International Journal of Civil Engineering & Geo-Environment 5 (2014) International Journal of Civil Engineering and Geo-Environment Journal home page: http://ijceg.ump.edu.my ISSN:21802742 Investigation of Parameters Affecting Discrete Vapour Cavity Model Hamid Shamloo 1, Reyhaneh Norooz 1*, Maryam Mousavifard 1 1 Department of Civil Engineering, K.N. Toosi University of Technology, Tehran 19697, Iran. A R T I C L E I N F O Keywords: Column separation Discrete vapour cavity model Pipelines Quasi-steady friction model Unsteady friction model Water hammer A B S T R A C T Rapid valve closure or pump shutdown causes a fast transient flow with column separation which causes problems such as hydraulic equipment damage and cavitation. This paper presents a numerical study of water hammer with column separation in a simple reservoirpipeline-valve system. The governing equations for two-phase transient flow in pipes have been solved based on the method of characteristics using a discrete vapour cavity model (DVCM) which has been coupled with four different friction models including the Brunone, Zielke, Vardy and Brown, and quasi-steady friction. The DVCM may produce unrealistic pressure pulses (spikes) due to the collapse of multi-cavities. Studies of the DVCM have suggested that using a weighting factor close to 1.0, using unsteady friction term in the DVCM and restricting the number of reaches improve the computational results and the present paper explores this suggestion in detail. The simulation results clearly show that using unsteady friction term in the DVCM reduces the unrealistic pressure pulses. It has been observed that using a weighting factor close to 1.0 and restricting the number of reaches do not improve the computational results in all cases. Instead, proper selection of these parameters reduces the unrealistic pressure pulses. 1. Introduction Vapourous cavitation occurs when the liquid pressure drops to the liquid vapour pressure. It may happen in two different types of cavitating flow: (1) a localized vapour cavity (large void fraction) and (2) distributed vapourous cavitation (small void fraction). The collapse of a large vapour cavity and the propagation of the shock wave through the vapourous cavitation zone cause the vapour change to liquid. When vapour cavities change to liquid, large pressures with steep wave fronts may happen. As an outcome, fluid transients may lead to severe accidents (De Almeida, 1991). Therefore, it is important to predict the commencement and amount of cavitation occurring in order to improve the performance and reliability of systems. Various types of vapourous cavitation models have been introduced (Wylie and Streeter, 1993; Bergant et al., 2006) including discrete cavity and interface models. The discrete vapour cavity model (DVCM) (Wylie and Streeter, 1978; 1993) is the most popular model for column separation and distributed cavitation in recent years (Bergant et al., 2006). The DVCM may produce unrealistic pressure pulses (spikes) due to the collapse of multi-cavities (Bergant et al., 2007). Studies of the DVCM have suggested that using a weighting factor close to 1.0, restricting the number of reaches (Bergant and Simpson, 1994) and using an unsteady friction term in the DVCM improve the computational results. Practical implications of column separation led to intensive laboratory and field research starting at the end of 19th century (Joukowsky, 1900). Wylie and Streeter (1978, 1993) have described the DVCM in detail and they provided a computer code for its implementation. Researchers have attempted to incorporate a number of *Corresponding author. Email address: reyhaneh_blueparto@yahoo.com
