Ocean Systems Engineering

  • 제2권3호
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  • Pages.229-238
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  • 2012
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  • 2093-6702(pISSN)
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  • 2093-677X(eISSN)
테크노프레스 (Techno-Press)
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Bi-stability in a vertically excited rectangular tank with finite liquid depth

  • Spandonidis, Christos C. (School of Naval Architecture and Marine Engineering, National Technical University of Athens) ;
  • Spyrou, Kostas J. (School of Naval Architecture and Marine Engineering, National Technical University of Athens)
  • 투고 : 2012.01.16
  • 심사 : 2012.09.08
  • 발행 : 2012.09.25
⟨ 이전 논문 다음 논문 ⟩

초록

We discuss the bi - stability that is possibly exhibited by a liquid free surface in a parametrically - driven two-dimensional (2D) rectangular tank with finite liquid depth. Following the method of adaptive mode ordering, assuming two dominant modes and retaining polynomial nonlinearities up to third-order, a nonlinear finite-dimensional nonlinear modal system approximation is obtained. A "continuation method" of nonlinear dynamics is then used in order to elicit efficiently the instability boundary in parameters' space and to predict how steady surface elevation changes as the frequency and/or the amplitude of excitation are varied. Results are compared against those of the linear version of the system (that is a Mathieu-type model) and furthermore, against an intermediate model also derived with formal mode ordering, that is based on a second - order ordinary differential equation having nonlinearities due to products of elevation with elevation velocity or acceleration. The investigation verifies that, in parameters space, there must be a region, inside the quiescent region, where liquid surface instability is exhibited. There, behaviour depends on initial conditions and a wave form would be realised only if the free surface was substantially disturbed initially.

