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.
@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}, volume = {20}, year = {2013}, pages = {139-173}, doi = {10.5802/ambp.324}, zbl = {06299067}, mrnumber = {3112242}, language = {en}, url = {http://dml.mathdoc.fr/item/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) pp. 139-173. doi : 10.5802/ambp.324. http://gdmltest.u-ga.fr/item/AMBP_2013__20_1_139_0/
[1] A penalization method to take into account obstacles in incompressible viscous flows, Numer. Math., Tome 81 (1999), pp. 497-520 | Article | MR 1675200 | Zbl 0921.76168
[2] Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces, J. Comp. Phys., Tome 227 (2008) no. 8, pp. 3896-3920 | Article | MR 2403872 | Zbl 1146.76044
[3] Numerical stability of the finite element immersed boundary method, M3AS, Tome 17 (2007), pp. 1479-1505 | MR 2359913 | Zbl 1186.76661
[4] 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 2303686 | Zbl pre05165537
[5] Weak force stalls protrusion at the leading edge of the lamellipodium, Biophys. J., Tome 90 (2006), pp. 1810-1820 | Article
[6] A second-order cut-cell method for the numerical simulation of 2D flows past obstacles, Computers and Fluids, Tome 65 (2012), pp. 80-91 | Article | MR 2966539
[7] Computational modeling of solid tumor growth: the avascular stage, SIAM Journal on Scientific Computing, Tome 32 (2010) no. 4, pp. 2321-2344 | Article | MR 2678103 | Zbl 1214.92039
[8] Quelques méthodes de paramètre d’ordre avec applications à la modélisation de processus cancéreux, ESAIM: Proceedings, Tome 18 (2007), pp. 163-180 | Article | MR 2404904 | Zbl pre05213264
[9] Passive control around the two-dimensional square back Ahmed body using porous devices, J. Fluids Eng., Tome 130 (2008) | Article
[10] A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows, J. Comp. Phys., Tome 124 (1996), pp. 449-464 | Article | MR 1383769 | Zbl 0847.76048
[11] 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., Tome 229 (2010), pp. 1043-1076 | Article | MR 2576238 | Zbl pre05668157
[12] Vortex sheet approximation of boundary layers, J. Comput. Phys., Tome 27 (1978) | Article | Zbl 0387.76040
[13] Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape, Computers and Fluids, Tome 35 (2006) no. 6, pp. 607-623 | Article | Zbl 1160.76369
[14] A Vortex Method for Bi-phasic Fluids Interacting with Rigid Bodies, Arxiv preprint math, LMC-IMAG (2006)
[15] A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies, J. Comput. Phys., Tome 227 (2008) | Article | MR 2463201 | Zbl 1146.76038
[16] Parametric resonance in immersed elastic boundaries, SIAM Journal on Applied Mathematics, Tome 65 (2004) no. 2, pp. 494-520 | Article | MR 2123067 | Zbl 1074.74024
[17] A vortex immersed boundary method for bluff body flows, ASME Summer Meeting, Montreal, Tome FEDSM-ICNMM2010-30787 (2010)
[18] Vortex Methods: Theory and Practice (2000) | MR 1755095
[19] A level-set formulation of immersed boundary methods for fluid-structure interaction problems, C. R. Math., Tome 338 (2004) no. 7, pp. 581-586 | Article | MR 2057034 | Zbl 1101.74028
[20] A level set method for fluid-structure interactions with immersed surfaces, Math. Models Meth. Appl. Sci., Tome 16 (2006) no. 3, pp. 415-438 | Article | MR 2238758 | Zbl 1088.74050
[21] Eulerian formulation and level set models for incompressible fluid-structure interaction, ESAIM-Math. Model. Numer. Anal., Tome 42 (2008), pp. 471-492 | Article | Numdam | MR 2423795 | Zbl 1163.76040
[22] Vortex simulation of active control strategies for transitional backward-facing step flows, Computers & Fluids, Tome 38 (2009) | Article | MR 2645733 | Zbl 1242.76226
[23] Combined immersed-boundary finite difference methods for three-dimensional complex flow simulations, J. Comput. Phys., Tome 161 (2000), pp. 35-60 | Article | MR 1762073 | Zbl 0972.76073
[24] On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems, J. Comp. Phys., Tome 208 (2005), pp. 75-105 | Article | MR 2144693 | Zbl 1115.76386
