在河川及海洋中有許多水利工程建設，如橋墩、海底油管等等，底床變形會對其結構物安全產生危害，像是橋墩會因底床變形崩塌、海底油管會因底床變形破裂；在水庫方面，臺灣地區坡陡流急，因此水庫泥沙淤積問題嚴重，研究底床變形機制，對水庫清淤或許有所助益。臺灣四面環海，底床變形對於國土保育及國土規劃亦有關連，透過研究底床變形，將可預測出海岸線的增長與後退。本研究主要目的即為研究在長時間水流與水波作用下之軟質孔隙彈性介質底床變形，延續郭遠錦(2011)之研究。因在郭遠錦(2011)，其數值模擬僅為簡單情境之邊界條件與實際物理問題尚有差異，並且其水體設置數值海綿層(sponge layer)將擾動流速吸收以防止水波反射回內部水體導致數值發散，但在實際問題中，並無設置海綿層這類機制，故需考慮這樣的處理方式對數值模擬的影響。又在郭遠錦(2011)展示之結果中，其底床變形模擬時間略短，其原因可歸咎於時距設定偏短上的限制，故尚無法有效模擬較真實底床變形之問題。研究中，水體主要根據勢流理論建立其控制方程式與邊界條件，而土體之控制方程式則採用 Biot(1956a、1956b、1962) 所建立之多孔隙彈性介質理論加以描述。底床固體位移以流函數(stream function)表示其底床固體位移，並以量階(order)分析簡化土體控制方程式為拉普拉斯方程式(Laplace Equation)與雙調和方程式(Biharmonic Equation)，提出水波與沙波間時間尺度差異顯著關係，並利用邊界元素法(boundary element method)模擬水流與水波作用下之軟質孔隙彈性介質底床變形。

本研究之主要貢獻敘述如下：(1.) 利用水波分散關係式與沙波分散關係式，繪製週波數與角頻率之關係圖，探討時間尺度差異顯著關係。(2.) 提出一無因次參數，為控制底床變形參數，其主要控制因子為流速與拉美第一彈性常數。(3.) 改善計算時距，使較能反應出在水流與水波作用下軟質孔隙彈性介質底床之變形趨勢，並運用數值技巧在水體上下游設置可使擾動流速通過之邊界條件，免除數值海綿層的影響使結果較貼近實際物理現象，並對其原邊界條件計算上之處理進行改善，以減小數值運算上的誤差。(4.) 在研究中亦建構不同情境之邊界條件，諸如土體透水與不透水邊界、加入懸浮載(suspended load)以及加入底床載(bed load)，使模擬結果在實際面上更具參考價值。最後，本研究之數值模擬結果與施宛平(1998)之數值模擬結果，以及李鴻源等人(1990)之實驗結果進行比對，驗證底床變形趨勢一致。

There are lots of hydraulic engineering buildings in rivers and the oceans, such as bridge pier and the seabed oil pipes, etc. Bed form deformation will destroy the hydraulic engineering buildings safety, like bridge pier will collapse and the seabed oil pipes will crack by bed form deformation. On the part of reservoir, Taiwan area is high gradient slope and large flow velocity, thus, there is serious impact of sediment deposition for Taiwan reservoirs, studying the problem of bed form deformation may have benefit for Taiwan reservoirs desilting. Taiwan is a small island surrounded by seas, bed form deformation relates to territory protection and plan. Through studying bed form deformation can predict that coastline grow and reduce. The aim of this research is to study the flow and water waves effects on bed form deformation of porous elastic media in the long time scale situation, and follow Kuo (2011)’s study. Because of Kuo (2011), the numerical simulation is only for simple situations of boundary conditions, there still have differences between actual physical problems. Furthermore, Kuo(2011) sets numerical sponge layer in water body on downstream for absorbing disturbance flow velocity to avoid water waves reflexing to inner water body which can lead to numerical divergence. However, there has no sponge layer mechanism in real problem; therefore, this process should be considered what effect for numerical simulation. In Kuo (2011), the bed form deformation time seems too short, and it can impute to the limitation of small time step; so that can’t be effective in simulating the actual bed form deformation problem. In this research, water body is based on potential theory to build up the governing equations and boundary conditions, and the soil body is based on Biot(1956a, 1956b, and 1962) which build up the porous elastic media theory to set up the governing equations. Utilizing the stream function to express the bed solid displacement, and using order analysis to simplify the soil body’s governing equations to Laplace equation and biharmonic equation. Bring up the significant difference between time scale of water waves and soil waves, and using boundary element method to simulate soft porous elastic media bed form deformation which is affected by flow and water waves.
