Smoothed Particle Hydrodynamics (SPH) has intrinsic advantages over the Finite Volume (FV) method for predicting multiphase flows, since the need for fluid phase interface reconstructions is avoided due to the natural advection of those interfaces based on the Lagrangian description of the control volumes. However, permeable boundary conditions are less straightforward to handle with SPH, even though domain inlets and outlets are needed in most engineering problems. In this work we propose a new framework for permeable boundary conditions for the Weakly Compressible SPH based on ghost particles. The major characteristics of our approach are the application of Navier Stokes Characteristic Boundary Conditions (NSCBC) and a particle mass generation and removal procedure at the boundary. This approach enables the application of non-reflective boundary conditions and redundantizes the need for advecting ghost particles from an external buffer zone to the internal domain. Additionally, the particle mass variation can exactly be balanced with the mass flux defined at the boundary. Furthermore, we are proposing a novel approach to set the ghost particle state based on an extrapolation of the state of particles located in the internal domain and on the domain boundary. We validate the proposed framework for permeable boundaries with one- and two-dimensional test cases. The major findings are as follows. Due to the particle mass generation and removal at the domain boundary, spurious oscillations are introduced due to a non-optimal particle distribution. Those oscillations are reduced in our work to an acceptable level ($1\times {10}^{-5}$ to $1\times {10}^{-4}$ relative to the values of the fluid dynamics variables) using existing stabilization schemes as well as the aforementioned novel methods for setting ghost particle states and particle masses at the permeable boundaries. We observe a very good agreement between the solutions obtained with the presented approach and analytical as well as numerical solutions obtained with the FV method. The proposed approach for setting the ghost particle states is indispensable for achieving a perfect balance between particle mass variation and mass flux across the permeable interface as well as the steady state and the temporal accuracy of the solution. Additionally, the pressure reflectivity at the permeable boundary is found to be in a very good agreement with the theory of the NSCBC method. Furthermore, for a SPH-FV coupled simulation we could demonstrate an excellent agreement compared to the FV method as well as the capability of our approach to handle significant flow unsteadiness and gradients at the permeable coupling interfaces without the introduction of spurious flow behavior.

平滑粒子流体动力学 (SPH) 比有限体积 (FV) 方法在预测多相流方面具有内在优势，因为基于控制体积的拉格朗日描述的那些界面的自然平流，避免了对流体相界面重建的需要。然而，即使在大多数工程问题中都需要域入口和出口，但渗透边界条件不太容易用 SPH 处理。在这项工作中，我们提出了一个新的框架，用于基于幽灵粒子的弱可压缩 SPH 的可渗透边界条件。我们方法的主要特征是纳维斯托克斯特征边界条件 (NSCBC) 的应用以及边界处的粒子质量生成和去除程序。这种方法使非反射边界条件的应用成为可能，并使鬼粒子从外部缓冲区平流到内部域的需求变得多余。此外，粒子质量变化可以与边界处定义的质量通量精确平衡。此外，我们提出了一种新的方法来设置鬼粒子状态，该方法基于对位于内部域和域边界上的粒子状态的外推。我们使用一维和二维测试用例验证了所提出的可渗透边界框架。主要发现如下。由于在域边界处粒子质量的产生和去除，由于非最佳粒子分布而引入了虚假振荡。这些振荡在我们的工作中被减少到可以接受的水平（粒子质量变化可以与边界处定义的质量通量精确平衡。此外，我们提出了一种新的方法来设置鬼粒子状态，该方法基于对位于内部域和域边界上的粒子状态的外推。我们使用一维和二维测试用例验证了所提出的可渗透边界框架。主要发现如下。由于在域边界处粒子质量的产生和去除，由于非最佳粒子分布而引入了虚假振荡。这些振荡在我们的工作中被减少到可以接受的水平（粒子质量变化可以与边界处定义的质量通量精确平衡。此外，我们提出了一种新的方法来设置鬼粒子状态，该方法基于对位于内部域和域边界上的粒子状态的外推。我们使用一维和二维测试用例验证了所提出的可渗透边界框架。主要发现如下。由于在域边界处粒子质量的产生和去除，由于非最佳粒子分布而引入了虚假振荡。这些振荡在我们的工作中被减少到可以接受的水平（我们提出了一种新的方法来设置鬼粒子状态，该方法基于位于内部域和域边界上的粒子状态的外推。我们使用一维和二维测试用例验证了所提出的可渗透边界框架。主要发现如下。由于在域边界处粒子质量的产生和去除，由于非最佳粒子分布而引入了虚假振荡。这些振荡在我们的工作中被减少到可以接受的水平（我们提出了一种新的方法来设置鬼粒子状态，该方法基于位于内部域和域边界上的粒子状态的外推。我们使用一维和二维测试用例验证了所提出的可渗透边界框架。主要发现如下。由于在域边界处粒子质量的产生和去除，由于非最佳粒子分布而引入了虚假振荡。这些振荡在我们的工作中被减少到可以接受的水平（$1\times {10}^{-5}$ 到 $1\times {10}^{-4}$相对于流体动力学变量的值）使用现有的稳定方案以及上述用于在可渗透边界设置幽灵粒子状态和粒子质量的新方法。我们观察到用所提出的方法获得的解与用 FV 方法获得的解析和数值解之间非常吻合。所提出的设置鬼粒子状态的方法对于实现粒子质量变化和穿过可渗透界面的质量通量以及解决方案的稳态和时间精度之间的完美平衡是必不可少的。此外，发现渗透边界处的压力反射率与 NSCBC 方法的理论非常吻合。此外，