Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method

In the smoothed particle hydrodynamics (SPH) discretisation method for the Navier–Stokes equations the most widespread method to solve for pressure and mass conservation is the weakly compressible assumption (WCSPH). This includes hydraulics applications and leads to some drawbacks such as severe artificial pressure fluctuations and a limitation to very small time steps related to the WCSPH Mach number and explicit method. This paper presents comparisons of a semi-implicit and truly incompressible SPH (ISPH) algorithm with the classical WCSPH method, showing how some of the problems encountered in WCSPH have been resolved by using ISPH to simulate incompressible flows. Mathematical models are presented before describing SPH formalism. Several standard boundary conditions are introduced and special attention is given to tracking the surface particles. The lid-driven cavity flow (Re = 400 and 1000) is performed as a benchmarking test. A bluff body test case (a square cylinder in a closed channel, Red = 20 and 100 based on the cylinder diameter) shows that pressure fields extracted from WCSPH are very unreliable whereas ISPH predict pressures and forces in closer agreement with classical finite volume CFD methods. Dam-breaking cases, with dry or wet beds downstream, are then presented to highlight free-surface flow and rapid dynamics effects. The WCSPH and ISPH results are generally verified with reference data from experiment and/or another numerical method. All the comparisons show improvement with ISPH and good agreement in general.