Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T15:27:35.217Z Has data issue: false hasContentIssue false
  • English
  • Français

Spatial and temporal characteristics of modulated waves in the circular Couette system

Published online by Cambridge University Press:  20 April 2006

M. Gorman  and
Harry L. Swinney
M. Gorman
Affiliation:
Department of Physics, The University of Texas, Austin, Texas 78712 Present address: Department of Physics, University of Houston, Houston, TX 77004.
Harry L. Swinney
Affiliation:
Department of Physics, The University of Texas, Austin, Texas 78712
Get access Rights & Permissions [Opens in a new window]

Abstract

We have used flow-visualization and spectral techniques to study the spatial and temporal properties of the flow that precedes the onset of weak turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating (the circular Couette system). The first three flow regimes encountered as the Reynolds number is increased from zero are well-known – Couette flow, Taylor-vortex flow, and wavyvortex flow. The present study concerns the doubly periodic regime that follows the (singly periodic) wavy-vortex-flow regime. Wavy-vortex flow is characterized by a single frequency f1, which is the frequency of travelling azimuthal waves passing a point of observation in the laboratory. The doubly periodic regime was discovered in studies of power spectra several years ago, but the fluid motion corresponding to the second frequency f2 was not identified. We have found that f2 corresponds to a modulation of the azimuthal waves; the modulation can be observed visually as a periodic flattening of the wavy-vortex outflow boundaries. Moreover, in addition to the previously observed doubly periodic flow state, we have discovered 11 more doubly periodic flow states. Each state can be labelled with two integers m and k, which are simply related to physical characteristics of the flow: m is the number of azimuthal waves, and k is related to the phase angle between the modulation of successive azimuthal waves by Δϕ = 2πk/m. This expression for the phase angle was first conjectured from the flow-visualization measurements and then tested to an accuracy of 0·01π in spectral measurements. Recently Rand (1981) has used dynamical-systems concepts and symmetry considerations to derive predictions about the space–time symmetry of doubly periodic flows in circularly symmetric systems. He predicted that only flows with certain space–time symmetries should be allowed. The observed flow states are in agreement with this theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 117 , April 1982 , pp. 123 - 142
Copyright
© 1982 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

Belyaev, Yu. N., Monakov, A. A., Sherbakov, S. A. & Yavorskaya, I. M. 1979 JETP Lett. 29, 329.
Benjamin, T. B. 1978 Proc. R. Soc. Lond. A 359, 27.
Bouabdallah, A. & Cognet, G. 1980 Laminar — turbulent transition in Taylor — Couette flow. In Laminar — Turbulent Transition (ed. R. Eppler & H. Fasel), p. 368. Springer.
Coles, D. 1965 J. Fluid Mech. 21, 385.
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.
Donnelly, R. J., Park, K., Shaw, R. & Walden, R. W. 1980 Phys. Rev. Lett. 44, 987.
Fenstermacher, P. R., Swinney, H. L. & Gollub, J. P. 1979 J. Fluid Mech. 94, 103.
Fultz, D., Long, R. R., Owens, G. V., Bohan, W., Kaylor, R. & Weil, J. 1959 Met. Monographs 4, 1.
Gollub, J. P. & Swinney, H. L. 1975 Phys. Rev. Lett. 35, 927.
Gorman, M., Reith, L. & Swinney, H. L. 1980 Ann. N.Y. Acad. Sci. 357, 10.
Gorman, M. & Swinney, H. L. 1979 Phys. Rev. Lett. 43, 1871.
Gorman, M., Swinney, H. L. & Rand, D. 1981 Phys. Rev. Lett. 46, 992.
Gregg, W. D. 1977 Analog and Digital Communication. Wiley.
Hide, R. 1953 Quart. J. R. Met. Soc. 79, 161.
Hide, R. 1958 Phil. Trans. R. Soc. Lond. A 250, 441.
Hide, R. & Mason, P. J. 1975 Adv. Phys. 24, 47.
Kuznetsov, E. A., Lvov, V. S., Predtechenskii, V. S., Sobolev, V. S. & Utkin, E. N. 1980 Sov. Phys. JETP Lett. 30, 207.
Lanford, O. E. 1981 Strange attractors and turbulence. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub), p. 7. Springer.
Lorenz, E. N. 1963a J. Atmos. Sci. 20, 130.
Lorenz, E. N. 1963b J. Atmos. Sci. 20, 448.
Mobbs, F. R., Preston, S. & Ozogan, M. S. 1979 An experimental investigation of Taylor vortex waves. In Taylor Vortex Flow Working Party, Leeds, p. 53.
Munson, B. R. & Menguturk, M. 1975 J. Fluid Mech. 69, 705.
Pfeffer, R. L., Buzyna, G. & Kung, R. 1980 J. Atmos. Sci. 37, 2129.
Pfeffer, R. L. & Chiang, Y. 1967 Mon. Weather Rev. 95, 75.
Pfeffer, R. L. & Fowlis, W. W. 1968 J. Atmos. Sci. 25, 361.
Pfister, G. & Gerdts, U. 1981 Phys. Lett. 83A, 23.
Rand, D. 1981 Arch. Rat. Mech. Anal. (to appear).
Ruelle, D. 1973 Arch. Rat. Mech. Anal. 51, 136.
Ruelle, D. & Takens, F. 1971 Commun. Math. Phys. 20, 167.
Sawatzki, O. & Zierep, J. 1970 Acta Mech. 9, 13.
Snyder, H. A. 1970 Int. J. Nonlinear Mech. 5, 659.
Swift, J., Gorman, M. & Swinney, H. L. 1981 Phys. Lett. (to appear).
Taylor, G. I. 1923 Phil. Trans. R. Soc. Lond. A 223, 289.
Walden, R. & Donnelly, R. J. 1979 Phys. Rev. Lett. 42, 301.
White, D. & Koschmieder, E. L. 1981 Geophys. Astrophys. Fluid Dyn. 18, 301.
Wimmer, M. 1976 J. Fluid Mech. 78, 317.
Wimmer, M. 1981 J. Fluid Mech. 103, 117.
Yahata, H. 1978 Prog. Theor. Phys. Suppl. 64, 176.
Yahata, H. 1979 Prog. Theor. Phys. 61, 791.
Yahata, H. 1980 Prog. Theor. Phys. 64, 782.
Yavorskaya, I. M., Belyaev, Yu. N., Monakhvo, A. A., Astaf'eva, N. M., Scherbakov, S. A. & Vredenskaya, N. D. 1981 (to be published).