Integrating Equation of Gradually Varied Flow

Patil, R. G.; Diwanji, V. N.; Khatsuria, R. M.

doi:10.1061/(asce)0733-9429(2001)127:7(624)

Cited by 17 publications

(5 citation statements)

References 4 publications

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“…When the left endpoint η a = 0, ε a = 0; at the same time, when ε a = 0, the derivation of 1/f (η a ) is 1. For Equation (20), when the left endpoint is η a = 0, we get ε a = 0; when the right endpoint is η a = 2, the value of η a is greater than the critical point of the transient mixed free-surface-pressure flow in the process of water diversion, so neglect η a = 2. Meanwhile, when η a = 0, the derivation of ε a = f (η a ) is 1.…”

Section: Arched Sectionsmentioning

confidence: 99%

“…Shirley [18] provided the procedure to compute critical depth in compound conduit sections. Recently, Patil et al [19] developed a generalized subroutine to compute the normal and critical depths for all types of conduit shapes using the gradually varied flow computation theory developed by Patil et al [20] and Chow [21]. However, all of these studies have focused mainly on regular channel sections.…”

Section: Introductionmentioning

confidence: 99%

“…(20) where ya is the critical depth in an arched section, m.Additionally, θ = arccos(ηa − 1), so Equation (8) can be written as:…”

mentioning

confidence: 99%

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Explicit Solution for Critical Depth in Closed Conduits Flowing Partly Full

Shang

Zhang

et al. 2019

Water

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Critical depth is an essential parameter for the design, operation, and maintenance of conduits. Circular, arched, and egg-shaped sections are often used in non-pressure conduits in hydraulic engineering, irrigation, and sewerage works. However, equations governing the critical depth in various sections are complicated implicit transcendental equations. The function model is established for the geometric features of multiple sections using the mathematical transform method and while considering non-dimensional parameters. Then, revised PSO algorithms are implemented in MATLAB, and the right solution’s formula for the critical depths in various non-pressure conduit sections is established through optimization. The error analysis results show that the established formula has broad applicability. The maximum relative errors of the formula for critical depths are less than 0.182%, 0.0629%, and 0.170% in circular, arched, and egg-shaped sections, respectively, which are more accurate than those of existing formulas; the form of the formula proposed in this work is also more compact than that of the existing formulas. The results of this research may be useful in design, operation, and maintenance in conduit engineering.

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Section: Arched Sectionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Explicit Solution for Critical Depth in Closed Conduits Flowing Partly Full

Shang

Zhang

et al. 2019

Water

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“…In other words, water depths at two cross-sections of the same profile are given while the distance between these two sections (L) is unknown. According to the literature review, various attempts for solving this problem may be categorized into several groups based on their methods: (1) semianalytical methods [1,2], (2) analytical solutions [3][4][5][6][7], (3) numerical schemes [8][9][10][11], (4) artificial intelligence (AI) models [12], and (5) optimization techniques [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Assessment of Artificial Intelligence Models for Estimating Lengths of Gradually Varied Flow Profiles

2021

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The study of water surface profiles is beneficial to various applications in water resources management. In this study, two artificial intelligence (AI) models named the artificial neural network (ANN) and genetic programming (GP) were employed to estimate the length of six steady GVF profiles for the first time. The AI models were trained using a database consisting of 5154 dimensionless cases. A comparison was carried out to assess the performances of the AI techniques for estimating lengths of 330 GVF profiles in both mild and steep slopes in trapezoidal channels. The corresponding GVF lengths were also calculated by 1-step, 3-step, and 5-step direct step methods for comparison purposes. Based on six metrics used for the comparative analysis, GP and the ANN improve five out of six metrics computed by the 1-step direct step method for both mild and steep slopes. Moreover, GP enhanced GVF lengths estimated by the 3-step direct step method based on three out of six accuracy indices when the channel slope is higher and lower than the critical slope. Additionally, the performances of the AI techniques were also investigated depending on comparing the water depth of each case and the corresponding normal and critical grade lines. Furthermore, the results show that the more the number of subreaches considered in the direct method, the better the results will be achieved with the compensation of much more computational efforts. The achieved improvements can be used in further studies to improve modeling water surface profiles in channel networks and hydraulic structure designs.

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“…Analytical solutions can be used as reference solutions for verifying numerical solutions and numerical programming [17,18]. Patil et al [19] introduced an improved direct method for the Chow method in prismatic channels to solve the GVF equation. The hydraulic parameters in their integration were considered to be variable, unlike the Chow method that the hydraulic parameters were constant at all depths.…”

Section: Introductionmentioning

confidence: 99%

Simple Semi-analytical Solutions Using the Perturbation Method for Gradually Varied Flow Profile in Triangular Channels

Sanayei¹,

Nasiri²

2020

MMEP

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In hydraulic engineering, the steady non-uniform flow in a channel with the gradual changes at the water surface level is introduced as the Gradually Varied Flow (GVF). For the design of open channels, it is necessary to calculate the GVF profile along the channel flow. The GVF profile is described by a nonlinear Ordinary Differential Equation (ODE). Because this equation is strongly nonlinear, providing new analytical and/or semi-analytical solutions for this equation without any simplifications and/or linearizations would be necessary and helpful. In this research, the Perturbation Method (PM) is proposed to present a semi-analytical solution for solving the GVF equation in the prismatic triangular channel. A total of two cases are studied in this paper. In case 1, the Manning equation and in case 2, the Chezy equation are applied as the resistance equations. The GVF profiles in the two cases are compared with the Finite Difference Method (FDM) profiles. Also, the effect of the summation truncation in the PM is studied for these cases. The results show that by increasing the terms approximation in the PM, the GVF profile converges to the FDM profile. A reference solution for efficiency assessment of numerical techniques can be provided by presented semi-analytical solutions in this paper. Furthermore, the proposed method in this paper can be used as a new idea in providing semi-analytical solutions to other open channel works.

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Integrating Equation of Gradually Varied Flow

Cited by 17 publications

References 4 publications

Explicit Solution for Critical Depth in Closed Conduits Flowing Partly Full

Explicit Solution for Critical Depth in Closed Conduits Flowing Partly Full

Assessment of Artificial Intelligence Models for Estimating Lengths of Gradually Varied Flow Profiles

Simple Semi-analytical Solutions Using the Perturbation Method for Gradually Varied Flow Profile in Triangular Channels

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