[1] We report laboratory experiments on intense bed-load driven by turbulent open-channel flows. Using high-speed cameras and a laser light sheet, we measured detailed profiles of granular velocity and concentration near the sidewall. The profiles provide new information on transport layer structure and its relation to the applied Shields stress. Contrary to expectations, we find that intense bed-load layers respond to changes in flow conditions by adjusting their granular concentration at the base, slightly above the bed. Two mechanisms account for the resulting behavior: stresses generated by immersed granular collisions, and equilibration of the otherwise unstable shear layer by density stratification. Without parameter adjustment, the deduced constitutive relations capture the responses of velocity, concentration, and layer thickness to a ten-fold increase in Shields stress.

[2] Intense bed-load, or sheet flow, occurs when a rapidly sheared transport layer, multiple grains thick, develops along a loose granular bed due to the tractive action of an overlying layer of free water (supplemented by gravity when the bed is inclined). This mode of transport can be driven by oscillatory flows [Asano, 1995], or by unidirectional flows in closed ducts and open channels [Sumer et al., 1996]. These three conditions are respectively of interest to coastal, dredging, and river engineers. We focus here on transport layers driven by steady uniform, turbulent open channel flows over inclined beds. Much is known about that case from theoretical investigations [Bagnold, 1956; Jenkins and Hanes, 1998], laboratory experiments [Smart, 1984; Sumer et al., 1996], and numerical simulations [Hsu et al., 2004; Yeganeh-Bakhtiary et al., 2009]. Nevertheless, we still lack constitutive relations accounting for internal layer structure in addition to layer-integrated quantities [Benda et al., 2005; Frey and Church, 2009]. To address this challenge, we measured velocity and concentration profiles in flume experiments. In contrast with previous hypotheses [Hanes and Bowen, 1985; Wilson, 2005; Berzi, 2011], the concentration profiles measured at increasing transport rates do not pivot around a fixed basal value, but adjust instead their value at the base. Accounting for this degree of freedom, general assumptions from fluid and granular mechanics lead to a simple set of constitutive relations.

2. Experiments

[3] To investigate intense bed-load, we conducted experiments with flowing water and a light sediment analogue in a close-circuit flume (Figure 1). The solid particles are PVC cylinders of diameter 3.24 ± 0.08 mm (mean ± standard deviation), height 2.64 ± 0.26 mm, and equivalent spherical diameter D = 3.35 mm. The granules have specific density s = ρ_{S}/ρ_{W} = 1.51, fall velocity w = 0.18 m s^{−1}, and critical angle of internal friction α_{0} = 31 degrees at very slow shear rate. Steady uniform free-surface flows were observed in a glass-walled open channel (length 5 m, depth 300 mm) of changeable width W. A pump and pipeline continuously re-circulated the total flow rate Q = Q_{W} + Q_{S} (water + solids) from outlet to inlet. An overshot weir placed at the downstream end of the channel caused a stationary bed deposit to form underneath the flows. For each width W and flow rate Q, we adjusted the weir height and inclination β until a bed deposit of uniform thickness was reached, and could be maintained without further adjustment for several minutes. We focused our attention on the sheet flow range [Sumer et al., 1996; Abrahams, 2003] 0.3 < < 3, where is the steep slope dimensionless Shields stress = HRS/(D(s − 1)(R − S)) [Smart, 1984]. Here R = tanα_{0}, S = tanβ, and H is the flow depth. This form of the Shields number takes into account the direct influence of gravity on the inclined transport layer, in addition to the shear stress exerted by the free water layer (see auxiliary material for a derivation). Within the above range, we conducted 48 runs at channel widths W_{1} to W_{3} equal to 88, 200 and 350 mm, target discharges Q_{1} to Q_{3} equal to 5, 10 and 15 l s^{−1}, and channel inclinations β between 0.5 and 5 degrees. Although bed-load can be driven by laminar flow [Mouilleron et al., 2009; Ouriemi et al., 2009], or co-exist with suspended transport [Sumer et al., 1996], our experiments concern only contact load driven by turbulent flow. Taking u_{*} = as shear velocity, we have 0.07 < u_{*} < 0.22 m s^{−1}. The ratio of fall velocity to shear velocity satisfies w/u_{*} > 0.8, while the particle Reynolds number satisfies u_{*}D/ν > 200.

