# Pressure Transient Model of Water-Hydraulic Pipelines with Cavitation
## 5 Conclusions  
To reveal the mechanism of cavitation growth and collapse both on upstream and downstream of the water-hydraulic pipeline, this paper proposes the finite difference method (FDM) for determining the transient pressure to estimate pipeline pressure transients caused by sudden changes in fluid velocity. Firstly, the dynamic model of cavitation volume was derived during pressure transients. Then, the cavitation appearance durations and volume changes were analyzed. Furthermore, the frictional force model with the steady and frequency-dependent unsteady items were constructed in the proposed dynamic model. By referring to experimental results in [35; 37], the simulation results of two computation methods were verified to indicate that the proposed FDM for transient pressure estimation has the following two advantages:  
(i) The FDM is consistent with experimental results, which is improvement over the MOC in terms of the phase differences and magnitudes of the pressure peaks.  
(ii) The FDM estimates not only the magnitudes of the pressure peaks but also the changes in cavitation volume to adapt the transient pressure both with and without cavitation. By statistical results, the IAD values of the FDM are much more favorable than that of the MOC.  
However, the aforementioned discussion assumes that no air is released during cavitation. In fact, water usually contains some dissolved air or gas. If the pressure declines under the saturation pressure, especially under vapor pressure with agitation, then a certain amount of air will be released as gas bubbles. Thus, these effects of gas bubbles on pressure transients with cavitation will be investigated in the near future.  
**Acknowledgments:** The authors are grateful to the anonymous reviewers and the editor for their constructive comments. This research was supported by the National Natural Science Foundation of China (51205045, 61305092, 51775089) and the Open Foundation of the State Key Laboratory of Fluid Power & Mechatronic Systems (GZKF-201515).  
**Author Contributions:** [PERSON] conceived and designed the structure of this paper; [PERSON] and [PERSON] performed the simulation; [PERSON] reviewed the literature.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Abbreviation**  
IAD & Integral of absolute difference  
\(B_{eff}\) (Pa) & Effective bulk modulus  
\(c_{0}\) (m/s) & Acoustic velocity in the fluid  
\(f\) & Coefficient of Darcy-Weisbach  
\(F_{0}\) (N) & Steady friction  
\(F(q)\) (N) & Friction  
\(m_{i}\) & Weighting constant  
\(n_{i}\) & Weighting constant  
\(p\) & Vector of pressures at nodes  
\(p_{A},p_{B},p_{P}\) (Pa) & Pressure at points \(A\), \(B\), and \(P\)  
\(p_{resu}\) (Pa) & Pressure in the upstream tank  
\(p_{resd}\) (Pa) & Pressure in the downstream tank  
\(p^{\prime}\) & New vector of pressures at nodes  
\(p_{Lexp}\) & Experimental results of pressure transients at the valve  
\(p_{Lss}\) & Steady-state pressure at the valve  
\(p_{Lth}\) & Simulation results of pressures transients at the valve  
\(q\) & Vector of flow rate at nodes\(q_{A},q_{B},q_{P}\) (m\({}^{3}\)/s) Flow rate at points \(A\), \(B\), and \(P\)  
\(q^{\prime}\) New vector of flow rate at nodes  
\(q_{in}\) (m\({}^{3}\)/s) Inflow rate  
\(q_{out}\) (m\({}^{3}\)/s) Outflow rate  
\(r_{0}\) (m) Radius of the pipeline  
\(v\) (m/s) Velocity in the fluid  
\(v_{0}\) (m/s) Initial velocity in the fluid  
\(V_{env}\) (m\({}^{3}\)) Cavitation volume  
\(Y_{i}\) (N) Weighting function  
\(\rho\) (kg/m\({}^{3}\)) Density of fluid  
\(\mu\) (Pa s) Viscosity of fluid