Immersed boundary methods for the numerical simulation of incompressible aerodynamics and fluid-structure interactions
[Méthodes de frontière immergée pour la simulation numérique en aérodynamique incompressible et interactions fluide-structure.]
Annales mathématiques Blaise Pascal, Tome 20 (2013) no. 1, pp. 139-173.

Dans ce travail, trois méthodes de frontière immergée sont décrites et validées pour la simulation numérique en aérodynamique incompressible et interactions fluide-structure. Ces trois approches sont : une méthode Cut Cell, une méthode Vortex-Penalisation et une méthode de forçage. Les deux premières techniques sont validées pour l’écoulement autour d’un obstacle cylindrique. La dernière est utilisée pour prédire les déformations d’une membrane élastique immergée dans un fluide. Ce papier confirme la capacité de cette famille de schémas numériques à simuler les écoulements incompressibles de manière précise et robuste.

In this work three branches of Immersed Boundary Methods (IBM) are described and validated for incompressible aerodynamics and fluid-structure interactions. These three approaches are: Cut Cell method, Vortex-Penalization method and Forcing method. The first two techniques are validated for external bluff-body flow around a circular obstacle. The last one is used to predict the deformations of an elastic membrane immersed in a fluid. The paper confirms the ability of this family of numerical schemes for accurate and robust simulation of incompressible flows.

DOI : 10.5802/ambp.324
Classification : 74F10, 65M06, 76D05
Mots clés : Immersed boundary method, Momentum forcing method, Vortex penalization method, Cut-cell method, Incompressible viscous flows, Complex geometry
James, Nicolas 1 ; Maitre, Emmanuel 2 ; Mortazavi, Iraj 3

1 LMA Université de Poitiers UMR CNRS 7348 Téléport 2 - BP 30179 Bd Marie et Pierre Curie 86962 Chasseneuil FRANCE
2 LJK Université de Grenoble UMR CNRS 5224 Tour IRMA, BP 53 51, rue des Mathématiques 38041 Grenoble Cedex 9 FRANCE
3 IMB Université de Bordeaux UMR CNRS 5251 MC 2 INRIA Bordeaux Sud-Ouest 351, cours de la libération 33405 Talence FRANCE
@article{AMBP_2013__20_1_139_0,
     author = {James, Nicolas and Maitre, Emmanuel and Mortazavi, Iraj},
     title = {Immersed boundary methods for the numerical simulation of incompressible aerodynamics and fluid-structure interactions},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {139--173},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {20},
     number = {1},
     year = {2013},
     doi = {10.5802/ambp.324},
     zbl = {06299067},
     mrnumber = {3112242},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ambp.324/}
}
TY  - JOUR
AU  - James, Nicolas
AU  - Maitre, Emmanuel
AU  - Mortazavi, Iraj
TI  - Immersed boundary methods for the numerical simulation of incompressible aerodynamics and fluid-structure interactions
JO  - Annales mathématiques Blaise Pascal
PY  - 2013
SP  - 139
EP  - 173
VL  - 20
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - http://www.numdam.org/articles/10.5802/ambp.324/
DO  - 10.5802/ambp.324
LA  - en
ID  - AMBP_2013__20_1_139_0
ER  - 
%0 Journal Article
%A James, Nicolas
%A Maitre, Emmanuel
%A Mortazavi, Iraj
%T Immersed boundary methods for the numerical simulation of incompressible aerodynamics and fluid-structure interactions
%J Annales mathématiques Blaise Pascal
%D 2013
%P 139-173
%V 20
%N 1
%I Annales mathématiques Blaise Pascal
%U http://www.numdam.org/articles/10.5802/ambp.324/
%R 10.5802/ambp.324
%G en
%F AMBP_2013__20_1_139_0
James, Nicolas; Maitre, Emmanuel; Mortazavi, Iraj. Immersed boundary methods for the numerical simulation of incompressible aerodynamics and fluid-structure interactions. Annales mathématiques Blaise Pascal, Tome 20 (2013) no. 1, pp. 139-173. doi : 10.5802/ambp.324. http://www.numdam.org/articles/10.5802/ambp.324/

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

[2] Beale, J. T.; Strain, J. Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces, J. Comp. Phys., Volume 227 (2008) no. 8, pp. 3896-3920 | DOI | MR | Zbl

[3] Boffi, D.; Gastaldi, L.; Heltai, L. Numerical stability of the finite element immersed boundary method, M3AS, Volume 17 (2007), pp. 1479-1505 | MR | Zbl

[4] Boffia, D.; Gastaldi, L.; Heltai, L. Stability results and algorithmic strategies for the finite element approach to the immersed boundary method, Proceeding of the Sixth European Conference on Numerical Mathematics and Advanced Applications (2005), pp. 557-566 (preprint available on http://www.ing.unibs.it/~gastaldi/paper.html) | MR

