Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-29T15:04:25.150Z Has data issue: false hasContentIssue false
  • English
  • Français

Multimodal analysis of weakly nonlinear sloshing in a spherical tank

Published online by Cambridge University Press:  19 February 2013

Odd M. Faltinsen  and
Alexander N. Timokha
Odd M. Faltinsen*
Affiliation:
Centre for Ships and Ocean Structures and Department of Marine Technology, Centre for Autonomous Marine Operations and Systems, Norwegian University of Science and Technology, NO-7091, Trondheim, Norway
Alexander N. Timokha
Affiliation:
Centre for Ships and Ocean Structures and Department of Marine Technology, Centre for Autonomous Marine Operations and Systems, Norwegian University of Science and Technology, NO-7091, Trondheim, Norway
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Sloshing in a spherical tank due to horizontal excitation is studied by using the nonlinear multimodal method which involves the analytically approximate sloshing modes by Faltinsen & Timokha (J. Fluid Mech., vol. 703, 2012, pp. 391–401). General fully and weakly nonlinear modal equations are derived but an emphasis is on the Moiseev–Narimanov asymptotic modal system which implies that the forcing frequency is close to the lowest natural sloshing frequency and there are no secondary resonances in the forcing frequency range leading to a nonlinear resonant amplification of double and triple harmonics in higher modes. The Moiseev–Narimanov modal system is used to construct an asymptotic time-periodic solution and, thereby, classify the corresponding steady-state wave regimes appearing as stable and unstable planar waves and swirling. The results on the stability boundaries are compared with experiments by Sumner & Stofan (1963, Tech. Rep. TN D-1991, NASA Technical Note) and Sumner (1966, Tech. Rep. TN D-3210, NASA). A good agreement is established for $0. 2\leq h\lesssim 1$. Discrepancy for higher liquid depths $1\lesssim h\lt 2$ are explained by secondary resonance.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
Papers
Information
Journal of Fluid Mechanics , Volume 719 , 25 March 2013 , pp. 129 - 164
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ansari, M. R., Firouz-Abadi, R. D. & Ghasemi, M. 2011 Two phase modal analysis of nonlinear sloshing in a rectangular container. Ocean Engng 38 (11–12), 1277–1282.CrossRefGoogle Scholar
Antuono, M., Bouscasse, B., Colagrossi, A. & Lugni, C. 2012 Two-dimensional modal method for shallow-water sloshing in rectangular basins. J. Fluid Mech. 700, 419–440.CrossRefGoogle Scholar
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
Bateman, H. 1944 Partial Differential Equations of Mathematical Physics. Dover.Google Scholar
Budiansky, B. 1958 Sloshing of liquids in circular canals and spherical tanks. Tech. Rep. Lockheed Aircraft Corporation, Missile System Division, Sunnyvale, California.Google Scholar
Budiansky, B. 1960 Sloshing of liquid in circular canals and spherical tanks. J. Aerosp. Sci. 27 (3), 161–172.CrossRefGoogle Scholar
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 Technol. 130, 0418011.Google Scholar
Eastham, M. 1962 An eigenvalue problem with parameter in the boundary condition. Q. J. Maths 13, 304–320.CrossRefGoogle Scholar
Faltinsen, O. M., Firoozkoohi, R. & Timokha, A. N. 2011 Effect of central slotted screen with a high solidity ratio on the secondary resonance phenomenon for liquid sloshing in a rectangular tank. Phys. Fluids 23, 062106.CrossRefGoogle Scholar
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2003 Resonant three-dimensional nonlinear sloshing in a square base basin. J. Fluid Mech. 487, 1–42.CrossRefGoogle Scholar
