f.daviaud spec, cea/saclay, france in collaboration with a. chiffaudel,
DESCRIPTION
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. Turbulent von Karman flow. Axisymmetry R π symmetry / radial axis R c =100 mm - PowerPoint PPT PresentationTRANSCRIPT
1Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
F.Daviaud SPEC, CEA/Saclay, France
In collaboration with A. Chiffaudel,
B. Dubrulle, L. Marié, F. Ravelet, P. Cortet
Transition to turbulence and turbulent bifurcation in a von Karman flow
VKS team
2Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Turbulent von Karman flow • Axisymmetry• Rπ symmetry / radial axis
• Rc=100 mm• H = 180 mm• f = 2-20 Hz• Re = 2π Rc f2 / ν = 102 – 106
• fluid: water and glycerol-water
f2
f1
Inertial stirringTM60 propellersVelocity regulation
+ -
3Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Mean on 60 s500 f -1
Mean on 1/20 s1/2 f -1
Mean on : 1/500 s1/50 f -1
3 « scales »
4Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Re = 90Stationary axisymmetric
meridian plane: poloïdalrecirculation
Tangent plane : shear layer
Re = 185m = 2 ; stationary
Re = 400m = 2 ; periodic
First bifurcations and symmetry breaking
5Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Time spectra as a function of Re
Re = 330 Re = 380 Re = 440
220 225 230
0.2
0.1
Periodic Quasi-Periodic Chaotic
6Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Time spectra as a function of Re
Re = 1000 Re = 4000TurbulentChaotic 2000 < Re < 6500
Bimodal distribution :
signature of the turbulent shear
7Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Rec= 330Ret = 3300
Developed turbulence
Globally supercritical transition via a Kelvin-Helmholtz type instability of the shear layer and secondary bifurcations
Transition to turbulence: Azimuthal kinetic energy fluctuations
Ravelet et al. JFM 2008
8Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Multiplicity of solutions
Rec= 330Ret = 3300
Kp= Torque/ρRc5 (2π f)2
Re-1
9Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Turbulent Bifurcation of the mean flow
Bifurcated flow (b) :no more shear layer broken symmetry
two cellsone state
one celltwo states
Symmetry broken: 2 different mean flows exchange stability.
Re = 3.105
10Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Turbulent Bifurcation
Re = 3.105
Ravelet et al. PRL 2004
• Kp = Torque/ρRc5 (2π f)2 ΔKp= Kp1- Kp2
• θ = (f2-f1) / (f2+f1)• Re = (f1+f2)1/2
11Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Stability of the symmetric state
Statistics on 500 runs fordifferent θ
Cumulative distributionfunctions of bifurcation time tbif:
P(tbif>t)=A exp(-(t-t0)/τ)
• t0f ~ 5• τ : characteristic bif. time
12Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Stability of the symmetric state
• symmetric state marginally stableτ → ∞ when θ → 0
exponent = -6
13Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Mutistability = f(Re)
14Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Forbidden zone with velocity regulation
θ = (F1-F2)/(F1+F2)
=
(K
p1-K
p2)/
(Kp
1+
Kp
2)
1 cell (velocity)2 cells (velocity)
Forbidden zone for stationary regimes
15Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Intermittent states (i)
θ = (F1-F2)/(F1+F2)
=
(K
p1-K
p2)/
(Kp
1+
Kp
2)
intermittent states (i)
(s)
(b2)
(b1)1 cell (velocity)
1 cell (torque)2 cells (torque)
Forbidden zone with torque regulation
16Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
, fKp
1 ~ Kp
2,
“intermittence”between 2 states
Kp1 Kp
2 1 cell
Kp1 Kp
2 2 cells
Torque regulation: stochastic transitions 1cell state → 2 cells states
17Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Meridional annulus
+
Small curvature, diameter 3/4
VKS dynamo experiment
PropellersTM73
18Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
0 0.5-0.5 1-1
Position of the shear layer = f(θ)
SPIV measurements
+
θ = 0.05, without annulus
Po
sitio
n z
of t
he s
ep
arat
rix θc = 0.09
θc = 0.175
without annulus
with annulus
θ = (F1-F2)/(F1+F2)The annulus stabilizes the separatrix
● stagnation point
◦ r = 0.7
19Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Transition 1 cell - 2 cells at θc= ± 0.175
+
θ = (F1-F2)/(F1+F2)
- quasi-continuous transition- small hysteresis- small Kp difference
very different from TM60 propellers
<K
p>
<Δ
Kp
> =
<K
p1>
- <
Kp
2>
20Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Transition at θc= ± 0.175
+
Mean ΔKp is continuous
θ = (F1-F2)/(F1+F2)
stochastic transition 1 →2 cells
(s)
(b)(s)… … (b) …
State (b)
State (s)
10 min. acquisitions
○ : θ increasing
● : θ decreasing
<Δ
Kp
>
Time (sec.)
Time (sec.)
21Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
θ = (F1-F2)/(F1+F2)
<Δ
Kp
>
+
Measurements : 2 – 200 min
Transition at θc= ± 0.175
22Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
+
θ = (F1-F2)/(F1+F2)Pro
bab
ility
of p
rese
nce
p(s
), p
(b)
θc = 0.174
p(s
)/p(b
)
θ = (F1-F2)/(F1+F2)
Transition at θc
Cf. de la Torre & BurguetePRL 2007 and Friday talk
23Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Origin of erratic field reversals observed in VKS experiment?
θ = 0.17
24Wall Bounded Shear Flows, Newton Institute, Cambridge, September 2008
Bifurcations in turbulent flows: theory?
1. von Karman : turbulent bifurcation
2. VKS: dynamo action and B reversals
3. Couette flows: turbulent stripes and spirals
Prigent, Dauchot et al.PRL 2002