1932

Abstract

Taylor-Couette flow, the flow between two coaxial co- or counter-rotating cylinders, is one of the paradigmatic systems in the physics of fluids. The (dimensionless) control parameters are the Reynolds numbers of the inner and outer cylinders, the ratio of the cylinder radii, and the aspect ratio. One key response of the system is the torque required to retain constant angular velocities, which can be connected to the angular velocity transport through the gap. Whereas the low–Reynolds number regime was well explored in the 1980s and 1990s of the past century, in the fully turbulent regime major research activity developed only in the past decade. In this article, we review this recent progress in our understanding of fully developed Taylor-Couette turbulence from the experimental, numerical, and theoretical points of view. We focus on the parameter dependence of the global torque and on the local flow organization, including velocity profiles and boundary layers. Next, we discuss transitions between different (turbulent) flow states. We also elaborate on the relevance of this system for astrophysical disks (quasi-Keplerian flows). The review ends with a list of challenges for future research on turbulent Taylor-Couette flow.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-fluid-122414-034353
2016-01-03
2024-06-12
Loading full text...

Full text loading...

/deliver/fulltext/fluid/48/1/annurev-fluid-122414-034353.html?itemId=/content/journals/10.1146/annurev-fluid-122414-034353&mimeType=html&fmt=ahah

Literature Cited

  1. Ahlers G. 1974. Low temperature studies of the Rayleigh-Bénard instability and turbulence. Phys. Rev. Lett. 33:1185–88 [Google Scholar]
  2. Ahlers G, Bodenschatz E, He X. 2014. Logarithmic temperature profiles of turbulent Rayleigh-Bénard convection in the classical and ultimate state for a Prandtl number of 0.8. J. Fluid Mech. 758:436–67 [Google Scholar]
  3. Ahlers G, Grossmann S, Lohse D. 2009. Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev. Mod. Phys. 81:503–37 [Google Scholar]
  4. Andereck CD, Liu SS, Swinney HL. 1986. Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164:155–83 [Google Scholar]
  5. Avila M. 2012. Stability and angular-momentum transport of fluid flows between co-rotating cylinders. Phys. Rev. Lett. 108:124501 [Google Scholar]
  6. Behringer RP. 1985. Rayleigh-Bénard convection and turbulence in liquid helium. Rev. Mod. Phys. 57:657–87 [Google Scholar]
  7. Bilson M, Bremhorst K. 2007. Direct numerical simulation of turbulent Taylor-Couette flow. J. Fluid Mech. 579:227–70 [Google Scholar]
  8. Bodenschatz E, Pesch W, Ahlers G. 2000. Recent developments in Rayleigh-Bénard convection. Annu. Rev. Fluid Mech. 32:709–78 [Google Scholar]
  9. Borrero-Echeverry D. 2014. Subcritical transition to turbulence in Taylor-Couette flow PhD Thesis, Georgia Inst. Technol., Atlanta [Google Scholar]
  10. Borrero-Echeverry D, Schatz MF, Tagg R. 2010. Transient turbulence in Taylor-Couette flow. Phys. Rev. E 81:025301 [Google Scholar]
  11. Brauckmann HJ, Eckhardt B. 2013a. Direct numerical simulations of local and global torque in Taylor-Couette flow up to Re = 30 000. J. Fluid Mech. 718:398–427 [Google Scholar]
  12. Brauckmann HJ, Eckhardt B. 2013b. Intermittent boundary layers and torque maxima in Taylor-Couette flow. Phys. Rev. E 87:033004 [Google Scholar]
  13. Büchel P, Lücke M, Roth D, Schmitz R. 1996. Pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem: Taylor vortices in axial flow. Phys. Rev. E 53:4764–77 [Google Scholar]
  14. Burin MJ, Czarnocki CJ. 2012. Subcritical transition and spiral turbulence in circular Couette flow. J. Fluid Mech. 709:106–22 [Google Scholar]
  15. Busse FH. 1967. The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30:625–49 [Google Scholar]
