Wall Overflow to Free Overfall: Revisiting the Discharge Characteristics of Curvilinear Flows Over Thin Weirs

The two-dimensional free-surface curvilinear flow over a thin weir was of much interest for early pioneers in fluid mechanics, due to the suitability of the problem to potential flow analysis and its practical application within the realm of hydraulic engineering for flow measurement. It is a complex problem, representing an amalgam of several canonical flows. 1D hydrostatic uniform open-channel flow exists upstream of the weir, and then separates into regions of rotational corner flow and convergent flow through an orifice, and finally transitions to a free-falling zero-pressure-gradient jet. More than a century onwards, we examine the same problem using a combination of experimental techniques and numerical simulations. These new data are analyzed in conjunction with the available historical data to examine the discharge characteristics of curvilinear weir flows in the limits of the wall overflow (i.e. where the weir height is very large) to the free overfall. Dimensional analysis reveals that the discharge coefficient in the classical form of the weir-discharge equation is best understood as a weir Froude number, Fr_h. The interaction between the coupled pressure and velocity fields is examined to elucidate the balance between inertial and gravitational effects as Fr_h varies. Based upon this, a new weir-discharge equation is proposed, where a logistic relationship is deduced to estimate Fr_h using readily measured hydraulic parameters. This yields a single monotonic curve over the full range of possible flows. Practical limitations on predicting weir discharge are set forward, defining regimes for scale effects due to viscosity and surface tension, as well as the transition from weir flow to sill flow, and ultimately the free overfall. The findings also hold potential for broader application to other common weir f low types.