Enhanced resolution of the continuity equation in explicit weakly compressible SPH simulations of incompressible free‐surface fluid flows

Abstract

This paper is dedicated to improved resolution of the continuity equation and thus more precise satisfaction of incompressibility conditions in explicit weakly compressible SPH (Smoothed Particle Hydrodynamics) simulations of incompressible free-surface fluid flows. In the explicit weakly compressible SPH methods, including δ-SPH with a widely adopted particle shifting scheme (or the so-called δ-plus-SPH), two incompressibility conditions corresponding to the invariant density condition and divergence-free velocity condition are not well resolved. To achieve enhanced resolution of the continuity equation as well as satisfaction of both incompressibility conditions, this paper presents two novel schemes, namely, Velocity-divergence Error Mitigating (VEM) and Volume Conservation Shifting (VCS) schemes. The VEM scheme corresponds to an additional term in the momentum equation that mitigates the velocity divergence error at every computational time step. The VEM term is derived by linking the instantaneous velocity divergence error to the undesired density time variations and accordingly to an error mitigating pressure pVEM obtained through an Equation of State (EOS). The corresponding pressure gradient term or acceleration leads to attenuation of numerical noise in the velocity divergence field. The VCS acts as a distinct particle shifting scheme to enforce local and thus global volume conservation. This distinct shifting is obtained through calculation of the gradient of a corresponding pressure, namely, pVCS, obtained through efficient solution of a Poisson Pressure Equation (PPE) derived from the concept of projection-based particle methods. By adopting both proposed schemes, the explicit SPH method is shown to provide enhanced satisfaction of both incompressibility conditions, i.e., divergence-free velocity and invariant density conditions, in simulation of incompressible free-surface fluid flows.