# Investigations of the Windage Losses of a High-Speed Shrouded Gear via the Lattice Boltzmann Method
## 1 Introduction  
High-speed gear systems are a crucial component of helicopter transmission, whose maximum input speed inside the main reducer easily outpaces 20,000 rpm, accompanied by significant no-load power losses. To mitigate the stress imposed on the cooling system, it is imperative to minimize these power losses. No-load power losses mainly consist of churning, windage, and pumping losses. Windage losses can make up as much as 50% of the total losses when the linear velocity of gears exceeds 125 m/s [1]. It is indicated that the windage behavior of high-speed gears should not be disregarded, thereby predicting windage losses and then adopting effective measures are essential for suppressing the windage effects. Based on existing research fruits, windage losses are proportional to the cube of the angular velocity [2; 3; 4]. The principal physical process that causes windage losses is the pressure field. Additionally, it is impossible to ignore viscous effects as a supplementary cause of windage losses [5]. Windage losses of various gears have been widely investigated. [PERSON] et al. [6] developed two analytical methods based on dimensional analysis and fluid flow analysis to predict windage losses of a spur gear. The effectiveness of the two methods was proven by the comparison of experimental results. Additionally, [PERSON] et al. [7], based on the work of [PERSON] et al. [8; 9], proposed a theoretical model to estimate windage losses of a spiral bevel gear, which includes power losses at the circumference, toe/heel, and within the tooth space. In addition to the theoretical study of individual gears, meshed gears have also been studied. According to [PERSON] et al. [10], the overall windage losses of meshed spur gears are equal to the product of the losses incurred by pocketing/squeezing at the gear mesh interface and the windage losses of individual gears. [PERSON] et al. [11]first exploited the particle image velocimetry technique to observe the flow field produced by the gear under ambient conditions. [PERSON] et al. [12; 13] experimentally studied the windage losses occurring in the meshed gears with centers far apart and found that the overall windage losses were marginally lower than the total of the individual windage losses. This is probably because the airflow produced by one gear facilitates the rotation of the other. [PERSON] et al. [14] also observed the same behavior through an unstructured overset grid method. Based on the ejected volumetric flow rate, [PERSON] et al. [15] utilized a computational fluid dynamics (CFD) code to estimate windage losses of an isolated spur gear. Additionally, [PERSON] et al. [16] developed the gear tooth model to swiftly examine the flow field close to the gear tooth in order to streamline the simulation model and increase the effectiveness of simulation calculation. [PERSON] et al. [17] applied various meshes combined with the finite volume method and turbulence model to simulate spur gears rotating in pure air. [PERSON] et al. [18] used the CFD method to study the impact of helix angle and face width on windage losses and proposed a modified analytical model. Two strategies have been investigated to reduce windage power losses. The first method focuses on modifications of the gear surface geometry. [PERSON] et al. [19] modified the tooth tip by adding a little chamfer on the leading edge, accompanied by a reduction in windage losses of about 6 percent. [PERSON] et al. [20] highlighted the presence of an ideal ramp value, at which WPL reduction reaches its peak. Modifying the gear teeth could indeed reduce windage losses, but with a decrease in gear tooth strength. Beyond that, implementing a shroud is an alternative strategy to lessen gear windage losses. [PERSON] et al. [21] established a theoretical method based on the idea of the swept volume, although it is only useful for forecasting load-independent losses of closely shrouded meshed gears with the minimum axial and radial clearances. [PERSON] et al.