Immersed boundary methods for the numerical simulation of incompressible aerodynamics and fluid-structure interactions
Annales mathématiques Blaise Pascal, Volume 20 (2013) no. 1, pp. 139-173.

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.

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.

DOI: 10.5802/ambp.324
Classification: 74F10, 65M06, 76D05
Keywords: Immersed boundary method, Momentum forcing method, Vortex penalization method, Cut-cell method, Incompressible viscous flows, Complex geometry
Nicolas James 1; Emmanuel Maitre 2; Iraj Mortazavi 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
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     journal = {Annales math\'ematiques Blaise Pascal},
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Nicolas James; Emmanuel Maitre; Iraj Mortazavi. Immersed boundary methods for the numerical simulation of incompressible aerodynamics and fluid-structure interactions. Annales mathématiques Blaise Pascal, Volume 20 (2013) no. 1, pp. 139-173. doi : 10.5802/ambp.324.

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

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

[4] D. Boffia; L. Gastaldi; L. Heltai 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 | MR

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

[6] F. Bouchon; T. Dubois; N. James 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] D. Bresch; T. Colin; E. Grenier; B. Ribba; O. Saut 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] D. Bresch; Th. Colin; E. Grenier; B. Ribba; O. Saut; O. Singh; C. Verdier 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] Ch. -H. Bruneau; I. Mortazavi; P. Gilliéron Passive control around the two-dimensional square back Ahmed body using porous devices, J. Fluids Eng., Volume 130 (2008) | DOI

[10] Y.C. Chang; T.Y. Hou; B. Merriman; S. Osher 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] Y. Cheny; O. Botella 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] A.J. Chorin Vortex sheet approximation of boundary layers, J. Comput. Phys., Volume 27 (1978) | DOI | Zbl

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

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

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

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

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

[23] 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 | DOI | MR | Zbl

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

[26] J. Kim; D. Kim; H. Choi 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] L. Lee; R.J. Leveque 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] R. J. LeVeque; Z. Li 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] R. J. LeVeque; Z. Li 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] N. Matsunaga; Y. Yamamoto Superconvergence of the shortley-weller approximation for dirichlet problems, J. Comp. Appl. Math., Volume 116 (2000), pp. 263-273 | DOI | MR | Zbl

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

[32] R. Mittal; H. Dong; M. Bozkurttas; F. M. Najjar; A. Vargas; A. V. Loebbecke 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] R. Mittal; G. Iaccarino Immersed Boundary Methods, Annual Review of Fluid Mechanics, Volume 37 (2005), pp. 239-261 | DOI | MR | Zbl

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

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

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

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

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

[40] S. Osher; J. A. Sethian 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] C. S. Peskin 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] C. S. Peskin The immersed boundary method, Acta Numerica, Volume 11 (2002), pp. 1-39 | DOI | MR | Zbl

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

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

[45] P. Ploumhans; G. S. Winckelmans 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] E. M. Saiki; S. Biringen 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] J. Stockie 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] P. G. Tucker; Z. Pan A cartesian cut-cell method for incompressible viscous flow, Appl. Math. Model., Volume 24 (2000), pp. 591-606 | DOI | Zbl

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

[50] T. Ye; R. Mittal; H. S. Udaykumar; W. Shyy. 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] N. Zhang; Z. C. Zheng An Improved Direct-Forcing Immersed Boundary Method for Finite Difference Applications, J. Comput. Phys., Volume 221 (2007), pp. 250-268 | DOI | MR | Zbl

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