Granular Matter (2011) 13:39–51DOI 10.1007/s10035-010-0221-8

Falling process of a rectangular granular step

Hsien-Ter Chou · Ching-Fang Lee

Received: 27 September 2009 / Published online: 26 September 2010© Springer-Verlag 2010

Abstract This study experimentally investigates the fallingprocess of a dry granular step in a transparent plexiglass chuteby particle image analysis. Three types of uniform sphericalbeads and one type of quartz sand were piled up with var-ious bed slopes and widths to elucidate their flow charac-teristics. The surface angles during the early slipping phaseare close to the failure angles that are associated with theactive earth pressure, based on the Mohr-Coulomb frictionlaw. For a given size of particles (d) and slope (θ), the retreat-ing upper granular surface follows a theoretical curve, anddimensionless mobile length decreases as the dimensionlesstime parameter t∗ increases. Velocity profiles measured at theside wall exhibit an exponential-like tail close to the staticregion at the bottom of the chute. As determined by the con-servation of mass and momentum, the relationship betweenthe characteristic velocity and the characteristic depth is lin-ear in the transient flow.

Keywords Granular flows · Falling process ·Granular step · Avalanches · Side wall effect

1 Introduction

Granular flows frequently occur in natural fields and engi-neering applications, including rock avalanches, debris flows,pyroclastic flows, the breaking of retaining walls or silos,and industrial granular transport. Granular matter exhibits themultiphase properties of solids, liquids and gases under var-ious external conditions. In most cases, the static propertiesand dynamic characteristics of granular matter are closely

H.-T. Chou (B) · C.-F. LeeDepartment of Civil Engineering, National Central University,Chung-Li 32001, Taiwan, ROCe-mail: htchou@cc.ncu.edu.tw

related to the natural process and artificial actions. Theavalanching process consists of both the increasing andthe decreasing phases in rectangle granular piles [1]. Boththe flow velocity and the flow depth increase rapidly to amaximum in the initial stage of an avalanche, slowly decay-ing thereafter. The final angle of a granular pile approachesthe angle of repose. In the initial stage, the falling process isindependent of both the type of granular matter and the side-wall friction, according to experimental results and numericalsimulations [2–8]. With respect to the collapse process, Lubeet al. [9] proposed that both the final avalanching time t f andthe height h∞ are independent of the granular material andthe roughness of the bottom of the chute. Based on experi-mental observations, the collapse of a granular column pro-ceeds through three stages: the first collapse stage involvesfree-fall behavior, which is solely controlled by gravity. Thesecond stage is characterized by lateral motion until the inter-face between the static and dynamic regimes reaches thefree surface. The sidewall effect is negligible in the first twostages. However, the effect of friction between particles andside walls plays a predominant role in the final stage, in whichthe flow layer is thinner than in the other stages and the ava-lanching process is slower.

A stony debris avalanche typically occurs on a steep slopewith exposed regolith. The failure is composed of two pro-cesses—the fast avalanching and the relatively slow surfaceflow with slope relaxation. The entire process has a shortduration, and it involves non-uniform flow behavior. Regard-ing the rapid falling of granular material, several questionshave not been answered, such as those concerning the defi-ciency of general theories and governing equations. Somestudies have focused on the relationship between the initialgranular column (geometric effect) and the pattern of thefinal deposition. While only a few attempts have been madeto elucidate the collapsing process, this work experimentally

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40 H.-T. Chou, C.-F. Lee

Fig. 1 Experimental setup (Gray region represents instantaneous flow-ing layer)

investigates the collapse of a granular step. Additionally, theeffects of side walls on the falling process, the transient veloc-ity profiles and the evolution of flow rate throughout the gran-ular avalanche are examined. Finally, a scaling law for thefalling process will be presented and confirmed with experi-mental results. The proposed conceptual model may promotethe understanding of the retreating process of granular slopes.

