本研究為在雷諾數 Re = 700, 1400, 2800, 5600, 10595 下，用 D3Q19 多鬆弛時間晶格波茲曼法模擬週期性山坡紊流流場。其中，山坡表面的部分採用了 BFL 曲面邊界修正法，外力部分則是採用了郭照立等人提出的作用力模型來驅動流場。模擬結果在不同雷諾數和不同位置橫截面下與 Breuer 等人的結果進行了比較，並得到了一致的觀察結果和發現。簡言之，週期性山坡紊流流場以晶格波茲曼法進行了完整的重現。此外，本研究應用了 CUDA 及 MPI 平行運算的架構在高速圖形顯示卡叢集上進行運算，並且進行了優化來獲得更高的運算效率。

In this work, turbulent channel flows over 3D periodic hills were simulated using the D3Q19 multiple-relaxation-time lattice Boltzmann method (MRT-LBM) at Reynolds numbers Re = 700, 1400, 2800, 5600, 10595. The BFL bounce-back rule for curved boundaries was implemented on hill surfaces, and Guo et al.’s body force model was added to drive the flow. Several observations and investigations were made that showed good agreement with Breuer et al.’s findings of with different Re numbers and streamwise locations. In brief, the LBM scheme for flows over periodic hills was well constructed. Additionally, the simulations were conducted on a multi-GPU cluster with both CUDA and MPI implemented, and some optimizations were applied for higher efficiency.

[1] U. Frisch, B. Hasslacher, and Y. Pomeau, “Lattice-gas automata for the navier-stokes equation,” Physical review letters, vol. 56, no. 14, p. 1505, 1986.
[2] S. Wolfram, “Cellular automaton fluids 1: Basic theory.”
[3] P. B. E. G. M. Krook, P. Bhatnagar, and E. Gross, “A model for collision processes in gases,” Phy. Rev, vol. 94, pp. 511–524, 1954.
[4] S. Chen, H. Chen, D. Martnez, and W. Matthaeus, “Lattice boltzmann model for simulation of magnetohydrodynamics,” vol. 67, pp. 3776–3779.
[5] Y. H. Qian, D. D’Humières, and P. Lallemand, “Lattice bgk models for navier-stokes equation,” EPL (Europhysics Letters), vol. 17, no. 6, p. 479, 1992.
[6] S. Chen and G. D. Doolen, “Lattice boltzmann method for fluid flows,” Annual Review of Fluid Mechanics, vol. 30, no. 1, pp. 329–364, 1998.
[7] S. Hou, Lattice Boltzmann Method for Incompressible, Viscous Flow. PhD thesis, KANSAS STATE UNIVERSITY., 1995.
[8] D. d’Humieres, “Generalized lattice-boltzmann equations,” Progress in Astronautics and Aeronautics, pp. 450–458, American Institute of Aeronautics and Astronautics, Jan. 1994.
[9] Lallemand and Luo, “Theory of the lattice boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability,” Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, vol. 61, pp. 6546–6562, June 2000.
[10] J. Latt and B. Chopard, “Lattice boltzmann method with regularized pre-collision distribution functions,” Mathematics and Computers in Simulation, vol. 72, no. 2, pp. 165 – 168, 2006. Discrete Simulation of Fluid Dynamics in Complex Systems.
[11] S. Ansumali and I. V. Karlin, “Entropy function approach to the lattice boltzmann method,” Journal of Statistical Physics, vol. 107, pp. 291–308, Apr 2002.
[12] H. Chen, S. Chen, and W. H. Matthaeus, “Recovery of the navier-stokes equations using a lattice-gas boltzmann method,” Physical Review A, vol. 45, no. 8, p. R5339, 1992.
[13] D. A. Wolf-Gladrow, “Lattice gas cellular automata and lattice boltzmann models,” 2000.
[14] G. Almeida, D. Durão, and M. Heitor, “Wake flows behind two-dimensional model hills,” Experimental Thermal and Fluid Science, vol. 7, no. 1, pp. 87 – 101, 1993.
[15] C. P. Mellen, J. Frölich, and W. Rodi, “Large Eddy Simulations of the flow over periodic hills,” in IMACS World Congress (M. Deville and R. Owens, eds.), 2000.
[16] M. Breuer, N. Peller, C. Rapp, and M. Manhart, “Flow over periodic hills– numerical and experimental study in a wide range of reynolds numbers,” Computers & Fluids, vol. 38, no. 2, pp. 433–457, 2009.
[17] L. S. Lin, H. W. Chang, and C. A. Lin, “Multi relaxation time lattice boltzmann simulations of transition in deep 2d lid driven cavity using gpu,” Computers & Fluids, vol. 80, no. Supplement C, pp. 381 – 387, 2013. Selected contributions of the 23rd International Conference on Parallel Fluid Dynamics ParCFD2011.
