Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T08:46:44.412Z Has data issue: false hasContentIssue false
  • English
  • Français

Modulated and unmodulated travelling azimuthal waves on the toroidal vortices in a spherical Couette system

Published online by Cambridge University Press:  21 April 2006

Koichi Nakabayashi  and
Yoichi Tsuchida
Koichi Nakabayashi
Affiliation:
Department of Mechanical Engineering, Nagoya Institute of Technology, Nagoya 466, Japan
Yoichi Tsuchida
Affiliation:
Department of Mechanical Engineering, Nagoya Institute of Technology, Nagoya 466, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

We have investigated the modulated and unmodulated travelling azimuthal waves appearing on the toroidal Taylor–Görtler (TG) vortices in a fluid contained between two concentric spheres with the inner sphere rotating. For smaller-clearance cases, toroidal TG vortices appear at the equator, just as in the flow between two concentric cylinders with the inner cylinder rotating. When the Reynolds number of the flow increases quasi-statically, spiral TG vortices appear in addition to toroidal TG vortices, and no modulation occurs, even if the Reynolds number further increases quasi-statically. However, when the Reynolds number is increased from zero to a particular value with a specific acceleration of the inner sphere, modulated wavy toroidal TG vortices appear. We found that the necessary condition for occurrence of modulation is the prevention of spiral TG vortices. Using simultaneous flow-visualization and spectral techniques, and measuring the fluctuation of sinks and sources of vortex boundaries, we obtained the frequency f1 of travelling azimuthal waves passing a fixed point in the laboratory and the modulation frequencies f2 and f′2 of these waves, as determined by an observer in the laboratory and an observer fixed in a reference frame that rotates in phase with the travelling azimuthal waves, respectively. The relationship among the characteristic frequencies, f1, f2 and f′2, obtained by modal analysis and the experimental results, is (f′2 + kf1/m)/f2 = − 1, where k and m are a modulation parameter and the wavenumber of travelling azimuthal waves, respectively.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 195 , October 1988 , pp. 495 - 522
Copyright
© 1988 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bartels F. 1982 Taylor vortices between two concentric rotating spheres. J. Fluid Mech. 119, 1.Google Scholar
Bouabdallah, A. & Cognet G. 1980 Laminar-turbulent transition in Taylor-Couette flow. In Laminar-Turbulent Transition (ed. R. Eppler & H. Pasel), p. 368. Springer.
Bühler, K. & Zierep J. 1983 Transition to turbulence in a spherical gap. In Proc. 4th Intl Symp. on Turbulent shear flows. Karlsruhe (ed. L. J. S. Bradbury, F. Durst. B. G. Launder. F. W. Schmidt & J. H. Whitelaw). Springer.
Bühler, K. & Zierep J. 1984 New secondary flow instabilities for high Re-number flow between two rotating spheres. In Laminar-Turbulent Transition (ed. V. V. Kozlov), p. 677. Springer.
Coles D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385.Google Scholar
Dennis, S. C. R. & Quartapelle L. 1984 Finite difference solution to the flow between two rotating spheres. Computers and Fluids 12, 77.Google Scholar
DiPrima, R. C. & Swinney H. L. 1981 Instabilities and transition in flow between concentric rotating cylinders. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub). p. 139. Springer.
Fenstermacher P. R., Swinney, H. L. & Gollub J. P. 1979 Dynamical instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech. 94, 103.Google Scholar
Gorman, M. & Swinney H. L. 1982 Spatial and temporal characteristics of modulated waves in the circular Couette system. J. Fluid Mech. 117, 123.Google Scholar
King G. P., Li Y., Lee, W. & Swinney H. L. 1984 Wave speeds in wavy Taylor-vortex flow. J. Fluid Mech. 141, 365.Google Scholar
Krause E. 1980 Taylor-Görtler vortices in spherical gaps. Comp. Fluid Dyn. 2, 81.Google Scholar
Mobbs F. R., Preston, S. & Ozogan M. S. 1979 An experimental investigation of Taylor vortex waves. Taylor Vortex Flow Working Party, Leeds, p. 53.
Munson, B. R. & Menguturk M. 1975 Viscous incompressible flow between concentric rotating spheres. Part 3. Linear stability and experiments. J. Fluid Mech. 69, 705.Google Scholar
Nakabayashi K. 1978 Frictional moment of flow between two concentric spheres, one of which rotates. Trans. ASME I: J. Fluids Engng 100, 281.Google Scholar
Nakabayashi K. 1983 Transition of Taylor-Görtler vortex flow in spherical Couette flow. J. Fluid Mech. 132, 209.Google Scholar
Nakabayashi, K. & Tsuchida Y. 1988 Spectral study of the laminar-turbulent transition in spherical Couette flow. J. Fluid Mech. 194, 101.Google Scholar
Ohji M., Shionoya, S. & Amagai K. 1986 A note on modulated wavy disturbances to circular Couette flow. J. Phys. Soc. Japan 55, 1032.Google Scholar
Rand D. 1981 Dynamics and symmetry, predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79, 1.Google Scholar
Sawatzki, O. & Zierep J. 1970 Das Stromfeld im Spalt zwischen zwei konzentrischen Kugelflächen, von denen die innere rotiert. Acta Mech. 9, 13.Google Scholar
Schrauf G. 1986 The first instability in spherical Taylor-Couette flow. J. Fluid Mech. 166, 287.Google Scholar
Schrauf, G. & Krause E. 1984 Symmetric and asymmetric Taylor vortices in a spherical gap. In Laminar-Turbulent Transition (ed. V. V. Kozlov), p. 659. Springer.
Swift J., Gorman, M. & Swinney H. L. 1982 Modulated wavy vortex flow in laboratory and rotating reference frames. Phys. Lett. 87 A, 457.Google Scholar
Waked, A. M. & Munson B. R. 1978 Laminar-turbulent flow in a spherical annulus. Trans. ASME I: J. Fluids Engng 100, 281.Google Scholar
Walden, R. W. & Donnelly R. J. 1979 Re-emergent order of chaotic circular Couette flow. Phys. Rev. Lett. 42, 301.Google Scholar
Wimmer M. 1976 Experiments on a viscous fluid flow between concentric rotating spheres. J. Fluid Mech. 78, 317.Google Scholar
Yavorskaya I. M., Belyaev Yu, N., Monakhov, A. A., Astaf'eva, N. M., Scherbakov, S. A. & Vvedenskaya, N. D. 1980 Stability, nonuniqueness and transition to turbulence in the flow between two rotating spheres. Rep. 595. Space Research Institute of the Academy of Science, USSR.Google Scholar