# CFD Investigations of Transient Cavitation Flows in Pipeline Based on Weakly-Compressible Model

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Governing Equations of Weakly Compressible Fluid

#### 2.2. Solution of RANS Equation Based on the Weakly Compressible Model

#### 2.3. Cavitation Model

#### 2.4. Computational Domains and Grid Model

_{l}and S

_{2}of the pipeline were respectively located at 9.0 m and 27.0 m from upstream inlet, and the position of the pressure sensor S

_{3}was 0.02 m upstream to the valve. The ball valve was modeled using a rotating cylinder with the same diameter of the pipe.

#### 2.5. Computational Scheme and Boundary Conditions

## 3. Results and Discussion

#### 3.1. Grid Independence Verification

_{3}monitoring position are shown in Figure 4.

#### 3.2. Time Step Independence Verification

_{3}monitoring position with different time steps (0.00001 s, 0.0005 s, 0.0001 s, and 0.001 s) are shown in Figure 5. It can be seen from Figure 5 that the maximum water hammer pressure and pressure fluctuation cycle are hardly affected by the time step. When the time step is smaller than 0.0001 s, the results are hardly affected by the time step. In order to ensure the numerical calculation efficiency and calculation accuracy, the final time step was chosen to be 0.0001 s. The C-F-L criterion was fulfilled with the selected time step size in this paper.

#### 3.3. Analysis of Transient Non-Cavitation Flow Results

#### 3.3.1. Analysis of the Pressure Fluctuation

_{l}, S

_{2}, and S

_{3}monitoring positions of each time point are shown in Figure 6, and compared to the experimental data.

_{3}monitoring point), the first peak value of the pressure fluctuation can be accurately predicted. However, the attenuation of the pressure in the simulations is lighter than the experimental result, after 0.465 s, the peak value of the experimental pressure head decreases from 55.33 m to 49.77 m due to the frictional resistance, and the calculated pressure head peak is 55.51 m. The discrepancy between the CFD results and experimental data after the first peak value of the pressure fluctuation is because the energy dissipation is underestimated during the transient process, which may be because the pipe friction effect in the transient process cannot be well simulated in the calculations or the turbulent dissipation is underestimated in the used turbulence model. One reason for the discrepancy may come from the unsteady friction models (i.e., the convolutional integral model and the instantaneous acceleration model) effects in the 1D model which are also not revealed well in the CFD simulations due to their complex mechanisms. Another reason for the discrepancy may be the fact that the 2D simulation cannot reveal well the 3D water hammer decay. The results at the other monitoring points are similar to those at the valve. And the simulation results can match well with the experimental data at each monitoring position.

#### 3.3.2. Analysis of the Pressure and Velocity Fields

#### 3.3.3. Analysis of the Velocity Distribution Change

_{1}monitoring point, with marked time nodes ${T}_{0}\sim {T}_{11}$ (different from time nodes in Figure 6b). The velocity distribution at S

_{1}monitoring position and the corresponding pressure field on the axis plane of the pipeline at these time nodes are shown and analyzed in five stages, and the stages are divided according to the pressure wave propagation directions. The change of average axial velocity $\overline{U}$ and pressure head $H$ at S

_{1}position are recorded and analyzed.

_{1}monitoring position of different time, and Figure 12 is the pressure field evolution on the axis plane of the pipeline in this stage. Before the valve is closed, the velocity at ${T}_{0}$ = 0 s time at S

_{1}monitoring position exhibits a fully developed steady-state velocity distribution, with $\overline{U}$ = 0.239 m/s, the pressure field at this time in Figure 12a shows the steady-state pressure field, with $H$ = 24.24 m, and local pressure distribution at S

_{1}monitoring position in Figure 11a shows that the pressure gradient on this line is almost 0. When the valve is closed, the pressure increase wave propagates from the valve to the upstream reservoir (Figure 12b, ${T}_{1}=0.044\text{}\mathrm{s}=Tc+0.71\times L/a$, $H$ = 37.29 m), the flow velocity in the pipeline gradually decreases (Figure 10b, $\overline{U}$ = 0.137 m/s); the local pressure distribution at S

_{1}monitoring position (Figure 11b) shows that when the pressure wave passes, the pressure at the upper part of the pipe is bigger, while the pressure at the lower part of the pipe is smaller with the biggest pressure difference 27.7 Pa, but this value is still very small compared to the average pressure 365,126 Pa on S

_{1}monitoring position and the biggest relative pressure difference is 27.7/365126 = 0.008%, thus, the 1D and quasi-2D model assumption that pressure on the cross-section is constant is still valid. After the pressure increase wave passes (Figure 10c and Figure 12c, ${T}_{2}=0.05\text{}\mathrm{s}=Tc+0.92\times L/a$), the pressure at S

_{1}position reaches the maximum (pressure head $H$ = 55.1 m), the average flow velocity is zero, there is a significant reverse flow near the wall and the velocity gradient is large, the velocity distribution of the core region remains the same shape as the initial steady state, and the local pressure at S

_{1}monitoring position is almost constant at this time (Figure 11c).

