The flow depths and velocities along the broad crested weir were analyzed for the condition of two converging walls of ϕ = 9.9° and the stepped chute. Investigations addressing inclined chute contractions were also carried out, as presented by, determining the role of the bottom slope on the flow patterns of the standing waves. Numerical simulations were also undertaken by solving the two-dimensional shallow-water equations or the three-dimensional Reynolds-averaged Navier–Stokes equations. More recent experimental research concerned the evaluation of shock surfaces, velocity profiles and/or turbulent kinetic energy, along with wave diffractors. Their experimental investigations and theoretical developments, including the application of the method of characteristics, were relevant for understanding the wave patterns for a range of Froude numbers and the establishment of design criteria in order to reduce or even eliminate the standing waves. Pioneering research conducted in the 1940s and 1950s on horizontal channels with lateral contraction was carried out by, among others.
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. 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. 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 overall development of the experimental data on flow depths, velocity profiles, and standing wave widths was generally well predicted by the numerical simulations. 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. 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.
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