Based on the MOC (Method of Characteristics), the applicabilities of both the elastic models of the SF (Steady Friction), the CB-UF (Convolution-Based Unsteady Friction) and the MIAB-UF (Modified Instantaneous Acceleration-Based UF), and the viscoelastic models of the SF-VE (SF-Viscoelasticity), the CB-UF-VE (Convolution-Based Unsteady Friction-Viscoelasticity) and the MIAB-UF-VE (Modified Instantaneous Acceleration-Based UF-Viscoelasticity), are investigated for hydraulic transients induced by a downstream rapid valve closure in a polymeric pipeline. The predicted results by the elastic models are very different from the experimental data, whereas the predicted pressure peaks and phases by the viscoelastic models agree well with the experimental data because considering the pipe–wall creep effect, the CB-UF-VE and the MIAB-UF-VE generate better results than the SF-VE. The creep effects near the first pressure peak are captured well by the viscoelastic models. The analyses of the contributions of different factors to pressure attenuation show that for transient flows in polymeric pipelines, the effect of VE is greater than that of the CB-UF and MIAB-UF. The spectrum analyses show that the pressure amplitudes and harmonic frequencies by the elastic models match badly with the experimental data, whereas those by the viscoelastic models match well with the experimental data. The harmonic frequencies by the MIAB-UF-VE are the best, followed by the CB-UF-VE, and are worst by the SF-VE.

  • The applicabilities of elastic models and viscoelastic models are investigated based on the MOC for the hydraulic transients in a viscoelastic pipe.

  • The pressure peaks and phases are captured.

  • The creep effects near the first pressure peak are captured.

  • The effects of VE and UF on pressure attenuation are analyzed.

  • The pressure amplitudes and harmonic frequencies by the viscoelastic models match well the experimental data.

Graphical Abstract

Graphical Abstract
Graphical Abstract

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.

Viscoelastic model

The one-dimensional transient flow governing equations in viscoelastic pipes are the continuity and momentum equation, respectively (Gally et al. 1979; Covas et al. 2005):
formula
(1)
formula
(2)
where t is the time, x is the coordinate along the pipeline axis, H is the piezometric head, Q is the flow rate, a is the wave speed, g is the gravitational acceleration, A is the pipe cross-sectional area, hf is the hydraulic loss per unit length and εr is the circumferential retarded strain of the viscoelastic pipe which needs to be further closed.
The retarded strain-rate term can be obtained based on the pressure and the corresponding strain of the viscoelastic pipe (Gally et al. 1979; Covas et al. 2005; Soares et al. 2009). First, when subjected to a certain stress, the viscoelastic PE pipe exhibits an immediate elastic response and a retarded-viscous response, so the total strain can be decomposed into an instantaneous elastic strain and a retarded strain :
formula
(3)
The total circumferential strain of the PE pipe corresponding to a circumferential-stress of is
formula
(4)
where J0 is the instantaneous creep compliance and J(t′) is the creep function at t′ time.
When the pipe is subjected to an internal pressure of , and the circumferential-stress is obtained by , the total circumferential strain is
formula
(5)
where is the pipe–wall constraint coefficient, is the pipe inner diameter, is the pipe wall thickness, these terms are assumed to be constant in the calculations; and is the pressure at time , and is the initial pressure.
In the generalized K–V model, the creep function can be in good agreement with the experimental creep curve (Weinerowska-Bords 2006). So, the generalized K–V model (Figure 1) of viscoelastic materials is used in the hydraulic transient computations and the creep function is
formula
(6)
where J0 is the creep compliance of the first spring (that is the instantaneous creep compliance of the viscoelastic material) and J0=1/E0; E0 is the Young's modulus of elasticity of the pipe; Jk is the creep compliance of the kth element of the K–V model and Jk=1/Ek; Ek is the modulus of elasticity of the spring of the kth element; is the retardation time of the dashpot of the kth element and ; is the viscosity of the dashpot the kth element; and is the number of the K–V elements. Parameters and are usually calculated by using the Inverse Transient Analysis (ITA) from the experimental data (Covas et al. 2005; Soares et al. 2009; Covas & Ramos 2010).
Figure 1

Generalized K–V model.

Figure 1

Generalized K–V model.

