二维双曲守恒律方程的熵稳定数值方法研究
发布时间：2019-01-04 09:14
【摘要】：多维双曲守恒律问题是目前计算流体力学领域的重要研究内容之一。求解双曲守恒律方程的熵稳定数值格式具有较强的物理背景,能够有效地避免一些非物理现象的产生。本文详细研究了熵稳定格式的构造方法和主要思想,并根据熵稳定格式的构造方法构造三角形网格下二维双曲守恒律的熵稳定格式,并通过二维Euler方程圆柱绕流问题的数值模拟,说明了格式不会产生红斑现象。主要工作有:(1)通过对熵守恒、熵稳定、熵相容格式理论研究的基础上,直接将几种格式推广到二维问题,并对几个典型算例进行了计算,得到了较好的结果,表明这种推广是可行的。(2)直接在非结构网格下构造熵稳定格式。首先通过离散得到非结构网格下二维双曲守恒律方程的半离散数值格式,接着以二维Euler方程为例讲述在这个半离散格式的基础之上构造熵稳定格式的具体过程。先构造熵守恒通量函数,然后在此基础之上添加一个Roe型的耗散项,得到了熵稳定格式。由于它只具有一阶精度,为了改进格式的精度,本文采用在计算区域相邻单元的交界面上重构熵变量的方法构造出了一种新的高精度熵稳定格式,并对二维Euler方程圆柱绕流问题进行了数值模拟。(3)通过数值算例验证构造的三角形网格下熵稳定格式的有效性,体现其高精度、高分辨率等特点。
[Abstract]:Multidimensional hyperbolic conservation law problem is one of the important research contents in computational fluid dynamics. The entropy stable numerical scheme for solving hyperbolic conservation law equations has a strong physical background and can effectively avoid some non-physical phenomena. In this paper, the construction method and main idea of entropy stable scheme are studied in detail, and the entropy stable scheme of two dimensional hyperbolic conservation law under triangular mesh is constructed according to the construction method of entropy stable scheme. The numerical simulation of the flow around a cylinder of two dimensional Euler equation shows that the scheme will not produce erythema phenomenon. The main works are as follows: (1) based on the research of entropy conservation, entropy stability and entropy compatible scheme, several schemes are directly extended to two-dimensional problems, and some typical examples are calculated, and good results are obtained. It is shown that this generalization is feasible. (2) Entropy stable schemes are constructed directly under unstructured meshes. First, the semi-discrete numerical scheme of two-dimensional hyperbolic conservation law equation under unstructured grid is obtained by discretization. Then, taking the two-dimensional Euler equation as an example, the concrete process of constructing entropy stable scheme based on this semi-discrete scheme is described. First, the entropy conservation flux function is constructed, and then a dissipative term of Roe type is added on this basis, and the entropy stable scheme is obtained. Because it has only the first order precision, in order to improve the accuracy of the scheme, a new high precision entropy stable scheme is constructed by using the method of reconstructing entropy variables at the interface of the adjacent units in the calculation area. Numerical simulation of the flow around a cylinder of two-dimensional Euler equation is carried out. (3) A numerical example is given to verify the effectiveness of the entropy stability scheme constructed under triangular meshes, which embodies the characteristics of high precision and high resolution.
【学位授予单位】：长安大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
本文编号：2400090
[Abstract]:Multidimensional hyperbolic conservation law problem is one of the important research contents in computational fluid dynamics. The entropy stable numerical scheme for solving hyperbolic conservation law equations has a strong physical background and can effectively avoid some non-physical phenomena. In this paper, the construction method and main idea of entropy stable scheme are studied in detail, and the entropy stable scheme of two dimensional hyperbolic conservation law under triangular mesh is constructed according to the construction method of entropy stable scheme. The numerical simulation of the flow around a cylinder of two dimensional Euler equation shows that the scheme will not produce erythema phenomenon. The main works are as follows: (1) based on the research of entropy conservation, entropy stability and entropy compatible scheme, several schemes are directly extended to two-dimensional problems, and some typical examples are calculated, and good results are obtained. It is shown that this generalization is feasible. (2) Entropy stable schemes are constructed directly under unstructured meshes. First, the semi-discrete numerical scheme of two-dimensional hyperbolic conservation law equation under unstructured grid is obtained by discretization. Then, taking the two-dimensional Euler equation as an example, the concrete process of constructing entropy stable scheme based on this semi-discrete scheme is described. First, the entropy conservation flux function is constructed, and then a dissipative term of Roe type is added on this basis, and the entropy stable scheme is obtained. Because it has only the first order precision, in order to improve the accuracy of the scheme, a new high precision entropy stable scheme is constructed by using the method of reconstructing entropy variables at the interface of the adjacent units in the calculation area. Numerical simulation of the flow around a cylinder of two-dimensional Euler equation is carried out. (3) A numerical example is given to verify the effectiveness of the entropy stability scheme constructed under triangular meshes, which embodies the characteristics of high precision and high resolution.
【学位授予单位】：长安大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
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