Diapositive 1

1
Transition to turbulence and turbulent
bifurcation in a von Karman flow
F.Daviaud SPEC, CEA/Saclay, France In
collaboration with A. Chiffaudel, B. Dubrulle,
L. Marié, F. Ravelet, P. Cortet
VKS team
2
Turbulent von Karman flow

• Axisymmetry
• Rp symmetry / radial axis
• Rc100 mm
• H 180 mm
• f 2-20 Hz
• Re 2p Rc f2 / ? 102 106
• fluid water and glycerol-water

f1
-

Inertial stirring TM60 propellers Velocity
regulation
f2
3
3  scales 
Mean on 60 s 500 f -1
Mean on 1/20 s 1/2 f -1
Mean on 1/500 s 1/50 f -1
4
First bifurcations and symmetry breaking
meridian plane poloïdal recirculation
Re 90 Stationary axisymmetric
Re 185 m 2 stationary
Re 400 m 2 periodic
Tangent plane shear layer
5
Time spectra as a function of Re
0.2
0.1
220
225
230
Re 330
Re 380
Re 440
Periodic
Quasi-Periodic
Chaotic
6
Time spectra as a function of Re
Re 1000
Re 4000
2000 lt Re lt 6500
Turbulent
Chaotic
Bimodal distribution signature of the turbulent
shear
7
Transition to turbulence Azimuthal kinetic
energy fluctuations
Developed turbulence
Globally supercritical transition via a
Kelvin-Helmholtz type instability of the shear
layer and secondary bifurcations
Ravelet et al. JFM 2008
Rec 330
Ret 3300
8
Multiplicity of solutions
Kp Torque/?Rc5 (2p f)2
Rec 330
Ret 3300
Re-1
9
Turbulent Bifurcation of the mean flow
Symmetry broken 2 different mean flows exchange
stability.
Bifurcated flow (b) no more shear layer broken
symmetry
Re 3.105
two cells one state
one cell two states
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Turbulent Bifurcation

• Kp Torque/?Rc5 (2p f)2 ?Kp Kp1- Kp2
• ? (f2-f1) / (f2f1)
• Re (f1f2)1/2

Re 3.105
Ravelet et al. PRL 2004
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Stability of the symmetric state
Statistics on 500 runs for different ?

• Cumulative distribution
• functions of bifurcation time tbif
• P(tbifgtt)A exp(-(t-t0)/t)
• t0f 5
• t characteristic bif. time

12
Stability of the symmetric state

• symmetric state
• marginally stable
• t ? 8 when ? ? 0

exponent -6
13
Mutistability f(Re)
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Forbidden zone with velocity regulation
Forbidden zone for stationary regimes
g (Kp1-Kp2)/(Kp1Kp2)
1 cell (velocity) 2 cells (velocity)
? (F1-F2)/(F1F2)
15
Forbidden zone with torque regulation
(b2)
Intermittent states (i)
g (Kp1-Kp2)/(Kp1Kp2)
(s)
intermittent states (i)
1 cell (velocity) 1 cell (torque) 2 cells
(torque)
(b1)
? (F1-F2)/(F1F2)
16
Torque regulation stochastic transitions
1cell state ? 2 cells states
Kp1 ? Kp2 ? 1 cell
Kp1 Kp2, intermittence between 2 states
q , f
Kp1 ? Kp2 ? 2 cells
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VKS dynamo experiment
Propellers TM73
Small curvature, diameter 3/4
Meridional annulus
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Position of the shear layer f(?)
SPIV measurements
?c 0.09
without annulus
? r 0.7
Position z of the separatrix
? stagnation point
?c 0.175
with annulus
? 0.05, without annulus
? (F1-F2)/(F1F2)
The annulus stabilizes the separatrix
19
Transition 1 cell - 2 cells  at ?c 0.175

• quasi-continuous transition
• small hysteresis
• small Kp difference
• very different from TM60 propellers

ltKpgt
lt?Kpgt ltKp1gt - ltKp2gt
? (F1-F2)/(F1F2)
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Transition at ?c 0.175
Mean ?Kp is continuous
Time (sec.)
(s)
(b)
(s)
(b)
? (F1-F2)/(F1F2)
stochastic transition 1 ?2 cells
Time (sec.)
21
Transition at ?c 0.175
Measurements 2 200 min
22
Transition at ?c
?c 0.174
p(s)/p(b)
Cf. de la Torre Burguete PRL 2007 and Friday
talk
? (F1-F2)/(F1F2)
23
Origin of erratic field reversals observed in
VKS experiment?
? 0.17
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Bifurcations in turbulent flows theory?

• von Karman turbulent bifurcation
• VKS dynamo action
• and B reversals
• 3. Couette flows turbulent stripes and spirals

Prigent, Dauchot et al. PRL 2002