## 2 Description of Experimental Setup
### Instrumentation and Measurement Techniques  
In order to characterize the fluid regenerator interaction from thermal and dynamic perspectives, temperatures, pressures, and velocities were measured as indicated in Figures 3 and 4. Figure 1: Experimental test bench composed of 1: electrical motor, 2: cylinder and crank shaft, 3: pneumatic cylinder, 4 and 5: connecting tubes, 6: hot heat exchanger (HHEX), 7: cold heat exchanger (CHEX), and 8: regenerator under test. Figure 3: Schematic of the alternating flow regenerator test bench. Figure 2: Drawings of different regenerators tested. (**a**) Front view of Str_30%. (**b**) Structure close-up of Str_30%. (**c**) Isometric view of Str_30%. (**d**) Front view of 35% and 40%. (**e**) Isometric view of 35%. (**f**) Isometric view of Str_40%. TR: microthermocouples planned. For temperature measurements, homemade 12.7 \(\upmu\)m diameter K type (Chromel-Alumel) microthermocoules were used (Label 3 on Figure 4). They were placed at both ends of the regenerator, and at five spots that were manufactured on the regenerator (Figure 2). In this paper, we only present the temperature evolution measured at both ends of the regenerator. Thermocoules were characterized under static and dynamic states (Lanzetta and Gavignet [32]). Their accuracy is \(\pm\)0.1 \({}^{\circ}\)C and their cut-off frequency \(Fr_{c}\) is 30 Hz for forced convection. The alternate flows have a frequency range between 0.65 to 5.88 Hz. Therefore, no temperature correction is required as the cut-off frequency is higher than that of the flow \((Fr<Fr_{c})\). Local instantaneous pressures were measured using XTL-140M-5 BARA Kulite sensors (Label 2 in Figure 4). The sensors were calibrated and placed at both ends of the regenerator. Two hot-wire sensors (TSI, Model: 1201) were used for measurements of the instantaneous axial velocity at each end of the regenerator (Label 1 in Figure 4). The hot-wire sensor was calibrated using a TSI IFA-300 system. The velocity signal was digitized using an A/D converter card and the data were processed via Thermal-Pro software. Sensors were placed on each regenerator side in the same section as pressure and temperature sensors. The hot-wire sensor only measures the velocity of the flow, i.e., an absolute velocity value. As the flow oscillates, a sign should be added to that value in order to give information regarding flow direction. Finally, the angular position of the piston \(\theta\) was obtained using a magnetic proximity sensor (FESTO SME-8-K-LED-24), with a response time of less than 5 ms. The proximity sensor was positioned on the cylinder (Label 4 on Figure 4) in such way that it delivered a signal of maximum value for the initial position of the piston \(A\) (Figure 5). Figure 4: Positions of the different sensors 1: hot-wire sensors, 2: pressure sensors, 3: microthermocoules, 4: magnetic proximity sensor. The calculation of the temporal angular position of the piston \(\theta\) was conducted theoretically. It consists of calculating the complete cycle time that corresponds to the two successive positions of port \(A\) and \(B\) (Figure 5). The compression phase \((A\to B)\) corresponds to the first half of the cycle and the expansion phase \((B\to A)\) corresponds to the other half. The formula used to obtain the transient angular position of the piston \(\theta\) is given by:  
\[\theta=360\frac{t(i)-t(max)}{\Delta t}, \tag{5}\]  
where \(t(i)\) is the variation of time during a cycle, \(t(max)\) is the time corresponding to the second position of the maximum dead center of the piston, and \(\Delta t\) is the time between the two successive positions of the maximum dead center of the piston. Since the flow alternates, it is necessary to define a temporal reference in order to describe the parameters' temporal evolutions. Based on sensor position detection, the zero-time reference for a single cycle matches the detection of a piston's lower spatial position. For the study of successive cycles, zero time matches the lower piston position of the first cycle of the series.