Vector penalty-projection methods for the solution of unsteady incompressible flows - Archive ouverte HAL Access content directly
Conference Papers Year : 2008

## Vector penalty-projection methods for the solution of unsteady incompressible flows

Philippe Angot

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Jean-Paul Caltagirone
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Pierre Fabrie
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#### Abstract

A new family of methods, the so-called two-parameter {\em vector penalty-projection} (VPP$_{r,\eps}$) methods, is proposed where an original {\em penalty-correction step} for the velocity replaces the standard scalar pressure-correction one to calculate flows with divergence-free velocity. This allows us to impose the desired boundary condition to the end-of-step velocity-pressure variables without any trouble. The counterpart to pay back is that in these methods, the cons\-traint on the discrete divergence of velocity is only satisfied approximately as $\cO(\eps\dt)$ within a {\em penalty-correction step} and the penalty parameter $0<\eps\leq 1$ must be decreased until the resulting splitting error is made negligible compared to the time discretization error. However, the crucial issue is that the linear system associated with the projection step can be solved all the more easily as $\eps\dt$ is smaller. Finally, the {\em vector penalty-projection method} (VPP$_{r,\eps}$) has several nice advantages: the Dirichlet or open boundary conditions are not spoiled through a scalar pressure-correction step. Moreover, this method can be generalized in a natural way for variable density or viscosity flows and we show that the vector correction step can be made quasi-independent on the density or viscosity variables (and also on the non-linear terms) if $\eta=\eps\dt$ is taken sufficiently small. These terms can be then neglected in practical schemes.

### Dates and versions

hal-00483583 , version 1 (14-05-2010)

### Identifiers

• HAL Id : hal-00483583 , version 1

### Cite

Philippe Angot, Jean-Paul Caltagirone, Pierre Fabrie. Vector penalty-projection methods for the solution of unsteady incompressible flows. 5th International Symposium on Finite Volumes for Complex Applications, Jun 2008, Aussois, France. pp.169-176. ⟨hal-00483583⟩

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