FINITE-ELEMENT SOLUTION OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS BY A HELMHOLTZ VELOCITY DECOMPOSITION

Finite element solution methods for the incompressible Navier-Strokes equations in primitive variables form are presented. To provide the necessary coupling and enhance stability, a dissipation in the form of a pressure Laplacian is introduced into the continuity equation. The recasting of the problem in terms of pressure and an auxiliary velocity demonstrates how the error introduced by the pressure dissipation can be totally eliminated while retaining its stabilizing properties. The method can also be formally interpreted as a Helmholtz decomposition of the velocity vector. The governing equations are discretized by a Galerkin weighted residual method and, because of the modification to the continuity equation, equal interpolations for all the unknowns are permitted. Newton linearization is used and at each iteration the linear algebraic system is solved by a direct solver. Convergence of the algorithm is shown to be very rapid. Results are presented for two-dimensional flows in various geometries.