A novel finite element framework is proposed for the numerical simulation of two phase flows with surface tension. The Level-Set (LS) method with piece-wise quadratic (P2) interpolation for the liquid-gas interface is used in order to reach higher-order convergence rates in regions with smooth interface. A balanced-force implementation of the continuum surface force model is used to take into account the surface tension and to solve static problems as accurately as possible. Given that this requires a balance between the discretization used for the LS function, and that used for the pressure field, an equal-order P2/P2/P2 scheme is proposed for the Navier-Stokes and LS advection equations, which are strongly coupled with each other. This fully implicit formulation is stabilized using the residual-based variational multiscale framework. In order to improve the accuracy and obtain optimal convergence rates with a minimum number of elements, an anisotropic mesh adaption method is proposed where the unstructured mesh is kept as fine as possible close to the zero iso-value of the P2 LS function. Elements are automatically stretched in regions with flat interface in order to keep the complexity fixed during the simulation. The accuracy and efficiency of this approach are demonstrated for two and three dimensional simulations of a rising bubble.

中文翻译：

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具有表面张力的两相流非结构化各向异性网格自适应的高阶有限元方法

提出了一种新的有限元框架，用于对具有表面张力的两相流进行数值模拟。液-气界面采用带分段二次 (P2) 插值的水平集 (LS) 方法，以便在具有光滑界面的区域中达到更高阶的收敛速度。连续表面力模型的平衡力实现用于考虑表面张力并尽可能准确地解决静态问题。鉴于这需要在用于 LS 函数的离散化和用于压力场的离散化之间取得平衡，针对 Navier-Stokes 和 LS 对流方程提出了等阶 P2/P2/P2 方案，它们与彼此。这种完全隐式的公式使用基于残差的变分多尺度框架来稳定。为了提高精度并以最少的单元数获得最佳收敛速度，提出了一种各向异性网格自适应方法，其中非结构化网格尽可能精细地保持在 P2 LS 函数的零等值附近。元素在具有平坦界面的区域中自动拉伸，以在模拟过程中保持复杂性不变。这种方法的准确性和效率在上升气泡的二维和三维模拟中得到了证明。元素在具有平坦界面的区域中自动拉伸，以在模拟过程中保持复杂性不变。这种方法的准确性和效率在上升气泡的二维和三维模拟中得到了证明。元素在具有平坦界面的区域中自动拉伸，以在模拟过程中保持复杂性不变。这种方法的准确性和效率在上升气泡的二维和三维模拟中得到了证明。