# zbMATH — the first resource for mathematics

A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating a flexible filament in an incompressible flow. (English) Zbl 1381.74085
Summary: A momentum exchange-based immersed boundary-lattice Boltzmann method, which is used to solve the fluid-flexible-structure-interaction problem, is introduced in this paper. The present method, overcoming the drawback of the conventional penalty method employing a user-defined spring parameter for calculating the interaction force induced by the immersed boundary, uses a concept of momentum exchange on the boundary to calculate the interaction force. Numerical examples, including a laminar flow past a circular cylinder, a filament flapping in the wake of the cylinder, a single filament with the upstream end fixed flapping in a uniform flow field and the interaction of two filaments flapping in the flow, are provided to validate the present method and to illustrate its capability of dealing with the fluid-flexible-structure-interaction problem. Particularly, with considering the filament mass effects, a single filament with a fixed centre point undergoing a bending transition in the flow is firstly studied in the present paper. Our numerical results compare qualitatively well to experimental results. For a single filament with a fixed centre point, it is found that the flexure modulus has a significant effect on the final state of the filament: for a larger flexure modulus, the filament reaches the ‘quasi-steady’ state finally; for a small flexure modulus, the filaments will be flapping like two filaments.

##### MSC:
 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74S20 Finite difference methods applied to problems in solid mechanics 76M28 Particle methods and lattice-gas methods
Full Text:
##### References:
 [1] Zhang, J.; Childress, S.; Libchaber, A.; Shelley, M., Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind, Nature, 408, 6814, 835-839, (2000) [2] Alben, S.; Shelley, M.; Zhang, J., Drag reduction through self-similar bending of a flexible body, Nature, 420, 6915, 479-481, (2002) [3] Alben, S.; Shelley, M.; Zhang, J., How flexibility induces streamlining in a two-dimensional flow, Phys. Fluids, 16, 1694, (2004) · Zbl 1186.76020 [4] Jia, L.; Li, F.; Yin, X.; Yin, X., Coupling modes between two flapping filaments, J. Fluid Mech., 581, 1, 199-220, (2007) · Zbl 1176.76044 [5] Ristroph, L.; Zhang, J., Anomalous hydrodynamic drafting of interacting flapping flags, Phys. Rev. Lett., 101, 19, 194502, (2008) [6] Hron, J.; Turek, S., A monolithic FEM/multigrid solver for an ALE formulation of fluid-structure interaction with applications in biomechanics, (2006), Springer · Zbl 1323.74086 [7] Hübner, B.; Walhorn, E.; Dinkler, D., A monolithic approach to fluid-structure interaction using space-time finite elements, Comput. Methods Appl. Mech. Engrg., 193, 23, 2087-2104, (2004) · Zbl 1067.74575 [8] Walhorn, E.; Kölke, A.; Hübner, B.; Dinkler, D., Fluid-structure coupling within a monolithic model involving free surface flows, Comput. Struct., 83, 25, 2100-2111, (2005) [9] Le Tallec, P.; Mouro, J., Fluid structure interaction with large structural displacements, Comput. Methods Appl. Mech. Engrg., 190, 24, 3039-3067, (2001) · Zbl 1001.74040 [10] Bathe, K. U.R.; Zhang, H.; Ji, S., Finite element analysis of fluid flows fully coupled with structural interactions, Comput. Struct., 72, 1, 1-16, (1999) · Zbl 1072.74545 [11] Farhat, C.; Lesoinne, M., Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems, Comput. Methods Appl. Mech. Engrg., 182, 3, 499-515, (2000) · Zbl 0991.74069 [12] R. L O Hner, C. Yang, J. Cebral, J.D. Baum, H. Luo, D. Pelessone, C. Charman, Fluid-structure-thermal interaction using a loose coupling algorithm and adaptive unstructured grids, 1998. · Zbl 0875.73165 [13] Piperno, S.; Farhat, C., Partitioned procedures for the transient solution of coupled aeroelastic problems—part II: energy transfer analysis and three-dimensional applications, Comput. Methods Appl. Mech. Engrg., 190, 24, 3147-3170, (2001) · Zbl 