A practical finite difference scheme for the Navier-Stokes equation on curved surfaces in R-3

Abstract

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. (C) 2020 Elsevier Inc. All rights reserved.