Prediction of the discharges within exponential and generalized trapezoidal channel cross-sections using three velocity points

A free over-fall can be used as a flow metering hydraulic structure with a single depth measurement of the end. Due to that, theoretical and experimental research has been undertaken on free over-falls with various approaches for different cross-sectional shapes. This paper presents a new and relatively simple theoretical approach for computing the end depth ratio (EDR) and the end depth discharge (EDD) relationships for exponential and generalized trapezoidal channel cross-sections at free over-falls for sub-critical flow regime. The exponential channel is a general channel cross-section which is defined mathematically with a single exponential equation where five widely known prismatic shapes can be generated (rectangular, parabolic, semi-parabolic, triangular and semi-triangular). Similarly, the generalized trapezoidal channel is a geometric shape that is defined mathematically with the 2nd degree equation where seven widely known prismatic channel cross-sectional shapes can be generated (rectangular, triangular, semi-triangular, trapezoidal, semi-trapezoidal, inverted triangular (∆) and semi-inverted triangular). This suggested theoretical approach uses three velocity points at the end (brink) section and the continuity equation to obtain the EDR. To verify the proposed EDR and EDD relationships, relevant experimental and theoretical study results were gathered and statistically examined through the percentage difference and the correlation coefficient (R2). The computed results show very close agreements with the earlier works. For engineers in practice, simple equations are also generated to estimate the direct discharges (Q) using the end depth (ye) for each of the above mentioned channel cross-sections.