對管道中的周期性山丘進行數值模擬，並分析產生二次運動的因素。研究中模擬了 Re = 50、200、220 和 700 的具有周期性山丘的管道中的層流和湍流以及 Re = 50、700 的具有周期性山丘的通道中的層流和湍流。 根據瞬時流向速度的觀測，臨界雷諾數約為Re = 220。根據不同雷諾數下和不同位置二次流的比較，最顯著的變化發生在Y = 2h。 為了闡明二次流的貢獻，使用雷諾平均Navier-Stokes方程將流速分解為黏性、湍流、對流和幾何項。 在
Y = 2h 處截面的分解速度，對流項在上半部和下半部分別為負和正，而幾何項則呈現相反的趨勢。 此外，流項在大多數情況下為負值，粘性項的輪廓與截面形狀有關。對雷諾平均 Navier-Stokes 方程進行積分以獲得摩擦係數和阻力係數。 與局部摩擦係數相關的幾何和對流項表現出相反的趨勢，在山坡上具有極值，尤其是在迎風面。全域阻力係數隨著雷諾數的增加而減小，在相同雷諾數下，渠道流的值略高於管道流的值。

Numerical simulations of periodic hills in a duct are performed to analyze the factors that produce secondary motion. Specifically, laminar and turbulent flow in a duct with periodic hills for Re = 50, 200, 220, and 700 and in a channel with periodic hills for Re = 50, 700 are simulated. According to the observations of the instantaneous streamwise velocity, the critical Reynolds number is approximately Re = 220. According to the comparison of the secondary flow in different locations at different Reynolds numbers, the most notable change occurs at Y = 2h. To clarify the contribution of the secondary flow, the flow velocity is decomposed into viscous, turbulent, convection, and geometric terms by using the Reynolds-averaged Navier–Stokes equation. For the decomposed velocities in the cross-section at Y = 2, the convection term is negative and positive in the upper and lower halves, respectively, whereas the geometric term exhibits the opposite trend. Additionally, the turbulent term is negative in most cases, and the profile of the viscous term is related to the shape of the section. The Reynolds-averaged Navier–Stokes equation is integrated to obtain the friction and drag coefficients. The geometric and convective terms associated with the local friction coefficients exhibit opposing tendencies, with extreme values on the hill, especially on the windward side. The global drag coefficient decreases with increasing Reynolds number, and the value for the channel flow is slightly higher than that for the duct flow at the same Reynolds number.

1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Theory of LBM . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Flow over periodic hills . . . . . . . . . . . . . . . . . . . 3
1.2.3 Secondary flow . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 4
1.2.5 GPU implementation . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Methodology 8
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . .8
2.1.2 The BGK approximation . . . . . . . . . . . . . . . . . . . . 9
2.1.3 The low-Mach-number approximation . . . . . . . . . . . . . . 11
2.2 Discretization of Boltzmann equation . . . . . . . . . . . . . .12
2.2.1 Discretization of time . . . . . . . . . . . . . . . . . . . .12
2.2.2 Discretization of phase space . . . . . . . . . . . . . . . . 13
3 Numerical algorithm 15
3.1 Multiple-relaxation-time lattice Boltzmann method . . . . . . . 15
3.2 Forcing term . . . . . . . . . . . . . . . . . . . . . . . . . .19
3.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 20
3.3.1 Halfway bounce-back boundary condition . . . . . . . . . . . .20
3.3.2 BFL scheme . . . . . . . . . . . . . . . . . . . . . . . . . .21
3.4 GPU implementation . . . . . . . . . . . . . . . . . . . . . . .23
4 Numerical results and discussion 26
4.1 The geometry of periodic hill . . . . . . . . . . . . . . . . . 26
4.2 The flow over the periodic hill in the channel . . . . . . . . .29
4.2.1 Variables in main flow direction . . . . . . . . . . . . . . .29
4.3 The flow over the periodic hill in the duct . . . . . . . . . . 33
4.3.1 The flow field of the periodic hill in the duct . . . . . . . 33
4.3.2 Critical Reynolds number . . . . . . . . . . . . . . . . . . .33
4.3.3 Comparison of the secondary flow of the periodic hill flow . .34
4.4 The role of secondary motions . . . . . . . . . . . . . . . . . 41
4.4.1 Contributions to streamwise velocity . . . . . . . . . . . . .41
4.4.2 Contributions to the friction coefficient . . . . . . . . . . 42
5 Conclusion and Future work 54
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .54
5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . 55