International Journal of Civil Engineering & Geo-Environment 5 (2014) unsteady friction models into DVCM. Shuy and Apelt (1983) performed numerical analyses with a number of friction models including steady, quasi-steady and unsteady friction models. For the case of water hammer (no cavitation) they found little differences in the results of the models, but for the case with column-separation (two-phase flow) large discrepancies occurred. Brunone et al. (1991) used the DVCM in combination with an instantaneous-acceleration unsteady friction model. Significant discrepancies between experiment and theory were found for all runs when using a quasi-steady friction term. Bergant and Simpson (1994) investigated the performance of quasi-steady and unsteady friction models similar to those used by Shuy and Apelt (1983). The instantaneous-acceleration and convolution-based unsteady friction models gave the best fit with experimental data for the case of water hammer. Bughazem and Anderson (2000) developed the DVCM with an instantaneous-acceleration unsteady friction term and found good agreement between theory and experiment. Numerical studies by Bergant and Tijsseling (2001) have shown that unsteady friction may cause a significant damping of the pressure spikes observed in measurements. Shu (2003) extended a two-phase flow model with Zielke s (1968) friction model. In this study, water hammer with column separation has been modelled using the DVCM coupled with the Zielke (1968) and Vardy and Brown unsteady friction models and the quasi-steady friction model. The results and the efficiency of the friction models have been compared with each other and sensitivity of this method to some of the parameters affecting the DVCM have been investigated. First, the structure of the model is explained and then the model is considered to study the column separation in a reservoir-pipeline-valve system. 2. Discrete Vapour Cavity Model In the DVCM, cavities are allowed to form at the computational sections if the computed pressure becomes less than the liquid vapour pressure. The classical water hammer solution is no longer valid at a vapour pressure section. Pure liquid with a constant wave speed c is assumed to occupy between computational sections. Each discrete vapour cavity is fully governed by two water hammer compatibility equations, one continuity equation for the vapour cavity volume and the constant vapour head. The compatibility equations (Wylie and Streeter, 1993) and continuity equation for the cavity volume (Wylie, 1984), written in a finite-difference form for the i- th computational section within the diamond grid, are: a) Water hammer compatibility equation along the C + characteristic line (Δx/Δt = c): c f x H H Q Q Q Q ga 2gDA 2 n n1 n, u n1, d n, u n1, d i i1 i i1 i i1 0 b) Water hammer compatibility equation along the C - characteristic line (Δx/Δt = -c): c f x H H Q Q Q Q ga 2gDA 2 n n 1 n, d n 1, u n, d n 1, u i i1 i i1 i i1 0 c) Continuity equation for the cavity volume: n n2 n2, d n2, u n, d n, u vap vap i i i i 1 Q Q Q Q 2t i i Where x = distance along the pipe, c = liquid wave speed, f = Darcy-Weisbach friction factor, D = internal pipe diameter, g = gravitational acceleration, H = pressure head, A = cross-sectional flow area, Q u = upstream discharge, Q d = downstream discharge, vapour cavity volume and ψ = weighting factor. vap = discrete The weighting factor ψ takes values between 0 and 1.0. The cavity collapses when its calculated volume becomes negative and the one-phase liquid flow is re-established and the water hammer solution 2.1 Friction model The friction term in one-dimensional transient flow is expressed as the sum of the unsteady part f u and the quasisteady part f q: f f f q Setting f u = 0 leads to the quasi-steady friction model. There are several friction models which have been introduced by many researchers to consider the unsteady part. The Brunone model (Brunone et al., 1991) has special popularity because of its simplicity and accuracy. This model was improved by Vitkovsky in 1998 (Bergant and Simpson, 1994; Brunone et al., 1995): f u KD V V csignv V V t x Where, K is the Brunone s friction coefficient and x is the distance along the pipe and sign (V) = (+1 for V 0 or -1 for V < 0). Vardy and Brown (1996) proposed the following empirical relationship to derive the Brunone s coefficient analytically (Bergant et al., 2001; Vardy and Brown, 1996): u (1) (2) (3) (4) (5)
