# Numerical Study on Single-Bubble Contraction-Rebound Characteristics in Cryogenic Fluids
## 1 Introduction  
For long-distance delivery and large-scale storage of cryogenic fluids, liquid storage and delivery are the most widely used methods in favor of high security and cost effectiveness [1; 2; 3; 4]. Among the cryogenic fluids, liquid hydrogen (normal boil point is 20.37 K) and liquid oxygen (normal boil point is 90.19 K) have been widely used in rocket engines [5; 6]. Taking liquid hydrogen as an example, its delivery cost is only 0.181 EUR/kg [7] and energy consumption is only 0.37 kWh/kg [8], which is much lower than 0.63-0.76 EUR/kg and 2.43 kWh/kg for high-pressure hydrogen delivery. In the transportation of cryogenic fluids, substantial sloshing cannot be avoided easily due to the jounce of transport tankers and LNG carrier ships. This results in a large variation in local pressure for the delivered cryogenic fluids. Under this condition, cavitation bubbles grow when the local pressure drops below the saturated pressure of liquids [9; 10]. When the operating pressure increases suddenly, the bubbles contract and rebound rapidly. During this contraction-rebound process, there is noise, vibration, pressure pulsation and damage to the fluid machinery due to the high pressure and temperature of the bubbles [11]. Single-bubble growth in ordinary fluids has been preliminarily studied. The single bubbles growth in water and freon has been focused on in many experiments [12; 13; 14]. Compared with bubble experiments in ordinary fluids, cavitation experiments in cryogenicfluids are relatively rare. Among cryogenic research, experiments of liquid nitrogen and hydrogen flow around a hydrofoil and an ogive were carried out by [PERSON] in the 1970s [15; 16; 17; 18]. The temperature and pressure distributions along the cavitation region of the hydrofoil surface were measured experimentally. Moreover, [PERSON] et al. [19] experimentally investigated liquid hydrogen cavitation characteristics in various inducers. The experimental results provided a reference for the design of an inducer for cryogenic fluids. Recently, [PERSON] et al. [20] studied the difference in cavitation characteristics between liquid nitrogen and water in an inducer using a visualization experiment. The experimental results indicated that the bubble size in liquid nitrogen was much smaller than that in water. In addition to cavitation flow experiments in fluid machinery, single-bubble growth and collapse in cryogenic fluids are the basis for the study of the cavitation mechanism in cryogenic fluids. However, there is relatively little experimental research in the open literature due to its extremely rapid growth and collapse process. [PERSON] et al. [21] carried out the bubble growth experiment in liquid nitrogen under depressurization and studied the bubble radius growth process. Due to the limitations of harsh low-temperature conditions, the pressure and temperature inside the nitrogen bubble were not investigated in the experiments. In addition to experimental studies, there have also been theoretical and numerical results focused on the heat and mass transfer process at the bubble interface. The isothermal hypothesis is often adopted in cavitation models of ordinary fluids. Transport cavitation models based on the Rayleigh-Plesset equation are used widely. Among these models, the Zwart-Gerber-Belamri model [22], Schnerr-Sauer model [23] and Singhal model [24] are applied in commercial CFD software. The growth and heat and mass transfer between the bubble and liquid are predicted using the single-bubble model [25; 26]. The bubble radius growth results calculated with the model are verified with data from experiments in water and freon. For ordinary fluids, the thermodynamic temperature drop is small and the liquid vapor density ratio is large. Additionally, the transition time of bubble growth from the dynamic growth stage to the thermal growth stage is equivalent to the bubble existence time. Therefore, the isothermal assumption is prone to be suitable for the cavitation in ordinary fluids.