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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
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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

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