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Discrete Dynamics in Nature and Society Volume 2017 ,2017-06-28
Vertical Distribution of Suspended Sediment under Steady Flow: Existing Theories and Fractional Derivative Model
Research Article
Shiqian Nie 1 , 2 HongGuang Sun 2 Yong Zhang 3 Dong Chen 4 Wen Chen 2 Li Chen 5 Sydney Schaefer 3
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DOI:10.1155/2017/5481531
Received 2017-03-28, accepted for publication 2017-05-24, Published 2017-05-24
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摘要

The fractional advection-diffusion equation (fADE) model is a new approach to describe the vertical distribution of suspended sediment concentration in steady turbulent flow. However, the advantages and parameter definition of the fADE model in describing the sediment suspension distribution are still unclear. To address this knowledge gap, this study first reviews seven models, including the fADE model, for the vertical distribution of suspended sediment concentration in steady turbulent flow. The fADE model, among others, describes both Fickian and non-Fickian diffusive characteristics of suspended sediment, while the other six models assume that the vertical diffusion of suspended sediment follows Fick’s first law. Second, this study explores the sensitivity of the fractional index of the fADE model to the variation of particle sizes and sediment settling velocities, based on experimental data collected from the literatures. Finally, empirical formulas are developed to relate the fractional derivative order to particle size and sediment settling velocity. These formulas offer river engineers a substitutive way to estimate the fractional derivative order in the fADE model.

授权许可

Copyright © 2017 Shiqian Nie et al. 2017
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

图表

Numerical results of different models in describing the vertical distribution of sediment suspension. Von Karman constant κ=0.4, shear velocity u∗=0.10, sediment settling velocity ω=0.01, water depth h=1.0, a=0.05h, and α=0.9 in the fADE model. For description simplicity, units are not used for parameters in this figure.

Numerical results of different models in describing the vertical distribution of sediment suspension. Von Karman constant κ=0.4, shear velocity u∗=0.10, sediment settling velocity ω=0.04, water depth h=1.0, a=0.05h, and α=0.9 in the fADE model. For description simplicity, units are not used for parameters in this figure.

Numerical results of different models in describing the vertical distribution of sediment suspension. Von Karman constant κ=0.4, shear velocity u∗=0.15, sediment settling velocity ω=0.01, water depth h=1.0, a=0.05h, and α=0.9 in the fADE model. For description simplicity, units are not used for parameters in this figure.

Results of the different models in describing the experimental data of Einstein and Chien [27]. (a) S-2 (d=1.3 mm), the fractional derivative order α=0.98, and the reference height a=0.1h. (b) S-3 (d=1.3 mm), the fractional derivative order α=0.94, and the reference height a=0.1h.

Results of the different models in describing the experimental data of Einstein and Chien [27]. (a) S-2 (d=1.3 mm), the fractional derivative order α=0.98, and the reference height a=0.1h. (b) S-3 (d=1.3 mm), the fractional derivative order α=0.94, and the reference height a=0.1h.

Results of the different models in describing the experimental data of Einstein and Chien [27]. (a) S-8 (d = 0.94 mm), the fractional derivative order α=0.88, and the reference height a = 0.1h. (b) S-9 (d = 0.94 mm), the fractional derivative order α=0.88, and the reference height a = 0.1h.

Results of the different models in describing the experimental data of Einstein and Chien [27]. (a) S-8 (d = 0.94 mm), the fractional derivative order α=0.88, and the reference height a = 0.1h. (b) S-9 (d = 0.94 mm), the fractional derivative order α=0.88, and the reference height a = 0.1h.

Results of the different models in describing the experimental data of Lyn [26]. (a) 1965EQ (d = 0.19 mm), the fractional derivative order α=0.93, and the reference height a = 0.1h. (b) 2565EQ (d = 0.24 mm), the fractional derivative order α=0.97, and the reference height a = 0.1h.

Results of the different models in describing the experimental data of Lyn [26]. (a) 1965EQ (d = 0.19 mm), the fractional derivative order α=0.93, and the reference height a = 0.1h. (b) 2565EQ (d = 0.24 mm), the fractional derivative order α=0.97, and the reference height a = 0.1h.

Results of the Rouse model in describing the experimental data of Einstein and Chien [27] with β=1.75, S-2, S-3 (d = 1.3 mm), and S-8, S-9 (d = 0.94 mm).

The relationship between the fractional index α in the fADE model and the particle size d. The best-fit functions are obtained using the least square fitting.

The relationship between the fractional index α in the fADE model and the settling velocity ω. The best-fit functions are obtained using the least square fitting.

通讯作者

HongGuang Sun.State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu 210098, China, hhu.edu.cn.shg@hhu.edu.cn

推荐引用方式

Shiqian Nie,HongGuang Sun,Yong Zhang,Dong Chen,Wen Chen,Li Chen,Sydney Schaefer. Vertical Distribution of Suspended Sediment under Steady Flow: Existing Theories and Fractional Derivative Model. Discrete Dynamics in Nature and Society ,Vol.2017(2017)

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