*1.3. Recent Progress in Cavitation Modeling in Metal Pipes*

Liu et al. [26] analyzed cavitation that can take place in long-distance transport pipelines. The water hammer due to the collapse of air cavities in the pipeline was discussed when the pump unit is shut down due to an incident. The theoretical and numerical analysis pointed out that it is very important to prevent the occurrence of large water hammer loads due to the collapse of flow interruption in such a system. Santoro et al. [27] tested the DVCM model, writing the continuity equation in terms of mass balance instead of volume balance. Such an assumption allowed calculations with appropriate computational fine grids. Additionally, the flow field was assumed to be two-dimensional (2D axial-symmetric flow), in order to evaluate unsteady friction without the need of parameters calibration. This research pointed out that one-dimensional (1D) models are weakly sensitive to grid size, whereas 2D model results are practically gridindependent, and in the opinion of the authors, the 2D model performs better than the 1D ones. Shankar et al. [28] studied the optimal operation of centrifugal pumps to avoid the major harmful issues as cavitation and water hammering. These authors built a system with a cascade parallel pumping setup. The extensive experimental study reveals that the preferable operating region enhances reliability as well as reduces the occurrence of

faults. This paper can serve as a reference to VFD pumping systems and paves the way for sensor-less control. Zhao et al. [29] built an experimental test stand to realize a water hammer event with multipoint collapse. The influencing factors and laws of the cavity length and water hammer pressure have been summarized using the experimental data. They also reveal that the initial flow rate and valve-closing speed greatly affect the water hammer pressure rise and cavity length. In their next work [30], the authors presented a new water hammer velocity formula, a new cavity model, and introduced a floating grid method. An in-house program written in C++ confirmed that the simulation results of the new model matched the measured values.

Sun et al. [31] proposed a quasi-two-dimensional transient model coupled with DVCM which, according to the authors' analysis, can provide a better fit than classic 1D solutions. Warda et al. [32] performed three-dimensional computer fluid dynamics (CFD) simulations based on the finite volume numerical approach. The cavitation was modeled with the use of two models: the Volume of Fluid (VOF) and Schnerr–Sauer. They concluded that the 3D model that was adopted is "deemed physically superior to the existing 1D models as it removes the restriction of the 1D models that vapor cavities, when formed, fill the whole cross-section of the pipe without radial variation". Sanín-Villa et al. [33] considered the influence of the convective terms in the momentum and continuity equations (which standardly are neglected). The cavitation problem has been evaluated by use of the DVCM model. In conclusion, they stated that the influence of the convective term is small compared with a simple model where those terms are neglected. Tang et al. [34] used Fluent software to investigate the cavitation flow in the pipeline. A density–pressure model has been implemented into the continuity equation by using the further development of a userdefined function, which gives the possibility of studying the effects of the variable wave speed on the transient cavitation flow. The weakly compressible fluid RANS model (CFD) results agree well with the measured results. Saidani et al. [35] analyzed the temperature effect (in a range from 4 to 95 ◦C) on unsteady flow with cavitation. These authors simulated single-phase and two-phase transient flows in a hydraulic copper pipe system. The DVCM and DGCM models were used. From the performed simulations, it was evident that the water hammer is considerably sensitive to the temperature, and its proper value needs to be considered at the design stage of hydraulic systems. Yang et al. [36] used a uniform cavitation distribution model in which the critical flow velocity gradients are calculated both in front and at the back of the section and are the sufficient condition to define water column separation. Dynamic meshes were applied for tracking the change of vaporous cavitation. However, multidimensional models are computationally expensive.
