FURTHER INVESTIGATION OF PARAMETERS AFFECTING WATER HAMMER WAVE ATTENUATION, SHAPE AND TIMING PART 2: CASE STUDIES

FURTHER INVESTIGATION OF PARAMETERS AFFECTING WATER HAMMER WAVE ATTENUATION, SHAPE AND TIMING PART 2: CASE STUDIES by Anton Bergant 1, Arris Tijsseling 2, John Vítkovský 3, Dídia Covas 4, Angus Simpson 3, Martin Lambert 3 1 Litostroj E.I. d.o.o, 1000 Ljubljana, Slovenia. 2 Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands. 3 The University of Adelaide, Adelaide SA 5005, Australia. 4 Instituto Superior Técnico, 1049-001 Lisboa, Portugal. ABSTRACT This paper further investigates parameters that may affect water hammer wave attenuation, shape and timing (Bergant and Tijsseling 2001). New sources that may affect the waveform predicted by classical water hammer theory include viscoelastic behaviour of the pipe-wall material, blockage and leakage in addition to the previously discussed unsteady friction, cavitation and fluid-structure interaction. These discrepancies are based on the same basic assumptions used in the derivation of the water hammer equations for the liquid unsteady pipe flow, i.e. the flow is considered to be one-dimensional (cross-sectionally averaged velocity and pressure distributions), the pressure is higher than the liquid vapour pressure, the pipe-wall and liquid have a linear elastic behaviour, unsteady friction losses are approximated to steady state losses, the amount of free gas in the liquid is negligible, fluid-structure coupling is weak (precursor wave pressure changes are much smaller than the water hammer pressures), the pipe is straight and of uniform shape (no blockage) and there is no lateral outflow (leakage) or inflow (pollution). Part 1 of this paper (Bergant et al. 2003) presented the mathematical tools needed to model the affect of the various parameter identified in the preceding paragraph. Part 2 of the paper presents a number of new case studies showing how these parameters (i.e., unsteady friction, gaseous cavitation, non-linear elastic behaviour of the pipe-wall material, blockage and leakage) affect pressure traces in a simple reservoir-pipeline-valve system. Each case study is compared to the results obtained by the standard (reference) water hammer model. Based on the affect of each parameter in the case study, conclusions are made concerning the transient behaviour of real systems. 1. INTRODUCTION A number of new case studies are presented to show how the effects of unsteady friction (convolution-based model), gaseous cavitation, viscoelastic behaviour of pipe wall, leakage and blockage change the water hammer waveform in a simple reservoir-pipeline-valve system. Each case is simulated by a standard water hammer model based on Eqs (4) and (5), and by a corresponding MOC numerical model that incorporates unsteady friction, gaseous cavitation, viscoelastic behaviour of pipe wall, leakage and blockage, or a combination of these as described in Part 1 of the paper (Bergant et al. 2003). The standard water hammer model is herein referred to as the 'reference or classic model'. In addition, the numerical results from unsteady friction and viscoelastic behaviour of pipe walls are compared with the results of measurements that have been performed in laboratories of the University of Adelaide, Adelaide, Australia and Imperial College, London, United Kingdom. The example piping system used for investigating water hammer wave forms comprises a metal (copper) pipeline of length 37.2 m, 22 mm internal diameter and 1.6 mm wall thickness that is upward sloping (Fig. 1). The transient event is generated by a rapid closure of the downstream end valve. The apparatus is installed in 1

