## 2 Theoretical Background
### 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 leveee 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 leveee 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 \[\frac{\partial}{\partial x}(uA_{x})+\frac{\partial}{\partial y}\big{(}vA_{y} \big{)}+\frac{\partial}{\partial z}(wA_{z})=\frac{RSOR}{\rho}\] (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 \[\frac{\partial u}{\partial t}+\frac{1}{V_{f}}\bigg{\{}uA_{x}\frac{\partial u}{ \partial x}+vA_{y}\frac{\partial u}{\partial y}+wA_{z}\frac{\partial u}{ \partial z}\bigg{\}}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+F_{x}- \frac{RSOR}{\rho V_{f}}u\] (4) \[\frac{\partial v}{\partial t}+\frac{1}{V_{f}}\bigg{\{}uA_{x}\frac{\partial v }{\partial x}+vA_{y}\frac{\partial v}{\partial y}+wA_{z}\frac{\partial v}{ \partial z}\bigg{\}}=-\frac{1}{\rho}\frac{\partial p}{\partial y}+F_{y}- \frac{RSOR}{\rho V_{f}}v\] (5) \[\frac{\partial w}{\partial t}+\frac{1}{V_{f}}\bigg{\{}uA_{x}\frac{\partial w }{\partial x}+vA_{y}\frac{\partial w}{\partial y}+wA_{z}\frac{\partial w}{ \partial z}\bigg{\}}=-\frac{1}{\rho}\frac{\partial p}{\partial z}+F_{z}- \frac{RSOR}{\rho V_{f}}w\] (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 leveee 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.