Bifurcations and multistability in turbulence

Bifurcations and multistability in turbulence VKS team A. Chiffaudel L. Marié F. Ravelet F.Daviaud CEA/Saclay, France O. Dauchot A. Prigent Conceptual Aspects of turbulence, Vienna, February 28 1

Bifurcations and multistability in turbulence Bifurcations in turbulent flows: 1. von Karman : bifurcation between mean flows 2. Couette flows: turbulent spiral and stripes 3. VKS: - dynamo action - multistability Conceptual Aspects of turbulence, Vienna, February 28 2

Turbulent von Karman flow f 1 Axisymmetry Rπ symmetry / radial axis R c =1 mm H=18 mm f=2-2 Hz Re= 2π R c f 2 / ν = 1 2 1 6 fluid: water and glycerol-water + - f 2 Inertial stirring Conceptual Aspects of turbulence, Vienna, February 28 3

.9.8 3 «scales» z/r Mean on 6 s 5 f -1.9 1 1 r/r.8 Mean on 1/2 s 1/2 f -1 Mean on : 1/5 s 1/5 f -1 Conceptual Aspects of turbulence, Vienna, February 28 4

First bifurcations and symmetry breaking Re < 175 m = 175 < Re < 33 m = 2, fixed meridian plane: poloïdal recirculation Re > 33 m = 2, rotating π/2 π 3π/2 2π Re = 9 Stationary axisymmetric Re = 185 m = 2 ; stationary Re = 4 m = 2 ; periodic Tangent plane : shear layer Conceptual Aspects of turbulence, Vienna, February 28 5

Time spectra as a function of Re 1 1 1.8.2.8.8.6.6.6.4.2.1 22 225 23.4.2.4.2 V θ V θ V θ.2.4.2.2.4.2.4.6.8 1 1 2 3 4 5 6 Time (f 1 ) 1 3.1.1 22 225 23.6.8 1 1 2 3 4 5 6 7 8 Time (f 1 ) Re = 33 Re = 38 Re = 44 Periodic 1 3 Quasi-Periodic.6.8 1 2 4 6 8 1 Time (f 1 ) 1 3 Chaotic 1 2 1 1 1 1 1 1 2 1 3 1 4 1 2 1 1 1 1 1 1 2 1 3 1 4 1 2 1 1 1 1 1 1 2 1 3 1 4 1 5 1 3 1 2 1 1 1 1 1 1 2 f a /f 1 5 1 3 1 2 1 1 1 1 1 1 2 Conceptual Aspects of turbulence, f a /f Vienna, February 28 1 5 1 3 1 2 1 1 1 1 1 6 1 2 f /f a

Time spectra as a function of Re V θ 1.8.6.4.2.2.4.6 V θ 1.8.6.4.2.2.4.6 1 1 1 2 1 3 1 4.8 1 1 2 3 4 5 6 7 8 Time (f 1 ) 1 2.8 1 2 4 6 8 1 12 Time (f 1 ) Re = 1 Re = 4 Chaotic 1 3 Turbulent 1 5 1.5 1.5.5 1 1.5 V θ 2 < Re < 65 1 1 1 1 1 1 2 1 5/3 1 2 1 1 1 1 1 1 2 1 5/3 Bimodal distribution : signature of the turbulent shear 1 3 1 3 1 4 1 2 1 1 1 f a /f 1 1 1 2 1 4 1 3 1 2 1 1 1 1 1 1 2 f Conceptual Aspects of turbulence, a /f Vienna, February 28 7

.35 Transition to turbulence: fluctuations kinetic energy.3 Re t ( ) Developped turbulence V θ 2 rms.25.2.15.1.5 Re c ( ) V θ 2 rms.3.2.1 5 1 (Re Re c ).5 Globally supercritical transition via a Kelvin- Helmoltz type instability of the shear layer and secondary bifurcations 1 3 5 7 Re Re c = 33 Re t = 33 Conceptual Aspects of turbulence, Vienna, February 28 Ravelet et al. JFM 28 8

z/r.9 Turbulent Bifurcation of the mean flow 1 Symmetry broken: 2 different mean flows exchange stability..9 two cells one state 1 r/r 1 Bifurcated flow (b) : no more shear layer broken symmetry.9.2.7.6 z/r.7.5 Kp1.4.3.2.1 3 6 9 12 t.f one cell two states.9 1 r/r Conceptual Aspects of turbulence, Vienna, February 28 9 1.4

Turbulent Bifurcation: memory effect? Kp= Torque/ρR c 5 (2π f) 2 θ =(f 2 -f 1 ) / (f 2 +f 1 ) Re = (f 1 +f 2 ) 1/2 Re=3.1 5.8.6 (b 1 ) (b 2 ).2.1 (b 2 ) Kp.4 Kp (s).2 (s).1 (b ) 1 1.5 θ.5 1.2 1.5 θ.5 1 Ravelet et al. PRL 24 Conceptual Aspects of turbulence, Vienna, February 28 1

