Penalized direct forcing and projection schemes for Navier–Stokes

Michel Belliard; Clarisse Fournier

doi:10.1016/j.crma.2010.09.016

Mathematical Problems in Mechanics

Penalized direct forcing and projection schemes for Navier–Stokes
[Méthode de forçage direct pénalisé et schémas de projection pour Navier–Stokes]

Présenté par : Olivier Pironneau

Michel Belliard ¹ ; Clarisse Fournier ²

¹ CEA, département d'études des réacteurs, DEN, DER/SSTH/LDAL - Cadarache, 13108 St Paul-lez-Durance cedex, France
² CEA, département d'études des réacteurs, DEN, DER/SSTH/LDAL, 38054 Grenoble, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1133-1136.

Résumé
Abstract

Cette Note présente une nouvelle méthode de forçage directe pour prendre en compte des obstacles dans un écoulement incompressible. Elle mélange méthodes de projection et de pénalisation $L^{2}$ des vitesses. La ventilation du terme de forçage direct pénalisé dans les équations de prédiction et de correction conduit à un traitement naturel des conditions aux limites pour la correction de pression aux bords des obstacles. Une expérience numérique est présentée à titre d'illustration.

This Note presents a new method of direct forcing to deal with obstacles in incompressible flows. It mixes projection schemes and velocity $L^{2}$ penalty schemes. The penalized direct forcing term is distributed in the velocity prediction and the correction equations. It leads to a natural treatment in the correction equation of the boundary conditions in pressure around obstacles. A numerical experiment provided an illustration of the method.

Reçu le : 2010-04-20
Accepté le : 2010-09-16
Publié le : 2010-10-01

DOI : 10.1016/j.crma.2010.09.016

Affiliations des auteurs :

Michel Belliard ¹ ; Clarisse Fournier ²

¹ CEA, département d'études des réacteurs, DEN, DER/SSTH/LDAL - Cadarache, 13108 St Paul-lez-Durance cedex, France
² CEA, département d'études des réacteurs, DEN, DER/SSTH/LDAL, 38054 Grenoble, France

@article{CRMATH_2010__348_19-20_1133_0,
     author = {Michel Belliard and Clarisse Fournier},
     title = {Penalized direct forcing and projection schemes for {Navier{\textendash}Stokes}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1133--1136},
     publisher = {Elsevier},
     volume = {348},
     number = {19-20},
     year = {2010},
     doi = {10.1016/j.crma.2010.09.016},
     language = {en},
}

TY  - JOUR
AU  - Michel Belliard
AU  - Clarisse Fournier
TI  - Penalized direct forcing and projection schemes for Navier–Stokes
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 1133
EP  - 1136
VL  - 348
IS  - 19-20
PB  - Elsevier
DO  - 10.1016/j.crma.2010.09.016
LA  - en
ID  - CRMATH_2010__348_19-20_1133_0
ER  -

%0 Journal Article
%A Michel Belliard
%A Clarisse Fournier
%T Penalized direct forcing and projection schemes for Navier–Stokes
%J Comptes Rendus. Mathématique
%D 2010
%P 1133-1136
%V 348
%N 19-20
%I Elsevier
%R 10.1016/j.crma.2010.09.016
%G en
%F CRMATH_2010__348_19-20_1133_0

Michel Belliard; Clarisse Fournier. Penalized direct forcing and projection schemes for Navier–Stokes. Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1133-1136. doi : 10.1016/j.crma.2010.09.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.09.016/

Bibliographie
Cité par

[1] Ph. Angot; Ch.-H. Bruneau; P. Fabrie A penalization method to take into account obstacles in incompressible viscous flows, Numer. Math., Volume 81 (1999), pp. 497-520

[2] A. Chorin Numerical solution of the Navier–Stokes equations, Math. Comp., Volume 22 (1968), pp. 745-762

