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Transition to Turbulence in Von Karman Flow: Bifurcations & Instabilities

Study on turbulent bifurcations, flow symmetries, and transition mechanisms in Von Karman flow using specific experimental setups and propellers. The research explores multiple flow states and their stability under various conditions. Findings provide insight into turbulent flow behaviors and potential applications.

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Transition to Turbulence in Von Karman Flow: Bifurcations & Instabilities

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  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 • 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 f1 - + Inertial stirring TM60 propellers Velocity regulation f2

  3. 3 « scales » Mean on60 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 < Re < 6500 Chaotic Turbulent 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/ρRc5 (2π 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

  10. Turbulent Bifurcation • Kp = Torque/ρRc5 (2π f)2 ΔKp= Kp1- Kp2 • θ = (f2-f1) / (f2+f1) • Re = (f1+f2)1/2 Re = 3.105 Ravelet et al. PRL 2004

  11. 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

  12. Stability of the symmetric state • symmetric state • marginally stable • τ→ ∞ when θ→ 0 exponent = -6

  13. Mutistability = f(Re)

  14. Forbidden zone with velocity regulation Forbidden zone for stationary regimes g = (Kp1-Kp2)/(Kp1+Kp2) 1 cell (velocity) 2 cells (velocity) θ = (F1-F2)/(F1+F2)

  15. Forbidden zone with torque regulation (b2) Intermittent states (i) g = (Kp1-Kp2)/(Kp1+Kp2) (s) intermittent states (i) 1 cell (velocity) 1 cell (torque) 2 cells (torque) (b1) θ = (F1-F2)/(F1+F2)

  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

  17. VKS dynamo experiment Propellers TM73 Small curvature, diameter 3/4 Meridional annulus

  18. Position of the shear layer = f(θ) -1 -0.5 0 0.5 1 SPIV measurements θc = 0.09 without annulus ◦r = 0.7 Position z of the separatrix ● stagnation point θc = 0.175 with annulus θ = 0.05, without annulus θ = (F1-F2)/(F1+F2) The annulus stabilizes the separatrix

  19. Transition 1 cell - 2 cells  at θc= ± 0.175 • quasi-continuous transition • small hysteresis • small Kp difference • very different from TM60 propellers <Kp> <ΔKp> = <Kp1> - <Kp2> θ = (F1-F2)/(F1+F2)

  20. 10 min. acquisitions ○ : θ increasing ● : θ decreasing State (b) <ΔKp> State (s) Transition at θc= ± 0.175 Mean ΔKp is continuous Time (sec.) (s) (b) (s)… … (b) … θ = (F1-F2)/(F1+F2) stochastic transition 1 →2 cells Time (sec.)

  21. Transition at θc= ± 0.175 <ΔKp> θ = (F1-F2)/(F1+F2) Measurements : 2 – 200 min

  22. Transition at θc Probability of presence p(s), p(b) θ = (F1-F2)/(F1+F2) θc= 0.174 p(s)/p(b) Cf. de la Torre & Burguete PRL 2007 and Friday talk θ = (F1-F2)/(F1+F2)

  23. Origin of erratic field reversals observed in VKS experiment? θ= 0.17

  24. 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

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