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Isothermal spherical Couette flow

  • Markus Junk
  • Christoph Egbers
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 549)

Abstract

We summarise different types of instabilities and flow patterns in isothermal spherical Couette flows as a function of the aspect ratio. The flow of a viscous incompressible fluid in the gap between two concentric spheres was investigated for the case, that only the inner sphere rotates and the outer one is stationary. Flow visualisation studies were carried out for a wide range of Reynolds numbers and aspect ratios to determine the instabilities during the laminar-turbulent transition and the corresponding critical Reynolds numbers as a function of the aspect ratio. It was found, that the laminar basic flow loses its stability at the stability threshold in different ways. The instabilities occurring depend strongly on the aspect ratio and the initial conditions. For small and medium aspect ratios (β ≤ 0.25), experiments were carried out as a function of Reynolds number to determine the regions of existence for basic flow, Taylor vortex flow, supercritical basic flow. For wide gaps, however, Taylor vortices could not be detected by quasistationary increase of the Reynolds number. The first instability manifests itself as a break of the spatial symmetry and non-axisymmetric secondary waves with spiral arms appear depending on the Reynolds number. For β = 0.33, spiral waves with an azimuthal wave number m = 6, 5and 4 were found, while in the gap with an aspect ratio of β = 0.5spiral waves with m = 5, 4 and 3 spiral arms exist. For β = 1.0, we could detect spiral waves with m = 4, 3 and 2 arms. We compare the experimental results for the critical Reynolds numbers and wave numbers with those obtained by numerical calculations. The flow modes occurring at the poles look very similar to those found in the flow between two rotating disks. Effects of non-uniqueness and hysteresis are observed in this regime.

