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数值方法-时线插值特征线法计算恒定流和非恒定流的统一方法。为了做到这一点的插值误差进行了理论分析，通过调查波的特性，网格，插值和边界条件之间的关系。将有限差分方程的线性稳定性分析推广到具有特定边界条件的有限区域。在新的稳定性分析中，引入多项式传递矩阵，通过求解与边界条件相关联的多项式，得到复放大因子，从而分析误差特性。通过施加质量和能量连续性条件，可以很容易地导出反映连接、分支等边界条件的整个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。对插值误差的非粘性分析表明，在简单管道中，插值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，插值误差随频率的增加而单调增加;在无插值的粘性分析中，离散误差有可能使最低频率模态叠加到最高频率模态上而产生过阻尼，尽管数值精度的顺序确实有很大影响。数值仿真结果验证了所分析的误差特性。本文还论证了定常流时间推进法的理论基础，即通过最大化离散误差，快速衰减所有频率模态，并提出了一种计算大型管网定常流的实用方法。

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.