2 Experimental setup

The experimental chutes consist of transparent acrylic plates(Fig. 1) with a length Li of 60 cm and a height H of 40cm. Four widths W (2.6, 5, 7.6, and 10 cm, respectively) areadopted to evaluate the effect of side-wall friction. The ini-tial height of each granular pile is about 10 cm. To preventthe slipping of particles at the bottom of the chute, the samegranular matter was glued to the bed of the chute. The chuteis filled three times with dry granular material. Each fillingis approximately 3.3 cm thick with random closed packingyielding a three-dimensional volume fraction frcp ∼ 0.64(2-D frcp ∼ 0.84 measured from the side wall). The experi-mental container is horizontally shaken for 10 s to level thegranular surface. A cord was attached to the front gate, whichis suddenly released at the beginning of each run. The gran-ular material collapses freely under the influence of grav-ity as soon as the gate is opened. To evaluate the effects of

particle size and shape, natural quartz sand (diameter, d =2 ± 0.2 mm, angle of repose, ϕ = 35.2o) and three dif-ferent granular materials are used in this work (Table 1).There are (1) glass beads (d = 2 ± 0.1 mm, ϕ = 18.1o),(2) mill stone (d = 3.5 ± 0.3 mm, ϕ = 24.6o), and (3)polystyrene beads (d = 5.8 ± 0.1 mm, ϕ = 20.8o). Theangles of repose of these materials were measured three timesby discharging the material vertically from a hopper onto awider transparent cylinder (D = 12 cm). Images of the finalstatic deposition angles were then obtained by using pack-age ImageJ (available from http://rsb.info.nih.gov/ij/) in fourradial directions and the average value was taken. The fric-tion coefficient between granular particles and the sidewallwas determined by performing an adjustable inclined platetest. A high-speed CCD camera (IDT XS-3) with a samplingrate of 200 fps is installed above the side wall of the chute.The capture region is illuminated by two continuous halogenlamps at both the upstream and downstream ends. In each run,all processes are recorded from the gate opening to the endof the avalanches. Particle tracking velocimetry (PTV) basedon the Voronoi cell method [10] and particle image veloci-metry (UraPIV, available from http://urapiv.wordpress.com/)are utilized to make velocity measurements with high spatialand temporal resolutions. The viewing window of the cross-sectional CCD camera is set to capture a region of interestof around 50×50 cm, which contains 1,350 squared pixels(1 cm ≈ 27 pixel widths). Figure 2 schematically depicts thegranular slopes at different stages. The bed slope of the chuteθ is adjusted from 0o to 10o, and the initial height of eachgranular pile hi = 10 cm.Li and L f represent the initial andfinal upper surface lengths, respectively. At any elapsed timet, the distance between the brink (nick point) and the finalstatic region (L f ) along the upper surface is given by L .

3 Results and discussion

3.1 Flow phenomenon

Lajeunesse et al. [3] proposed a characteristic time scale τc =(h/g)1/2, which represents the time of free-fall above thebase of a block through a height of h [3]. When the elapsedtime t exceeds τc(τc = 0.1s in this work), the slumping grad-ually transforms into fully developed flow. Figure 3 presents

Table 1 Properties of fourgrains used in experiments

* Coefficient of restitution

Media Diameter d(mm) Density ρ(g/cm3) e∗ Angle of repose ϕ(deg.)

Plastic bead 2.00 2.33 0.80 18.1Mill stone 3.46 2.77 0.53 24.6Polystyrene bead 5.80 2.10 0.32 20.8Quartz sand 2.00 2.65 0.20 35.2

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Falling process of a rectangular granular step 41

g

Lf

hi

L

Li

S

Vs

(I). flowing layer(II). temporal static layer(III). final static zone

(II)

(III)

h M

(I)s

fhout

Fig. 2 Initial and final granular slopes

four images of the flowing layer and the static zone in the hor-izontal chute (θ = 0o) after the gate was opened. The dashedlines denote the boundaries between two flow regimes. Whent < τc, the particle movements are typical en mass move-ments along the failure line. In contrast, when t > τc, thegranular motions are surface flows. As displayed in Fig. 3,the initial sliding stage is only a small fraction of the over-all collapse duration. Therefore, the sliding phase is muchshorter than the surface-flow phase.

