## 1 Introduction  
The rapid development and application of precooling technology to supersonic aero-engines demand high requirements of the sealing technology [1, 2]. Gas face seals are potential candidates in such engines, due to their suitability in harsh working conditions, low leakage and high reliability [3, 4, 5, 6]. In the supercritical helium precool loop of a synergetic air-breathing rocket engine, seals are used to separate the low-temperature refrigerant and high-temperature lubricating oil, resulting in operating conditions of large ambient temperature gradients, and challenging their seal reliability [7, 8, 9]. However, research has illustrated that the ambient temperature gradient of seal rings leads to a larger thermal deformation for both the seal rings and the gas film under high-temperature conditions, thereby damaging the sealing performance [10]. Meanwhile, the thermal deformation and sealing performance of supercritical helium face seals under low-temperature conditions have yet to be explored, in particularly the large ambient temperature gradient conditions. Generally, the temperature gradient generated by the friction, heat convection, and dissipation in seal rings significantly affects the thermal deflections of face seals [11, 12, 13, 14]. Considering the thermal effect, [PERSON] et al. [15, 16] established an analytical model to investigate the thermo-hydrodynamic behaviors of a face seal under high-pressure conditions, and they found that an obvious temperature gradient existed, and a converging gap was formed correspondingly to this. More extensive research into the thermo-hydrodynamic characteristics of gas face seals was then carried out [17; 18; 19; 20]. Focusing on the shape and parameters of microgrooves, [PERSON] et al. [21], [PERSON] et al. [22], and [PERSON] et al. [23], respectively, analyzed the sealing behaviors of gas face seals with different dynamic pressure grooves, such as spiral grooves and inclined-ellipse dimples. Results showed that the temperature gradient of gas film between the inlet and outlet of the seals ranged from 10 K to 50 K, resulting in a divergent deformation. In addition to the dynamic pressure grooves, the operating condition is another important factor affecting the thermo-hydrodynamic characteristics of gas face seals [24; 25; 26]. Under high temperatures and high rotational speed conditions, [PERSON] et al. [27] numerically analyzed the thermal hydrodynamic behaviors of a supercritical CO\({}_{2}\) T-grooved face seal, and found that the maximal temperature difference of the gas film could reach 140 K; a convergent sealing gap was formed simultaneously when the sealing temperature was 600 K and the linear rotational velocity exceeded 400 mm/s. Furthermore, with an ambient temperature gradient exerted on the gas face seal, [PERSON] et al. [10] explored the thermal distortion and sealing performance of a N\({}_{2}\) face seal compared with the condition without an ambient temperature gradient. It was found that the extra temperature gradient contributed to generating a sharp divergent deformation of the face seal, decreasing its load capacity, and increasing its leakage. The research on thermo-hydrodynamic behaviors of face seals under low-temperature conditions has mainly focused on liquid oxygen and liquid nitrogen, which is seriously inadequate for the supercritical helium face seal [28; 29; 30]. Helium is an inert refrigerant, and widely selected in precooled circulation systems, owing to its advantages of high specific heat, low cyclic-pressure ratio, and safety [1; 31; 32]. However, the thermophysical properties of helium in its supercritical state change significantly with respect to temperature and pressure, especially its viscosity [33; 34]. Taking the real gas effect into account, [PERSON] et al. [35] numerically explored the thermo-hydrodynamic characteristics of a T-grooved face seal operating with supercritical helium under low-temperature and high-pressure conditions.