## 4 Conclusions  
In this study, the Levee Overtopping Risk Curve is proposed using dimensionless variables such as the ratio of superelevation to levee height (\(\frac{\mathrm{A}y}{H}\)), the ratio of water depth to levee height (\(\frac{\mathrm{h}}{H}\)), and the Froude number to predict the risk of levee overtopping more effectively in river curves. The main conclusions are as follows:  
* By examining the interrelationships among dimensionless parameters related to levee overtopping, this study introduces a Levee Overtopping Risk Curve. This approach demonstrates the advantage of scale invariance, indicating that it can be applied to levees of various sizes and under different flow conditions. Furthermore, numerical modeling on two levees of different scales confirmed that the proposed curve can  
Figure 10: Area fraction according to the distribution ratio of water depth to levee height.  
Figure 9: Distribution of water depth to levee height ratios based on simulation results: (**a**) Results of small levee (No. 1); (**b**) Results of big levee (No. 2).  
clearly distinguish between safe and risky states. This finding provides evidence of the predictive reliability and applicability of the dimensionless parameters in designing the curve.
* In addition, by incorporating the flow characteristics of river bends (e.g., superelevation and local hydraulic structures), which have been relatively overlooked in previous studies, this research elucidates how superelevation in bends contributes to increased overtopping risk. This outcome offers fundamental data for preemptively identifying leveee overtopping hazards during the design and maintenance of bend sections.
* Moreover, the proposed Levee Overtopping Risk Curve can serve as a practical hydraulic engineering design guideline, offering efficient and objective decision-making support in levee design and flood risk management processes. In conclusion, this study establishes a methodological framework for consistently predicting leveee overtopping risk under diverse conditions, thereby enhancing the effectiveness of levee stability assessments and flood risk management in river bends.  
Although this study presented the levee overtopping risk curve through numerical analysis, further validation with diverse field data and additional numerical experiments is necessary. Moreover, since levee breaches involve complex mechanisms such as soil characteristics, erosion, and internal seepage, incorporating geotechnical factors into future research could enable more precise risk predictions. Considering climate change scenarios, which are characterized by increasing variability in rainfall and flow rates, could further enhance the effectiveness and scalability of the overtopping risk curve.  
The levee overtopping risk curve proposed in this study serves as a standardized and quantitative tool for assessing levee overtopping risk in river curves. By integrating geotechnical and climate change factors in future research, a more comprehensive and accurate flood risk management system can be established, thereby reducing flood damage and enhancing the efficiency of levee design and maintenance.  
[PERSON] and [PERSON] performed methodology, numerical analysis, writing/editing the paper. S.O.L. contributed to funding, the aim of the paper, research progress. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Korea Environment Industry & Technology Institute (KEITI) through Research and Development on the Technology for Securing the Water Resources Stability in Response to Future Change Program, funded by Korea Ministry of Environment (MOE) (RS-2024-00332877).  
Not applicable.  
The raw data supporting the conclusion of this article will be made available by the corresponding author on request.  
The authors declare no conflicts of interest.