## Abstract

Discrete multiple-cavity models coupled with quasi-two-dimensional (quasi-2D) friction models are effective solutions to simulating transient cavitation pipe flows. The simulation accuracy of such models hinges upon the understanding of key parameters of the models, which remains elusive so far. To address such an open issue, this paper employs the discrete vapor cavity model (DVCM) and the discrete gas cavity model (DGCM), combined with the quasi-2D friction model, with a particular focus on revealing the sensitivity of these models to the key parameters such as grid number and weighting parameters. Based on the quantitative analysis and pressure fluctuation history, a method is developed to evaluate the accuracy of numerical results. Results show that the inclusion of the quasi-2D friction model improves the accuracy of predicting time of cavity formation and collapse; however, it does not affect the selection of grid number. Meanwhile, numerical results are sensitive to the weighting parameter of the viscous term in the quasi-2D friction model except for the case of low-intensity cavitation and its value of 1 is suggested for all cases. From the practical point of view, our finding is helpful to understand the feature of discrete multiple-cavity models and improve the simulating accuracy of transient cavitation pipe flows.

## HIGHLIGHTS

For the two-dimensional discrete vapor cavity model (2D-DVCM) and the two-dimensional discrete gas cavity model (2D-DGCM), the selection of grid number is independent of the inclusion of the friction model and only depends on the assumption of the cavitation model.

The 2D-DVCM and 2D-DGCM can accurately predict the time of cavity formation and collapse.

A procedure for evaluating the accuracy of numerical results is proposed considering the complexity of pressure fluctuation.

### Graphical Abstract

## NOMENCLATURES

The following symbols are used in this paper:

*A*Cross-sectional area of the pipe (m

^{2})*a*Wave speed (m/s)

*C*Constant value

*C*_{M}Characteristic constant

*C*_{P}Characteristic constant

*D*Pipe diameter (m)

- g
Acceleration of gravity (m/s

^{2})*H*Pressure head (m)

*H*_{b}Barometric pressure head (m)

*H*_{v}Relative vapor pressure head (m)

*K*_{n}Known values

*K*_{p}Known values

*L*Pipe length (m)

*M*_{g}Mass of free gas (kg)

*N*_{r}Radial grid number

*N*_{x}Axial grid number

*p*Pressure (Pa)

Absolute vapor pressure (Pa)

*Q*Discharge (m

^{3}/s)*R*Radius of pipe (m)

*R*_{g}Gas constant of free gas

*r*Distance in radial direction from the pipe axis (m)

*r*_{j}Coordinates of boundary points of reaches in radial direction (m)

*r*_{c j}Coordinates of middle points of reaches in radial direction (m)

*T*Absolute temperature (K)

*t*Computational time (s)

Time interval (s)

*u*Axial velocity (m/s)

*V*_{cav}Cavity volume (m

^{3})*V*_{g}Volume of free gas (m

^{3})*V*_{m}Volume of mixture (m

^{3})*v*Radial velocity components (m/s)

*x*Distance in axial direction of pipe (m)

*z*Elevation of pipe (m)

## GREEK SYMBOLS

## SUPERSCRIPTS

## SUBSCRIPTS

## ABBREVIATIONS

## INTRODUCTION

Hydraulic transients caused by pump failure or valve rapid closure could induce pressure surges in pipeline systems (Essaidi & Triki 2021). When the pressure drops to the saturated vapor pressure of pure liquid, vaporization would happen in the pipeline. Subsequently, the liquid column would be separated by vapor bubbles. This phenomenon, commonly known as column separation, has an influence on subsequent transient responses in the systems (Wylie *et al.* 1993; Adamkowski & Lewandowski 2012). Especially, cavities would be formed at some specified locations, such as the knee and closed end of the pipeline. After the collapse of the cavity, rejoining of the separated columns may result in instantaneous pressure rise, which even exceeds the Joukowsky pressure rise (Bergant *et al.* 2006). Therefore, it is essential to predict accurately the pressure oscillation of transient flows with column separation.