_{1}monitoring position of different time, and Figure 15 is the pressure field evolution on the axis plane of the pipeline in this stage. When the pressure decrease wave generated at the upstream reservoir propagates to the downstream valve (Figure 15a, ${T}_{3}=0.058\text{}\mathrm{s}=Tc+1.21\times L/a$), the pressure head at S

_{1}position decreases to 41.29 m, the velocity direction is from the valve to the reservoir (Figure 13a, $\overline{U}$ = −0.102 m/s), the reverse flow develops from the wall region and then diffuses to the core region, and the local pressure distribution in Figure 14a shows that the pressure at the upper part of the section is bigger while the pressure at the lower part of the section is smaller with a biggest pressure difference of 22.5 Pa, but the average pressure on S

_{1}monitoring position at this time is 410,492 Pa and the biggest relative pressure difference is 22.5/410492 = 0.005% (still very small). After the pressure wave passes (Figure 15b, ${T}_{4}=0.064\text{}\mathrm{s}=Tc+1.42\times L/a$), the pressure head decreases to 24.46 m, the average axial velocity $\overline{U}$ = −0.236 m/s, and the pressure at S

_{1}monitoring position at this time is almost constant with a very small pressure gradient (Figure 14b). In this process, the velocity distribution of the core region at S

_{1}position stays the same shape as the initial state, and the velocity magnitude at the wall region is bigger than that of the core region.

_{1}monitoring position of different time, and Figure 18 is the pressure field evolution on the axis plane of the pipeline in this stage. At ${T}_{5}=0.092\text{}\mathrm{s}=Tc+2.42\times L/a$, the pressure head at S

_{1}position is 24.08 m, and the velocity distribution in Figure 16a exhibits that the velocity at the wall region firstly begins to change (with cross-sectional average velocity of $\overline{U}$ = −0.233 m/s), and the velocity change gradually diffuses to the core region, at ${T}_{6}=0.1\text{}\mathrm{s}=Tc+2.7\times L/a$, $\overline{U}$ = −0.14 m/s. After the pressure wave passes (Figure 18c, ${T}_{7}=0.108\text{}\mathrm{s}=Tc+2.99\times L/a$), the pressure head at S

_{1}monitoring position decreases to the minimum, −5.57 m, and the axial average velocity is 0 m/s, the velocity distribution in Figure 16c is similar to that in Figure 10c. In the process, the velocity distribution of the core region remains the same to the initial shape, but the velocity peak clearly spreads to the core region. The local pressure distribution shows that the pressure difference on the S

_{1}monitoring position is very small before the pressure wave passes (Figure 17a), the pressure difference is bigger (15.3 Pa) when the pressure wave passes (Figure 17b) but the average pressure on S

_{1}monitoring position at this time is 114,364 Pa and the biggest relative pressure difference is 15.3/114364 = 0.01% (still very small), and the pressure difference is also not big after the pressure wave passes (Figure 17c).

_{1}monitoring position of different time, and Figure 21 is the pressure field evolution on the axis plane of the pipeline in this stage. When the pressure increase wave generated at the upstream reservoir in the fourth stage reaches the S

_{1}monitoring position (Figure 21a, ${T}_{8}=0.116\text{}\mathrm{s}=Tc+3.27\times L/a$), the pressure head at S

_{1}position is 12.91 m, the velocity at the wall region is positive Figure 19a, with average axial velocity $\overline{U}$ = 0.148 m/s; the local pressure difference is 14.5 Pa (Figure 20a), the average pressure on S

_{1}monitoring position at this time is 126,411 Pa, and the biggest relative pressure difference is 14.5/126411 = 0.01% (still very small). After the pressure wave passes (Figure 21b, ${T}_{9}=0.124\text{}\mathrm{s}=Tc+3.56\times L/a$), the pressure head is 24.15 m, $\overline{U}$ = 0.233 m/s, and the local pressure on S

_{1}monitoring position is almost constant (Figure 20b). Finally, the velocity distribution in Figure 19b will be returned to the initial steady state.

