Demystifying the discharge coefficient for flow over thin weirs and sills

A revised derivation of the discharge coefficient for flows over thin weirs and sills in the limits of wall overflow to a free overfall is given. Using dimensional analysis, we show that the discharge coefficient, Cd , in the classical weir-discharge equation is best understood as a weir Froude number, Frₕ , which accounts for the combined effects of inertia, contraction and viscous energy losses within the flow field. A comprehensive set of experimental data from historical studies is complimented by new data from the authors, featuring both laboratory flume experiments and three-dimensional numerical simulations of weir flows. Synthesis of these data elucidates the interaction between the coupled pressure and velocity fields, and the balance between inertial and contraction effects as Frₕ varies. Analysis of the vertical pressure gradient reveals that the thickness of the nappe initially widens with increasing inertia, but then contracts again towards the free overfall limit due to diminishing flow separation at the base of the weir. These insights allow for a physical explanation of the transition between weir and sill flows using the channel Froude number. Practical limitations on predicting weir discharge and a description of characteristic flow regimes are also set forward.