# 臺灣博碩士論文加值系統

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 在本研究中,吾人使用速度-渦度法(velocity-vorticity method)來得到一些著名問題之解析解,如柏傑斯漩渦(Burgers vortex),庫蒂流(Couette flow),海根-蒲休葉流(Hagen-Poiseuille flow),以及史托克斯第二問題(Stokes* second problem)等等.當然,控制方程式即是奈維爾-史托克斯方程式(Navier-Stokes equations),而其中的對流項(convective terms)也許可以拿掉,或者對時間的導數可以忽略,或是壓力梯度已經完全給定. 藉由渦度傳輸方程式(vorticity transport equations),不管壓力已知還是未知,速度和渦度場可以被決定.如此一來,不但可以顯示出速度-渦度法的效用而且能較方便地以數值方法來決定其他更複雜的流場. 同時,應該要強調的是吾人所得到的解析解與傳統的方法, 例如主要變數法(primitive variable method),所得到的是一樣的. 至於將來,則是希望利用速度-渦度法找出二維和三維更複雜流場的解析解,如方形穴室流(square cavity flows),或圓形穴室流(circular cavity flows),等.
 In this study, we make use of the velocity-vorticity method to obtain the exact solutions of some well-known problems, for instance the Burgers vortex, Couette flow, Hagen-Poiseuille flow, and Stokes* second problem, etc. Of course, the governing equations are, namely, the Navier-Stokes equations, for which the convective terms may be dropped out, or the time derivative can be neglected, or the pressure gradient is given exactly. By virtue of the vorticity transport equations, the velocity and vorticity fields can be determined regardless of whether the pressure is known or not. It not only shows the effect of the velocity-vorticity method but makes it convenient to determine other more complicated flow fields by numerical methods. Meanwhile, it should be emphasized that the solutions we obtained are exactly the same as those solved by the conventional methods, such as the primitive variable method. In the future, we are in expectation of finding exact solutions for more complex two dimensional and three dimensional flow fields by virtue of the velocity-vorticity method, such as square and circular cavity flows.
 CHAPTER 1. INTRODUCTION**************.1 1.1. Prelude**.******************.**.1 1.2. Motives and Purposes*.****************2 1.3. Literature Review ****************..*.*2 CHAPTER 2. THEORETICAL DERIVATION AND GOVERNING EQUATIONS****************5 2.1. The Navier-Stokes Equations**************..5 2.2. Derivation of Vorticity Transport Equations********..11 CHAPTER 3. SOME EXACT SOLUTIONS FOR NAVIER-STOKES EQUATIONS**********..***..19 3.1. The Burgers Vortex**.****************19 3.2. Couette Flow due to Pressure Gradient between a Fixed and a Moving Plate*************..31 3.3. Hagen-Poiseuille Flow*****************.39 3.4. Pulsating Flow between Parallel Surfaces****.*****..46 3.5. Steady Flow of a Viscous Fluid under Gravity Sliding down an Inclined Plane ************.52 3.6. Stokes* Second Problem******.**********..61 CHAPTER 4. CONCLUSIONS***************..68 REFERENCES*************..********...70
 劉英宏 ,"利用速度渦度法解析三維黏性不可壓縮流場", 國立台灣大學土木工程學研究所碩士論文,1998.Archeson, D. J. (1990). Elementary Fluid Dynamics , Oxford UniversityPress.Aris, R. (1989). Vector, Tensors, and the Basic Equations of FluidMechanics, Dover Publications, Inc..Batchelor, G. K. (1967) , An Introdution to Fluid Dynamics, CambridgeUniversity Press.Chorin, T. P. (1994) , Vorticity and Turbulence , Springer-Verlag.Courant, C., and John, F. (1965), Introduction to Calculus and AnalysisVolume 1, Springer-Verlag, New York.Currie, I. G. (1993) , Fundamental Mechanics of Fluids, MGoda, K. (1979) , A Multistep Technique with Implicit DifferenceSchemes for Calculating Two- or Three -Dimensional CavityFlows, J. Comput. Phys. , 30, pp. 76-95.Holmes, P. , Lumley, J. C. and Berkooz, G. (1996), Turbulence, CoherentStructures, Dynamical Systems and Symmetry, Cambridge UniversityPress.Ku, H. C. , Hirsh, R. S. , and Taylor, T. D. (1986) , A PseudospectralMethod for Solution of the Three - Dimensional IncompressibleNavier-Stokes Equations , J. Comput. Phy. , 70, pp. 439-462.Landau, L. D., and Lifshitz, E. M. (1959). Fluid Mechanics, J. B. Sykesand W. H. Reid, translators, Pergamon Press, Oxford, England.Liggett, J. A. (1994).Fluid Mechanics , McGraw-Hill, Inc. .Liu, Y. H. and Young, D. L. (1998). Velocity-Vorticity Method for theThree-Dimensional Incompressible Viscous Flows, Proceedings of the22nd National Conference on Theoretical and Applied Mechanics,Tainan, Taiwan.Logan, J. D. (1994). An Introduction to Nonlinear Partial DifferentialEquations. John Wiley & Sons, Inc..Munson, B. R., Young, D. F., and Okiishi, T. H. (1994). Fundamental ofFluid Mechanics, John Wiley & Sons, Inc..O*Neil, P. V. (1992). Advanced Engineering Mathematics, PWS-KENTPublishing Company.Smirnov, M. M. (1964).Second-Order Partial DifferentialEquations .translated by Scripta Technica Ltd. , King*sCollege,London.Sokolnikoff, I. S. (1964). Tensor Analysis . John Wiley & Sons, Inc..Trim, D. W. (1990). Applied Partial Differential Equations . PWS-KENTPublishing Company.
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 1 以基本解方法求解對流-擴散、柏格斯及奈維爾-史托克斯方程式 2 利用速度-渦度法解析三維黏性不可壓縮流場

 1 8、梁志輝：〈清代嘉義地區之社會變遷〉，《史聯雜誌》第23期，1993年11月，頁77-98。 2 5、賴子清：〈古今嘉義市詩文社〉，《嘉義市文獻》第3期，嘉義，嘉義市文獻委員會，1986年4月，頁37-43。 3 4、邱奕松：〈鄉賢錄〉，《嘉義市文獻》第1期，嘉義，嘉義市文獻委員會，1983年2月，頁120-125。 4 5、葉石濤：《臺灣文學史綱》，高雄，文學界雜誌社，1993年，再版。

 1 三物種競爭系統之行波解 2 速度渦度法配合有限元素法解析二維不可壓縮流場 3 論洛特卡-佛爾特拉方程組之解 4 明渠中不穩定現象-滾波之研究 5 一維變密度土石流模式 6 不恆定明渠流方程式之型式與穿臨界流 7 多顆粒泥砂躍移模式之研究 8 應用疊代近乎無偏估計法釐定觀測量方差之研究 9 台灣西南部GPS地殼變形監測網之靈敏度分析 10 含三維人造建物之數值影像正射糾正 11 小波理論於數值地形模型之多重解析度分析 12 平行子結構有限元素之網格分割最佳化研究 13 建築工程資訊管理物件架構之設計與實作 14 以三維CAD模型探討建築初步設計 15 工程合約相關資訊共享之研究

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