Avenue, , Lisbon, Portugal c Instituto Superior Técnico, Technical University of Lisbon (TULisbon), Rovisco Pais

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1 This article was downloaded by: [IAHR ] On: 3 April 212, At: 2:11 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Hydraulic Research Publication details, including instructions for authors and subscription information: Transient vaporous cavitation in viscoelastic pipes Alexandre K. Soares a, Dídia I.C. Covas b & Nelson J.G. Carriço c a School of Civil Engineering, Federal University of Goiás, Praça Universitária, Goiânia, Brazil b Instituto Superior Técnico, Technical University of Lisbon (TULisbon), Rovisco Pais Avenue, 149-1, Lisbon, Portugal c Instituto Superior Técnico, Technical University of Lisbon (TULisbon), Rovisco Pais Avenue, 149-1, Lisbon, Portugal Available online: 3 Apr 212 To cite this article: Alexandre K. Soares, Dídia I.C. Covas & Nelson J.G. Carriço (212): Transient vaporous cavitation in viscoelastic pipes, Journal of Hydraulic Research, 5:2, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Journal of Hydraulic Research Vol. 5, No. 2 (212), pp International Association for Hydro-Environment Engineering and Research Research paper Transient vaporous cavitation in viscoelastic pipes ALEXANDRE K. SOARES (IAHR Member), Assistant Professor, School of Civil Engineering, Federal University of Goiás, Praça Universitária, Goiânia, Brazil. aksoares@gmail.com (author for correspondence) DÍDIA I.C. COVAS, Associate Professor, Instituto Superior Técnico, Technical University of Lisbon (TULisbon), Rovisco Pais Avenue, Lisbon, Portugal. didia.covas@civil.ist.utl.pt Downloaded by [IAHR ] at 2:11 3 April 212 NELSON J.G. CARRIÇO, PhD Student, Instituto Superior Técnico, Technical University of Lisbon (TULisbon), Rovisco Pais Avenue, Lisbon, Portugal. nelson.carrico@gmail.com ABSTRACT The research focuses on the analysis of transient cavitating flow in polyethylene pipes, which are characterized by a viscoelastic rheological behaviour. A hydraulic transient solver incorporating the description of the rheological mechanical behaviour of these pipes and the cavitating pipe flow has been developed. Both the discrete vapour cavity model and the discrete gas cavity model have been used to describe transient cavitating flow. These assume that discrete air cavities are formed at fixed pipe sections and consider a constant wave speed in the pipe reaches between the cavities. An experimental programme was conducted in a pipe-rig composed of a high-density polyethylene (HDPE) pipeline. Numerical results have been compared with collected data resulting in a good agreement. Effects related to the viscoelasticity of HDPE pipes and to the occurrence of vapour pressures during the transient events are also discussed. Keywords: Cavitation; fluid transient; polyethylene pipe; viscoelasticity; water hammer 1 Introduction The prediction of maximum transient pressures in pipe systems is typically used to verify if pipe materials, pressure classes, and wall thicknesses are sufficient to withstand pressure loads to avoid pipe rupture or system damage. A verification of the minimum allowable pressures is important to prevent cavitation and column separation, to avoid pipe collapse. Two main types of transient cavitation occur in fluid systems (Wylie and Streeter 1993): gaseous cavitation, in which the flow is characterized by the presence of micro-bubbles of free gas distributed along the pipe so that the wave speed is pressure-dependent (Kranenburg 1974, Wiggert and Sundquist 1979, Wylie 198, 1992, Chaudhry et al. 199, Pezzinga 23, Cannizzaro and Pezzinga 25), and vaporous cavitation, in which the liquid flow is completely separated by its vapour phase as the cavity is formed due to a pressure drop to vapour pressure as the basis of column separation (Wylie 1984, Simpson and Wylie 1991, Simpson and Bergant 1994a, Bergant and Simpson 1999, Shu 23). Actually, both phenomena can occur during low pressure transients (Wylie 1992). The response may involve column separation and subsequent rejoining, vaporization and condensation, air release, dispersion of wave fronts, and shock waves. Cavitating pipe flows have been studied by using data from laboratory (the majority) and field tests; yet, it has hardly been studied in plastic pipes. Mitosek (2) experimentally investigated the factors influencing pressure changes by transient vaporous cavitation in medium density polyethylene and in unplasticized polyvinyl chloride pipes. Borga et al. (24) described transient cavitating flows in high-density polyethylene (HDPE) pipes by incorporating reduction parameters due to the non-elastic behaviour of pipe walls and variable celerity. The same test rig was used by Soares et al. (28a), by incorporating a viscoelastic (VE) transient solver. Hadj-Taïeb and Hadj-Taïeb (29) proposed a mathematical model numerically solved by the two-step Lax-Wendroff scheme to describe the vaporous cavitation in VE pipes; the mixture density is calculated by means of Revision received 21 February 212/Open for discussion until 31 October 212. ISSN print/issn online 228

