在此研究中，我們採用了 Fakhari所提出的三維兩相流晶格波茲曼模型，但
不同的是此研究使用的為單鬆弛時間而非多鬆弛時間的晶格波茲曼模型，利用
多圖形顯示卡叢集來進行數值模擬。為了展示此研究所提出方法的能力，
我們進行了一些數值模擬來驗證此晶格波茲曼法的可行性。這些模擬包含由重力驅
動之方管流、一靜止水滴的壓力分佈(即拉普拉斯定理 )、一中間有矩形空間之
空氣的水滴放置在一旋轉流場、萊利-泰勒不穩定性與一液滴撞擊液體薄膜之模
擬。重力驅動方管流用來測試此方法是否可運行在較極端之情形且與解析解之
結果吻合，拉普拉斯定理之模擬結果符合理論上之結果，矩形槽之旋轉水滴驗證了此研究採用之兩相流介面捕捉模型 Allen-Cahn較為準確與穩定。在萊利 -泰
勒不穩定性之模擬中泡泡與尖刺之位置與He之結果完美符合，但在介面之速
度相較於Zhang之結果有些震盪產生。最終液滴撞擊液體薄膜之模擬似乎因單
鬆弛時間晶格波茲曼法在雷諾數的限制，無法完全與Fakhari所採用的多鬆弛時
間晶格波茲曼法吻合。

In this thesis, we adopted the three-dimensional lattice Boltzmann model for two-phase flow from Fakhari [22] but we used the single-relaxation time LBM model (SRT-LBM) instead of using the multiple-relaxation time LBM model (MRT-LBM) on the graphic processing
units (GPUs) cluster platform. To demonstrate the capability of the proposed method, several numerical simulations were done for the validation of the present lattice Boltzmann equation (LBE) method. These simulations included square duct flow driven by gravity, the
pressure distribution of a stationary droplet (Laplace Law), the evolution of the interface of a slotted sphere droplet in rotational
flow field, Rayleigh-Taylor instability (RTI) and finally a droplet hitting a thin liquid layer. Square duct flow driven by gravity was tested to prove the validation of the presented method under extreme situations and was in a good agreement with the analytical solutions. The results for stationary droplets at different pressures at the droplet interface for various surface tensions were good compared with the theoretical solution based
on Laplace law. The results of Zalesak's disk showed that the interface capturing equation of the Allen-Cahn model was more accurate and more stable, thus the focus of this thesis was
about Allen-Cahn model. In the simulations of RTI, the results of the positon of bubble and spike were in a good agreement with He et al. [42], but there were some oscillation in velocity compared with Zhang et al. [43]. Finally, the results of milk crown problem seemed to face the limitation of the Raynold's number for SRT-LBM model. The results failed to meet the same as Fakhari [50].

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 Rayleigh Taylor Instability . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Milk Crown Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 GPU implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Objective and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Theory and governing equations 10
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 The BGK approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 The low-Mach-number approximation . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Discretization of the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1 Discretization of space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.2 Discretization of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 The free-energy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Lattice Boltzmann model for multiphase flow . . . . . . . . . . . . . . . . . . . . 16
2.6.1 The governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6.2 Discrete Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6.3 Interface capturing equation . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Numerical algorithm 23
3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 Distribution function of Allen-Cahn model . . . . . . . . . . . . . . . . . 24
3.1.2 Distribution functions of Cahn-Hilliard model . . . . . . . . . . . . . . . 24
3.2 Gradient treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 GPU implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1 Memory access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.2 Multi-GPU implementation . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Numerical results 33
4.1 Duct
ow driven by gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Laplace law of a stationary drop . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Rigid-body rotation of Zalesak's disk . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4 Rayleigh-Taylor instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4.1 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5 Droplet impact on a thin liquid film . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.5.1 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.5.2 Evolution of the droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5.3 Crown radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 Future works 61

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