## 3 Results and Discussion
### Simulation Results  
A high frequency regenerator at low temperature, operating at 50 Hz with a cold end temperature of 80 K, and a warm end temperature of 285 K, is simulated. The flow field and heat transfer in the oscillating processes are discussed in detail. With the aid of Lagrangian representation applied in the model, the trajectories of selected gas parcels inside the regenerator can easily be obtained, as well as such time-dependent changes of gas parcel parameters as temperature, pressure, density and velocity. The trajectories of gas parcels inside the regenerator are illustrated in Figure 7. The gas parcels are numbered on the basis of their positions at the initial time, with indices increasing sequentially along the direction from warm- to cold-ends. The five parcels numbered 25, 50, 75, 100 and 125 are selected for tracking purposes. It can be seen from the trajectories of gas parcels that the parcels' backward and forward motion over time shows an oscillatory pattern of movement, and that the parcels close to the warm end would be expelled from the regenerator at the second half cycle. From the figure, it can be observed that the highest and lowest points during the time period of the gas parcels' trajectories decrease gradually, which indicates the gas parcels approaching the warm end in the first few cycles of the simulation. The variations of temperature, pressure, density and velocity with the positions of the five gas parcels tracked in the regenerator are shown in Figure 8. It can be established from the figures that each gas parcel's trajectory consists of three parts of curves, which is consistent with the three cycles of the simulation, indicating that gas parcels move in a reciprocating pattern. The temperature variations shown in Figure 8a indicate that the temperature decreases when gas parcels move from the warm end to the cold end and the temperature rises when moving from the cold end to the warm end. The average temperature of the first half of the cycle is higher than that of the second half, which clearly illustrates the regenerator's cooling mechanism; the parcel's average temperature decreases with the increase of its index that is ordered from the warm end to the cold end. These gas parcels form a series of heat exchange units at different temperatures from 285 to 80 K. These results help to improve our understanding of how regenerators work from the perspective of gas parcels. Figure 7: The trajectories of selected gas parcels in the regenerator. Figure 8b shows the pressure variations of selected gas parcels as a function of position. The parcels' pressure firstly increases and then decreases at a rather slow speed when moving from the warm end to the cold end, while the opposite is true when the process is in reverse, moving from the cold end to the warm end. The average pressure in the first half cycle is higher than that in the second half. Under the influence of temperature and pressure, the parcels' density shown in Figure 8c rises constantly during the motion from the warm end to the cold end and decreases when moving in the opposite direction. The average density of the cooling process is higher than the heating process. The variations of parcel velocity, as shown in Figure 8d, follow a trend similar to the pressure. When combining the results from the four figures, it is obvious that the variations of the parcels' temperature, pressure, density and velocity in different cycles decrease with the increase of parcel index: in other words, gas parcels near the warm end change most dramatically in the first three cycles. This may result from the fact that the parcel's velocity is larger when mass flow is constant with a rather smaller gas density under the higher temperature at the warm end. The larger velocity can result in a larger range of motion within the same time cycle, thus causing larger resistance and pressure differences, and which therefore leads to dramatic velocity variations.