Theoretical Determination of Modal Damping Ratio of Sloshing Using a Variational Method

This paper investigates a variational method for theoretically determining the first damping ratio of sloshing in cylindrical and arbitrary axisymmetric tanks. In this method, a virtual work expression for the viscous terms in the Navier–Stokes equations is transformed into the first-mode damping term, thereby extending Hamilton’s principle for nonviscous sloshing to a variational principle of viscous sloshing. By applying the Galerkin method to the variational principle, a computationally efficient analysis is conducted. For arbitrary axisymmetric tanks, a method for reducing the influence of the change in the fluid velocity boundary condition on the damping estimation is investigated. The proposed method provides theoretical foundations for past empirical results for cylindrical and spherical tanks.