Several numerical models have been developed, such as discrete multiple-cavity models (Simpson & Bergant 1994; Bergant & Simpson 1999; Adamkowski & Lewandowski 2009), distributed vaporous cavitation model (Pezzinga & Cannizzaro 2014), and so on (Bergant *et al.* 2006). Among these models, the discrete multiple-cavity models are commonly used, such as the discrete vapor cavity model (DVCM) and the discrete gas cavity model (DGCM). The DVCM is a classical model, which is solved by the method of characteristics (MOC). It can depict the essential characteristic of the transient cavitating flows (Simpson & Bergant 1994). However, the DVCM may generate some numerical oscillations and unrealistic pressure spikes (Bergant & Simpson 1999). As an alternative model, the DGCM has been developed. It assumes that a small quantity of free air is concentrated at each computational section and the volume of cavity satisfies the ideal gas law. Some studies have shown that the DGCM could reduce numerical oscillations and avoid some unrealistic pressure spikes (Soares *et al.* 2015). However, the one-dimensional traditional DGCM (1D-DGCM) is developed using the steady or quasi-steady friction models, which could not estimate the effect of wall shear stress exactly. To solve those problems, 1D unsteady friction models have been incorporated into the solution process of transient cavitating flows (Adamkowski & Lewandowski 2012; Mosharaf-Dehkordi & Firoozabadi 2018). Some studies have indicated that the DGCM combined with the 1D unsteady friction models could improve the accuracy of the simulation results (Urbanowicz *et al.* 2012; Soares *et al.* 2017; Duan *et al.* 2020; Zhou *et al.* 2021).

As another effective method to calculate the wall shear stress accurately, transient cavitation models coupled with two-dimensional (2D) or quasi-2D friction models have been developed. Pezzinga & Cannizzaro (2014) and Pezzinga & Santoro (2017) combined different cavitation models with the quasi-2D friction model. Those cavitation models are the distributed vaporous cavitation model, bubble flow model, and shallow-water cavitation model, respectively. Wang *et al.* (2016) combined the classic DVCM with a 2D computational fluid dynamics model. Jang *et al.* (2016) combined the DGCM and a quasi-2D friction model. Numerical results show that the proposed 2D model predicts the maximum pressure heads more accurately than the corresponding 1D model. Santoro *et al.* (2018) presented the improved DVCMs with both the 1D quasi-steady friction model and the quasi-2D friction model, respectively.

The simulation accuracy of above-mentioned models hinges upon the understanding of key parameters of the models; thus, some researchers have studied the impact of key parameters on numerical results of the models. Bergant *et al.* (2008) analyzed the influence of the amount of free gas on numerical results calculated by the 1D-DGCM. They pointed out that the volume fractions smaller than 10^{−7} have a slight effect on the numerical results. Based on the above-mentioned Pezzinga & Cannizzaro (2014) cavitation model, Gao *et al.* (2018) investigated the sensitivity of grid numbers and the selection of turbulence models. The optimal ratio of axial and radial grid numbers is obtained. Meanwhile, they indicated that the numerical results obtained using the two-region turbulence model are similar to those calculated using the five-region turbulence model. Santoro *et al.* (2018) compared the results computed by the improved 1D and 2D DVCMs on different grid sizes. The sensitivity of the 1D-improved DVCMs to the grid size has a significant difference in different cases, whereas the 2D-improved DVCMs perform more regularly. Sun *et al.* (2020) investigated effects of weighting coefficients in the classic DVCM with the quasi-2D friction model (2D-DVCM) on the numerical results.

To utilize the 2D-DVCM and 2D-DGCM efficiently and accurately, this paper investigates the sensitivity of numerical results to key parameters, such as grid numbers and weighting parameters. Meanwhile, a procedure for evaluating the accuracy of numerical results is proposed because of the complexity of pressure fluctuation, such as existing of multiple cavities and pressure oscillation after cavities collapse. The pressure fluctuation history is compared with the experimental data provided by the literature (Bergant & Simpson 1999). Zhu *et al.* (2018) investigated the effects of unsteady friction, pipe-wall viscoelasticity, and air content on the transient pressure.

