Bifurcations and multistability in turbulence

Transcription

1 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

2 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

3 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 = fluid: water and glycerol-water + - f 2 Inertial stirring Conceptual Aspects of turbulence, Vienna, February 28 3

4 «scales» z/r Mean on 6 s 5 f 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

5 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

6 Time spectra as a function of Re V θ V θ V θ Time (f 1 ) Time (f 1 ) Re = 33 Re = 38 Re = 44 Periodic 1 3 Quasi-Periodic Time (f 1 ) 1 3 Chaotic f a /f Conceptual Aspects of turbulence, f a /f Vienna, February f /f a

7 Time spectra as a function of Re V θ V θ Time (f 1 ) Time (f 1 ) Re = 1 Re = 4 Chaotic 1 3 Turbulent V θ 2 < Re < / /3 Bimodal distribution : signature of the turbulent shear f a /f f Conceptual Aspects of turbulence, a /f Vienna, February 28 7

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

9 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 z/r.7.5 Kp t.f one cell two states.9 1 r/r Conceptual Aspects of turbulence, Vienna, February

10 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= (b 1 ) (b 2 ).2.1 (b 2 ) Kp.4 Kp (s).2 (s).1 (b ) θ θ.5 1 Ravelet et al. PRL 24 Conceptual Aspects of turbulence, Vienna, February 28 1

11 Stability of the symmetric state Kp (s) 6 t.f t bif. f (b 2 ) θ =.24 Kp (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 = t.f t f ~ 5 τ: characteristic bif. time Conceptual Aspects of turbulence, Vienna, February 28 11

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

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

14 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 Conceptual Aspects of turbulence, Vienna, February 28 Re 14

15 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

16 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

17 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

18 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 η o i R TC η TC = ( 1 + ) 2 η R Conceptual Aspects of turbulence, Vienna, February 28 18

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

20 Transition from laminar to turbulent flow Response to localized & instantaneous finite amplitude perturbation A Long transient Sustained coexistence Viscous5 damping Ru Rg R Re Dauchot Daviaud POF 95 Conceptual Aspects of turbulence, Vienna, February 28 2

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

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

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

24 Turbulent stripes in plane Couette flow λ x /h Comparison with Taylor-Couette flow when W i = -W o TCF PCF 1 8 λ z /h TCF PCF R TC,Cp R TC,Cp 75 2 Conceptual Aspects of turbulence, Vienna, February

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

26 I.L. (u.a.) From Turbulence to Spiral Turbulence,3,2 < A >² R o = ,1 86 R i ,3 < A >² R o = 12.4, z (mm) ~ supercritical bifurcation, Conceptual Aspects of turbulence, Vienna, February R i

27 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