exploring the physics of sediment transport in non-uniform super...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

Exploring the physics of sediment transport in non-uniform

super-critical flows using a large dataset of particle trajectories

J. Heyman1,3, P. Bohorquez2 and C. Ancey1

Abstract. In this article, we present 10 experiments carried out in a steep, shallow wa-ter flume made of an erodible bed of natural, uniform gravel. We simultaneously recordedbed load particle motion, bed and water elevation using two high-speed cameras. Par-ticle trajectories were reconstructed with the help of a robust, original tracking algorithm,available on demand. Added together, the trajectories obtained represent more than 30 days’worth of data, with sample every 0.005 s. We share this data with the research commu-nity (https://goo.gl/p4GbsR). Because the flow was supercritical and sediment trans-port was moderate, two-dimensional bed morphologies spontaneously developed and mi-grated in the granular bed. The local flow and bed conditions over which bed load trans-port occurred were thus continuously changing, both spatially and temporally, preclud-ing the use of traditional transport rate analysis. Instead, we propose an original sta-tistical analysis that allows us to estimate how dependent particle velocity, particle ac-celeration, particle diffusivity, particle entrainment, and deposition rates are upon thevariations of local bed shear stress and local bed slope. We found that: (i) particle ve-locity was linearly dependent upon the shear velocity; (ii) particle streamwise acceler-ation suggested the existence of two equilibrium particle velocities separated by an un-stable equilibrium; (iii) particle diffusivity grew linearly with the shear velocity; (iv) theparticle deposition rate was proportional to the inverse of the shear stress, and it de-creased linearly with downward slopes; and (v) the entrainment rate was strongly cor-related to the immediate proximity of other moving particles, but only weakly to theshear stress and bed slope. These original results could be used as inputs or parametersfor modern bed load transport models and numerical simulations.

1. Introduction

The inherent complexity of bed load transport makes theprediction of sediment transport rates difficult in both natu-ral rivers and the laboratory. This complexity is particularlyobvious when bed load transport occurs on steep bed slopes(θ > 2%) and/or close to incipient motion [Recking , 2013].These are the particular conditions—frequently encounteredin mountainous areas—considered in this paper.

None of the existing bed load transport theories is to-tally satisfactory for the prediction of sediment transportrates under those conditions. For instance, formulas basedon the equilibrium between flow power and sediment trans-port dissipation [Bagnold , 1956], failed to accurately predictthe transport rates at low discharge levels and on arbitrarilysloping beds [Seminara et al., 2002; Recking , 2013]. On theother hand, traditional empirical formulas, such as Meyer-Peter and Muller [1948], are unsuitable for steep slopes andfail also to accurately predict transportation rates. Otherapproaches, e.g., Lagrangian analysis of particle saltation[Sekine and Kikkawa, 1992; Sekine and Parker , 1992; Niñoand Garcia, 1994; Abbott and Francis, 1977], may be more

1Laboratory of Environmental Hydraulics, School ofArchitecture, Civil and Environmental Engineering, ÉcolePolytechnique Fédérale de Lausanne, Switzerland

2Área de Mecánica de Fluidos, Departamento deIngenieŕıa Mecánica y Minera, CEACTierra, Universidad deJaén, Campus de las Lagunillas, 23071, Jaén, Spain

3E-dric Hydraulic Engineering, Ch. du Rionzi 54,CH-1052 Le Mont-sur-Lausanne, Switzerland

Copyright 2015 by the American Geophysical Union.0148-0227/15/$9.00

appropriate for steep flows, but they often require numericalsimulation in order to obtain average results.

Models considering the processes of entrainment and de-position separately [Charru and Hinch, 2006a, b; Anceyet al., 2008] have been shown to perform relatively well inmildly sloping beds [Lajeunesse et al., 2010]. Their proba-bilistic counterpart was proven to accurately describe sta-tistical moments in transport rates over steep slopes [Anceyet al., 2008; Heyman et al., 2014]. However, they rely on thequantification of several physical phenomena, such as theentrainment and deposition processes [Papanicolaou et al.,2002; Valyrakis et al., 2010] and particle transport down aslope [Nikora et al., 2002; Ancey et al., 2003; Martin et al.,2012]; these specificities are not yet fully understood in thecase of steep slopes.

Some of the specificities of bed load transport in one-dimensional, shallow water flows over steep slopes are asfollows [Niño et al., 1994; Ancey et al., 2006; Church andZimmermann, 2007; Ancey et al., 2008; Ferguson, 2012;Heyman et al., 2013, 2014]: (1) significant fluctuations inbed load transport rates; (2) non-negligible interactions be-tween moving particles and the granular bed; (3) the effectsof slope angle on particle dynamics; (4) and the rich varietyof upper flow regime bedforms, such as standing waves, an-tidunes, breaking antidunes, cyclic steps, chutes and pools,and slug flows.

In steep rivers, the observation and characterization ofbed load transport is particularly hazardous [Yager et al.,2015, 2012; Church and Zimmermann, 2007; Rickenmannet al., 2012; Chiari and Rickenmann, 2010]; observation hasrequired the design of simplified experimental studies [Böhmet al., 2004; Heyman et al., 2013]. The use of high-speed pho-tography and video recording have been mentioned in sev-eral previous studies [Abbott and Francis, 1977; Niño et al.,1994; Drake et al., 1988; Radice et al., 2006; Böhm et al.,

1

X - 2 HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES

2004; Lajeunesse et al., 2010; Roseberry et al., 2012; Heymanet al., 2014; Heays et al., 2014; Frey , 2014], as have modernimage analysis techniques [Heyman and Ancey , 2014]. Theyoffer a good opportunity to observe and quantify bed loaddynamics in non-stationary and non-uniform conditions.

Technological limitations have meant that until today,filmed samples of bed load particle motion have been shortand have only covered small surface areas; this may have bi-ased statistical results. When flow is close to particle incip-ient motion, when interactions between moving and restingparticles exist and when bed forms develop, the necessaryfilm acquisition time and window length may be consider-able [Heyman et al., 2014].

Moreover, most of these pioneering studies only relatedthe sediment characteristics (entrainment rate, depositionrate, and velocity, for instance) to average values of shearstress and bed slope over the whole experimental flume andduring the entire acquisition time. The effects of local andshort-lived variations in shear stress and bed slopes, com-mon in steep flows, have not been investigated.

In this article, we present an original experimental setupcapable of quantifying sediment transport rates at the localscale, together with the local flow and bed characteristics.These were obtained by taking high-speed films of sedimenttransport in a steep flume, and subsequently digitally pro-cessing the images using an efficient, original algorithm.

This study’s first objective was thus to contribute to thebody of experimental data on particle dynamics under wa-ter at spatial and temporal resolutions that, to the best ofour knowledge, have not been previously reached or do notappear in the literature. Given the numerous analytical pos-sibilities afforded by the substantial amount of informationgathered (approximately 1,000 km of particle trajectories),we provide access to the raw data as well as to the treatmentroutines used, should any interested researchers wish to usethem.

The study’s second objective was to offer a first statisticaloverview of these data and compare the results to previousfindings. Most theories, experiments, and simulations agreethat bed load particles are transported with a velocity pro-portional to the shear velocity of the flow—that is, to thesquare root of the shear stress [Shim and Duan, 2015; Duránet al., 2012; Lajeunesse et al., 2010; Niño et al., 1994; Ab-bott and Francis, 1977; Bagnold , 1956]. Entrainment rateshave been found to be linearly dependent on shear stressclose to incipient motion [Durán et al., 2012; Lajeunesseet al., 2010], whereas deposition rates have sometimes beenfound to be independent of shear stress [Lajeunesse et al.,2010; Ancey et al., 2008], but also to increase linearly withshear velocity [Parker et al., 2003]. How dependent otherbed load transport variables—such as the particle diffusiv-ity (defined in section 3.2.3)—are on flow variation has notbeen investigated deeply as yet [Drake et al., 1988; Furbishet al., 2012a; Heyman et al., 2014; Ramos et al., 2015]. Theseestimates nevertheless showed that bed load discharge in-creased with shear stress at the power 2/3, which is in agree-ment with many empirical and physical formulas [Meyer-Peter and Muller , 1948; Bagnold , 1956; van Rijn, 1984; La-jeunesse et al., 2010]. Close to the incipient particle motionthreshold, however, this ratio is questionable and needs to bestudied further. At those low mobility regimes, the energytransfer between moving and resting particles is also likelyto have a strong influence on transport rates. Furthermore,the particularities of particle transport with regard to lo-cal changes in bed slope and flow conditions (e.g., transportover bed forms), are not fully understood yet.

This paper thus considers two questions. Firstly, do clas-sical bed load scaling laws apply on steep sloping beds closeto the incipient particle motion threshold? Secondly, how isbed load transport influenced by the non-uniformities cre-ated by local changes in bed slope and/or flow conditions?The latter question is of particular relevance since numerical

Figure 1. From top to bottom: initial image, estimatedforeground, bed and water level estimation, particle de-tection and tracking.

models require that bed load transport equations be validat the numerical mesh scale, which is sometimes no biggerthan a few grain diameters [Chiari et al., 2010; Balmforthand Vakil , 2012; Bohorquez and Ancey , 2015].

The article is organized as follows. We begin by describ-ing the experimental apparatus used, the numerical algo-rithm developed, and the data obtained. This is followed byan analysis of the kinematics of moving particles as a func-tion of local variables. Next, we explore the specificities ofparticle entrainment and deposition processes and their de-pendence on local flow conditions. Finally, we highlight theimportance of the interactions between resting and movingparticles.

