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Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders


Turbulent Taylor–Couette flow with arbitrary rotation frequencies ω1, ω2 of the two coaxial cylinders with radii r1 < r2 is analysed theoretically. The current Jω of the angular velocity ω(x,t) = uϕ(r,ϕ,z,t)/r across the cylinder gap and and the excess energy dissipation rate ϵw due to the turbulent, convective fluctuations (the ‘wind’) are derived and their dependence on the control parameters analysed. The very close correspondence of Taylor–Couette flow with thermal Rayleigh–Bénard convection is elaborated, using these basic quantities and the exact relations among them to calculate the torque as a function of the rotation frequencies and the radius ratio η = r1/r2 or the gap width d = r2r1 between the cylinders. A quantity σ corresponding to the Prandtl number in Rayleigh–Bénard flow can be introduced, . In Taylor–Couette flow it characterizes the geometry, instead of material properties of the liquid as in Rayleigh–Bénard flow. The analogue of the Rayleigh number is the Taylor number, defined as Ta ∝ (ω1 − ω2)2 times a specific geometrical factor. The experimental data show no pure power law, but the exponent α of the torque versus the rotation frequency ω1 depends on the driving frequency ω1. An explanation for the physical origin of the ω1-dependence of the measured local power-law exponents α(ω1) is put forward. Also, the dependence of the torque on the gap width η is discussed and, in particular its strong increase for η → 1.

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van den Berg, T. H., Doering, C., Lohse, D. & Lathrop, D. 2003 Smooth and rough boundaries in turbulent Taylor–Couette flow. Phys. Rev. E 68, 036307.
Bradshaw, P. 1969 The analogy between streamline curvature and buoyancy in turbulent shear flow. J. Fluid Mech. 36, 177191.
Breuer, M., Wessling, S., Schmalzl, J. & Hansen, U. 2004 Effect of inertia in Rayleigh–Bénard convection. Phys. Rev. E 69, 026302.
Chandrasekhar, S. 1953 The instability of a layer of fluid heated from below and subject to Coriolis forces. Proc. R. Soc. Lond. A 217, 306327.
Doering, C. R. & Constantin, P. 1992 Energy-dissipation in shear driven turbulence. Phys. Rev. Lett. 69, 16481651.
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, 379386.
Eckhardt, B., Grossmann, S. & Lohse, D. 2000 Scaling of global momentum transport in Taylor–Couette and pipe flow. Eur. Phys. J. B 18, 541544.
Eckhardt, B., Grossmann, S. & Lohse, D. 2005 Energy and dissipation balances in rotating flows. In Progress in Turbulence (ed. KittelJ. Peinke, A. J. Peinke, A., Barth, S. & Oberlack, M.), pp. 4750. Springer.
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 Fluxes and energy dissipation in thermal convection and shear flows. Europhys. Lett. 78, 24001.
Esser, A. & Grossmann, S. 1996 Analytic expression for Taylor–Couette stability boundary. Phys. Fluids. 8, 18141819.
Funfschilling, D. & Ahlers, G. 2004 Plume motion and large scale circulation in a cylindrical Rayleigh–Bénard cell. Phys. Rev. Lett. 92, 194502
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: A unifying view. J. Fluid. Mech. 407, 2756.
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 33163319.
Grossmann, S. & Lohse, D. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.
Grossmann, S. & Lohse, D. 2004 Fluctuations in turbulent Rayleigh–Bénard convection: The role of plumes. Phys. Fluids. 16, 44624472.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.
Lathrop, D. P. 1992 Turbulent drag and transport in high Reynolds number flow. PhD thesis, University of Texas at Austin.
Lathrop, D. P., Fineberg, J. & Swinney, H. S. 1992 a Turbulent flow between concentric rotating cylinders at large Reynolds numbers. Phys. Rev. Lett. 68, 15151518.
Lathrop, D. P., Fineberg, J. & Swinney, H. S. 1992 b Transition to shear-driven turbulence in Couette-Taylor flow. Phys. Rev. A 46, 63906405.
Lewis, G. S. & Swinney, H. L. 1999 Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette-Taylor flow. Phys. Rev. E 59, 54575467.
Lim, T. T. & Tan, K. S. 2004 A note on power-law scaling in a Taylor–Couette flow. Phys. Fluids 16, 140144, erratum Phys. Fluids 16, 2712.
Marcus, P. S. 1984 Simulation of Taylor–Couette flow. Part 1. Numerical methods and comparison with experiment. J. Fluid Mech. 146, 4564.
Marcus, P. S. 1984 Simulation of Taylor–Couette flow. Part 2. Numerical results for wave-vortex flow with one travelling wave. J. Fluid Mech. 146, 65113.
Nickerson, E. C. 1969 Upper bounds on the torque in cylindrical Couette flow. J. Fluid Mech. 38, 807815.
Pohlhausen, E. 1921 Der Wärmeaustausch zwischen Festkörpern und Flüssigkeiten mit kleiner Reibung und kleiner Wärmeleitung. Z. Angew. Math. Mech. 1, 115121.
Prandtl, L. 1905 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In Verhandlungen des III. Int. Math. Kongr., Heidelberg 1904, pp. 484491. Leipzig: Teubner.
Smith, G. P. & Townsend, A. A. 1982 Turbulent Couette flow between concentric cylinders at large Taylor numbers. J. Fluid Mech. 123, 187217.
Taylor, G. I. 1936 a Fluid friction between rotating cylinders. I. Torque measurements. Proc. R. Soc. Lond. A 157, 546564.
Taylor, G. I. 1936 b Fluid friction between rotating cylinders. II. Distribution of velocity between concentric cylinders when outer one is rotating and inner one is at rest. Proc. R. Soc. Lond. A 157, 565578.
Tong, P., Goldburg, W. I., Huang, J. S. & Witten, T. A. 1990 Anisotropy in turbulent drag reduction. Phys. Rev. Lett. 65, 27802783.
Wendt, F. 1933 Turbulente Strömungen zwischen zwei rotierenden Zylindern. Ing. Arch. 4, 577595.
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Journal of Fluid Mechanics
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  • EISSN: 1469-7645
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