本研究使用D3Q19 模型和 D3Q27 模型之多鬆弛係數晶格波茲曼方法，模擬週期性山坡在平板內的層流與紊流流場，計算之雷諾數分別為 100,1400,2800 三種。週期性邊界包含主流場方向和側向，山坡曲面邊界的部分則採用了根據反彈格式做線性內插的 BFL 格式。然而使用BFL 曲面修正方法會造成質量流失和數值不穩定，因此本研究採用了多種針對分布函數做修正的方式來消除 BFL 導致的問題。另外，此研究的數值模擬使用了高速圖形顯示卡運算，將程式以二維切割的方式做平行運算，以達到加速運算的效果。

Simulation of laminar and turbulent flow on periodic hills in the channel at Reynolds number 100,1400,2800 is adopted in this study by using multiple-relaxation-time lattice Boltzmann method.
The D3Q19 model and D3Q27 model are adopted in the works. As most cases in the LBM works, the uniform mesh is used in present works. In the present research, the periodic boundary, the halfway bounce-back scheme, and the BFL scheme are adopted. However, the BFL scheme could make the mass of the flow field instability, which means the mass leakage happens. Hence, the distribution function correct strategies would be applied to solve the problem and compare the different results.
For accelerating the efficiency of the simulation, the multi-GPU cluster and two-dimensional decomposition would be adopted with the overlapping strategy.

[1]Frisch, U., Hasslacher, B., & Pomeau, Y. (1986). Lattice-gas automata for the Navier-Stokes equation. Physical review letters, 56(14), 1505.

[2]Wolfram, S. (1986). Cellular automaton fluids 1: Basic theory. Journal of statistical physics, 45(3-4), 471-526.

[3]Bhatnagar, P. L., Gross, E. P., & Krook, M. (1954). A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Physical review, 94(3), 511.

[4]Almeida, G. P., Durao, D. F. G., & Heitor, M. V. (1993). Wake flows behind two-dimensional model hills. Experimental Thermal and Fluid Science, 7(1), 87-101.

[5]Mellen, C. P., Fr¨ohlich, J., & Rodi, W. (2000, August). Large eddy simulation of the flow over periodic hills. In 16th IMACS world congress (pp. 21-25).

[6]Breuer, M., Peller, N., Rapp, C., & Manhart, M. (2009). Flow over periodic hills-numerical and experimental study in a wide range of Reynolds numbers. Computers & Fluids, 38(2),
433-457.

[7]McNamara, G. R., & Zanetti, G. (1988). Use of the Boltzmann equation to simulate lattice-gas automata. Physical review letters, 61(20), 2332.

[8]Higuera, F. J., & Jim´enez, J. (1989). Boltzmann approach to lattice gas simulations. EPL (Europhysics Letters), 9(7), 663.

[9]He, X., & Luo, L. S. (1997). Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Physical Review E, 56(6), 6811.

[10]He, X., & Luo, L. S. (1997). A priori derivation of the lattice Boltzmann equation. Physical Review E, 55(6), R6333.

[11]Kono, K., Ishizuka, T., Tsuda, H., & Kurosawa, A. (2000). Application of lattice Boltzmann model to multiphase flows with phase transition. Computer physics communications, 129(1-3), 110-120.

[12]Hou, S., Shan, X., Zou, Q., Doolen, G. D., & Soll, W. E. (1997). Evaluation of two lattice Boltzmann models for multiphase flows. Journal of Computational Physics, 138(2), 695-713.

[13]He, X., Chen, S., & Zhang, R. (1999). A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of RayleighTaylor instability. Journal of computational physics, 152(2), 642-663.

[14]Shih, C. H., Wu, C. L., Chang, L. C., & Lin, C. A. (2011). Lattice Boltzmann simulations of incompressible liquidgas systems on partial wetting surfaces. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 369(1945), 2510-2518.

