zbMATH — the first resource for mathematics

An immersed boundary method based on the lattice Boltzmann approach in three dimensions, with application. (English) Zbl 1225.76249
Summary: The immersed boundary (IB) method originated by Peskin has been popular in modeling and simulating problems involving the interaction of a flexible structure and a viscous incompressible fluid. The Navier-Stokes (N-S) equations in the IB method are usually solved using numerical methods such as FFT and projection methods. Here in our work, the N-S equations are solved by an alternative approach, the lattice Boltzmann method (LBM). Compared to many conventional N-S solvers, the LBM can be easier to implement and more convenient for modeling additional physics in a problem. This alternative approach adds extra versatility to the immersed boundary method. In this paper we discuss the use of a 3D lattice Boltzmann model (D3Q19) within the IB method. We use this hybrid approach to simulate a viscous flow past a flexible sheet tethered at its middle line in a 3D channel and determine a drag scaling law for the sheet. Our main conclusions are: (1) the hybrid method is convergent with first-order accuracy which is consistent with the immersed boundary method in general; (2) the drag of the flexible sheet appears to scale with the inflow speed which is in sharp contrast with the square law for a rigid body in a viscous flow.

76M28 Particle methods and lattice-gas methods
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
PDF BibTeX Cite
Full Text: DOI
[1] Weihs, D., Hydromechanics of fish schooling, Nature, 241, 290-291, (1973)
[2] Koehl, M.A.R., The interaction of moving water and sessile organisms, Scientific American, 247, 124-132, (1982)
[3] Fung, Y.C., Biomechanics: mechanical properties of living tissues, (1993), Springer New York
[4] Zhang, X.; Schmidt, D.; Perot, B., Accuracy and conservation properties of a three-dimensional unstructured staggered mesh scheme for fluid dynamics, J. comput. phys., 175, 764-791, (2002) · Zbl 1018.76036
[5] Zhang, X.; Ni, S.Z.; He, G.W., A pressure-correction method and its applications on an unstructured Chimera grid, Comput. fluids, 37, 993-1010, (2008) · Zbl 1237.76095
[6] Peskin, C.S., Flow patterns around heart valves: a numerical method, J. comput. phys., 25, 220, (1977)
[7] Peskin, C.S., The immersed boundary method, Acta numer., 11, 479, (2002) · Zbl 1123.74309
[8] Hughes, T.J.R.; Liu, W.; Zimmerman, T.K., Lagrangian – eulerian finite element formulation for incompressible viscous flows, Comput. methods appl. mech. eng., 29, (1981)
[9] Donea, J.; Giuliani, S.; Halleux, J.P., An arbitrary lagrangian – eulerian finite element method for transient dynamic fluid structure interactions, Comput. methods appl. mech. eng., 33, 689, (1982) · Zbl 0508.73063
[10] Sulsky, D.; Chen, Z.; Schreyer, H.L., A particle method for history-dependent materials, Comput. mech. appl. mech. eng., 118, 179-197, (1994) · Zbl 0851.73078
[11] Sulsky, D.; Zhou, S.J.; Schreyer, H.L., Application of a particle-in-cell method to solid mechanics, Comput. phys. commun., 87, 136-152, (1994)
[12] Glowinski, R.; Pan, T.; Periaux, J., A fictitious domain method for Dirichlet problem and applications, Comp. methods appl. mech. eng., 111, (1994) · Zbl 0845.73078
[13] Glowinski, R.; Pan, T.; Periaux, J., A fictitious domain method for external incompressible viscous flow modeled by navier – stokes equations, Comp. methods appl. mech. eng., 112, (1994) · Zbl 0845.76069
[14] Glowinski, R.; Pan, T.; Hesla, T.; Joseph, D.; Periaux, J., A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, J. comput. phys., 169, 363, (2001) · Zbl 1047.76097
[15] LeVeque, R.J.; Li, Z.L., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 31, 1019-1044, (1994) · Zbl 0811.65083
[16] LeVeque, R.J.; Li, Z.L., Immersed interface methods for Stokes flows with elastic boundaries or surface tension, SIAM J. sci. comput., 18, 709-735, (1997) · Zbl 0879.76061
[17] Li, Z.L.; Lai, M.C., Immersed interface methods for navier – stokes equations with singular forces, J. comput. phys., 171, 822-842, (2001) · Zbl 1065.76568
[18] Li, Z.L., The immersed interface method – numerical solutions of PDEs involving interfaces and irregular domains, (2006), SIAM press Philadelphia