[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] Krafczyk, M., Schulz, M., & Rank, E. (1998). Lattice-gas simulations of two-phase flow in porous media. Communications in numerical methods in engineering, 14(8), 709-717.
[14] 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.
[15] Hashimoto, Y., & Ohashi, H. (1997). Droplet dynamics using the lattice-gas method. International Journal of Modern Physics C, 8(04), 977-983.
[16] 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.
[17] Ansumali, S., & Karlin, I. V. (2002). Entropy function approach to the lattice Boltzmann method. Journal of Statistical Physics, 107(1-2), 291-308.
[18] Karlin, I. V., Gorban, A. N., Succi, S., & Boffi, V. (1998). Maximum entropy principle for lattice kinetic equations. Physical Review Letters, 81(1), 6.
[19] Luo, L. S. (2000). Theory of the lattice Boltzmann method: Lattice Boltzmann models for nonideal gases. Physical Review E, 62(4), 4982.
[20] d’Humi ́eres, D. (2002). Multiple–relaxation–time 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.
[21] 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.
[22] 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.
[23] 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.
[24] Long, R. R. (1959). The motion of fluids with density stratification. Journal of Geophysical Research, 64(12), 2151-2163.
[25] Jackson, P. S., Hunt, J. C. R. (1975). Turbulent wind flow over a low hill. Quarterly Journal of the Royal Meteorological Society, 101(430), 929-955.
[26] 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.
[27] 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
[28] Rapp, C., & Manhart, M. (2011). Flow over periodic hills: an experimental study. Experiments in fluids, 51(1), 247-269.
[29] 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.
[30] 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.
[31] Brundrett, E., & Baines, W. D. (1964). The production and diffusion of vorticity in duct flow. Journal of Fluid Mechanics, 19(3), 375-394.
[32] Gavrilakis, S. (1992). Numerical simulation of low-Reynolds-number turbulent flow through a straight square duct. Journal of Fluid Mechanics, 244, 101-129.
[33] Huser, A., & Biringen, S. (1993). Direct numerical simulation of turbulent flow in a square duct. Journal of Fluid Mechanics, 257, 65-95.
[34] Fukagata, K., Iwamoto, K., & Kasagi, N. (2002). Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows. Physics of Fluids, 14(11), L73- L76.
[35] Modesti, D., Pirozzoli, S., Orlandi, P., & Grasso, F. (2018). On the role of secondary motions in turbulent square duct flow. Journal of Fluid Mechanics, 847.
[36] 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.
[37] Skordos, P. A. (1993). Initial and boundary conditions for the lattice Boltzmann method. Physical Review E, 48(6), 4823.
[38] Inamuro, T., Yoshino, M., & Ogino, F. (1995). A non-slip boundary condition for lattice Boltzmann simulations. Physics of fluids, 7(12), 2928-2930.
[39] Zou, Q., & He, X. (1997). On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Physics of fluids, 9(6), 1591-1598.
[40] Chen, S., Martinez, D., & Mei, R. (1996). On boundary conditions in lattice Boltzmann methods. Physics of fluids, 8(9), 2527-2536.
[41] Filippova, O., & H ̈anel, D. (1997). Lattice-Boltzmann simulation of gas-particle flow in filters. Computers & Fluids, 26(7), 697-712.
[42] Bouzidi, M. H., Firdaouss, M., & Lallemand, P. (2001). Momentum transfer of a Boltzmann-lattice fluid with boundaries. Physics of fluids, 13(11), 3452-3459.
[43] 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.
[44] 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.
[45] Li . (2020). Simulations of flows over periodic hills with mass conserving lattice Boltzmann method on multi-GPU cluster , National Tsing Hua University, Department of Power Mechanical Engineering, Master’s thesis.
[46] 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.
[47] 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.
[48] 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.
[49] 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.
[50] 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.
[51] 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.
[52] 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.
[53] Davidson, L., & Peng, S. H. (2003). Hybrid LES-RANS modelling: a one-equation SGS model combined with a k-ω model for predicting recirculating flows. International Journal for Numerical Methods in Fluids, 43(9), 1003-1018.
[54] von Terzi, D., Hinterberger, C., Garc ́ıa-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).
[55] 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.
[56] 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.
[57] 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.
[58] 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).
[59] 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).
[60] Beck, A. D., Bolemann, T., Flad, D., Frank, H., Gassner, G. J., Hindenlang, F., & Munz, C. D. (2014). High-order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations. International Journal for Numerical Methods in Fluids, 76(8), 522-548.
[61] Gloerfelt, X., & Cinnella, P. (2019). Large eddy simulation requirements for the flow over periodic hills. Flow, Turbulence and Combustion, 103(1), 55-91.
[62] 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).
[63] 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 analysis–the flow over periodic hills. Flow, Turbulence and Combustion, 95(2-3), 193-209.
[64] Gombosi, T. I., & Gombosi, A. (1994). Gaskinetic theory (No. 9). Cambridge University Press.
[65] Suga, Kazuhiko, et al.(2015). A D3Q27 multiple-relaxation-time lattice Boltzmann method for turbulent flows. Computers & Mathematics with Applications 69.6 : 518-529.
[66] Huang, X. Y. (2017). Simulations of turbulent flow over periodic hills with multiple-relaxation-time Lattice Boltzmann method on multi-GPU cluster (Unpublished master dissertation). National Tsing Hua University, Hsinchu, Taiwan.