2. Experiments

2.1. Setup

Experiments were carried out in a steep, narrow flumeat the École Polytechnique Fédérale de Lausanne, Switzer-land. The flume is 2.5 m long and 3.5 cm wide. The slopeand water discharge are manually adjustable. Water dis-charge was continuously monitored using an electromagneticflow meter. The sediment constituting the granular bed wasmade up of natural gravel particles with a narrow grain-size distribution: mean diameter was d50 = 6.4 mm, andd90 = 6.7 mm. Choosing a narrow grain-size distribution fa-cilitated particle tracking and limited grain sorting effects.Sediment was fed into the top of the flume at a constantbut adjustable rate via conveyor belt. After each experi-ment, sediment was collected at the downstream end of thechannel, dried, and weighed.

Two cameras with 1280×200 pixel resolution were placedside by side at the downstream end of the channel andtook pictures through the transparent side wall at a rate of200 frames per second (fps). The total length of the obser-vation window was slightly less than 1 m (giving a precisionof about 0.4 mm/pixel).

HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES X - 3

Table 1. Average experimental parameters: tan θ, bed slope; Sf , friction slope; u∗, shear velocity; ws, settlingvelocity; τ∗b ; Shields stress; Fr, Froude number; ū, depth-averaged velocity; h, water depth; q

∗s , dimensionless sed-

iment discharge (tracking algorithm); q∗s,in, dimensionless sediment discharge at the flume inlet; γ∗, dimensionless

particle activity; up, average particle velocity; r↓p∗, dimensionless particle deposition rate; r↑

∗, dimensionlessareal entrainment rate.

tanθ(%) Sf (%) u∗/ws τ∗b Fr ū/ws h̄/d50 q

∗s q

∗s,in γ

∗ up/ws r↓∗p

r↑∗ Frames

(a) 3.5 3.8 0.170 0.093 1.18 1.37 4.1 0.003 0.002 0.005 0.56 0.021 1.0 · 10−4 4 × 3 · 104(b) 3.4 2.9 0.172 0.095 1.22 1.42 4.2 0.003 0.005 0.006 0.56 0.021 1.0 · 10−4 2 × 3 · 104(c) 3.5 3.2 0.182 0.106 1.28 1.47 4.0 0.004 0.005 0.006 0.60 0.018 1.1 · 10−4 14 × 4 · 103(d) 4.5 4.7 0.189 0.115 1.28 1.49 4.2 0.002 0.002 0.004 0.57 0.020 0.8 · 10−4 4 × 3 · 104(e) 4.9 4.8 0.188 0.114 1.32 1.37 3.3 0.002 0.002 0.003 0.56 0.023 0.7 · 10−4 4 × 3 · 104(f) 4.7 4.4 0.176 0.099 1.28 1.31 3.2 0.002 0.003 0.004 0.59 0.031 0.9 · 10−4 23 × 4 · 103(g) 4.9 4.4 0.180 0.104 1.31 1.34 3.2 0.005 0.007 0.010 0.59 0.020 2.0 · 10−4 8 × 3 · 104(h) 5.1 5.2 0.190 0.116 1.37 1.34 3.0 0.002 0.004 0.004 0.60 0.019 0.7 · 10−4 8 × 3 · 104(i) 5.1 4.9 0.194 0.121 1.39 1.40 3.1 0.004 0.005 0.007 0.63 0.017 1.3 · 10−4 4 × 3 · 104(j) 5.2 4.9 0.175 0.098 1.32 1.25 2.7 0.004 0.005 0.007 0.62 0.020 1.2 · 10−4 8 × 3 · 104

Ten experiments were performed using various mean bedslopes, mean water discharge rates, and mean sediment in-puts (Table 1). For each experiment, we filmed four to eightsequences of 150 s at 200 fps (i.e., 30,000 frames). This corre-sponded to the maximum random access memory availableon the computer used. In some experiments, we acquiredmore sequences of only 4,000 frames (i.e., 20 s) at a time.

2.2. Tracking Procedure

Image processing and automatic particle tracking weresubsequently performed on the images obtained (Fig. 1).The processing steps from the raw images to particle trajec-tories were the following:

1. The raw images were processed using the powerful yetsimple method of median background subtraction [Yilmazet al., 2006; Radice et al., 2006]. This permits a distinctionbetween a static background image, B (made up of parti-cles resting on the bed), and a moving foreground, F (themoving particles and water surface).

2. Bed and water surface elevations were estimated auto-matically from the vertical gradients in images pixel inten-sity. As the cameras were at a slight angle to the channelbed plane, the perspective effect was corrected by adding apositive 2.5 mm shift to the water surface level.

3. An algorithm detected the centroid position of themoving particles in the foreground images. This wasachieved after thresholding the foreground image and com-puting the properties of connected regions (e.g., area,barycentre, and eccentricity).

4. A trajectory number was then attributed to each mov-ing particles. This trajectory number was transmittedthrough frames from the closest particle in the previousframe. In case of conflict (e.g., if two particles receive thesame trajectory number, for instance), the trajectory wasconsidered to be broken and new trajectory numbers wereattributed to the conflicting particles.

5. To reconstruct broken trajectories, a Kalman filter wasapplied to each missing measurement in order to predictmissing positions. The motion model used for the Kalmanfilter assumed constant acceleration (i.e., constant externalforces on a particle). A global optimization algorithm, theso-called Hungarian algorithm [Munkres, 1957], was thenused to connect overlapping trajectories.

Although the algorithm presented here is fairly efficientat detecting and tracking moving particles (demonstrationvideo available online), it cannot guarantee a complete re-construction of every trajectory. When the moving particleconcentration is locally large, and thus particles frequentlyocclude each other, a trajectory may end before the particlereally stops or, on the contrary, an ending trajectory maybe erroneously associated with the incipient motion of a dif-ferent particle. The tracking algorithms overall performancethus depends on the local concentration of moving particles.

Thanks to the large amount of data available, we believethat these reconstruction errors are not statistically signifi-cant. In appendix A1, we compare the bed load dischargeobtained by the tracking algorithm to fluxes that were moni-tored simultaneously using an acoustic sensor and processedusing a method described in Heyman et al. [2013]. For themonitored experiments, the two estimates agreed well, sug-gesting a good accuracy of the tracking algorithm.

We summarize the algorithm in appendix A1 and providethe Matlab source code online.

2.3. Tracking variables

For each experiment, the data available consisted ofa set of particle positions, velocities, and accelerations(~xp(t), ~vp(t),~ap(t))k, where k is the particle index and t is theframe number, projected on a two-dimensional (~x, ~z) planeapproximately parallel to the flume walls. The horizontaland vertical components of the state vectors are denoted as~xp = (xp, zp), ~vp = (up, wp) and ~ap = (ax,p, az,p). Parti-cle angular velocities were also obtained, but their analysisis beyond the scope of the present paper. Added together,with samples every 0.005 s, the trajectories obtained rep-resent more than 30 days of data. In other words, the cu-mulative distance traveled by particles and recorded by thecamera was about 1,000 km.

Local particle activity γ(x, y, t)(i.e., the number of mov-ing particles per unit bed area) can be estimated from theposition of moving particles. In this study, we did not trackparticles in the transverse direction y, so results were simplyaveraged over the flume width and particle activity per unitflume width was denoted as γ(x, t) [particles/m2]. Becausethe tracked particles are points in the space plane, γ(x, t)must be computed using a kernel estimation technique

γ(x, t) =

∫∫Kw(x− xp(t))dS, (1)

where S is the bed surface and Kw is a smoothing kernel ofbandwidth w [m] [Diggle, 2014; Botev et al., 2010]. Similarly,local sediment discharge [m2/s] is defined as the product ofparticle activity and particle velocity:

qs(x, t) = Vp

∫∫Kw(x− xp(t))up(t)dS, (2)

with Vp as the particle volume. At low transport rates, thecorrelations between up(t) and γ are negligible, and the av-erage solid discharge is simply 〈qs〉 = Vp 〈up〉〈γ〉, where theangular brackets are used to symbolize an averaging proce-dure (either temporal, spatial, or over all the experiments).

In addition to these trajectories, the bed elevation b(x, t),the local bed slope tan θ(x, t) = ∂xb(x, t), the water surfaceelevation w(x, t), and its local slope tanα(x, t) = ∂xw(x, t)


Figure 2. Six runs of experiment (g) in the space–time plane. Particle trajectories are represented bycolored filaments, whereas water surface and bed elevation are represented by interpolated surfaces.

were also measured. Note that in contrast to the usual con-

ventions in bed load transport, we took the angles to be

negative for downward slopes and positive for adverse slopes.

An example of the typical data obtained is shown in Fig. 2

for experiment (g).

It is worth remembering that the definition of “local”

quantities, such as bed slope or bed elevation, is neither ob-

vious nor unique. They depend intimately upon the scale at

which they are observed, in a similar manner to which the

length of Britains coast depends on the size of the ruler used

to measure it. As the bed is made of rough gravel, at the

limit of an infinitely small ruler, bed slopes alternating be-

tween + and -90◦ are likely to be observed, though bed load

may not be influenced much by such small asperities. As a

consequence, some of the results obtained may depend on

the spatial averaging scale chosen. The exact averaging pro-

cedure is detailed in appendix A2 and, whenever possible,we have provided results for several averaging scales.

2.4. Additional variables

In addition to the variables previously defined, we esti-mated the water depth, the depth-averaged flow velocity,and the bed shear stress as follows. Water depth is simplyh(x, t) = w(x, t) − b(x, t). For stationary flows, the depth-averaged flow velocity reads ū(x, t) = Qw/(Bh(x, t)), withQw as the prescribed water discharge and B as the channelwidth.

Bed shear stress is more difficult to estimate precisely. Inuniform, steady conditions for infinite-width channels, thebottom shear stress must balance the tangential componentof a unit fluid weight so that τ = %ghSf , where h is the waterdepth and Sf is the friction slope. However, in the presentcase, the bed slope and flow depth change locally so thatthese forces may not balance everywhere. Moreover, thechannels lateral glass walls and the relatively small channel-width-over-water-depth ratio (0.5 to 2) precluded any use


of the infinite channel width assumption. In such extremelynarrow channels, secondary currents are strong and can sig-nificantly modify the shear stress repartition between thelateral walls and the bed [Guo and Julien, 2005].

