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On the spin-up and spin-down of a rotating fluid. Part 1. Extending the Wedemeyer model

Published online by Cambridge University Press:  11 April 2006

Patrick D. Weidman
Patrick D. Weidman
Affiliation:
Department of Aerospace Engineering, University of Southern California, Los Angeles
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Abstract

The Wedemeyer model describing the spin-up of a fluid in a rotating cylinder is generalized to include the case of spin-down. Attention is focused on spin-up and spin-down at a finite (constant) acceleration for small Ekman numbers En. It is found that when the full nonlinear Ekman suction is included for spin-up from rest, the characteristics near the propagating wave front intersect, thus yielding an O(1) velocity discontinuity for the inviscid model. This anomalous behaviour is limited to only a small range of Rossby numbers and does not appear in any of the spin-down solutions. Transient spin-down velocity profiles in the O(E1/4Ω) shear layer at the cylindrical wall are calculated for the quasi-steady flow which occurs for sufficiently small decelerations. Results for impulsive spin-up and spin-down between infinite parallel plates are presented and compared with the asymptotic solutions given by Greenspan & Weinbaum and Benton. Finally, characteristic spin-up and spin-down times are computed for both the contained cylinder and the infinite-parallel-plates geometry.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 77 , Issue 4 , 22 October 1976 , pp. 685 - 708
Copyright
© 1976 Cambridge University Press

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References

Abramowitz, M. & Stegun, A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Penton, E. R. 1966 On the flow due to a rotating disk, J. Fluid Mech. 4, 781–800.Google Scholar
Benton, E. R. 1968 A composite Ekman layer problem. Tellus, 20, 667–672.Google Scholar
Benton, E. R. 1973 Nonlinear hydrodynamic and hydromagnetic spin-up driven by Ekman-Hartmann boundary layers. J. Fluid Mech. 57, 337–360.Google Scholar
Benton, E. R. & Clark, A. 1974 Ann. Rev. Fluid Mech. 6, 257–280.Google Scholar
Briley, W. R. & Walls, H. A. 1971 A numerical study of time dependent rotating flows in a cylindrical container at low and moderate Reynolds numbers. In Proc. 2nd Int. Conf. Numerical Methods in Fluid Dyn., Berkeley, pp. 377–384. Springer.CrossRefGoogle Scholar
Carrier, G. F. 1971 Swirling flow boundary layers. J. Fluid Mech. 49, 133–144.Google Scholar
Courant, R. 1936 Differential and Integral Calculus, vol. 2. Interscience.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Greenspan, H. P. & Howard, L. N. 1963 On a time dependent motion of a rotating fluid. J. Fluid Mech. 17, 385–404.Google Scholar
Greenspan, H. P. & Weinbaum, 8. 1965 On nonlinear spin-up of a rotating fluid. J. Fluid Mech. 44, 66–85.Google Scholar
Ingersoll, A. P. & Venezian, G. 1968 Nonlinear spin-up of a contained fluid. Dept. Geolog. Sci., Calif. Inst. Tech., Pasadena, Contribution no. 1612.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, 617–631.Google Scholar
Stewartson, K. 1957 On almost rigid rotations. J. Fluid Mech. 3, 17–26.Google Scholar
Venezian, O. 1969 Spin up of a contained fluid. Topics in Ocean Engng, 1, 212–223.Google Scholar
Venezian, G. 1970 Non-linear spin-up. Topics in Ocean Engng, 2, 87–96.Google Scholar
Watkins, W. B. & Hussey, R. G. 1973 Spin-up from rest: limitations of the Wedemeyer model. Phys. Fluids, 16, 1530–1531.Google Scholar
Watkins, W. B. & Hussey, R. G. 1976 Spin-up from rest in a cylinder. Phys. Fluids (in press).Google Scholar
Wedemeyer, E. H. 1964 The unsteady flow within a spinning cylinder. J. Fluid Mech. 20, 383–399.Google Scholar