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研究生:洪立昌
研究生(外文):Li ChangHung
論文名稱:利用速度渦度法求一些奈維爾-史托克斯方程式之解析解
論文名稱(外文):Some Analytic Solutions of the Navier-Stokes Equations by the Velocity-Vorticity Formulation
指導教授:楊 德 良
指導教授(外文):Der Liang Young
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:土木工程學研究所
學門:工程學門
學類:土木工程學類
論文種類:學術論文
論文出版年:1999
畢業學年度:87
語文別:中文
論文頁數:78
中文關鍵詞:速度-渦度法對流項渦度傳輸方程式奈維爾-史托克斯方程式解析解
外文關鍵詞:velocity-vorticity methodconvective termvorticity transport equationsNavier-Stokes equationsexact solutions
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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

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