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Lattice BGK Model for Incompressible Axisymmetric Flows

Published online by Cambridge University Press:  20 August 2015

Ting Zhang ,
Baochang Shi ,
Zhenhua Chai  and
Fumei Rong
Ting Zhang*
Affiliation:
State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, China
Baochang Shi*
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
Zhenhua Chai*
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
Fumei Rong*
Affiliation:
State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, China
*
Email:[email protected]
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
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Abstract

In this paper, a lattice Boltzmann BGK (LBGK) model is proposed for simulating incompressible axisymmetric flows. Unlike other existing axisymmetric lattice Boltzmann models, the present LBGK model can eliminate the compressible effects only with the small Mach number limit. Furthermore the source terms of the model are simple and contain no velocity gradients. Through the Chapman-Enskog expansion, the macroscopic equations for incompressible axisymmetric flows can be exactly recovered from the present LBGK model. Numerical simulations of the Hagen-Poiseuille flow, the pulsatile Womersley flow, the flow over a sphere, and the swirling flow in a closed cylindrical cavity are performed. The results agree well with the analytic solutions and the existing numerical or experimental data reported in some previous studies.

Keywords

47.11-j02.60.CbLattice BGK modelincompressible axisymmetric
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 5 , May 2012 , pp. 1569 - 1590
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Qian, Y. H., Succi, S. and Orszag, S. A., Recent advances in lattice Boltzmann computing, Annu. Rev. Comput. Phys., 3 (1995), 195–242.Google Scholar
[2]Benzi, R., Succi, S. and Vergassola, M., The lattice Boltzmann equation-theory and applications, Phys. Rep., 222 (1992), 145–197.CrossRefGoogle Scholar
[3]Chen, S. and Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30 (1998), 329–364.Google Scholar
[4]Aidun, C. K. and Clausen, J. R., Lattice-Boltzmann method for complex flows, Annu. Rev. Fluid Mech., 42 (2010), 439–472.Google Scholar
[5]Succi, S., The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford University Press, 2001.CrossRefGoogle Scholar
[6]Zou, Q., Hou, S., Chen, S. and Doolen, G. D., An improved incompressible lattice Boltzmann model for time-independent flows, J. Stat. Phys., 81 (1995), 35–48.Google Scholar
[7]Lin, Z., Fang, H. and Tao, R., Improved latticel Boltzmann model for incompressible two-dimensional steady flows, Phys. Rev. E, 54 (1997), 6323–6330.CrossRefGoogle Scholar
[8]Chen, Y. and Ohashi, H., Lattice-BGK methods for simulating incompressible fluid flows, Int. J. Mod. Phys. C, 8 (1997), 793–803.CrossRefGoogle Scholar
[9]He, X. and Luo, L. S., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J. Stat. Phys., 88 (1997), 927–944.Google Scholar
[10]Guo, Z. L., Shi, B. C. and Wang, N. C., Lattice BGK model for incompressible Navier-Stokes equation, J. Comput. Phys., 165 (2000), 288–306.Google Scholar
[11]Artoli, A. M., Hoekstra, A. G. and Sloot, P. M. A., 3D pulsatile flow in the lattice Boltzmann BGK method, Int. J. Mod. Phys. C, 13 (2002), 1119–1134.Google Scholar
[12]Inamuro, T., Tomita, R. and Ogino, F., Lattice Boltzmann simulations of drop deformation and breakup in shear flows, Int. J. Mod. Phys. B, 17 (2003), 21–26.Google Scholar
