An Unsteady Implicit SMAC Scheme for Two-Dimensional Incompressible Navier-Stokes Equations

Abstract

A finite-difference method based on the SMAC (Simplified Marker and Cell) scheme for analyzing two-dimensional unsteady incompressible viscous flows is developed. The fundamental equations are the incompressible Navier-Stokes equations of contravariant velocities and the elliptic pressure equation in general curvilinear coordinates. With application of the Crank-Nicholson scheme, unsteady flow is calculated iteratively by the Newton method at each time step, and the elliptic pressure equation is solved by the Tschebyscheff SLOR method with alternating the computational directions. Therefore, the elliptic character of incompressible flow is well described. The present implicit scheme is stable under the proper boundary conditions, since spurious error and numerical instabilities can be suppressed by employing the staggered grid and upstream differences such as the modified QUICK scheme. Numerical results for two-dimensional flows through a decelerating cascade at high Reynolds numbers are shown. Some computed results of the surface pressure coefficient are in satisfactory agreement with the experimental data.