# Effect of Height and Geometry of Stepped Spillway on Inception Point Location
## 3 Results and Discussions
### Effect of Step Height on Inception Length Using Flat Stepped Spillways  
Four stepped spillway models with different step heights (0.10, 0.09, 0.05, and 0.04 m) were simulated to predict the changes in inception length with the decrease in step height. The inception point is closer to the spillway crest with a larger step height under the same discharge. The results also reveal that the inception point length increases with the discharge in all stepped spillway models. The comparison between the VOF results of the inception point lengths with different step heights when the discharge is 0.8 m\({}^{2}\)/s is shown in Figure 4. It is seen clearly that with the decrease in step height, the inception point moves downward under the same discharge. The inception lengths (_Li_) with respect to discharge, step height, and Froude surface roughness (\(F\)) are listed in Table 2. With increase of discharge, the inception length increases under the conditions of the same number of steps. With the increase in the number of steps, the inception length also increases under the same discharge. The Froude surface roughness, defined with respect to surface roughness (ks = _h_cos\(\theta\)), which depends on unit discharge (\(q\)) and step height, is illustrated as Equation (8). The symbols in the equation are listed at the end of the text. Figure 4: Water volume fraction contours of different step heights for stepped spillway models in the central plane at discharge 0.8 m\({}^{2}\)/s: (**a**) flat stepped spillway with step height 0.10 m; (**b**) flat stepped spillway with step height 0.09 m; (**c**) flat stepped spillway with step height 0.05 m; and (**d**) flat stepped spillway with step height 0.04 m.  
\[F=\frac{q}{\sqrt{g\sin\theta_{\mathrm{Ks}}s}} \tag{8}\]  
The variation of the inception length (\(Li\)) with respect to the discharge at different step heights is shown in Figure 5. The trend of the figure shows that at the highest discharge, the inception length is the maximum under all the conditions of different step heights. With the decrease in step height, the inception length increases under the same discharge. The relationship between inception length (\(Li\)) and Froude surface roughness (\(F\)) at different step heights is shown in Figure 6. With the increase in Froude surface roughness, the inception length increases. According to Equation (8), Froude surface roughness (\(F\)) increases with the decrease in step height and the increase in discharge. It is also apparent from Figure 6 that the Froude surface roughness (\(F\)) values increase with the decrease of the step height. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Unit Discharge** & **Step Height** & **Number of** & **Froude Surface** & **Inception Point** & **Surface** & \\ \(q\) (m\({}^{2}\)/s) & **(m)** & **Steps N** & **Roughness \(F\)** & **Length \(Li\) (m)** & **Roughness \(L\)/ks** \\ \hline
0.80 & 0.10 & 9 & 17.0 & 0.90 & 0.097 & 9.23 \\
1.00 & 0.10 & 9 & 22.0 & 1.35 & 0.097 & 13.84 \\
1.20 & 0.10 & 9 & 27.0 & 1.57 & 0.097 & 16.10 \\
1.40 & 0.10 & 9 & 31.0 & 1.80 & 0.097 & 18.46 \\
0.80 & 0.09 & 11 & 19.0 & 1.10 & 0.090 & 12.17 \\
1.00 & 0.09 & 11 & 24.0 & 1.47 & 0.090 & 16.26 \\
1.20 & 0.09 & 11 & 29.0 & 1.84 & 0.090 & 20.35 \\
1.40 & 0.09 & 11 & 34.0 & 2.20 & 0.090 & 24.34 \\
0.80 & 0.05 & 19 & 50.0 & 1.27 & 0.048 & 26.45 \\
1.00 & 0.05 & 19 & 62.5 & 1.70 & 0.048 & 35.41 \\
1.20 & 0.05 & 19 & 75.0 & 2.14 & 0.048 & 44.38 \\
1.40 & 0.05 & 19 & 90.0 & 2.34 & 0.048 & 75.26 \\
0.80 & 0.04 & 24 & 72.0 & 1.68 & 0.038 & 51.08 \\
1.00 & 0.04 & 24 & 90.0 & 2.19 & 0.038 & 59.30 \\
1.20 & 0.04 & 24 & 109.0 & 2.53 & 0.038 & 64.55 \\
1.40 & 0.04 & 24 & 127.0 & 2.86 & 0.038 & 75.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of unit discharge (_q_), step height, Froude surface roughness (\(F\)), inception point length (\(Li\)), surface roughness (ks) and normalized (\(Li\)). Figure 5: Variation of inception length (\(Li\)) with discharge rate at different step heights of flat stepped spillway models.