Alternatively, bed shear stress may be estimated usingdimensional analysis :

τ(x, t)

%=f

8ū(x, t)2, (3)

where f is a function of dimensionless numbers character-izing the flow. For instance, f = 8κ2/ ln(11Rh/ks)

2 in the

Keulegan formula, and f = 8gR−1/3h /K

2 with a Manning–Strickler formula, where κ is the von Karman constant, Rhis the hydraulic radius, ks is the equivalent roughness, andK is the Strickler coefficient. The difference in flow re-sistance between the lateral glass walls and the granularbed can be modelled by separating the hydraulic radius Rhinto a bed hydraulic radius (Rb) and a “wall” hydraulic ra-dius (Rw), with each of them bearing a part of the totalshear stress [Guo, 2014a]. Semi-analytical expressions forbed shear stress have also been proposed specially for suchnarrow flumes [Guo and Julien, 2005].

The applicability of different bed friction laws is studiedin appendix B. Their predictions were roughly equivalentin our channel. For simplicity, we chose a friction factor fcomputed from Keulegan’s formula, with ks = 2.0d50; wealso separated bed- and wall-borne shear stress. Below, weonly considered bed-borne shear stress to be relevant for bedload transport.

2.5. Orders of magnitude and dimensions

The problem of sediment transport can be convenientlystudied using dimensional analysis. Some of the dimension-less numbers useful in the study of sediment transport aresummarized in Table 2, along with their average values inour experiments.

In addition to the numbers defined in Table 2, thesediment discharge qs is made dimensionless by q

∗s =

qs/√gd350(s− 1) ≈ 2 − 6 · 10−3, the bed shear stress

[kg/(ms2)] by τ∗ = τ/(%(s − 1)gd50) ≈ 0.09 − 0.12,also known as the Shields stress. Particle activity γ[particles/m2] is made dimensionless by γ∗ = γd250 ≈ 3−11 ·10−3. With regard to Brown and Lawler [2003], we foundthat the settling velocity of particles was ws ≈ 0.55 m/s.

Table 2. Dimensionless numbers governing flow and sedi-ment transport. Rh, hydraulic radius; ū, depth-average watervelocity; ν, fluid kinematic viscosity; g gravity acceleration;h, water depth; d50, particle mean diameter; s = %s/%, den-sity ratio between sediment and fluid; u∗, shear velocity; B,channel width; ks, bed equivalent roughness; κ, von Karmanconstant; ws = 0.55m/s, settling velocity determined fromequation Eq. (37) in Brown and Lawler [2003].

Variable Name Expression ValueRe Reynolds number 4Rhū/ν 3.5 · 104Fr Froude number ū/

√gh 1.1–1.3

d∗ dimensionless graindiameter

d50((s− 1)g/ν2

)1/3157

Rep particle Reynoldsnumber

d50u∗/ν 630

channel aspect ra-tio

B/h 0.5–1.5

Relative roughness ks/h 0.25–0.65

ws∗ dimensionless set-tling velocity

ws/((s− 1)gν)1/3 22.5

P Rouse number ws/(κu∗) 13St Stokes number d50wss/(9ν) 977

By using the orders of magnitude obtained, we were ableto refine the characterization of the sediment transport inour experiments. First, the particle Reynolds number sug-gested a rough hydrodynamic bed boundary, common togravel bed streams. Second, the fairly high value of theRouse number allowed us to exclude transport by suspen-sion. The sediment in these experiments was thus onlytransported as bed load. Third, the relatively high Stokesnumber suggested that particle-particle interactions throughcollisions are not damped by fluid viscosity, allowing therestitution coefficient to approach 1, as in aeolian bed loadtransport [Joseph et al., 2001].

The particle Reynolds number, together with the di-mensionless grain diameter, enabled the estimation of thevalue of the critical shear stress for incipient particle mo-tion. Buffington and Montgomery [1997] suggested thatτ∗cr = 0.02 − 0.08, pointing out that a high relative rough-ness may decrease the shear stress available at the bed forsediment transport and shift the apparent incipient motionthreshold to higher values. Thus, the transport stage in thepresent experiments (i.e., τ∗/τ∗cr) theoretically lay between1 and 6 for the maximum and minimum incipient motionthresholds. In practice, the transport stages in all the ex-periments were very close to, if not below, 1 (τ∗ < τ∗cr). Mostof time, when the sediment feeding system was stopped atthe end of an experiment, sediment transport in the chan-nel also ceased rapidly, proving that the flow alone was oftennot sufficiently strong to entrain new particles.

It should be noted that critical shear stress is ill-definedin gravel-bed streams. Indeed, gravel size distribution islarge, particle exposure to the flow is varied, and turbulentflow itself is intermittent; thus particles are not entrainedsimultaneously past a given shear stress threshold. On thecontrary, the rate of entrainment of particles by the flowslowly increases from 0 at low shear stress to larger valuesat higher shear, where it becomes limited by the energy dis-sipation due to sediment transport itself. Nevertheless, theconcept of critical shear stress still gives a useful referencepoint for the characterization of the transport regime.

Sediment transport rates were fairly low, as judged bythe small dimensionless particle activity: on average, only0.3% to 1.1% of the bed was covered by moving particles.In comparison, Lajeunesse et al. [2010] experimentally ex-plored regimes where at least 5% of the bed was moving;Durán et al. [2012] numerically simulated bed load regimeswhere at least 8% of the bed was moving. In the range ex-plored in this study, a bed load shear layer did not developedand particles moved sporadically by jumping. This resultwas confirmed by the relatively low sediment discharge q∗srecorded in all experiments, confirming that the transportstage was τ∗/τ∗cr ≈ 1.

3. Results

3.1. Preliminary remarks

As can be inferred by looking at Fig. 2, bed and water el-evations varied considerably in time and space, even thoughthe water and sediment inputs were kept constant. Indeed,bed forms developed naturally and migrated in the experi-mental flume, locally modifying bed topography, flow, andsediment transport. It thus made little sense to try to relatethe statistics of transport processes to averages made overeach experimental run.

In contrast, we envisioned the whole set of experimentsas representative of various local and instantaneous bound-ary conditions that may affect sediment transport: we con-sequently ignored the individual particularities of each ex-periment (mainly the prescribed flume angle, sediment in-put, and water discharge). Statistics concerning particle


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(c)0.81Fr− 0.47(a)

k = 4d50k = 10d50k = 40d50

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Figure 3. (a) Average particle velocity function of particle elevation: violet triangles scaled with u∗,and black circles scaled with ū. (b) Distribution of particle velocity and theoretical fits: red and blackdashed lines are Gaussian; violet dashed lines are exponential. (c) Particle velocity function of local shearvelocity. Inset: Local particle velocity is a function of the Froude number. (d) Particle velocity function

of local bed slope: u∗,cr = u∗,0√

sin (θ + α) / sinα with u∗,0/ws = 0.2 and α = 47◦. Various averaging

scales are encoded by point type: crosses (2d50); squares (5d50); and circles (10d50). Black dots indicatescaling using u∗, whereas violet dots indicate scaling using u∗ − u∗,cr. θ > 0 corresponds to a downwardsloping bed.

transport were estimated by pooling the whole experimentaldataset (about 6.5 million values), without regard to specificrun conditions.

Most of the flow variables—such as bed slope, and wa-ter depth and velocity—can be estimated accurately locally(see appendix A2). In contrast, shear stress is evaluated us-ing flow resistance laws derived in uniform flow conditions;this may thus yield inaccurate values locally. Indeed, pre-dicting the depth of a free surface flow over a given bedtopography by merely using flow resistance laws would be achallenging task, one beyond the scope of the present study.We were, therefore, obliged to assume that, even at the lo-cal scale, shear stress was sufficiently well estimated by flowresistance laws, regardless of the local topography.

First, we consider an analysis of the motion of bed loadparticles without taking into account their entrainment anddeposition. Second, we focus on the mass exchanges betweenthe granular bed and the moving particles.

3.2. Kinematics

Once entrained, bed load particles were transporteddownstream by the water flow. Their motion consisted ofvarious alternating stages: free flight in the fluid column,rebound, and rolling over the granular bed. Particle veloci-ties, therefore, fluctuated considerably during transport, asthey were subjected to variable forces: turbulent flow drag,

collision with other particles, and/or granular friction. Con-sequently, moving particles tend to disperse in space whilethey are transported, resulting in a diffusive effect on parti-cle activity [Furbish et al., 2012a; Ancey and Heyman, 2014;Heyman et al., 2014].

In this subsection, we first verify how dependent the dis-tribution of particle velocities is on the local flow variables.We then study the instantaneous acceleration of particlesand relate this to forces acting on them during their trans-port. Finally, we quantify the dispersion of moving particlesand relate this to the local flow characteristics.

It should be noted that when estimating how dependenta variable related to particle motion (e.g., velocity, acceler-ation, or entrainment rate) is on a flow characteristic (e.g.,shear velocity, Froude number, or bed slope), the variationswith respect to all other flow characteristics are automati-cally averaged over the whole experimental dataset. For in-stance, when estimating the dependence of particle velocityon u∗, the variations in relation to other flow characteristicsare averaged so that up(u∗) =

∫F(u∗, θ,Fr, ..)θ dθFr dFr ....

3.2.1. VelocityWe first consider the average particle velocity 〈up〉 as

function of the particle elevation zp above the gravel bed(Fig. 3(a)). Note that the instantaneous particle elevationcould not be known exactly: first, because the gravel bedlevel cannot be defined exactly; and second, because the


particles vertical position is not uniquely defined in the im-age projection plane (the camera was at a slight angle tothe flume). This is why, as shown in Fig. 3(a), particleswere observed moving at elevations as low as −2d50. zp stillprovides a rough estimate of the particles vertical positionin the fluid column. When averaged over numerous trajec-tories, individual elevation errors should cancel each otherout.