Robin hydraulic laboratory of the Department of Civil and Environmental Engineering at the University of Adelaide. The initial flow velocity in our case studies is V 0 = 0.2 m/s, static head in the tank 2, H T = 32 m, valve closure time t c = 09 s, and water hammer wave speed a = 1319 m/s (Bergant et al. 2001). The number of reaches for each computational run is N = 32. The experimental apparatus is fully described by Bergant et al. (2001). Tank 1 2.03 m Tank 2 m Valve Pipeline DATUM Fig. 1 Experimental apparatus with copper pipeline (pipe length L = 37.2 m; pipe diameter D = 22. mm) An experimental facility with a single plastic pipeline installed at the Department of Civil and Environmental Engineering, Imperial College (Covas et al., 2001; Covas, 2003) is used for investigation of viscoelastic behaviour of pipe walls. The pipe is made of high-density polyethylene (SDR11 PE100 NP16) with 63 mm outside diameter (ID=50.6 mm) and a length of 277 m (see Fig. 2). The pipe-rig includes a pump and a pressurised tank at the upstream end and a globe valve at the downstream end. The globe valve was used for generating transient conditions. The steady-state flow is measured with an electromagnetic flow meter. The data acquisition system is composed of an acquisition board, four strain-gauge type pressure transducers, three strain gauges and a computer. The acquisition board has a maximum sampling rate of 9600 Hz per channel. The pressure transducers have pressure ranges of 0 to 10 bar and accuracy of ±0.3% of the full range. The strain gauges have one grid, 1 cm length and a resistance of 350Ω with an accuracy of ±0.2%. GV T1 SG1 12 m 8 m BV2 T8 SG8 Reservoir PUMP T5 SG5 CV IV1 T3 FM IV2 AIR VESSEL BV2 - B all valve CV - C heck valve GV - G lobe valve IV1,IV2 - I solating valves FM - Flowmeter T - Pressure Transducer SG - Strain Gauge T3 T5/SG5 T8/SG8 T1/SG1 GV 0 117 199 272.5 273 Fig. 2 Experimental apparatus with high-density polyethylene pipeline (pipe length L = 277. m; pipe diameter D = 50.6 mm) Various data sets were collected with a sampling rate of 600 Hz. Four pressure transducers and three strain gauges were located in the sites shown in Fig. 2. These tests covered a wide range of flows from laminar (Q=56 l/s; Re=1400) to smooth turbulent (Q=2.0 l/s; Re=50000). Transient events were generated by the sudden closure of the globe valve. 2

2 UNSTEADY FRICTION The effect of unsteady friction on water hammer wave forms is investigated using the copper pipeline shown in Fig. 1. Numerical results from the reference model (steady friction) and the unsteady friction model (pure convolution-based model (Zielke 1968) and convolution-based model with momentum correction factor β 0 = 1.019 (Chen 1992)) are compared with the measured results from the experimental run with identical system parameters to the numerical model. The Reynolds number of the initial flow is 4,369 making the initial flowstate turbulent. The smooth-pipe turbulent Vardy-Brown weighting function formulae are used (Vardy and Brown 2003). The results are compared at the valve ( ) and at the midpoint ( ) and are presented in Fig. 3. 2 2-2 2-2 2 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 with Momentum Correction Factor -2 2-2 2 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 with Momentum Correction Factor e) -2 0.2 0.4 0.6 0.8 f) -2 0.2 0.4 0.6 0.8 Fig. 3 Comparison of heads at the valve ( ) and at the midpoint ( ) in copper pipeline (Fig. 1); V 0 = 0.2 m/s. The simulation results show that the steady friction model underestimates the damping observed in the experimental results. Additionally, the steady friction model does not predict the evolution of the shape of the pressure oscillations, demonstrating steady friction s inability to model frequency-dependent attenuation. The convolution-based model better predicts both the shape and damping of the pressure oscillations, but becomes increasingly out-of-phase with the experimental data. Finally, the convolution-based unsteady friction model including the momentum correction factor performs the best, closely matching the frequency-dependent damping and the phase of the pressure oscillations. There remain some small differences between the model results and the experimental data mainly with respect to the high frequency components of the response, i.e. the simulations produce an overly sharp response whereas the experimental data are more rounded. This difference points towards a high frequency damping phenomena that is not being adequately modelled currently. 3

3. GASEOUS CAVITATION The effect of gaseous cavitation on the transient wave form is presented for the selected case of rapid closure of the valve positioned at the downstream end of the copper pipeline apparatus as shown in Fig. 1. Computational runs were performed using the discrete gas cavity model (DGCM) (Wylie 1984; Wylie and Streeter 1993) for two distinct flow situations: (1) Gaseous cavitation in liquid with free gas distributed at all computational sections (simulation of pressure waves in bubbly flow). (2) Trapped gas pocket at the midpoint of the pipeline. A weighting factor of ψ = 1.0 has been used in the DGCM model. Six distinct values of gas void fraction at atmospheric conditions α g0 = {10-7 ; 10-6 ; 10-5 ; 10-4 ; 10-3 and 10-2 } were selected in the analysis for both flow situations (initial flow velocity V 0 = 0.2 m/s, static head in the tank 2, H T = 32 m, valve closure time t c = 09 s). The aim of the analysis is to identify the effect of the quantity of free gas on transients. Computational results from the DGCM model are compared with 'reference model' results at the valve ( ) and at the midpoint ( ) and are depicted in Figs. 4 and 5 respectively. 2 =10-7 2 =10-7 -2 2 0 0.20 0.40 0.60 0.80 =10-5 -2 2 0 0.20 0.40 0.60 0.80 =10-5 -2 2 0 0.20 0.40 0.60 0.80 =10-3 -2 2 0 0.20 0.40 0.60 0.80 =10-3 e) -2 0 0.20 0.40 0.60 0.80 f) -2 0 0.20 0.40 0.60 0.80 Fig. 4 Comparison of heads at the valve ( ) and at the midpoint ( ) in copper pipeline (Fig. 1); V 0 = 0.2 m/s. Liquid with Free Gas Distributed at all Computational Sections The DGCM model results with very low gas void fraction α g0 = 10-7 (Figs. 4( and 4() perfectly match the 'reference model' results. Similar results were obtained with gas void fraction α g0 = 10-6. DGCM model is 4

