## 1 Introduction  
Stepped spillways are spillways with successive steps on the spillway chute. These steps create bottom rotations beneath descending water, which help with energy dissipation and eliminate the need for further energy dissipators downstream. Simultaneously, the flow turbulence created by the steps improves free-surface air-water transfer, reducing the possibility of cavitation. The stepped spillways are generally uniform in width along the flow channel. With increasing discharge, three distinct flow regimes of nappe, transition and skimming have been found, and their flow performances, energy dissipation efficiencies, pressure characteristics and air-water flow properties have been thoroughly investigated [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Another type of stepped spillway includes converging sidewalls that cause the chute width to decrease continually throughout the flow path. These converging stepped spillways can accommodate some unique geological and topographical limits while also improving discharge capacity. The hydraulic performance of the air-water flow over the steps is affected and complicated by sidewall convergence in converging stepped spillways. One typical flow phenomenon occurring on the converging stepped spillways is the creation of standing waves near the sidewalls, especially during the skimming flow regime [11, 12, 13, 14, 15]. [PERSON] et al. [13] experimentally measured the depth of standing waves over a converging stepped spillway with a mild bottom slope of \(\theta=18.4^{\circ}\) and found when the sidewall convergence angle \(\Phi=15^{\circ}\), \(30^{\circ}\) and \(52^{\circ}\), the maximum standing wave depth can reach \(1.0\lambda_{c}\), \(1.75\lambda_{c}\) and \(3.0\lambda_{c}\), respectively. Herein, \(h_{c}=(q^{2}/g)^{1/3}\) represents the critical flow depth at the spillway entrance, \(q\) represents the unit discharge at the spillway entrance, and \(g=9.81\) m\(\cdot\)s\({}^{-2}\) is the gravity constant. As comparison, the flow depth of the region further away from the converging sidewalls should be smaller than \(h_{c}\) because the flow pattern is supercritical with a Froude number \(F_{r}>1\). [PERSON] et al. [14] experimentally found that under a steep slope of \(\theta=45^{\circ}\), the maximum flow depth near the converging sidewall ranges from \(2.35\lambda_{c}\) to \(5.55\lambda_{c}\), which is significantly higher than the values reported by [PERSON] et al. [13] under the milder slope (\(\theta=18.4^{\circ}\)). [PERSON] et al. [15] experimentally studied hydraulic characteristics within the standing waves of skimming flows over a converging stepped spillway of \(\theta=48.3^{\circ}\). They considered the standing wave to be the disturbed region by the sidewall, while the region outside the standing wave to be the undisturbed region, which has the same hydraulic characteristics as the flow over the stepped spillway with uniform-width; they found that the disturbed region has higher flow velocity and lower air concentration than the undisturbed region, and the width of the disturbed region increases with increasing convergence angle. For the differences between the converging and uniform-width stepped spillways, the recent studies mainly focused on the features and width of standing waves above the free-surface [16; 17; 18]. The impact of converging sidewalls on the mainstream and bottom rotations of the skimming flow, on the other hand, is rarely explored. To the best of our understanding, whether the regions of the mainstream and the bottom rotations disturbed by the converging sidewall have the same widths as the standing wave still remains unclear. The disturbed widths of the converging sidewall for the mainstream and bottom rotations should vary depending on the incoming flow discharge and the geometry parameters of the converging stepped spillway, although no research or knowledge exists.