Comprehensive application analyses of elastic models and viscoelastic models in transient flows in polymeric pipelines

With the development of polymer material technology, high polymer pipes have been widely used due to relevant advantages such as good flexibility, corrosion resistance, less fouling and convenient installation. The representative high polymer pipe materials are PVC (polyvinyl chloride), PE (polyethylene) and PP (polypropylene). Strong positive and negative pressures caused by water hammer may also damage high polymer pipes due to rapid valve closure/opening or improper operation, power failure and pump shutdown. Compared with elastic pipes such as metal pipes, polymer pipes exhibit both elastic behavior and viscous behavior when subjected to hydraulic pressure in pipes, and the pipe walls behave viscoelastically. However, the viscoelastic behavior is not usually properly considered in pipe system design as transient events are evaluated either by empirical rules or by a classical transient method of characteristics (MOC), which influences the mechanical performance of polymers. Therefore, the understanding of pipeline viscoelastic behavior and related model assumptions is particularly important for the hydraulic pipeline design.

Friction factor affects the transient flows greatly both in elastic pipes and in viscoelastic pipes. Traditionally, the investigations of transient flows in the elastic pipelines using the elastic water hammer model (Wylie & Streeter 1983; Chaudhry 2014) employ the Steady Friction (SF) model which cannot predict the pressure decay well in transient flows; on the contrary, Unsteady Friction (UF) models provide relatively accurate pressure predictions and thus attract more attention (Ghidaoui et al. 2005; Martins et al. 2017). Zielke (1968) derived the Convolution-Based Unsteady Friction (CB-UF) model based on the analytical solution of the transient laminar flow, and the wall shear stress is divided into two parts, i.e. the steady part and the unsteady one equal the weighted sum of the historical acceleration of the fluid. But the CB-UF model by Zielke (1968) is time-consuming because it needs huge computer storage. Trikha (1975) further developed the model for laminar flow by Zielke (1968) by determining an approximate expression for the convolution-based friction such that the computation of this expression requires much less computer storage or computation time than the computation of the exact expression. Also, the comparisons between the calculated results for a Reservoir–Pipe–Valve system with a 36 m straight copper pipe and the experimental data show that the approximate expression predicts accurately the pressure evolution. Based on an approximate representation of a turbulent pipe flow as a reasonable composition of the laminar annulus and a uniform core in the moderate range of Reynolds Numbers, a new weighting function model of transient turbulent pipe friction at moderate Reynolds Numbers was developed by Vardy & Hwang (1993), and its accuracy was validated by confirming close agreement with the inverse numerical evaluations. Also, then a family of weighting function curves was provided based on experimental data and Zielke's curve is identified as the appropriate upper bound. Another family of the instantaneous acceleration-based unsteady friction (IAB-UF) model assumes that the UF term is a function of the instantaneous local and convective accelerations (Brunone et al. 1995; Ghidaoui et al. 2005). Pezzinga (2000) proposed the modified MIAB-UF model by introducing the quantity as a factor in the spatial derivative, which considers well additional dissipations related to convective acceleration.

Although these unsteady frictions have achieved some success in elastic pipes, it is necessary to further study their applicability in transient flow in viscoelastic pipes. By using the Acoustic Doppler Velocimeter Profiler (ADVP), Brunone et al. (2000) experimentally investigated the pressure and the velocity profiles at the downstream end section for two transient events caused by downstream valve closure in a 352 m PE pipe test rig. The velocity profiles clearly show regions of flow recirculation and flow reversal. The comparisons between the experimental pressures and the results predicted by Brunone's IAB-UF model (1995) demonstrated that this model employs an extremely high decay coefficient in order to match well with the experimental data and is able to track the decay of pressure peak after the first cycle. However, the UF model cannot accurately reproduce the shape of the experimental pressure curve during the pressure decay process.

