δ-SPH model for multi-phase flow: how to correctly select the sound speeds of the different phases

In the present work the multi-phase SPH model presented in Grenier et al (2019) [9] is considered and extended
in order to include a diffusive term in the continuity equation. The latter based on the δ-SPH model of Antuono et al. (2012)
[1], allows to improve the evaluation of the pressure field, removing numerical noise and improving also the particles spatial distribution. Beside the δ-SPH model, also the Riemann-SPH version is considered, indeed both these models can handle multi-
phase flow in presence of a free-surface. The time stepping and the choice of the speeds of sound for the different phases are discussed, showing that this choice is driven not only by physical consideration but also by numerical constraints linked to the stability of the scheme.In this work we show that the two SPH variants, δ− and Riemann- SPH, have different regions of stability when a multi-phase flow is considered. It is interesting that for some density and speed of sound ratios the δ-SPH model allows for bigger time steps, but in other conditions it is the Riemann-SPH model to be more convenient. This is just a preliminary study on the stability constraints linked to multi-phase flow solved by SPH models. The topic deserves a further investigation in order to analyse
theoretically the stability limits and derive possible solutions to go beyond the intrinsic stability boundaries of a specific SPH
model. In the last section, the simulation of a water-entry problem with air entrainment is presented in order to show how the proposed SPH method is able to treat such kind of violent multi-phase flows.