# Stepped Spillways and Energy Dissipation: A Non-Uniform Step Length Approach
## 3 Results and Discussion
### Non-Uniform Stepped Spillways  
In this section, the effect of non-uniform step lengths on the energy dissipation for convex, concave, random, and semi-uniform configurations is discussed. The fraction of water flow over non-uniform configuration models investigated in this study can be seen in Figure 12. In the concave spillway, the extent of air entrainment was less in the water flow at the beginning of the steps and it increased towards the end of the chute, contrary to the convex spillway. The air entrainment in the concave configuration was larger compared to the convex configuration. The differences in the increase in the air entrainment could be explained by the flow disturbances resulting from an increase in the length of \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Reference** & **0** & **H\({}_{\text{dam}}\)** & **N** & **h** & **L** & **Q** & **m\({}^{3}\)/sec** & **d\({}_{\text{d}}\)/h** & \multicolumn{3}{c}{**\(\Delta\)E/E**} & **Error \%** \\ \cline{5-10}  & & & & & & & **Experiment** & **Numerical** & \\  & & & & & & & **Results** & **Results** & **Results** & \\ \hline  & & & & & & 0.057 & 0.7 & 0.61 & 0.622 & 1.967 \\  & & & & & & 0.083 & 0.9 & 0.63 & 0.602 & 4.444 \\ Zhang and & & & & & & 0.098 & 1 & 0.63 & 0.605 & 3.968 \\  & 45 & 1.2 & 12 & 0.1 & 0.112 & 1.1 & 0.62 & 0.601 & 3.065 \\  & & & & & & 0.145 & 1.3 & 0.60 & 0.572 & 4.667 \\  & & & & & & 0.179 & 1.5 & 0.57 & 0.534 & 6.316 \\  & & & & & & 0.216 & 1.7 & 0.53 & 0.501 & 5.472 \\ \hline  & & & & & & 0.056 & 0.69 & 0.76 & 0.71 & 6.579 \\  & & & & & & 0.116 & 1.11 & 0.64 & 0.636 & 0.625 \\ Felder [2] & 26.56 & 1 & 10 & 0.1 & 0.2 & 0.143 & 1.28 & 0.63 & 0.616 & 2.222 \\  & & & & & 0.161 & 1.38 & 0.66 & 0.592 & 10.303 \\  & & & & & & 0.180 & 1.49 & 0.58 & 0.577 & 0.517 \\  & & & & & & 0.202 & 1.61 & 0.57 & 0.550 & 3.509 \\ \hline  & & & & & & 0.095 & 0.97 & 0.65 & 0.629 & 3.23 \\ Felder [2] & 26.56 & 1 & 15 & h & L & 0.116 & 1.11 & 0.62 & 0.616 & 0.645 \\  & & 10 steps (h = 0.05 m) & & 0.1 & 0.143 & 1.28 & 0.56 & 0.599 & 6.964 \\  & & 5 steps (h = 0.1 m) & & 0.2 & 0.161 & 1.38 & 0.56 & 0.582 & 3.928 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The experimental and numerical energy dissipation results of the last step for different stepped spillways. the steps toward the bottom of the spillway. In the random configurations, the air entrainment did not provide better aeration performance compared to uniform configurations. Figure 10: Effect of uniform step length on the rate of energy dissipation. Figure 9: Air–water flow on uniform stepped spillways with different slopes for \(d_{o}\)/\(\hbar\) = 1.3: (**a**) uniform stepped spillway with step length 0.15 m; (**b**) uniform stepped spillway with step length 0.20 m; (**c**) uniform stepped spillway with step length 0.22 m; (**d**) uniform stepped spillway with step length 0.25 m; (**e**) uniform stepped spillway with step length 0.30 m and (**f**) uniform stepped spillway with step length 0.35 m.  
The effect of non-uniform step length stepped spillways on the energy dissipation can be seen in Figure 13. Energy losses in the concave stepped spillway were approximately 9% more than the convex stepped spillway, and very similar to the uniform stepped spillway with the slope 1V:2.2H. For the concave spillway, a large vortex area occurs at the end of the lower half of the chute. The large vortex reduces the velocity, resulting in an increase of the energy dissipation. The small steps at the lower half of the convex spillway led to the formation of small vortices that contributed to a lower velocity reduction, resulting in less energy dissipation (Figure 14).