The water hammer problem and its numerical analysis

Abstract

This dissertation involves mathematical models of water flow in a pipeline system. The topic has been studied for a long time, such that water distribution systems have evolved in the past few decades, and mathematical models also evolved from linear to nonlinear models, from incompressible to compressible models and also from systems with infinite space domain to systems with boundary conditions, etc. In my research, I managed to analyze the water flow by a 1d Euler equation with dynamical boundary conditions, where the dynamical boundary conditions are in terms of the valve in the middle of the pipe that can be fast closed, causing hydraulic shock which we call the water hammer inside the pipe. This Euler equation system with dynamical boundary conditions is new and is inspired by switch differential algebraic equations (sDAE) proposed by Prof. Stephan Trenn, which is a system consisting of ordinary differential equations (ODE) and algebraic equations. The valve closing and opening can be modeled as the switching signals of different systems, and accordingly in the partial differential equation (PDE) system they can be modeled as changing boundary conditions. Therefore, throughout my research, I performed the analysis by separating the cases that are before and after the valve closure, both in theoretical and numerical ways.