[4] To measure granular velocities and concentrations, two synchronized cameras (resolution 1024 × 1024 pixels, scale ≈3 pixels/mm, frame rate 500 Hz over duration 3.07 s) were rigidly affixed to the flume. The first camera recorded side images of the flow (Figure 2a) in order to capture granular velocities by particle tracking velocimetry [Capart et al., 2002]. The second camera, oriented obliquely, recorded the deformation of a laser stripe projected onto the near-wall grains by a transverse laser light sheet (see Figure 1). About 10^{6} velocity vectors and 10^{5} laser hits were acquired for each run, and collected into grids of non-overlapping bins to obtain distributions of velocity and distance to wall (Figures 2b–2d). Contrasting with the stationary bed, the bed-load layer is characterized by very high velocity gradients du/dz (Figure 2b), up to 50 s^{−1}. At the base of the bed-load layer, we observe a distinct transition across which slowly sheared, densely packed grains become highly agitated and loosely dispersed. This is manifested by a marked increase in spread of the normal-to-bed velocity distribution (Figure 2c), and a jump in the number of laser hits detected away from the side wall (Figure 2d).

[5] The frequency of recorded laser hits in relation to distance from the wall is used to estimate granular concentration, following the measurement principle proposed by Spinewine et al. [2011]. The idea is that laser hits away from the wall become more frequent when the concentration decreases. More precisely, the granular concentration can be related to the mean free path μ of light rays in the dispersion using the formula [Fraccarollo and Marion, 1995]

where particles are approximated by spheres of diameter D (see auxiliary material for an analysis of shape and orientation effects). In Spinewine et al. [2011], the mean free path μ was determined from the mean distance to wall of observed laser hits, an approach well suited for moderate to high concentrations. For low concentrations, however, light attenuation in the liquid starts to affect the mean distance to wall. In that case, a better indicator is provided by the frequency m/n of laser hits per image frames detected within a certain distance y_{max} from the wall. To combine the ranges of both indicators, we estimate the concentration c as the value maximizing the likelihood of jointly observing mean (restricted to data y < y_{max}) and frequency m/n (see auxiliary material for details). The resulting concentration profiles c(z) cover both the dilute upper part and dense lower part of the transport layer. Only in the underlying bed do statistics become poor: there the stationary laser sheet illuminates motionless particles instead of averaging over multiple configurations of moving particles.

[6] We present in Figure 3 the measured profiles of mean longitudinal velocity u(z) and volumetric granular concentration c(z) for a series of 11 runs conducted at width W_{2} and target discharge Q_{2}. Profile data were obtained from non-overlapping bins of thickness Δz = 3.5 mm. Estimated from three repeat runs conducted under identical conditions, root-mean-square errors on velocity and concentration are e_{u} = 0.019 m/s and e_{c} = 0.060 m^{3}/m^{3} respectively. For clarity, levels z are shown relative to bed level z_{0}, defined as the elevation where granular motion ceases [Hanes and Inman, 1985]. This is obtained in practice as the level beneath which the shear rate = du/dz drops below 2% of the maximum shear rate of the profile. Bed and water levels determined from sidewall images were checked against periscope and point gauge measurements acquired at the centerline (see Figure 1).

[7] The measured velocity and concentration profiles can first be used to obtain layer-integrated quantities. The bed-load flux per unit width q_{S} can be calculated from

and we checked the resulting estimates against values q_{S} = Q_{S}/W derived from trap measurements of granular discharge Q_{S} at the flume outlet (see auxiliary material). In Figure 4a, dimensionless bed-load transport rates _{S} = Q_{S}/W are plotted against Shields stress . A second quantity of interest is the concentration integral

From this follows the normal immersed weight of transported grains per unit bed area, (ρ_{S} − ρ_{W})g cosβh_{S}, as well as dimensionless bed load parameter = h_{S}/D introduced by Bagnold [1956]. As predicted by Bagnold, this parameter varies linearly with Shields stress (Figure 4b). As argued by Hanes and Bowen [1985], however, ratio / is better approximated by the tangent of the critical angle of internal friction, tanα_{0} = 0.60, than by Bagnold's proposed value tanα = 0.32.