K C 2 (1) A. e W / C (8) The Vardy s shear decay coefficient C* from Vardy and Brown (1996) is: with a) laminar flow: C 0.00476 (2) b) turbulent flow: C R 7.41 log 14.3/ R 0.05 in which R = Reynolds number (R = VD/υ). The original version of Zielke s model (Zielke, 1968) is used in this paper. The model was analytically developed for transient laminar flow. The unsteady part of friction term is related to the weighted past velocity changes at a computational section: k1 32 ( f ) V V W (( k j) t) (4) u i, k i, j1 i, j1 DVi, k Vi, k j1 (3) A 1 12.86 15.29 0.2821; ; log10 0.0567 2 c R (14) R According to the authors this model is valid for the initial Reynolds numbers R < 10 8 and for smooth pipes only. 3. Experimental Apparatus The computational results are compared with the results of experimental studies conducted by Bergant and Simpson (1995) which were carried out using a long horizontal pipe with length of 37.20 m and inner diameter of 0.0221 m that connects upstream and downstream reservoirs (see Figure 1). The water hammer wave speed was experimentally determined as c = 1319 m/s. A transient event is initiated by a rapid closure of the ball valve. Pressures measured at the valve (H v) and at the midpoint (H mp) are presented in this paper. i 0.02 : W( ) e 5 i1 n (5) 0.02 : W ( ) 6 i1 m e i i2 / 2 (6) 4 2 k jt D in which j and k = multiples of the time step Δt, W = weights for past velocity changes, υ = kinematic viscosity, τ = dimensionless time, and coefficients {n i, i = 1,..., 5} = {-26.3744, -70.8493, -135.0198, -218.9216, 322.5544} and {m i, i = 1,..., 6} = {0.282095, -1.25, 1.057855, 0.937500, 0.396696, -0.351563}. The velocity profile analyses for turbulent unsteady flow allows o state that the relation (Eqn. (4)) proposed by Zielke is correct for turbulent unsteady flows if only a weighting function W would be related to the Reynolds number (Vardy and Brown, 1995). The Vardy and Brown obtained R-dependent weighting function W is: (7) Figure 1: Experimental set up 4. Comparison of Numerical Models In order to investigate the performance of the DVCM and the effects of different friction models on accuracy of the results, the numerical and experimental results are compared. Computational runs are performed for a rapid closure of the valve positioned at the downstream end of the horizontal pipe at the downstream reservoir (see Fig). The initial velocity is V 0 = 0.30 m/s and the constant static head in the upstream reservoir and the vapour pressure head are H ur = 22 m and H vap = -10.25 m. The rapid valve closure begins at time t = 0 s. To study the effects of mesh size, two numbers of reaches were selected, N x = {32, 128}. The weighting factor ψ in Eqn. was chosen 0.5 and 1.0 in order to investigate its impact on the accuracy of
International Journal of Civil Engineering & Geo-Environment 5 (2014) the computational results. The maximum measured head at the valve is 95.6 m for t = 0.1766 s. Comparison of computational and measured results for the unsteady and quasi-steady friction models with an initial flow velocity V 0 = 0.30 m/s is presented in agrees well till 0.22 sec. The discrepancies between the results are magnified at later. The maximum computed heads predicted by the models are: (1) DVCM-Brunone s unsteady friction model: H v,max = {N x = 32, ψ = 1, 110.10 m at t = 0.173 s} (2) DVCM-Zielke s unsteady friction model: H v,max = {N x = 32, ψ = 1, 104.25 m at t = 0.169 s} (3) DVCM-Vardy & Brown s unsteady friction model: H v,max = {N x = 32, ψ = 1, 105.12 m at t = 0.169 s} (4) DVCM-quasi-steady friction model: H v,max = {N x = 32, ψ = 1, 118.49 m at t = 0.250 s} Figure 2: Comparison of computed and measured heads at the valve (H v) and at the midpoint (H mp): V 0 = 0.3 m/s, ψ = 1, N x = 32.
