A fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier-Stokes problems

Philippe Angot; Jean-Paul Caltagirone; Pierre Fabrie

hal-00667121, version 1

A fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier-Stokes problems

Philippe Angot ( Auteur à contacter de préférence ) ¹, Jean-Paul Caltagirone () ^2, 3, Pierre Fabrie () ⁴

Applied Mathematics Letters 25, 11 (2012) 1681--1688

Résumé : We present a new {\em fast vector penalty-projection method (VPP$_{\eps}$)} to efficiently compute the solution of unsteady Navier-Stokes problems governing incompressible multiphase viscous flows with variable density and/or viscosity. 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 {\em two-step approximate divergence-free vector projection} yielding a velocity divergence vanishing as $\cO(\eps\,\dt)$, $\dt$ being the time step, with a penalty parameter $\eps$ as small as desired until the machine precision, {\em e.g.} $\eps=10^{-14}$, whereas the solution algorithm can be extremely fast and cheap. Indeed, the proposed {\em vector correction step} typically requires only a few iterations of a suitable preconditioned Krylov solver whatever the spatial mesh step. The method is numerically validated on three benchmark problems for non-homogeneous or multiphase flows 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 very robust method running faster than usual methods and being able to efficiently and accurately compute 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 Mécanique et d'Ingénierie de Bordeaux (I2M)
Université Sciences et Technologies - Bordeaux I – CNRS : UMR5295 – Arts et Métiers ParisTech – Institut Polytechnique de Bordeaux
4 : Institut de Mathématiques de Bordeaux (IMB)
CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II

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Contributeur : Philippe Angot <>
Soumis le : Mardi 7 Février 2012, 00:01:53
Dernière modification le : Vendredi 20 Juillet 2012, 19:12:27

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DOI : 10.1016/j.aml.2012.01.037

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%% hal-00667121, version 1
%% http://hal.archives-ouvertes.fr/hal-00667121
@article{angot:hal-00667121,
    hal_id = {hal-00667121},
    url = {http://hal.archives-ouvertes.fr/hal-00667121},
    title = {{A fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier-Stokes problems}},
    author = {Angot, Philippe and Caltagirone, Jean-Paul and Fabrie, Pierre},
    abstract = {{We present a new {\em fast vector penalty-projection method (VPP$\_{\eps}$)} to efficiently compute the solution of unsteady Navier-Stokes problems governing incompressible multiphase viscous flows with variable density and/or viscosity. 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 {\em two-step approximate divergence-free vector projection} yielding a velocity divergence vanishing as $\cO(\eps\,\dt)$, $\dt$ being the time step, with a penalty parameter $\eps$ as small as desired until the machine precision, {\em e.g.} $\eps=10^{-14}$, whereas the solution algorithm can be extremely fast and cheap. Indeed, the proposed {\em vector correction step} typically requires only a few iterations of a suitable preconditioned Krylov solver whatever the spatial mesh step. The method is numerically validated on three benchmark problems for non-homogeneous or multiphase flows 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 very robust method running faster than usual methods and being able to efficiently and accurately compute sharp test cases whatever the density, viscosity or anisotropic permeability jumps, whereas other methods crash.}},
    keywords = {Vector penalty-projection method ; Divergence-free penalty-projection ; Penalty method ; Splitting prediction-correction scheme ; Navier-Stokes equations ; incompressible non-homogeneous or multiphase flows},
    language = {Anglais},
    affiliation = {Laboratoire d'Analyse, Topologie, Probabilit{\'e}s - LATP , Transferts, {\'e}coulements, fluides, {\'e}nerg{\'e}tique - TREFLE , Institut de M{\'e}canique et d'Ing{\'e}nierie de Bordeaux - I2M , Institut de Math{\'e}matiques de Bordeaux - IMB},
    pages = {1681--1688},
    journal = {Applied Mathematics Letters},
    volume = {25},
    number = {11 },
    note = {8 pages },
    audience = {internationale },
    doi = {10.1016/j.aml.2012.01.037 },
    year = {2012},
    month = Nov,
    pdf = {http://hal.archives-ouvertes.fr/hal-00667121/PDF/AML2\_ACF11.pdf},
}

%F hal-00667121, version 1
%0 Journal Article
%U http://hal.archives-ouvertes.fr/hal-00667121
%2 MATH:MATH_NA;MATH:MATH_AP;PHYS:MECA:MEFL;PHYS:PHYS:PHYS_FLU-DYN;PHYS:PHYS:PHYS_COMP-PH;SPI:MECA:MEFL;SPI:FLUID
%T A fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier-Stokes problems
%A Angot, Philippe
%A Caltagirone, Jean-Paul
%A Fabrie, Pierre
%+ Laboratoire d'Analyse, Topologie, Probabilités - LATP , Transferts, écoulements, fluides, énergétique - TREFLE , Institut de Mécanique et d'Ingénierie de Bordeaux - I2M , Institut de Mathématiques de Bordeaux - IMB
%Z EDP
%X We present a new {\em fast vector penalty-projection method (VPP$_{\eps}$)} to efficiently compute the solution of unsteady Navier-Stokes problems governing incompressible multiphase viscous flows with variable density and/or viscosity. 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 {\em two-step approximate divergence-free vector projection} yielding a velocity divergence vanishing as $\cO(\eps\,\dt)$, $\dt$ being the time step, with a penalty parameter $\eps$ as small as desired until the machine precision, {\em e.g.} $\eps=10^{-14}$, whereas the solution algorithm can be extremely fast and cheap. Indeed, the proposed {\em vector correction step} typically requires only a few iterations of a suitable preconditioned Krylov solver whatever the spatial mesh step. The method is numerically validated on three benchmark problems for non-homogeneous or multiphase flows 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 very robust method running faster than usual methods and being able to efficiently and accurately compute sharp test cases whatever the density, viscosity or anisotropic permeability jumps, whereas other methods crash.
%G Anglais
%2 2011-02-25
%R 10.1016/j.aml.2012.01.037
%J Applied Mathematics Letters
%2 internationale
%8 2012-11-00
%2 2012-02-09
%V 25
%N 11
%P 1681--1688
%K Vector penalty-projection method ; Divergence-free penalty-projection ; Penalty method ; Splitting prediction-correction scheme ; Navier-Stokes equations ; incompressible non-homogeneous or multiphase flows
%Z MSC[2010]: 35Q30; 35Q35; 49M30; 65M08; 65M85; 65N08; 65N85; 74F10; 76D05; 76D45; 76M12; 76R10; 76T10
%Z 8 pages

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