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研究生： 盧韋勳
Lu, Wei-Shiun
論文名稱： 複數格網之應用-黏性圓柱流場之間隙效應分析
Application of complex-grid method to a viscous flow past cylinders of gap effects
指導教授： 唐啟釗
Tang, Chii-Jau
學位類別： 碩士
Master
系所名稱： 工學院 - 水利及海洋工程學系
Department of Hydraulic & Ocean Engineering
論文出版年： 2010
畢業學年度： 98
語文別： 中文
論文頁數： 105
中文關鍵詞： 複數級數解 、流函數-渦度模式 、有限解析離散法 、震盪流
外文關鍵詞： complex series solution, stream function and vorticity formulation, Finite-analytic method, oscillatory flow
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    • 本文以流函數-渦度模式求解Navier-Stokes方程組，可數值模擬二維、非穩態、不可壓縮之黏性流場並分析近壁鈍體間隙之影響。應用圓定理之複數級數解(Milne-Thomson (1968))來解析求得雙圓柱勢流場，進一步以此解建構可控制密度之正交數值格網，並作為計算時之初始條件，推算黏性流場。流場控制方程式採用有限解析(FA)離散法，並以三對角矩陣逐線超鬆弛疊代(LSOR)法，求得所有變數之收斂解。文中計算三種流況：(1)均勻流通過兩相近雙圓柱流況。(2)均勻流通過單一圓柱緊鄰半無限長平板邊壁流況。(3)微小振幅波引致震盪流經緊鄰底床圓柱流場的變化。依據不同之流況，適度調整格網密度，並討論不同之雷諾數、間隙比來探討其對流場的變化影響。

    With a streamfunction-vorticity numerical model for solution of the Navier-Stokes equations, this dissertation analyzed the gap effects between a blunt object and other solid boundary submerged in a two-dimensional unsteady incompressible viscous flow field. A complex series solution obtained by circle theorem (Milne-Thomson (1968)) was applied to obtain the analytic solution of potential flow for a uniform flow past two parallel circular cylinders. Meanwhile, this solution was not only used to construct a controllable orthogonal grid system but to specify an initial condition to calculate the corresponding viscous flow. The government equations after discretized by the finite-analytic (FA) method were sought for their convergent flow solutions by tridiagonal matrix method with linewise successive over-relaxation (LSOR) iterative algorithm. Three flow problems were then considered here:
    (1) A uniform flow past a pair of circular cylinders by side-by-side arrangement;
    (2) A uniform flow passing a circular cylinder near a plane wall; and
    (3) The oscillatory flow induced by small-amplitude waves around a circular cylinder near a plane wall.
    The suitable justification of grid resolution respectively for various flow problems thus enables us to investigate those variation effects of Reynolds numbers and gap ratios on the flow characteristics.

