【作者】 蒋光彪；

【导师】 何永森；

【作者基本信息】 湘潭大学 ， 计算数学， 2009， 博士

【摘要】 对紊流的数值仿真是流动、传热传质中最基本的课题.何永森等研究者通过一系列数值实验发现,用DLR型k—ε紊流模型·BFC法(边界拟合曲线坐标变换法),能够对总扩散角为8~0、扩散度为4的锥形渐扩管路内完全发展的不可压粘性紊流场较精确地数值仿真(又称数值模拟,计算机仿真)。多重网格方法(Multi-Grid Method,简称MGM)也称为多格子方法或多层网格法,是求解偏微分方程(组)大规模离散化方程最有效的方法,它一般可以分为几何多重网格方法(Geometric Multigrid Method,简称CMC)和代数多重网格方法(Algebraic multigrid method,简称AMG)。近年来,由于大型计算机的迅速发展和功能的日趋完善,从而使得多重网格方法做为最优的算法将理论用于现实。由于实际应用问题的错综复杂性,以及数值商业软件对“即插即用”型求解器的要求,使得几何多重网格方法的应用变得越来越困难,而代数多重网格方法的高效性和稳健性(robustness“鲁棒性”)使之成为了当今多重网格方法的研究热点.本文针对边界拟合曲线坐标贴体网格系统下代数多重网格方法在紊流数值预测中的应用问题,在何永森教授工作的基础上,详细探讨了代数多重网格方法在数值预测紊流中的实施过程,并围绕如何提高锥形渐扩管路内紊流数值仿真效率与精度展开研究,进行了一系列数值实验及与物理实验结果的比较,得到了一些有意义的成果,丰富和充实了代数多重网格方法,拓宽了代数多重网格方法在一些领域中的应用。具有理论和工程应用价值.主要内容和成果包括:(一)原来的程序软件判断数值收敛与否是根据数值计算结果的情况分次沿时间步推进求解并以人工方式进行.本文对这一情况进行了改进,通过添加了一个判断收敛的子程序,变人工方式为自适应控制收敛方式.数值实验表明,用这种方式控制迭代次数,当迭代结束时,各种流动参数的误差(前后两时间步的差)曲线处于平稳并且很小,计算结果与实验结果较好符合。(二)将AMG方法引入到紊流数值预测的有限差分计算领域,详细探讨了代数多重网格方法在紊流数值预测中的实施过程.研究了不同有限差分格式下,紊流模型离散得到的大规模代数系统对应系数矩阵的整合及边界条件的嵌入方法,讨论了系数矩阵的“三元组”压缩存储方式,利于节约内存及便于利用AMG方法求解。编制了代数多重网格方法数值求解紊流模型离散得到的大规模代数离散系统的程序接口;将代数多重网格方法与DLR、DHR型k—ε紊流模型.BFC法结合,应用编制的程序DLRAMG和DHRAMG对锥形渐扩管路内紊流进行了数值预测,将数值计算结果与物理实验结果进行了比较,多重网格方法的计算结果与实验结果符合较好。并且与POINT-SOR方法相比,可以节约近三分之一的CPU时间,提高了数值预测效率。(三)针对原来十三点格式所用的模板节点多,方程组系数矩阵的带宽大,非零元多,数值求解费时问题,设计了一种基于五节点模板的新五点差分格式,并将其与AMG方法结合,进一步提高了紊流数值预测的效率.数值实验结果表明,在AMG方法求解的条件下,新五点差分格式比原十三点差分格式可以节约近三分之一的CPU机时。(四)本文提出了一种可以同时实现控制网格正交性和任意控制边界网格间距的一种BFC网格生成的新方法。该方法可对生成的网格边界间距大小任意控制,同时生成的BFC网格还具有边界及内部较好的正交性.应用实例的计算结果表明,该方法能够对复杂边界的单连通域或多连通域生成较理想的BFC网格。(五)另外,研究了k-ε紊流模型对数值预测正弦波壁流动的应用。基于有限体积法结合非正交同位网格系统,压力与速度耦合采用SIMPLE方法,对该种流动进行了数值预测,得到了与实验结果符合较好的计算结果。更多还原

【Abstract】 Numerical predictions(or numerical simulations) for turbulent flow is the base project in fluid flow and heat transform.He yongsen et al find that fully developed incompressible turbulent flow in a conical diffuser having a total divergence of 8~0 and an area ratio of 4:1 can be simulated by a DLR turbulent model and it’s BFC(Boundary-Fitted Coordinates) method,but the predictions for turbulent flow have low efficiency.Algebraic multi-grid methods are by far the most,efficient methods for solving large scale algebraic systems arising from discretizations of PDEs or a system of PDEs.Generally speaking,there are two types of multigrid methods:geometric-based approach and algebraic approach.The large computers have developed very fast and the function of computer have been improved in recent years,which make the multigrid methods vary from optimal algorithm theoretically to optimal algorithm in practice.Since the complexities for practical application problems and the requirements for the "plug and play" solvers in numerical business softwares,it is difficult to construct a sequence of nested discretizations or meshes needed for geometric multigrid method. The algebraic multigrid method(AMG) has become the hotspot due to the high performance and robustness.In this paper,we make some in-depth studies for applying the AMG algorithms in numerical predictions for turbulent flow.A detailed discussion of implement procession of applying AMG methods to simulate the turbulent flow.Focus on improving the efficiency of simulation,we put forward some feasible methods.A number of numerical experiments have been performed.The computational results are compared with the results of experimental results,and some crucial numerical results are obtained.These researches will make the AMG algorithms richer and apply the AMG methods to more researching fields.The main contents and results are listed as followings:1.A convergence criterion is posed in numerical computation and thus the steps of time are made by the mode of self-adaptive control.Compared with experimental results,the numerical results can be accepted.In the picture about the curve of convergent history,the errors of time step n+1 and time step n keep little variation.2.By the precondition of finite difference method,We use the AMG methods in the field of numerical prediction of turbulent flow.A detailed implement process of the AMG method is given in this paper.The process of generating coefficient matrix of different difference schemes is also given in this paper and the method is introduced in detail.The method about how the boundary conditions are embed in the algebraic systems is studied,we develop the joint programs of the AMG method and the program DLRAMG、DHRAMG.Under the same computational condition and control precision,the AMG method is more efficient,than the SOR method which was often used in the past and about one-third of the total CPU time can be saved.3.Focus on the problem that the number of nodes of 13 points difference scheme is much and the computation cost much time,we design a new 5 points difference scheme.This new 5 points scheme has less non-zero elements than the 13 points scheme in the their systems and then the cost time for computation can be dccrcascd.Numerical experiments show that one third of CPU time can be saved.4.In this paper,a new method is posed to gcncratc the BFC gird with the adjustable boundary mesh intervals and orthogonality.The real examples show that the grids can be generated by the method for the simple connected and mutli-connected regions with complicated boundary.5.In addition,in this paper,a body-fitted non-orthogonal collocated grid system is generated. The standardκ-εmodel and the wall function are adopted.Numerical simulation is done for turbulent flow around 2-D single sinusoidal hill and two sinusoidal hills.Non-orthogonal collocated grid-based SIMPLE algorithm is adopted for solving the coupling system of the velocity and pressure equations.Simulation results agree well with the cxperimental data.更多还原