Avalanching behaviors are captured from ten successivelong-exposure images (Fig. 4a–d). The initial solid sliding(t < τc) phase is independent the change in the bed slope(Fig. 4a, b), but the increase in the bed slope considerablyenhances the retreating rate of the upper surface in the surfaceflow phase (t >> τc), as shown in Fig. 4c, d. The flowingthickness also increases with the bed slope in the surface flowstage. The observed sliding failure surface is consistent withthe well-known Mohr-Coulomb failure criterion so the criti-cal angle of inclination, θ f , between the slip surface and thebed plane in the active state is

θ f = 45 + ϕ

2[deg .] (1)

where ϕis the angle of repose of the granular material. Whenthe front gate was suddenly opened, the granular assemblyis in Rankine’s active state. Most particle trajectories abovethe sliding failure surface are parallel to the slip surface, aspresented in Fig. 5a. Subsequently in the surface flow phase(t > τc), a series of saltations above the granular surfaceare observed, as shown in Fig. 4c, d. When the angle of theslip surface is less than θ f as shown in Fig. 5b, the saltationbehavior of mobile material (ex. glass beads) seems to beintensified.

Material properties significantly affect the evolution of agranular surface, as displayed in Fig. 6a–d. A few local humpson the inclined surface are formed because of local instabil-ities in the surface flow stage. Granular and non-sphericalparticles with greater angles of repose (such as sand, Fig. 6d)have a steeper surface slope than the smooth glass beads(Fig. 6a) for a particular bed slope. Increasing the bed slopeextends the motion of the upper surface and reduces the final

collapsing time. Notably, the angles of the final surface slopesexceed the original angles of repose owing to the sidewallfriction in the narrow chute.

3.2 Scaling law for the retreating upper granular surface

Based on experimental observations, we assert that theretreating upper granular surface during the collapsing pro-cess (Fig. 2) is composed of two co-existing modes: an initialfast solid-like sliding and a slow surface granular flow. Therolling rate associated with the surface flow is assumed tobe constant (∼ Cr ), while the avalanching mode of a granu-lar step depends on the sliding scale and it also decays withtime. The governing equation for the retreating upper gran-ular surface, in terms of the distance L , as shown in Fig. 2,is thus

d L

dt= Cr − L

t + ta(2)

where t is the elapsed time; ta is a constant, and t f is the finalavalanching time. Equation 2 can be solved by integration toyield

L = Cr

2(t + ta) + C2

(t + ta)(3)

where Cr and C2, are constants that are determine from theboundary conditions: at the beginning of the collapse pro-cess (t = 0), L = La(La = Li − L f ), and at the end of thecollapse process (t = t f ), L = 0.

Cr = 2(−Lata)

(ta + t f )2 − t2a

(4a)

C2 = Lata(ta + t f )2

(ta + t f )2 − t2a

(4b)

A new dimensionless sliding time scale, K = ta/(ta +t f ),and a dimensionless time χ = (t + ta)/(ta + t f ) are defined,then one can normalize the mobile length L according to theinitial mobile length La as follows

L∗ = L

La= K

1 − K 2

1 − χ2

χ(5)

The dimensionless time χ is replaced by a new dimensionlessparameter, t∗, defined as

t∗ = t

t f= 1

1 − K(χ − K ) (6)

Substituting Eq. 6 into Eq. 5 yields a formula for the retreat-ing of the upper granular surface as

L

La= K

1 − K 2

1 − (t∗ + K (1 − t∗))2

(t∗ + K (1 − t∗))(7)

The experimental results obtained with various slopes forplastic beads, mill stone, polypropylene beads, and sand are

