Dimension Splitting Method for the Three Dimensional Rotating Navier-Stokes Equations

• Published in: Acta Mathematicae Applicatae Sinica 2012-07, Vol.28 (3), p.417-442
• Main Authors: Li, Kai-tai, Yu, Jia-ping, Shi, Feng, Huang, Ai-xiang
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Math Applications in Computer Science; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Navier-Stokes方程; Theoretical; 三维旋转; 几何形状; 分裂法; 尺寸; 差分近似; 法线方向; 近似方法
• ISSN: 0168-9673; 1618-3932
• DOI: 10.1007/s10255-012-0161-7

In this paper, we propose a dimensional splitting method for the three dimensional （3D） rotating Navier-Stokes equations. Assume that the domain is a channel bounded by two surfaces and is decomposed by a series of surfaces i into several sub-domains, which are called the layers of the flow. Every interface i between two sub-domains shares the same geometry. After establishing a semi-geodesic coordinate （S-coordinate） system based on i, Navier-Stoke equations in this coordinate can be expressed as the sum of two operators, of which one is called the membrane operator defined on the tangent space on i, another one is called the bending operator taking value in the normal space on i. Then the derivatives of velocity with respect to the normal direction of the surface are approximated by the Euler central difference, and an approximate form of Navier-Stokes equations on the surface i is obtained, which is called the two-dimensional three-component （2D-3C） Navier-Stokes equations on a two dimensional manifold. Solving these equations by alternate iteration, an approximate solution to the original 3D Navier-Stokes equations is obtained. In addition, the proof of the existence of solutions to 2D-3C Navier-Stokes equations is provided, and some approximate methods for solving 2D-3C Navier-Stot4es equations are presented.