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

  • HAMID ALEMI ARDAKANI (a1) and THOMAS J. BRIDGES (a1)

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.

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Corresponding author

Email address for correspondence: t.bridges@surrey.ac.uk

References

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Abbott, M. B. 1979 Computational Hydraulics: Elements of the Theory of Free-Surface Flows. Pitman Publishers.
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. 413426. Springer.
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.
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.
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.
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.
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.
Alemi Ardakani, H. & Bridges, T. J. 2009 f Surge–sway simulations with additional detail. Tech. Rep. Department of Mathematics, University of Surrey.
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.
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, 3173.
Billingham, J. 2002 Nonlinear sloshing in zero gravity. J. Fluid Mech. 464, 365391.
Bridges, T. J. 1987 Secondary bifurcation and change of type for three-dimensional standing waves in finite depth. J. Fluid Mech. 179, 137153.
Bridges, T. J. 2009 Wave breaking and the surface velocity field for three-dimensional water waves. Nonlinearity 22, 947953.
Buchner, B. 2002 Green water on ship-type offshore structures. PhD thesis, Technical University of Delft.
Caglayan, I. & Storch, R. L. 1982 Stability of fishing vessels with water on deck: a review. J. Ship Res. 26, 106116.
Cariou, A. & Casella, G. 1999 Liquid sloshing in ship tanks: a comparative study of numerical simulation. Marine Struct. 12, 183198.
Chen, Y. G., Djidjeli, K. & Price, W. G. 2009 Numerical simulation of liquid sloshing phenomena in partially filled containers. Comput. Fluids 38, 830842.
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, 321330.
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.
Dingemans, M. W. 1997 Water Wave Propagation Over Uneven Bottoms. Part 2: Nonlinear Wave Propagation. World Scientific.
Disimile, P. J., Pyles, J. M. & Toy, N. 2009 Hydraulic jump formation in water sloshing within an oscillating tank. J. Aircraft 46, 549556.
Evans, D. J. 1977 On the representation of orientation space. Mol. Phys. 34, 317325.
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2003 Resonant three-dimensional nonlinear sloshing in a square basin. J. Fluid Mech. 487, 142.
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, 359397.
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.
Faltinsen, O. M. & Timokha, A. N. 2003 An adaptive multimodal approach to nonlinear sloshing in a rectangular tank. J. Fluid Mech. 432, 167200.
Faltinsen, O. M. & Timokha, A. N. 2009 Sloshing. Cambridge University Press.
Gerrits, J. 2001 Dynamics of liquid-filled spacecraft. PhD thesis, University of Groningen, The Netherlands.
Hanson, A. J. 2006 Visualizing Quaternions. Morgan-Kaufmann/Elsevier.
Huang, Z. 1995 Nonlinear shallow water flow on deck and its effect on ship motion. PhD thesis, Technical University of Nova Scotia, Halifax, Canada.
Huang, Z. J. & Hsiung, C. C. 1996 Nonlinear shallow water on deck. J. Ship Res. 40, 303315.
Huang, Z. J. & Hsiung, C. C. 1997 Nonlinear shallow-water flow on deck coupled with ship motion. In 21st Symposium on Naval Hydrodynamics, pp. 220234. National Academies Press.
Ibrahim, R. A. 2005 Liquid Sloshing Dynamics. Cambridge University Press.
Kidambi, R. & Shankar, P. N. 2004 The effects of the contact angle on sloshing in containers. Proc. R. Soc. Lond. A 460, 22512267.
Kim, Y. 2001 Numerical simulation of sloshing flows with impact load. Appl. Ocean Res. 23, 5362.
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. 981992, ASME.
Kobine, J. J. 2008 Nonlinear resonant characteristics of shallow fluid layers. Phil. Trans. R. Soc. Lond. A 366, 11311346.
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. 383390. ASME.
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, 280310.
La Rocca, M., Sciortino, G. & Boniforti, M. A. 2000 A fully nonlinear model for sloshing in a rotating container. Fluid Dyn. Res. 27, 2352.
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, 1020.
Lee, S. K., Surendran, S. & Lee, G. 2005 Roll performance of a small fishing vessel with live fish tank. Ocean Engng 32, 18731885.
Leendertse, J. 1967 Aspects of a computational model for long-period water wave propagation. Tech. Rep. RM-5294-PR. Rand Corporation.
Leimkuhler, B. & Reich, S. 2004 Simulating Hamiltonian Dynamics. Cambridge University Press.
Leubner, C. 1981 Correcting a widespread error concerning the angular velocity of a rotating rigid body. Am. J. Phys. 49, 232234.
Liu, D. & Lin, P. 2008 A numerical study of three-dimensional liquid sloshing in tanks. J. Comput. Phys. 227, 39213939.
McIntyre, M. E. 2003 Potential vorticity. InEncyclopedia of Atmospheric Sciences (ed. Holton, J. R., Pyle, J. A. & Curry, J. A.), vol. 2. Academic/Elsevier.
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143165.
Murray, R. M., Lin, Z. X. & Sastry, S. S. 1994 A Mathematical Introduction to Robotic Manipulation. CRC Press.
Ockendon, J. R. & Ockendon, H. 1973 Resonant surface waves. J. Fluid Mech. 59, 397413.
O'Reilly, O. M. 2008 Intermediate Dynamics for Engineers: A Unified Treatment of Newton–Euler and Lagrangian Mechanics. Cambridge University Press.
Pantazopoulos, M. S. 1987 Numerical solution of the general shallow water sloshing problem. PhD thesis, University of Washington, Seattle, WA.
Pantazopoulos, M. S. 1988 Three-dimensional sloshing of water on decks. Marine Technol. 25, 253261.
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.
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, 254284.
Peregrine, D. H. 1998 Surf-zone currents. Theor. Comput. Fluid Dyn. 10, 295309.
Peregrine, D. H. 1999 Large-scale vorticity generation by breakers in shallow and deep water. Eur. J. Mech. B/Fluids 18, 403408.
Pomeau, Y., Le Berre, M., Guyenne, P. & Grilli, S. 2008 a Wave breaking and generic singularities of nonlinear hyperbolic equations. Nonlinearity 21, T61T79.
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, 18511866.
Pratt, L. J. 1983 On internal flow over topography. Part 1. Semigeostrophic adjustment to an obstacle. J. Fluid Mech. 131, 195218.
Rapaport, D. C. 1985 Molecular dynamics simulations using quaternions. J. Comput. Phys. 60, 306314.
Rebouillat, S. & Liksonov, D. 2010 Fluid–structure interaction in partially filled liquid containers: a comparative review of numerical approaches. Comput. Fluids 39, 739746.
Romero, I. 2008 Formulation and performance of variational integrators for rotating bodies. Comput. Mech. 42, 825836.
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, 2733.
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.
Taylor, G. I. 1953 An experimental study of standing waves. Proc. R. Soc. Lond. A 218, 4459.
Titterton, D. H. & Weston, J. L. 2004 Strapdown Inertial Navigation Technology, 2nd edn. Institute of Electrical Engineers.
Vanyo, J. P. 1993 Rotating Fluids in Engineering and Science. Butterworth-Heineman.
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, 8299.
aus der Wiesche, S. 2003 Computational slosh dynamics: theory and industrial application. Comput. Mech. 30, 374387.
Wiggins, S. 1987 Chaos in the quasiperiodically forced Duffing oscillator. Phys. Lett. A 124, 138142.
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, 500510.
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, 337355.
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.
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Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions

  • HAMID ALEMI ARDAKANI (a1) and THOMAS J. BRIDGES (a1)

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