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 (ks)—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 (ks = 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 δ/ksn with n~0.4. Likewise, the dissipation rate of TKE follows a similar collapse with n~0.3. For the turbulent eddy viscosity, an exponent of n~0.5 was adopted based on dimensional analysis, although the values for the smoothest configuration deviate from the curve.