Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T18:13:40.440Z Has data issue: false hasContentIssue false
  • English
  • Français

Development of Lattice Boltzmann Flux Solver for Simulation of Incompressible Flows

Published online by Cambridge University Press:  03 June 2015

C. Shu ,
Y. Wang ,
C. J. Teo  and
J. Wu
C. Shu*
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Y. Wang
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
C. J. Teo
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
J. Wu
Affiliation:
Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
*
*Corresponding author. Email: [email protected]
Get access

Abstract

A lattice Boltzmann flux solver (LBFS) is presented in this work for simulation of incompressible viscous and inviscid flows. The new solver is based on Chapman-Enskog expansion analysis, which is the bridge to link Navier-Stokes (N-S) equations and lattice Boltzmann equation (LBE). The macroscopic differential equations are discretized by the finite volume method, where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers. The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh, tie-up of mesh spacing and time interval, limitation to viscous flows. LBFS is validated by its application to simulate the viscous decaying vortex flow, the driven cavity flow, the viscous flow past a circular cylinder, and the inviscid flow past a circular cylinder. The obtained numerical results compare very well with available data in the literature, which show that LBFS has the second order of accuracy in space, and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.

Keywords

20B40Chapman-Enskog analysisflux solverincompressible flowNavier-Stokes equationlattice Boltzmann equation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 4 , August 2014 , pp. 436 - 460
Copyright
Copyright © Global-Science Press 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] Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 135 (1997), pp. 118–125.CrossRefGoogle Scholar
[2] Chorin, A. J., Numerical solution of the Navier-Stokes equation, Math. Comput., 22 (1971), pp. 745–762.CrossRefGoogle Scholar
[3] Patankar, S. V. and Spadling, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Trans., 15 (1972), pp. 1787–1806.CrossRefGoogle Scholar
[4] Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1981.Google Scholar
[5] Van Doormaal, J. P. and Raithby, G. D., Enhancement of the simple method for predicting incompressible fluid flows, Numer. Heat Trans., 7 (1984), pp. 147–163.Google Scholar
[6] Issa, R. I., Solution of the implicit discretized fluid flow equations by operator splitting, J. Comput. Phys., 62 (1985), pp. 40–65.CrossRefGoogle Scholar
[7] Kim, J. and Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), pp. 308–323.CrossRefGoogle Scholar
[8] Bell, J. B., Colella, P. and Glaz, H. M., A second order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85 (1989), pp. 257–283.CrossRefGoogle Scholar
[9] Brown, D. L., Cortez, R. and Minion, M. L., Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168 (2001), pp. 464–499.CrossRefGoogle Scholar
[10] Chen, S., Chen, H., Martinez, D. O. and Matthaeus, W. H., Lattice Boltzmann model for simulation of magnetohydrodynamics, Phys. Rev. Let., 67 (1991), pp. 3776–3779.CrossRefGoogle ScholarPubMed
[11] Qian, Y. H., D’Humieres, D. and Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett., 17 (1992), pp. 479–484.CrossRefGoogle Scholar
[12] He, X., Luo, L. S. and Dembo, M., Some progress in lattice Boltzmann method: Part I. Nonuniform mesh grids, J. Comput. Phys., 129 (1996), pp. 357.CrossRefGoogle Scholar
[13] Chen, S. and Doolen, G., Lattice Boltzmann method for fluidflows, Ann. Rev. Fluid Mech., 30 (1998), pp. 329–364.CrossRefGoogle Scholar
[14] Luo, L. S., Unified theory of the lattice Boltzmann models for nonideal gases, Phys. Rev. Lett., 81 (1998), pp. 1618–1621.CrossRefGoogle Scholar
[15] Mei, R., Luo, L. S. and Shyy, W., An accurate curved boundary treatment in the lattice Boltzmann method, J. Comput. Phys., 155 (1999), pp. 307–330.CrossRefGoogle Scholar
[16] Lallemand, P. and Luo, L. S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 61 (2000), pp. 6546–6562.CrossRefGoogle ScholarPubMed
[17] Guo, Z. L., Shi, B. C. and Wang, N. C., Lattice BGK model for incompressible Navier-Stokes equation, J. Comput. Phys., 165 (2000), pp. 288–306.CrossRefGoogle Scholar
[18] Shu, C., Chew, Y. T. and Niu, X. D., Least square-based LBM: a meshless approach for simulation of flows with complex geometry, Phys. Rev. E, 64 (2001), 045701.CrossRefGoogle Scholar
[19] Shu, C., Niu, X. D. and Chew, Y. T., Taylor series expansion- and least square-based lattice Boltzmann method: two-dimensional formulation and its applications, Phys. Rev. E, 65 (2002), 036708.CrossRefGoogle Scholar
[20] Succi, S., Mesoscopic modeling of slip motion at fluid-solid interfaces with heterogeneous catalysis, Phys. Rev. Lett., 89 (2002), 064502.CrossRefGoogle ScholarPubMed
[21] Feng, Z. and Michaelides, E., The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems, J. Comput. Phys., 195 (2004), pp. 602–628.CrossRefGoogle Scholar
