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標題: 尤拉方程數值離散先天困境之通化分析
Generalized Genetic Analysis of Numerical Discretization Dilemma of Euler Equation
作者: 鍾佑政
Jong, Yow-Jeng
關鍵字: Numerical Discretization Dilemma
Euler Equations
Bernoulli Equation
出版社: 機械工程學系所
引用: 1.Lee, H. J., et al., “The Life and Death of Euler, Bernoulli, Navier-Stokes Equations and Associated CFD for So-called Incompressible Fluid Flow”, anticipated to appear. 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). 4.Stokes, G., On the Theory of the Internal Friction of Fluids in Motion, Transactions of the Cambridge Philosophical Society (1845). 5.Caughey, D. A. and Hafez, M. M., Frontiers of Computational Fluid Dynamics 1998, World Scientific, Singapore, p.172 (1998). 6.Anderson, J. D., Computational Fluid Dynamics, McGraw-Hill, Singapore, p.145 (1995). 7.Deconinck, H. and Koren, B., Euler and Navier-Stokes Solvers Using Multi-Dimensional Upwind Schemes and Multigrid Acceleration, Vieweg and Sohn, Wiesbaden (1997). 8.Gallavotti, G., Foundations of Fluid Dynamics, Springer-Verlag, Heidelberg (2002). 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). 15.Wesseling, P., Principles of Computational Fluid Dynamics, Springer-Verlag, Berlin (2001). 16.Tokaty, G. A., A History and Philosophy of Fluid Mechanics, Dover Publications, New York, pp.73-80 (1971).
摘要: 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.
其他識別: U0005-3107200812281900
Appears in Collections:機械工程學系所



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