3 Journal of Hydraulic Research Vol. 5, No. 2 (212) Transient vaporous cavitation in VE pipes 229 Downloaded by [IAHR ] at 2:11 3 April 212 a non-linear expression of the liquid volume fraction instead of the linear Henry law. This work presents numerical results obtained by combining three dynamic effects in hydraulic transients, namely, unsteady friction (UF), pipe wall viscoelasticity, and cavitation. Physical data were collected from an experimental HDPE pipeline, assembled at Instituto Superior Técnico, Lisbon, Portugal. A transient solver incorporating the above three effects has been developed and the collected data were compared with the numerical results. Conclusions are drawn concerning the importance of considering UF, pipe wall viscoelasticity, and distributed cavitation models during cavitating flows in plastic pipes. 2 Mathematical models 2.1 Basic equations One-dimensional transient flows in VE pipes are described by the momentum and the continuity equations (Almeida and Koelle 1992, Covas et al. 25, Soares et al. 28b) dh dt 1 dq + g H A dt x + gh f = (1) + a2 Q ga x + 2a2 dε r = g dt (2) where x is the coordinate along the pipe axis, t is time, H the piezometric head, Q the discharge, a the elastic wave speed, g the gravity acceleration, A the pipe cross-sectional area, ε r the retarded strain, and h f the head loss per unit length. If subjected to a constant stress, plastic material presents an instantaneous elastic strain followed by a gradual time-dependent retarded strain, which describes the creep compliance function of the material. The strain stress equation (i.e. the total strain generated by a continuous application of a stress σ t )is t ε(t) = J σ(t) + σ(t t ) J (t ) dt (3) t in which J is the instantaneous creep compliance and J (t ) the creep function at time t. The creep function is described by a mathematical expression, which is incorporated in the hydraulic transient equations by the generalized Kelvin Voigt mechanical model of a VE solid as N KV J (t) = J + J k (1 e t/τ k ) (4) k=1 where J is the creep compliance of the first spring (J = 1/E, with E as Young s modulus of pipe elasticity), J k the creep compliance of the spring of the Kelvin Voigt k-element (J k = 1/E k, with E k as the modulus of elasticity of k-th spring element), τ k = μ k /E k the retardation time of the dashpot of element k, with μ k as the corresponding viscosity, and N KV as the number of Kelvin Voigt elements. The method of characteristics (MOC) has been used to solve the system of Eqs. (1) (3) (Covas et al. 25 or Soares et al. 28b). 2.2 Cavitating pipe flow models Discrete vapour cavity model The discrete vapour cavity model (DVCM) is based on the column separation hypothesis. It assumes that discrete air cavities are formed at fixed pipe sections by considering that a constant wave speed in the pipe reaches between these cavities (Bergant et al. 26). This model is particularly adequate if the pipe profile has sections with higher elevation, where air tends to form air pockets, or if only a portion of the system is subjected to vapour pressure. MOC is used to solve basic equations with a fixed grid and, whenever a particular section of the pipe reaches a pressure below vaporization pressure, it is treated as an internal boundary condition with the absolute pressure in a cavity set equal to the vapour pressure p = p v. The up- and downstream discharges Q Pu and Q P at a cavity are computed from the compatibility relations by integrating Eqs. (1) and (2) and ignoring mass transfer during cavitation; its volume follows from d v dt = Q P Q Pu (5) which is numerically approximated in the MOC with a staggered grid by ( v ) t P = ( v) t 2 t P +[ψ(qp t Qt Pu ) + (1 ψ)(qt 2 t P Q t 2 t Pu )]2 t (6) Here ( v ) t P and ( v) t 2 t P are the cavity volumes at the current time step and at 2 t time steps earlier, respectively, and ψ is a numerical weighting factor. The cavity collapses when its