There are followings the main contribution of this study : (1.) Utilizing the dispersion relation of water waves and soil waves to plot angular frequency and wave number diagram, and investigating significant difference between time scales. (2.) Proposed a dimensionless parameter for the control of bed form deformation, the main controlling factors are flow velocity and shear modulus of the soil. (3.) Improved calculation of time step, that can be able to reflect soft porous elastic media bed form deformation trends on the flow and water waves affect. Using numerical skill to set up boundary condition which can let disturbance flow velocity pass in water body on upstream and downstream, in order to eliminate the impact of numerical sponge layer to make the results closer to actual physical phenomenon. And the calculation of its original boundary conditions of the process to make improvements to reduce the error of numerical simulation. (4.) Constructed different situations boundary conditions in this study, such as soil body permeable and impermeable boundary on upstream and downstream, adding suspended load, and adding bed load, so that numerical simulations are useful in reality. Finally, numerical simulation results of this study will verify with Shi (1998) and Li, et al (1990).

摘要 I
Abstract III
圖目錄 IX
表目錄 XIII
符號說明 XIV
第一章 導論 1
1.1. 研究動機與目的 1
1.2. 文獻回顧 1
1.3. 研究內容與方法 5
1.4. 章節介紹 6
第二章 理論基礎 7
2.1. 控制方程式 7
2.1.1. 均勻水體之控制方程式 7
2.1.2. 多孔隙彈性介質底床之控制方程式 8
2.2. 邊界條件 10
2.2.1. 自由液面之邊界條件 10
2.2.2. 底床表面之邊界條件 10
2.2.3. 上下游邊界條件 12
2.2.4. 底床底部邊界條件 13
第三章 時間尺度與邊界值問題簡化 14
3.1. 時間尺度 14
3.1.1. 郭遠錦(2011)水波驅動沙波之解析結果 15
3.1.2. 郭遠錦(2011)沙波驅動水波之解析結果 16
3.1.3. 郭遠錦(2011)探討水波與沙波間之時間尺度關係 17
3.2. 簡化控制方程式 18
3.2.1. 簡化均勻水體之控制方程式 18
3.2.2. 簡化多孔隙彈性介質底床之控制方程式 18
3.3. 簡化邊界條件 21
3.3.1. 簡化自由液面之邊界條件 21
3.3.2. 簡化底床表面之邊界條件 22
3.4. 無因次化參數 24
第四章 數值方法 25
4.1. 邊界元素法 26
4.1.1. 拉普拉斯方程式(Laplace Equation)之邊界積分式 26
4.1.2. 雙調和方程式(Biharmonic Equation)之邊界積分式 27
4.2. 邊界條件給定 29
4.2.1. 均勻水體擾動流速勢能 ϕ1 之邊界條件 30
4.2.2. 底床孔隙擾動水壓力 p2 之邊界條件 32
4.2.3. 底床固體位移流函數 ψ2 之邊界條件 34
4.3. 初始條件給定與計算時距 39
4.4. 演算流程 42
4.5. 本研究模式之限制 45
第五章 本研究模式驗證與應用 46
5.1. 以定值平均流速之模擬應用與驗證 48
5.1.1. 土體上下游均透水之應用與驗證 48
5.1.2. 土體上游不透水之應用 58
5.1.3. 土體下游不透水之應用 60
5.1.4. 加入懸浮載(suspended load)之應用 62
5.1.5. 加入河床載(bed load)之應用與驗證 64
5.2. 渠道漸縮之模擬應用 69
5.2.1. 土體上下游均透水之應用 70
5.2.2. 土體上游不透水之應用 72
5.2.3. 土體下游不透水之應用 74
5.2.4. 加入懸浮載(suspended load)之應用 76
5.2.5. 加入河床載(bed load)之應用 78
5.3. 渠道漸寬之模擬應用 80
5.3.1. 土體上下游均透水之應用 81
5.3.2. 土體上游不透水之應用 83
5.3.3. 土體下游不透水之應用 85
5.3.4. 加入懸浮載(suspended load)之應用 87
5.3.5. 加入河床載(bed load)之應用 89
第六章 結論與建議 91
6.1. 結論 91
6.2. 建議 92
參考文獻 93
附錄A 郭遠錦(2011)水波驅動沙波之解析解 98
附錄B 郭遠錦(2011)沙波驅動水波之解析解 102
附錄C 提供不同計算流程之概論 106

1.Anderson, A. G., 1953, “The charateristics of sediment waves formed by flow in open channels”, 3rd Midwest. Conference on Fluid Mechanics, University of Minnesota, Minneapolis, March, pp. 379-395.
