Abstract

We examine the use of natural boundary conditions and conditions of the Sommerfeld type for finite element simulations of convective transport in viscous incompressible flows. We show that natural boundary conditions are superior in the sense that they always provide a correct boundary condition, as opposed to the Sommerfeld‐type conditions, which can lead to a singular formulation and a great loss of accuracy. For the Navier–Stokes equations, the natural boundary conditions must be combined with a simple method to eliminate perturbations on the pressure at the open boundary, which is the source of most errors.

Abstract

The Navier–Stokes equations written in Laplace form are often the starting point of many numerical methods for the simulation of viscous flows. Imposing the natural boundary conditions of the Laplace form or neglecting the viscous contributions on free surfaces are traditionally considered reasonable and harmless assumptions. With these boundary conditions any formulation derived from integral methods (like finite elements or finite volumes) recovers the pure Laplacian aspect of the strong form of the equations. This approach has also the advantage of being convenient in terms of computational effort and, as a consequence, it is used extensively. However, we have recently discovered that these resulting Laplacian formulations violate a basic axiom of continuum mechanics: the principle of objectivity. In the present article we give an accurate account about these topics. We also show that unexpected differences may sometimes arise between Laplace discretizations and divergence discretizations.