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Centrifugal instabilities during spin-down to rest in finite cylinders. Numerical experiments

Published online by Cambridge University Press:  20 April 2006

G. P. Neitzel
Affiliation:
Department of Mechanical and Energy Systems Engineering, Arizona State University, Tempe, Arizona 85281
Stephen H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60201

Abstract

A cylinder filled with a viscous, incompressible fluid is in an initial state of rigid-body rotation about its axis of symmetry. If the container is brought to rest impulsively, the resulting unsteady spin-down flow may be subject to sidewall instabilities due to an imbalance between centrifugal and pressure gradient forces. These instabilities are examined numerically using a finite-difference simulation to integrate the axisymmetric Navier–Stokes equations for a variety of aspect ratios and Reynolds numbers. The Taylor–Görtler vortex-wavelength spectrum, the torque and the angular momentum histories are calculated. Criteria for the onset time for instability and the spin-down time are given. The effects of the enhanced mixing due to instability on the spin-down characteristics and torque are discussed. The results are compared with experiment.

Type
Research Article
Copyright
© 1981 Cambridge University Press

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References

Briley, W. R. & Walls, H. A. 1971 A numerical study of time-dependent rotating flow in a cylindrical container at low and moderate Reynolds numbers. Proc. 2nd Int. Conf. Numer. Meth. Fluid Dyn. Lecture Notes in Physics, vol. 8, p. 377. Springer.
Davis, S. H. 1971 Finite amplitude instability of time-dependent flows. J. Fluid Mech. 45, 33.Google Scholar
Euteneuer, G.-A. 1969 Störwellenlängen-Messung bei Längswirbeln in laminaren Grenzschichten an konkav gekrümmten Wänden. Acta Mech. 7, 161.Google Scholar
Euteneuer, G.-A. 1970 Einige Ergebnisse experimenteller Untersuchungen an instationâren Görtler-Taylor-Wirbeln. Z. angew. Math. Mech. 50, 177.Google Scholar
Eufeneuer, G.-A. 1972 Die Entwicklung von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkaven Wänden. Acta Mech. 13, 215.Google Scholar
Euteneuer, G.-A., Heynatz, T. J., and Siedenkersting, H. 1968 Der Bodeneinfluss bei Anlauf- und bei Bremsströmungen zäher Flüssigkeiten in rotierenden Zylindergefässen. Z. angew. Math. Mech. 48, 190.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
Innes, G. 1973 An experimental study of the Ekman layer in a contained rotating fluid. preprint.
Kitchens, C. W. 1980 Navier-Stokes solutions for spin-up in a filled cylinder. A.I.A.A.J. 18, 929.Google Scholar
Maxworthy, T. 1971 Turbulence Measurements in Liquids (ed. G. K. Paterson & J. L. Zakin), p. 32. Dept. Chem. Eng., Univ. Missouri-Rolla.
Michaelidis, K. 1977 Reibungsmomente der instationären Bremsströmung am Zylindermantel eines flüssigkeitsgefüllten zylindrischen Gefässes. Acta Mech. 26, 1.Google Scholar
Neitzel, G. P. 1979 Centrifugal instability of decelerating swirl-flow within finite and infinite circular cylinders. Ph.D. dissertation, The Johns Hopkins University.
Neitzel, G. P. & Davis, S. H. 1980 Energy stability theory of decelerating swirl flows Phys. Fluids 23, 432.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Scientific Papers, vol. 6, 447. Cambridge University Press.
Tillmann, W. 1967 Development of turbulence during the build-up of a boundary layer at a concave wall. Phys. Fluids Suppl. 10, 108.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.Google Scholar
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.Google Scholar