Two-dimensional Vorticity Dynamics

Two-dimensional Vorticity Dynamics

u =

u(x, y, t)v(x, y, t)

0

ω =

00

ω(x, y, t)

ω =∂v

∂x− ∂u

∂y

Consider the stream function ψ(x, y)

A = ψ(x, y)ez u = ∇× A

ω = −∆ψez

ψ(x, y, t) = − 12 π

ω(x, y, t) ln(R) dx dy + ψp

R2 = (r − r)2 = (x− x)2 + (y − y)2

u =1

2 π

ω(x, y, t)ez × (r − r)

|r − r|2 dx dy +∇φ

Biot and Savart Law: 2D Flows

For large r it decreases algebraically Γ/(2πr)

Dω

Dt= ν∆ω

Cancellation of Vorticity

Diffusion of Vorticity

ν = 0Viscous Flow

∂ω

∂t= ν

∂2ω

∂y2

Vorticity diffuses by Viscous Effect

A unique vortex layer

ω ∼ Γδ(t)

ω =Γ√4νt

exp(− y2

4νt)

Circulation is conserved

t = 0

ω = ω0 sin(ky)An array of vortex layers

∂ω

∂t= ν

∂2ω

∂y2

ω(y, t) = ω0 exp(−k2t) sin(ky)

Vorticity Cancellation by Viscous Effect

a2 ≡

ω | CM |2 dS ω dS

Γ =

ω dS

C : Position of the vorticity centroid

Size of total vorticity repartition

a2 = a20 + 4 ν t

OC =

ω OM dS ω dS

The total vorticity field ν = 0

a2 ≡

ω | CM |2 dS ω dS

Γ =

ω dS

C : Position of the vortex center the centroid

Size of the vortex

a2 = a20 + 4 ν t

OC =

ω OM dS ω dS

Unique Vortex ν = 0

Conservation of vorticity pointwise

ν = 0

Size of any given vortex (only for a uniform vorticity)

Conservation of circulation of any given vortex

Inviscid vortex flow

⇒

⇒

In an assembly of vortices this means :

|r| << |r|

ln(R) =12

ln |r − r|2 =

12

lnr2

1−

2r2

r · r

∼ ln(r)−r · r

r2+ O

a2

L2

ψ = −ln(r)2 π

ω dx dy

Γ

+r

2 π r2

r ω dx dy

rC=0

+O

a2

r2

OC =

ω OM dS ω dS

Potential vortex flow

ψ ∼ −Γ2 π

ln(r) point vortex

+O

a2

r2

At a point far away from a vortex location, the velocity field produced by a vortex is similar to a

point-vortex velocity ω = Γδ(x− xc)δ(y − yc)

ur(r) =1r

∂ψ

∂θ= 0, uθ(r) = −∂ψ

∂r=

Γ2 π r

Potential vortex flow

For the velocity field at M, only two quantities matter

The distance OM to the vortex centroid O

The vortex circulation Γ

Γj

rj(xj , yj)

r(x, y)

M

uVelocity due to vortex j at point r(x, y)

uj =Γ

2 π |r − rj |2

−(y − yj)

x− xj

0

xi

yi

=

j =i

Γj

2 π R2ij

−(yi − yj)

xi − xj

, R2

ij = |ri − rj |2

A system of point vortices : Dynamical system

H = − 14 π

N

k=1

j =k

Γk Γj ln(Rkj)

This system is an Hamiltonian system

pi ≡ Γi yiqi ≡ xi

qi =∂H

∂pi, pi = −∂H

∂qi

H is a conserved energy

Interaction energy

Kinetic energy is infinite

xC

yC

=

1Γ

N

i=1

Γi

xi

yi

A centroid

A dispersion radius

d2 =1Γ

N

i=1

Γi [(xi − xC)2 + (yi − yC)2]

Other Conserved Quantities

System of Two point vortices

r1

r2Γ1

Γ2

Γ1 x1 + Γ2 x2 = cteΓ1 y1 + Γ2 y2 = cte

3 Conserved quantities

4 dynamical variables x1, y1, x2, y2

The dispersion radius gives again R12 = |r1 − r2| = Cst

H = − 12 π

Γ1 Γ2 ln(R12) = Cst ⇒ R12 = |r1 − r2| = Cst

Two point vortices Γ1Γ2 > 0

C is located between 1 and 2.

The system rotates around C at angular velocity

Ω =Γ1

2 π R12

Γ1 + Γ2

Γ1 R12=

Γ1 + Γ2

2 π R212

Two point vortices Γ1Γ2 < 0

C is located on the side of the more intense vortex

Two Point Vortices: a Dipole Γ1 + Γ2 = 0

The dipole does not rotate, and translates at velocity

Γ/(2π R12)streamlines in the comoving Frame

(a) (b)

streamlines in the Laboratory

Trapping and Transporting Fluid

Three Point Vortices

Simple dynamics : an integrable motion

Triangle shape oscillates with a period T1

Triangle rotates with a period T2

T1

T2is rational number (initial condition) ⇒ periodic

T1

T2

is in general not in an irrational number

⇒ Quasi-periodic Motion

Four or more point Vortices

The dynamical system is non integrable: chaotic

4 Conserved quantities

8 dynamical variables

Point Vortices with a Wall

(a) (b)

A unique vortex along a wall A Dipole along a wall

Image theory

Infinite Row of Point Vortices

Pairing instability

From an Infinite Row of Point Vorticestowards

Vortex Sheet

Vortex Sheet

self-similar spiralling

Roll-up behind a wing

Vortex sheet near trailing edges of a Delta wing

A First Vortex sheet with zero total circulation

Another Vortex sheet with non nul vortex circulation

EXTENDED 2D VORTICES

Vortex axisymmetrization

Vorticity Streamfunction

Stripping

Vortex in a Potential Strain field

Vortex along a wall or a Dipole

Gerris (S.Popinet)

Two-dimensional Turbulence : Inverse Cascade

Merging of Vortices

Vortex Merging

Josserand and Rossi 2005

Vortex Dipole

Vortex In a bath

ω(y, z, t = 0) = ω(+)(y, z) + ω(−)(y, z)

ω(±)(y, z) = ± Γ0

πa20

exp− (y ∓ b0/2)2 + z2

a20

!0

a0

b0y

z

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

1.2

t

!

I

!h

Cancellation of Vorticity

t = 0 20 40

50 90 1000

1 0.5 0 0.5 11

0.5

0

0.5

1

1.5

2

2.5

3t = 0 t = 8

Strain dipole in the Plane

Numerical Simulation Shallow Water equations (Code Gerris S.Popinet)