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研究生:謝蒼仁
研究生(外文):Tsang-Jen Hsieh
論文名稱:隱式逆擴散加權基本不震盪算則的發展及其應用
論文名稱(外文):THE DEVELOPMENT AND APPLICATIONS OF IMPLICIT ANTI-DIFFUSIVE WEIGHTED ESSENTIALLY NON-OSCILLATORY SCHEMES
指導教授:王興華
學位類別:博士
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:2009
畢業學年度:97
語文別:中文
論文頁數:201
中文關鍵詞:逆擴散WENO算則預調矩陣高解析算則
外文關鍵詞:anti-diffusive WENO schemespre-conditioned matrixhigh resolutuin scheme
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本文發展應用於可壓縮流之逆擴散加權基本不震盪算則,在時間離散方面,採用LU-SGS(Lower-Upper Symmetric-Gauss-Seidel)隱式解法以增強數值之穩定性及加速程式之收歛,同時亦採用多區塊有限體積網格法以彈性處理複雜幾何邊界問題。在空間離散方面,無黏性通量採用原始WENO (Weighted Essentially Non-Oscillatory)與逆擴散通量修正之WENO (Anti-diffusive WENO)高解析算則,黏性通量則採用中央差分法。紊流流場計算採用Spalart-Allmaras單方程式紊流模式。
本文採用數種不同之算則進行一系列可壓縮流的測試及驗證,算例包含了震波撞擊楔型體之繞射、第四型(type IV)的多震波干涉模擬、穿音速流經NACA 0012 及 RAE 2822 翼剖面、穿音速紊流流經ONERA M6 翼、超音速紊流流經三角翼之分析以及F16戰機在超音速流場之分析等,除了詳細比較原始WENO算則與逆擴散通量修正之WENO、逆擴散通量修正之Mapped WENO、逆擴散通量之修正平滑指示器(Modified Smoothness Indicator ; MSI) WENO算則等計算結果外,並與實驗值及其他數值分析結果比較。
一般可壓縮流數值分析方法應用於自由流馬赫數低於0.3時,數值收斂困難。因此,本文另一主要目的為擴展前述已發展成熟之可壓縮流程式,參考Weiss 及 Smith預調矩陣法,將原始WENO算則以及Anti-diffusive WENO算則轉化應用於預調矩陣系統,使得程式也可以有效的處理低速不可壓縮流的流場問題,期望所建構的分析工具可初步滿足未來氣動力設計需求。分析案例包含了二維低速拖曳方穴流場、三維低速高攻角(0.069馬赫、攻角27度)三角翼流場分析以及三維低速高攻角(0.1馬赫、攻角40度)三維戰機流場分析。
經由一系列的測試及驗證,顯示本文所採用之Anti-diffusive WENO算則除了具有良好的收斂性能與高階精度外,在相同的網格下對不連續接觸面的解析比原始WENO算則更為陡峭。在所有的測試案例中,不論是二維或三維,無黏性及紊流流場的模擬均能獲得精確的結果,並且與相關流場問題的數值解及實驗結果的比較也相當的吻合。
A class of lower-upper symmetric-Gauss-Seidel (LU-SGS) implicit anti-diffusive WENO schemes for solving the two- and three-dimensional compressible Navier-Stokes equations with Spalart-Allmaras one equation turbulence model is presented. Weighted essentially non-oscillatory spatial operator with and without anti-diffusive flux is employed for inviscid fluxes and central differencing for viscous fluxes. A numerical flux of WENO scheme in flux limiter form is adopted, which consists of first-order and high-order fluxes and allows for a more flexible choice of first order dissipative methods. Numerical experiments with several variants of the original WENO schemes, including anti-diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator WENO scheme are tested for the two- and three-dimensional inviscid/viscous flows. Computations of unsteady oblique shock wave diffraction over a wedge and steady flows over NACA 0012, RAE 2822 airfoils, ONERA M6 wing, delta wing and F16 fighter are presented to test and compare the methods.
