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Lambert W-function Solution for Uniform Flow Depth Problem

Author

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  • Ahmed A. Lamri

    (University of Mohamed Khider)

  • Said M. Easa

    (Toronto Metropolitan University)

Abstract

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.

Suggested Citation

  • Ahmed A. Lamri & Said M. Easa, 2022. "Lambert W-function Solution for Uniform Flow Depth Problem," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 36(8), pages 2653-2663, June.
    • Handle: RePEc:spr:waterr:v:36:y:2022:i:8:d:10.1007_s11269-022-03167-4
    DOI: 10.1007/s11269-022-03167-4
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    References listed on IDEAS

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    1. Pin-Chun Huang & Kuo-Lin Hsu & Kwan Tun Lee, 2021. "Improvement of Two-Dimensional Flow-Depth Prediction Based on Neural Network Models By Preprocessing Hydrological and Geomorphological Data," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 35(3), pages 1079-1100, February.
    2. Majid Niazkar & Seied Afzali, 2015. "Optimum Design of Lined Channel Sections," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 29(6), pages 1921-1932, April.
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