Keywords

Reynolds Number Aspect Ratio Spiral Wave Critical Reynolds Number Outer Sphere 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    C.D. Andereck, S.S. Liu, H.L. Swinney: Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155–183 (1986)CrossRefADSGoogle Scholar
  2. 2.
    K. Araki, J. Mizushima, S. Yanase: The nonaxisymmetric instability of the widegap spherical Couette flow. Phys. Fluids 9, 1197–1199 (1997)CrossRefADSGoogle Scholar
  3. 3.
    N.M. Astafyeva: Nonlinear shear flow in rotating spherical layers and global atmosphere motion modelling (in Russian). Izv. Vusov PND 5, 3–30 (1997)Google Scholar
  4. 4.
    F. Bartels: Taylor vortices between two concentric rotating spheres. J. Fluid Mech. 119, 1–25 (1982)zbMATHCrossRefADSGoogle Scholar
  5. 5.
    P. Bar-Yoseph, A. Solan, R. Hillen, K.G. Roesner: Taylor vortex flow between eccentric coaxial rotating spheres. Phys. Fluids A 2, 1564–1573 (1990)CrossRefADSGoogle Scholar
  6. 6.
    P. Bar-Yoseph, K.G. Roesner, A. Solan: Vortex breakdown in the polar region between rotating spheres. Phys. Fluids A 4, 1677–1686 (1992)CrossRefADSGoogle Scholar
  7. 7.
    Yu.N. Belyaev, A.A. Monakhov, I.M. Yavorskaya: Stability of spherical Couette flow in thick layers when the inner sphere revolves. Fluid Dyn. 13, 162–163 (1978)CrossRefADSGoogle Scholar
  8. 8.
    Yu.N. Belyaev, A.A. Monakhov, G.N. Khlebutin, I.M. Yavorskaya: Investigation of stability and nonuniqueness of the flow between rotating spheres (in Russian). Rep. No. 567, Space Research Institute of the Academy of Sciences, Moscow, USSR (1980)Google Scholar
  9. 9.
    Yu.N. Belyaev, A.A. Monakhov, S.A. Scherbakov, I.M. Yavorskaya: Some routes to turbulence in spherical Couette Flow. in: V.V. Kozlov (ed.): Laminar-Turbulent Transition. IUTAM-Symp. Novosibirsk/USSR, Springer (1984)Google Scholar
  10. 10.
    J.P. Bonnet, T. Alziary de Roquefort: Ecoulement entre deux spheres concentriques en rotation. J. Mec. 13, 373 (1976)ADSGoogle Scholar
  11. 11.
    Yu.K. Bratukhin: On the evaluation of the critical Reynolds number for the flow of fluid between two rotating spherical surfaces. PMM 25, 858–866 (1990)Google Scholar
  12. 12.
    K. Bühler: Strömungsmechanische Instabilitäten zäher Medien im Kugelspalt. VDI-Berichte, Reihe7: Strömungstechnik Nr.96 (1985)Google Scholar
  13. 13.
    K. Bühler: Symmetric and asymmetric Taylor vortex flow in spherical gaps. Acta Mechanica 81, 3–38 (1990)CrossRefMathSciNetGoogle Scholar
  14. 14.
    K. Bühler, J. Zierep: New secondary instabilities for high Re-number flow between two rotating spheres. in: Kozlov, V.V. (ed.): Laminar-Turbulent Transition. IUTAM-Symp. Novosibirsk/USSR, Springer (1984)Google Scholar
  15. 15.
    K. Bühler, J. Zierep: Dynamical instabilities and transition to turbulence in spherical gap flows. in: G. Comte-Bellot, J. Mathieu: Advances in turbulence. Proc. 1st Europ. Turb. Conf., Lyon, France, Springer (1986)Google Scholar
  16. 16.
    S.C.R. Dennis, L. Quartapelle: Finite difference solution to the flow between two rotating spheres. Comp. Fluids. 12, 77–92 (1984)zbMATHCrossRefADSGoogle Scholar
  17. 17.
    G. Dumas: The spherical Couette flow and its large-gap stability by spectral simulations. Proceedings of CFD 94, Canadian Society of CFD, Toronto, Canada, 67–75 (1994)Google Scholar
  18. 18.
    G. Dumas, A. Leonard: A divergence-free spectral expansions method for threedimensional flows in spherical-gap geometries. J. Comput. Phys. 111, 205–219 (1994)zbMATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    C. Egbers: Zur Stabilität der Strömung im konzentrischen Kugelspalt. Dissertation, Universität Bremen (1994)Google Scholar
  20. 20.
    C. Egbers, H.J. Rath: The existence of Taylor vortices and wide-gap instabilities in spherical Couette flow. Acta Mech. 111, 125–140 (1995)CrossRefGoogle Scholar
  21. 21.
    C. Egbers, H.J. Rath: LDV-measurements on wide-gap instabilities in spherical Couette flow. Developments in Laser Techniques and Applications to Fluid Mechanics (Eds.: R.J. Adrian, D.F.G. Durao, F. Durst, M.V. Heitor, M. Maeda, J.H. Whitelaw, Springer, 45–66 (1996)Google Scholar
  22. 22.
    R. Hollerbach: Time-dependent Taylor vortices in wide-gap spherical Couette flow. Phys. Rev. Lett. 81, 3132–3135 (1998)CrossRefADSGoogle Scholar
  23. 23.
    R. Hollerbach: A spectral solution of the magnetoconvection equations in spherical geometry. Int. J. Num. Meth. Fluids 32, 773–797 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    G.N. Khlebutin: Stability of fluid motion between a rotating and a stationary concentric sphere. Fluid Dyn. 3, 31–32 (1968)CrossRefADSGoogle Scholar
