We analyze numerical approximations for axisymmetric two-phase flow in the arbitrary Lagrangian-Eulerian (ALE) framework. We consider a parametric formulation for the evolving fluid interface in terms of a one-dimensional generating curve. For the two-phase Navier-Stokes equations, we introduce both conservative and nonconservative ALE weak formulations in the 2d meridian half-plane. Piecewise linear parametric elements are employed for discretizing the moving interface, which is then coupled to a moving finite element approximation of the bulk equations. This leads to a variety of ALE methods, which enjoy either an equidistribution property or unconditional stability. Furthermore, we adapt these introduced methods with the help of suitable time-weighted discrete normals, so that the volume of the two phases is exactly preserved on the discrete level. Numerical results for rising bubbles and oscillating droplets are presented to show the efficiency and accuracy of these introduced methods.

翻译：本文在任意拉格朗日-欧拉框架下分析轴对称两相流的数值近似方法。我们针对随时间演变的流体界面，采用基于一维生成曲线的参数化表述。针对两相纳维-斯托克斯方程，我们在二维子午半平面内引入守恒型与非守恒型ALE弱形式。采用分段线性参数单元离散移动界面，并将其与体方程的运动有限元近似相耦合，从而衍生出一系列兼具等分布性质或无条件稳定性的ALE方法。此外，我们借助适当的时间加权离散法向量对这些方法进行改进，使得两相体积在离散层面得到精确保持。通过上升气泡与振荡液滴的数值结果展示了所提出方法的有效性与精度。