1015.74009 [14] Rugonyi, S.; Bathe, K. J., On finite element analysis of fluid flows fully coupled with structural interactions, CMES Comput. Model. Eng. Sci., 2, 2, 195-212, (2001) [15] Wall, W. A.; Genkinger, S.; Ramm, E., A strong coupling partitioned approach for fluid-structure interaction with free surfaces, Comput. Fluids, 36, 1, 169-183, (2007) · Zbl 1181.76147 [16] Küttler, U.; Wall, W. A., Vector extrapolation for strong coupling fluid-structure interaction solvers, J. Appl. Mech., 76, 2, (2009) [17] Hirt, C. W.; Amsden, A. A.; Cook, J. L., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys., 14, 3, 227-253, (1974) · Zbl 0292.76018 [18] Souli, M.; Ouahsine, A.; Lewin, L., ALE formulation for fluid-structure interaction problems, Comput. Methods Appl. Mech. Engrg., 190, 5, 659-675, (2000) · Zbl 1012.76051 [19] Peskin, C. S., Flow patterns around heart valves: a numerical method, J. Comput. Phys., 10, 2, 252-271, (1972) · Zbl 0244.92002 [20] Stockie, J. M.; Green, S. I., Simulating the motion of flexible pulp fibres using the immersed boundary method, J. Comput. Phys., 147, 1, 147-165, (1998) · Zbl 0935.76065 [21] D.M. McQueen, C.S. Peskin, Heart simulation by an immersed boundary method with formal second-order accuracy and reduced numerical viscosity. 2000. [22] Zhu, L.; Peskin, C. S., Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, J. Comput. Phys., 179, 2, 452-468, (2002) · Zbl 1130.76406 [23] Zhu, L.; Peskin, C. S., Interaction of two flapping filaments in a flowing soap film, Phys. Fluids, 15, 1954, (2003) · Zbl 1186.76611 [24] Zhu, L., Viscous flow past a flexible fibre tethered at its centre point: vortex shedding, J. Fluid Mech., 587, 217, (2007) · Zbl 1141.76374 [25] Zhu, L.; Chin, R. C., Simulation of elastic filaments interacting with a viscous pulsatile flow, Comput. Methods Appl. Mech. Engrg., 197, 25, 2265-2274, (2008) · Zbl 1158.76458 [26] Zhu, L., Interaction of two tandem deformable bodies in a viscous incompressible flow, J. Fluid Mech., 635, 455, (2009) · Zbl 1183.76709 [27] Zhu, L.; Peskin, C. S., Drag of a flexible fiber in a 2D moving viscous fluid, Comput. Fluids, 36, 2, 398-406, (2007) · Zbl 1177.76305 [28] He, X.; Luo, L., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J. Stat. Phys., 88, 3-4, 927-944, (1997) · Zbl 0939.82042 [29] Chen, S.; Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30, 1, 329-364, (1998) · Zbl 1398.76180 [30] He, X.; Chen, S.; Doolen, G. D., A novel thermal model for the lattice Boltzmann method in incompressible limit, J. Comput. Phys., 146, 1, 282-300, (1998) · Zbl 0919.76068 [31] 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, 2, 642-663, (1999) · Zbl 0954.76076 [32] He, X.; Li, N., Lattice Boltzmann simulation of electrochemical systems, Comput. Phys. Commun., 129, 1, 158-166, (2000) · Zbl 0976.76066 [33] Xu, K.; He, X., Lattice Boltzmann method and gas-kinetic BGK scheme in the low-Mach number viscous flow simulations, J. Comput. Phys., 190, 1, 100-117, (2003) · Zbl 1236.76052 [34] Shu, C.; Liu, N.; Chew, Y. T., A novel immersed boundary velocity correction-lattice Boltzmann method and its application to simulate flow past a circular cylinder, J. Comput. Phys., 226, 2, 1607-1622, (2007) · Zbl 1173.76395 [35] Niu, X. D.; Shu, C.; Chew, Y. T., A thermal lattice Boltzmann model with diffuse scattering boundary condition for micro thermal flows, Comput. Fluids, 36, 2, 273-281, (2007) · Zbl 1177.76321 [36] Guo, Z.; Liu, H.; Luo, L.; Xu, K., A comparative study of the LBE and GKS methods for 2D near incompressible laminar flows, J. Comput. Phys., 227, 10, 4955-4976, (2008) · Zbl 1388.76291 [37] Niu, X.; Yamaguchi, H.; Yoshikawa, K., Lattice Boltzmann model for simulating temperature-sensitive ferrofluids, Phys. Rev. E, 79, 4, 046713, (2009) [38] Rong, F.; Guo, Z.; Chai, Z.; Shi, B., A lattice Boltzmann model for axisymmetric thermal flows through porous media, Int. J. Heat Mass Tran., 53, 23, 5519-5527, (2010) · Zbl 1201.80046 [39] Wang, J.; Wang, D.; Lallemand, P.; Luo, L., Lattice Boltzmann simulations of thermal convective