Refined Solutions of Externally Induced Sloshing in Half-Full Spherical Containers

Abstract

A mathematical model is developed for calculating liquid sloshing effects such as hydrodynamic pressures and forces in half-full spherical containers under arbitrary external excitation. The velocity potential is expressed in a series form, where each term is the product of a time function and the associated spatial function. Because of the spherical configuration, the problem is not separable and the associated spatial functions are nonorthogonal. Application of the boundary conditions results in a system of coupled nonhomogeneous ordinary linear differential equations. The system is solved numerically, implementing a typical fourth-order Runge-Kutta integration scheme. The proposed simple methodology is capable of predicting sloshing effects in half-full spherical containers under arbitrary external excitation in an accurate manner. Hydrodynamic pressures and horizontal forces on the wall of a spherical container are calculated for real earthquake ground motion data. Dissipation effects are included in the present formulation, and their influence on the response is examined. Finally, it is shown that for the particular case of harmonic excitation, the system of ordinary differential equations results in a system of linear algebraic equations, which yields an elegant semianalytical solution.