Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T07:14:59.040Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave speeds in wavy Taylor-vortex flow

Published online by Cambridge University Press:  20 April 2006

Gregory P. King ,
Y. Li, W. Lee ,
Harry L. Swinney  and
Philip S. Marcus
Gregory P. King
Affiliation:
Department of Physics, University of Texas, Austin TX 78712 Present address: Royal Signals and Radar Establishment, Great Malvern, WR 14 3PS, England.
Y. Li, W. Lee
Affiliation:
Department of Physics, University of Texas, Austin TX 78712
Harry L. Swinney
Affiliation:
Department of Physics, University of Texas, Austin TX 78712
Philip S. Marcus
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Present address: Division of Applied Sciences, Aiken Hall, Harvard University, Cambridge, MA 02138.
Get access Rights & Permissions [Opens in a new window]

Abstract

The speed of travelling azimuthal waves on Taylor vortices in a circular Couette system (with the inner cylinder rotating and the outer cylinder at rest) has been determined in laboratory experiments conducted as a function of Reynolds number R, radius ratio of the cylinders η, average axial wavelength $\overline{\lambda}$, number of waves m1 and the aspect ratio Γ (the ratio of the fluid height to the gap between the cylinders). Wave speeds have also been determined numerically for axially periodic flows in infinite-length cylinders by solving the Navier-Stokes equation with a pseudospectral technique where each Taylor-vortex pair is represented with 32 axial modes, 32 azimuthal modes (in an azimuthal angle of 2π/m1) and 33 radial modes. Above the onset of wavy-vortex flow the wave speed for a given η decreases with increasing R until it reaches a plateau that persists for some range in R. In the radius-ratio range examined in our experiments we find that the wave speed in the plateau region increases monotonically from 0.14Ω at η = 0.630 to 0.45Ω at η = 0.950 (where the wave speed is expressed in terms of the rotation frequency Ω of the inner cylinder). There is a much weaker dependence of the wave speed on $\overline{\lambda}$, m1 and Γ. For three sets of parameter values (R, $\overline{\lambda}$, η and m1) the wave speeds have been measured, extrapolated to infinite aspect ratio, and compared with the numerically computed values. For each of these three cases the agreement is within 0.1 %.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 141 , April 1984 , pp. 365 - 390
Copyright
© 1984 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