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42 H.-T. Chou, C.-F. Lee

Fig. 3 Sequence ofphotographs from experimentsfor which W = 5 cm andθ = 0o: a d = 2 mm, glass bead,b d = 3.5 mm, mill stone,c d = 5.8 mm, polystyrene bead,d d = 2 mm, sand. White dashedline represents interface betweenflowing layer and static zone

Fig. 4 Long-exposureexperimental images of surfaceavalanching at different timesand bed slopes (d = 2 mm, glassbeads, W = 5 cm

=0o =5o =10o

(a) t=τc

(b) t=2τc

(c) t=5τc

(d) t=10τc

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Falling process of a rectangular granular step 43

flowing layer saltating layer

(a) (b)

f f

the critical boundary between flowing layer and static region .

θf =45o+ϕ /2 θf <45o+ϕ /2

Fig. 5 a Solid sliding and b saltating layer. Red dashed line representscritical boundary between flowing layer and static region

shown in Fig. 7. The observed histograms for different col-lapsing process for various experiments agree closely withEq. 7. The coefficient of the characteristic time scale K isapproximately 0.07 in all cases.

3.3 Effect of side walls

Taberlet et al. [16] found that the avalanching angle (θs)

is related to side-wall friction, channel width and flowingthickness. Grasselli et al. and Zhou et al. [17,18] proposedthe following relationship based on experimental results andnumerical simulations

θs = ϕ(1 + α1e−κW ) (8)

where θs is the surface angle measured from the side wall;ϕ = angle of repose, and α1 and κ are coefficients that spec-ify the effect of the sidewall. Experimental results obtainedat a constant width reveal that the magnitude of κ is indepen-dent of granular materials (or particle sizes) (Grasselli et al.[17]). However, Zhou et al. [18] found that the coefficient κ

decreases as the particle size increases based on numericalsimulations. du Pont et al. [25] proposed a non-exponentialrelationship between scaled surface angle (θs −ϕ)/ϕ and thecontainer width W . The relationship between (θs −ϕ)/ϕ andW/d is shown in Fig. 8 based on experimental results fromother studies and the present investigation. For a particularwidth, the surface slope is inversely proportional to particlesize for both spherical and irregular particles. The effect ofthe side walls on the surface angle is proposed as

θs − ϕ = ϕ

1 + α2(W/d)(9)

where α2 is a fitting constant (= 0.345 for all materials). For anextremely narrow channel, θs may reach twice angle of reposeϕ(W ∼ 0, θs ∼ 2ϕ). For a wide channel, θs approximates

ϕ(W ∼ ∞, θs ∼ ϕ). The effect of the side wall friction onthe inclined surface angle should be taken into account as longas W/d < 20, as depicted in Fig. 8. The experimental dataindicate that the correlation between κ and d is insignificant.The value of κ is not only controlled by particle size but it isalso affected by wall friction and the coefficient of restitution.Further studies on this issue are required.

3.4 Transient density and velocity profiles

The evolution of the granular surface is transient and non-uniform, so the profiles of granular velocity and density weremeasured at the middle point of the inclined free surface dur-ing the collapsing process (Fig. 2, point M). Based on ten suc-cessively captured images, the local 2D solid area fraction isassumed to be the ratio of area occupied by the particles tothe area of the segment [11]

Cs = nπd2

4A(10)

where n, d and A are the number of beads in each segment,the diameter of the beads, and the area of the rectangular bin,respectively. As shown in Fig. 9a–c, the changes in the solidfraction profiles (C = Ci − Cs , the difference between theinitial and transient concentrations, Ci and Cs; Ci in all casesvaries from 0.68 to 0.76) for the three materials and slopesfollow the same trend, which C varies with flowing depth.Although the granular packing in this study is 3-D geometry,but the profile of solid fraction is measured from the trans-parent side wall. The particles behind the first layer near theimmediate wall are negligible. Consequently, the range ofthe solid fraction profiles obtained here may yield the 2-Dgeometry instead of 3-D. The most dilute part correspondsto the fast flowing layer close to the free surface of the gran-ular assembly, the densest part is beneath the flowing layer.In all experiments, the solid fraction increases with depth tothe quasi-static region. The increase of the bed slope doesnot influence the solid fraction profile, but it increases thethickness of the flow layer instead. The average solid vol-ume fraction Ci in the static region is around 0.68 ∼ 0.76(2-D frcp ∼ 0.84) and the limits between the upper and lowerbounds are close to the values for random loose packing [12].Orpe et al. (2007) [11] proposed that the upper limit on Cs

