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A Simplified Lattice Boltzmann Method without Evolution of Distribution Function

  • Z. Chen (a1), C. Shu (a1), Y. Wang (a1), L. M. Yang (a2) and D. Tan (a1)...

In this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order of accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.

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*Corresponding author. Email: (C. Shu)
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[1] S. Chen , H. Chen , D. Martnez and W. Matthaeus , Lattice Boltzmann model for simulation of magnetohydrodynamics, Phys. Rev. Lett., 67 (1991), 3776.

[2] Y. Qian , D. D’Humiéres and P. Lallemand , Lattice BGK models for Navier-Stokes equation, EPL (Europhysics Letters), 17 (1992), 479.

[3] D. D’Humiéres , Multiplerelaxationtime lattice Boltzmann models in three dimensions, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 360 (2002), pp. 437451.

[4] R. Mei , L.-S. Luo and W. Shyy , An accurate curved boundary treatment in the lattice Boltzmann method, J. Comput. Phys., 155 (1999), pp. 307330.

[5] Z. Guo , C. Zheng and B. Shi , An extrapolation method for boundary conditions in lattice Boltzmann method, Phys. Fluids, 14 (2002), pp. 20072010.

[6] P. Lallemand and L.-S. Luo , Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 61 (2000), 6546.

[7] X. He and G. Doolen , Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder, J. Comput. Phys., 134 (1997), pp. 306315.

[8] H. Chen , S. Chen and W. H. Matthaeus , Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method, Phys. Rev. A, 45 (1992), R5339.

[9] S. Chen , D. Martinez and R. Mei , On boundary conditions in lattice Boltzmann methods, Phys. Fluids, 8 (1996), pp. 25272536.

[10] Z. Guo , B. Shi and N. Wang , Lattice BGK model for incompressible NavierStokes equation, J. Comput. Phys., 165 (2000), pp. 288306.

[11] C. Shu , X. Niu and Y. Chew , Taylor series expansion and least squares-based lattice Boltzmann method: three-dimensional formulation and its applications, Int. J. Modern Phys. C, 14 (2003), pp. 925944.

[12] Z.-G. Feng and E. E. Michaelides , The immersed boundary-lattice Boltzmann method for solving fluidparticles interaction problems, J. Comput. Phys., 195 (2004), pp. 602628.

[13] Y.-H. Zhang , X.-J. Gu , R. W. Barber and D. R. Emerson , Capturing Knudsen layer phenomena using a lattice Boltzmann model, Phys. Rev. E, 74 (2006), 046704.

[15] C. Lim , C. Shu , X. Niu and Y. Chew , Application of lattice Boltzmann method to simulate microchannel flows, Phys. Fluids, 14 (2002), pp. 22992308.

[16] X. He , S. Chen and G. D. Doolen , A novel thermal model for the lattice Boltzmann method in incompressible limit, J. Comput. Phys., 146 (1998), pp. 282300.

[17] Y. Peng , C. Shu and Y. Chew , Simplified thermal lattice Boltzmann model for incompressible thermal flows, Phys. Rev. E, 68 (2003), 026701.

[18] G. Tang , W. Tao and Y. He , Lattice Boltzmann method for gaseous microflows using kinetic theory boundary conditions, Phys. Fluids, 17 (2005), 058101.

[19] X. He , S. Chen and R. Zhang , A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of RayleighTaylor instability, J. Comput. Phys., 152 (1999), pp. 642663.

[20] X. Shan and H. Chen , Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E, 47 (1993), 1815.

[21] T. Inamuro , T. Ogata , S. Tajima and N. Konishi , A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Comput. Phys., 198 (2004), pp. 628644.

[22] X. Shan , Simulation of Rayleigh-Bénard convection using a lattice Boltzmann method, Phys. Rev. E, 55 (1997), 2780.

[24] T. Liszka and J. Orkisz , The finite difference method at arbitrary irregular grids and its application in applied mechanics, Comput. Struct., 11 (1980), pp. 8395.

[25] C. B. Lee , New features of CS solitons and the formation of vortices, Phys. Lett. A, 247 (1998), pp. 397402.

[26] C. Lee , Possible universal transitional scenario in a flat plate boundary layer: Measurement and visualization, Phys. Rev. E, 62 (2000), 3659.

[28] C. K. Aidun and J. R. Clausen , Lattice-Boltzmann method for complex flows, Ann. Rev. Fluid Mech., 42 (2010), pp. 439472.

[29] S. Chen and G. D. Doolen , Lattice Boltzmann method for fluid flows, Ann. Rev. Fluid Mech., 30 (1998), pp. 329364.

[30] X. He and L.-S. Luo , Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E, 56 (1997), 6811.

[31] G. R. McNamara and G. Zanetti , Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett., 61 (1988), 2332.

[33] Y. Wang , C. Shu and C. Teo , Development of LBGK and incompressible LBGK-based lattice Boltzmann flux solvers for simulation of incompressible flows, Int. J. Numer. Methods Fluids, 75 (2014), pp. 344364.

[34] Y. Wang , C. Shu and C. Teo , Thermal lattice Boltzmann flux solver and its application for simulation of incompressible thermal flows, Comput. Fluids, 94 (2014) pp. 98111.

[35] Y. Wang , C. Shu , H. Huang and C. Teo , Multiphase lattice Boltzmann flux solver for incompressible multiphase flows with large density ratio, J. Comput. Phys., 280 (2015), pp. 404423.

[36] Y. Wang , L. Yang and C. Shu , From lattice Boltzmann method to lattice Boltzmann flux solver, Entropy, 17 (2015), pp. 77137735.

[37] F. M. White , Fluid Mechanics, McGraw-Hill, New York, 2003.

[38] R. Benzi , S. Succi and M. Vergassola , The lattice Boltzmann equation: theory and applications, Phys. Reports, 222 (1992), pp. 145197.

[40] T. Inamuro , M. Yoshino and F. Ogino , Accuracy of the lattice Boltzmann method for small Knudsen number with finite Reynolds number, Phys. Fluids, 9 (1997), pp. 35353542.

[41] Z. Guo and C. Shu , Lattice Boltzmann Method and Its Applications in Engineering, World Scientific, 2013.

[42] J. Kim and P. Moin , Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), pp. 308323.

[44] J. D. Sterling , S. Chen , Stability analysis of lattice Boltzmann methods, J. Comput. Phys., 123 (1996), pp. 196206.

[45] X. Niu , C. Shu , Y. Chew and T. Wang , Investigation of stability and hydrodynamics of different lattice Boltzmann models, J. Stat. Phys., 117 (2004), pp. 665680.

[46] U. Ghia , K. N. Ghia and C. Shin , High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48 (1982), 387411.

[47] R. Mei , L.-S. Luo , P. Lallemand and D. D’Humiéres , Consistent initial conditions for lattice Boltzmann simulations, Comput. Fluids, 35 (2006), pp. 855862.

[48] R. Mei and W. Shyy , On the finite difference-based lattice Boltzmann method in curvilinear coordinates, J. Comput. Phys., 143 (1998), pp. 426448.

[49] Z. Guo , C. Zheng and B. Shi , Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E, 65 (2002), 046308.

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