LOW-GRAVITY PROPELLANT SLOSH ANALYSIS USING SPHERICAL COORDINATES

Abstract

Low-gravity sloshing in a convex axisymmetrical container subjected to lateral excitation is formulated by a variational principle and is solved analytically by a modal analysis method. The use of spherical coordinates enables us: (i) to determine analytically the system of characteristic functions for an arbitrary convex container, for which time-consuming and expensive numerical methods have been used in the past; (ii) to express the liquid surface and its dynamical displacement in the form of a single-valued function even when the liquid surface curves strongly, due to surface tension; and (iii) to satisfy the compatibility condition for the liquid surface displacement at the container wall. The variational principle is transformed into a frequency equation in the form of a standard eigenvalue problem by the Galerkin method, in which admissible functions for the velocity potential and the liquid surface displacement are determined analytically in terms of the Gauss hypergeometric series. Since this process is analytical and the dimension of the eigenvalue problem for obtaining a sufficiently converged solution is very low, little computation time and cost are involved. Numerical results show that: (a) the influences of the Bond number on the eigenfrequency are different for high and low liquid-filling levels; (b) neglecting the surface tension underestimates the magnitude of the surface oscillation; and (c) the liquid depth yielding the maximum slosh force and moment increases with decreasing Bond number.