## MATHEMATICAL MODEL

### Governing equations

*et al.*2005). Meanwhile, the radial velocity and its derivatives can be neglected in the governing equations. This is because the radial velocity is much smaller than the axial velocity and pressure by orders of magnitude. Therefore, the 2D governing equations for transient pipe flows can be expressed as follows (Vardy & Hwang 1991):where

*x*is the distance in the axial direction of the pipe,

*r*is the distance in the radial direction from the pipe axis,

*t*is the time,

*ρ*is the density of liquid,

_{l}*H*is the pressure head,

*u*is the axial velocity,

*a*is the wave speed,

*g*is the gravitational acceleration, and

*τ*is the total shear stress.

### Discrete multiple-cavity model

Both DVCM and DGCM assume that there is a pure liquid column between the adjacent grid points. In the DVCM, a cavity is allowed to form at any of the grid points whenever the pressure drops below the saturated vapor pressure of liquid. At this moment, the pressure head in a cavity is equal to the vapor pressure head of the liquid. Meanwhile, the volume of cavity is calculated by , where *Q* and *Q _{u}* are the discharges at the downstream and the upstream section of the cavity, respectively, and Δ

*t*is the time interval (Wylie

*et al.*1993).

*et al.*1993). The mass of the gas cavity is calculated as follows:where

*M*is the mass of free gas,

_{g}*R*is the gas constant of free gas,

_{g}*T*is the absolute temperature, is the gas void fraction at reference absolute pressure, (), and is the volume of mixture.

*z*is the elevation of the pipe,

*H*is the relative vapor pressure head, , is the absolute vapor pressure, and

_{v}*H*is the barometric pressure head.

_{b}*Q*and

*Q*, is given by the following equation (Wylie

_{u}*et al.*1993):

The volume of the cavity will no longer increase once the upstream flow adjacent to cavity becomes zero, and it decreases when the flow is in the direction of the original flow (Simpson 1986). The cavity will eventually disappear when the upstream flow is equal to the downstream flow. As a result, the liquid column returns to continuous and the solution for water hammer without column separation can be used.

## NUMERICAL SCHEME

*et al.*2020). For the 1D-DGCM, the solution for transient flows is calculated using the MOC. The characteristic forms of 1D transient flow equations are given as follows:where

*B*

*=*

*a/*(

*gA*).

*n*Δ

*t*and (

*n*

*+*1)Δ

*t*at the downstream and upstream sides of the gas cavity, the discretized forms of Equation (10) can be expressed as follows:where the superscript

*n*denotes the time step index, the characteristic constants

*C*and

_{P}*C*are written as follows:

_{M}*ψ*is a weighting coefficient (0 <

*ψ*≤ 1),

*t*is the computational time, and the subscript

*u*denotes the upstream side at a cavity.

By solving Equation (17), may be obtained (Jang *et al.* 2016).

For the 2D-DGCM, the computational grids for the discretization of characteristic equations are shown in Figure 2. The pipe is discretized into *N _{r}* cylinders with varying thicknesses in the radial direction. The pipe length

*L*is divided into

*N*reaches of constant length Δ

_{x}*x*in the axial direction. The downstream and upstream axial velocities,

*u*and

*u*, are located in the middle of each reach. The turbulent viscosity

_{u}*ν*is located at the boundaries of each reach in the radial direction. The five-layer turbulent model is used to confirm the turbulent viscosity

_{T}*ν*.

_{T}For the quasi-2D model, when there is a cavity or gas cavity at the grid point, the volume of the cavity at time (*n* + 1)Δ*t* is computed by Equation (15), in which the discharges *Q _{u}* and

*Q*are obtained by integrating the axial velocity distribution of the upstream and downstream side of the cavity. Also, the radial velocity at the cavity or gas cavity is ignored.

