## 1. Introduction

[2] Bed load transport, which results from the motion of particles rolling, sliding or traveling in a succession of low jumps or “saltations” along the bed of a stream, is of fundamental importance for river morphodynamics. It may indeed represent an important fraction of the total sediment flux transported in a river (up to 60%), especially in gravel bed rivers [*Métivier et al.*, 2004; *Meunier et al.*, 2006; *Liu et al.*, 2008]. Many aspects of morphologic changes in rivers are governed by bed load transport, including bank erosion, bed forms and the rate at which the river incises relief [*Yalin*, 1977; *Dietrich and Smith*, 1984; *Gomez*, 1991; *Graf and Altinakar*, 1996; *Yalin and Ferreira da Silva*, 2001]. Despite the large number of works addressing the problem (and summarized hereafter), bed load transport remains poorly understood to this day with two important consequences for earth sciences: (1) landscape evolution models still rely on empirical laws with no physical basis [*Crave and Davy*, 2001; *Braun and Sambridge*, 1997; *Carretier and Lucazeau*, 2005]; (2) estimates of denudation rates from transport data often neglect the contribution of bed load or involve arbitrary assumptions about its importance [*Ahnert*, 1970; *Summerfield and Hulton*, 1994].

[3] Most of the laws for bed load transport proposed in the literature consist of semiempirical equations derived from a fit of data acquired in flume experiments, with few consideration of the physics at the grain scale. Accordingly, our objective in this paper is to describe bed load transport at the grain scale. This was achieved by developing an experimental apparatus allowing the investigation of the motion of bed load particles under steady and spatially uniform turbulent flow above a flat sediment bed of uniform grain size. As discussed in the following, the originality of our approach with respect to previous investigations is that (1) our experiments are interpreted within the frame of an erosion-deposition model proposed by *Charru* [2006], thus determining the choice of the measured quantities, the measurement methods, and the way to analyze the data, and (2) contrarily to the majority of bed load transport laws proposed in the literature, the model of *Charru* [2006] accounts for a relaxation effect which strongly influences the development of bed forms.

[4] One way to formulate the problem of sediment transport in rivers and to identify the relevant controlling parameters is to proceed to dimensional analysis. The volumetric sediment transport rate per unit river width *q*_{s} is expected to depend on eight parameters involving three dimensions (length, time, mass): the fluid density *ρ*, the sediment density *ρ*_{s}, the kinematic viscosity of water *ν*, the gravitational acceleration *g*, the flow depth *H*, the bed slope *S*, the characteristic sediment diameter *D* (usually the median diameter of the sediment bed), and the shear velocity *u**, defined from the shear stress exerted by the fluid flow on the river bed by *τ* = *ρu**^{2}. Dimensional analysis leads to the following relation for the dimensionless sediment transport rate

where

[5] The Shields number *τ** is a dimensionless shear stress [*Shields*, 1936], and *Re*_{s} is a settling Reynolds number defined with the characteristic settling velocity *V*_{s} = . Note that the particle Reynolds number *Re*_{*} = *u***D*/*ν* is related to the above numbers by *Re*_{*} = *Re*_{s}.

[6] Determining the function *f* which relates *q** to the set of parameters *τ**, *Re*_{s}, *R*, *S* and *H*/*D* has been the goal of a huge number of works either theoretical [*Duboys*, 1879; *Einstein*, 1950; *Bagnold*, 1956, 1973; *Ashida and Michiue*, 1973; *Engelund and Fredsoe*, 1976; *Bridge and Dominic*, 1984], experimental [*Meyer-Peter and Müller*, 1948; *Fernandez-Luque and Van Beek*, 1976; *Wong*, 2003; *Recking et al.*, 2009] or based on the analysis of field data [*Bagnold*, 1980; *Gomez*, 1991]. Commonly cited bed load transport laws are listed in Table 1. Although no relationship has gained universal acceptance, some general features are accepted.

