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研究生:林岳霆
研究生(外文):Yueh-TingLin
論文名稱:應用階梯近似法模擬黏性波於任意底床之散射
論文名稱(外文):Using Step Approximation for Simulating Viscous Wave Scattering over Arbitrary Topography
指導教授:許泰文許泰文引用關係
指導教授(外文):Tai-Wen Hsu
學位類別:博士
校院名稱:國立成功大學
系所名稱:水利及海洋工程學系碩博士班
學門:工程學門
學類:河海工程學類
論文種類:學術論文
論文出版年:2012
畢業學年度:100
語文別:中文
論文頁數:178
中文關鍵詞:黏性平面波近似法階梯近似法黏性效應底床黏著程度
外文關鍵詞:viscous plane wave approximationstep methodthe effect of molecular viscositybottom sliding coefficient
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本文主要是利用黏性平面波近似法(viscous plane wave approximation,VPWA)之概念與連續微小階梯連結積分方式,並結合包含壓力、剪力與表面張力之黏性線性自由液面邊界條件(free surface viscous boundary conditions)與可滑動底床邊界條件(slipping bottom boundary conditions),並以Navier Stokes equation為基礎建立VPWA模式,藉以模擬真實水波或者是具有黏性之波浪傳遞時與任意海底地形交互作用之變形效應,同時反射率與透過率的變化亦會被探討。文中亦就不同型態階梯近似法(step method)之理論基礎做一完整描述與比較。而不同型態階梯近似法諸如直接特徵函數展開法(DEMM)、轉換矩陣法(TMM)與間接特徵函數展開法(IEMM)等。然而階梯近似法中包含了前進模態(propagating mode)與振盪模態(evanescent mode)兩種模式,而本文僅考慮前進模態以簡化理論之解析。本文利用陳(1996)對真實流體一階線性解之解析,將黏性效應與底床黏著程度藉由Navier Stokes equation與自由表面條件、底床滑動條件的聯合求解來一併考慮至分散關係式中。然而在勢能流之情況在底床無剪力存在而且是光滑也是完全滑動之情況,本模式VPWA在不考慮黏性時可以完全退化為傳統之PWA(plane wave approximation),而且本模式亦可以完全描述任意底床上至自由液面處之完整流速剖面。接著模式驗證之部分,若以水波之邊界層內部來論述,本模式之水平速度式與Longuet-Higgins(1958) 與Phillips(1977)等人之解析一致。而且VPWA於邊界層內所計算之流速超射現象與林等人(1989、1995)之實驗數據呈現一致。另外,由VPWA所計算出之摩擦因子與雷諾數呈現線性關係,而且屬於層流邊界層之範疇,此與往昔實驗數據與經驗公式之結果一致。接著藉由VPWA探討真實流體或者是黏性流體的波動,在考慮目前已存在之黏性流體情況下,分別以純水、SAE-30機油與甘油為例,探討黏性波浪通過幾何形狀底床,如斜坡、潛堤、窪地等地形之交互作用,同時亦探討擾變底床之布拉格共振效應。由模擬結果中發現就理想流情況下本模式對於窪地計算之結果與Bender與Dean(2003) 以及Jung等人(2008)之解析結果相符。然而就布拉格共振效應而言,本模式之反射率計算結果與各階梯近似法一致,2k/K1.0處均發生主頻共振,唯次諧波共振2k/K2.0處,含振盪模態相較僅含前進模態可再給與修正。另外,就不同黏性流體而言,流體黏滯性與底床黏著程度增強時,反射率與透過率會受兩者之聯合阻滯效應而使其相對降低,此時能量亦相對耗損,而且黏性效應相對於底床效應對其之影響則是較為顯著。然而,由結果來看,底床黏著程度在深水時影響已漸式微,但淺水區時仍然具有一定程度之影響,但值得注意的是底床黏著程度的增加會對反射率與透過率的趨勢造成些微的相位偏移。

