有限容积法对流项离散格式的稳定性与有界性研究

Study on the Stability and Boundedness of Discrete Schemes for Convection Term Based on Finite Volume Method

对于流动传热问题,控制方程对流项处理不当会导致模拟结果出现无物理意义的振荡或越界现象。针对有限容积法对流项离散格式的稳定性和有界性开展深入研究,并提出了一种新的高阶有界格式。在对流项离散格式稳定性方面,指出在均分网格条件下,边界相邻节点的对流稳定性条件与内部节点不同,并给出了边界相邻节点处常见对流项离散格式的稳定性条件;指出绝对稳定的SGSD格式权重系数计算方法不唯一,并给出了两种适合于强对流问题的权重系数表达式。在对流项离散格式有界性方面,提出了高阶有界格式的通用分段线性形式——GHB格式(general high-order bounded scheme),可提高程序的通用性;高阶有界格式计算精度、收敛速率和健壮性等性能互相制约,根据规正变量图中不同区域线型对这些性能的影响,提出了含有两个可调参数的性能可控的高阶有界格式——AHB格式(adjustable high-order bounded scheme);在此基础上,通过优化参数得到了计算精度、有界性、收敛速率和健壮性兼顾的ABC格式(accuracy,boundedness and convergence rate-balanced scheme),并通过多个典型算例验证了该格式计算性能。

For the flow and heat transfer problems,improper discretization of the convection term in the governing equation may cause the simulation results to be out of bounds or oscillate unphysically.In this paper,the stability and boundedness of discrete schemes for the convection term based on the finite volume method were studied,and a new high-order bounded scheme was proposed.For the stability of discrete schemes for the convection term,the convection stability conditions of nodes adjacent to boundary differ from those of internal nodes based on the uniform grid,and the stable range is presented for the common discrete schemes.Besides,the calculation method of the weight coefficient of the absolutely stable SGSD scheme is not unique.To this end,two calculation expressions of the weight coefficient suitable for the strong convection problem were provided.Regarding the boundedness of discrete schemes for the convection term,a general piecewise linear form of the high-order bounded scheme named GHB scheme(general high-order bounded scheme)was proposed,favoring improving the universality of the program.Besides,based on the idea that the computational accuracy,convergence rate,and robustness of the higher-order bounded scheme are mutually restricted and related to the line segment trends in the normalized variable diagram,an adjustable high-order bounded scheme(AHB scheme)with two adjustable parameters was proposed.Furthermore,an ABC scheme(accuracy,boundedness,and convergence rate-balanced scheme)balancing accuracy,boundedness,convergence rate,and robustness is obtained by optimizing the parameters in the AHB scheme,and its computational performance is proven by solving some classical problems.