Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations

The two-dimensional incompressible Navier-Stokes equations are solved in a time-accurate manner, using the method of pseudocompres sibility. Using this method, subiterations in pseudotime are required to satisfy the continuity equation at each time step. An upwind differencing scheme, based on flux-difference splitting, is used to compute the convective terms. The upwind differencing is biased, based on the sign of the local eigenvalue of the Jacobian matrix of the convective fluxes. Both third-order and fifth-order differencing schemes are used on the convective fluxes throughout the grid's interior. The equations are solved using an implicit line relaxation scheme. This solution scheme is stable and is capable of running at large time steps in pseudo-time, leading to fast convergence for each physical time step. A variety of computed results are presented to validate the present scheme. Results for the flow over an oscillating plate are compared with the exact analytic solution, and good agreement is seen. Excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure. Flow solutions over a circular cylinder with vortex shedding are also presented. Finally, the flow past an airfoil at —90° angle of attack is computed.