[25] Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, Tome 12 (1965), pp. 2182-2189 | Article | Zbl 1180.76043
[26] An immersed-boundary finite volume method for simulation of flow in complex geometries, J. Comput. Phys., Tome 171 (2001), pp. 132-150 | Article | MR 1843643 | Zbl 1057.76039
[27] An immersed interface method for incompresible Navier-Stokes equations, SIAM J. Sci. Comp., Tome 25 (2003) no. 3, pp. 832-856 | Article | MR 2046114 | Zbl 1163.65322
[28] The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources, SIAM J. Numer. Anal., Tome 31 (1994), pp. 1019-1044 | Article | MR 1286215 | Zbl 0811.65083
[29] Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., Tome 18 (1997) no. 3, pp. 709-735 | Article | MR 1443639 | Zbl 0879.76061
[30] Superconvergence of the shortley-weller approximation for dirichlet problems, J. Comp. Appl. Math., Tome 116 (2000), pp. 263-273 | Article | MR 1750921 | Zbl 0952.65082
[31] Formulation eulerienne du couplage fluide structure, analyse mathématique et applications en biomécanique, Thèse de l’Université de Grenoble (2008)
[32] A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, J. Comput. Phys., Tome 227 (2008), pp. 4825-4852 | Article | MR 2414837 | Zbl pre05276051
[33] Immersed Boundary Methods, Annual Review of Fluid Mechanics, Tome 37 (2005), pp. 239-261 | Article | MR 2115343 | Zbl 1117.76049
[34] Combined immersed-boundary/B-Spline methods for simulations of flow in complex geometries, NASA Ames Research Center/Stanford University (1997), pp. 317-327
[35] The simulation of vortex dynamics downstream of a plate separator using a vortex-finite element method, Int. J. Fluid Dynamics, Tome 5 (2001)
[36] A divergence-free interpolation scheme for the immersed boundary method, Int. J. Numer. Method Fluid, Tome 56 (2008), pp. 1845-1884 | Article | MR 2397813 | Zbl pre05312566
[37] 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, Tome 13 (1999) | Article
[38] Modelling of the Actin-cytoskeleton in symmetric lamellipodial fragments, Cell Adhesion and Migration, Tome 2 (2008) no. 2, pp. 117-126 | Article
[39] Level set methods and Dynamic Implicit Surfaces, Springer (2003) | MR 1939127 | Zbl 1026.76001
[40] Fronts Propagating with Curvature Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations, J. Comput. Phys., Tome 79 (1988) no. 1, pp. 12-49 | Article | MR 965860 | Zbl 0659.65132
[41] The fluid dynamics of heart valves: experimental, theoretical, and computational methods, Ann. Rev. Fluid Mech., Tome 14 (1982), pp. 235-259 | Article | MR 642539 | Zbl 0488.76129
[42] The immersed boundary method, Acta Numerica, Tome 11 (2002), pp. 1-39 | Article | MR 2009378 | Zbl 1123.74309
[43] Numerical Analysis of Blood Flow in the Heart, J. Comp. Phys., Tome 25 (1977), pp. 220-252 | Article | MR 490027 | Zbl 0403.76100
[44] Improved volume conservation in the computation of flows with immersed boundaries, J. Comput. Phys., Tome 105 (1993), pp. 33-46 | Article | MR 1210858 | Zbl 0762.92011
[45] Vortex methods for high-resolution simulations of viscous flow past bluff bodies of general geometry, Journal of Computational Physics, Tome 165 (2000) | Article | MR 1807293 | Zbl 1006.76068
[46] Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method, J. Comput. Phys., Tome 123 (1996), pp. 450-465 | Article | Zbl 0848.76052
[47] Analysis of Stiffness in the immersed boundary method and implications for time-stepping schemes, J. Comp. Phys., Tome 154 (1999), pp. 41-64 | Article | Zbl 0953.76070
[48] A cartesian cut-cell method for incompressible viscous flow, Appl. Math. Model., Tome 24 (2000), pp. 591-606 | Article | Zbl 1056.76059
[49] Recent advances in the immersed boundary method, ECCOMAS CFD (2006)
[50] Numerical Simulation of two-dimensional flows over a circular cylinder using the immersed boundary method, J. Comp. Phys., Tome 156 (1999), pp. 209-240 | Article | Zbl 0957.76043
[51] An Improved Direct-Forcing Immersed Boundary Method for Finite Difference Applications, J. Comput. Phys., Tome 221 (2007), pp. 250-268 | Article | MR 2290571 | Zbl 1108.76051