Numerical Approach for Predicting Levee Overtopping in River Curves Through Dimensionless Parameters

Abstract

Recent climate changes have led to an increase in flood intensity, often resulting in frequent levee overtopping, which causes significant human and property damage. High vulnerability to such breaches is expected in general, especially at river curves. This study aims to predict the occurrence of levee overtopping at these critical points and to suggest a curve, the levee overtopping risk curve, to assess overtopping probabilities. For this purpose, several dimensionless parameters, such as superelevation relative to levee height ($∆$y/H) and the channel’s Froude number, were examined. Based on dimensional analysis, a relationship was developed, and the levee overtopping curve was finally proposed. The accuracy of this curve was validated through numerical analysis using a selected levee case, which clearly distinguished between safe and risky conditions for levee overtopping. The curve is designed for immediate integration into the hydraulic design processes, providing engineers with a reliable method for optimizing levee design to mitigate overtopping risks. It also serves as a critical decision-making tool in flood risk management, particularly for urban planning and infrastructure development in areas prone to flooding.

1. Introduction

Levees are major civil engineering structures built along rivers to protect lives and property from flood damage. Primarily constructed from soil and other fill materials, levees play a key role in safeguarding adjacent areas. Their installation has led to a substantial increase in population and assets in nearby regions. However, if the flood defense fails, these areas become more exposed to flood impacts, resulting in even greater damage.

With the recent increase in both the scale and frequency of rainfall due to climate change, levee safety has become a critical concern, and actual levee failures have resulted in severe flood damage. In July 2018, the collapse of reservoirs in Hiroshima, Fukuoka, Hyogo, and Okayama in Japan caused Shinkansen train services to be suspended and roads to be cut off. In October 2019, Typhoon Hagibis led to the breach of a levee along the Chikuma River in Nagano, Japan, resulting in widespread flooding and evacuation orders. In April 2024, Australia’s Warragamba Dam, which supplies water to Sydney, exceeded its storage capacity and overflowed, while the levee along the Cooks River in southwest Sydney collapsed, forcing road closures. In June 2024, a levee near Fischach in southern Germany failed, resulting in flooding that forced residents to evacuate the town. Similarly, in July of the same year, the Dongting Lake levee in China breached over a 100-m stretch, inundating residential areas. These accidents highlight the critical role of levee design and management in mitigating disaster-induced losses.

Understanding the mechanism of levee failure requires a grasp of failure characteristics that depend on both the properties of the river and the levee. Variations in flow conditions and levee characteristics can lead to different failure patterns each time, making it challenging to identify the unique features of levee collapses precisely. To address this issue, diverse research efforts have analyzed and compared levee failure characteristics through past case studies, numerical analyses, and hydraulic experiments.

Cristofano [1] conducted an early empirical investigation into levee breach mechanisms by assuming a fixed breach width and analyzing the relationship between flow discharge and erosion. Subsequent studies by Lee and Han [2] employed numerical simulations to investigate the interaction between main channel and floodplain flows during overtopping, emphasizing the importance of breach width in determining outflow. Other researchers, including Han et al. [3], Kim et al. [4], and Yoon et al. [5], extended the analysis to probabilistic risk assessment, geotechnical stability, and hydraulic classifications of levee failures. Further contributions by Kim et al. [6,7], Yang et al. [8], Chaudhry [9], Kim and Kwak [10] and Lee et al. [11] explored breach development, discharge variability, and reinforcement effects through numerical and physical modeling.

