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研究生:許朝敏
研究生(外文):Chao-Min Hsu
論文名稱:波流場中波浪變形之研究
論文名稱(外文):Wave Transformations over Varying Bottom Topography with Currents
指導教授:林銘崇林銘崇引用關係
指導教授(外文):Ming-Chung Lin
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:工程科學及海洋工程學研究所
學門:工程學門
學類:綜合工程學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:中文
論文頁數:116
中文關鍵詞:水流波浪變形布拉格反射布斯尼斯克方程式
外文關鍵詞:Boussinesq equationsBragg reflectioncurrentsWave transformations
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本文以理論及數值探討波浪、水流與地形變化三者間交互作用的現象,分別以變分及水深方向以多項式近似假設之方法推導波流場之控制方程式,此二種方法可以容易的將一複雜三維問題轉換成為二維問題。Isobe (1994) 運用變分原理推導一組波浪控制方程式,模擬波浪通過緩變地形之變化。本研究利用其方法,將水流對波浪的影響效應含括考慮,假設水流在鉛直方向變化不大,近乎均勻分佈 (uniform distributions) 之流場,藉由不同波浪運動水深分佈函數之假設,可使此方程式與其它現存描述波浪運動之方程式相契合。若能獲得波流場之水深方向分佈函數,則此方程式對於波流場內波浪變形的模擬,理應可以得到不錯的成果。
以數學操作另行推導出高階非線性含流之布斯尼斯克方程式(Boussinesq equations),與前人研究的不同處,在於推導的過程中,所有有關非線性參數(振幅與水深的比值) 的項不作任何的簡化;而對於分散性參數(水深與波長的比值)方面,僅保留至二階,其餘較高階有關分散性的項則被忽略,故此方程式分散性可以準確至二階,可以處理包含較強非線性及弱分散性的問題。依據 Schäffer and Madsen (1995) 的建議,此方程式可以擴展適用至相對水深較深的區域。
隨後以擴展型布斯尼斯克方程式 (extended Boussinesq equations) 數值模擬波浪通過潛堤之變化情形,結果以實驗值相互驗證。波浪通過潛堤時會衍生出高階諧和波,結果顯示有部分的能量從主頻波被轉移至高階諧和波上,而在非線性效應較強的條件下,更多能量會被轉移至高階諧和波。研究中也發現列發現順向流會增加諧和波之間能量的轉換,而逆向流則會減少諧和波之間能量的轉換。在數值計算中也發現,當波遭遇足夠強度之逆向流時,會產生波阻 (wave blocking) 的現象。
最後探討波流場中布拉格反射的現象,同時針對不同形狀沙漣底床(正弦、半正弦)、水流、沙漣數目與沙漣厚度對反射係數及接近共振時相對波長的位置作一系列之分析研究。結果顯示,由於線性假設的理論解對於此一非線性的現象無法正確的模擬,所以應用高階非線性布斯尼斯克方程式來模擬波浪、水流及地形變化,可以更精準的預估波浪的變形。
Wave propagating over various bottom topography with currents are examined theoretically and numerically. Two different theoretical methods are performed in the paper. The advantage of the two methods is easily to transfer a three-dimensional problem into two dimension. Isobe (1994) applied the method of variation to derive a set of equations describing wave propagating over the mild slope bottom topography with no currents. In this paper, the current effects are considered with the same method. The current is assumed to be uniformly distributed in vertical direction. Based on the assumption of wave functions in the vertical direction, various governing equations can be obtained. The derived equations should simulate well in the wave transformations, if the vertical distribution functions of waves in the wave-current field are well known.
A set of Boussinesq-type equations including current effects was also derived to examine wave transformations. Different from previous researcher, the nonlinear terms are not simplified during derivation. In the aspect of frequency dispersion (the water depth to wavelength ratio), the terms with orders higher than second order are neglected. Using the method proposed by Schäffer and Madsen (1995), the derived equations can be applied to a larger relative water depth.
The extended Boussinesq equations were used to study waves traveling over a submerged obstacle. The results were verified by physical experiments. Super harmonic waves are generated as waves passing the obstacles. The results show that the energy of primary waves is partially transferred to its super harmonics. The amount of energy transferred increases as wave nonlinearity gets stronger. It is also found the currents affect the amount of transferred energy. The energy transferred to super harmonics increases as wave interacting with follow flows and decreases as waves encountered adverse flows. The phenomenon of wave blocking was also clearly observed in numerical experiments.
Finally, the phenomena of the Bragg reflections were discussed. The effects of the shape of the sand ripples, current velocities, the number of the ripples and the thickness of the ripples on the reflection coefficients and relative wave-lengths were also investigated. A comparison between experimental, numerically and theoretical results indicates that the assumptions of linearity are not a sufficiently accurate basis for simulating the nonlinear variations. Strongly nonlinear equations should be used to predict the interactions among waves, ripples and currents.
中文摘要 ..I
英文摘要 .III
表 目 錄 ..V
圖 目 錄 .VI
符號說明 …XI
第一章 緒論 ...1
1-1 研究目的 …1
1-2 相關文獻回顧 …2
1-2-1 緩波方程式 …3
1-2-2 布斯尼斯克方程式 …4
1-2-3 波浪通過潛堤的變化 …8
1-2-4 布拉格共振反射現象 …9
1-3 本文組織 …..11
第二章 波流場中波浪變形控制方程式之推導 …13
2-1 基本假設 .…13
2-2 變分法推導波流場之控制方程式 .14
2-2-1 以變分法解析波流場 .14
2-2-2 控制方程式之變形與討論 .17
2-3 含流效應之高階非線性布斯尼斯克方程式推導 .21
2-3-1 控制方程式之推導 .21
2-3-2 分散關係式之討論 .26
第三章 擴展型布斯尼斯克方程式之探討之探討 …30
3-1 擴展型布斯尼斯克方程式基本物理特性 .30
3-1-1 分散關係式 .31
3-1-2 群波速 .33
3-1-3 淺化梯度 .34
3-2 數值方法 .35
3-2-1 方程式數值差分化 .35
3-2-2 邊界條件 .39
3-2-3 穩定條件 .40
3-2-4 邊界條件數值測試 .40
3-3 應用擴展型布斯尼斯克方程式分析波浪通過潛堤之波場變化 .41
3-3-1 潛堤及波浪物理特性條件 .42
3-3-2 計算結果與討論 .43
第四章 波流場中布斯尼斯克方程式方程式之數值驗證 …57
4-1 無流狀況下之數值驗證 .58
4-1-1 入射波邊界條件或造波邊界條件 (source function) .58
4-1-2 波浪通過潛堤之變形 .60
4-2 波流場之數值驗證 .61
4-2-1 等水深之波流交互作用情形 .61
4-2-2 高階諧和波與流之作用現象 .66
4-2-3 波浪通過潛堤與流之作用現象 .68
第五章 波流場中布拉格反射現象之研究 …76
5-1 前言 .76
5-2 存波時布拉格共振之數值計算 .76
5-2-1 正弦沙漣底床 .76
5-2-2 半正弦沙漣底床 .84
5-3 波流共存時布拉格共振之數值計算 .87
5-3-1 正弦沙漣底床 .89
5-3-2 半正弦沙漣底床 .96
第六章 結論與建議 ..107
6-1 結論 107
6-2 建議 109
參考文獻 ...110
謝 誌 ...116
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