Particle velocity can be scaled either by using the depth-averaged velocity ū or the shear velocity u∗, and the parti-cle elevation by using either the flow depth h or the graindiameter d50. As shown in Fig. 3(a), both options gavesimilar results. However, retaining both scaling options al-lowed us to distinguish two motion regimes: at low eleva-tions (zp ≤ d50), particle velocity was close to the shearvelocity, whereas at higher elevations (zp ≥ 0.5h), averageparticle velocity reached the depth-averaged flow velocity.

In other words, the rolling and sliding motion of particlesin contact with the bed occurred at a velocity close to theshear velocity, whereas the particles saltating high enoughin the fluid column reached the depth-averaged flow velocity.

The distribution of particle elevations P (zp), also plottedin Fig. 3(a), suggested that, most of the time, particles movebetween −1 and 2 particle diameters under/above the zerobed level. At those heights, particles travel at very differentvelocities.

The presence of these two regimes was also apparentwhen looking at the distribution of velocities, as shown inFig. 3(b). The distribution of particle velocities locatedat or under the bed level was approximately exponential,whereas the distribution of particle velocities located twograin diameters or above the bed level was clearly Gaus-sian. The overall distribution was a composite of both, andmay be fairly well approximated by a truncated Gaussiandistribution.

Some authors have reported exponential distributions ofparticle velocity [Lajeunesse et al., 2010; Furbish et al.,2012b], but others have reported Gaussian and truncatedGaussian distributions [Martin et al., 2012; Ancey and Hey-man, 2014]. Fig. 3(b) suggests that depending on the dom-inant type of particle motion—low rolling/sliding or highsaltation—the shape of the particle velocity distributionsmay differ: exponentially distributed velocities for low el-evation particle motion and Gaussian distribution for highparticle flights.

The relationship between particle velocity and u∗ or ū isshown in Fig. 3(c). Here, all velocities were made dimen-sionless by using settling velocity ws.

As is often seen in literature, the average particle velocitywas linearly dependent on the shear velocity, with a coeffi-cient of 4.2 (Fig. 3(c)). This is dimensionally consistentwith Bagnold [1973] law, ūs/ws ∝

√τ∗, which is widely

used in sedimentation engineering: see Section 1.4 in Yalinand Ferreira da Silva [2001]. For comparison, Abbott andFrancis [1977] and Martin et al. [2012] reported coefficientsbetween 13 and 14 over fixed beds, whereas Lajeunesse et al.[2010] and Niño et al. [1994] reported coefficients of 4.4 and5.6, respectively, for flows over erodible beds—this is closerto our findings. Using direct numerical simulations, Duránet al. [2012] also reported a linear relationship between 〈up〉and u∗, with a coefficient of approximately 3.

The linear relationship is even more pronounced when upis plotted against ū: the height at which particles salatate ishigh in comparison to the water depth, and particle velocitymay be more influenced by ū than by u∗ [Bagnold , 1973].Taking

〈up〉 ≈ 4.2u∗ − 0.16ws , (4)

and then using (3) and dividing by√gh, we also have

〈up〉√g h

= 4.2

√f

8Fr− 0.16 ws√

g h. (5)

Thus, if f does not vary much with h, then 〈up〉 /√gh ∝ βFr.

Chatanantavet et al. [2013] obtained 0.6 ≤ β ≤ 0.8 forbedrock channels. We found β = 0.81 (inset of Fig. 3(c)).

Interestingly, the ratio of particle velocity to shear veloc-ity 〈up〉 /u∗ depends almost linearly on the local bed slope θ(Fig. 3(d)). High velocities are encountered on steep slopes,whereas adverse slopes lead to a small particle-shear veloc-ity ratio. Note that the averaging scale at which bed slope iscomputed influences the relationship: the larger the ruler,the more dependent particle velocity is on the bed slope.This increase may be explained by two factors: (i) parti-cles take a finite time to adapt their velocities to change inslope; and (ii) changes in bed slopes that are smaller thanthe average particle jump length are “felt” less by saltatingparticles.

Dependence on θ may instead be modelled by divid-ing 〈up〉 by an excess shear velocity u∗ − u∗,cr, wherethe critical shear velocity u∗,cr varies with the bed slope.Based on the balance of forces acting on a particle, Fer-nandez Luque and van Beek [1976] proposed that u∗,cr =u∗,0

√sin (θ + α) / sinα, where u∗,0 is a reference shear ve-

locity at zero slope and α is a local angle of repose, takento be 47◦.

3.2.2. Acceleration

The study of particle acceleration provides some insightinto the magnitude and nature of the forces driving theirmotion. We first present average particle streamwise accel-eration 〈ax,p〉 and vertical acceleration 〈az,p〉, as functionsof the particle elevation zp in Fig. 4(c).

The change of sign in vertical particle acceleration atabout zp = d50/2 tells us that most of the particles locatedbelow d50/2 rose up in the fluid column, whereas above thiselevation they were mainly settling down, most likely be-cause of gravity. As previously noted with the distributionof particle elevations, the particles are thus mainly trans-ported at an elevation of zp ≈ d50/2. The interesting featurein Fig. 4(c) is that both vertical and streamwise particle ac-celerations change sign at the same elevation, and alwaysmaintain their signs in a opposite direction. When parti-cles are below d50/2, their average vertical acceleration ispositive but their streamwise acceleration is negative, sug-gesting that they mainly decelerate owing to bed friction.In contrast, when particles are above d50/2, their averagevertical acceleration is always negative but their streamwiseacceleration is positive, due to the flow drag.

Without losing too much generality, the equations of lon-gitudinal particle motion (in the streamwise direction) canbe formulated as

dxpdt

= up, (6)

mpdupdt

= f1(up, zp, ū, . . . ) + f2(up, zp, ū, . . . )ξ(t), (7)

where f1 and f2ξ(t) are an average and a random force act-ing on the particle, respectively, and ξ is white noise [Furbishet al., 2012a; Ancey and Heyman, 2014; Fan et al., 2014].f1 is the average drift force felt by any particle in motionand is the result of both the average flow drag and the av-erage friction over the bed, whereas f2ξ(t) simultaneouslyrepresents all the force fluctuations arising during particlemotion, mainly the randomness of collision angles and ve-locities with the bed and fluctuations in the turbulent flowdrag.

The first and second moments of longitudinal particle ac-celeration are thus related to the average particle forces:〈ax,p(t)〉 = 〈f1〉 /mp and

〈a2x,p

〉− 〈ax,p〉2 = 〈f2〉2 δ(0)/m2p,


-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

〈ax,p〉/

g

up/ū

0.40ū

1.25ū

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

(〈a2 x,p

〉 −〈a

x,p〉2)/g2

up/ū

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.1 0.120.140.160.18 0.2 0.220.240.260.28 0.3

〈Du∗ 〉

u∗/ws

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-2

-1

0

1

2

3

4

5

z p/h

z p/d50

〈ax,p〉 /g

〈az,p〉 /g

P (zp)

(a)

(b)

〈Du∗〉 = 3.3u∗/ws − 0.25

(d)

(c)

Figure 4. (a) First and (b) second moments of instantaneous particle acceleration as a function ofparticle velocity. (c) Average streamwise and instantaneous vertical acceleration as a function of particleelevation zp. (d) Dimensionless particle diffusivity function of shear velocity.

respectively. In Figs. 4(a) and (b), we plotted the estimatesof the first and second moments of longitudinal particle ac-celeration dependent on the particle velocity.

The expected force on a particle changed depending onup, as seen in Fig. 4(a). Within the velocity ranges [0, 0.4ū]and [1.25ū,∞], 〈ax,p〉 < 0, and thus particle velocity tendedto decrease: forces acted against their motion. In contrast,for velocities between 0.4 and 1.25ū, the particles were accel-erated: forces acted in the direction of their motion. Whenup < 0, 〈ax,p〉 < 0, so, in absolute value, particle velocitytends to decrease to the null velocity.

The change of sign of 〈ax,p〉 suggests that two stable equi-librium particle velocities exist, namely 0 and 1.25ū, andthat these are separated by an unstable equilibrium parti-cle velocity at up = 0.4ū. On average, all particles with avelocity below 0.4ū came to rest, whereas the others tendedto reach a velocity approximately equal to 1.25ū. The lat-ter may correspond to the maximum flow velocity in thechannel.

Curiously, Quartier et al. [2000] showed that under cer-tain conditions, a cylinder rolling down a rough plane mayreach two limit velocities, depending on the initial condi-tions, and can thus present a mean acceleration diagramsimilar to Fig. 4(a). This hysteretic behavior comes fromthe fact that a moving grain must always rise above thegrains beneath it in order to maintain its motion, an area ofpotential trapping wells for a moving particle without suf-ficient kinetic energy [Riguidel et al., 1994; C. Ancey et al.,1996; Dippel et al., 1996]. We have good reasons to believethat we face a similar phenomenon for grain motion underwater, even though the details of the forces involved might

be different and the rolling motion might be interrupted bylonger saltations.

Interestingly, this phenomena was not observed by Fur-bish et al. [2012b] in their experiments using shallower slopesand smaller grains. Furthermore, the shape of 〈ax,p〉 isclearly non-linear, contrasting with the results of Furbishet al. [2012b], who observed that 〈ax,p〉 was a linearly de-creasing function of up.

The second moment of particle acceleration is more diffi-cult to interpret (Fig. 4(b)). Unlike in Furbish et al. [2012b],there was no clear linear trend. Significant fluctuations inparticle acceleration were mainly observed at relatively lowor high particle velocities. We distinguished two local min-ima, at up = 0 and up = 1.0ū, as well as a local maximaat up = 0.4; these seem to correlate with the equilibriumvelocities defined previously.

