# Pressure Transient Model of Water-Hydraulic Pipelines with Cavitation
## 4 Simulation Results
### Case 3: Pressure Transients with Cavitation on the Upstream Side of the Valve  
[PERSON] et al. [37] also provided parameters of tested pipelines in the case of transient pressure pulsations in a horizontal upstream pipeline, as listed in Table 6. Compared with Case 2 listed in Table 4, the parameters of the test pipeline are the same, except for the values of the upstream pressure and the initial velocity. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Upstream pressure \(p_{resu}\) (bar) & 4.90325 \\ Downstream pressure \(p_{resd}\) (bar) & 0.98065 \\ Pipe radius \(r_{0}\) (mm) & 7.6 \\ Pipe length \(L\) (m) & 200 \\ Water density \(\rho\) (kg/m\({}^{3}\)) & 1000 \\ Initial velocity \(v_{0}\) (m/s) & 1.45 \\ Viscosity of fluid \(c_{0}\) (cP) & \(1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of pressure transients with cavitation on the upstream side of the valve. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Times** & **Method** & \begin{tabular}{c} **MOC** \\ **(case 1)** **(case 1)** \\ \end{tabular} & \begin{tabular}{c} **FDM** \\ **(case 2)** \\ \end{tabular} & \begin{tabular}{c} **MOC** \\ **(case 2)** \\ \end{tabular} & \begin{tabular}{c} **FDM** \\ **(case 2)** \\ \end{tabular} & \
\begin{tabular}{c} **MOC** \\ **(case 3)** \\ \end{tabular} &
\begin{tabular}{c} **FDM** \\ **(case 3)** \\ \end{tabular} \\ \hline \multirow{4}{*}{1 st time} & start time (s) & & 0.40 & 0.38 & 0.60 & 0.65 \\  & end time (s) & & 3.77 & 3.87 & 1.21 & 1.24 \\  & duration (s) & – & – & 3.37 & 3.49 & 0.61 & 0.59 \\  & maximum volume time (s) & & 2.32 & 2.11 & 1.08 & 1.21 \\  & maximum volume (m\({}^{3}\)) & & \(1.401\times 10^{-4}\) & \(1.372\times 10^{-4}\) & \(1.305\times 10^{-5}\) & \(3.955\times 10^{-6}\) \\ \hline \multirow{4}{*}{2 nd time} & start time (s) & & 4.34 & 4.37 & 2.05 & 1.98 \\  & end time (s) & & 6.14 & 6.03 & 2.17 & 2.23 \\  & duration (s) & – & 1.80 & 1.66 & 0.12 & 0.25 \\  & maximum volume time (s) & & 5.12 & 5.73 & 2.15 & 2.21 \\  & maximum volume (m\({}^{3}\)) & & \(4.155\times 10^{-5}\) & \(1.892\times 10^{-5}\) & \(1.281\times 10^{-6}\) & \(6.785\times 10^{-7}\) \\ \hline \multirow{4}{*}{3 rd time} & start time (s) & & 6.72 & 6.54 & \\  & end time (s) & & 7.71 & 7.62 & \\  & duration (s) & – & 0.99 & 1.08 & – & – \\  & maximum volume time (s) & & 7.46 & 6.76 & \\  & maximum volume (m\({}^{3}\)) & & \(1.009\times 10^{-5}\) & \(1.535\times 10^{-6}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Trends of cavitation volume. Figure 7: Simulation results of the cavitation volume in the first element on the downstream side of the valve using the MOC and the FDM. Figure 8 demonstrates the sequence of pressures with cavitation caused by instant valve closure. Compared with the pressure pulsations in the downstream pipeline, after a sudden valve closure, the fluid is brought to rest, firstly causing a high pressure peak at the upstream side of the valve. Experimental results from Sanada (the solid line in Figure 8) show that the initial pressure at the valve is about \(16\times 10^{5}\) Pa when the valve is closed. It then reduces to the vapor pressure and keeps a steady state until about 0.5 s. Upon collapse of the cavitation, another pressure wave is generated at the valve. The subsequent pressure peak is reduced because of the friction force in the pipeline. The figure also shows the simulation results from the MOC and the FDM. For the case of the upstream side of the valve, the final steady-state pressure at the valve is equal to the upstream tank pressure (4.90 bar). As listed in Table 3, the IADs of the MOC and the FDM are 12.39% and 10.84%. It is clear that the results of the FDM has much better consistency with the experimental results than those using the MOC.