Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-21T05:42:51.942Z Has data issue: false hasContentIssue false

A Unified Wall-Boundary Condition for the Lattice Boltzmann Method and its Application to Force Evaluation

Published online by Cambridge University Press:  01 December 2014

S.-Y. Lin
Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan
Y.-H. Chin
Marketing & Supply Chain Management, Overseas Chinese University Taichung, Taiwan
F.-L. Yang
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan
J.-F. Lin*
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan
J.-J. Hu
Department of Information Management, Shu-Te University Kaohsiung, Taiwan
C.-S. Chen
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan
S.-H. Hsieh
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan
* Corresponding author (
Get access


A unified wall-boundary condition for the pressure-based lattice Boltzmann method (LBM) is proposed. The present approach is developed from the direct-forcing technique in the immersed boundary method and is derived from the equilibrium pressure distribution function. The proposed method can handle many kinds of wall boundaries, such as fixed wall and moving wall boundaries, in the same way. It is found that the new method has the following advantages: (1) simple in concept and easy to implement, (2) higher-order accuracy, (3) mass conservation, and (4) a stable and good convergence rate. Based on this wall-boundary condition, if a solid wall is immersed in a fluid, then by applying Gauss's theorem, the formulas for computing the force and torque acting on the solid wall from fluid flow are derived from the volume integrals over the solid volume instead of from the surface integrals over the solid surface. Based on the pressure-based LBM, inlet and outlet boundary conditions are also proposed. The order of accuracy of the proposed boundary condition is demonstrated with the errors of the velocity field, wall stress, and gradients of velocity and pressure. The steady flow past a circular cylinder is simulated to demonstrate the efficiency and capabilities of the proposed unified method.

Research Article
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 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.)