[19] Liu, X.; Fedkiw, R.; Kang, M., A boundary condition capturing method for poisson’s equation on irregular domain, J. comput. phys., 160, 151-178, (2000) · Zbl 0958.65105
[20] Kang, M.; Fedkiw, R.; Liu, X., A boundary condition capturing method for multiphase incompressible flow, J. sci. comput., 15, 323-360, (2000) · Zbl 1049.76046
[21] Nguyen, D.; Fedkiw, R.; Kang, M., A boundary condition capturing method for incompressible flame discontinuities, J. comput. phys., 176, 205-227, (2002)
[22] Zhang, L.; Gersternberger, A.; Wang, X.; Liu, W.K., Immersed finite element method, Comput. methods appl. mech. eng., 193, 2051, (2004) · Zbl 1067.76576
[23] Liu, W.K.; Kim, D.K.; Tang, S., Mathematical foundations of the immersed finite element method, Comput. mech., (2005)
[24] Wang, X.; Liu, W.K., Extended immersed boundary method using FEM and RKPM, Comput. methods appl. mech. eng., 193, 12-14, 1305, (2004) · Zbl 1060.74676
[25] Wang, X., From immersed boundary method to immersed continuum method, Int. J. multiscale comput. eng., 4, 1, 127-145, (2006)
[26] Wang, X. Sheldon, An iterative matrix-free method in implicit immersed boundary/continuum methods, Comput. struct., 85, 739-748, (2007)
[27] Hou, T.Y.; Li, Z.L.; Osher, S.; Zhao, H.K., A hybrid method for moving interface problems with application to the Hele-Shaw flow, J. comput. phys., 134, 236-252, (1997) · Zbl 0888.76067
[28] Cottet, G.H.; Maitre, E., A level set formulation of immersed boundary methods for fluid – structure interaction problems, C.R. acad. sci. Paris, ser. I, 338, 581-586, (2004) · Zbl 1101.74028
[29] Xu, J.; Li, Z.; Lowengrub, J.; Zhao, H., A level set method for interfacial flows with surfactant, J. comput. phys., 212, 2, 590-616, (2006) · Zbl 1161.76548
[30] Cottet, G.H.; Maitre, E., A level set method for fluid – structure interactions with immersed interfaces, Math. models methods appl. sci., 16, 415-438, (2006) · Zbl 1088.74050
[31] Udaykumar, H.S.; Shyy, W.; Rao, M.M, A mixed eulerian – lagrangian method for fluid flows with complex and moving boundaries, Int. J. numer. methods fluids, 22, 691, (1996) · Zbl 0887.76059
[32] Ye, T.; Mittal, R.; Udaykumar, H.S.; Shyy, W., An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries, J. comput. phys., 156, 209, (1999) · Zbl 0957.76043
[33] Mittal, R.; Iaccarino, G., Immersed boundary methods, Ann. rev. fluid mech., 37, 239-261, (2005) · Zbl 1117.76049
[34] Borazjani, I.; Ge, L.; Sotiropoulos, F., Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies, J. comput. phys., 227, 16, 7587-7620, (2008) · Zbl 1213.76129
[35] C.S. Peskin, Flow patterns around heart valves: a digital computer method for solving the equations of motion. Ph.D. Thesis. Physiol., Albert Einstein Coll. Med, Univ. Microfilms. 378: 72-30, 1972.
[36] Peskin, C.S.; McQueen, D.M., Computational biofluid dynamics, Contemp. math., 141, 161, (1993) · Zbl 0786.76108
[37] Peskin, C.S.; McQueen, D.M., A general method for the computer simulation of biological systems interacting with fluids, Sympos. soc. exp. biol., 49, 265, (1995)
[38] Peskin, C.S.; McQueen, D.M., Fluid dynamics of the heart and its valves, (), 309
[39] McCracken, M.F.; Peskin, C.S., A vortex method for blood flow through heart valves, J. comput. phys., 35, 183-205, (1980) · Zbl 0428.92010
[40] Peskin, C.S.; Printz, B.F., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. comput. phys., 105, 33, (1993) · Zbl 0762.92011
[41] Rosar, M.E.; Peskin, C.S., Fluid flow in collapsible elastic tubes: a three-dimensional numerical model, New York J. math., 7, 281-302, (2001) · Zbl 1051.76016
[42] Roma, A.M.; Peskin, C.S.; Berger, M.J., An adaptive version of the immersed boundary method, J. comput. phys., 153, 509-534, (1999) · Zbl 0953.76069
[43] Lai, M.C.; Peskin, C.S., An immersed boundary method with formal second order accuracy and reduced numerical viscosity, J. comput. phys., 160, 705, (2000) · Zbl 0954.76066
[44] D.M. McQueen, C.S. Peskin, L. Zhu, The immersed boundary method for incompressible fluid – structure interaction, in: Proceedings of the First M.I.T. Conference on Computational Fluid and Solid Mechanics, June 2001.