*n*Δ

*t*and (

*n*

*+*1)Δ

*t*at the downstream and upstream sides of the cavity, the discretized forms of Equation (21) can be expressed as follows:withwhere

*ε*is the weighting coefficient of the viscous term in the characteristic equations (0.5 ≤

*ε*≤ 1), the subscripts

*i*and

*j*indicate the axial and radial step indexes, respectively,

*r*and

_{j}*r*are the coordinates of the boundary and middle points of reaches in the radial direction, respectively. The source terms

_{cj}*K*and

_{pi,j}*K*are the known parameters, whose elements depend on

_{ni,j}*H*and

*u*at the previous time level. The unknown parameters of Equations (22) and (23) are the axial velocities at the downstream and upstream sides of the cavity at time (

*n*

*+*1)Δ

*t*,

*u*

^{n+}^{1}, and

*u*

_{u}^{n+}^{1}, and these solutions are calculated using the Thomas algorithm (Anderson 2002). After that, the wall shear stress is calculated using Equation (5), and the discharges

*Q*and

_{u}*Q*at time (

*n*

*+*1)Δ

*t*are obtained using Equations (11) and (12), respectively.

## NUMERICAL RESULTS AND DISCUSSION

Based on experimental data provided by Bergant & Simpson (1999), several numerical cases are considered in this section. Primarily, in order to check whether the DVCM and DGCM generate excessive numerical errors related to the essence of these models, discrete multiple-cavity models (DVCM and DGCM) without friction terms are tested. Then, the influences of grid numbers *N _{r}* and

*N*and weighting coefficients on the numerical results of the 2D-DGCM and 2D-DVCM are investigated, in which weighting coefficients include the weighting coefficient of the viscous term in the quasi-2D friction model and the weighting coefficient of calculating cavity volumes in discrete multiple-cavity models. A flowchart that illustrates the methodology of analysis is shown in Figure 3, where two quantitative analysis indicators, RMSE

_{x}_{p}and

*t*, are introduced to evaluate the accuracy of the simulation and the detailed meanings of them are given in the following discussion. Finally, the performance comparison of 2D-DVCM and 2D-DGCM focuses on two key variables, including pressure head and cavity volume.

_{cav}As above mentioned, numerical results are compared with results of measurements by Bergant & Simpson (1999). The experimental system is a typical reservoir–pipe–valve system and it composes of copper straight sloping pipe. The transient flows are caused by a rapid downstream valve closure. The pipe length is 37.23 m and the pipe diameter is 22.1 mm. The upward slope of the pipe is 3.2°. The upstream reservoir pressure height is 22 m. The calculated wave speed is 1,319 m/s, which is assumed to be constant in the section of the pure liquid column. The sampling frequency for the experimental apparatus is 5,000 Hz. The sampling frequency used in the numerical calculation is close to the experimental one, which ensures the pressure spikes are captured. Three different steady-state velocities, *V*_{0}, of 0.30, 0.71, and 1.40 m/s are studied in following sections. The severity of cavitation is introduced (Pezzinga & Santoro 2017) and the above three tests are, respectively, defined as the cases of low-intensity cavitation, moderate-intensity cavitation, and high-intensity cavitation.

### Comparison of numerical results for frictionless case

Figure 4 shows the numerical results of both DVCM and DGCM without friction term for three cases in terms of pressure head at the valve. The grid number *N _{x}* used in both models is 16 and the Courant number of 1 is used. As can be seen in Figure 4, there are obvious unrealistic pressure peaks in numerical results of the DVCM. As an improved model, numerical results of the DGCM avoid some unrealistic pressure peaks. For the case of

*V*

_{0}= 0.30 m/s, pressure peaks calculated by both models are obviously higher than experimental data in the middle and late stages of transient events. For cases of

*V*

_{0}= 0.71 m/s and

*V*

_{0}= 1.40 m/s, the DVCM and DGCM perform poorly in predicting the formation and collapse of multiple cavities in the transient process. The classical DVCM is only slightly sensitive to grid number

*N*(Santoro

_{x}*et al.*2018). As shown in Figure 4(d), the vapor fractions of the largest cavity calculated by both models are within the limitation of 10% of each pipe section.