International Journal of Civil Engineering & Geo-Environment 5 (2014) The four models slightly overestimate the measured maximum heads. Brunone model yields better conformance with the experimental data while the other three models yield poor results, and it also gives a better timing of the transient event than the other three models. The Zielke model estimates the maximum head more accurately than the other three models. The Zielke, and Vardy and Brown models produce similar computational results and the time of maximum head is the same for both of them. The quasi-steady model estimates the maximum head and its time wrongly. In the unsteady friction models, unrealistic pressure pulses do not exist while in the quasi-steady friction model, the unrealistic pulses are distinguishable obviously. The four models exhibit an incapability to reproduce the experimental oscillations and disregard them and just reproduced them with sufficient accuracy in a short time immediately after closing the valve. In order to investigate sensitivity of this method to the numbers of reaches, two different numbers of reaches N x = {32, 128} were selected. Comparison of computational and measured results for the unsteady and quasi-steady friction models with an initial flow velocity V 0 = 0.30 m/s and weighting factor ψ = 1 is presented in Figures 3 to 6. The maximum computed heads predicted by the models are: 1) DVCM-Brunone s unsteady friction model: H v,max = {N x = 32, 110.10 m at t = 0.173 s} H v,max = { N x = 128, 110.31 m at t = 0.175 s} 2) DVCM-Zielke s unsteady friction model: H v,max = { N x = 32, 104.25 m at t = 0.169 s} H v,max = { N x = 128, 103.95 m at t = 0.171 s} 3) DVCM-Vardy & Brown s unsteady friction model: H v,max = { N x = 32, 105.12 m at t = 0.169 s} H v,max = { N x = 128, 103.95 m at t = 0.171 s} 4) DVCM-quasi-steady friction model: H v,max = { N x = 32, 118.49 m at t = 0.250 s} H v,max = { N x = 128, 109.68 m at t = 0.176 s Figure 3: Comparison of computed heads by the Brunone model with N x = 32, 128 at the valve (H v) and at the midpoint (H mp): V 0 = 0.3 m/s, ψ = 1
Figure 4: Comparison of computed heads by the Zielke model with N x = 32, 128 at the valve (H v) and at the midpoint (H mp): V 0 = 0.3 m/s, ψ = 1 Figure 5: Comparison of computed heads by the Vardy and Brown model with N x = 32, 128 at the valve (H v) and at the midpoint (H mp): V 0 = 0.3 m/s, ψ = 1
Figure 6: Comparison of computed heads by the quasi-steady friction with N x = 32, 128 at the valve (H v) and at the midpoint (H mp): V 0 = 0.3 m/s, ψ = 1 In both different numbers of reaches, both quasi-steady model and unsteady models (Figures 3 to 5) slightly overestimate the maximum heads. Choosing different numbers of reaches has various effects on the four models. In the Brunone model, the larger number of reaches has become more successful to predict the time of the transient event but the smaller one estimates the maximum head more accurate and exhibits a capability to reproduce the experimental oscillations while the larger number of reaches disregards them. In the Zielke, Vardy and Brown, and quasi-steady models, the larger number of reaches has better agreement in terms of simulating the maximum head and its time. In the Zielke, and Vardy and Brown models, the smaller number of reaches is capable to predict the oscillations which exist in the experimental results. In the quasisteady model, the smaller number of reaches produces less unrealistic pressure pulses. In order to investigate the effects of a weighting factor on accuracy of the results, two different values of weighting factor ψ = {0.5, 1.0} were selected. Comparison of computational and measured results for unsteady and quasi-steady models with an initial flow velocity V 0 = 0.30 m/s and number of reaches N x = 32 is presented in Figures 7 to 10. The maximum computed heads predicted by models are: 1) DVCM-Brunone s unsteady friction model: H v,max = {ψ = 1, 110.10 m at t = 0.173 s} H v,max = {ψ = 0.5, 110.10 m at t = 0.173 s} 2) DVCM-Zielke s unsteady friction model: H v,max = {ψ = 1, 104.25 m at t = 0.169 s} H v,max = {ψ = 0.5, 105.91 m at t = 0.171 s} 3) DVCM- Vardy & Brown s unsteady friction model: H v,max = {ψ = 1, 105.12 m at t = 0.169 s} H v,max = {ψ = 0.5, 106.86 m at t = 0.172 s} 4) DVCM-quasi-steady friction model: H v,max = {ψ = 1, 118.49 m at t = 0.250 s} H v,max = {ψ = 0.5, 118.46 m at t = 0.250 s}
Figure 7: Comparison of computed heads by the Brunone model with ψ = 0.5, 1 at the valve (H v) and at the midpoint (H mp): V 0 = 0.3 m/s, N x = 32 Figure 8: Comparison of computed heads by the Zielke model with ψ = 0.5, 1 at the valve (H v) and at the midpoint (H mp): V 0 = 0.3 m/s, N x = 32