  16. Cadot O, Couder Y, Daerr A, Douady S, Tsinober A. 1997. Energy injection in closed turbulent flows: stirring through boundary layers versus inertial stirring. Phys. Rev. E 56:427–33 [Google Scholar]
  17. Chandrasekhar S. 1981. Hydrodynamic and Hydromagnetic Stability New York: Dover [Google Scholar]
  18. Chossat P, Iooss G. 1994. The Couette-Taylor Problem New York: Springer [Google Scholar]
  19. Chouippe A, Climent E, Legendre D, Gabillet C. 2014. Numerical simulation of bubble dispersion in turbulent Taylor-Couette flow. Phys. Fluids 26:043304 [Google Scholar]
  20. Coles D. 1965. Transition in circular Couette flow. J. Fluid Mech. 21:385–425 [Google Scholar]
  21. Coughlin K, Marcus PS. 1996. Turbulent bursts in Couette-Taylor tlow. Phys. Rev. Lett. 77:2214–17 [Google Scholar]
  22. Cross MC, Hohenberg PC. 1993. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65:851–1112 [Google Scholar]
  23. DiPrima RC, Swinney HL. 1981. Instabilities and transition in flow between concentric rotating cylinders. Hydrodynamic Instabilities and the Transition to Turbulence HL Swinney, JP Gollub 139–80 New York: Springer [Google Scholar]
  24. Doering C, Constantin P. 1994. Variational bounds on energy dissipation in incompressible flow: shear flow. Phys. Rev. E 49:4087–99 [Google Scholar]
  25. Dong S. 2007. Direct numerical simulation of turbulent Taylor-Couette flow. J. Fluid Mech. 587:373–93 [Google Scholar]
  26. Dong S. 2008. Turbulent flow between counter-rotating concentric cylinders: a direct numerical simulation study. J. Fluid Mech. 615:371–99 [Google Scholar]
  27. Donnelly RJ. 1991. Taylor-Couette flow: the early days. Phys. Today 44:1132–39 [Google Scholar]
  28. Drazin P, Reid WH. 1981. Hydrodynamic Stability Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  29. Dubrulle B, Dauchot O, Daviaud F, Longgaretti PY, Richard D, Zahn JP. 2005. Stability and turbulent transport in Taylor-Couette flow from analysis of experimental data. Phys. Fluids 17:095103 [Google Scholar]
  30. Dubrulle B, Hersant F. 2002. Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection. Eur. Phys. J. B 26:379–86 [Google Scholar]
  31. Eckhardt B, Grossmann S, Lohse D. 2000. Scaling of global momentum transport in Taylor-Couette and pipe flow. Eur. Phys. J. B 18:541–44 [Google Scholar]
  32. Eckhardt B, Grossmann S, Lohse D. 2007a. Fluxes and energy dissipation in thermal convection and shear flows. Europhys. Lett. 78:24001 [Google Scholar]
  33. Eckhardt B, Grossmann S, Lohse D. 2007b. Torque scaling in turbulent Taylor-Couette flow between independently rotating cylinders. J. Fluid Mech. 581:221–50 [Google Scholar]
  34. Eckhardt B, Schneider T, Hof B, Westerweel J. 2007c. Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39:447–68 [Google Scholar]
  35. Edlund EM, Ji H. 2014. Nonlinear stability of laboratory quasi-Keplerian flows. Phys. Rev. E 89:021004 [Google Scholar]
  36. Esser A, Grossmann S. 1996. Analytic expression for Taylor-Couette stability boundary. Phys. Fluids 8:1814–19 [Google Scholar]
  37. Hersant F, Dubrulle B, Huré J-M. 2005. Turbulence in circumstellar disks. Astron. Astrophys. 429:531–42 [Google Scholar]
  38. Fardin MA, Perge C, Taberlet N. 2014. “The hydrogen atom of fluid dynamics”: introduction to the Taylor-Couette flow for soft matter scientists. Soft Matter 10:3523–35 [Google Scholar]
  39. Fenstermacher PR, Swinney HL, Gollub JP. 1979. Dynamical instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech. 94:103–28 [Google Scholar]
  40. Gebhardt T, Grossmann S. 1993. The Taylor-Couette eigenvalue problem with independently rotating cylinders. Z. Phys. B 90:475–90 [Google Scholar]
  41. Grossmann S. 2000. The onset of shear flow turbulence. Rev. Mod. Phys. 72:603–18 [Google Scholar]
  42. Grossmann S, Lohse D. 2000. Scaling in thermal convection: a unifying view. J. Fluid. Mech. 407:27–56 [Google Scholar]
