Supercomputing and the Finite Element Approximation of the Navier-Stokes Equations for Incompressible Viscous Fluids

We discuss in this paper the numerical simulation of unsteady incompressible flows modeled by the Navier-Stokes equations, concentrating most particularly on flows at Reynold number of the order of 10^3 to 10^4. The numerical methodology described here is of modular type and well suited to super computers; it is based on time discretization by operator splitting, and space discretization by low order finite element approximations, leading to highly sparse matrices. The Stokes subproblems originating from the splitting are treated by an efficient Stokes solver, particularly efficient for flow at high Reynold numbers; the nonlinear subproblems associated with the advection are solved by a least squares/preconditioned conjugate gradient method. The methodology discussed here is then applied to the simulation of jets in a cavity, using a CRAY X-MP supercomputer. Various visualizations of the numerical results are presented, in order to show the vortex dynamics taking place in the cavity