Hostname: page-component-5d84bcc8dc-7mb8q Total loading time: 0 Render date: 2026-08-10T23:55:00.954Z Has data issue: false hasContentIssue false
  • English
  • Français

Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions

Published online by Cambridge University Press:  14 January 2011

HAMID ALEMI ARDAKANI  and
THOMAS J. BRIDGES
HAMID ALEMI ARDAKANI
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK
THOMAS J. BRIDGES*
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK
*
†Email address for correspondence: t.bridges@surrey.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

New shallow-water equations (SWEs), for sloshing in three dimensions (two horizontal and one vertical) in a vessel which is undergoing rigid-body motion in 3-space, are derived. The rigid-body motion of the vessel (roll–pitch–yaw and/or surge–sway–heave) is modelled exactly and the only approximations are in the fluid motion. The flow is assumed to be inviscid but vortical, with approximations on the vertical velocity and acceleration at the surface. These equations improve previous shallow-water models. The model also extends to three dimensions the essence of the Penney–Price–Taylor theory for the highest standing wave. The surface SWEs are simulated using a split-step alternating direction implicit finite-difference scheme. Numerical experiments are reported, including comparisons with existing results in the literature, and simulations with vessels undergoing full 3-D rotations.

JFM classification

Geophysical and Geological Flows: Rotating flows Waves/Free-surface Flows: Surface gravity waves

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 667 , 25 January 2011 , pp. 474 - 519
Copyright
Copyright © Cambridge University Press 2011