  43. Grossmann S, Lohse D. 2001. Thermal convection for large Prandtl number. Phys. Rev. Lett. 86:3316–19 [Google Scholar]
  44. Grossmann S, Lohse D. 2002. Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66:016305 [Google Scholar]
  45. Grossmann S, Lohse D. 2004. Fluctuations in turbulent Rayleigh-Bénard convection: the role of plumes. Phys. Fluids 16:4462–72 [Google Scholar]
  46. Grossmann S, Lohse D. 2011. Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23:045108 [Google Scholar]
  47. Grossmann S, Lohse D. 2012. Logarithmic temperature profiles in the ultimate regime of thermal convection. Phys. Fluids 24:125103 [Google Scholar]
  48. Grossmann S, Lohse D, Sun C. 2014. Velocity profiles in strongly turbulent Taylor-Couette flow. Phys. Fluids 26:025114 [Google Scholar]
  49. He W, Tanahashi M, Miyauchi T. 2007. Direct numerical simulation of turbulent Taylor-Couette flow with high Reynolds number. Advances in Turbulence XI: Proceedings of the 11th EUROMECH European Turbulence Conference, June 25–28, 2007, Porto, Portugal JMLM Palma, A Silva Lopes 215–17 New York: Springer [Google Scholar]
  50. He X, Funfschilling D, Bodenschatz E, Ahlers G. 2012a. Heat transport by turbulent Rayleigh-Bénard convection for Pr = 0.8 and : ultimate-state transition for aspect ratio Γ = 1.00. New J. Phys. 14:063030 [Google Scholar]
  51. He X, Funfschilling D, Nobach H, Bodenschatz E, Ahlers G. 2012b. Transition to the ultimate state of turbulent Rayleigh-Bénard convection. Phys. Rev. Lett. 108:024502 [Google Scholar]
  52. Howard LN. 1972. Bounds on flow quantities. Annu. Rev. Fluid. Mech. 4:473–94 [Google Scholar]
  53. Hristova H, Roch S, Schmid P, Tuckerman L. 2002. Transient growth in Taylor-Couette flow. Phys. Fluids 14:3475–84 [Google Scholar]
  54. Huisman SG, Lohse D, Sun C. 2013a. Statistics of turbulent fluctuations in counter-rotating Taylor-Couette flows. Phys. Rev. E 88:063001 [Google Scholar]
  55. Huisman SG, Scharnowski S, Cierpka C, Kähler C, Lohse D, Sun C. 2013b. Logarithmic boundary layers in strong Taylor-Couette turbulence. Phys. Rev. Lett. 110:264501 [Google Scholar]
  56. Huisman SG, van der Veen RCA, Sun C, Lohse D. 2014. Multiple states in highly turbulent Taylor-Couette flow. Nat. Commun. 5:3820 [Google Scholar]
  57. Huisman SG, van Gils DPM, Grossmann S, Sun C, Lohse D. 2012. Ultimate turbulent Taylor-Couette flow. Phys. Rev. Lett. 108:024501 [Google Scholar]
  58. Hultmark M, Vallikivi M, Bailey SCC, Smits AJ. 2012. Turbulent pipe flow at extreme Reynolds numbers. Phys. Rev. Lett. 108:094501 [Google Scholar]
  59. Ji H, Balbus S. 2013. Angular momentum transport in astrophysics and in the lab. Phys. Today 66:827 [Google Scholar]
  60. Ji H, Burin M, Schartman E, Goodman J. 2006. Hydrodynamic turbulence cannot transport angular momentum effectively in astrophysical disks. Nature 444:343–46 [Google Scholar]
  61. Kadanoff LP. 2001. Turbulent heat flow: structures and scaling. Phys. Today 54:34–39 [Google Scholar]
  62. Koschmieder EL. 1993. Bénard Cells and Taylor Vortices Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  63. Kraichnan RH. 1962. Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5:1374–89 [Google Scholar]
  64. Landau LD, Lifshitz EM. 1987. Fluid Mechanics Oxford: Pergamon [Google Scholar]
  65. Lathrop DP, Fineberg J, Swinney HS. 1992a. Transition to shear-driven turbulence in Couette-Taylor flow. Phys. Rev. A 46:6390–405 [Google Scholar]
  66. Lathrop DP, Fineberg J, Swinney HS. 1992b. Turbulent flow between concentric rotating cylinders at large Reynolds numbers. Phys. Rev. Lett. 68:1515–18 [Google Scholar]
  67. Lewis GS, Swinney HL. 1999. Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette-Taylor flow. Phys. Rev. E 59:5457–67 [Google Scholar]