In the case of a uniform flow, Ancey and Heyman [2014]showed that the distribution of particle velocities was Gaus-sian if f1 was proportional to the difference in the velocitiesbetween the flow and the particle and f2 was constant. Al-though this Gaussian law is a good approximation of thedata presented here in the positive range of particle ve-locities (Fig. 3(b)), the force model that they chose onlyhold for particle velocities close to up = 1.25baru. Furbishet al. [2012b] proposed that both f1 and f2 were linearly pro-portional to up. They found that particle velocities shouldbe exponentially distributed, which agreed with their data.However, this formula does not apply to the present studyeither.3.2.3. Diffusivity


As shown previously, particle velocity fluctuations favorthe spread of particles through space, with some movingfaster or slower than the bulk average velocity. This diffu-sive effect tends to smooth out fluctuations in particle activ-ity. Dispersion of bed load particles was previously shown toresult in a macroscopic diffusion effect in the particle flux,with a diffusivity Du [m

2/s] [Furbish et al., 2012a]. Thetotal streamwise sediment discharge is thus the sum of anadvective and a diffusive flux:

qs = Vs

(upγ −Du

∂γ

∂x

). (8)

There is little previous experimental evidence of the keyrole of bed particle diffusion involving weak and partial bedload transport [Drake et al., 1988; Nikora et al., 2002; Fur-bish et al., 2012a; Ancey and Heyman, 2014; Heyman et al.,2014]. The average particle diffusivity may be obtained bycomputing the mean squared displacement of particles. Thelatter can be shown to grow asymptotically linearly withtime at a rate proportional to Du [Taylor , 1922].Du may be a function of the particle shape and size, mean

velocity, and flow and bed characteristics. Below, we explorediffusivitys dependence on shear velocity.

As flow and bed conditions are continuously changingthrough space in our experiments, local diffusivity cannot beestimated using the mean squared displacement of particles.To estimate a local Du we used the connection between theLagrangian integral time scale of particle velocity τL (i.e.,the integral of the particle velocity autocorrelation ρup(t)),the variance of particle velocity σ2up , and the diffusivity ofparticles Du [Taylor , 1922; Furbish et al., 2012a]:

Du = τLσ2up . (9)

The same problem arises when estimating τL: the autocor-relation must be computed over a sufficiently long period,even though the flow and bed conditions encountered bythe particle can change. We avoid this by assuming that theautocorrelation of particle velocity is well modeled by thedecaying exponential [Martin et al., 2012]

ρup(t) ≈ e−t/τL . (10)

Practically, τL is estimated by using (10) with only the firstslags of the discrete time autocorrelation values, where theflow and conditions can be considered as approximately con-stant.

We introduce the dimensionless particle diffusivity Du∗ =

Du/(d50ws) and plot its variations against the shear veloc-ity in Fig. 4(d). Interestingly, Du

∗ scales linearly withu∗/ws, taking values from 0.2 at low shear to 1 at highershear rates. By comparison, Furbish et al. [2012a] obtainedDu∗ ≈ 0.7 − −1.9 using a similar calculation. In con-

trast, using the mean squared displacement method, Hey-man et al. [2014] found a much larger effective diffusivity(Du

∗ = 2.8−5) as did Ramos et al. [2015] (Du∗ ≈ 3.6).Thisdifference may be explained by the different methods used toestimate Du

∗: the mean squared displacement considers thediffusion limit over large time periods while the Lagrangiantime scale is estimated over a shorter duration and tends tounderestimate long-term diffusion.

Fig. 4(d) sheds new light on the relative magnitudeof particle diffusivity with respect to turbulence diffusiv-ity in weak bed load transport. Characteristic values forthe particle diffusion shown in Fig. 10 scale as Du ≈3.3 d50 u∗ [m

2/s]. This can be written as a function ofthe macroscopic length scale h by using the flow-depth tograin-size ratio h/d50 from Table 1, which yields 0.8 ≤Du/(hu∗) ≤ 1. Turbulent dispersion of passive scalars,also known as stochastic Lagrangian dispersion, has been

widely studied since the 1980s because of its relevance inpollution—for instance, in atmospheric dispersion [Wilsonand Sawford , 1996]. Under a uniform regime, Brouwers[2012] recently showed that turbulent diffusivity is given byDu = 2σ

2/(C0 �), where the Kolmogorov constant is C0 = 6,the energy dissipation rate � ≈ 2.5u2∗/h, and the Reynoldsstress is approximately σ ≈ 3/2u2∗. Consequently, in a uni-form, steady flow, we can establish that the turbulent dif-fusivity is Du = 0.3u∗ h. This value must be increased inunsteady flow, as well as under non-uniform flow conditions;see equation (67) in Brouwers [2012]. A similar scaling wasfound in these experiments but the constant of proportion-ality was higher in bed load transport than in turbulentdiffusion.

The introduction of a Péclet number, Pe= up∆x/Du,may be useful to quantify the relative importance of ad-vection versus the diffusion of particles at a particular scale∆x. Taking Du

∗ = Du/(d50ws) ≈ 0.5, ws = 0.55 m/s andup ≈ 0.3m/s, we have Pe ≈ 1.1∆x/d50. Thus, we can dis-tinguish two spatial (and equivalent temporal) scales: (i) for∆x� d50 (or ∆t� d50/up), the transport of moving parti-cles is mainly advective (Pe� 1); and (ii) for ∆x� d50 (or∆t� d50/up), the transport is mainly diffusive (Pe� 1).

3.3. Mass exchanges

At the incipient stage of bed load transport, particle mo-tion is intermittent. From its resting position in the gran-ular bed sheared by the flow, a particle is occasionally setin motion when the resultant of external forces exceeds itsown weight—a process called entrainment. It subsequentlycomes to a halt further downstream, where a bed asperityis sufficiently deep to retain it—a process called deposition.The relative importance of these two processes conditionsbed aggradation or erosion.3.3.1. Definitions and averages

The entrainment of a particle (E) is assumed to be arandom event that occurs if two conditions are met in theexperimental particle trajectories time series: (i) the par-ticle velocity magnitude exceeds a given velocity thresholdvth; and (ii) the distance from the particle mass center tothe estimated bed elevation does not exceed zmax. Althoughthe first condition is usual in tracking experiments [Campag-nol et al., 2013; Radice et al., 2006], the second is aimed atreducing overestimations of particle entrainment rates dueto the existence of broken trajectories.

In a similar manner, the deposition of a particle (D) isassumed to be a random event that occurs if: (i) the parti-cle velocity magnitude falls below a given velocity thresholdvth; and (ii) the distance from the particle mass center to theestimated bed elevation does not exceed zmax. In practice,we fixed vth = 0.01ws and zmax = d50.

We denote P (D,∆t) as the probability of a moving par-ticle being deposited during the short time span ∆t. Theparticle deposition rate r↓p [s

−1] is then directly r↓p =P (D,∆t)/∆t. Note the use of the subscript p to indicatethat the rate refers to a single particle. Assuming that allparticles are similar, the areal deposition rate (e.g., the num-ber of particles deposited per square meter per second) issimply r↓ = r↓pγ.

Similarly, P (E,∆t) is the probability of a resting par-ticle being entrained during ∆t. This gives the particleentrainment rate r↑p = P (E,∆t)/∆t [s

−1]. Assuming aconstant areal density of particles resting on the bed ψ,the probability of any resting particle being entrained inthe infinitesimal time interval ∆t, in a bed surface A, isP (Ẽ,∆t) = ψAP (E,∆t). Thus, the areal entrainment rateof particles per unit bed area, denoted by r↑ [particles m

−2

s−1], is r↑ = ψr↑p.


0

0.005

0.01

0.015

0 0.005 0.01 0.015

〈r↑∗〉/

〈r↓p

∗ 〉

〈γ∗〉

abc

de f

g

h

ij

1:1

Figure 5. Average particle activity compared to averagebed exchange rate (r↑/r↓p).

We used the dimensionless entrainment and depositionrates, r↑

∗ = r↑d350/ws, r↑p

∗ = r↑pd50/ws, r↓∗ = r↓d

350/ws

and r↓p∗ = r↓pd50/ws, respectively.

The average areal entrainment rate 〈r↑〉 is obtained bydividing the number of particles that were entrained duringa period T , inside a bed surface A, by both T and A. Theaverage particle deposition rate 〈r↓p〉 is obtained by assum-ing that moving particles always have the same probabilityq of being deposited (or the probability 1− q of continuingtheir movement), independently of their position or theirtrajectory. Thus, 〈r↓p〉 simply equals to the total numberof particle depositions divided by the cumulative time of alltrajectories. We found 〈r↑∗〉 = 1.14 ·10−4 and 〈r↓p∗〉 = 0.02.

Few data exist on entrainment and particle depositionrates, thus any comparison with the literature must be han-dled carefully. Lajeunesse et al. [2010] found r↓p

∗ ≈ 0.094(but 0.052 when using the settling velocity computed byBrown and Lawler [2003]) and r↑

∗ ≈ 3.2 · 10−3 for atransport stage τ/τcr ≈ 2. Ancey et al. [2008] obtainedr↓p∗ ≈ 0.05 and r↑∗ ≈ 7 · 10−4 for the same transport stage.On the whole, entrainment rates in the present study were

smaller than in previous ones, partly because the transportstage was smaller (we estimated τ/τcr ≈ 1). Depositionrates were also smaller than those found in the literature. Itis worth stressing that such differences may originate in thedependence of entrainment and deposition rates on the cho-sen vth, but also in the trajectories missed by the trackingalgorithm.

Entrainment and deposition rates are linked to bed eleva-tion changes via the simple Exner equation [Einstein, 1950]

(1− ξp)∂b

∂t= Vs (r↓ − r↑) = Vs (r↓pγ − r↑) , (11)

where ξp is the bed porosity. In a stationary, uniformcase in which the bed is neither aggradating nor eroding(∂b/∂t = 0), eq. (11) implies that 〈r↑〉 / 〈r↓p〉 = γeq. As canbe verified in Fig. 5, the present studys experiments showedno net aggradation or erosion.3.3.2. Local estimates

A difficulty arises when estimating the effects of local flowand bed conditions on particle entrainment and depositionrates. Indeed, the local flow characteristics are by definitioncontinuous spatio-temporal fields, whereas entrainment ordeposition occur at discrete locations (Fig. 6). This problemis recurrent in the analysis of point processes and is gener-ally referred to as a non-parametrical estimation (since thedependence is a-priori unknown) of dependence on a covari-ate continuous field (spatio-temporal in our case) [Baddeleyet al., 2012].

