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研究生:劉英宏
研究生(外文):Liu, Ying-Hong
論文名稱:利用速度-渦度法解析三維黏性不可壓縮流場
論文名稱(外文):Velocity-Vorticity Method For The Three-Dimensional Incompressible Viscous Flows
指導教授:楊德良楊德良引用關係---
指導教授(外文):Young,Der-Liang
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:土木工程學系
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:1998
畢業學年度:86
語文別:中文
論文頁數:6
中文關鍵詞:三維奈維爾-司徒克斯方程式速度-渦度法有限元素法邊界元素法穴室流
外文關鍵詞:three-dimensionNavier-Stokes equationsvelocity-vorticity methodfinite element methodboundary element methodcavity flow
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本文利用速度(Velocity)-渦度(Vorticity)法搭配有限元素法(Finite
element method) ,邊界元素法(Boundary element method)及二階朗吉
庫特法(Runge-Kutta method)求解三維慣性座標和非慣性座標不可壓縮黏
性流場,文中以簡單之三維箱型穴室流(cavity flow)作為驗證程式的依
據。

在驗證過程中,先對雷諾茲數趨近於零的司徒克斯(Stokes)流場作分析,
確定收斂後,再對奈維爾-司徒克斯(Navier-Stokes) 方程式在中低雷諾
茲數流場作分析,以瞭解流場變化和渦度變化的關係。由驗證的題目中發
現邊界的黏滯力會引發渦度的改變,而渦度會擴散至整個流場中,驅動整
個流場變化,而流場變化又相繼改變渦度,如此相互反制影響。

在雷諾茲數100、400、1000的驗證中,雖本模式網格為,但對整個流場的
變化,均有全貌的表現。如再把網格增為,和Ku以及Goda的研究中之速度
場做比較時,發現整個速度分佈的趨勢是一致的。但此二人所用的網格數
較本模式多,可見得速度-渦度法對於流場物理特性的掌握,有其獨特的
一面。尤其在雷諾茲數400、1000,流場之非線性項之作用逐漸增強時,
本模式計算收斂性強,且在速度梯度變化劇烈之處,均有詳細的描敘,這
是因為速度-渦度法是直接計算渦度,而渦度是速度的一次微分,故能以
較少的網格數,來表現速度的劇烈變化。

在非慣性座標的驗證,如忽略向心力的作用之下,可由速度-渦度法中去
計算出科氏力效應的作用。初步驗證的結果,和運動方程式所表現的運動
行為是相吻合的。
In this study, we use the velocity-vorticity method, in
combination with finite element method, boundary element method
and Runge-Kutta method to solve the three-dimensional inertial
and non-inertial unsteady incompressible viscous flows. The
correctness of the model is verified by a simple driven cavity
flow by comparing with existing literature.

As far as flow verification is concerned, we first analyze
the Stokes flow of which Reynolds number tends to zero. After
convergence is performed, we then analyze the flows satisfying
the Navier-Stokes equations with low Reynolds numbers. In
computation, one can understand the relation between the change
of flow velocity and vorticity. By the process of verification,
we know the fact that the viscous force on the boundary induces
the generation of the vorticity, and interactively the
vorticity would diffus
e into the whole flows, and cause the variation of the flows.
However the variation of the flows also will change the
vorticity. Finally, they are influenced and coupled each other.

On verifying Reynolds numbers of 100, 400, 1000, although
the mesh of the model is , we obtain good performance of the
change of the whole flows. By increasing the mesh size to , in
comparison of the flows with the study cases of Ku and Goda, we
find that the trend of the flows is the same as their results.
However the mesh in their studies are much more than the model
of this study. So the velocity-vorticity method has shown much
better control in the physics of the flows and also the
accuracy in the proc
ess of computation than the velocity-pressure method. In
particularly, at Reynolds number 400, 1000,as the influence of
the non-linear terms is enchanced, the convergence of this
model is much faster and better, and it relates in detail of
the flows at the big variation of the velocity gradient.

In verifying the flows in the non-inertial coordinate. We
ignore the centrifugal force, and use the velocity-vorticity
method to compute the effect of Coriolis force. The result has
revealed more complex behavior of motion for the more general
Navier-Stokes equations with the consideration of external
forces, such as the Coriolis force.
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