在本研究中使用Lee與Lin所提出的晶格波茲曼模型進行三維的多相流模擬計算，並在多張圖形顯示卡叢集上運行以獲得液珠撞擊液膜的結果。經由一系列的三維基準測試，驗證目前使用的晶格波茲曼方程方法，包括靜止液珠、單顆液珠撞擊液膜、質量守恆定律與在旋轉流場中開槽球體介面的演化過程。對於各種不同的表面張力下，靜止液珠所產生的內外壓差的數值結果與拉普拉斯定律的理論結果也相符合。而液珠撞擊液膜的模擬，發現擴散半徑隨著時間演變得過程也與理論預測符合。此外，為了提高計算效率，除了將二維平行切割應用在多圖形顯示卡叢集，還採用Allen-Cahn方程的晶格波茲曼模型，得知使用Allen-Cahn方程的晶格波茲曼模型更穩定。

This thesis presents a simulation of a single droplet impact on a thin liquid film using Lee and Lin's three-dimensional two-phase lattice Boltzmann model on the graphics processing unit (GPU) cluster platform. In a series of 3D benchmarks, tests were conducted for validation of the present lattice Boltzmann equation (LBE) method, including a stationary droplet, the impact of a single droplet on a liquid film, the law of conservation of mass and the evolution of an interface in the form of a slotted sphere in a rotational flow field. The numerical results for stationary droplets at different pressures at the droplet interface for various surface tensions agree well with the theoretical solution based on Laplace law. In the simulation of a droplet impact, the time evolution of the crown radius is in good agreement with theoretical predictions. Also, in order to improve computational efficiency, the LBM for the Allen-Cahn equation was conducted. The results show that the LB model for the Allen-Cahn equation is more accurate and more stable.

1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Lattice Boltzmann method . . . . . . . . . . . . . . . . . 1
1.1.2 Multiphase fluid systems . . . . . . . . . . . . . . . . . 2
1.1.3 Graphics processing unit . . . . . . . . . . . . . . . . . 3
1.2 Literature survey . . . . . . . .. . . . . . . . . . . . . . 4
1.2.1 Lattice Boltzmann multiphase model . . . . . . . . . . . . 4
1.2.2 Drop impact . . . . . . . . . . .. . . . . . . . . . . . . 6
1.2.3 GPU implementation . . . . . . . . . . . . . . . . . . . . 7
1.3 Objective and motivation . . . . . . . . . . . . . . . . . . 9
2 Theory and governing equations . . . . . . . . . . . . . . . 10
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . 10
2.2 The BGK approximation . . . . . . . . . . . . . . . . . . . 11
2.3 The low-Mach-number approximation . . . . . . . . . . . . . 13
2.4 Discretization of the Boltzmann equation . . . . . . . . . 14
2.4.1 Discretization of phase space . . . . . . . . . . . . . . 14
2.4.2 Discretization of time . . . . . . . . . . . . . . . . . 15
2.5 The free-energy model . . . . . . . . . . . . . . . . . . . 16
2.6 Lattice Boltzmann model for multiphase flow . . . . . . . . 17
2.6.1 The governing equations . . . . . . . . . . . . . . . . . 17
2.6.2 Discrete Boltzmann equation . . . . . . . . . . . . . . . 18
2.6.3 Interface capturing equation . . . . . . . . . . . . . . .22
3 Numerical algorithm . . . . . . . . . . . . . . . . . . . . .26
3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . .26
3.1.1 Modied distribution functions with Cahn-Hilliard equation . 27
3.1.2 Modied distribution functions with Allen-Cahn equation . . . 28
3.2 Gradient treatments . . . . . . . . . . . . . . . . . . . . 29
3.3 Boundary condition . . . . .. . . . . . . . . . . . . . . . 30
3.4 GPU implementation . . . . . . . . . . . . . . . . . . . . 31
3.4.1 Memory access . . . . . . . . . . . . . . . . . . . . . . 31
3.4.2 Multi-GPU implementation . . . . . . . . . . . . . . . . .32
4 Numerical results 36
4.1 Laplace law of a stationary drop . . . . . . . . . . . . . .36
4.2 Droplet impact on a thin liquid lm . . . . . . . . . . . . .37
4.2.1 Simulation setup . . . . . . . . . . . . . . . . . . . . .37
4.2.2 Various outcomes of droplet impact . . . . . . . . . . . .39
4.2.3 Crown radius . . . . . . . . . . . . . . . . . . . . . . .39
4.3 Mass conservation . . . . . . . . . . . . . . . . . . . . . 40
4.4 Rigid-body rotation of Zalesak's disk . . . . . . . . . . . 41
5 Conclusions and future works . . . . . . . . . . . . . . . . 60

[1] U. Frisch, B. Hasslacher, and Y. Pomeau, "Lattice-gas automata for the Navier-Stokes equation," Phys. Rev. Lett. 56, 1505,(1986).
