## 2 Materials and Methods
As the bottom slope \(\theta\) is an important geometry parameter of stepped spillways and should influence the skimming flow, three stepped spillway models with \(\theta=11.3^{\circ}\), \(30.0^{\circ}\) and \(48.3^{\circ}\) were designed and tested, and they were connected to the inlet basin with a WES profile. The mild slope model of \(\theta=11.3^{\circ}\) was designed based on an undergoing drainage project in Shanghai according to Froude similarity with the scale of 1:33, the width of the spillway entrance \(W_{i}=1500\) mm, the total step number \(N=15\) and the step height \(B=30.0\) mm. For the moderate slope model, the dimensions were \(\theta=30.0^{\circ}\), \(W_{i}=1500\) mm, \(N=30\) and \(B=60.0\) mm. The steep slope model of \(\theta=48.3^{\circ}\) was designed according to the experimental model of converging stepped spillway studied by [PERSON] et al. [15]  
Figure 1: Computational domain, boundary conditions and mesh distribution of converging stepped spillway, inlet basin and outlet channel (with \(\theta=48.3^{\circ}\), \(B=22.5\) mm and \(\Phi=22.6^{\circ}\) as an example). with \(W_{i}\) = 1000 mm, \(N\) = 65 and \(B\) = 22.5 mm. Four sidewall convergence angles of \(\Phi\) = 0\({}^{\circ}\), 12\({}^{\circ}\), 18.8\({}^{\circ}\) and 22.6\({}^{\circ}\), as well as three dimensionless inlet critical flow depths of \(h_{c}\)/\(B\) = 3.36, 3.69 and 4.02 were investigated for each stepped spillway model. The Weber number _We_ ranged from 51 to 173 and the Reynold number _Re_ ranged from 6.5 \(\times\) 10\({}^{4}\) to 1.23 \(\times\) 10\({}^{6}\) at the spillway outlet. Table 1 summarizes the detailed parameters of the total 33 cases that were tested. Because the converging sidewall limits the total length of the spillway and the milder spillway has a longer step length than the steeper spillway, the total step number \(N\) decreases as the spillway slope \(\theta\) increases. To verify the accuracy of the whole numerical procedure, an additional case of the skimming flow over a stepped spillway was created according to the experiment of [PERSON] et al. [23] with \(W_{i}\) = 1800 mm, \(B\) = 38 mm, \(\theta\) = 14.04\({}^{\circ}\), \(N\) = 40, \(\Phi\) = 0\({}^{\circ}\) and \(h_{c}\)/\(B\) = 3.53. This simulation was repeated using Grid sets 1 to 3 for a grid sensitivity analysis. All grid sets are intensive close to the steps and the converging sidewalls. Grid sets 1 to 3 feature total grid numbers of 1.25, 2.50 and 5.00 million, respectively, with the cell sizes near the steps of 3.3 mm \(\times\) 3.3 mm, 2.2 mm \(\times\) 2.2 mm and 1.5 mm \(\times\) 1.5 mm, respectively. Figure 2 compares the computed velocity profiles of \(V\) along flow depth \(h\) on the edge of several steps and the experiment data from [PERSON] et al. [23]. The velocity profiles of Grid set 2 and 3 were found to be indistinguishable and to correlate well with experimental measurements on steps of \(n\) = 3, 6 and 12, demonstrating the viability of the existing numerical model in simulating the skimming flow over stepped spillways. Grid creation for the scenarios indicated in Table 1 was referred to Grid set 2 in terms of calculation precision and efficiency. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Cases No.**} & **Initial Width \(W_{i}\) (mm)** & **Step Height B (mm)** & **Spillway Slope \(\theta\) (\(\circ\))** & **Step Number \(N\) (\(\circ\))** & **Inlet Critical Flow Depth (\(h_{c}\)/\(B\) (\(\downarrow\))**)** & **Convergence Angle \(\boldsymbol{\hat{\Phi}}\) (\({}^{\circ}\))** \\ \hline \hline
1–12 & 1000 & 22.5 & 48.3 & 65 & 3.36 & 0 \\  & & & & & 3.69 & 12 \\  & & & & & 4.02 & 22.6 \\ \hline
13–21 & 1500 & 60 & 30 & 30 & 3.36 & 0 \\  & & & & & 3.69 & 12 \\  & & & & & 4.02 & 18.8 \\ \hline
22–33 & 1500 & 30 & 11.3 & 15 & 3.36 & 0 \\  & & & & & 3.69 & 12 \\  & & & & & 4.02 & 18.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Tested cases. Figure 2: Comparisons of velocity profiles on various steps between our numerical simulation and experiment data from [PERSON] et al. [23].