[13]Bhaumik, S. K. and Lakshmisha, K. N., Lattice Boltzmann simulation of lid-driven swirling flow in confined cylindrical cavity, Comput. Fluids, 36 (2007), 1163–1173.Google Scholar
[14]Halliday, I., Hammond, L. A., Care, C. M., Good, K. and Stevens, A., Lattice Boltzmann equation hydrodynamics, Phy. Rev. E, 64 (2001), 011208.Google Scholar
[15]Premnath, K. N. and Abraham, J., Lattice Boltzmann model for axisymmetric multiphase flows, Phys. Rev. E, 71 (2005), 056706.Google Scholar
[16]Mukherjee, S. and Abraham, J., Lattice Boltzmann simulations of two-phase flow with high density ratio in axially symmetric geometry, Phys. Rev. E, 75 (2007), 026701.Google Scholar
[17]Lee, T. S., Huang, H. and Shu, C., An axisymmetric incompressible lattice Boltzmann model for pipe flow, Int. J. Mod. Phys. C, 17 (2006), 645–661.Google Scholar
[18]Reis, T. and Phillips, T. N., Modified lattice Boltzmann model for axisymmetric flows, Phys. Rev. E, 75 (2007), 056703.Google Scholar
[19]Reis, T. and Phillips, T. N., Numerical validation of a consistent axisymmetric lattice Boltz-mann model, Phys. Rev. E, 77 (2008), 026703.CrossRefGoogle ScholarPubMed
[20]Zhou, J. G., Axisymmetric lattice Boltzmann method, Phys. Rev. E, 78 (2008), 036701.CrossRefGoogle ScholarPubMed
[21]Chen, S., Tölke, J, Geller, S. and Krafczyk, M., Lattice Boltzmann model for incompressible axisymmetric flows, Phys. Rev. E, 78 (2008), 046703.Google Scholar
[22]Peng, Y., Shu, C., Chew, Y. T. and Qiu, J., Numerical investigation of flows in Czochralski crystal growth by an axisymmetric lattice Boltzmann method, J. Comput. Phys., 186 (2003), 295–307.Google Scholar
[23]Niu, X. D., Shu, C. and Chew, Y. T., An axisymmetric lattice Boltzmann model for simulation of Taylor-Couette flows between two concentric cylinders, Int. J. Mod. Phys. C, 14 (2003), 785–796.Google Scholar
[24]McCracken, M. E. and Abraham, J., Simulations of liquid break up with an axisymmetric, multiple relaxation time, index-function lattice Boltzmann model, Int. J. Mod. Phys. C, 16 (2005), 1671–1692.Google Scholar
[25]Lee, T. S., Huang, H. and Shu, C., An axisymmetric incompressible lattice BGK model for simulation of the pulsatile flow in a circular pipe, Int. J. Numer. Meth. Fluids, 49 (2005), 99–116.Google Scholar
[26]Huang, H., Lee, T. S. and Shu, C., Hybrid lattice Boltzmann finite difference simulation of axisymmetric swirling and rotating flows, Int. J. Numer. Methods Fluids, 53 (2007), 1707–1726.Google Scholar
[27]Huang, H., Lee, T. S. and Shu, C., Lattice Boltzmann simulation of gas slip flow in long microtubes, Int. J. Numer. Meth. Heat Fluid Flow, 17 (2007), 587–607.Google Scholar
[28]Mukherjee, S. and Abraham, J., Investigations of drop impact on dry walls with a lattice-Boltzmann model, J. Colloid Interface Sci., 312 (2007), 341–354.Google Scholar
[29]Mukherjee, S. and Abraham, J., Crown behavior in drop impact on wet walls, Phys. Fluids, 19 (2007), 052103.Google Scholar
[30]Chen, S., Tölke, J., Geller, S. and Krafczyk, M., Simulation of buoyancy-driven flows in a vertical cylinder using a simple lattice Boltzmann model, Phys. Rev. E, 79 (2009), 016704.Google Scholar
[31]Guo, Z. L., Han, H. F., Shi, B. C. and Zheng, C. G., Theory of the lattice Boltzmann equation: lattice Boltzmann model for axisymmetric flows, Phys. Rev. E, 79 (2009), 046708.Google Scholar