It is known that most of the numerical algorithms developed for compressible flows are often inefficient or even inaccurate at low Mach numbers. Therefore, the other objective of this study is to extend the developed compressible flow solver for imcompressible flows computation. The Weiss-Smith preconditioned scheme is adopted and three computations are performed for validation, one is the two dimensional cavity flow, the second one is the three dimensional low subsonic flow over delta wing with high angle of attack and the other one is the three dimensional low subsonic flow over F16 fighter with high angle of attack.
By using the WENO scheme with anti-diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high order accuracy is maintained and in particular the contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. The present solutions are also compared with experimental data and other computational results and exhibit good agreement.
目錄
誌 謝 I
摘 要 II
ABSTRACT IV
目錄 VI
表目錄 X
圖目錄 XI
符號說明 XVIII
第 一 章 緒 論 1
1.1 引言 1
1.2 文獻回顧 5
1.3 研究目的 11
1.4 本文內容 12
第二章 統御方程式 15
2.1 守恆定律 15
2.2 二維尤拉方程式 17
2.3 三維尤拉方程式 19
2.4 二維雷諾平均奈維爾-史托克方程式 22
2.5 三維雷諾平均奈維爾-史托克方程式 25
第 三 章 高解析算則 27
3.1 ENO 算則 28
3.2 WENO 算則 34
3.3 差分式WENO算則 38
3.4 Mapped WENO算則 42
3.5 修正平滑指示器WENO5算則 43
3.6 逆擴散通量修正WENO5算則 44
第 四 章 預調式矩陣高解析度WENO算則 46
4.1 預調式矩陣 46
4.2 預調系統WENO3算則 49
4.3 預調系統WENO5算則 50
4.4 預調系統逆擴散通量修正WENO5算則 52
第 五 章 二維無黏性流場之應用 54
5. 1 統御方程式 54
5.2. 數值方法 55
5.2.1 空間離散(space discretization) 55
5.2.2 時間離散(time discritization) 55
5.3 邊界條件 58
5.4 初始條件 62
5.5 二維無黏性可壓縮流的應用 63
5.5.1 穿音速流經NACA 0012 翼剖面 64
5.5.2 穿音速流經RAE 2822 翼剖面 65
5.5.3 震波撞擊楔型體之繞射 67
5.6 結語 68
第 六 章 二維黏性紊流流場之應用 70
6.1 統御方程式 70
6.2 數值方法 71
6.2.1 空間離散(space discretization) 71
6.2.2 時間離散(time discretization) 72
6.3 邊界條件 72
6.4 結果與討論 73
6.4.1 次音速紊流流經NACA 0012 翼剖面 73
6.4.2 穿音速紊流流經RAE 2822翼剖面 76
6.4.3 第四型(type IV)多震波干涉模擬 78
6.5 結語 80
第 七 章 三維黏性紊流複雜外形流場分析 82
7.1 三維黏性可壓縮流之統御方程式 82
7.2 數值方法 82
7.2.1 時間離散法 83
7.3 三維黏性流場之應用 84
7.3.1 穿音速流流經ONERA M6翼 85
7.3.2 超音速流流經橢圓截面三角翼 86
7.3.3 F16戰機超音速流場之分析 88
7.4 結語89
第 八 章 預調式矩陣WENO算則應用於低馬赫數流場模擬 90
8.1 統御方程式 90
8.2 數值方法 90
8.2.1 時間積分—虛擬時步內部疊代法 91
8.3 邊界條件 92
8.4 預調式矩陣高解析度WENO算則之應用 93
8.4.1 正方柱背風面分離渦漩流場 93
8.4.2 三維低速高攻角三角翼流場 95
8.4.3 F-16戰機低馬赫數高攻角流場 96
8.5 結語 7
第 九 章 結論與展望 98
9.1 結論98
9.2 展望100
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