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 $$k{-}\epsilon$$ k - ϵ 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.

中文翻译：

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SPH 方法中的加气建模：具有开放边界的两相混合物配方

水中夹带空气是水利工程中流动的一个常见特征——溢出水坝、堤坝上的波浪等——其精确建模是更好地设计此类结构的关键。平滑粒子流体动力学 (SPH) 方法似乎是对这种高度扭曲的流动进行建模的一种自然方式。为了避免与气泡或液滴的完全离散化相关的计算成本过高，首先开发并验证了高密度比流动的混合模型，该模型依赖于具有相间相对速度的基于体积的公式。（第 13 届国际 SPHERIC 研讨会论文集。爱尔兰戈尔韦，2018 年；Int J Multiph Flow 111:158–174，2019 年。https://doi.org/10.1016/j.ijmultiphaseflow.2018.11.007）。不像在多流体 SPH 中那样有一个一劳永逸的分配阶段，现在，每个粒子都通过其各自的体积分数携带两个相。在目前的工作中，为了处理实际的加气应用案例，Ferrand 等人描述的开放边界公式。（Comput Phys Commun 210:29–44, 2017. https://doi.org/10.1016/j.cpc.2016.09.009）适用于这种混合模型。然后，在通过 $$k{-}\epsilon$$ k - ϵ 模型引入湍流后，提出了对气泡相对速度的特定闭合，包括斯托克斯阻力项和湍流扩散。然后将该模型应用于两种加气情况：用于界面曝气的阶梯式溢洪道和用于局部曝气的插入式射流。最后考虑了 La Coche 发电厂（法国）的放电控制结构的 3D 工业测试案例。