[3] E.A. Fadlun; R. Verzicco; P. Orlandi; J. Mohd-Yusof Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. Comput. Phys., Volume 161 (2000), pp. 35-60

[4] Th. Gallouët; L. Gastaldo; R. Herbin; J.-C. Latché An unconditionally stable pressure correction scheme for the compressible barotropic Navier–Stokes equations, M2AN Math. Model. Numer. Anal., Volume 42 (2008), pp. 303-331

[5] T. Ikeno; T. Kajishima Finite-difference immersed boundary method consistent with wall conditions for incompressible turbulent flow simulations, J. Comput. Phys., Volume 226 (2007), pp. 1485-1508

[6] M. Jobelin; B. Piar; Ph. Angot; J.-C. Latché Une méthode de pénalité-projection pour les écoulements dilatables, European J. Comput. Mech., Volume 17 (2008), pp. 453-480

[7] J. Mohd-Yusof, Combined immersed boundaries/B-splines methods for simulations of flows in complex geometries, CTR Annual Research Briefs, NASA Ames/Stanford University, 1997.

[8] C. Peskin Numerical analysis of blood flow in heart, J. Comput. Phys., Volume 25 (1977), pp. 220-252

[9] J. Shen On error estimates of some higher order projection and penalty-projection methods for Navier–Stokes equations, Numer. Math., Volume 62 (1992), pp. 49-73

[10] R. Temam Sur l'approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires, Arch. Ration. Mech. Anal., Volume 32 (1969), pp. 135-153

[11] http://www-trio-u.cea.fr

Georis Billo; Michel Belliard; Pierre Sagaut Comparison of several interpolation methods to reconstruct field data in the vicinity of a finite element immersed boundary, Computers Mathematics with Applications, Volume 123 (2022), pp. 123-135 | DOI:10.1016/j.camwa.2022.08.002 | Zbl:1524.76133
Georis Billo; Michel Belliard; Pierre Sagaut, INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2020, Volume 2425 (2022), p. 020009 | DOI:10.1063/5.0082404
Georis Billo; Michel Belliard; Pierre Sagaut A finite element penalized direct forcing immersed boundary method for infinitely thin obstacles in a dilatable flow, Computers Mathematics with Applications, Volume 99 (2021), pp. 292-304 | DOI:10.1016/j.camwa.2021.08.005 | Zbl:1524.76132
Adrien Drouillet; Guillaume Bois; Romain Le Tellier; Raphaël Loubère; Mathieu Peybernes Mathematical Modeling and Associated Numerical Simulation of Fusion/Solidification Front Evolution in the Context of Severe Accident of Nuclear Power Engineering, Mathematics, Volume 10 (2021) no. 1, p. 116 | DOI:10.3390/math10010116
Arthur Sarthou; Stéphane Vincent; Jean-Paul Caltagirone Consistent Velocity–Pressure Coupling for Second-Order L2-Penalty and Direct-Forcing Methods, Fluids, Volume 5 (2020) no. 2, p. 92 | DOI:10.3390/fluids5020092
Michel Belliard Numerical modeling of an in-vessel flow limiter using an immersed boundary approach, Nuclear Engineering and Design, Volume 330 (2018), p. 437 | DOI:10.1016/j.nucengdes.2018.01.027
C. Introïni; M. Belliard; C. Fournier A second order penalized direct forcing for hybrid Cartesian/immersed boundary flow simulations, Computers and Fluids, Volume 90 (2014), pp. 21-41 | DOI:10.1016/j.compfluid.2013.10.044 | Zbl:1391.76419
Wim-Paul Breugem; Vincent van Dijk; René Delfos Flows Through Real Porous Media: X-Ray Computed Tomography, Experiments, and Numerical Simulations, Journal of Fluids Engineering, Volume 136 (2014) no. 4 | DOI:10.1115/1.4025311

Cité par 8 documents. Sources : Crossref, zbMATH

Commentaires - Politique