AN UNSTEADY IMPLICIT SMAC SCHEME FOR 2-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

A finite-difference method based on the SMAC (Simplified Marker and Ce ll) scheme for analyzing two-dimensional unsteady incompressible visco us flows is developed. The fundamental equations are the incompressibl e Navier-Stokes equations of contravariant velocities and the elliptic pressure equation in general curvilinear coordinates. With applicatio n of the Crank-Nicholson scheme, unsteady flow is calculated iterative ly by the Newton method at each time step, and the elliptic pressure e quation is solved by the Tschebyscheff SLOR method with alternating th e computational directions. Therefore, the elliptic character of incom pressible flow is well described. The present implicit scheme is stabl e under the proper boundary conditions, since spurious error and numer ical instabilities can be suppressed by employing the staggered grid a nd upstream differences such as the modified QUICK scheme. Numerical r esults for two-dimensional flows through a decelerating cascade at hig h Reynolds numbers are shown. Some computed results of the surface pre ssure coefficient are in satisfactory agreement with the experimental data.