Close modal
According to this viscoelastic model, the retarded strain-rate in Equation (1) can be calculated by summing the retarded strain-rate of each of the K–V elements:
formula
(7)
and
formula
(8)
where is the retarded strain of the last time step and .
The retarded strain of the current time step is
formula
(9)

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

UF models in transient flows

For fast transients in pipes, the head loss term in Equation (2) is decomposed into two components, the steady-state component , and the unsteady-state component (Ghidaoui et al. 2005; Martins et al. 2017):
formula
(10)
The steady-state hydraulic loss component is calculated as follows:
formula
(11)
where f is the Darcy–Weisbach friction factor.

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.

Convolution-based unsteady friction model

Based on Zielke's (1968) UF model, Trikha's (1975) CB-UF equation is derived and simplified as follows:
formula
(12)
where Yi (i = 1, 2, 3) is expressed as:
formula
(13)
where m1 = 40, m2 = 8.1, m3 = 1, n1 = −8,000, n2 = −200, and n3 = −26.4; is the kinematic viscosity of the fluid.

Modified IAB-UF model

The IAB-UF model is developed by many researchers (Pezzinga 2000; Brunone et al. 2004; Ramos et al. 2004). In the modified instantaneous acceleration-based unsteady friction (MIAB-UF) model of Pezzinga (2000), the unsteady-state component is
formula
(14)
where is Brunone friction coefficient, Sign is the sign function and is defined as follows:
formula
(15)

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 Brunone friction coefficient is calculated from the shear attenuation coefficient :
formula
(16)
The shear attenuation coefficient formula proposed by Vardy & Brown (2003) is
formula
(17)
where is the Reynolds number.

Method of characteristics

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 the SF model, the ordinary differential equations are valid on the and characteristic lines:
formula
(18)
formula
(19)
For Trikha's (1975) CB-UF model, the MOC equations are:
formula
(20)
formula
(21)

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

when ,
formula
(22)
formula
when ,
formula
(23)
formula
(24)

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.

Table 1

Main parameters of different experimental schemes

Case no.Initial velocity V0 (m/s)Initial Reynolds number Re0
0.0278 1,400 
0.4973 25,000 
0.9946 50,000 
Case no.Initial velocity V0 (m/s)Initial Reynolds number Re0
0.0278 1,400 
0.4973 25,000 
0.9946 50,000 

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 Jk 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.

Analysis of the transient pressure evolution

The case of the medium initial velocity

Because experimental transient pressures at three different positions of T1 (upstream the valve), T2 and T3 are available in the literature for Case 2 (V0 = 0.4973 m/s), comprehensive and detailed analyses for Case 2 are carried out, the investigations of Cases 1 and 3 focus on the comparison of different characteristics. For Case 2, the pressure head changes upstream the valve (T1) during the transient flow generated by the downstream rapid valve closure predicted by the three elastic models and the experimental data are shown in Figure 2 and the pressure head changes by the three viscoelastic models in Figure 3. The comparisons between the experimental data and the predicted show that the SF, the CB-UF and the MIAB-UF cannot reproduce the pressure transient process, whereas the SF-VE, the CB-UF-VE and the MIAB-UF-VE can accurately capture the time-dependent pressure among which the MIAB-UF-VE has the most accurate prediction results from the whole transient perspective. It can be seen from Figure 2 that the predicted pressure waveform is mainly caused by the joint action of the reduction of flow velocity during valve closing and the hydraulic friction, but the influence degree of the closing process and the friction model is different for different elastic models. At the same time, it also shows that the main reason for the prediction error is caused by the elastic model, whereas the hydraulic friction model only improves the corresponding calculated results to some extent, which shows that the viscoelastic model plays a major role in the predictions and the friction model plays a secondary role. It can be seen from Figure 3 that the creep effect of the viscoelastic model is the key factor for accurately predicting pressure transient, and in the stage of pressure quick rise during the first period, the valve closing and the elasticity of water and pipe play an important role, whereas the pipe hydraulic resistance and the pipe–wall VE play little role, and therefore, the pressure quickly reaches the maximum value. Before the rarefaction wave returns, the maximum pressure firstly decreases a little and rapidly from 0.26 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 increases a little due to the line packing effect, i.e. the effect caused by the velocity decrease greater than that caused by the retarded strain, as can be seen in Figures 4(a), 5(a) and 6(a), which indicate 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 as can be seen in Figures 4(b), 5(b) and 6(b). In addition, Figures 4(a), 5(a) and 6(a) also demonstrate the phase relationship of the pressure head, the retarded strain and the wave speed. It is shown in viscoelastic pipes, when subjected to transient pressure, the pipe–wall behaves in a creep mode and the retarded strain lags behind the pressure phase. Also, Figures 4(b), 5(b) and 6(b) indicate the synchronization of the pressure head with the elastic strain in the total strain. At the same time, it can be seen from Figures 4,5 to 6 that the elastic strain amplitudes in the process of pressure transient are much greater than the retarded strain amplitudes, and the corresponding time-dependent resultant strains are illustrated in Figure 7.
Figure 2