### Grid analysis

RMSE is employed to show the deviation between the numerical and the experimental results. In this part, RMSE_{p} is defined as RMSE calculated by the peak value of each water hammer period at the valve. A smaller RMSE_{p} value means a good consistency of pressure peaks of numerical results and the experimental data. The RMSE_{p} investigation covers the first five periods of the water hammer with column separation after the downstream-valve closed, which is considered to be capable of reflecting the main characteristics of transient processes.

Axial grid number *N _{x}* = 16 is suggested for the classical discrete multi-cavity models in these cases in the literature of Bergant & Simpson (1999). Meanwhile, Simpson & Bergant (1994) suggested that the maximum length of a cavity should be below 10% compared to each computational section. Therefore, for the 2D-DVCM and 2D-DGCM, the

*N*values of 8, 16, 32, and 48 (several times of 16) are taken into account in this section. Also, the

_{x}*N*values of 20, 40, 60, 80, 100, and 120 are tested in the 2D-DGCM.

_{r}Figure 5(b) shows the RMSE_{p} calculated using the results of the 2D-DGCM with different *N _{x}* and

*N*values for the case of

_{r}*V*

_{0}= 0.30 m/s. As the

*N*value increases, the RMSE

_{r}_{p}decreases when the

*N*value is specified. When

_{x}*N*is greater than 100, the difference is insignificant. Therefore,

_{r}*N*= 100 is used in the 2D-DGCM for all tests. As shown in Figure 5(a), a similar procedure has been carried out in the 2D-DVCM and the value of

_{r}*Nr*is set to 60 for all tests.

Figure 6 presents the results obtained by the 2D-DVCM and 2D-DGCM with different values of axial grid number *N _{x}* and reports RMSE

_{p}values for three testing cases. For the case of

*V*

_{0}= 0.30 m/s, the RMSE

_{p}of the 2D-DVCM and 2D-DGCM decreases with the increase of grid number. Meanwhile, the RMSE

_{p}values for the 2D-DGCM are relatively smaller than the corresponding ones of the 2D-DVCM. For the case of

*V*

_{0}= 0.71 m/s, the minimum RMSE

_{p}values of both models can be obtained at

*N*= 16. Compared with the other two cases, for the case of

_{x}*V*

_{0}= 1.40 m/s, RMSE

_{p}values of both models fluctuate greatly with the change of grid number

*N*. The sensitivity of the 2D-DVCM and 2D-DGCM to the grid number is obviously affected by cavitation conditions.

_{x}_{p}values cannot represent the accuracy of the calculated results completely. In addition to values of pressure peaks, shapes of pressure pulse and time of cavity formation and collapse are also important indicators to describe transient cavitating flows accurately. For this reason, the pressure fluctuations computed by the two models are compared with the corresponding experimental data in the following works, in which two grid numbers with smaller deviations from the measurement are selected. A quantitative analysis of time of cavity formation is conducted by defining parameter

*t*to estimate the degree of agreement with the experimental data. The parameter

_{cav}*t*is given as follows:where and

_{cav}*t*is the time of formation of the

_{ie}*i*-th cavity based on the experimental data, and

*t*is the time of formation of the

_{in}*i*th cavity based on the numerical results. The lower value of

*t*means that the discrepancy between numerical and experimental results is smaller. Figure 7 shows

_{cav}*t*values of different grid numbers for two test cases of repeated cavity formation and collapse. As can be seen in this figure, the

_{cav}*t*values for the 2D-DVCM with

_{cav}*N*= 16 are both smaller for two flow conditions. Compared with the 2D-DVCM, numerical results calculated by the 2D-DGCM are closer to experimental data in terms of time of cavity formation for different testing cases.