  68. Lohse D, Xia KQ. 2010. Small-scale properties of turbulent Rayleigh-Bénard convection. Annu. Rev. Fluid Mech. 42:335–64 [Google Scholar]
  69. Lorenz EN. 1963. Deterministic nonperiodic flow. J. Atmos. Sci. 20:130–41 [Google Scholar]
  70. Maretzke S, Hof B, Avila M. 2014. Transient growth in linearly stable Taylor-Couette flows. J. Fluid Mech. 742:254–90 [Google Scholar]
  71. Marques F, Lopez J. 1997. Taylor-Couette flow with axial oscillations of the inner cylinder: Floquet analysis of the basic flow. J. Fluid Mech. 348:153–75 [Google Scholar]
  72. Marusic I, McKeon BJ, Monkewitz PA, Nagib HM, Smits AJ, Sreenivasan KR. 2010. Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues. Phys. Fluids 22:065103 [Google Scholar]
  73. Merbold S, Brauckmann HJ, Egbers C. 2013. Torque measurements and numerical determination in differentially rotating wide gap Taylor-Couette flow. Phys. Rev. E 87:023014 [Google Scholar]
  74. Meseguer A. 2002. Energy transient growth in the Taylor-Couette problem. Phys. Fluids 14:1655–60 [Google Scholar]
  75. Mullin T, Cliffe KA, Pfister G. 1987. Unusual time-dependent phenomena in Taylor-Couette flow at moderately low Reynolds numbers. Phys. Rev. Lett. 58:2212–15 [Google Scholar]
  76. Nordsiek F, Huisman SG, van der Veen RCA, Sun C, Lohse D, Lathrop DP. 2015. Azimuthal velocity profiles in Rayleigh-stable Taylor-Couette flow and implied axial angular momentum transport. J. Fluid Mech. 744:342–62 [Google Scholar]
  77. Ostilla-Mónico R, Huisman SG, Jannink TJG, Van Gils DPM, Verzicco R. et al. 2014a. Optimal Taylor-Couette flow: radius ratio dependence. J. Fluid Mech. 747:1–29 [Google Scholar]
  78. Ostilla-Mónico R, Stevens RJAM, Grossmann S, Verzicco R, Lohse D. 2013. Optimal Taylor-Couette flow: direct numerical simulations. J. Fluid Mech. 719:14–46 [Google Scholar]
  79. Ostilla-Mónico R, van der Poel EP, Verzicco R, Grossmann S, Lohse D. 2014b. Boundary layer dynamics at the transition between the classical and the ultimate regime of Taylor-Couette flow. Phys. Fluids 26:015114 [Google Scholar]
  80. Ostilla-Mónico R, van der Poel EP, Verzicco R, Grossmann S, Lohse D. 2014c. Exploring the phase diagram of fully turbulent Taylor-Couette flow. J. Fluid Mech. 761:1–26 [Google Scholar]
  81. Ostilla-Mónico R, Verzicco R, Grossmann S, Lohse D. 2014d. Turbulence decay towards the linearly stable regime of Taylor-Couette flow. J. Fluid Mech. 748:R3 [Google Scholar]
  82. Paoletti MS, Lathrop DP. 2011. Angular momentum transport in turbulent flow between independently rotating cylinders. Phys. Rev. Lett. 106:024501 [Google Scholar]
  83. Paoletti MS, van Gils DPM, Dubrulle B, Sun C, Lohse D, Lathrop DP. 2012. Angular momentum transport and turbulence in laboratory models of Keplerian flows. Astron. Astrophys. 547:A64 [Google Scholar]
  84. Pfister G, Rehberg I. 1981. Space dependent order parameter in circular Couette flow transitions. Phys. Lett. 83:19–22 [Google Scholar]
  85. Pirro D, Quadrio M. 2008. Direct numerical simulation of turbulent Taylor-Couette flow. Eur. J. Mech. B Fluids 27:552–66 [Google Scholar]
  86. Ravelet F, Delfos R, Westerweel J. 2010. Influence of global rotation and Reynolds number on the large-scale features of a turbulent Taylor–Couette flow. Phys. Fluids 22:055103 [Google Scholar]
  87. Rayleigh L. 1917. On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93:148–54 [Google Scholar]
  88. Richard D. 2001. Instabilités hydrodynamiques dans les écoulements en rotation différentielle PhD Thesis, Univ. Paris-Diderot [Google Scholar]
  89. Richard D, Zahn J. 1999. Turbulence in differentially rotating flows: what can be learned from the Couette-Taylor experiment. Astron. Astrophys. 347:734–38 [Google Scholar]
  90. Roche PE, Gauthier G, Kaiser R, Salort J. 2010. On the triggering of the ultimate regime of convection. New J. Phys. 12:085014 [Google Scholar]
