Skip to main content Skip to navigation

Analytic stage discharge formulae for flow in straight trapezoidal open channels


Analytic stage–discharge formulae are derived for flow in straight trapezoidal channels, based on the 2D analytic velocity distribution in open channels given by Shiono and Knight [Shiono K, Knight DW, Turbulent open-channel flows with variable depth across the channel. J Fluid Mech 1991;222:617–46]. A simple hand-calculation method is provided. Legendre incomplete elliptic integrals of the first and second kinds and a binomial series expansion are used in the derivation of these analytic formulae, together with physically based hydraulic parameters, such as local friction factor (f), dimensionless eddy viscosity (λ) and secondary flow (Γ). The stage–discharge results obtained from the formulae are shown to be in good agreement with experimental data, as are the corresponding analytic velocity and boundary shear stress distributions. The influences of f, λ and Γ on the stage–discharge relationship are also discussed.


Huasheng Liao - State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, Sichuan 610065, China

Donald W. Knight -

Department of Civil Engineering, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK



Advances in Water Resources - Volume 30 Issue 11, November 2007, Pages 2283-2295


Keywords: Boundary shear; Channel flow; Mathematical models; River flow; Turbulent flow; Velocity distribution