Detailed study on the extension of the delta-SPH model to multi-phase flow

In the present work the multi-phase SPH model presented in Grenier et al. (2009) is considered and extended through the inclusion of a diffusive term in the continuity equation. The latter based on the ?-SPH model of Antuono et al. (2012), allows to improve the evaluation of the pressure field, removing numerical noise and improving also the particles spatial distribution. 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. To this aim, comparisons are provided to a Riemann-SPH multi-phase model. In particular we show that the proposed ?-SPH multi-phase solver has a different stability region than its Riemann-SPH counterpart, allowing for bigger time steps for some density and speed of sound ratios, while in other conditions the Riemann-SPH multi-phase model is more convenient. A series of validating tests are carried out over different benchmarks widely used in the SPH literature. As a final test-case the water entry of a corrugated panel involving the entrapment of an air cavity is considered to show how the proposed multi-phase ?-SPH method is able to accurately treat complex water impact events.