To better predict the transient flow characteristics in viscoelastic pipes, it is a lack of generality to simply adjust the parameters of a certain model empirically, and the viscoelastic model should further consider a retarded viscoelastic response. The retarded behavior is modeled by an additional time-dependent term in the continuity equation which reasonably allows for the pipe–wall creep coefficients, and the pressure head variation of transient flows can be predicted well. Gally et al. (1979) determined the creep function by dynamic tests of the PE pipe–wall material, and the verification of the viscoelastic model with SF by the experimental pressure and circumferential strain in a 43.1-m-long PE pipe demonstrated that a slight disagreement in the strain data and numerical results was observed, and the experimental data showed that the axial strain was very small relative to the circumferential strain in the case of the pipe axially fixed. Soares et al. (2008) investigated the transient flows induced by the rapid closure of the downstream valve in a PVC pipeline, and concluded that the UF effects are negligible compared to pipe–wall viscoelasticity (VE) in PVC pipes through the series analysis and comparison of experimental data and calculation results by a viscoelastic model with and without UF. In addition, Soares et al. (2009) firstly analyzed the transient flows in a high-density polyethylene (HDPE) pipeline by using the viscoelastic model with SF, and the predicted pressure evolution was in good agreement with the experimental data. The DVCM and DGCM cavitation models with and without the VE model were used to further study transient cavitation flow, respectively, the comparisons between the numerical results and the experimental data showed that the time-dependent pressure by the DGCM together with the VE model was more consistent than other model combinations. Duan et al. (2010b) derived the energy equations of transient flows in viscoelastic pipelines and carried out the energy analysis. The numerical results showed that the work done by the fluid on the viscoelastic pipe wall is bigger than that done by the viscoelastic pipe wall on the fluid, and the related viscoelastic dissipation only occurred in the pipe–wall absorbing/accumulating the energy from the fluid.

Bergant et al. (2011) experimentally and numerically analyzed the water hammer experiments in a Reservoir–Pipe–Valve system with a 275.2-m-long DN250 PVC pipeline, and the comparisons showed that the pressure obtained by only the UF model was not in good agreement with the experimental data, whereas the incorporation of both the UF model and viscoelastic model could capture well the pressure decay and the relevant prediction was in good agreement with the experimental data. The investigations of Ramos et al. (2004) concentrated on the analyses of pressure attenuation in transient flows generated by the rapid flow-rate change for several single pipeline systems with different characteristics including pipe materials (plastic and metal), diameters, pipe–wall thickness, length and so on. The two-friction coefficient MIAB-UF model was used to simulate the transient flows in plastic pipelines, the predicted pressure attenuation peaks could agree with the experimental data, but the pressure evolution curve shape could not match well with the experimental data. The pressure peak evolutions predicted by the two formulas for pressure peak attenuation corresponding to elastic and non-elastic pipes, respectively, are correspondingly accurate. Meniconi et al. (2014) investigated transient energy dissipation and pressure decay in viscoelastic pipes with an in-line valve, analyzed the effects of the partially-closed valve opening, VE and UF on the transient pressure and confirmed that the relation between the decay of dimensionless pressure peaks and dimensionless time starting from the fifth characteristic time of the pipe followed exponential laws. Zhu et al. (2018) investigated air–water mixing transient flows in viscoelastic pipes using VE and UF, and demonstrated that the existence of air content in water flows may reduce greatly the influence of the VE effect and the effect of UF on transient pressure attenuation is greatly enhanced with the increase of air content.

In addition, the water hammer governing equations were transformed into the frequency domain by using the Fourier Transform and solved by the transfer matrix method, and the Frequency Response Function Method (FRFM) was widely used in the transient analysis (Lee et al. 2006). Duan et al. (2012) used the extended FRFM for leak detection in a viscoelastic pipe system, the extended FRFM is validated from numerical results by one-dimensional viscoelastic transient models in the time domain and the investigations showed that the amplitude damping and phase shift of the pressure wave were influenced by the pipe–wall VE, which has little influence on the leak-induced patterns of pressure peaks in transient system frequency responses. Gong et al. (2015) employed the FRF to investigate the transient flows in a viscoelastic pipeline, where the transfer matrix of a uniform viscoelastic pipeline was derived using the generalized multi-element Kelvin–Voigt (K–V) model. The frequency response diagrams (FRDs) for the transient flows in the viscoelastic pipeline were compared with the FRDs of an elastic pipeline under conditions of steady and unsteady frictions, respectively, and the comparisons showed that the pipe–wall VE led to frequency-dependent shifting of the resonant frequencies, which lacks the comparison with the experimental data.

