## 3 Results and Discussion
### Simulation Results
Meanwhile, gas volume is considerably large when the gas density is small, since the mass of each gas parcel at the initial time is the same, thus increasing the heat exchange area between parcels and solid matrix and enlarging the temperature differences. The large variations of parcels' temperature and pressure yield a large variation of parcel density. According to the Reynolds transport theorem, Lagrangian representation can be converted to Eulerian representation. Figure 9 shows the distribution of parameters inside the regenerator from the perspective of a Eulerian representation. It can be seen from the distribution of gas temperature that the temperature declines from the warm end to the cold end. With the initial value of 285 K,  
Figure 8: Parameter variations of selected gas parcels as a function of position: (**a**) Temperature; (**b**) Pressure; (**c**) Density; (**d**) Velocity. the heat exchange increases, and the gas temperature decreases gradually in the process of calculation. It is noticeable from Figure 9a that the temperature drops dramatically near the cold end during the simulation of the cooling-down processes. This is because the temperature of the gas entering the regenerator at the cold end is set to 80 K during the second half cycle of the oscillating flow. In the early stages, the large temperature difference between gas parcels and solid matrix results in dramatic temperature change. As the oscillating flow continues, the distribution of temperature develops into a linear pattern. As shown in Figure 9b, the distribution of pressure in the regenerator changes periodically over the cycle. During the first half cycle of the oscillating gas flow, pressure declines from the warm end to the cold end, while during the second half cycle, the pressure distribution is reversed. The distribution of density shown in Figure 9c is similar to the temperature distribution on the existence of dramatic changes, which can result from the temperature variations. As shown in Figure 9d, the velocity seems nearly linear, with apparent fluctuations near the cold end. This phenomenon can be explained by the reversals of gas parcels. There are gas parcels with different speeds and directions flowing into both ends of the regenerator during the entire cycle of the oscillating flow, which generates an unstable pressure in the process of parcel velocity reversal, and therefore leads to the presence of velocity fluctuations. Since the parcels from the warm end have higher average velocity, the velocity reversal occurs near the cold end, which can be observed in Figure 9d. Figure 9: Distribution of gas parameters in the regenerator: (**a**) Temperature; (**b**) Pressure; (**c**) Density; (**d**) Velocity. The thermal performance of the regenerator is usually measured by the ineffectiveness, which is as follows [26]:  
\[\lambda\equiv\frac{\left(\int_{0}^{\tau_{\mathrm{w}}}mhtht\right)_{\mathrm{C}}}{ \left(\int_{0}^{\tau_{\mathrm{w}}}mhtht\right)_{\mathrm{W}}+\left(\int_{\tau_{ \mathrm{w}}}^{\tau}mhtht\right)_{\mathrm{C}}}=\frac{H}{H_{\mathrm{W}}-H_{ \mathrm{C}}} \tag{10}\]  
where \(0-\tau_{\mathrm{w}}\) is the time period when fluid flows into the warm end, and \(\tau_{\mathrm{w}}-\tau\) is the time period when fluid flows into the cold end. \(H\) represents the enthalpy loss at the cold end during an entire cycle, and \(H_{\mathrm{w}}-H_{\mathrm{c}}\) represents the maximum heat transfer that can be achieved in the regenerator. The purpose of the regenerator is to deliver as much of this acoustic power to the cold end as possible with minimum loss.