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2005a Classification of three-dimensional nonlinear sloshing in a square-base tank with finite depth. J. Fluids Struct. 20 (1), 81–103.CrossRefGoogle Scholar
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2005b Resonant three-dimensional nonlinear sloshing in a square base basin. Part 2. Effect of higher modes. J. Fluid Mech. 523, 199–218.CrossRefGoogle Scholar
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2006 Transient and steady-state amplitudes of resonant three-dimensional sloshing in a square base tank with a finite fluid depth. Phys. Fluids 18, 012103.CrossRefGoogle Scholar
Faltinsen, O. M. & Timokha, A. N. 2001 Adaptive multimodal approach to nonlinear sloshing in a rectangular tank. J. Fluid Mech. 432, 167–200.CrossRefGoogle Scholar
Faltinsen, O. M. & Timokha, A. N. 2009 Sloshing. Cambridge University Press.Google Scholar
Faltinsen, O. M. & Timokha, A. N. 2012 Analytically approximate natural sloshing modes for a spherical tank shape. J. Fluid Mech. 703, 391–401.CrossRefGoogle Scholar
Gavrilyuk, I., Hermann, M., Lukovsky, I., Solodun, O. & Timokha, A. 2008 Natural sloshing frequencies in rigid truncated conical tanks. Engng Comput. 25 (6), 518–540.CrossRefGoogle Scholar
Gavrilyuk, I., Lukovsky, I. & Timokha, A. N. 2000 A multimodal approach to nonlinear sloshing in a circular cylindrical tank. Hybrid Methods Engng 2 (4), 463–483.Google Scholar
Gavrilyuk, I., Lukovsky, I., Trotsenko, Yu. & Timokha, A. 2007 Sloshing in a vertical circular cylindrical tank with an annular baffle. Part 2. Nonlinear resonant waves. J. Engng Maths 57, 57–78.CrossRefGoogle Scholar
Gavrilyuk, I. P., Lukovsky, I. A. & Timokha, A. N. 2005 Linear and nonlinear sloshing in a circular conical tank. Fluid Dyn. Res. 37, 399–429.CrossRefGoogle Scholar
Hargneaves, R. 1908 A pressure-integral as kinetic potential. Phil. Mag. 16, 436–444.CrossRefGoogle Scholar
Hasheminejad, S. M. & Aghabeigi, M. 2009 Liquid sloshing in half-full horizontal elliptical tanks. J. Sound Vib. 324, 332–349.CrossRefGoogle Scholar
Hysing, T. 1976 Loads related to sloshing in spherical tanks. Tech. Rep. Det Norske Veritas, Hvik, Norway.Google Scholar
Ikeda, T. & Ibrahim, R. A. 2005 Nonlinear random responses of a structure parametrically coupled with liquid sloshing in a cylindrical tank. J. Sound Vib. 284, 75–102.CrossRefGoogle Scholar
Ikeda, T., Ibrahim, R. A., Harata, Y. & Kuriyama, T. 2012 Nonlinear liquid sloshing in a square tank subjected to obliquely horizontal excitation. J. Fluid Mech. 700, 304–328.CrossRefGoogle Scholar
Karamanos, S. A., Papaprokopiou, D. & Platyrrachos, M. A. 2006 Sloshing effects on the seiesmic design of horizontal-cylindrical and spherical industrial vessels. J. Pressure Vessel Technol. 128, 328–340.CrossRefGoogle Scholar
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 Technol. 131 (5)051301.CrossRefGoogle Scholar
La Rocca, M., Scortino, M. & Boniforti, M. A. 2000 A fully nonlinear model for sloshing in a rotating container. Fluid Dyn. Res. 27, 225–229.CrossRefGoogle Scholar
Love, J. S. & Tait, M. J. 2010 Nonlinear simulation of a tuned liquid damper with damping screens using a modal expansion technique. J. Fluids Struct. 26 (7–8), 1058–1077.CrossRefGoogle Scholar
Love, J. S. & Tait, M. J. 2011 Non-linear multimodal model for tuned liquid dampers of arbitrary tank geometry. Intl J. Non-Linear Mech. 46 (8), 1065–1075.CrossRefGoogle Scholar
Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397.CrossRefGoogle Scholar
