本研究以D3Q19晶格波茲曼法模擬二相流問題，主要比較兩種介面捕捉公式（Cahn-Hilliard equation 與 Allen-Cahn equation）之差異。以靜止液珠的測試為例，兩種介面捕捉公式造成的假性速度相差不大，但Cahn-Hilliard模型模擬的結果會出現氣液組成率超出上限（純液體）的情形，進而導致液珠收縮的現象；單顆液珠震盪與兩顆液珠融合的測試結果顯示兩種介面捕捉公式的模擬結果與解析解一致。不過Allen-Cahn模型無法模擬旋節分解（Spinodal decomposition），只能用Cahn-Hilliard模型表現隨機生成的氣液組成率到最後發展分離成氣液雙相。本研究利用圖形顯示卡叢集搭配一維切割進行平行運算，並使用多個stream將邊界點的傳值時間隱藏在內部點的計算時間中進行加速計算。

This thesis presents the simulation of the three-dimensional two-phase lattice Boltzmann model on the graphics processing unit (GPU) cluster. In this thesis, the objective is to develop a program according to Lee's model with the Allen-Cahn equation, then comparing the results to Lee's model with the Cahn-Hilliard equation. Take one stationary drop as an example, the difference of the generated spurious velocity between these two interface capturing equations are not much different. However, the drop shrinkage phenomenon only occurs in the result of the Cahn-Hilliard model. The numerical results of the Allen-Cahn model both in single drop oscillation and two drops merging are identical to the theoretical solution. However, the Allen-Cahn model is unable to simulate the spinodal decomposition phenomenon, so the phase decomposition from the initial random liquid-vapor composition only can be shown in the result of the Cahn-Hilliard model. This thesis uses one-dimensional decomposition to implement parallel programming and uses the overlapping strategy which hiding the boundary exchange time in the computing time to improve computational efficiency.

1 Introduction. . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Motivation and research backgrounds . . . . . . . . . . . . 1
1.1.2 Lattice Boltzmann method . . . . . . . . . . . . . . . . . . 2
1.1.3 Graphics processing unit . . . . . . . . . . . . . . . . . . 3
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Lattice Boltzmann two-phase model . . . . . . . . . . . . . . 3
1.2.2 GPU implementation . . . . . . . . . . . . . . . . . . . . . 5
1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Discrete Boltzmann equation with the external force . . . . . . 8
2.3 DBE for the pressure evolution and the momentum equations . . . 9
2.4 DBE for the phase-field models . . . . . . . . . . . . . . . . . 12
2.4.1 Cahn-Hilliard model . . . . . . . . . . . . . . . . . . . . . 12
2.4.2 Allen-Cahn model . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Gradient treatments . . . . . . . . . . . . . . . . . . . . . . 15

3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . 16
3.2 Periodic boundary condition . . . . . . . . . . . . . . . . . . 17
3.3 GPU implementation . . . . . . . . . . . . . . . . . . . . . . 18

4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 Stationary drop . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.1 Spurious velocity . . . . . . . . . . . . . . . . . . . . . . 23
4.1.2 Drop shrinkage phenomenon . . . . . . . . . . . . . . . . . . 24
4.2 Zalesak’s disk . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Single Drop oscillation . . . . . . . . . . . . . . . . . . . . 29
4.4 Two drops merging . . . . . . . . . . . . . . . . . . . . . . 31
4.5 Spinodal decomposition . . . . . . . . . . . . . . . . . . . . 33

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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