沉浸邊界法在這裡用來解決流體中的運動物體，其座標系使用固定不動的直角座標系，時間方面使用分步法。此方法首先應用於探討低雷諾數下的二維振翅運動，將結果與實驗進行比較，得到很好的一致性。
將此方法推廣於計算三維的壓力驅動的波蘇拉流，及上邊界帶動的庫艾特方管流，其模擬的邊界條件為主流場方向週期性邊界條件。為加速計算，使用快速複立葉變換可使三維的壓力波松方程簡化為二維的矩陣計算。儘管如此，平行計算效率仍在48顆CPU後效率下降，原因乃是波松方程計算過程中資進行所有資料收集的動作耗掉大部分的時間。因此舒康補矩陣解法被採用來減少計算處理器間的資料交換並增加計算效率，然而儘管資料交換的時間減少了，但計算單元內的計算卻增加了，因此得到更長的計算時間。
最後，為了更多減少三維物體於流體中的計算時間，移動計算範圍法因而在此發展出來。此方法是當物體運動時，計算網格跟著物體增加及減少。二維的反覆振翅運動被首先用來檢驗此方法的準確性，而計算範圍為原計算域的四分之一。與實驗以及其他模擬結果比較，本方法結果顯出相當地準確性。三維的自由掉落球體也被進一步地模擬出來，其預測的阻力係數對不同雷諾數的結果也與統計的實驗結果相符合。最後，一於重力下自由掉落的橢球體也進一步地被模擬，而雷諾數從44.5到1617。在此雷諾數範圍下，其運動軌跡為二維運動，而反覆擺盪的運動模式在所有雷諾數中都存在，除了在低雷諾數44.5下垂直掉落的模式外。模擬結果與先前文獻的研究結果皆符合，而當此橢球擺盪時，不同的渦流結構產生於其分支尾端，其中也包含著名的髮夾形狀的渦流。

The immersed-boundary method (IBM) is used here to solve flows with dynamically moving objects on fixed Cartesian grids using the fractional step method. The method is first applied to investigate flows of a 2D flapping wing at low Re. The results is validated with the benchmark solutions, and have good agreement with them.
Simulations are further applied to compute 3D Poiseuille, and Couette duct flows with periodic boundary in the streamwise direction. The solution of the 3D pressure Poisson can be reduced to the 2D equation using fast Fourier transform. However, parallel efficiency deteriorates at 48 CPU processors, due to the requirement of all-to-all communications in the pressure Poisson equation. Therefore the Schur complement method (SCM) is adopted to reduce the communications between parallel processors and enhance the efficiency. However, the communications reduce but calculation inside the processors increase, and hence receiving the opposite results.
Finally, in order to reduce the computational cost of flows with an embedded moving object, a moving domain method is developed, where grids are generated or deleted based on the location of the object centroid. A 2D flapping wing flow is first examined using the quarter size of the original fixed computational domain. Comparing with the benchmark solutions, the simulation results show compatible results. Further, sphere settling under gravity is also simulated. The predicted drag coefficients at different Re agree reasonably with measurements. Finally, an oblate spheroid descending under gravity is further simulated with Re at 44.5 to 1617. For the Re examined, the motions are 2D, and the fluttering motions exist in all the Reynolds numbers investigated except at Re = 44.5, in which steady falling prevails. This is consistent with previous investigations for falling disks. As the spheroid flutters, different vortex branches are generated in the trailing edge direction, in which hairpin-like vortices occur.

Abstract
iii
Contents
xii
1
Introduction
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Unsteady
ying mechanisms of insect
ight . . . . . . . . . . . . . . .
2
1.2.1
Clap-and-Fling
. . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.2
Delayed Stall
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.3
Rotational Circulation . . . . . . . . . . . . . . . . . . . . . .
4
1.2.4
Wake Capture . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Parallelization and matrix solvers . . . . . . . . . . . . . . . . . . . .
4
1.4
Moving domain method and freely falling problem . . . . . . . . . . .
6
1.5
Motivations and Objectives
. . . . . . . . . . . . . . . . . . . . . . .
9
2
Numerical Methods
12
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2
Mathematical formulation
. . . . . . . . . . . . . . . . . . . . . . . .
13
2.3
Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3.1
Spatial discretization . . . . . . . . . . . . . . . . . . . . . . .
15
2.3.2
Temporal discretization . . . . . . . . . . . . . . . . . . . . . .
16
2.3.3
Forcing strategies . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.3.4
Rigid body motion . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3.5
Rotational velocity . . . . . . . . . . . . . . . . . . . . . . . .
21
2.4
The Pressure Poisson Equation
. . . . . . . . . . . . . . . . . . . . .
23
2.4.1
Direct fast Fourier transform solver . . . . . . . . . . . . . . .
25
2.4.2
Schur Complement Method
. . . . . . . . . . . . . . . . . . .
27
2.4.3
Other matrix solvers
. . . . . . . . . . . . . . . . . . . . . . .
34
2.5
Moving domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.6
Complete solution procedure . . . . . . . . . . . . . . . . . . . . . . .
42
3
Results and Discussions
44
3.1
Immersed Boundary Method
. . . . . . . . . . . . . . . . . . . . . .
45
3.1.1
Wing Motion
. . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.1.2
Code Validation . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.2
2D MPI Parallelization
. . . . . . . . . . . . . . . . . . . . . . . . .
50
3.2.1
Decaying Vortex
. . . . . . . . . . . . . . . . . . . . . . . . .
52
3.2.2
Laminar Poiseuille Duct Flow . . . . . . . . . . . . . . . . . .
53
3.2.3
Laminar Couette Duct Flow . . . . . . . . . . . . . . . . . . .
54
3.2.4
Calculation Eciency . . . . . . . . . . . . . . . . . . . . . . .
54
3.3
Schur Complement Method
. . . . . . . . . . . . . . . . . . . . . . .
56
3.4
Moving Domain Method . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.4.1
Flapping wing . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.4.2
Simulation of a sphere settling under gravity . . . . . . . . . .
68
3.4.3
Freely falling object . . . . . . . . . . . . . . . . . . . . . . . .
79
4
Conclusions and Future Work
95
4.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
5
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100

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