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Abbott, M. B. 1979 Computational Hydraulics: Elements of the Theory of Free-Surface Flows. Pitman Publishers.Google Scholar
Adee, B. H. & Caglayan, I. 1982 The effects of free water on deck on the motions and stability of vessels. In Proceedings of the Second International Conference Stab. Ships and Ocean Vehicles, Tokyo, Japan, vol. 218, pp. 413–426. Springer.Google Scholar
Alemi Ardakani, H. & Bridges, T. J. 2009 a Dynamic coupling between shallow-water sloshing and a vehicle undergoing planar rigid- body motion. Tech. Rep. Department of Mathematics, University of Surrey.Google Scholar
Alemi Ardakani, H. & Bridges, T. J. 2009 b Review of the 3–2–1 Euler angles: a yaw–pitch–roll sequence. Tech. Rep. Department of Mathematics, University of Surrey.Google Scholar
Alemi Ardakani, H. & Bridges, T. J. 2009 c Review of the Dillingham, Falzarano & Pantazopoulos three-dimensional shallow-water equations. Tech. Rep. Department of Mathematics, University of Surrey.Google Scholar
Alemi Ardakani, H. & Bridges, T. J. 2009 d Review of the Huang–Hsiung three-dimensional shallow-water equations. Tech. Rep. Department of Mathematics, University of Surrey.Google Scholar
Alemi Ardakani, H. & Bridges, T. J. 2009 e Shallow water sloshing in rotating vessels: details of the numerical algorithm. Tech. Rep. Department of Mathematics, University of Surrey.Google Scholar
Alemi Ardakani, H. & Bridges, T. J. 2009 f Surge–sway simulations with additional detail. Tech. Rep. Department of Mathematics, University of Surrey.Google Scholar
Alemi Ardakani, H. & Bridges, T. J. 2010 Dynamic coupling between shallow water sloshing and horizontal vehicle motion. Eur. J. Appl. Math., doi:10.1017/S0956792510000197.CrossRefGoogle Scholar
Barnes, R. T. H., Hide, R., White, A. A. & Wilson, C. A. 1983 Atmospheric angular momentum fluctuations, length-of-day changes and polar motion. Proc. R. Soc. Lond. A 387, 31–73.Google Scholar
Billingham, J. 2002 Nonlinear sloshing in zero gravity. J. Fluid Mech. 464, 365–391.Google Scholar
Bridges, T. J. 1987 Secondary bifurcation and change of type for three-dimensional standing waves in finite depth. J. Fluid Mech. 179, 137–153.CrossRefGoogle Scholar
Bridges, T. J. 2009 Wave breaking and the surface velocity field for three-dimensional water waves. Nonlinearity 22, 947–953.Google Scholar
Buchner, B. 2002 Green water on ship-type offshore structures. PhD thesis, Technical University of Delft.Google Scholar
Caglayan, I. & Storch, R. L. 1982 Stability of fishing vessels with water on deck: a review. J. Ship Res. 26, 106–116.Google Scholar
Cariou, A. & Casella, G. 1999 Liquid sloshing in ship tanks: a comparative study of numerical simulation. Marine Struct. 12, 183–198.Google Scholar
Chen, Y. G., Djidjeli, K. & Price, W. G. 2009 Numerical simulation of liquid sloshing phenomena in partially filled containers. Comput. Fluids 38, 830–842.CrossRefGoogle Scholar
Chen, Y.-H., Hwang, W.-S. & Ko, C.-H. 2000 Numerical simulation of the three-dimensional sloshing problem by boundary element method. J. Chinese Inst. Engng 23, 321–330.CrossRefGoogle Scholar
Dillingham, J. T. & Falzarano, J. M. 1986 Three-dimensional numerical simulation of green water on deck. In 3rd International Conference on Stability of Ships and Ocean Vehicles, Gdansk, Poland, (ed. Kobylinski, L.). Tech. U. Gdansk.Google Scholar
Dingemans, M. W. 1997 Water Wave Propagation Over Uneven Bottoms. Part 2: Nonlinear Wave Propagation. World Scientific.Google Scholar
Disimile, P. J., Pyles, J. M. & Toy, N. 2009 Hydraulic jump formation in water sloshing within an oscillating tank. J. Aircraft 46, 549–556.CrossRefGoogle Scholar
Evans, D. J. 1977 On the representation of orientation space. Mol. Phys. 34, 317–325.CrossRefGoogle Scholar
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2003 Resonant three-dimensional nonlinear sloshing in a square basin. J. Fluid Mech. 487, 1–42.CrossRefGoogle Scholar
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2006 a Resonant three-dimensional nonlinear sloshing in a square-base basin. Part 3. Base ratio perturbations. J. Fluid Mech. 551, 359–397.Google Scholar
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2006 b Transient and steady-state amplitudes of resonant three-dimensional sloshing in a square base tank with finite fluid depth. Phys. Fluids 18, 012103.CrossRefGoogle Scholar
Faltinsen, O. M. & Timokha, A. N. 2003 An adaptive multimodal approach to nonlinear sloshing in a rectangular tank. J. Fluid Mech. 432, 167–200.Google Scholar
Faltinsen, O. M. & Timokha, A. N. 2009 Sloshing. Cambridge University Press.Google Scholar
Gerrits, J. 2001 Dynamics of liquid-filled spacecraft. PhD thesis, University of Groningen, The Netherlands.Google Scholar
Hanson, A. J. 2006 Visualizing Quaternions. Morgan-Kaufmann/Elsevier.Google Scholar
Huang, Z. 1995 Nonlinear shallow water flow on deck and its effect on ship motion. PhD thesis, Technical University of Nova Scotia, Halifax, Canada.Google Scholar
Huang, Z. J. & Hsiung, C. C. 1996 Nonlinear shallow water on deck. J. Ship Res. 40, 303–315.Google Scholar
Huang, Z. J. & Hsiung, C. C. 1997 Nonlinear shallow-water flow on deck coupled with ship motion. In 21st Symposium on Naval Hydrodynamics, pp. 220–234. National Academies Press.Google Scholar
Ibrahim, R. A. 2005 Liquid Sloshing Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Kidambi, R. & Shankar, P. N. 2004 The effects of the contact angle on sloshing in containers. Proc. R. Soc. Lond. A 460, 2251–2267.Google Scholar
Kim, Y. 2001 Numerical simulation of sloshing flows with impact load. Appl. Ocean Res. 23, 53–62.Google Scholar
Kleefsman, K. M. T., Erwin Loots, G., Veldman, A. E. P., Buchner, B., Bunnik, T. & Falkenberg, E. 2005 The numerical simulation of green water loading including vessel motions and the incoming field. In Proceedings of the 24th International Conference on Offshore Mechanics and Arctic Engineering (OMAE'05), vol. OMAE2005-67448, pp. 981–992, ASME.Google Scholar