in the dense region, for hexagonal closed packing, is 0.866.The evolution of both surface velocity and the thickness

of the flowing layer, respectively, at the midpoint along theinclined surface are shown in Fig. 10a, b. The experimentalresults of Siavoshi et al. [1] are also included for compari-son. These curves demonstrate that both surface velocity andflow thickness reach their peaks at around t = τc, in the initialcollapsing stage (t∗ < 0.1; where the definition of t∗ is givenby Eq. 6. The appearance of maximum velocity at t = τc

reveals that the sidewall effect in the initial free-fall stage

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44 H.-T. Chou, C.-F. Lee

X [cm]

20 30 40 50 60

s [c

m]

0

2

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X [cm]

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s [c

m]

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s [c

m]

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t=0.5∼5.0τc

t=10.0τc

t=tfθ=0o

θ=5o

θ=10o

20 30 40 50 60

s [c

m]

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X [cm]

20 30 40 50 60

s [c

m]

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t=5.0∼5.0τc

t=10.0τc

t=tfθ=0o

θ=5o

X [cm]

20 30 40 50 60

s [c

m]

0

2

4

6

8

10

θ=10o

(a) (b)

20 30 40 50 60

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m]

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m]

0

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t=0.5∼5.0τc

t=10.0τc

t=tfθ=0o

θ=5o

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X [cm]

20 30 40 50 60

s [c

m]

0

2

4

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10

20 30 40 50 60

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m]

0

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20 30 40 50 60

s [c

m]

0

2

4

6

8

10

t=0.5∼5.0τc

t=10.0τc

t=tfθ=0o

θ=5o

θ=10o

(c) (d)

Fig. 6 Evolution of flowing surface for various slopes with W =5 cm: a d = 2.0 mm, glass beads b d = 3.5 mm, mill stonesc d = 5.8 mm, polystyrene beads d d = 2.0 mm, sand. The time inter-

val between black solid lines downward from the top is 0.5τc(i.e., t =0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, and 5.0τc). The height, s, andthe length, X , are defined in Fig. 2

is negligible. The dimensionless flowing thickness decreasesas the granular size increases, and it increases slightly as theangle of the bed slope increases.

The down-slope velocity profiles were measured at themidpoint of the inclined free surface (point M in Fig. 2)from ten images captured successively through the transpar-ent side wall. Numerous exponential velocity profiles, foruniform and steady flows [13–15,19–21] and non-uniformand unsteady flows [8,9], have been presented. Figure 11a–dshow the mean velocity profiles for three bed slopes (0o, 5o,and 10o). du Pont et al. [22] proposed the following velocityprofiles for transient surface slumping in a rotating drum as

v (y, t) = vs (t) exp (−y/λ) (11)

where y represents the downward distance from the inclinedfree surface (y = 0) and vs represents the surface velocityat the side wall. λ is the characteristic length scale for gran-ular avalanches. Except in the early stage of surface flow(t∗ ∼ 0.5), the exponential velocity profiles as displayed inFig. 11a–l, are reasonable. The evolution of transient veloc-ity profiles for glass beads of sizes 0.2 < d < 3 mm is givenby λ/d = 3 ± 1 [22]. The mean values of λ/d among allexperiments are 4 ± 1 (glass beads, d = 2 mm), 3 ± 0.5(mill stone, d = 3.5 mm), 2 ± 0.3 (polystyrene beads,d = 5.8 mm) and 3 ± 1 (sand, d = 2 mm), respectively.Komastu et al. [23] found that the characteristic length scaleof the exponential velocity profile for the creep motion isapproximately the granular size. Figure 12 compares the

