移動式液態儲存槽內自由液面的擺盪關聯到很多工程上應用的問題，例如，高速公路上的油罐車、受到地震作用時，儲油槽內液體擺盪、遠洋航行的船隻上裝載液態儲存槽、還有飛機或太空梭上燃料的擺盪等等。本研究的目的是建立一個三維的數值水槽來研究液體擺盪以及液體與內嵌結構物，例如隔板 （baffle），之間的交互作用。用已發展的三維時間獨立之有限差分法來分析計算當隔板水槽受到六個自由度震動時，槽內液體擺盪的情況。數值水槽之數值模式主要是求解三維的Navier-Storkes方程式和非線性的自由液面邊界條件以及假設流體為不可壓縮流體，並轉換慣性座標系統至移動座標系統下來進行運算。複雜的固-液交界處乃採用一維的虛擬網格法（one dimensional ghost cell approach）來處理。為了減少數值運算之時間以及考慮到大量記憶體之消耗，建立叢集電腦 ( PC-cluster)以及撰寫平行語法程式(Message Passing Interface)來執行平行計算。平行計算之軟體乃採用MPICH2。

為了驗證數值方法的正確性，本模式經過一連串的嚴謹的驗證，不僅比對文獻上已發表的解析、實驗以及數值的結果，為了更進一步驗證本模式的精準度，自行建立起實驗設備來加以比對模擬的結果。所有的比對結果顯示本模式有很好的一致性跟精準度。在考慮無內嵌結構物水槽內液體擺盪，模擬耦合的surge和sway 運動方式下，伴隨著不同的水平震動角度、震動頻率以及不同的靜水深度下，擺盪波(sloshing waves)的分類、振幅、擺盪造成的作用力(sloshing-induced forces) 以及擺盪波之能量轉換關係均有詳細的討論。擺盪波種類被分類成六種，分別為單一方向波(single-directional waves)、對角線波(diagonal waves)、類矩形波(square-like waves)、類旋轉波(swirling-like waves)、旋轉波(swirling waves)、以及不規則波(irregular waves). 這些波的出現跟水槽的震動頻率有密切的關係。水平震動角度的不同，對擺盪波的影響之研究與分析也詳加討論，尤其是對旋轉波的探討。透過波譜分析，對上述幾種擺盪波的形成與擺盪波自然震動頻率的波型之間的關係作分析，證明擺盪波的形成與自然震動頻率間有密不可分的關係。對於旋轉波轉換旋轉方向的機制，分別對旋轉流的情況、瞬間的水面變化、重力加速度的影響、以及瞬間外力作用的方向來進行探討。本模式也有詳加探討液體擺盪在對於水槽受到耦合的surge-sway-heave運動方式情況下，並發現當heave motion的震動頻率是水槽自然震動頻率的2倍時，對擺盪波會有不穩定的影響出現。Heave的運動方式對上述幾種擺盪波的影響也在本文中探討，結果顯示除了不規則波以外，heave的運動方式會轉變其他種波的類型至旋轉波。

二維以及三維的液態阻尼水槽(Tuned liquid damper)之研究也包含在本研究之中。對二維隔板水槽(baffled tank)，隔板垂直鑲嵌在水槽底部中央，綜合性的討論分析包括隔板的高度對水槽自然共振頻率的影響、漩渦的發展以及剝離現象、渦流剝離頻率跟擺盪波震動頻率之間的關係、隔板附近生成漩渦的大小尺寸、漩渦之間的交互作用等等。結果發現，隔板的高度明顯的改變的水槽的第一個共振頻率分布，而且水深的影響也扮演個很重要的角色，換而言之，不同水深下因為不同隔板高度造成水槽共振頻率的轉變也因此而不同。當考慮兩個隔板鑲嵌在水槽底部，隔板之間的距離為0.2倍的水槽長度是個有效的工具，不僅可以減少擺盪波的振幅，也可以有效的轉變擺盪波的共振頻率。擺盪振幅很明顯的受到內嵌隔板數目的影響。內嵌隔板數目越多，擺盪波振幅就越小。隔板尖端渦流的演變過成可分成四個步驟，分離剪力層流的形成和漩渦的生成，垂直噴射流的形成和渦流分離，剝落的漩渦和擺盪流(sloshing flow)之間的交互作用（彎曲流的形成）以及彎曲流(snaky flow)和擺盪波之間的交互作用。當隔板水槽的震動頻率接近其第一個自然震動頻率，因為強烈的垂直噴射流所造成的隔板尖端渦流分離現象發生。生成漩渦的大小和隔板的高度有極密切的關係。

