Listen
graphic
View largeDownload slide

Numerous studies on the pressure fluctuations and cavity volume variations of a transient cavitation flow in viscoelastic pipes are available in the literature. However, the effect of water temperature on the cavity volume and energy conversion has been studied less often. This paper employs the discrete vapor cavity model using quasi-steady friction and quasi-two-dimensional friction models to calculate the cavity volume for different water temperatures and investigates the effects of water temperature on the appearance of the first cavitation at the downstream valve, and on the pressure damping in a tank-piping-valve system using an integrated energy analysis approach. The results show that the differences between the pressure and energy variations of the transient cavitation flow simulated using different models were minimal under different water temperature conditions. Moreover, as the water temperature increased, the appearance time of the cavity is postponed, and the volume of the cavity decreases. The energy dissipation increases continuously with an increase in the volume of the cavitation and water temperature in viscoelastic pipes. This study provides valuable insights into the variation pattern of the cavity and the effect of vapor cavities on the rise and decay of the pipeline pressure in different situations.

Listen

  • This study investigated the effect of different water temperatures on the calculated results of transient cavitation flow in viscoelastic pipes.

  • The energy variations during transient flow and transient cavitation flow in viscoelastic pipes were compared.

  • The concept of the conversion ratio from kinetic energy to elastic energy was proposed, where a smaller conversion ratio may result in a larger energy dissipation.

energy analysis, transient cavitation flow, viscoelastic pipeline, water temperature
Listen

The following symbols are used in this paper:

A

area of the pipe (m2)

Ak

characteristic constant

a

wave speed (m/s)

B

constant value

Bk

characteristic constant

Ck

characteristic constant

CM

characteristic constant

CP

characteristic constant

D

pipe diameter (m)

Df

energy of frictional effect

F

characteristic constant

f

Darcy–Weisbach friction factor

G

instantaneous kinetic energy

g

acceleration of gravity (m/s2)

H

pressure head (m)

J0

the instantaneous elastic creep

Jk

the creep compliance of the spring of k-th element

M

the elastic energy of the total internal energy

m

the number of grid points

N

the number of K-V elements

the experimentally measured pressure at the valve (N)

the pressure at the valve calculated by the model (N)

Q

flow rate (m3/s)

Qi

the downstream flow rate of the cavity (m3/s)

Qui

the upstream flow rate of the cavity (m3/s)

the radial flow per unit of the pipe length

r

radial distance along the pipe from the pipe axis (m)

x

the distance in the axial direction of the pipe (m)

t

the time coordinate (s)

u

axial velocity

V

average velocity in section (m/s)

V*

the volume of cavity (m3)

Vi

the downstream flow velocity of the cavity (m/s)

Vui

the upstream flow velocity of the cavity (m/s)

v

the radial velocity

WR

the power on the pipe wall

WL

the power from the ends of the pipeline

Listen

the density

τk, Jk

the retarded components (k = 1, …, N and N = number of Kelvin–Voigt elements)

τw

the total wall shear stress (N)

εr

the retarded strain

Listen
n

time step index

*

volume

Listen
exp

experimental result

i

axial step index

j

radial step index

mod

simulated result

N

number of Kelvin–Voigt elements

n

time step index

Listen
1D

quasi-steady

DVCM

discrete vapor cavity model

K-V model

Kelvin–Voigt model

MOC

method of characteristics

quasi-2D

quasi-two-dimensional

Listen

Water hammer with column separation is a common hydraulic transient phenomenon caused by sudden power failures or the improper manual operation of a pressurized pipeline (Tijsseling 1996; Bergant et al. 2006), which can cause a rapid increase in pressure fluctuations. Consequently, the pressure can propagate through an entire pipeline, posing a threat to the safe operation of hydraulic machinery and piping systems (Hadj-Taleb & Hadj-Taleb 2009). Moreover, the viscoelastic properties of pipes can affect the pressure of transient cavitation flow, thereby contributing to the safe operation of piping systems. The viscoelastic properties of pipes can be greatly influenced by the water temperature, directly impacting the transient cavitation flow in viscoelastic pipes.

Various numerical models have been developed for the simulation of transient cavitation flow – such as the discrete cavity and continuous cavity models – among which the discrete cavity model is the more widely used (Simpson & Bergant 1994). Commonly used discrete cavity models include the discrete vapor cavity model (DVCM) and the discrete gas cavity model (DGCM). The classical DVCM describes the essential characteristics of transient cavitation flow and is widely used by researchers (Bergant & Simpson 1999). However, the DVCM combined with the quasi-steady friction model generates virtual pressure oscillations when simulating pressure fluctuations. Consequently, some researchers have used the unsteady friction model instead of the quasi-steady friction model to solve this problem and avoid unrealistic pressure peaks. Soares et al. (2017) combined an unsteady friction model with cavitation models to simulate transient cavitation flow in a horizontal copper pipe. The results showed excellent agreement between the numerical results and experimental data, the unsteady friction model also having an attenuating effect on the pressure generated after cavity collapse. Pezzinga & Santoro (2017) and Santoro et al. (2018) combined the DVCM with one-dimensional (1D) steady-state and quasi-two-dimensional (quasi-2D) friction models in elastic pipes and found that the quasi-2D friction model could better reproduce the water hammer pressure profile. Based on the experimental and simulation results, the DVCM was more suitable for low-cavitation phenomena or the first stage of cavitation. By comparing the solution of the water hammer equation considering unsteady friction and viscoelasticity with the classical numerical solution, the introduction of viscoelasticity and unsteady friction improved the accuracy of the simulation (Seck 2020). Additionally, Soares et al. (2009) combined the DVCM with 1D unsteady friction models to investigate cavitation in polyethylene pipes. They found that the effect of unsteady friction on the pressure was small in viscoelastic pipes, but viscoelasticity had a considerable effect on the results.

