## 1 Introduction  
Hydrogen is regarded as a source of green energy, because it is non-toxic, pollution-free, and makes zero contribution to the greenhouse effect. At present, the utilization of hydrogen energy mainly consists of the following aspects: hydrogen-powered vehicles, hydrogen power generation, molten carbonate fuels, and solid oxide batteries [1; 2; 3; 4]. Related technologies are drawing increasing attention [1]. For the storage and large-scale utilization of hydrogen energy, liquid-state hydrogen is indispensable, compared to gaseous and solid storage states, due to its high storage efficiency and high energy density [5; 6; 7]. For hydrogen liquefaction, the Brayton cycle and the Claude cycle are often used. The optimized design of the liquefaction cycle consists of pre-cooling, throttling, and circulation processes [8; 9]. The hydrogen turbo-expander is the main piece of refrigeration equipment employed in the process of hydrogen liquefaction. The cycle efficiency and yield of the hydrogen liquefaction process are greatly affected by the performance of the hydrogen turbo-expander [10]. As a key component, the performance of high-speed gas bearings is significant factor influencing the reliability of turbo-expander [11]. Depending on their working mechanism, gas bearings can be categorized into aerostatic bearings and aerodynamic bearings. With aerostatic bearings, external air is used to provide the required bearing load capacity. For aerodynamic bearings, there is no external gas supply. Their load capacity is generated by the viscosity of the gas and the high-speed shear effect in the wedge-shaped clearance. Generally speaking, aerodynamic bearings have good adaptability. Compared with other aerodynamic journal bearings with rigid surfaces, herringbone grooved journal bearingshave high stability and adaptability due to the pumping effect of the grooves on gas [12]. Currently, herringbone grooved journal bearings have important applications in lightweight rotating machinery and high-precision equipment, and are widely used in gyroscopes, automotive turbochargers, turbo-compressors for refrigeration, etc. [13]. With respect to aerodynamic bearings, their performance with respect to aspects such as static load and dynamic stability are the most important research topics [14; 15]. [PERSON] et al. found that friction torque, stiffness and punching performance were deeply affected by the parameters of herringbone grooved gas bearings. The bearing load capacity can be improved by increasing the groove angle, decreasing the width-to-ridge ratio, or increasing the groove circumferential ratio [16]. [PERSON] et al. explored the influence of geometric parameters on the performance parameters of aerodynamic bearings based on the CFD model. A new groove shape was proposed that was able to effectively improve the static and dynamic performance of the bearing [17]. [PERSON] et al. analyzed the influence of ambient pressure on the dynamic characteristics of gas foil bearings. It was found that the direct stiffness \(K_{xx}\) and \(K_{yy}\) and the direct damping \(D_{xx}\) and \(D_{yy}\) of the gas film increased with increasing ambient pressure [18]. [PERSON] et al. reviewed the challenges presented by gas bearings operating at low temperatures. On the basis of experiments and analysis, several solutions were proposed to overcome these difficulties [19]. In recent years, more practical models and applications have been widely studied, including the real gas lubrication model, the rarefaction or turbulence model, and other physical models [13]. At present, most of the research on turbo-expanders is carried out with air as the working medium [20]. Conversely, there have been few studies on hydrogen turbo-expanders [21]. In the design process of cryogenic turbo-expanders, it is necessary to consider the influence of different lubricating media on the performance of gas-lubricated journal bearings. [PERSON] et al. performed numerical calculations for bearings with different gaseous lubricants.