最简燃烧模型的两类典型初值问题
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摘要
本文研究了最简燃烧模型的两类典型初值问题:最简Chapman-Jouguet燃烧模型的广义Riemann问题—初始束缚能的扰动、最简Zeldovich-von Neumann-D(?)ring燃烧模型的激波与化学反应区的相互作用问题。
首先,应用特征分析法,考察了最简Chapman-Jouguet燃烧模型的广义Riemann问题—初始束缚能的扰动。对该问题的研究能够反应燃烧波由于初始束缚能的变化而引起的不稳定性。利用文[65,77]所给出的熵条件,在物理平面(x-t平面)原点附近(t充分小),对流函数f(u)为凸和非凸两种情况,本文分别构造出了该广义Riemann问题的唯一解。对比广义Riemann解和相应的Riemann解的结构,可以观察到:对于大多数情况,初始束缚能扰动之后,Riemann解的结构是稳定的;而对于一些典型的情况,初始束缚能的扰动会引起Riemann解的结构发生本质的变化。例如,不同类型的爆轰波之间的相互转化,不同类型的爆燃波之间的相互转化。更重要地,爆轰波到爆燃波的转化,该现象曾在数值模拟[44]中出现;爆燃波到爆轰波的转换,它是著名的物理现象-DDT现象。
其次,应用细致的数学分析和特征分析,本文研究了最简Zeldovich-von Neumann-D(?)ring燃烧模型的激波和化学反应区的相互作用问题。对该问题的研究能够从内部机理了解激波对燃烧过程的影响。该工作分成两部分:第一部分,通过化学反应区中的特征分析,构造性地获得了激波和有限宽度爆轰区相互作用的全局解。分析全局解,可以发现在某些情况下,激波能加速反应阵面。同时,当反应速率趋于无穷大时,本文证明了该全局解的极限恰好是相应CJ燃烧模型激波和爆轰波相互作用的解。第二部分,同样通过化学反应区中的特征分析,构造性地获得了激波和有限宽度爆燃区相互作用的全局解,并且发现在某些条件下,反应区内部分正在燃烧的气体将在有限的时刻被熄灭。类似爆轰情形,本文证明了该全局解在反应速率趋于无穷大时的极限,恰好是相应CJ燃烧模型激波和爆燃波相互作用的解。
This dissertation is concerned with two kinds of initial value problems for the simplest combustion model. The first one is to consider the generalized Riemann problem for the simplest Chapman-Jouguet combustion model. The second one is to discuss the inteaction problem of shock and chemical reaction zone for the simplest Zeldovich-von Neumann-D(o|¨)ring combustion model.
First of all, with the method of characteristic analysis, we consider the generalized Riemann problem for the simplest Chapman-Jouguet combustion model—the perturbation on initial binding energy. The problem is able to exhibit the instability for combustion wave due to the perturbation on initial binding energy. Under the entropy conditions given by [65] and [77], the solutions are obtained constructively in a neighborhood of the origin (t is small enough) on the physical plane(x-t plane). We accomplish the discussion according to the convexity and nonconvexity of flux function. Comparing the structure of the solutions to the generalized Riemann problem and the corresponding Riemann problem, we obtain the following results. Combustion waves in the corresponding Riemann solutions are able to retain their forms after perturbation on initial binding energy. However, the perturbation may bring essential changes to the combustion waves for some typical cases, such as the transition between different types of detonation and the transition between different types of deflagration. Furthermore, we can observe two important phenomena. The one is the transition from detonation to deflagration, which also appears in the numerical solutions in [44]. The other is the transition from deflagration to detonation (DDT), which has been one of the core problems in gas dynamic combustion.
Second, with the refined mathematical analysis and characteristic analysis, we consider the interaction problem of shock and chemical reaction zone for the simplest Zeldovich-von Neumann-D(o|¨)ring combustion model. The problem helps us understand the influence of the shock on reaction processes on the basis of internal mechanism. Our work consists of two parts. In partⅠ, by analyzing the characteristics in chemical reaction zone, we obtain the global solutions of the interactions of shock and detonation zone of finite width constructively and find that the shock speeds up the reaction front in some cases. Moreover, we prove that the limits of the global solutions as reactive rate k tends to infinity are just the solutions of interactions of shock and detonation for the simplest Chapman-Jouguet combustion model. In partⅡ, the interactions of shock and deflagration zone of finite width are considered. The global solutions of the problem arc constructed by analyzing characteristics in the chemical reaction zone. We observe some burning gas in the reaction zone will be extinguished at a finite time in some cases. By studying the limits of the solutions as the reactive rate goes to infinity, we obtain that the limits are the solutions of the corresponding initial value problem for the simplest Chapman-Jouguet combustion model.