In this thesis, a viscous plane wave approximation (VPWA) based on integration method with sequent small steps is used to simulate viscous wave propagation over a arbitrary topography. The effect of bottom sliding coefficient and molecular viscosity on wave reflection and reflection coefficient was studied. Chen’s(1996) analytical solution is implemented in the present model to account for air pressure, shear stress, and surface tension of the dynamic free surface boundary conditions for solving the Navier-Stokes equation.
Different types of step method, such as IEMM(indirect eigenfunction matching method), TMM(transfer-matrix method) and DEMM(direct eigenfunction matching method) were used to describe viscous plane wave transformation. The numerical results were compared and discussed. Although the inclusion of evanescent modes in DEMM may raise the solution more accurate for wave scattering of potential flow, all evanescent modes are negligible in the theoretical formulation for simplicity.
The present theory describes the entire velocity profile from bottom to free surface for wave propagation over an arbitrary bottom for the real fluid. The formulation of the VPWA can be analytically reduced to the traditional formulation by Lamb when the bottom sliding coefficient and molecular are neglected. Similarly, the dispersion relation is simplified to that obtained from linear wave theory(LWT). Typical examples of waves of an ideal fluid, pure water, SAE-30 oil, and glycerin propagating over an uneven bottom were demonstrated by VPWA. The theory was validated by previous analytical solutions of Longuet-Higgins(1958) and Phillips(1977) for velocity profiles inside the boundary layer. Numerical results of velocity profiles with overshoot are in good agreement with laboratory measurements(Lin et al., 1989, 1995). Based on the calculated velocity profile gradient, the wave friction factor was obtained by VPWA as a function of Reynolds number which follows experimental data and empirical formula. The effect of molecular viscosity on wave scattering over trench of different shapes was investigated. Numerical results were compared fairly well with Bender and Dean (2003) and Jung et al. (2008). Wave energy loss becomes larger when the molecular effect increases. Bragg scattering of waves travelling over sinusoidal undulation was also studied using VPWA. Interesting results were found and compared with experimental data.

中文摘要 I
英文摘要 III
目錄 V
表目錄 VIII
圖目錄 X
符號說明 XIV
第一章 緒論 1
1-1 研究動機與目的 1
1-2 前人研究 3
1-2-1 波浪通過台階與斜坡底床之研究 6
1-2-2 波浪通過潛堤與窪地之研究 7
1-2-3 波浪通過沙漣與系列潛堤底床之研究 11
1-2-4 真實流體波動之相關研究 14
1-3 本文組織 17
第二章 理論基礎 19
2-1 控制方程式組 19
2-2 邊界條件 20
2-3 黏性分散關係式 22
2-4 波數受黏滯性與底床性質之影響 25
第三章 微小階梯連結技巧 28
第四章 模式驗證 37
4-1 波浪通過幾何形狀斜坡之驗證 37
4-1-1 Roseau斜坡 37
4-1-2 Porter斜坡 46
4-2波浪通過水平擾變底床之驗證 56
4-2-1 正弦形狀底床 56
4-2-2 複合式正弦底床 64
4-3波浪通過系列潛堤之驗證 73
4-3-1 拱型潛堤 73
4-3-2 系列潛體 78
4-4波浪通過幾何形狀窪地之驗證 86
4-4-1對稱型梯形窪地 86
4-4-2對稱型拋物線形窪地 93
4-4-3對稱高斯分佈型窪地 99
4-4-4 斜率不對稱型之梯形窪地 104
4-4-5前後水深不對稱型之梯形窪地 109
第五章 黏性波通過幾何地形之分析 116
5-1 水平底床之波形分析 116
5-2 反射率與透過率受黏性之影響 119
5-2-1斜坡底床 119
5-2-2拱型潛體 122
5-2-3梯形窪地 123
5-2-4擾變水平底床 125
5-3 能量耗損之分析 126
第六章 水波通過幾何地形之底床效應分析 131
6-1邊界層特性分析 131
6-1-1水平流速分析 131
6-1-2底床摩擦與剪力之分析 135
6-1-3水平流速受黏著度之影響 137
6-2反射率與透過率受底床效應之分析 140
6-2-1 斜坡底床 140
6-2-2 拱形潛堤 142
6-2-3 梯形窪地 143
6-2-4 擾變底床 145
6-3 能量損失 146
第七章結論與建議 150
7-1 結論 150
7-2建議 152
參考文獻 154
附錄 A 水深函數之推導 162
附錄 B 黏性分散關係式之推導 164
附錄 C 間接特徵函數法(IEMM) 168
附錄 D 轉換矩陣法(TMM) 174
附錄 E 直接特徵函數法(DEMM) 177


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