Abstract Herein, we present a phase-field model and its efficient numerical method for incompressible single and binary fluid flows on arbitrarily curved surfaces in a three-dimensional (3D) space. An incompressible single fluid flow is governed by the Navier–Stokes (NS) equation and the binary fluid flow is governed by the two-phase Navier–Stokes–Cahn–Hilliard (NSCH) system. In the proposed method, we use a narrow band domain to embed the arbitrarily curved surface and extend the NSCH system and apply a pseudo-Neumann boundary condition that enforces constancy of the dependent variables along the normal direction of the points on the surface. Therefore, we can use the standard discrete Laplace operator instead of the discrete Laplace–Beltrami operator. Within the narrow band domain, the Chorin’s projection method is applied to solve the NS equation, and a convex splitting method is employed to solve the Cahn–Hilliard equation with an advection term. To keep the velocity field tangential to the surface, a velocity correction procedure is applied. An effective mass correction step is adopted to preserve the phase concentration. Computational results such as convergence test, Kevin–Helmholtz instability, and Rayleigh–Taylor instability on curved surfaces demonstrate the accuracy and efficiency of the proposed method.

中文翻译：

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3D空间任意曲面上两相流的相场模型及其有效数值方法

摘要 在此，我们提出了一种相场模型及其有效的数值方法，用于三维 (3D) 空间中任意曲面上不可压缩的单流体和二元流体流动。不可压缩的单一流体流动由 Navier-Stokes (NS) 方程控制，二元流体流动由两相 Navier-Stokes-Cahn-Hilliard (NSCH) 系统控制。在所提出的方法中，我们使用窄带域来嵌入任意曲面并扩展 NSCH 系统，并应用伪诺依曼边界条件，强制沿表面上点的法线方向的因变量保持不变。因此，我们可以使用标准离散拉普拉斯算子代替离散拉普拉斯-贝尔特拉米算子。在窄带域内，采用Chorin投影法求解NS方程，采用凸分裂法求解带对流项的Cahn-Hilliard方程。为了保持速度场与表面相切，应用了速度校正程序。采用有效的质量校正步骤来保持相浓度。曲面上的收敛测试、凯文-亥姆霍兹不稳定性和瑞利-泰勒不稳定性等计算结果证明了所提出方法的准确性和效率。