应用积分方程法研究对流传热问题

发布时间：2019-06-29 19:57

【摘要】：许多自然现象和工程领域均涉及对流传热问题,由于传热介质结构的复杂性和实际问题中边界条件的多样性,我们一般无法得到对流传热问题的解析解,所以数值方法成为研究对流传热的重要途径。目前,研发高效率高精度的数值方法成为对流传热问题研究的一大热点。对于给定流场的对流传热现象,其控制方程为对流扩散方程,因此研发数值求解对流扩散方程的新算法具有重要的应用价值。本文基于级数形式的格林函数,将偏微分方程形式的对流传热方程转化为积分方程形式,再利用正交多项式的性质将其简化为常微分方程组方程组,数值求解该方程组可得对流扩散方程的数值解。作为一种无需网格划分的方法,积分方程法的精度主要取决于正交多项式的截断项数,经过实际应用发现,对于大多数算例,采用较少的截断项数,便可获得比较好的精度,从而避免了为增加精度而增加网格数量所带来的计算量和存储量的增加,因此相比于需要网格划分的方法,大大提高了计算效率。其次,该方法在求解非线性对流扩散方程时也展示了一定的优势。本文首先简要介绍了研究背景和国内外研究现状,并回顾了文中方法所需要的一部分数学知识。之后我们应用积分方程法分别求解了不同边界条件下的对流扩散方程,并与精确解进行了比较。这些数值实验表明积分方程法在求解对流扩散方程时具有高精度和高效率的优点。
[Abstract]:Many natural phenomena and engineering fields are involved in convective heat transfer. Because of the complexity of heat transfer medium structure and the diversity of boundary conditions in practical problems, we can not get the analytical solution of convective heat transfer problem, so numerical method has become an important way to study convective heat transfer. At present, the research and development of high efficiency and high precision numerical methods has become a hot spot in the study of convective heat transfer. For the convective heat transfer phenomenon of a given flow field, the governing equation is the convective diffusion equation, so it is of great application value to develop a new algorithm for solving the convective diffusion equation. In this paper, based on the Green's function in the form of series, the convective heat transfer equation in the form of partial differential equation is transformed into the form of integral equation, and then it is simplified into a set of ordinary differential equations by using the properties of orthogonal multinomial. The numerical solution of the convective diffusion equation can be obtained by solving the equations. As a method without grid division, the accuracy of integral equation method mainly depends on the number of truncated terms of orthogonal multinomial. Through practical application, it is found that for most examples, better accuracy can be obtained by using less number of truncated terms, thus avoiding the increase of computation and storage caused by increasing the number of grid in order to increase the accuracy, so compared with the method of grid division, The calculation efficiency has been greatly improved. Secondly, this method also shows some advantages in solving nonlinear convective diffusion equations. In this paper, the research background and research status at home and abroad are briefly introduced, and some mathematical knowledge needed by the method in this paper is reviewed. Then we use the integral equation method to solve the convective diffusion equations under different boundary conditions and compare them with the exact solutions. These numerical experiments show that the integral equation method has the advantages of high accuracy and high efficiency in solving convective diffusion equations.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.5;O551.3

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