*1.2. Transient Cavitating Flow in Plastic Pipes*

Among the phenomena accompanying transient flows, the most important are (1) unsteady friction (UF, other name: skin friction), (2) cavitation (CAV), (3) viscoelastic property of pipe deformation (flow in plastic conduits) associated with the retarded strain (RS), and (4) mutual fluid–structure interaction (FSI) of the flowing liquid with the vibrations of the pipe walls. In this work, we will focus in detail on the first three phenomena. They will be implemented in the revised DBCM model. The continuity equation with the retarded strain term was originally proposed by Rieutord and Blanchard [9]. Cavitation was modeled by Güney [10] using the column-separation modeling assumption proposed by Swaffield [11] and Safwat [12]. Another scientist examining the effect of cavitation occurring during transient flow in PE and PVC pipes was Mitosek [13,14], who showed that an increased pressure reduction is accompanied with gas desorption (reduced pressure oscillations

with the increase time period of their existence). Hadj-Taïeb with Taïeb [15] proposed initially a numerical model based on the conservative finite difference method to solve the nonlinear system of hyperbolic partial differential equations describing the transient flow in which the degasification takes place (according to Henry's law). Their study showed that the degasification area is strongly connected with the wall elasticity. The same two authors [16] proposed an alternative modified mathematical model that includes retarded strain and cavitation, which was solved with the second-order finite difference scheme. The mixture density in this model was expressed by means of a non-linear expression of the liquid volume fraction. Borga et al. [17,18] conducted several transient tests with localized gas cavities in around 200 m long HDPE pipe and concluded that the presence of the leak (or air valves) in cavitating flow induces a greater damping and dispersion of transient pressure waves. Soares et al. [19,20] continued the research of Borga (which was done under the supervision of H. Ramos) and compared the effect of used cavitation models DVCM and DGCM for the prediction of transient flows with cavitation in HDPE pipes. The results indicated that the assumption of the ideal gas law (DGCM) is more appropriate than a simple adoption of vapor pressure when the pressure reaches vapor pressure (DVCM) and induces more attenuation and dispersion of transient pressures. For flows with cavitation, a new set of pipe-wall viscoelastic parameters was determined (calibration technique). The unsteady friction losses, pipe-wall viscoelasticity, and wave speed variation due to the formation of localized gas cavities were described only by the creep function. Such an approach lumped all these important phenomena in the coefficients of the creep function. Keramat et al. [21] utilized DVCM and modeled RS uses a modified Kelvin–Voigt model to study the unsteady flow with cavitation in plastic pipes. His model did not include at that time the unsteady friction effects. The main conclusion from the presented simulations (compared to simulated results with Covas [22] and Soares data [19]) is that viscoelastic pipes strongly diminish the dangers of column separation: "First, cavity opening and collapse occur only one or two times instead of tens of times (as inelastic pipes)". Two years later, Keramat and Tijsseling [23] were first to present a numerical model that included all four important phenomena that take place in transient pipe flows: UF, CAV, RS, and FSI. Unfortunately, to date, there are no experimental results that are conducted to check this interesting model in the full extent. In 2018, Urbanowicz and Firkowski developed the foundations of the model presented in this paper [24]. A year later, Urbanowicz et al. [25] compared the DACM and DBCM models, which had not been used before, for the analysis of cavitation flows in plastic pipes, indicating that they both model the phenomena in a similar way, despite the fact that they are characterized by a significantly different mathematical notation.