[8] Beyond layer integrals, the velocity and concentration profiles of Figure 3 provide new information on transport layer structure and its relation to the applied Shields stress. For this and other series, the bed-load layer thickness grows from a few to more than a dozen grain diameters as the Shields stress increases from = 0.41 to = 2.49. The bed-load layer accounts for about 20% of the flow depth at Shields stress = 0.41, rising to approximately 80% of the flow depth at = 2.49. For all runs, bed-load layers are spanned by nearly linear profiles of longitudinal velocity and granular concentration over most of their thickness. Velocity profiles depart from linearity at the top and bottom. At the top, they taper off slightly as the pure water layer is approached. At the base, they gradually ramp down toward vanishing velocity and velocity gradient at bed elevation z_{0}. Previous investigators [Hanes and Bowen, 1985; Wilson, 2005] envisioned linear concentration profiles rotating around a pivot of fixed granular concentration near the bed. We find instead that the concentration profiles form a striking set of approximately parallel lines, displaced outward as the Shields stress increases. Gradient breaks occur at the base, beneath which concentrations rise sharply toward bed concentration c_{0} = 0.66. Concentration values in the motionless bed are characterized by large fluctuations because there the stationary laser sheet cannot average over multiple particle configurations. Consistent with recent measurements by Matoušek [2009], the granular concentration slightly above the bed is less than the bed concentration value. It also varies systematically from run to run, increasing with the Shields stress . Measured profiles for the other series (see auxiliary material) exhibit similar trends.

[9] To examine these trends in a quantitative way, we introduce the following set of layer structure parameters (Figures 5a and 5b). Following Hsu et al. [2004], we define the top of the bed-load transport layer as the elevation where the granular concentration drops to value c = 0.08, at which particle interdistance becomes equal to the particle diameter. The velocity at the top of the bed-load layer is then obtained by interpolating measured profile u(z) at elevation . The thickness of the bed-load transport layer is defined as δ = − z_{0}. Distinct from bed level z_{0}, we define the elevation of the base of the transport layer as the zero intercept of a linear fit to the velocity profile in the range 0.1 < u(z)/ < 0.9. The difference ɛ = − z_{0} is the thickness of the gradual ramp transition connecting the linear part of the velocity profile to the motionless bed. Finally, the basal granular concentration is obtained by extrapolating to level a linear fit to the concentration profile c(z) in the range 0.08 < c < 0.4. Together with bed level z_{0}, the structure variables ɛ, , and δ capture the main features of our measured profiles. In Figures 5c–5f, we plot the four structure variables against Shields stress for all 48 experimental runs.

3. Constitutive Relations

[10] We now attempt to derive constitutive relations accounting for the above layer structure observations. As suspected by Hanes and Inman [1985], a transitional sub-layer of small thickness ɛ = − z_{0} appears to form between the rigid bed and the region where dispersed grains randomly collide. Invoking dense granular rheology [Jop et al., 2006], the dependence of thickness ɛ on Shields stress can be approximated by (see Figure 5c)

Here tanα_{0} and tanα_{∞} are internal friction coefficients at zero and high shear rates, respectively, for which dry granular flow experiments [Jop et al., 2006] indicate tanα_{∞}/tanα_{0} ≈ 1.7. Thickness ɛ increases linearly with Shields stress . It can thus be expected to become more important at steeper slopes, as debris flow conditions are approached. In the present experiments, it already accounts for up to 20% of the transport layer thickness. To obtain simple constitutive relations, however, we now assume that the concentration drops from bed value c_{0} to basal value across a sharp interface. We thus neglect thickness ɛ and approximate ≈ z_{0}. Likewise, we suppose that velocity and concentration profiles in the layer are linear, and neglect the elevation difference needed for the concentration to drop from 0.08 to 0 at the top of the bed-load layer.

[11] We make the following physical hypotheses (see auxiliary material for a more detailed derivation): 1) Throughout the bed-load layer, the collisional normal stress σ′ balances the submerged weight of overlying grains [Bagnold, 1956]. At the base, therefore, = (ρ_{S} − ρ_{W})gδ cosβ. 2) Fluid turbulence contributes to shear stresses elsewhere in the layer. At the base, however, collisions alone transmit the total shear stress = (ρ_{W}H + (ρ_{S} − ρ_{W})δ)g sinβ to the bed [Berzi and Jenkins, 2008]. From kinetic theory [Savage and Jeffrey, 1981; Jenkins and Hanes, 1998], therefore,

where f(c) = c(2 − c)^{1/2}/(1 − c)^{3/2} represents the influence of granular concentration on the frequency of collisions. The influence of liquid inertia on immersed collisions is introduced via added mass a(c) = (1 + 2c)/(1 − c) [Armanini et al., 2005]. 3) At the bed, entrainment of static grains into the bed-load layer is controlled by the Coulomb yield criterion = R, where R = tanα_{0} [Hanes and Bowen, 1985]. 4) Within the bed-load layer, density and velocity gradients adjust to an equilibrium state, characterized by a definite value of the gradient Richardson number [Turner, 1973; Pugh and Wilson, 1999]

Turner [1973] proposed that gravity-driven stratified shear flows self-adjust in this way near the threshold of instability, and obtained value G ≈ 0.058 (adopted hereafter) from density current experiments [Ellison and Turner, 1959]. Similar values of the gradient Richardson number were later measured in sheet flow experiments [Pugh and Wilson, 1999].