Additionally, water temperature not only directly affects the saturation vapor pressure of a liquid but can also influence the intrinsic parameters of the viscoelastic pipe (Ghidaoui et al. 2005). Several studies have investigated this topic. Saidani et al. (2022) simulated the pressure fluctuations in a copper pipe system in a temperature range of 4–95 °C, and found that the higher the temperature, the higher the cavity volume in the copper pipe. Mousavifard (2022) discussed the effect of water temperature on k-w model parameters and transient flow profiles in a viscoelastic pipe. The results showed that the higher the temperature, the lower the velocity gradient near the wall, and the slower the propagation of turbulence to the inner layer. Hadj-Taieeb & Hadj-Taieeb (2009) noted a viscoelastic effect on pressure attenuation and dispersion in their simulations. They pointed out that the higher the temperature, the more evident the pipe creep effect, and the shorter the cavity duration.

However, the aforementioned studies were based on collecting transient pressure signals at specific locations, which could reflect the pipeline system properties. However, these studies did not provide comprehensive information on the entire pipe (Pan et al. 2020). An energy analysis method was introduced to study and evaluate the state of pipeline systems. Duan et al. (2010) used an integrated energy analysis method to analyze the attenuation of pressure by viscoelastic effects and unsteady friction. The results showed that the viscoelastic properties of a pipe could affect the peaks of later pressure fluctuations (Duan et al. 2017). Additionally, the viscoelastic effect proved to be more important in long pipe systems compared with the effect of unsteady friction, while unsteady friction dominated energy dissipation in pipes of small diameter and shorter pipe systems (Duan et al. 2012). Lee (2013) employed an integrated energy analysis approach to evaluate the accuracy of a transmission line model. Kashif et al. (2023) employed an energy analysis approach to illustrate that pressure vessels greatly affected water hammer protection. Meniconi et al. (2014) developed the relationship between pressure dissipation and kinetic and elastic energies in viscoelastic pipes using integrated energy analysis. Andrade et al. (2023) explained the energy variations of transient flow in viscoelastic pipes. However, this requires further investigation as there was no clear explanation of the effect of water temperature on pressure attenuation and cavity volume in transient cavitation flow in viscoelastic pipes.

Consequently, this study investigated the effects of 1D quasi-steady friction and quasi-2D models on the numerical simulation of transient cavitation flow in viscoelastic pipes at different water temperatures, the impact of water temperature on the cavity volume and energy variations in the entire pipe being analyzed. This study provides valuable insights into the influence of water temperature on the transient cavitation flow in viscoelastic pipes.

Listen

Governing equations

Listen
The quasi-2D governing equations of transient flow in viscoelastic pipes can be expressed as follows (Pezzinga & Santoro 2017):
(1)
(2)
where H denotes the piezometric head, Q denotes the flow rate, x denotes the distance in the axial direction of the pipe, r denotes the radial distance along the pipe from the pipe axis, t denotes the time coordinate, A denotes the area of the pipe, a denotes the wave speed, g denotes gravitational acceleration, denotes the pipe wall stress, u denotes the axial velocity, v denotes the radial velocity, and denotes the retarded strain.
By integrating Equations (1) and (2) over the cross-sectional area of the pipe, the 1D governing equations can be obtained as follows (Covas et al. 2004b):
(3)
(4)
where D denotes the pipe diameter. The wall shear stress can be calculated using the Darcy–Weisbach equation:
(5)
where f denotes the Darcy–Weisbach friction factor, and V(t) denotes the average cross-sectional flow velocity.

Numerical discretization of the retarded strain

Listen
The creep behavior of viscoelastic pipes can be described using the Kelvin–Voigt (K-V) model, which consists of an elastic element (spring) and a viscous element (dashpot), as shown in Figure 1 (Covas et al. 2004a, 2005):
(6)
where Jk denotes the creep compliance of the spring of the k-th element, denotes the retarded time of the k-th element, and N denotes the number of K-V elements.
Figure 1
Schematic diagram of K-V model.
View largeDownload slide

Schematic diagram of K-V model.

Figure 1
Schematic diagram of K-V model.
View largeDownload slide

Schematic diagram of K-V model.

Close modal
When the K-V model with N elements is employed, the time derivative of the retarded strain can be expressed as follows:
(7)
Finally, algebraic expression of Equation (7) is shown as (Covas et al. 2005):
(8)
(9)
(10)

Discrete vapor cavity model

Listen
The DVCM assumes that when the pressure at the computational points drops to the saturated vapor pressure at a given temperature, a cavity is formed at the specified points. The cavities separate the continuous liquid column, the liquid columns between adjacent points exhibiting a constant pressure wave velocity. The pressure in the cavity is equal to the vaporization pressure of the liquid.
(11)
The cavity volume at grid point i can be calculated from the upstream and downstream flow rates, as shown in Figure 2:
(12)
Figure 2
The calculation of volume of cavity.
View largeDownload slide

The calculation of volume of cavity.

Figure 2
The calculation of volume of cavity.
View largeDownload slide

The calculation of volume of cavity.

Close modal
The numerical integration of Equation (12) yields:
(13)
where Vi* denotes the cavity volume at grid point i, Qui denotes the upstream flow rate of the cavity at grid point i, Qi denotes the downstream flow rate of the cavity, Vui denotes the upstream flow velocity of the cavity, Vi denotes the downstream flow velocity of the cavity, ⁠, ⁠, and denote the known values at time ⁠, ⁠, ⁠, and denote the unknown values at time ⁠, Hv denotes the steam pressure head, and denotes the weight coefficient of calculating the cavity volume, taking the value range of 0.5 ∼ 1.