[45] Griffith, B.E.; Peskin, C.S., On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficient smooth problems, J. comput. phys., 208, 1, 75-105, (2005) · Zbl 1115.76386
[46] Zhu, L.; Peskin, C.S., Simulation of a flexible flapping filament in a flowing soap film by the immersed boundary method, J. comput. phys., 179, 2, 452-468, (2002) · Zbl 1130.76406
[47] Kim, Y.; Peskin, C.S., Penalty immersed boundary method for an elastic boundary with mass, Phys. fluids, 19, 5, (2007), article number 053103 · Zbl 1146.76441
[48] Tu, C.; Peskin, C.S., Stability and instability in the computation of flows with moving immersed boundaries: a comparison of three methods, SIAM J. sci. statist. comput., 13, 1361-1376, (1992) · Zbl 0760.76067
[49] Mayo, A.A.; Peskin, C.S., An implicit numerical method for fluid dynamics problems with immersed elastic boundaries, (), 261-277 · Zbl 0787.76055
[50] Fauci, L.J.; Fogelson, A.L., Truncated Newton methods and the modeling of complex elastic structures, Comm. pure appl. math., 46, 787, (1993) · Zbl 0741.76103
[51] Taira, K.; Colonius, T., The immersed boundary method: a projection approach, J. comput. phys., 225, 2, 2118-2137, (2007) · Zbl 1343.76027
[52] Y. Mori, C.S. Peskin, Implicit second-order immersed boundary method with boundary mass, J. Comput. Phys. (2006) (submitted). · Zbl 1158.74533
[53] Atzberger, P.J.; Kramer, P.R.; Peskin, C.S., A stochastic immersed boundary method for biological fluid dynamics at microscopic length scale, J. comput. phys., 224, 2, 1255-1292, (2007) · Zbl 1124.74052
[54] Atzberger, P.J.; Kramer, P.R., Error analysis of a stochastic immersed boundary method incorporating thermal fluctuations, Math. comput. simul., 79, 3, 379-408, (2008) · Zbl 1159.65006
[55] Y.H. Qian, Lattice gas and lattice kinetic theory applied to the Navier-Stokes equations, Ph.D. Thesis. University Pierre et Marie Curie, Paris, 1990.
[56] Chen, S.Y.; Chen, H.D.; Martinez, D.; Matthaeus, W., Lattice Boltzmann model for simulation of magnetohydrodynamics, Phys. rev. lett., 67, 3776, (1991)
[57] S. Hou, Lattice Boltzmann method for incompressible viscous flow, Ph.D. Thesis. Kansas State Univ., Manhattan, Kansas.
[58] He, X.; Luo, L.-S., Theory of lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys. rev. E, 56, 6811, (1997)
[59] He, X.; Luo, L.-S., A priori derivation of the lattice Boltzmann equation, Phys. rev. E, 55, 6, (1997)
[60] Luo, L.-S., Unified theory of the lattice Boltzmann models for nonideal gases, Phys. rev. lett., 81, 1618, (1998)
[61] He, X.; Chen, S.; Zhang, R., A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of rayleigh – taylor instability, J. comput. phys., 152, 642-663, (1999) · Zbl 0954.76076
[62] Wolf-Gladrow, D.A., Lattice-gas cellular automata and lattice Boltzmann models — an introduction, (2000), Springer Berlin · Zbl 0999.82054
[63] Luo, L.-S., Theory of the lattice Boltzmann method: lattice Boltzmann models for nonideal gases, Phys. rev. E, 62, 4, 4982-4996, (2000)
[64] Succi, S., The lattice Boltzmann equation, (2001), Oxford Univ. Press Oxford
[65] Chen, S.Y.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Ann. rev. fluid mech., 30, 329, (1998) · Zbl 1398.76180
[66] Feng, Z.G.; Michaelides, E.E., The immersed boundary-lattice Boltzmann method for solving fluid – particles interaction problems, J. comput. phys., 195, 602-628, (2004) · Zbl 1115.76395
[67] Feng, Z.G.; Michaelides, E.E., Proteus: a direct forcing method in the simulations of particulate flows, J. comput. phys., 202, 20-51, (2005) · Zbl 1076.76568
[68] Niu, X.D.; Shu, C.; Chew, Y.T.; Peng, Y., A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows, Phys. lett. A, 354, 173-182, (2006) · Zbl 1181.76111
[69] Ladd, A.J.C., J. fluid mech., 271, 285, (1994)
