Semi-analytical modeling of sediment-laden open-channel flows with the effects of stratification, hindered settling, and eddy viscosities

Abstract

This study proposes semi-analytical models for simultaneous distribution of fluid velocity and suspended sediment concentration in an open-channel turbulent flow using three kinds of eddy viscosities. Apart from the classical parabolic eddy viscosity which is based on a log-law velocity profile, we consider two recently proposed eddy viscosities based on the concept of velocity and length scales. To deal with the flows with high sediment concentration, several turbulent features such as the hindered settling mechanism and the stratification effect are incorporated in the model. The governing system of highly nonlinear differential equations is solved using the homotopy analysis method (HAM), which produces solutions in the form of convergent series. Numerical and theoretical convergence analyses are provided for all three types of eddy viscosities. The effects of parameters on the derived models are discussed physically. Experimental data on both dilute and non-dilute flows are considered to verify the HAM-based solutions. The effects of the stratification correction factor (β) and the turbulent Schmidt number (α) reveal that they should be determined optimally for applicability of the proposed models in terms of accurate prediction with data. This optimal procedure required further investigation of these parameters, and, thus, an analysis of β and α is carried out, which linked them with the particle diameter through particle settling velocity, reference fluid velocity, and reference sediment concentration by proposing regression equations. Furthermore, using the optimal values of the parameters, the proposed models corresponding to the eddy viscosities based on the exponentially decreasing turbulent kinetic energy function and von Karman's similarity hypothesis are seen to be superior to the model corresponding to a parabolic eddy viscosity. Finally, a comment on the HAM is made where it is observed that the method can remove the numerical singularity of the governing equations at the water surface, which arises because of the consideration of vanishing eddy viscosity thereat.

Reinvestigating the Kinetic Model for the Suspended Sediment Concentration in an Open Channel Flow

“…Equation (28) shows that the upward concavity/convexity of the concentration profiles, which is associated with the sign of ∂ 2 C ∂y 2 , is related to the sign of ∂R ∂y…”

“…Before applying these models in real situations, one-dimensional vertical (1-DV) models are validated by comparisons to laboratory measurements of suspended sediment concentrations in open channel flows. Since the pioneer studies of Rouse [8], Hunt [9], and Einstein and Chien [10], different research studies were conducted during recent decades (Elata and Ippen [11]; Coleman [12]; Parker and Coleman [13]; Umeyama [14]; Cao et al [15]; Chiu et al [16]; Guo and Julien [17]; Graf and Cellino [18]; Cao et al [19]; Wright and Parker [20]; Herrmann and Madsen [21]; Toorman [22]; Absi [23]; Pittaluga [24]; Dey et al [25]; Sun et al [26]; Ghoshal et al [27]; Kumbhakar et al [28]) to understand the distribution of sediments in suspension. Despite this large amount of research, most available mathematical models are from the classical approach based on the advection diffusion equation, an ordinary differential Equation (ODE) where concentrations of suspended sediment result from the balance between upward mixing and downward settling fluxes.…”