Numerical scheme

Listen
The 1D and quasi-2D governing equations can be solved using the method of characteristics (MOC), as shown in Figures 3 and 4, respectively. The partial differential equations may be transformed into ordinary differential equations as follows:
(14)
(15)
Figure 3
Rectangular grid system.
View largeDownload slide

Rectangular grid system.

Figure 3
Rectangular grid system.
View largeDownload slide

Rectangular grid system.

Close modal
Figure 4
Difference grid system for the quasi-2D model.
View largeDownload slide

Difference grid system for the quasi-2D model.

Figure 4
Difference grid system for the quasi-2D model.
View largeDownload slide

Difference grid system for the quasi-2D model.

Close modal

Numerical solution of the 1D model

Listen
By integrating Equation (14) along the characteristic line in the time range of and ⁠, its discrete form yields:
(16)
Combined with Equation (8), the 1D characteristic equations can be expressed as follows:
(17)
(18)
(19)
(20)
where ⁠. When a cavity is formed at the grid node, the pressure at the cavity can be calculated using Equation (11). Then, substituting the known pressure at into Equations (17) and (18), the upstream and downstream flow rates at time can be calculated, the volume of the cavity being calculated using Equation (13).

Numerical solution of quasi-2D model

Listen
Integrating Equation (15) along the characteristic lines at and time yields the following (Sun et al. 2022):
(21)
(22)
(23)
(24)
(25)
where and denote the weighting factors, denotes the radial flux, the subscripts i and j denote the axial and radial step indices, respectively, and and denote the coordinates of the boundary and middle points of the reaches in the radial direction, respectively.
When there is a cavity at node i, substituting Equation (11) into Equations (21) and (22), the unknown parameters and at time can be obtained using Thomas algorithm (Vardy & Hwang 1991). The wall shear stress at time can then be calculated as follows:
(26)
where denotes the turbulent viscosity.

Finally, and can then be solved using Equations (17) and (18), respectively. The cavity volume can be calculated using Equation (13).

Listen

Experimental data

Listen
The laboratory data obtained at different temperatures by Gally et al. (1979) were in a typical tank-pipe-valve system, as shown in Figure 5, comprising a low-density polyethylene pipe of total length of 43.1 m, an inner diameter of 45.8 mm, and a wall thickness of 4.2 mm. The experiments were conducted by heating the fluid in the tank using a heating- and temperature-control device. The valve closure time was 12 ms and three high-precision pressure sensors were installed upstream, midstream, and downstream of the pipe. The experimental parameters were as listed in Tables 1 and 2.
Table 1

Parameters of transient cavitating flows experiments

Temperature (°C)Initial pressure (×105 Pa)Initial mean velocity (m/s)Kinematic viscosity (×10−6 m2/s)Reynolds numberBulk modulus (×109 Pa)Density (kg/m³)
13.8 2.99 1.27 1.17 45155 2.14 999.3 
25 2.99 1.27 0.892 59229 2.24 997.1 
31 2.99 1.27 0.784 67389 2.27 995.3 
38.5 2.99 1.27 0.675 78270 2.295 992.6 
Temperature (°C)Initial pressure (×105 Pa)Initial mean velocity (m/s)Kinematic viscosity (×10−6 m2/s)Reynolds numberBulk modulus (×109 Pa)Density (kg/m³)
13.8 2.99 1.27 1.17 45155 2.14 999.3 
25 2.99 1.27 0.892 59229 2.24 997.1 
31 2.99 1.27 0.784 67389 2.27 995.3 
38.5 2.99 1.27 0.675 78270 2.295 992.6 
View Large
Table 2

Parameters of transient flows experiments

Temperature (°C)Initial pressure (×105 Pa)Initial mean velocity (m/s)Kinematic viscosity (×10−6 m2/s)Reynolds numberBulk modulus (×109 Pa)Density (kg/m³)
13.8 1.06 0.49 1.17 17 422 2.14 999.3 
25 1.07 0.55 0.892 25 650 2.24 997.1 
31 1.07 0.57 0.784 30 245 2.27 995.3 
38.5 1.06 0.56 0.675 34 513 2.295 992.6 
Temperature (°C)Initial pressure (×105 Pa)Initial mean velocity (m/s)Kinematic viscosity (×10−6 m2/s)Reynolds numberBulk modulus (×109 Pa)Density (kg/m³)
13.8 1.06 0.49 1.17 17 422 2.14 999.3 
25 1.07 0.55 0.892 25 650 2.24 997.1 
31 1.07 0.57 0.784 30 245 2.27 995.3 
38.5 1.06 0.56 0.675 34 513 2.295 992.6 
View Large
Figure 5
Schematic diagram of the experimental setup.
View largeDownload slide

Schematic diagram of the experimental setup.

Figure 5
Schematic diagram of the experimental setup.
View largeDownload slide

Schematic diagram of the experimental setup.

Close modal

Parameters selection of DVCM

Listen
The DVCM with the quasi-2D model (2D-DVCM) and quasi-steady friction model (1D-DVCM) were employed to simulate the pressure variations of transient flow in viscoelastic pipes at different water temperatures. Friction models include frictionless, 1D quasi-steady friction, and quasi-2D models, respectively. The effect of the number of grids and weighting coefficient (Ψ) of the cavity volume to accurately simulate pressure fluctuations were investigated. Moreover, a quantitative analysis index (tcav) was introduced to evaluate the accuracy of the simulation results. According to the literature’ suggestions (Simpson & Bergant 1994), for the 1D-DVCM and 2D-DVCM, axial grid numbers of 50, 100, 200, and 300 were taken into account separately. In the 2D-DVCM, the radial grid numbers ranged from 20 to 100, at intervals of 20. The root mean square error (RMSE) was used to reflect the deviation between the simulated and experimental results. A smaller RMSE value indicates better consistency between the peak pressures of the numerical results and the experimental data. Consequently, the RMSE calculated using each grid number was analyzed, as shown in Figures 6 and 7, to quantify the difference between the calculated and experimental pressure values at the valve.
Figure 6
RMSE values of 2D-DVCM at different temperatures for different Nx and Nr values. (a) In the case of 13.8 °C, (b) in the case of 25 °C, (c) in the case of 31 °C, and (d) in the case of 38.5 °C.
View largeDownload slide

RMSE values of 2D-DVCM at different temperatures for different Nx and Nr values. (a) In the case of 13.8 °C, (b) in the case of 25 °C, (c) in the case of 31 °C, and (d) in the case of 38.5 °C.