While these studies have significantly advanced the understanding of levee breach processes, they primarily focus on post-overtopping failure mechanisms. However, in practice, accurately predicting the onset of overtopping is critical for timely risk mitigation. Azhar et al. [12] developed a regional-scale, data-driven model to estimate levee failure probability due to overtopping, incorporating factors such as overtopping depth, duration, and erosion resistance. Their approach highlighted the need for large-scale flood risk assessments beyond site-specific analyses, particularly under climate change scenarios. Pang and Onda [13] applied a 3D numerical model with a boundary-fitted grid to simulate lateral overtopping in a curved channel. Their model was validated through comparisons with experimental data, confirming its reliability in capturing water level and velocity distributions near a side weir. Taylor-Burns et al. [14] demonstrated that horizontal levees, as a nature-based adaptation strategy, can reduce overtopping risk by up to 30% while providing ecological benefits, particularly in scenarios of sea level rise in estuarine environments.

In contrast, studies related to river bends include that of Furbish [15], who measured velocities in a bend of the Beaton River in Canada and observed increases in velocity with bend length and curvature. Xu et al. [16] examined how channel aspect ratio, sinuosity, and curvature ratio affect flow, showing that as the aspect ratio grows, velocity variations along the main flow direction decrease markedly, whereas lateral velocity patterns remain less affected. They also highlighted the significant role of the curvature ratio in redistributing the primary flow. Moreover, Boghdady et al. [17] investigated the influence of bend geometry on flow patterns and morphological changes, developing a new equation to predict bend scour based on secondary flow analysis and the depth of erosion.

Despite these diverse studies, most remain focused on prismatic straight channel flow. In reality, rivers feature not only straight reaches but also bends, where three-dimensional (3D) flow characteristics can profoundly alter levee failure mechanisms. Because factors such as river curvature have not yet been fully incorporated into levee stability assessments, research findings obtained from straight channels cannot fully capture the complexities of actual levee breaches in curved rivers. On the other hand, work specifically targeting river bends often focuses on internal flow or sediment processes rather than extensively addressing levee failures.

Consequently, although various approaches to levee breach mechanisms have been explored, studies that address the overtopping characteristics of levees in river bends remain limited. Moreover, many investigations focus on specific levee conditions or narrow scenarios, thereby limiting the applicability of their results to broader design or operational contexts. Therefore, it is essential to elucidate overtopping processes particular to river bends and establish an analytical framework that applies to a range of scales and conditions. Such research will ultimately facilitate a comprehensive understanding of levee behavior in meandering rivers and contribute to more effective design and risk management of levees.

Most levees are constructed using natural materials, such as soil, which significantly affects their structural stability. Soil-based levees can fail if water seeps in or overtopping occurs, especially under flood conditions, where levees sustain severe damage and may easily collapse. These considerations are critical in levee design and maintenance. Figure 1 shows the relative frequencies of different levee failure modes, categorized into overtopping, piping, and slope instability. As shown in Figure 1, among these, overtopping accounts for about 70% of failures. Overtopping mainly occurs during floods and is often observed in curved river sections where superelevation can arise [18].

For effective and economic structural measures, accurate estimation of levee overtopping in river curves plays a pivotal role in mitigating potential flood damage. In general, measures to minimize flood damage from overtopping can be categorized into two main approaches: structural and non-structural. A primary structural measure involves raising the levee crest height to prevent overtopping, whereas non-structural measures rely on legal frameworks or early warning systems rather than physical modifications. Although uncertainties associated with climate change continue to rise, structural projects such as constructing or upgrading dams and other river facilities demand extensive timelines and substantial budgets, making flexible responses to climate change challenging [23]. In addition to external drivers such as climate change, uncertainties in model input parameters—such as inflow conditions, levee material properties, and breach thresholds—can significantly impact the accuracy of overtopping predictions. Recent studies have emphasized the importance of accounting for such uncertainties to enhance model robustness and support practical flood risk management [24,25]. Accordingly, this study aims to predict levee overtopping risk in river curves by examining correlations among dimensionless parameters related to levees. Since this dimensionless framework is applicable regardless of levee scale, it is expected to be highly valuable for designing and assessing levees of various sizes.