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참고문헌

  1. Abramson, H.N. (1966), The dynamics of liquids in moving containers, NASA Report SP 106.
  2. Benjamin, T.B. and Ursell, F. (1954), "The stability of the plane free surface of a liquid in a vertical periodic motion", Proceedings of the Royal Society A 225.
  3. Bredmose, H., Brocchini, M., Peregrine, D.H. and Thais, L. (2003), "Experimental investigation and numerical modelling of steep forced water waves", J. Fluid. Mech., 490, 217-49. https://doi.org/10.1017/S0022112003005238
  4. Chen, W., Haroun, M.A. and Liu, F. (1996) "Large amplitude liquid sloshing in seismically excited tanks", Earthq. Eng. Struct. D., 25(7), 653-669. https://doi.org/10.1002/(SICI)1096-9845(199607)25:7<653::AID-EQE513>3.0.CO;2-H
  5. Chern, M.J., Borthwick, A.G.L. and Taylor, R. (1999), "A pseudospectral s-transformation model of 2-D nonlinear waves", J. Fluid. Struct., 13, 607-630. https://doi.org/10.1006/jfls.1999.0221
  6. Craik, A.D.D. and Armitage, J. (1995), "Faraday excitation, hysteresis and wave instability in a narrow rectangular wave tank", Fluid Dyn. Res., 15(3), 129-143. https://doi.org/10.1016/0169-5983(94)00037-Z
  7. Decent, S.P. (1995), Hysteresis and mode competition in Faraday waves, Ph.D. thesis, University of St Andrews, Scotland.
  8. Decent, S.P. and Craik, A.D.D. (1995), "Hysteresis in Faraday resonance", J. Fluid Mech., 293, 237-268. https://doi.org/10.1017/S0022112095001704
  9. Dhooge, A., Govaerts,W., Kuznetsov, Y.A., Mestrom,W., Riet, A.M. and Sautois, B. (2003) MATCONT and CL_MATCONT: Continuation Toolboxes for MATLAB, Report of Gent and Utrecht Universities.
  10. Dodge, F.T. (1966), "Verticall excitation of propellant tanks. In the book: the dynamics of liquids in moving containers, NASA Report, SP 106.
  11. Faltinsen, O.M. (1974), "A nonlinear theory of sloshing in rectangular tanks", J. Ship Res., 18(4), 224-241.
  12. Faltinsen, O.M. and Rognebakke, O.F. (2000), "Sloshing. Keynote lecture". Proceedings of the International Conference on Ship and Shipping Research (NAV2000), Venice, September.
  13. Faltinsen, O.M., Rognebakke, O.F., Lukovsky, I.A. and Timokha, A.N. (2000), "Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth", J. Fluid Mech., 407, 201-234. https://doi.org/10.1017/S0022112099007569
  14. Faltinsen, O.M. and Timokha, A.N. (2001), "Adaptive multimodal approach to nonlinear sloshing in a rectangular tank", J. Fluid Mech., 432, 167-200.
  15. Faltinsen, O.M and Timokha, A.N. (2002), "Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth", J. Fluid Mech., 470, 319-357.
  16. Faltinsen, O.M. and Timokha, A.N. (2009), Sloshing, ISBN: 978-0-521-88111-1, Cambridge University Press, New York.
  17. Faraday, M. (1831), "On a peculiar class of acoustical figures, and on certain forms assumed by groups of particles upon vibrating elastic surfaces", Philos. T. R. Soc., 121, 299-340. https://doi.org/10.1098/rstl.1831.0018
  18. Frandsen, J.B. and Borthwick, L. (2003), "Simulation of sloshing motions in fixed and vertically excited containers using a 2-D inviscid transformed finite difference solver", J. Fluid. Struct., 18(2), 197-214. https://doi.org/10.1016/j.jfluidstructs.2003.07.004
  19. Frandsen, J.B. (2004), "Sloshing motions in the excited tanks", J. Comput. Phys., 196(1), 53-87. https://doi.org/10.1016/j.jcp.2003.10.031
  20. Ibrahim, R.A. (2005), Liquid Sloshing Dynamics, ISBN: 978-0-521-83885-6, Cambridge University Press, New York.
  21. Fultz, D. (1962), "An experimental note on finite-amplitude standing gravity waves", J. Fluid Mech., 13(2), 193-212. https://doi.org/10.1017/S0022112062000622
  22. Kim, Y., Nam, B.W., Kim, D.W. and Kim, Y.S. (2007), "Study on coupling effects of ship motion and sloshing", Ocean Eng., 34(16), 2176-2187. https://doi.org/10.1016/j.oceaneng.2007.03.008
  23. Konstantinov, A., Mikityuk, Y. and Pal'ko, L.S. (1978), "Study of the dynamic stability of a liquid in a cylindrical cavity, in dynamics of elastic systems with continuous-discrete parameters", Naukova Dumka, 69-72.
  24. Lewis, D.J. (1950), "The instability of liquid surfaces when accelerated in a direction perpendicular to their planes", Proceedings of the Royal Society A, A 202.
  25. Mathiessen, L. (1870), "Uber die transversal-schwingungen tonender tropharer und elastisher flussigkeiten", Annalen der Physik, 141, 375-393.
  26. Miles, J.W. (1994), "Faraday waves: rolls versus squares", J. Fluid Mech., 269, 353-371. https://doi.org/10.1017/S002211209400159X
  27. Narimanov, G.S. (1957), "Movement of a tank partly filled by a fluid: the taking into account of non smallness of amplitude", J. Appl. Math. Mech. (PMM), 21, 513-524 (in Russian).
  28. Perlin, M. and Schultz, W.W. (1996), "On the boundary conditions at an oscillating contact-line: a physical/numerical experimental program", Proceedings of the NASA 3rd Microgravity Fluid Physics Conference, Cleveland, OH, 615-620.
  29. Rayleigh, J.W.S. (1883a.), "On the crispations of fluid resting upon a vibrating support", Philos. Mag., 15, 229-235. https://doi.org/10.1080/14786448308627342
  30. Rayleigh, J.W.S. (1883b), "On maintained vibrations", Philos. Mag., 15, 229-235. https://doi.org/10.1080/14786448308627342
  31. Rayleigh, J.W.S. (1887), "On the maintenance of vibrations by forces of double frequency and on the propagation of waves through a medium endowed with a periodic structure", Philos. Mag., 24, 145-159. https://doi.org/10.1080/14786448708628074
  32. Simoneli, F. and Gollup, J.P. (1989), "Surface wave mode interactions: effects of symetry and degeneracy", J. Fluid Mech., 199, 471-494. https://doi.org/10.1017/S0022112089000443
  33. Spandonidis, C. (2010), Parametric sloshing in a 2D rectangular tank, Postgraduate Thesis, National Technical University of Athens (in Greek).
  34. Spandonidis, C. and Spyrou, K.J. (2011), "Parametric Sloshing in A 2D Rectangular Tank with finite liquid depth", Proceedings of the 14th International Congress of the International Maritime Association of the Mediterranean (IMAM), Genova, September.
  35. Spyrou, K.J., Tigkas, I., Scanferla, G., Pallikaropoulos, N. and Themelis, N. (2008) "Prediction potential of the parametric rolling behaviour of a post-panamax containership", Ocean Eng., 35(11-12), 1235-1244. https://doi.org/10.1016/j.oceaneng.2008.03.013
  36. Spyrou, K.J., and Tigkas, I. (2011), "Nonlinear surge dynamics of a ship in astern seas: continuation analysis of periodic states with hydrodynamic memory", J. Ship Res., 55, 19-28.
  37. Takizawa, A. and Kondo, S. (1995), "Computer discovery of the mechanism of flow-induced sloshing. fluid-sloshing and fluid-structure interaction", Proceedings of the ASME Pressure Vessel Piping Conference, PVP-314.
  38. Taylor, G.I. (1950), "The instability of liquid surfaces when accelerated in a direction perpendicular to their planes", Proceedings of the Royal Society, A 201.
  39. Turnbull, M.S., Borthwick, A.G.L. and Eatock Taylor, R. (2003), "Numerical wave tank based on a stransformed finite element inviscid flow solver", Int. J. Numer. Meth. Fl., 42, 641-663. https://doi.org/10.1002/fld.539
  40. Wu, G.X. (2007), "Second -order resonance of sloshing in a tank", Ocean Eng., 34, 2345-2349. https://doi.org/10.1016/j.oceaneng.2007.05.004