_{x}Figure 8 shows the comparisons between experimental results and numerical results of the 2D-DVCM and 2D-DGCM with different axial grid numbers for the case of *V*_{0} = 0.30 m/s. As shown in this figure, the combination with the quasi-2D friction model effectively avoids the appearance of unrealistic pressure peaks. After the collapse of the cavity, for the 2D-DVCM, some pressure peaks of numerical results appear at a different time from experimental results (as shown in p4 and p5 in Figure 8(a)). For both models, different *N _{x}* values have only a slight effect on the pressure peak.

Figure 9 shows the comparison of numerical results computed by the 2D-DVCM and 2D-DGCM with different *N _{x}* values for the case of

*V*

_{0}= 0.71 m/s. As shown in Figure 9(a), for the 2D-DVCM, both pressure peak and cavity formation are sensitive to grid number and only the first three cavitation events are predicted well. Meanwhile, based on the results of quantitative analysis in Figure 7, the numerical results of 2D-DVCM with

*Nx*= 16 are in better agreement with the experimental results. In Figure 9(b), the results of 2D-DGCM show a better agreement with experimental data in the period of existing cavities. Some discrepancies of pressure pulse occur after cavities collapse and results of the 2D-DGCM with

*N*= 16 are better.

_{x}The inclusion of the quasi-2D friction model significantly improves the prediction of time of repeated cavity formation and collapse, especially in the case of high-intensity cavitation. As shown in Figure 10, pressure oscillations computed by the 2D-DVCM and 2D-DGCM are compared with the experimental results for *V*_{0} = 1.40 m/s. The 2D-DVCM is also sensitive to the grid number in this case. Figure 10(a) indicates that the duration of cavity computed by the 2D-DVCM with *N _{x}* = 32 is underestimated compared with the experimental results. In Figure 10(b), the numerical results of the 2D-DGCM with

*N*= 16 and

_{x}*N*= 48 are consistent with the experimental results. The 2D-DGCM allows a better reproduction of time of cavity formation and collapse with respect to the 2D-DVCM. Based on the results of quantitative analysis in Figure 7, for the case of high-intensity cavitation, both 2D-DVCM and 2D-DGCM with

_{x}*N*= 16 are capable of getting the satisfactory results.

_{x}### Effects of weighting coefficients *ψ* and *ε*

The performance of the 2D-DGCM using different weighting coefficients *ψ* and *ε* is investigated in this part. The analysis is conducted in the cases of low-intensity cavitation (*V*_{0} = 0.30 m/s) and high-intensity cavitation (*V*_{0} = 1.40 m/s). A similar analysis of the 2D-DVCM has been carried out in the author's previous paper (Sun *et al.* 2020).

Figure 11 shows the pressure head calculated by using different weighting coefficients of the 2D-DGCM for the case of *V*_{0} = 0.30 m/s. In Figure 11(a), the weighting coefficient *ψ* is set to 0.6 and the effect of different weighting coefficient *ε* values is studied. There are slight differences among several numerical results. The time of cavity formation and collapse can be reproduced well by using different values of weighting coefficient *ε*. The pressure peaks calculated by using *ε* = 0.6 and 1.0 are more consistent with the experimental results than that using *ε* = 0.8. In Figure 11(b), the value of *ε* is designated as 1.0, and it can be found that the value of *ψ* affects pressure peaks and the phase of successive amplitudes. The results of the 2D-DGCM with *ψ* = 0.6 have the best agreement in simulating column separation and free oscillations. The maximum pressure head occurs at *t* = 0.184 s as a short-duration pressure peak, which is caused by the superposition of the pressure head from the collapsed cavity and the reservoir wave head reflected from the closed valve. The short-duration pressure peak of numerical results is sensitive to the value of *ψ* and further discussion is presented as follows.

