We propose and analyze unfitted finite element approximations for the two-phase incompressible Navier--Stokes flow in an axisymmetric setting. The discretized schemes are based on an Eulerian weak formulation for the Navier--Stokes equation in the 2d-meridian halfplane, together with a parametric formulation for the generating curve of the evolving interface. We use the lowest order Taylor--Hood and piecewise linear elements for discretizing the Navier--Stokes formulation in the bulk and the moving interface, respectively. We discuss a variety of schemes, amongst which is a linear scheme that enjoys an equidistribution property on the discrete interface and good volume conservation. An alternative scheme can be shown to be unconditionally stable and to conserve the volume of the two phases exactly. Numerical results are presented to show the robustness and accuracy of the introduced methods for simulating both rising bubble and oscillating droplet experiments.

翻译：针对轴对称设置下的不可压两相Navier-Stokes流动，我们提出并分析了非拟合有限元逼近方法。离散格式基于二维子午半平面内Navier-Stokes方程的欧拉弱形式，并结合了移动界面生成曲线的参数化表述。我们分别采用最低阶Taylor-Hood元与分段线性元对体区域内的Navier-Stokes方程和移动界面进行离散化。本文讨论多种格式，其中一种线性格式在离散界面上具有均匀分布特性和良好的体积守恒性。另一种替代格式被证明是无条件稳定的，并能精确保持两相体积。数值结果展示了所提方法在模拟气泡上升和液滴振荡实验中的稳健性与精度。