Towards an understanding of the mechanisms leading to air entrainment in the skimming flow over stepped spillways

Modern turbulence models allow for detailed numerical solutions, involving millions of data points and numerous degrees of freedom; these solutions have the potential to provide notable insights into the flows past hydraulic structures. In this paper, we present new analyses of a Detached-eddy simulation of the flow past stepped spillways in three dimensions, in order to investigate the coherent structures conducive to the phenomenon of air entrainment. The analyses focus on the spatial distributions of vorticity and velocity, as well as time series of the vorticity component in the transverse direction. A new index, Vn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_n$$\end{document}, is proposed in order to represent the spatial location of patches of vorticity magnitude, showing by definition that time-averaged values of such index (V¯n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{V}_n$$\end{document}) constitute the fraction of time in which the vorticity exceeds ns-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\;{\hbox {s}}^{-1}$$\end{document}. When such average values are plotted for the central plane of the spillway, they strongly agree with plots of turbulent kinetic energy, conclusively connecting the vorticity patches with the turbulence intensities. The spatial evolution of velocity and vorticity components in a curved surface located at the experimental values of the thickness of the boundary layer indicates an important development of turbulence, manifested by large instantaneous values of the main flow variables. Three-dimensional plots of iso-surfaces of constant V¯n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{V}_n$$\end{document} and of turbulent kinetic energy show a similar growth rate, providing further evidence of the interconnection of variables. Finally, these results suggest that steps “compensate” the decay of turbulence by generating vorticity patches in between the steps, which they then become released to the flow and reach positions close to the free surface.