# Investigations of the Windage Losses of a High-Speed Shrouded Gear via the Lattice Boltzmann Method
## 1 Introduction
[22] proposed a method based on the gear architecture that involves summing the windage effects on the tooth surface, toe/heel, conical surface, and circumferential surfaces to determine the windage losses of spiral bevel gears. The prediction model of windage losses under jet lubrication was also studied [23]. As a sufficiently versatile and accurate tool, CFD techniques have been widely used. According to research by [PERSON] et al. [24], rather than the air mass flow rate pushed by the teeth, the amount of fluid enclosed within the casing controls the reduction of windage losses with decreasing casing dimensions. [PERSON] et al. [25] showed through the simulation results that the axial baffle mostly affects the power losses brought on by the viscous effect, whereas the radial baffle primarily lowers the pressure losses. In terms of windage loss reduction, radial baffles outperform axial baffles when there is less clearance. When compared to a non-baffle arrangement, windage losses are reduced by 16% when the radial and axial baffles are combined. The windage losses of meshing level gears under various shroud configurations were investigated by [PERSON] et al. [26] and [PERSON] et al. [27]. The findings demonstrated that the gear pair's windage losses were reduced to the lowest when the meshing angle was 30\({}^{\circ}\) and the shroud clearance was 1 mm. Similarly, the windage losses when using jet lubrication on meshing level gears with shrouds were studied by the CFD method [28; 29]. [PERSON] et al. [30] proved through experimental results that a loose shroud on the gears with a very high pitch line velocity still effectively reduces windage power losses. [PERSON] et al. [31] conducted experiments to study the impact of single-side or double-side baffles on the drag losses of helical and spur gears. For spur gears, symmetrical flanges always lower windage losses, regardless of the amplitude of the clearance. Windage losses can be altered by a single axial flange. For helical gears, a flange close to the discharge side reduces windage losses significantly less than one on the intake side. In addition to the common spur gear and spiral bevel gear, herringbone gears with shrouds were also studied [32]. In comparison to conventional computational fluid dynamics (CFD) methods, the Lattice Boltzmann method is characterized by high computational efficiency and the ability to simulate complex geometries. Currently, several researchers have employed the Lattice Boltzmann method to simulate the phenomenon of gear splash lubrication. [PERSON] et al. [33] investigated oil flow in a vertically oriented gearbox. [PERSON] et al. [34] analyzed the internal oil distribution and power losses in an automotive reducer under varying oil levels. [PERSON] et al. [35] simulated fluid flow and heat transfer at various gear speeds in a cone-cylinder two-stage gearbox. Furthermore, [PERSON] et al. [36] and [PERSON] et al. [37] investigated jet lubrication of gears. Compared to the traditional finite volume method, the fast and simple Lattice Boltzmann method was exploited to examine a shrouded spur gear's windage losses in various configurations and verified the outcome with experimental results. Considering the radial and axial clearance parameters, a dimensionless torque containment factor integrating the air compressibility at high Mach numbers was introduced to improve the theoretical formula of windage losses for shrouded spur gear.