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

本研究的目的是建立一种统一的计算定常和非定常流动的方法，该方法是基于弹性水柱模型的数值方法之一。为此，通过研究波浪特性、网格、插值和边界条件之间的关系，对插值误差进行了理论分析。推广有限差分方程的线性稳定性分析，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，并用复放大因子分析误差特性，即与边界条件相关的多项式解。通过施加质量和能量连续性条件，可以很容易地推导出反映连接、分支等边界条件的总系统的多项式传递矩阵。它使我们能够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，插补误差随频率的增加而单调增加，而对于串联管道、分支管道等一般管道系统则不一定如此。在不加插值的粘性分析中，尽管数值精度的顺序对离散误差有显著影响，但离散误差有可能出现最低频率模态加最高频率模态的过阻尼。通过数值模拟验证了所分析的误差特性。验证了时间推进法求解稳定流的理论基础，该方法通过使离散误差最大化来快速衰减各频率模态，并提出了一种计算大型管网稳定流的实用方法。

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.