[2] S. Wolfram, "Cellular automaton fluids 1: Basic theory," J. Stat. Phys. 45, 471, (1986).
[3] F. J. Higuera, S. Sussi, and R. Benzi, "3-dimensional
flows in complex geometries with the lattice Boltzmann method," Europhys. Lett. 9, 345, (1989).
[4] F. J. Higuera, and J. Jemenez, "Boltzmann approach to lattice gas simulations," Europhys. Lett. 9, 663, (1989).
[5] P. L. Bhatnagar, E. P. Gross, and M. Grook, "A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems," Phys. Rev. E 94, 511, (1954).
[6] S. Harris, "An introduction to the theory of the Boltzmann equation," Holt, Rinehart and Winston, New York, (1971).
[7] U. Frisch, D. d'Humieres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.P. Rivet, "Lattice gas hydrodynamics in two and three dimensions," Complex Syst. 1, 649, (1987).
[8] D. O. Martinez, W. H. Matthaeus, S. Chen, and D. C. Montgomery, "Comparison of a spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics," Phys. Fluids. 6, 1285, (1994).
[9] R. Scardovelli, and S. Zaleski, "Direct numerical simulation of free-surface and intercal flow," Annu.Rev. Fluid Mech. 31, 567, (1999).
[10] S. Osher, and R. P. Fedkiw, "Level set method: An overview and some recent results," J. Comput. Phys. 169, 463, (2001).
[11] D. M. Anderson, G. B. McFadden, and A. A. Wheeler, "Diffuse-interface
methods in fluid mechanics," Annu. Rev. Fluid Mech. 30, 139-65, (1998).
[12] T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, "Boundary integral methods for multicomponent fluids and multiphase materials," J. Comput.Phys. 169, 302, (2001).
[13] P. Y. Hong, L. M. Huang, L. S. Lin, and C.A. Lin, "Scalable multi-relaxation-time lattice Boltzmann simulations on multi-GPU cluster," Computers &
Fluids. 110, 1-8, (2015).
[14] H. Liang, B. C. Shi, Z.L. Guo, Z. H. Chai, "Free energy of a nonuniform system. i. interfacial free energy," Phys.Rev. E 89, 053320, (2014).
[15] H. Ding, P. D. M. Spelt, and C. Shu, "Diffuse interface model for incompressible two-phase flows with large density ratios," J. Comput. Phys. 226, 2078,(2007).
[16] Y. Q. Zu, and S. He, "Phase-eld-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts," Phys.Rev. E 87, 043301, (2013).
[17] J. W. Cahn, J. E. Hilliard, "Free energy of a nonuniform system. i. interfacial free energy," J. Comput. Phys. 28, 258, (1958).
[18] S. M. Allen, J. W. Cahn, "Mechanisms of phase transformations within the miscibility gap of Fe-Rich Fe-Al alloys," Acta Metall. 24, 425, (1976).
