一种用于具有三 junctions 的多相流的变分前向追踪方法
A variational front-tracking method for multiphase flow with triple junctions
AI总结：
本文提出了一种变分前向追踪方法，用于多相流中具有三 junctions 的尖锐界面模型，通过参数化和欧拉形式化结合，实现精确体积保持和无条件稳定性。
摘要：
我们提出并分析了一种变分前向追踪方法，用于多相流的尖锐界面模型。不同相之间的流体界面在二维空间中用曲线网络表示，在三维空间中用表面簇表示，其中三个界面交汇于三 junctions，以及界面与固定平面边界交汇的边界点/线。该模型由不压缩的纳维-斯托克斯方程在本体域中描述，具有经典界面条件，以及适当在三 junctions 和边界点/线上的边界条件。我们提出了一种弱形式化，结合了用于演化的界面的参数化形式和用于本体方程的欧拉形式化。我们采用不适应的离散化耦合形式，以获得完全离散的有限元方法，其中在弱假设下可以证明解的存在性和唯一性。所构造的方法在离散能量方面具有无条件稳定性结果。此外，我们适应了引入的方法，以确保每个相的精确体积保持性。对于三相流和四相流的数值示例展示了所引入方法的稳健性和准确性。
英文摘要：
We present and analyze a variational front-tracking method for a sharp-interface model of multiphase flow. The fluid interfaces between different phases are represented by curve networks in two space dimensions (2d) or surface clusters in three space dimensions (3d) with triple junctions where three interfaces meet, and boundary points/lines where an interface meets a fixed planar boundary. The model is described by the incompressible Navier--Stokes equations in the bulk domains, with classical interface conditions on the fluid interfaces, and appropriate boundary conditions at the triple junctions and boundary points/lines. We propose a weak formulation for the model, which combines a parametric formulation for the evolving interfaces and an Eulerian formulation for the bulk equations. We employ an unfitted discretization of the coupled formulation to obtain a fully discrete finite element method, where the existence and uniqueness of solutions can be shown under weak assumptions. The constructed method admits an unconditional stability result in terms of the discrete energy. Furthermore, we adapt the introduced method so that an exact volume preservation for each phase can be achieved for the discrete solutions. Numerical examples for three-phase flow and four-phase flow are presented to show the robustness and accuracy of the introduced methods.
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