Figure 6
RMSE values of 2D-DVCM at different temperatures for different Nx and Nr values. (a) In the case of 13.8 °C, (b) in the case of 25 °C, (c) in the case of 31 °C, and (d) in the case of 38.5 °C.
View largeDownload slide

RMSE values of 2D-DVCM at different temperatures for different Nx and Nr values. (a) In the case of 13.8 °C, (b) in the case of 25 °C, (c) in the case of 31 °C, and (d) in the case of 38.5 °C.

Close modal
Figure 7
RMSE values of 1D-DVCM for different Nx values.
View largeDownload slide

RMSE values of 1D-DVCM for different Nx values.

Figure 7
RMSE values of 1D-DVCM for different Nx values.
View largeDownload slide

RMSE values of 1D-DVCM for different Nx values.

Close modal
The RMSE value can be defined as follows:
(27)
where denotes the experimentally measured pressure at the valve, denotes the pressure at the valve calculated using the model, and m denotes the number of grid points.

Figure 6 shows the RMSE calculated using the 2D-DVCM model for different Nx and Nr values at different water temperatures. When Nx remains constant, an increase in the value of Nr results in a decrease in the RMSE. However, the difference is negligible as the variations in the RMSE values are extremely small, especially for the (water temperature) cases of 13.8 and 25 °C. For the Nx value, at 13.8 °C, the RMSE values increase considerably with increasing Nx, while at other temperatures, the RMSE values exhibit the same trend. This trend is also evident in the 1D-DVCM model.

Figure 7 shows the RMSE values obtained using the 1D-DVCM model for different grid numbers at various temperatures. Similarly, for the 1D-DVCM, as the Nx value increases at different temperatures, the RMSE values gradually increase.

Additionally, the influence of the weighting coefficient of the cavity volume on pressure fluctuations (Ψ) with a range of values between 0.5 and 1 was investigated, through the results of the simulations using different Ψ values being compared. It is evident that the differences between simulation results are slight, but the result is the most accurate when Ψ = 1. Consequently, this study adopted a weighting coefficient Ψ = 1, an axial grid number of 50, and a radial grid number for the 2D-DVCM of 80, respectively.

Comparison of pressure fluctuations calculated using different friction models

Listen
Figure 8 shows a comparison between the pressure fluctuations calculated using the 1D-DVCM, no-friction DVCM (NF-DVCM), and 2D-DVCM at four water temperatures. As shown in the figure, both the 1D-DVCM and 2D-DVCM are capable of accurately simulating the peak and phase of the pressure fluctuations at different water temperatures. The first peak pressure simulated by both models has a difference of 0.2 bar from the experimental data at 13.8 °C, but at the other three temperatures, the corresponding value computed by both models coincides with the experimental data. However, the 2D-DVCM model generates excessive damping in the later pressure waves; consequently, it is always lower than the experimental pressure at the second and third peaks. Additionally, both models show a slight lag in phase compared to the experimental data, which is evident in the comparison of pressure fluctuations at 31 and 38.5 °C. Therefore, both models perform well in simulating the peak pressure and phase of pressure fluctuations at different water temperatures.
Figure 8
Pressure head simulated by different friction models at four temperatures. (a) In the case of 13.8 °C, (b) in the case of 25 °C, (c) in the case of 31 °C, and (d) in the case of 38.5 °C.
View largeDownload slide

Pressure head simulated by different friction models at four temperatures. (a) In the case of 13.8 °C, (b) in the case of 25 °C, (c) in the case of 31 °C, and (d) in the case of 38.5 °C.

Figure 8
Pressure head simulated by different friction models at four temperatures. (a) In the case of 13.8 °C, (b) in the case of 25 °C, (c) in the case of 31 °C, and (d) in the case of 38.5 °C.
View largeDownload slide

Pressure head simulated by different friction models at four temperatures. (a) In the case of 13.8 °C, (b) in the case of 25 °C, (c) in the case of 31 °C, and (d) in the case of 38.5 °C.

Close modal
However, the RMSE does not completely represent the accuracy of the computed results. The formation and collapse times of cavities are also important indicators for describing the accuracy of the simulated transient cavitation flows. Thus, the cavity formation times obtained using the 1D-DVCM and 2D-DVCM were compared with the experimental data. The parameter tcav can be defined to quantitatively analyze the consistency between the simulation results and the experimental data in terms of the cavity formation time, as follows (Sun et al. 2022):
(28)
⁠. Where texp and tmod denote the formation times of the ith cavity in the experimental and numerical results, respectively.
Figure 9 shows a comparison between the accuracy of the three friction models in predicting the cavitation formation time at different temperatures. It is evident that the results obtained using the 2D-DVCM are closer to the experimental results than those obtained using the 1D-DVCM. However, the difference in the tcav value computed using both models is less than 3%. Moreover, as the temperature increases, the accuracy of all three models in simulating the cavitation formation time also improves. At 38.5 °C, the difference in the cavitation formation times calculated using the 2D-DVCM model is less than 1%, demonstrating that the friction model is necessary to simulate the formation time of the cavity and that the numerical simulation results of the 2D friction model are better than those of the 1D friction model.
Figure 9
tcav value calculated using three frictional models at different temperatures.
View largeDownload slide

tcav value calculated using three frictional models at different temperatures.

