# Pressure Transient Model of Water-Hydraulic Pipelines with Cavitation
## 1 Introduction  
Transient pressure pulsations generated by the rapid closure of a valve would easily cause hydraulic pipeline systems to burst because the pressure pulsations exceed the safe operating range of such pipelines. Violent pressure pulsations result in cavitation growth and collapse in these systems. To study the mechanism of cavitation during pressure transient pulsations, it is necessary to investigate the cavitation appearance, its volume evaluation, and the effect on the pipeline and hydraulic systems. Pressure transients with cavitation in pipelines have been investigated by many researchers. [PERSON] et al. [1] presented the gas-nonbubbly flow model to predict pressure increments, which involved cavitation on the downstream side of the pipeline as a valve was instantaneously closed. They used a water-glycol mixture and an oil/water emulsion fluid including mineral oil as working fluids and compared the computed pressure pulsations with experimental results. [PERSON] et al. [2; 3] then proposed a MacCormack scheme and a Gabutti scheme for pressure transient analysis, which was verified both in computed simulation and experimental studies. Although modeling accuracy was achieved, discrepancies in the pressure magnitudes between simulations and experiments were found. Transient pressure pulsations often lead to unexpected chatter, overshooting, and a zero bias of tracking error in the electrohydraulic control system [4; 5; 6; 7]. [PERSON] et al. [8; 9; 10] developed a vaporous cavitation model that used a two-phase homogeneous equilibrium to simulate pipeline pressure transients with upstream, midstream, and downstream cavitation. [PERSON] et al. [11; 12] discussed three cavitation models: the discrete vapor cavity model (DVCM), the discrete gas cavity model (DGCM), and the generalizedinterface vaporous cavitation model (GIVCM). The comparative results of the three cavitation models indicated that the GIVCM was able to directly obtain the regions of vaporous cavitation occurrence. [PERSON] et al. [13; 14; 15; 16], via genetic algorithms, developed the parametric identification of the gas bubble model and the frequency-dependent friction model. Parametric identification and noise suppression are also addressed in mechanical ventilation [17; 18; 19; 20]. [PERSON] et al. [21] recently studied water hammers with cavitation in a simple reservoir-pipeline-valve system and a pumping station. [PERSON] et al. [22] verified the robustness of the DGCM via analysis of the experimental results. [PERSON] et al. [23] performed a detailed study of the actual behavior of different valves (both air intake and exhaust) and described the mathematical characterization of different commercial valves. [PERSON] et al. [24] presented a numerical modeling of pipelines with air pockets and air valves to study the behavior of the air inside pipes as the air was expelled through air valves. [PERSON] et al. [25] investigated the unsteady flow of a non-Newtonian fluid due to the instantaneous valve closure in a pipeline. Comparison revealed a remarkable deviation in pressure history and velocity profile with respect to the water hammer in Newtonian fluids. [PERSON] et al. [26] adopted a second-order finite volume method for cavitation in the water column separation of pipelines to capture vapor cavities and predict their growth and collapse. [PERSON] et al. [27] adopted a two-dimensional CFD model to characterize liquid column separation. The simulation results revealed the formation of an intermediate cavity and both the location and shape of the region undergoing distributed vaporous cavitation. [PERSON] [28] also studied water hammers with column separation as a one-dimensional flow. The volume of the cavity was determined by [PERSON]'s method, and the air bubbles were considered to affect the speed of sound.