flows in two dimensions, Comput. Math. Appl., (2012) [40] Feng, Z.; Michaelides, E. E., The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems, J. Comput. Phys., 195, 2, 602-628, (2004) · Zbl 1115.76395 [41] Feng, Z.; Michaelides, E. E., $$\langle i \rangle$$ proteus: a direct forcing method in the simulations of particulate flows, J. Comput. Phys., 202, 1, 20-51, (2005) · Zbl 1076.76568 [42] 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, 3, 173-182, (2006) · Zbl 1181.76111 [43] Geller, S.; Tölke, J.; Krafczyk, M., Lattice-Boltzmann method on quadtree-type grids for fluid-structure interaction, (Fluid-Structure Interaction, (2006), Springer), 270-293 · Zbl 1323.76079 [44] An explicit model for three-dimensional fluid-structure interaction using LBM and p-FEM, (Geller, S.; Kollmannsberger, S.; El Bettah, M.; Krafczyk, M.; Scholz, D.; Düster, A.; Rank, E., Fluid Structure Interaction II, (2010), Springer), 285-325 · Zbl 1213.76148 [45] Hao, J.; Zhu, L., A lattice Boltzmann based implicit immersed boundary method for fluid-structure interaction, Comput. Math. Appl., 59, 1, 185-193, (2010) · Zbl 1189.76407 [46] Zhu, L.; He, G.; Wang, S.; Miller, L.; Zhang, X.; You, Q.; Fang, S., An immersed boundary method based on the lattice Boltzmann approach in three dimensions, with application, Comput. Math. Appl., 61, 12, 3506-3518, (2011) · Zbl 1225.76249 [47] Cheng, Y.; Zhang, H., Immersed boundary method and lattice Boltzmann method coupled FSI simulation of mitral leaflet flow, Comput. Fluids, 39, 5, 871-881, (2010) · Zbl 1242.76372 [48] Vahidkhah, K.; Abdollahi, V., Numerical simulation of a flexible fiber deformation in a viscous flow by the immersed boundary-lattice Boltzmann method, Commun. Nonlinear Sci. Numer. Simul., 17, 3, 1475-1484, (2012) · Zbl 1364.76191 [49] Tian, F.; Luo, H.; Zhu, L.; Liao, J. C.; Lu, X., An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments, J. Comput. Phys., 230, 19, 7266-7283, (2011) · Zbl 1327.76106 [50] Connell, B. S.; Yue, D. K., Flapping dynamics of a flag in a uniform stream, J. Fluid Mech., 581, 1, 33-67, (2007) · Zbl 1124.76011 [51] EiFi, B., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E, 65, 046308, (2002) · Zbl 1244.76102 [52] Aidun, C. K.; Clausen, J. R., Lattice-Boltzmann method for complex flows, Annu. Rev. Fluid Mech., 42, 439-472, (2010) · Zbl 1345.76087 [53] Gottlieb, S.; Shu, C., Total variation diminishing Runge-Kutta schemes, Math. Comput. Amer. Math. Soc., 67, 221, 73-85, (1998) · Zbl 0897.65058 [54] A.J. Ladd, Numerical simulations of particulate suspensions via a discretized boltzmann equation part i. Theoretical Foundation, 1993, arXiv preprint comp-gas/9306004. · Zbl 0815.76085 [55] Ladd, A. J., Numerical simulations of particulate suspensions via a discretized Boltzmann equation. part 2. numerical results, J. Fluid Mech., 271, 1, 311-339, (1994) · Zbl 0815.76085 [56] Sui, Y.; Chew, Y.; Roy, P.; Low, H., A hybrid immersed-boundary and multi-block lattice Boltzmann method for simulating fluid and moving-boundaries interactions, Int. J. Numer. Meth. Fl., 53, 11, 1727-1754, (2007) · Zbl 1110.76042 [57] Shi, X.; Phan-Thien, N., Distributed Lagrange multiplier/fictitious domain method in the framework of lattice Boltzmann method for fluid-structure interactions, J. Comput. Phys., 206, 1, 81-94, (2005) · Zbl 1087.76543 [58] Xu, S.; Wang, Z. J., An immersed interface method for simulating the interaction of a fluid with moving boundaries, J. Comput. Phys., 216, 2, 454-493, (2006) · Zbl 1220.76058 [59] Russell, D.; Jane Wang, Z., A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, J. Comput. Phys., 191, 1, 177-205, (2003) · Zbl 1160.76389 [60] Gao, T.; Tseng, Y.; Lu, X., An improved hybrid Cartesian/immersed boundary method for fluid-solid flows, Int. J. Numer. Meth. Fl., 55, 12, 1189-1211, (2007) · Zbl 1127.76045 [61] Silva, L. E.; Silveira-Neto, A.; Damasceno, J., Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method, J. Comput. Phys., 189, 2, 351-370, (2003) · Zbl 1061.76046
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.