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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
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
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.
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
∂+ ∇ = ∇
∂
dΓ
(*)
(*) = 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)
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
tν
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
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)
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≡ ∇ −
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)
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 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
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
Vortex in periodic shear flow
________________________________
Laboratory visualisation of the formation of
manifolds:manifolds:
manifolds.mpgmanifolds.mpg
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