Figure 9
tcav value calculated using three frictional models at different temperatures.
View largeDownload slide

tcav value calculated using three frictional models at different temperatures.

Close modal

Effect of water temperature on the cavity volume

Listen
The water temperature affects the saturation vapor pressure of the liquid and can influence the formation and duration of cavities. Table 3 lists the variations in cavity volume at different temperatures. As is evident from Table 3, with an increase in water temperature, the formation time of the cavitation increases, meanwhile, the duration of the cavity decreases continuously. Figure 10 shows the variation in the maximum volume of the cavity at different temperatures. It is evident that the maximum volume of the cavities at the valve decreases with increasing water temperature.
Table 3

Variation of the volume of the cavity at different temperatures

Temperature (°C)Time of cavity appearance (s)Cavity duration (s)Maximum volume of cavity (10−5 m³)Percentage of maximum cavity (%)Appearance time of maximum volume of cavity (s)
13.8 0.392
1.439 		0.546
0.18 		4.913 3.56 0.922 
25 0.493 0.495 3.447 2.50 0.926 
31 0.552 0.467 2.641 1.91 0.981 
38.5 0.706 0.38 1.135 0.82 1.076 
Temperature (°C)Time of cavity appearance (s)Cavity duration (s)Maximum volume of cavity (10−5 m³)Percentage of maximum cavity (%)Appearance time of maximum volume of cavity (s)
13.8 0.392
1.439 		0.546
0.18 		4.913 3.56 0.922 
25 0.493 0.495 3.447 2.50 0.926 
31 0.552 0.467 2.641 1.91 0.981 
38.5 0.706 0.38 1.135 0.82 1.076 
View Large
Figure 10
Maximum volume of cavity at the valve.
View largeDownload slide

Maximum volume of cavity at the valve.

Figure 10
Maximum volume of cavity at the valve.
View largeDownload slide

Maximum volume of cavity at the valve.

Close modal
Listen
This section compares the variations in the different energy terms of transient no cavitation and cavitation flows, employing integrated energy analysis to investigate the effects of cavity volume and water temperature on the conversion and damping of energy. The energy equation of the transient flow can be expressed as follows (Duan et al. 2017; Pan et al. 2022):
(29)
where M denotes the elastic energy, G denotes the instantaneous kinetic energy, denotes the energy of the frictional effect, denotes the energy of the viscoelastic behavior, and denotes the power from both ends of the pipeline, respectively.
The remaining terms in the 1D energy equation of transient flow in viscoelastic pipes can be expressed as follows:
(30)
(31)
(32)
(33)
(34)
In the quasi-2D energy equation of the transient flow in a viscoelastic pipe, the energy equation of the viscoelastic behavior and elastic energy equation are shown in Equations (33) and (34), the other parts being expressed as follows (Wu et al. 2022):
(35)
(36)
(37)

Analysis of kinetic-elastic energy variations of different friction models

Listen
Figure 11 compares the kinetic and elastic energies of the two friction models during transient cavitation flow in viscoelastic pipes. As shown in Figure 11, the main differences are in the formation time and volume of the cavity. As shown in Figure 9, the 2D-DVCM predicts the formation time of the cavity more accurately than the 1D-DVCM. Consequently, the following energy analysis uses the numerical results computed by the 2D-DVCM model.
Figure 11
Energy conversion of different friction models. (a) Test with cavitation at 25 °C and (b) test with cavitation at 38.5 °C.
View largeDownload slide

Energy conversion of different friction models. (a) Test with cavitation at 25 °C and (b) test with cavitation at 38.5 °C.

Figure 11
Energy conversion of different friction models. (a) Test with cavitation at 25 °C and (b) test with cavitation at 38.5 °C.
View largeDownload slide

Energy conversion of different friction models. (a) Test with cavitation at 25 °C and (b) test with cavitation at 38.5 °C.

Close modal

Kinetic and elastic energy variations in transient flow

Listen
Figure 12 shows the variations in the kinetic and elastic energies in the transient flow at different temperatures. Each point in Figure 12 represents the energy state of the pipeline system at a given time. At the beginning of the transient flow, kinetic energy begins to be converted into elastic energy. Owing to the influence of frictional dissipation, viscoelastic properties, and fluid flow patterns, a complex curve is formed. The curve starts at point A (t = 0) and decays toward the origin in the direction of the arrows. From Figure 12(a)–12(d), it is evident that the variation in water temperature does not affect the shape of the energy graph. However, water temperature influences the inclination angle of the zigzag waveform. The discussions in detail are as follows.
Figure 12
Conversion between kinetic and elastic energy in transient cavitation flow. (a) Test no cavitation 13.8 °C, (b) test no cavitation 25 °C, (c) test no cavitation 31 °C, and (d) test no cavitation 38.5 °C. Please refer to the online version of this paper to see this figure in colour: https://dx.doi.org/10.2166/hydro.2023.231.
View largeDownload slide

Conversion between kinetic and elastic energy in transient cavitation flow. (a) Test no cavitation 13.8 °C, (b) test no cavitation 25 °C, (c) test no cavitation 31 °C, and (d) test no cavitation 38.5 °C. Please refer to the online version of this paper to see this figure in colour: https://dx.doi.org/10.2166/hydro.2023.231.

