Penalty Function Solution of Steady-State Navier-Stokes Equations

The addition of a penalized term to the Galerkin form of the momentum equations in the finite element solution to the Navier-Stokes equations for steady, incompressible, two dimensional viscous flow is discussed. The implementation of this penalty function finite element method has been examined using three types of rectangular elements. The eight-noded quadratic (serendipity) element is found to show no advantages over the others except that it can be implemented using reduced integration in both matrices without introducing hourglass modes. Biquadratic elements with nine nodes (Lagrangian) are shown to provide a significantly better accuracy than four-noded bilinear elements in such situations as driven cavity flow. The penalty method has been found to provide excellent mass conservation, with stream functions calculated from the computed velocity field virtually path independent.