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In this paper, a new algorithm based on differential geometry viewpoint tosolve the 3D rotating Navier-Stokes equations with complex Boundary isproposed, which is called Bi-parallel algorithm. For xample, it can be appliedto passage flow between two blades in impeller and circulation flow throughaircrafts with complex geometric shape of boundary. Assume that a domain in$R^3$ can be decomposed into a series sub-domain, which is called "flow layer",by a series smooth surface $\Im_k, k=1,...,M$. Applying differential geometrymethod, the 3D Navier-Stokes operator can be split into two kind of operator:the "Membrane Operator" on the tangent space at the surface $\Im_k$ and the"Bending Operator" along the transverse direction. The Bending Operators areapproximated by the finite different quotients and restricted the 3DNaver-Stokes equations on the interface surface $\Im_k$, a Bi-Parallelalgorithm can be constructed along two directors: "Bending" direction and"Membrane" directors. The advantages of the method are that: (1) it can improvethe accuracy of approximate solution caused of irregular mesh nearly thecomplex boundary; (2) it can overcome the numerically effects of boundarylayer, whic is a good boundary layer numerical method; (3) it is sufficiency tosolve a two dimensional sub-problem without solving 3D sub-problem.

Geometry Method for the Rotating Navier-Stokes Equations With Complex Boundary and the Bi-Parallel Algorithm