A dimension split method for the incompressible Navier–Stokes equations in three dimensions


Correspondence to: K. Li, Department of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China.

E-mail: ktli@mail.xjtu.edu.cn


In this paper, we describe a new method for the three-dimensional steady incompressible Navier–Stokes equations, which is called the dimension split method (DSM). The basic idea of DSM is that the three-dimensional space is split up into a cluster of two-dimensional manifolds and then the three-dimensional solution is approximated by the solutions on these two-dimensional manifolds. Through introducing some technologies, such as SUPG stabilization, multigrid method, and such, we firstly make DSM feasible in the computation of real flow. Because of split property of DSM, all computation is carried out on these two-dimensional manifolds, namely, a series of two-dimensional problems only need to be solved in the computation of three-dimensional problem, which greatly reduces the difficulty and the computational cost in the mesh generation. Moreover, these two-dimensional problems can be computed simultaneously and a coarse-grained parallel algorithm would be constructed, whereas the two-dimensional manifold is considered as the computation unit. In the last, we explore the behavior and the accuracy of the proposed method in two numerical examples. Firstly, error estimates, performance of multigrid method, and parallel algorithm are well-demonstrated by the known analytical solution case. Secondly, the computations of three-dimensional lid-driven cavity flows with different Reynolds numbers are compared with other numerical simulations. Results show that the present implementation is able to exhibit good stability and accuracy properties for real flows. Copyright © 2013 John Wiley & Sons, Ltd.