In this paper, several combined models, i.e. the SF, the CB-UF, the MIAB-UF, the SF-VE, the CB-UF-VE and the MIAB-UF-VE are used to investigate the transient flow characteristics in the viscoelastic HDPE pipeline in the experimental rigs of Covas et al. (2004, 2005). The pressure evolutions produced by the different models are compared with each other and with the experimental data, and the pressure evolutions by the viscoelastic models are analyzed together with the corresponding retarded strain evolutions. The effects of the UF and VE on pressure peak attenuation are investigated by calculating the pressure peak differences of different models. The pressure evolutions predicted by the elastic models and the viscoelastic models and the experimental data are transformed to the frequency domain by using FFT, and the influence of the UF and the VE on harmonic frequency change is analyzed.

In the traditional transient elastic model, the transient flows are described by the continuity and momentum equation, and the pressure wave speed which takes into account the elasticity of the fluid and pipe wall is included in the continuity equation. But for the transient flows in viscoelastic pipes, the response of the pipe wall to its internal hydraulic pressure exhibits an instantaneous elastic effect and a hysteretic effect where the pipe–wall strain lags behind the stress exerted on it. Thus, in the transient viscoelastic models, the retarded strain term is added to the continuity equation to model the VE of the pipeline.

Generally, Equations (1), (2) and (8) constitute the governing equations of the transient flows in viscoelastic pipes.

According to different descriptions of the generation mechanism of the UF term, the existing UF models can be divided into three categories: the CB-UF models, the IAB-UF models, and the models directly derived from irreversible thermodynamic processes (Axworthy et al. 2000). The CB-UF model and the MIAB model are adopted because they are relatively mature and have relatively more applications.

This model provides good results in the transient flows in elastic pipelines. The further applicability of this model is analyzed here in the viscoelastic HDPE pipeline.

The MOC is used to transform the partial differential equations of (1) and (2) into the ordinary differential equations. The SF model, Trikha's (1975) CB-UF model, and Pezzinga's (2000) MIAB model are used to model the transient flows in the viscoelastic pipeline, and their MOC equations are as follows.

For Pezzinga's (2000) MIAB-UF model, the MOC equations are determined by the values of the Sign function:

The Reservoir–Pipe–Valve viscoelastic pipeline transient flow system is taken from the literature of Covas et al. (2004, 2005). This system consists of a constant head reservoir, an HDPE pipeline with length L = 271.5 m between the vessel and the downstream globe valve, inner diameter of 56 mm, wall thickness of 6.2 mm, and a globe valve at the pipe end. The water hammer events are generated by the rapid closure of the downstream valve. Pressure transducers T1, T2 and T3 are arranged at 0.5 m, 74.57 m (0.27L), and 155.08 m (0.57L) away from the downstream valve, respectively, to collect the transient pressure data. The initial steady-states of the three experimental schemes are shown in Table 1.

The pressure wave speed of 395 m/s was determined experimentally and the corresponding instantaneous creep compliance J₀ = 0.7 GPa⁻¹. The more the number of the K–V elements, the closer the creep function is to the creep curve obtained from the experiment, but the more the number of the K–V elements does not make the transient solution more accurate (providing simply a different combination of J_k parameters) (Covas et al. 2005). A three-element generalized K–V model (τ₁ = 0.05 s, J₁ = 0.0804 GPa⁻¹, τ₂ = 0.5 s, J₂ = 0.1113 GPa⁻¹, τ₃ = 10 s, J₃ = 0.5456 GPa⁻¹) is accurate enough to model the pipe–wall VE in the transient flows.

Several combined models based on the above models are systematically applied to study the transient flow characteristics in the pipe, i.e. the SF, the CB-UF, the MIAB-UF, the SF-VE, the CB-UF-VE and the MIAB-UF-VE, respectively. When the VE effect is ignored in the simulations, the strain-rate terms in Equations (18)–(24) are deleted.

