臺灣博碩士論文加值系統

數值分析時常常遇到格點取決的問題，當格點取少時，所得的結果誤差大，但是格點取多時所需之電腦計算時間較長或產生浮點運算誤差(floating point errors)，故本文採用非等距格點在性質變化區如速度、壓力梯度較小區域取較疏之格點，而在變化較大區採較密之格點，以期節省運算時間。
本文以自撰的FORTRAN程式(採用Patankar SIMPLER method數值方法) 以非等距(Non-Uniform Grid)的格點配置分析內流場中設置障礙物後之流速、流線、及迴流區域變化，其使了解不同障礙物分佈對平行板層流場所造成的影響，並與等距(Uniform Grid)的格點配置做一驗證及比較，結果證明非等距的格點配置能以較少的格點數及運算時間得到同等正確甚至更佳的結果。
本文首先分別以入口為完全發展流的解析解及入口為均勻流之經驗公式與本文之數值解作比較，皆可得到極佳結果，如此可以證明本程式在解決平板層流內流場的正確性。再控制不同的分佈變化率、阻尼因子去調配最佳格點，以最少的格點求得準確的解。舉例來說，本文在計算雷諾數為100的流場中，可減少約95%之格點，其節省的運算時間更達原等距格點之1/20，其提升的運算效率頗為驚人，故本文探討非等距格點有它一定的發展潛力。
障礙物對平行板層流場的影響，本文所求之流場分佈情況與相關文獻比較，可得到不錯的結果。由本文結果可得相同入口流量的之完全發展流或均勻流對流經相同障礙物後形成的迴流區流場無影響。不同障礙物高度及厚度對迴流區之影響經由模擬結果可發現障礙物高度對整體流場之影響遠大於厚度。不同障礙物分佈對流場的影響，可視為流體流經不同區域的變化情形，又因交錯式障礙物排列較為學者所接受，故本文探討交錯式障礙物排列對流場的影響。結果發現交錯式障礙物排列方式會對流場有加速的狀況產生。

關鍵字：雷諾數、非等距格點、障礙物

The selection of the total number of grids & grid size is one of the most important issues in the field of numerical analysis. A non-uniform grids approach with finer grids in the regions of greater property gradient is applied in an attempt of cutting down computation time.
A computer program written in FORTRAN using Patankar’s SIMPLER Algorithm with Non-Uniform grid set-up is developed to analyze the entrance region of a 2-D flow between parallel plates. Flow fields with obstructions are studied in order to understand the effects of different configurations to its velocity, pressure, & recirculation. Comparisons between Uniform & Non-Uniform Grid set-ups are made. It is found that Non-Uniform Grid set-up could use much less number of grid and save computation time.
Numerical results of this study are compared with those of analytical or empirical correlations in the entrance region in order to validate the accuracy of the program. Then the program is used to examine the entrance region with higher with Non-Uniform Grid. It is found, for example, a total saving of umbers of grid & computation time up to 95% at =100. Case studies of up to 2,000 is easily reached with non-uniform grid set-up in this study.
The effects of Obstructions Thickness & Weight to the flow field are studied. The effect of its height has much stronger effect to its flow field. Flow of arrays of staggered obstructions is analyzed and its effects to forced convection is positive & further studies to its energy equation are recommended.

Keywords：Reynolds Number、Non-Uniform Grids、Block

中文摘要…………………………………………………………………I
英文摘要……………………………………………………………… II
目錄…………………………………………………………………… IV
表目錄………………………………………………………………… VI
圖目錄…………………………………………………………………VII
符號說明…………………………………………………………………X
第一章 緒論…………………………………………………………… 1
1-1 前言…………………………………………………………… 1
1-2 研究動機……………………………………………………… 1
1-3 文獻回顧……………………………………………………… 2
第二章 理論分析與數值方法………………………………………… 3
2-1 CFD數值模擬方法簡介…………………………………………3
2-2 SIMPLER之理論基礎 ………………………………………… 5
2-2-1 基本假設………………………………………………5
2-2-2 統御方程式……………………………………………5
2-2-3 方程式無因次化………………………………………6
2-3 格點配置…………………………………………………… 6
2-3-1 交錯式格點於邊界上之配置…………………………7
2-3-2 非等格距參數…………………………………………7
2-4 SIMPLER之數值方法……………………………………………… 8
2-4-1 動量離散方程式…………………………………………… 8
2-4-2 壓力離散方程式……………………………………………10
2-4-3 SIMPLER求解步驟………………………………………… 11
2-5 加速收殮的方法………………………………………………… 11
2-6 邊界條件………………………………………………………… 12
2-7 障礙物邊界處理………………………………………………… 13
2-7-1具障礙物邊界條件解題步驟……………………………… 13
2-8 代數方程式求解方法…………………………………………… 14
2-9 誤差標準設定…………………………………………………… 14
第三章 結果與討論……………………………………………………15
3-1 Bench mark之測試…………………………………………15
3-1-1完全發展流測試………………………………………15
3-1-2 入口均勻流測試…………………………………… 16
3-1-3等距格點之標準建立…………………………………16
3-1-4 神奇的格子………………………………………… 17
3-1-5較高雷諾數測試………………………………………20
3-2 具障礙物分佈之等距格點與非等距格點研究與比較……22
3-2-1有障礙物分佈之流場格點……………………………22
3-3 障礙物造成分離流場之研究與比較………………………23
3-3-1 不同入口速度場之結果比較……………………… 23
3-3-2 單一障礙物厚度對迴流區之影響………………… 23
3-3-3 單一障礙物高度對迴流區之影響………………… 24
3-4 不同障礙物分佈對流場之影響之比較與研究……………24
3-4-1 週期性不對稱障礙物擺設對流場之影響………… 24
第四章 結論與展望……………………………………………………26
4-1 結論…………………………………………………………26
4-2 展望…………………………………………………………27
參考文獻……………………………………………………………… 28
附圖…………………………………………………………………… 30
附錄 A 二維動量離散方程式推導……………………………………50
B 二維壓力修正值離散方程式推導……………………………52
C 二維壓力虛擬值離散方程式推導……………………………54
D TDMA (TriDiagonal-Matrix Algorithm)
代數方程式推導………………………………………………56

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