## 1. Introduction

[2] The wide range of grain sizes found in most rivers, especially gravel bed rivers, complicates the problem of the prediction of bedload transport rate. As stated by *Wilcock and Kenworthy* [2002], grain size influences sediment transport in two different ways. For given flow conditions above a bed of homogeneous sediment, transport is controlled by the absolute size of grains, small grains being more mobile than large ones. However, when the sediment bed is a mixture of different grain sizes, relative size effects tend to increase the transport rate of larger grains and decrease the transport rate of smaller grains [*Wilcock*, 1993, 2001; *Wilcock and Kenworthy*, 2002]. This effect is very sensitive to the composition of the mixture which can change during transport and in response to variations in flow and sediment supply. Relative size effects thus influence the transport rate of each individual size and, consequently, the overall transport rate in a gravel bed river [*Kuhnle and Southard*, 1988; *Wilcock and Crowe*, 2003; *Parker*, 2008].

[3] A first approach to tackle this difficult problem consists of avoiding part of the difficulties associated with specifying individual size distribution by predicting the total transport rate as a function of a single representative grain size *D*, usually the median diameter *D*_{50}of the grain-size distribution [*Meyer-Peter and Müller*, 1948]. Within the framework of this approach, the volumetric transport rate per unit river width *q*_{s} is related to the flow shear stress *τ* by the *Meyer-Peter and Müller* [1948] equation,

where *c* is a dimensionless coefficient, is a dimensionless transport rate called the Einstein number and *θ* is a dimensionless shear stress called the Shields number. These two dimensionless numbers are defined by

where *g* is the gravitational acceleration, *ρ* is the fluid density, *ρ*_{s} is the sediment density and *R* = (*ρ*_{s} − *ρ*)/*ρ.* The critical Shields number *θ*_{c} is the value of *θ* below which sediment transport ceases [*Shields*, 1936; *Meyer-Peter and Müller*, 1948]. Equation (1) is practical because the only sediment data required are the representative size and the value of *θ*_{c}. It is however unable to predict changes of grain size distribution in the river.

[4] A second approach consists of discretizing the grain-size distribution into a finite number*N* of populations of characteristic grain size *D*_{i} and investigating how the transport rate of each population, *q*_{s,i}, depends on the shear stress and the grain-size distribution at the bed surface [*Parker*, 2008]. It is usually observed that 1) *q*_{s,i} increases with the proportion of grains belonging to the ith population within the surface layer, *f*_{i}; 2) far from the threshold of motion, where *θ*_{i} = *τ*/*ρRgD*_{i} is the Shields number calculated for the grain size *D*_{i} [*Liu et al.*, 2008]. Consequently, many authors hypothesized a relation of the following form [*Wilcock*, 1988; *Parker*, 2008]:

where is the dimensionless transport rate of the grains of size *D*_{i} and *F*(*θ*_{i},*θ*_{c,i}) is a function capturing the complexity introduced by the grain-size distribution. Among other things,*F*(*θ*_{i},*θ*_{c,i}) depends on the critical Shields number of incipient motion associated with the grains of size *D*_{i}, *θ*_{c,i}, which depends in turn on the grain-size distribution at the bed surface. Within the framework of this approach, authors have therefore focused on the determination of both*F*(*θ*_{i},*θ*_{c,i}) and *θ*_{c,i} from experimental data and theoretical modeling.

[5] *Parker et al.* [1982], for example, proposed that

where *a*, *b*, *l*, *m* and *p* are adjustable coefficients depending on the configuration investigated [*Parker et al.*, 1982; *Parker*, 1990; *Wilcock and Kenworthy*, 2002; *Wilcock and Crowe*, 2003; *Powell et al.*, 2001, 2003]. As the Shields number approaches the value of incipient sediment motion, sediment transport becomes sensitive to local bed heterogeneities and pavement effects, which may not be captured by (5). This led to the formulation of several transport laws specifically designed to describe sediment transport in the vicinity of the threshold [*Parker et al.*, 1982; *Parker*, 1990; *Wilcock and Kenworthy*, 2002; *Wilcock and Crowe*, 2003].

[6] The prediction of the critical Shields number of the grains of size *D*_{i}, *θ*_{c,i}, rests on a qualitatively well observed phenomenon: the ability of a grain of size *D*_{i} to move with respect to some representative grain size *D*_{c} increases with the ratio *D*_{i}/*D*_{c} [*Parker*, 2008]. This is usually formulated by a relation of the form

where *θ*_{c,c} = *τ*_{c,c}/*ρgD*_{c} is the critical Shields number of the representative grain size, *τ*_{c,c} is the value of the shear stress at which grains of size *D*_{c} start moving and *G*is the so-called hiding function which incorporates relative size effects [*Kirchner et al.*, 1990; *Buffington et al.*, 1992].