[70] Sui, Y.; Chew, Y.T.; Roy, P.; Low, H.T., A hybrid immersed-boundary and multi-block lattice Boltzmann method for simulating fluid and moving-boundaries interactions, Int. J. numer. methods fluids, 53, 11, 1727-1754, (2007) · Zbl 1110.76042
[71] Filippova, O.; Succi, S.; Mazzocco, F.; Arrighetti, C.; Bella, G.; Hanel, D., Multiscale lattice Boltzmann schemes with turbulence modeling, Comput. sci. eng., 170, 2, 812-829, (2001) · Zbl 1012.76073
[72] Yu, H.; Girimaji, S.S.; Luo, L.S., DNS and LES of decaying isotropic turbulence with and without frame rotation using lattice Boltzmann method, J. comput. phys., 209, 599, (2005) · Zbl 1138.76373
[73] Peng, Y.; Luo, L.-S., A comparative study of immersed-boundary and interpolated bounce-back methods in LBE, Prog. comput. fluid. dyn., 8, 1-4, 156-167, (2008) · Zbl 1388.76306
[74] Le, G.; Zhang, J., Boundary slip from the immersed boundary lattice Boltzmann models, Phys. rev. E, 79, 026701, (2009)
[75] Sui, Y.; Chew, Y.T.; Roy, P.; Cheng, Y.P.; Low, H.T., Dynamical motion of red blood cells in a simple shear flow, Phys. fluids, 20, 112106, (2008) · Zbl 1182.76730
[76] Feng, Z.G.; Michaelides, E.E., Robust treatment of no-slip boundary condition and velocity updating for the lattice-Boltzmann simulation of particulate flows, Comput. fluids, 38, (2009) · Zbl 1237.76137
[77] Y. Cheng, H. Zhang, Immersed boundary method and lattice Boltzmann method coupled FSI simulation of mitral leaflet flow, Comput. Fluids (2010), in press (doi:10.1016/j.compfluid.2010.01.003). · Zbl 1242.76372
[78] Guo, Z.; Zheng, C.; Shi, B., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. rev. E, 65, 4, 046308, (2002) · Zbl 1244.76102
[79] Steinburg, V., Bend and survive, Nature, 420, 473, (2002)
[80] Alben, S.; Shelley, M.; Zhang, J., Drag reduction through self-similar bending of a flexible body, Nature, 420, 6915, (2002)
[81] Alben, S.; Shelley, M.; Zhang, J., How flexibility induces streamlining in a two-dimensional flow, Phys. fluids, 16, 5, 1694, (2002) · Zbl 1186.76020
[82] Zhu, L.; Peskin, C.S., Drag of a flexible fiber in a 2D moving viscous fluid, Comput. fluids, 36, 398-406, (2007) · Zbl 1177.76305
[83] Zhu, L., Viscous flow past an elastic fibre tethered at its center point: vortex shedding, J. fluid mech., 587, 217-234, (2007) · Zbl 1141.76374
[84] Zhu, L., Scaling laws for drag of a compliant body moving in an incompressible viscous fluid, J. fluid mech., 607, 387-400, (2008) · Zbl 1146.76020
[85] Batchelor, G.K., An introduction to fluid dynamics, (1967), Cambridge Univ. Press Cambridge · Zbl 0152.44402
[86] Bhatnagar, P.L.; Gross, E.P.; Krook, M., A model for collision processes in gases, I: small amplitude process in charged and neutral one-component system, Phys. rev., 94, 511, (1954) · Zbl 0055.23609
[87] Mei, R.; Shyy, W.; Yu, D.; Luo, L.S., Lattice Boltzmann method for 3-D flows with curved boundary, J. comput. phys., 161, 680-699, (2000) · Zbl 0980.76064
[88] Chen, S.Y.; Martinez, D.; Mei, R.W., On boundary conditions in lattice Boltzmann methods, Phys. fluids, 8, 9, 2527-2536, (1996) · Zbl 1027.76630
[89] Yang, X.L.; Zhang, X.; Li, Z.L.; He, G.W., A smoothing discrete delta function approach to reduce force oscillations in moving boundary simulations, J. comput. phys., 228, 20, 7821-7836, (2009) · Zbl 1391.76590
[90] D.W. Qi, Private communications, 2009.
[91] Najjar, F.M; Vanka, S.P., Effects of intrinsic three-dimensionality on the drag characteristic of a normal flat plate, Phys. fluids, 7, 10, 2516-2518, (1995) · Zbl 1026.76513
[92] Carruthers, A.C.; Filippone, A., Aerodynamic drag of streamers and flags, J. aircraft, 42, 4, 976-982, (2005)
[93] Yoshizawa, A., Drag of a finite flat plate set parallel to a uniform flow, J. phys. soc. Japan, 32, 1677, (1972)
[94] Taherzadeh, D., Biofilms and drag, Biotechnol. bioeng., 105, 3, 600-610, (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.