calculated volume becomes less than zero. The liquid phase is re-established and the standard water hammer procedure is valid again. Although the DVCM is easily implemented, it has deficiencies such as (Shu 23): (i) restrictions to avoid the prediction of negative cavity sizes, resulting in unrealistically large pressure spikes; (ii) the internal boundary condition allows for vapour cavities to be formed only at computing nodes, so that the results are biased according to where the computing nodes are located; (iii) the model is limited in its ability to model cavitation because the cavity size and its mass transfer are ignored; and (iv) a discharge discontinuity is assumed at each computing node so that two predicted discharges result. The difference between these values increases for a low number of computing nodes. However, if many computing nodes are used, a corresponding number of discontinuities leads to an internally inconsistent mathematical model. Simpson and Bergant (1994b) recommend that the

4 23 A.K. Soares et al. Journal of Hydraulic Research Vol. 5, No. 2 (212) Downloaded by [IAHR ] at 2:11 3 April 212 maximum volume of discrete cavities should be less than 1% of the reach volume. Discrete gas cavity model Assuming a homogeneous gas liquid mixture with gas bubbles uniformly distributed, and the pressure within the bubbles independent of surface tension and vapour pressure, results in expressions for the wave velocity. For typical values of dissolved air in the fluid (<1%), fluid density tends to the liquid density (ρ m = ρ l ), so that its wave speed is given by (Wylie and Streeter 1993) a m = a 1 + (α ρ l a 2 /p g ) (7) where subscripts m, l, and g denote mixture, liquid, and gas, respectively, a is the wave speed, p the absolute pressure, α the pressure-dependent volumetric ratio of gas in the mixture, α = g / m, with g as the gas cavity volume and m the mixture volume. As the wave speed varies, there are two methods for solving the characteristic equations by using a: (i) variable characteristic grid model or (ii) constant grid with space or time interpolations. The latter is easier to handle water column separation in multi-pipe systems. An alternative to model free gas distributed throughout the liquid in a homogeneous mix is achieved by lumping the mass of free gas at computing sections, leading to the discrete gas cavity model () involving the MOC to solve the equations. Each isolated small volume of gas isothermally expands and contracts as the pressure varies, according to the perfect gas law (Wylie and Streeter 1993) ( g ) t P (H t P z P H v ) = (H z H v )α A x (8) where H is the reference piezometric head, α the void fraction at H, z the pipe elevation, and H v the gauge vapour pressure head of the liquid. In, the total absolute pressure in the gas cavity corresponds to the sum of partial pressures of gas and vapour. The continuity equation for the gas cavity volume is ( g ) t P = ( g) t 2 t P +[ψ(qp t Qt Pu ) + (1 ψ)(qt 2 t P Q t 2 t Pu )]2 t (9) As in DVCM, between each computing section and concentrated gas volume, pure liquid with a constant wave speed is assumed. The is also able to simulate vaporous cavitation by utilizing a low initial gas void fraction (α 1 7 ) (Wylie 1984, Simpson and Bergant 1994b). 3 Experimentation Experiments were performed in a set-up composed of HDPE pipes (Fig. 1). The facility includes a single pipe of 23 m total length, of 44 mm inner diameter, of 3 mm wall thickness, an upstream air vessel, and a downstream free discharge valve. The pipe was on the ground as a coil with small differences in elevation of.2 m, except for the downstream end discharging to a tank 1 m above the ground. An electromagnetic flow meter was used to measure the steady-state discharge to ±.2%. The data acquisition system was composed of an acquisition board Pico Figure 1 Scheme of experimental set-up with HDPE pipes

