Parameters affecting water-hammer wave attenuation, shape and timing—Part 1: Mathematical tools

Abstract

This twin paper investigates key parameters that may affect the pressure waveform predicted by the classical theory of water-hammer. Shortcomings in the prediction of pressure wave attenuation, shape and timing originate from violation of assumptions made in the derivation of the classical water-hammer equations. Possible mechanisms that may significantly affect pressure waveforms include unsteady friction, cavitation (including column separation and trapped air pockets), a number of fluid-structure interaction (FSI) effects, viscoelastic behaviour of the pipewall material, leakages and blockages. Engineers should be able to identify and evaluate the influence of these mechanisms, because first these are usually not included in standard waterhammer software packages and second these are often "hidden" in practical systems.This Part 1 of the twin paper describes mathematical tools for modelling the aforementioned mechanisms. The method of characteristics (MOC) transformation of the classical water-hammer equations is used herein as the basic solution tool. In separate additions: a convolution-based unsteady friction model is explicitly incorporated; discrete vapour and gas cavity models allow cavities to form at computational sections; coupled extended water-hammer and steel-hammer equations describe FSI; viscoelastic behaviour of the pipe-wall material is governed by a

Effect of drop in pipe wall stiffness on water-hammer speed and attenuation

“…The interaction between the structural wall and the confined water influences the transient flow behaviour providing signatures of specific characteristics along the conduit (Bergant et al 2008a). The changes in transient pressures generated by a wall-leaking crack have been extensively studied using hydraulic-based monitoring techniques (Ferrante and Brunone 2003, Covas et al 2005, Hunaidi 2006.…”

Detection of Local Wall Stiffness Drop in Steel-Lined Pressure Tunnels and Shafts of Hydroelectric Power Plants Using Steep Pressure Wave Excitation and Wavelet Decomposition

“…The differentiation and combination of these two conservation equations leads to the acoustic plane wave equation expressed as (Bergant et al 2008)…”

Multiple scales analysis of water hammer attenuation

“…For convenience, these parameters are defined in the analysis as the equally contributing parameters C 1 C 4 = C 1 c 4 2 , C 4 = c 4 2 and C 5 = c 5 2 with C 1 , c 4 , and c 5 being of order one. Once again, although the convective term C 1 c 4 2 is typically neglected, as in [4], it is retained here since it may have a comparable magnitude to c 4 2 and c 5 2 . A natural choice for the artificial scaling constant used here is 2 = O(1/c p ).…”

“…The convective term V V X is typically neglected (e.g. as reviewed in [4]) but is retained here because C 1 is O(1); this is also true for the steady velocity, 12 ), considered in the water main application in Section 4.3. The continuity parameters C 1 C 4 , C 4 , and C 5 are presumed to be small and inversely proportional to the square of the celerity c p 1, and C 1 is order one.…”

“…Standard eigenfunction methods have also been used, and in [16] this approach was applied to the pressure-wave transmission through a conduit between two pressure-vessels. The method of characteristics has been useful to numerically investigate the effects of including such specialized aspects as cavitation, column separation, and leakage/blockage [4,5]. It was found that Poisson-coupling, a very specific type of fluid-structure interaction, can lead to a 'beat' phenomenon in certain cases.…”