當水槽受到耦合的surge-sway激盪時候，兩種三維的液態阻尼水槽，分別為隔板水槽以及垂直板水槽也在本研究中加以探討。因為隔板的影響，對角線波以及單一方向波均轉換成旋轉波。當水槽受到對角線方向激盪時候，本研究發現類矩型波和旋轉波以及不規則波和旋轉波共同出現的情況。本研究所使用的隔板或是垂直板，鑲嵌方向和水槽寬度平行，當水平震動角度介於0度和10度之間的時候，這種設計能有效的減少擺盪波振幅，並且隨著震動角度增加，在縱軸方向的擺盪波的振幅明顯的增大。對於隔板造成擺盪波自然震動頻率的偏移，這種設計的隔板主要是影響x(橫軸)方向自然震動頻率的轉移。而垂直板的長度，不僅對水槽自然震動頻率有明顯的影響，也改變了擺盪波的類型。但是對於不規折波，垂直板的影響似乎很小，但透過頻譜分析下，出現的主要頻率個數多於隔板水槽分析後的結果。

Liquid sloshing with unrestrained free surface in a moving container is associated with various engineering problems, such as tankers on highways, liquid oscillations in large storage tanks caused by earthquakes, sloshing of liquid cargo in ocean-going vessels, and the motion of liquid fuel in aircraft and spacecraft. The purpose of this study is to develop a three-dimensional (3D) numerical wave tank with or without internal structures to investigate the mechanism of liquid sloshing and the interaction between the fluid and internal structures. The developed 3D time-independent finite difference method is applied on solving liquid sloshing in tanks with or without the influence of baffles under the ground motion of six-degrees of freedom. The 3D Navier-Stokes equations were solved and transformed to a tank-fixed coordinate system, and the fully nonlinear kinematic and dynamic free surface boundary conditions for fluid sloshing in a rectangular tank with a square base were considered. The fluid is assumed incompressible in this study. The complicated interaction in the vicinity of the fluid-structure interface was solved by implementing one dimensional ghost cell approach and the stretching grid technique near the fluid-structure boundaries were used to catch the detailed evolution of local flow field. A PC-cluster was established by linking several single computers to reduce the computational times due to the implementation of the 3D numerical model. The Message Passing Interface (MPI) parallel language and MPICH2 software were utilized to code the computer codes and to carry out the circumstance of parallel computation, respectively.

The developed numerical scheme was verified by rigorous benchmark tests. Not only the reported analytical, numerical and experimental studies were compared with the present numerical results, the experimental investigation was also involved in the present work to further validate the accuracy of the numerical scheme. All the benchmark tests of this study showed excellent accuracy of the developed numerical scheme. For a tank without internal structures, the coupled motions of surge and sway are simulated with various excitation angles, excitation frequencies and water depths. The characteristics of sloshing waves are dissected in terms of the classification of sloshing wave types, sloshing amplitude, beating phenomenon, sloshing-induced forces and energy transfer of sloshing waves. Six types of sloshing waves, named single-directional, diagonal, square-like, swirling-like, swirling and irregular waves, were found and classified in the present study and the occurrence of these waves are tightly in connection with the excitation frequency of the tank. The effect of excitation angle on the characteristics of sloshing waves is explored and discussed, especially for swirling waves. The spectral analyses of sloshing displacement of various sloshing waves are examined and a clear evidence of the correlation between sloshing wave patterns and resonant modes of sloshing waves are demonstrated. The mechanism of switching direction of swirling waves is discussed by investigating the situation of circulatory flow, the instantaneous free surface, the gravitational effect and the instantaneous direction of external forcing. The coupling effects of heave, surge and sway motions were also included in this study and the result showed an unstable influence of heave motion on the kinematic and dynamic characteristics of sloshing waves when the vertical excitation frequency of the tank is twice as large as the fundamental natural frequency. Except irregular waves, the other types of sloshing waves are converted into swirling waves due to the effect of heave motion.

