【作者】 魏涛；

【导师】 许明田；

【作者基本信息】 山东大学 ， 热能工程， 2016， 博士

【摘要】 对流扩散方程和Navier-Stokes方程是流体力学中的两类基本方程,它们在物理、化学和工程中有着广泛的应用。在简化的情形下通过求解这两类方程还能得到解析的结果,但当遇到复杂问题时就很难求得解析解,这时数值求解成为一种非常有效的手段,于是构造精确、稳定和高效的数值方法成为研究这两类问题的重要内容。近来,无网格方法,特别是基于积分方程理论提出的精度高和计算量少的算法正引起一些学者的兴趣。本文提出了一种积分方程法用于数值求解对流扩散方程和Navier-Stokes方程,通过和其它方法的比较,该方法展现出了一些优势。在研究稳态对流扩散问题时,首先引入一个关于格林函数的拉普拉斯方程,将格林函数展开成级数形式并代入拉普拉斯方程,可将拉普拉斯方程变为一代数方程组,求解该方程组可得到级数形式的格林函数。利用格林函数的性质,可将对流扩散方程转化为积分方程。此时,再用相同的正交多项式把所要求解的未知量写成级数形式,然后利用多项式的正交性质,可把积分方程化为一个代数方程组,求解该方程组即可得到对流扩散方程级数形式的近似解。最后,借助Chebyshev多项式和Fourier级数,应用积分方程法求解了非齐次边界条件的一维对流扩散问题和齐次边界条件的二维对流扩散问题。在与有限体积法、有限元法和迎风差分法的比较中积分方程法都表现出了很高的精度,尤其在处理对流占优的对流扩散问题时更是展示出了很好的稳定性。对于非稳态对流扩散方程,由于与稳态情形相比,方程中多了时间变量,因此,需要考虑时间变量的离散方式。这里选用Crank-Nicolson方法对方程的时间变量进行离散,这种方法既保证了离散格式的简洁,又使结果达到了较好的精度。对时间变量离散后得到的方程可以看做是稳态形式的方程,这时构造与此方程相关的关于格林函数的拉普拉斯方程,并利用格林函数的性质,就可以将离散后的对流扩散方程转化为关于空间变量的积分方程。通过把格林函数和未知变量展开成级数形式,可进一步将积分方程简化为一个代数方程组。求解该方程组就可以得到用级数形式的有限和表示的非稳态对流扩散问题的近似解。在数值实验部分,用两个一维非稳态对流扩散问题和四个二维非稳态对流扩散问题检验了该方法。在一维问题中,给出的左右边界条件一侧为第一类边界条件,另一侧为第二类边界条件。在二维问题中,给出的既有对流占优的问题,也有非常数对流速度的问题,并将计算结果与多种方法进行了比较。在与变分多尺度方法的比较中,两种方法都在网格数较小时就达到了很高的精度,但积分方程法的精度(尤其是计算效率)要明显好于变分多尺度方法。在与有限体积元法的比较中,当网格数较少时,积分方程法的优势并不明显,但随着网格数的增加和时间步长的减小,积分方程法误差减小的速度显著快于有限体积元法。在求解流体力学中的Navier-Stokes方程时,如何处理速度-压力的耦合是一棘手的问题。通过比较现有的各种处理该问题的方法,并考虑到积分方程法的特点,本文将采用投影法处理Navier-Stokes方程中的速度-压力耦合问题。为了将投影法离散后的方程转化为积分方程,文中根据离散方程的形式分别引入了格林函数满足的拉普拉斯方程。然后利用格林函数的性质,将离散后的Navier-Stokes方程转化为积分方程。由于所研究的Navier-Stokes方程中速度场满足的是一般性的边界条件,并且投影法中的中间变量满足的是Neumann边界条件,所以为了计算的方便,统一采用了Chebyshev多项式对格林函数和未知变量进行展开。应用Chebyshev多项式的性质可以将所要求解的方程离散化为代数方程组,求解这些方程组就可以得到相应方程的解。最后,本文给出了一个算例用于检验该方法。计算结果表明与分数步法相比,积分方程法具有很好的精确性和收敛性,并且对于相同节点数,所用的CPU时间要明显少于分数步法。文中还专门讨论了周期边界条件下的对流扩散方程和Navier-Stokes方程。在这种边界条件下,由于可以采用Fourier级数对格林函数和未知变量进行展开,所以计算格式相对简单。这里也是首先构造格林函数的拉普拉斯方程,再利用格林函数的性质将对流扩散方程或者Navier-Stokes方程转化为积分方程,并进一步利用级数的正交性,将积分方程简化为一个常微分方程组,最后应用TVDRunge-Kutta方法对该方程组进行数值求解。计算结果表明,在求解对流扩散问题时,积分方程法的精度以及收敛性要好于局部间断Galerkin方法。在求解具有不同Reynolds数的Navier-Stokes方程时,积分方程法和投影法相比也有很高的精度。尤其对于高Reynolds数的不可压缩流动问题,积分方程法的优势更加明显。更多还原

