# Effect of Height and Geometry of Stepped Spillway on Inception Point Location
## 4 Conclusions  
The inception point location of the stepped spillways depends on the discharge, step height, and step geometry. The distance of the inception point from the crest (_Li_) has a positive correlation with the discharge. Under the same discharge, the step height has a negative correlation with _Li_, while the number of steps shows a positive correlation with it. The inception point location is closer to the spillway crest in the stepped spillways with higher step height. The value of _Li_ changes with the Froude surface roughness (_F_). _Li_ increases with \(F\), while \(F\) increases with the unit discharge but decreases with the step height.  
Various simulations and comparison among the three kinds of spillway forms (round, pooled, and flat) were conducted and analyzed. It was shown that under conditions of the same discharge and number of steps, the location of the inception point in a round stepped spillway is closest to the spillway crest, while it is farthest in a flat stepped spillway. As for the non-aerated flow zone, it decreases with the stepped height, as does the probability of cavitation. Under the same boundary conditions, the round stepped spillway shows the least non-aerated flow zone, while the flat form shows the most. Therefore, the minimum probability of cavitation damage may occur in the round stepped spillway, and thus it is better to use the round stepped spillway to prevent cavitation.  
Conceptualization, W.W.; methodology, A.R.; validation, A.R., and X.C.; formal analysis, A.R.; investigation, A.R.; resources, W.W.; writing--original draft preparation, A.R.; writing--review and editing, W.W., and X.C.; visualization, A.R., and X.C.; supervision, W.W.; funding acquisition, W.W. This work was supported by the National Key R&D Program of China (2018 YFC0407203), the National Natural Science Foundation of China (Nos. 51779216, 51279175) and the Zhejiang Provincial Natural Science Foundation of China (Grant No. LZ16E090001). The authors declare no conflict of interest.  
The authors declare  
\begin{tabular}{l l} \(G_{b}\) & generation of k due to buoyancy (kg/(ms\({}^{3}\))) \\ \(G_{k}\) & generation of k due to fluid Shear \(G_{k}=\mu_{i}S^{2}\) (kg/ms\({}^{3}\)) \\ \(g\) & gravitational acceleration (m/s\({}^{2}\)) \\ \(h\) & step height (m) \\ \(hp\) & height of Pool (m) \\ \(k\) & turbulent kinetic energy (m\({}^{2}\)/s\({}^{2}\)) \\ \(ks\) & roughness height, \(ks=h\)cos\(\theta\) (m) \\ \(l\) & step length (m) \\ \(Li\) & distance of inception point location from spillway crest (m) \\ \(Li/k_{s}\) & normalized distance from spillway crest to Inception Point \\ M1-M6 & number of stepped spillway models \\ \(N\) & number of steps \\ \(S\) & modulus of mean rate of tensor \(S=\sqrt{2S_{ij}S_{ij}}\) \\ \(S_{k}\) & source term of kinetic energy (kg/ms\({}^{3}\)) \\ \(S_{e}\) & source term of dissipation rate (kg/ms\({}^{4}\)) \\ \(S_{ij}\) & mean rate of deformation \\ \(p\) & pressure in (N/m\({}^{2}\)) \\ \(\mathrm{q}\) & unit discharge (m\({}^{2}\)/sec) \\ \(W\) & width of the step (m) \\ \(Y_{m}\) & effect of compressibility on turbulence (kg/ms\({}^{3}\)) \\ \(\sigma_{k}\) & turbulent Prandtl number \\ \(\sigma_{e}\) & turbulent Prandtl number \\ \(u_{i}\) & velocity in \(x_{i}\) direction (m/s) \\ \(u_{j}\) & velocity in \(x_{j}\) direction (m/s) \\ \(\alpha_{a}\) & volume fraction of air (\%) \\ \(\alpha_{w}\) & volume fraction of water (\%) \\ \(\varepsilon\) & turbulent dissipation rate (m\({}^{2}\)/s\({}^{3}\)) \\ \(\theta\) & spillway slope (\({}^{\circ}\)) \\ \(\mu\) & molecular dynamic viscosity (kg/ms) \\ \(\mu_{t}\) & turbulent dynamic viscosity (kg/ms) \\ \(\rho\) & cell density (kg/m\({}^{3}\)) \\ \(\rho_{a}\) & density of air (kg/m\({}^{3}\)) \\ \(\rho_{w}\) & density of water (kg/m\({}^{3}\)) \\ \(t\) & time (s) \\ \(v\) & kinematic viscosity (m\({}^{2}\)/s) \\ \(\omega\) & angular velocity \\ \end{tabular}