2. Theoretical Background

2.1. Superelevation

Curvature is a common feature in natural channels, and the flow in curved reaches is generally non-uniform and more complex to analyze than in straight channels, primarily due to the presence of acceleration along the bend [26]. Consequently, understanding and controlling flow behavior in and around channel bends often becomes necessary. In highly sinuous rivers, as water levels rise, the flow rotation in bends can become helicoidal. This secondary flow, superimposed on the downstream main flow, destabilizes the channel and creates a surface elevation difference between the inner and outer banks, known as superelevation, causing the water level on the outer bank to rise. Such increased water levels along the outer bank of a bend can lead to levee overtopping damage and additional erosion caused by secondary currents.

In curved channels, centrifugal forces induce a rise in the water surface and produce helical secondary flows. As shown in Figure 2, this leads to flow concentration either toward the inner or outer side. The water level on the outer bank tends to rise, while that on the inner bank falls, generating a cross-sectional water surface slope. By assuming concentric streamlines and a hydrostatic pressure distribution in a curved channel, and then applying the Euler equation in the transverse direction, the water surface slope can be expressed as a function of flow velocity and radius of curvature, as given in Equation (1).

$${\displaystyle \frac{dh}{dr}}={\displaystyle \frac{V^2}{gr}}$$

(1)

where, $h$ represents the water level, $r$ denotes the radius of curvature, $V$ is the mean velocity at water depth, and $g$ is the gravitational acceleration. If the velocity and water surface slope are constant in a rectangular meandering channel, the superelevation in the bend is governed by the ratio of velocity head to channel width and radius of curvature and is given by Equation (2).

$$∆h={\displaystyle \frac{B{V}^2}{g{r}_c}}$$

(2)

where $∆h$ represents the water level difference between the inner and outer sides of the bend, $B$ denotes the channel width, $V$ is the mean velocity, and $r_c$ is the radius of curvature at the channel center.

2.2. Numerical Model

In this study, the three-dimensional numerical model FLOW-3D, developed by Flow Science, Inc. in Los Alamos, New Mexico, was employed [28]. It is a general-purpose computational fluid dynamics (CFD) program that supports a variety of functions. In addition to turbulence, shallow water, and solidification analyses, it is extensively used in the water resources field as well as in general industrial processes such as casting and inkjet applications.

For turbulence modeling, a range of models, including the mixing-length model, turbulence energy model, RNG (Re-Normalization Group) model, and LES (Large Eddy Simulation) model, can be applied Among these, the RNG model is known for its longer computation time but provides higher accuracy for complex turbulent flows [29]. Hence, the RNG model was selected for this study.

The RNG model offers robust methods for simulating complex free-surface flows, most notably its volume-of-fluid (VOF) approach and FAVOR (Fractional Area–Volume Obstacle Representation) technique, which allow for accurate representation of solid boundaries. These features enable detailed analysis of three-dimensional flow processes, including overtopping and levee breach dynamics, at relatively high precision. However, because the model relies on rectangular grids, capturing curvilinear boundaries or intricate terrain remains challenging, often requiring finer mesh resolutions or additional modeling steps. To overcome the limitations associated with the rectangular grid system, the mesh resolution was adjusted to allow for a more accurate representation of complex geometries. In particular, the vertical (z) direction of the computational domain was discretized into 15 layers, enabling a finer approximation of terrain elevation changes and curved surfaces. This improvement in resolution contributed to a more accurate reproduction of the levee slope geometry.

The governing equations in this model are the continuity equation and momentum equations, which are given by Equations (3)–(6) [28].