2.Bagnold, R. A., 1936, “The Movement of the Desert Sand”, Poceedings, Royal Society, London, A. No. 892, Vol. 157, pp. 594-620.
3.Biot, M. A., 1956a, “Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency Range”, The Journal of the Acoustical Society of America, Vol. 28, pp.168-178.
4.Biot, M. A., 1956b, “Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Low-frequency Range”, The Journal of the Acoustical Society of America, Vol. 28, pp.179-191.
5.Biot, M. A., 1962, “Mechanics of deformation and acoustic propagation in porous media”, The Journal of the Acoustical Society of America, Vol. 33, pp. 1482-1498.
6.Brebbia, C. A., Telles, J., and Wrobel, L., 1984, Boundary Element Techniques, Springer-Verlag.
7.Dalrymple, R. A. and Liu, P.L.-F., 1978, “Waves over soft muds: A two-layer fluid model”, Journal of Physical Oceanography, Vol. 8, pp.1121-1131.
8.Darwin, G. H., 1883-84, “On the Formation of Ripple-Marks”, Proceedins, Royal Society, London.
9.DuBuat, P. M., 1879, Principes d’Hydraulique, De L’Imprimerie de Monsieur, Paris.
10.Einstein, H. A., 1950, “The bed-load function for sediment transportation in open channel flows”, U.S. Department of Agriculture, Soil Conservation Service, Technical Bulletin, no. 1026.
11.Engelund, F., 1970, “Instability of erodible beds”, Journal of Fluid Mechanics, Vol.42, pp.225-244.
12.Exner, F. M., 1925, “Uber die Wechselwirkung Zwischen Wasser und Geschiebe in Flussen”, Sitzenberichte der Akademie der Wissenchaften, Wien, Heft 3-4.
13.Fratantonio, M. and Rencis, J. J., 2000, “Exact boundary element integration for two-dimensional Laplace equation”, Engineering Analysis with Boundary Elements, Vol. 24, pp. 325-342.
14.Graf, W. H., 1971, Hydraulics of Sediment Transport, McGraw-Hill Book Company.
15.Huang, L. H. and Chwang, A. T., 1990, “Trapping and absorption of sound waves II a sphere covered with a porous layer”, Wave Motion, Vol. 12, pp.401-414.
16.Huang, L. H. and Song, C. H., 1993, “Dynamic Response of Poroelastic Bed to Water Waves”, J. of Hydraul. Engrg., ASCE, Vol. 119, No.9, pp. 1003-20.
17.Jaswon, M. A., 1963, “Integral equation methods in potential theory. I.”, Proceedings of the Royal Society of London, Series A, Vol. 275, pp.23-32.
18.Karageorghis, A. and Fairweather, G., 1988, “The Almansi method of fundamental solutions for solving biharmonic problems”, International Journal for Numerical Methods in Engineering, Vol. 26, pp.1665-1682.
19.Kennedy, J. F., 1963, “The mechanics of dunes and antidunes in erodible-bed channels” Journal of Fluid Mechanics, Vol. 16, part 4, pp.521-545
20.Kennedy, J. F., 1969, “The formation of sediment ripples, dunes and antidunes”, Annual Review of Fluid Mechanics, Vol. 1, pp.147-168.
21.Liu, P. L.-F., 1973, “Damping of water waves over porous bed”, Journal of the Hydraulics division, Vol. 99, pp.2263-2271.
22.Liu, K. F. and Mei, C. C., 1993, “Long waves in shallow water over a layer of Bingham-plastic fluid-mud-I. Physical aspects”, International Journal of Engineering and Science, Vol.31, pp.125-144.
