# Pressure Transient Model of Water-Hydraulic Pipelines with Cavitation
## 4 Simulation Results
### Case 2: Pressure Transients with Cavitation on the Downstream Side of Valve  
The case of transient pressure pulsations with cavitation in the horizontal downstream pipeline was also investigated. Some experimental parameters from [PERSON] et al. [37] are listed in Table 4. The corresponding experimental results are shown as the solid line in Figure 6. As the valve is quickly closed, the pressure reduces and stays at vapor pressure for about 3 s. The pressure then drops again and stays at vapor pressure for about 1.5 s. For the third time, the pressure falls and stays at vapor pressure for about 1 s.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & \(p_{Lss}\) **(bar)** & **IAD of MOC** & **IAD of FDM** \\ \hline
1 & 4.22 & 0.91\% & 0.05\% \\
2 & 0.98065 & 2.75\% & 2.47\% \\
3 & 4.90325 & 12.39\% & 10.84\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: The integral of absolute difference (IAD) of the method of characteristics (MOC) and the finite difference method (FDM).  
The results obtained from the MOC and the FDM are also shown in Figure 6. It is clear that obvious differences exist between the MOC simulation and the experimental results, especially in terms of the phase differences of the subsequent peaks. However, the results of the FDM via Matlab/Simulink Platform are consistent with the experimental results.  
As listed in Table 3, for Case 2, the final steady-state pressure at the valve is 0.98065 bar, which is equal to the downstream tank pressure. Different from Case 1, the two predictive methods have similar effects (the IADs of the two predictive methods are 2.75% and 2.47%).  
The corresponding cavitation volumes in the element close to the valve predicted by the FDM and the MOC are shown in Figure 7. The trends of vaporous cavitation volume under three cases are listed in Table 5, which includes the cavitation start time, the end time, the duration, the maximum volume, and the corresponding time points.  
Similar to the MOC, the FDM is also able to track the trends of cavitation volume. The results indicate that the computed pressure peak declines to the saturated vapor pressure after the valve is rapidly closed and after cavitation forms. However, this new cavitation collapses at 3.87 s (FDM) and 3.77 s (MOC). The maximum volumes of cavitation first are \(1.372\times 10^{-4}\) m\({}^{3}\) (FDM) and \(1.401\times 10^{-4}\) m\({}^{3}\) (MOC). When the pressure declines again, cavitation is generated again, but it is much smaller (\(1.892\times 10^{-5}\) m\({}^{3}\) from the FDM and \(4.155\times 10^{-5}\) m\({}^{3}\) from the MOC) than the first instance. Once again, cavitation collapses at the arrival of the third pressure peak. The durations of the third cavitation is 1.08 s (FDM) and 0.99 s (MOC). Thus, over this short period (12 s), the cavitation demonstrates generation and collapse three times.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Upstream pressure \(p_{resu}\) (bar) & 6.55164 \\ 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) & 1.5 \\ Viscosity of the fluid \(c_{0}\) (cP) & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters of pressure transients with cavitation on the downstream side of the valve.  
Figure 6: Simulation results and experimental data of pressure transients with cavitation on the downstream side of the valve.