Finite-Volume Solutions to the Water-Hammer Equations in Conservation Form Incorporating Dynamic Friction Using the Godunov Scheme

Abstract

Although derived from the principles of conservation of mass and momentum, the water-hammer equations integrating dynamic friction are almost never expressed in conservative form. This is because the pressure and volume discharge are used as variables but these are not conserved quantities, especially when the one-dimensional velocity profile is distorted from its assumed steady state shape due to the large accelerations imposed on the fluid particles across the cross section. This paper presents the derivation of the water-hammer equations in conservation form incorporating dynamic friction. With the dynamic friction taken into account, a source term appears in the basic partial differential equations as presented by Guinot. The numerical algorithm implements the Godunov approach to one-dimensional hyperbolic systems of conservation laws on a finite-volume stencil. Two case studies are used to illustrate the influence of the various formulations. A comparative study between the analytical solution, the numerical solution with quasi-steady friction only, the numerical solution with dynamic friction, and the measurements has been presented. The results indicate that the dynamic friction formulation reduces the peak water hammer pressures when compared with a quasi-steady representation.