5 Journal of Hydraulic Research Vol. 5, No. 2 (212) Transient vaporous cavitation in VE pipes 231 (with four channels), four pressure transducers WIKA PN25 with an absolute pressure range from to 25 bar and an accuracy of ±.5% of the full range, and a laptop computer. The transient pressure data were collected with a frequency of 5 Hz at four pipe sections: at the air vessel (T1), downstream the ball valve located at the upstream pipe end (T4), at the pipe centre (T2), and at downstream pipe end (T3). The ball valve immediately downstream of the air vessel was used to generate the transient cavitating flow. Pressure data collected at T1 and T3 have been used for steady-state calibrations and T3 for noncavitating flow calibrations. 4 Analysis of results 4.1 Calibration Numerical Results (VE) Numerical Results (VE + UF) 1 t (s) 2 J(t): VE Downloaded by [IAHR ] at 2:11 3 April 212 The creep function was calibrated for non-cavitating flow considering unsteady friction (UF) as described by Vítkovský et al. (2) with a decay coefficient of k 3 =.3. A set of VE parameters was determined by an inverse model based on the two-step procedure, namely, the Genetic Algorithm and the Levenberg Marquardt search method (Soares et al. 211). The calculated elastic wave speed was 315 m/s and, according to the recommendation of Covas et al. (25) for HDPE pipes, three Kelvin Voigt elements were determined via inverse calculations (τ 1 =.18 s, J 1 =.256 GPa 1, τ 2 =.5 s, J 2 =.238 GPa 1, and τ 3 = 3. s, J 3 =.211 GPa 1 for only VE; τ 1 =.18 s, J 1 =.256 GPa 1, τ 2 =.5 s, J 2 =.21 GPa 1, and τ 3 = 3. s, J 3 =.2 GPa 1 for VE + UF). Parameters J k and τ k were estimated by using transient pressure data collected at transducer T3. The numerical results are shown in Fig. 2. Creep functions considering and neglecting UF indicate minor differences, which proves that UF effect on transient dissipation and the phase shift is negligible if compared with the VE effect in this pipe rig. The obtained creep function assuming UF is an approximation of the actual VE mechanical pipe behaviour. Uncertainties, including the axial and circumferential pipe constraints (pipe anchorage), confining soils for buried pipes, temperature, the stress time history of the pipe, and fluid inertia effects can modify the numerically-estimated creep. Even for identical discharge, the creep functions varied slightly from test to test; as successive transients were carried out, the pipe tended to relax not having sufficient time for complete recovery. In conclusion, calibrated creep functions may not represent the accurate rheological behaviour of an HDPE pipe wall, but the combination of mechanical behaviour of the pipe material in the system and of other damping effects (Soares et al. 28b, 211). 4.2 Verification The DVCM and the creep function calibrated for non-cavitating flows were used to describe cavitating flows. All simulations involve a staggered grid with computational sections spaced by x = 2 m. The numerical results from the DVCM taking into account the VE behaviour are compared with the experimental J (GPa -1 ) 1 J(t): VE + UF 5 t (s) 1 Figure 2 Non-cavitating tests numerical results and collected data at transducer T3, calibrated creep functions with and without UF for Q = 2.72 l/s, R 8,, a = 315 m/s data in Fig. 3 for location T4. The results indicate that this cavitation model cannot describe the attenuation and dispersion of observed transient pressures. Additionally to the deficiencies stated by Shu (23), discrepancies between the numerical results and the collected data, even when including the VE model, are due to the assumption of equal absolute pressure in the gas cavities and the vapour pressure causing unrealistic overpressures as well as to the energy dissipation during the gas expansion and contraction. The was tested to analyse if the results were better to describe the system behaviour, considering a small initial void fraction of α = 1 8. These numerical results and the creep function calibrated for non-cavitating flows are shown in Fig. 3. It was found that: (i) a better adjustment was obtained when using than DVCM, (ii) the assumption of the ideal gas law is more appropriate than the simple use of vapour pressure as it influences the energy dissipation during the expansion and contraction of gas cavities, (iii) an exponent of 1 in the polytropic gas was assumed to obtain explicit equations; yet, the resulting implicit formulation is recommended given the highly non-linear response, (iv) free gas is assumed to behave isothermally, which is valid for tiny bubbles; in this study, large bubbles were formed on the upper portion of the pipe cross-section and growth along the near-horizontal pipeline, (v) shock waves may form due to the steepening of pressure waves, but does not account for shock-wave front modelling as interface models