[5] Bohnet, S.; Ananthakrishnan, R.; Mogilner, A.; Meister, J.-J.; Verkhovsky, A. Weak force stalls protrusion at the leading edge of the lamellipodium, Biophys. J., Volume 90 (2006), pp. 1810-1820 | DOI

[6] Bouchon, F.; Dubois, T.; James, N. A second-order cut-cell method for the numerical simulation of 2D flows past obstacles, Computers and Fluids, Volume 65 (2012), pp. 80-91 | DOI | MR

[7] Bresch, D.; Colin, T.; Grenier, E.; Ribba, B.; Saut, O. Computational modeling of solid tumor growth: the avascular stage, SIAM Journal on Scientific Computing, Volume 32 (2010) no. 4, pp. 2321-2344 | DOI | MR | Zbl

[8] Bresch, D.; Colin, Th.; Grenier, E.; Ribba, B.; Saut, O.; Singh, O.; Verdier, C. Quelques méthodes de paramètre d’ordre avec applications à la modélisation de processus cancéreux, ESAIM: Proceedings, Volume 18 (2007), pp. 163-180 | DOI | MR

[9] Bruneau, Ch. -H.; Mortazavi, I.; Gilliéron, P. Passive control around the two-dimensional square back Ahmed body using porous devices, J. Fluids Eng., Volume 130 (2008) | DOI

[10] Chang, Y.C.; Hou, T.Y.; Merriman, B.; Osher, S. A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows, J. Comp. Phys., Volume 124 (1996), pp. 449-464 | DOI | MR | Zbl

[11] Cheny, Y.; Botella, O. The LS-STAG method: A new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties, J. Comp. phys., Volume 229 (2010), pp. 1043-1076 | DOI | MR

[12] Chorin, A.J. Vortex sheet approximation of boundary layers, J. Comput. Phys., Volume 27 (1978) | DOI | Zbl

[13] Chung, M.-H. Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape, Computers and Fluids, Volume 35 (2006) no. 6, pp. 607-623 | DOI | Zbl

[14] Coquerelle, M.; Allard, J.; Cottet, G. -H.; Cani, M. -P. A Vortex Method for Bi-phasic Fluids Interacting with Rigid Bodies, Arxiv preprint math, LMC-IMAG (2006)

[15] Coquerelle, M.; Cottet, G. -H. A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies, J. Comput. Phys., Volume 227 (2008) | DOI | MR | Zbl

[16] Cortez, R.; Peskin, C.S.; Stockie, J.M.; Varela, D. Parametric resonance in immersed elastic boundaries, SIAM Journal on Applied Mathematics, Volume 65 (2004) no. 2, pp. 494-520 | DOI | MR | Zbl

[17] Cottet, G. -H.; Gallizio, F.; Magni, A.; Mortazavi, I. A vortex immersed boundary method for bluff body flows, ASME Summer Meeting, Montreal, Volume FEDSM-ICNMM2010-30787 (2010)

[18] Cottet, G. -H.; Koumoutsakos, P. Vortex Methods: Theory and Practice, 2000 | MR

[19] Cottet, G.-H.; Maitre, E. A level-set formulation of immersed boundary methods for fluid-structure interaction problems, C. R. Math., Volume 338 (2004) no. 7, pp. 581-586 | DOI | MR | Zbl

[20] Cottet, G.-H.; Maitre, E. A level set method for fluid-structure interactions with immersed surfaces, Math. Models Meth. Appl. Sci., Volume 16 (2006) no. 3, pp. 415-438 | DOI | MR | Zbl

[21] Cottet, G.-H.; Maitre, E.; Milcent, T. Eulerian formulation and level set models for incompressible fluid-structure interaction, ESAIM-Math. Model. Numer. Anal., Volume 42 (2008), pp. 471-492 | DOI | Numdam | MR | Zbl

[22] Creusé, E.; Giovannini, A.; Mortazavi, I. Vortex simulation of active control strategies for transitional backward-facing step flows, Computers & Fluids, Volume 38 (2009) | DOI | MR | Zbl

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

[24] Griffith, B.E.; Peskin, C.S. On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems, J. Comp. Phys., Volume 208 (2005), pp. 75-105 | DOI | MR | Zbl

[25] Harlow, F. H.; Welch, J. E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, Volume 12 (1965), pp. 2182-2189 | DOI | Zbl

[26] Kim, J.; Kim, D.; Choi, H. An immersed-boundary finite volume method for simulation of flow in complex geometries, J. Comput. Phys., Volume 171 (2001), pp. 132-150 | DOI | MR | Zbl

[27] Lee, L.; Leveque, R.J. An immersed interface method for incompresible Navier-Stokes equations, SIAM J. Sci. Comp., Volume 25 (2003) no. 3, pp. 832-856 | DOI | MR | Zbl

[28] LeVeque, R. J.; Li, Z. The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources, SIAM J. Numer. Anal., Volume 31 (1994), pp. 1019-1044 | DOI | MR | Zbl