For the case of *V*_{0} = 0.30 m/s, Figure 12 indicates the difference between short-duration pressure peaks calculated by the 2D-DVCM and 2D-DGCM with different *ψ* values and the corresponding value of experimental data. In this figure, the ratio |*H _{sd,num}*–

*H*|/

_{sd,exp}*H*is employed as an index for evaluating the accuracy, where

_{sd,exp}*H*is the short-duration pressure peak of numerical results and

_{sd,num}*H*is the short-duration pressure peak of experimental data. As shown in Figure 12, the percentage of difference between the

_{sd,exp}*H*value and the

_{sd,exp}*H*values calculated by the 2D-DVCM is always less than 5%. However, the

_{sd,num}*H*values computed by the 2D-DGCM are more sensitive to the weighting coefficient

_{sd,num}*ψ*values. The

*H*calculated by the 2D-DGCM with

_{sd,num}*ψ*= 0.6 is in good agreement with the experimental results, but the

*H*values of another two cases differ from the experimental results by up to 20–25%.

_{sd,num}Figure 13 shows the pressure head traces obtained by using the 2D-DGCM with different weighting coefficients for the case of *V*_{0} = 1.40 m/s. In Figure 13(a), *ψ* is set to 0.6 and the effect of varying *ε* values is studied. The pressure heads computed by *ε* = 0.6 and *ε* = 0.8 have higher dissipations. The time of cavity collapse predicted by the 2D-DGCM with *ε* = 0.6 and *ε* = 0.8 is ahead of the experimental data. In the case of *ε* = 1.0, the simulation results are closer to the experimental data than the other two cases. The comparison among simulation results with weighting coefficient *ψ* values of 0.6, 0.8, and 1.0 is shown in Figure 13(b) and the weighting coefficient *ε* is set to 1.0. The pressure fluctuations predicted by the 2D-DGCM with *ψ* = 1.0 are quite different from the experimental results. Based on the above analysis, it can be shown that the 2D-DGCM is more sensitive to the values of weight coefficients *ψ* and *ε* in the case of high-intensity cavitation.

### Comparison of the 2D-DVCM and 2D-DGCM

To display the performance of discrete multiple-cavity models combined with the quasi-2D friction model, comparisons between numerical results of the 2D-DVCM and 2D-DGCM are carried out in Figures 14 and 15. The cases of *V*_{0} = 0.71 m/s and *V*_{0} = 1.40 m/s are tested in this section because discrete multiple-cavity models behave poorly in the prediction of time of repeated cavity formation and collapse (Bergant *et al.* 2006). According to the above-mentioned analysis, proper values of parameters in each model have been used. The time of cavity formation and collapse predicted by the 2D-DGCM is considerably closer to the experimental data than that calculated by the 2D-DVCM. The different results between the two models can be attributed to discrepancy in the assumptions of the models, that is, the equations governing the generation of vapor or gas cavities are different. The DVCM allows cavities to be formed whenever the pressure drops to vapor pressure. However, for the DGCM, the cavity volume is the sum of volume of gas cavity and vapor cavity, where the gas cavity volume increases as the pressure reduces and the vapor cavity is formed when the pressure drops to vapor pressure. Therefore, the DVCM is a limited case of the DGCM.

As shown in Figure 15, vapor fraction is obtained by the 2D-DGCM and the 2D-DVCM for the cases of *V*_{0} = 0.71 m/s and *V*_{0} = 1.40 m/s. It can be observed that the vapor fractions of the maximum cavity computed by both models are below 10% of the computational section of the pipeline, which is suggested by the literature (Simpson & Bergant 1994).

### Discussions of numerical simulation and experimental data

Generally, the numerical results of the 2D-DVCM and 2D-DGCM are in good agreement with the experimental results as shown in the above figures. The pressure peaks and cavity formation of the column separation phenomenon could be well reproduced. Some numerical oscillations appear in numerical results for the case of moderate-intensity cavitation (as shown in Figure 9, *t*3–*t*6), which are strongly affected by the *N _{x}* value and are caused by the reflection and superposition of pressure waves. In the current research, these questions are considered to be reasonable. This is because, from the engineering point of view, this research pays more attention to the prediction of dangerous pressure peaks and the reproduction of the cavity formation process.

