Three‐dimensional arbitrary Lagrangian–Eulerian numerical prediction method for non‐linear free surface oscillation

Abstract

SUMMARYA numerical prediction method has been proposed to predict non-linear free surface oscillation in an arbitrarily-shaped three-dimensional container. The liquid motions are described with Navier -Stokes equations rather than Laplace equations which are derived by assuming the velocity potential. The profile of a liquid surface is precisely represented with the three-dimensional curvilinear co-ordinates which are regenerated in each computational step on the basis of the arbitrary Lagrangian -Eulerian (ALE) formulation. In the transformed space, the governing equations are discretized on a Lagrangian scheme with sufficient numerical accuracy and the boundary conditions near the liquid surface are implemented in a complete manner. In order to confirm the applicability of the present computational technique, numerical simulations are conducted for the free oscillations of viscid and inviscid liquids and for highly non-linear oscillation. In addition, non-linear sloshing motions caused by horizontal and vertical excitations and a transition from non-linear sloshing to swirling are numerically predicted in three-dimensional cylindrical containers. Conclusively, it is shown that these sloshing motions associated with high non-linearity are reasonably predicted with the present numerical technique.

Sloshing motions in excited tanks

“…Ferrant and Le Touze [15] applied an inviscid pseudo-spectral model to predict 3-D free sloshing. Ushijima [38] used an arbitrary Lagrangian-Eulerian method on boundary-fitted grids to analyse viscous sloshing and swirling effects in a 3-D cylindrical fixed tank.…”

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

“…Using inviscid fluid finite elements, Wu et al focused on near resonance cases in tanks excited by both sway and surge motions. Furthermore, Ushijima (1998) used an arbitrary Lagrangian-Eulerian method on boundary-fitted grids to analyse viscous sloshing and swirling effects in a 3D cylindrical tank.…”

Numerical visualization of free surface oscillation predicted with arbitrary Lagrangian-Eulerian method

“…The unit vectors ni and ti are normal and tangential to the free surface and the stress tensor sij is defined by (8) where p0 is the atmospheric pressure and m is the dynamic viscosity. The detailed forms of these boundary conditions in the transformed space are indicated by Ushijima (1998). In the present paper, it is assumed that the atmospheric pressure equals zero and that the effect of surface tension can be neglected.…”