Influence of Step Height on Turbulence Statistics in the Non-Aerated Skimming Flow in Steep-Stepped Spillways

The classical assumption of self-similarity in flow velocities and turbulence statistics has been successfully validated for fully developed flows in open channels, pipes, and boundary layers. However, its application in developing boundary-layer flows in channels with steep slopes and large roughness elements has not yet been thoroughly scrutinized. This study investigates whether turbulence statistics exhibit self-similar behavior when properly scaled in steep-stepped spillways. Specifically, it explores the influence of roughness height ((Formula presented.))—representing the cavity size of a steep-stepped spillway—on turbulence statistics in the non-aerated skimming flow region. Numerical simulations, extensively validated against experimental data, were conducted for a stepped spillway with a fixed slope angle of 51.34°, using five roughness heights ((Formula presented.) = 6.25, 3.12, 1.56, 0.78 and 0.39 cm), corresponding to step height-to-length ratios of 10:8, 5:4, 2.5:2, 1.25:1 and 0.625:0.5, respectively. The results show that the dimensionless profiles of turbulent kinetic energy (TKE) at the step edges collapse onto a single curve when rescaled by a factor of (Formula presented.) with (Formula presented.). Likewise, the dissipation rate of TKE follows a similar collapse with (Formula presented.). For the turbulent eddy viscosity, an exponent of (Formula presented.) was adopted based on dimensional analysis, although the values for the smoothest configuration deviate from the curve.