It can be seen from Figure 7 that the periodic decay evolutions of the total strain are predicted. The comparisons between the calculated total strain and the experimental show that the SF-VE underestimates the peak in the first three periods, overestimates the peak in the subsequent periods and the corresponding difference gradually tends to increase, whereas the CB-UF-VE and the MIAB-UF-VE underestimate the peak at each period, and the three viscoelastic models predict the phase well in the first two periods, but then the predicted phase advances and the corresponding phase difference gradually increases. The maximum total strain of the experimental data, predicted by the SF-VE, the CB-UF-VE and the MIAB-UF-VE are 8.37E-4, 7.57E-4, 7.84E-4 and 7.56E-4, respectively, and the corresponding relative errors are 9.54, 6.34 and 9.70%, respectively.

The maximum pressure relative errors at T1 by the SF, the CB-UF and the MIAB-UF are 3.66, 4.82 and 3.64%, respectively, and the pressure peak relative errors increase as the transient lasts for these three models. The maximum pressure relative errors at T1 by the SF-VE, the CB-UF-VE and the MIAB-UF-VE are 0.03, 0.18 and 0.18%, respectively, much more accurate than those by the three elastic models, and the pressure peak relative errors by the three viscoelastic models are all very small and the maximum errors are 1.24, 2.12 and 1.99%, respectively. Except that the SF-VE predicts better the pressure evolution in the first two periods than the CB-UF-VE and the MIAB-UF-VE do in Figure 3, the prediction by the SF-VE is not in good agreement with the experimental data including the peaks and the phase a little ahead of the experimental data, and the CB-UF-VE and the MIAB-UF-VE produce better the pressure head evolution.

The at T1 by the SF-VE, the CB-UF-VE and the MIAB-UF-VE are 4.7, 2.8 and 2.3%, respectively, which indicates that the prediction by the MIAB-UF-VE is the most accurate, followed by the CB-UF-VE and worse by the SF-VE.

In general, the transient viscoelastic models can reproduce the pressure head change law in the entire pipeline. The maximum pressure relative errors at T2 by the SF-VE, the CB-UF-VE and the MIAB-UF-VE are 0.003, 0.75 and 0.08%, respectively, and the pressure peak relative errors at T2 by the three viscoelastic models are all very small and the maximum errors are 1.09, 2.04 and 1.67%, respectively. On the whole, the at T2 by the SF-VE, the CB-UF-VE and the MIAB-UF-VE are 6.78, 3.79 and 3.01%, respectively. The maximum pressure relative errors at T3 by the SF-VE, the CB-UF-VE and the MIAB-UF-VE are 0.05, 0.35 and 0.02%, respectively, and the pressure peak relative errors at T3 by the three viscoelastic models are all very small and the maximum errors are 0.86, 1.6 and 1.03%, respectively. On the whole, the at T3 by the SF-VE, the CB-UF-VE and the MIAB-UF-VE are 4.98, 2.82 and 2.35%, respectively.

However, for Case 1, there are still differences in details, in particular the pressure response caused by creep is very different. Before the first rarefaction wave returns, the maximum pressure decreases a little and rapidly from 0.23 to 1.56 s due to the joint action of the pipe hydraulic resistance and the pipe–wall VE of a stress release with a retarded strain increase, and then there is no slight pressure increase due to the line packing effect like in Case 2, as can be seen in Figures 22(a), 23(a) and 24(a). As can be seen in Table 4, compared to the numerical results for Case 2, the maximum pressure relative errors at T1 predicted by the elastic models are smaller as the initial velocity decreases for Case 1 in Figure 18, and the pressure peak relative errors increase with time for these three models; the maximum pressure relative errors at T1 by the SF-VE, the CB-UF-VE and the MIAB-UF-VE change a little, the predictions are much more accurate than those by the three elastic models. For Case 1, the pressure peak relative errors by the SF-VE increase as the transient evolves, whereas the pressure peak relative errors by the CB-UF-VE and the MIAB-UF-VE are very small and the maximum errors are 0.06 and 0.04%, respectively. Except that the SF-VE can accurately predict the pressure phase in the first period, its pressure phase prediction error is large and gradually increases with time, whereas the CB-UF-VE and the MIAB-UF-VE can accurately predict the pressure phase in the whole transient process. On the whole, the at T1 shown in Table 5 indicate that the prediction by the CB-UF-VE is the most accurate, followed by the MIAB-UF-VE and worse by the SF-VE.