[22] Zhang, Y. H., Gu, X. J., Barber, R. W. and Emerson, D. R., Capturing Knudsen layer phenomena using a lattice Boltzmann model, Phys. Rev. E, 74 (2006), 046704.CrossRefGoogle ScholarPubMed
[23] Shan, X., Yuan, X. F. and Chen, H., Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation, J. Fluid Mech., 550 (2006), pp. 413–441.CrossRefGoogle Scholar
[24] Aidun, C. K. and Clausen, J. R., Lattice-Boltzmann method for complex flows, Ann. Rev. Fluid Mech., 42 (2010), pp. 439–472.CrossRefGoogle Scholar
[25] Swift, M. R., Orlandini, E., Osborn, W. and Yoemans, J. M., Lattice Boltzmann simulations ofliquid-gas and binary fluid systems, Phys. Rev. E, 54 (1996), pp. 5041–5052.CrossRefGoogle ScholarPubMed
[26] He, X., Chen, S. and Doolen, G. D., A novel thermal model for the lattice Boltzmann method in incompressible limit, J. Comput. Phys., 146 (1998), pp. 282–300.CrossRefGoogle Scholar
[27] Kataoka, T. and Tsutahara, M., Lattice Boltzmann method for the compressible Navier-Stokes equations with flexible specific-heat ratio, Phys. Rev. E, 69 (2004), R035701.Google Scholar
[28] Zhang, Y. H., Qin, R. and Emerson, D. R., Lattice Boltzmann simulation of rarefied gas flows in microchannels, Phys. Rev. E, 71 (2005), 047702.CrossRefGoogle ScholarPubMed
[29] Guo, Z. L., Asinari, P. and Zheng, C. G., Lattice Boltzmann equation for microscale gas flows of binary mixtures, Phys. Rev. E, 79 (2009), 026702.CrossRefGoogle ScholarPubMed
[30] Mendoza, M., Boghosian, B.M., Herrmann1, H. J. and Succi, S., Fast lattice Boltzmann solver for relativistic hydrodynamics, Phys. Rev. Lett., 105 (2010), 014502.CrossRefGoogle ScholarPubMed
[31] Wang, M. R. and Kang, Q. J., Modeling electrokinetic flows in microchannels using coupled lattice Boltzmann method, J. Comput. Phys., 229 (2010), pp. 728–744.CrossRefGoogle Scholar
[32] Wu, J. and Shu, C., A solution-adaptive lattice Boltzmann method for two-dimensional incompressible viscous flows, J. Comput. Phys., 230 (2011), pp. 2246–2269.CrossRefGoogle Scholar
[33] Li, X. J., Zhao, R. G. and Zhong, C. W., Novel immersed boundary-lattice Boltzmann method based on feedback law, Transactions of Nanjing University of Aeronautics and Astronautics, 29 (2012), pp. 179–186.Google Scholar
[34] Huang, H. B., Wu, Y. F. and Lu, X. Y., Shear viscosity of dilute suspensions of ellipsoidal particles with a lattice Boltzmann method, Phys. Rev. E, 86 (2012), 046305.CrossRefGoogle ScholarPubMed
[35] Zhao, L. Q., Sun, J. H. and Xu, C. Y., Flow field analyses of plane jet at low Reynolds number using lattice Boltzmann method, Transactions of Nanjing University of Aeronautics and Astronautics, 29 (2012), pp. 199–206.Google Scholar
[36] Liu, H. H., Zhang, Y. H. and Valocchi, A. J., Modeling and simulation of thermocapillary flows using lattice Boltzmann method, J. Comput. Phys., 231 (2012), pp. 4433–4453.CrossRefGoogle Scholar
[37] Xu, K., Gas-kinetic schemes for unsteady compressible flow simulations, VKI Report, 1998–03 (1998).Google Scholar
[38] Tang, H. Z. and Xu, K., A high-order gas-kinetic method for multidimensional ideal magnetohy-drodynamics, J. Comput. Phys., 165 (2000), pp. 69–88.CrossRefGoogle Scholar
[39] Xu, K., A gas-kinetic BGK scheme for the Navier-Stokes equations, and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171 (2001), pp. 289–335.CrossRefGoogle Scholar
[40] Chen, S. Z., Xu, K., Li, C. B. and Cai, Q. D., A unified gas kinetic scheme with moving mesh and velocity space adaptation, J. Comput. Phys., 231 (2012), pp. 6643–6664.CrossRefGoogle Scholar
[41] Ghia, U., Chia, K. N. and Shin, C. T., High-Re solutions for incompressible flow using the Navier-Stokes equations a multigrid method, J. Comput. Phys., 48 (1982), pp. 387–411.CrossRefGoogle Scholar
[42] Dennis, S. C. R. and Chang, G. Z., Numerical solutions for steady flow past a circular cylinder at Reynolds number up to 100, J. Fluid Mech., 42 (1970), pp. 471–489.CrossRefGoogle Scholar
[43] Nieuwstadt, F. and Keller, H. B., Viscous flow past circular cylinders, Comput. Fluids, 1 (1973), pp. 59–71.CrossRefGoogle Scholar
[44] He, X. and Doolen, G. D., Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder, J. Comput. Phys., 134 (1997), pp. 306–315.CrossRefGoogle Scholar
[45] Shukla, R. K., Tatineni, M. and Zhong, X., Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations, J. Comput. Phys., 224 (2007), pp. 1064–1094.CrossRefGoogle Scholar
[46] Park, J., Kwon, K. and Choi, H., Numerical solutions of flow past a circular cylinder at Reynolds number up to 160, KSME Int. J., 12 (1998), pp. 1200.CrossRefGoogle Scholar
[47] Braza, M., Chassaing, P. and Minh, H. H., Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder, J. Fluid Mech., 165 (1986), pp. 79–130.CrossRefGoogle Scholar
[48] Benson, M. G., Bellamy-Knights, P. G., Gerrard, J. H. and Gladwell, I., A viscous splitting algorithm applied to low Reynolds number flows round a circular cylinder, J. Fluids Struct., 3 (1989), pp. 439–479.CrossRefGoogle Scholar
[49] Ding, H., Shu, C., Yeo, K. S. and Xu, D., Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method, Comput. Meth. Appl. Mech. Eng., 193 (2004), pp. 727–744.CrossRefGoogle Scholar