[18] W. Xian and A. Takayuki, “Multi-gpu performance of incompressible flow computation by lattice boltzmann method on gpu cluster,” Parallel Computing, vol. 37, no. 9, pp. 521 – 535, 2011. Emerging Programming Paradigms for Large-Scale Scientific Computing.
[19] C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, “Scalable lattice boltzmann solvers for cuda gpu clusters,” Parallel Computing, vol. 39, no. 6, pp. 259 – 270, 2013.
[20] F. Kuznik, C. Obrecht, G. Rusaouen, and J. J. Roux, “Lbm based flow simulation using gpu computing processor,” Computers & Mathematics with Applications, vol. 59, no. 7, pp. 2380 – 2392, 2010. Mesoscopic Methods in Engineering and Science.
[21] C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, “A new approach to the lattice boltzmann method for graphics processing units,” Computers & Mathematics with Applications, vol. 61, no. 12, pp. 3628 – 3638, 2011. Mesoscopic Methods for Engineering and Science Proceedings of ICMMES-09.
[22] G. R. McNamara and G. Zanetti, “Use of the boltzmann equation to simulate lattice-gas automata,” Phys. Rev. Lett., vol. 61, pp. 2332–2335, Nov 1988.
[23] F. J. Higuera and J. Jiménez, “Boltzmann approach to lattice gas simulations,” EPL (Europhysics Letters), vol. 9, no. 7, p. 663, 1989.
[24] X. He and L. S. Luo, “Theory of the lattice boltzmann method: From the boltzmann equation to the lattice boltzmann equation,” Physical Review E, vol. 56, no. 6, p. 6811, 1997.
[25] X. He and L. S. Luo, “A priori derivation of the lattice boltzmann equation,” vol. 55, pp. R6333–R6336, June 1997.
[26] K. Kono, T. Ishizuka, H. Tsuda, and A. Kurosawa, “Application of lattice boltzmann model to multiphase flows with phase transition,” Computer Physics Communications, vol. 129, no. 1, pp. 110 – 120, 2000.
[27] S. Hou, X. Shan, Q. Zou, G. D. Doolen, and W. E. Soll, “Evaluation of two lattice boltzmann models for multiphase flows,” vol. 138, pp. 695–713, 12 1997.
[28] X. He, S. Chen, and R. Zhang, “A lattice boltzmann scheme for incompressible multiphase flow and its application in simulation of rayleigh-taylor instability,” Journal of Computational Physics, vol. 152, no. 2, pp. 642 – 663, 1999.
[29] C. H. Shih, C. L. Wu, L. C. Chang, and C. A. Lin, “Lattice boltzmann simulations of incompressible liquid-gas systems on partial wetting surfaces,” Philosophical Transactions of the Royal Society of London Series A, vol. 369, pp. 2510–2518, June 2011.
[30] M. Krafczyk, M. Schulz, and E. Rank, “Lattice-gas simulations of two-phase flow in porous media,” Communications in Numerical Methods in Engineering, vol. 14, no. 8, pp. 709–717, 1998.
[31] J. Bernsdorf, G. Brenner, and F. Durst, “Numerical analysis of the pressure drop in porous media flow with lattice boltzmann (bgk) automata,” Computer Physics Communications, vol. 129, no. 1, pp. 247 – 255, 2000.
[32] D. M. Freed, “Lattice-boltzmann method for macroscopic porous media modeling,” International Journal of Modern Physics C, vol. 9, pp. 1491–1503, 1998.
[33] Y. Hashimoto and H. Ohashi, “Droplet dynamics using the lattice-gas method,” International Journal of Modern Physics C, vol. 8, pp. 977–983, 1997.
[34] H. Xi and C. Duncan, “Lattice boltzmann simulations of three-dimensional single droplet deformation and breakup under simple shear flow,” vol. 59, pp. 3022–3026, Mar. 1999.
[35] D. d’Humières and P. Lallemand, “Numerical simulations of hydrodynamics with lattice gas automata in two dimensions,” Complex Systems, vol. 1, 1987.
[36] R. Cornubert, D. d’Humières, and D. Levermore, “A knudsen layer theory for lattice gases,” Physica D: Nonlinear Phenomena, vol. 47, no. 1, pp. 241 – 259, 1991.
[37] X. He, Q. Zou, L.S. Luo, and M. Dembo, “Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice boltzmann bgk model,” Journal of Statistical Physics, vol. 87, pp. 115–136, Apr 1997.
[38] P. A. Skordos, “Initial and boundary conditions for the lattice boltzmann method,” Phys. Rev. E, vol. 48, pp. 4823–4842, Dec 1993.
[39] T. Inamuro, M. Yoshino, and F. Ogino, “A nonslip boundary condition for lattice boltzmann simulations,” Physics of Fluids, vol. 7, no. 12, pp. 2928–2930, 1995.