Table 5

at T1 by the three viscoelastic models for Cases 1–3

Case no. .	at T1 by the three viscoelastic models (–) .
	SF-VE (%) .	CB-UF-VE (%) .	MIAB-UF-VE (%) .
1	9.2	2.5	2.6
2	4.7	2.8	2.3
3	5.7	4.6	3.6

Case no. .	at T1 by the three viscoelastic models (–) .
	SF-VE (%) .	CB-UF-VE (%) .	MIAB-UF-VE (%) .
1	9.2	2.5	2.6
2	4.7	2.8	2.3
3	5.7	4.6	3.6

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For Case 3, the maximum pressure relative errors by the SF, the CB-UF and the MIAB-UF are still large, as can be seen in Table 3 and Figure 20. As can be seen in Figure 21, in the first period, the pressure heads are over-predicted by using the SF-VE, the CB-UF-VE and the MIAB-UF-VE, respectively. Except that the predicted pressure at the second peak is in good agreement with the experimental, there are some differences at other peaks. The predicted pressure phase in the first two periods is in good agreement with the experimental, and from the third period, the phase difference between the predicted and the experimental gradually increases. The shown in Table 4 indicate that the prediction by the MIAB-UF-VE is the most accurate, followed by the CB-UF-VE and worse by the SF-VE.

Generally, from the perspective of Cases 1, 2 and 3, the pressure head evolution can be predicted better by the viscoelastic models compared to the elastic models, and the effect of the pipe wall VE is much larger than the friction effects in viscoelastic pipelines. The CB-UF-VE and the MIAB-UF-VE can generate relatively better results than the SF-VE from the perspective of the whole transient flow process. Because the pressure maximum is particularly important for transient flow studies, the comparative analyses between the predicted and the experimental for Case 1, Case 2 and Case 3 show that the viscoelastic models predict well when the relative pressure is high, such as the maximum pressure head 78.3 m for Case 3, and the prediction is more accurate when the relative pressure is low, such as the maximum pressure head 49.4 m for Case 1 and 63.1 m for Case 2, which indicates that the viscoelastic models have certain pressure application range.

The UF and the VE are key factors in the pressure head evolution in transient flows in viscoelastic pipelines, and therefore it is necessary to study their effects on pressure attenuation.

For Case 1, it can be seen from Figure 30(b) that at the first peak is −3.48% and is −2%, and the peak predicted by the SF-VE is the most accurate, followed by the prediction by the MIAB-UF-VE, and the prediction by the CB-UF-VE is relatively poor as can be seen from Figure 3. At the second pressure peak, of 15% is smaller than of 22.5%, which indicates that the effect caused by the CB model just considering the instantaneous acceleration is smaller than that by the MIAB model considering both the instantaneous local and advection acceleration, but the prediction results by the CB-UF-VE model are in best agreement with the experimental, followed by the MIAB-UF-VE model and worse by the SF-VE model, as can be seen from Figure 3. At the third pressure peak, is equal to ⁠. From the fouth to the seventh pressure peak, is smaller than and both decrease, and the contribution by the VE to the pressure attenuation relatively increases as the pressure decays, whereas the contributions both by the CB-UF and by the MIAB-UF relatively decrease. The pressure peaks from the third to the seventh predicted by the MIAB-UF-VE model are in best agreement with the experimental, followed by the CB-UF-VE model and worse by the SF-VE model, as can be seen from Figure 3.

