臺灣博碩士論文加值系統

本文主旨在於應用數值方法,來計算低雷諾數條件下,圓球於管流中所受阻力情形,並與其他學者的實驗及理論結果作驗證,藉以得到圓球表面所受之阻力與雷諾數及管徑的關係,最終並獲得一關係式,使其能夠充分描述圓球在流場中的所受的阻力與其他無因次參數間的關係。
在數值方法上，採用虛擬壓縮性法(pseudo-compressibility)並配合有限體積法(finite volume formulation)來計算不可壓縮流奈維爾-史托克方程之數值解,在空間離散方面,採用TVD2 雙曲型守恆率之高解析算則,時間離散方面,則採用LUSSOR隱式解法。
在流場計算上採用33*17,65*33,129*65三種不同的網格數目,並配合七種不同之管徑及六種不同的雷諾數,且利用Richardson 外插法,藉以得到一較高階準確的結果,最後並與其他的實驗結果、數值解及理論推出之解析解作比對,經比較後,其趨勢與結果均相當吻合,利用此最終所得之結果,經曲線擬合後,即可推導出與管徑及雷諾數相關的阻力方程式,而此式不但可以分析在有限管壁下物體的受力情形,亦可將邊界推廣至遠域之流場中,與傳統無限域之stokes’formula 來做比較,進而評估一般將計算域截斷於有限遠處所引起的誤差。

In this thesis, the drag force on the sphere moving constantly in a low-Reynolds-number pipe flow is investigated via numerical calculation. The incompressible Navier-Stokes equations are formulated in a pseudo-compressibility form. The numerical scheme makes use of a finite volume strategy and the numerical flux term are evaluated using the Total-Variation Diminishing (TVD) technology commonly applied to the compressible flow. Steady solution is obtained by marching (iterating) in time until the artificial time derivative of pressure term in the continuity equation drops to zero.
In the final calculation, six different Reynolds numbers (Re) ranging from 0.1 to 1 and 7 different pipe diameter and sphere diameter ratios (D/d) are selected. In each case, the drag force on the sphere is calculated and the results are compared with the existing approximate theoretical values derived from correcting the Stokes’ formula. Both results agree well in trends, the slight deviation is due to the fact that theoretical value were obtained based on the linearized Navier-Stokes equations (Stokes creeping-flow equations), while the fully nonlinear form of the Navier-Stokes equations are adopted in the present calculations. Finally, a least-squared regression technique is applied to collapse the calculated result into a single expression exhibiting the functional relationship between the drag force and the Reynolds numbers (Re), and the pipe- sphere diameter ratio parameter (D/d).

第一章緒論
1.1前言 ………………………………………………………………1
1.2文獻回顧 …………………………………………………………2
1.3研究動機 …………………………………………………………5
1.4研究方法 …………………………………………………………5
第二章統御方程式
2.1黏性不可壓縮流之統御方程式 …………………………………8
2.2含虛擬壓縮性之統御方程式……………………………………10
第三章數值方法
3.1格點產生簡介……………………………………………………16
3.2空間離散…………………………………………………………17
3.2.1二維非線性系統之TVD修正通量算則……………………17
3.3時間離散法………………………………………………………22
3.3.1 LUSSOR隱式時間解法……………………………………22
3.4邊界條件的處理…………………………………………………27
3.4.1壁面之邊界條件 ……………………………………………27
3.4.2對稱之邊界條件 ……………………………………………28
3.4.3管流上游入口處邊界條件 …………………………………28
3.4.4下游出口處之邊界條件 ……………………………………29
第四章計算結果分析與討論
4.1收斂條件…………………………………………………………30
4.2虛擬壓縮因子……………………………………………………31
4.3基本算例…………………………………………………………32
4.4圓球阻力的計算與比較分析……………………………………33
4.5圓球阻力與雷諾數、管徑之分析………………………………35
第五章結論與未來研究方向
5.1結論………………………………………………………………38
5.2未來研究方向……………………………………………………39
參考文獻 ……………………………………………………………40
附圖 …………………………………………………………………42
附表 …………………………………………………………………51

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