  25. 25.
    M. Liu, C. Blohm, C. Egbers, P. Wulf, H.J. Rath: Taylor vortices in wide spherical shells. Phys. Rev. Lett. 77, 286–289 (1996)CrossRefADSGoogle Scholar
  26. 26.
    C.K. Mamun, L. Tuckerman: Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids 2, 1564–1573 (1995)MathSciNetGoogle Scholar
  27. 27.
    P. Marcus, L. Tuckerman: Simulation of the flow between concentric rotating spheres. J. Fluid Mech. 185, 1–65 (1987)zbMATHCrossRefADSGoogle Scholar
  28. 28.
    A.A. Monakhov: Limit of main flow stability in spherical layers. Fluid Dynamics 31, 535–538 (1996)CrossRefADSGoogle Scholar
  29. 29.
    B.R. Munson, M. Menguturk: Viscous incompressible flow between concentric rotating spheres. Part 3: Linear stability and experiments. J. Fluid Mech. 69, 705–719 (1975)zbMATHCrossRefADSGoogle Scholar
  30. 30.
    K. Nakabayashi: Transition of Taylor-Görtler vortex flow in spherical Couette flow. J. Fluid Mech. 132, 209–230 (1983)CrossRefADSGoogle Scholar
  31. 31.
    K. Nakabayashi, Y. Tsuchida: Spectral study of the laminar-turbulent transition in spherical Couette flow. J. Fluid Mech., 194, 101–132 (1988)CrossRefADSGoogle Scholar
  32. 32.
    K. Nakabayashi, W. Sha: Vortical structures and velocity fluctuations of spiral and wavy vortices in the spherical Couette flow. this book (2000)Google Scholar
  33. 33.
    G. Nicolis: Introduction to nonlinear science. Cambridge University Press (1995)Google Scholar
  34. 34.
    O. Sawatzki, J. Zierep: Das Stromfeld im Spalt zwischen zwei konzentrischen Kugelflächen, von denen die innere rotiert. Acta Mech. 9, 13–35 (1970)CrossRefGoogle Scholar
  35. 35.
    G. Schrauf: The first instability in spherical Taylor-Couette flow. J. Fluid Mech. 166, 287–303 (1986)zbMATHCrossRefADSGoogle Scholar
  36. 36.
    M.P. Sorokin, G.N. Khlebutin, G.F. Shaidurov: Study of the motion of a liquid between two rotating spherical surfaces. J. Appl. Mech. Tech. Phys. 6, 73–74 (1966)ADSGoogle Scholar
  37. 37.
    G.I. Taylor: Stability of a viscous liquid contained between two rotating cylinders. Phil.Trans. A 223, 289–293 (1923)CrossRefADSGoogle Scholar
  38. 38.
    A.M. Waked, B.R. Munson: Laminar turbulent flow in spherical annulus. J. Fluids Eng. 100, 281–286 (1978)CrossRefGoogle Scholar
  39. 39.
    M. Wimmer: Experiments on a viscous fluid between concentric rotating spheres. J. Fluid Mech. 78, 317–335 (1981)CrossRefADSGoogle Scholar
  40. 40.
    M. Wimmer: Experiments on the stability of viscous flow between two concentric rotating spheres. J. Fluid Mech. 103, 117–131 (1981)CrossRefADSGoogle Scholar
  41. 41.
    P. Wulf, C. Egbers. H.J. Rath: Routes to chaos in wide-gap spherical Couette flow. Phys. Fluids 11, 1359–1372 (1999)CrossRefADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    V.I. Yakushin: Instability of the motion of a liquid between two rotating spherical surfaces. Fluid Dyn. 5, 660–661 (1970)CrossRefADSGoogle Scholar
  43. 43.
    H. Yamaguchi, J. Fujiyoshi, H. Matsui: Spherical Couette flow of a viscoelastic fluid. Part 1: Experimental study of the inner sphere rotation. J. Non-Newtonian Fluid Mech. 69, 29–46 (1997)CrossRefGoogle Scholar
  44. 44.
    H. Yamaguchi, H. Matsui: Spherical Couette flow of a viscoelastic fluid. Part 2: Numerical study for the inner sphere rotation. J. Non-Newtonian Fluid Mech. 69, 47–70 (1997)CrossRefGoogle Scholar
  45. 45.
    H. Yamaguchi, B. Nishiguchi: Spherical Couette flow of a viscoelastic fluid. Part 3: A study of outer sphere rotation. J. Non-Newtonian Fluid Mech. 69, 47–70 (1997)CrossRefGoogle Scholar
  46. 46.
    R.-J. Yang: A numerical procedure for predicting multiple solutions of a spherical Taylor-Couette flow. Int. J. Numer. Methods Fluids 22, 1135–1147 (1996)zbMATHCrossRefADSGoogle Scholar
  47. 47.
    I.M. Yavorskaya, Yu.N. Belyaev, A.A. Monakhov: Experimental study of a spherical Couette flow. Sov. Phys. Dokl. 20, 256–258 (1975)ADSGoogle Scholar
  48. 48.
    I.M. Yavorskaya, Yu.N. Belyaev, A.A. Monakhov, N.M. Astaf'eva, S.A. Scherbakov, N.D. Vvedenskaya: Stability, non-uniqueness and transition to turbulence in the flow between two rotating spheres. IUTAM-Symposium, Toronto, Canada (1980)Google Scholar
  49. 49.
    O.Yu. Zikanov: Symmetry breaking bifurcations in spherical Couette flow. J. Fluid Mech. 310, 293–324 (1996)zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Markus Junk
    • 1
  • Christoph Egbers
    • 1
  1. 1.Centre of Applied Space Technology and Microgravity (ZARM)University of BremenBremenGermany

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