Comparative study on the accuracy and stability of SPH schemes in simulating energetic free-surface flows

Free-surface flows are of significant interest in Computational Fluid Dynamics (CFD). However, modelling them especially when the free-surface breaks or impacts on solid walls can be challenging for many CFD techniques. Smoothed Particle Hydrodynamics (SPH) has been reported as a robust and stable method when applied to these problems. In modelling incompressible flows using the SPH method an equation of state with a large sound speed is typically used. This weakly compressible approach (WCSPH) results in a stiff set of equations with a noisy pressure field and stability issues at high Reynolds number. As a remedy, the incompressible SPH (ISPH) technique was introduced, which uses a pressure projection technique to model incompressibility. Although the pressure field calculated by ISPH is smooth, the stability of the technique is still an open discussion. An alternative approach is to use an acoustic Riemann solver and replace the particle velocities and pressures by pressures and velocities determined from a Riemann solver. This technique is equivalent to the Godunov method in Eulerian techniques and so will be called the Godunov SPH method (GSPH). However, since the acoustic Riemann solver is a first order approximation of the Riemann solution, it is highly dissipative and cannot be employed in energetic free-surface flows without modification. In this paper, the GSPH method is modified by using the HLLC (Harten Lax and van Leer-Contact) Riemann solver. The accuracy of the modified GSPH technique is further improved by utilising the MUSCL (Monotone Upstream-centred Schemes for Conservation Laws) scheme with Slope-Limiter. This modified GSPH method along with the WCSPH and ISPH techniques are used to study non-linear sloshing flow. The accuracy, stability and efficiency of the techniques are assessed and the results are compared with experimental data.