## 4 Results and Analysis
### Energy Dissipation  
Effective energy dissipation is essential for the protection of downstream riverbeds and structures. To estimate the residual energy downstream of the spillway, the averaged flow velocity at the chute end \(V_{T}\) is analyzed. \(V_{T}\) is an indicator of the energy dissipation. If more energy is dissipated by the steps, both the residual kinetic energy and \(V_{T}\) are smaller. \(V_{T}\) is dependent on \(H_{T}\), \(q\), \(h\), \(l\), \(\theta\), and \(g\). It is expressed as  
\[V_{T}=f_{3}(H_{T},q,h,l,\theta,g) \tag{16}\]  
Based on Buckingham Pi theory, Equation (16) is rewritten in a dimensionless form as  
\[\frac{V_{T}}{\left(gH_{T}\right)^{0.5}}=f_{4}\left(\frac{q^{2}}{gH_{T}^{3}}, \frac{h}{l},\theta\right) \tag{17}\]  
Define \(V_{T}/(gH_{T})^{0.5}\) = velocity ration, and \(q^{2}/\left(gH_{T}^{3}\right)\) = drop number. For a given spillway chute, \(h/l\) represents its slope and is a constant. Therefore, the velocity ratio is a function of the drop number and inclination angle. Figure 8a illustrates the variation of \(V_{T}/(gH_{T})^{0.5}\) with \(q^{2}/\left(gH_{T}^{3}\right)\) for different \(\theta\). The simulations indicate that at, for example, \(q\) = 0.31 m\({}^{2}\)/s, \(V_{T}\) ranges from 4.64 m/s at \(\theta\) = \(-20^{\circ}\) to 3.21 m/s at \(\theta\) = \(20^{\circ}\). Moreover, at a given \(\theta\) value, the velocity ratio exhibits an almost linear upward trend with an increase in the drop number, showing that a larger discharge gives rise to higher residual energy. At a fixed drop number, the velocity ratio descends with an increase in \(\theta\). Shifting from \(\theta\) = \(-20^{\circ}\) to \(20^{\circ}\), the velocity ratio drops on average by 44.0%. Compared to the upward layouts, the residual energy of the downward ones is larger. Herein, based on Equation (17), an empirical equation is developed to estimate the velocity ratio. It is  
\[\frac{V_{T}}{\left(gH_{T}\right)^{0.5}}=\lambda_{1}\ln\left(\frac{q^{2}}{gH_{ T}^{3}}\right)+\lambda_{2}\theta+\lambda_{3} \tag{18}\]  
where \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) are coefficients. Their values are listed in Table 4. In comparison with the experimental data on an upward inclined stepped chute [10], Figure 8b shows the predicted velocity ratios against the simulated ones. Equation (18) gives an acceptable estimation of the velocity ratio, with \(R^{2}>0.98\). Nevertheless, it generally overestimates the physical results, which is likely associated with the difference in the chute slope. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{\(P_{\min}\)} & \multicolumn{2}{c}{\(P_{\max}\)} \\ \cline{2-5}  & \(a\) & \(b\) & \(a\) & \(b\) \\ \hline Horizontal face & \(-0.7016\) & \(0.4501\) & \(0.0410\) & \(0.2244\) \\ Vertical face & \(-0.8408\) & \(0.4225\) & \(-0.4377\) & \(0.6136\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Coefficients in Equation (15). The relative energy loss, defined as \(E_{L}\)/\(E_{0}\), is another parameter to quantify the energy dissipation of a stepped spillway, in which \(E_{L}\) = energy loss between two cross-sections and \(E_{0}\) = total energy head at the upper cross-section. Together with the \(\theta=0^{\circ}\) results by [PERSON] et al. [11], Figure 8c shows the variation of \(E_{L}\)/\(E_{0}\) with \(d_{c}\)/\(h\). The agreement is generally good. Similar to the findings by [PERSON] et al. [26], [PERSON] and [PERSON] [27] and [PERSON] and [PERSON] [28], \(E_{L}\)/\(E_{0}\) exhibits a downward trend with an increasing \(d_{c}\)/\(h\). For the \(\theta=-20^{\circ}\) chute, \(E_{L}\)/\(E_{0}\) is much smaller than for the other layouts, which is ascribed to the smaller bottom friction elements. Under the same flow conditions, a progressive transition from a down- to upward step surface enhances the roughness effect and boosts the energy dissipation. This is because the inclined steps obstruct the flow, producing more spray and trapping recirculation vortices in the step cavities.