# Investigations of the Windage Losses of a High-Speed Shrouded Gear via the Lattice Boltzmann Method
## 3 Results and Discussion
### Torque Containment Factor  
Based on the above analysis, the windage power losses brought on by a loose shroud on a high-speed spur gear (e.g., \(\zeta_{r}\geq 0.5\) and \(\zeta_{a}\geq 1\)) are close to the gear power losses in free configurations. When a tight shroud (e.g., \(\zeta_{r}\leq 1.0\) and \(\zeta_{a}\leq 0.3\)) approaches the spur gear, the generated windage losses are significantly suppressed. Even though the windage losses are greatly reduced to 15% of the power losses of the gear in free air, as a tight shroud is fairly close to the gear (e.g., \(\zeta_{r}\approx 0.005\), \(\zeta_{a}\approx 0.045\); \(\zeta_{r}\approx 0.1\), \(\zeta_{a}\approx 0.045\)), the narrow gap between the shroud and the gear is not suitable from the point of gear cooling performance. Therefore, a tighter shroud configuration (e.g., \(\zeta_{r}\leq 1.0\) and \(\zeta_{a}\leq 0.3\)) needs to be further analyzed for the gearbox body design, especially for the prediction of the torque containment factor \(C_{\zeta}\). According to [PERSON] et al. (2014), the torque containment factor \(C_{\zeta}\) is determined by the ratio \(V_{r}\) of the fluids enclosed in the gear in the shroud and the gear, as well as the Reynolds number \(Re\). The torque containment factor \(C_{\zeta}\) is expressed through a fitting process, as follows:  
Figure 10: Dimensionless torque \(C_{T}\) against Mach number \(M\) with different clearances: (**a**) \(\zeta_{r}=0.2\); (**b**) \(\zeta_{r}=0.5\); (**c**) \(\zeta_{r}=1.0\). \[C_{\xi}=\frac{1}{1+kRe^{n_{1}}{V_{r}}^{n_{2}}} \tag{8}\]  
With  
\[Re=\frac{\rho_{air}\omega{R_{p}}^{2}}{\mu} \tag{9}\]  
And  
\[V_{r}=\frac{V_{Fluid}}{V_{Gear}} \tag{10}\]  
The decreased degree of a shrouded gear's windage losses in comparison to a free gear is represented by Equations (8)-(10). Considering the analysis above, the torque containment factor \(C_{\xi}\) should be related to the Reynolds number \(Re\), the axial clearance ratio \(\zeta_{a}\), and the radial clearance ratio \(\zeta_{r}\). We extracted the torque containment factor \(C_{\xi}\) from the results according to Equations (6) and (7), and the change rules of the torque containment factor against rotating speed with different configurations are presented in Figure 11. It is indicated that the curves of the containment factor are nearly straight when the distance between the gear and tight shroud is very small. As is described in [PERSON] et al. (2014), the torque containment factor \(C_{p}\), as the ratio of the windage power losses of a shrouded gear to the losses of a gear in free air, has taken into account the special impact of the air compressibility at high speed. However, air compressibility has been solely considered as in Equations (6) and (7), so this decay factor of torque \(C_{\xi}\) has little to do with the rotating speed. Furthermore, an empirical equation is obtained through a fitting process to explain the relationship between the clearance ratios \(\zeta_{a}\), \(\zeta_{r}\), and the containment factor \(C_{\xi}\) and described by Equations (11) and (12). The positive correlation between CFD data and empirical equations is highlighted by the coefficient of determination (\(R^{2}\)) of 0.9812 and the sum of squared residuals (\(SSE\)) of 0.004008. \[C_{\xi}=p_{0}+p_{1}\zeta_{r}+p_{2}\zeta_{a}+p_{3}{\zeta_{r}}^{2}+p_{4}\zeta_{r }\zeta_{a}+p_{5}\zeta_{a}{\zeta_{a}}^{2} \tag{11}\]  
With  
\[\left\{ \begin{array}{l}p_{0}=-0.1726\\ p_{1}=0.2043\\ p_{2}=3.505\\ p_{3}=-0.3025\\ p_{4}=2.268\\ p_{5}=-9.275\end{array} \right. \tag{12}\]  
Considering the above-developed model of the torque containment factor integrating the free torque coefficient, the theoretical windage power losses can be determined compared with numerical ones, as shown in Figures 12 and 13. The theoretical calculations agree well with the numerical findings, with the relative difference of the windage power losses of no more than 8.76%.