23.Liu, K. F. and Mei, C. C., 1993, “Long waves in shallow water over a layer of Bingham-plastic fluid-mud-II. Mathematical derivation of long wave equations”, International Journal of Engineering and Science, Vol.31, pp.145-155.
24.Liu, P. L.-F. and Chan, I. C., 2007, “A note on the effects of a thin visco-elastic mud layer on small amplitude water-wave propagation”, Coastal Engineering, Vol. 54, pp.233-247.
25.Madsen, O. S., 1978, “Wave-induced pore pressures and effective stresses in a porous bed”, Geotechnique, Vol. 28, pp.377-393.
26.Mase, H., Oki, S. I. and Takeba, K., 1995, “Wave equation over permeable rippled bed and analysis of Bragg scattering of surface gravity waves”, Journal of Hydraulic Research, Vol. 33, pp.789-812.
27.Mei, C. C., Krotov, M., Huang, Z. H., and Huhe, A., 2010, “Short and long waves over a muddy seabed”, Journal of Fluid Mechanics, Vol. 643, pp.33-58.
28.Moshagen, H. and Torum, A., 1975, “Wave Induced Pressures in Permeable Seabeds”, J. of the Waterway, Harbors, and Coastal Eng., ASCE, 101, No. WW1, pp. 49-57.
29.Moin, P., 2010, Fundamentals of Engineering Numerical Analysis, Second Edition, Cambridge university press.
30.Putnam, J. A., 1949, “Loss of wave energy due to percolation in a permeable sea bottom”, Trans. Am. Geophys. Union, 30: pp. 349-356.
31.Reid, R. O. and Kajiura, K., 1957, “On the damping of gravity waves over a permeable sea bed”, Trans. Am. Geophys. Union, 30: pp. 662-666.
32.Richards, K. J., 1980, “The formation of ripples and dunes on an erodible bed”, Journal of Fluid Mechanics, Vol. 99, pp.597-618.
33.Sayre, W. W., 1969, “Dispersion of Silt Particles in Open Channel Flow. ” J. Hydr. Div., ASCE, Vol. 95, 3, pp.1009-1038.
34.Selvadurai, A. P. S., 2000, Partial differential equations in mechanics, Berlin, New York, Springer.
35.Shields, A., Ott, W. P. and van Uchelen, J. C., 1936, Application of Similarity Principles, and Turbulence Research to Bed-Load Movement, California Institute of Technology, Pasadena.
36.Sleath, J. F. A., 1970, “Wave induced pressure in bed of sand”, J. Hydr. Div., ASCE, 96: pp. 367-378.
37.Sumer, B. M. and Bakioglu, M., 1984, “On the formation of ripples on an erodible bed”, Journal of Fluid Mechanics, Vol.144, pp.177-190.
38.Verruijt, A., 1969, Elastic Storage of Aquifers, Flow through Porous Media, DeWiest, R.J.M. (Ed.), Academic Press.
39.White, C. M., 1940, “The equilibrium of grains on the bed of an alluvial channel”, Proceedings of the Royal Society of London, Series A, Vol. 174, pp.332-338.
40.Yamamoto, T., Koning, H. L., Sellmeijer, H. and Hijum, E. V., 1978, “On the response of a poro-elastic bed to water waves”, Journal of Fluid Mechanics, Vol. 87, pp. 193-206.
41.李鴻源、楊德良、黃良雄、顏清連，1990，北區防洪設施跨河構造物及河道穩定與抽砂作業相互影響之研究，國立臺灣大學水工試驗所，研究報告第119號, 中華民國八十年六月。
42.郭遠錦，2011，水下長時間尺度之底床變形研究，國立臺灣大學土木工程學研究所博士論文。
43.江宜玲，1994，渠道底床變形之研究, 國立臺灣大學土木工程學研究所碩士論文。
44.江百祥，1996，多孔彈性明渠底床線性瞬時變形之研究，國立臺灣大學土木工程學研究所碩士論文。
45.謝平城，1999，軟孔隙彈性介質底床之流體動力反應，國立臺灣大學土木工程學研究所博士論文。
46.施宛平，1998，河川水面暨多孔彈性底床瞬時變形之研究，國立臺灣大學土木工程學研究所碩士論文。
47.宋長虹，1993，水波作用下多孔彈性底床動力反應之研究，國立臺灣大學土木工程學研究所博士論文。