*Article* **Modeling Transient Pipe Flow in Plastic Pipes with Modified Discrete Bubble Cavitation Model**

**Kamil Urbanowicz 1,\*, Anton Bergant 2,3, Apoloniusz Kodura 4, Michał Kubrak 4, Agnieszka Malesi ´nska 4, Paweł Bury <sup>5</sup> and Michał Stosiak <sup>5</sup>**


**Abstract:** Most of today's water supply systems are based on plastic pipes. They are characterized by the retarded strain (RS) that takes place in the walls of these pipes. The occurrence of RS increases energy losses and leads to a different form of the basic equations describing the transient pipe flow. In this paper, the RS is calculated with the use of convolution integral of the local derivative of pressure and creep function that describes the viscoelastic behavior of the pipe-wall material. The main equations of a discrete bubble cavity model (DBCM) are based on a momentum equation of two-phase vaporous cavitating flow and continuity equations written initially separately for the gas and liquid phase. In transient flows, another important source of pressure damping is skin friction. Accordingly, the wall shear stress model also required necessary modifications. The final partial derivative set of equations was solved with the use of the method of characteristics (MOC), which transforms the original set of partial differential equations (PDE) into a set of ordinary differential equations (ODE). The developed numerical solutions along with the appropriate boundary conditions formed a basis to write a computer program that was used in comparison analysis. The comparisons between computed and measured results showed that the novel modified DBCM predicts pressure and velocity waveforms including cavitation and retarded strain effects with an acceptable accuracy. It was noticed that the influence of unsteady friction on damping of pressure waves was much smaller than the influence of retarded strain.

**Keywords:** retarded strain; cavitation; water hammer; unsteady friction; method of characteristics
