Discontinuous Galerkin methods for two-phase flows with soluble surfactants

研究的总体目标是开发和实现一个高精度的数值求解器，用于计算包括相界面和界面输运方程在内的多相流。具体来说，我们打算将新开发的DG库BoSSS （Bounded Support Spectral Solver，有界支持谱求解器）扩展为单相流，采用结合切割单元法和非光滑基函数的界面跟踪方法。为了精确计算被相界面分割的单元的表面积分和体积积分，提出了切割单元法。此外，诸如Heaviside函数之类的非光滑基函数允许在不同流体性质之间进行明确的划分。在这方面，界面跟踪方法基于水平集和流体体积，或者很可能是两者的结合，以确保局部质量保存和高度精确的前方位置。最后，界面方程打算以类似于水平集公式的方式求解，其中界面是一个二维流形，嵌入并在三维流形中求解。与水平集方法的带符号距离函数类似，需要建立一定的嵌入条件，将二维流形上的界面输运方程耦合到三维流形中。关键的目标是，后一种方法的结合可以避免经典方案的严重缺点，如界面上的质量缺陷或由于表面曲率计算中的数值误差而人为诱导的流动，仅举几例。

The general research objective is to develop and implement a high-accurate numerical solver for the computation of multiphase flows including phase interfaces and interfacial transport equations with the Discontinuous Galerkin (DG) method. Specifically, it is intended to extend the newly developed DG library BoSSS (Bounded Support Spectral Solver) for single phase flows by an interface tracking method combined with the cut-cell method and non-smooth basis functions. The cut-cell method is proposed to accurately compute surface and volume integrals for those cells which are split by the phase interface. Further the non-smooth basis functions such as the Heaviside function allow for a sharp delimitation between different fluid properties. In this connection the interface tracking method is based on level-set and volume-of-fluid, or most likely a combination of both, to ensure both local mass preservation and a highly accurate front position. Finally, the interfacial equation is intended to be solved in an analogous fashion to the level-set formulation where the interface, a two-dimensional manifold, is embedded and solved in a manifold of dimension three. Similar to the signed-distance function for the level-set method certain embedding conditions are to be developed to couple the interfacial transport equations, which live on a two-dimensional manifold, into a three-dimensional manifold. The key goal is that the combination of the latter methods may avoid serious shortcomings of classical schemes such as mass deficit at the interface or artificially induced flows due to numerical errors in the computation of surface curvature to name only a few.