## 6 Conclusions  
This paper focuses on the variable tooth thickness gear as the research object, conducting transient CFD simulations of the flow field near the nozzle under different gear speeds and pitch cone angles. By integrating the VOF multiphase flow technology with the Fluent UDF program, accurate simulations and calculations of the lubricating oil volume fraction on the gear surface were performed. Based on the fitting analysis of injection speed curves under different gear structural parameters, this paper further proposes a comprehensive optimization guide for injection lubrication of variable tooth thickness gears. The main conclusions are as follows:  
1. The simulation analysis results indicate that the variation of oil volume fraction with injection speed exhibits different patterns under the nine combinations of pitch cone angles and rotational speeds. The findings of this study provide a basis for predicting and analyzing the optimal injection speed of variable tooth thickness gears, with significant reference values for practical applications under similar conditions. 2. Gas phase analysis reveals that when the pitch cone angle remains constant, the internal pressure of the gearbox gradually increases with the increase in gear speed. Conversely, when the gear speed remains constant, the pressure in the engagement area decreases significantly with the gradual increase in pitch cone angle. 3. Under the speed conditions of 8000 rpm and 10,000 rpm, the gear oil volume fraction increases gradually with the increase in injection speed, reaching its maximum at an injection speed of 22.5 m/s, though it has not yet reached the optimal injection speed. Under the 6000 rpm speed condition, the oil volume fraction shows a trend of first increasing and then decreasing with injection speed, achieving optimal injection speed at 20 m/s. 4. At various speed conditions, with the increase in gear pitch cone angle, the contact area during gear engagement correspondingly expands, leading to a gradual increase in the overall oil volume fraction of the gear system. The study indicates that the oil volume fraction reaches its optimal value when the pitch cone angle is 4\({}^{\circ}\). To promote the further development of oil injection lubrication systems, future research should focus on several key directions. Firstly, it is necessary to conduct in-depth research on MQL technology and evaluate its applicability and effectiveness in existing methods. Secondly, it is necessary to explore the applicability of the proposed numerical model in bio-based lubricants and lubricants containing nano-additives to meet the needs of new lubricating materials. At the same time, expanding the numerical model to consider the temperature distribution of machine tools and studying thermal and multiphase effects (including ventilation) to improve the accuracy of the model are required. In addition, experimental research will be conducted to evaluate the application of AI and IoT technologies in real-time control of lubrication parameters, achieving intelligent lubrication management. Finally, we must analyze the relationship between transmission errors and oil distribution patterns in the gear contact area to improve lubrication design and enhance mechanical performance. These research directions will provide comprehensive theoretical support and practical guidance for the optimization and innovation of oil injection lubrication systems. Conceptualization, H.Z. ; methodology, Y.L. ; software, Q.C. ; validation, Y.L. ; formal analysis, H.Z. ; investigation, H.Z. ; resources, Y.L. ; data curation, Y.L. ; writing--original draft preparation, H.Z. ; writing--review and editing, Y.L. ; visualization, Q.C. ; supervision, Y.L. ; project administration, H.Z. ; funding acquisition, Y.L. 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 number 52265008; and the Key R&D Program Project of Gansu Province, grant number 24 YFGA021. Not applicable.