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Analytically approximate natural sloshing modes for a spherical tank shape

Published online by Cambridge University Press:  12 June 2012

Odd M. Faltinsen  and
Alexander N. Timokha
Odd M. Faltinsen*
Affiliation:
Centre for Ships and Ocean Structures & Department of Marine Technology, Norwegian University of Science and Technology, NO-7091, Trondheim, Norway
Alexander N. Timokha
Affiliation:
Centre for Ships and Ocean Structures & Department of Marine Technology, Norwegian University of Science and Technology, NO-7091, Trondheim, Norway
*
†Email address for correspondence: [email protected]
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Abstract

The multimodal method requires analytical (exact or approximate) natural sloshing modes that exactly satisfy the Laplace equation and boundary condition on the wetted tank surface. When dealing with the nonlinear sloshing problem, the modes should also allow for an analytical continuation throughout the mean free surface. Appropriate analytically approximate modes were constructed by Faltinsen & Timokha (J. Fluid Mech., vol. 695, 2012, pp. 467–477) for the two-dimensional circular tank. The present paper extends this result to the three-dimensional, spherical tank shape and, based on that, establishes specific properties of the linear liquid sloshing.

JFM classification

Mathematical Foundations: Computational methods Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 703 , 25 July 2012 , pp. 391 - 401
Copyright
Copyright © Cambridge University Press 2012

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References

1. Barnyak, M., Gavrilyuk, I., Hermann, M. & Timokha, A. 2011 Analytical velocity potentials in cells with a rigid spherical wall. Z. Angew. Math. Mech. 91 (1), 38–45.CrossRefGoogle Scholar
2. Budiansky, B. 1958 Sloshing of liquids in circular canals and spherical tanks. Tech. Rep. Lockheed Aircraft Corporation, Missile System Division, Sunnyvale, California.Google Scholar
3. Budiansky, B. 1960 Sloshing of liquid in circular canals and spherical tanks. J. Aerosp. Sci. 27 (3), 161–172.CrossRefGoogle Scholar
4. Curadelli, O., Ambrosini, D., Mirasso, A. & Amani, M. 2010 Resonant frequencies in an elevated spherical container partially filled with water: FEM and measurement. J. Fluids Struct. 26 (1), 148–159.CrossRefGoogle Scholar
5. Dassios, G. & Fokas, A. S. 2008 Methods for solving elliptic PDSs in spherical coordinates. SIAM J. Appl. Math. 68, 1080–1096.CrossRefGoogle Scholar
6. Drodos, G. C., Dimas, A. A. & Karabalis, D. L. 2008 Discrete modes for seismic analysis of liquid storage tanks of arbitrary shape and height. J. Pressure Vessel Technology 130, 0418011.Google Scholar
7. Faltinsen, O. M. & Timokha, A. N. 2009 Sloshing. Cambridge University Press.Google Scholar
8. Faltinsen, O. M. & Timokha, A. N. 2010 A multimodal method for liquid sloshing in a two-dimensional circular tank. J. Fluid Mech. 665, 457–479.CrossRefGoogle Scholar
9. Faltinsen, O. M. & Timokha, A. N. 2012 On sloshing modes in a circular tank. J. Fluid Mech. 695, 467–477.CrossRefGoogle Scholar
10. Karamanos, S. A., Papaprokopiou, D. & Platyrrachos, M. A. 2006 Sloshing effects on the seismic design of horizontal–cylindrical and spherical industrial vessels. J. Pressure Vessel Technology 128, 328–340.CrossRefGoogle Scholar
11. Karamanos, S. A., Papaprokopiou, D. & Platyrrachos, M. A. 2009 Finite element analysis of externally-induced sloshing in horizontal–cylindrical and axisymmetric liquid vessels. J. Pressure Vessel Technology 131 (5), Paper No. 051301.CrossRefGoogle Scholar
12. Komarenko, A. 1980 Asymptotic expansion of eigenfunctions of a problem with a parameter in the boundary conditions in a neighborhood of angular boundary points. Ukrainian Math. J. 32 (5), 433–437.CrossRefGoogle Scholar
13. Komarenko, A. 1998 Asymptotics of solutions of spectral problems of hydrodynamics in the neighborhood of angular points. Ukrainian Math. J. 50 (6), 912–921.CrossRefGoogle Scholar
14. Kondratiev, V. 1967 Boundary problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16, 227–313.Google Scholar
15. Kozlov, V. & Kuznetsov, N. 2004 The ice-fishing problem: the fundamental sloshing frequency versus geometry of holes. Math. Meth. Appl. Sci. 27, 289–312.CrossRefGoogle Scholar
16. Kulczycki, T. & Kuznetsov, N. 2011 On the high spots of fundamental sloshing modes in a trough. Proc. R. Soc. A 467 (2132), 2427–2430.CrossRefGoogle Scholar
17. Kulczycki, T. & Kwaśnicki, M. 2012 On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains. Proc. Lond. Math. Soc., in press.CrossRefGoogle Scholar
18. Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
19. Lukovsky, I. A., Barnyak, M. Y. & Komarenko, A. N. 1984 Approximate Methods of Solving the Problems of the Dynamics of a Limited Liquid Volume. Naukova Dumka (in Russian).Google Scholar
20. McIver, P. 1989 Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth. J. Fluid Mech. 201, 243–257.CrossRefGoogle Scholar
21. Patkas, L. & Karamanos, S. A. 2007 Variational solution of externally induced sloshing in horizontal–cylindrical and spherical vessels. J. Engng Mech. 133 (6), 641–655.Google Scholar
22. Polyanin, A. D. 2001 Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman and Hall/CRC.CrossRefGoogle Scholar
23. Rebouillat, S. & Liksonov, D. 2010 Fluid–structure interaction in partially filled liquid containers: a comparative review of numerical approaches. Comput. Fluids 39, 739–746.CrossRefGoogle Scholar
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