## 4 Results and Analysis
### Vortex Formation and Velocity Distribution  
Along the symmetry plane, Figure 2a showcases, at \(d_{c}/h=4.8\), the flow fields at different \(\theta\) values. The illustrated steps are no. 42 and 43. Over the steps, the flow exhibits a coherent structure with vortex formation in the step cavities. The flow patterns of the step layouts are similar. However, the vortex structures in the cavities differ in extent. The extent of the vortex flow becomes larger with an increase in \(\theta\). A larger vortex structure enhances the momentum exchange and energy dissipation. The vortices almost disappear if the step inclination gets larger downwards (e.g., \(\theta=-20^{\circ}\)), which is due to the fact that the roughness effects becomes smaller. The vortex characteristics affect the degree of energy loss of the flow [14]. To visualize the vortex structures, the \(Q\)-criterion, \(Q_{c}\), is introduced  
\[Q_{c}=\frac{1}{2}\Big{(}\big{\|}\Omega_{ij}\big{\|}^{2}-\big{\|}S_{ij}\big{\|} ^{2}\Big{)} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c} \hline  & \(X/L=0.10\) & \(X/L=0.25\) & \(X/L=0.40\) & \(X/L=0.60\) \\ \hline \(r_{21}\) & 1.14 & 1.14 & 1.14 & 1.14 \\ \(r_{32}\) & 1.31 & 1.31 & 1.31 & 1.31 \\ \(\phi_{1}\) & 0.0926 & \(-0.0663\) & 0.1066 & 0.6643 \\ \(\phi_{2}\) & 0.0933 & \(-0.0658\) & 0.1063 & 0.6644 \\ \(\phi_{3}\) & 0.0928 & \(-0.0661\) & 0.1086 & 0.6675 \\ \(p\) & 0.63 & 1.38 & 1.33 & 0.85 \\ GCI (\%) & 10.68 & 4.00 & 1.70 & 0.04 \\ \hline \end{tabular}
\end{table} 
Table 1: Showcased calculations of grid convergence index (GCI) in \(P/(pgd_{c})\). \begin{table}
\begin{tabular}{c c c c c} \hline  & \(y/y_{\max}=0.2\) & \(y/y_{\max}=0.4\) & \(y/y_{\max}=0.6\) & \(y/y_{\max}=0.8\) \\ \hline \(r_{21}\) & 1.14 & 1.14 & 1.14 & 1.14 \\ \(r_{32}\) & 1.31 & 1.31 & 1.31 & 1.31 \\ \(\phi_{1}\) & 0.5807 & 0.7790 & 0.9 in which  
\[\Omega_{ij}=\frac{1}{2}\bigg{(}\frac{\partial u_{i}}{x_{j}}-\frac{\partial u_{j}}{ x_{i}}\bigg{)} \tag{10}\]  
\[S_{ij}=\frac{1}{2}\bigg{(}\frac{\partial u_{i}}{x_{j}}+\frac{\partial u_{j}}{x_{ i}}\bigg{)} \tag{11}\]  
\(\Omega_{ij}\) = rate of rotation tensor, and \(S_{ij}\) = rate of strain tensor. They represent pure rotational motion and irrotational motion, respectively. In the fluid, if \(Q_{c}>0\), rotation is the dominant motion. If \(Q_{c}<0\), deformation is the main motion pattern. However, due to experimental error and viscous diffusion, \(Q_{c}=0\) cannot indicate the existence of vortex. Based on this method, the iso-surface of the vortex is identified in each layout, shown in Figure 2b (\(d_{c}/h\) = 4.8, \(Q_{c}\) = 100). The primary vortex characteristics are captured by \(Q\)-criterion. The vortex structures appear immediately downstream the vertical step faces. They cover an approximately cylindrical area in the step cavities and span across the channel width. There is a small region below the step vertices without vortex, which is possibly caused by the collision between the overlaying flow and up-climbing circulating flow. To compare the scale of the vortex, the area of its cross-section \(A\) is evaluated. Generally, \(A\) increases from downward to upward cases. For instance, compared with the conventional model, a \(20^{\circ}\) downward inclination leads to a 78% decrease, while a \(20^{\circ}\) upward inclination results in a 59% increase. Figure 2: Flow fields at the steps at different \(\theta\) values (for \(d_{c}/h\) = 4.8 at steps no. 42 and 43): (**a**) velocity vectors, and (**b**) vortex structures (\(Q_{c}\) = 100). The flow over the steps features a developing boundary layer, above which there is an ideal flow region [23]. In this region down the spillway, the flow accelerates due to gravity and the velocity varies significantly. For the boundary layer, an analytical approach is used to predict the velocity distribution [24; 25].