## 1. Introduction

[2] Turbulent flows can be characterized using time mean flow parameters and corresponding fluctuating intensities, so turbulence effects on sediment transport may generally comprise those due to the time mean flow as well as those associated with the turbulence intensity. However, previous studies of sediment transport are largely based on the information of the mean flow behavior such that most equations for bed load transport rate are only associated with the time mean bed shear stress or flow velocity. This may be because many sediment transport experiments are conducted under uniform flow conditions, where the variation of the near-bed turbulence intensity is usually limited. For example, in uniform open channel flows, the fluctuation or turbulence intensity of the streamwise velocity, expressed as the ratio of its RMS to time mean values, varies from 12% to 32% [*Nezu and Nakagawa*, 1993] and thus turbulence effects on the particle motion may not be easily noticed in such laboratory experiments.

[3] In comparison, many practical situations are associated with widely varying turbulence intensities due to flow unsteadiness or nonuniformity. For example, turbulence in the surf zone is enhanced due to wave breaking, and thus bed sediment particles can be picked up and transported in obviously differing manners from those observed in laboratory open channels. Another example is the process of local scouring around hydraulic structures such as bridge piers. For these two cases, the local turbulence structure is altered significantly such that corresponding sediment transport phenomena are much more complex. Unfortunately, relevant research is lacking in the literature.

[4] *Grass* [1970] appeared the first to directly address the influence of turbulence on instability of individual bed particles. *Girgis* [1977] used fine sand with a mean sieve diameter of 0.143 mm for observing bed load transport on a flat bed, which was induced by turbulent and laminar water flows, respectively. An interesting finding obtained by Girgis was that for the same bed shear stress, the sediment transport rate induced by turbulent flows increased by 30–100% in comparison with that generated by laminar flows [see also *Grass and Ayoub*, 1982]. This result is qualitatively consistent with *Yalin and Karahan*'s [1979] observations on the incipient motion of sediment particles in laminar flows. For the same particle Reynolds number, which varied approximately from 0.7 to 10, Yalin and Karahan found the critical shear stress in laminar flows to be generally larger than that obtained in turbulent flows. This implies that the turbulence fluctuation effectively enhances sediment transport at the incipient condition.

[5] Recently, *Sumer et al.* [2003] performed a laboratory study with controllable shear stress fluctuations to investigate turbulence effects on bed load transport. Their experimental results obtained for the two bed conditions, flat bed and ripple-covered bed, show that the sediment transport rate increases markedly with increasing turbulence levels. In particular, for the flat bed condition, the increased sediment transport rate was found to be closely related to the bed shear stress fluctuation.

[6] This study attempts to perform an analysis of turbulence effects on sediment transport with stochastic considerations. It is noted that stochastic approaches have been adopted previously by several researchers including *Einstein* [1950], *Gessler* [1970], and *Paintal* [1971]. However, these studies actually failed to include variations of turbulence intensity because the relative fluctuation of different random variables, which is defined as the ratio of the RMS to time mean value, was always taken to be constant, for example, 0.5 for the lift force [*Einstein*, 1950], 0.57 [*Gessler*, 1970] and 0.5 [*Paintal*, 1971] for the bed shear stress, and 0.36 for the near-bed velocity [*Cheng and Chiew*, 1998].

[7] For simplicity, we first consider bed load transport in laminar flows. For this extreme condition, the bed shear stress fluctuation reduces to zero, and therefore the sediment transport rate is only subject to randomness related to bed particle arrangement. Then, we assume that the bed load function derived for laminar flows is applicable for computing instantaneous sediment transport in turbulent flows. Given the dimensionless transport rate for laminar flows denoted by ϕ_{L}, and the probability density function of turbulent bed shear stress denoted by f(τ), the sediment transport rate for turbulent flows can be thus expressed as

where τ_{min} and τ_{max} represent the range of the bed shear stress variation. Equation (1) was previously adopted by *Girgis* [1977] [see also *Grass and Ayoub*, 1982], but the proposed approaches for evaluating the two functions, f(τ) and ϕ_{L}, were empirical, which finally caused the verification of equation (1) incomplete. Other formulations similar to equation (1) have also been proposed by *Lopez and Garcia* [2001] for computing the risk of sediment erosion and by *Garcia et al.* [1999] for investigating navigation-induced sediment resuspension.

[8] In this note, equation (1) is analytically evaluated. First, theoretical formulations of f(τ) and ϕ_{L }are introduced so that turbulence effects on bed load transport can be theoretically explored using equation (1). Then, the analytical results obtained are compared with experimental data. To facilitate all analyses conducted in this study, the condition to be considered is limited to a flat bed, which is comprised of uniform sand particles subject to unidirectional flows.