Skip to main content

The Taylor-Couette Flow: The Hydrodynamic Twin of Rayleigh-Bénard Convection

  • Chapter
Dynamics of Spatio-Temporal Cellular Structures

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 207))

Abstract

There is a strong analogy between Rayleigh-Bénard convection and the Taylor-Couette system. This analogy is well known when dealing with the primary instability, and is based on the existence of an unstable stratification in both systems. We show that the analogy can be extended beyond the primary instability modes, to the weakly non-linear regime and even further to the fully turbulent one.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Couette, “Sur un nouvel appareil pour l’étude du frottement des fluides”, Ann. Chim. Phys. (6) 21, 433–510, 70 (1890).

    Google Scholar 

  2. A. Mallock, “Determination of the viscosity of water”, Phil. Trans. Roy. Soc. A 187, 41–56 (1896).

    Article  ADS  Google Scholar 

  3. L. Rayleigh, “On the dynamics of revolving fluids”, Proc. R. Soc. London, Ser. A 93, 148–54 (1916).

    ADS  Google Scholar 

  4. L. Rayleigh, “On convective currents in a horizontal layer of fluid, when the higher temperature is on the under side”, Phil. Mag (6) 32, 529–46 (1916).

    Google Scholar 

  5. G.I. Taylor, “Stability of a viscous liquid contained between two rotating cylinders”, Phil. Trans. Roy. Soc. London, Ser. A 223, 289–343 (1923).

    Article  ADS  Google Scholar 

  6. H. Jeffreys, “Some cases of instability in fluid motion”, Proc. R. Soc. London, Ser. A 118, 195–208 (1928).

    Article  ADS  Google Scholar 

  7. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford (1961).

    MATH  Google Scholar 

  8. F.H. Busse, “Bounds for turbulent shear flow”, J. Fluid Mech. 41, 219 (1970).

    Article  MATH  ADS  Google Scholar 

  9. E. Guyon, J.P. Hulin, and L. Petit, Hydrodynamique Physique, Savoirs actuels, Paris (2002).

    Google Scholar 

  10. E.L. Koscmieder, Bénard Cells and Taylor Vortices, Cambridge University Press, Cambridge (1993).

    Google Scholar 

  11. B.J. Bayly, “Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows”, Phys. Fluids A 31(1), 56–64 (1988).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. I. Mutabazi, C. Normand, and J.E. Wesfreid, “Gap size effects on centrifugally and rotaionally driven instabilities”, Phys. Fluids A 4, 1199–1205 (1992).

    Article  MATH  ADS  Google Scholar 

  13. P. Manneville, Structures Dissipatives, Chaos et Turbulence, Aléa Saclay, Saclay (1991).

    Google Scholar 

  14. A. Esser and S. Grossmann, “Analytic expression for Taylor-Couette stability boundary”, Phys. Fluids 8, 1814 (1996).

    Article  MATH  ADS  Google Scholar 

  15. B. Dubrulle, O. Dauchot, F. Daviaud, F. Hersant, P.-Y. Longaretti, D. Richard, and J-P. Zahn,“Stability and turbulent transport in rotating shear flows: prescription from analysis of cylindrical and plane Couette flows”, submitted to Phys. Fluids (2004).

    Google Scholar 

  16. D.K. Lezius and J.P. Johnston, “Roll-cell instabilities in rotating laminar and turbulent channel flows”, J. Fluid Mech. 77, 573 (1976).

    Article  Google Scholar 

  17. O.J.E. Matsson and P.H. Alfredsson, “Curvature-and rotation-induced instabilities in channel flow”, J. Fluid Mech. 210, 537–563 (1990).

    Article  ADS  Google Scholar 

  18. J.E. Wesfreid, Y. Pomeau, M. Dubois, C. Normand, and P. Bergé, “Critical effects in Rayleigh-Bénard convection”, J. Physique (Paris) 39, 725 (1978).

    Google Scholar 

  19. H. Yahata, “Slowly-varying amplitude of the Taylor vortices near the instability point”, Prog. Theor. Phys. 57, 347 (1977).

    Article  ADS  Google Scholar 

  20. P. Tabeling, “Dynamics of the phase variable in the Taylor vortex system”, J. Phys. Lett. 44, 16 (1983).

    Article  Google Scholar 

  21. P. Hall, “Evolution equations for Taylor vortices in the small-gap limit”, Phys. Rev. A 29, 2921 (1984).

    Article  ADS  Google Scholar 

  22. Y. Demay and G. Iooss, “Calcul des solutions bifurquées pour le problème de Couette-Taylor avec les 2 cylindres en rotation”, J. Méca. Théor. Appl., numéro spécial, 193 (1984).

