Lambert W-function Solution for Uniform Flow Depth Problem

Using the logarithmic law of resistance under the rivers' uniform flow involves implicit governing equations in the normal depth. Therefore, many authors have developed approximate solutions for determining the normal depths for wide rectangular and cosine-shaped sections. This paper presents new, more accurate direct solutions for predicting the normal depths of these sections using the Lambert W-Function and doubly infinite expanded series. The equations are expressed in dimensionless roughness and viscosity, which are functions of five physical parameters: flow discharge, roughness, kinematic viscosity, channel width, and angle of repose of the bed material. The novelty of this paper is as follows: (1) the proposed equations are in the form of fast converging power series, (2) the normal depth equations are developed for the rough and smooth flow regimes, and the transition region between them, and (3) the maximum relative error of the proposed solutions range from 0.01% to 0.8%, compared with 2% to 14% for the existing solutions.

Automated summary

This paper presents new and accurate direct solutions for predicting the normal depths of wide rectangular and cosine-shaped sections using the Lambert W-Function and doubly infinite expanded series. The proposed equations are applicable for both rough and smooth flow regimes as well as the transition region, and have the characteristics of fast convergence.