123

Falling process of a rectangular granular step 45

t*

0.0 0.2 0.4 0.6 0.8 1.0

L*

0.0

0.2

0.4

0.6

0.8

1.0

d=2.0 mm(θ=0o

)d=2.0 mm(θ=5

o

)d=2.0 mm(θ=10

o

)d=3.5 mm(θ=0

o

)d=3.5 mm(θ=5

o

)d=3.5 mm(θ=10

o

)d=5.8 mm(θ=0

o

)d=5.8 mm(θ=5

o

)d=5.8 mm(θ=10

o

)d=2.0 mm, sand(θ=0

o

)d=2.0 mm, sand(θ=5

o

)d=2.0 mm, sand(θ=10

o

)Eq.(7), K=0.07

Fig. 7 Dimensionless length, L/La , against dimensionless time scalefor various slopes with glass beads, mill stones, polypropylene beadsand sand

W/d0 20 40 60

(θs−

ϕ)/ϕ

0.0

0.2

0.4

0.6

0.8d=2.0 mm, glass beadd=3.5 mm, mill stoned=5.8 mm, polystyrene beadd=2.0 mm, sandd=180 μm, sand(Grasselli, 1997)d=250 μm, sand(Grasselli, 1997)

d=200 μm, glass bead(Grasselli, 1997)d=3.0 mm, glass bead(du Pont, 2003)d=5.0 mm, glass bead(du Pont, 2003)

Fig. 8 Ratio (θ − ϕ)/ϕ as a function of normalized width W/d. Solidcurve is best-fit regression given by equation: (θs − ϕ) = ϕ/(1 +α2(W/D)) where α2 = 0.345. The values of k for experimental materialare −0.16(d = 2.0 mm), −0.19(d = 3.5 mm), −0.11(d = 5.8 mm),and −0.10(d = 2.0 mm, sand)

characteristic lengths λ/d for instantaneous avalanches inrotating drums and chute experiments [24–26]. Although thefluctuation increases, λ/d tends to decrease as the particlesize d increases. The values of λ/d obtained from the chuteexperiments exceed those in the rotating drums for the sameparticle size. Notably, the particle with a higher coefficientof restitution exhibits greater mobility. Therefore, the λ/dratio for the glass bead is larger than that of sand at the samediameter (d = 2 mm).

As plotted in Fig. 10, the dimensionless velocities v/√

gdat given time t∗ increase continuously with the bed slope butdecrease as the particle size increases. The transient velocityprofiles always decrease with time in the surface flow regime.The shear rate γ in the upper flowing layer for glass beads,being obtained from Fig. 11(a), is approximately 0.38

√g/d.

Midi [15] reported the value of shear rate for the steady gran-ular flow is around 0.5

√g/d). Other profiles (Fig. 11b–d)

exhibit a decay trend in shear rate which is affected by thebed slope and time duration. The shear rate of unsteady gran-ular flows changes slightly with time and it resembles surfaceflows in a narrow channel [8]. The well-known power-lawvelocity profile is also applicable to steady, uniform chuteflows as follows

v (y)/vs = 1 − (1 − y/h)η (12)

where vs is the flowing velocity at the free surface. TheBagnold velocity profile (η = 3/2) is not applicable whenthe side-wall effect is evident (i.e., sidewall-stabilized heaps(SSH)) [27]. This difference exists even though granularflows are very thin. Since the flow configuration in thiswork is strongly affected by side walls, SSH rheologyshould be more applicable than Bagnold rheology. In thisunsteady regime, the following modified power-law profileis observed at the side wall instead of the Bagnold velocityprofile

v (y)/vs = (1 − y/h)η (13)