capable of simulating water hammer events in 'pure' liquid. The DGCM model results with low gas void fraction α g0 = 10-5 (Figs. 4( and 4() reveal a weak effect of localised small amounts of free gas on pressure traces. Larger amounts of free gas α g0 = {10-4 ; 10-3 and 10-2 } significantly affect pressure wave attenuation, shape and timing. The discrepancies between the DGCM model results and the 'reference model' results increase with increasing gas void fraction. Figs. 4(e) and 4(f) present DGCM model results with large gas void fraction α g0 = 10-3. The actual flow situation represents a gaseous cavitation case with a maximum head significantly smaller than the water hammer head (at the valve - DGCM model:,max = 52.6 m; 'reference model':,max = 58.9 m). Trapped Air Pocket at the Midpoint of the Pipeline Trapped gas pockets may cause severe operational problems in piping systems (Wylie and Streeter 1993). In our case study, the gas pocket is trapped at the midpoint of the copper pipeline. Again six computational runs with void fraction of the trapped gas pocket α g0,mp = {10-7 ; 10-6 ; 10-5 ; 10-4 ; 10-3 and 10-2 } have been performed. The corresponding gas volume is small in comparison to the total water volume. The DGCM model has been slightly modified with a void fraction at the midpoint α g0,mp different from the value of void fraction in other computational sections. A void fraction of 10-7 has been selected in all other computational sections with 'pure' liquid. 2 =10-6 2 =10-6 -2 2 0 0.20 0.40 0.60 0.80 =10-4 -2 2 0 0.20 0.40 0.60 0.80 =10-4 -2 2 0 0.20 0.40 0.60 0.80 =10-2 -2 2 0 0.20 0.40 0.60 0.80 =10-2 e) -2 0 0.20 0.40 0.60 0.80-2 f) 0 0.20 0.40 0.60 0.80 Fig. 5 Comparison of heads at the valve ( ) and at the midpoint ( ) in copper pipeline (Fig. 1); V 0 = 0.2 m/s. The DGCM model results with very small trapped gas pocket - α g0,mp = 10-6 (Figs. 5( and 5() match the 'reference model' results. The very small volume trapped gas pocket does not affect the pressure wave front induced by rapid closure of the downstream end valve. The DGCM model results with small trapped gas pocket - α g0,mp = 10-4 (Figs. 5( and 5() show a weak effect of trapped pocket on pressure traces. Larger amounts of free gas in the trapped pocket at the midpoint (α g0,mp = {10-3 and 10-2 }) significantly affect pressure wave 5