    目錄 中文摘要 I 英文摘要 II 誌謝 III 目錄 IV 圖目錄 VI 表目錄 VIII 符號說明 IX 第一章　緒論 1 1.1 概說 1 1.2 研究目的 2 1.2.1 複數格網解析法與差分計算幾何係數誤差比較 5 1.2.2 雙圓柱複數格網應用 10 1.3 本文概述 10 第二章　數學模式與相關條件 12 2.1 控制方程式 12 2.2 相關邊界條件 15 第三章　數值方法 17 3.1 格網生成及座標轉換 17 3.1-1 格網生成 17 3.1.2 複數解析幾何係數轉換 22 3.2 有限解析離散法 24 3.3 三對角矩陣求解 27 3.4 超鬆弛收斂法 29 3.5 複合格網計算 29 第四章 結果討論與分析 32 4.1 模式驗證 33 4.1.1 均勻流通過單一圓柱情形 33 4.1.2 均勻流通過雙圓柱之流場 38 4.1-3 均勻流通過圓柱緊鄰平板邊壁 41 4.2 低雷諾數雙圓柱黏性流場 44 4.2.1 低雷諾數穩定流況 46 4.2.2 低雷諾數非穩定流況 55 4.2.3 兩圓間隙流量之分析 60 4.2.4 間隙流量之阻塞效應 64 4.3 低雷諾數圓柱緊鄰邊壁黏性流場 66 4.3.1 低雷諾數穩定流流況 68 4.3.2 低雷諾數非穩定流流況 71 4.3.3 間隙比與流體通過柱壁面間流通量分析 78 4.4 微小振幅波浪引致震盪流對邊壁與圓柱流場之影響 83 4.4.1 底床速度剖面驗證 85 4.4.2 近底床圓柱受振盪流場渦流現象討論 87 第五章 結論 92 參考文獻 95 附錄一　模式因次關係 99 附錄二　座標轉換 101 附錄三　柱體上流函數之推導 103 自述 105 圖目錄 圖1.1 複數流網格點分佈情形 6 圖1.2 使用差分法與解析法計算格網轉換係數所得流函數誤差分佈圖 9 圖2.1 均勻流通過緊鄰邊壁之圓柱示意圖 15 圖3.1 雙圓柱案例之格網示意圖 20 圖3.2 調整後雙圓柱影像之複數級數解格網 21 圖3.3 不同解析度度格網交界示意圖 30 圖4.1 均勻流通過圓柱之驗證(Re=26) 32 圖4.2 各數值實驗結果比較 35 圖4.3 Bouard &Coutanceau (1980)對流場特性之分類 36 圖4.4 α與β現象之實驗數值對照 38 圖4.5 應用圓定理分析均勻流通過雙圓柱之勢流解流線圖 40 圖4.6 黏性流通過雙圓柱(Re=0.011，G/D=0.2) 40 圖4.7 黏性流通過雙圓柱之驗證(Re=0.011，G/D=0.2) 40 圖4.8 圓柱緊鄰邊壁間隙流量變化 42 圖4.9 黏性流通單圓柱緊鄰邊壁之驗證(Re=0.011，G/D=0.1) 42 圖4.10 計算雙圓柱流場格網佈置圖 44 圖4.11 雙圓柱流場計算流程圖 45 圖4.12 三種低雷諾數流況之分類情形 47 圖4.13 Re=30、20及10對應G/D=0.8、0.7及0.6之流線變化關係 52 圖4.14 Re=26在G/D=0.8、0.6、0.4、0.2之渦度分佈圖 54 圖4.15 Re=100在各間隙比下之渦度場變化 58 圖4.16 各種流況與間隙比和雷諾數之對應關係 59 圖4.17 Re=100各間隙比圓上流函數及流量時間發展歷程 62 圖4.18 Re=26 各間隙比圓上流函數及流量時間發展歷程 63 圖4.19 Re=1，G/D=0.2上圓起始圓上流函數發展歷程 65 圖4.20 g/D=0.5 圓柱緊鄰邊壁格網 67 圖4.21 黏性流(Re=26)通過緊臨邊壁圓柱之等值流線分佈 69 圖4.22 黏性流(Re=26)通過緊臨邊壁圓柱之等值渦度線分佈 70 圖4.23 黏性流(Re=100)通過緊臨邊壁圓柱之等值流線分佈 72 圖4.24 黏性流(Re=100)通過緊臨邊壁圓柱之等值渦度線分佈 73 圖4.25 黏性流(Re=100，g/D=1.5)等值渦度線完整週期變化 75 圖4.26 黏性流(Re=100，g/D=0.9)等值渦度線完整週期變化 76 圖4.27 不同g/D值影響圓後方觀察點之流函數時間歷程圖 77 圖4.28 圓上流函數時間歷程圖 81 圖4.29 Re=26、100； 圓上流函數時間發展歷程回歸 82 圖4.30 週期波所引致震盪流場與近底床柱體作用示意圖 83 圖4.31 波浪引致震盪流計算之流程 84 圖4.32 邊界層內速度剖面數值解與理論解之比較 86 圖4.33 等值渦線分佈(g=0.25,σt=π/2) 87 圖4.34 等值渦線分佈(g=0.25,σt=π) 88 圖4.35 等值渦線分佈(g=0.05,σt=π/2) 89 圖4.36 等值渦線分佈(g=0.05,σt=π) 90 圖4.37 昇力係數隨不同波浪相位演變 90 表目錄 表1.1 解析法及差分法最大誤差量 7 表4.1 勢能流與無滑移圓上流函數之比較 64 表4.2 勢能流與黏性流各間隙下流函數比較 78 表4.3 模擬不同柱體受震盪波流作用之無因次參數 85

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