_{1}monitoring position and Figure 23 is the corresponding local pressure distribution. At (${T}_{10}=0.148\text{}\mathrm{s}=Tc+4.41\times L/a$), the pressure head at S

_{1}position is 24.85 m, a little bigger than that of the original steady state due to the effect of the first part of the wave speed, and the average axial velocity $\overline{U}$ is 0.228 m/s. After the wave passes (Figure 24b, ${T}_{11}=0.164\text{}\mathrm{s}=Tc+4.98\times L/a$), the pressure head reaches the maximum, 51.89 m, and the average axial velocity $\overline{U}$ = 0 m/s. The local pressure distribution in Figure 23a,b show that the pressure gradient at S

_{1}monitoring position at both time are not big because both time nodes are not pressure wave passing time.

_{1}monitoring position, two different regions can be identified: a wall region with a fast flow change and a high gradient, and a core region related to the velocity distribution of the past time history. These regions behave differently, because they depend on different flow characteristics: the wall region depends on the viscosity, and the core region depends on the fluid inertia. During the propagation of pressure wave, the change of velocity distribution is mainly due to diffusion, and the distribution always spread from the wall region to the core region, which is also the main reason for the change of turbulent condition in the pipe. There is a local pressure difference when the pressure wave passes the monitoring position, but the pressure difference is very small and negligible compared to the local average pressure; at the time before and after the pressure wave passes, the pressure on the monitoring position is almost constant; thus, the 1D and quasi 2D model assumption that pressure on the cross-section during the pressure wave propagation is constant is still valid.

_{1}monitoring position local region at the initial time of T = 0 s and at T = 0.082 s, and it shows that during the transient process, the turbulence dissipation rate has a relatively big value on the wall region than on the core region.

#### 3.4. Analysis of Transient Cavitation Flow Results

#### 3.4.1. Analysis of the Pressure Fluctuation

_{l}, S

_{2}, and S

_{3}monitoring positions are compared with the experimental data in Figure 26. The wave speed $a$ in the first and second stage of the first pressure wave cycle is constant ($a$ = 1280 m/s) due to the material in the pipeline is single phase water, while the wave speed in the third and fourth stage of the first pressure wave cycle is variable due to cavitation. The average pressure wave speed ${a}_{1}$ in the third and fourth stage can be evaluated by the pressure head change at S

_{l}monitoring position and with equation as follows:

_{1}monitoring position.

_{1}monitoring position in the wave speed of ${a}_{1}$ during the third stage (with cavitation) of the first wave cycle. The arrival time of the pressure decrease wave at S

_{1}monitoring position, $Tv$, is identified with the rarefaction wave contours, as shown in Figure 27 and Figure 28. If $Tv$ = 0.0774 s is identified as the rarefaction wave arrival time, the result is ${a}_{1}$ = 1277 m/s, while if $Tv$ = 0.08 s is identified as the rarefaction wave arrival time, the result is ${a}_{1}$ = 1136.84 m/s. Thus, there are different values of for the rarefaction wave average speed, and because the pressure gradient is bigger when T = 0.08 s than when T = 0.0774 s, T = 0.08 s is chosen as the rarefaction wave arrival time, and the value of ${a}_{1}$ = 1136.84 m/s is chosen as the wave speed of the rarefaction wave and it is used in the analysis of the pressure and velocity field in Section 3.4.2, which indicates the pressure wave speed in the cavitation flow is smaller than in the non-cavitation flow.

_{3}monitoring position (0.02 m upstream of the valve) during the transient event. The instance when maximum vapor volume fraction forms correspond to the moment of the minimum pressure head. The vapor volume fraction peak value reduces faster than the pressure head, and when the second cavitation occurs at T = 0.02359 s, the vapor volume fraction value increases again, but never reach the maximum value in the first vapor volume fraction peak. Later on, the vapor volume fraction diminishes, although there is still a pressure head change in the pipeline.

#### 3.4.2. Analysis of the Vapor Volume Fraction and Pressure and Velocity Fields

_{3}monitoring position, with marked time nodes ${T}_{0}\sim {T}_{6}$ of several typical moments in the first cavity formation-development-crash evolution process. The vapor phase volume fraction of these typical moments at the valve upstream local region on the axial plane of the pipeline and the corresponding pressure and velocity fields are analyzed in two stages: the cavity region growth stage (Figure 30, Figure 31 and Figure 32) and the cavity region reduction stage (Figure 33, Figure 34 and Figure 35).