Lukovsky, I. A. 1975 Nonlinear Sloshing in Tanks of Complex Geometrical Shape. Naukova Dumka (in Russian).Google Scholar
Lukovsky, I. A. 1990 Introduction to Nonlinear Dynamics of Rigid Bodies with the Cavities Partially Filled by a Fluid. Naukova Dumka (in Russian).Google Scholar
Lukovsky, I. A., Ovchynnykov, D. V. & Timokha, A. N. 2012 Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: modal equations. Nonlinear Oscillations 14 (4), 512–525.CrossRefGoogle Scholar
Lukovsky, I. A. & Timokha, A. N. 1995 Variational methods in nonlinear problems of the dynamics of a limited liquid volume. Institute of Mathematics of NASU, ISBN 5-7702-1184-9 (in Russian).Google Scholar
Lukovsky, I. A. & Timokha, A. N. 2002 Modal modelling of nonlinear sloshing in tanks with non-vertical walls. Non-conformal mapping technique. Intl J. Fluid Mech. Res. 29 (2), 216–242.Google Scholar
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
McIver, P. 1989 Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth. J. Fluid Mech. 201, 243–257.CrossRefGoogle Scholar
Miles, J. W. 1962 Stability of forced oscillations of a spherical pendulum. Q. Appl. Maths 20, 21–32.CrossRefGoogle Scholar
Miles, J. W. 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75 (3), 419–448.CrossRefGoogle Scholar
Miles, J. W. 1984a Internally resonant surface waves in circular cylinder. J. Fluid Mech. 149, 1–14.CrossRefGoogle Scholar
Miles, J. W. 1984b Resonantly forces surface waves in circular cylinder. J. Fluid Mech. 149, 15–31.CrossRefGoogle Scholar
Moiseev, N. N. 1958 On the theory of nonlinear vibrations of a liquid of finite volume. J. Appl. Math. Mech. 22 (5), 860–872.CrossRefGoogle Scholar
Moore, R. E. & Perko, L. M. 1964 Inviscid fluid flow in an accelerating cylindrical container. J. Fluid Mech. 22, 305–320.CrossRefGoogle Scholar
Morand, J.-P. & Ohayon, R. 1995 Fluid Structure Interaction. Applied Numerical Methods. John Wiley & Sons.Google Scholar
Narimanov, G. S. 1957 Movement of a tank partly filled by a fluid: the taking into account of non-smallness of amplitude. Prikl. Math. Mech. 21, 513–524.Google Scholar
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
Perko, L. M. 1969 Large-amplitude motions of liquid–vapour interface in an accelerating container. J. Fluid Mech. 35, 77–96.CrossRefGoogle Scholar
Stofan, A. J. & Armsted, A. L. 1962 Analytical and experimental investigation of forces and frequencies resulting from liquid sloshing in a spherical tank. Tech. Rep. D-1281. NASA, Lewis Research Center, Clevelend, Ohio.Google Scholar
Sumner, I. E. 1966 Experimental investigations of stability boundaries for planar and nonplanar sloshing in spherical tanks. Tech. Rep. TN D-3210, NASA.Google Scholar
Sumner, I. E. & Stofan, A. J. 1963 An experimental investigation of the viscous damping of liquid sloshing in spherical tanks. Tech. Rep. TN D-1991, NASA Technical Note.Google Scholar
Takahara, H., Hara, K. & Ishida, T. 2012 Nonlinear liquid oscillation in a cylindrical tank with an eccentric core barrel. J. Fluids Struct. 35, 120–132.CrossRefGoogle Scholar
Takahara, H. & Kimura, K. 2012 Frequency response of sloshing in an annular cylindrical tank subjected to pitching excitation. J. Sound Vib. 331 (13), 3199–3212.CrossRefGoogle Scholar
Utsumi, M. 2011 Theoretical determination of modal damping ratio of sloshing using a variational method. J. Pressure Vessel Technol. 133, 011301-1.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.Google Scholar
Supplementary material: File

Faltinsen et al. supplementary material

Supplementary material

Download Faltinsen et al. supplementary material(File)
File 718.3 KB