# Effect of Height and Geometry of Stepped Spillway on Inception Point Location
## 2 Materials and Methods  
Numerical modelling is a widely applied technique performed with the aid of a computer to solve the Navier-Stokes equation, which is based upon the conservation of mass, energy, and momentum. CFD software is used to simulate the stepped spillway models. Fluent software uses the finite volume method (FVM) to discretize the Navier-Stokes equation. The FVM method is the most commonly used method to discretize the dynamic equations in CFD. The control volume technique converts the governing equation into algebraic equations rather than solving the equations numerically. Integration equations for the control volume are solved in an implicit form.  
Six kinds of stepped spillway models with different step sizes and configurations used for numerical simulation are listed in Table 1. The models varying according to step height with flat, pooled, and round steps, are shown in Figure 2. All the six models were simulated respectively, using VOF as the multiphase model and the realizable k-\(\epsilon\) model as the turbulence model. The slope of all the models is held constant.  
The inlet boundary condition of the stepped spillway model is represented by the discharge. In the initial state, there is no water in the spillway and the spillway surface is directly connected to the atmosphere. The outlet boundary condition of the spillway is set by pressure. All the walls are stationary with no slip velocity. The three-dimensional grid models and boundary conditions used for computation are shown in Figure 3. The results of the three-dimensional model are better than those of the two-dimensional model because the three-dimensional model consider the air phase more accurately than the two-dimensional model.  
The calculation domain is discretized into an unstructured grid with 194,447 mesh elements and the number of nodes used for computation is 215,760. The inflation layer is applied to the steps to yield better results. The area around the steps is particularly considered to achieve better results. Therefore, 10 layers of inflation are applied on the steps to better study the flow of the spillway.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Model** & **Step Geometry** & **Step Height(m)** & **Step Length(m)** & **Slope\({}^{\circ}\)** \\ \hline M1 & Flat & 0.10 & 0.45 & 12.52\({}^{\circ}\) \\ M2 & Flat & 0.09 & 0.37 & 13.67\({}^{\circ}\) \\ M3 & Flat & 0.05 & 0.21 & 13.39\({}^{\circ}\) \\ M4 & Flat & 0.04 & 0.17 & 13.24\({}^{\circ}\) \\ M5 & Pooled (\(h_{p}=0.01\) m) & 0.04 & 0.15 & 14.93\({}^{\circ}\) \\ M6 & Round & 0.04 & 0.16 & 14.03\({}^{\circ}\) \\ \hline \end{tabular}
\end{table}
Table 1: Design parameters of different stepped spillways for numerical simulation.  
Figure 3: Numerical stepped spillway model and boundary conditions: (**a**) three-dimensional numerical stepped spillway model, (**b**) boundary conditions applied to the model.  
Figure 2: Sketch of stepped spillway configuration: (**a**) M1 with Flat Steps (\(h\) = 0.1 m, \(l\) = 0.45 m); (**b**) M2 with flat steps (\(h\) = 0.09 m, \(l\) = 0.37 m); (**c**) M3 with flat Steps (\(h\) = 0.05 m, \(l\) = 0.21 m); (**d**) M4 with flat steps (\(h\) = 0.04 m, \(l\) = 0.17 m); (**e**) M5 with pooled steps; (**f**) M6 with round steps.