_{3}monitoring point is firstly detected to fall below the vapor pressure. As can be seen in Figure 33a, a local cavity is formed at the top of the pipe wall (the vapor phase volume fraction value is small), and the vapor zone develops upstream along the top of the pipe wall, which is different from the traditional DVCM and DGCM model assumption that “the vapor cavity occupies the entire calculated section and the position remains the same”; in the subsequent time, due to the pressure decrease wave propagates from the valve to the upstream reservoir, the local cavity volume fraction at the S

_{3}monitoring point gradually increases and the vapor region continues to develop along the top of the pipe wall toward the upstream reservoir (${T}_{1}=0.0761\text{}\mathrm{s}=Tc+2L/a+0.12L/{a}_{1}$). The maximum length of the cavity region along the pipe is 3.73 m, and it occurs at ${T}_{2}=0.078s=Tc+2L/a+0.18L/{a}_{1}$. The cavity region growth stage is from T

_{0}= 0.0757 s to T

_{2}= 0.078 s, it lasts 0.0023 s, during this time, the pressure decrease wave generated at the valve in the third stage of the pressure wave cycle propagates from the valve to the upstream reservoir, as can be seen in Figure 31, and velocity field in Figure 32 shows that the cross-section average velocity in the pipe gradually decreases to 0.

_{3}= 0.087 s to T

_{6}= 0.1354 s, and it lasts for 0.0484 s.

#### 3.4.3. Analysis of the Variable Sound Speed Field in the Transient Cavitation Flow

_{2}= 0.078 s, T

_{3}= 0.087 s and T

_{4}= 0.097 s, the average pressure wave speed values are all 1278 m/s, which is almost equal to the pressure wave speed value of 1277 m/s, when the rarefaction wave arrives at the S

_{1}monitoring position at $Tv$ = 0.0774 s (illustrated in Section 3.4.1). However, the average wave speed from Figure 36 at T

_{0}= 0.0757 s, T

_{1}= 0.0761 s and T

_{6}= 0.1354 s are 1280 m/s because of the small vapor volume fraction at these times. Therefore, the pressure wave speed in Figure 36 calculated with Equation (16) are almost identical to the values calculated with the arrival time of the rarefaction wave.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**Comparison of the cross-sectional average pressure fluctuations at the S

_{3}position in different grids.

**Figure 5.**Comparison of cross-sectional average pressure fluctuations at the S

_{3}position of different time steps.

**Figure 6.**Comparison of experimental data and numerical simulation results of pressure head fluctuation at each monitoring point. (

**a**) S

_{1}, (

**b**) S

_{2}, (

**c**) S

_{3}.

**Figure 9.**Experimental data and numerical simulation results of pressure head fluctuation at S

_{1}monitoring point.

**Figure 10.**Axial velocity distribution at different time (${T}_{0}$ = 0 s, ${T}_{1}$ = 0.044 s, and ${T}_{2}$ = 0.05 s) of the S

_{1}monitoring point.

**Figure 11.**Local pressure distribution at S

_{1}monitoring position of different time (${T}_{0}$ = 0 s, ${T}_{1}$ = 0.044 s, and ${T}_{2}$ = 0.05 s).

**Figure 12.**Pressure field evolution on the axis plane of the pipeline (${T}_{0}$ = 0 s, ${T}_{1}$ = 0.044 s, and ${T}_{2}$ = 0.05 s).

**Figure 13.**Axial velocity distribution at different time (${T}_{3}$ = 0.058 s and ${T}_{4}$ = 0.064 s) of the S

_{1}monitoring point.

**Figure 14.**Local pressure distribution at S

_{1}monitoring position of different time (${T}_{3}$ = 0.058 s and ${T}_{4}$ = 0.064 s).

**Figure 15.**Pressure field evolution on the axis plane of the pipeline (${T}_{3}$ = 0.058 s and ${T}_{4}$ = 0.064 s).

**Figure 16.**Axial velocity distribution at different time (${T}_{5}$ = 0.092 s, ${T}_{6}$ = 0.1 s, and ${T}_{7}$ = 0.108 s) of the S

_{1}monitoring point.

**Figure 17.**Local pressure distribution at S

_{1}monitoring position of different time (${T}_{5}$ = 0.092 s, ${T}_{6}$ = 0.1 s, and ${T}_{7}$ = 0.108 s).

**Figure 18.**Pressure field evolution on the axis plane of the pipeline (${T}_{5}$ = 0.092 s, ${T}_{6}$ = 0.1 s, and ${T}_{7}$ = 0.108 s).

**Figure 19.**Axial velocity distribution at different time (${T}_{8}$ = 0.116 s and ${T}_{9}$ = 0.124 s) of S

_{1}monitoring point.

