Citation: Nóbrega, J.D.; Matos, J.; Schulz, H.E.; Canelas, R.B. Smooth and Stepped Converging Spillway Modeling Using the SPH Method. Water 2022, 14, 3103. https:// doi.org/10.3390/w14193103 Academic Editor: Giuseppe Oliveto Received: 6 August 2022 Accepted: 26 September 2022 Published: 2 October 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). water Article Smooth and Stepped Converging Spillway Modeling Using the SPH Method Juliana D. Nóbrega 1, *, Jorge Matos 2 , Harry E. Schulz 3,4 and Ricardo B. Canelas 5 1 School of Civil and Environmental Engineering, Federal University of Goiás, Goiânia 74605-220, GO, Brazil 2 Civil Engineering Research and Innovation for Sustainability, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal 3 Department of Hydraulics and Sanitation, São Carlos School of Engineering, University of São Paulo, São Carlos 13566-590, SP, Brazil 4 Hydro Engineering Solutions, Auburn, AL 36830, USA 5 Bentley Systems, 1990-208 Lisbon, Portugal * Correspondence: juliana.nobrega@ufg.br Abstract: Three-dimensional (3D) simulations using the smoothed particle hydrodynamics (SPH) method were performed for smooth and stepped spillways with converging walls, in order to evaluate the influence of the wall deflection and the step macro-roughness on the main non-aerated flow properties. The simulations encompassed a 1V:2H sloping spillway, wall convergence angles of 9.9 ◦ and 19.3 ◦ , and discharges corresponding to skimming flow regime, in the stepped chute. The overall development of the experimental data on flow depths, velocity profiles, and standing wave widths was generally well predicted by the numerical simulations. However, larger deviations in flow depths and velocities were observed close to the upstream end of the chute and close to the pseudo-bottom of the stepped invert, respectively. The results showed that the height and width of the standing waves were significantly influenced by the wall convergence angle and by the macro-roughness of the invert, increasing with a larger wall deflection, and attenuated on the stepped chute. The numerical velocity and vorticity fields, along with the 3D recirculating vortices on the stepped invert, were in line with recent findings on constant width chutes. Keywords: non-aerated flow region; skimming flow; smooth spillway; smoothed particle hydrodynamics; standing wave; stepped spillway 1. Introduction Converging transitions along discharge channels, i.e., having a terminal structure narrower than the crest, may be imposed by site or economy constraints [1]. In light of the scenarios of climate change and urbanization, converging transitions have also been applied in dam rehabilitation projects to increase the discharge capacity of spillways, by extending the spillway crest width [2,3]. A larger crest width presents the advantage of reducing the hydraulic head over the crest, and thus the pressure over the slab for identical discharge [4]. However, the adverse effects of the wall deflection on channels with supercritical flows comprise the occurrence of oblique standing waves, also named oblique hydraulic jumps or oblique shock waves [5,6]. Therefore, in converging chutes, special attention should be given to the design of training walls in order to prevent wave run-up, because of the significant increase in the flow depths near the walls [1,5]. Besides, concerns may arise with the efficiency of the energy dissipation in the stilling basin at the toe of the dam [3,7]. Pioneering research conducted in the 1940s and 1950s on horizontal channels with lateral contraction was carried out by [4,5,8,9], among others. Their experimental investiga- tions and theoretical developments, including the application of the method of character- istics, were relevant for understanding the wave patterns for a range of Froude numbers Water 2022, 14, 3103. https://doi.org/10.3390/w14193103 https://www.mdpi.com/journal/water