(a) Pressure head changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the elastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Figure 2

(a) Pressure head changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the elastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Close modal
Figure 3

(a) Pressure head changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Figure 3

(a) Pressure head changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Close modal
Figure 4

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the SF-VE. (b) Pressure head and elastic strain changes.

Figure 4

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the SF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 5

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the CB-UF-VE. (b) Pressure head and elastic strain changes.

Figure 5

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the CB-UF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 6

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the MIAB-UF-VE. (b) Pressure head and elastic strain changes.

Figure 6

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the MIAB-UF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 7

(a) Total strain changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models and the experimental data. (b) The corresponding enlarged views of the areas labeled by circle.

Figure 7

(a) Total strain changes upstream the valve (T1) for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models and the experimental data. (b) The corresponding enlarged views of the areas labeled by circle.

Close modal

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.

To further analyze the prediction accuracy by the viscoelastic models on the whole and based on the experimental data at the monitoring point, the averaged relative time-dependent error of the pressure head is calculated as:
formula
(25)
where and are the predicted and the experimental pressure head at the moment, respectively, m is the total number of the time moments.

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.

The pressure head evolutions by the SF-VE, the CB-UF-VE and the MIAB-UF-VE in Figures 8 and 9, the corresponding retarded strain and velocity evolutions in Figures 10(a), 11(a), 12(a), 13(a), 14(a) and 15(a), the elastic strain evolutions in Figures 10(b), 11(b), 12(b), 13(b), 14(b) and 15(b) at T2 and T3 are similar to that at T1 for Case 2, respectively. As can be seen in Figures 3(b) and 8(b), the main difference between the transients at T2 and T1 is that the predicted pressure transient near the maximum at T2 behaves like the transient flow inside an elastic pipeline, although the viscoelastic model also gives the corresponding retarded strain, but the experimental one at T2 behaves like the retarded strain transient in the viscoelastic pipeline. Both the predicted pressure transient characteristics and the experimental one near the maximum pressure at T3 are more like the transient flow in an elastic pipeline. Also, the difference among T1, T2 and T3 is determined mainly by the relative magnitudes of both the corresponding elastic strain and retarded strain. The maximum retarded strain and the maximum elastic strain by the three viscoelastic models at T1, T2 and T3 are shown in Table 2, at T1 are 35, 36 and 34% by the SF-VE, the CB-UF-VE and the MIAB-UF-VE, respectively, at T2 by the three viscoelastic models are all 32%, and at T3 by the three viscoelastic models are all 26%. The maximum total strains at T2 produced by the SF-VE, the CB-UF-VE and the MIAB-UF-VE are 7.11E-4, 7.28E-4 and 7.1E-4, respectively, in Figure 16. The maximum total strains at T3 produced by the SF-VE, the CB-UF-VE and the MIAB-UF-VE are 6.47E-4, 6.54E-4 and 6.47E-4, respectively, in Figure 17. The comparisons of the maximum total strains at T1, T2 and T3 show that the farther away from the downstream valve, the smaller the maximum total strains, which is consistent with the maximum pressure at the corresponding positions.
Table 2

Maximum retarded strain and maximum elastic strain by the three viscoelastic models at T1, T2 and T3 (–)

Viscoelastic modelsT1
T2
T3
SF-VE 2.00E-04 5.77E-04 1.72E-04 5.40E-04 1.33E-04 5.18E-04 
CB-UF-VE 2.06E-04 5.79E-04 1.76E-04 5.54E-04 1.34E-04 5.23E-04 
MIAB-UF-VE 2.00E-04 5.79E-04 1.72E-04 5.39E-04 1.33E-04 5.17E-04 
Viscoelastic modelsT1
T2
T3
SF-VE 2.00E-04 5.77E-04 1.72E-04 5.40E-04 1.33E-04 5.18E-04 
CB-UF-VE 2.06E-04 5.79E-04 1.76E-04 5.54E-04 1.34E-04 5.23E-04 
MIAB-UF-VE 2.00E-04 5.79E-04 1.72E-04 5.39E-04 1.33E-04 5.17E-04 
Figure 8

(a) Pressure head changes at T2 for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Figure 8

(a) Pressure head changes at T2 for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Close modal
Figure 9

(a) Pressure head changes at T3 for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Figure 9

(a) Pressure head changes at T3 for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Close modal
Figure 10

(a) Pressure head, retarded strain and velocity changes at T2 for Case 2 (V0 = 0.4973 m/s) by the SF-VE. (b) Pressure head and elastic strain changes.

