Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 5481531, 11 pages

https://doi.org/10.1155/2017/5481531

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

^{1}College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing, Jiangsu 210098, China^{2}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^{3}Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA^{4}Key Laboratory of Water Cycle and Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China^{5}Division of Hydrologic Sciences, Desert Research Institute, Las Vegas, NV 89119, USA

Correspondence should be addressed to HongGuang Sun

Received 28 March 2017; Revised 10 May 2017; Accepted 24 May 2017; Published 28 June 2017

Academic Editor: Hengfei Ding

Copyright © 2017 Shiqian Nie et al. 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.

#### 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.

#### 1. 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 [1–3]. The physical mechanism of a steady sediment suspension distribution is a dynamic equilibrium of vertical fluxes between downward sediment settling and upward turbulent diffusion. Nonequilibrium of the two opposing processes may result in erosion or deposition, which consequently brings environmental issues such as water and soil erosion and reservoirs deposition.

The theories of gravity, diffusion, mixing, energy dissipation, similarity, and stochastic models have been applied to study the vertical diffusion of sediment [4–7]. The theories of gravity and diffusion, among others, are the most frequently used. Yet, the gravity theory has been questioned due to the ambiguous definition of “suspension energy” [8]. The diffusion theory has been more widely applied in recent decades since it explains many observations in laboratory and field settings [9–11]. In addition, the two-phase flow system, which has been applied to investigate a variety of problems of interest in fluvial hydraulics, may be described in terms of either macroscopic or microscopic methods [12–18].

The diffusion theory is derived from the traditional advection-diffusion equation (ADE) model by assuming Fick’s first law for sediment diffusion in turbulence:where is the sediment volumetric concentration, is the vertical coordinate, is the sediment settling velocity, and is the sediment turbulent diffusion coefficient along the direction. When downward sediment settling and upward turbulent diffusion reach the balanced state, it results in Here , and is the fluid eddy viscosity and based on the Karman-Prandtl logarithmic velocity profile. The analytical solution of (2) yields the well-known Rouse vertical concentration profile [19]: where is the relative sediment concentration, is a reference concentration at a given height above the riverbed, [] is the water depth, [] is the reference height, [dimensionless] is the Von Karman constant, [] is the shear velocity, and [dimensionless] is a proportionality coefficient related to diffusion of sediment particles.

As a milestone in the history of sediment transport, the Rouse formula (3) has been widely used for decades [4, 8, 28]. However, limitations of this theory are also obvious: the sediment concentration is calculated as zero at the water surface and infinity at the riverbed. Vanoni also stated, in his classical manual [29], that the Rouse formula can only represent the shape of the distribution not the actual values in a predictive sense. Therefore, many researchers have put forward modifications or improvements on the Rouse formula [8, 16, 20–23]. Besides the formula investigated in the next section, there are many valuable works on this topic. For example, Greimann et al. [24, 30] offered both numerical and analytical expressions for concentration profiles, based on two-phase flow analysis. Bombardelli and Jha [31] and Jha and Bombardelli [32–34] established a framework composed of the complete two-fluid model, a partial two-fluid model, and a standard sediment-transport model and further discussed different models in describing three datasets. Toorman [35] derived Eulerian equations for the vertical flux and momentum of suspended particles in dilute sediment-laden open-channel flow in equilibrium using the two-fluid approach.

Recently, the fractional advection-dispersion equation (fADE) model has been developed to describe anomalous diffusion of sediment [21, 36–38]. 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. However, the advantages and parameter determination method of the fADE model in describing suspended sediment transport are still unclear [36–43]. In this study, we investigate the Rouse formula and six improved models (including the fADE model) with explicit expressions. A comparison of these models in describing previously published experimental data has been presented for illustrative purposes. Furthermore, this study develops empirical formulas for accessing the fractional derivative order in terms of particle sizes or sediment settling velocities. The aim is to help river engineers in estimating the vertical distribution of suspended sediment concentration via the fADE model in real-world applications.

#### 2. Improved Models Based on the Rouse Formula

##### 2.1. Model 1 (M1)

To obtain a sediment concentration distribution formula, which can be applied throughout the flow region along the depth in natural rivers, Zhang [20] developed a vertical distribution formula of suspended sediment by solving the diffusion equation accompanied by the velocity distribution formula in sediment-laden flow:

##### 2.2. fADE Model

Numerous studies have shown that particle dynamics in turbulent flow exhibits anomalous diffusive characteristics and can be well described using the fractional Fick’s law [44]. Chen et al. [21] pointed out that turbulence bursting sometimes plays a key role in sediment diffusion and further proposed a fADE model using the fractional derivative to characterize nonlocal particle movement in steady turbulence:where is the fractional derivative order . Considering that the anomalous diffusion occurs in an entire water body, they replaced the depth-dependent coefficient with a depth-averaged diffusivity in (5). New is obtained by integrating Rouse’s [29] expression of from the reference height () to the water surface :In model (5), the definition of fractional derivative is expressed as follows [45]:By employing the theoretical technique of fractional diffusion equation and property of Mittag-Leffler function, the analytical solution of (5) for sediment suspension can be written as follows [39]:in which the Mittag-Leffler function is expressed as

##### 2.3. Model 2 (M2)

The Van Rijn formula is a frequently used method to describe the vertical distribution of suspended sediment. It is a piecewise distribution function for vertical sediment concentration, which considers the interaction between flow and sediment [46]. Zheng et al. [22] further developed a continuous sediment concentration distribution formula by modifying the Van Rijn formula. Then, the vertical distribution formula of suspended sediment can be written aswhere the suspension index .

##### 2.4. Power Law Model

This model was introduced from the turbulent energy theory approach, which yields a sediment distribution expressed as follows [23]: If we employ Prandtl’s logarithmic velocity profile () in formula (11), then the Rouse formula (3) can be achieved. But if we employ the power law type velocity profile in steady flow, where is the maximum velocity and the exponent relates to the Reynolds number and the drag coefficient. In a general case, equals 1/6~1/10, such as [6]. We then obtain a new expression of the vertical sediment distribution by substituting formula (12) into (11):where the exponent .

##### 2.5. Wang Model

A typical theoretical model for the particle concentration distribution is given by Wang and coworkers using the kinetic theory for two-phase flow [14]: If and , the above formula is converted into which is named as the Lane-Kalinske formula.

##### 2.6. Two-Phase Flow Model

Although some two-phase flow models describe the dynamics of suspended sediment well, they have no explicit expression [32, 33]. For small particles and dilute concentrations in sediment-laden flow (i.e., and ), we can use a simplified model of two-phase flow [24, 30]:where , is the reference height, , and is the coefficient accounting for the crossing-trajectory and continuity effects.

#### 3. Comparison and Discussion

A summary of seven existing models including the Rouse formula and its extensions is shown in Table 1. Based on the observation of Table 1, it is clear that the six extended models improve the applicability of the Rouse formula and overcome the main description drawbacks at the riverbed and surface (except for the power law model and the two-phase flow model at the surface). As shown in Table 1, M1, M2, and Wang model have less parameters, while the fADE model, the power law model, and two-phase flow model contain one more parameter than the other models. M2 is obtained by modifying the Van Rijn model, among others. 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 then a theoretical model for the particle concentration distribution is derived from the kinetic theory. The power law model and M1 are obtained by a combination of the velocity distribution formula. In the two-phase flow model, a theoretical expression is obtained by assuming dilute concentrations and small particles. The fADE model is achieved by employing a fractional derivative to characterize anomalous diffusion in turbulence.