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:我们提出并分析两阶段压缩型导航-斯托克斯在轴线设置中流动的不适的有限元素近似值。离散方案基于2D半平面的纳维-斯托克斯方程式的Eularian弱配方,以及不断演变的界面曲线的参数配方。我们使用最低顺序Taylor-Hood和单向线性元素分别将散装和移动界面的纳维尔-斯托克斯配制分离。我们讨论了各种计划,其中包括一个在离散界面上具有均衡分布属性的线性计划,以及良好的体积保护。另一个方案可以无条件地保持稳定,并精确地保存两个阶段的体积。提出了数字结果,以显示采用的方法的稳健性和准确性,用以模拟不断上升的泡泡泡和振动的滴子实验。</s>