本文關(guān)鍵詞：氣體動(dòng)力學(xué)歐拉方程組流近似下的狄拉克激波和真空　出處：《云南大學(xué)》2016年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 等熵歐拉方程組 非等熵歐拉方程組 零壓流 零壓流型系統(tǒng) Chaplygin氣體 廣義Chaplygin氣體 修正Chaplygin氣體 黎曼問(wèn)題 狄拉克激波 真空 流近似 數(shù)值模擬

【摘要】：本文研究流體力學(xué)中的等熵歐拉方程組和非等熵歐拉方程組流近似下的集中和氣穴現(xiàn)象,以及狄拉克激波和真空的形成。首先構(gòu)造流擾動(dòng)零壓流的黎曼解,發(fā)現(xiàn)了參數(shù)化的狄拉克激波和常密度狀態(tài).證明了當(dāng)流擾動(dòng)消失時(shí),參數(shù)化的狄拉克激波和常密度狀態(tài)分別收斂到零壓流的狄拉克激波和真空狀態(tài)。然后在不同的壓力律下構(gòu)造帶流擾動(dòng)的等熵歐拉方程組和非等熵歐拉方程組的黎曼解,并分析了當(dāng)包含壓力的流擾動(dòng)趨于零時(shí),狄拉克激波和真空的形成機(jī)制。結(jié)論表明,不同的流近似對(duì)狄拉克激波和真空的形成具有不同的影響；狄拉克激波和真空對(duì)于流擾動(dòng)是穩(wěn)定的。這些研究結(jié)果進(jìn)一步豐富了狄拉克激波和真空的理論。第一章介紹狄拉克激波的研究現(xiàn)狀和本文的主要研究工作。第二章簡(jiǎn)要回顧零壓流的黎曼解。第三章研究帶流擾動(dòng)的等熵歐拉方程組。首先,解流擾動(dòng)零壓流系統(tǒng)的黎曼問(wèn)題,我們發(fā)現(xiàn)了參數(shù)化的狄拉克激波和常密度狀態(tài),并證明了當(dāng)流擾動(dòng)趨于零時(shí),這兩類解分別收斂到零壓流的狄拉克激波和真空狀態(tài)。然后,構(gòu)造流擾動(dòng)等熵歐拉方程組的黎曼解,并證明了當(dāng)包含壓力的流擾動(dòng)趨于零時(shí),任何二激波黎曼解收斂到了零壓流的狄拉克激波解；任何二疏散波黎曼解趨于了零壓流的真空解。最后,對(duì)上述極限過(guò)程進(jìn)行數(shù)值模擬。第四章研究帶流擾動(dòng)的非等熵歐拉方程組。我們嚴(yán)格證明了當(dāng)包含壓力的流擾動(dòng)消失時(shí),帶流擾動(dòng)的非等熵歐拉方程組任何包含兩個(gè)激波和可能的1-接觸間斷的黎曼解收斂到了零壓流的狄拉克激波解；任何包含兩個(gè)疏散波和可能的1-接觸間斷的黎曼解趨于零壓流的真空解。數(shù)值結(jié)果和理論分析完全一致。第五章考慮帶流擾動(dòng)的等熵Chaplygin氣體方程組.我們證明了包含壓力的流擾動(dòng)趨于零時(shí),帶流擾動(dòng)的等熵Chaplygin氣體方程組任何二激波黎曼解和任何參數(shù)化的狄拉克激波解集中到零壓流的狄拉克激波解；任何二疏散波黎曼解趨于零壓流的真空解.我們還證明了一種單參數(shù)流擾動(dòng)消失時(shí),帶流擾動(dòng)的等熵Chaplygin氣體方程組任何滿足特定初值的二激波黎曼解和任何參數(shù)化的狄拉克激波解收斂到等熵Chaplygin氣體方程組的狄拉克激波解。數(shù)值結(jié)果與理論分析吻合。第六章考慮帶流擾動(dòng)的修正Chaplygin氣體方程組。首先,我們證明了當(dāng)包含壓力的三參數(shù)流擾動(dòng)消失時(shí),帶流擾動(dòng)的修正Chaplygin氣體方程組任何二激波黎曼解收斂到了零壓流的狄拉克激波解；任何二疏散波黎曼解趨于了零壓流的真空解。其次,我們證明了當(dāng)一種二參數(shù)流擾動(dòng)消失時(shí),帶流擾動(dòng)的修正Chaplygin氣體方程組部分二激波黎曼解收斂到了廣義Chaplygin氣體方程組的狄拉克激波解。再次,我們還證明了當(dāng)一種單參數(shù)流擾動(dòng)消失時(shí),帶流擾動(dòng)的廣義Chaplygin氣體方程組部分二激波黎曼解和任何參數(shù)化的狄拉克激波解收斂到了廣義Chaplygin氣體方程組的狄拉克激波解。最后,對(duì)以上極限過(guò)程進(jìn)行數(shù)值模擬。第七章研究帶流擾動(dòng)的零壓流型系統(tǒng)。通過(guò)構(gòu)造單參數(shù)流擾動(dòng)零壓流型系統(tǒng)的黎曼解,我們得到參數(shù)化的狄拉克激波和廣義常密度狀態(tài),證明了當(dāng)流擾動(dòng)趨于零時(shí),參數(shù)化的狄拉克激波和廣義常密度狀態(tài)分別趨于零壓流型系統(tǒng)的狄拉克激波和真空狀態(tài)。然后,求解二參數(shù)流擾動(dòng)零壓流型系統(tǒng)的黎曼問(wèn)題,并證明了當(dāng)包含壓力的流擾動(dòng)消失時(shí),任何二激波黎曼解和任何二疏散波黎曼解分別收斂到零壓流型系統(tǒng)的狄拉克激波解和真空解。
[Abstract]:In this paper, we study the centralization and cavitation phenomena of the isentropic Euler equations in fluid mechanics and the non isentropic Euler equations, and the formation of Dirac shock and vacuum. First, we construct the Riemann solution of the flow disturbed zero pressure flow, and find the parameterized Dirac shock and the constant density state. It is proved that when the flow disturbance vanishes, the parameterized Dirac shock and the constant density state converge to the Dirac shock and the vacuum state of the zero pressure flow respectively. Then we construct the Riemann solution of the isentropic Euler equations and the non isentropic Euler equations under different pressure laws, and analyze the formation mechanism of Dirac shock and vacuum when the flow disturbance contains pressure. The results show that different flow approximations have different effects on the formation of Dirac shock wave and vacuum, and the Dirac shock wave and vacuum are stable for the flow disturbance. The results of these studies further enrich the theory of Dirac's shock wave and vacuum. The first chapter introduces the research status of Dirac shock wave and the main research work of this paper. In the second chapter, the Riemann solution of zero pressure flow is briefly reviewed. In the third chapter, the isentropic Euler equations with flow disturbance are studied. First, the Riemann problem of the zero pressure flow system disturbed by the solution is found. We find the parameterization of the Dirac shock and the constant density state, and prove that when the flow disturbance tends to zero, the two solutions converge