[29] LeVeque, R. J.; Li, Z. Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., Volume 18 (1997) no. 3, pp. 709-735 | DOI | MR | Zbl

[30] Matsunaga, N.; Yamamoto, Y. Superconvergence of the shortley-weller approximation for dirichlet problems, J. Comp. Appl. Math., Volume 116 (2000), pp. 263-273 | DOI | MR | Zbl

[31] Milcent, T. Formulation eulerienne du couplage fluide structure, analyse mathématique et applications en biomécanique, Thèse de l’Université de Grenoble, 2008

[32] Mittal, R.; Dong, H.; Bozkurttas, M.; Najjar, F. M.; Vargas, A.; Loebbecke, A. V. A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, J. Comput. Phys., Volume 227 (2008), pp. 4825-4852 | DOI | MR

[33] Mittal, R.; Iaccarino, G. Immersed Boundary Methods, Annual Review of Fluid Mechanics, Volume 37 (2005), pp. 239-261 | DOI | MR | Zbl

[34] Mohd-Yusof, J. Combined immersed-boundary/B-Spline methods for simulations of flow in complex geometries, NASA Ames Research Center/Stanford University (1997), pp. 317-327

[35] Mortazavi, I.; Giovannini, A. The simulation of vortex dynamics downstream of a plate separator using a vortex-finite element method, Int. J. Fluid Dynamics, Volume 5 (2001)

[36] Muldoon, F.; Acharya, S. A divergence-free interpolation scheme for the immersed boundary method, Int. J. Numer. Method Fluid, Volume 56 (2008), pp. 1845-1884 | DOI | MR

[37] Noca, F.; Shiels, D.; Jeon, D. A comparison of methods for evaluating time-dependent fluid dynamic forces on bodies, using only velocity fields and their derivatives, Journal of Fluids and Structures, Volume 13 (1999) | DOI

[38] Olz, D.; Schmeiser, C.; Small, V. Modelling of the Actin-cytoskeleton in symmetric lamellipodial fragments, Cell Adhesion and Migration, Volume 2 (2008) no. 2, pp. 117-126 | DOI

[39] Osher, S.; Fedkiw, R. P. Level set methods and Dynamic Implicit Surfaces, Springer, 2003 | MR | Zbl

[40] Osher, S.; Sethian, J. A. Fronts Propagating with Curvature Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations, J. Comput. Phys., Volume 79 (1988) no. 1, pp. 12-49 | DOI | MR | Zbl

[41] Peskin, C. S. The fluid dynamics of heart valves: experimental, theoretical, and computational methods, Ann. Rev. Fluid Mech., Volume 14 (1982), pp. 235-259 | DOI | MR | Zbl

[42] Peskin, C. S. The immersed boundary method, Acta Numerica, Volume 11 (2002), pp. 1-39 | DOI | MR | Zbl

[43] Peskin, C.S. Numerical Analysis of Blood Flow in the Heart, J. Comp. Phys., Volume 25 (1977), pp. 220-252 | DOI | MR | Zbl

[44] Peskin, S.; Printz, B.F. Improved volume conservation in the computation of flows with immersed boundaries, J. Comput. Phys., Volume 105 (1993), pp. 33-46 | DOI | MR | Zbl

[45] Ploumhans, P.; Winckelmans, G. S. Vortex methods for high-resolution simulations of viscous flow past bluff bodies of general geometry, Journal of Computational Physics, Volume 165 (2000) | DOI | MR | Zbl

[46] Saiki, E. M.; Biringen, S. Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method, J. Comput. Phys., Volume 123 (1996), pp. 450-465 | DOI | Zbl

[47] Stockie, J. Analysis of Stiffness in the immersed boundary method and implications for time-stepping schemes, J. Comp. Phys., Volume 154 (1999), pp. 41-64 | DOI | Zbl

[48] Tucker, P. G.; Pan, Z. A cartesian cut-cell method for incompressible viscous flow, Appl. Math. Model., Volume 24 (2000), pp. 591-606 | DOI | Zbl

[49] Tullio, M. De; Cristallo, A.; Balaras, E.; Pascazio, G.; Palma, P. De; Iaccarino, G.; Napolitano, M.; Verzicco, R.; Wesseling, P.; Oñate, E.; Périaux, J. Recent advances in the immersed boundary method, ECCOMAS CFD (2006)

[50] Ye, T.; Mittal, R.; Udaykumar, H. S.; Shyy., W. Numerical Simulation of two-dimensional flows over a circular cylinder using the immersed boundary method, J. Comp. Phys., Volume 156 (1999), pp. 209-240 | DOI | Zbl

[51] Zhang, N.; Zheng, Z. C. An Improved Direct-Forcing Immersed Boundary Method for Finite Difference Applications, J. Comput. Phys., Volume 221 (2007), pp. 250-268 | DOI | MR | Zbl

Cité par Sources :