Analysis of the incompressibility constraint in the smoothed particle hydrodynamics method

SUMMARY Smoothed Particle Hydrodynamics (SPH) is a particle‐based, fully Lagrangian method for fluid‐flow simulations. In this work, fundamental concepts of the method are first briefly recalled. Then, we present a thorough comparison of two different incompressibility treatments in SPH: the weakly compressible approach, where a suitably chosen equation of state is used, and the truly incompressible method (in two basic variants), where the velocity field projection onto a divergence‐free space is performed. A noteworthy aspect of the study is that in each incompressibility treatment, the same boundary conditions are used (and further developed) that allows a direct comparison to be made. Two‐dimensional and three‐dimensional validation cases are studied. Problems associated with the numerical setup are discussed, and an optimal choice of the computational parameters is proposed and verified. The efficiency issues of the incompressibility treatments are considered, and the speed‐up features are highlighted. The results show that the present state‐of‐the‐art truly incompressible methods (based on a velocity correction) suffer from density accumulation errors. To address this issue, an algorithm, based on a correction for both particle velocities and positions, is presented. The usefulness of this density correction is examined and demonstrated.