For the pressure amplitudes in Case 2, as shown in Figure 33(a), the pressure amplitudes predicted by the elastic models are in poor agreement with the experimental data, in which the amplitudes predicted by the CB-UF are closest to the experimental, the first four order amplitudes by the CB-UF are 12.77, 3.19, 1.71 and 1.01 m, respectively, but the first four order amplitudes obtained from the experimental are 5.82, 1.20, 0.55 and 0.31 m, respectively. The pressure amplitudes predicted by the viscoelastic models in Figure 34(a) are in good agreement with the experimental, among which the amplitudes predicted by the MIAB-UF-VE are closest to the experimental, and the relative errors of the first four amplitudes are 6.49, 1.58, 2.87 and 9.05%, respectively. The relative errors of the first four amplitudes predicted by the CB-UF-VE are 7.04, 9.83, 11.92 and 14.71%, respectively. The relative errors of the first four amplitudes predicted by the SF-VE are 13.27, 8.54, 1.63 and 4.98%, respectively.

For the spectrum analyses in Case 2, the first 11 order harmonic frequencies obtained from the experimental in Figure 33(b) are 0.30, 1.00, 1.70, 2.40, 3.10, 3.85, 4.60, 5.25, 5.90, 6.55 and 7.30 (unit: Hz), respectively. The harmonic frequencies by the frictionless elastic model and the SF are correspondingly equal, and the first 11 harmonic frequencies are 0.35, 1.10, 1.80, 2.55, 3.30, 4.00, 4.75, 5.45, 6.20, 6.90 and 7.65 (unit: Hz), respectively, which are all larger than the corresponding experimental data, the relative error of the seventh  harmonic frequency is the smallest (3.26%) and the relative error of the first harmonic frequency is the largest (16.67%). Except that there is certain difference between the fifth harmonic frequency (3.25 Hz) by the CB-UF and that (3.3 Hz) by the frictionless elastic model, the other harmonic frequencies by the CB-UF are equal to the corresponding harmonic frequencies by the frictionless elastic model. There are some differences between all the harmonic frequencies by the CB-UF and the corresponding experimental data, with the smallest relative error of 3.26% of the seventh  harmonic frequency and the largest relative error of 16.67% of the fundamental frequency. The first 11 harmonic frequencies by the MIAB-UF are 0.35, 1.10, 1.80, 2.50, 3.20, 3.95, 4.65, 5.35, 6.10, 6.80 and 7.50 (unit: Hz), respectively. There are some differences between other harmonic frequencies by the MIAB-UF and the corresponding experimental data, the smallest relative error of the ninth harmonic frequency is 1.09%, and the largest relative error of the first harmonic frequency is 16.67%. On the whole, the harmonic frequencies predicted by the elastic models are not in good agreement with the experimental.

For Case 2, as can be seen in Figure 34(b), the first four harmonic frequencies predicted by the SF-VE and the CB-UF-VE are, respectively, equal, which are 0.35, 1.05, 1.75 and 2.45 (unit: Hz), respectively, all the predicted harmonic frequencies are larger than the corresponding experimental data, and the relative errors are 16.67, 4.96, 2.94 and 2.08%, respectively. The first four harmonic frequencies predicted by the MIAB-UF-VE are 0.35, 1.00, 1.70 and 2.40 Hz, respectively, and except for the fundamental frequency (relative error 16.67%), the other harmonic frequencies are equal to the corresponding experimental data. In general, the harmonic frequencies predicted by the viscoelastic models are in good agreement with the experimental, in which the predictions by the MIAB-UF-VE are most accurate, and those by the SF-VE and the CB-UF-VE have the same accuracy.