• Continuity equation

  $${\displaystyle \frac{\partial }{\partial x}}\left(u{A}_x\right)+{\displaystyle \frac{\partial }{\partial y}}\left(v{A}_y\right)+{\displaystyle \frac{\partial }{\partial z}}\left(w{A}_z\right)={\displaystyle \frac{RSOR}{\rho }}$$

  (3)

  where $x, y, z$ represent the velocity components in the $u,v,w$ dimensions, respectively; $A_x,A_y,A_z$ denote the area fraction for fluid flow; $\rho$ is the density; and $RSOR$ represents the mass source/sink term.
• Momentum equations

  $${\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{1}{V_f}}\left\{u{A}_x{\displaystyle \frac{\partial u}{\partial x}}+v{A}_y{\displaystyle \frac{\partial u}{\partial y}}+w{A}_z{\displaystyle \frac{\partial u}{\partial z}}\right\}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x}}+F_x-{\displaystyle \frac{RSOR}{\rho V_f}}u$$

  (4)

  $${\displaystyle \frac{\partial v}{\partial t}}+{\displaystyle \frac{1}{V_f}}\left\{u{A}_x{\displaystyle \frac{\partial v}{\partial x}}+v{A}_y{\displaystyle \frac{\partial v}{\partial y}}+w{A}_z{\displaystyle \frac{\partial v}{\partial z}}\right\}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial y}}+F_y-{\displaystyle \frac{RSOR}{\rho V_f}}v$$

  (5)

  $${\displaystyle \frac{\partial w}{\partial t}}+{\displaystyle \frac{1}{V_f}}\left\{u{A}_x{\displaystyle \frac{\partial w}{\partial x}}+v{A}_y{\displaystyle \frac{\partial w}{\partial y}}+w{A}_z{\displaystyle \frac{\partial w}{\partial z}}\right\}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial z}}+F_z-{\displaystyle \frac{RSOR}{\rho V_f}}w$$

  (6)

  where $A_x,A_y,A_z$ represent the area fraction for fluid flow; $V_f$ stands for the volume fraction; $\rho$ is the density; $RSOR$ represents the mass source/sink term; and $F_x,F_y,F_z$ represents the total force, including gravitational and inertial forces, in each respective direction.

While the theoretical formulations and modeling techniques outlined above are grounded in established hydraulic principles, the present study advances their application by integrating them into a unified dimensionless framework for levee overtopping risk assessment. Unlike previous studies that have treated superelevation and flow behavior in isolation or within specific case studies, this work introduces a generalized risk curve validated through 3D numerical simulation. This integration not only enhances the understanding of flow behavior in curved channels but also provides a scalable, predictive tool that can support practical levee design across various geometries and conditions.

3. Dimensional Analysis

3.1. Dimensionless Parameters

In this study, the key parameters responsible for levee overtopping are reviewed, and standardized criteria for levee design and stability assessment are proposed based on these parameters. To achieve this, dimensional analysis is applied to examine levees of various scales under a unified framework. By defining levee characteristics as dimensionless variables, dimensional analysis enables consistent evaluations across diverse river conditions and levee design scenarios.

The derivation of dimensionless variables offers a tool for quantitatively assessing the likelihood of levee overtopping and comparing the performance of levees with different sizes and design conditions. This enables the establishment of consistent guidelines in levee design and management, as well as more accurate predictions of overtopping risks under actual flood conditions.

The need for dimensional analysis in the context of levee overtopping underscores the importance of systematic approaches in levee design and stability assessment. Through dimensional analysis, this study generalizes levee properties and incorporates complex flow conditions in river curves to predict the risk of overtopping. This contributes to reducing flood damage and enhancing the efficiency of levee design and maintenance.

Parameters affecting levee overtopping include channel width, velocity, curvature, levee height, water surface elevation, and superelevation. In particular, in river curves, velocity increases and superelevation occurs, resulting in a higher probability of levee overtopping. Equations (7) and (8) are the superelevation estimation formulas proposed by the Arizona Department of Water Resources (ADWR) in 1985 [30]. Accordingly, in 2009, the Korea Water Resources Association (KWRA) adopted an intermediate superelevation coefficient of 0.75 (within the allowable range of 0.5 to 1.0) from Equation (7). It used this to present a revised superelevation formula, as shown in Equation (9) [31].

