# Pressure Transient Model of Water-Hydraulic Pipelines with Cavitation
## 4 Simulation Results  
### Case 1: Pressure Transients without Cavitation on the Downstream Side of Valve  
The experimental results of the transient pressure pulsations close to the valve in the horizontal downstream pipeline are given by [PERSON] et al. [35] and the related parameters of the tested pipeline are listed in Table 2. Here, the element number \(n\) is selected as 30 in the simulation. The sensitivity of this element number \(n\) has been discussed in [13]. The corresponding experimental results of transient pressure pulsations close to the valve are shown as the solid line in Figure 5.  
Figure 5 denotes that, in the downstream pipeline, the pressure at the vicinity of the valve is reduced quickly when the valve is closed. At the same time, the negative pressure wave propagates to  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Upstream tank pressure \(p_{resu}\) (bar) & 4.25 \\ Downstream tank pressure \(p_{resd}\) (bar) & 4.22 \\ Pipe radius \(r_{0}\) (mm) & 11.05 \\ Pipe length \(L\) (m) & 37.2 \\ Water density \(\rho\) (kg/m\({}^{3}\)) & 1000 \\ Initial velocity \(v_{0}\) (m/s) & 0.3 \\ Acoustic velocity in the fluid \(c_{0}\) (m/s) & 1319 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of pressure transients without cavitation on the downstream side of the valve.  
Figure 5: Simulation and experimental pressure transients without cavitation on the downstream side of the valve.  
the downstream tank. Then, the positive pressure wave is reflected from the downstream tank and travels back to the valve, which leads to the first positive pressure peak. This process may be repeated several times before the fluid energy is dissipated due to the frictional force of the pipeline.  
As shown in the experimental results, from 0 to 1 s, the attenuated peaks of the pressure pulsations are decreased very slowly. Because of the lower frictional force, the magnitudes of the pressure peaks decay slower in water, which is different from the corresponding pressure transient results with the working fluid as hydraulic oil ([PERSON] et al. [13]). Thus, if the pressure pulsations are always greater than the saturated vapor pressure, no cavitation forms.  
For comparison, the simulation results of MOC are illustrated as the dash-dotted line in Figure 5a. The simulation results of the FDM platform are presented as the dashed line in Figure 5b. It can be seen that, from 0.4 to 1 s, the phase difference between the MOC simulation and the experimental results is more obvious. However, the simulation results of the FDM are still consistent with the experimental pressure results.  
To further compare the two predictive methods, the error between the simulation and the experimental results was evaluated by the percentage of the integral of absolute difference (IAD) as follows ([PERSON] et al. [36]):  
\[IAD=\frac{\int\limits_{0}^{T}\left|p_{Lth}-p_{L\exp}\right|dt}{p_{L\text{ss}}T }\times 100\%. \tag{21}\]  
The IAD results of the two predictive methods in the three cases are listed in Table 3. In Case 1, the final steady-state pressure at the valve is equal to the downstream tank pressure (4.22 bar), and the IAD of the FDM is about 0.05%. Thus, the simulation of the FDM is consistent with the experimental result, which is superior to the MOC (IAD = 0.91%).