    Google Scholar 

  23. Y. Demay, G. Iooss, and P. Laure, “Wave patterns in the small gap Couette-Taylor problem”, Eur. J. Mech. B 11, 621 (1992).

    MATH  MathSciNet  Google Scholar 

  24. M.A. Dominguez-Lerma, G. Ahlers, and D. S. Cannell, “Marginal stability curve and linear growth rate for rotating Couette-Taylor flow and Rayleigh-Bénard convection”, Phys. Fluids 27, 856 (1984).

    Article  MATH  ADS  Google Scholar 

  25. G. Ahlers, “Experiments on bifurcation and one-dimensional patterns in nonlinear systems far from equilibrium”, in Lectures in the Sciences of Complexity, ed. by D.L. Stein, Addison-Wesley, Redwood City, CA (1989).

    Google Scholar 

  26. Y. Pomeau and P. Manneville, “Stability and fluctuations of a spatially periodic convective flow”, J. Phys. Lett. 40, 610 (1979).

    Article  MathSciNet  Google Scholar 

  27. J.E. Wesfreid and V. Croquette, “Forced Phase Diffusion in Rayleigh-Bénard Convection”, Phys. Rev. Lett. 45, 6340 (1980).

    Article  Google Scholar 

  28. H. Paap and H. Riecke, “Drifting vortices in ramped Taylor vortex flow. Quantitative results from phase equation”, Phys. Fluids A 3, 1519 (1991).

    Article  MATH  ADS  Google Scholar 

  29. M. Wu and C.D. Andereck, “Phase modulation of Taylor vortex flow”, Phys. Rev. A 43, 2074 (1991).

    Article  ADS  Google Scholar 

  30. M. Wu and C.D. Andereck, “Phase dynamics of wavy vortex flow”, Phys. Rev. Lett. 67, 1258 (1991).

    Article  ADS  Google Scholar 

  31. M. Wu and C.D. Andereck, “Phase dynamics in the Taylor-Couette system”, Phys. Fluids A 4, 2432 (1992).

    Article  ADS  Google Scholar 

  32. D. Coles, “Transition in circular Couette flow”, J. Fluid Mech. 21, 385–425 (1965).

    Article  MATH  ADS  Google Scholar 

  33. C.D. Andereck, S.S. Liu, and H.L. Swinney, “Flow regimes in a circular Couette system with independently rotating cylinders”, J. Fluid Mech. 164, 155–183 (1986).

    Article  ADS  Google Scholar 

  34. R. Tagg, W.S. Edwards, and H.L. Swinney, “Nonlinear standing waves in Couette-Taylor flow”, Phys. Rev. A 39, 3734 (1989).

    Article  ADS  Google Scholar 

  35. R. Tagg, “The Couette-Taylor problem”, Nonlinear Science Today 4, 1 (1994).

    Article  MathSciNet  Google Scholar 

  36. F.H. Busse, “Transition to turbulence in Rayleigh-Bénard convection”, in Hydrodynamic Instabilities and the Transition to Turbulence eds. H.L. Swinney and J.P. Gollub, Springer, New York (1981).

    Google Scholar 

  37. B. Dubrulle and F. Hersant, “Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection”, Eur. Phys. J. B 26, 379 (2002).

    ADS  Google Scholar 

  38. A. Schlüter, D. Lortz, and F. Busse, “Transition to turbulence in Rayleigh-Bénard convection”, J. Fluid Mech. 23, 129 (1965).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  39. J.K. Platten and J.C. Legros, Convection in liquids, Springer, New York (1984).

    MATH  Google Scholar 

  40. D.P. Lathrop, J. Fineberg, and H.L. Swinney, “Transition to shear-driven turbulence in Couette-Taylor flow”, Phys. Rev A 46, 6390 (1992).

    Article  ADS  Google Scholar 

  41. G.P. King, Y. Li, W. Lee, H.L. Swinney, and P.S. Marcus, “Wave speeds in wavy Taylor vortex flow”, J. Fluid Mech. 41, 365 (1984).

