## 1 Introduction  
The high-speed roller bearing is the core of the main shaft supporting the structure of high-speed rotating machines like aero-engine, and its stable and reliable operation is crucial to ensure the normal operation of the entire engine [1]. With the improvement of the thrust-to-weight ratio and economy performance of aero-engines, the main-shaft rotating speed is also increasing, leading to a higher bearing DN (rotating speed (rpm) \(\times\) inner diameter (mm)) value and larger bearing heat generation [2]. The challenges of bearing lubrication and cooling are increasing in the bearing and oil system design. Oil shortages lead to insufficient lubrication, increased mechanical friction, bearing overheating, and even oil ignition [3]. On the contrary, too much oil supply could not only increase the system weight and load but also cause severe fluid viscous friction [4]. Therefore, it is necessary to accurately understand the behavior of oil within the bearing so as to complete the precise design of the lubrication system and main-shaft bearing lubrication scheme and ensure the performance, lifespan, and reliability of the aero-engine. Traditionally, the main-shaft roller bearing in aero-engines commonly uses jet lubrication [5]. However, as the bearing DN value increases, the centrifugal force, together with high-speed rotating rolling elements, makes it difficult for the oil to enter the bearing cavity [6], so under-race lubrication is developed. In under-race lubrication, oil is ejected from several nozzles and carried by oil scoops connecting with the inner ring. With centrifugal force caused by high-speed rotation, oil is supplied into the bearing cavity through radial holes on the inner ring, achieving lubrication and cooling of the bearing. Compared to jet lubrication, under-race lubrication provides better lubrication and cooling effects for high-speed conditions and has been increasingly adopted in modern aero-engine [7]. In high-speed roller bearings, the interaction between the lubricating fluid and the moving elements causes complex oil-gas two-phase flow within the bearing cavity [8]. Many scholars have conducted extensive research on this topic. As early as 1974, [PERSON] et al. [9] experimentally studied the operating characteristics of ball bearings with contact angles of 20\({}^{\circ}\) and 24\({}^{\circ}\) under DN values of 3.0 \(\times\) 10\({}^{6}\). The results showed that with the increase in the total amount of lubricant oil and inner ring cooling flow, the temperatures of both the inner and outer rings of under-race lubrication bearings significantly decreased. [PERSON] et al. [10; 11] conducted an experimental study on the operating characteristics of ball bearings and cylindrical roller bearings with under-race lubrication. They found a good correlation between the predicted power loss values and the experimental test values. [PERSON] [12] indirectly derived an estimation expression for the average oil volume percent inside the ball bearing with jet lubrication based on experimental studies of thermodynamic characteristics. [PERSON] et al. [13] modified [PERSON]'s [12] formula for under-race lubrication conditions, taking into account the rotating speed of oil and changing the inner ring rotating speed to the relative speed between the cage and inner ring. [PERSON] et al. [14] used the CFD method to study the oil-gas two-phase flow in the jet lubrication ball bearing. They found that the oil phase distribution inside the bearing is uneven. The increase in rotating speed would decrease the average oil phase volume fraction, whereas the effect of the oil flow rate is adverse. [PERSON] et al. [15] investigated the influence of the location of oil on the total friction torque of a roller bearing with a vertical rotating axis. [PERSON] et al. [16] found that as the jet velocity increases, the churning torque decreases, while as the bearing speed increases, the churning torque increases. [PERSON] et al.