Kobine, J. J. 2008 Nonlinear resonant characteristics of shallow fluid layers. Phil. Trans. R. Soc. Lond. A 366, 1131–1346.Google Scholar
Laranjinha, M., Falzarano, J. M. & Guedes Soares, C. 2002 Analysis of the dynamical behaviour of an offshore supply vessel with water on deck. In Proceedings of 21st International Conference Offshore Mechanics and Arctic Engineering, Paper No. OMAE2002-OFT28177, vol. 1, pp. 383–390. ASME.Google Scholar
La Rocca, M., Mele, P. & Armenio, V. 1997 Variational approach to the problem of sloshing in a moving container. J. Theor. Appl. Fluid Mech. 1, 280–310.Google Scholar
La Rocca, M., Sciortino, G. & Boniforti, M. A. 2000 A fully nonlinear model for sloshing in a rotating container. Fluid Dyn. Res. 27, 23–52.Google Scholar
Lee, S. J., Kim, M. H., Lee, D. H., Kim, J. W. & Kim, Y. H. 2007 The effects of LNG-tank sloshing on the global motions of LNG carriers. Ocean Engng 34, 10–20.Google Scholar
Lee, S. K., Surendran, S. & Lee, G. 2005 Roll performance of a small fishing vessel with live fish tank. Ocean Engng 32, 1873–1885.CrossRefGoogle Scholar
Leendertse, J. 1967 Aspects of a computational model for long-period water wave propagation. Tech. Rep. RM-5294-PR. Rand Corporation.Google Scholar
Leimkuhler, B. & Reich, S. 2004 Simulating Hamiltonian Dynamics. Cambridge University Press.Google Scholar
Leubner, C. 1981 Correcting a widespread error concerning the angular velocity of a rotating rigid body. Am. J. Phys. 49, 232–234.Google Scholar
Liu, D. & Lin, P. 2008 A numerical study of three-dimensional liquid sloshing in tanks. J. Comput. Phys. 227, 3921–3939.Google Scholar
McIntyre, M. E. 2003 Potential vorticity. InEncyclopedia of Atmospheric Sciences (ed. Holton, J. R., Pyle, J. A. & Curry, J. A.), vol. 2. Academic/Elsevier.Google Scholar
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143–165.Google Scholar
Murray, R. M., Lin, Z. X. & Sastry, S. S. 1994 A Mathematical Introduction to Robotic Manipulation. CRC Press.Google Scholar
Ockendon, J. R. & Ockendon, H. 1973 Resonant surface waves. J. Fluid Mech. 59, 397–413.Google Scholar
O'Reilly, O. M. 2008 Intermediate Dynamics for Engineers: A Unified Treatment of Newton–Euler and Lagrangian Mechanics. Cambridge University Press.Google Scholar
Pantazopoulos, M. S. 1987 Numerical solution of the general shallow water sloshing problem. PhD thesis, University of Washington, Seattle, WA.Google Scholar
Pantazopoulos, M. S. 1988 Three-dimensional sloshing of water on decks. Marine Technol. 25, 253–261.Google Scholar
Pantazopoulos, M. S. & Adee, B. H. 1987 Three-dimensional shallow water waves in an oscillating tank. In Proc. ASCE Speciality Conference on Coastal Hydrodynamics, (ed. Dalrymple, R. A.) pp. 399–412.Google Scholar
Penney, W. G. & Price, A. T. 1952 Part II. Finite periodic stationary gravity waves in a perfect fluid. Phil. Trans. R. Soc. Lond. A 244, 254–284.Google Scholar
Peregrine, D. H. 1998 Surf-zone currents. Theor. Comput. Fluid Dyn. 10, 295–309.Google Scholar
Peregrine, D. H. 1999 Large-scale vorticity generation by breakers in shallow and deep water. Eur. J. Mech. B/Fluids 18, 403–408.Google Scholar
Pomeau, Y., Le Berre, M., Guyenne, P. & Grilli, S. 2008 a Wave breaking and generic singularities of nonlinear hyperbolic equations. Nonlinearity 21, T61–T79.Google Scholar
Pomeau, Y., Jamin, T., Le Bars, M., Le Gal, P. & Audoly, B. 2008 b Law of spreading of the crest of a breaking wave. Proc. R. Soc. Lond. A 464, 1851–1866.Google Scholar
Pratt, L. J. 1983 On internal flow over topography. Part 1. Semigeostrophic adjustment to an obstacle. J. Fluid Mech. 131, 195–218.Google Scholar
Rapaport, D. C. 1985 Molecular dynamics simulations using quaternions. J. Comput. Phys. 60, 306–314.Google Scholar
Rebouillat, S. & Liksonov, D. 2010 Fluid–structure interaction in partially filled liquid containers: a comparative review of numerical approaches. Comput. Fluids 39, 739–746.Google Scholar
Romero, I. 2008 Formulation and performance of variational integrators for rotating bodies. Comput. Mech. 42, 825–836.Google Scholar
Ruponen, P., Matusiak, J., Luukkonen, J. & Ilus, M. 2009 Experimental study on the behavior of a swimming pool onboard a large passenger ship. Marine Technol. 46, 27–33.Google Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Taylor, G. I. 1953 An experimental study of standing waves. Proc. R. Soc. Lond. A 218, 44–59.Google Scholar
Titterton, D. H. & Weston, J. L. 2004 Strapdown Inertial Navigation Technology, 2nd edn. Institute of Electrical Engineers.CrossRefGoogle Scholar
Vanyo, J. P. 1993 Rotating Fluids in Engineering and Science. Butterworth-Heineman.Google Scholar
Veldman, A. E. P., Gerrits, J., Luppes, R., Helder, J. A. & Vreeburg, J. P. B. 2007 The numerical simulation of liquid sloshing on board spacecraft. J. Comput. Phys. 224, 82–99.Google Scholar
aus der Wiesche, S. 2003 Computational slosh dynamics: theory and industrial application. Comput. Mech. 30, 374–387.Google Scholar
Wiggins, S. 1987 Chaos in the quasiperiodically forced Duffing oscillator. Phys. Lett. A 124, 138–142.Google Scholar
Wu, C.-H. & Chen, B.-F. 2009 Sloshing waves and resonance modes of fluid in a 3D tank by a time-independent finite difference method. Ocean Engng 36, 500–510.Google Scholar
Wu, G. S., Ma, Q. W. & Eatock Taylor, R. 1998 Numerical simulation of sloshing waves in a 3D tank based on a finite element method. Appl. Ocean Res. 20, 337–355.Google Scholar
Zhou, Z., De Kat, J. O. & Buchner, B. 1999 A nonlinear 3D approach to simulate green water dynamics on deck. In Proceedings of the 7th International Conference Numer. Ship Hydro. (ed. Piquet, J.), Nantes, France, vol. 7. DTIC.Google Scholar