SOME NUMERICAL SCHEMES USING CURVILINEAR COORDINATE GRIDS FOR INCOMPRESSIBLE AND COMPRESSIBLE NAVIER-STOKES EQUATIONS

In this review paper some numerical schemes recently developed by the authors and their coworkers for analysing the cascade flows of turboma chinery are described. These schemes use the curvilinear coordinate gr id and solve the momentum equations of contravariant velocities (volum e flux). The compressible flow schemes are based on the delta-form app roximate-factorization finite-difference scheme, and are improved by u sing the diagonalization, the flux difference splitting and the TVD sc hemes to save computational effort and to increase stability and resol vability. Furthermore, using higher-order compact TVD MUSCL schemes, w e can capture not only shock waves but also contact surfaces very shar ply. On the other hand, the incompressible flow schemes are based on t he well-known SMAC scheme, and are extended to the curvilinear coordin ate grid and further to the implicit scheme to reduce computations. Th ese schemes, like the SMAC scheme, satisfy the continuity condition id entically, and suppress the occurrence of spurious errors. In both the compressible and incompressible schemes, for the turbulent flow the k appa-epsilon turbulence model with the law of the wall or considering the low Reynolds number effects is employed, and for the unsteady flow the Crank-Nicholson method is employed and the solution at each time step is obtained by the Newton iteration. Use of the volume flux inste ad of the physical velocity is inevitable for the MAC type schemes, an d makes it easy to impose boundary conditions. Finally, some calculate d results using the present schemes are shown.