Figure 12
Conversion between kinetic and elastic energy in transient cavitation flow. (a) Test no cavitation 13.8 °C, (b) test no cavitation 25 °C, (c) test no cavitation 31 °C, and (d) test no cavitation 38.5 °C. Please refer to the online version of this paper to see this figure in colour: https://dx.doi.org/10.2166/hydro.2023.231.
View largeDownload slide

Conversion between kinetic and elastic energy in transient cavitation flow. (a) Test no cavitation 13.8 °C, (b) test no cavitation 25 °C, (c) test no cavitation 31 °C, and (d) test no cavitation 38.5 °C. Please refer to the online version of this paper to see this figure in colour: https://dx.doi.org/10.2166/hydro.2023.231.

Close modal

At the beginning of the transient flow (point A), the kinetic energy reaches its maximum value. The elastic potential energy is not zero but is not easily distinguishable (as is evident in Figure 12(d)). Subsequently, kinetic energy is converted into elastic potential energy. To better evaluate the energy variations, the concept of conversion ratio can be introduced in the following analysis: The conversion ratio is the ratio of the maximum elastic energy to the maximum kinetic energy. Additionally, the conversion ratio is reflected in the inclination angle of the zigzag waveform shown in the graph, the inclination angle being positively proportional to the conversion ratio. A smaller conversion ratio indicates greater energy dissipation, which is consistent with the results of a study conducted by Pan et al. (2022).

The conversion ratio decreases over time, however, the decrease is not significant, as shown in Figure 12(c). Moreover, the water temperature has a major influence on the conversion ratio – that is, as the water temperature increases, the conversion ratio decreases – primarily due to the increase in the pipeline's viscoelastic properties at higher temperatures, which allows for better energy storage and dissipation, resulting in increased energy dissipation and decreased energy retention in the system.

Variations of kinetic and elastic energies of transient cavitation flow

Listen
Figure 13 depicts the variations in the kinetic and elastic energies in the transient cavitation flow. The conversion ratio of kinetic energy to elastic energy with temperature variation follows the same pattern as that in transient flow. However, the presence of cavities greatly affects the energy variation in the pipeline. The energy conversion in the cavity is indicated by the red line, and it is evident from Figure 13(a) that the presence of cavities (at approximately 0.5–1 s) varies the original curve trend, sharply reducing the conversion ratio of kinetic energy to elastic energy. As the cavity grows, the conversion ratio continues to decrease, reaching its lowest value at point M when the cavity is at its maximum. The energy variation becomes even more complex and turbulent when the cavity collapses, as shown by the irregular wave-like shape of the M-P segment in the graph. By comparing Figure 13(a)–13(d), it is evident that a larger cavity volume may cause a lower conversion ratio of kinetic energy to elastic energy.
Figure 13
Conversion between kinetic and elastic energy in transient cavitation flow. (a) Test with cavitation 13.8 °C, (b) test with cavitation 25 °C, (c) test with cavitation 31 °C, and (d) test with cavitation 38.5 °C. Please refer to the online version of this paper to see this figure in colour: https://dx.doi.org/10.2166/hydro.2023.231.
View largeDownload slide

Conversion between kinetic and elastic energy in transient cavitation flow. (a) Test with cavitation 13.8 °C, (b) test with cavitation 25 °C, (c) test with cavitation 31 °C, and (d) test with cavitation 38.5 °C. Please refer to the online version of this paper to see this figure in colour: https://dx.doi.org/10.2166/hydro.2023.231.

Figure 13
Conversion between kinetic and elastic energy in transient cavitation flow. (a) Test with cavitation 13.8 °C, (b) test with cavitation 25 °C, (c) test with cavitation 31 °C, and (d) test with cavitation 38.5 °C. Please refer to the online version of this paper to see this figure in colour: https://dx.doi.org/10.2166/hydro.2023.231.
View largeDownload slide

Conversion between kinetic and elastic energy in transient cavitation flow. (a) Test with cavitation 13.8 °C, (b) test with cavitation 25 °C, (c) test with cavitation 31 °C, and (d) test with cavitation 38.5 °C. Please refer to the online version of this paper to see this figure in colour: https://dx.doi.org/10.2166/hydro.2023.231.

Close modal

Analysis of the effect of cavity volume on energy variation

Listen

The conversion ratios in the transient flow were 31.69, 22.29, 15.96, and 12.33%, respectively, as the water temperature increasing. Similarly, when cavities were formed, the conversion ratios were 22.43, 19.53, 17.24, and 14.29% separately, as the water temperature increasing. The conversion ratios of the kinetic and elastic energies at different water temperatures indicated that as the temperature increased, the conversion ratios decreased.

Furthermore, by comparing the initial kinetic energy at point A in Figures 12 and 13, it is evident that a higher initial flow velocity is capable of generating a greater initial kinetic energy, with temperature changes having little effect on the initial kinetic energy. Additionally, elastic energy is the energy produced by the coupling between the fluid and pipeline. Therefore, the higher the initial flow velocity, the greater the initial elastic energy, as reflected in the red circle section of Figure 12(d) and the corresponding parts in Figures 12 and 13.

Figure 14 compares the variations of kinetic and elastic energies in transient cavitation flow and in transient flow at 13.8 °C. From the values of the elastic and kinetic energies, as shown in Figure 14, it is evident that elastic energy and kinetic energy are not convertible to each other – that is, the elastic energy cannot be converted into kinetic energy with a higher numerical value over time. When considering other energy terms, the kinetic energy originates from the ends of the pipeline. Furthermore, by comparing the two graphs in Figure 14, it is evident that the sum of the kinetic and elastic energies in the transient flow dissipates uniformly. In transient cavitation flow, the energy dissipates much faster, particularly in the cavity (0.5–1 s). Moreover, the rate of energy dissipation is also reflected in the number of sawtooth waves, as shown in Figures 12 and 13. Faster dissipation reflects fewer sawtooth waves – that is, a sparser graph – which is consistent with the conclusions that can be drawn from Figure 14.
Figure 14
Comparison of variation of kinetic-elastic energy work at 13.8 °C. (a) Test with cavitation and (b) test no cavitation.
View largeDownload slide

Comparison of variation of kinetic-elastic energy work at 13.8 °C. (a) Test with cavitation and (b) test no cavitation.

