Analytical Solution for Laminar Water Hammer With Frequency-Dependent Friction

Abstract

An analytic solution for the classical water hammer problem for unsteady laminar pipe flow involving Zielke's frequency-dependent friction model is derived. By means of separation of variables, the solution of the initial boundary value problem is compiled in the form of an infinite series representing its modal expansion. In this way, solving the damped wave equation reduces to root-finding of scalar algebraic equations for the complex eigenfrequencies. An analysis of this dispersion relation underlines the frequency-dependent nature of the solution in comparison with less sophisticated friction models.