Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing

Numerical and analytical results are presented for fluid sloshing, of a two-layer inviscid, incompressible and immiscible fluid with thin layers and a rigid lid, coupled to a vessel which is free to undergo horizontal motion governed by a nonlinear spring. Exact analytical results are obtained for the linear problem, giving the natural frequencies and the resonance structure, particularly between the fluid and vessel. A numerical method for the linear and nonlinear equations is developed based on the high-resolution f-wave-propagation finite volume methods due to Bale et al. (2002) [SIAM Journal on Scientific Computing 24, 955–978], adapted to include the pressure gradient at the rigid-lid, and coupled to a Runge–Kutta solver for the vessel motion. The numerical simulations in the linear limit are compared with the exact analytical solutions. The coupled nonlinear numerical solutions with simulations near the internal 1:1 resonance are presented. Of particular interest is the partition of energy between the vessel and fluid motion.