Linear Solution for Liquid Sloshing in an Upright Elliptical Cylindrical Container with an Eccentric Core Barrel

Abstract

A three-dimensional hydrodynamic analysis founded on the linear water wave theory and separation of variables in elliptical coordinates is undertaken to investigate the free sloshing characteristics in the doubly connected fluid domain of an upright (rigid-walled) cylindrical vessel of elliptical cross section that contains an internal (nonconfocal) elliptical core barrel of arbitrary size, aspect ratio, location, and orientation. The lateral tank boundary (impenetrability) condition is imposed by using the addition theorems for Mathieu functions, and a (4N×4N) truncated set of algebraic equations is eventually obtained after application of the relevant orthogonality relations. Extensive numerical experiments include plots of the first five sloshing wave numbers as functions of the (normalized) barrel eccentricity, for selected tank/barrel aspect ratios, barrel angular locations/orientations, and size. Also, the first four dimensionless natural sloshing frequencies are tabulated for selected liquid elevations and a wide range of geometrical configurations. Moreover, certain two-dimensional images of the liquid free-surface oscillation mode shapes are graphically portrayed. The prominent (moderate) effects of container cross-sectional ellipticity and barrel eccentricity (barrel aspect ratio and angular location/orientation) on the liquid sloshing characteristics are noted. In particular, in the circular tank configuration, several frequency clusters are formed among contiguous frequency branches that progressively merge together as the barrel eccentricity decreases (problem symmetry increases), leading to repeated doublet frequencies. Also, at certain barrel eccentricities, frequency crossovers (mode crossings) between different frequency branches are observed. Accuracy of simulations is confirmed by using a standard finite-element package in addition to comparisons with the existing literature data.