Abernathy, F. H., Bertschy, J. R., Chin, R. W. & Keyes, D. E. 1980 Polymer induced fluctuations in high strain-rate laminar flows. J. Rheol. 24, 647.Google Scholar
Ahlers, G., Cannell, D. S. & Dominguez Lerma, M. A. 1983 Possible mechanism for transitions in wavy Taylor-vortex flow. Phys. Rev. A27, 1225.Google Scholar
Andereck, C. D., Dickman, R. & Swinney, H. L. 1983 New flows in a circular Couette system with co-rotating cylinders. Phys. Fluids 26, 1395.Google Scholar
Andereck, C. D., Liu, S. S. & Swinney, H. L. 1984 Flow regimes in a circular Couette system with independently rotating cylinders. Submitted to J. Fluid Mech.Google Scholar
Barcilon, A., Brindley, J., Lessen, M. & Mobbs, F. R. 1979 Marginal instability in Taylor-Couette flows at a very high Taylor number. J. Fluid Mech. 94, 453.Google Scholar
Batchelor, G. K. 1960 Appendix to Donnelly, R. J. & Simon, N. J. ‘An empirical torque relation for supercritical flow between rotating cylinders’. J. Fluid Mech. 7, 401.Google Scholar
Benjamin, T. B. 1978 Bifurcation phenomena in steady flows of a viscous fluid. II. Experiments. Proc. R. Soc. Lond. A 359, 27.Google Scholar
Berman, N. S. 1978 Drag reduction by polymers. Ann. Rev. Fluid Mech. 10, 47.Google Scholar
Brandstäter, A., Pfister, G. & Schulz-DuBois, E. O. 1982 Excitation of a Taylor vortex mode having resonant frequency dependence of coherence length. Phys. Lett. 88A, 407.Google Scholar
Burkhalter, J. E. & Koschmieder, E. L. 1973 Steady supercritical Taylor vortex flow. J. Fluid Mech. 58, 547.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Cole, J. A. 1976 Taylor-vortex instability and annulus-length effects. J. Fluid Mech. 75, 1.Google Scholar
Cole, J. A. 1981 Wavy vortex onset and cylinder radius ratio. Taylor Vortex Flow Working Party. Second meeting, Tufts University, Medford, Mass., June 1981 (unpublished).
Cole, J. A. 1983 The effect of cylinder radius ratio on wavy vortex onset. Taylor Vortex Flow Working Party, Third meeting, Nancy, France, 1983 (unpublished).
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385.Google Scholar
Davey, A., Di Pima, R. C. & Stuart, J. T. 1968 On the instability of Taylor vortices. J. Fluid Mech. 31, 17.Google Scholar
Di Prima, 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 Early nonperiodic transitions in Couette flow. Phys. Rev. Lett. 44, 987.Google Scholar
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
Feynman, R. P., Leighton, R. B. & Sands, M. 1964 The Feynman Lectures on Physics, vol. II, p. 41–11. Addison-Wesley.
Gorman, M., Reith, L. A. & Swinney, H. L. 1980 Modulation patterns, multiple frequencies, and other phenomena in circular Couette flow. Ann. NY Acad. Sci. 357, 10.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
Gorman, M., Swinney, H. L. & Rand, D. A. 1981 Doubly periodic circular Couette flow: experiments compared with predictions from dynamics and symmetry. Phys. Rev. Lett. 46, 992.Google Scholar
Jeffery, G. B. 1923 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161.Google Scholar
Jones, C. A. 1981 Nonlinear Taylor vortices and their stability. J. Fluid Mech. 102, 249.Google Scholar
King, G. P. & Swinney, H. L. 1983 Limits of stability and irregular flow patterns in wavy vortex flow. Phys. Rev. A27, 1240.Google Scholar
Lorenzen, A., Pfister, G. & Mullin, T. 1983 End effects on the transition to time-dependent motion in the Taylor experiment. Phys. Fluids 26, 1.Google Scholar
L'vov, V. S., Predtechensky, A. A. & Chernykh, A. I. 1981 Bifurcation and chaos in a system of Taylor vortices: a natural and numerical experiment. Sov. Phys. JETP 53. 562.Google Scholar
Marcus, P. S. 1981 Effects of truncation in modal representations of thermal convection. J. Fluid Mech. 103, 241.Google Scholar
Marcus, P. S. 1984a Simulation of Taylor-Couette flow - numerical methods and comparison with experiment. Submitted to J. Fluid Mech.Google Scholar
Marcus, P. S. 1984b Simulation of Taylor-Couette flow - numerical results for wavy vortex flow with one travelling wave. Submitted to J. Fluid Mech.Google Scholar
Mullin, T. & Benjamin, T. B. 1980 Transition to oscillatory motion in the Taylor experiment. Nature 288, 567.Google Scholar
Park, K., Crawford, G. L. & Donnelly, R. J. 1983 Characteristic lengths in the wavy vortex state of Taylor-Couette flow. Phys. Rev. Lett. 51, 1352.Google Scholar
Robinson, J. L. 1967 Finite amplitude convection cells. J. Fluid Mech. 30, 577.Google Scholar
Shaw, R. S., Andereck, C. D., Reith, L. A. & Swinney, H. L. 1982 Superposition of travelling waves in the circular Couette system. Phys. Rev. Lett. 48, 1172.Google Scholar
Smith, G. P. & Townsend, A. A. 1982 Turbulent Couette flow between concentric cylinders at large Taylor numbers. J. Fluid Mech. 123, 187.Google Scholar
Snyder, H. A. 1969 Change in wave-form and mean flow associated with wavelength variations in rotating Couette flow. Part 1. J. Fluid Mech. 35, 337.Google Scholar
Snyder, H. A. 1970 Waveforms in rotating Couette flow. Intl J. Non-Linear Mech. 5, 659.Google Scholar
Zakin, J. L., Ni, C. C., Hansen, R. J. & Reischman, M. N. 1977 Laser Doppler velocimetry studies of early turbulence. Phys. Fluids Suppl. 20, S85.Google Scholar
Zhang, L. H. & Swinney, H. L. 1984 Wavy modes in turbulent Taylor vortex flow. Submitted to J. Fluid Mech.Google Scholar