气体动力学欧拉方程组流近似下的狄拉克激波和真空

发布时间：2017-12-28 02:32

本文关键词：气体动力学欧拉方程组流近似下的狄拉克激波和真空　出处：《云南大学》2016年博士论文　论文类型：学位论文

【摘要】：本文研究流体力学中的等熵欧拉方程组和非等熵欧拉方程组流近似下的集中和气穴现象,以及狄拉克激波和真空的形成。首先构造流扰动零压流的黎曼解,发现了参数化的狄拉克激波和常密度状态.证明了当流扰动消失时,参数化的狄拉克激波和常密度状态分别收敛到零压流的狄拉克激波和真空状态。然后在不同的压力律下构造带流扰动的等熵欧拉方程组和非等熵欧拉方程组的黎曼解,并分析了当包含压力的流扰动趋于零时,狄拉克激波和真空的形成机制。结论表明,不同的流近似对狄拉克激波和真空的形成具有不同的影响；狄拉克激波和真空对于流扰动是稳定的。这些研究结果进一步丰富了狄拉克激波和真空的理论。第一章介绍狄拉克激波的研究现状和本文的主要研究工作。第二章简要回顾零压流的黎曼解。第三章研究带流扰动的等熵欧拉方程组。首先,解流扰动零压流系统的黎曼问题,我们发现了参数化的狄拉克激波和常密度状态,并证明了当流扰动趋于零时,这两类解分别收敛到零压流的狄拉克激波和真空状态。然后,构造流扰动等熵欧拉方程组的黎曼解,并证明了当包含压力的流扰动趋于零时,任何二激波黎曼解收敛到了零压流的狄拉克激波解；任何二疏散波黎曼解趋于了零压流的真空解。最后,对上述极限过程进行数值模拟。第四章研究带流扰动的非等熵欧拉方程组。我们严格证明了当包含压力的流扰动消失时,带流扰动的非等熵欧拉方程组任何包含两个激波和可能的1-接触间断的黎曼解收敛到了零压流的狄拉克激波解；任何包含两个疏散波和可能的1-接触间断的黎曼解趋于零压流的真空解。数值结果和理论分析完全一致。第五章考虑带流扰动的等熵Chaplygin气体方程组.我们证明了包含压力的流扰动趋于零时,带流扰动的等熵Chaplygin气体方程组任何二激波黎曼解和任何参数化的狄拉克激波解集中到零压流的狄拉克激波解；任何二疏散波黎曼解趋于零压流的真空解.我们还证明了一种单参数流扰动消失时,带流扰动的等熵Chaplygin气体方程组任何满足特定初值的二激波黎曼解和任何参数化的狄拉克激波解收敛到等熵Chaplygin气体方程组的狄拉克激波解。数值结果与理论分析吻合。第六章考虑带流扰动的修正Chaplygin气体方程组。首先,我们证明了当包含压力的三参数流扰动消失时,带流扰动的修正Chaplygin气体方程组任何二激波黎曼解收敛到了零压流的狄拉克激波解；任何二疏散波黎曼解趋于了零压流的真空解。其次,我们证明了当一种二参数流扰动消失时,带流扰动的修正Chaplygin气体方程组部分二激波黎曼解收敛到了广义Chaplygin气体方程组的狄拉克激波解。再次,我们还证明了当一种单参数流扰动消失时,带流扰动的广义Chaplygin气体方程组部分二激波黎曼解和任何参数化的狄拉克激波解收敛到了广义Chaplygin气体方程组的狄拉克激波解。最后,对以上极限过程进行数值模拟。第七章研究带流扰动的零压流型系统。通过构造单参数流扰动零压流型系统的黎曼解,我们得到参数化的狄拉克激波和广义常密度状态,证明了当流扰动趋于零时,参数化的狄拉克激波和广义常密度状态分别趋于零压流型系统的狄拉克激波和真空状态。然后,求解二参数流扰动零压流型系统的黎曼问题,并证明了当包含压力的流扰动消失时,任何二激波黎曼解和任何二疏散波黎曼解分别收敛到零压流型系统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.
【学位授予单位】：云南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O35

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