本文使用單鬆弛晶格波茲曼法結合Cahn-Hilliard模型，來計算液滴碰撞過程中的能量變化，以及濃度對剪切流中的液滴對凝聚的影響。透過對能量變化的研究，我們發現直接使用耗散能公式無法完美捕捉號散能，這是因為在碰撞過程中，界面區域發生了快速且複雜的變化，只能透過能量守恆公式來間接獲得耗散能。在研究液滴濃度的影響時，將液滴濃度分為三個方向：x方向、y方向和z方向，分析只聚焦在x方向和y方向的濃度效應。對於給定的x方向的濃度，增加y方向的濃度會使液滴在x方向的速度增加，使它們更容易合併。相反地，對於給定的y方向濃度，增加x方向的濃度會導致液滴在x方向的速度減少，使得凝聚更具挑戰性。然而，當x方向的濃度超過0.8並且顯著降低液滴的水平速度時，液滴聚結現象實際上變得更容易發生。

In this thesis, the single-relaxation lattice Boltzmann method combined with the Cahn-Hilliard model is employed to calculate the energy evolution during droplet collisions and the influence of concentration on droplet coalescence under shear flow. Through our investigation of energy variations, we uncover the challenge of directly obtaining the complete dissipation energy using the dissipation energy formula. This difficulty arises due to rapid and complex changes in the interfacial region during the collision process, making it only possible to obtain the dissipation energy indirectly through the energy conservation. When investigating the impact of droplet concentration, the droplet concentration is divided into three directions: x-direction, y-direction, and z-direction. The analysis only focuses on the effects of concentration in the x-direction and y-direction. For a given x-direction concentration, an increase in y-direction concentration results in an increase in the droplet's velocity in the x-direction, making it easier for them to merge. Conversely, for a given y-direction concentration, an increase in x-direction concentration leads to a decrease in the droplet's velocity in the x-direction, making the coalescence more challenging. However, when the x-direction concentration exceeds 0.8 and significantly reduces the droplet's velocity, the coalescence actually becomes easier.

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 Experiment about the droplet coalescence . . . . . . . . . . . . . . . . . 4
1.2.2 Numerical simulation about droplet coalescence . . . . . . . . . . . . . . 5
1.2.3 Energy evolution of droplet collision . . . . . . . . . . . . . . . . . . . . . 7
1.2.4 GPU implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.5 Previous work in laboratory . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Objective and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Methodology 12
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The BGK and low-Mach number approximation . . . . . . . . . . . . . . . . . . 13
2.3 Discretization of the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Discretization of space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Discretization of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Lattice Boltzmann model for multi-phase flow . . . . . . . . . . . . . . . . . . . 16
2.4.1 The free-energy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 The governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.3 Discrete Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.4 Interface capturing equation . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Summary of lattice Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . 22
2.6 GPU implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Formulas of droplet energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Results and Discussion 28
3.1 Energy conservation testing of single droplet translation . . . . . . . . . . . . . . 28
3.2 Energy evolution for droplet collision . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 Collision regime map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 Energy budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Deformation testing of a single droplet under shear flow . . . . . . . . . . . . . . 36
3.4 Influence of droplet concentration on droplet coalescence in shear flow . . . . . . 37
3.4.1 Droplet concentration of x-direction . . . . . . . . . . . . . . . . . . . . . 38
3.4.2 Droplet concentration of y-direction . . . . . . . . . . . . . . . . . . . . . 39
3.4.3 Analysis of the influence of Ψx and Ψy . . . . . . . . . . . . . . . . . . . 40
4 Conclusion and Future Work 52

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