Development of an artificial compressibility methodology with implicit LU-SGS method

A numerical method has been developed to solve the steady and unsteady inco mpressible Navier-Stokes equations in a two-dimensional, curvilinear coordi nate system. The solution procedure is based on the method of artificial co mpressibility and uses a third-order flux-difference splitting upwind diffe rencing scheme for convective terms and second-order center difference for viscous terms. A time-accurate scheme for unsteady incompressible flows is achieved by using an implicit real time discretization and a dual-time appr oach, which introduces pseudo-unsteady terms into both the mass conservatio n equation and momentum equations. An efficient fully implicit algorithm LU -SGS, which was originally derived for the compressible Eulur and Navier-St okes equations by Jameson and Yoon [1], is developed for the pseudo-compres sibility formulation of the two dimensional incompressible Navier-Stokes eq uations for both steady and unsteady flows. A variety of computed results a re presented to validate the present scheme. Numerical solutions for steady flow in a square lid-driven cavity and over a backward facing step and for unsteady flow in a square driven cavity with an oscillating lid and in a c ircular tube with a smooth expansion are respectively presented and compare d with experimental data or other numerical results.