In this article, we propose a practical and highly efficient finite difference approach for two-phase fluid simulations on three-dimensional (3D) surfaces. The hydrodynamically coupled interfacial motion is captured by using the conservative Allen–Cahn–Navier–Stokes (CACNS) equations. By adopting the closest point method and the pseudo-Neumann boundary condition, the direct computations on curved surfaces are transferred to the 3D simulations in a narrow band domain embedding the surface. The projection method with pressure correction is used to decouple the computations of velocity and pressure. The operator splitting method is used to split the calculation of conservative Allen–Cahn equation into subproblems and the nonlinear part can be analytically solved. Therefore, the whole computation in each time iteration is highly efficient and easy to implement. The numerical experiments on various 3D curved surfaces are investigated to show the good performance of the proposed method.

在本文中，我们提出了一种实用且高效的有限差分方法，用于三维 (3D) 表面上的两相流体模拟。通过使用保守的 Allen-Cahn-Navier-Stokes (CACNS) 方程捕获流体动力学耦合的界面运动。通过采用最近点法和伪诺依曼边界条件，将曲面上的直接计算转移到嵌入曲面的窄带域中的 3D 模拟中。带有压力校正的投影方法用于解耦速度和压力的计算。算子分裂法用于将保守的 Allen-Cahn 方程的计算拆分为子问题，非线性部分可以解析求解。所以，每次迭代的整个计算效率高且易于实现。研究了各种 3D 曲面上的数值实验，以表明所提出方法的良好性能。