## 3 Results and Discussion
### Pressure-Drop and Velocity Phase Shift  
In this subsection, the evolution of pressure drops \(\Delta P\) and velocities are compared. The pressure drop is calculated using the following formula:  
\[\Delta P=|P_{hot\ side}(\theta)-P_{cold\ side}(\theta)|, \tag{17}\]  
where \(P_{hot\ side}(\theta)\) and \(P_{cold\ side}(\theta)\) are pressures at the hot and cold inlet sides of the regenerator, respectively. In Figure 14, different evolutions are plotted together as a function of the angle of rotation \(\theta\) for the three regenerator structures (Str_30%, Str_35%, and Str_40%). As fluid velocity shows an almost constant plateau around the maximum velocity \(u_{max}\) over a significant rotation angle interval, measuring phase shift between the pressure drop and velocity maximum appears hazardous. Substantial uncertainties in the calculation of the angular phase difference between the head loss and velocity are probable. Therefore, it appears to be more justified to determine this phase shift between the minimum values. Furthermore, as experiments were conducted for different frequencies and regenerators, we gathered results on a single phase shift versus kinetic Reynolds number \(Re_{\omega}\) (defined in the Introduction) pattern. Those evolutions in phase shift \(\phi(\Delta P,\ u)\) are presented for all regenerators regarding:  
* Strokes of the piston (\(C_{1}\) = 64.4 mm and \(C_{2}\) = 128.4 mm) in (Figure 15a). * Temperature gradient (\(\Delta T_{1}\) = 70 \({}^{\circ}\)C/10 \({}^{\circ}\)C and \(\Delta T_{2}\) = 40 \({}^{\circ}\)C/10 \({}^{\circ}\)C) in (Figure 15b). * Regenerator length (\(L_{1}\) = 60 mm and \(L_{2}\) = 70 mm) in (Figure 15c). Figure 13: Evolution of friction factor at maximum pressure drop of regenerator ((**a**): Str_30%, (**b**): Str_35%, (**c**): Str_40%. \(L_{1}\) = 60 mm, \(C_{1}\) = 64.4 mm) versus kinetic Reynolds number \(Re_{\omega}\). Figure 14: Evolution of the pressure drop \(\Delta P\) and the velocity \(u\) as a function of the angle of rotation \(\theta\) of a regenerator (**(a)**: Str_30%, **(b)**: Str_35%, **(c)**: Str_40%. \(L_{1}=60\) mm, \(C_{1}=64.4\) mm) versus the phase for the temperature gradient \(\Delta T_{1}=70\)\({}^{\circ}\)C/10 \({}^{\circ}\)C.  
Figure 15 shows that the phase shift between the pressure drop and velocity is significant and can reach 71\({}^{\circ}\) at high Reynolds frequency numbers for any structures. We can also observe that the phase shift increases linearly with the kinetic Reynolds number, regardless of the configuration. This means that there are negligible effects of displacement, regenerator length, and temperature gradient on the phase shift in our investigated parameter area. This confirms the theoretical and experimental results of [PERSON] and [PERSON] (2011); [PERSON] (2011); [PERSON] and [PERSON] (2018), and [PERSON] et al. (2013). However, the above results show that the phase shift depends on the structure of the regenerator. Thus, all the results have been plotted for each regenerator for all the corresponding experiments independently of other parameters (length, stroke's motion, and temperature gradient). The obtained graphs are presented in Figure 16.  
In each graph in Figure 16, the phase difference appears as a growing linear function of the oscillating Reynolds number \(Re_{\omega}\) for each regenerator. Seeking a law confirms this observation, as the regression curves added to the figure proves it. Regressions are given in Table 3, assuming that at zero frequency a phase shift should be null.