Figure 10

(a) Pressure head, retarded strain and velocity changes at T2 for Case 2 (V0 = 0.4973 m/s) by the SF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 11

(a) Pressure head, retarded strain and velocity changes at T2 for Case 2 (V0 = 0.4973 m/s) by the CB-UF-VE. (b) Pressure head and elastic strain changes.

Figure 11

(a) Pressure head, retarded strain and velocity changes at T2 for Case 2 (V0 = 0.4973 m/s) by the CB-UF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 12

(a) Pressure head, retarded strain and velocity changes at T2 for Case 2 (V0 = 0.4973 m/s) by the MIAB-UF-VE. (b) Pressure head and elastic strain changes.

Figure 12

(a) Pressure head, retarded strain and velocity changes at T2 for Case 2 (V0 = 0.4973 m/s) by the MIAB-UF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 13

(a) Pressure head, retarded strain and velocity changes at T3 for Case 2 (V0 = 0.4973 m/s) by the SF-VE. (b) Pressure head and elastic strain changes.

Figure 13

(a) Pressure head, retarded strain and velocity changes at T3 for Case 2 (V0 = 0.4973 m/s) by the SF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 14

(a) Pressure head, retarded strain and velocity changes at T3 for Case 2 (V0 = 0.4973 m/s) by the CB-UF-VE. (b) Pressure head and elastic strain changes.

Figure 14

(a) Pressure head, retarded strain and velocity changes at T3 for Case 2 (V0 = 0.4973 m/s) by the CB-UF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 15

(a) Pressure head, retarded strain and velocity changes at T3 for Case 2 (V0 = 0.4973 m/s) by the MIAB-UF-VE. (b) Pressure head and elastic strain changes

Figure 15

(a) Pressure head, retarded strain and velocity changes at T3 for Case 2 (V0 = 0.4973 m/s) by the MIAB-UF-VE. (b) Pressure head and elastic strain changes

Close modal
Figure 16

Total strain changes at T2 for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models.

Figure 16

Total strain changes at T2 for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models.

Close modal
Figure 17

Total strain changes at T3 for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models.

Figure 17

Total strain changes at T3 for Case 2 (V0 = 0.4973 m/s) by the viscoelastic models.

Close modal

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.

The case of the low and high initial velocity

The predicted pressure evolutions at T1 for Case 1 (V0 = 0.0278 m/s) by the three elastic models and by the three viscoelastic models shown in Figures 18 and 19, respectively, and the corresponding experimental data are similar to those for Case 2, but the corresponding pressure amplitudes decrease because the initial velocity decreases; the predicted pressure transients at T1 for Case 3 (V0 = 0.9946 m/s) by the three elastic models and by the three viscoelastic models shown in Figures 20 and 21, respectively, and the corresponding experimentals are also similar to those for Case 2, but the corresponding amplitudes increase due to higher initial velocity. Similarly, the comparisons show that the SF, the CB-UF and the MIAB-UF cannot capture the pressure transient, whereas the SF-VE, the CB-UF-VE and the MIAB-UF-VE can do so accurately. The phase relations between the pressure head, the retarded strain and the velocity changes at T1 predicted by the three viscoelastic models in Figures 22(a), 23(a), 24(a), 25(a), 26(a) and 27(a), and the phase synchronization of the pressure head and the elastic strain in Figures 22(b), 23(b), 24(b), 25(b), 26(b) and 27(b) are similar to those for Case 2. Figures 28 and 29 show that the periodic decay evolutions of the predicted total strain for Cases 1 and 3 are similar to those for Case 2, but the maximum strains shown in Table 3 are correspondingly much smaller due to lower initial velocity for Case 1, and much larger due to higher initial velocity for Case 3.
Table 3

