Sloshing of a liquid fuel in toroidal tanks with account for capillary effect

A numerical approach is proposed to solve the linear sloshing problem of an incompressible inviscid liquid with account for surface tension effects, which are predominant in the low-gravity environment. A variational formulation is derived by the linearization of motion equations for the liquid near its initial equilibrium state with consideration of a pressure drop on the free surface and a free-end boundary condition on the contact line. The continuous problem domain is discretized by the finite element method. After discretization, the classical generalized eigenvalue problem is obtained, whose solutions are the natural frequencies and mode shapes. Several examples show the effect of the Bond number and the fluid-filled volume on the liquid behavior in toroidal tanks. A comparison of numerical results with experimental measurements under ground conditions reveals that under microgravity condition, the surface tension force and the boundary condition on the contact line play an important role when determining the natural frequencies and mode shapes of the liquid sloshing. Each fluid-filled volume has its own characteristic Bond number, above which the natural frequencies approximate to the experimental values obtained under ground conditions. The presented results can be used in the coupling dynamic analysis of a spacecraft with propellant tanks. The author is grateful to the supervisor associate professor A.N. Temnov for help in formulating the problem and discussion of the results of the work.