$$∆y=C{\displaystyle \frac{B{V}^2}{g{r}_c}}; (for {\displaystyle \frac{B}{r_c}}\ge 0.33)$$

(7)

$$∆y=C{\displaystyle \frac{B{V}^2}{g{r}_c}}+0.25{\displaystyle \frac{V^2}{2g}}; (for {\displaystyle \frac{B}{r_c}}<0.33)$$

(8)

$$∆y=0.75{\displaystyle \frac{B{V}^2}{g{r}_c}}$$

(9)

where $∆y$ is the superelevation, $B$ is the channel width, $V$ is the channel velocity, $r_c$ is the radius of curvature at the channel center, $g$ is the gravitational acceleration, and $C$ is the superelevation coefficient, which typically ranges from 0.5 to 1.0.

Accordingly, a functional relationship for levee overtopping is defined as shown in Equation (10), and dimensionless parameters relevant to overtopping are identified.

$$P_o=f(∆y, V, g, H, h, \alpha )$$

(10)

where $P_o$ represents the probability of levee overtopping, $∆y$ is the superelevation, $V$ is the velocity, $g$ is the gravitational acceleration, $H$ is the levee height, $h$ is the water depth, and $\alpha$ is the curvature.

Using five parameters, a functional relationship between the probability of levee overtopping $\left(P_o\right)$ and the dimensionless variables was derived as shown in Equation (11). The first variable is the ratio of superelevation $(∆y)$ to levee height $\left(H\right)$, which indicates the risk of levee overtopping in river curves. The second variable is the ratio of water depth (h) to levee height (H), allowing for an intuitive assessment of overtopping risk based on river water depth. The final dimensionless variable is the Froude number $\left(Fr\right)$, which is widely used to examine hydraulic phenomena by considering the velocity $\left(V\right)$ and the influence of gravity $\left(g\right)$.

$$P_o=f\left({\displaystyle \frac{∆y}{H}},{\displaystyle \frac{h}{H}},Fr\right)$$

(11)

Dimensionless analysis is a crucial method that enables the consistent definition of their characteristics regardless of their size. This approach is especially beneficial for comparing and evaluating the performance of levees of different scales, and it can enhance the understanding of various levee designs while contributing to the integration of design standards.

• Ratio of superelevation to levee height, ${\displaystyle \frac{∆y}{H}}$

Using the ratio of superelevation to levee height is one of the key indicators for assessing the risk of levee overtopping in river curves. Superelevation refers to the phenomenon where water levels rise on the outer side and decrease on the inner side of a river curve due to centrifugal forces. The magnitude of superelevation is determined by the radius of curvature and the velocity of the river.

The ratio of superelevation to levee height indicates how much higher the superelevation is compared to the levee height, and it varies with the radius of curvature and flow velocity. In river curves with larger radii of curvature and higher velocities, superelevation becomes more pronounced, thereby increasing the likelihood of overtopping. This ratio plays a crucial role in quantitatively evaluating the influence of river curves in levee design and in predicting overtopping risks in advance.

• Ratio of water depth to levee height, ${\displaystyle \frac{h}{H}}$

The ratio of water depth to levee height is another dimensionless variable that can be used to assess the likelihood of levee overtopping intuitively. Water depth reflects the flow characteristics of the river, and the higher the water depth relative to levee height, the greater the probability of overtopping. This ratio is helpful in levee design for easily comparing overtopping risks based on water depth conditions and for determining appropriate levee heights. Particularly in river curves, variations in water depth can strongly influence overtopping, as the curved channel geometry often leads to significant velocity gradients and uneven flow distribution. Consequently, analyzing the ratio of water depth to levee height provides a more systematic approach to evaluating overtopping stability in these areas.

• Froude number, $Fr$

The Froude number is a representative dimensionless variable that characterizes the interaction between velocity and gravity, and it is widely used in fluid dynamics to evaluate flow characteristics. The Froude number is expressed by Equation (12), which compares the inertial forces to gravitational forces to determine whether the flow is subcritical, critical, or supercritical.

$$Fr={\displaystyle \frac{V}{\sqrt{gh}}}$$

(12)

where $V$ is the velocity, $g$ is the gravitational acceleration, and $h$ is the water depth.