Figure 9: Comparison of computed heads by the Vardy and Brown model with ψ = 0.5, 1 at the valve (H v) and at the midpoint (H mp): V 0 = 0.3 m/s, N x = 32 Figure 10: Comparison of computed heads by the quasi-steady friction with ψ = 0.5, 1 at the valve (H v) and at the midpoint (H mp): V 0 = 0.3 m/s, N x = 32
International Journal of Civil Engineering & Geo-Environment 5 (2014) Two different values of weighting factors affect the four models differently. In the quasi-steady models (Figure 10), two different values of weighting factor predict the maximum head and its time similarly. In the Brunone model, the weighting factor ψ = 0.5 reproduces the experimental oscillations but in the quasi-steady model (Figure ), it produces unrealistic pressure pulses. For the Zielke, and Vardy and Brown models (Figure ), the value of ψ = 0.5 has better agreement in terms of simulating the time of the maximum head and has less phase shift while the value of ψ = 1 predicts the maximum heads more accurately. 5. Conclusion Column separation occurs when the liquid pressure decreases to the liquid vapour pressure. When vapour cavities change to liquid, large pressures with steep wave fronts may happen. The DVCM is the most popular model for column separation and distributed cavitation in recent years. Unrealistic pressure pulses (spikes) in the DVCM due to the collapse of multi-cavities can be reduced by using unsteady friction term in the DVCM or proper selection of the number of computational reaches and the values of weighting factor. In this study, water hammer with column separation has been modelled using the Brunone, Zielke and Vardy and Brown unsteady friction models and quasi-steady friction model. The numerical results were compared with the experimental data for validation purposes, and the comparison indicated that: The unrealistic pressure pulses in the quasi-steady model are more than unsteady models. The Zielke and Vardy and Brown models produce more phase shift than the other two models. The Brunone model correlated better with the experimental data than the other two models. It has been observed that using a weighting factor close to 1.0 and restricting the number of reaches do not improve the computational results in all cases. Otherwise proper selection of the number of computational reaches and the values of weighting factor significantly improves the computational results. In the unsteady models, the smaller number of reaches exhibits a capability to reproduce the experimental oscillations while in the quasi-steady model, the unrealistic pressure pulses are produced rather better than expected. In the Brunone model, the weighting factor ψ = 0.5 reproduces the experimental oscillations but it produces unrealistic pressure pulses in the quasisteady, Zielke and Vardy and Brown models. For the Zielke and Vardy and Brown models, the value of ψ = 0.5 has less phase shift. References Bergant, A. and Simpson, A. R. 1994. Estimating unsteady friction in transient cavitating pipe flow. Water pipeline systems, pp. 3-16, Mechanical Engineering Publications, London, UK. Bergant, A. and Simpson, A. R. 1995. Water Hammer and Column Separation Measurements in an Experimental Apparatus. Department of Civil and Environmental Engineering, Report R128. The University of Adelaide, Adelaide, Australia. Bergant, A. and Tijsseling, A. S. 2001. Parameters affecting water hammer wave attenuation, shape and timing. in Proceedings of the 10th International Meeting of the IAHR Work Group on the Behavior of Hydraulic Machinery under Steady Oscillatory Conditions, C2, pp. 12, Trondheim, Norway. Bergant, A., Simpson, A. R. and Tijsseling, A.S. 2006. Water Hammer with Column Separation: A Historical Review. Journal of Fluids and Structures Elsevier 22(2), 135-171. Bergant, A., Simpson, A. R. and Vitkovsky, J. 2001. Developments in unsteady pipe flow friction modelling. Journal of Hydraulic Research IAHR, 39(3), 249 257. Bergant, A., Tijsseling, A. S., and Vitkovsky, J. 2007. Discrete Vapour Cavity Model with Improved Timing of Opening and Collapse of Cavities. in Proceedings of 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems. pp. 117-128, Romania. Brunone, B., Golia, U. M. and Greco, M. 1991. Some remarks on the momentum equation for fast transients. in Proceedings of 9th International Meeting of the IAHR Work Group on Hydraulic Transients with Column Separation, pp. 140-148, Valencia, Spain. Brunone, B., Golia, U. M. and Greco, M. 1995. Effects of two-dimensionality on pipe transients modelling. Journal of Hydraulic Engineering ASCE, 121(12), 906-912.
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