Gauss theorem for tensor of any order, such as scalar, vector and second order tensor, is presented with a tensor analysis technique. A general integral conservation form of Navier-Stokes equations in any three-dimensional curvilinear coordinate is derived. A timemarching algorithm coupled with finite volume is applied to discretization of the governing equations. A CFD code is developed to simulate a three dimensional rotating viscous flow field inside an NASA low-speed centrifugal compressor (LSCC) impeller with vaneless diffuser. Numerical algorithm and general integral conservation form of N-S equations are validated with experimental data. It provides a study basis for complex physical region.

The Legendre and adjoint Legendre functions are used to study numerically the steady-state and unsteady, axisymmetric incompressible flow between two concentrically rotating spheres.