## 4 Conclusions  
In this paper, the problem of oil-gas two-phase flow inside the under-race lubrication ball bearing is studied using numerical simulation. The effects of factors such as bearing rotating speed, oil flow rate, oil viscosity, and oil density on the oil-gas two-phase distribution and average oil volume percent inside the bearing are investigated. Finally, a numerical analysis-based calculation method for the average oil volume percent inside the ring-race lubrication ball bearing is developed. The following conclusions are drawn:  
1. In the under-race lubrication ball bearing, the oil phase exhibits a periodic distribution along the circumference, with a period that is equal to the number of supply holes. The oil phase is more abundant near the supply holes and decreases in abundance as it moves away from them. In the radial direction, due to the centrifugal force and the outer ring guiding cage, the oil phase is mainly concentrated near the outer ring.
2. With the increase of bearing rotating speed, the oil phase tends to distribute more evenly in both the circumferential and radial directions inside the bearing. As the oil flow rate increases, the oil within the bearing exhibits greater fluctuation in both the circumferential and radial directions. As the oil viscosity increases, the oil distributes more evenly in the circumferential direction, but the radial distribution uniformity decreases. Oil density does not significantly alter the oil distribution characteristics within the bearing.  
Figure 26: Oil fraction results.  
3. With the increase of bearing rotating speed, the oil phase volume fractions near the inner and outer rings of the bearing gradually decrease. The increase in oil flow rate will gradually increase the volume fraction of the oil phase near the inner and outer rings of the bearing, with more oil accumulating around the outer ring compared to the inner ring. The increase of oil viscosity will also increase the accumulation of oil phase near the inner and outer rings of the bearing, but the difference in oil phase between the two regions experiences a process of first decreasing and then increasing. The oil density basically does not change the oil phase distribution near the inner and outer rings. 4. The average oil volume percent in the bearing ring cavity is primarily influenced by three parameters: bearing rotating speed, oil flow rate, and oil viscosity. The change in oil density has a relatively small range and can be neglected when estimating the average oil volume percent.  
Conceptualization, Y.L. ([PERSON]) and Y.L. ([PERSON]); methodology, Y.L. ([PERSON]); software, Y.L. ([PERSON]); validation, Y.L. ([PERSON]), Y.L. ([PERSON]), L.J. and C.L.; formal analysis, Y.L. ([PERSON]); investigation, [PERSON] ([PERSON]); resources, [PERSON]; data curation, [PERSON] ([PERSON]); writing--original draft preparation, [PERSON] ([PERSON]); writing--review and editing, [PERSON] ([PERSON]); visualization, [PERSON] ([PERSON]); supervision, [PERSON] ([PERSON]); project administration, [PERSON] ([PERSON]); funding acquisition, [PERSON] ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
The authors thank the supporting of the National Science and Technology Major Project of China (Grant number: J2019-III-0023-0067) and the Natural Science Foundation of China (Grant number: 52005409).  
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The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.