6 232 A.K. Soares et al. Journal of Hydraulic Research Vol. 5, No. 2 (212) 4 DVCM 4 DVCM t (s) t (s) Downloaded by [IAHR ] at 2:11 3 April t (s) 2 Figure 3 Numerical results taking into account pipe wall VE versus experimental data at transducer T4 DVCM, for Q = 4. l/s, R 12,, a = 315 m/s and bubble flow models do, (vi) small pipe displacement during transients or a free discharge outlet at the downstream pipe end lead to more uncertainties on the system behaviour. Elongated large bubbles were formed at the pipe top, increasing the complexity if pure liquid and bubbles are moving with different velocities. 4.3 Calibration of creep function A new creep function was calibrated for cavitating flow considering both DVCM and. UF was neglected as its effect on transient pressures was minor as compared with the present pipe wall VE (Section 4.1). The considered absolute vapour pressure head at 25 C was.33 m. The elastic wave speed was kept at 315 m/s and three Kelvin Voigt elements were determined via inverse calculations (τ 1 =.1 s, J 1 =.1 GPa 1, τ 2 =.5 s, J 2 =.695 GPa 1, and τ 3 = 3. s, J 3 =.69 GPa 1 ) by using transient pressure data collected at transducer T4. The numerical results shown in Fig. 4 indicate that neither DVCM nor describe the attenuation and dispersion of the observed transient pressures, even at the elastic wave speed of 315 m/s. By analysing the experimental elastic wave speed (Section 4.4), a final set of VE parameters was calibrated considering only. The elastic wave speed was set at 287 m/s and three Kelvin Voigt elements were determined, defining the creep function for cavitating flows J (τ 1 =.1 s, J 1 =.6 GPa 1, τ 2 =.5 s, J 2 =.35 GPa 1, and τ 3 = 3. s, J 3 =.5 GPa 1 ) t (s) 2 Figure 4 Numerical results taking into account pipe wall VE versus experimental data at transducer T4 DVCM, for Q = 4. l/s; R 12, ; a = 315 m/s Figure 5 shows the numerical results for Q = 4. l/s. The creep function J was also validated for Q = 3. l/s (Fig. 6). Results from the numerical simulations indicate that J is higher than that for non-cavitating flows (Fig. 7). This is due to two main reasons, namely, (1) the pipe rig was not rigidly fixed to the ground and during the transients a small axial movement was observed, (2) the air cavity was formed at the upstream end as negative transient pressure propagated to downstream and dissolved as the positive pressure wave returned; this cavity neither occupied the full pipe cross-section nor did it instantaneously and completely dissolve in the fluid as the pressure increased above vapour pressure; some inevitable dissolved air bubbles remained in the fluid even for positive pressures. Both the axial pipe displacement and the additional dissolved air delayed the transient pressure wave propagation, numerically described by the high value of estimated creep function. J does not represent the actual rheological behaviour of the HDPE material, but the combination of UF, the pipe wall VE, pipe constraints, and additional dissolved air. Despite these effects of different physical origin and their behaviour depending on the type of flow (Duan et al. 21), an artificial creep function describes them numerically. 4.4 Wave speed analysis The validity of the elastic wave speed calibrated can be analysed by scrutinizing the transient pressures observed in the

7 Journal of Hydraulic Research Vol. 5, No. 2 (212) Transient vaporous cavitation in VE pipes J (GPa -1 ) 2 1 J*(t): VE (cavitating flow) J(t): VE (non-cavitating flow) J(t): VE + UF (non-cavitating flow) t (s) 2 5 t (s) 1 Figure 7 Calibrated creep functions 2 4 T4 t* =.355 s L = 12 m a 287 m/s Downloaded by [IAHR ] at 2:11 3 April t (s) 2 Figure 5 numerical results taking into account pipe wall VE versus experimental data at transducers T4, T2 for Q = 4. l/s; R 12, ; a = 287 m/s t (s) 3. T2 T2 t* =.32 s L = 11 m a 316 m/s t (s) 2 T t (s) 3. Figure 8 Wave speed calculation by tracking pressure wave experimental data with cavitation, experimental data without cavitation for Q = 4. l/s t (s) 2 Figure 6 numerical results taking into account pipe wall VE versus experimental data at transducers T4, T2 for Q = 3. L/s; R 87, ; a = 287 m/s experiments. Pressure head data obtained for transducers T2 (pipe centre) and T4 (upstream of pipe) during cavitating tests are shown in Fig. 8. The initial pressure wave reflection obtained by upstream ball valve closure at T4 travels over the pipe and reaches T2 (about 12 m away from T4) after.355 s, resulting in a wave speed of 287 m/s (note that in cavitation simulations the elastic wave speed was assumed equal to 287 m/s). This lower wave speed value is due to air dissolved in the fluid that, even in a minute percentage, can strongly decrease the wave speed (e.g..1% of air dissolved decreases celerity by 1%), and that has been dissolved in the fluid in the previous cavitating flow tests.