Bayes’ theorem can provide a simple solution to this es-timation problem. To illustrate this, let us study a particlespropensity to deposit with respect to the value of the lo-cal bed slope it moves on. Naturally, the following methodequally applies to other estimation cases. The local bedslope θ(x, t) may be taken to be a random field drawn froma random variable Θ, with probability density fΘ(θ). Letus consider the conditional probability P (D|Θ ∈ [θ, θ+dθ]),i.e., the probability that a moving particle deposits during∆t knowing that it is currently moving over a local bed slopecomprised in between θ and θ + dθ. We have

P (D|Θ ∈ [θ, θ + dθ]) = P (Θ ∈ [θ, θ + dθ]|D)P (D)P (Θ ∈ [θ, θ + dθ]) ,

dθ→0= P (D)

fΘ|D(θ)

fΘ(θ), (12)

where the first line uses Bayes’ theorem and the secondline introduces fΘ|D(θ), the probability density of the lo-cal bed slopes at deposition sites. Dividing (12) by ∆t,gives r↓p(θ) = 〈r↓p〉 fΘ|D/fΘ. 〈r↓p〉 was estimated previ-ously, whereas fΘ|D can be estimated empirically using theslope samples taken at particle deposition sites. In contrast,fΘ is simply estimated empirically by sampling all slopesencountered by moving particles.

The evaluation of the dependence of the local areal en-trainment rate can be determined in the same way as thedeposition rate. A subtlety exists, however, in that the prob-ability fΘ, required to compute P (E|Θ ∈ [θ, θ + dθ]), is notestimated in the same way as the particle deposition rate.For deposition, fΘ was estimated by sampling all the slopesencountered by moving particles since, by definition, onlymoving particles can deposit. On the contrary, all particlesresting on the bed can be entrained, so that fΘ must beestimated with samples of all the possible slopes (and notonly slopes on which particles are moving).3.3.3. Excess shear stress

First, we present the dependence of r↑ and r↓p on theexcess Shields stress τ∗− τ∗cr, where τ∗cr is computed accord-ing to the Fernandez Luque and van Beek [1976] formula:τ∗cr = τ

∗0 sin (θ + α) / sinα, with α = 47 and τ

∗0 = 0.035, a

reference Shields stress at zero slope.The result is shown in Fig. 7(a) and (b) for several spa-

tial averaging scales (4, 10 and 20d50). First, it appearsclear that the choice of averaging scale does not significantlychange the results for either the entrainment or depositionrates.

Fig. 7(a) also shows that the areal entrainment rate onlyhas a weak positive dependence on τ∗−τ∗cr. For excess shearstresses from 0.05 to 0.1, r↑

∗ is even a decreasing functionof τ∗− τ∗cr. Lajeunesse et al. [2010] reported a clear positivelinear dependence on τ∗ − τ∗cr for bed load transport overgentle slopes, whereas experiments by Ancey et al. [2008],also carried out over steep slopes, showed no strong correla-tion between flow strength and entrainment rates. We willshow that this absence of correlation can be explained bythe fact that entrainment is essentially triggered by othermoving particles rather than by the flow itself.

One alternative is to estimate the distribution of excessshear stresses as shown above for the particle depositionrate (i.e., with shear stress samples taken where particlesare moving), thus somehow conditioning r↑ on γ ≥ 0. Thismodified entrainment rate is noted as r̃↑∗ and plotted onthe same figure. r̃↑∗ shows a weak but clearer positive lin-ear dependence on the excess Shields stress.

Another interesting feature of Fig. 7(a) is that at low ex-cess Shields, the entrainment rate does not become zero (aswould be expected in presence of a critical threshold for theinitiation of motion). Again, this phenomenon may orig-inate from the particle entrainment facilitated by movingparticles coming from upstream.


tan θ

τ∗0

0.2

0.4

0.6

0.8

1

Distance

(m)

-0.3

-0.2

-0.1

0

0.1

0 50 100 150Time (s)

0

0.2

0.4

0.6

0.8

1

Distance

(m)

0

0.05

0.1

0.15

0.2

Figure 6. Estimated local bed slope tan θ (color), particle entrainment (white dot) and particle de-position (black dot). Continuous fields are spatially averaged over 10d50 using the method described inappendix A2

0.01

0.02

0.03

0.04

0 0.05 0.1 0.15 0.2

r ↓p∗

τ∗ − τ∗cr

0

0.5

1

1.5

2

r ↑∗

(10−4)

0

0.5

1

1.5

2

r ↑∗

(10−4)

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1tan θk

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1tan θk

r↓p∗(4d50)r↓p∗(10d50)r↓p∗(40d50)

0.013[τ∗c,0/(τ

∗ − τ∗cr) + 1]

(b)

r̃↑∗(10d50)

(a)

r↑∗(4d50)r↑∗(10d50)r↑∗(40d50)

(a)

r↓p∗(4d50)r↓p∗(10d50)r↓p∗(40d50)

(d) 0.042 tanθ4 + 0.0210.054 tanθ10 + 0.0220.105 tanθ40 + 0.024

(d)

r̃↑∗(10d50)

(c)

r↑∗(4d50)r↑∗(10d50)r↑∗(40d50)

(c)

Figure 7. (a) Particle entrainment rate and (b) deposition rate functions of the excess Shields stressτ∗ − τ∗cr, with τ∗cr = τ∗0 sin (θ + α) / sinα, α = 47◦ and τ∗0 = 0.035. (c) Particle entrainment rate and (d)deposition rate functions of the local bed slope tan θk, where k is the averaging scale.

Fig. 7(b) shows a strong inverse correlation between thedeposition rate and the excess Shields stress. At high excessShields stresses, the deposition rate is approximately equal

to 0.013, whereas at low excess Shields stress, its value con-


γ∗

0 50 100 150Time (s)

0

0.2

0.4

0.6

0.8

1

Distance

(m)

0

0.01

0.02

0.03

0.04

Figure 8. Estimated local activity γ∗ (color), particle entrainment (white dot) and particle deposition(black dot). Estimation of the local density of particles is made using a Gaussian kernel with a fixedspatial bandwidth of size 5d50.

stantly increases. This result may be interpreted intuitivelyby the fact that a particle will stop more easily in a weakflow than in a strong one. The simple equation

r↓p∗ = r↓

∗p,∞

(τ∗0

τ∗ − τ∗cr+ 1

), (13)

fits the deposition rate well, where r↓∗p,∞ ≈ 0.013 is a limit

dimensionless deposition rate. To the best of the authorsknowledge, a deposition rates dependence on excess Shieldsstress has never been reported before. The local flow con-ditions, not previously taken into consideration in the lit-erature, may thus have an influence on bed load transportrates through an alteration of particle deposition rates.

The dependence of r↑∗ and r↓p

∗ on local bed slope θ ispresented in Fig. 7(c–d). In contrast to the local excessShields stress, the averaging scale of local bed slope doesinfluence the results. This is particularly apparent for thedeposition rate in Fig. 7(d): the larger the averaging scale,the stronger the dependence of r↓p

∗ on θ. In other words,a moving particle “feels” the bed slope beneath it more ifthe slope is constant over a long distance rather than if it iscontinually changing.

Fig. 7(c) shows that r↑∗ does not change significantly

with bed slope, supporting the idea that the particle en-trainment mechanism cannot be explained using classic bedload transport theories. In contrast, Fig. 7(d) shows thatr↓p∗ depends approximately linearly on the local bed slope.

3.4. Feedback effects of the particle activity

3.4.1. Preliminary remarksAncey and Heyman [2014] showed that particle activity

obeys the general mass conservation equation

∂γ

∂t+∂qs∂x

= r↑ − r↓. (14)

As mentioned above, the areal deposition rate r↓ equalsthe individual particle deposition rate r↓p times the par-ticle activity γ. In contrast, the literature has often as-sumed that the areal entrainment rate r↑ was independentof γ [Lajeunesse et al., 2010; Parker et al., 2003]. However,an increasing number of studies have suggested that parti-cle entrainment depends on particle activity via a kind offeedback loop mechanism. This behavior has been observedexperimentally on steep slopes with low to moderate mobil-ity regimes [Ancey et al., 2008; Heyman et al., 2013, 2014].More specifically, this behavior was shown to favor the mo-tion of particles in groups rather than individually [Drakeet al., 1988; Heyman et al., 2014] and to generate high fluc-tuations in bed load discharge [Ancey et al., 2008; Heyman

0

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04

r ↑∗ ,r ↓

∗(10−4)

γ∗

r↓∗

r↑∗

r↑∗ ≃ 0.024γ∗r↓∗ ≃ 0.025γ∗

Figure 9. Entrainment and deposition rates as a func-tion of local particle activity.

et al., 2013]. Ancey et al. [2008] proposed that

r↑∗ = r0 + r1γ

∗, (15)

where r0 and r1 are two scalars, thus introducing a feedbackloop mechanism into the entrainment process. In their pa-per, r1 ≡ µd50/ws, where µ [s−1] is referred to as a collectiveentrainment rate.

Physically, the high Stokes number of moving particlessuggests that particle collisions are weakly damped by thefluid. The momentum transfer during collisions betweenmoving and static particles may, therefore, be important.In the case of low mobility regimes, this may be sufficientto trigger particle entrainment. It is also likely that locally,several particles may be entrained simultaneously by coher-ent turbulent eddies. In this case, the entrained particlesalso move in groups so that the transport also appears to beclustered.

