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.
A simple numerical verification method for differential equations based on infinite dimensional sequential iteration