[15]Krafczyk, M., Schulz, M., & Rank, E. (1998). Latticegas simulations of twophase flow in porous media. Communications in numerical methods in engineering, 14(8), 709-717.

[16]Bernsdorf, J., Brenner, G., & Durst, F. (2000). Numerical analysis of the pressure drop in porous media flow with lattice Boltzmann (BGK) automata. Computer physics communications, 129(1-3), 247-255.

[17]Freed, D. M. (1998). Lattice-Boltzmann method for macroscopic porous media modeling. International Journal of Modern Physics C, 9(08), 1491-1503.

[18]Hashimoto, Y., & Ohashi, H. (1997). Droplet dynamics using the lattice-gas method. International Journal of Modern Physics C, 8(04), 977-983.

[19]Xi, H., & Duncan, C. (1999). Lattice Boltzmann simulations of three-dimensional single droplet deformation and breakup under simple shear flow. Physical Review E, 59(3), 3022.

[20]Karlin, I. V., Gorban, A. N., Succi, S., & Boffi, V. (1998). Maximum entropy principle for lattice kinetic equations. Physical Review Letters, 81(1), 6.

[21]Ansumali, S., & Karlin, I. V. (2002). Entropy function approach to the lattice Boltzmann method. Journal of Statistical Physics, 107(1-2), 291-308.

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

[23]d’Humi´eres, D. (1992). Generalized lattice-Boltzmann equations. Rarefied gas dynamics.

[24]Premnath, K. N., Pattison, M. J., & Banerjee, S. (2009). Generalized lattice Boltzmann equation with forcing term for computation of wall-bounded turbulent flows. Physical Review E, 79(2), 026703.

[25]Suga, K., Kuwata, Y., Takashima, K., & Chikasue, R. (2015). A D3Q27 multiple-relaxation-time lattice Boltzmann method for turbulent flows. Computers & Mathematics with Applications, 69(6), 518-529.

[26]Yu, H., Girimaji, S. S., & Luo, L. S. (2005). DNS and LES of decaying isotropic turbulence with and without frame rotation using lattice Boltzmann method. Journal of Computational Physics, 209(2), 599-616.

[27]Kang, S. K., & Hassan, Y. A. (2013). The effect of lattice models within the lattice Boltzmann method in the simulation of wall-bounded turbulent flows. Journal of Computational Physics, 232(1), 100-117.

[28]Lee, Y. H., Huang, L. M., Zou, Y. S., Huang, S. C., & Lin, C. A. (2018). Simulations of turbulent duct flow with lattice Boltzmann method on GPU cluster. Computers & Fluids, 168, 14-20.

[29]Temmerman, L., Leschziner, M. A., Mellen, C. P., & Fr¨ohlich, J. (2003). 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, 24(2), 157-180.

[30]Fr¨ohlich, J., Mellen, C. P., Rodi, W., Temmerman, L., & Leschziner, M. A. (2005). Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions. Journal of Fluid Mechanics, 526, 19.

[31]Rapp, C., & Manhart, M. (2011). Flow over periodic hills: an experimental study. Experiments in fluids, 51(1), 247-269.

[32]Chang, P. H., Liao, C. C., Hsu, H. W., Liu, S. H., & Lin, C. A. (2014). Simulations of laminar and turbulent flows over periodic hills with immersed boundary method. Computers & Fluids, 92, 233-243.

[33]Su, C. W. (2018). MRT-LBM simulations of turbulent flows over periodic hills at different Reynolds numbers(Unpublished master dissertation). National Tsing Hua University, Hsinchu, Taiwan.

[34]Davidson, L., & Peng, S. H. (2003). Hybrid LESRANS modelling: a oneequation SGS model combined with a k-ω model for predicting recirculating flows. International Journal for Numerical Methods in Fluids, 43(9), 1003-1018.

