Mixture model for two-phase flows with high density ratios: A conservative and realizable SPH formulation

Abstract

The numerical modelling of two-phase mixture flows with high density ratios (e.g. water/air) is challenging. Multiphase averaged models with volume fraction representation encompass a simple way of simulating such flows: mixture models with relative velocity between phases. Such approaches were implemented in SPH (Smoothed Particle Hydrodynamics) using a mass-weighted definition of the mixture velocity, but with limited validation. Instead, to handle high density ratios, a mixture model with a volumetric mixture velocity is developed in this work. To avoid conservation issues raised by the discretization of the relative material displacement contribution in the volume fraction equation, a formulation on phase volumes is derived following a finite volume reasoning. Conservativity, realizability, limit behaviour for single-phase flow are the leading principles of this derivation.Volume diffusion is added to prevent development of instabilities due to the colocated nature of SPH. This model is adapted to the semi-analytical SPH wall boundary conditions. Running on GPU, this approach is successfully applied to the separation of phases in a settling tank with low to high density ratios. An analytical solution on a two-phase mixture Poiseuille flow is also used to check the accuracy of the numerical implementation. Then, a Rayleigh-Taylor instability test case is performed to compare with multi-fluid SPH. Finally, a comparison with experimental and numerical data is made on a sand dumping case; this highlights some limits of this mixture model.

Air Entrainment Modeling in the SPH Method: A Two-Phase Mixture Formulation with Open Boundaries

“…To preserve the symmetry of Fig. 1: Control volume in a two-component flow (gas phase in red, liquid phase in blue), the corresponding volume fractions and velocity fields [21,22].…”

“…SPH particles are no longer assigned to a specific phase: each individual particle carries the different phases with their respective volume fractions and are moved with a unique mixture velocity field. In this work, we aim at testing the performances of the SPH mixture model developed in [21,22] for air-water flows. Nevertheless, this modeling requires to tackle several questions, addressed in this work:…”

“…However, few works considered multifluid open boundaries. We choose here the approach developed in [19] and extend it to the mixture model detailed in [21,22] to allow distinct phases or mixtures to enter and leave the domain.…”

“…The Smoothed Particle Hydrodynamics (SPH) method appears as a natural way to model such highly distorted flows. To avoid computationally prohibitive costs related to the full discretization of bubbles or drops, a mixture model for high density ratio flows relying on a volume-based formulation with relative velocity between phases has first been developed and validated in [21,22]. Instead of having a once and for all assigned phase as in multifluid SPH, each particle now carries both phases through their respective volume fractions.…”

A Robust Method for Wetting Phenomena Within Smoothed Particle Hydrodynamics

“…In the case of multiphase systems the USAW method can be used, e.g. a recent work on the SPH mixture model [55]. Since it was developed with a different purpose in mind, it does not have any specific feature which would make it tempting for interfacial flows modelling.…”