Comparison of incompressible and weakly-compressible SPH models for free-surface water flows

Free surface flows represent a significant problem in computational fluid dynamics (CFD). The difficulties are multiplied when the free surfaces overturn or impact against solid surfaces, especially if air is entrapped in the process. Although many models capable of simulating such flows have been reported in the literature, particle-based methods such as SPH are arguably the most appealing conceptually and intuitively. In this paper, we model water flows with free surfaces. We assume that the flows are incompressible. There are many different implementations of SPH but there are two main approaches for modelling incompressible flows. In weakly-compressible SPH (WCSPH) computations of water flows, fluid pressure is related to particle density using a stiff equation of state. An alternative is incompressible SPH (ISPH). Here, a Poisson equation is solved to determine the pressure in an approach based upon the orthogonal decomposition method frequently used in grid-based methods. Recent work by Lee et al. (2008) has shown that, under certain circumstances, ISPH performs better than WCSPH for several flows. In this paper, we compare WCSPH and ISPH results for two standard dam-break problems and for regular water waves impacting against a vertical wall. Results are compared with experimental data where possible and show qualitative and quantitative agreement. In our version of WCSPH, MLS or Shephard filtering density 'renormalisation' methods are used and a variety of boundary condition formulations are tested and it is concluded that in the optimum configuration, WCSPH performs at least as well as ISPH, and in some respects clearly performs better.