[35]von Terzi, D., Hinterberger, C., Garca-Villalba, M., Fr¨ohlich, J., Rodi, W., & Mary, I.(2005). LES with downstream RANS for flow over periodic hills and a model combustor flow. In EUROMECH colloquium (Vol. 469, pp. 6-8).

[36]Ziefle, J., Stolz, S., & Kleiser, L. (2008). Large-eddy simulation of separated flow in a channel with streamwise-periodic constrictions. AIAA journal, 46(7), 1705-1718.

[37]ˆSari´c, S., Jakirli´c, S., Breuer, M., Jaffr´ezic, B., Deng, G., Chikhaoui, O., ... & Peller, N.(2007). Evaluation of detached eddy simulations for predicting the flow over periodic hills. In ESAIM: proceedings (Vol. 16, pp. 133-145). EDP Sciences.

[38]Hickel, S., Kempe, T., & Adams, N. A. (2008). Implicit large-eddy simulation applied to turbulent channel flow with periodic constrictions. Theoretical and Computational Fluid Dynamics, 22(3-4), 227-242.

[39]Xia, Z., Shi, Y., Hong, R., Xiao, Z., & Chen, S. (2013). Constrained large-eddy simulation of separated flow in a channel with streamwise-periodic constrictions. Journal of Turbulence, 14(1), 1-21.
[40] Chaouat, B., & Schiestel, R. (2013). Hybrid RANS/LES simulations of the turbulent flow over periodic hills at high Reynolds number using the PITM method. Computers & Fluids, 84, 279-300.

[41]Diosady, L. T., & Murman, S. M. (2014). Dns of flows over periodic hills using a discontinuous galerkin spectral-element method. In 44th AIAA Fluid Dynamics Conference (p. 2784).

[42]Balakumar, P., Park, G. I., & Pierce, B. (2014). DNS, LES, and wall-modeled LES of separating flow over periodic hills. In Proceedings of the Summer Program (pp. 407-415).

[43]Beck, A. D., Bolemann, T., Flad, D., Frank, H., Gassner, G. J., Hindenlang, F., & Munz,C.D. (2014). Highorder discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations. International Journal for Numerical Methods in Fluids, 76(8), 522-548.

44]Gloerfelt, X., & Cinnella, P. (2019). Large eddy simulation requirements for the flow over periodic hills. Flow, Turbulence and Combustion, 103(1), 55-91.

[45]Schr¨oder, A., Schanz, D., Roloff, C., & Michaelis, D. (2014, July). Lagrangian and Eulerian dynamics of coherent structures in turbulent flow over periodic hills using time-resolved tomo PIV and 4D-PTV Shake-the-box 17th Int. In Symp. on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal (pp. 07-10).

[46]Schr¨oder, A., Schanz, D., Michaelis, D., Cierpka, C., Scharnowski, S., & K¨ahler, C. J.(2015). Advances of PIV and 4D-PTV Shake-The-Box for turbulent flow analysisthe flow over periodic hills. Flow, Turbulence and Combustion, 95(2-3), 193-209.

[47]K¨ahler, C. J., Scharnowski, S., & Cierpka, C. (2016). Highly resolved experimental results of the separated flow in a channel with streamwise periodic constrictions. Journal of Fluid Mechanics, 796, 257-284.

[48]Ladd, A. J. (1994). Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. Journal of fluid mechanics, 271,
285-309.

[49]Skordos, P. A. (1993). Initial and boundary conditions for the lattice Boltzmann method. Physical Review E, 48(6), 4823.

[50]Inamuro, T., Yoshino, M., & Ogino, F. (1995). A nonslip boundary condition for lattice Boltzmann simulations. Physics of fluids, 7(12), 2928-2930.

[51]Chen, S., Martinez, D.,& Mei, R. (1996). On boundary conditions in lattice Boltzmann methods. Physics of fluids, 8(9), 2527-2536.

[52]Zou, Q., & He, X. (1997). On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Physics of fluids, 9(6), 1591-1598.

