Variational generalization of the Green–Naghdi and Whitham equations for fluid sloshing in three-dimensional rotating and translating coordinates

Abstract

This paper derives an averaged Lagrangian functional for dynamic coupling between rigid-body motion and its interior shallow-water sloshing in three-dimensional rotating and translating coordinates; with a time-dependent rotation vector. A new set of variational shallow-water equations (SWEs) and generalized Green–Naghdi equations for the interior fluid sloshing with 3-D rotation vector and translations, and also the equations of motion for the linear momentum and angular momentum of the rigid-body containing shallow water, are derived from the averaged Lagrangian functional, which describes a columnar motion, by using Hamilton’s principle and the Euler–Poincaré variational framework. The generalized Green–Naghdi equations have a form of potential vorticity (PV) conservation, which can be obtained from the particle-relabelling symmetry, and is a combination of the PV derived by Miles and Salmon (1985) and the PV derived by Dellar and Salmon (2005) for geophysical fluid dynamics problems, where the rotation vector varies spatially. By applying the assumption of zero-potential-vorticity flow to the averaged Lagrangian functional, a new set of Boussinesq-like evolution equations are derived, which are a generalization of the Whitham equations for fluid sloshing in three-dimensional rotating and translating coordinates. Moreover, the new variational principles are appended to Luke’s variational principle to present a unified variational framework for the hydrodynamic problem of interactions between gravity-driven potential-flow water waves and a freely floating rigid-body, dynamically coupled to its interior weakly dispersive nonlinear shallow-water sloshing in three dimensions.