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

Abstract

Air entrainment within water is a common feature of flows over hydraulic works – spill over a dam, wave breaking on a dike, etc. – and its accurate modeling is a key to better design such structures. 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 Fonty et al. (Proceedings of 13th international SPHERIC workshop. Galway, Ireland, 2018; Int J Multiph Flow 111:158–174, 2019. https://doi.org/10.1016/j.ijmultiphaseflow.2018.11.007 ). 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. In the present work, in order to handle practical air entrainment application cases, the open boundary formulation described in Ferrand et al. (Comput Phys Commun 210:29–44, 2017. https://doi.org/10.1016/j.cpc.2016.09.009 ) is adapted to this mixture model. Then, after introducing turbulence through a $$k{-}\epsilon$$ model, a specific closure on the air bubbles relative velocity is proposed including a Stokesian drag term and turbulent diffusion. This model is then applied to two cases of air entrainment: a stepped spillway for interfacial aeration and a plunging jet for local aeration. Finally a 3D industrial test case of discharge-control structure at the La Coche power plant (France) is considered. While valuable insights are obtained for the volume fraction field, further investigations are required to improve the modeling of the flow dynamics.