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

Case no.Maximum total strains predicted by the three viscoelastic models (–)
SF-VECB-UF-VEMIAB-UF-VE
4.05E-5 4.21E-5 4.04E-5 
7.57E-4 7.84E-4 7.56E-4 
1.63E-3 1.68E-3 1.62E-3 
Case no.Maximum total strains predicted by the three viscoelastic models (–)
SF-VECB-UF-VEMIAB-UF-VE
4.05E-5 4.21E-5 4.04E-5 
7.57E-4 7.84E-4 7.56E-4 
1.63E-3 1.68E-3 1.62E-3 
Figure 18

(a) Pressure head changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the elastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Figure 18

(a) Pressure head changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the elastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Close modal
Figure 19

(a) Pressure head changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the viscoelastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Figure 19

(a) Pressure head changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the viscoelastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Close modal
Figure 20

(a) Pressure head changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the elastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Figure 20

(a) Pressure head changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the elastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Close modal
Figure 21

(a) Pressure head changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the viscoelastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Figure 21

(a) Pressure head changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the viscoelastic models. (b) The corresponding enlarged views of the areas labeled by circle.

Close modal
Figure 22

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the SF-VE. (b) Pressure head and elastic strain changes.

Figure 22

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the SF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 23

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the CB-UF-VE. (b) Pressure head and elastic strain changes.

Figure 23

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the CB-UF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 24

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the MIAB-UF-VE. (b) Pressure head and elastic strain changes.

Figure 24

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the MIAB-UF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 25

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the SF-VE. (b) Pressure head and elastic strain changes.

Figure 25

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the SF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 26

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the CB-UF-VE. (b) Pressure head and elastic strain changes.

Figure 26

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the CB-UF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 27

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the MIAB-UF-VE. (b) Pressure head and elastic strain changes.

Figure 27

(a) Pressure head, retarded strain and velocity changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the MIAB-UF-VE. (b) Pressure head and elastic strain changes.

Close modal
Figure 28

Total strain changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the viscoelastic models.

Figure 28

Total strain changes upstream the valve (T1) for Case 1 (V0 = 0.0278 m/s) by the viscoelastic models.

Close modal
Figure 29

Total strain changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the viscoelastic models.

Figure 29

Total strain changes upstream the valve (T1) for Case 3 (V0 = 0.9946 m/s) by the viscoelastic models.

Close modal

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 4

Maximum pressure relative errors at T1 by the three elastic and three viscoelastic models for Cases 1–3

Case no.Maximum pressure relative errors by the elastic and viscoelastic models (–)
SF (%)CB-UF (%)MIAB-UF (%)SF-VE (%)CB-UF-VE (%)MIAB-UF-VE (%)
0.19 0.29 0.19 0.041 0.049 0.044 
3.66 4.82 3.64 0.03 0.18 0.18 
10.75 12.63 10.73 3.23 4.9 3.07 
Case no.Maximum pressure relative errors by the elastic and viscoelastic models (–)
SF (%)CB-UF (%)MIAB-UF (%)SF-VE (%)CB-UF-VE (%)MIAB-UF-VE (%)
0.19 0.29 0.19 0.041 0.049 0.044 
3.66 4.82 3.64 0.03 0.18 0.18 
10.75 12.63 10.73 3.23 4.9 3.07 
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 (%)
9.2 2.5 2.6 
4.7 2.8 2.3 
5.7 4.6 3.6 
Case no. at T1 by the three viscoelastic models (–)
SF-VE (%)CB-UF-VE (%)MIAB-UF-VE (%)
9.2 2.5 2.6 
4.7 2.8 2.3 
5.7 4.6 3.6 

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.

Effects of different factors on pressure attenuation

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.

Based on the comparisons between the calculated results and the experimental data, the analysis method similar to Duan et al. (2010a) and Zhu et al. (2018) is employed to investigate the relative importance of the UF and the VE to pressure attenuation. The analyses are based on the pressure peaks because the peaks and the corresponding valleys attenuate approximately symmetrically. The CB-UF contribution to the pressure peak attenuation is expressed by the difference between the corresponding time sequential pressure peaks at T1 predicted by the SF-VE model and by the CB-UF-VE model, and similarly, the MIAB-UF contribution to the pressure peak attenuation is expressed by the difference between the pressure peaks at T1 predicted by the SF-VE model and by the MIAB-UF-VE model, shown in Figures 30(a), 31(a) and 32(a) for Case 1, Case 2 and Case 3, respectively, and it can be seen that the pressure attenuation contributed by the CB-UF model is not bigger than that 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. Also, therefore, compared to the VE contribution to the pressure peak attenuation expressed by the difference between the pressure peaks at T1 predicted by the CB-UF model and by the CB-UF-VE model, the relative importance of the CB-UF to pressure attenuation is characterized by , and the corresponding relative importance of the MIAB-UF, as shown in Figures 30(b), 31(b) and 32(b) for the three cases, respectively, and at the same time, on the whole, it can be seen from Figures 30(b), 31(b) and 32(b) that the contribution of VE to pressure attenuation is greater than that of the CB-UF and that of the MIAB-UF.
Figure 30

