Exact Riemann solution for the Euler equations with nonconvex and nonsmooth equation of state
Exact Riemann solution for the Euler equations with nonconvex and nonsmooth equation of state
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具有非凸非光滑状态方程的欧拉方程的精确黎曼解
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
W. Dahmen
中科院分区：
文献类型：
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作者：
A. Voss;W. Dahmen
本文研究非凸非光滑状态方程的Euler方程的Riemann问题的解。主要目标是通过波曲线组合来开发精确的黎曼解，该波曲线组合考虑了物理相关效应，例如复合波或跨相边界的分裂波。一个经常被考虑的实验配置是激波管，其中流体的两种状态最初被隔膜分开。瞬时移除隔膜导致嵌入在管内的移动分离区域中的各种物理状态的发展，其中流体以特定方式被压缩或稀薄。在标准的设置，只有冲击波和稀疏波存在的情况下，所谓的BZT流体具有相空间中的负非线性区域。这导致了一种额外的波型，由激波和稀疏部分组成，或者膨胀激波仍然满足热力学第二定律。当我们另外考虑相变时，即，在我们的上下文中，从蒸气到液体的转变，或者反之亦然，可能发生进一步的非经典效应，例如在相边界处的波分裂。一般来说，激波管问题在时空平面上的解可以看作是波曲线在相空间中的投影。简而言之，每条波浪曲线收集了从其原点在某种意义上可以达到的状态。如果给出了初始数据，那么找到激波管问题的解意味着确定波曲线和中间状态，使得波曲线形成从一个初始状态到另一个初始状态的路径。一旦构造出波曲线，由于系统的标度不变性，激波管问题的解就一直是已知的。这里提到的关于激波管问题的解的大多数方面都是针对一般双曲守恒律方程组一对一地出现的。总之，本文的内容包括：（1）通过合成波解析构造激波管问题（Liu）熵解的构造原理，包括波曲线穿越相边界和负非线性区域的可能性，（2）构造原理中出现的附加合成波曲线类型的存在性和唯一性证明;（III）嵌入包含精确黎曼解算器的库中的算法，以数值方式计算激波管问题的解;（IV）对各种构型的几种数值格式的解与相应的精确解和问题的答案进行了比较，该方案是否能够捕捉到所有新的波现象，如波分裂。
This thesis is concerned with the solution of the Riemann problem for the Euler equations with nonconvex and nonsmooth equation of state. The main goal is to develop an exact Riemann solution by means of wave curve composition, which accounts for physically relevant effects such as composite waves or split waves across phase boundaries. A frequently considered experimental configuration is a shock tube, where two states of a fluid are initially separated by a diaphragm. Instantaneously removing the diaphragm leads to a development of various physical states embedded in moving separated regions inside the tube, where the fluid is compressed or rarefied in a specific manner. In contrast to the standard setting, where only shock waves and rarefaction waves exist, so-called BZT-fluids possess a region of negative nonlinearity in phase space. This leads to an additional wave type, composed of shock and rarefaction parts or to expansion shocks still satisfying the second law of thermodynamics. When we additionally take into account phase transitions, i.e., in our context transitions from vapor to liquid or vice versa, further non-classical effects such as wave splitting at the phase boundary may occur. In general the solution of a shock tube problem in space–time plane can be considered as a projection of wave curves in phase space. In short, each wave curve collects states which can be reached in a certain sense from its origin. If initial data is given, finding a solution to the shock tube problem means to determine the wave curves and the intermediate states, such that the wave curves form a way from one initial state to the other. Once the wave curves are constructed, the solution of the shock tube problem is known for all times due to the scale-invariance of the system. Here most of the mentioned aspects concerning solutions of a shock tube problem arise one–to–one for general hyperbolic systems of conservation laws. In summary, this thesis contains: (I) a construction principle in order to analytically construct the (Liu) entropy solution for the shock tube problem by composing waves, including the possibility of wave curves crossing phase boundaries and regions with negative nonlinearity; (II) an existence and uniqueness proof for the additional composite wave curve type occurring in the construction principle; (III) algorithms embedded in a library containing the exact Riemann solver to compute the solution for a shock tube problem numerically; (IV) comparisons of solutions of several numerical schemes for various configurations with the respective exact solution and the answer of the question, whether the schemes are capable to catch all new wave phenomena such as wave splitting.