# Investigations of the Windage Losses of a High-Speed Shrouded Gear via the Lattice Boltzmann Method
## 3 Results and Discussion
### Influence of the Shroud  
A spur gear's windage losses can be calculated by multiplying the dimensionless torque \(C_{T}\) by the air density \(\rho_{air}\), the angular speed \(\omega\) to the power of three, and the pitch radius \(R_{p}\) to the power of five, namely, \(P_{wi}=0.5C_{T}\rho_{air}\omega^{3}R_{p}\)\({}^{5}\). When the gear is in free configuration, the formulas of the dimensionless torque \(C_{T\_{free}}\) have been theoretically developed and verified by experiments and CFD simulation by [PERSON] et al. [6]. Considering the laminar \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Group**} & **Global** & **Local Refined** & **Total Number of** & **Wall Clock Time** & **Numerical Windage** \\  & **Resolution (mm)** & **Resolution (mm)** & **Lattice Nodes (-)** & **(min)** & **Losses (W)** \\ \hline
1 & 5 & 2.5 & 639,504 & 310 & 443 \\
2 & 6 & 3 & 370,080 & 196 & 475 \\
3 & 7 & 3.5 & 235,456 & 154 & 425 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical results against different lattice sizes. Figure 3: Comparison of numerical and experimental results for \(\zeta_{r}\approx 0.1009\) and \(\zeta_{a}\approx 1.0644\): (**a**) windage losses \(P_{wi}\)[30]; (**b**) dimensionless torque \(C_{T}\)[30]. or turbulent flows, the dimensionless torque coefficient \(C_{T-free}\) for the unrestricted scenario is given as follows:  
\[ \begin{split} C_{T-free}&=\underbrace{\frac{2n_{L}\pi}{5 -2m_{L}}\frac{1}{\text{Re}^{*n_{L}}}\left(\frac{R^{*}}{R_{p}}\right)^{5}}_{ \text{Laminar zone}}+\underbrace{\frac{2n_{T}\pi}{5-2m_{T}}\left[\frac{1}{ \text{Re}^{n_{T}}}-\frac{1}{\text{Re}^{*n_{T}}}\left(\frac{R^{*}}{R_{p}} \right)^{5}\right]}_{\text{Turbulent zone}}\\ &\quad+\tfrac{z-\epsilon_{a}}{4}\tfrac{b_{x}}{R_{p}}\left[1+ \tfrac{2(1+z_{a})}{z}\right]^{4}(1-\cos\phi)(1+\cos\phi)^{3}\end{split}  \tag{2}\]  
As is known to all, the static pressure and windage losses would be rapidly reduced when the gear is in a shrouded configuration. A torque containment factor \(C_{\xi}\) is introduced to represent the torque-reducing degree with regard to the free case for the shrouded gear. The dimensionless torque coefficient \(C_{T-shroud}\) for the enclosed scenario is given as follows:  
\[C_{T-shroud}=C_{\xi}C_{T-free} \tag{3}\]  
In addition, the compressibility of air cannot be ignored once the Mach number exceeds 0.3. Then, the fluid density will be reduced, which can be calculated as follows:  
\[\rho_{air}^{\prime\prime}=\left(1+\frac{\gamma-1}{2}\bigg{(}\frac{\pi n_{g}R_ {p}}{30V_{S}}\bigg{)}^{2}\right)^{-1/(\gamma-1)}\rho_{air} \tag{4}\]  
where \(\gamma\) represents the gas constant, \(\gamma=1.4\). \(V_{S}\) represents the sound speed, \(V_{S}\) equals 340 m/s at 15 \({}^{\circ}\)C and 1 atm. Put Equation (4) into Equation (3), and then the torque coefficient \(C_{T-shroud}\) for the enclosed case can be expressed as follows:  
\[C_{T-shroud}=C_{\xi}C_{T-free}\Bigg{(}1+\frac{\gamma-1}{2}\bigg{(}\frac{\pi n_ {g}R_{p}}{30V_{S}}\bigg{)}^{2}\Bigg{)}^{-1/(\gamma-1)} \tag{5}\]  
Namely,  
\[C_{T-shroud}=C_{\xi}C_{T-free}^{\prime\prime} \tag{6}\]  
With  
\[C_{T-free}^{\prime\prime}=C_{T-free}\Bigg{(}1+\frac{\gamma-1}{2}\bigg{(}\frac{ \pi n_{g}R_{p}}{30V_{S}}\bigg{)}^{2}\Bigg{)}^{-1/(\gamma-1)} \tag{7}\]  
This paper covers only the cylindrical shroud used for spur gears, and the axial and radial dimensions (or the ratio of radial clearance \(\zeta_{r}=C_{r}/R_{p}\) and axial clearance \(\zeta_{a}=C_{a}/b_{g}\)) are the major concern in examining how it affects windage losses. Depending on the LBM method's numerical configuration, the windage losses of the gear in different shrouded configurations are further explored. The numerical dimensional windage torque \(C_{T-shroud}\) is plotted in Figures 4-6 for different configurations, and the spur gear's theoretical windage losses in its unconstrained configuration calculated based on [PERSON]'s formulas [6] are also superimposed. Some findings obtained from the numerical investigations in Figures 4-6 are concluded as follows:  
1.