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International Symposium FVCA6, Prague : Tchèque, République (2011)

A Spectacular Vector Penalty-Projection Method for Darcy and Navier-Stokes Problems

Philippe Angot (  Corresponding author ) 1, Jean-Paul Caltagirone 2, Pierre Fabrie 3

(2011-06)

We present a new fast vector penalty-projection method (VPP$_{\eps}$), issued from noticeable improvements of previous works [7, 3, 4], to efficiently compute the solution of unsteady Navier-Stokes/Brinkman problems governing incompressible multiphase viscous flows. The method is also efficient to solve anisotropic Darcy problems. The key idea of the method is to compute at each time step an accurate and curl-free approximation of the pressure gradient increment in time. This method performs a two-step approximate divergence-free vector projection yielding a velocity divergence vanishing as $\mathcal{O}(\eps\, \dt)$, $\dt$ being the time step, with a penalty parameter $\eps$ as small as desired until the machine precision, e.g. $\eps= 10^{−14}$, whereas the solution algorithm can be extremely fast and cheap. The method is numerically validated on a benchmark problem for two-phase bubble dynamics where we compare it to the Uzawa augmented Lagrangian (UAL) and scalar incremental projection (SIP) methods. Moreover, a new test case for fluid-structure interaction problems is also investigated. That results in a robust method running faster than usual methods and being able to efficiently compute accurate solutions to sharp test cases whatever the density, viscosity or anisotropic permeability jumps, whereas other methods crash.

1:	Laboratoire d'Analyse, Topologie, Probabilités (LATP)
	CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III
2:	Transferts, écoulements, fluides, énergétique (TREFLE)
	CNRS : UMR8508 – Université Sciences et Technologies - Bordeaux I – École Nationale Supérieure de Chimie et de Physique de Bordeaux (ENSCPB) – Arts et Métiers ParisTech
3:	Institut de Mathématiques de Bordeaux (IMB)
	CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II

Subject

:

Mathematics/Numerical Analysis Engineering Sciences/Mechanics/Fluids mechanics Physics/Mechanics/Mechanics of the fluids

Keyword(s): Vector penalty-projection – Penalty method – Splitting method – Multiphase Navier-Stokes/Brinkman – Anisotropic Darcy problem – Incompressible flows

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From: Philippe Angot <>

Submitted on: Saturday, 23 July 2011 12:08:56

Updated on: Saturday, 23 July 2011 13:08:42