## 4 Conclusions and Outlook  
The skimming flow over the converging stepped spillway is investigated in this research using a numerical simulation. The results reveal that the converging sidewall has considerable impact on the free-surface, mainstream and bottom rotations. For the free-surface, the standing wave is formed from the first step, and a vortex zone is seen within the standing wave, causing air entrainment and a relatively high aeration region. For the mainstream, the streamlines near the converging sidewall are almost parallel to the sidewall, and as the distance from the sidewall rises, they gradually move to parallel with the central plane. For the bottom rotations, 2-dimensional rotations, cone-shape rotations and chaotic rotations are observed in turn with decreasing distance to the converging sidewall. The streamlines of 2-dimensional rotation and cone-shape rotation are nearly parallel to the central plane. Near the converging sidewall, the bottom rotations are squeezed, and the angles between the rotation streamlines and the central plane are visible. The widths of the disturbed regions for free-surface, mainstream and bottom rotations are defined, obtained and compared. For the free-surface, the dimensionless disturbed region width \(W_{\!f}/h_{c}\) increases to the maximum and then slightly decreases along the spillway. The increasing rate and maximum of \(W_{\!f}/h_{c}\) increase with increasing convergence angle \(\Phi\) and decreasing spillway slope \(\theta\). When the critical flow depth \(h_{c}/B\) increases, the maximum of \(W_{\!f}/h_{c}\) increases, and the maximum point travels to the spillway upstream. For the mainstream, \(W_{\!m}/h_{c}\) continues to increase along the spillway, peaking on the last step. The maximum of \(W_{\!m}/h_{c}\) increases with increasing of \(\Phi\) and decreasing \(h_{c}/B\). For bottom rotations, \(W_{\!b}/h_{c}\) increases sharply to the maximum and then remains constant for \(\theta=48.3^{\circ}\), but \(W_{\!b}/h_{c}\) slightly decreases after reaching the maximum for \(\theta=11.3^{\circ}\) and \(30.0^{\circ}\). The maximum of \(W_{\!b}/h_{c}\) increases with increasing \(\Phi\) and decreasing \(h_{c}/B\), and the maximum point moves upstream with the increase of \(\Phi\) and \(h_{c}/B\). The prediction equations are established for the maximum disturbed region widths of \(\max(W_{\!f}/h_{c})\), \(\max(W_{\!m}/h_{c})\) and \(\max(W_{\!b}/h_{c})\) for free-surface, mainstream and bottom rotations of skimming flow. It is stated that \(\max(W_{\!b}/h_{c})>\max(W_{\!m}/h_{c})>\max(W_{\!f}/h_{c})\) always exists, and \(\max(W_{\!b}/h_{c})\) is nearly 1.5 times as large as \(\max(W_{\!m}/h_{c})\) and 3 times as large as \(\max(W_{\!f}/h_{c})\). Based on the achievements from our numerical study and the experiments of [PERSON] et al. [15] and [PERSON] et al. [18], the disturbed region of skimming flow generated by the converging sidewall potentially causes risks for engineering. The standing waves placing higher demands on the sidewall height design, and the disturbed region has higher flow velocity and lower air concentration than the undisturbed flow region, presenting a challenge to cavitation control and energy dissipation. Our study provides knowledge about the range of the disturbed regions; therefore, appropriate engineering measurements can be taken to provide protection for this region during the spillway design, construction and operation. Furthermore, because spillways are often used to discharge excess flows from a sediment basin [25], the standing waves, the squeezed mainstream and the distorted bottom rotations can change the sediment transport capacity of the skimming flow, and the sediment deposition in the converging stepped spillway and its downstream is noteworthy for future study. Conceptualization, Y.C., S.Q. and H.X. ; methodology, Y.C. and G.L. ; Data curation, Y.C. and G.L. ; Formal analysis, J.F. and X.W. ; Validation, Y.C. and G.L. ; Writing--original draft preparation, Y.C. and G.L. ; Writing--review and editing, S.Q. and H.X.