An SPH Projection Method

A new formulation is introduced for enforcing incompressibility in Smoothed Particle Hydrodynamics (SPH). The method uses a fractional step with the velocity field integrated forward in time without enforcing incompressibility. The resulting intermediate velocity field is then projected onto a divergence-free space by solving a pressure Poisson equation derived from an approximate pressure projection. Unlike earlier approaches used to simulate incompressible flows with SPH, the pressure is not a thermodynamic variable and the Courant condition is based only on fluid velocities and not on the speed of sound. Although larger time-steps can be used, the solution of the resulting elliptic pressure Poisson equation increases the total work per time-step. Efficiency comparisons show that the projection method has a significant potential to reduce the overall computational expense compared to weakly compressible SPH, particularly as the Reynolds number, Re, is increased. Simulations using this SPH projection technique show good agreement with finite-difference solutions for a vortex spin-down and Rayleigh–Taylor instability. The results, however, indicate that the use of an approximate projection to enforce incompressibility leads to error accumulation in the density field.