Novel method for calculating flow depth in open channels combining the iteration theory and the curve-fitting technique
DOI:
https://doi.org/10.25165/ijabe.v%25vi%25i.9037Keywords:
critical depth, trapezoidal cross section, direct solutions, combined iteration-curve-fitting method, high-degree polynomial equationAbstract
Calculation of critical depth in open channels or closed conduits is a prerequisite for efficient hydraulic design, operation, and maintenance of irrigation channels and drainage ditches. Determination of critical depth in the trapezoidal cross section is of particular significance as it is one of the most widely used channel sections throughout the world, while no closedform analytical solutions exist. Based on the novel combined iteration-curve-fitting method, the existing equations were unified in the same function model, and two new equations were proposed for directly calculating critical depth in trapezoidal open channels. The maximum absolute relative errors of the two proposed equations are 0.004 94% and 0.165%, respectively, in wide application ranges. Comparison and evaluation of the proposed and existing equations for calculating critical depth in trapezoidal open channels were also presented. The introduction and application of the novel method could make the process of function model establishment much more efficient, which provides more insights into the hydraulics calculation of channels and ditches. Moreover, this paper provides reference for the problems related to the empirical equations of high-degree polynomial equations. Keywords: critical depth, trapezoidal cross section, direct solutions, combined iteration-curve-fitting method, high-degree polynomial equation DOI: 10.25165/j.ijabe.20251804.9037 Citation: Chen C. Novel method for calculating flow depth in open channels combining the iteration theory and the curvefitting technique. Int J Agric & Biol Eng, 2025; 18(4): 190–194.References
Zhao Y F, Wang Z Z, Lu Q. Simplified calculation formulas for critical water depth of horseshoe cross section. Transactions of the CSAE, 2011; 27(2): 28–32. (in Chinese)
Zhang K D, Lyu H X, Zhao Y F. Direct calculation for normal depth and critical depth of circular section tunnel under free flow. Transactions of the CSAE, 2009; 25(3): 1–5. (in Chinese)
Akan A O, Iyer S S. Open Channel Hydraulics (2nd edition). ButterworthHeinemann, Oxford. 2021; 448p.
Swamee P K. Critical depth equations for irrigation canals. Journal of Irrigation and Drainage Engineering, 1993; 119(2): 400–409.
Liu J L, Wang Z Z, Leng C J, Zhao Y F. Explicit equations for critical depth in open channels with complex compound cross sections. Flow Measurement and Instrumentation, 2012; 24: 13–18.
Raikar R V, Reddy M S S, Vishwanadh G K. Normal and critical depth computations for egg-shaped conduit sections. Flow Measurement and Instrumentation, 2010; 21(3): 367–372.
Vatankhah A R, Easa S M. Explicit solutions for critical and normal depths in channels with different shapes. Flow Measurement and Instrumentation, 2011; 22(1): 43–49.
Vatankhah A R. Explicit solutions for critical and normal depths in trapezoidal and parabolic open channels. Ain Shams Engineering Journal, 2013; 4(1): 17–23.
Vatankhah A R. Critical and normal depths in semielliptical channels. Journal of Irrigation and Drainage Engineering, 2015; 141(10). doi: 10.1061/(ASCE)IR.1943-4774.000088.
Varandili S A, Arvanaghi H, Ghorbani M A, Yassen Z M. A novel and exact analytical model for determination of critical depth in trapezoidal open channels. Flow Measurement and Instrumentation, 2019; 68: 101575.
Arvanaghi H, Mahtabi G, Rashidi M. New solutions for estimation of critical depth in trapezoidal cross section channel. Journal of Materials and Environmental Science, 2015; 6(9): 2453–2460.
Cheng T J, Wang J, Sui J. Calculation of critical flow depth using method of algebraic inequality. Journal of Hydrology and Hydromechanics, 2018; 66(3): 316–322.
Prabhata K S, Pushpa N R. Exact equations for critical depth in a trapezoidal canal. Journal of Irrigation and Drainage Engineering, 2005; 131(5): 474–476.
Wang Z Z. Formula for calculating critical depth of trapezoidal open channel. Journal of Hydraulic Engineering, 1998; 124(1): 90–91.
Wang Z Z, Yuan S, Wu C L. A final inquiry on a formula for calculating critical depth of open channel with trapezoidal cross section. Journal of Hydraulic Engineering, 1999; 4: 14–17. (in Chinese).
Vatankhah A R. Uniform flow depth in trapezoidal open channels. Flow Measurement and Instrumentation, 2023; 94: 102458.
Vatankhah A R. General solution of conjugate depth ratio (power-law channels). Journal of Irrigation and Drainage Engineering, 2017; 143(9): 06017009.
Davey K, Al-Tarmoom A, Sadeghi H. A two-experiment approach to hydraulic jump scaling. European Journal of Mechanics B-Fluids, 2025; 111: 215–228.
Fenton J D. Velocity distributions in open channels and the calculation of discharge. Journal of Irrigation and Drainage Engineering, 2025; 151(2): 04025002.
Han Y C, Chu P P, Liang M Y, Tang W, Gao X P. Explicit iterative algorithm of normal water depth for trapezoid and parabolic open channels under ice cover. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2018; 34(14): 101–106. (in Chinese).
Amara L, Achour, B. Delta-perturbation expansion for critical flow depth problem in trapezoidal channels. Flow Measurement and Instrumentation, 2023; 91: 102362.
Vatankhah A R, Jalali M. Comments on “Delta-perturbation expansion for critical flow depth problem in trapezoidal channels”. Flow Measurement and Instrumentation, 2023; 94: 102467.
Lamri A A, Easa S M, Bouziane M T, Bijankhan M, Han Y C. Direct solutions for uniform flow parameters of wide rectangular and triangular sections. Journal of Irrigation and Drainage Engineering, 2021; 147(7): 06021005.
Lamri A A. Easa S M, ASEC M. Closure to “Direct solutions for uniform flow parameters of wide rectangular and triangular sections” by Ahmed A. Lamri, Said M. Easa, Mohamed T. Bouziane, Mohammad Bijankhan, and Yan-Cheng Han. Journal of Irrigation and Drainage Engineering, 2023; 149(1). doi: 10.1061/(ASCE)IR.1943-4774.000172,
Lamri A A, Easa S M. Lambert W-Function solution for uniform flow depth problem. Water Resources Management, 2022; 36: 2653–2663.
Lamri A A, Easa S M, Asec M. Closure to “Explicit solution for pipe diameter problem using Lambert W-Function”. Journal of Irrigation and Drainage Engineering, 2023; 149(7). doi: 10.1061/JIDEDH.IRENG-10141.
Akan A O. Open Channel Hydraulics (First edition). Butterworth Heinemann, Oxford, 2006; 302p.
Elhakeem M. Explicit solution for flow depth in open channels of trapezoidal cross-sectional area: Classic problem of interest. Journal of Irrigation and Drainage Engineering, 2017; 143(7). doi: 10.1061/ (ASCE)IR.1943-4774.0001179.
Dai S B, Yang J J, Ma Y L, Jin S. Explicit formulas of normal, alternate and conjugate depths for three kinds of parabola-shaped channels. Flow Measurement and Instrumentation, 2020; 74: 101753.
Dai S B, Ma Y L, Jin S. Direct calculation formulas for normal depths of four kinds of parabola-shaped channels. Flow Measurement and Instrumentation, 2019; 65: 180–186.
Downloads
Published
How to Cite
Issue
Section
License
IJABE is an international peer reviewed open access journal, adopting Creative Commons Copyright Notices as follows.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
