Coupled shallow-water fluid sloshing in an upright annular vessel

The coupled motion of shallow-water sloshing in a horizontally translating upright annular vessel is considered. The vessel's motion is restricted to a single space dimension, such as for Tuned Liquid Damper systems. For particular parameters, the system is shown to support an internal 1:1 resonance, where the frequency of coupled sloshing mode which generates the vessel's motion is equal to the frequency of a sloshing mode which occurs in a static vessel. Using a Lagrangian Particle Path formation, the fully nonlinear motion of the system is simulated using an efficient numerical symplectic integration scheme. The scheme is based on the implicit-midpoint rule which conserves energy and preserves the energy partition between the fluid and the vessel over many time-steps. Linear and nonlinear results are presented, including those showing the system transitioning to higher-frequency eigenmodes as the fluid depth is reduced.