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.
翻译:提出一个新的有限要素框架,用于模拟有表面张力的两阶段流的数值模拟。为液态气体接口采用“水平-Set”(LS)法,配有小片四方形(P2)干涉法,以便在有平滑界面的区域达到更高水平的趋同率;为平衡实施连续地表力模型,以考虑到表面张力,并尽可能准确地解决静态问题。鉴于这需要在LS函数使用的离散和压力字段使用的平衡之间,为导航-Stokes和LS倾斜方程式提出了等级P2/P2/P2/P2方案,这些方程式相互紧密结合。这种完全隐含的配方正在使用基于残余的变异性多尺度框架稳定下来。为了提高准确性,并尽可能以最小数量的元素获得最佳的趋同率,建议一种不结构型图象适应方法,尽可能接近P2 LS函数的零值。在两个区域自动展展展,在模拟过程中,以平坦的精度方式使这种精确度的界面在两个区域进行。