The study related to tuned liquid damper (TLD) in 2D and 3D tanks were considered. A comprehensive investigation for a 2D tank with vertically tank bottom-mounted baffles (baffled tank) are demonstrated and discussed with respect to the influence of baffle height on the natural mode of the tank, the evolution of vortices and vortex shedding phenomenon, the relationship between the vortex shedding frequency and the excitation frequency of the tank, the vortex size generated in the vicinity of the baffle tip, the interaction of vortices inside the tank. The baffle height shows a significant influence on the shift of the first natural frequency of the baffled tank and the liquid depth also plays an important part in determining this influence. In other words, the shift of the first natural mode due to various baffle height is varied with water depths. The design of two baffles separated by 0.2 times the tank breadth is an efficient tool to not only reduce the sloshing amplitude but switch the first natural frequency of the tank. The sloshing displacement is affected distinctly by different numbers of baffles mounted vertically on the tank bottom. The more baffles mounted onto the tank bottom, the smaller the sloshing displacement is presented in both the transient and steady-state periods. The processes of the evolution of vortices near the baffle tip are categorized into four phases: the formation of separated shear layer and generation of vortices, the formation of a vertical jet and shedding of vortices, the interaction between shedding vortices and sloshing flow (the generation of snaky flow) and the interaction between snaky flow and sloshing waves. Vortex shedding phenomenon due to stronger vertical jets occurs when the excitation frequency is close to the first natural mode of the baffled tank. The size of the vortex generated near the baffle tip is discussed and the vortex size is closely correlated with the baffle height.

Two types of 3D tuned liquid dampers, a vertically tank bottom-mounted baffle and a vertical plate, are discussed for a tank under coupled surge-sway motions. The wave types of diagonal and single-directional waves switch to the swirling type due to the influence of the baffle. The phenomenon of square-like waves or irregular waves coexisting with swirling waves is found in the baffled tank under diagonal excitation. The baffle and the vertical plate mounted parallel to the east (west) wall of the tank can effectively reduce the sloshing amplitude when the excitation angle is between 0 degree and 10 degree and the corresponding sloshing displacement in the sway (z) direction becomes more dominant with the increase of the excitation angle. The shift of the first natural mode of the baffled tank due to various baffle heights in the x direction is dominated in this design of baffled tank. The length of the plate can cause a significant influence on not only the variation of the natural frequencies but the type of the sloshing waves. The influence of the vertical plate on the irregular waves is insignificant and several peaks appear in the spectral analysis of the sloshing displacement for the irregular waves and the numbers of peaks are more than that of the baffled tank.

Contents
Chinese Abstract.........................................................................................................I
English Abstract........................................................................................................IV
Contents..................................................................................................................VIII
Notation..................................................................................................................XIII
Figure caption.........................................................................................................F-1
List of tables...........................................................................................................F-25

Chapter 1 Introduction

1.1 Backgrounds...................................................................................................1-1
1.2 Literature review...........................................................................................1-2

1.2.1 Resonant sloshing waves...................................................................1-6
1.2.2 Transient and steady state sloshing waves....................................1-8
1.2.3 Faraday waves.....................................................................................1-10
1.2.4 Tuned liquid damper (TLD) .............................................................1-11
1.2.5 The present study..............................................................................1-12

1.3 Summary of chapters...................................................................................1-14

Chapter 2 Mathematical formulation.....................................................2-1

2.1 Governing equations in a tank-fixed coordinate system...........2-1
2.2 Boundary conditions.............................................................................2-4

2.2.1 Free surface boundary conditions...................................................2-4
2.2.2 Solid wall boundary conditions........................................................2-7

2.3 The conservations of momentum and energy.....................................2-8

2.3.1 The conservation of momentum.....................................................2-9
2.3.2 The conservation of energy............................................................2-11

2.4 The coordinate transformation..............................................................2-13
2.5 Dimensionless equations.........................................................................2-18
2.6 Natural modes of a 3D tank....................................................................2-20

Chapter 3 Computational Algorithm......................................................3-1

3.1 Finite difference method...........................................................................3-1
3.2 One-dimensional ghost cell approach....................................................3-6
3.3 Iterative procedures....................................................................................3-9
3.4 Computational flow chart........................................................................3-11

Chapter 4 Introduction of Message Passing Interface (MPI) .............4-1

4.1 Hardware and software for setting up a PC Cluster............................4-1
4.2 A brief introduction of program parallelization.................................4-4

Chapter 5 Stability analysis and benchmark tests...................................5-1

