## 2 Materials and Methods  
In this paper, the computational fluid dynamics (CFD) software _Fluent_ was performed to solve the Reynolds-averaged Navier-Stokes (RANS) equations given below:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}\Big{(} \rho u_{i}u_{j}\Big{)}=f_{i}-\frac{\partial p}{\partial x_{i}}+\frac{\partial }{\partial x_{i}}\bigg{[}\mu\bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+ \frac{\partial u_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u _{i}}{\partial x_{j}}\bigg{)}\bigg{]}+\frac{\partial}{\partial x_{j}}\Big{(}- \rho\overline{u^{\prime}_{i}u^{\prime}}_{j}\bigg{)} \tag{2}\]  
where \(t\) is the time, \(u_{i}\) and \(u_{j}\) are time-averaged components of velocity, \(p\) is the pressure, \(\rho\) is the fluid density, \(\mu\) is the dynamic viscosity, \(\delta_{ij}\) is the Kronecker delta (\(\delta_{ij}=1\) if \(i=j\) and \(\delta_{ij}=0\) if \(i\
eq j\)), and \(\Big{(}-\rho\overline{u^{\prime}_{i}u^{\prime}}_{j}\Big{)}\) is the Reynolds stress and represents the effect of turbulence. The volume of fluid (VOF) method was used to characterize the interface between air and water. In each computational cell, the sum of the volume fractions of air, \(\alpha_{a}\), and water, \(\alpha_{w}\), is unity and can be given as:  
\[\alpha_{w}+\alpha_{a}=1 \tag{3}\]  
where both \(\alpha_{a}\) and \(\alpha_{w}\) are equal to or larger than 0. The skimming flow over converging stepped spillways has a high Reynolds number and is characterized by strong streamline curvature and distorted bottom rotations. The velocity and turbulent energy of the standing wave and the mainstream are high, and there is strong shear stress between the mainstream and bottom rotations. The realizable \(k\)-\(\varepsilon\) model is a high Reynolds model, and it can well capture the mean flow of complex structures and is good for flows involving rotation, boundary layers under strong adverse pressure gradients, separation and recirculation, according to Fluent User's Guide [19]. Therefore, the realizable \(k\)-\(\varepsilon\) model is adopted in this study. [PERSON] et al. [20] compared four turbulence models (i.e., realizable \(k\)-\(\varepsilon\) model, SST \(k\)-\(\varepsilon\) model, \(\
u^{2}\)-\(f\) model and LES model) and considered the realizable \(k\)-\(\varepsilon\) model as the most efficient in simulating skimming flows over stepped spillways. The realizable \(k\)-\(\varepsilon\) model was also successfully applied to simulating the skimming flow over a V-shaped stepped spillway by [PERSON] et al. [21] and the stepped spillway with chamfers and cavity blockages by [PERSON] et al. [22]. The software _Gambit_ was used for geometric modeling and meshing. Figure 1 illustrates the computational domain, which includes an inlet basin, a stepped spillway model and an outlet channel. The inlet basin is 2000 mm long and 355 mm deep to ensure that the incoming flow to the stepped spillway is stable. The outlet channel is 2000 mm long, allowing for uniform unrestricted drainage. The sidewall is 300 mm above the spillway and channel bottom to avoid overflow. Because converging stepped spillways are symmetrical, only half of the geometry was simulated, and the _symmetry_ boundary (marked yellow) was set at the central plane. The upstream boundary of the inlet basin and the downstream boundary of the outlet channel were determined as the _mass flow inlet_ (marked green) and _pressure outlet_ (marked blue) boundaries, respectively. The top boundary of the stepped spillway model was defined as the _pressure outlet_ (marked dark blue) boundary with zero relative pressure, and all the solid geometry surface was set as the no-slip _wall_ boundary with standard wall functions.