Stability of the symmetric state Kp 1 1.8.6.4.2 (s) 6 t.f 12 1 1 t bif. f (b 2 ) θ =.24 Kp 2.8.6.4.2 (s) 6 t.f 12 θ =.163 (b 2 ) Statistics on 5 runs for different θ Cumulative distribution functions of bifurcation time t bif : 1 2 P(t bif >t)=a exp(-(t-t )/τ) 1 3 θ =.267 12 24 36 48 t.f t f ~ 5 τ: characteristic bif. time Conceptual Aspects of turbulence, Vienna, February 28 11

Stability of the symmetric state 1 5 fit: power law exponent = -6 1 4 symmetric state marginally stable τ when θ τ. f 1 3 1 2 1 1.7.1.15.2.25.3.35 θ Conceptual Aspects of turbulence, Vienna, February 28 12

.2.1 Mutistability = f(re).2.1 (b 2 ) K p.1 K p.1 (s).2 1.5.5 1 θ (b 1 ).2 1.5.5 1 θ Conceptual Aspects of turbulence, Vienna, February 28 13

Multiplicity of solutions Kp= Torque/ρR c 5 (2π f) 2 1 Re c = 33 Re t = 33 ( ) b Kp ( ) s 1 1 Fit: Re -1 (+) s 1 1 1 2 1 3 1 4 1 5 1 6 Conceptual Aspects of turbulence, Vienna, February 28 Re 14

Torque regulation: stochastic transitions 1cell state 2 cells states Kp 1 Kp 2 1 cell Kp 1 ~ Kp 2, oscillation between 2 states θ, f Kp 1 Kp 2 2 cells Conceptual Aspects of turbulence, Vienna, February 28 15

U Couette flows d Ω o Ω i L z L x L d r i Γ x = L x / d Γ z = L z / d η = r i / r o Γ θ = π (r i + r o ) / d Γ z = L / d Conceptual Aspects of turbulence, Vienna, February 28 16

Plane Couette flow setup d = 2h = 7, 3.5, 1.5 mm L x = 578 mm L z = 255 mm Γ x = L x / d = 385 Γ z = L z / d = 17 1 control parameter : Uh R Cp = υ z y d x Daviaud PRL 92 glass sheet Conceptual Aspects of turbulence, Vienna, February 28 17

Cylindrical Couette setup Ω o Ω i L d η = r i / r o Γ θ = π (r i + r o ) / d Γ z = L / d Setup r i (mm) d (mm) r i η Γ z Γ θ 2 control parameters : R i Ωi ri d = and υ η R R o Ωo rod = υ Physical control parameter : TC η1 49.9.87.983 431 358 o i R TC η2 48.11 1.85.963 23 167 TC = ( 1 + ) 2 η R Conceptual Aspects of turbulence, Vienna, February 28 18

Taylor-Couette flow visualization mirror neon UV Light transmission through Kalliroscope flakes ~3 neon UV 27 cm ~35 132 cm Uniform lighting avoiding parasitic reflections CCD camera Conceptual Aspects of turbulence, Vienna, February 28 Prigent et al. PRL 22 19

Transition from laminar to turbulent flow Response to localized & instantaneous finite amplitude perturbation A 5 45 4 35 3 25 Long transient Sustained coexistence 2 15 1 Viscous5 damping 35 31 315 32 325 33 335 34 345 35 Ru Rg R Re Dauchot Daviaud POF 95 Conceptual Aspects of turbulence, Vienna, February 28 2

Transition from turbulent to laminar flow in plane Couette flow Conceptual Aspects of turbulence, Vienna, February 28 21

Transition from turbulent to laminar flow in Taylor Couette flow Conceptual Aspects of turbulence, Vienna, February 28 22

z Spiral Turbulence in extended geometry θ Conceptual Aspects of turbulence, Vienna, February 28 23 Andereck, Liu, Swinney, J. Fluid Mech (1986)

Turbulent stripes in plane Couette flow 155 135 λ x /h Comparison with Taylor-Couette flow when W i = -W o TCF PCF 1 8 λ z /h TCF PCF 115 6 95 4 R TC,Cp R TC,Cp 75 2 Conceptual Aspects of turbulence, Vienna, February 28 24 335 345 355 365 375 385 395 45 415 335 345 355 365 375 385 395 45 415

v z (ms 1 ).2 TC flow: LDV measurements Turbulence Spiral Laminar.2 -.2 55 6 65 55 6 65 55 6 65 v rms (ms 1 ).1 time (s) time (s) time (s).5 55 6 6555 6 6555 6 65 time (s) time (s) Conceptual Aspects of turbulence, Vienna, February 28 time (s) 25

1.6 1.4 I.L. (u.a.) From Turbulence to Spiral Turbulence,3,2 < A >² R o = 85 1.2 1.8,1 86 R i 6 65 7 75 8 85.6,3 < A >² R o = 12.4,2.2 1 2 3 4 z (mm) ~ supercritical bifurcation,1 5 6 7 8 9 Conceptual Aspects of turbulence, Vienna, February 28 26 857 R i

Couette flows: summary ε ε c A discontinuous transition from laminar to turbulent flow (unstable finite amplitude solutions) A continuous transition from turbulent to laminar flow (Ginzburg-Landau equations + noise) Conceptual Aspects of turbulence, Vienna, February 28 27