Authors | Transport Rate q* | Comments |
---|---|---|

- a
This list is not exhaustive and other transport formulas can be found in the work by *García*[2006].
| ||

Meyer-Peter and Müller [1948] | 8 (τ* − τ*_{c}) | Derived from a fit of experimental data. |

Wong [2003] | 3.97 (τ* − τ*_{c}) | Derived from a fit of experimental data. |

Einstein [1950] | 12f (τ* − τ*_{c}) | Theoretical derivation; f is a fitting parameter. |

Bagnold [1973] | (τ* − τ*_{c}) | Theoretical derivation; μ is a friction coefficient and V is the average particle velocity. |

Ashida and Michiue [1973] | 17 (τ* − τ*_{c}) ( − ) | Theoretical derivation and fit of experimental data. |

Fernandez-Luque and Van Beek [1976] | 5.7 (τ* − τ*_{c}) | Derived from a fit of experimental data. |

Engelund and Fredsoe [1976] | 18.74 (τ* − τ*_{c}) ( − 0.7) | Theoretical derivation. |

Bridge and Dominic [1984] | (τ* − τ*_{c}) ( − ) | Theoretical derivation and fit of experimental data; μ is a friction coefficient and α is a fitting parameter. |

[7] 1. Most formulas involve a threshold value of the Shields number *τ**_{c} below which no sediment is transported, depending on *Re*_{s} and *S* [*Shields*, 1936; *Wiberg and Smith*, 1987; *Lamb et al.*, 2008].

[8] 2. They all predict the same dependence *q** ∝ *τ**^{3/2} sufficiently far from the threshold i.e., in the limit *τ** ≫ *τ**_{c}.

[9] Two main groups can be distinguished. The first one predicts *q** ∝ (*τ** − *τ**_{c})^{3/2} [*Meyer-Peter and Müller*, 1948; *Einstein*, 1950; *Fernandez-Luque and Van Beek*, 1976; *Wong*, 2003] whereas the second one proposes *q** ∝ (*τ** − *τ**_{c}) ( − ) [*Ashida and Michiue*, 1973; *Engelund and Fredsoe*, 1976; *Bridge and Dominic*, 1984]. These laws provide similar predictions far from threshold but exhibit significant differences close to threshold.

[10] An alternative way to consider the problem of bed load transport is to consider that *q*_{s} can be written

where *n* (dimensions [*L*]^{−2}) is the surface density of moving particles i.e., the number of moving particles per unit bed area, *V* is their averaged velocity, and *δv* is the volume of an individual particle. A better insight into the problem of bed load transport can be gained from the separate modeling of *V* and *n*, that is, the determination of their dependence on the parameters *τ**, *Re*_{s}, *R*, *S* and *H*/*D*.

[11] This approach has motivated several investigations of bed load transport at the grain scale. *Francis* [1973], *Abbott and Francis* [1977] and later *Hu* [1996] and *Lee and Hsu* [1994] investigated experimentally the trajectory of an isolated grain propelled by a water stream over a *nonerodible bed*. A second group of investigators studied the trajectories of bed load particles over an *erodible bed* [*Fernandez-Luque and Van Beek*, 1976; *Van Rijn*, 1984; *Nino and Garcia*, 1994; *Charru et al.*, 2004]. All these authors concluded that small saltation jumps are the main form of bed load motion and tried to characterize the variation with *D* and *τ** of the average saltation height *h*_{s}, saltation length *L*_{s} and particle velocity *V*. Their results are summarized in Table 2. Their conclusions about *h*_{s} and *L*_{s} are quite different. But, with the exception of *Lee and Hsu* [1994], they all found the same dependency for *V*

where *u**_{c} is the threshold shear velocity, and *a* and *b* are positive coefficients related to the effective friction coefficient and an effective fluid velocity at the height of the grains. The values of *a* and *b* differ from one author to the other (see Table 2). *V* is positive so that *b* ⩽ 1. Note that the case *b* = 1 implies that the particle velocity cancels at the onset of sediment transport whereas *b* < 1 corresponds to a nonzero velocity *a*(1 − *b*)*u**_{c} at threshold. Particle motion has also been investigated from numerical integration of the equation of motion of a particle, with simple models for the hydrodynamic forces. These lead to the same qualitative results [see e.g., *Wiberg and Smith*, 1989].

