On the fundamental equations of water hammer

Abstract

The wide spectrum of practical problems that warrant water-hammer modelling has increased the importance of careful formulation of the fundamental equations of water hammer and critical analysis of their assumptions. To this end, this paper reviews the relation between state equations and wave speeds in single as well as multiphase and multicomponent transient flows, formulates the various forms of one- and two-dimensional water-hammer equations and illuminates the assumptions inherent in these equations. The derivation of the one- and two-dimensional water-hammer equations proceeds from the three-dimensional Navier - Stokes equations for a compressible fluid. The governing equations for turbulent water-hammer flows are obtained by applying ensemble averaging to the simplified form of the Navier - Stokes equations for water-hammer problems. Unlike time averaging, ensemble averaging is applicable to unsteady flows where the time scale of the transient is often much smaller than the time scales of the turbulence. Order of magnitude analysis, physical understanding and recent research findings are used throughout the paper to evaluate the accuracy of the assumptions made in the derivation of the one- and two-dimensional water-hammer equations.