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

Published online by Cambridge University Press:  22 May 2007

BRUNO ECKHARDT ,
SIEGFRIED GROSSMANN  and
DETLEF LOHSE
BRUNO ECKHARDT
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, Renthof 6, D-35032 Marburg, [email protected]; [email protected]
SIEGFRIED GROSSMANN
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, Renthof 6, D-35032 Marburg, [email protected]; [email protected]
DETLEF LOHSE
Affiliation:
Department of Applied Physics, University of Twente, 7500 AE Enschede, The [email protected]
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Abstract

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 = r2 − r1 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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 581 , 25 June 2007 , pp. 221 - 250
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

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.Google ScholarPubMed
Bradshaw, P. 1969 The analogy between streamline curvature and buoyancy in turbulent shear flow. J. Fluid Mech. 36, 177–191.CrossRefGoogle Scholar
Breuer, M., Wessling, S., Schmalzl, J. & Hansen, U. 2004 Effect of inertia in Rayleigh–Bénard convection. Phys. Rev. E 69, 026302.Google ScholarPubMed
Chandrasekhar, S. 1953 The instability of a layer of fluid heated from below and subject to Coriolis forces. Proc. R. Soc. Lond. A 217, 306–327.Google Scholar
Doering, C. R. & Constantin, P. 1992 Energy-dissipation in shear driven turbulence. Phys. Rev. Lett. 69, 1648–1651.CrossRefGoogle ScholarPubMed
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, 379–386.CrossRefGoogle Scholar
Eckhardt, B., Grossmann, S. & Lohse, D. 2000 Scaling of global momentum transport in Taylor–Couette and pipe flow. Eur. Phys. J. B 18, 541–544.CrossRefGoogle Scholar
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. 47–50. Springer.CrossRefGoogle Scholar
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 Fluxes and energy dissipation in thermal convection and shear flows. Europhys. Lett. 78, 24001.CrossRefGoogle Scholar
Esser, A. & Grossmann, S. 1996 Analytic expression for Taylor–Couette stability boundary. Phys. Fluids. 8, 1814–1819.CrossRefGoogle Scholar
Funfschilling, D. & Ahlers, G. 2004 Plume motion and large scale circulation in a cylindrical Rayleigh–Bénard cell. Phys. Rev. Lett. 92, 194502CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: A unifying view. J. Fluid. Mech. 407, 27–56.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 3316–3319.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.Google ScholarPubMed
Grossmann, S. & Lohse, D. 2004 Fluctuations in turbulent Rayleigh–Bénard convection: The role of plumes. Phys. Fluids. 16, 4462–4472.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.Google Scholar
Lathrop, D. P. 1992 Turbulent drag and transport in high Reynolds number flow. PhD thesis, University of Texas at Austin.Google Scholar
Lathrop, D. P., Fineberg, J. & Swinney, H. S. 1992 a Turbulent flow between concentric rotating cylinders at large Reynolds numbers. Phys. Rev. Lett. 68, 1515–1518.CrossRefGoogle Scholar
Lathrop, D. P., Fineberg, J. & Swinney, H. S. 1992 b Transition to shear-driven turbulence in Couette-Taylor flow. Phys. Rev. A 46, 6390–6405.CrossRefGoogle ScholarPubMed
Lewis, G. S. & Swinney, H. L. 1999 Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette-Taylor flow. Phys. Rev. E 59, 5457–5467.Google ScholarPubMed
Lim, T. T. & Tan, K. S. 2004 A note on power-law scaling in a Taylor–Couette flow. Phys. Fluids 16, 140–144, erratum Phys. Fluids 16, 2712.CrossRefGoogle Scholar
Marcus, P. S. 1984 Simulation of Taylor–Couette flow. Part 1. Numerical methods and comparison with experiment. J. Fluid Mech. 146, 45–64.CrossRefGoogle Scholar
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, 65–113.CrossRefGoogle Scholar
Nickerson, E. C. 1969 Upper bounds on the torque in cylindrical Couette flow. J. Fluid Mech. 38, 807–815.CrossRefGoogle Scholar
Pohlhausen, E. 1921 Der Wärmeaustausch zwischen Festkörpern und Flüssigkeiten mit kleiner Reibung und kleiner Wärmeleitung. Z. Angew. Math. Mech. 1, 115–121.CrossRefGoogle Scholar
Prandtl, L. 1905 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In Verhandlungen des III. Int. Math. Kongr., Heidelberg 1904, pp. 484–491. Leipzig: Teubner.Google Scholar
Smith, G. P. & Townsend, A. A. 1982 Turbulent Couette flow between concentric cylinders at large Taylor numbers. J. Fluid Mech. 123, 187–217.CrossRefGoogle Scholar
Taylor, G. I. 1936 a Fluid friction between rotating cylinders. I. Torque measurements. Proc. R. Soc. Lond. A 157, 546–564.Google Scholar
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, 565–578.Google Scholar
Tong, P., Goldburg, W. I., Huang, J. S. & Witten, T. A. 1990 Anisotropy in turbulent drag reduction. Phys. Rev. Lett. 65, 2780–2783.CrossRefGoogle ScholarPubMed
Wendt, F. 1933 Turbulente Strömungen zwischen zwei rotierenden Zylindern. Ing. Arch. 4, 577–595.CrossRefGoogle Scholar