Two-dimensional magnetohydrodynamic turbulence in the limits of ...
Low dimensional turbulence - oca.euLow dimensional turbulence F. Daviaud in collaboration with A....
Transcript of Low dimensional turbulence - oca.euLow dimensional turbulence F. Daviaud in collaboration with A....
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Low dimensional turbulence
F. Daviaud
in collaboration with A. Chiffaudel, B. Dubrulle
P.-P. Cortet, P. Diribarne, D. Faranda, E. Herbert, L. Marié, R. Monchaux, F. Ravelet, B. Saint-Michel,
C. Wiertel and V. Padilla, Y . Sato (Hokkaido University)
VKS team & SHREK team
Laboratoire SPHYNX Service de Physique de l’Etat Condensé
CEA Saclay - CNRS UMR 3680 Université Paris Saclay
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Turbulence: 2 non reconcilable viewpoints
• Landau (1944): Turbulence = superposition of a growing number of modes with incommensurate oscillation frequencies, resulting from an infinite number of bifurcations with increasing Re • Ruelle and Takens (1971): Turbulent states = described by a small number of degrees of freedom, i.e. by a low dimensional "strange attractor" on which all turbulent motions concentrates in a suitable phase space
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Lorenz attractor (63)
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Turbulence and symmetry breaking (1)
• Transition to turbulence: succession of symmetry
breakings/bifurcations of the flow
→ 2 main types of transitions: • supercritical, continuous Ex: Rayleigh Bénard convection or Taylor Couette flow • subcritical, discontinuous, finite amplitude solutions Ex: plane Couette or Taylor Couette flow
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Transition to turbulence in Taylor Couette
laminar
stationary pattern
oscillating pattern
chaos
turbulence
……
End of story? Andereck et al. JFM 86 Re
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Transition to turbulence in Taylor Couette
Laminar/mean state
stationary pattern
oscillating pattern
chaos
turbulence
……
End of story?
?
Andereck et al. JFM 86 Re 6
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Turbulence and symmetry breaking (2)
• at “large” Reynolds number Re:
Turbulence is fully developed (K41, intermittency..)
Symmetries are statistically restored (Frisch 95)
• However, at large Re, observation of new symmetry
breakings which concern the flow statistics :
Turbulent bifurcations, instabilities…chaos?
→ in natural systems
→ in laboratory experiments
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Atmospheric circulation
Zonal Blocked
Weeks et al. Science
278, 1598 (1997)
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Persistent states, abrupt transitions. Scales: ~500 km, 2-5 years. Known since 1960s (Taft 1972).
M. Kawabe, JPO, 25, 3103 (1995)
Oceanic persistent states : “Kuroshio” stream
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Global oceanic circulation and D/O events
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Erratic inversions of the magnetic field
Earth VKS experiment 11
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Natural systems characterized by : • strongly turbulent flows • several mean large scale states • transitions/bifurcations towards or between these
mean states on temporal scales >> fluctuations.
Simple model experiments: • stability of the turbulent flow ? • transitions: → low dimensional dynamical systems ? → chaos?
Large scale bifurcations in turbulence
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Turbulent bifurcations in lab experiments
• geophysical experiments: Weeks et al. Science (1997)
• RB convection at high Ra : Chilla et al., EPJB (2004) - Roche et al. NJP (2010) - Grossmann and
Lohse, POF (2011) - Alhers et al. NJP (2011) - van der Poel et al. (2011)
• turbulent rotating RB convection: Stevens et al. PRL (2009) - Weiss et al., PRL (2010)
• spherical Couette flow Zimmermann et al. (2011)
• Taylor-Couette flow Lathrop et Mujica (2006)
• von Karman flows VKS (2007)
de la Torre and Burguete (2007, 2012)
• wake behind a sphere Cadot et al. (2013)
• ……
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Turbulence or chaos?