[53]Filippova, O., & H¨anel, D. (1997). Lattice-Boltzmann simulation of gas-particle flow in filters. Computers & Fluids, 26(7), 697-712.

[54]Mei, R., Luo, L. S., & Shyy, W. (1999). An accurate curved boundary treatment in the lattice Boltzmann method. Journal of computational physics, 155(2), 307-330.

[55]Bouzidi, M. H., Firdaouss, M., & Lallemand, P. (2001). Momentum transfer of a Boltzmann-lattice fluid with boundaries. Physics of fluids, 13(11), 3452-3459.

[56]Lin, K. H., Liao, C. C., Lien, S. Y., & Lin, C. A. (2012). Thermal lattice Boltzmann simulations of natural convection with complex geometry. Computers & Fluids, 69, 35-44.

[57]Lallemand, P., & Luo, L. S. (2003). Lattice Boltzmann method for moving boundaries. Journal of Computational Physics, 184(2), 406-421.

[58]Chen, H., Teixeira, C., & Molvig, K. (1998). Realization of fluid boundary conditions via discrete Boltzmann dynamics. International Journal of Modern Physics C, 9(08), 1281-1292.

[59]Kao, P. H., & Yang, R. J. (2008). An investigation into curved and moving boundary treatments in the lattice Boltzmann method. Journal of Computational Physics, 227(11), 5671-5690.

[60]Le Coupanec, E., & Verschaeve, J. C. (2011). A mass conserving boundary condition for the lattice Boltzmann method for tangentially moving walls. Mathematics and Computers in Simulation, 81(12), 2632-2645.

[61]Sanjeevi, S. K., Zarghami, A., & Padding, J. T. (2018). Choice of no-slip curved boundary condition for lattice Boltzmann simulations of high-Reynolds-number flows. Physical Review E, 97(4), 043305.

[62]T¨olke, J., & Krafczyk, M. (2008). TeraFLOP computing on a desktop PC with GPUs for 3D CFD. International Journal of Computational Fluid Dynamics, 22(7), 443-456.

[63]T¨olke, J. (2010). Implementation of a Lattice Boltzmann kernel using the Compute Unified Device Architecture developed by nVIDIA. Computing and Visualization in Science, 13(1), 29.

[64]Obrecht, C., Kuznik, F., Tourancheau, B., & Roux, J. J. (2013). Scalable lattice Boltzmann solvers for CUDA GPU clusters. Parallel Computing, 39(6-7), 259-270.

[65]Chang, H. W., Hong, P. Y., Lin, L. S., & Lin, C. A. (2013). Simulations of flow instability in three dimensional deep cavities with multi relaxation time lattice Boltzmann method on graphic processing units. Computers & Fluids, 88, 866-871.

[66]Lin, L. S., Chang, H. W., & Lin, C. A. (2013). Multi relaxation time lattice Boltzmann simulations of transition in deep 2D lid driven cavity using GPU. Computers & Fluids, 80, 381-387.

[67]d’Humi´eres, D. (2002). Multiplerelaxationtime lattice Boltzmann models in three dimensions. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 360(1792), 437-451.

[68]Luo, L. S. (2000). Theory of the lattice Boltzmann method: Lattice Boltzmann models for nonideal gases. Physical Review E, 62(4), 4982.

[69]Xian, W., & Takayuki, A. (2011). Multi-GPU performance of incompressible flow computation by lattice Boltzmann method on GPU cluster. Parallel Computing, 37(9),
521-535.

[70]Gombosi, T. I., & Gombosi, A. (1994). Gaskinetic theory (No. 9). Cambridge University Press.

[71]Fakhari, A., Bolster, D., & Luo, L. S. (2017). A weighted multiple-relaxation-time lattice Boltzmann method for multiphase flows and its application to partial coalescence cascades. Journal of Computational Physics, 341, 22-43.

[72]Dellar, P. J. (2002). Nonhydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations. Physical Review E, 65(3), 036309.