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研究生:鍾佑政
研究生(外文):Yow-Jeng Jong
論文名稱:尤拉方程數值離散先天困境之通化分析
論文名稱(外文):Generalized Genetic Analysis of Numerical Discretization Dilemma of Euler Equation
指導教授:莊書豪莊書豪引用關係李興軍李興軍引用關係
學位類別:博士
校院名稱:國立中興大學
系所名稱:機械工程學系所
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:2008
畢業學年度:96
語文別:英文
論文頁數:35
中文關鍵詞:數值離散尤拉公式伯努利公式
外文關鍵詞:Numerical Discretization DilemmaEuler EquationsBernoulli Equation
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尤拉方程式自西元1755年推導出來後,一直是流體分析的基本核心公式。對不可壓、非黏流性而言,此統御公式乍看似乎相當簡單,但除了在我們先前發表的論文“生與死…”和“揭開…”中提到作為動量方程式的嚴重純真性問題外,在數值計算時亦會產生嚴重的存在、唯一、穩定與振盪等問題。以擴管與縮管流為例,看似無辜的人工假設壁壓等於當地偏前之流向壓力,在尤拉公式推導過程中卻導致數值離散後與伯努利公式比較時的嚴重誤差,並使已經不出現在尤拉公式中的壁壓 ,在數值離散時卻仍陰魂不散地繼續進一步造成嚴重的存在、唯一、穩定及振盪問題,甚而違反基本能量原理。雖然有限元素法結合餘式配重法在處理固體力學相關之偏微分方程時,表現出強而有力的態勢,可是當碰到計算流體力學的偏微分方程問題時,卻遭遇極大的困難。因此近幾十年來計算流體力學學者常設計各種各樣求解不可壓流的專案補救之處理手法,諸如流線上風法、人工黏滯等,以期跳脫其純真性與數值解的困境。但這些權宜手段究竟並非正本清源之道,反而常造成治絲益棼的窘境。故本文為追根究柢,乃參照回歸高精準微積分計算,從基本面重新詳細檢視尤拉方程式的原始推導過程,並對照極簡的伯努利因式分解,以徹底解析尤拉公式在數值離散時造成嚴重困境之真正先天性原因。更有趣而重要的是:由於同樣的先天因素,不論對不可壓或可壓流,或是穩態/非穩態流而言,增加了黏滯項的耐威爾-史托克公式皆無法避免此一推導過程的致命性問題。
Euler equations derived in 1755 are among the most important equations for general fluid analysis. Although this fundamental governing equation looks simple at first glance for incompressible, inviscid fluid flow, nevertheless except for its critical nongenuiness as momentum equation discussed in our earlier “The life and death…” and “Unveiling…” papers[1,2], it also does lead to crucial numerical discretization dilemma related to existence, uniqueness, stability and oscillation problems, thus constitutes one of the most difficult problems in FD/CFD. Take both gradual pipe expansion and contraction flows for instances, in the process of deriving Euler equation, the seemingly innocent artificial equalization of crosswise and forward-biased streamwise pressures will give serious discretization error compared with Bernoulli equation, i.e. the wall pressure although already disappeared from Euler equation will still later ghostly (or more vividly say 陰魂不散地) bring about troublesome problems, and even violate the fundamental energy principle. Although finite element method together with method of weighted residuals constitute a powerful tool to solve PDE in solid mechanics, yet while coming to the scenarios of solving PDE in CFD, it turns out to be much difficult. To escape from this discretization and more important intrinsic nongenuiness dilemma, various kinds of ad hoc treatments such as streamline upwinding, artificial viscosity, and so on are devised awkwardly as make-shift measures in an attempt to somehow manipulate some energy dissipation and meanwhile almost inevitably contaminate the flow field. In this research, for the sake of finding the real culprit, we have to go back to the original high-precision calculus and re-scrutinize the original deriving process of Euler equation. Then we can insightfully compare the discretized Euler equation with factorized Bernoulli equation, in order to find the inherent true pathology leading to these serious numerical discretization dilemmas of Euler equation. More interestingly and importantly, due to like genetic nature, the associated Navier-Stokes equations with additional viscous terms are not immune to this kind of derivation fatality either, no matter for incompressible/compressible, steady/unsteady flows.
Abstract I
Table of Contents V
List of Figures VII

Chapter 1 Introductioon 1

 1.1 Backguound 1
 1.2 Thesis Outline 2

Chapter 2 Reviewing Derivation Process of Traditional Euler Equation 4

 2.1 Derivation Process of Traditional Euler Equation 4

Chapter 3 Inherent Discretization Dilemma Analysis of Euler Equation for Pipe Expansion Flow 8

 3.1 Discretization Dilemma Due to Assuming Equality of Crosswise and Forward-Biased Streamwise Pressures for pipe expansion Flow 8
 3.2 Discretization Dilemma Due to Assuming Equality of Crosswise and Backward-Biased Streamwise Pressures for Pipe Expansion Flow 11
 3.3 Discretization Analysis of Assuming Equality of Crosswise and Average Streamwise Pressures for Pipe Expansion Flow 14
 3.4 Deficiency of Differential Rule of General Calculus for Pipe Expansion Flow 17

Chapter 4 Inherent Discretization Dilemma Analysis of Euler Equation for Pipe Contraction Flow 20

 4.1 Discretization Dilemma Due to Assuming Equality of Crosswise and Forward-Biased Streamwise Pressures for Pipe Contraction Flow 20
 4.2 Discretization Dilemma Due to Assuming Equality of Crosswise and Backward-Biased Streamwise Pressures for Pipe Contraction Flow 23
 4.3 Discretization Analysis of Assuming Equality of Crosswise and Average Streamwise Pressures for Pipe Contraction Flow 26
 4.4 Deficiency of Differential Rule of General Calculus for Pipe Contraction Flow 28

Chapter 5 Conclusions 31

References 33
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2.Chiang, J. S, Chung, S. H., and Lee, H. J., Unveiling the Crucial Role of Impact Energy Loss for So-called Incompressible Fluid Flow, Journal of Marine Science and Technology, Vol. 12, No. 1, pp. 45-52, (2004).
3.Navier, Claude Louis M. H., Mémoirs de l’Académie des Sciences de l’Institut de France. t. VI, Paris (1822).
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9.Gresho, P. M., Sani, R. L. and Engelman, M. S., Incompressible Flow and the Finite Element Method, John Wiley, Chichester (1998).
10.Habashi, W. G. and Hafez, M. M., Computational Fluid Dynamics Techniques, Gordon and Breach Publishers, SA, p.367 (1995).
11.Jiang, B.-N., The Least-Squares Finite Element Method, Springer-Verlag, Heidelberg (1998).
12.Marchioro, C. and Pulvirenti, M., Mathematical Theory of Incompressible Nonviscous Fluids, (1994), Springer-Verlag, New York.
13.Turek, S., Efficient Solvers for Incompressible Flow Problems, Springer-Verlag, Berlin (1999).
14.Wendt, J. F., Computational Fluid Dynamics - an Introduction, Springer-Verlag, New York, p.6 (1992).
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16.Tokaty, G. A., A History and Philosophy of Fluid Mechanics, Dover Publications, New York, pp.73-80 (1971).
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