**3. Experimental Verification of New Model**

In order to demonstrate the effectiveness of the newly presented model, in this section, the results of simulation tests will be compared with the experimental results presented by Güney [10]. The Güney experimental test stand located at the INSA research center in Lyon (France) was a simple system consisting of three main components: reservoir–pipe–valve (Figure 2).

In the analyzed RPV system, in steady flow, water flowed directly into the atmosphere. The pipe had a total length of *L* = 43.1 m and an internal diameter *D* = 0.0416 m (the wall thickness of the pipe was *e* = 0.0042 m). The test pipe was made of low-density polyethylene (LDPE). The experimental tests of the water hammer forced by the sudden (momentary) closure of the valve (shutting off the flow) have been carried out for five different temperatures of the flowing liquid (water). In Table 1, the parameters required to simulate the analyzed unsteady flows with cavitation are tabulated. It can be seen that although the initial flow velocity was similar, due to the change in viscosity, the value of the Reynolds number increased with the temperature increase (almost twice as high for *Case* 05 : *Re*<sup>05</sup> ≈ 82, 000 than for *Case* 01 : *Re*<sup>01</sup> ≈ 45, 500). After the temperature change, not only do the parameters related to the flowing liquid change (Table 1) but also the values

of the parameters representing the mechanical properties of the pipe; thus, it is necessary to compare their values (*J* creep compliances and *τ* retardation time coefficients values are presented in Table 2).

**Figure 2.** Schematic diagram of Güney's experimental test stand: 1—booster pump, 2—temperature stabilization system, 3—test stand supply pump, 4—thermometer, 5—reservoir, 6—LDPE pipe, 7, 8, and 9—pressure transducers, 10—quick-closing valve.

Güney used the time–temperature superposition principle (also known as time– temperature reducibility) to derive his creep compliance functions for different temperatures. During initial simulations, complete Güney creep compliance functions were used (three exponential terms) that can be found in the works [10,52]. As the initially obtained simulation results indicated that this creep function is a source of simulation error, we had a detailed look at the original coefficients. We noticed that the corresponding creep compliance values of small retardation times (*τ* < 1.5Δ10−<sup>4</sup> [*s*]–original *J*<sup>1</sup> and *τ*<sup>1</sup> coefficients) are out of the frequency range of the used dynamic viscoelastometer RHEOVIBRON. Filtering out this coefficient for small retardation times (rejecting from the analysis original *J*<sup>1</sup> and *τ*<sup>1</sup> coefficients without changing all other experimentally defined creep coefficients) helped to receive corrected comparisons results.

The creep functions for LDPE have different characteristics (Figure 3) than those for the typical currently used plastic material, namely HDPE. The LDPE material has higher values of creep compliance than the HDPE material. Additionally, we may see (Figure 3) that an increase in temperature increases the creep compliance values. The HDPE traces which are presented for comparison in Figure 3 were obtained experimentally by Covas et al. [22].

**Figure 3.** Creep functions for two different PE pipes.



*J*

*i*

—creep-compliance

coefficients;

*τ*

*i*

—retardation

 times.

The pressure wave speeds were estimated based on the empirically observed duration of the first pressure amplitudes. Their values summarized in Table 1 enabled the determination of *J*<sup>0</sup> (see Table 2) from the transformed formula of the pressure wave speed:

$$J\_0 = \frac{1}{\rho \Xi c^2} - \frac{1}{K\_l \Xi} \tag{56}$$

where *ξ* = 0.97; Ξ = *<sup>D</sup> <sup>e</sup> ξ* = 9.61.

The method of characteristics was used with a constant number of reaches *N* = 64. The selected number of reaches meets the computational compliance criteria discussed in paper [53], i.e., *N* > 10. Extra simulation studies performed during the preparation of this paper whose purpose was to investigate the impact of the number of reaches showed that there are no significant differences between the results of *N* = 16, 32, and the selected 64. A finer grid is favorable in the case of instantaneous valve closure. The time steps are calculated on the basis of the Courant–Friedrichs–Lewy (CFL) stability condition *Cn* = (*c*·Δ*t*) <sup>Δ</sup>*<sup>x</sup>* ≤ 1. In order to keep the value of the CFL number equal to one, appropriate values of the time steps should be determined from Δ*t* = Δ*x*/*c* (wave speeds *c* are given in Table 1). In the MOC Δ*x* = *L*/*N* i.e., Δ*x* ≈ 0.67 m. Then, the following time steps are obtained for the five cases: Δ*tG*<sup>01</sup> = 0.0022 s; Δ*tG*<sup>02</sup> = 0.0025 s; Δ*tG*<sup>03</sup> = 0.0027 s; Δ*tG*<sup>04</sup> = 0.0029 s; and Δ*tG*<sup>05</sup> = 0.0031 s, respectively. The results of the simulation tests compared with the experimental data are presented in Figure 4.

**Figure 4.** Computed and measured results for different cases: (**a**) Case 01 (13.8 °C); (**b**) Case 02 (25 °C); (**c**) Case 03 (31 °C); (**d**) Case 04 (35 °C); (**e**) Case 05 (38.5 °C).

The qualitative analysis of the obtained results (Figure 4) indicates the following:


**Figure 5.** Enlargement of (**a**) top of first amplitude and (**b**) early stage of second amplitude.
