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2D TURBULENCE and COHERENT VORTEX STRUCTURES GertJan van Heijst Herman Clercx Fluid Dynamics Laboratory Dept of Physics Eindhoven University of Technology The Netherlands www.fluid.tue.nl

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2D TURBULENCE

and

COHERENT VORTEX STRUCTURES

GertJan van Heijst

Herman Clercx

Fluid Dynamics Laboratory

Dept of Physics

Eindhoven University of Technology

The Netherlands

www.fluid.tue.nl

atmospheric depression

Oceanic vortices

Geophysical flows (vortices) are

quasi-two-dimensional

due todue to

- planetary background rotation

- density stratification

- geometrical confinement (thin shell)

2D flow dynamics

(inviscid flow)

with ( , ,0)v u v=

1( . )

vv v p

t ρ

∂+ ∇ = − ∇

vorticity

vorticity equation:

→( . ) 0vt

ωω

∂+ ∇ =

∂0

D

Dt

ω=

(0,0, )vω ω≡ ∇× =

2D turbulence

Conservation of vorticity

or( . ) 0vt

ωω

∂+ ∇ =

∂0

D

Dt

ω=

implies conservation of

kinetic energy E ~ ½ v2

enstrophy V ~ ½ ω2

t∂ Dt

2D turbulence: conservation of E and V

• kinetic energy E:

• enstrophy V:

→ spectral flux to smaller k-values

2D turbulence

• kinetic energy: smaller → larger scales

(inverse energy cascade)

• enstrophy: larger → smaller scales

(direct enstrophy cascade)

• weakly dissipative

Self-organisation

Numerical simulation of 2D turbulence

Santangelo & Benzi (1989)

Statistical mechanics theories based on the Euler equation were

developed by e.g.

- Kraichnan, JFM 67 (1975)

- Pointin & Lundgren, PoF 19 (1976)

- Miller, PRL 65 (1990)

- Robert & Sommeria, JFM 229 (1991)

aimed at describing (final) equilibrium states on bounded or

double-periodic domains.

Numerical simulations

walls: sources of vorticity (filaments)

Numerical simulations

walls: sources of vorticity (filaments)

Circulation Γ

contour C

.C

v dlΓ ≡ ∫

• double-periodic: Γ = 0

• stress-free: Γ = ??

• no-slip: Γ = 0

C

Vorticity:

( . )D D

dxdy v dxdyt

ωω

∂+ ∇ =

∂∫ ∫

21

Re( . )v

t

ωω ω

∂+ ∇ = ∇

dΓ $ 21

Re( . )

D D

dv n ds dxdy

dtδ

ω ωΓ

= + = ∇∫ ∫

Vorticity:

( . )D D

dxdy v dxdyt

ωω

∂+ ∇ =

∂∫ ∫

21

Re( . )v

t

ωω ω

∂+ ∇ = ∇

(*)

(*) = 0 for no-slip b.c. (v = 0)

= 0 for stress-free b.c. (ω = 0)

= 0 for periodic b.c.

$ 21

Re( . )

D D

dv n ds dxdy

dtδ

ω ωΓ

= + = ∇∫ ∫

“leakage” of vorticity through ∂D→ change in Γ:

net vorticity flux through ∂D

$21 1

Re Re.

D D

ddxdy n ds

dt δ

ω ωΓ

= ∇ = ∇∫ ∫

“leakage” of vorticity through ∂D→ change in Γ:

net vorticity flux through ∂D

$21 1

Re Re.

D D

ddxdy n ds

dt δ

ω ωΓ

= ∇ = ∇∫ ∫

• no-slip:

• stress-free:

• periodic:

0 & 0d

dt

ΓΓ = =

0d

dt

Γ≠

0 0d

dt

ΓΓ = → =

Laboratory experiment in a stratified fluid

Maassen, Clercx, van Heijst – PoF 14 (2002)

& JFM 495 (2003)

Laboratory experiment Re* ≈ 5000, L0 ≈ 0

Laboratory experiment: vorticity

Laboratory experiment decaying quasi-2D turbulence

kinetic energy & enstrophy (both normalised)

Laboratory experiment: angular momentum |L/Lsb|

– – – L0 ≈ 0 —— |L0| > 0

Angular momentum

square →

ˆ.( )D

L k r v dA= × =∫

[ ] 2xv yu dxdy dxdyψ= − =∫∫ ∫∫

circle O →

1 2

0 0

ˆ.( )k r v rdrd

π

θ×∫ ∫

Rate of change of L:

[ ] [ ]ˆ ˆ. ( . ) .k r v v dA k r p dA= − × ∇ − ×∇∫ ∫

ˆ.( )D

dL dk r v dA

dt dt= ×∫

[ ] [ ]ˆ ˆ. ( . ) .D D

k r v v dA k r p dA= − × ∇ − ×∇∫ ∫

21 ˆ.( )Re

D

k r v dA+ ×∇∫

on a physical, bounded domain with no-slip boundaries

(Γ = 0):

1 2ˆ. ( . )

Re ReD D

dLpr d s r n ds

dtω

∂ ∂

= + − Γ∫ ∫

1dL∫ ∫

torque by (inviscid) torque by viscous

pressure forces (normal and shear)

(normal stress) stresses

1ˆ. ( . )

