A Moment-Based Chezy Formula for Bed Shear Stress in Varied Flow

Elgamal, Mohamed

doi:10.3390/w13091254

Cited by 2 publications

(6 citation statements)

References 22 publications

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“…The polynomial coefficients depend notably on two velocity scales-the depth-averaged velocity and moment-based integral velocity-alongside the dimensionless Chezy coefficient. The moment concept facilitates the recovery of vital velocity profile details overlooked by conventional depth-averaged models, all while avoiding the computational complexity of 2D vertical models [23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Mapping Mean Velocity Field over Bed Forms Using Simplified Empirical-Moment Concept Approach

Elgamal

2023

Water

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The log-wake law was successful in mapping velocity fields for uniform flow over flat surfaces, even in cases of wake effects (velocity dips, wall effects, and secondary currents). However, natural riverbeds with undulations and bedforms challenge these models. This study introduces a moment-based empirical method for rough estimation of the velocity fields over stationary 2D bedforms. It proposes three polynomial velocity profile templates (first, fifth, and eighth orders) with coefficients deduced analytically while taking into account an array of flow conditions and assumptions, including slip velocity at the bed, mass and moment of momentum conservations, imposing inviscid potential flow near the water surface, and incorporation of near-bed shear stress utilizing a moment-based Chezy formula. Remarkably, the coefficients of these polynomials are primarily reliant on two crucial velocity scales, the depth-averaged velocity (uo) and the moment-derived integral velocity (u1), along with the dimensionless reattachment coefficient (Kr). Validation of the proposed approach comes from ten lab experiments, spanning Froude numbers from 0.1 to 0.32, offering empirical data to validate the obtained velocity profiles and to establish the relationship of the spatial variation in the normalized u1 velocity along bedforms. This study reveals that the assumption of a slip boundary condition at the bed generally enhances the accuracy of predicted velocity profiles. The eighth-order polynomial profile excels within the eddy zone and close to reattachment points, while the fifth-order profile performs better downstream, approaching the crest. Importantly, the efficacy of this approach extends beyond water flow to encompass airflow scenarios, such as airflow over a negative step. The research findings highlight that linear velocity, as employed in Vertically Averaged and Moment models (VAM), exhibits approximately 70% less velocity mismatch compared to constant Vertically Averaged (VA) models. Moreover, the utilization of the fifth-order and eighth-order velocity profiles results in substantial improvements, reducing velocity mismatch by approximately 86% and 90%, respectively, in comparison to VA models. The insights gained from this study hold significant implications for advancing vertically averaged and moment-based models, enabling the generation of approximate yet more realistic velocity fields in scenarios involving flow over bedforms. These findings directly impact applications related to sediment transport and mixing phenomena.

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Section: Introductionmentioning

confidence: 99%

Mapping Mean Velocity Field over Bed Forms Using Simplified Empirical-Moment Concept Approach

Elgamal

2023

Water

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“…On the other hand, the models overlook the vertical velocity details due to the depth-averaging approximation and, thus, these models only use one velocity scale, the depth-averaged velocity, in each flow direction. As a result, depth-averaged models were not able to accurately map the spatial variation in the flow and turbulence structure over a varying bed terrain, such as the case of shallow water flowing over a train of bedforms and dunes [3,4].…”

Section: Introductionmentioning

confidence: 99%

“…The coefficient α in Equation ( 25) provides the ratio between the integral velocity, u 1 , and the mean velocity, U o , in the case of uniform flow over a flat bed [3]. It is known that the log-law could be used to predict the velocity profile in the inner region in the case of uniform flow.…”

Section: Introductionmentioning

confidence: 99%

“…When the flow is accelerating, the shape of the velocity becomes more uniform and consequently, u 1 becomes small compared to its corresponding value in the case of decelerating flow, Figure 2. For more details, refer to [3].…”

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confidence: 99%

“…uniform and consequently, 𝑢 becomes small compared to its corresponding value in the case of decelerating flow, Figure 2. For more details, refer to [3]. As it will be discussed later on, the new integral-moment velocity scale 𝑢 will have an important role in the development of the new k-ε model.…”

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confidence: 99%

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A Moment-Based Depth-Averaged K-ε Model for Predicting the True Turbulence Intensity over Bedforms

Elgamal

2022

Water

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Turbulence models are critical for depth-averaged flow models in at least two ways: (i) as closures for momentum equations and (ii) as indicators of the spatial variability in the turbulence intensity field, which is crucial for sediment transport and bedform evolutions. This paper introduces a novel moment-based depth-averaged k-ε turbulence (MDAKE) model that could be considered as a revised version for the standard k-ε Rastogi–Rodi (SDAKE) model and can be used to estimate the true values for the depth-averaged turbulence kinetic energy in more complex and varied flow conditions with accelerating–decelerating flow fields. The study in hand shows that the SDAKE model tends to overestimate the true depth-averaged turbulent kinetic energy () by 50 to 130% in the benchmark case of uniform flow over a flatbed. Further, the SDAKE model assumes that the bed shear velocity is an appropriate scale for the generation terms of both turbulent kinetic energy and dissipation. When bed topographic features vary, a shear flow zone is formed and the assumption is invalid. Since most of the turbulence is generated by shear flow zones away from the bed, the SDAKE model’s estimates for the depth-averaged turbulent kinetic energy field are out of phase with measurements for the flow over a train of bedforms. Therefore, a newly developed depth-averaged KE model based on the moment concept (MDAKE) is presented here. The model replaces bed shear velocity with the integral moment velocity scale (). The calibrated MDAKE model is used to predict turbulent kinetic energy over a train of bedforms. The results of the MDAKE model are in phase and generally in reasonable agreement with the measurements.

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A Moment-Based Chezy Formula for Bed Shear Stress in Varied Flow

Cited by 2 publications

References 22 publications

Mapping Mean Velocity Field over Bed Forms Using Simplified Empirical-Moment Concept Approach

Mapping Mean Velocity Field over Bed Forms Using Simplified Empirical-Moment Concept Approach

A Moment-Based Depth-Averaged K-ε Model for Predicting the True Turbulence Intensity over Bedforms

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