**Figure 20.**Local pressure distribution at S

_{1}monitoring position of different time (${T}_{8}$ = 0.116 s and ${T}_{9}$ = 0.124 s).

**Figure 21.**Pressure field evolution on the axis plane of the pipeline (${T}_{8}$ = 0.116 s and ${T}_{9}$ = 0.124 s).

**Figure 22.**Axial velocity distribution at different time (${T}_{10}$ = 0.148 s and ${T}_{11}$ = 0.164 s) of the S

_{1}monitoring point.

**Figure 23.**Local pressure distribution at S

_{1}monitoring position of different time (${T}_{10}$ = 0.148 s and ${T}_{11}$ = 0.164 s).

**Figure 24.**Pressure field evolution on the axis plane of the pipeline (${T}_{10}$ = 0.148 s and ${T}_{11}$ = 0.164 s).

**Figure 25.**Turbulence dissipation on S

_{1}monitoring position local region at T = 0 s and T = 0.082 s.

**Figure 26.**Comparison between Computational Fluid Dynamics (CFD) simulation results and experimental data.

**Figure 27.**T = 0.0774 s Pressure field on the axis plane of the pipeline (

**a**) and local pressure distribution around S

_{1}monitoring position (

**b**).

**Figure 28.**T = 0.08 s Pressure field on the axis plane of the pipeline (

**a**) and local pressure distribution around S

_{1}monitoring position (

**b**).

**Figure 29.**Pressure head change and vapor volume fraction change at S

_{3}monitoring position (0.02 m upstream of the valve).

**Figure 30.**Evolution of vapor volume fraction on the axis plane of the pipeline in the cavity growth stage.

**Figure 33.**Evolution of vapor volume fraction on the axis plane of the pipeline in the cavity collapse stage.

Experiment No. | Initial Velocity (m/s) | Initial Reynolds Number | Water Level of Upstream Reservoir (m) |
---|---|---|---|

Experiment 1 | 0.239 | 4531 | 24.30 |

Experiment 2 | 0.332 | 6294 | 23.41 |

Case No. | Initial Velocity (m/s) | Pressure of Inlet (Pa) | Pressure of Outlet (Pa) | Valve Closing Time (ms) |
---|---|---|---|---|

Case 1 | 0.239 | 237,739.7 | 235,960.4 | 24 |

Case 2 | 0.332 | 229,060.1 | 225,776.9 | 16 |

Time Node | Time(s) | In Terms of $\mathit{T}\mathit{c}$ and $\mathit{L}\mathbf{/}\mathit{a}$ |
---|---|---|

${T}_{1}$ | 0.024 | $Tc+0\times L/a$ |

${T}_{2}$ | 0.036 | $Tc+0\times L/a$ |

${T}_{3}$ | 0.052 | $Tc+1\times L/a$ |

${T}_{4}$ | 0.06 | $Tc+1.28\times L/a$ |

${T}_{5}$ | 0.072 | $Tc+1.71\times L/a$ |

${T}_{6}$ | 0.08 | $Tc+1.99\times L/a$ |

${T}_{7}$ | 0.086 | $Tc+2.2\times L/a$ |

${T}_{8}$ | 0.098 | $Tc+2.63\times L/a$ |

${T}_{9}$ | 0.108 | $Tc+2.99\times L/a$ |

${T}_{10}$ | 0.12 | $Tc+3.41\times L/a$ |

${T}_{11}$ | 0.13 | $Tc+3.77\times L/a$ |

${T}_{12}$ | 0.136 | $Tc+3.98\times L/a$ |

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**MDPI and ACS Style**

Tang, X.; Duan, X.; Gao, H.; Li, X.; Shi, X.
CFD Investigations of Transient Cavitation Flows in Pipeline Based on Weakly-Compressible Model. *Water* **2020**, *12*, 448.
https://doi.org/10.3390/w12020448

**AMA Style**

Tang X, Duan X, Gao H, Li X, Shi X.
CFD Investigations of Transient Cavitation Flows in Pipeline Based on Weakly-Compressible Model. *Water*. 2020; 12(2):448.
https://doi.org/10.3390/w12020448

**Chicago/Turabian Style**

Tang, Xuelin, Xiangyu Duan, Hui Gao, Xiaoqin Li, and Xiaoyan Shi.
2020. "CFD Investigations of Transient Cavitation Flows in Pipeline Based on Weakly-Compressible Model" *Water* 12, no. 2: 448.
https://doi.org/10.3390/w12020448