Resonant three-dimensional nonlinear sloshing in a square-base basin

An asymptotic modal system is derived for modelling nonlinear sloshing in a rectangular tank with similar width and breadth. The system couples nonlinearly nine modal functions describing the time evolution of the natural modes. Two primary modes are assumed to be dominant. The system is equivalent to the model by Faltinsen et al. for the two-dimensional case. It is validated for resonant sloshing in a square-base basin. Emphasis is on finite fluid depth but the behaviour with decreasing depth to intermediate depths is also discussed. The tank is forced in surge/sway/roll/pitch with frequency close to the lowest degenerate natural frequency. The theoretical part concentrates on periodic solutions of the modal system (steady-state wave motions) for longitudinal (along the walls) and diagonal (in the vertical diagonal plane) excitations. Three types of solutions are established for each case: (i) 'planar'/'diagonal' resonant standing waves for longitudinal/diagonal forcing, (ii) 'swirling' waves moving along tank walls clockwise or counterclockwise and (iii) 'square'-like resonant standing wave coupling in-phase oscillations of both the lowest modes.