Construction of Numerical Stability Analysis in Systems and Error Estimate on Water Hammer Analysis in Pipeline Systems

本文的研究目的是建立一种基于弹性水柱模型的特征线法和时间线插值法计算定常和非定常水流的统一方法。为此，通过研究波动特征、网格、内插和边界条件之间的关系，从理论上分析了内插误差。推广了有限差分方程线性稳定性分析，使其适用于具有特定边界条件的有限区域。在新的稳定性分析中引入了多项式传递矩阵，误差特性可以用复放大因子来分析，它是与边界条件相关的多项式的解。通过施加质量和能量连续条件，可以很容易地导出反映连接、分支等边界条件的总系统的多项式传递矩阵。它使我们有可能准确地分析系统中的数值误差。对插补误差的非粘性分析表明，在简单管道中，插补误差随着频率的增加而单调增加，而对于一般的管道系统，如串联管道、分支管道系统等，插补误差并不一定适用。对于没有任何内插的粘性分析，离散误差有可能使最低频率的振型在最高频率的振型上附加过阻尼，但数值精度的阶数对其影响很大。通过数值仿真验证了所分析的误差特性。证明了恒定流时间推进法的理论基础，即通过最大化离散化误差来快速抑制所有频率模式，并提出了一种计算大型管网恒定流的实用方法。

The aim of this study is to establish a unified method of calculating steady and unsteady flows using the method of characteristics with time-line interpolations, which is one of numerical methods based on the elastic water column model. To do this the interpolation errors are theoretically analyzed by investigating the relationships among the wave characteristics, grids, interpolations, and boundary conditions. The linear stability analysis of finite difference equations is extended so that it holds for a finite region with specified boundary conditions. Within the new stability analysis the polynomial transfer matrix is introduced, and the error characteristics can be analyzed by the complex amplify factors, which is the solution of the polynomial associated with the boundary conditions. The polynomial transfer matrix of the total system, which reflects the boundary conditions such as connections, branches, and so on, is easily derived by imposing the mass and energy continuity conditions. It makes us possible to exactly analyses the numerical errors in systems. The non-viscous analysis of the interpolation errors reveals that the errors increases monotonously as the frequency increases in a simple pipe while it does not necessarily hold for general pipe system such as series pipes, branching piping systems, and so on. With the viscous analysis without any interpolations the discretization errors have a possibility of over-damping of the lowest frequency mode adding to the highest frequency one, though the order of numerical accuracy does influence significantly. The error characteristics analyzed are examined rightly by the numerical simulations. The theoretical basis of the time marching approach for steady flows is also justified, which quickly damps out all frequency modes by maximizing the discretization errors, and a practical method of calculating the steady flows in large network pipelines is proposed.