by the curvature B.

No curvature = no geometric phase.

Appearance in general relativity (parallel transport), quantum mechanics(Berrys phase, holonomic quantum computing), integrable systems(Hannays angles), classical mechanics (falling cat, Foucault pendulum),etc.



Th Li l b f D ff


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The Lie algebra ofDiffvol. . .

The space Diffvol is (formally) a Lie group: multiplication is given bycomposition, and the unit is the identity map:

= and e = id.

Lie algebra: TeDiffvol = Xvol, the algebra of divergence-free vectorfields on F0 which are tangent to F0, with the Jacobi bracket:

[X,Y]X = [X,Y].

The minus sign is due to the right action ofDiffvol on itself.



d i d l X


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...and its dual Xvol

An naive candidate for Xvol is 1(F0), with pairing

,X =

F0

(X) d2x.

However, this pairing is degenerate: for all X Xvol,

df,X =F0

df(X) d2x =F0

f X n dl F0

f X d2x = 0.

Hence, the right dual is given by

Xvol =1(F0)

d0(F0)= d1(F0),

where the last isomorphism is given by [] d.



Th i d Diff i i


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The momentum map associated to Diffvol: vorticity

The particle relabeling symmetry induces a momentum map

J : T(Emb(F0,R2) SE(2)) Xvol

given by (recall that u =

1

)

J(, g; , g) = d(u).

In classical notation, this becomes J = ( u); the vorticity of thefluid in the reference configuration is the momentum map associatedto the the particle relabeling symmetry.



S l ti d ti ith t t Diff i t d i


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Symplectic reduction with respect to Diffvol: introducingpoint vortices

Form the quotient J1()/G. If is a regular value, then this is amanifold with a reduced symplectic form characterized by

= .

In the context of fluid dynamics, working on a level set J1() meansmaking an assumption about the vorticity of the system.

Here: N point vortices of strength i, i = 1, . . . , N:

=

Ni=1

i(x xi)dx dy

What does J1()/G look like?



S i l t t b dl d ti


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Special case: cotangent bundle reduction

The reduced symplectic manifold is related to

J1()/G T(Q/G) Q/G Q/G;

the diffeomorphism depends on the choice of A.

The reduced symplectic form is given by

= can

where is a magnetic 2-form on Q/G given by Q,G

= d,A.

Using the Cartan structure formula

dA = B [A,A],

we can relate to the curvature B.



Cotangent bundle reduction: point vortices


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Cotangent bundle reduction: point vortices

The isotropy group Diffvol, consists of all diffeomorphisms preserving

: = . For point vortices, Diffvol, iff

(xi) = xi (for i = 1, . . . ,N).

The quotient space Q/G is in this case

(Emb(F0,R2) SE(2))/Diffvol, = R

2N SE(2).

The reduced cotangent bundle is hence

T(Q/G) Q/G Q/G = T(SE(2)) R2N;

product of a cotangent bundle and a co-adjoint orbit.



The reduced symplectic form has the vortex rigid body


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The reduced symplectic form has the vortex-rigid bodyinteraction

The magnetic two-form is a two-form on R2N

SE(2) and can bewritten as

(x1, . . . , xN; g)((v1, . . . , vN, g1), (w1, . . . ,wN, g2))

=

Ni=1

i dx(vi,wi) + d(AdA)(g1, vi; g2,wi).

Second term: encodes the vortex-rigid body interaction.

Ni=1 idx(vi,wi) is the Konstant-Kirillov-Souriau form on theco-adjoint orbit R2N. The stream functions A are harmonic conjugates to A (elementary

velocity potentials corresponding to the A-direction in SE(2)).


Reduction by stages Euclidian symmetry

Reduction with respect to SE (2) yields the BMR Poisson


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Reduction with respect to SE(2) yields the BMR Poissonstructure

The rigid body and point vortices are invariant under the action of SE(2)on TSE(2) R2N. Factoring out this symmetry boils down to rewritingthe equations of motion in body coordinates.

Do Poisson reduction to reduce the magnetic symplectic structure on

TSE(2) R2N to the following Poisson structure on se(2) R2N:

{f, k}int =f

k

{f|P, k|P}R2N

f

,k

f

k +

k

f,

In coordinates, this yields precisely the BMR Poisson structure.


Reduction by stages Euclidian symmetry

The momentum map provides the link with the SMBK


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The momentum map provides the link with the SMBKPoisson structure

Let X(SE(2) R2N) be the infinitesimal generator corresponding to se(2). Define the Bg-potential : SE(2) R2N se(2) by

i = d, .

Looks like a momentum map, but for the magnetic two-form .

Induces a shift map S : se(2) R2N se(2) R2N, with

S(, x) = ( (e, x), x).

This is a Poisson map:

{f S, g S}BMR = {f, g}SMBK S.


Example: circulation

Rigid body with circulation


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Rigid body with circulation

Circulation: realized by having a point vortex at the center of mass.

Symplectic reduction gives the Poisson bracket for a rigid body withcirculation:

{f, g} = {f, g}se(2)

f

Px

g

Py

g

Px

f

Py

curvature term.

The resulting dynamics describes the motion of a rigid body underthe influence of a gyroscopic force due to the curvature term. This isthe familiar Kutta-Joukowski force, similar to the Lorentz force.

= 0Px = Py/I Py/m

Py = Px/I + Px/m,

F

V



Conclusions


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Conclusions

Geometric reduction yields the equations of motion for a rigid bodyinteracting with point vortices.

Symplectic reduction wrt to Diffvol brings in the vorticity;

The momentum map associated to the residual SE(2) action gives ashift map between BMR and SMBK.

Classical fluid dynamical effects have a geometric interpretation.

The Kutta-Joukowski force;

Analogy with magnetodynamics.



Open questions


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Open questions

This is probably only the tip of the iceberg. . . General distributions of vorticity: do reduction at a different value

of J.

Controllability of a fluid through rigid body motions. Stirring?

Geometric integrators for fluid-solid interactions. Find a discreteanalogue of this procedure of reduction by stages.

Relative equilibria and their stability, chaos. For one vortex + body,the reduced phase space is a symplectic leaf O in se(2) R2, so

dim O = 4. Independent commuting integrals: energy and rotationsaround symmetry axis.



References


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References

V. I. Arnold and B. A. Khesin, Topological methods in

hydrodynamics, Applied Mathematical Sciences, vol. 125,Springer-Verlag, New York, 1998.

B. N. Shashikanth, J. E. Marsden, J. W. Burdick, and S. D. Kelly,The Hamiltonian structure of a two-dimensional rigid circular cylinder

interacting dynamically with N point vortices, Phys. Fluids 14(2002), no. 3, 12141227.

A. V. Borisov, I. S. Mamaev, and S. M. Ramodanov, Motion of acircular cylinder and n point vortices in a perfect fluid, Regul. ChaoticDyn. 8 (2003), no. 4, 449462.

J. Vankerschaver, E. Kanso, and J. Marsden, The Geometry andDynamics of Interacting Rigid Bodies and Point Vortices, Submittedto J. Geom. Mech., 2008.


Joris Vankerschaver- The Motion of Solid Bodies in Perfect Fluids: A Geometric Outlook

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