# Investigations of the Windage Losses of a High-Speed Shrouded Gear via the Lattice Boltzmann Method
## 4 Conclusions  
The Lattice Boltzmann method (LBM) was utilized to forecast the windage phenomenon of a spinning gear in order to examine the windage power losses of a shrouded spur gear. The primary goal of this research was to empirically demonstrate a relationship between the dimensionless torque (windage power losses) and the radial and axial clearances from a cylindrical shroud to a spur gear.  
First of all, the CFD model modeling by the Lattice Boltzmann Method was verified by reproducing experiments in the scientific literature, where the windage losses from a rotating spur gear with a tight cylindrical shroud surrounding it were measured. Numerical predictions achieve very good agreement with experimental findings with a relative error not exceeding 6.7%.  
In order to evaluate the containment effect of the shroud on windage losses, the CFD approach was subsequently expanded to simulate the gear inside various cylindrical shrouds. It is well known that the windage effects would be reduced when the gear is enclosed by suppressing the air angular momentum driven by the rotating speed. Additionally, the results indicated that the decrease in windage losses is related to the shroud dimensions, including the radial and axial clearances. The impact of a somewhat loose shroud on windage losses is minimal. In the case of small radial clearance, the continuous reduction of axial clearance may increase windage losses.  
A torque containment factor integrating the air compressibility at high Mach numbers is introduced to represent the reduction in torque (windage power losses) for the shrouded gear compared to the free case. The demonstration of a direct relationship between the shroud clearances and the torque containment factor is studied. Based on the above findings, the theoretical method to predict the windage power losses of a gear in free air has been improved according to this connection relationship that integrates the air compressibility effect. This improved method can now estimate a gear's windage losses in an enclosed cylindrical arrangement, obtain high goodness of fit and satisfactory effect with the numerical results, and be applied to the preliminary design stage.  
Conceptualization, Y.D. and C.Y.; methodology, X.Z.; software, C.Y.; validation, Y.D. and X.Z.; formal analysis, X.Z.; investigation, C.Y.; resources, Y.D.; data curation, C.Y.; writing--original draft preparation, X.Z.; writing--review and editing, C.Y.; visualization, X.Z.; supervision, X.Z. and Y.D.; project administration, Y.D.; funding acquisition, Y.D. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 52305081 and 52475078), the Natural Science Foundation of Changsha City of China (Grant No. kq2208275), and the National Defense Pre-Research Foundation of China (Grant No. KY-1044-2023-0451).  
Not applicable.  
The original contributions presented in this study are included in the article, and further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.