A high Froude number indicates a relatively fast velocity, and inertial forces dominate over gravity. In river curves, a higher Froude number tends to cause the flow to shift outward, which can accelerate levee overtopping and erosion. Therefore, the Froude number plays a crucial role in quantitatively assessing the flow characteristics of rivers and the risk of levee overtopping, as well as in establishing design criteria for these applications.

3.2. Levee Overtopping Risk Curve

The levee overtopping risk curve utilizes the Froude number, which is the ratio of superelevation to levee height, and the water depth to levee height as its primary variables. This study examines the relationships between these dimensionless variables to predict the risk of levee overtopping. The Froude number is shown on the x-axis, while the ratio of superelevation to levee height is on the y-axis. Overtopping is considered likely when the sum of the ratio of superelevation to levee height and the water depth to levee height exceeds 1.

Figure 3 illustrates the levee overtopping risk curve, showing that as the ratio of superelevation to levee height or the Froude number increases, the risk of levee overtopping also rises. For example, suppose the ratio of superelevation to levee height is 0.3 and the ratio of water depth to levee height is 0.6. In that case, the sum of these two variables is 0.9, indicating a low risk of overtopping. However, if the ratio of water depth to levee height increases to 0.8, the sum becomes 1.1, suggesting a high risk of overtopping.

The overtopping risk curve presented in this study serves as a standardized tool for predicting overtopping conditions across levees of various sizes and characteristics. This curve enables the quantitative assessment of the likelihood of overtopping under specific flow conditions during levee design. By providing a consistent basis for comparing different levee designs and operational conditions, the curve enhances the efficiency of the design process. By utilizing this overtopping risk curve, the risk of levee overtopping can be consistently assessed regardless of the levee’s scale. This tool is instrumental in preemptively identifying conditions that may lead to overtopping, thereby helping to prevent potential damage. Through this curve, the safety of each levee can be quantitatively evaluated, and effective response strategies can be developed to minimize damage.

3.3. Application of Numerical Analysis

To effectively utilize the levee overtopping risk curve, it is essential to validate its accuracy and general applicability. For this purpose, the FLOW-3D model was employed to validate superelevation, and the scale effects on levees were assessed to ensure that the model accurately reproduces levee behavior across various scales. By simulating levees of different sizes, the influence of scale on flow dynamics and overtopping risk was identified and quantified. This validation phase plays a crucial role in establishing the reliability and applicability of the levee overtopping risk curve for predicting levee performance under diverse conditions.

3.3.1. Configurations of the Numerical Model

To ensure that the flow simulation of the model is accurately reproduced, a hypothetical levee with a 90-degree curvature, as shown in Figure 4, was constructed. This setup allowed for the simulation of flow in a curved levee and the reproduction of superelevation phenomena. Figure 5 presents the simulation results in the curved section.

The boundary conditions applied in the three-dimensional numerical model are summarized in

Table 1. Boundary condition of the model.

Location	Boundary Condition	Location	Boundary Condition
X Min	Wall	X Max	Outflow
Y Min	Flowrate	Y Max	Wall
Z Min	Wall	Z Max	Pressure (fraction = 0)

. A wall condition was assigned to the bottom of the domain (Z min) and the left bank boundary (X min and Y max). The top boundary (Z max), which is in contact with the atmosphere, was set as a pressure boundary with no active flow, and the flow fraction was specified as zero. The upstream boundary at the levee inlet (Y min) was defined using a volume flow rate condition. The downstream boundary (X max) was specified as outflow.