to the Dirac shock and the vacuum state of the zero pressure flow respectively. Then, we construct the Riemann solution of the isentropic Euler equations of flow disturbance, and prove that when the flow disturbance with pressure tends to zero, any two shock Riemann solution converges to the Dirac shock solution of the zero pressure flow. Any two evacuation wave Riemann solution tends to the zero pressure flow vacuum solution. Finally, the numerical simulation of the above limit process is carried out. In the fourth chapter, the non isentropic Euler equations with flow disturbance are studied. We strictly prove that when the pressure flow disturbance disappears, Dirac shock with non turbulent entropy Euler equations contain two shock and any possible 1- Riemann contact discontinuity solution converges to zero pressure flow solution; any contains two possible 1- rarefaction wave and contact discontinuity solution to Riemann vacuum zero pressure flow solution. The numerical results are in perfect agreement with the theoretical analysis. The fifth chapter take the isentropic gas flow Chaplygin equations. We prove that the pressure flow disturbance tends to zero, with turbulent Dirac shock entropy Chaplygin gas equations of any two shock Riemann solution and any parametric solution to Dirac shock zero pressure flow solution; any two rarefaction wave Riemann solution the vacuum tends to zero pressure flow solution. We also prove that a single parameter flow disturbance disappears, with turbulent Dirac Dirac Riemann two shock shock shock entropy Chaplygin gas equations satisfy the specific solutions and any initial value of any parametric solutions converge to the entropy of Chaplygin gas equations. The numerical results are in agreement with the theoretical analysis. In the sixth chapter, the modified Chaplygin gas equation group with flow disturbance is considered. First, we prove that when the three parameter flow disturbance containing pressure is disappearing, the Riemann shock solution of any two shock wave converges to the Dirac shock solution of the zero pressure flow, and any two evacuation wave Riemann solution tends to the vacuum solution of the zero pressure flow when the flow perturbation of the modified Chaplygin gas equation vanishes. Secondly, we prove that when a two parameter flow disturbance vanishes, the partial two shock Riemann solution of the modified Chaplygin gas equations with flow perturbation converges to the Dirac shock solution of the generalized Chaplygin gas equations. Thirdly, we also prove that when a single parameter flow perturbation vanishes, the partial two shock Riemann solution and any parameterized Dirac shock solution of the generalized Chaplygin gas equations with flow perturbation converge to the Dirac shock solution of the generalized Chaplygin gas equations. Finally, the numerical simulation of the above limit process is carried out. In the seventh chapter, the zero pressure flow pattern system with flow disturbance is studied. Riemann zero pressure flow disturbance solution of the system by constructing a single parameter flow, we obtain the parameters of the shock wave and the generalized Dirac density, prove that when the flow tends to zero, the shock wave and the generalized parametric Dirac constant density state tends to zero pressure flow system were Dirac shock and vacuum state. Then, the Riemann problem of the two parameter flow perturbation zero pressure flow system is solved. It is proved that when the flow disturbance containing pressure is disappearing, any two shock Riemann solution and any two evacuation wave Riemann solution converge to the Dirac shock solution and the vacuum solution of the zero pressure flow type system respectively.
【學(xué)位授予單位】：云南大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O35

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