Abstract
A new measure of entropy based on incomplete information theory has been proposed in the literature for deriving velocity profiles in open-channel flow. This entropy is a one-parameter generalization of the Shannon entropy, which can be recovered through the entropy index value as unity. The approach considered a specific range for the entropy index and determined its value close to 1 by carrying out a data analysis procedure. However, this choice for the index may not justify the applicability of this entropy over the Shannon entropy as it is a particular case for this entropy. The present study extended this approach to both one- and two-dimensional cases for wide and narrow open channels, by considering the index as a varying parameter and calculating it using the second-order moment constraint, which physically represents the hydrodynamic transport of momentum. The derived velocity profile was validated using laboratory and field data and compared with the existing equation. The proposed velocity profile showed a slight improvement over the existing one in the case of wide channels, while for narrow channels, it shows superiority to the other entropy-based equations. The slight improvement for the one-dimensional case can be attributed to the available formulae for the momentum distribution coefficient that are used. Indeed, the proposed approach conceptually justifies the applicability of the modified entropy over the Shannon entropy, as the index takes on non-unity values. The effects of the entropy index on the wide and narrow channel velocity profiles are discussed, which reveals the possible physical explanation of the index through its relationship with shear stress values. Also, for practical purposes, regression equations are proposed, which relate the index with the momentum distribution coefficient. Moreover, the analytical derivation of the velocity profile is based on the series approximation of the Lambert function, which is verified for the considered data sets.
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Data Availability Statement
All the turbulence flow data used in this study are available upon request.
Acknowledgments
The first two authors thank the National Science and Technology Council of Taiwan for financially supporting this research under Contract Nos. MOST 108-2221-E-002-011-MY3 and MOST 111-2221-E-002-057-MY3. We express our sincere thanks to the reviewers for their technical comments, which helped improve the quality of the paper.
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Journal of Hydrologic Engineering
Volume 28 • Issue 10 • October 2023
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© 2023 American Society of Civil Engineers.
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Received: Dec 16, 2022
Accepted: May 15, 2023
Published online: Aug 15, 2023
Published in print: Oct 1, 2023
Discussion open until: Jan 15, 2024
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