[32]Wang, L., Guo, Z. L. and Zheng, C. G., Multi-relaxation-time lattice Boltzmann model for axisymmetric flows, Comput. Fluids, 39 (2010), 1542–1548.Google Scholar
[33]Guo, Z. L., Shi, B. C. and Zheng, C. G., A coupled lattice BGK model for the Boussinesq equations, Int. J. Numer. Meth. Fluids, 39 (2002), 325–342.Google Scholar
[34]Ladd, A. J. C. and Verberg, R., Lattice-Boltzmann simulations of particle-fluid suspensions, J. Stat. Phys., 104 (2001), 1191–1251.Google Scholar
[35]Guo, Z. L., Zheng, C. G. and Shi, B. C., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E, 65 (2002), 046308.CrossRefGoogle ScholarPubMed
[36]Filippova, O. and Hänel, D., Grid refinement for lattice-BDK models, J. Comput. Phys., 147 (1998), 219–228.Google Scholar
[37]Mei, R., Luo, L. S. and Shyy, W., An accurate curved boundary treatment in the lattice Boltz-mann method, J. Comput. Phys., 155 (1999), 307–330.Google Scholar
[38]Guo, Z. L., Zheng, C. G. and Shi, B. C., An extrapolation method for boundary conditions in lattice Boltzmann method, Phys. Fluids, 14 (2002), 2007–2010.Google Scholar
[39]Rimon, Y. and Cheng, S. I., Numerical solution of a uniform flow over asphereat intermediate Reynolds number, Phys. Fluids, 12 (1969), 949–959.Google Scholar
[40]Roos, F. W. and Willmarth, W. W., Some experimental results on sphere and disk drag, AIAA J., 9 (1971), 285–291.Google Scholar
[41]Clift, R., Grace, R. J. and Weber, E. W., Bubbles, Drops and Particles, Academic Press, New York, 1978.Google Scholar
[42]Shirayama, S., Flow past a sphere: topological transitions of the vorticity field, AIAA J., 30 (1992), 349–358.Google Scholar
[43]Mittal, R., A Fourier Chebyshev spectral collocation method for simulation flow past sphere-sand spheroids, Int. J. Numer. Methods Fluids, 30 (1999), 921–937.Google Scholar
[44]Johnson, T. A. and Patel, V. C., Flow past a sphere up to a Reynolds number of 300, J. Fluid Mech., 378 (1999), 19–70.Google Scholar
[45]Wang, X. Y., Yeo, K. S., Chew, C. S. and Khoo, B. C., A SVD-GFD scheme for computing 3D incompressible viscous fluid flows, Comput. Fluids, 37 (2008), 733–746.Google Scholar
[46]Kim, J., Kim, D. and Choi, H., An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys., 171 (2001), 132–150.Google Scholar
[47]Kim, D. and Choi, H., Characteristics of laminar flow past a sphere in uniform shear, Phys. Fluids, 17 (2005), 103602.CrossRefGoogle Scholar
[48]Mei, R. W., Shyy, W., Yu, D. Z. and Luo, L. S., Lattice Boltzmann method for 3-D flows with curved boundary, J. Compt. Phys., 161 (2000), 680–699.Google Scholar
[49]Escusier, M. P., Observations of the flow produced in a cylindrical container by a rotating endwall, Exp. Fluids, 2 (1984), 189–196.Google Scholar
[50]Lugt, H. J. and Haussling, H. J., Axisymmetric vortex breakdown in rotating fluid within a container, J. Appl. Mech., 49 (1982), 921–923.Google Scholar
[51]Lugt, H. J. and Abboud, M., Axisymmetric vortex breakdown with and without temperature effects in a container with a rotating lid, J. Fluid Mech., 179 (1987), 179–200.Google Scholar
[52]Lopez, J. M., Axisymmetric vortex breakdown part 1: confined swirling flow, J. Fluid Mech., 221 (1990), 533–552.Google Scholar
[53]Brown, G. L. and Lopez, J. M., Axisymmetric vortex breakdown part 2: physical mechanisms, J. Fluid Mech., 221 (1990), 553–576.Google Scholar
[54]Fujimura, K., Koyama, H. S. and Hyun, J. M., Time-dependent vortex breakdown in a cylinder with a rotating lid, Trans. ASME. J. Fluids Eng., 119 (1997), 450–453.Google Scholar