在本研究中使用Lee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高效率的結果。其中驗證的例子有消除氣泡周圍由於數值誤差而產生的假性速度。在單球上升的模擬中，討論在不同無因次參數bond number和Morton number下，對於氣泡上升過程的變形過程以及終端速度的影響，不論是模擬的變形過程以及雷諾數都與實驗結果相符合。而雙球上升的模擬中，無論初始氣泡是在同軸或非同軸的狀態，都可以觀察到下方氣泡由於上方氣泡產生的尾流而加速上升的現象。經由不同氣泡大小的設定，也能發現體積大的氣泡受到流場的影響也較大。此外，將二維平行切割應用在多圖形顯示卡叢集，效能測試的結果顯示即使使用多張圖形顯示卡計算，依舊能維持高平行效率。

In this thesis, a three-dimensional two-phase lattice Boltzmann model [28] is adopted on multi graphic processing unit (GPU) cluster. Such a binary system at high density ratio is carried out with high performance of computation. In the study, The spurious velocity caused by the force imbalance near the two-phase interface can be successfully suppressed in the simulation. A single bubble rising in a rectangular domain is discussed with different regimes of Bond number (Bo) and Morton number (Mo). The terminal Reynolds number (Re) and the deformed shape are consistent with both experimental results [23] and simulation results obtained by Lee et al. [28]. The phenomenon of the trailing bubble catching up the leading bubble is observed. Further, an efficient multi-GPU cluster implementation with two
dimensional decomposition is examined and the program maintains good scalability even in the case with high GPU number.

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 . . . . . . . . . . . . . . . . . 2
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Lattice Boltzmann multiphase model . . . . . . . . . . . . 3
1.2.2 Bubble dynamics. . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 GPU implementation . . . . . . . . . . . . . . . . . . . . 6
1.3 Objective and motivation . . . . . . . . . . . . . . . . . . 7
2 Theory and governing equations 8
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . 8
2.2 The BGK approximation. . . . . . . . . . . . . . . . . . . . 9
2.3 The low-Mach-number approximation . . . . . . . . . . . . . 11
2.4 Discretization of the Boltzmann equation. . . . . . . . . . 12
2.4.1 Discretization of phase space . . . . . . . . . . . . . . 12
2.4.2 Dicretization of time . . . . . . . . . . . . . . . . . . 13
2.5 The free-energy model . . . . . . . . . . . . . . . . . . . 14
2.6 Lattice Boltzmann model for multiphase flow . . . . . . . . 15
2.6.1 The governing equations . . . . . . . . . . . . . . . . . 15
2.6.2 Discrete Boltzmann equation . . . . . . . . . . . . . . . 15
2.6.3 Interface capturing equation. . . . . . . . . . . . . . . 17
3 Numerical algorithm 20
3.1 Simulation procedure. . . . . . . . . . . . . . . . . . . . 20
3.2 Gradient treatments . . . . . . . . . . . . . . . . . . . . 21
3.3 Boundary condition. . . . . . . . . . . . . . . . . . . . . 22
3.4 GPU implementation. . . . . . . . . . . . . . . . . . . . . 23
3.4.1 Memory access . . . . . . . . . . . . . . . . . . . . . . 23
3.4.2 Multi-GPU implementation. . . . . . . . . . . . . . . . . 24
4 Numerical results 28
4.1 Spurious velocity elimination . . . . . . . . . . . . . . . 28
4.2 One bubble rising . . . . . . . . . . . . . . . . . . . . . 29
4.3 two bubbles rising. . . . . . . . . . . . . . . . . . . . . 30
4.3.1 In-line case. . . . . . . . . . . . . . . . . . . . . . . 31
4.3.2 Off-line case . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Performance of GPU implementation . . . . . . . . . . . . . 33
5 Conclusions 45

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