With the introduction of the quasi-2D friction model, the difference between the DVCM and the DGCM in simulation accuracy is reduced. However, the accuracy of the 2D-DVCM is strongly affected by grids numbers, while the 2D-DGCM is more robust, which can be observed especially for the case of *V*_{0} = 1.40 m/s. Discrepancies in equations governing the cavity formation of these two models are the main reason for these differences.

Another issue that needs to be discussed is the reliability of experimental data. This research is only conducted based on the experimental data of the above-mentioned typical reservoir–pipe–valve system. Although a sampling device with a sufficiently high frequency is used, there may still be errors between the experimental data and the real pressure value. Therefore, the average error of the first few periods is more important than the exact values. Meanwhile, more wide-ranging verifications are necessary for transient simulation, such as based on experimental data obtained by different configuration pipeline systems.

## CONCLUSIONS

This paper investigates the effects of different grid numbers and weighting coefficients on the accuracy of the 2D-DVCM and 2D-DGCM in order to reproduce transient cavitating pipe flow more precisely. Meanwhile, a procedure for evaluating the accuracy of numerical results is carried out. First, the accuracy of the pressure peak is checked by the RMSE_{p}, then the time of cavity formation is examined through the *t _{cav}* value, and finally, the calculated duration of cavity is verified through the comparison of the pressure fluctuation history and experimental data. The main conclusions can be summarized as follows:

The differences in RMSE

_{p}computed by the 2D-DVCM and 2D-DGCM are slight when finer radial grids are used. The axial grid number of*N*= 16 is recommended to adopt in the testing cases, which is the same as that used in the classical DVCM and DGCM (without friction term). Therefore, the selection of axial grid number is independent of the inclusion of the quasi-2D friction model and only depends on the assumption of the cavitation model._{x}With respect to the DGCM without friction term and the 2D-DVCM, the 2D-DGCM allows a better reproduction of transient cavitation pipe flow. The 2D-DGCM can get the satisfactory simulation results of both pressure peaks and time of cavity formation and collapse as long as a grid number with a smaller value of RMSE

_{p}is used. However, for numerical results of the 2D-DVCM, the time of cavity formation and collapse is sensitive to grid number.The value of the short-duration pressure peak calculated by the 2D-DGCM is sensitive to the value of weighting coefficient

*ψ*, whereas the weighting coefficient*ψ*value in the 2D-DVCM affects the corresponding short-duration pressure peak slightly. The weighting coefficient*ε*= 1 is an appropriate selection for all testing cases.The 2D-DGCM has good agreement with experimental results in predicting the time of repeated cavity formation and collapse. The 2D-DVCM can better reproduce the pressure rise after the cavity collapses.

The agreement between experimental results and numerical results of the 2D-DVCM and 2D-DGCM depends on the values of grid number and weighting coefficients; however, the improvement in both pressure peaks and the time of cavity formation and collapse cannot be obtained at the same time in some cases. Further research on more advanced friction models should be considered, and new improved discrete multiple-cavity models need to be proposed.

## ACKNOWLEDGEMENTS

This work was financially supported by the Fundamental Research Funds for the Central Universities (No. 2572021BJ02), the National Natural Science Foundation of China (Grant Nos 51808102 and 51978202), the Science Foundation of Harbin University of Commerce (No. 18XN068), and the Natural Science Fund in Heilongjiang Province (Nos LH2019E111 and LH2021E003).

## CONFLICT OF INTEREST

The authors declare no conflict of interest.

## DATA AVAILABILITY STATEMENT

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

## REFERENCES

*Large Water Hammer Pressures due to Column Separation in Sloping Pipes*

*PhD Thesis*