## 3 Results
### Widths of Disturbed Regions for Free-Surface, Mainstream and Bottom Rotations  
Because the converging sidewall has various effects on the free-surface, mainstream and bottom rotations of the skimming flow, the disturbed regions for free-surface, mainstream, and bottom rotations should be different as well. The boundary of the disturbed regions by converging sidewall are defined as follows for quantitative analysis and comparison. The disturbed region boundary for the free-surface is where the water depth increases at the fastest rate in the width direction (see Figure 4b). The disturbed region boundary for the mainstream is when the streamlines begin to run parallel to the central plane (see Figure 5c,d). The disturbed region boundary for bottom rotations is defined as the point at which the streamline of bottom rotations with a distance of (0.5 Btan \(\theta\)) from the step vertical face is exactly parallel to the converging sidewall (see Figure 6f). Based on these definitions, the widths of the disturbed regions for free-surface, mainstream and bottom rotations are represented by \(W_{f}\), \(W_{m}\) and \(W_{b}\) in Figures 4-6, respectively. The disturbed region width increases from free-surface to mainstream to bottom rotations, implying that \(W_{f}<W_{m}<W_{b}\) persists at all times. Figure 7 shows the dimensionless disturbed region width of \(W_{f}/h_{c}\) for free-surface varying along the spillway. In general, with increasing step number \(n\), \(W_{f}/h_{c}\) climbs fast and  
Figure 6: Bottom rotations above the: (**a**) uniform-width stepped spillway; (**b**) converging stepped spillway with \(\theta=48.3^{\circ}\), \(B=22.5\) mm, \(h_{c}/B=3.36\) and \(\Phi=22.6^{\circ}\). Different shapes of bottom rotation: (**c**) 2-dimensional rotations; (**d**) cone-shape rotations; (**e**) chaotic-shape rotations; (**f**) definition of disturbed region boundary for bottom rotations. reaches its maximum within the first half of the spillway; then, it remains almost constant or begins to fall as the spillway approaches its exit. In scenarios with steep spillway slope and big convergence angle (\(\theta=48.3^{\circ}\) and \(\Phi=22.6^{\circ}\)), \(W_{f}/h_{c}\) grows continually along the spillway and reaches a maximum at the spillway exit. \(W_{f}/h_{c}\) is observed to be higher when the spillway slope \(\theta\) is milder and the convergence angle \(\Phi\) is larger. \(W_{f}/h_{c}\) increases somewhat as \(h_{c}/B\) decreases, indicating that \(h_{c}/B\) has a limited effect on \(W_{f}/h_{c}\). Figure 8 shows the dimensionless disturbed region width of \(W_{m}/h_{c}\) for mainstream. For all cases, \(W_{m}/h_{c}\) increases constantly with increasing \(n\) and reaches its maximum at the spillway outflow. With decreasing spillway slope \(\theta\) and dimensionless critical flow depth \(h_{c}/B\) as well as rising convergence angle \(\Phi\), the maximum \(W_{m}/h_{c}\) increases. For the mild and moderate spillways with \(\theta=11.3^{\circ}\) and \(30.0^{\circ}\), the growing rate of \(W_{m}/h_{c}\) is large when close to the spillway entrance and gradually reduces with increasing \(n\). For the steep spillways with \(\theta=48.3^{\circ}\), the increasing rate of \(W_{m}/h_{c}\) is small when \(n\leq 30\), but grows significantly further downstream. The reason for the different increasing rate of \(W_{m}/h_{c}\) is that the spread of the sidewall disturbance in the spillway width direction is related and limited by the inertial force of flow that increases with flow velocity, while it is promoted by the vortex generated from the free-surface by the sidewall squeezing (shown in Figure 4b). For the mild and moderate spillways with \(\theta=11.3^{\circ}\) and \(30.0^{\circ}\), the flow velocity and inertial force are minimal when close to the spillway entry and gradually rise with increasing \(n\), which is opposed to the growing rate of \(W_{m}/h_{c}\).