An adaptive multimodal approach to nonlinear sloshing in a rectangular tank

Abstract

Two-dimensional nonlinear sloshing of an incompressible fluid with irrotational flow in a rectangular tank is analysed by a modal theory. Infinite tank roof height and no overturning waves are assumed. The modal theory is based on an infinite-dimensional system of nonlinear ordinary differential equations coupling generalized coordinates of the free surface and fluid motion associated with the amplitude response of natural modes. This modal system is asymptotically reduced to an infinite-dimensional system of ordinary differential equations with fifth-order polynomial nonlinearity by assuming sufficiently small fluid motion relative to fluid depth and tank breadth. When introducing inter-modal ordering, the system can be detuned and truncated to describe resonant sloshing in different domains of the excitation period. Resonant sloshing due to surge and pitch sinusoidal excitation of the primary mode is considered. By assuming that each mode has only one main harmonic an adaptive procedure is proposed to describe direct and secondary resonant responses when Moiseyev-like relations do not agree with experiments, i.e. when the excitation amplitude is not very small, and the fluid depth is close to the critical depth or small. Adaptive procedures have been established for a wide range of excitation periods as long as the mean fluid depth h is larger than 0.24 times the tank breadth l. Steady-state results for wave elevation, horizontal force and pitch moment are experimentally validated except when heavy roof impact occurs. The analysis of small depth requires that many modes have primary order and that each mode may have more than one main harmonic. This is illustrated by an example for h/l = 0.173, where the previous model by Faltinsen et al. (2000) failed. The new model agrees well with experiments.