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研究生:臧淞
研究生(外文):Sung Tsang
論文名稱:兩非平行板間之二維週期性暫態黏性流動
論文名稱(外文):The Periodic Two-Dimensional Radial Flow of Viscous Fluid Between Two Inclined Plane Walls with an Oscillating Source
指導教授:孫珍理
口試委員:黃振康林怡均
口試日期:2013-06-14
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:機械工程學研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2013
畢業學年度:101
語文別:中文
論文頁數:82
中文關鍵詞:擴散器噴嘴震盪源流阻抗徑向週期性流動非平行板相似變換魏爾斯特拉斯橢圓函數
外文關鍵詞:diffusernozzleoscillating sourceflow impedanceperiodic radial flowinclined plane wallssimilarity transformationWeierstrassian elliptic function
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本研究針對兩非平行板間,具黏性及不可壓縮性之二維週期性暫態黏性流動問題,既半週期為噴嘴流,另半週期為擴張流,利用similarity transformation,由物理模型 (Navier Stokes equations) 推導出一非線性ordinary differential equation (ODE) 及壓力方程式,再利用elliptic function求解流場結構,並透過值級分析探討壓力梯度與速度分布間之可能性關係。在後續分析部分,藉著類比AC電路概念得到流場流阻抗Zf,並利用流阻抗比評估暫態擴散器之流動特性。當中改變之參數有驅動源震盪角頻率、入出口長度比h/R、半角及Reynolds number Re。另一方面,我們利用壓降及體積流率Q,計算所需之平均功率,並探討平均功率隨各參數而改變之關係。由值級分析,得知當無因次中心速度F0夠大,且震盪角頻率夠小時,擴散器之壓力降將為負,這意味著無法在給定任一弦波壓降的情況下求解速度場,只能由已知速度反向求解壓力。由流阻抗Zf的觀點,我們得知當半角及Reynolds number Re增加時,流阻抗比具有兩種不同之變化趨勢。首先當震盪角頻率較低時,流阻抗比先隨半角或Reynolds number Re增加而下降,直到一最佳點後,流阻抗比才隨半角或Reynolds number Re的增加而上升;反之,在高震盪角頻率情況下,流阻抗比隨半角或Reynolds number增加而下降。接著當入出口長度比h/R上升或震盪角頻率上升時,流場將轉而由非穩態項所主導,且隨著非穩態項逐漸變強,流阻抗比將隨入出口長度比h/R上升或震盪角頻率上升而上升。此外,在改變入出口長度比h/R及震盪角頻率的過程中,我們還發現流阻抗比具有兩個平坦之區域,分別為當Womersley number (Wo) 小於0.1時的準穩態區及Womersley number (Wo) 大於10的非穩態區。另一方面,對於流場平均功率來說,我們發現在低震盪角頻率情況下,平均功率先隨半角增加而下降,直到一最佳點後,平均功率才隨半角增加而上升;反之在高震盪角頻率情況下,平均功率隨半角增加而上升。但是不管是在低或高震盪角頻率,若改變Reynolds number Re時,平均功率則是隨Reynolds number Re增加而上升。而對於入出口長度比h/R及震盪角頻率來說,擴散器之壓降則分別隨入出口長度比h/R上升而下降及隨震盪角頻率上升而上升,故平均功率隨入出口長度比h/R上升而下降及隨震盪角頻率上升而上升。


In this thesis, we employ similarity transformation and elliptic functions to solve the two dimensional radial flow of viscous fluid between two inclined plane walls with an oscillating source. Before the solving process, we carry out a scaling analysis to identify the allowable pressure gradient and maximum velocity. Moreover, we use the AC-circuit analogy to obtain the flow impedance Zf. By determining the ratio of flow impedance in opposite flowing direction, physical characteristics of the flow field are investigated. We also calculate the average power from instantaneous pressure drop Δp and flow rate Q. Influences of angular frequency w, slendness h/R, half angle a and Reynolds number Re in the ratio of flow impedance and average power are studied.
Through scaling analysis, we find that the pressure gradient across a diffuser must be positive when the angular frequency w is sufficiently small while the non-dimensional central velocity F0 is sufficiently large. As a result, there is a upper maximum of pressure drop Δp above which velocity solution cannot be found and we can only calculate pressure drop Δp for a given velocity oscillation. At low angular frequency w, there exists an optimal Reynolds number Re and half angle a that minimize the ratio of flow impedance. For Re < Reopt or a < aopt, the ratio of flow impedance decreases with Reynolds number Re or half angle a. For Re > Reopt or a > aopt, the ratio of flow impedance increases with Reynolds number Re or half angle a. In contrast, the ratio of flow impedance increases monotonically with both increasing half angle a and Reynolds number Re at high angular frequency w. As slendness h/R and angular frequency w augment (or Womersley number Wo), the effect of unsteady term amplifies. This leads to an increase in the ratio of flow impedance with both slendness h/R and angular frequency w. The influence of Womersley number Wo in ratio of flow impedance can be categorized into three regions: quasi steady region (Wo < 0.1), transition region (0.1 ≤ Wo ≤ 10) and unsteady region (Wo > 10). On the other hand, at low angular frequency w, there exists an optimal half angle a that minimize the average power. For a < aopt, the average power decreases with half angle a. For a > aopt, the average power increases with half angle a. In contrast, the average power increases monotonically with increasing half angle a at high angular frequency w. Moreover, for Reynolds number Re, the average power increases monotonically with increasing Reynolds number Re either high angular frequency w or low angular frequency w. As slendness h/R and angular frequency w augment (or Womersley number Wo), pressure drop Δp decreases with increasing slendness h/R while increases with increasing angular frequency w. This leads to a decrease in the average power with increasing slendness h/R and an increase in the average power with increasing angular frequency w.


目錄
摘要 i
abstract iii
符號索引 vii
圖表索引 x
第一章 緒論 1
1.1 前言 1
1.2 文獻回顧 1
1.3 研究動機與目的 6
第二章 流場統御方程式 7
2.1 方程式推導 7
2.2 主要函數之基本性質 9
2.2.1 判別式Δ (discriminant) 9
2.2.2 體積流率Q (flow rate) 16
2.3 值級分析 (scaling analysis) 18
第三章 壓力分布與速度分布 20
3.1 由已知壓力求解速度場 20
3.1.1 半角α對臨界壓降之影響 20
3.1.2 入出口長度比h/R對臨界壓降之影響 21
3.1.3 作動震盪角頻率ω對臨界壓降之影響 21
3.2 由已知速度求解壓力分布 22
3.3 邊界層分離 22
第四章 流阻抗 (flow impedance) 24
4.1 流阻 (flow resistance) 與流感抗 (flow inductance) 24
4.2 流阻抗比η (ratio of flow impedance) 25
4.2.1 半角α之影響 26
4.2.2 震盪角頻率ω之影響 29
4.2.3 入出口長度比h/R之影響 32
4.2.4 Reynolds number Re之影響 37
4.3 流阻抗比η反轉 40
第五章 功率 (power) 42
5.1 半角α之影響 42
5.2 入出口長度比h/R之影響 43
5.3 震盪角頻率ω之影響 45
5.4 Reynolds number Re之影響 46
第六章 結論 47
參考文獻 50


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