[19] L. Zheng, T. Lee, Z. Guo and D. Rumschitzki, "Shrinkage of bubbles and drops in the lattice Boltzmann equation method for nonideal gases," Phys. Rev. E. 89, 033302, (2014).
[20] Y. Sun, C. Beckermann, "Sharp interface tracking using the phase-field equation," J. Comput. Phys. 220, 626-653,(2007).
[21] P.-H. Chiu, Y.-T. Lin, "A conservative phase eld method for solving incompressible two-phase flows," J. Comput. Phys. 230, 185-204, (2011).
[22] Andrew K. Gunstensen and Daniel H. Rothman, "Lattice Boltzmann model of immiscible fluids," Phys. Rev. 43, 4320-4327, (1991).
[23] Daniel H. Rothman and Jeffrey M. Keller, "Immiscible cellular-automaton fluids," J. Stat. Phys. 52(3), 1119-1127, (1988).
[24] D. Grunau, S. Y. Chen, and K. Eggert, "A lattice Boltzmann model for multiphase fluid flows," Phys. Fluids A 5, 2557, (1993).
[25] X. Shan and H. Chen, "Lattice Boltzmann model for simulating flows with multiple phases and components," Phys. Rev. E. 47, 1815-1819, (1993).
[26] X. Shan and H. Chen, "Simulation of Nonideal Gases and Liquid-GasPhase Transitions by the Lattice Boltzmann Equation," Phys. Rev. E. 49, 2941-2948,
(1994).
[27] X. Shan, and G. D. Doolen, "Multicomponent Lattice-Boltzmann Model With Interparticle Interaction," J. Stat. Phys. 52, 379-393, (1995).
[28] M. R. Swift, W. R. Osborn, J. M. Yeomans "Lattice Boltzmann simulation of nonideal fluids," Phys. Rev. Lett. 75(5), 830-833, (1995).
[29] M. R. Swift, W. R. Osborn, J. M. Yeomans "Lattice Boltzmann simulations of liquid-gas and binary-fluid systems," Phys. Rev. E,54, 5041-5052, (1996).
[30] X. He. Shan, G.D. Doolen, "A discrete Boltzmann equation model for non-ideal gases," Phys. Rev. 57, R13, (1998).
[31] X. He, S. Chen, R. Zhang, "A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of RayleighTaylor instability," J. Comput. Phys. 152, 642, (1999).
[32] P. Yuan, L. Schaefer, "Equations of state in a lattice Boltzmann model," Phys. Fluids 18, 042101,(2006).
[33] T. Lee, "Effects of incompressibility on the elimination of parasitic currents in the lattice Boltzmann equation method for binary fluids," Comput. Math. Appl.58, 987-994, (2010).
[34] T. Lee, and P. F. Fischer, "Eliminating parasitic currents in the lattice Boltzmann equation method for nonideal gases," Phys. Rev.E. 74, 046709, (2006).
[35] D. Jacqmin, "Calculation of two-phase Navier-Stokes
flows using phase-field modeling," J. Comput. Phys. 155, 96-127, (1999).
[36] T. Lee, and C. L. Lin, "A stable discretization of the lattice Boltzmann
equation for simulation of incompressible two-phase flows at high density ratio," J. Comput. Phys. 206, 16-47, (2005).
[37] X. He, S. Chen, R. Zhang, "A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability," J. Comput. Phys. 152(2), 642-663, (1999).
[38] T. Lee, and C. L. Lin, "Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces," J. Comput. Phys. 229, 8045-8063, (2010).
[39] A. L. Yarin and D. A. Weiss, "Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity," J. Fluid Mech. 283, 141, (1995).
[40] M. F. Trujillo and C. F. Lee, "Impact of a drop onto a wetted wall: description of crown formation and propagation," J. Fluid Mech. 472, 373, (2002).