[12] Together, these assumptions suffice to derive constitutive relations for intense bed-load. Introducing dimensionless thickness = δ/D and velocity = /, we deduce the three simple expressions

The bed-load layer thickness, top velocity, and Shields stress can thus be calculated explicitly from parameter , the basal granular concentration. Conversely, one can deduce from by finding the root of an implicit equation. The resulting theoretical curves for , , and are plotted in Figures 5d–5f, together with the experimental data. For the top velocity (Figure 5d), we obtain from (8) and (9) the relation = ^{1/2}. Relations for the basal concentration (Figure 5e) and thickness (Figure 5f), on the other hand, are not expressible in power law form. Only for moderate Shields stresses do they approximate scalings ∝ ^{1/2} and ∝ ^{1/2}. For more intense bed-load, the relations for and deviate inward and outward, respectively. We emphasize that the curves represent predictions, not fits, since the constitutive relations include no tuning coefficient. For all three variables , , and , calculated curves are in good agreement with the experimental data. For thickness (Figure 5f), the predicted curve also concurs with previous experiments [Sumer et al., 1996] and with results from numerical simulations [Hsu et al., 2004].

[13] Finally, integrating the product of linear profiles c(z) and u(z) over layer thickness δ yields result _{S} = for the dimensionless bed-load transport rate _{S} = Q_{S}/W. Substituting relations (7)–(9) into this formula, we deduce

where is a constant coefficient. This result for turbulent flow contrasts with the laminar case, for which the particle flux scales with the Shields stress to the power 3 [Leighton and Acrivos, 1986]. Neglecting threshold of motion effects ( ≫ 0.047), equation (10) takes the same form as the empirical law of Meyer-Peter and Müller [1948], obtained from experiments at lower Shields stresses. Bagnold [1956] also proposed a law of this form. In addition to the exponent 3/2, we derive for the coefficient the value = (2/(9GR^{3}))^{1/2} ≈ 4.2, close to the value obtained by Wong and Parker [2006] from their re-analysis of the Meyer-Peter and Müller experiments. In Figure 4a, the predicted curve is compared with our measurements for PVC (s = 1.51), and with flume data for natural gravel (s = 2.65) reported by Smart [1984] and Rickenmann [1991]. For s = 1.51, transport rates are somewhat under-predicted at high Shields stresses, possibly because the suspension threshold is approached (ratio of fall velocity to shear velocity w/u_{*} ≈ 0.8). This is supported by Figure 4b, where data for dimensionless bed load are slightly higher than expected assuming contact load alone.

4. Conclusion

[14] In this letter, new measurements of granular velocity and concentration were used to characterize transport layer structure in intense bed-load. Transport layers were observed to accommodate changes in flow conditions by adjusting their granular concentration at the base. Accounting for this degree of freedom, we derived simple constitutive relations based on two physical mechanisms: equilibration between shear rate and density stratification, and stress production by granular collisions. The basal concentration intervenes in both mechanisms by controlling the density gradient across the sheared bed-load layer, and the frequency of collisions with the bed. Without parameter adjustment, the proposed constitutive relations are found to capture the responses of velocity, concentration, and layer thickness to a ten-fold increase in Shields stress. Their compatibility with the Meyer-Peter and Müller relation also suggests a continuity of behavior between bed-load and intense bed-load. Further work is needed to clarify the equilibration mechanism, explain profile shapes, and test the constitutive relations in more general flow conditions (unsteady, non-uniform flows).

Acknowledgments

[15] Michele Larcher, Benoît Spinewine, and Kristian Toigo helped with the experimental measurements, performed at the Laboratory of Hydraulics of the University of Trento. Marwan Hassan and three anonymous referees provided valuable feed-back. The work was supported by UDT, NTU, the EU (Call Outgoing 4, PAT, Trento), and the National Science Council, Taiwan.

[16] The Editor thanks the anonymous reviewers for their assistance in evaluating this paper.