  91. Schartman E, Ji H, Burin MJ. 2009. Development of a Couette-Taylor flow device with active minimization of secondary circulation. Rev. Sci. Instrum. 80:024501 [Google Scholar]
  92. Schartman E, Ji H, Burin MJ, Goodman J. 2012. Stability of quasi-Keplerian shear flow in a laboratory experiment. Astron. Astrophys. 543:A94 [Google Scholar]
  93. Siggia ED. 1994. High Rayleigh number convection. Annu. Rev. Fluid Mech. 26:137–68 [Google Scholar]
  94. Smith GP, Townsend AA. 1982. Turbulent Couette flow between concentric cylinders at large Taylor numbers. J. Fluid Mech. 123:187–217 [Google Scholar]
  95. Smits AJ, Marusic I. 2013. Wall-bounded turbulence. Phys. Today 66:25–30 [Google Scholar]
  96. Smits AJ, McKeon BJ, Marusic I. 2010. High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43:353–75 [Google Scholar]
  97. Srinivasan S, Kleingartner JA, Gilbert JB, Cohen RE, Milne AJB, McKinley GH. 2015. Sustainable drag reduction in turbulent Taylor-Couette flows by depositing sprayable superhydrophobic surfaces. Phys. Rev. Lett. 114:014501 [Google Scholar]
  98. Strogatz SH. 1994. Nonlinear Dynamics and Chaos New York: Perseus [Google Scholar]
  99. Tagg R. 1994. The Couette-Taylor problem. Nonlinear Sci. Today 4:31–25 [Google Scholar]
  100. Taylor GI. 1923. Stability of a viscous liquid contained between two rotating cylinders. Philo. Trans. R. Soc. A 223:289–343 [Google Scholar]
  101. Taylor GI. 1936. Fluid friction between rotating cylinders. Proc. R. Soc. Lond. A 157:546–64 [Google Scholar]
  102. Tong P, Goldburg WI, Huang JS, Witten TA. 1990. Anisotropy in turbulent drag reduction. Phys. Rev. Lett. 65:2780–83 [Google Scholar]
  103. Trefethen L, Trefethen A, Reddy S, Driscol T. 1993. Hydrodynamic stability without eigenvalues. Science 261:578–84 [Google Scholar]
  104. Tuckerman LS. 2014. Taylor vortices versus Taylor columns. J. Fluid Mech. 750:1–4 [Google Scholar]
  105. van den Berg TH, Doering C, Lohse D, Lathrop D. 2003. Smooth and rough boundaries in turbulent Taylor-Couette flow. Phys. Rev. E 68:036307 [Google Scholar]
  106. van den Berg TH, Luther S, Lathrop DP, Lohse D. 2005. Drag reduction in bubbly Taylor-Couette turbulence. Phys. Rev. Lett. 94:044501 [Google Scholar]
  107. van den Berg TH, van Gils DPM, Lathrop DP, Lohse D. 2007. Bubbly turbulent drag reduction is a boundary layer effect. Phys. Rev. Lett. 98:084501 [Google Scholar]
  108. van Gils DPM, Bruggert GW, Lathrop DP, Sun C, Lohse D. 2011a. The Twente turbulent Taylor-Couette (T3C) facility: strongly turbulent (multi-phase) flow between independently rotating cylinders. Rev. Sci. Instrum. 82:025105 [Google Scholar]
  109. van Gils DPM, Huisman SG, Bruggert GW, Sun C, Lohse D. 2011b. Torque scaling in turbulent Taylor-Couette flow with co- and counter-rotating cylinders. Phys. Rev. Lett. 106:024502 [Google Scholar]
  110. van Gils DPM, Huisman SG, Grossmann S, Sun C, Lohse D. 2012. Optimal Taylor-Couette turbulence. J. Fluid Mech. 706:118–49 [Google Scholar]
  111. van Gils DPM, Narezo-Guzman D, Sun C, Lohse D. 2013. The importance of bubble deformability for strong drag reduction in bubbly turbulent Taylor-Couette flow. J. Fluid Mech. 722:317–47 [Google Scholar]
  112. van Hout R, Katz J. 2011. Measurements of mean flow and turbulence characteristics in high-Reynolds number counter-rotating Taylor-Couette flow. Phys. Fluids 23:105102 [Google Scholar]
  113. Wendt F. 1933. Turbulente Strömungen zwischen zwei rotierenden Zylindern. Ingenieurs-Archiv 4:577–95 [Google Scholar]
  114. Zagarola MV, Smits AJ. 1998. Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373:33–79 [Google Scholar]
/content/journals/10.1146/annurev-fluid-122414-034353
Loading
/content/journals/10.1146/annurev-fluid-122414-034353
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error