We present a practical finite difference scheme for the incompressible Navier–Stokes equation on curved surfaces in three-dimensional space. In the proposed method, the curved surface is embedded in a narrow band domain and the governing equation is extended to the narrow band domain. We use the standard seven-point stencil for the Laplace operator instead of a discrete Laplacian–Beltrami operator by using the closet point method and pseudo-Neumann boundary condition. The well-known projection method is used to solve the incompressible Navier–Stokes equation in the narrow band domain. To make the velocity field be parallel to the surface, a velocity correction step is used. Various numerical experiments, such as the divergence-free test, the convergence rate, and the energy dissipation, are performed on curved surfaces, which demonstrated that our proposed method is robust and practical.

我们为三维空间中的曲面上的不可压缩Navier–Stokes方程提供了一个实用的有限差分方案。在所提出的方法中，曲面被嵌入在窄带域中，并且控制方程被扩展到窄带域。通过使用隐蔽点法和伪Neumann边界条件，我们对拉普拉斯算子使用标准的七点模具，而不是离散的拉普拉斯–贝特拉米算子。众所周知的投影方法用于解决窄带域中不可压缩的Navier–Stokes方程。为了使速度场平行于表面，使用了速度校正步骤。在曲面上进行了各种数值实验，例如无散度测试，收敛速度和能量耗散，