【Abstract】 The convection-diffusion equation and Navier-Stokes equation are two basic equations in fluid dynamics. They have been widely applied in physics, chemistry and engineering. The analytical solutions of these equations are only available in some simple cases. Usually, it is difficult to get analytical solutions of the convection-diffusion equation and Navier-Stokes equation. Thus, numerical method becomes a powerful tool for solving these equations in domains with complex geometies. For the convection dominated convection-diffusion problems and the complex flows with high Reynolds number, it is of great value to construct accurate, stable and efficient numerical methods. Recently, the meshless methods, especially those based on the integral equation theory have been gaining interest for its high accuracy and reduction of computational costs. Encouraged by the successful use of these methods, in the present work, we attempt to propose an integral equation approach for solving the convection-diffusion equation and Navier-Stokes equation.When studying the steady state convection-diffusion equation, a Laplace equation about the Green’s function is firstly introduced. Then the Laplace equation is discretized into algebraic equations by expanding the Green’s function into series form. Solving this algebraic equation system and using the property of the Green’s function, the convection-diffusion equation can be transformed into integral equation. Expanding the unknown function with the same series and using its orthogonality, the integral equation is reduced into an algebraic equation system. Solving this algebraic equation system, we can obtain the approximate solution of the convection-diffusion equation. With the help of Chebyshev polynomial and Fourier series, the integral equation approach is used to approximate the one-dimensional convection-diffusion problems with nonhomogeneous boundary conditions and the two-dimensional convection-diffusion problems with homogeneous boundary conditions. The comparisons with the finite volume method, the finite element method and the upwind difference method show that the integral equation approach is more accurate and stable. The good stability is also illustrated by the convection-dominated convection-diffusion problems.In the unsteady state convection-diffusion equation, the time variable appears, so we need to consider the time discretization method. Here, we choose the Crank-Nicolson method for the time discretization, as it can not only give a simple discrete scheme, but also give rise to a good accuracy. Then by using the Green’s function of the Laplace equation in the series form, the convection-diffusion equation is transformed into an integral equation that is further converted into an algebraic equation system. Solving this algebraic equation system, we can get the approximate solution of the original problem in the series form. In the section of numerical experiments, two one-dimensional convection-diffusion problems and four two-dimensional convection-diffusion problems are used to examine this integral equation approach. For one-dimensional problem the Dirichlet boundary condition is prescribed on one side and the Neumann boundary condition on the other. In the two-dimensional cases, both convection-dominated convection-diffusion problem and convection-diffusion problem with non-constant convection velocity are considered. In comparison with the characteristic variational multiscale method, the integral equation approach is more efficient. And both methods can give rise to numerical solutions with a high accuracy even with small grid numbers. The comparison between the integral equation approach and the finite volume element method shows that the integral equation approach doesn’t have obvious advantage for the small number of grid points, but as increas’ng the number of the grid points and decreasing the time step, the integral eqation approach demosntrates a better accuacry. Furhermore, the convergence speed of the integral equation approach is much faster than that of the finite volume element method.The velocity-pressure linkage is one of the main difficulties for solving Navier-Stokes equation. Several mehtods are avaiable for dealing with this problem. Here, we choose the projection method to solve this problem by considering the feature of the integral equation approach. According to the form of the discrete equations given by the preject method, we give different Laplace equations about the Green’s function. By using the property of the Green’s function, the Navier-Stokes equation can be converted into integral equations. As the boundary condition of the Navier-Stokes equation is in the general form and the intermediate variable of the projection method gives a Neumann boundary condition, we expand the Green’s functions and the unknown functions by using the Chebyshev polynomial. With the help of the property of the Chebyshev polynomial, we can get three algebraic equation systems about the unknown functions. Solving these algebraic equation systems, we can obtain the numerical solution of the Navier-Stokes equation. Finally, the accuracy and performance of the integral equation approach are examined by solving an unsteady Navier-Stokes equation. In comparison with the fractional-step method it leads to more accurate results and takes less CPU-time.In this thesis we also investigate the convection-diffusion equation and Navier-Stokes equation with period boundary conditions. In this case, the Green’s function and the unknown function can be expanded in Fourier series form, so the discretization scheme is simpler than that for other boundary conditions. We also need to construct a Laplace equation about the Green’s function, and transform the convection-diffusion equation or the Navier-Stokes equation into the integral equation which is further reduced to the ordinary differential equations system. Then the TVD Runge-Kutta method is employed to numerically solve the equations system. In comparison with the local discontinuous Galerkin method the integral equation approach shows a better convergence, especially for the convection dominated and nonlinear convection-diffusion problems. Furthermore, the results given by the integral equation approach is much more accurate than that obtained by the projection method in simulating the Taylor vortices. Finally, the integral equation approach demonstrates a high accuracy in simulating the incompressible flows with high Reynolds number.更多还原