1.He, X. and Luo, L. S., “A Priori Derivation of the Lattice Boltzmann Equation,Physical Review E, 55, pp. 63336336 (1997).CrossRefGoogle Scholar
2.He, X. and Luo, L. S., “Theory of the Lattice Boltzrnann Equation: From Boltzmann Equation to Lattice Boltzmann Equation,Physical Review E, 56, pp. 68116817 (1997).CrossRefGoogle Scholar
3.Chen, H., Chen, S. and Matthaeus, W. H., “Recovery of the Navier-Stokes Equations Using a Lattice-Gas Boltzmann Method,Physical Review A, 45, pp. 53395342 (1992).CrossRefGoogle ScholarPubMed
4.He, X. and Luo, L. S., “Lattice Boltzmann Model for the Incompressible Navier-Stoke Equation,Journal of Statistical Physics, 88, pp. 927944 (1997).CrossRefGoogle Scholar
5.Ladd, A. J. C., “Numerical Simulation of Particular Suspensions via a Discretized Boltzmann Equation, Part 1. Theoretical Foundation,Journal of Fluid Mechanics, 271, pp. 285309 (1994).CrossRefGoogle Scholar
6.Ladd, A. J. C., “Numerical Simulation of Particular Suspensions via a Discretized Boltzmann Equation, Part 2. Numerical Results,Journal of Fluid Mechanics, 271, pp. 311339 (1994).CrossRefGoogle Scholar
7.Behrend, Q., “Solid-Fluid Boundaries in Particle Suspension Simulation via the Lattice Boltzmann Method,Physical Review E, 52, pp. 11641175 (1995).Google ScholarPubMed
8.Ginzbourg, I. and Alder, P. M., “Boundary Flow Condition Analysis for the Three-Dimensional Lattice Boltzmann Model,Journal of Physics II France, 4, pp. 191214 (1994).Google Scholar
9.Filippova, O. and Hanel, D., “Grid Refinement for Lattice-BGK Models,” Journal of Computational Physics, 147, pp. 219228 (1998).CrossRefGoogle Scholar
10.Mei, R., Luo, L. S. and Shyy, W., “An Accurate Curved Boundary Treatment in the Lattice Boltz-mann Method,” Journal of Computational Physics, 155, pp. 307330 (1999).CrossRefGoogle Scholar
11.Yu, D., “Viscous Flow Computations with the Lattice Boltzmann Equation Method,” Ph.D. Dissertation, Department of Mechanical and Aerospace Engineering, University of Florida, Florida, U.S. (2002).Google Scholar
12.Lai, M. C. and Peskin, C. S., “An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity,” Journal of Computational Physics, 160, pp. 705719 (2000).CrossRefGoogle Scholar
13.Fadlun, E. A., Verzicco, R., Orlandi, P. and Mohd-Yusof, J., “Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations,” Journal of Computational Physics, 161, pp. 3560 (2000)CrossRefGoogle Scholar
14.Lee, C., “Stability Characteristics of the Virtual Boundary Method in Three-Dimensional Applications,” Journal of Computational Physics, 184, pp. 559591 (2003).CrossRefGoogle Scholar
15.Feng, Z. G. and Michaelides, E. E., “The Immersed Boundary-Lattice Boltzmann Method for Solving Fluid-Particles Interaction Problems,” Journal of Computational Physics, 195, pp. 602628 (2004).CrossRefGoogle Scholar
16.Peskin, C. S., “Flow Patterns Around Heart Valves: A Numerical Method,” Journal of Computational Physics, 10, pp. 252271 (1972).CrossRefGoogle Scholar
17.Peskin, C. S., “Numerical-Analysis of Blood-Flow in Heart,” Journal of Computational Physics, 25, pp. 220252 (1977).CrossRefGoogle Scholar
18.Uhlmann, M., “An Immersed Boundary Method with Direct Forcing for the Simulation of Particulate Flows,” Journal of Computational Physics, 209, pp. 448476 (2005).CrossRefGoogle Scholar
19.Feng, Z. G. and Michaelides, E. E., “Proteus: A Direct Forcing Method in the Simulations of Particu-late Flows,” Journal of Computational Physics, 202, pp. 2051 (2005).CrossRefGoogle Scholar
20.Lin, S. Y., Chin, Y. H., Hu, J. J. and Chen, Y. C., “A Pressure Correction Method for Fluid-Particle Interaction Flow: Direct-Forcing Method and Sedimentation Flow,” International Journal for Numerical Methods in Fluids, 67, pp. 17711798 (2011).CrossRefGoogle Scholar
21.Lin, S. Y. and Lin, J. F., “Numerical Investigation of Lubrication Force on a Spherical Particle Moving to a Plane Wall at Finite Reynolds Numbers,” International Journal of Multiphase Flow, 53, pp. 4053 (2013).CrossRefGoogle Scholar
22.Lin, S. Y., Lin, C. T., Chin, Y. H. and Tai, Y. H., “A Direct-Forcing Pressure-Based Lattice Boltzmann Method for Solving Fluid-Particle Interaction Problems,” International Journal for Numerical Method in Fluid, 66, pp. 648670 (2011).CrossRefGoogle Scholar
23.Ziegler, D. P., “Boundary Conditions for Lattice Boltzmann Simulations,” Journal of Statistical Physics, 71, pp. 11711177 (1993).CrossRefGoogle Scholar
24.He, X., Luo, L. S. and Dembo, M., “Some Progress in the Lattice Boltzmann Method, Part 1, NonUniform Mesh Grids,” Journal of Computational Physics, 129, pp. 357363 (1996).CrossRefGoogle Scholar
25.Ghia, U., Ghia, K. N. and Shin, C. T., “High-Resolutions for Incompressible Flow Using the Na-vier- Stokes Equations and a Multigrid Method,” Journal of Computational Physics, 48, pp. 387411 (1982).CrossRefGoogle Scholar
26.Rogers, S. E., Kwak, D. and Kiris, C., “Numerical Solution of the Incompressible Navier-Stokes Equations for Steady-State and Time-Dependent Problems,” AIAA Journal, 89, p. 463 (1989).Google Scholar
27.Fornberg, B., “A Numerical Study of Steady Viscous Flow Past a Circular Cylinder,” Journal of Fluid Mechanics, 98, pp. 819855 (1980).CrossRefGoogle Scholar
28.Dennis, S. C. R. and Chang, G. Z., “Numerical Solutions for Steady Flow past a Circular Cylinder at Reynolds Numbers up to 100,” Journal of Fluid Mechanics, 42, pp. 471489 (1970).CrossRefGoogle Scholar
29.Nieuwstadt, F. and Keller, H. B., “Viscous Flow past Circular Cylinders,” Computations & Fluids, 1, pp. 5971 (1973).CrossRefGoogle Scholar
30.He, X., Doolen, G. D. and Clark, T., “Comparison of the Lattice Boltzmann Method and the Artificial Compressibility Method for Navier-Stokes Equations,” Journal of Computational Physics, 179, pp. 439451 (2002).CrossRefGoogle Scholar