Skip to main content Accessibility help
×
Home

Numerical Study of Stability and Accuracy of the Immersed Boundary Method Coupled to the Lattice Boltzmann BGK Model

  • Yongguang Cheng (a1), Luoding Zhu (a2) and Chunze Zhang (a1)

Abstract

This paper aims to study the numerical features of a coupling scheme between the immersed boundary (IB) method and the lattice Boltzmann BGK (LBGK) model by four typical test problems: the relaxation of a circular membrane, the shearing flow induced by a moving fiber in the middle of a channel, the shearing flow near a non-slip rigid wall, and the circular Couette flow between two inversely rotating cylinders. The accuracy and robustness of the IB-LBGK coupling scheme, the performances of different discrete Dirac delta functions, the effect of iteration on the coupling scheme, the importance of the external forcing term treatment, the sensitivity of the coupling scheme to flow and boundary parameters, the velocity slip near non-slip rigid wall, and the origination of numerical instabilities are investigated in detail via the four test cases. It is found that the iteration in the coupling cycle can effectively improve stability, the introduction of a second-order forcing term in LBGK model is crucial, the discrete fiber segment length and the orientation of the fiber boundary obviously affect accuracy and stability, and the emergence of both temporal and spatial fluctuations of boundary parameters seems to be the indication of numerical instability. These elaborate results shed light on the nature of the coupling scheme and may benefit those who wish to use or improve the method.

