Effects of Geometry on Strong Free-Surface Vortices in Subcritical Approach Flows

Strong free-surface vortices are employed extensively in the hydraulic engineering industry in areas such as flow regulation, energy dissipation, and energy generation. Despite their long history of use, the literature on strong free-surface vortices appears to lack detailed experimental investigations, particularly with regard to subcritical approach flows. This paper reports a comprehensive experimental program that was implemented on 12 scaled vortex chamber geometries to identify the key dependent hydraulic parameters. Two-dimensional (2D) laser particle tracking velocimetry (PTV) was employed to determine the field circulation, ${\mathrm{\Gamma }}_{\infty }$. It was found that the field circulation and, hence, the circulation number ($N_{\mathrm{\Gamma }}$) is strongly dependent on the approach flow geometry, which was characterized by a nondimensional approach flow factor, $y_{\alpha }$ comprising the approach flow, depth $h/d$, and geometric factor, $\alpha$. The discharge number ($N_Q$) varied inversely with the circulation number following relationships governed by two further empirical parameters: the constant ($k_{\alpha }$) and exponent ($n_{\alpha }$). Specific to each geometry, empirical models that related these terms to the approach flow geometry are presented. These findings collectively deliver an alternative simple model to determine the depth-discharge relationship in vortex flows. The values of the radial Reynolds (Rr) number and Weber number (W) in the experiments suggested that the model should be scalable according to the criteria of previous studies. This has been supported by a validation using two prototype systems reported in the literature producing errors of less than 15%. Finally, two new flow classes in describing vortex flows have been defined: transitionally subcritical, when $0.7<{\mathsf{F}}_i<1$; and unstably subcritical, when the nondimensional approach flow factor $\alpha <1$ for relatively large approach flow depths.