Bruno Piar

A finite element penalty-projection method for incompressible flows

The penalty-projection method for the solution of Navier-Stokes equations may be viewed as a projection scheme where an augmentation term is added in the first stage, namely the solution of the momentum balance equation, to constrain the divergence of the predicted velocity field. After a presentation of the scheme in the time semi-discrete formulation, then in fully discrete form for a finite element discretization, we assess its behaviour against a set of benchmark tests, including in particular prescribed velocity and open boundary conditions. The results demonstrate that the augmentation always produces beneficial effects. As soon as the augmentation parameter takes a significant value, the projection method splitting error is reduced, pressure boundary layers are suppressed and the loss of spatial convergence of the incremental projection scheme in case of open boundary conditions does not occur anymore. For high values of the augmentation parameter, the results of coupled solvers are recovered. Consequently, in contrast with standard penalty methods, there is no need for a dependence of the augmentation parameter with the time step, and this latter can be kept to reasonable values, to avoid to degrade too severely the conditioning of the linear operator associated to the velocity prediction step.

Une méthode de pénalité-projection pour les écoulements dilatables

Nous présentons dans cet article une nouvelle méthode de correction de pression pour les écoulements dilatables. Nommée « méthode de pénalité-projection », cette technique diffère des schémas de projection usuels par l’ajout dans l’étape de prédiction d’un terme de pénalisation, construit pour contraindre la vitesse à satisfaire le bilan de masse. Ce terme est multiplié par un coefficient r, dit paramètre de pénalisation. Nous montrons par des expériences numériques que ce schéma est bien plus précis que la méthode usuelle. L’erreur de fractionnement, dominante à fort pas de temps, est réduite à volonté en augmentant r ; à noter, toutefois, que l’usage d’une valeur trop importante dégrade le conditionnement de l’opérateur associé à l’étape de prédiction. Par ailleurs, les pertes de convergence de la méthode de projection usuelle en cas de conditions aux limites ouvertes sont corrigées, dès que r est non nul.

A Staggered Scheme with Non-conforming Refinement for the Navier-Stokes Equations

We propose a numerical scheme for the incompressible Navier-Stokes equations. The pressure is approximated at the cell centers while the vector valued velocity degrees of freedom are localized at the faces of the cells. The scheme is able to cope with unstructured non-conforming meshes, involving hanging nodes. The discrete convection operator, of finite volume form, is built with the purpose to obtain an \(L^2\)-stability property, or, in other words, a discrete equivalent to the kinetic energy identity. The diffusion term is approximated by extending the usual Rannacher-Turek finite element to non-conforming meshes. The scheme is first order in space for energy norms, as shown by the numerical experiments.