[40] Q. Zou and X. He, “On pressure and velocity boundary conditions for the lattice boltzmann bgk model,” Physics of Fluids, vol. 9, no. 6, pp. 1591–1598, 1997.
[41] S. Chen, D. Martínez, and R. Mei, “On boundary conditions in lattice boltzmann methods,” Physics of Fluids, vol. 8, no. 9, pp. 2527–2536, 1996.
[42] O. Filippova and D. Hänel, “Grid refinement for lattice-bgk models,” Journal of Computational Physics, vol. 147, no. 1, pp. 219–228, 1998.
[43] R. Mei, L.S. Luo, and W. Shyy, “An accurate curved boundary treatment in the lattice boltzmann method,” Journal of computational physics, vol. 155, no. 2, pp. 307–330, 1999.
[44] M. Bouzidi, M. Firdaouss, and P. Lallemand, “Momentum transfer of a boltzmann-lattice fluid with boundaries,” Physics of Fluids, vol. 13, no. 11, pp. 3452–3459, 2001.
[45] C. F. Ho, C. Chang, K. H. Lin, and C. A. Lin, “Consistent boundary conditions for 2d and 3d lattice boltzmann simulations,” vol. 44, pp. 137–155, 05 2009.
[46] L. Temmerman, M. A. Leschziner, C. P. Mellen, and J. Fröhlich, “Investigation of wall-function approximations and subgrid-scale models in large eddy simulation of separated flow in a channel with streamwise periodic constrictions,” International Journal of Heat and Fluid Flow, vol. 24, no. 2, pp. 157 – 180, 2003.
[47] J. FRÖHLICH, C. P. MELLEN, W. RODI, L. TEMMERMAN, and M. A.
LESCHZINER, “Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions,” Journal of Fluid Mechanics, vol. 526, p. 1966, 2005.
[48] R. Manceau, J. Bonnet, M. Leschziner, and F. Menter, “10th joint ercoftac (sig-15)/iahr/qnet-cfd workshop on refined flow modelling,” Universite de Poitiers,2002.
[49] J. Bolz, I. Farmer, E. Grinspun, and P. Schröoder, “Sparse matrix solvers on the gpu: Conjugate gradients and multigrid,” ACM Trans. Graph., vol. 22, pp. 917–924, July 2003.
[50] Z. Fan, F. Qiu, A. Kaufman, and S. Yoakum-Stover, “Gpu cluster for high performance computing,” in Supercomputing, 2004. Proceedings of the ACM/IEEE SC2004 Conference, pp. 47–47, IEEE, 2004.
[51] J. Krüger and R. Westermann, “Linear algebra operators for gpu implementation of numerical algorithms,” in ACM SIGGRAPH 2005 Courses, p. 234, ACM, 2005.
[52] J. Tölke, “Implementation of a lattice boltzmann kernel using the compute unified device architecture developed by nvidia,” Computing and Visualization in Science, vol. 13, p. 29, Jul 2008.
[53] E. Riegel, T. Indinger, and N. A. Adams, “Implementation of aălattice–boltzmann method for numerical fluid mechanics using the nvidia cuda technology,” Computer Science - Research and Development, vol. 23, pp. 241–247, Jun 2009.
[54] H. W. Chang, P. Y. Hong, L. S. Lin, and C. A. Lin, “Simulations of three-dimensional cavity flows with multi relaxation time lattice boltzmann method and graphic processing units,” Procedia Engineering, vol. 61, no. Supplement C,
pp. 94 – 99, 2013. 25th International Conference on Parallel Computational Fluid Dynamics.
[55] T. I. Gombosi, Gaskinetic Theory. July 1994.
[56] S. Harris, An introduction to the theory of the Boltzmann equation. Courier Corporation, 2004.
[57] C. Peng, “The lattice boltzmann method for fluid dynamics: theory and applications,” M. Math, Department of Mathematics, Ecole Polytechnique Federale de Lausanne, 2011.
[58] D. D’Humières, I. Ginzburg, M. Krafczyk, P. Lallemand, and L. S. Luo, “Multiple-relaxation-time lattice boltzmann models in three dimensions.,” Philosophical transactions. Series A, Mathematical, physical, and engineering sciences, vol. 360, pp. 437–451, Mar. 2002.
[59] Z. Guo and C. Zheng, “Analysis of lattice boltzmann equation for microscale gas flows: Relaxation times, boundary conditions and the knudsen layer,” International Journal of Computational Fluid Dynamics, vol. 22, no. 7, pp. 465–473, 2008.
[60] Z. Guo, C. Zheng, and B. Shi, “Discrete lattice effects on the forcing term in the lattice boltzmann method,” Phys. Rev. E, vol. 65, p. 046308, Apr 2002.