## 1 Introduction  
In intermediate and tail gearboxes in rotorcrafts or conventional automotive applications, the gears are partially submerged in lubricating oil, while the non-submerged parts are lubricated by oil droplets and ligaments flung off from the oil bath, known as dip or splash lubrication. The heat generated during the interaction of oil with rotating wheels is called churning loss, a type of no-load power loss independent of loading conditions. Another no-load loss related to air is called windage loss. Additionally, loss occurs due to the compression and expansion of both oil and air in the gear meshes, known as pumping loss. This study does not address pumping loss, as we focused on an isolated wheel, including a disc, spur gear, straight bevel gear, and orthogonal face gear (see Figure 1). Not only experimental and theoretical investigations on the interactions of a nonhomogeneous fluid of air-oil mixture with rotary wheels under splash lubrication, but also computational fluid dynamics (CFD) have been widely used to reproduce the complexity of two-phase flow. Many scientific papers have focused on the drag torque of rotary discs (e.g., [PERSON] [1], [PERSON] and [PERSON] [2], and [PERSON] [3]). Among them, some studies noted that the main difference in no-load power losses between the disc and gear lies in the windage losses of gear teeth [4]. To date, a substantial portion of the scientific literature has concentrated on investigating the churning losses of spur gears. [PERSON] [5] found that the gears churn the lubricant, giving rise to power losses that are related to the amount and properties of the lubricant. [PERSON] et al. [6; 7; 8] and [PERSON] et al. [9; 10] proposed a set of theoretical analytical models for predicting the churning drag torque of an isolated cylindrical gear or gear pair with or without a shroud. [PERSON] et al. [11] investigated the influence of oil aeration on churning losses. Furthermore, the transition state between windage and churning phenomena was experimentally and theoretically explored, respectively [12; 13]. Windage losses can be 25% greater for oil-immersion depths under the gear tip compared to those without oil. Leveraging CFD techniques, some studies ([PERSON] et al. [14; 15; 16]; [PERSON] and [PERSON] et al. [17; 18]; [PERSON] et al. [19]; [PERSON] and [PERSON] [20]; [PERSON] et al. [21]; [PERSON] et al. [22]; [PERSON] et al. [23]) estimated the churning drag torques and reproduced the oil flow pattern characteristics of splash-lubricated gears. The CFD technique applies not only to rotating gears but also to rolling bearings. [PERSON] et al. [24] presented experimental studies and numerical simulations that illustrate the influence of the oil quantity on the drag and churning losses. [PERSON] et al. [25] numerically predicted the aeration that influences lubrication mechanisms in rolling element bearings and compared the results with experimental data acquired on a dedicated test rig exploiting particle image velocimetry (PIV). [PERSON] [26] proposed some models to predict the no-load losses of gearboxes even if their accuracy and range of application are limited. [PERSON] et al. [27] and [PERSON] and [PERSON] [28] proposed a prediction model for gear windage losses, and the total power losses of a gear pair were also obtained, including the windage losses of individual gears as well as pumping losses [29; 30]. [PERSON] et al. [31] and [PERSON] et al. [32] numerically investigated the complexity of the actual airflow patterns aspirated in the axial direction and ejected radially and effectively estimated the gear windage power losses. [PERSON] et al. [33] measured the windage losses of a pseudo-gear pair using a test rig and indicated that the total windage losses were slightly less than the sum of the losses of each gear, similar to the foundation obtained by [PERSON] et al. [34] via CFD simulations. Spiral level gears are usually adopted in gear transmission systems in rotorcrafts. [PERSON] et al. [35] and [PERSON] et al.