## 4 Conclusions  
This study was focused on our understanding of the physical parameters characterizing alternating flows at different frequencies submitted to a temperature gradient and passing through different metallic regenerator structures that can be found in Stirling cycle thermal machines, and in so-called alternative machines. These investigations were experimentally performed using a specific test bench, allowing for the measurement of instantaneous pressures, velocities, and temperatures at both sides of the regenerator over several cycles.  
From the results presented above, the following main conclusions may be drawn:  
* The dynamic characteristics of the regenerator are finally determined from the instantaneous measurements of the pressures, velocities, and temperatures on each side of our regenerators. Temperature measurements required the development of microthermocouples of 12.7 \(\upmu\)m (type K). They were manufactured in our laboratory and characterized both statically and dynamically.
* The effect of the compressibility of the working fluid (variation of the density) in our experimental device is not negligible, which brings us closer to the conditions actually generated in thermal machines operating an alternating regime.
* The friction coefficient is calculated using a classical correlation at the time of maximum velocity. The friction factor \(f\) seems to decrease with the kinetic Reynolds number \(\ R\epsilon_{\omega}\) for a 30% porosity regenerator. For the other tested regenerators, we observe that it is almost constant with a little dispersion. To minimize the dispersion effects, another definition was proposed to calculate the friction factor at the time of maximum pressure drop. The results then show, for the three regenerators, a single trend that is clearer as a function of the frequency Reynolds number.
* A new correlation for the phase shift between the velocity and pressure drop of different regenerator structures was obtained. It was found that the effect of \(R\epsilon_{\omega}\) on the phase shift is more dominant than that of the hydraulic diameter to length ratio \(D_{h}/L\). This correlation equation is useful in the design of regenerators in a Stirling machine or alternating flow-based devices.  
In future, this experiment will be carried out for other regenerators, especially those with different porosities and shapes (mesh, grid, and channel). In order to do so, the test bench will be enhanced. We also aim to determine the energy balance on the regenerator and on each heat exchanger. The last improvement will be the temperature measurement inside and at the wall of the regenerators.  
Conceptualization, G.L. and F.L.; literature review and analysis, M.S.K.; methodology, all authors; investigation, all authors; writing--Original draft preparation, G.L.; writing--Review and editing, all authors. All authors have read and agreed to the published version of the manuscript. This work was supported by the EIPHI Graduate School (contract ANR-17-EURE-0002) and the Region Bourgogne-Franche-Comte, France. Data Availability Statement: Data sharing is not applicable. The authors declare no conflict of interest.