Implicit numerical scheme based on SMAC method for unsteady incompressible Navier-Stokes equations

Abstract

An implicit numerical scheme is developed based on the simplified marker and cell (SMAC) method to solve Reynolds-averaged equations in general curvilinear coordinates for three-dimensional (3-D) unsteady incompressible turbulent flow. The governing equations include the Reynolds-averaged momentum equations, in which contravariant velocities are unknown variables, pressure-correction Poisson equation and k-ɛturbulent equations. The governing equations are discretized in a 3-D MAC staggered grid system. To improve the numerical stability of the implicit SMAC scheme, the higherorder high-resolution Chakravarthy-Osher total variation diminishing (TVD) scheme is used to discretize the convective terms in momentum equations and k-ɛequations. The discretized algebraic momentum equations and k-ɛequations are solved by the time-diversion multiple access (CTDMA) method. The algebraic Poisson equations are solved by the Tschebyscheff SLOR (successive linear over relaxation) method with alternating computational directions. At the end of the paper, the unsteady flow at high Reynolds numbers through a simplified cascade made up of NACA65-410 blade are simulated with the program written according to the implicit numerical scheme. The reliability and accuracy of the implicit numerical scheme are verified through the satisfactory agreement between the numerical results of the surface pressure coefficient and experimental data. The numerical results indicate that Reynolds number and angle of attack are two primary factors affecting the characteristics of unsteady flow.