3.3.2. Validation of Superelevation in the Numerical Model

In this study, superelevation occurring in the curved section was simulated using the three-dimensional model, and the theoretical equations based on Equations (7) and (9) were compared with the results. Figure 6 shows the water surface elevation distribution along the curved section of the channel under overtopping conditions. The figure highlights the superelevation effect, where the outer bank exhibits a higher water level due to the centrifugal force generated in the curved flow. This pattern is consistent with theoretical expectations and previous experimental observations, supporting the accuracy of the numerical simulation in reproducing curved flow dynamics. Notably, the gradient across the cross section visually confirms the presence of secondary flow effects, which contribute to the asymmetric water surface profile.

Figure 7 displays a graph comparing the theoretical results with the simulation outcomes. The simulation results from FLOW-3D fell within the superelevation coefficient range of 0.5 to 1.0 as presented by the ADWR [30]. Additionally, when compared with the equations proposed by the KWRA [31], the average error rate of the simulation results was approximately 3%. This error rate was calculated based on the difference between the simulation results and the theoretical superelevation values obtained using a coefficient (C) of 0.75, which corresponds to the trapezoidal cross-section shown in Figure 4. The superelevation coefficient (C) varies depending on the cross-sectional shape and typically ranges between 0.5 and 1.0. The selection of C = 0.75 ensures consistency between the theoretical estimation and the model configuration. This indicates that the three-dimensional FLOW-3D model can appropriately reproduce flow conditions in river curves.

3.3.3. Validation of Levee Overtopping Risk Curve Using Numerical Analysis

As shown in Figure 3, an overtopping risk assessment was performed using three dimensionless parameters (the Froude number, the ratio of superelevation to levee height, and the ratio of water depth to levee height) regardless of a levee’s size or shape. To confirm the curve’s reliability, FLOW-3D simulations were conducted on two levees with heights of 0.25 m for levee No. 1 and 0.50 m for levee No. 2 under overtopping conditions (Figure 8), with a Froude number of 0.63, a superelevation ratio of 0.3, and a water depth ratio of 0.8. By comparing these numerical results with the overtopping risk curve, it was determined whether each levee would experience overtopping based on consistent dimensionless criteria.

In this study, two virtual levees were simulated using numerical analysis to examine overtopping behavior in a curved channel, thereby verifying the reliability of the levee overtopping risk curve. Consistent with the predictions of the overtopping risk curve, both levees experienced overtopping in the numerical simulations. Despite the difference in levee size, both exhibited similar overtopping patterns, enhancing confidence in the curve’s reliability. Additionally, the dimensionless characteristics of levees were examined by comparing the distribution of water depth relative to levee height, based on the overtopping risk curve. Figure 9 shows the distribution of water depth for a small levee (No. 1) and a large levee (No. 2), revealing similar flow characteristics in both cases. Furthermore, as illustrated in Figure 10, an area-ratio analysis demonstrated that the difference in area ratios between the two levees remained below 3.8%, indicating minimal variation.

These results confirm that the overtopping risk curve can be consistently applied to levees of different scales to predict overtopping risk. By adopting a standardized, dimensionless-based approach, this method facilitates efficient evaluation of levee performance under various conditions and ensures reliable safety measures. Despite the differences in absolute height, the two levees exhibited nearly identical values for ∆y/H and h/H, resulting in consistent overtopping behavior, as predicted by the risk curve. This suggests that the overtopping response is primarily governed by relative hydraulic conditions rather than absolute geometry.

Additionally, the slight difference in area ratios observed in Figure 10 suggests minimal scale-dependent variation in the flow structure. These results emphasize the robustness and applicability of the proposed dimensionless overtopping criterion across different levee sizes, supporting the use of the risk curve as a predictive tool in practical design scenarios.

4. Conclusions

In this study, the Levee Overtopping Risk Curve is proposed using dimensionless variables such as the ratio of superelevation to levee height $({\displaystyle \frac{∆y}{H}})$, the ratio of water depth to levee height $({\displaystyle \frac{h}{H}})$, 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 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 levee 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 levee 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.