The first 11 order harmonic frequencies by the elastic models and obtained from the experimental for Case 1 in Figure 35(b) and for Case 3 in Figure 37(b) are shown in Tables 7 and 10, respectively. The harmonic frequencies by the frictionless elastic model and the SF are, respectively, equal, and except that the predicted fundamental frequency is equal to the experimental, there are some differences between the other harmonic frequencies and the corresponding experimental data; for Case 1, the relative error of the ninth harmonic frequency is the smallest (1.64%), and the relative error of the 11th harmonic frequency is the largest (12.5%); for Case 3, the relative error of the third  harmonic frequency is the smallest (3.85%) and the relative error of the 11th harmonic frequency is the largest (15%). For Case 1, except that there is a certain difference between the fifth harmonic frequency (3.25 Hz) by the CB-UF and that by the frictionless elastic model (3.3 Hz), the other frequencies by the CB-UF are the same as those by the frictionless elastic model correspondingly; except that the fundamental frequency by the CB-UF is the same as the experimental, there are some differences between the other order frequencies and the corresponding experimental data, the smallest relative error is 1.64% (the ninth frequency) and the largest relative error is 12.5% (the 11th frequency). For Case 3, except that there is a certain difference between the 11th harmonic frequency (7.60 Hz) by the CB-UF and that (7.67 Hz) by the frictionless elastic model, the other harmonic frequencies by the CB-UF are the same as the corresponding harmonic frequencies by the frictionless elastic model; except that the fundamental frequency by the CB-UF is equal to the experimental, there are some differences between other frequencies by the CB-UF and the corresponding experimental data, the smallest relative error of the seventh  harmonic frequency is 2.13% and the largest relative error of the 11th harmonic frequency is 14%. For Case 1, the fundamental frequency and the ninth harmonic frequency by the MIAB-UF are the same as the corresponding experimental data, but there are some differences between the other frequencies and the corresponding experimental data, among which the relative error of the seventh  frequency is the smallest (1.09%), and the relative error of the 11th frequency is the largest (10.29%). For Case 3, the fundamental frequency by the MIAB-UF is the same as the experimental, but there are some differences between other harmonic frequencies and the corresponding experimental data, the smallest relative error of the seventh  harmonic frequency is 1.09%, and the largest relative error of the 11th harmonic frequency is 13%. On the whole, similar to Case 2, the frequencies predicted by the elastic models are not in good agreement with the experimental.

For Case 1, it can be seen from Figure 36(b) and Table 8 that the first four harmonic frequencies by the SF-VE and the CB-UF-VE are, respectively, equal, the first and third harmonic frequencies are the same as the corresponding experimental data, but the second and fourth harmonic frequencies are larger than the corresponding experimental, and the relative errors are 5 and 2.08%, respectively. Except for the third harmonic frequency (relative error 2.86%) by the MIAB-UF-VE, the other harmonic frequencies are equal to the corresponding experimental data. In general, the harmonic frequencies predicted by the viscoelastic models are in good agreement with the experimental, in which the predictions by the MIAB-UF-VE are most accurate, and those by the SF-VE and the CB-UF-VE have the same accuracy.

For Case 3, it can be seen from Figure 38(b) and Table 11 that the first three harmonic frequencies predicted by the SF-VE are equal to the corresponding experimental data, and the relative error between the fourth harmonic frequency by the SF-VE and the corresponding experimental is 2.78%. The first four harmonic frequencies predicted by the CB-UF-VE and the MIAB-UF-VE are all equal to the corresponding experimental data.

Overall, compared with the experimental data, the amplitudes and harmonic frequencies predicted by the elastic models are very poor, and accordingly, the predictions by the viscoelastic models are very good, in which the harmonic frequencies predicted by the MIAB-UF-VE are the best, followed by the CB-UF-VE and worse by the SF-VE.

Based on MOC, for the Reservoir-Viscoelastic Pipe–Valve system under three initial flow velocities of low, medium and high conditions, the elastic models of the SF, the CB-UF and the MIAB-UF and the viscoelastic models of the SF-VE, the CB-UF-VE and the MIAB-UF-VE were used to investigate the hydraulic transients in viscoelastic pipelines. There is an instantaneous elastic response (considered in the elastic wave speed) and a retarded viscoelastic response in the viscoelastic model, and the retarded behavior of the pipe–wall is represented by an additional retarded strain-rate term that is incorporated into the continuity equation. The accuracy of the elastic models and the viscoelastic models in predicting pressure evolution is evaluated by comparing the numerical results with the experimental data, including creep effects related to the pressure peaks. The contributions of the CB-UF, the MIAB-UF and the VE to pressure peak attenuation are further compared and analyzed, and based on FFT, the influence of the friction (steady and unsteady) and the VE on the pressure amplitudes and harmonic frequencies are investigated. The main conclusions are as follows:

• The elastic models greatly overestimate the peaks except for the first peak, although the UF models have a slight improvement on the calculation results, and the predicted phase difference gradually increases with time from the third peak due to the fact that the viscoelastic characteristics of the pipeline are not considered. However, the predicted pressure peaks and phases by the viscoelastic models are in good agreement with the experimental due to considering the pipe–wall creep effect, and the effect caused by the pipe–wall VE is much larger than those by the friction. The relative errors between the first peak at T1 predicted by the SF-VE, the CB-UF-VE and the MIAB-UF-VE and the experimental data are 0.041, 0.049 and 0.044%, respectively, for Case 1, the corresponding relative errors for Case 2 are 0.03, 0.18 and 0.18%, respectively, and the corresponding relative errors for Case 3 are 3.23, 4.9 and 3.07%, respectively. The relative errors between the first peak at T1 predicted by the SF, the CB-UF and the MIAB-UF and the experimental data are 0.19, 0.29 and 0.19%, respectively, for Case 1, the corresponding relative errors for Case 2 are 3.66, 4.82 and 3.64%, respectively, and the corresponding relative errors for Case 3 are 10.75, 12.63, 10.73%, respectively. It can also be concluded that the viscoelastic models are more relatively applicable for the low and medium initial velocity conditions than for the high initial velocity condition.

• The CB-UF-VE and the MIAB-UF-VE can generate relatively better results than the SF-VE from the perspective of the whole transient flow process. The averaged relative time-dependent error of the pressure head at T1 by the SF-VE, the CB-UF-VE and the MIAB-UF-VE are 9.2, 2.5 and 2.6%, respectively, for the low initial velocity, the corresponding for the medium initial velocity are 4.7, 2.8 and 2.3%, respectively, and the corresponding for the high initial velocity are 5.7, 4.6 and 3.6%, respectively.

• The pressure evolutions of transient flows in viscoelastic pipelines are greatly affected by the retarded strain. At T1, after the pressure quick rise induced by the rapid valve closure, the maximum pressure immediately afterwards decreases a little and rapidly, due to the joint action of the pipe hydraulic resistance and the pipe–wall VE of a stress release with a retarded strain increase before the rarefaction wave returns, which indicates that the pipe material does not behave in a linear elastic mode where the pipe–wall strain response has the same trend as the corresponding pressure rise. But the pressure transients near the pressure maximum at T2 and T3 behave more like the transient flow inside an elastic pipeline. In addition, the total strain predicted by the viscoelastic models is also in good agreement with the experimental for medium initial velocity. The evolutions of the elastic strain and the pressure predicted by viscoelastic models are consistent in phase, whereas there is certain phase difference between the evolutions of the retarded strain and the pressure.

• The contribution of VE to pressure peak attenuation is greater than that of the CB-UF and that of the MIAB-UF. From the fourth to the seventh pressure peak, is smaller than and both decrease, and the contribution by the VE to the pressure attenuation relatively increases as the pressure decays whereas the contributions both by the CB-UF and by the MIAB-UF relatively decrease. It is also revealed that the pressure attenuation contributed by the CB-UF model is not bigger than by the MIAB-UF model at the first two pressure peaks but bigger than that by the MIAB-UF model from the third to the seventh pressure peaks.

• For the pressure amplitudes in the frequency domain, the predictions by the elastic models are not in good agreement with the experimental data, but the amplitudes predicted by the viscoelastic models are in good agreement with the experimental. The harmonic frequencies of the experimental data are irregular, and each harmonic frequency predicted by the frictionless elastic model is equal to the corresponding frequency predicted by the SF and the predicted harmonic frequencies increase by odd multiples and there are some differences compared with the corresponding experimental data. The harmonic frequencies predicted by the CB-UF and the MIAB-UF do not increase by odd multiples and there are also some differences compared with the corresponding experimental data. On the whole, the frequencies predicted by the elastic models are not in good agreement with the experimental. But the harmonic frequencies predicted by the viscoelastic models are in good agreement with the experimental, in which the harmonic frequencies predicted by the MIAB-UF-VE are the best, followed by the CB-UF-VE and worse by the SF-VE.

This work was supported by the National Natural Science Foundation of China (Grant Nos 52079140 and 51779257).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.