The normalized velocity profiles for W = 5 cm are con-cave curves, as shown in Fig. 13a–d. The index, η decreasesslightly as elapsed time t∗ passes. The effect of the bed slopeon η is negligible here. All of the velocity profiles in thecollapsing stage are substantially different from those asso-ciated with Bagnold rheology with a convex profile (graysolid lines in Fig. 13). For SSH rheology, similar velocityprofiles have been obtained for polystyrene discs in 2D sim-ulation and for a 3D chute [16,27]. The Bagnold profile typ-ically appears in a wide channel with less frictional effectand steep bed slope, or a flowing layer of the entire depth indry granular system. Bi et al. [27] introduced a dimension-less number, ξ = (μwh)/(μi W ), to identify the rheology ofgranular flows. The flow is analogous to Bagnold rheologywhen ξ << 1. In contrast, SSH rheology may develop witha concave profile when ξ ∼ 1 or ξ >> 1. The relation-ship between the above two indices is reasonable as shownin Fig. 14, and the velocity profile would approach linear-ity (η ∼ 1) in a later stage. As the flow rate decreases, thethickness of the flowing layer becomes thinner than the ini-tial avalanche. The flow becomes shallow flow (h ∼ 1d)

on top of a quasi-static region at the end of the collapsingprocess. Accordingly, ξ must decrease for a given channelwidth, wall friction, and angle of repose of the material.Another flow rate based on SSH rheology, Q ∝ (h − hjam),

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46 H.-T. Chou, C.-F. Lee

Fig. 9 Profiles of the initialsolid volume fraction Ci and theflow-induced volume fractionchange C for mill stone(d = 3.5 mm) with W = 5 cm:a θ = 0o, b θ = 5o, c θ = 10o.The flow-induced solid volumefraction change is defined asC = Ci − Cs(Ci : initial solidvolume fraction denoted by graydashed lines; Cs the instantflow-induced solid volumefraction)

C

0.0 0.2 0.4 0.6 0.8

Ci

0.0 0.2 0.4 0.6 0.8

y/d

0

5

10

15

20

25

t=τc

t=2τc

t=5τc

t=10τc

Ci

C

0.0 0.2 0.4 0.6 0.8

Ci

0.0 0.2 0.4 0.6 0.80

5

10

15

20

25

t=τc

t=2τc

t=5τc

t=10τc

Ci

C

0.0 0.2 0.4 0.6 0.8

Ci

0.0 0.2 0.4 0.6 0.80

5

10

15

20

25

t=τc

t=2τc

t=5τc

t=10τc

Ci

(b) θ=5ο

Δ Δ Δ

(a) θ=0ο (c) θ=10ο

Ci=0.68~0.72 Ci=0.70~0.74 Ci=0.74~0.77

Fig. 10 a Dimensionlesssurface velocity Vs/(gd)0.5 andb flowing depth versus timescale for various granularmaterial (Vs depicts the surfacevelocity). Gray dashed curvesare best fits to four sets ofexperimental measurements.Two short-dashed linesdelineate the dimensionlesscharacteristic time for variousexperiments

t*

0.0 0.2 0.4 0.6 0.8 1.0

Vs/

(gd)

0.5

0

1

2

3

4

5d=2.0 mm(θ=0o)d=2.0 mm(θ=5o)d=2.0 mm(θ=10o)d=3.5 mm(θ=0o)d=3.5 mm(θ=5o)d=3.5 mm(θ=10o)d=5.8 mm(θ=0o)d=5.8 mm(θ=5o)d=5.8 mm(θ=10o)d=2.0 mm-sand(θ=0o)d=2.0 mm-sand(θ=5o)d=2.0 mm-sand(θ=10o)Siavoshi(2005)

τc/tf

t*

0.0 0.2 0.4 0.6 0.8 1.0

h/d

0

5

10

15

20

25d=2.0 mm(θ=0o)d=2.0 mm(θ=5o)d=2.0 mm(θ=10o)d=3.5 mm(θ=0o)d=3.5 mm(θ=5o)d=3.5 mm(θ=10o)d=5.8 mm(θ=0o)d=5.8 mm(θ=5o)d=5.8 mm(θ=10o)d=2.0 mm-sand(θ=0o)d=2.0 mm-sand(θ=5o)d=2.0 mm-sand(θ=10o)Siavoshi(2005)