However • no complete theory of these transitions • all tentative to find the strange attractor of a
turbulence state (atmosphere or climate) failed so far:
- Nicolis 84: yes - Grassberger 86: no - Lorenz 91: unlikely!
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Turbulence or chaos?
End of the story? abandon all hope to apply tools from dynamical systems theory to a turbulent flow, except for transition to turbulence? No! → in this talk: try to reconcile the 2 points of view in fully developed turbulence
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A model experiment: the von Karman flow
• flow between 2 rotating impellers •Re = 102 to 106 (water and water-glycerol) 107 (Sodium) and 108 (liquid Helium)
• inhomogeneous and intense turbulence
• different forcings: + -
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Reynolds number
Rotation number
(System asymmetry)
Experimental setup
2 control parameters
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• SPIV 3 velocity components in a vertical plane
up to 50 000 fields
freq max=15 Hz
resolution = 2 mm
Measurements
• Torque measurements 18
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Transition to turbulence in von Karman
laminar
stationary pattern
oscillating pattern
chaos
turbulence
Re
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Re = 90 Stationary axisymmetric
meridian plane:
poloïdal
recirculation
Tangent
plane :
shear
layer
Re = 185 m = 2 ; stationary
Re = 400 m = 2 ; periodic
First bifurcations and symmetry breaking
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Time spectra as a function of Re
Re = 330 Re = 380 Re = 440
220 225 230
0.2
0.1
Periodic Quasi-Periodic Chaotic
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Time spectra as a function of Re
Re = 1000 Re = 4000 Turbulent Chaotic
2000 < Re < 6500
Bimodal distribution : signature of the turbulent shear
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Rec= 330 Ret = 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 23
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Turbulent von Karman at Re> 5000
Instantaneous Flow Mean flow
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Transition to turbulence in von Karman
laminar
stationary pattern
oscillating pattern
chaos
turbulence
……
End of story?
Re
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Eckhaus type instability (1)
Energy spectrum (Re= 106)
Herbert et al. Phys. Fluids 2013
+ sense, fully turbulent regime, Kp constant, and yet…
Cut at kz=0
Evolution of kx with Re
m = 0 1 2? 3
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Eckhaus type instability (2)
→ parity change (m change) • cf. laminar/turbulent transition • related to the number of mean
turbulent vortices in the shear layer
Re= 104: von Karman turbulent flow « stable » (no m change up to Re = 106) …
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Transition to turbulence in von Karman
laminar
stationary pattern
oscillating pattern
chaos
turbulence
Eckhaus type instability
turbulence*
End of story?
Re
θ 28
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Transition of the turbulent mean state
→ response to an external symmetry breaking: θ≠0
• sense -: abrupt transition
• sense +: soft transition
Ravelet et al. Phys. Rev. Lett. 2004
Saint-Michel et al. Phys. Rev. Lett. 2013
NJP 2014
Thalabard et al. NJP 2015
Cortet et al. Phys. Rev. Lett. 2010
J. Stat. Mech. 2011
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Soft transition
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Global angular momentum I = f(θ)
Symmetry parameter:
Re = 120
Re = 67 000
Re = 890 000
)(.
)(1I(t)
21
2 ffR
trudzrdrd
V
Order parameter: kinetic momentum
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Susceptibility larger at intermediate Re
Re = 120 → χI = 0.24
Re = 67 000 → χI = 42
Re = 890 000 → χI = 9
Global angular momentum I = f(θ)
0 I
I
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Susceptibility: χ = f(Re)
Divergence of the susceptibility around Re = Rec ≈ 40 000 → phase transition?