Figure 14
Comparison of variation of kinetic-elastic energy work at 13.8 °C. (a) Test with cavitation and (b) test no cavitation.
View largeDownload slide

Comparison of variation of kinetic-elastic energy work at 13.8 °C. (a) Test with cavitation and (b) test no cavitation.

Close modal

Comparative analysis of energy terms of frictional effect and viscoelastic behavior

Listen
Figure 15 shows the dimensionless variation of the energy terms of the frictional effect and viscoelastic behavior at 13.8 °C. As shown in Figure 15(a) and 15(b), the energy term of the viscoelastic behavior in the transient flow is much higher than the corresponding value in the transient cavitation flow. Therefore, the conversion ratio is always lower in a transient cavitation flow than the corresponding value in a transient flow. Additionally, within the time period of 5–10 L/a, the existence of cavities not only reduces the energy of the viscoelastic behavior but also greatly attenuates the energy term of the frictional effect. This is because the presence of cavities reduces the energy term of the frictional effect, thus weakening the friction effect.
Figure 15
Comparison of the energy terms of frictional effect and viscoelastic behavior during transient flows. (a) Test with cavitation and (b) test no cavitation.
View largeDownload slide

Comparison of the energy terms of frictional effect and viscoelastic behavior during transient flows. (a) Test with cavitation and (b) test no cavitation.

Figure 15
Comparison of the energy terms of frictional effect and viscoelastic behavior during transient flows. (a) Test with cavitation and (b) test no cavitation.
View largeDownload slide

Comparison of the energy terms of frictional effect and viscoelastic behavior during transient flows. (a) Test with cavitation and (b) test no cavitation.

Close modal

From the above analysis, as the water temperature increases, the conversion ratios of the kinetic and elastic energies decrease considerably and the energy dissipation increases. The volume of the cavity also has a major influence on the conversion ratio of the energy and energy dissipation. The larger the volume of the cavity, the greater the energy dissipation in the cavity, and the faster the energy decay. Moreover, the variations between the energies in the cavity formation and collapse phases are complex and intense.

Listen

This study investigated the effect of different water temperatures on the calculated results of transient cavitation flow in viscoelastic pipes. The effects of water temperature on the volume and formation time of the cavities were examined. By employing an integrated energy analysis method, energy equations for 1D and quasi-2D transient cavitation flows in viscoelastic pipes were developed, and the energy variations during transient flow and transient cavitation flow in viscoelastic pipes were compared. Moreover, this study explored the effects of water temperature and cavity volume on kinetic energy, elastic energy, frictional energy, and viscoelastic behavior, respectively. The conclusions were as follows.

As the water temperature increased, the maximum volume and duration of cavitation at downstream valve gradually decreased. Additionally, the formation of the cavities and the time of maximum cavity volume occurred later.

The concept of the conversion ratio from kinetic energy to elastic energy was proposed, where a smaller conversion proportion could result in greater energy dissipation. The results indicated that an increase in the water temperature resulted in a decrease in the conversion ratio and an increase in energy dissipation.

The presence of cavities increased the energy dissipation for transient cavitation flow in viscoelastic pipes. Additionally, larger cavity volumes resulted in greater energy dissipation. During the cavity collapse phase, the energy variations became more complex and chaotic.

The differences between the 1D-DVCM and 2D-DVCM simulations in terms of the pressure peak, phase, and cavity appearance time were small. Consequently, it was reasonable to use a one-dimensional quasi-steady friction model to replace the quasi-2D friction model.

Listen

This work was financially supported by the Fundamental Research Funds for the Central Universities (2572021BJ02), the National Natural Science Foundation of China (51808102, 51978202), Young Innovative Talents Support Program of Harbin University of Commerce (2020CX07), and the Natural Science Fund in Heilongjiang Province (LH2020E028). The authors would like to thank Professor Kamil Urbanowicz from the West Pomeranian University of Technology for providing the experimental data.

Listen

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

Listen

The authors declare there is no conflict.

Andrade
D. M.
,
Rachid
F. B. D.
 &
Tijsseling
A. S.
2023
Fluid transients in viscoelastic pipes via an internal variable constitutive theory
.
Applied Mathematical Modelling
114
,
846
–
869
.
Google Scholar
Crossref
Search ADS
 
Bergant
A.
 &
Simpson
A. R.
1999
Pipeline column separation flow regimes
.
Journal of Hydraulic Engineering
125
(
8
),
835
–
848
.
Google Scholar
Crossref
Search ADS
 
Bergant
A.
,
Simpson
A. R.
 &
Tijsseling
A. S.
2006
Water hammer with column separation: a historical review
.
Journal of Fluids and Structures
22
(
2
),
135
–
171
.
Google Scholar
Crossref
Search ADS
 
Covas
D. I. C.
,
Stoianov
I.
,
Mano
J. F.
,
Ramos
H.
,
Graham
N.
 &
Maksimovic
C.
2004a
The dynamic effect of pipe-wall viscoelasticity in hydraulic transients. Part I – experimental analysis and creep characterization
.
Journal of Hydraulic Research
42
(
5
),
516
–
530
.
Google Scholar
Crossref
Search ADS
 
Covas
I. S.
,
Ramos
H.
,
Graham
N.
,
Maksimovic
C.
 &
Butler
D.
2004b
Water hammer in pressurized polyethylene pipes: conceptual model and experimental analysis
.
Urban Water Journal
1
(
2
),
177
–
197
.
Google Scholar
Crossref
Search ADS
 
