Depth-Averaged Specific Energy in Open-Channel Flow and Analytical Solution for Critical Irrotational Flow over Weirs

Free surface flow in open-channel transitions is characterized by distributions of velocity and pressure that deviate from uniform and hydrostatic conditions, respectively. Under such circumstances the widely used expressions in textbooks [e.g., $E=h+U^2/(2g)$ and $h_c={(q^2/g)}^{1/3}$] are not valid to investigate the changes in velocity and depth. A depth-averaged form of the Bernoulli equation for ideal fluid flows introduces correction coefficients to account for the real velocity and pressure distributions into the specific energy equation. The behavior of these coefficients in curvilinear motion at and in the neighbourhood of control sections was not documented in the literature. Herein detailed two-dimensional ideal fluid flow computations are used to characterize the entire velocity and pressure fields in typical channel controls involving transcritical flow, namely the round-crested weir, the transition from mild to steep slope and the free overfall. The detailed two-dimensional ideal fluid flow solution is used to study the behavior of the depth-averaged coefficients, and a novel generalized specific energy diagram is introduced using universal coordinates. The development is used to pursue a simplified critical flow theory for curved flow, relevant to water discharge measurement with circular weirs.