A finite element dimensional splitting method for 3D rotating Navier-Stokes equations

Abstract

In this paper, a finite element splitting method for rotating Navier-Stokes equations is established. It is well known that the channel in turbo-machinery is bounded by two adjacent blades, top surface (shroud), bottom surface (hub), inlet and outlet. The channel is divided into several sub-channels by a series of the surfaces which are isomorphic to the blades surface. Based on a special curvilinear coordinate system, the velocity is approximated by the Hermite finite element with 3 degree and the pressure is approximated by the Lagrange finite element with 1 degree on the rotating direction. The Navier-Stokes equations are expressed into the sum of membrane and bending operators. Applying finite element expansion for the velocity and pressure leads to the finite element algebraic equations whose unknown variable are the approximated function value of the velocity and pressure and derivative value of the velocity with respect to transverse variable at the nodes. Each surface represents a two dimensional manifold in the channel. There are 8 unknown functions on each two dimensional manifold and total unknown functions satisfy a system of partial differential equations. In particular, they are called boundary layer equations on positive blade surface and negative blade surface. The system can be solved by a two scale parallel algorithm which is proposed in this paper.