## 3 Dimensional Analysis  
### Dimensionless Parameters  
In this study, the key parameters responsible for leveee overtopping are reviewed, and standardized criteria for leveee 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 leveee characteristics as dimensionless variables, dimensional analysis enables consistent evaluations across diverse river conditions and leveee 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 leveee 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 leveee 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, leveee 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]. \[\Delta y=C\frac{BV^{2}}{g\tau_{c}};\ \left(for\ \frac{B}{r_{c}}\geq 0.33\right) \tag{7}\]  
\[\Delta y=C\frac{BV^{2}}{g\tau_{c}}+0.25\frac{V^{2}}{2g};\ \left(for\ \frac{B}{r_{c}}<0.33\right) \tag{8}\]  
\[\Delta y=0.75\frac{BV^{2}}{g\tau_{c}} \tag{9}\]  
where \(\Delta y\) 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 leveee overtopping is defined as shown in Equation (10), and dimensionless parameters relevant to overtopping are identified. \[P_{o}=f(\Delta y,\ V,\ g,\ H,\ h,\ \alpha) \tag{10}\]where \(P_{o}\) represents the probability of leveee overtopping, \(\Delta y\) is the superelevation, \(V\) is the velocity, \(g\) is the gravitational acceleration, \(H\) is the leveee height, \(h\) is the water depth, and \(\alpha\) is the curvature. Using five parameters, a functional relationship between the probability of leveee overtopping \((P_{o})\) and the dimensionless variables was derived as shown in Equation (11). The first variable is the ratio of superelevation \((\Delta y)\) to leveee height \((H)\), which indicates the risk of levee overtopping in river curves. The second variable is the ratio of water depth (h) to leveee height (H), allowing for an intuitive assessment of overtopping risk based on river water depth. The final dimensionless variable is the Froude number \((Fr)\), which is widely used to examine hydraulic phenomena by considering the velocity \((V)\) and the influence of gravity \((g)\). \[P_{o}=f\left(\frac{\Delta y}{H},\frac{h}{H},Fr\right) \tag{11}\]  
Dimensionless analysis is a crucial method that enables the consistent definition of their characteristics regardless of their size.