本研究採用耦合相場和水平集方法（CPFLS）來模擬一般壓力方程框架下的兩相流。幾個測試問題，例如單渦旋問題和液滴振盪，用於評估現有數值方法的能力。在不同的Morton和Eo值下單一氣泡上升的結果與過去的研究大致相同。此外，我們研究了具有不同氣泡尺寸的氣泡聚結。結果表明，當上下氣泡的直徑不同時，氣泡所需的結合時間增加。

The coupled phase-field and level set approach (CPFLS) is used in this study to simulate two- phase flows using the general pressure equation framework. Several test problems, such as the single vortex problem and elongated droplet oscillations, were used to evaluate the capacity of the existing numerical procedure. The results of a single bubble rising under different Morton and Eo values are consistent with previous findings. Furthermore, we investigated the bubble coalescence case with different bubble sizes and discovered that when the diameters of the leading and following bubbles differ, the duration of the coalescence increases.

Abstract
Contents
1 Introduction 1
1.1 Introduction 1
1.2 Literature survey 3
1.2.1 Algorithm 3
1.2.2 Interface methods 5
1.3 Motivation and objective 9
2 Methodology 10
2.1 Temporal discretization 10
2.2 Conservative phase-field method 11
2.2.1 Advection term 11
2.2.2 Diffusion and anti-diffusion 11
2.2.3 Boundary condition 12
2.3 Level set method 13
2.3.1 Advection term 13
2.3.2 Re-initialization 14
2.4 Navier-Stoke equation solver 15
2.4.1 Momentum equation 15
2.4.2 General pressure equation 17
3 Result 19
3.1 Validation of numerical methods 19
3.1.1 Validation of flow field solver 19
3.1.2 Validation of interface capturing 20

3.2 Two-phase flow 20
3.2.1 Droplet oscillation 20
3.2.2 Bubble rising 21
3.2.3 Bubble coalescence 23
4 Conclusion 51
4.1 Future work 52

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