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

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.

The streamwise velocity distribution in a two-stage channel with ice cover

“…where t xy ¼ rz yx @U=@y; z yx ¼ jHU Ã ; U Ã is the shear velocity; j is the eddy viscosity coefficient; H is the water depth (Nie et al 2017). The U Ãa , and U Ãb are shear velocity of ice cover and channel bed that can be obtained by solving follow equations (Parthasarathy & Muste 1994)…”

Evaluating Regime Change of Sediment Transport in the Jingjiang River Reach, Yangtze River, China

“…According to Fick's First Law, particles may make "local" jumps, as most particle jumps induced by turbulence are constrained to a small distance (∆y) in a given ∆t according to the Central Limit Theorem (CLT), and ∆y is characterized by the length of Representative Elementary Volume (REV) [27,28]. However, modern observations of coherent structures in turbulence have proven that high-speed currents may intermittently sweep the bed, carry much sediment, and eject directly into the upper part of the water column [27,29].…”

Development of Inverse Mathematical Model of Rouse Formula to Estimate Suspended Sediment Concentration Along Width of Channel

“…In 2017, seven mathematical models were reviewed including fractional Advection-Diffusion Equation (fADE) model and an innovative method for vertical distribution of concentration of suspended sediment was derived. Their approach correlates sensitivity of fractional index of fADE model with size and settling velocity of sediment particles [7]. In 2018, a method was derived appropriate expression for depth-averaged β; modified the expression of sediment and turbulent diffusion coefficient and also establish the relation between β and normalized settling velocity for finding suspended sediment concentration by selecting sediment diffusion coefficient on based of finite mixing length theory through turbulent mixing process [8] a 2-D numerical model for vertical distribution of concentration of suspended sediment from channel bed to water surface [9].…”