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標題: 尤拉公式數值離散之先天困境分析
Inherent Numerical Discretization Dilemma Analysis of Euler Equation
作者: 遲毓熙
Chih, Yu-Hsi
關鍵字: Euler equation
出版社: 機械工程學系所
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摘要: 尤拉方程式自西元1755年推導出來後,兩百多年來一直是流體動力學的基本核心公式,對不可壓、非黏流而言,此一公式乍看似乎相當簡單,但數值計算時卻會產生嚴重的存在、唯一、穩定與振盪等問題。以擴管流為例,尤拉公式在推導過程不經意令壁壓等於當地之流向壓力,此舉相當於暗中作出無衝擊能損的假設,已導致理論與實驗的嚴重誤差;而在數值離散時,已經不出現在尤拉公式中的壁壓 ,卻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,可是當用於流體力學問題時,卻造成CFD學界極大的困難,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、人工壓縮、交錯格點、非等階元素.....等,以期跳脫數值解的困境,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,乃參照回歸自然的李-江尤拉公式,對照極簡的伯努利因式分解,從基本的系統動量及能量觀念,重新詳細檢視推導尤拉方程式的原始過程,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。
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, it does lead to serious numerical discretization dilemma related to existence, uniqueness, stability and oscillation problems. Take gradual pipe expansion flow for instance, in the process of deriving Euler equation, the intrinsic negligence of impact energy loss by seemingly innocent artificial equalization of crosswise and streamwise fluid pressure will lead to serious error compared with experimental data. Moreover, the already disappeared wall pressure from Euler equation will further ghostly bring about crucial existence, uniqueness, stability and oscillation problems, and even violate fundamental energy principle. Although finite element method together with method of weighted residuals constitute a powerful tool to solve PDE in solid mechanics, while solving PDE in CFD, it turns out to be much difficult. To escape from this dilemma, various kinds of ad hoc treatments such as streamline upwinding, artificial viscosity, artificial compressibility, staggered grid, non-equal-order elements and so on are devised awkwardly in an attempt to somehow manipulate some energy dissipation when dealing with incompressible fluid flow. Nevertheless, these make-shift compensating measures are not really solving problem from the root up, and also result to a rather contaminated flow field. In this research, for the sake of finding the true culprit, we refer the back-to-nature Lee-Chiang Euler equation, basic momentum and energy concepts, and the extremely concise Bernoulli equation factorization to re-scrutinize the original deriving process of Euler equation, in order to find the inherent true pathology leading to these serious numerical discretization dilemma of Euler equation.
其他識別: U0005-2107200612345900
Appears in Collections:機械工程學系所



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