Covas
D. I. C.
,
Stoianov
I.
,
Mano
J. F.
,
Ramos
H.
,
Graham
N.
 &
Maksimovic
C.
2005
The dynamic effect of pipe-wall viscoelasticity in hydraulic transients. Part II – model development, calibration and verification
.
Journal of Hydraulic Research
43
(
1
),
56
–
70
.
Google Scholar
Crossref
Search ADS
 
Duan
H. F.
,
Ghidaoui
M. S.
,
Lee
P. J.
 &
Tung
Y. K.
2010
Unsteady friction and viscoelasticity in pipe fluid transients
.
Journal of Hydraulic Research
48
(
3
),
354
–
362
.
Google Scholar
Crossref
Search ADS
 
Duan
H. F.
,
Lee
P. J.
,
Ghidaoui
M. S.
 &
Tung
Y. K.
2012
System response function-based leak detection in viscoelastic pipelines
.
Journal of Hydraulic Engineering
138
(
2
),
143
–
153
.
Google Scholar
Crossref
Search ADS
 
Duan
H. F.
,
Meniconi
S.
,
Lee
P. J.
,
Brunone
B.
 &
Ghidaoui
M. S.
2017
Local and integral energy-based evaluation for the unsteady friction relevance in transient pipeflows
.
Journal of Hydraulic Engineering
143
(
7
),
04017015
.
Google Scholar
Crossref
Search ADS
 
Gally
M.
,
Guney
M.
 &
Rieutord
E.
1979
An investigation of pressure transients in viscoelastic pipes
.
Journal of Fluids Engineering
101
,
495
.
Google Scholar
Crossref
Search ADS
 
Ghidaoui
M. S.
,
Zhao
M.
,
McInnis
D. A.
 &
Axworthy
D. H.
2005
A review of water hammer theory and practice
.
Applied Mechanics Reviews
58
(
1
),
49
–
76
.
Google Scholar
Crossref
Search ADS
 
Hadj-Taleb
L.
 &
Hadj-Taleb
E.
2009
Numerical simulation of transient flows in viscoelastic pipes with vapour cavitation
.
International Journal of Modelling & Simulation
29
(
2
),
206
–
213
.
Google Scholar
Crossref
Search ADS
 
Kashif
M.
,
Boran
Z.
,
Fazal
E. J.
 &
Wuyi
W.
2023
Transient flow analysis for pumping system comprising pressure vessel using unsteady friction model
.
International Journal of Mechanical Sciences
244
,
108093
.
Google Scholar
 
Meniconi
S.
,
Brunone
B.
,
Ferrante
M.
 &
Massari
C.
2014
Energy dissipation and pressure decay during transients in viscoelastic pipes with an in-line valve
.
Journal of Fluids & Structures
45
,
235
–
249
.
Google Scholar
Crossref
Search ADS
 
Mousavifard
M.
2022
Turbulence parameters during transient cavitation flow in viscoelastic pipe
.
Journal of Hydraulic Engineering
148
(
4
),
04022004
.
Google Scholar
Crossref
Search ADS
 
Pan
B.
,
Duan
H. F.
,
Meniconi
S.
,
Urbanowicz
K.
,
Che
T.
 &
Brunone
B.
2020
Multistage frequency-domain transient-based method for the analysis of viscoelastic parameters of plastic pipes
.
Journal of Hydraulic Engineering
146
(
3
),
04019068
.
Google Scholar
Crossref
Search ADS
 
Pan
B.
,
Keramat
A.
,
Capponi
C.
,
Meniconi
S.
,
Brunone
B.
 &
Duan
H. F.
2022
Transient energy analysis in water-filled viscoelastic pipelines
.
Journal of Hydraulic Engineering
148
(
1
),
04021051
.
Google Scholar
Crossref
Search ADS
 
Saidani
A.
,
Fourar
A.
 &
Massouh
F.
2022
Influence of temperature on transient flow with cavitation in copper pipe-rig
.
Modeling Earth Systems and Environment
8
,
2449
–
2459
.
Google Scholar
Crossref
Search ADS
 
Simpson
A. R.
 &
Bergant
A.
1994
Numerical comparison of pipe-column-separation models
.
Journal of Hydraulic Engineering
120
(
3
),
361
–
377
.
Google Scholar
Crossref
Search ADS
 
Soares
A. K.
,
Covas
D. I. C.
,
Ramos
H. M.
 &
Reis
L. F. R.
2009
Unsteady flow with cavitation in viscoelastic pipes
.
International Journal of Fluid Machinery and Systems
2
(
4
),
269
–
277
.
Google Scholar
Crossref
Search ADS
 
Soares
A. K.
,
Martins
N.
 &
Covas
D. I. C.
2017
Transient vaporous cavitation in a horizontal copper pipe
.
Journal of Hydraulic Research
55
(
5
),
731
–
736
.
Google Scholar
Crossref
Search ADS
 
Sun
Q.
,
Hao
Y. J.
 &
Cheng
P. F.
2022
Parametric analysis of discrete multiple-cavity models with the quasi-two-dimensional friction model for transient cavitating pipe flows
.
Journal of Water Supply: Research and Technology – AQUA
71
(
2
),
355
–
337
.
Google Scholar
Crossref
Search ADS
 
Tijsseling
A. S.
1996
Fluid-structure interaction in liquid-filled pipe systems: a review
.
Journal of Fluids & Structures
10
(
2
),
109
–
146
.
Google Scholar
Crossref
Search ADS
 
Vardy
A. E.
 &
Hwang
K. L.
1991
A characteristics model of transient friction in pipes
.
Journal of Hydraulic Research
29
(
5
),
669
–
684
.
Google Scholar
Crossref
Search ADS
 
© 2023 The Authors
This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits copying, adaptation and redistribution, provided the original work is properly cited (http://creativecommons.org/licenses/by/4.0/).