[41] G. E. Cossali, M. Marengo, M. A. Coghe, and S. Zhdanov, "The role of time in single drop splash on thin lm," Exp. Fluids 36, 888, (2004).
[42] S. Mukherjee and J. Abraham, "Crown behavior in drop impact on wet walls," Phys. Fluids 19, 052103, (2007).
[43] J. Bolz, I. Farmer, E. Grinspun, and P. Schroder, \Sparse matrix solvers on the GPU: Conjugate gradients and multigrid," ACM Trans. Graph. (SIGGRAPH) 22, 917, (2003).
[44] F. A. Kuo, M. R. Smith, C. W. Hsieh, C. Y. Chou, and J. S. Wu, "GPU acceleration for general conservation equations and its application to several engineering problems," Comput. Fluids 45, 147, (2011).
[45] J. Tolke, "Implementation of a lattice Boltzmann kernel using the compute unied device architecture developed by nVIDIA," Comput. Visual Sci. 13, 29, (2008).
[46] J. Tolke, and M. Krafczyk, "TeraFLOP computing on a desktop PC with GPUs for 3D CFD," Int. J. Comput. Fluid D. 22, 443, (2008).
[47] C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, "A new approach to the lattice Boltzmann method for graphics processing units," Comput. Math. Appl. 61, 3628, (2011).
[48] X. Wang, T. Aoki, "Multi-GPU performance of incompressible flow computation by lattice Boltzmann method on GPU cluster," Parallel. Computing. 37, 521, (2011).
[49] J. Myre, S. D. C. Walsh, D. Lilja and M. O. Saar, "Performance analysis of single-phase, multiphase, and multicomponent lattice-Boltzmann fluid flow simulations on GPU clusters," Concurrency Comput.: Pract. and Exper. 23,
332-350, (2010).
[50] T. C. Huang, C. Y. Chang, and C. A. Lin, "Simulation of droplet dynamic with high density ratio two-phase lattice Boltzmann model on multi-GPU cluster," Comput. Fluids. 000, 1-8, (2018).
[51] Tamas I. Gombosi, "Gas kinetic theory," Cambridge University Press, (1994).
[52] 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, 6811-6817, (1997).
[53] D. A. Wolf-Gladrow, "Lattice-gas cellular automata and lattice Boltzmann models - an introduction," Springer, Lecture Notes in Mathematics, p.159, (2000).
[54] D. Jamet, O. Lebaigue, N. Coutris, J. M. Delhaye, "The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change," J. Comput. Phys. 169, 624-651, (2001).
[55] J. S. Rowlinson and B. Widom, "Molecular Theory of Capillarity, Clarendon," Oxford, (1989).
[56] V. M. Kendon, M. E. Cates, I. Pagonabarraga, J.C. Desplat, P. Bladon, "Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study," J. Fluid Mech. 440, 147-203, (2001).
[57] H. W. Chang, P .Y. Hong, L. S. Lin and C. A. Lin, "Simulations of flow instability in three dimensional deep cavities with multi relaxation time lattice Boltzmann method on graphic processing units," Comput. & Fluids. 88, 866-871, (2013).
[58] J. Myre, S. D. C. Walsh, D. Lilja and M. O. Saar, "Performance analysis of single-phase, multiphase, and multicomponent lattice-Boltzmann fluid flow simulations on GPU clusters," Concurrency Comput.: Pract. and Exper. 23, 332-350, (2010).
[59] Rieber M, and Frohn A ., "IA numerical study on the mechanism of splashing," Int J. Heat Fluid Flow 20,455-61, (1999).
[60] P. Yue, C. Zhou, and J. J. Feng, "Spontaneous shrinkage of drops and mass conservation in phase-eld simulations," J. Comput. Phys. 223, (2007).
[61] P. Yue, C. Zhou, and J. J. Feng, \Shrinkage of bubbles and drops in the lattice Boltzmann equation method for nonideal gases," Phys. Rev.E. 89,033-302,
(2014).