# Numerical Study on Single-Bubble Contraction-Rebound Characteristics in Cryogenic Fluids
## 4 Analysis of Periodicity and Attenuation Rate of Bubble Collapse  
From Section 3, all the physical parameters of the bubbles featured oscillation characteristics. The oscillation period and attenuation rate were influenced by the operating pressure and supercooling degree. In this section, the attenuation rate and periodicity of the liquid hydrogen bubble collapse process were studied and compared with the other fluids. Moreover, the relationship between the temperature attenuation rate inside the bubble and dimensionless number was drawn. The period length of the bubble oscillation was defined as the length of time between peak values in terms of radius, radius growth rate, temperature and pressure, respectively. The period of the different parameters was almost same. Therefore, the first period of the bubble radius oscillation was studied and is shown in Figure 15. The statistical results indicated that the bubble oscillation period was the shortest in liquid hydrogen and the longest in water. The period in liquid nitrogen was slightly longer than that in liquid methane. The period length of the bubble oscillation decreased with the operating pressure. Moreover, the increase in the supercooling degree also reduced the bubble oscillation period. In addition, the influence of supercooling degree on period was weaker than that of the operating pressure in the different fluids. Besides the period, the attenuation rate was also an important parameter. The attenuation rate was the dimensionless result of dividing the difference between the amplitude of the current period and that of the next period with the amplitude of the current period. The attenuation rate in the first thirty oscillation periods is shown in Figure 16. The attenuation rate of the bubble radius was suppressed with operating pressure and lessened with supercooling degree. The attenuation rate was almost not affected by operating pressure. The influence of the supercooling degree on the radius attenuation rate was not as great as that of the operating pressure. However, under the condition of smaller supercooling degree, the attenuation rate of the bubble radius almost did not vary with period. With larger supercooling degree, the slope of the radius attenuation rate increased obviously in both the ascending and descending parts. Figure 14: The relationship between the maximum peak pressure and Wb/Re. Furthermore, the attenuation rate of the bubble growth rate (_dR/dt_) was investigated in the first thirty periods. The attenuation rate with different operating pressures and supercooling degrees is shown in Figure 17. Similar to the influence of operating pressure  
Figure 16: Attenuation rate of liquid hydrogen bubble radii under different operating pressures (**a**) and different supercooling degrees (**b**) during collapse. Figure 15: Bubble period of liquid hydrogen (**a**), liquid nitrogen (**b**), liquid methane (**c**) and liquid water (**d**) during collapse. and supercooling degree on the attenuation rate of bubble radius, operating pressure greatly enlarged the attenuation rate _dR/dt_, while it was reduced slightly with higher supercooling degree. The difference was that the higher supercooling degree only raised the slope of the attenuation rate _dR/dt_ but did not change its monotonicity. Different from the attenuation rate mentioned above, the attenuation rate of the temperature inside the bubble (\(T_{v}\)) dropped significantly in the first thirty periods, as shown in Figure 18. Therefore, the period investigated was extended. The attenuation rate of \(T_{v}\) was like an \"S\" curve. Its slope reached a maximum in the middle 20 periods, while its decline was very slow at the beginning and ending. In addition to the \(T_{v}\) attenuation rate, its maximum slope was enlarged with the operating pressure. The influence of the supercooling degree on the maximum value, minimum value and slope of the \(T_{v}\) attenuation rate was slight.