    Article  ADS  Google Scholar 

  42. A. Barcilon and J. Brindley, “Organized structures in turbulent Taylor-Couette flows at a very high Taylor number”, J. Fluid Mech. 143, 429 (1984).

    Article  MATH  ADS  Google Scholar 

  43. F. Wendt, “Turbulente Stromungen zwischen zwei rotierenden konaxialen Zylindern”, Ingenieur-Archiv. 4, 577 (1933).

    Article  Google Scholar 

  44. G.I. Taylor,“Fluid friction between rotating cylinders I. — Torque measurements”, Proc. R. Soc. London A 157, 546 (1936).

    Article  ADS  Google Scholar 

  45. P. Tong, W.I. Goldburg, J.S. Huang, and T.A. Witten, “Anisotropy in turbulent drag reduction”, Phys. Rev. Lett. 65, 2780 (1990).

    Article  ADS  Google Scholar 

  46. G.S. Lewis and H.L. Swinney, “Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette-Taylor flow”, Phys Rev. E 59, 5457 (1999).

    Article  ADS  Google Scholar 

  47. F. Heslot, B. Castaing, and A. Libchaber, “Transitions to turbulence in helium gas”, Phys. Rev. A 36, 5870 (1987).

    Article  ADS  Google Scholar 

  48. B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X-Z. Wu, S. Zaleski, and G. Zanetti, “Scaling of hard thermal turbulence in Rayleigh-Bénard convection”, J. Fluid Mech. 204, 1 (1989).

    Article  ADS  Google Scholar 

  49. R. Kraichnan, “Mixing-length analysis of turbulent thermal convection at arbitrary Prandtl number”, Phys. Fluids 5, 1374 (1962).

    Article  ADS  Google Scholar 

  50. X. Chavanne, F. Chilla, B. Castaing, B. Hébral, B. Chabaud, and J. Chaussy, “Observation of the ultimate regime in Rayleigh-Bénard convection”, Phys. Rev. Lett. 79, 3648 (1997).

    Article  ADS  Google Scholar 

  51. B. Dubrulle, “Scaling laws predictions from a solvable model of turbulent thermal convection”, Europhys. Letters 51, 513 (2000).

    Article  ADS  Google Scholar 

  52. B. Dubrulle, “Logarithmic corrections to scaling in turbulent thermal convection”, Euro. Phys. J. B 21, 295 (2001).

    Article  ADS  Google Scholar 

  53. F.H. Busse, “Transition to turbulence in Rayleigh-Bénard convection”, in Nonlinear Physics of Complex Systems ed. G. Parisi, S.C. Muller, and W. Zimmermann, Lecture Notes in Physics 476, 1, Springer, New York (1996).

    Chapter  Google Scholar 

  54. F.H. Busse, “Transition to turbulence in Rayleigh-Bénard convection”, Arch. Rat. Mech. Anals 47, 28 (1972).

    MATH  MathSciNet  Google Scholar 

  55. J.J. Niemela, L. Skrbek, K.R. Sreenivasan, and R.J. Donnelly, “Turbulent convection at very high Rayleigh number”, Nature 404, 837 (2000).

    Article  ADS  Google Scholar 

  56. J.W. Deardoff and G.E Willis, “Investigation of turbulent thermal convection between horizontal plates”, J. Fluid Mech. 28, 675 (1967).

    Article  ADS  Google Scholar 

  57. E. Guyon and P. Pieranski, “Convective instabilities in nematic liquid crystals”, Physica 73, 184 (1974).

    Article  ADS  Google Scholar 

  58. I.H. Herron, “Stability criteria for flow along a convex wall”, Phys. Fluids A 3, 1825 (1991).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  59. D.J. Tritton and P.A. Davies, in Hydrodynamic Instabilities and the Transition to Turbulence, H.L. Swinney and J.P. Gollub (eds.), Springer, New York (1981).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer Science+Business Media, Inc.

About this chapter

Cite this chapter

Prigent, A., Dubrulle, B., Dauchot, O., Mutabazi, I. (2006). The Taylor-Couette Flow: The Hydrodynamic Twin of Rayleigh-Bénard Convection. In: Mutabazi, I., Wesfreid, J.E., Guyon, E. (eds) Dynamics of Spatio-Temporal Cellular Structures. Springer Tracts in Modern Physics, vol 207. Springer, New York, NY. https://doi.org/10.1007/978-0-387-25111-0_13

Download citation

Publish with us

Policies and ethics