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Taylor–Couette flow with independently rotating end plates

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Abstract

Results are presented from a combined numerical and experimental study of steady bifurcation phenomena in a modified Taylor–Couette geometry where the end plates of the flow domain are allowed to rotate independently of the inner cylinder. The ends rotate synchronously and the ratio between the rate of rotation of the ends ωe and the inner cylinder ωi defines a control parameter Ω:=ωe/ωi. Stationary ends favour inward motion along the end walls whereas rotating walls promote outward flow. We study the exchange between such states and focus on two-cell flows, which are found in the parameter range between Ω=0 and Ω=1 for Γ=2. Hence Ω is used as an unfolding parameter. A cusp bifurcation is uncovered as the organizing centre for the stability exchange between the two states. Symmetry breaking bifurcations, which lead to flows that break the mid-plane symmetry are also revealed. Overall, excellent agreement is found between numerical and experimental results.

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References

  1. Abshagen, J., Meincke, O., Pfister, G., Cliffe, K.A., Mullin, T.: Transient dynamics at the onset of Taylor vortices. J. Fluid Mech. 476, 335–343 (2003)

  2. Benjamin, T.B.: Bifurcation phenomena in steady flows of a liquid, 1. Theory, 2. Experiments. Proc. R. Soc. Lond. A 359, 1–27, 27–43 (1978)

  3. Benjamin, T.B., Mullin, T.: Anomalous modes in Taylor–Couette flow. Proc. R. Soc. Lond. A 377, 221–249 (1981)

  4. Benjamin, T.B., Mullin, T.: Notes on the multiplicity of flow in the Taylor experiment. J. Fluid Mech. A 121, 219–230 (1982)

  5. Cliffe, K.A.: Numerical calculations of two-cell and single-cell Taylor flows. J. Fluid Mech. 135, 219–233 (1983)

  6. Cliffe, K.A., Mullin, T.: A numerical and experimental study of anomalous modes in the Taylor experiment. J. Fluid Mech. 153, 243–258 (1985)

  7. Cliffe, K.A., Kobine, J.J., Mullin, T.: The role of anomalous modes in Taylor–Couette flow. Proc. R. Soc. Lond. A 439, 341–357 (1992)

  8. Cliffe, K.A., Spence, A., Tavener, S.J.: The numerical analysis of bifurcation problems with application to fluid mechanics. Acta Numerica 39, 39–131 (2000)

  9. Cliffe, K.A., Spence, A.: The calculation of high order singularities in the Taylor problem. In: Numerical methods for bifurcation problems. Kupper, T., Mittelmann, H.D., Weber, H. (Eds.) 129, 70. Birkauser Basel (1984)

  10. Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–899 (1993)

  11. Furukawa, H., Watanabe, T., Toya, Y., Nakamura, I.: Flow pattern exchange in the Taylor–Couette system with a very small aspect ratio. Phys. Rev. E 65, 036306 (2002)

  12. Lopez , J.M., Marques, F.: Small aspect ratio Taylor–Couette flow: Onset of a very-low-frequency three-torus state. Phys. Rev. E, 68(3): art. no. 036302, Part 2, Sep. 2003

  13. Marcus, P.S., Tuckerman, L.S.: Simulation of flow between concentric rotating spheres. 1. Steady states. J. Fluid Mech. 185, 1–30 (1987)

  14. Marcus, P.S., Tuckerman, L.S.: Simulation of flow between concentric rotating spheres. 2. Transitions. J. Fluid Mech. 185, 31–65 (1987)

  15. Moore, G., Spence, A.: The calculation of turning points of nonlinear equations. SIAM J. Numer. Anal. 17, 388–399 (1980)

  16. Mullin, T.: Mutation of steady cellular flows in the Taylor experiment. J. Fluid Mech. 121, 207–218 (1982)

  17. Mullin, T.: The Nature of Chaos. Oxford University Press Oxford (1993)

  18. Mullin, T., Blohm, C.: Bifurcation phenomena in a Taylor–Couette flow with asymmetric boundary conditions. Phys. Fluids 13(1), 136–140 (2001)

  19. Mullin, T., Toya, Y., Tavener, S.J.: Symmetry breaking and multiplicity of states in small aspect ratio Taylor–Couette flow. Phys. Fluids 14(8), 2778–2787 (2002)

  20. Pfister, G., Rehberg, I.: Space-dependent order parameter in circular Couette flow. Phys. Lett. A 83, 19–22 (1981)

  21. Pfister, G., Schmidt, H., Cliffe, K.A., Mullin, T.: Bifurcation phenomena in Taylor–Couette flow in a very short annulus. J. Fluid Mech. 191, 1–18 (1988)

  22. Pfister, G., Schulz, A., Lensch, B.: Bifurcations and a route to chaos of a one-vortex-state in Taylor–Couette flow. Eur. J. Mech. B/Fluids 10, 247–252 (1991)

  23. Schaeffer, D.G.: Qualitative analysis of a model for boundary effects in the Taylor problem. Math. Proc. Camb. Phil. Soc. 87, 307–337 (1980)

  24. Schulz, A., Pfister, G., Tavener, S.J.: The effect of outer cylinder rotation on Taylor–Couette flow at small aspect ratio. Phys. Fluids 15, 417–425 (2003)

  25. Swinney, H.L., Gollub, J.P.: Hydrodynamic Instabilities and the Transition to Turbulence. Springer Berlin Heidelberg (1981)

  26. Tagg, R.: The Couette-Taylor problem. Nonlinear Sci. Today4, 1–25 (1994)

  27. Tavener, S.J., Mullin, T., Cliffe, K.A.: Novel bifurcation phenomena in a rotating annulus. J. Fluid Mech. 229, 483–497 (1991)

  28. Taylor, G.I.: Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc. Lond. A 223, 289–343 (1923)

  29. Toya, Y., Nakamura, L., Yamashita, S., Ueki, Y.: An experiment on a Taylor vortex flow in a gap with small aspect ratio. Acta Mech. 102, 137 (1994)

  30. Werner, B., Spence, A.: The computation of symmetry-breaking bifurcation points. SIAM J. Numer. Anal. 21, 567–576 (1984)

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Authors and Affiliations

  1. Institute for Experimental and Applied Physics, University of Kiel, Kiel, 24098, Germany

    J. Abshagen, J. Langenberg & G. Pfister

  2. Institute for Marine Research, University of Kiel, 24105, Kiel, Germany

    J. Abshagen

  3. Serco Assurance, Harwell, OX11 0QJ, Didcot Oxon, UK

    K.A. Cliffe

  4. Manchester Centre for Nonlinear Dynamics, Oxford Road, M13 0PL, Manchester, UK

    T. Mullin

  5. Department of Mathematics, Colorado State University, CO 80623, Fort Collins, USA

    S.J. Tavener

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  1. J. Abshagen
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  2. K.A. Cliffe
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  3. J. Langenberg
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  4. T. Mullin
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  5. G. Pfister
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  6. S.J. Tavener
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Correspondence to T. Mullin.

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Communicated by

H.J.S. Fernando

PACS

47.20, 47.11, 47.54

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Abshagen, J., Cliffe, K., Langenberg, J. et al. Taylor–Couette flow with independently rotating end plates. Theor. Comput. Fluid Dyn. 18, 129–136 (2004). https://doi.org/10.1007/s00162-004-0135-3

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  • DOI: https://doi.org/10.1007/s00162-004-0135-3

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