# Stepped Spillways and Energy Dissipation: A Non-Uniform Step Length Approach
## 1 Introduction  
To prevent overtopping and possible failure of a dam, excess water in reservoirs is released by spillways. However, the released water, due to high kinetic energy, can cause erosion downstream [1; 2; 3]. In general, the effects of erosion are prevented by stilling basins, and hydraulic jumps are employed to protect the structures from erosion hazards [4; 5]. The length of the stilling basin located at the foot of the spillway should be kept as short as possible to minimize construction costs. In order to achieve this goal, one possible solution is to consider stepped spillways instead of the traditional smooth spillways. A stepped spillway has a profile made up of steps on the chute, which increases the energy dissipation and in turn helps to reduce the required size of the stilling basin. The energy dissipation occurs by flow recirculation, with or without air entrainment, under the pseudo-bottom caused by the flow on each step [6; 7]. Another important feature of stepped spillways is, because of the high turbulence of the flow and the absorbed air from the atmosphere, the risk of cavitation reduces [8; 9; 10]. The flow over a stepped spillway, with respect to the discharge rate, can be classified into three main regimes: Nappe flow for low discharge rates, skimming flow for higher discharge rates, and a transition regime that can be considered as the upper and lower boundaries of nappe and skimming flow, respectively. The flow regime can be affected by the channel slope and the spillway geometry, such as step height and length [11; 12; 13]. The amount of energy dissipation is directly related to the flow regime. The largest energy dissipation occurs under the nappe and transition flow regimes, however, these may cause flow instability [2]. Instable flows must be avoided as they can be dangerous due to causing unnecessary vibrations in the structures [11]. Therefore, in general, the hydraulic design of dams is based on the skimming flow regime [14; 15; 16]. For efficient design, the quantity of energy dissipation in stepped spillways, \(\Delta E/E_{t}\), at the end of the stepped spillway is determined, where \(\Delta E\) is the total head loss (\(\Delta E=E_{t}-H_{\text{res}}\)) and \(E_{t}\) is the maximum upstream head [2; 16]. \(H_{\text{res}}\) is the residual energy and can be calculated as:  
\[H_{\text{res}}=\int\limits_{y=0}^{y90}(1-C)dy\times\cos\theta+\frac{{q_{w}}^{ 2}}{2\times g\times\left(\int_{y=0}^{y90}(1-C)dy\right)^{2}}+Z \tag{1}\]  
where \(C\) is the air concentration, \(y\) is measured perpendicular to the pseudo-bottom formed by the step edges, \(y_{90}\) is the depth where the local \(C=90\%\), and \(Z\) is the step edge elevation above the datum. Based on the uniform flow condition the amount of energy dissipation of the flow over a step can be calculated [3; 15]. The uniform flow condition would imply a long chute with many steps, which may not be a typical form of stepped structure [13]. In the present study, due to the limited chute length, the energy dissipation was calculated in the absence of a uniform equilibrium flow condition. The effect of the slope on energy dissipation for stepped spillways with uniform flat horizontal steps has been investigated, mostly by experimental studies [2; 3; 7; 9; 13; 15; 17; 18; 19; 20; 21]. Experiments indicated that the energy loss due to the steps depends on the ratio of the critical flow depth (\(d_{c}\)) to the step height (\(lt\)), and the number of steps, \(N\). When the value of \(d_{c}\)/\(h\) approaches one, near the limit of the skimming flow, the energy dissipation will be significantly affected by the stepped surface [22]. For higher values of \(d_{c}\)/\(h\), the effect of \(N\) becomes appreciable and at a certain \(d_{c}\)/\(h\), and the energy dissipation improves as \(N\) increases [13; 22]. In the literature, there are few experimental studies that have focused on the flow over non-uniform stepped spillways. [PERSON] [23] studied the energy dissipation on a stepped spillway with non-uniform step heights.