We present a finite element based variational interface-preserving and conservative phase-field formulation for the modeling of incompressible two-phase flows with surface tension dynamics. The preservation of the hyperbolic tangent interface profile of the convective Allen-Cahn phase-field formulation relies on a novel time-dependent mobility model. The mobility coefficient is adjusted adaptively as a function of gradients of the velocity and the order parameter in the diffuse interface region in such a way that the free energy minimization properly opposes the convective distortion. The ratio of the convective distortion to the free energy minimization is termed as the convective distortion parameter, which characterizes the deviation of the diffuse interface profile from the hyperbolic tangent shape due to the convection effect. In the phase-field formulation, the mass conservation is achieved by enforcing a Lagrange multiplier with both temporal and spatial dependence on the phase-field function. We integrate the interface-preserving and conservative phase-field formulation with the incompressible Navier-Stokes equations and the continuum surface tension force model for the simulation of incompressible two-phase flows. A positivity preserving scheme designed for the boundedness and stability of the solution is employed for the variational discretization using unstructured finite elements. We examine the convergence and accuracy of the Allen-Cahn phase-field solver through a generic one-dimensional bistable convection-diffusion-reaction system in a stretching flow. We quantify and systematically assess the relative interface thickness error and the relative surface tension force error with respect to the convective distortion parameter. Two- and three-dimensional rising bubble cases are further simulated to examine the effectiveness of the proposed model on the volume-preserving mean curvature flow and the interface-preserving capability. Finally, we demonstrate the applicability of the proposed model for a case of two bubbles rising and merging with a free surface in an unstructured finite element mesh, which includes complex topological changes and surface tension dynamics.

我们提出了一种基于有限元的变分界面保持和保守相场公式，用于利用表面张力动力学建模不可压缩的两相流。对流艾伦-卡恩相场公式的双曲正切界面轮廓的保留依赖于新型的时变迁移模型。迁移率系数根据扩散界面区域中的速度梯度和阶跃参数进行自适应调整，以使自由能最小化正确地对抗对流畸变。对流变形与自由能最小化之比称为对流变形参数，该参数表征了由于对流效应而导致的扩散界面轮廓与双曲线切线形状的偏差。在相场公式中，通过执行一个Lagrange乘数来实现质量守恒，该乘数在时间和空间上都依赖于相场函数。我们将界面保留和保守相场公式与不可压缩的Navier-Stokes方程和连续表面张力模型相结合，以模拟不可压缩的两相流。使用解的有界性和稳定性设计的正保留方案用于使用非结构化有限元的变分离散化。我们通过拉伸流中的通用一维双稳态对流扩散反应系统来检验Allen-Cahn相场求解器的收敛性和准确性。我们量化并系统地评估相对对流变形参数的相对界面厚度误差和相对表面张力误差。进一步模拟了二维和三维上升气泡情况，以检验该模型对体积保持平均曲率流和界面保持能力的有效性。最后，我们证明了所提出的模型适用于在非结构化有限元网格中两个气泡上升并与自由表面合并的情况，其中包括复杂的拓扑变化和表面张力动力学。进一步模拟了二维和三维上升气泡情况，以检验该模型对体积保持平均曲率流和界面保持能力的有效性。最后，我们证明了所提出的模型适用于在非结构化有限元网格中两个气泡上升并与自由表面合并的情况，其中包括复杂的拓扑变化和表面张力动力学。进一步模拟了二维和三维上升气泡情况，以检验该模型对体积保持平均曲率流和界面保持能力的有效性。最后，我们证明了所提出的模型适用于在非结构化有限元网格中两个气泡上升并与自由表面合并的情况，其中包括复杂的拓扑变化和表面张力动力学。