Vertical Distribution of Suspended Sediment under Steady Flow: Existing Theories and Fractional Derivative Model

Shiqian Nie,Wen Chen,Dong Chen,Li Chen,Hongguang Sun,Yong Zhang,Sydney Schaefer

doi:10.1155/2017/5481531

Abstract

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.

Highlights

The vertical distribution of suspended sediment concentrations in steady turbulent flow is an important measure when evaluating the suspended flux in natural rivers and canals [1,2,3]
In the Wang model, a particle velocity distribution function is obtained in the equilibrium state or in a dilute steady state for a particle in two-phase flow, and a theoretical model for the particle concentration distribution is derived from the kinetic theory
The Rouse model, Model 1 (M1), the fractional advection-diffusion equation (fADE) model, Wang model, Model 2 (M2), the power law model, and the two-phase flow model offer a similar description of the vertical distribution between heights 0.05h and 0.9h

Summary

Introduction

The vertical distribution of suspended sediment concentrations in steady turbulent flow is an important measure when evaluating the suspended flux in natural rivers and canals [1,2,3]. The physical mechanism of a steady sediment suspension distribution is a dynamic equilibrium of vertical fluxes between downward sediment settling and upward turbulent diffusion. As a milestone in the history of sediment transport, the Rouse formula (3) has been widely used for decades [4, 8, 28] Limitations of this theory are obvious: the sediment concentration is calculated as zero at the water surface and infinity at the riverbed. The fractional advection-dispersion equation (fADE) model has been developed to describe anomalous diffusion of sediment [21, 36,37,38]. As an extension of the traditional advection-dispersion equation, the fADE model can describe the anomalous diffusive characteristics of suspended sediment in turbulence, for example, nonlocal displacement or superdiffusion in certain circumstances such as turbulence bursting. The aim is to help river engineers in estimating the vertical distribution of suspended sediment concentration via the fADE model in real-world applications

Improved Models Based on the Rouse Formula

Comparison and Discussion

Fractional Index of the fADE Model and Hydraulic Parameters

Conclusions