On general form of Navier-Stokes equations and implicit factored scheme

A general weak conservative form of Navier-Stokes equations expressed with respect to non-orthogonal curvilinear coordinates and with primitive variables was obtained by using tensor analysis technique, where the contravariant and covariant velocity components were employed. Compared with the current coordinate transformation method, the established equations are concise and forthright, and they are more convenient to be used for solving problems in body-fitted curvilinear coordinate system. An implicit factored scheme for solving the equations is presented with detoiled discussions in this paper. For n-dimensional flow the algorithm requires n-steps and for each step only a block tridiagonal matrix equation needs to be solved. It avoids inverting the matrix for large systems of equations and enhances the speed of arithmetic. In this study, the Beam-Warming's implicit factored schceme is extended and developed in non-orthogonal curvilinear coordinate system.