Authors | L_{s} | h_{s} | V | τ* | S | Re_{s} | Re_{*} | H/D |
---|---|---|---|---|---|---|---|---|

- a
Note that the equations of *Fernandez-Luque and Van Beek*[1976],*Abbott and Francis*[1977],*Sekine and Kikkawa*[1992], and*Nino and Garcia*[1994] rely on dimensional analysis whereas the formulas of*Lee and Hsu*[1994] and*Hu*[1996] result from a purely empirical fit of experimental data. The ranges of Shields numbers*τ**, bed slopes*S*, settling Reynolds numbers*Re*_{s}, particle Reynolds numbers*Re*_{*}, and*H*/*D*explored in these experiments are also indicated.
| ||||||||

Fernandez-Luque and Van Beek [1976] | = 16 | V = 11.5 (u* − 0.7u*_{c}) | 0.03–0.64 | 0.21–0.4 | 106–760 | 18–608 | 36–133 | |

Abbott and Francis [1977] | dependency with τ* | dependency with τ* and D | V = a (u* − u*_{c}) | 0.0076–1.556 | 0–0.0184 | 717–3538 | 62–4413 | 3–14 |

no dependency with D | a = 13.4–14.3 | |||||||

Sekine and Kikkawa [1992] | = 3000 · τ*^{1/4} ( − ) | V = 8 · (u*^{2} − u*_{c}^{2})^{1/2} | 0.043–0.233 | 120–3836 | 25–1852 | |||

Nino and Garcia [1994] | = 2.3 | dependency with τ*/τ*_{c} and D | V = a (u* − u*_{c}) | 0.09–0.14 | 0.03–0.07 | 7400–21,900 | 2220–8200 | 2.6–4.7 |

a = 6.8–8.5 | ||||||||

Lee and Hsu [1994] | = 196.3 · (τ* − τ*_{c})^{0.788} | = 14.3(τ* − τ*_{c})^{0.575} | V = 11.53 u* (τ* − τ*_{c})^{0.174} | 0.06–0.5 | 0.002–0.023 | 200–493 | 50–75 | 20–90 |

Hu [1996] | = 27.5 · ()^{0.94}τ*^{0.9} | = 1.78()^{0.86}τ*^{0.69} | V = 11.9 (u* − 0.44u*_{c}) | 0.07–1.67 | 0.001–0.014 | 58–1018 | 15–1315 | 21–60 |

[12] All the bed load transport laws discussed so far establish a relation between the local flow rate of particles and the local shear stress exerted by the fluid flow on the bed. These relations consider implicitly that the particle flux is in equilibrium with the shear stress, so that their use in nonequilibrium conditions, i.e., when the shear stress varies in space or time, is questionable. Indeed, the particle flux does not respond instantaneously to a change of shear stress but adjusts itself with a spatial or temporal delay due to particle inertia or particle settling. This so-called relaxation effect is now recognized to have a strong influence on the development of bed forms, especially ripples [*Charru*, 2006]. In the case of sand particles in air flow, such relaxation effects have been introduced by *Sauermann et al.* [2001] and *Andreotti et al.* [2002] from a phenomenological first-order differential equation for the sand flux, which accounts for an inertial relaxation through an acceleration length ℓ_{acc} = *ρ*_{s}/*ρD* of the grains. In water, whose density is much larger than that of air, *Charru* [2006] argued that this characteristic length would not be the most relevant one, and developed an erosion-deposition model of bed load transport for both viscous and turbulent flows. This model, which is described in section 2, was initially derived from viscous flow experiments [*Charru et al.*, 2004; *Charru and Hinch*, 2006], but has not been tested with turbulent flows. Indeed, despite the plethora of experimental investigations of bed load transport in a turbulent flow, measurements of relevant quantities such as *V* or *n* are either missing or inconsistent (see Table 2).

[13] In this paper, we investigate the motion of individual bed load particles entrained by a turbulent flow in a small experimental flume. Our objectives are to characterize the motion of individual bed load particles, to establish their relation to the macroscopic sediment transport and to analyze the measurements within the frame of the erosion-deposition model proposed by *Charru* [2006]. To this end, we performed systematic measurements of the velocity and surface density of the moving grains, together with the lengths and durations of the particle flights.

[14] The paper is organized as follows. In section 2, we briefly recall the derivation of the erosion-deposition model. Section 3 is dedicated to the description of the experimental setup and procedure. The experimental results are presented in section 4 and discussed in section 5. The paper ends with a summary of the results and conclusions.