二维N-S方程的EFG数值模拟及其应用研究

发布时间：2019-04-02 18:33

【摘要】：Navier-Stokes方程(简称N-S方程)具有高度的非线性特性,几乎无法获得精确解,因此采用数值方法来求解N-S方程则成为了研究的热点。目前主要的数值方法有:有限元法、有限体积法和有限差分法等,但这些方法因存在网格易在迭代计算过程中出现网格畸变、计算不稳定、数值振荡等问题,从而限制了其计算精度以及效率的提高,因此为N-S方程的求解提供一种高效稳定的数值计算方法是十分必要的。无网格Galerkin(Element-free Galerkin,EFG)法是一种新型且仅需要节点信息的计算方法,它可构造出高阶的场函数,能摆脱不连续性对问题的约束,具有更高的光滑性等优势,因此本文采用分离式的EFG法来求解N-S方程,并对其工程应用进行研究,其主要内容如下:(1)通过采用罚因子法来处理N-S方程中的压力项,得到了二维N-S方程的EFG法的离散矩阵形式,通过算例验证了所提算法的可行性,探讨了高斯点个数、缩放系数以及流体雷诺数对计算结果的影响规律,并与有限元结果进行了对比分析,所得结果显示EFG法具有较好的收敛速度和计算精度。(2)探讨了EFG法中不同本质边界条件处理方式即拉格朗日乘子法(LMM)、罚函数法(PM)、耦合有限元法(FEM—EFG)对N-S方程计算结果的影响,并与有限元结果进行比较可知,LMM法的计算结果与有限元解的相对误差最小,而PM法结果的相对误差最大,FEM—EFG法的结果与LMM法的结果十分接近,同时探讨了PM法施加边界条件时罚因子的选取对计算精度的影响,所得结果显示调整罚因子的值可有效提高模拟计算精度。(3)针对翼型和弹丸流场中对流项(惯性项)会引起数值不稳定问题,将EFG法仅需要节点信息的优势与迎风稳定化算法相结合,构造了一种计算点的偏心支持域,并将其应用到NACA0012翼型和弹丸流场问题的模拟计算中,有效地避免了数值振荡现象,并提高了计算精度。本文采用EFG法和采用偏心支持域的EFG法对二维N-S方程进行了求解和分析,得到的结果表明EFG法具有计算稳定、精度高、收敛快等特点,所得结论对流体研究及其工程应用具有一定的理论参考和应用价值。
[Abstract]:The Navier-Stokes equation (abbreviated as Nus equation) has a high degree of non-linearity and can hardly obtain the exact solution. Therefore, the numerical method for solving the Nus equation has become a hot topic in the field of research. At present, the main numerical methods are: finite element method, finite volume method and finite difference method, but these methods have many problems such as grid distortion, computational instability, numerical oscillation and so on. Therefore, it is necessary to provide an efficient and stable numerical calculation method for solving the Navier-Stokes equation, which limits the accuracy and efficiency of the calculation. Meshless Galerkin (Element-free Galerkin,EFG) method is a new computing method which only needs node information. It can construct higher-order field functions, get rid of the constraints of discontinuity to the problem, and has the advantages of higher smoothness and so on. The main contents of this paper are as follows: (1) the penalty factor method is used to deal with the pressure term in the Navier-Stokes equation, and its engineering application is studied in this paper. The main contents are as follows: (1) the penalty factor method is used to deal with the pressure term in the Nus equation. The discrete matrix form of the EFG method for the two-dimensional Navier-Stokes equation is obtained. The feasibility of the proposed algorithm is verified by an example. The effects of the number of Gao Si points, the scaling coefficient and the Reynolds number of the fluid on the calculated results are discussed. Compared with the finite element results, the results show that the EFG method has good convergence rate and calculation precision. (2) the different methods of dealing with the essential boundary conditions in the EFG method, that is, Lagrangian multiplier method (LMM), are discussed. The effect of penalty function method (PM), coupled finite element method (FEM-EFG) on the calculation results of the Nus equation is compared with that of the finite element method. It is shown that the relative error between the LMM method and the finite element solution is the smallest. The relative error of PM method is the largest, and the result of FEM-EFG method is very close to that of LMM method. At the same time, the influence of penalty factor on the accuracy of calculation is discussed when PM method is applied with boundary conditions. The results show that adjusting the penalty factor can effectively improve the simulation accuracy. (3) the convective term (inertia term) in the airfoil and projectile flow field will cause the numerical instability problem. Combining the advantage of EFG method which only needs node information and upwind stabilization algorithm, an eccentricity support domain for calculating points is constructed and applied to the simulation of NACA0012 airfoil and projectile flow field, which effectively avoids the phenomenon of numerical oscillation. The calculation precision is improved. In this paper, the EFG method and the EFG method with eccentricity support domain are used to solve and analyze the two-dimensional Navier-Stokes equation. The results show that the EFG method has the characteristics of stable calculation, high precision and fast convergence. The conclusion has certain theoretical reference and application value for fluid research and its engineering application.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82

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