Correction of the Manning equation for estimating the energy slope line in unsteady flow in rivers using Particle Swarm Optimization (PSO) algorithm
Year: 2026
Type: erratum
Abstract: Estimating the energy slope is of high importance in the analysis of unsteady flow. While the energy slope in steady free-surface flow can be calculated using the Manning equation, previous studies have applied this equation in unsteady flow analyses as well, which, according to the investigations of this study, leads to significant errors. Therefore, in the present research, using the Manning equation (S f \( = \frac{{{\text{n}}^{2} {\text{V}}^{2} }}{{{\text{R}}^{{{\raise0.5ex\hbox{$\scriptstyle 4$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 3$}}}} }} \) ), a cubic relationship for analyzing unsteady flow in coarse-grained porous media ( \(S_{f} = aV + bV^{2} + c\frac{dV}{{dt}}\) ), real flood data from the Karun River in Iran, and the Particle Swarm Optimization (PSO) algorithm, a novel equation is proposed for calculating the energy slope of unsteady flow in open channels (S f \( = \frac{{{\text{n}}^{2} {\text{V}}^{2} }}{{{\text{R}}^{{{\raise0.5ex\hbox{$\scriptstyle 4$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 3$}}}} }} + c\frac{{dV}}{{dt}} \) ) for the first time. In this approach, the coefficient c is optimized using the PSO algorithm as a function of Sf of the Manning equation in steady flow and separately optimized for the rising and falling limbs of the flood hydrograph. In other words, unlike previous studies that only examined the Manning roughness coefficient to improve the accuracy of the equation, this study calculates the energy slope using real flood data. Subsequently, S f values are computed using both the Manning equation and the proposed equation and compared with observed values. The results show that the mean relative error when using the Manning equation and the proposed equation are 7949% and 1.9%, respectively, indicating the high accuracy of the proposed equation.
Source: Scientific Reports
Authors: Hadi Norouzi, Jalal Bazargan
Institutions: University of Garmsar, University of Zanjan
Language: English
Subfield: Civil and Structural Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,690
OpenAlex ID: W7203829299