Some Steady-State Numerical Solutions for the Incompressible Navier-Stokes Equations Using Artificial Compressibility

摘要

The aim of this paper is the numerical prediction of steady and unsteady flows for an incompressible and viscous fluid in a planar geometry. A numerical solution of the two-dimensional constant-density Navier-Stokes equations for both steady-state and time-dependent problems is presented. Governing equations are written in a general curvilinear coordinates using primitive variables. The method is based on the artificial compressibility approach using an iterative process in pseudo-time at each physical time level to compute the time-accurate solution. The principle consists of introducing a term of pressure pseudo-time derivative in the continuity equation.A technique of splitting for differencing the convective terms is applied in the numerical code. So, the flux differencing is based on the construction of numerical fluxes that are third -order discretised using the Roe method. Concerning viscous terms, a second-order central-difference is adopted. The numerical scheme is obtained by applying the Pade formula for both the time and the pseudo time. By convenience, the scheme used for the pseudo time is fully implicit, whereas the code permits flexibility in choosing different numerical schemes in time. The system of equations is solved with an approximate LU factorization. Results mentioned in this paper include some well-known steady-state cases to validate the elaborated code: 1) a driven-cavity flow for the Reynolds numbers Re= 400, 3200 and 5000; 2) a flow across an asymmetric curvature with Re= 100 ; 3) the case of a flow across a sudden constriction.