# Numerical Study on Single-Bubble Contraction-Rebound Characteristics in Cryogenic Fluids
## 5 Conclusions and Discussions  
In this paper, based on the existing bubble growth model, a mathematical model suitable for the collapse process of a single bubble was proposed, considering the vapor compression and expansion process inside the bubble. The collapse-rebound process of bubbles in liquid hydrogen was studied. The variation in the period length and the attenuation rates of different parameters with operating pressure and supercooling degree were investigated using statistical methods and comparisons. The following conclusions can be drawn.  
1. The amplitude of physical parameters increased with the supercooling degree and decreased with the operating pressure. The influence of the supercooling degree on the bubble collapse process was larger than that of the operating pressure. However, the supercooling degree and the operating pressure had different effects on the attenuation rates of the physical parameters. The attenuation rate of every parameter was suppressed with higher operating pressure obviously, and slightly decreased with the supercooling degree. For bubbles in various fluids, the oscillation period was shortened under larger operating pressure and supercooling degree.
2. The amplitudes and frequencies of various parameters of bubbles in cryogenic media were higher than those in normal media. Specifically, the amplitude and frequency of every bubble parameter in liquid hydrogen were much larger than that in the other fluids. Moreover, it took the liquid hydrogen bubble the shortest time to reach equilibrium. For different liquids, the magnitude of the attenuation rates of parameters was roughly consistent with that of the standard temperature drop (\(\Delta T\)) and thermal effect parameters (\(\sum(T_{inf})\)). Moreover, all attenuation rates of bubbles increased with the ambient pressure and decreased with the supercooling degree of the fluid.
3. The maximum peak pressure and maximum attenuation rate of the temperature could be roughly predicted using dimensionless numbers. The maximum pressure peak increased monotonically with the dimensionless parameter (Wb/Re) and presented a roughly exponential relationship. Moreover, the maximum attenuation rate of temperature inside the bubble increased with the dimensionless parameter (\(\Delta T\)/\(T_{inf}\)) monotonously and presented a roughly logarithmic relationship.  
Figure 22: Relationship between attenuation rate of temperature inside the bubble and \(\Delta T\)/\(T_{inf}\).  
**Author Contributions:** Data curation, S.Y.; formal analysis, S.Y., M.Q., M.L., Y.L. and W.D.; resources, Q.Z.; supervision, T.L., Y.H.; validation, S.Y.; writing--original draft, S.Y.; writing--review and editing, S.Y. and T.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This project is supported by the National Natural Science Foundation of China (51976150), the China Postdoctoral Science Foundation (2021M692533) and the Youth Innovation Team of Shaanxi Universities.  
**Institutional Review Board Statement:** This paper does not involve humans or animals.  
**Informed Consent Statement:** This paper does not involve humans or animals.  
**Conflicts of Interest:** The authors declare no conflict of interest.