A NUMERICAL-METHOD FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS BASED ON AN APPROXIMATE PROJECTION

In this method we present a fractional step discretization of the time -dependent incompressible Navier-Stokes equations. The method is based on a projection formulation in which we first solve diffusion-convect ion equations to predict intermediate velocities, which are then proje cted onto the space of divergence-free vector fields. Our treatment of the diffusion-convection step uses a specialized second-order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high Reynolds number. in contras t to conventional projection-type discretizations that impose a discre te form of the divergence-free constraint, we only approximately impos e the constraint; i.e., the velocity held we compute is not exactly di vergence-free. The approximate projection is computed using a conventi onal discretization of the Laplacian and the resulting linear system i s solved using conventional multigrid methods. Numerical examples are presented to validate the second-order convergence of the method for E uler, finite Reynolds number, and Stokes now. A second example illustr ating the behavior of the algorithm on an unstable shear layer is also presented.