Below, we investigate both the areal entrainment and de-position rate dependence on local particle activity.3.4.2. Dependence on γ∗(x, t)

As described above, the local particle activity γ(x, t) isnot a well-defined continuous quantity since particles areconsidered as discrete points in space. Nevertheless, an es-timation of γ(x, t) can be computed using standard kerneldensity estimation techniques (Eq. (1)), and this is shownin Fig. 8. As previously, Bayes’ theorem can be used to es-timate the relationships for r↑

∗(γ) and r↓∗(γ). The results

are shown in Fig. 9.


Surprisingly, r↑ shows the same strong dependence on γas r↓. A tangent fitted for values of γ

∗ > 0.01 gives a slopeof 0.025 for r↓

∗, and of r1 = 0.024 for r↑∗. The former is

close to the average particle deposition rate 〈r↓p∗〉 (Table 1)confirming the reliability of the relationship r↓

∗ = r↓p∗γ∗.

The slope of the entrainment rate, r1, is only slightly smallerthan the deposition rate. r0, the y-intercept coordinate, isvanishingly small, suggesting that the main source of parti-cle entrainment comes from the feedback loop mechanism.

Rewriting Eq. (14) together with Eq. (15) and the esti-mated values ε = r↓p

∗ − r1 � 1 and r0 ≈ 0, we get

∂γ∗

∂t+∂q∗s∂x≈ εγ∗, (16)

where ε is very small. At the limit ε → 0, the equilibriumvalue γ∗eq is not uniquely determined by (16). In this limitingcase, and for any value of γ∗, the source term in the Exnerequation (11) is null, so that for any γ∗ the bed neither ag-grades nor erodes. γ∗ thus depends solely on the imposedupstream boundary condition in Eq. 16.

As described previously, a conveyor belt continuously in-jected moving particles into the channel, at a prescribed andconstant rate, one meter upstream of the tracking window.After a short transition period, the average sediment dis-charge stabilized to a constant value. Meanwhile, at theend of an experiment, we noticed that bed load transportfroze rapidly in the channel after stopping the feeding sys-tem. This corroborates previous findings that at low mo-bility regimes, sediment transport intensity (e.g., the parti-cle activity) is solely determined by the upstream boundarycondition and not by the flow intensity.

In more realistic scenarios, ε is very small but larger than0, so that it takes a very long distance for particle activityto relax to its equilibrium value (γeq = 0 if r0 = 0). Thisdistance—also called the saturation length—was calculatedtheoretically by Heyman et al. [2014] for the case of a feed-back entrainment mechanism, and it was found to be

`sat = d502`∗cPec

(√1 + 4Pe−2c − 1

)−1(17)

with Pec = 〈up〉 `∗c/(wsD∗u), a Péclet number computed atthe dimensionless correlation length given by `∗c =

√D∗u/ε.

Replacing the variables values by their estimates in our ex-periments yields a saturation length of `sat ≈ 3.5m. Thus, inchannels shorter than a few saturation lengths, the upstreamboundary condition controls sediment transport, even forε > 0.

It is also possible to compute the average bed aggradationspeed from (11). On average, and for the highest input sed-iment flux, ∂b/∂t = wsεVsγ

∗/(d350(1 − ξp)) ≈ 13cm/hour.However, no net aggradation was observed during experi-ments, even though they lasted for several hours. This canbe explained by the intermittent upstream migration of localscour holes (or antidunes) that eroded the bed. A numericalcoupling of (11) and (14) with the Saint-Venant equationsmay reveal such phenomena.

4. Conclusions

In the present article, we analyzed an original dataset ofmeasurements taken from 10 bed load transport experimentscarried out in steep, super-critical, shallow-water flows. Nat-ural gravel of uniform diameters was used to ensure a phys-ical similarity with steep mountainous torrents while main-taining simplified experimental conditions. Using two lat-eral cameras and a powerful, novel tracking algorithm, morethan 1,000 km of bed load particle trajectories were recon-structed. To the best of our knowledge, this is the firsttime that such a large dataset of particle trajectories has

been collected. Moreover, we simultaneously recorded bedand water elevations so that various flow and bed spatio-temporal variables could be reconstructed, such as the localbed shear stress and the local bed slope. The particle trajec-tory data, together with the tracking algorithm codes, areavailable online to any interested researchers.

We presented some statistical characteristics of particletrajectories and described their dependence upon local flowand bed variations. As bed morphologies spontaneously de-veloped in the experimental channel, flow and bed topogra-phy were continuously changing through time and space, in-fluencing bed load transport in their turn. Because of thesefluctuations, we chose to consider the group of experimentsas a combined set of various topographic and flow bound-ary conditions and to study their impact on the sedimenttransport process.

With regards to particle kinematics, the important resultsmay be summarized as follows:

1. The average streamwise particle velocity scaled withthe shear velocity, whereas 〈up〉 /

√gh was proportional to

the channel Froude number. The particle velocity distribu-tion was a combination of an exponential distribution, forparticles close to the bed, and a Gaussian distribution, forparticles transported higher in the fluid column.

2. The average streamwise particle acceleration suggestedthe presence of two stable equilibrium particle velocities sep-arated by an unstable one. The first stable equilibriumstate was zero velocity and this, on average, attracted par-ticles whose velocities were smaller than the unstable equi-librium at up = 0.4ū. The second stable equilibrium statewas found at velocities approximately equal to the depth-averaged flow velocity, and this suggested that particles setin motion might travel a long time without stopping.

3. The average streamwise particle diffusivity, character-izing the average spread of particles while they advecteddownstream, was found to be a linear function of the shearvelocity.

Concerning mass exchanges (e.g., entrainment and parti-cle deposition) the important results were:

4. The particle deposition rate was inversely correlatedwith the local Shields stress. In other words, the lower theShields stress, the higher the deposition rate. It was alsocorrelated with the bed slope: the steeper the bed slope,the smaller the deposition rate.

5. The areal entrainment rate did not show a simple lin-ear dependence on either the bed shear stress or the bedslope. In contrast, it was strongly dependent on the localparticle activity, in a similar manner to the areal depositionrate.

6. In contrast to previous experimental studies, our ex-periments were carried out at a transport stage below orclose to 1. Under these conditions, the “classic” entrainmentrate (r0) cancels out, whereas the “collective” entrainmentrate (r1) remains positive and close to the deposition rate.

The majority of these observations have not been previ-ously reported in the literature. They thus constitute poten-tially new working material with which to test and developmodern bed load theories.

The behavior of particle entrainment close to the incipi-ent motion threshold is particularly interesting. We showed,that for these low mobility regimes (τ/τcr ≈ 1), the entrain-ment of bed particles was mainly triggered by other movingparticles. As a consequence, the relationship between theflow power and sediment transport rates was not unique.In fact, the sediment transport rates depended strongly onthe upstream boundary conditions. In other words, varioussediment transport rates may be observed for the same bedslope and shear stress. In addition to the sediment sup-ply “shortage” effect, reducing bed load rates in gravel bed


rivers [Zimmermann et al., 2010; Recking , 2012; Travagliniet al., 2015], we conclude that there also exists a sedimentsupply “excess” effect that can increase transport rates overlong distances.

We also found that the averaging scale does not signif-icantly influence the dependence of transport rates on bedshear stress (or shear velocity). This result is important fornumerical models of bed load transport. It suggests thatthe closure equations for bed load transport should be validwhatever the numerical grid size chosen. In contrast, wefound that the bed load rates changed depending on theaveraging scale of bed slope. Generally, the longer the dis-tance that bed slopes are constant, the more influence theyhave on transport rates; rapidly changing slopes have lessinfluence on transport rates.

These results are new; it should thus be kept in mind thatsome of them may be biased by our experimental and nu-merical methodology. For instance, it is likely that particleentrainment and deposition rates were artificially increasedby the broken trajectories that the tracking algorithm couldnot reconstruct. However, it is unlikely that this would af-fect their dependence on the flow and bed conditions. Fur-ther experimental or numerical work should, in any case,seek to validate the results presented here.

There is an urgent need to develop algorithms to accu-rately track bed load transport particles and to estimatetheir accuracy. In order to reduce trajectory reconstructionerrors, adopting a tracking algorithm with a probabilisticframework may help to improve results: each measurementwould come with a confidence score from 0 to 1, dependingon the certainty of the numerical detection. The statisti-cal treatment of particle characteristics would then system-atically take those confidence scores into account, limitingpossible bias, and thus providing a better estimate.

Notation

〈•〉 averaging ,•∗ dimensionless variable,θ bed slope,

d50 mean grain diameter,

d90 90 percentile grain diameter,

Sf friction slope,

ws particle settling velocity,

τb bed shear stress,

Re channel Reynolds number,

Fr Froude number,

u∗ flow shear velocity,

u depth-averaged flow velocity,

h water depth,

qs sediment discharge,

qs,out sediment discharge at the flume outlet,

qs,int sediment discharge at the flume inlet,

γ particle activity,

up particle velocity,

~xp = (xp, zp) particle position vector,

~vp = (up, vp) particle velocity vector,

~ap = (ax,p, az,p) particle acceleration vector,

B channel width,

Vs particle volume,

% fluid density,

g gravity acceleration,

R hydraulic radius,

ks characteristic roughness,

f friction coefficient,

%s sediment density,

Rep particle Reynolds number,

St Stokes number,t time,

f1 average force,

f2ξ(t) random force,

Du particle diffusivity,κ von Karman constant,

τL Lagrangian integral time scale of par-ticle velocity,

ρup particle velocity autocorrelation function,

σup standard deviation of particle velocity,

zmax max elevation for a particle to depositor erode,

vth velocity threshold for a particle to de-posit or erode,

E particle entrainment event,

D particle deposition event,

Table 3. Tracking algorithm, 1st pass. Detection and track-ing of “simple” particle trajectories.