attenuation, shape and timing. The discrepancies between the DGCM model results and the 'reference model' results increase with increasing the volume of trapped gas pocket. Figs. 5(e) and 5(f) presents DGCM model results with large gas void fraction α g0,mp = 10-2. In this case, the actual flow situation represents a gaseous cavitation case with a maximum head larger than the water hammer head (at the valve - DGCM model:,max = 67.7 m; 'reference model':,max = 58.9 m). 4 VISCOELASTIC BEHAVIOUR OF PIPE WALL The viscoelastic transient solver requires, as input data, the description of the creep compliance function represented by the generalised Kelvin-Voigt model. This model is described by a set of parameters: the instantaneous elastic creep J 0 and the retarded components, τ k and J k. If the creep J(t) is unknown, this function has to be calibrated by adjusting the results of the numerical model to the transient data. The transient model was calibrated by adjusting the numerical results to collected data in the experimental apparatus with high-density polyethylene pipeline (see Fig. 2) neglecting unsteady friction. Calibrations were carried out for several transient tests from laminar to smooth-pipe turbulent flows (Q 0 =54 l/s to Q 0 =1.98 l/s). The creep function J(t) was represented by several combinations of Kelvin-Voigt elements (1 to 6 elements). The parameters τ k and J 0 were fixed before calibration. The parameters J k were estimated by minimising the Least Square Error (LSE) between the calculated and measured piezometric-head at T1 (at the downstream end see Fig. 2). The Levenberg-Marquardt algorithm was used to carry out the optimisation. Calibrated creep functions for the high-density polyethylene pipeline (Fig. 2) are presented in Fig. 6. Each combination of parameters is just one possible mechanical representation of the pipe creep. Different creep functions were obtained for each flow, though only one should theoretically exist. This is because unsteady friction (which is not accounted for in these simulations) has a similar dynamic effect on the pressure wave as pipe-wall viscoelasticity, as it attenuates the maximum pressure fluctuations and, to a smaller extent, increases the dispersion of pressure wave. The only numerical way that the optimisation algorithm has to describe friction is by including it in the creep function (increasing the total creep). In fact, there is a resemblance between the frequency-dependent friction and the frequency-dependent creep, as both depend on the past-time histories of the fluid. 1.30 J(t) (GPa -1 ) 1.20 1.10 1.00 0.90 Q = 0.253 l/s Q = 1.980 l/s Q = 0.105 l/s Q = 56 l/s Q = 56 l/s Trika (1973) T=5 s Q = 56 l/s Trika (1973) T=10 s Q = 56 l/s Trika (1973) T=20 s Q = 1.756 l/s Q = 0.756 l/s Q = 1.008 l/s Q = 0.500 l/s Q = 1.510 l/s 0.80 2.0 4.0 6.0 8.0 1 Fig. 6 Calibrated creep-functions for several flow-rates in high-density polyethylene pipeline (Fig. 2) (Covas et al., 2002) Numerical results obtained for Q 0 =1.0 l/s are compared with the collected piezometric-head time variation at Locations 1 and 5 corresponding to distances from the upstream end of 273 m and 117 m respectively (Fig. 7). Two numerical solutions for the piezometric-head are presented. The first (dashed-line) was calculated using the classic ( elastic ) water hammer equations and the second (continuous-thin-line) the viscoelastic equations. The classic solution shows large discrepancies in both the pressure amplitude and phase with experimental data, where as the viscoelastic solution fits perfectly with the collected data. 6

7 Location 1 Location 5 7 "Viscoelastic model" "Viscoelastic model" H H "Elastic model" "Elastic model" Fig. 7 Comparison of heads at the valve (Location 1 in Fig. 2) and at the midpoint (Location 5 in Fig. 2) in highdensity polyethylene pipeline, Q 0 =1.0 l/s Numerical Results Considering both Viscoelasticity and Unsteady Friction In an attempt to distinguish the viscoelastic effect of pipe-wall from the unsteady-friction effect, the creepfunction was calibrated for the laminar conditions (Q=56 l/s; Re=1400) considering unsteady frictional effects. Unsteady friction was described by the Zielke weighting function (Zielke 1968). The creep-function was calibrated for the sample size T=10 s (Fig. 6). Numerical results obtained are compared with collected data (Fig. 8). Four numerical solutions are presented: (i) classic solution; (ii) results of the implementation of Zielke (1968) formulation only; (iii) results of the implementation of pipe-viscoelasticity only; and (iv) a combination of these two effects. The latter results show good agreement with experimental data, which does not happen with the results of (i), (ii) and (iii). 49.6 49.1 Classic Solution 49.6 49.1 Zielke (1968) 48.6 48.6 H 48.1 H 48.1 47.6 47.6 47.1 47.1 49.6 49.1 Viscoelasticity 49.6 49.1 Zielke (1968) + Viscoelasticity 48.6 48.6 H 48.1 47.6 H 48.1 47.6 47.1 47.1 Fig. 8 Comparison of heads at the valve (Location 1) for laminar conditions in high-density polyethylene pipeline (Fig. 2), Q 0 =56 l/s Assuming a good approximation of the creep-function was achieved with the calibration for the laminar conditions, this function was used for smooth turbulent conditions (Q=1.008 l/s; Re=25000). Numerical results for six different cases are presented in Fig. 9: (i) classic solution; (ii) implementation of Zielke (1968) formulation only; (iii) implementation of viscoelasticity only; and (iv) Zielke (1968) and viscoelasticity. The following conclusions can be drawn: (i) unsteady friction cannot generate the total damping observed in pressure fluctuations as it can be seen in the solutions without viscoelasticity; (ii) results obtained combining unsteady 7

