Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T13:44:14.202Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical solutions for the spin-up of a stratified fluid

Published online by Cambridge University Press:  20 April 2006

Jae Min Hyun ,
William W. Fowlis  and
Alex Warn-Varnas
Jae Min Hyun
Affiliation:
N.A.S.A., Marshall Space Flight Center, Alabama 35812, U.S.A.
William W. Fowlis
Affiliation:
N.A.S.A., Marshall Space Flight Center, Alabama 35812, U.S.A.
Alex Warn-Varnas
Affiliation:
Naval Ocean Research and Development Activity, Bay St Louis, Mississippi 39520, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Numerical solutions for the impulsively started spin-up of a thermally stratified fluid in a cylinder with an insulating side wall are presented. Previous experimental and numerical work on stratified spin-up had not provided a comprehensive and accurate set of flow-field data. Further, comparisons of this work with theory showed, in general, a substantial discrepancy. The theory was scaled using the homogeneous meridional-flow spin-up time scale and thus viscous-diffusion effects were excluded from the interior. It was anticipated that these effects could only be significant on the larger viscous-diffusion time scale. However, the comparisons with theory showed a faster rate of decay for the measurements even over the shorter meridional-flow spin-up time scale. Previous workers had suggested a number of explanations but the cause of the discrepancy was still unresolved. To provide data to extend the previous work, a numerical model was used. The model was first checked against accurate experimental measurements of stratified spin-up made using a laser-Doppler velocimeter. New accurate results which cover ranges of Ekman number (5·92 × 10−4 ≤ E ≤ 7·24 × 10−4), Rossby number (0·019 ≤ ε ≤ 0·220), stratification parameter (0·0 ≤ Sa−1 ≤ 1·03), and Prandtl number (5·68 ≤ σ ≤ 7·10) are presented. These results show the radial and vertical structure of the decaying azimuthal and meridional flows. The inertial–internal gravity oscillations excited by the impulsive spin-up are clearly seen. By making use of conclusions from the previous work and the results presented in this paper, it is established that viscous diffusion in the interior is the cause of the discrepancy with theory. Stratification causes the meridional spin-up flow to be confined closer to the boundary disks. This results in non-uniform spin-up of the interior and hence flow gradients in the interior. These gradients introduce viscous diffusion into the interior sooner than anticipated by the theory. A previous suggestion that the faster decay rate is due to angular momentum being injected into the interior from an oscillation of the meridional corner-jet flow is shown to be untenable.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 117 , April 1982 , pp. 71 - 90
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

Barcilon, A., Lau, J., Piacsek, S. & Warn-Varnas, A. 1975 Numerical experiments on stratified spin-up. Geophys. Fluid Dyn. 7, 29.Google Scholar
Benton, E. R. & Clark, A. 1974 Spin-up. Ann. Rev. Fluid Mech. 6, 257.Google Scholar
Buzyna, G. & Veronis, G. 1971 Spin-up of a stratified fluid. J. Fluid Mech. 50, 579.Google Scholar
Fowlis, W. W. & Martin, P. J. 1975 A rotating laser Doppler velocimeter and some new results on the spin-up experiment. Geophys. Fluid Dyn. 7, 67.Google Scholar
Greenspan, H. P. 1969 The Theory of Rotating Fluids. Cambridge University Press.
Greenspan, H. P. & Howard, L. N. 1963 On a time-dependent motion of a rotating fluid. J. Fluid Mech. 17, 385.Google Scholar
Holton, J. R. 1965 The influence of viscous boundary layers on transient motions in a stratified rotating fluid. J. Atmos. Sci. 22, 402.Google Scholar
Lee, S. M. 1975 An investigation of stratified spin-up using a rotating laser — Doppler velocimeter. M.S. thesis, Florida State University.
Pedlosky, J. 1967 The spin-up of a stratified fluid. J. Fluid Mech. 28, 463.Google Scholar
Sakurai, T. 1969 Spin-down problem of rotating stratified fluid in thermally insulated circular cylinders. J. Fluid Mech. 37, 689.Google Scholar
Saunders, K. D. & Beardsley, R. C. 1975 An experimental study of the spin-up of a thermally stratified rotating fluid. Geophys. Fluid Dyn. 7, 1.Google Scholar
Wachpress, E. L. 1966 Iterative Solutions of Elliptic Systems. Prentice-Hall.
Walin, G. 1969 Some aspects of time-dependent motion of a stratified rotating fluid. J. Fluid Mech. 36, 289.Google Scholar
Warn-Varnas, A., Fowlis, W. W., Piacsek, S. & Lee, S. M. 1978 Numerical solutions and laser — Doppler measurements of spin-up. J. Fluid Mech. 85, 609.Google Scholar