The violation of objectivity in Laplace formulations of the Navier-Stokes equations

• Published in: International journal for numerical methods in fluids Vol. 54; no. 6-8; pp. 639 - 664
• Main Authors: Limache, A., Idelsohn, S., Rossi, R., Oñate, E.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 30.06.2007; Wiley
• Subjects: Computational methods in fluid dynamics; Exact sciences and technology; finite element method; Fluid dynamics; free surfaces; Fundamental areas of phenomenology (including applications); Hydrodynamic waves; natural boundary conditions; Navier-Stokes equations; objectivity; Physics; Laplace operator; Computational fluid dynamics; Digital simulation; Boundary conditions; natural boundary conditions; objectivity; Finite element method; Free surface flow; Modelling; Viscous fluids; Numerical convergence; Navier-Stokes equations; free surfaces
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.1480

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. Copyright © 2007 John Wiley & Sons, Ltd.