Simulation of sloshing motions in fixed and vertically excited containers using a 2-D inviscid σ-transformed finite difference solver

Abstract

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model simulates two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free surface and bed boundary is applied to map the moving free-surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. Predictions of small-amplitude free-surface motions in fixed and vertically excited tanks compare well with second order small perturbation theory. For stable steep waves in the vertically excited tank, the free surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow infinitely large, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep nonbreaking waves. It is computationally quick and accurate. The σ-transformation removes the need for free-surface smoothing for the cases considered herein.