Demystifying the Discharge Coefficient for Flow Over Thin Weirs

Weirs are ubiquitous in their application within hydraulic engineering for purposes of both stage regulation and flow measurement. Determination of the discharge passing over a weir structure relies upon the empirically-calibrated discharge coefficient (C_d). The necessity of this coefficient has long been understood due to the development of the weir discharge equation from inviscid orifice flow theory. However, its physical significance in relation to the jet dynamics of weir flow has remained unclear. High resolution computational fluid dynamics simulation and particle image velocimetry results presented here unravel the combined effects of contraction, kinetic energy, and viscous losses due to flow separation on the behavior of C_d over a range of flow conditions and inclined planar geometries. Further insights from pressure and velocity fields reveal the presence of a previously unknown regime of self-similarity in the weir jet dynamics. Implications for the more informed design and operation of future weir structures, and improved methods for flow estimation are discussed.