# Effect of Height and Geometry of Stepped Spillway on Inception Point Location
## 1 Introduction
A mathematical formula was developed to measure the length of the inception point from the spillway crest (\(Li\)) for determining the start of air entrainment with slope of 22\({}^{\circ}\)[20]. [PERSON] and [PERSON] [21] experimentally measured the inception point location of different stepped spillway configurations. In their study, 12 different empirical equations were derived based on the experimental results for different configurations of stepped spillways. [PERSON] et al. [22] experimentally measured the inception point of a stepped spillway model with a slope of 39.29\({}^{\circ}\). Different models of different step heights were used and an empirical formula was derived based on their experiments. Energy dissipation in a round-shaped stepped spillway increases by 3% compared with that in a flat stepped spillway. A round stepped spillway dissipates more energy due to very quick air entrainment of the skimming flow. Hence, the length of the training wall height in a round stepped spillway decreases by approximately 20%. The risk of cavitation in a round stepped spillway is also less than that of a flat stepped spillway [23]. [PERSON] and [PERSON] [24] conducted experiments on pooled and flat stepped spillways with slopes of 8.9\({}^{\circ}\) and 26.6\({}^{\circ}\). The results showed that energy dissipation in the pooled stepped spillway is larger than in the flat stepped spillway with a smaller slope. In the pooled stepped spillway, the mean air concentration is larger than that in the flat stepped spillway. [PERSON] and [PERSON] [25] performed 40 experiments on three different stepped spillway models to study the relationship among the  
Figure 1: Schematic of inception point location in a stepped spillway during the skimming flow regime. inception length, the discharge, and the chute angle. It was found that the inception length increases with the discharge but decreases with the chute angle. Physical modelling of a stepped spillway is very difficult and does not always yield accurate results to develop physical models for experiments. Fortunately, with the development of high-performance computers, the computational fluid dynamics (CFD) technique has been developed. [PERSON] et al. [26] found that CFD is a very authentic and reliable source to simulate the flow over a stepped spillway. The k-\(\varepsilon\) turbulence model was used to simulate turbulent flow in a stepped spillway. [PERSON] and [PERSON] [27] studied the energy dissipation in a stepped spillway by numerical modelling with different parameters and investigated the effects of these parameters on energy dissipation. The parameters involved in energy dissipation include the step height, the number of steps, the spillway slope and the unit discharge. [PERSON] and [PERSON] [28] used the well-known commercial software ANSYS Fluent with different numerical models. They used the volume of fluid (VOF) and the mixture model to simulate the flow in a stepped spillway and evaluated the performance of each model to determine which one accurately simulated the skimming flow over a stepped spillway. [PERSON] et al. [29] performed numerical simulation with the help of turbulent modelling and concluded that turbulent modelling is a very good and efficient method to determine the complexity of stepped spillway flow. In their study, Flow 3D software was used for numerical simulation. [PERSON] et al. [30] simulated a spillway model with the help of four different turbulence models: realizable k-\(\varepsilon\), k-\(\omega\), shear stress transport k-\(\omega\), and large eddy simulation (LES). Spillway models were simulated using these four turbulence models and the results compared. The results obtained with the realizable k-\(\varepsilon\) model were more accurate than the other three turbulence models. [PERSON] et al. [31] conducted a numerical simulation and developed a numerical model for determining the boundary layer in a stepped spillway with a steep slope. The numerical model was based on the implicit finite difference scheme. [PERSON] et al.