## 1 Introduction  
The use of very low temperatures in advanced technology for applications in medical science, electronic technics, computer technology, aerospace exploration and other fields requires high performance cryogenic systems of low cost, simple design, small vibrations, low noise production, long life and high efficiency [1]. The pulse tube refrigerator (PTR) has been widely used because of the absence of moving parts at cryogenic temperature [2]. As the most important component in a PTR system, the regenerator has the advantages of a large heat transfer area and enhanced fluid mixing [3]. Its oscillating flow characteristics play a critical role in the performance of a PTR, which has attracted extensive attention in both theoretical and experimental studies. [PERSON] et al. reported a series of numerical studies on the performance and optimization of regenerators of a 4 K cryogenic system [4; 5; 6; 7]. They used the commercial software Regen 3.2 in the simulation and design. They obtained the temperature curves along the regenerator which were later demonstrated to be consistent with the experimental results [7]. The comparative study of He-3 and He-4 systems showed that He-3 was more like an ideal gas at low temperature, which indicated that the substitution of He-3 for He-4 helped to improve the performance of regenerator [5]. Using Regen 3.3, the influence of porosity on the losses of the regenerator was analyzed, showing that the regenerator loss decreased with the decrease of porosity [4]. They also investigated the influence of the operating parameters, e.g., average pressure, pressure ratio and warm-end temperature; the regenerator parameters, e.g., material, shape and hydraulic diameter of the solid matrix; and the geometric parameters. Based on the optimization, the ineffectiveness was reduced to 0.36 and the second-law efficiency was enhanced to 25% [6]. [PERSON] et al. [8] investigated the effects of a local thermal non-equilibrium on thermally-developing forced convection in a porous medium in parallel plate channels with the classic Graetz method, and the momentum equation of porous medium was established using the Brinkman model. Their study revealed that the local Nusselt number was closely related to the Peclet number and the thermal conductivity of both solids and gas. [PERSON] et al. [9] used the commercial computational fluid dynamics code (CFD) Fluent to simulate convective heat transfer and thermal dispersion during laminar pulsating flow in porous media, wherein the gas pulsation was realized through the inlet boundary condition of sine waves. Their simulations covered the porosities range from 0.64 to 0.84, the frequency range from 0 to 100 Hz, and the Reynolds number range from 70 to 980, which showed that the heat transfer in the porous medium had a strong dependence on the porosity, Reynolds number and operating frequency. Based on the simulation results, empirical correlations of both the thermal diffusion term and the Nusselt number were proposed, respectively. [PERSON] et al. [10] carried out numerical simulations of laminar forced convection in horizontal pipes partially or completely filled with porous material, and obtained the effects of the geometry (cylindrical and annular) of the porous media and the placement in the pipe on the flow distribution. The influence of the external diameter of porous media and the Darcy number was also investigated. The results showed, that the average Nusselt number of the flow in the pipes with the placement of porous media increased, while the entrance length decreased; there existed a critical value for the external diameter of porous media, and the thermal performance degraded when the external diameter exceeded this value. [PERSON] et al. [11] performed both theoretical and experimental studies to reveal the influence of regenerator void volume on the performance of a precooled 4 K high frequency Stirling type PTR.