Dimension splitting method for the three dimensional rotating Navier-Stokes equations

Abstract

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 \(\Im \) and is decomposed by a series of surfaces \(\Im _i \) into several sub-domains, which are called the layers of the flow. Every interface \(\Im _i \) between two sub-domains shares the same geometry. After establishing a semi-geodesic coordinate (S-coordinate) system based on \(\Im _i \), 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 \(\Im _i \), another one is called the bending operator taking value in the normal space on \(\Im _i \). 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 \(\Im _i \) 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-Stokes equations are presented.