On a marching level-set method for extended discontinuous Galerkin methods for incompressible two-phase flows: Benchmark Data

In this work a solver for instationary two-phase flows on the basis of the extended Discontinuous Galerkin (extended DG/XDG) method is presented. The XDG method adapts the approximation space conformal to the position of the interface. This allows a sub-cell accurate representation of the incompressible Navier-Stokes equations in their sharp interface formulation. The interface is described as the zero set of a signed-distance level-set function and discretized by a standard DG method. For the interface, resp. level-set, evolution an extension velocity field is used and a two-staged algorithm is presented for its construction on a narrow-band. On the cut-cells a monolithic elliptic extension velocity method is adapted and a fast-marching procedure on the neighboring cells. The spatial discretization is based on a symmetric interior penalty method and for the temporal discretization a moving interface approach is adapted. A cell agglomeration technique is utilized for handling small cut-cells and topology changes during the interface motion. The method is validated against a wide range of typical two-phase surface tension driven flow phenomena including capillary waves, an oscillating droplet and the rising bubble benchmark.

本研究提出了一种基于扩展间断伽辽金（extended Discontinuous Galerkin，XDG/extended DG）方法的非定常两相流求解器。该XDG方法将逼近空间调整为与界面位置共形的形式，可实现不可压缩纳维-斯托克斯方程（Navier-Stokes equations）锐界面格式的亚单元精度数值表征。界面被描述为符号距离水平集（signed-distance level-set）函数的零水平集，并通过标准间断伽辽金（Discontinuous Galerkin, DG）方法进行离散。针对界面（即水平集）的演化过程，本研究采用扩展速度场，并提出了一种在窄带上构建该速度场的两阶段算法：在割单元上采用整体椭圆型扩展速度方法，在相邻单元上部署快速行进算法。空间离散采用对称内部罚函数法，时间离散则采用移动界面格式。本研究采用单元聚合技术处理界面运动过程中的小型割单元与拓扑结构变化。该方法通过一系列典型的表面张力驱动两相流动现象进行验证，涵盖毛细波、振荡液滴以及上升气泡基准算例。