## 3 Results  
### Flow Pattern  
Figure 3 shows the typical skimming flow pattern of the converging stepped spillway, including the free-surface (air concentration \(C\) = 0.9), streamlines in the mainstream (originating at the half flow depth at the spillway entry) and the bottom rotations between the mainstream and the step surface. For the free-surface, the standing wave is generated since the water depth along the converging walls rapidly increases. For the mainstream, the streamlines close to the sidewall are practically parallel to the sidewall, and their direction quickly changes as they move away from the sidewall, eventually becoming parallel to the central plane. The bottom rotations exhibit a 2-dimensional profile at the central plane, which is remarkably similar to the rotation structure seen on uniform-width stepped spillways [24]. The bottom rotations lengthen as they move away from the central plane, forming a cone shape with the sharp apex facing the sidewall. The bottom rotations become entirely deformed and exceedingly chaotic 3-dimensional profiles as they move closer to the sidewall. Figure 4a depicts the top view of the skimming flow through the converging stepped spillway, and the free-surface color is coded by the flow depth \(h\) perpendicular to the pseudo bottom (i.e., the line linking the step edges). Because of the increased flow velocity, the flow depth along the central plane falls throughout the spillway, and the flow pattern is similar to that of stepped spillways with uniform-width. The standing wave begins at the first step and continues to the spillway outlet, with the standing wave's height gradually increasing along the spillway. Figure 4b shows the cross sections of standing wave on the edge of various steps, and the cross section color is coded by the air concentration \(C\). The water depth grows slowly at first, then rapidly, and eventually gently as the distance to the converging walls decreases. The place with the highest growing rate of water depth can be considered the standing wave's boundary. Due to the increase of the unit discharge along the spillway and the squeezing of the sidewall, the standing wave generally rises taller and wider. A vortex is created from the free-surface near the standing wave boundary during steps 20 to 35, and it enters the standing wave and advances toward the sidewall as \(n\) increases. The vortex causes air entrainment and leads to a relatively high aeration region in the standing wave, which is supported by the experimental observation of [PERSON] et al. [15] on the converging stepped spillway. Figure 3: Typical skimming flow pattern of the converging stepped spillway with \(\theta\) = 48.3\({}^{\circ}\), \(B\) = 22.5 mm, \(h_{c}/B\) = 3.36 and \(\Phi\) = 22.6\({}^{\circ}\). Blue surface: free-surface; red lines: mainstream; green and blue rotating lines: bottom rotations. Figure 5 compares the streamlines of the mainstream over the converging stepped spillway and the stepped spillway with uniform-width. For the uniform-width stepped spillway, all streamlines are parallel to the central plane, while for the converging stepped spillway, the streamlines near the converging sidewall are roughly parallel to the sidewall and have considerable angles with the central plane. As the distance from the sidewall increases, the streamlines gradually shift from parallel with the sidewall to parallel with the central plane. Figure 6 compares the bottom rotations above the uniform-width stepped spillway (Figure 6a) and the converging stepped spillway (Figure 6b). Most of the bottom rotations in the uniform-width stepped spillway exhibit 2-dimensional properties such as a rotating disc parallel to the central plane and are not stretched in the width direction of the spillway.