1: Read first frame I;2: Initialize background B = I;3: loop Video frames4: Read frame pixel intensity Ik ∈ [0, 1];5: Detect bed and water elevation bk and wk;6: Track water surface velocity (PIV on {wk−1, wk});7: Compute foreground Fk ← |Ik −Bk|;8: Remove Fk pixels higher than w − 20;9: Dilate Fk of 2 pixels;

10: Erode Fk of 2 pixels;11: Compute connected regions of Fk > 0.2;12: loop Detected regions13: if Area > 40 pixels & Eccentricity < 0.95 then14: Save region barycentre i, j to array Ck;15: end if16: end loop17: Compute Euclidean distances D(k,k−1) between Ck to

Ck−1;18: Find non-conflicting associations with min(D(k,k−1));19: Update existing trajectories with non-conflicting

association;20: Create new trajectories with conflicting or unassigned Ck;21: Update background Bk ← 0.01sign(Ik −Bk) +Bk;22: end loop

Table 4. Tracking algorithm, 2nd pass. Reconstruction ofbroken trajectories. ∗: probability is estimated assuming thatthe particle follows a simple stochastic process of drift com-puted from the equation of motion with constant acceleration(~ap = ~ap,i) and constant diffusion (Dx = 30 cm

2 s−1 andDy = 10 cm2 s−1).

1: Read trajectory file obtained from Algorithm 3;2: loop Trajectory end points (~xp, t)i3: loop Trajectory start points (~xp, t)j4: Compute the probability for a particle to be in state5: (~xp, t)j knowing that it was in state (~xp, t)i,6: i.e., P ((~xp, t)j |(~xp, t)i)∗ ;7: Construct combination cost matrix from probabilities:8: C(i,j) = − ln [P ((~xp, t)j |(~xp, t)i)];9: end loop

10: end loop11: Estimate best trajectory combinations using the Hungarian

optimization algorithm (the maximum cost per combinationis set to 5);

12: Reconstruct trajectories from the obtained combinations;


r↓p particle deposition rate,

r↓ areal deposition rate,

r↑p particle entrainment rate,

r↑ areal entrainment rate,

r̃↑ areal entrainment rate (different computation),

Θ random variable of the bed slope,

fΘ probability density function of the bedslope,

fT probability density function of the bedShields stress,

α critical angle,

τcr critical shear stress,

u∗,cr critical shear velocity,

τ0 critical shear stress at zero slope,

u∗,0 critical shear velocity at zero slope,

ξp bed porosity,

r0, r1 entrainment rate coefficients,

ε difference between particle depositionrate and “collective” entrainment rate,

`sat saturation length,

`c correlation length,

Pec Péclet number taken at the correlationlength.

Appendix A: Numerical treatment

A1. Tracking Algorithm

We summarize the first pass of the tracking algorithm inTable 3. The second pass, useful for reconstructing brokentrajectories, is summarized in Table 4.

To estimate the efficiency of our algorithm, we comparedthe mean sediment discharge obtained from the trajectoryreconstruction to a discharge obtained from the accelerome-ters used by Heyman et al. [2013]. The comparison is shownin Fig. 10.

0

0.001

0.002

0.003

0.004

0.005

0.006

0 0.001 0.002 0.003 0.004 0.005 0.006

〈q∗ s〉

(TrackingAlg.)

〈q∗s 〉 (Accelerometers)

adh

ij

b

Figure 10. Comparison of dimensionless sediment dis-charge estimates from two methods: (1) the tracking al-gorithm and (2) the accelerometers used by Heyman et al.[2013].

Comparing methods, the sediment discharges agree rel-atively well. Together with possible reconstruction errorsin the tracking algorithm, it is worth pointing out that theacoustic measurement technique—being an indirect measureof sediment transport—may also be responsible for the scat-ter of points in Fig. 10. Note that the overall efficiency ofthe tracking algorithm cannot be estimated solely from thesediment discharge: other indicators need to be computed.Among these, as a measure of the deviation of a trackedtrajectory from a visual reference, the percentage of goodentrainment or deposition events detected may be an in-

teresting criterion with which to evaluate the efficiency oftracking algorithms. This, however, is beyond the scope ofthe present study.

A2. Computation of the average local bed level andbed slope.

Detection of bed and water levels were obtained by thedetection of high vertical gradients in the raw images. Ascan be observed in Fig.1, the bed level estimate fluctuatessignificantly because of the finite diameter of the grains andtheir various shapes. In other words, the values for the bedlevel, water depth, and bed slope ultimately depend on thescale at which they are observed; that is, on the size andthe type of the averaging operation chosen. We should,therefore, expect the statistics obtained at a given averagingscale to differ from those obtained at another scale. This isparticularly true for the definition of bed slope, where thesame location can yield totally different values depending onwhether bed slope is defined at scales in the order of particlediameters, in the order of the typical bed form length, or atthe scale of a stretch of river.

The computation of these averages is performed as fol-lows. Given a scale kd50 (k grain diameters), the bed level,and the bed slope at a location xb are computed by fittinga linear regression curve y = αx+ β to bed elevation points(xi, yi) located at a distance smaller than k/2d50 from x.The bed slope at xb is then tan

−1(α) and the bed elevationis yb = αxb + β. The same averaging procedure is used tocalculate the water depth and its slope. Issues may arise inthe image boundary as a decreasing number of points areavailable in the near neighborhood of xb to estimate the re-gression coefficients. It is interesting to note that higherorder estimates of bed elevation and slope may be obtainedby fitting the neighboring points to a second or higher orderpolynomial. In doing so, we can obtain information aboutthe local convexity of the bed surface. Boundary issues arestronger in non-linear regressions so that we chose the linearregression in this study.

Appendix B: Estimation of the bed shearstress

In uniform, steady conditions, bed shear stress can bederived from the force necessary to balance tangential fluidweight:

τb = %gRbSf , (B1)

where Rb is the bed hydraulic radius (i.e., the hydraulic ra-dius corresponding to the gravel bed) and Sf the energyslope.

B1. Bed hydraulic radius

For infinitely large channels, Rb = h, is the flow depth;whereas for narrow rectangular channels of uniform rough-ness, Rb = Bh/(B + 2h). In our experiments, the chan-nel was made of lateral glass walls and a gravel bed, soit could not be considered of uniform roughness. Further-more, being fairly narrow, the lateral walls still dissipated anon-negligible part of the flow energy so that Rb < h andRb > Bh/(B+2h). These two bed hydraulic radii constitutethe natural boundaries for the real bed shear stress.

The Einstein–Johnson [Einstein, 1934, 1942; Johnson,1942] method can be used to estimate the part of the shearstress carried by the bed and the part carried by the sidewalls [Guo, 2014b]. The main hypothesis is that the entirecross section of the channel can be divided into two areas:one supported by the walls (Aw) and one supported by the


0

0.1

0.2

0.3

1 2 3 4 5 6 7

f

Rb/d50

(a)

0.20.5

1.0

ks/d50 = 1.5

2.0

2.5

3.0 3.5 4.5 5.5 6.5

0

5

10

15

0 0.2 0.4 0.6 0.8 1

τ

Distance [m]

̺gRbSf , Rb = h(1− 2Rw/B)Keulegan (ks = 2d50)

Manning (K = 43)Guo (z0 = ks/30)

(b)

Figure 11. (a) Average friction coefficient f as a function of average bed hydraulic radius Rb, withRb = h (circles), Rb = Bh/(B + 2h) (down triangles) and Rb = h (1− 2Rw/B) (up triangles). Averageswere taken over the whole flume for each experimental run. Diamonds represent the Guo and Julien[2005] parameters. For comparative purposes, f in Keulegan’s formula has been plotted for various val-ues of ks (dotted lines). (b) Examples of predictions using different resistance laws for an experimentalrun, shown in the photo-montage from the two cameras in the middle panel. The vertical distanceshave been distorted for better visualization. The velocity profiles are those calculated using Guo’s pa-rameters (Eqs. (B8)) while the dashed lines are the detected water and bed elevation. The yellowband in the bottom graph stands for the natural shear stress boundaries calculated using Rb = h andRb = Bh/(B + 2h).

bed (Ab). These two areas can be assumed to form two

independent parallel channels, working side by side, with

the same average flow velocity and slope, but with different

roughness and hydraulic radii. The hydraulic radius of the

channel corresponding to the wall friction, noted Rw, can

be determined taking a Darcy-Weibach-type law [Johnson,

1942] with a friction coefficient fw:

τw%

=fw8u2f = gRwSf . (B2)

fw itself can be estimated using the von-Karman-Prandtl

friction law

1√fw

= 2 log(

Rew√fw)− 0.8, Rew =

4Rwufν

. (B3)

The pair of equations, (B2) and (B3), uniquely determine

Rw and fw, although not explicitly (one thus needs to iterate

to reach the solution).

By definition, Rw = Aw/(2h) and Ab = Bh−Aw, allow-ing us to determine the bed hydraulic radius

Rb =AbB

= h

(1− 2Rw

B

). (B4)

B2. Friction coefficient

The bed shear stress may also be estimated using friction

laws derived from dimensional analysis, such as the Darcy-

Weibach formula

τb(x, t)

%=f

8ūf (x, t)

2, (B5)

where f is the friction coefficient and may be a function of di-

mensionless numbers, such as the channel Reynolds number

Re= 4Rhū/ν, the roughness Reynolds number k+s = ksu∗/ν,

and the relative roughness size ks/h, for example.


Keulegan’s [1938] formula, in particular, derives from thelog law and reads

f =8κ2

ln2(11Rb(x, t)/ks)(B6)

where ks is an equivalent roughness height. Eq. (B6) isassumed to hold locally in all conditions—for unsteady aswell as spatially variable flows—although it was originallyderived for flows in equilibrium. Another well-known rela-tionship, the so-called Gauckler–Manning–Strickler formulareads,

f = 8g

K2R1/3b

, (B7)

with K, the Strickler coefficient, inversely proportional tothe roughness of the bed. Both of these laws are functionsof the bed hydraulic radius.

For flows in smooth rectangular channels, the semi-analytical formula of Guo and Julien [2005] can also beused. Its application to non-homogeneous cross sections isdescribed in Guo [2014a]. By studying the intensity and di-recti

exploring the physics of sediment transport in non-uniform super...

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