我們採用了預處理不可壓縮多相體模式加上界面壓縮模型來模擬氣泡上升的問題，本文中的人工壓縮法是採用Chorin所發表方法再進一步推展到不可壓縮多相Navier-Stokes雙曲線方程式。為了界面更加明確，我們在模擬氣泡上升的界面處理結合了THINC介面壓縮函數，使其結果使界面更加明確清晰。我們成功使用上述方法來研究氣泡上升在不同邦德(Bond)數和阿基米德(Archimedes)數對終端速度及氣泡外型的影響。

This paper first applies pre-conditioning incompressible multi-fluid flow model based on sharp interface technique to simulate the bubble rising flow problems. Here, the preconditioning matrix modified from Chorin’s artificial compressibility concept is used to make the incompressible multi-fluid Navier-Stokes equations to be hyperbolic. Therefore, a simple convection-pressure flux-splitting method with the THINC formulation for volume fraction function compressed reconstruction is used to maintain the preservation of sharp interface evolutions of bubble rising flow simulation. The revolution of a rising bubble at different Bond number and Archimedes number is shown and discussed.

Table of Contents
Nomenclature	iii
List of Figure	vi
1.	Introduction	1
1.1.	Background	1
1.2.	Numerical Algorithm Review	2
1.3.	Bubble Rising Review	5
2.	Numerical Models	11
2.1.	Governing Equations	11
2.2.	Numerical Methodology	15
2.3.	Time Discretization	16
2.4.	Numerical Flux	19
2.5.	Interface Compression	22
2.6.	Surface Tension	24
3.	Result of 2D Bubbles Rising in Viscous Liquids	34
3.1.	Grid Independence	35
3.2.	Comparison with Experiment	37
3.3.	The Simulated Bubble Shapes	41
3.4.	Bubble Rising Velocity	42
3.5.	The Evolution of Bubble Rising	44
4.	Conclusions	45
5.	References	46
Appendix	58
Figure 2.1 The diagrams for the interfaces across grid points	26
Figure 2.2 Computational domain for surface tension	27
Figure 2.3 Diagrams of the curvature of interface	29
Figure 2.4 Numerical discretization of volume fraction for diagram (c)	32
Figure 2.5 (a) Simulated surface tension by J. U. Brackbill et al.[60] (b) the current simulated vector of surface tension (Bo=166)	33
Figure 3.1 The regime map of experimental rising bubble shape in liquids [4]	35
Figure 3.2 Effect of mesh sizes on the simulated bubble shape respectively.	37
Figure 3.3 Comparison of terminal bubble shapes observed in experiments [1] and predicted under various Reynolds and Bond numbers. (Re=Ar x U*)	39
Figure 3.4 Comparison of terminal bubble shapes observed in experiments [1] and predicted under various Reynolds and Bond numbers (Re=Ar x U*)	40
Figure 3.5 The predicted bubble shapes at different Archimedes and Bond numbers. The ratios of density and viscosity are pl/pg  = 10 and ul/ug = 10,	42
Figure 3.6 The rising velocity vs. time at the different Archimedes number. (Bo=116)	43
Figure 3.7 The rising velocity vs. time at different Bond numbers. (Ar=20)	44
Figure 3.8 The evolution of bubble rising	45

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