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Numerical study of the oscillations of axially excited liquid annuli with rotational symmetry enclosed in revolving circular cylindrical containers

Published online by Cambridge University Press:  26 April 2006

M. Ehmann  and
J. Siekmann
M. Ehmann
Affiliation:
Chair of Mechanics, University GH Essen, 45 117 Essen, Germany
J. Siekmann
Affiliation:
Chair of Mechanics, University GH Essen, 45 117 Essen, Germany
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Abstract

In this paper we investigate numerically the periodic, axisymmetric response of a liquid annulus enclosed in a revolving cylinder and subject to a periodic axial excitation within the range of the natural frequencies. We use a description which employs as solution variables the transverse component of the vorticity vector, the scalar stream function and the transverse velocity component, as well as the position of the free surface. The acceleration due to gravity is neglected, whereas friction and surface tension are taken into account. At the fixed walls the no-slip condition is fulfilled except at points on the contact line. At the contact line the slip condition is applied. The solution of this problem is achieved using the spectral method in the time direction and finite differences in the space direction, whereby surface-adapted coordinates are utilized. Far away from the walls the computational results show good agreement with results obtained from a linearized theory assuming an inviscid liquid, while at the walls boundary layers are generated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 297 , 25 August 1995 , pp. 215 - 230
Copyright
© 1995 Cambridge University Press

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References

Anderson, D. A., Tannehill, J. C. & Pletcher, R. H. 1984 Computational Fluid Mechanics and Heat Transfer. Hemisphere.
Balasubramanian, R. 1990 Numerical simulation of unsteady viscous free surface flow. J. Comput. Phys. 90, 396–430.Google Scholar
Bauer, H. F. 1980 Schwingungen nichtmischbarer Flüssigkeiten im rotierenden Kreiszylinder. Z. Angew. Math. Mech. 60, 653–661.Google Scholar
Bauer, H. F. 1981 Freie Schwingungen nichtmischbarer Flüssigkeiten im rotierenden Kreiszylinder unter Berücksichtigung der Oberflächenspannung. Forsch. Ing.-Engng Res. 47 6, 190–198.Google Scholar
Bauer, H. F. 1982a Rotating finite liquid systems under zero-gravity. Forsch. Ing.-Engng Res. 48 6, 169–200.Google Scholar
Bauer, H. F. 1982b Coupled oscillations of a solidly rotating liquid bridge. Acta Astronautica 9, 547–563.Google Scholar
Bauer, H. F. 1984 Natural damped frequencies of an infinitely long column of immiscible viscous liquids. Z. Angew. Math. Mech. 64, 475–490.Google Scholar
Becker, E. & Büurger, W. 1975 Kontinuumsmechanik. Teubner.
Canuto, C., Hussanini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.
Carruthers, R. J. & Grasso, M. 1972 Studies of floating liquid zones in simulated zero gravity. J. Appl. Phys. 43, 436–445.Google Scholar
Chun, Ch.-H., Ehmann, M., Siekmann, J. & Wozniak, G. 1987 Vibrations of rotating menisci. In Proc. 6th European Symp. on Materials Sciences under Microgravity Conditions, Bordeaux, France, February 1987, ESA-256, pp. 226–234.
Daly, B. J. 1969 Technique for including surface-effects in hydrodynamic calculations. J. Comput. Phys. 4, 97–117.Google Scholar
Ehmann, M. 1991 Numerische Berechnung der Schwingungen axial angeregter rotations-symmetrischer Flüssigkeitsannuli in rotierenden zylindrischen Behältern. Dissertation, Universität GH Essen.
Fletcher, C. A. J. 1984 Computational Galerkin Methods. Springer.
Gillis, J. 1961 Stability of a column of rotating viscous liquid. Proc. Camb. Phil. Soc. 57, 152–159.Google Scholar
Greenspan, H. P. 1969 The Theory of Rotating Fluids. Cambridge University Press.
Gresho, P.M. & Lee, R. L. 1981 Don't suppress the wiggles - they're telling you something!. Computers Fluids 9, 223–253.Google Scholar
Harlow, F. H. & Welch, J. E. 1965 Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182–2189.Google Scholar
Hearn, A. C. 1983 REDUCE User's Manual, Version 3.0. Rand Publication CP78.
Hirt, C. W., Amsden, A. A. & Cook, J. L. 1974 An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14, 227–253.Google Scholar
Hirt, C. W. & Nichols, B. D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225.Google Scholar
Hocking, L. M. 1960 The stability of a rigidly rotating column of liquid. Mathematika 1 7, 1–9.Google Scholar
Hocking, L. M. & Michael, D. H. 1959 The stability of a column of a rotating liquid. Mathematika 25 6, 25–32.Google Scholar
Hulzen, Van J. A. & Calmet, J. 1982 Computer algebra systems. In Computer Algebra Symbolic and Algebraic Computation (ed. B. Buchberger, G. E. Collins & R. Loos, in cooperation with R. Albrecht). Springer.
Kollmann, F. G. 1962 Freie Schwingungen eines rotierenden Flüssigkeitsringes. Ing. Arch. XXXI, 250–257.Google Scholar
Landau, L. D. & Lifschitz, E. M. 1987 Fluid Mechanics, 2nd end. Pergamon.
Mason, G. 1970 An experimental determination of the stable length of cylindrical liquid bubbles. J. Colloid Interface Sci. 32, 172–176.Google Scholar
Miles, J. W. 1959 Free surface oscillations in a rotating liquid. Phys. Fluids 2, 297–305.Google Scholar
Miles, J. W. & Troesch, B. A. 1961 Surface oscillations of a rotating liquid. J. Appl. Mech. 83, 491–496.Google Scholar
Peyret, R. & Taylor, Th. D. 1983 Computational Methods for Fluid Flow. Springer.
Plateau, J. A. F. 1873 Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires. Gauthier-Villars.
Raake, D. 1991 Untersuchung axial erregter Eigenschwingungen nichtmischbarer Fluide in einer schnell rotierenden zylindrischen Küvette. Dissertation, Universität GH Essen.
Rayleigh, Lord 1892 On the instability of cylindrical fluid surfaces. Phil. Mag. (5) 34, 177–180.Google Scholar
Rayna, G. 1987 REDUCE Software for Algebraic Computation. Springer.
Ryskin, G. & Leal, L. G. 1984 Numerical solution of free-boundary problems in fluid mechanics. Part 1. The finite-difference technique. J. Fluid Mech. 148, 1–17.Google Scholar
Seebold, J. G. & Reynolds, W. C. 1965 Configuration and stability of a rotating axisymmetric meniscus at low g. Tech. Rep. LG-4. Thermosciences Division, Department of Mechanical Engineering, Stanford University.
Veldman, A. E. P. & Vogels, M. E. S. 1984 Axisymmetric liquid sloshing under low-gravity conditions. Acta Astronautica 11, 641–649.Google Scholar