[7] The frequently cited forms of the hiding function *G*(*D*_{i}/*D*_{c}) are listed in Table 1. Most of them reduce to

with 0 ≤ *γ* ≤ 1. Equation (7) presents two interesting limiting cases. The first one corresponds to *γ* = 0 which implies *θ*_{c,i}/*θ*_{c,c} = 1. In this case, a grain of given size *D*_{i} within a mixture has exactly the same critical Shields number as it would have if the bed was composed entirely of grains of size *D*_{i}. In such a scenario, the initiation of transport of sediment mixtures is highly selective, based on grain size. The second limiting case corresponds to *γ* = 1 which implies *θ*_{c,i}/*θ*_{c,c} = *D*_{i}/*D*_{c} so that *τ*_{c,i} = *τ*_{c,c} where *τ*_{c,i} is the value of the shear stress at which grains of size *D*_{i} start moving. In this limiting case, the effect of the mixture is to equalize the threshold of motion so that all grains are mobilized at the same absolute boundary shear stress, a configuration referred to as “equal mobility”. In practice, sediment mixtures appear to behave in between these two limiting cases.

Hiding Function | Reference | Remark |
---|---|---|

- a
*Egiazaroff*[1965] and*Ashida and Michiue*[1973] define*D*_{c}as the arithmetic mean surface grain size whereas*Parker et al.*[1982],*Parker*[1990] and*Wilcock and Crowe*[2003] used the geometric mean surface grain size.
| ||

Egiazaroff [1965] | Theoretical modeling | |

for | Ashida and Michiue [1973] | Based on experimental data |

for | ||

Parker et al. [1982] | Based on field data | |

Powell et al. [2001, 2003] | Based on field data | |

with | Wilcock and Crowe [2003] | Based on experimental data |

[8] The transport laws discussed so far rely on the fit of empirical sediment transport rate curves. Consequently the physical meaning of the coefficients involved in equations (5) and (7) is unclear. An alternative approach to the problem of bedload transport is to consider that *q*_{s,i} can be written

where *n*_{i} (dimensions [*L*]^{−2}) is the surface density of moving particles of the *i*th population defined as the number of particles of size *D*_{i} in motion per unit bed area, *V*_{i} is their averaged velocity and *δv*_{i} is the volume of an individual particle. A better insight in the problem of bedload transport can be gained from the separate measurements of *V*_{i} and *n*_{i} and the determination of their dependence on the control parameters. This approach has motivated several experimental investigations of bedload transport at the grain scale and the development of statistical transport models [*Einstein*, 1937, 1950; *Francis*, 1973; *Fernandez-Luque and Van Beek*, 1976; *Abbott and Francis*, 1977; *Lee and Hsu*, 1994; *Ancey et al.*, 2008; *Ganti et al.*, 2009; *Ancey*, 2010; *Furbish et al.*, 2012a, 2012b, 2012c; *Roseberry et al.*, 2012].

[9] *Lajeunesse et al.* [2010], in particular, reported the results of an experimental investigation of the motion of bedload particles, under steady and spatially uniform turbulent flow above a flat sediment bed of uniform grain size. Using a high-speed video imaging system to investigate the trajectories of the moving particles, they found that: (1) the surface density of moving particles increases linearly with (*θ* − *θ*_{c}),

where *α* = 4.6 ± 0.2 is a coefficient related to the erosion and deposition rate of bedload particles and *σ* = 1/*D*^{2} is the number of particles at repose per unit surface of the bed; (2) the average particle velocity increases linearly with , with a finite nonzero value at threshold,

where is a characteristic sedimentation velocity and *β* = 4.4 ± 0.2 and *V*_{c}/*V*_{s} = 0.11 ± 0.03 are two fitting coefficients.

[10] As far as we know, experiments similar to those of *Lajeunesse et al.* [2010] have been carried out with homogeneous sediment beds only. In this paper, our objective is therefore to extend the experimental approach of *Lajeunesse et al.* [2010] to the case of a mixture of sediments of different sizes. To this end, we investigate experimentally bedload transport above a bed composed of a bimodal mixture of small and large grains sheared by a turbulent flow in a small experimental flume. We focus on the measurement of the surface density of moving particles and of their average velocity for each size fraction. The experimental results allow us to characterize how sediment transport depends on the proportion of small and large grains on the bed.