ReD D

dLpr d s r n ds

dtω

∂ ∂

= +∫ ∫

Forced 2D turbulence

• on a square domain D

• with no-slip walls: v = 0 on ∂D

1v∂←forcing

Molenaar, Clercx & vH – Physica D 196 (2004)

vH, Clercx & Molenaar – JFM 554 (2006)

21( . )

vv v p v f

ρ

∂+ ∇ = − ∇ + ∇ +

Vorticity equation

with2( . )v q

t

ωω ν ω

∂+ ∇ = ∇ +

∂.q k f= ∇×

initial condition:

.v u

k vx y

ω∂ ∂

= ∇× = −∂ ∂

( 0) 0tω = =

Angular momentum

rate of change:

with

21

2( ) .( ) ( , )

D D

L t k r v dA r r t dAω= × = −∫ ∫

21( . ) ( )

dLr n ds r ds M t

ων ω ν

∂= − +∫ ∫ with

numerical simulations: M(t) << [leading terms]

(two orders of magnitude)

21

2( . ) ( )

D D

dLr n ds r ds M t

dt n

ων ω ν

∂ ∂

∂= − +

∂∫ ∫

21

2( ) ( , )

D

M t r q r t dA= ∫

Angular momentum

L*

time t

normalised: L*=L(t)/Lu(t), with Lu(t) the angular momentum of

solid-body rotation with energy E

vorticity

t = 800 t = 900 t = 1000

Competition between two effects:

(a) selforganisation→ formation of one large cell

(“spontaneous spin-up”): L > 0

(b) erosion of cell by vorticity filaments (wall boundary (b) erosion of cell by vorticity filaments (wall boundary

layers): L → 0

• forcing weaker: (a) > (b) → cell ( L > 0)

• forcing stronger: (a) < (b) → no organised flow

(L ≈ 0)

angular momentum L & kinetic energy E

enstrophy Z

vorticity ω and Weiss function Qw

t=800 t=900 t=1000

> 0: strain-dominated (hyperbolic)

< 0: vorticity-dominated (elliptic)

2 22( )wQ v ω≡ ∇ −

What happens at higher Re-values?

Numerical simulations at Re = 20,000

(Geert Keetels)

(simulations by Werner Kramer)

Conclusions confined 2D turbulence

Solid (no-slip) walls play an important role in the

evolution of decaying and forced 2D turbulence:

• they exert forces (normal & shear stresses) thus

promoting the selforganisation process, leading to promoting the selforganisation process, leading to

“spontaneous spin-up”

• they act as sources of vorticity filaments (enstrophy),

which affect the evolution of the interior flow (e.g.

spectral characteristics, erosion of large-scale

circulation cells)

Numerical simulation of 2D turbulence

weak forcing

Classification• monopolar vortex

- circular / elliptical

- net angular momentum

• dipolar vortex• dipolar vortex- net linear momentum

- translation

- also: a-symmetric

(curved trajectory)

• tripolar vortex- net angular momentum

- rotation around core centre

Vortex interactions and deformation

• merging of vortices

when close enough, like-signed vortices

may merge into one bigger vortex, while

producing filamentary structuresproducing filamentary structures

• vortex in shear / strain fields

shear / strain leads to deformation of vortex

structures: elliptical core + vorticity filaments

Vortex merging: laboratory experiment

Vortex merging experiments

Vortex merging experiments

Vortex merging experiments

Vortex merging experiments

Laboratory experiment on vortex merger

Interaction of atmospheric vortices

Vortex in (periodic) shear flow

point vortex model

rv

π

γθ

2=vortex:

y

x

•γ yvx α=shear:

shear

strength: )cos1()( 0 tt sωεαα +=

Point vortex in linear shear

Steady (unperturbed)

shear-vortex flow:

shear-vortex flow:

• two stagnation points

• separatrix•

Steady shear-vortex flow

For a steady flow the stable and unstable manifolds coincide:

p+p+ p+

p- p-p-

Perturbed shear-vortex flow

For a perturbed flow the stable and unstable manifolds

no longer coincide:

p+p+ p+

p- p-p-

Lobe dynamics

Lobes provide a mechanism for fluid

exchange between vortex interior and

exterior:

entrainment lobedetrainment lobe

Lobe area µ→ amount of fluid

exchanged per cycle

vortex exterior vortex interior

Entrainment

lobe area µ/ε vs. 1/ωs

Entrainment

lobe area µ/ε as function of 1/ωs and α0

Contour length stretch

length stretch λ = l(t)/l(0) of a contour, initially placed around the

fixed point p+ (with fs = ωs/2π)

Particle transport

Poincaré map after 50 cycles of

of 2 × 1631 particles initially in

the entrainment lobes

E

E'

Perturbation frequency

ωs = 1.6π

Tracer transport

Poincaré sections for different perturbation

frequencies ωs

ωs = 0.8π ωs = 2.0π ωs = 3.2π

Laboratory experiments

• Side view: ΩΩΩΩ

• Top view:

Dye visualisation of the vortex interior

Vortex in periodic shear flow

Vortex in periodic shear flow

________________________________

Laboratory visualisation of the formation of

manifolds:manifolds:

manifolds.mpgmanifolds.mpg

General conclusions

• 2D turbulence → emergence of coherent vortex

structures

• Confinement (no-slip walls):

→ vorticity filaments & forces

→ selforganisation / breakdown organised state

• Interaction of vortices (merging, shear)