首先,应用特征分析法,考察了最简Chapman-Jouguet燃烧模型的广义Riemann问题—初始束缚能的扰动。对该问题的研究能够反应燃烧波由于初始束缚能的变化而引起的不稳定性。利用文[65,77]所给出的熵条件,在物理平面(x-t平面)原点附近(t充分小),对流函数f(u)为凸和非凸两种情况,本文分别构造出了该广义Riemann问题的唯一解。对比广义Riemann解和相应的Riemann解的结构,可以观察到:对于大多数情况,初始束缚能扰动之后,Riemann解的结构是稳定的;而对于一些典型的情况,初始束缚能的扰动会引起Riemann解的结构发生本质的变化。例如,不同类型的爆轰波之间的相互转化,不同类型的爆燃波之间的相互转化。更重要地,爆轰波到爆燃波的转化,该现象曾在数值模拟[44]中出现;爆燃波到爆轰波的转换,它是著名的物理现象-DDT现象。
其次,应用细致的数学分析和特征分析,本文研究了最简Zeldovich-von Neumann-D(?)ring燃烧模型的激波和化学反应区的相互作用问题。对该问题的研究能够从内部机理了解激波对燃烧过程的影响。该工作分成两部分:第一部分,通过化学反应区中的特征分析,构造性地获得了激波和有限宽度爆轰区相互作用的全局解。分析全局解,可以发现在某些情况下,激波能加速反应阵面。同时,当反应速率趋于无穷大时,本文证明了该全局解的极限恰好是相应CJ燃烧模型激波和爆轰波相互作用的解。第二部分,同样通过化学反应区中的特征分析,构造性地获得了激波和有限宽度爆燃区相互作用的全局解,并且发现在某些条件下,反应区内部分正在燃烧的气体将在有限的时刻被熄灭。类似爆轰情形,本文证明了该全局解在反应速率趋于无穷大时的极限,恰好是相应CJ燃烧模型激波和爆燃波相互作用的解。
This dissertation is concerned with two kinds of initial value problems for the simplest combustion model. The first one is to consider the generalized Riemann problem for the simplest Chapman-Jouguet combustion model. The second one is to discuss the inteaction problem of shock and chemical reaction zone for the simplest Zeldovich-von Neumann-D(o|¨)ring combustion model.
First of all, with the method of characteristic analysis, we consider the generalized Riemann problem for the simplest Chapman-Jouguet combustion model—the perturbation on initial binding energy. The problem is able to exhibit the instability for combustion wave due to the perturbation on initial binding energy. Under the entropy conditions given by [65] and [77], the solutions are obtained constructively in a neighborhood of the origin (t is small enough) on the physical plane(x-t plane). We accomplish the discussion according to the convexity and nonconvexity of flux function. Comparing the structure of the solutions to the generalized Riemann problem and the corresponding Riemann problem, we obtain the following results. Combustion waves in the corresponding Riemann solutions are able to retain their forms after perturbation on initial binding energy. However, the perturbation may bring essential changes to the combustion waves for some typical cases, such as the transition between different types of detonation and the transition between different types of deflagration. Furthermore, we can observe two important phenomena. The one is the transition from detonation to deflagration, which also appears in the numerical solutions in [44]. The other is the transition from deflagration to detonation (DDT), which has been one of the core problems in gas dynamic combustion.
Second, with the refined mathematical analysis and characteristic analysis, we consider the interaction problem of shock and chemical reaction zone for the simplest Zeldovich-von Neumann-D(o|¨)ring combustion model. The problem helps us understand the influence of the shock on reaction processes on the basis of internal mechanism. Our work consists of two parts. In partⅠ, by analyzing the characteristics in chemical reaction zone, we obtain the global solutions of the interactions of shock and detonation zone of finite width constructively and find that the shock speeds up the reaction front in some cases. Moreover, we prove that the limits of the global solutions as reactive rate k tends to infinity are just the solutions of interactions of shock and detonation for the simplest Chapman-Jouguet combustion model. In partⅡ, the interactions of shock and deflagration zone of finite width are considered. The global solutions of the problem arc constructed by analyzing characteristics in the chemical reaction zone. We observe some burning gas in the reaction zone will be extinguished at a finite time in some cases. By studying the limits of the solutions as the reactive rate goes to infinity, we obtain that the limits are the solutions of the corresponding initial value problem for the simplest Chapman-Jouguet combustion model.
引文
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