(a) The contributions by the CB-UF and the MIAB-UF for Case 1. (b) Comparisons of and .

Figure 30

(a) The contributions by the CB-UF and the MIAB-UF for Case 1. (b) Comparisons of and .

Close modal

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 Case 2, the evolutions of and shown in Figure 31(b) are similar to those for Case 1. As can be seen from Figure 10, the first pressure 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; the second pressure peak predicted by the SF-VE is the most accurate, followed by the prediction by the CB-UF-VE, and the prediction by the MIAB-UF-VE is relatively poor; the third and fourth pressure peaks predicted by the SF-VE are in best agreement with the experimental, followed by the MIAB-UF-VE and worse by the CB-UF-VE; the pressure peaks from the fifth 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.
Figure 31

(a) The contributions by the CB-UF and the MIAB-UF for Case 2. (b) Comparisons of and .

Figure 31

(a) The contributions by the CB-UF and the MIAB-UF for Case 2. (b) Comparisons of and .

Close modal
For Case 3, as shown in Figure 32(b), is negative at the first pressure peak, which is similar to those in Case 1 and Case 2, but is positive, and the peak predicted by the MIAB-UF-VE is the most accurate, followed by the prediction by the CB-UF-VE, and the prediction by the SF-VE is relatively poor as can be seen from Figure 21. Also, the evolutions of and at other pressure peaks are similar to those for Case 1 and Case 2. As can be seen from Figure 21, the second pressure peak predicted by the CB-UF-VE is in best agreement with the experimental, followed by the SF-VE and worse by the MIAB-UF-VE; from the third pressure peak on, the prediction results by the SF-VE are in best agreement with the experimental, followed by the MIAB-UF-VE and worse by the CB-UF-VE.
Figure 32

(a) The contributions by the CB-UF and the MIAB-UF for Case 3. (b) Comparisons of and .

Figure 32

(a) The contributions by the CB-UF and the MIAB-UF for Case 3. (b) Comparisons of and .

Close modal

Analysis of transient frequency

The UF and the VE also influence the frequency in the transient flows (Lee et al. 2006; Duan et al. 2012; Gong et al. 2015). Therefore, the time-dependent pressure head changes for the three cases at T1 are transformed to the frequency domain by using Fast Fourier Transform (FFT), and the corresponding amplitude-frequency diagrams are shown in Figures 33(a), 34(a), 35(a), 36(a), 37(a) and 38(a). It can be seen from the amplitude-frequency diagrams that the 12th harmonic amplitude accounts for 2.7% of the first harmonic amplitude and 62% of the 11th harmonic amplitude, and therefore, only the first 11 order harmonic frequencies of both the prediction by the elastic models and the experimental data are shown in Figures 33(b), 35(b) and 37(b). Based on the same analyses, the fifth harmonic amplitude accounts for 4.2% of the first harmonic amplitude and 53% of the fourth harmonic amplitude, Figures 34(b), 36(b) and 38(b) show only the first four harmonic frequencies predicted by the viscoelastic models and the experimental, respectively.
Figure 33

(a) Pressure head amplitudes at T1 for Case 2 (V0 = 0.4973 m/s) in the frequency domain produced by different elastic models and the experimental data. (b) Corresponding first 11 harmonic frequencies.

Figure 33

(a) Pressure head amplitudes at T1 for Case 2 (V0 = 0.4973 m/s) in the frequency domain produced by different elastic models and the experimental data. (b) Corresponding first 11 harmonic frequencies.

Close modal
Figure 34

(a) Pressure head amplitudes at T1 for Case 2 (V0 = 0.4973 m/s) in the frequency domain produced by different viscoelastic models and the experimental data. (b) Corresponding first four harmonic frequencies.