friction and viscoelasticity (calibrated for laminar conditions) have a slightly higher dampening than the actual observed data. This can be due to an error associated with Zielke (1968) formula for turbulent flows, or simplifications of the mathematical model, like assuming constant cross-section parameters. 7 Classic Solution 7 Zielke (1968) H H 7 Viscoelasticity 7 Zielke (1968) + Viscoelasticity H H Fig. 9 Comparison of heads at the valve (Location 1) for smooth turbulent conditions in high-density polyethylene pipeline (Fig. 2), Q 0 =1.008 l/s 5 LEAKAGE AND BLOCKAGE Both the leak and the blockage were located at 13.95 m upstream of the valve (at the 3/8 th point from the valve) in the copper pipeline system (Fig. 1). Each pipeline fault was sized such that ratio of the steady state leak discharge and blockage head loss to the characteristic system pipe flow or reservoir head, for leaks and blockages respectively, were equal for the leak and blockage. The leak is of size C d A O = 6.1 10 8 m 2 with a diameter of 0.33 mm. This equates to an orifice area to pipe area ratio of 2%. The ratio of the flow loss across the orifice to the steady-state flow through the pipeline is 2.01%. Fig. 10 shows the effect of the leak on the pressure response at the valve and at the midpoint using both the reference model (steady state friction model) and the convolution-based unsteady friction model. The presence of a leak serves to increase the damping in the system. Additionally, the pressure response is more complicated due to the reflections from the leak. The use of the convolution-based unsteady friction model damps the leak-induced reflections as the simulation time progresses, which is to be expected in an unsteady friction dominant system. The blockage is of size C d A O = 2.1 10 5 m 2 with a diameter of 6.21 mm and is located at the same position in the pipeline as the leak. This equates to an orifice area to pipe area ratio of 8.04%. The ratio of the head loss across the orifice to the steady-state head in the pipeline is 2.01% (similar to the leak). Fig. 11 shows the pressure response for the blocked pipeline system. The blockage effect can be thought of as the opposite of the leak effect with regards to the shape of the pressure response. Again, the inclusion of the convolution-based unsteady friction model damps the reflections from the blockage. 8

2 with Leakage 2 with Leakage -2 2 0.2 0.4 0.6 0.8 with Leakage -2 2 0.2 0.4 0.6 0.8 with Leakage -2 0.2 0.4 0.6 0.8-2 0.2 0.4 0.6 0.8 Fig. 10 Comparison of heads at the valve ( ) and at the midpoint ( ) in copper pipeline (Fig. 1); V 0 = 0.2 m/s. 2 with Blockage 2 with Blockage -2 2 0.2 0.4 0.6 0.8 with Blockage -2 2 0.2 0.4 0.6 0.8 with Blockage -2 0.2 0.4 0.6 0.8-2 0.2 0.4 0.6 0.8 Fig. 11 Comparison of heads at the valve ( ) and at the midpoint ( ) in copper pipeline (Fig. 1); V 0 = 0.2 m/s. The presence of leaks and blockages creates a frequency-dependent attenuation of the pressure response with little frequency-dependent dispersion. In this respect their effect is to damp and complicate the pressure response without the phase of the response. The addition of unsteady friction in both cases accelerates the attenuation, making the leak- and blockage-induced pressure response shapes less defined. In both cases the size of the leak and blockage are minor; however, despite their small size, they contribute largely to the overall damping. 9

