An Approximation of the Improved Rouse Equation

Abstract

Rouse equation, which was derived from the diffusion theory, is well known in the study of steady state suspended sediment transport over erodible beds. Although this equation being regarded as Rouse law could be applied effectively, it is unrealistic that the concentration at the free surface is always zero. In addition, for deriving the depth-averaged concentration, the numerical integration or the table lookup has to be performed. Bose and Dey[1] improved the Rouse equation using a modified sediment diffusivity in order to overcome the zero value concentration, but this equation can not be integrated analytically yet. In this paper, according to two equilibrium profiles respect to constant and linear diffusion coefficients, an approximate solution of the improved Rouse equation is given using a general weight-averaged method in order to be integrated analytically. Through verification with experimental data, the results show that the approximation of the improved Rouse equation behave generally better than itself, as well as the Rouse equation and van Rijn equation over the whole water depth. It is revealed that, nevertheless some empirical, this approximation is reasonable, and has higher accuracy. Moreover it can be integrated analytically.

Sediment Flux and Its Environmental Implications

“…Ni and his colleagues also demonstrated that most well-known formulas such as those proposed by Rouse (1937), Lane and Kalinske (1941), Hunt (1954), Ananian and Gerbashian (1965), Zagustin (1968), Laursen (1980), and Itakura and Kishi (1980) were merely special cases of the generalized formula under different conditions. This stimulated further studies which are still on-going (Cheng et al, 2013;Kundu and Ghoshal, 2014), the aim being to extend the general expression to an increasingly wide range of applicability.…”

“…Ni and his colleagues also demonstrated that most well-known formulas such as those proposed by Rouse (1937), Kalinske (1941), Hunt (1954), Ananian and Gerbashian (1965), Zagustin (1968), Laursen (1980), andItakura andKishi (1980) were merely special cases of the generalized formula under different conditions. This stimulated further studies which are still on-going (Cheng et al, 2013;Kundu and Ghoshal, 2014), the aim being to extend the general expression to an increasingly wide range of applicability.Given that the suspended sediment transport rate is determined by the integral product of the sediment velocity and concentration over the flow depth, a plethora of formulas have been derived from different mathematical expressions for velocity and concentration profiles (Buyevich, 1990;Rasteiro et al, 1993;Davis and Gecol, 1994;Cheung et al, 1996;Xue and Sun, 2003;Deng et al, 2008;Bai and Duan, 2014).However, various further aspects must also be considered when calculating sediment transport in unsteady flow, in a non-straight channel or in a human disturbed river system (Lenzi and Marchi, 2000;Sammori et al, 2004;Francke et al, 2008;Marttila and Kløve, 2010;Gao and Puckett, 2012;Kabir et al, 2014). For example, although sediment-discharge hysteresis loops have been much analyzed in order to facilitate a better understanding of sediment transport processes, it remains unclear how to characterize accurately the hysteresis using indices (Aich et al, 2014).…”