晶格波茲曼法為一較新且適合作高速計算解流體力學的數值方法。本研究中為利用Lee 的多相流模型並加以實做在多張圖形顯示卡叢集上以模擬複雜且具高密度差之流體並得到高效率的結果。其中驗證的例子有消除靜止液珠介面上因力量不平衡所造成的假性速度、液珠的融合震盪、二相的分離現象以及不同的液珠碰撞結果。而在碰撞過程中主要為調查韋伯數與碰撞參數對於結果的影響，液珠的碰撞融合與碰撞分離現象也與理論實驗相符合。此外，在本研究中對於多圖形顯示卡施作部分也提出了有效率的資料傳遞型態，結果顯示即便是使用多張圖形顯示卡計算，仍舊能維持高平行效率。

Lattice Boltzmann method (LBM) is a novel and relative new approach in the field of hydrodynamics and well-suited to efficient high-performance computing implementation. In this thesis, a three-dimensional two-phase lattice Boltzmann model is adopted based on Lee et al. on multi graphic processing unit (GPU) cluster. Such a complex binary system at high density ratio is achieved easily and yields high performance. Here, several numerical validation are examed. First, the spurious velocity appearing near the two-phase interface caused by the force imbalance is successfully eliminated in a stationary droplet case. Droplets oscillation is validated and the results are in good agreement with analytical solutions. The behaviours of two droplets collisions are also investigated. "Coalescence" and "stretching separation" phenomena are observed and studied in terms of two dimensionless variables, Weber number We and impact parameter B, showing good consistency to benchmark. Further, an efficient multi-GPU cluster implementation is proposed and achieves 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 . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Graphics processing unit . . . . . . . . . . . . . . . . . . . . . 3
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Lattice Boltzmann multiphase model . . . . . . . . . . . . . . 4
1.2.2 GPU implementation . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Objective and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Theory and governing equations 9
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 The BGK and the low-Mach-number approximation . . . . . . . . . . 10
2.2.1 The BGK approximation . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 The low-Mach-number approximation . . . . . . . . . . . . . . 12
2.3 Discretization of the Boltzmann equation . . . . . . . . . . . . . . . . 13
2.3.1 Discretization of phase space . . . . . . . . . . . . . . . . . . . 13
2.3.2 Dicretization of time . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 The free-energy model . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 The free-energy function . . . . . . . . . . . . . . . . . . . . . 16
2.5 Lattice Boltzmann model for multi phase flow . . . . . . . . . . . . . . 17
2.5.1 The governing equations . . . . . . . . . . . . . . . . . . . . . 17
2.5.2 Discrete Boltzmann equation . . . . . . . . . . . . . . . . . . 18
2.5.3 Interface capturing equation . . . . . . . . . . . . . . . . . . . 19
3 Numerical algorithm 21
3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Gradient treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Periodic boundary condition . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 GPU implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.1 Memory access . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.2 Multi-GPU implementation . . . . . . . . . . . . . . . . . . . 26
4 Numerical results 28
4.1 A single droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.1 Spurious velocity elimination . . . . . . . . . . . . . . . . . . . 28
4.1.2 Droplet oscillation . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Droplets merging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.1 Phase separation . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.2 Two droplets merging . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Droplets collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.1 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.2 Various outcomes of droplet collision . . . . . . . . . . . . . . 38
4.4 Performance of GPU implementation . . . . . . . . . . . . . . . . . . 41
5 Conclusions 45

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