Rec
Rec
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Mean field critical divergence
Control parameter: turbulence « temperature »
Castaing, J. Phys. II (1996)
Analogy with ferromagnetic systems
Magnetization M angular momentum I Applied field H symmetry param. θ Temperature T Reynolds number Re
with Rec = 40 000
T<Tc T=Tc T> Tc Magnetization
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Abrupt transition
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Turbulent Bifurcation: 2 different mean flows exchange stability. A symmetry is broken
Bifurcated flow (b) : no more shear layer
two cells
one state
one cell
two states
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coexistence of the 3 states only in turbulent regimes
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VK flow: multiplicity of solutions
Rec= 330 Ret = 3300
Kp= Torque/ρRc5 (2π f)2
Re-1
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Stability of the symmetric state
Statistics on 500 runs for different θ
Cumulative distribution functions of bifurcation time tbif: P(tbif>t)=A exp(-(t-t0)/τ) • t0f ~ 5 • τ : characteristic bif. time
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Stability of the symmetric state
• symmetric state marginally stable τ → ∞ when θ → 0
exponent = -6
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Turbulent bifurcation
Torque difference Kinetic momentum
(s) (b1) (b2)
(s)
(b2)
(b1)
Hysteresis for Re > 16 000
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Evolution of ΔKp for different Reynolds
Re= 800 - 3000 – 5800 - 15300 - 195000 42
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Ferromagnetism
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Evolution of ΔKp for different Reynolds
Re= 800 - 3000 – 5800 - 15300 - 195000
ΔKp,0 Δθr
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Ferroturbulence
cf. magnetization M0 at H=0
cf. coercitive field Bc at M=0
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In liquid Helium (SHREK experiment)
Collaboration with ENS Lyon, SBT, Neel, LEGI, LUTh (EuHIT)
Superfluid He
Liquid He
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Transition to turbulence in von Karman
laminar
stationary pattern
oscillating pattern
chaos
turbulence
Eckhaus type instability
Turbulence*
End of story?
Re
θ Transitions
forcing 47
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Speed control Asymmetry control: θ
Conjugate parameter: γ
Turbulence and forcing protocols
Torque control Asymmetry control: γ
Conjugate parameter: θ
21
21 CC
CC
21
21 ff
ff
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Turbulent bifurcation: speed control
Speed control: forbidden γ zone 49
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Turbulent bifurcation: torque control
torque control: new mean states accessible
torque speed
50
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Turbulent bifurcation: torque control
3 states (attractors) (s*) (b*) (i*) in the θ pdf and the joint (f1,f2) pdf
(s*)
(i*)
(b*)
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Dynamical regimes: torque control
Impeller speed:
f1
f2
γ = -0.016
γ = -0.046
γ = -0.067
γ = -0.089
γ = -0.091
γ = -0.010 Re=500 000
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Torque control: mean velocity fields
Stability of turbulent states depends on forcing protocol! 57
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Experimental time series
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Classical chaos Stochastic chaos
Instantaneous velocity Bifurcation = Breaking of rotational symmetry Chaos Attractor N=dimension of attractor
Mean velocity Bifurcation = Breaking of R symmetry of mean flow Chaos with noise Stochastic Attractor N=dimension of attractor+
dimension of noise
Torque control: chaos?
Yes but shift of paradigm to stochastic chaos
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Modelisation: stochastic Duffing attractor
minimal dynamical system model (not reducible to SDE) - autonomous oscillator at frequency f0 - dynamics of γ(t) induced by the turbulent fluctuations
represented by a stochastic force. - θ → - θ symmetry excludes the presence of a quadratic non-
linearity
Stochastic Duffing equations, with two variables 𝑥 (exp: θ ) and y = 𝑥 , with random forcing 𝑧 (exp: dynamics of γ(t) ) obeying:
Control parameter: μ (exp: γ)
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Comparison Turbulent & Duffing attractor
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Comparison with Duffing attractor
Effective dimension: - Turbulent attractor =10 - Duffing attractor = 9
Lyapounov exponents (≠ deterministic Duffing)
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Low dimensional turbulence?
→ in turbulent von Karman flows: - instability of turbulent states f(Re, symmetry,forcing) - multiplicity of solutions, depending on forcing protocol - possible transitions between these solutions: 1st or 2nd order like - emergence of low dimensional dynamics - turbulent chaotic attractors
Chaotic turbulence?
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