Figure 34

(a) Pressure head amplitudes at T1 for Case 2 (V0 = 0.4973 m/s) in the frequency domain produced by different viscoelastic models and the experimental data. (b) Corresponding first four harmonic frequencies.

Close modal

The case of the medium initial velocity

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 case of the low and high initial velocity

For the pressure amplitudes in Cases 1 and 3, similar to Case 2, it can be seen from Figures 35(a) and 37(a) that the predictions by the elastic models are not in good agreement with the experimental data, in which the amplitudes by the CB-UF are closest to the experimental; for Case 1, the first four order amplitudes by the CB-UF are 0.96, 0.24, 0.13 and 0.08 m, respectively, but the first four order amplitudes obtained from the experimental are 0.33, 0.07, 0.03 and 0.02 m, respectively; for Case 3, 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. Figures 36(a) and 38(a) show that the amplitudes predicted by the viscoelastic models are generally in good agreement with the experimental compared to those by the elastic models. As can be seen in Table 6, the amplitudes predicted by the CB-UF-VE are closest to the experimental for Case 1. Also, the relative errors by the three viscoelastic models for Case 3 are shown in Table 9.
Table 6

Relative errors of the first four order pressure amplitudes in the frequency domain at T1 by the three viscoelastic models for Case 1

OrderRelative errors by the viscoelastic models (–)
SF-VE (%)CB-UF-VE (%)MIAB-UF-VE (%)
32.90 5.01 5.86 
26.72 1.57 15.97 
20.05 1.66 11.35 
8.88 1.95 2.49 
OrderRelative errors by the viscoelastic models (–)
SF-VE (%)CB-UF-VE (%)MIAB-UF-VE (%)
32.90 5.01 5.86 
26.72 1.57 15.97 
20.05 1.66 11.35 
8.88 1.95 2.49 
Table 7

First 11 order harmonic frequencies of transient pressure at T1 by the elastic models and the experimental data for Case 1

OrderPredicted frequencies and experimental data (Hz)
FrictionlessSFCB-UFMIAB-UFExperimental data
0.35 0.35 0.35 0.35 0.35 
1.10 1.10 1.10 1.10 1.00 
1.80 1.80 1.80 1.80 1.75 
2.55 2.55 2.55 2.50 2.40 
3.30 3.30 3.25 3.20 3.10 
4.00 4.00 4.00 3.95 3.90 
4.75 4.75 4.75 4.65 4.60 
5.45 5.45 5.45 5.40 5.30 
6.20 6.20 6.20 6.10 6.10 
10 6.90 6.90 6.90 6.80 6.50 
11 7.65 7.65 7.65 7.50 6.80 
OrderPredicted frequencies and experimental data (Hz)
FrictionlessSFCB-UFMIAB-UFExperimental data
0.35 0.35 0.35 0.35 0.35 
1.10 1.10 1.10 1.10 1.00 
1.80 1.80 1.80 1.80 1.75 
2.55 2.55 2.55 2.50 2.40 
3.30 3.30 3.25 3.20 3.10 
4.00 4.00 4.00 3.95 3.90 
4.75 4.75 4.75 4.65 4.60 
5.45 5.45 5.45 5.40 5.30 
6.20 6.20 6.20 6.10 6.10 
10 6.90 6.90 6.90 6.80 6.50 
11 7.65 7.65 7.65 7.50 6.80 
Figure 35

(a) Pressure head amplitudes at T1 for Case 1 (V0 = 0.0278 m/s) in the frequency domain produced by different elastic models and the experimental data. (b) Corresponding first 11 harmonic frequencies.

Figure 35

(a) Pressure head amplitudes at T1 for Case 1 (V0 = 0.0278 m/s) in the frequency domain produced by different elastic models and the experimental data. (b) Corresponding first 11 harmonic frequencies.

Close modal
Figure 36

(a) Pressure head amplitudes at T1 for Case 1 (V0 = 0.0278 m/s) in the frequency domain produced by different viscoelastic models and the experimental data. (b) Corresponding first four harmonic frequencies.

Figure 36

(a) Pressure head amplitudes at T1 for Case 1 (V0 = 0.0278 m/s) in the frequency domain produced by different viscoelastic models and the experimental data. (b) Corresponding first four harmonic frequencies.

Close modal
Figure 37

(a) Pressure head amplitudes at T1 for Case 3 (V0 = 0.9946 m/s) in the frequency domain produced by different elastic models and the experimental data. (b) Corresponding first 11 harmonic frequencies.