8 234 A.K. Soares et al. Journal of Hydraulic Research Vol. 5, No. 2 (212) Downloaded by [IAHR ] at 2:11 3 April 212 The elastic wave speed for cavitation tests can be compared with that for non-cavitation tests. Figure 8 shows pressure head data obtained for transducers T2 and T3 (downstream of pipe) during non-cavitating tests. After the downstream ball valve closure at T3, a pressure wave propagates over the pipe reaching T2 (about 11 m away from T3) after.32 s, resulting in an elastic wave speed of 316 m/s, in accordance with that used in simulations without cavitation. 5 Conclusions This research presents experimental tests and numerical analyses of water hammer with cavitation in HDPE pipes. Pressure data in smooth-wall turbulent conditions were collected during transient events caused by valve closure. A hydraulic transient solver accounting for UF, pipe wall VE behaviour, and cavitating flow has been developed. The collected data were used to calibrate and compare two developed models for cavitating flows: DVCM and. The numerical results indicate that the DVCM is imprecise to describe the hydraulic system behaviour. The assumption of the ideal gas law inherent in the is more appropriate than the simple adoption of vapour pressure as the pressure reaches vapour pressure, and induces more attenuation and dispersion of transient pressures. Despite the excellent adjustment obtained between the numerical results and the pressure data for non-cavitating flow considering both viscoelasticity and UF, it should be noted that: the contribution of UF for the transient dissipation is negligible as compared with the viscoelasticity effect, as close calibrated creep functions resulted in this study; and the pipe movement and the additional dissolved air introduced in the fluid during the cavitating flow (even during positive pressures) introduce an additional pressure damping and phase shift that were described by a calibrated artificial creep function; as a result this function was almost the double of the actual creep of the HDPE pipe. In conclusion, dissolved air delays the transient events and, no matter how accurate the cavitation model is, there are always inevitable uncertainties associated with the physical system artificially described by parameters of the numerical model. Considering these analyses, cavitating flows in pressurized plastic pipe systems have to be further investigated. Future research work should consider other numerical methods, such as two-dimensional models coupled with the MOC or computational fluid dynamic models for the description of cavitating flows transients in plastic pipes. Acknowledgements The authors gratefully acknowledge the financial support of Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, Brazil), and the Portuguese Foundation for Science and Technology (FCT), who provided a postdoctoral scholarship to the first author. Notation A = pipe cross-sectional area (m 2 ) a = elastic wave speed (m/s) E = Young s modulus of pipe elasticity (Pa) E k = Young s modulus of spring elasticity (Pa) g = gravity acceleration (m/s 2 ) H = piezometric head (m) h f = head loss per unit length ( ) J, J = creep compliance (Pa 1 ) J = instantaneous or elastic creep compliance (Pa 1 ) J k = creep of springs of Kelvin Voigt elements (Pa 1 ) k 3 = UF decay coefficient ( ) L = pipe length (m) N KV = number of Kelvin Voigt elements ( ) p = absolute pressure (Pa) p v = vapour pressure (Pa) p/γ = pressure head (m) Q = discharge (m 3 /s) Q = steady-state discharge (m 3 /s) R = Reynolds number ( ) t, t = time (s) x = coordinate along pipe axis (m) z = elevation (m) α = pressure-dependent volumetric ratio of gas in mixture (void fraction) ( ) ε r = retarded strain (m/m) t = time-step increment (s) x = space-step increment (m) μ = viscosity of dashpots (kg/sm) ρ = fluid density (kg/m 3 ) σ = stress (Pa) τ k = retardation time of Kelvin Voigt elements (s) ψ = numerical weighting factor ( ) = volume of cavity (m 3 ) Subscripts g = gas l = liquid m = mixture gas liquid v = vapour References Almeida, A.B., Koelle, E. (1992). Fluid transients in pipe networks. Computational Mechanics Publications, Elsevier Applied Science, Southampton UK. Bergant, A., Simpson, A.R. (1999). Pipeline column separation flow regimes. J. Hydraulic Eng. 125(8), Bergant, A., Simpson, A.R., Tijsseling, A. (26). Water hammer with column separation: A historical review. J. Fluids Struct. 22(2), Borga, A., Ramos, H., Covas, D., Dudlik, A., Neuhaus, T. (24). Dynamic effects of transient flows with cavitation in pipe

9 Journal of Hydraulic Research Vol. 5, No. 2 (212) Transient vaporous cavitation in VE pipes 235 Downloaded by [IAHR ] at 2:11 3 April 212 systems. Proc. 9th Int. Conf. Pressure Surges 2, BHR Group, Bedfordshire UK. Cannizzaro, D., Pezzinga, G. (25). Energy dissipation in transient gaseous cavitation. J. Hydraulic Eng. 131(8), Chaudhry, M.H., Bhallamudi, S.M., Martin, C.S., Naghash, M. (199). Analysis of transient pressures in bubbly, homogeneous, gas-liquid mixtures. J. Fluids Eng. 112(2), Covas, D., Stoianov, I., Mano, J., Ramos, H., Graham, N., Maksimovic, C. (25). The dynamic effect of pipe-wall viscoelasticity in hydraulic transients 2: Model development, calibration and verification. J. Hydraulic Res. 43(1), Duan, H.F., Ghidaoui, M., Lee, P.J., Tung, Y.K. (21). Unsteady friction and visco-elasticity in pipe fluid transients. J. Hydraulic Res. 48(3), Hadj-Taïeb, L., Hadj-Taïeb, E. (29). Numerical simulation of transient flows in viscoelastic pipes with vapour cavitation. Int. J. Modelling and Simulation 29(2), Kranenburg, C. (1974). Gas release during transient cavitation in pipes. J. Hydraulics Div. ASCE 1(HY1), Mitosek, M. (2). Study of transient vapor cavitation in series pipe systems. J. Hydraulic Eng. 126(12), Pezzinga, G. (23). Second viscosity in transient cavitating pipe flows. J. Hydraulic Res. 41(6), Shu, J.J. (23). Modelling vaporous cavitation on fluid transients. Int. J. Pressure Vessels and Piping 8(3), Simpson, A.R., Bergant, A. (1994a). Numerical comparison of pipe-column-separation models. J. Hydraulic Eng. 12(3), Simpson, A.R., Bergant, A. (1994b). Developments in pipeline column separation experimentation. J. Hydraulic Res. 32(2), Simpson, A.R., Wylie, E.B. (1991). Large water-hammer pressures for column separation pipelines. J. Hydraulic Eng. 19(5), Soares, A.K., Covas, D.I.C., Ramos, H.M., Reis, L.F.R. (28a). Unsteady flow with cavitation in viscoelastic pipes. Proc. 24 th IAHR Symp. Hydraulic Machinery and Systems, Foz do Iguassu, Brazil 152, 1 13 (CD-ROM). Soares, A.K., Covas, D.I.C., Reis, L.F.R. (28b). Analysis of PVC pipe-wall viscoelasticity during water hammer. J. Hydraulic Eng. 134(4), Soares, A.K., Covas, D.I.C., Reis, L.F.R. (211). Leak detection by inverse transient analysis in an experimental PVC pipe system. J. Hydroinformatics 13(2), Vítkovský, J.P., Lambert, M.F., Simpson, A.R. (2). Advances in unsteady friction modelling in transient pipe flow. Proc. 8 th Int. Conf. Pressure Surges, , A. Anderson, ed. BHR Group, Suffolk UK. Wiggert, D.C., Sundquist, M.J. (1979). The effect of gaseous cavitation on fluid transients. J. Fluids Eng. 11(1), Wylie, E.B. (198). Free air in liquid transient flow. Proc. 3 rd Int. Conf. Pressure Surges, BHRA, Cantenbury UK. Wylie, E.B. (1984). Simulation of vaporous and gaseous cavitation. J. Fluids Eng. 16(3), Wylie, E.B. (1992). Low void fraction two-component twophase transient flow. Unsteady flow and fluid transients, 3-9, R. Bettess, J. Watts, eds. Balkema, Rotterdam NL. Wylie, E.B., Streeter, V.L. (1993). Fluid transients in systems. Prentice Hall, Englewood Cliffs NJ.