# Investigations of the Windage Losses of a High-Speed Shrouded Gear via the Lattice Boltzmann Method
## 3 Results and Discussion
### Influence of the Shroud
The numerical gear power losses with a large clearance (\(\zeta_{r}\geq 0.5\) and \(\zeta_{a}\geq 1.0\)) are close to the maximum theoretical power losses of an unshrouded gear. In other words, a loose shroud has no obvious containment effect on the gear power losses. Compared to Figure 3, a tight shroud would significantly reduce the gear power losses. It has been reduced by at least 50% of windage losses for \(\zeta_{r}=0.1009\) and \(\zeta_{a}=1.0644\) with respect to the case of \(\zeta_{r}=0.5\) and \(\zeta_{a}=1.0\). 2. Even with a shroud, the windage losses are proportional to the cube of the rotating speed. The dimensionless torque \(C_{T}\) has little change when the rotating speed falls below about 5900 rpm and the Mach number is 0.3. Once the Mach number increases, the dimensionless value will decrease obviously. The most important cause of this is the increasingly apparent compressible flow phenomenon of the air. 3. Windage losses or dimensionless torques are increasing with the increases of the axial clearance of the shroud compared to Figure 4 to Figure 5. The losses are approaching the theoretical maximum values, even exceeding them. This changing trend is reasonable and acceptable considering the deficiencies in both numerical and theoretical methods. 4. As for the same axial clearance \(\zeta_{a}\) of the shroud, the radial clearance changed \(\zeta_{r}\) from 0.5 to 1.5, and the windage losses had no significant changes. It is a similar finding to a 2021 study that claimed that the high-speed gear mostly generates airflow that is between 1.0 and 1.5 times the gear's radius in a radial direction [14]. This scope is governed by the gear teeth. As for the same radius clearance \(\zeta_{r}\), the radial clearance changed \(\zeta_{r}\) from 1.0 to 2.0, and the variation in windage losses is obvious. Figure 4: Numerical findings against the rotating speed or Mach number for \(\zeta_{a}\approx 1.0\): (**a**) windage losses \(P_{\text{u}zi}\); (**b**) dimensionless torque \(C_{T}\). Figure 5: Numerical findings against the rotating speed or Mach number for \(\zeta_{a}\approx 1.5\): (**a**) windage losses \(P_{\text{u}zi}\); (**b**) dimensionless torque \(C_{T}\). Given that these observations are obtained mainly from the cases with larger clearances, smaller clearances are further investigated to discuss the shroud's containment impact. The axial clearance between the spur gear and the tight shroud is set to 3 mm, 6 mm, and 9 mm, namely, \(\zeta_{a}\) is about 0.1, 0.2, and 0.3. Meanwhile, the ratio of radial clearance is set to 0.2, 0.5, and 1.0, respectively. The numerical results including windage power losses and dimensionless torque values are presented in Figures 7-10 for different configurations, and the theoretical dimensionless torque of a spur gear in free configuration is also superimposed. Some findings from the numerical analysis are also concluded, as follows:  
1. It is apparent that the gear's windage losses with a tight shroud (\(\zeta_{r}\leq 1.0\) and \(\zeta_{a}\leq 0.3\)) still rapidly increase with the rotating speed, as shown in Figures 7 and 9. As in the incompressible flow (\(M\leq 0.3\)), the dimensionless torque \(C_{T}\) changes little with the increase in the Mach number. As the Mach number continues to grow (\(M\geq 0.3\)), the torque coefficient \(C_{T}\) declines (see Figures 8 and 10). 2. In contrast to a loose shroud (\(\zeta_{r}\geq 0.5\) and \(\zeta_{a}\geq 1\)), a tight shroud (\(\zeta_{r}\leq 1.0\) and \(\zeta_{a}\leq 0.3\)) obviously restrains the generation of gear windage power losses. Especially for \(\zeta_{r}=0.2\), \(\zeta_{a}\approx 0.3\), the windage losses decrease more than 83.3%. It is the same as the previous conclusion by [PERSON] [38]. In general, the windage losses decrease as the radial or axial clearances reduce. However, in the case of small radial clearance (\(\zeta_{r}=0.2\)), continuous reduction of axial clearance may become counter-productive.