Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T16:12:31.413Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  03 June 2015

Yongguang Cheng ,
Luoding Zhu  and
Chunze Zhang
Yongguang Cheng*
Affiliation:
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, P.R. China
Luoding Zhu*
Affiliation:
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, IN 46202, USA
Chunze Zhang
Affiliation:
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, P.R. China
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

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.

Keywords

76M2874F10Immersed boundary methodlattice Boltzmann methodfluid-structure interactionflexible boundarycomplex boundaryaccuracystabilityverification.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 1 , July 2014 , pp. 136 - 168
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[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), 72–102.Google Scholar
[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.Google Scholar
[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), 56–60.Google Scholar
[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), 109–117.Google Scholar
[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), 1095–1110.Google Scholar
[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), 137–177.Google Scholar
[7]Fauci, L. and Peskin, C.S., A computational model of aquatic animal locomotion, J.Comput. Phys., 77 (1988), 85–108.CrossRefGoogle Scholar
[8]Givelberg, E., Modeling elastic shells immersed in fluid, PhD thesis, Courant Institute of Mathematical Sciences, New York University, September (1997).Google Scholar
[9]Wang, N.T. and Fogelson, A.L., Computational methods for continuum models of platelet aggregation, J. Comput. Phys., 151 (1999), 649–675.Google Scholar
[10]Sulsky, D. and Brackbill, J.U., A numerical method for suspension flow, J. Comput. Phys. 96(1991): 339–368.Google Scholar
[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), 19–45.Google Scholar
[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), 402–440.CrossRefGoogle Scholar
[13]Bottino, D.C., Modeling viscoelastic networks and cell deformation in the context of the immersed boundary method, J. Comput. Phys., 147 (1998), 86–113.Google Scholar
[14]Miller, L.A. and Peskin, C.S., Flexible fling in tiny insect flight, J. Exp. Biol., 212(19) (2009), 3076–3090.CrossRefGoogle ScholarPubMed
[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), 927–942.Google Scholar
[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.Google Scholar
[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.Google Scholar
[18]Kim, Y. and Peskin, C.S., 2-D parachute simulation by the immersed boundary method, SIAM J. Sci. Comput., 28(6) (2006), 2294–2312.CrossRefGoogle Scholar
[19]Li, Z.L. and Lai, M.C., Immersed interface methods for Navier-Stokes equations with singular forces, J. Comput. Phys., 171 (2001), 822–842.Google Scholar
[20]Cortez, R. and Minion, M., The blob projection method for immersed boundary problems, J. Comput. Phy., 161 (2000), 428–453.Google Scholar
[21]Wang, X.S., From immersed boundary method to immersed continuum method, Int. J. Multiscale Compu. Eng., 4(1) (2006), 127–145.Google Scholar
[22]Liu, W.K., Kim, D.K., and Tang, S., Mathematical foundations of the immersed finite element method, Computational Mechanics, 39(3) (2007), 211–222.Google Scholar
[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), 33–49.Google Scholar
[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), 10–49.Google Scholar
[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), 705–719.Google Scholar
[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), 452–468.Google Scholar
[27]Kim, Y. and Peskin, C.S., Penalty immersed boundary method for an elastic boundary with mass, Phys. of Fluids, 19(5) (2007), 053103.Google Scholar
[28]Hou, T.Y. and Shi, Z., An efficient semi-implicit immersed boundary method for the Navier-Stokes equations, J. Comput. Phys., 227 (2008), 8968–8991.Google Scholar
[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), 871–881.Google Scholar
[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), 285–295.Google Scholar
[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), 47–55.Google Scholar
[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), 3506–3518.Google Scholar
[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), 3485–3505.Google Scholar
[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), 602–628.Google Scholar
[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), 460–478.Google Scholar
[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), 156–166.Google Scholar
[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), 1607–1622.Google Scholar
[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), 4486–4498.Google Scholar
[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), 173–182.Google Scholar
[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), 629–641.Google Scholar
[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), 185–193.Google Scholar
[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), 7266–7283.Google Scholar
[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), 1375–1396.Google Scholar
[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), 629–641.Google Scholar
[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), 75–105.Google Scholar
[46]Le, G. and Zhang, J., Boundary slip from the immersed boundary lattice Boltzmann models, Phys. Rev. E, 79(2) (2009), 026701.Google Scholar
[47]Newren, E.P., Enhancing the immersed boundary method: stability, volume conservation and implicit solvers, PhD dissertation of the University of Utah, USA, 2007.Google Scholar
[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), 702–719.Google Scholar
[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.Google Scholar
[50]Wu, J. and Shu, C., Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications, J. Compu. Phys., 228 (2009), 1963–1979.Google Scholar
[51]Bringley, T.T., Analysis of the immersed boundary method for Stokes flow, PhD dissertation, New York University, 2008.Google Scholar
[52]Chen, S. and Doolen, G.D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30 (1998), 329–364.Google Scholar
[53]Peskin, C.S., The immersed boundary method, Acta Numerica, 11 (2002), 479–517.Google Scholar
[54]Pozrikidis, C., Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, Oxford, New York, 1997.Google Scholar
[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), 454–493.Google Scholar