A consistent second order ISPH for free surface flow

Abstract

The Incompressible Smoothed Particle Hydrodynamics (ISPH) is now a popular numerical method for modelling free surface flows, in particular the breaking waves and violent wave-structures interaction. The ISPH requires the projection approach, leading to solving a pressure Poisson's equation (PPE). Although the accuracy and convergence of the numerical scheme to discretise the Laplacian operator involved in PPE is critical for securing a satisfactory solution of the PPE, the overall performance of the ISPH is also influenced by other key numerical implementations, including (1) estimation of the viscous terms; (2) calculation of the velocity divergence; (3) discretisation of the boundary conditions for the PPE; and (4) evaluation of the pressure gradient. In our previous paper [29], the quadratic semi-analytical finite difference interpolation scheme (QSFDI), which has a leading truncation error at third order derivatives, has been adopted to discretise the Laplacian operator. In this paper, the QSFDI will be adopted, not only for discretising the Laplacian operator, but also for approximating viscous terms, velocity divergence, boundary conditions and pressure gradient. The performance of the newly formulated consistent second order ISPH is assessed by various cases including the oscillating liquid drop, the wave propagation, and the liquid sloshing. The results do not only demonstrate a second order convergence over a limited range of conditions and a higher computational efficiency, i.e., requiring less computational time to achieve the same accuracy, but also show a better mass/energy conservation property and capacity of reproducing a smooth pressure field, than other ISPH models considered in this study.