Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T09:35:58.876Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical solutions for spin-up from rest in a cylinder

Published online by Cambridge University Press:  20 April 2006

Jae Min Hyun ,
Fred Leslie ,
William W. Fowlis  and
Alex Warn-Varnas
Jae Min Hyun
Affiliation:
Clarkson College of Technology, Department of Mechanical and Industrial Engineering, Potsdam, New York 13676
Fred Leslie
Affiliation:
Space Science Laboratory, NASA Marshall Space Flight Center, Huntsville, Alabama 35812
William W. Fowlis
Affiliation:
Space Science Laboratory, NASA Marshall Space Flight Center, Huntsville, Alabama 35812
Alex Warn-Varnas
Affiliation:
Naval Ocean Research and Development Activity, Bay St Louis, Mississippi 39520
Get access Rights & Permissions [Opens in a new window]

Abstract

Numerical solutions for the impulsively started spin-up from rest of a homogeneous fluid in a cylinder for small Ekman numbers are presented. The basic analytical theory for this spin-up flow is due to Wedemeyer (1964). Wedemeyer's solution shows that the interior flow is divided into two regions by a moving front which propagates radially inward across the cylinder. The fluid ahead of the front remains non-rotating, while the fluid behind the front is being spun up. Experimental observations have shown that Wedemeyer's model captures the essential dynamics of the azimuthal flow, but that it is not a quantitative model. Wedemeyer made several assumptions in formulating an Ekman compatibility condition, and inconsistencies exist between these assumptions and his solution. Later workers attempted to improve the analytical theory, but their work still included the same basic assumptions made by Wedemeyer.

No previous work has provided a comprehensive and accurate set of three-dimensional flow-field data for this spin-up problem. We chose to acquire such data using a numerical model based on the Navier–Stokes equations. This model was first checked against accurate laser-Doppler measurements of the azimuthal flow for spin-up from rest. New flow-field data over a range of Ekman numbers 9·18 × 10−6 [les ] E [les ] 9·18 × 10−4 are presented. Diagnostic studies, which reveal the various contributions to spin-up of the separate inviscid and viscous terms as functions of radius and time, are also presented. The plots of the viscous-diffusion term reveal the moving front, which is identified as a layer of enhanced local viscous activity. Immediately after the impulsive start, viscous diffusion is seen to be the major contributor to spin-up, then the nonlinear radial advection term takes over, and, finally, when spin-up is well progressed, the linear Coriolis force dominates. In the vicinity of the front, the inward radial flow is a maximum, and the vertical velocity is very small. Strong radial gradients of the vertical velocity are observed across the front and behind the front at the edge of the Ekman layer, and the azimuthal flow behind the front shows strong departures from solid-body rotation. These results enable us to fill in details of the flow not accurately given by Wedemeyer's model and its extensions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 127 , February 1983 , pp. 263 - 281
Copyright
© 1983 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

Benton, E. R. 1979 Vorticity dynamics in spin-up from rest Phys. Fluids 22, 1250.Google Scholar
Benton, E. R. & Clark, A. 1974 Spin-up Ann. Rev. Fluid Mech. 6, 257.Google Scholar
Greenspan, H. P. 1968 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
Hyun, J. M., Fowlis, W. W. & WARN-VARNAS, A. 1982 Numerical solutions for the spin-up of a stratified fluid J. Fluid Mech. 117, 71.Google Scholar
Kitchens, C. W. 1979 Navier-Stokes solutions for spin-up from rest in a cylindrical container. ARBRL-TR-02193, Ballistic Research Laboratory, Aberdeen Proving Ground, Maryland.
Kitchens, C. W. 1980 Navier-Stokes solutions for spin-up in a filled cylinder A.I.A.A. J. 18, 929.Google Scholar
Rogers, M. H. & Lance, G. N. 1960 The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk J. Fluid Mech. 7, 617.Google Scholar
Venezian, G. 1969 Spin-up of a contained fluid Topics in Ocean Engng 1, 212.Google Scholar
Venezian, G. 1970 Nonlinear spin-up Topics in Ocean Engng 2, 87.Google Scholar
Watkins, W. B. & Hussey, R. G. 1973 Spin-up from rest: limitations of the Wedemeyer model. Phys. Fluids 16, 1530.
Watkins, W. B. & Hussey, R. G. 1977 Spin-up from rest in a cylinder Phys. Fluids 20, 1596.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
Wedemeyer, E. H. 1964 The unsteady flow within a spinning cylinder J. Fluid Mech. 20, 383.Google Scholar
Weidman, P. D. 1976a On the spin-up and spin-down of a rotating fluid. Part 1. Extending the Wedemeyer model. J. Fluid Mech. 77, 685.
Weidman, P. D. 1976b On the spin-up and spin-down of a rotating fluid. Part 2. Measurements and stability. J. Fluid Mech. 77, 709.