Copyright

Corresponding author

Corresponding author.Email:ygcheng@whu.edu.cn

References

Hide All
[1]Peskin, C.S., Flow patterns around heart valves: a digital computer method for solving the equations of motion, PhD thesis, Physiol., Albert Einstein Coll. Med., Univ. Microfilms, 378 (1972), 72102.
[2]Peskin, C.S. and McQueen, D.M., A general method for the computer simulation of biological systems interacting with fluids-Biological fluid dynamics, C. P., Ellington et al. (Ed.), The company of biologists limited, Cambridge, 1995.
[3]McQueen, D.M. and Peskin, C.S., A three-dimensional computer model of the human heart for studying cardiac fluid dynamics, Computer Graphics, 34 (2000), 5660.
[4]Lemmon, J.D. and Yoganathan, A.P., Three-dimensional computational model of left heart diastolic function with fluid-structure interaction, J. Biomech. Eng., 122 (2000), 109117.
[5]McQueen, D.M., Peskin, C.S., and Yellin, E.L., Fluid dynamics of the mitral valve: physiological aspects of a mathematical model, Am. J. of Physiol., 242 (1982), 10951110.
[6]Griffith, B.E., Luo, X., McQueen, D.M., and Peskin, C.S., Simulating the fluid dynamics of natural and prosthetic heart valves using the immersed boundary method, Int. J. App. Mech., 1(1) (2009), 137177.
[7]Fauci, L. and Peskin, C.S., A computational model of aquatic animal locomotion, J.Comput. Phys., 77 (1988), 85108.
[8]Givelberg, E., Modeling elastic shells immersed in fluid, PhD thesis, Courant Institute of Mathematical Sciences, New York University, September (1997).
[9]Wang, N.T. and Fogelson, A.L., Computational methods for continuum models of platelet aggregation, J. Comput. Phys., 151 (1999), 649675.
[10]Sulsky, D. and Brackbill, J.U., A numerical method for suspension flow, J. Comput. Phys. 96(1991): 339368.
[11]Jung, E. and Peskin, C.S., 2-D simulation of valveless pumping using the immersed boundary method, SIAM J. Sci. Comput., 23(1) (2001), 1945.
[12]Arthurs, K.M., Moore, L.C., Peskin, C.S. et al., Modeling arteriolar flow and mass transport using the immersed boundary method, J. Comput. Phys., 147 (1998), 402440.
[13]Bottino, D.C., Modeling viscoelastic networks and cell deformation in the context of the immersed boundary method, J. Comput. Phys., 147 (1998), 86113.
[14]Miller, L.A. and Peskin, C.S., Flexible fling in tiny insect flight, J. Exp. Biol., 212(19) (2009), 30763090.
[15]Kim, Y., Lim, S., Raman, S.V. et al., Blood flow in a compliant vessel by the immersed boundary method, Ann. Biomed. Eng., 37(5) (2009), 927942.
[16]Teran, J., Fauci, L.J., and Shelley, M., Viscoelastic fluid response can increase the speed and efficiency of a free swimmer, Phys. Rev. Lett., 104 (2010), 038101.
[17]Naji, A., Atzberger, P.J., and Brown, F.L., Hybrid elastic and discrete-Particle approach to biomembrane dynamics with application to the mobility of curved integral membrane proteins, Phys. Rev. Lett., 102(13) (2009), 138102.
[18]Kim, Y. and Peskin, C.S., 2-D parachute simulation by the immersed boundary method, SIAM J. Sci. Comput., 28(6) (2006), 22942312.
[19]Li, Z.L. and Lai, M.C., Immersed interface methods for Navier-Stokes equations with singular forces, J. Comput. Phys., 171 (2001), 822842.
[20]Cortez, R. and Minion, M., The blob projection method for immersed boundary problems, J. Comput. Phy., 161 (2000), 428453.
[21]Wang, X.S., From immersed boundary method to immersed continuum method, Int. J. Multiscale Compu. Eng., 4(1) (2006), 127145.
[22]Liu, W.K., Kim, D.K., and Tang, S., Mathematical foundations of the immersed finite element method, Computational Mechanics, 39(3) (2007), 211222.
[23]Peskin, C.S. and Printz, B.F., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. Comp. Phys., 105 (1993), 3349.
[24]Griffith, B.E., Hornung, R.D., McQueen, D.M., and Peskin, C.S., An adaptive, formally second order accurate version of the immersed boundary method, J. Comput. Phys., 223(1) (2007), 1049.
[25]Lai, M.C. and Peskin, C.S., An immersed boundary method with formal second order accuracy and reduced numerical viscosity, J. Comput. Phys., 160 (2000), 705719.
[26]Zhu, L. and Peskin, C.S., Simulation of a flexible flapping filament in a flowing soap film by the immersed boundary method, J. Comput. Phys., 179(2) (2002), 452468.
[27]Kim, Y. and Peskin, C.S., Penalty immersed boundary method for an elastic boundary with mass, Phys. of Fluids, 19(5) (2007), 053103.
[28]Hou, T.Y. and Shi, Z., An efficient semi-implicit immersed boundary method for the Navier-Stokes equations, J. Comput. Phys., 227 (2008), 89688991.
[29]Cheng, Y.G. and Zhang, H., Immersed boundary method and lattice Boltzmann method coupled FSI simulation of mitral leaflet flow, Comput. & Fluids, 39(5) (2010), 871881.
[30]Zhang, J., Johnson, P.C., and Popel, A.S., An immersed boundary lattice Boltzmann approach to simulate deformable liquid capsules and its application to microscopic blood flows, Phys. Biol., 4 (2007), 285295.
[31]Zhang, J., Johnson, P.C., and Popel, A.S., Red blood cell aggregation and dissociation in shear flows simulated by lattice Boltzmann method, J. Biomech., 41 (2008), 4755.
[32]Zhu, L., He, G., Wang, S. et al., An immersed boundary method based on the lattice Boltzmann approach in three dimensions, with application, Comput. & Math. with Applications, 61 (2011), 35063518.
[33]Krueger, T., Varnik, F., and Raabe, D., Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method, Comput. & Math. with Applications, 61 (2011), 34853505.
[34]Feng, Z.G. and Michaelides, E.E., The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems, J. Comput. Phys., 195 (2004), 602628.
[35]Peng, Y., Shu, C. et al., Application of multi-block approach in the immersed boundary-lattice Boltzmann method for viscous fluid flows, J. Comput. Phys., 218 (2006), 460478.
[36]Peng, Y. and Luo, L.-S., A comparative study of immersed-boundary and interpolated bounce-back methods in LBE, Progress in Comput. Fluid Dyn., 8(1-4) (2008), 156166.
[37]Shu, C., Liu, N. and 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 (2007), 16071622.
[38]Dupuis, A., Chatelain, P., and Koumoutsakos, P., An immersed boundary-lattice Boltzmann method for the simulation of the flow past an impulsively started cylinder, J. Comput. Phys., 227 (2008), 44864498.
[39]Niu, X.D., Shu, C., Chew, Y.T., and Peng, Y., A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows, Phys. Lett. A, 354 (2006), 173182.
[40]Cheng, Y.G. and Li, J.P., Introducing unsteady non-uniform source terms into the lattice Boltzmann model, Int. J. Num. Meth. Fluids, 56 (2008), 629641.
[41]Hao, J. and Zhu, L., A lattice Boltzmann based implicit immersed boundary method for fluid-structure-interaction, Computers and Mathematics with Applications, 59 (2010), 185193.
[42]Tian, F.B., Luo, H., Zhu, L. et al., An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments, J. Comput. Phys., 230(19) (2011), 72667283.
[43]Cheng, Y.G., Zhang, H., and Liu, C., Immersed Boundary-Lattice Boltzmann Coupling Scheme for Fluid-Structure Interaction with Flexible Boundary, Commun. Comput. Phys., 9(5) (2011), 13751396.
[44]Kang, S.K. and Hassan, Y.A., A comparative study of direct-forcing immersed boundary-lattice Boltzmann methods for stationary complex boundaries, Int. J. Num. Meth. Fluids, 66(9) (2011), 629641.
[45]Griffith, B.E. and Peskin, C.S., On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems, J. Comput. Phys., 208 (2005), 75105.
[46]Le, G. and Zhang, J., Boundary slip from the immersed boundary lattice Boltzmann models, Phys. Rev. E, 79(2) (2009), 026701.
[47]Newren, E.P., Enhancing the immersed boundary method: stability, volume conservation and implicit solvers, PhD dissertation of the University of Utah, USA, 2007.
[48]Newren, E.P., Fogelson, A.L., Guy, R.D., and Kirby, R.M., Unconditionally stable discretizations of the immersed boundary equations, J. Comput. Phys., 222 (2007), 702719.
[49]Guo, Z., Zheng, C. and Shi, B., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E, 65(4) (2002), 046308.
[50]Wu, J. and Shu, C., Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications, J. Compu. Phys., 228 (2009), 19631979.
[51]Bringley, T.T., Analysis of the immersed boundary method for Stokes flow, PhD dissertation, New York University, 2008.
[52]Chen, S. and Doolen, G.D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30 (1998), 329364.
[53]Peskin, C.S., The immersed boundary method, Acta Numerica, 11 (2002), 479517.
[54]Pozrikidis, C., Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, Oxford, New York, 1997.
[55]Xu, S., Wang, Z.J., An immersed interface method for simulating the interaction of a fluid with moving boundaries, J. Comput. Phys., 216 (2006), 454493.

Keywords

Related content

Powered by UNSILO

Numerical Study of Stability and Accuracy of the Immersed Boundary Method Coupled to the Lattice Boltzmann BGK Model

  • Yongguang Cheng (a1), Luoding Zhu (a2) and Chunze Zhang (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.