Smoothed Particles Hydrodynamics (SPH) has been extensively applied in ocean/coastal engineering. In general, the SPH method can be divided into weakly compressible SPH (WCSPH) and incompressible SPH (ISPH). The former is often used due to its higher computational efficiency compared with the latter. The standard weakly compressible SPH has the issue of pressure fluctuations. Two typical WCSPH models developed to eliminate the pressure fluctuations are the δ-SPH and Riemann–SPH. The δ-SPH introduces a diffusive term in the continuity equation and Riemann–SPH uses a Riemann solver which determines the interaction between particles by a simple limiter to decrease the inherent numerical dissipation. In the present work, three schemes (Standard- WCSPH, δ-SPH and Riemann–SPH) are firstly considered to simulate the case of an oscillating drop, as a classical free-surface flow benchmark. The comparison between numerical simulation and theoretical solution on the time variation of energy and semi-major axis are conducted. The detailed pressure field is also discussed. Then test cases of still water and violent sloshing are conducted to further compare three schemes. The numerical results are compared to available analytical and experimental data.


Smoothed Particle Hydrodynamics (SPH) is a mesh-free Lagrangian method, which was firstly developed for studying the astrophysics problems by Gingold and Monaghan (1977) and Lucy (1977), and became very popular in simulating fluid flow for its flexibility in adapting to complex geometries and describing free surface flow (Liu and Liu, 2003). In general, the SPH method can be categorized into two different frameworks of WCSPH and ISPH. The Weakly Compressible SPH limits fluid compressibility by imposing a large speed of sound to the equation of state (Monaghan, 1994), while ISPH (Shao and Lo, 2003) achieves the fluid incompressibility by solving a Poisson Pressure Equation (PPE). The former is often used due to its faster computational time when compared with the latter.

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