5.1 A 3D tank without internal structures.......................................................5-1

5.1.1 Convergence study .............................................................................5-2
5.1.2 Conservations of mass, momentum, and energy..................5-6
5.1.3 Benchmark tests..................................................................................5-8
5.1.4 Experiment investigation.................................................................5-9
5.1.5 Reynolds number effect..................................................................5-14


5.2 A tank with baffles (Tuned liquid damper) ............................................5-19

5.2.1 A surface-piercing baffle at the middle of a 2D tank.....5-20
5.2.2 A vertically tank bottom-mounted baffle in a 2D and 3D tank……………………………………………………………………5-22
5.2.3 A 3D tank with a vertical plate....................................................5-27

Chapter 6 Horizontal motion...........................................................................6-1

6.1 Classification of sloshing waves...................................................................6-3

6.1.1 Diagonal waves and single-directional waves..............................6-3
6.1.2 Square-like waves................................................................................6-7
6.1.3 Irregular waves.....................................................................................6-9
6.1.4 Swirling-like waves............................................................................6-11
6.1.5 Swirling waves....................................................................................6-13

6.2 The effects of excitation angles and water depths on the sloshing waves. ..............................................................................................................6-17

6.2.1 Single-directional waves and diagonal waves............................6-18
6.2.2 Square-like waves..............................................................................6-21
6.2.3 Irregular waves...................................................................................6-25
6.2.4 Swirling waves....................................................................................6-29
6.2.5 The classification of sloshing waves..............................................6-39

6.3 Sloshing wave patterns and resonant wave modes.............................6-43

6.3.1 The Fast Fourier Transform (FFT) analysis of various sloshing waves....................................................................................................6-43
6.3.2 The relationship between the free surface profile and the resonant wave modes………………………………………………………..6-52

6.4 The evolution of sloshing-induced forces of various sloshing waves ……………………………………..6-56

6.4.1 Single-directional waves and diagonal waves............................6-58
6.4.2 Square-like waves..............................................................................6-60
6.4.3 Irregular waves.................................................................................6-66
6.4.4 Swirling waves..................................................................................6-68
6.4.5 Hydrodynamic force coefficient...................................................6-77

6.5 The kinematic and dynamic characteristics of sloshing waves.......6-81

6.5.1 Single-directional waves and diagonal waves..............................6-81
6.5.2 Square-like waves................................................................................6-86
6.5.3 Irregular waves....................................................................................6-88
6.5.4 Swirling waves.....................................................................................6-90

6.6 The limitation of the present numerical model..................................6-102

6.7 Summary........................................................................................................6-104

Chapter 7 Surge-Sway-Heave motion..........................................................7-1

7.1 The effect of heave motion on various sloshing waves.........................7-1

7.1.1 Heave motion effect on single-directional waves and diagonal waves.........................................................................................................7-5
7.1.2 Heave motion effect on square-like waves.......................................7-9
7.1.3 Heave motion effect on irregular waves.........................................7-11
7.1.4 Heave motion effect on swirling-like waves...................................7-13
7.1.5 Heave motion effect on swirling waves...........................................7-16

7.2 Summary..........................................................................................................7-18

Chapter 8 Tuned liquid damper (TLD) in a 2D tank....................................8-1

8.1 The influence of vertically tank-bottom mounted baffles on the natural frequency of tanks under various water depths . ....................8-1

8.1.1 One baffle..................................................................................................8-1
8.1.2 Two baffles..............................................................................................8-13
8.1.3 Four baffles.............................................................................................8-23

8.2 The evolution of vortices in the baffled tank ........................................8-29

8.2.1 Evolution of vortices and vortex shedding.....................................8-29
8.2.2 Vortex shedding versus excitation frequency................................8-63
8.2.3 Relationship between vortex size and baffle height....................8-71
8.2.4 Hydrodynamic interaction of vortices in a multi-baffled tank.8-101

8.3 Summary........................................................................................................8-111


Chapter 9 Tuned liquid damper in a 3D tank...............................................9-1

9.1 Effect of a vertically tank bottom-mounted baffle on sloshing waves ............................................................................................................9-1
9.2 Effect of a vertical plate on sloshing waves.......................................9-23
9.3 Summary.....................................................................................................9-45

Chapter 10 Conclusions ..................................................................................10-1

10.1 Conclusion................................................................................................10-1
10.2 Recommendation for further research ..........................................10-8

Appendix A B C...................................................................................................A-1

Reference................................................................................................................R-1

List of publications……………………………………...…………………..…P-1

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