7. CONCLUSIONS Experiments even in simple pipelines show that transient modelling of observed events is significantly more complex than traditional waterhammer analysis suggests. The transients are damped, dissipated and lagged by a variety of phenomena that can be relatively easily identified if they act alone. For example, an air pocket established along a pipeline could be reasonably well modelled and located, provided it is known that an air pocket alone is the sole source of any changes in the transient signal. If on the other hand, the cause of the discrepancy between the classic theory is unknown, then the identification of exactly what phenomenon is present in the pipeline becomes problematic because of the similarities in effect between the many different phenomenon. For example, the distinction between frictional and mechanical damping in plastic pipes is difficult because the visco-elastic behaviour of the pipe has a dissipative and dispersive effect on the pressure wave similar to unsteady friction and gaseous cavitation. This paper illustrates the effects of the following phenomena on transient modelling 1. The impact of three-dimensional nature of the flow and fluid turbulence 2. Modification of fluid properties gas entrainment 3. Non-elastic response of the containing pipe 4. Local changes in pipeline cross-section or geometry blocks or leaks and highlights the similarities and differences between them. ACKNOWLEDGEMENT AND DISCLAIMER The Surge-Net project (see http://www.surge-net.info) is supported by funding under the European Commission s Fifth Framework Growth Programme via Thematic Network Surge-Net contract reference: G1RT-CT-2002-05069. The authors of this paper are solely responsible for the content and it does not represent the opinion of the Community, the Community is not responsible for any use that might be made of data therein. APPENDIX I. REFERENCES Bergant, A., and Tijsseling, A. (2001). Parameters affecting water hammer wave attenuation, shape and timing. Proceedings of the 10th International Meeting of the IAHR Work Group on the Behaviour of Hydraulic Machinery under Steady Oscillatory Conditions, Trondheim, Norway, Paper C2, 12 pp. Bergant, A., Simpson, A., and Vítkovský, J. (2001). Developments in Unsteady Pipe Flow Friction Modelling. Journal of Hydraulic Research, IAHR, 39(3), 249-257. Bergant A., Tijsseling, A., Vítkovský, J.P., Covas, D., Simpson, A., and Lambert, M. (2003). Further investigation of parameters affecting water hammer wave attenuation, shape and timing. Part 1: mathematical tools Proceedings of the 11th International Meeting of the IAHR Work Group on the Behaviour of Hydraulic Machinery under Steady Oscillatory Conditions, Stuttgart, Germany, 12 pp. Chen, C.-L. (1992). Momentum and Energy Coefficients Based on Power-Law Velocity Profile. Journal of Hydraulic Engineering, ASCE, 118(11), 1571-1584. Covas, D. (2003). "Inverse Transient Analysis for Leak Detection and Calibration of Water Pipe Systems - Modelling Special Dynamic Effects." PhD, Imperial College of Science, Technology and Medicine, University of London, London, UK. Covas, D., Stoianov, I., Graham, N., Maksimovic, C., Ramos, H., and Butler, D. (2001). "Leakage Detection in Pipelines Systems by Inverse Transient Analysis: from Theory to Practice." Water Software Systems: Theory and Applications, Eds. Ulanicki, B., Coulbeck, B., and Rance, J., Pub. Research Studies Press, Proceedings of the International Conference on Computing and Control for the Water Industry, CCWI 2001, De Montford University, Leicester, UK, 3-16. Covas, D., Stoianov, I., Graham, N., Maksimovic, C., Ramos, H., and Butler, D. (2002). "Hydraulic Transients in Polyethylene Pipes.", Proceedings of 1st Annual Environmental & Water Resources Systems Analysis (EWRSA) Symposium, A.S.C.E. EWRI Annual Conference, Roanoke, Virginia, USA. Trikha, A.K. (1975). An Efficient Method for Simulating Frequency-Dependent Friction in Transient Liquid Flow. Journal of Fluids Engineering, Transactions of the ASME, 97, 97-105. Vardy, A.E., and Brown, J.M.B. (2003). Transient Turbulent Friction in Smooth Pipe Flows. Journal of Sound and Vibration, 259(5), January, 1011-1036. Wylie, E.B. (1984). Simulation of vaporous and gaseous cavitation. Journal of Fluids Engineering, ASME, 106(3), 307-311. Wylie, E.B., and Streeter, V.L. (1993). Fluid Transients in Systems. Prentice-Hall Inc., Englewood Cliffs, New Jersey, USA. 10

Zielke, W. (1968). Frequency-Dependent Friction in Transient Pipe Flow. Journal of Basic Engineering, Transactions of the ASME, 90(1), 109-115. APPENDIX II. NOTATION The following symbols are used in this paper: a = liquid wave speed; A O = cross-sectional orifice area; C d = orifice discharge coefficient; D = inner pipe diameter; H = head; J, J 0 = creep-compliance, instantaneous or elastic creep-compliance; J k = creep of the springs of the Kelvin-Voigt elements; L = pipe length; N = number of computational reaches; Q = discharge (flow) or node downstream-end discharge; Re = Reynolds number; t = time; t c = valve closure time; V = average velocity; α g = gas void fraction; β = momentum correction factor; T = sample time; τ k = retardation time of the dashpot of k-element; ψ = pipe constraint coefficient, weighting factor; Subscripts: g = gas; mp = midpoint; O = relating to an orifice; ve = valve; 0 = based on steady-state or reference conditions; elastic component Abbreviations: DGCM = discrete gas cavity model; 11