Figure 37

(a) Pressure head amplitudes at T1 for Case 3 (V0 = 0.9946 m/s) in the frequency domain produced by different elastic models and the experimental data. (b) Corresponding first 11 harmonic frequencies.

Close modal
Figure 38

(a) Pressure head amplitudes at T1 for Case 3 (V0 = 0.9946 m/s) in the frequency domain produced by different viscoelastic models and the experimental data. (b) Corresponding first four harmonic frequencies.

Figure 38

(a) Pressure head amplitudes at T1 for Case 3 (V0 = 0.9946 m/s) in the frequency domain produced by different viscoelastic models and the experimental data. (b) Corresponding first four harmonic frequencies.

Close modal

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.

Table 8

First four order harmonic frequencies of transient pressure at T1 by the viscoelastic models and the experimental data for Case 1

OrderPredicted frequencies and experimental data (Hz)
SF-VECB-UF-VEMIAB-UF-VEExperimental data
0.35 0.35 0.35 0.35 
1.10 1.10 1.10 1.00 
1.80 1.80 1.80 1.75 
2.55 2.55 2.50 2.40 
OrderPredicted frequencies and experimental data (Hz)
SF-VECB-UF-VEMIAB-UF-VEExperimental data
0.35 0.35 0.35 0.35 
1.10 1.10 1.10 1.00 
1.80 1.80 1.80 1.75 
2.55 2.55 2.50 2.40 
Table 9

Relative errors of the first four order pressure amplitudes in the frequency domain at T1 by the three viscoelastic models for Case 3

OrderRelative errors by the viscoelastic models (–)
SF-VE (%)CB-UF-VE (%)MIAB-UF-VE (%)
2.53 8.17 6.10 
5.07 6.88 3.35 
13.07 0.71 2.14 
3.55 8.91 0.48 
OrderRelative errors by the viscoelastic models (–)
SF-VE (%)CB-UF-VE (%)MIAB-UF-VE (%)
2.53 8.17 6.10 
5.07 6.88 3.35 
13.07 0.71 2.14 
3.55 8.91 0.48 

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.

Table 10

First 11 order harmonic frequencies of transient pressure at T1 by the elastic models and the experimental data for Case 3

OrderPredicted frequencies and experimental data (Hz)
FrictionlessSFCB-UFMIAB-UFExperimental Data
0.33 0.33 0.33 0.33 0.33 
1.07 1.07 1.07 1.07 1.00 
1.80 1.80 1.80 1.80 1.73 
2.53 2.53 2.53 2.53 2.40 
3.27 3.27 3.27 3.20 3.13 
4.00 4.00 4.00 3.93 3.80 
4.73 4.73 4.73 4.67 4.53 
5.47 5.47 5.47 5.33 5.13 
6.20 6.20 6.20 6.13 5.73 
10 6.93 6.93 6.93 6.80 6.27 
11 7.67 7.67 7.60 7.53 6.67 
OrderPredicted frequencies and experimental data (Hz)
FrictionlessSFCB-UFMIAB-UFExperimental Data
0.33 0.33 0.33 0.33 0.33 
1.07 1.07 1.07 1.07 1.00 
1.80 1.80 1.80 1.80 1.73 
2.53 2.53 2.53 2.53 2.40 
3.27 3.27 3.27 3.20 3.13 
4.00 4.00 4.00 3.93 3.80 
4.73 4.73 4.73 4.67 4.53 
5.47 5.47 5.47 5.33 5.13 
6.20 6.20 6.20 6.13 5.73 
10 6.93 6.93 6.93 6.80 6.27 
11 7.67 7.67 7.60 7.53 6.67 

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.

Table 11

First four order harmonic frequencies of transient pressure at T1 by the viscoelastic models and the experimental data for Case 3

OrderPredicted frequencies and experimental data (Hz)
SF-VECB-UF-VEMIAB-UF-VEExperimental data
0.33 0.33 0.33 0.33 
1.00 1.00 1.00 1.00 
1.73 1.73 1.73 1.73 
2.47 2.40 2.40 2.40 
OrderPredicted frequencies and experimental data (Hz)
SF-VECB-UF-VEMIAB-UF-VEExperimental data
0.33 0.33 0.33 0.33 
1.00 1.00 1.00 1.00 
1.73 1.73 1.73 1.73 
2.47 2.40 2.40 2.40 

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.

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