Abstract
The motion of an ideal fluid in a rectangular tank is studied, under conditions in which the tank is subjected to horizontal sinusoidal periodic forcing. A novel technique is presented for solving the problem; it makes use of a Fourier-series representation in which the time-dependent coefficients are shown to obey a system of forced nonlinear ordinary differential equations. Time-periodic solutions are computed using further Fourier-series representations and Newton’s method to find the doubly subscripted arrays of coefficients. It is shown that the linearized solution describes the motion reasonably well, except near the regions of linearized resonance. A weakly nonlinear theory near resonance is presented, but is found to give a poor description of the motion. Extensive nonlinear results are shown which reveal intricate behaviour near resonance. A method is given for computing the stability of time-periodic solutions; it reveals that the solution branch corresponding to linearized theory is stable, but that additional unstable periodic solution branches may also be present. Further quasi-periodic and chaotic solutions are detected.
Similar content being viewed by others
References
Virella JC, Prato CA, Godoy LA (2008) Linear and nonlinear 2D finite element analysis of sloshing modes and pressures in rectangular tanks subject to horizontal harmonic motions. J Sound Vib 312: 442–460
Hermann M, Timokha A (2005) Modal modelling of the nonlinear resonant fluid sloshing in a rectangular tank I: a single-dominant model. Math Models Methods Appl Sci 15: 1431–1458
Ibrahim RA, Pilipchuk VN, Ikeda T (2001) Recent advances in liquid sloshing dynamics. ASME Appl Mech Rev 54: 133–199
Frandsen JB (2004) Sloshing motions in excited tanks. J Comput Phys 196: 53–87
Chester W (1968) Resonant oscillations of water waves. I. Theory. Proc R Soc Lond A 306: 5–22
Chester W, Bones JA (1968) Resonant oscillations of water waves. II. Experiment. Proc R Soc Lond A 306: 23–39
Ockendon JR, Ockendon H (1973) Resonant surface waves. J Fluid Mech 59: 397–413
Seydel R (1994) Practical bifurcation and stability analysis: from equilibrium to Chaos, second edition. Springer-Verlag Inc, New York
Hill DF (2003) Transient and steady-state amplitudes of forced waves in rectangular basins. Phys Fluids 15: 1576–1587
Hill D, Frandsen J (2005) Transient evolution of weakly nonlinear sloshing waves: an analytical and numerical comparison. J Eng Math 53: 187–198
Gardarsson SM, Yeh H (2007) Hysteresis in shallow water sloshing. J Eng Mech ASCE 133: 1093–1100
Hermann M, Timokha A (2008) Modal modelling of the nonlinear resonant fluid sloshing in a rectangular tank II: secondary resonance. Math Models Methods Appl Sci 18: 1845–1867
Amundsen DE, Cox EA, Mortell MP (2007) Asymptotic analysis of steady solutions of the KdVB equation with application to resonant sloshing. Z Angew Math Phys 58: 1008–1034
Bredmose H, Brocchini M, Peregrine DH, Thais L (2003) Experimental investigation and numerical modelling of steep forced water waves. J Fluid Mech 490: 217–249
Chen B-F, Nokes R (2005) Time-independent finite difference analysis of fully non-linear and viscous fluid sloshing in a rectangular tank. J Comput Phys 209: 47–81
Wang CZ, Khoo BC (2005) Finite element analysis of two-dimensional nonlinear sloshing problems in random excitations. Ocean Eng 32: 107–133
Celebi MS, Akyildiz H (2002) Nonlinear modeling of liquid sloshing in a moving rectangular tank. Ocean Eng 29: 1527–1553
Armenio V, La Rocca M (1996) On the analysis of sloshing of water in rectangular containers: numerical study and experimental validation. Ocean Eng 23: 705–739
Cariou A, Casella G (1999) Liquid sloshing in ship tanks: a comparative study of numerical simulation. Mar Struct 12: 183–198
Liu D, Lin P (2008) A numerical study of three-dimensional liquid sloshing in tanks. J Comput Phys 227: 3921–3939
Faltinsen OM, Rognebakke OF, Lukovsky IA, Timokha AN (2000) Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. J Fluid Mech 407: 201–234
Faltinsen OM, Timokha AN (2002) Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth. J Fluid Mech 470: 319–357
Ikeda T (2007) Autoparametric resonances in elastic structures carrying two rectangular tanks partially filled with liquid. J Sound Vib 302: 657–682
Forbes LK, Chen MJ, Trenham CE (2007) Computing unstable periodic waves at the interface of two inviscid fluids in uniform vertical flow. J Comput Phys 221: 269–287
Forbes LK, Hocking GC (2007) Unsteady draining flows from a rectangular tank. Phys Fluids 19(082104), 14 pp
Atkinson KA (1978) An introduction to numerical analysis. Wiley, New York
Gradshteyn IS, Ryzhik IM (2000) Tables of integrals, series and products, 6th edn. Academic Press, San Diego
Abramowitz M, Stegun IA (eds) (1972) Handbook of mathematical functions. Dover Publications, Inc, New York
Kreyszig E (2006) Advanced engineering mathematics, 9th edn. Wiley, Singapore
Moore DW (1979) The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc R Soc Lond A 365: 105–119
Cowley SJ, Baker GR, Tanveer S (1999) On the formation of Moore curvature singularities in vortex sheets. J Fluid Mech 378: 233–267
Forbes LK (2009) The Rayleigh–Taylor instability for inviscid and viscous fluids. J Eng Math. doi:10.1007/s10665-009-9288-9
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Forbes, L.K. Sloshing of an ideal fluid in a horizontally forced rectangular tank. J Eng Math 66, 395–412 (2010). https://doi.org/10.1007/s10665-009-9296-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-009-9296-9