Solving Trapezoidal Free Overfall by Inserting a Brink Pressure Effect into Sharp-Crested Weir Theory

Free overfalls offer a simple approach to determining a discharge using a single depth measurement at the end section (brink) of a channel. Fundamental equations based on the sharp-crested weir theory, both continuity and energy, were used to determine the end-depth ratio (EDR) and the end-depth discharge relationships for generalized trapezoidal free overfalls in both subcritical and supercritical flow regimes. Streamline curvature and the brink pressure effects were also incorporated into the aforementioned equations to better describe the flow phenomenon. In subcritical flow regimes, the EDR is the ratio between the water depth at the brink and the critical depth. For trapezoidal channel cross sections, the EDR cannot be expressed as a constant value even when the flow is in a subcritical regime, unlike cases for rectangular ($y_b=0.714y_c$) and triangular ($y_b=0.800y_c$) channel cross sections. In supercritical flow regimes, the EDR is the ratio between the water depth at the brink and the normal (uniform) depth. Hence, the upstream Froude number (${\mathsf{F}}_n$) was involved as a predictive variable. The conventional form of Manning’s equation was used to express the discharge as a function of the depth of water at the brink and the geometric characteristics of the channel cross section. Simple empirical discharge equations were generated for practicing engineers. Verification of the theory with existing theoretical and experimental data was also performed.