A fourth-order C-bracket scheme for nonlinear shallow-water sloshing in an oscillating tank with non-periodic inflow-outflow boundary conditions and biharmonic dissipation.

Abstract

The second-order energy- and potential-enstrophy-conserving numerical scheme introduced by Arakawa & Lamb (1981) for the shallow-water equations with periodic boundary conditions, and extended by Salmon (2004) in the context of Hamiltonian Poisson-bracket discretisation, is further extended to a fourth-order discretisation for the problem of nonlinear shallow-water sloshing over a corrugated bottom surface in a rectangular rigid basin, with symmetric and non-symmetric porous solid side walls and periodic and non-periodic prescribed inflow-outflow boundary conditions, undergoing a prescribed coupled surge-sway motion. Adaptation to a finite domain with periodic and non-periodic inflow-outflow boundary conditions requires a new approach to the boundary conditions at porous solid boundaries, and the ghost cell grid point approximations in the context of the fourth-order finite-difference discretisation on the Arakawa C-grid. Theoretical Poisson-bracket arguments are used to define the required boundary conditions. In addition, standard biharmonic dissipation is incorporated into the numerical modelling to prevent potential enstrophy from accumulating at the smallest resolved scales, which improves the quality and stability of the approximate solutions, enabling long-term integrations to be carried out. The scheme is implemented, shown to preserve the total mass, energy, and potential enstrophy over long-time integration with bounded fluctuations. The presented higher-order C-bracket sloshing integrator provides a robust, stable, fast and precise building block for long-time computational modelling of floating ocean wave energy devices with flexible components, like a flexible bottom surface or membrane for wave energy extraction.