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研究生:陳永昱
研究生(外文):Yung-Yu Chen
論文名稱:環度守恆法特性研究與擴散渦漩法之比較
論文名稱(外文):An Exploration of Properties of Circulation Conserved Scheme and its Comparison with Other Viscous Diffusion Models
指導教授:黃美嬌黃美嬌引用關係
指導教授(外文):Mei-Jiau Huang
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:2002
畢業學年度:90
語文別:中文
論文頁數:171
中文關鍵詞:渦漩方法無格點擴散速度環度守恆
外文關鍵詞:Vortex methodGrid freeDiffusion velocityCirculation conservation
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本論文將以實際的數值模擬來探討環度守恆法 (Circulation Conserved Scheme, CCS) 的基本性質,由於環度守恆法是對擴散渦漩法 (Viscous Diffusion Model, VDM) 的修正,故也將比較環度守恆法與擴散渦漩法的模擬結果。
本論文將測試環度守恆法的穩定性、收歛能力以及準確度。影響穩定性與收歛能力的變因有模擬中渦漩粒子的個數、渦漩粒子的密度以及環度守恆質面積的大小。在準確度方面,將以高斯分佈渦度場以及圓盤渦度場此二有解析解的軸對稱流場進行分析,並與 Leonard 渦泡方法及擴散渦漩法比較。
模擬的結果發現,雖然環度守恆法比之擴散渦漩法更準確,且二者的渦泡寬度都確實可以擴散地比 Leonard 渦泡方法慢,但當環度守恆法中考慮環度守恆的質面積過大時,此法將不穩定;解決的方法是縮小質面積。當質面積大小趨近於零時,模擬結果不但對準確度影響不大,而可以大大的改善穩定性,同時也大量減低了計算量。

This thesis investigates will discuss the basic properties about Circulation Conserved Scheme (CCS) by performing numerical simulations. Because CCS does the correction to Viscous Diffusion Method (VDM), results of the two methods will also be compared and discussed.
The stability, convergence and accuracy will be the issue of interest. Factors that dominate stability and convergence include the number and density of vortex particles, and size of material surface that conserves circulation in CCS. Two axisymmetrical flow fields, the Gaussian distributed and disk patch vorticity field, will be simulated and compared and discussed by using VDM, CCS and Leonard vortex blob methods.
Simulation results show that the growth of core sizes of vortex particles is slower in all methods employing the viscous diffusion velocity compared to Leonard's. The CCS is more accurate than VDM methods, but because less stable as the the size of material surface are large, choosing material surfaces of zero size can not only solve this problem, but also significantly reduces the computations, and does little harm on the accuracy.

1 緒論
2 數值方法
2.1 渦泡方法概論
2.2 Leonard 渦泡方法
2.3 VDM (擴散渦漩法)
2.4 CCS (環度守恆法)
2.4.1 環度在有限面積上的守恆
2.4.2 矩陣物理意義
2.4.3 環度在極小面積上的守恆
2.5 模擬程式的設計重點
3 渦度列測試
3.1 收歛測試
3.1.1 三顆渦泡
3.1.2 五顆渦泡
3.2 收歛測試之結果分析
3.2.1 渦泡數目
3.2.2 渦泡密度
3.3 \sigma 變化分析
3.4 結論
4 高斯分佈渦度場測試
4.1 分析方法
4.2 以 18 顆渦泡構成流場
4.2.1 初始條件
4.2.2 y_{{ij}_{max}} 參數影響下的收歛測試
4.2.3 \alpha 參數影響下的收歛測試
4.2.4 y_{{ij}_{max}}=6.0 時,\alpha 參數影響下的收歛測試
4.2.5 各種渦泡方法間的比較
4.3 以 36 顆渦泡構成流場
4.3.1 初始條件
4.3.2 y_{{ij}_{max}} 參數影響下的收歛測試
4.3.3 \alpha 參數影響下的收歛測試
4.3.4 y_{{ij}_{max}}=6.0 時,\alpha 參數影響下的收歛測試
4.3.5 各種渦泡方法間的比較
4.4 結論
5 渦度圓盤測試
5.1 流場說明
5.2 模擬設定
5.3 各種方法的比較
5.3.1 準確度
5.3.2 特徵寬度 \sigma
5.4 結論
6 結論與展望

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[5] J. Goodman, “Convergence of the random vortex method.” Commun. Pure Appl.
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[6] D.-G. Long, “Convergence of the random vortex method in two dimensions.” J. Amer.
Math. Soc., Vol. 1, 1988, 779-
[7] C. Greengard (1985), “The core spreading vortex method approxmates the wrong
equation.” Journal of Computational Physics 61, 345-348 (1985).
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that is both deterministic and convergent.” SIAM J. Sci. Comput. Vol. 17, No. 2,
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51
52 ¢Z¤
[10] P. Degond & S. Mas-Gallic (1989), “The weighted particle method for convectiondi
®usion equations. Part I: the case of an isotropic viscosity.” Mathamatics of Computation,
volume 53, number 188, October 1989, 485-507.
[11] D. Fishelov (1990), “A new vortex scheme for viscous flow.” Journal of Computational
Physics 86, 211-224 (1990).
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model — the di®usion velocity method.” Computers & Fluid Vol. 19, No. 3/4, 433-441,
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[15] 黃美嬌 (2002), “Circulation Conserved Diffusion Vortex Method”, 中國航空太空學會會刊, 第三十四卷, 第二期, 民國九十一年六月.

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