A three-dimensional fully linear finite difference approach to predict waves in tanks excited by ship motions

We developed a three-dimensional (3D) linear frequency domain approach using the finite difference method (FDM) to predict shallow water waves in a partially filled tank or a swimming pool excited by ship motions. This approach assumed that the exciting tank motions are so small that the water motions in the tank are proportional to the exciting tank motions. The dependence of fluid motions (relative to the tank) on the vertical coordinate was assumed as known; hence, calculations could be reduced from three to two spatial dimensions, reducing the computational complexity of this approach. Also, non-rectangular tank shapes can be modeled. Moving side walls (piston-type actuators), which may be installed to dampen or to excite waves in the pool, are taken into account by specific boundary conditions. The bottom friction is accounted for by a non-dimensional parameter which is determined empirically. To validate the FDM approach for a model tank without moving sidewalls, we consider harmonic and irregular tank motions, comparing results of the FDM with results of commercial Computational Fluid Dynamics (CFD) software and of model tests. By solving the unsteady Reynolds-averaged Navier-Stokes (URANS) equations combined with the volume of fluid (VOF) technique, our CFD simulations are carried out over a broad frequency range including sloshing eigenfrequencies of the model tank. Thus, we demonstrate that the FDM approach yields predictions of acceptable accuracy efficiently except near resonance. Based on the comparative CFD simulations, the limits of our FDM for tanks without moving sidewalls are also discussed.