# Effect of Height and Geometry of Stepped Spillway on Inception Point Location
## 3 Results and Discussions
### Effect of Step Geometry on Inception Length Using Different Step Shapes  
In order to analyze the effect of step geometry on inception length, three different step geometry models (flat, pooled, and round) with the same number of steps were simulated to obtain the inception length in these three different stepped spillways. Conditions of the water volume fraction contours of the flat, pooled, and round stepped spillway models with the same number of steps and unit discharge rate of 0.8 m\({}^{2}\)/s and 1.0 m\({}^{2}\)/s were simulated. The inception point locations are shown in Figure 8. The location of the inception point in the round stepped spillway model is near the spillway crest. In the pooled stepped spillway model, the location of the inception point is further away from the spillway crest compared with the round stepped, but nearer than in the flat stepped spillway model. The location of the inception point in the flat stepped spillway model is far away from the spillway  
Figure 6: Variation of inception length (\(Li\)) with Froude surface roughness (\(F\)) at different stepped heights of flat stepped spillway models.  
Figure 7: Normalized \(Li\) with respect to Froude surface roughness.  
crest compared with the round or the pooled stepped spillway models under the same discharge. Thus, the location of the inception point moves toward the spillway crest with changes in step geometry from the flat to the pooled, to the round stepped spillway. The distances of the inception point from the crest _Li_ of the stepped spillway models with different step geometry under the same number of steps and discharge are listed in Table 3.  
\(Li\) increases when the stepped geometry changes from round to pooled, and further increases when the geometry changes from pooled to flat, as shown in Figure 9. The round stepped spillway has the lowest _Li_ compared with both the pooled stepped and the flat stepped spillways. The _Li_ of the flat stepped spillway is greater than that of the pooled stepped spillway under the same unit discharge. For all the stepped spillway geometries, _Li_ increases with unit discharge.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Discharge \(q\) (m\({}^{2}\)/s)** & **Step Geometry** & **Number of Steps \(N\)** & _Li_ **(m)** \\ \hline
0.8 & Flat & 24 & 1.68 \\
1.0 & Flat & 24 & 2.19 \\
1.2 & Flat & 24 & 2.53 \\
1.4 & Flat & 24 & 2.86 \\
0.8 & Pooled & 24 & 1.50 \\
1.0 & Pooled & 24 & 1.85 \\
1.2 & Pooled & 24 & 2.19 \\
1.4 & Pooled & 24 & 2.53 \\
0.8 & Round & 24 & 1.18 \\
1.0 & Round & 24 & 1.37 \\
1.2 & Round & 24 & 1.53 \\
1.4 & Round & 24 & 1.69 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of unit discharge (\(q\)), step geometry, and inception point (_Li_).  
Figure 8: Water volume fraction contours of round, pooled, and flat stepped spillway models in the central plane with same number of steps \(N=24\) at unit discharge 0.8 m\({}^{2}\)/s and 1.0 m\({}^{2}\)/s (**a**) flat stepped spillway (\(N=24\)) at unit discharge 0.8 m\({}^{2}\)/s; (**b**) pooled stepped spillway (\(N=24\)) at unit discharge 0.8 m\({}^{2}\)/s; (**c**) round stepped spillway (\(N=24\)) at unit discharge 0.8 m\({}^{2}\)/s; (**d**) flat stepped spillway (\(N=24\)) at discharge 1.0 m\({}^{2}\)/s; (**e**) pooled stepped spillway (\(N=24\)) at unit discharge 1 m\({}^{2}\)/s; and (**f**) round stepped spillway (\(N=24\)) at unit discharge 1 m\({}^{2}\)/s.