## 6 Conclusions  
This study systematically analyzed the composition mechanism and variation characteristics of the hydraulic radial force on the impeller of a centrifugal pump using a data-centric approach based on CFD datasets. The research aimed to provide theoretical guidance for suppressing the hydraulic radial force and improving the operational stability of centrifugal pumps. Below are the key findings and highlights:  
1. Periodic Behavior and Dominant Frequency: The hydraulic radial force exhibited periodic behavior with a dominant frequency corresponding to the blade passing frequency (BPF). This finding aligns with previous studies and confirms the significance of BPF in unsteady flow phenomena.
2. Force Decomposition and Symmetry: The hydraulic radial force on the impeller could be decomposed into the contributions from each blade, with the main parts of the fitting functions of centrally symmetric blades canceling each other out. This left only the constant terms and fitting residuals, which determined the overall force curve. This discovery offers a deeper understanding of the force composition and provides a foundation for targeted blade optimization.
3. Flow Condition Impact: The study demonstrated that the hydraulic radial force varied significantly with flow conditions. At the rated flow rate, the force components showed minimal fluctuations, while deviations from the rated flow rate led to increased fluctuations. This insight is crucial for designing pumps that operate efficiently across a range of flow conditions.
4. Data-Driven Insights: The application of FFT and trigonometric function fitting revealed detailed frequency characteristics and potential variation patterns in the hydraulic radial force. This data-driven approach enhances the ability to predict and mitigate unsteady flow phenomena, offering a robust framework for future research.
5. Optimization Implications: The findings suggest that focusing on the constant terms and fitting residuals in the force components can lead to more effective optimization of blade structures. This targeted approach can significantly help to reduce the hydraulic radial force and improve the operational stability and efficiency of centrifugal pumps.  
The following supporting information can be downloaded at: [[https://doi.org/10.4121/0413e5](https://doi.org/10.4121/0413e5) ef-ee3e-4c2a-9474-943 ec664 dcf8]([https://doi.org/10.4121/0413e5](https://doi.org/10.4121/0413e5) ef-ee3e-4c2a-9474-943 ec664 dcf8).  
Conceptualization, H.Z.; methodology, H.Z.; software, K.L. and J.H.; validation, T.L. and Y.L.; formal analysis, T.L.; investigation, K.L. and Y.L.; resources, H.Z.; data curation, H.Z.; writing--original draft preparation, H.Z.; writing--review and editing, K.L. and J.H.; supervision, Q.Z. and L.J.; project administration, H.Z.; funding acquisition, L.J. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research is supported by the Natural Science Foundation of Hunan Province, China, grant number 2023 JJ50240.  
**Data Availability Statement:** The raw data supporting the conclusions of this article will be made available by the authors on request.  
**Acknowledgments:** The authors would like to acknowledge the constructive reviews given by anonymous reviewers.  
**Conflicts of Interest:** The authors declare no conflict of interest.