τc/tf

(a) (b)

that differs from Bagnold rheology, Q ∝ h5/2, is verified.The dimensionless flow rate Q/(gd3)0.5 evolves spontane-ously toward decreasing linearly in all experimental runs areshown in Fig. 15, and the jammed flowing depth, hjam, is4.4 d, being less than that in previous study [27]. This resultmay come from differences in both the flowing regimes andthe configurations. As shown in Figs. 14 and 15, the velocityprofiles in unsteady, narrow granular flows are not consistentwith the Bagnold profiles. In our experimental configura-tion, the thickness of the quasi-static region always exceedsthat of the flowing layer, even in the final stage. Moreover,the sidewall effect and a thin flowing layer immediatelyabove the thick quasi-static bottom may also promote SSHrheology.

3.5 Characteristic velocity and flow depth for the transientflow

The flow rate per unit widthQ for homogeneous granularmatter can be calculated by integrating velocity profiles asfollows

Q =∞∫

0

vdy ≈ U · H (14)

where U and H are the characteristic velocity and thecharacteristic flow depth for the transient flow, respectively.Similarly, the unit-width momentum M can be obtained byintegrating over the whole profile

123

Falling process of a rectangular granular step 47

v/(gd)0.5

0 1 2 3 4 5

0

5

10

15

20

25

t*=0.1(θ=10o)

t*=0.3(θ=10o)

t*=0.5(θ=10o)

(c)

v/(gd)0.5

0 1 2 3 4 5

0

5

10

15

20

25

t*=0.1(θ=5o)

t*=0.3(θ=5o)

t*=0.5(θ=5o)

(b)

v/(gd)0.5

0 1 2 3 4 5

h/d

0

5

10

15

20

25

t*=0.1(θ=0o)

t*=0.3(θ=0o)

t*=0.5(θ=0o)

(a)

v/(gd)0.5

0 1 2 3

h/d

0

2

4

6

8

10

12

14

16

t*=0.1(θ=0o)

t*=0.3(θ=0o)

t*=0.5(θ=0o)

(d)

v/(gd)0.5

0 1 2 3

0

2

4

6

8

10

12

14

16

t*=0.1(θ=5o)

t*=0.3(θ=5o)

t*=0.5(θ=5o)

(e)

v/(gd)0.5

0 1 2 3

0

2

4

6

8

10

12

14

16

t*=0.1(θ=10o)

t*=0.3(θ=10o)

t*=0.5(θ=10o)

(f)

v/(gd)0.5

0 1 2

h/d

0

2

4

6

8

t*=0.1(θ=0o)

t*=0.3(θ=0o)

t*=0.5(θ=0o)

(g)

v/(gd)0.5

0 1 2

0

2

4

6

8

t*=0.1(θ=5o)

t*=0.3(θ=5o)

t*=0.5(θ=5o)

(h)

v/(gd)0.5

0 1 2

0

2

4

6

8

t*=0.1(θ=10o)

t*=0.3(θ=10o)

t*=0.5(θ=10o)

(i)

v/(gd)0.5

0 1 2 3

h/d

0

5

10

15

20

25

t*=0.1(θ=0o)

t*=0.3(θ=0o)

t*=0.5(θ=0o)

(j)

v/(gd)0.5

0 1 2 3

0

5

10

15

20

25

t*=0.1(θ=5o)

t*=0.3(θ=5o)

t*=0.5(θ=5o)

(k)

v/(gd)0.5

0 1 2 3

0

5

10

15

20

25

t*=0.1(θ=10o)

t*=0.3(θ=10o)

t*=0.5(θ=10o)

(l)

Fig. 11 Scaled velocity profiles from experiments for which W =5 cm at the middle point of the flowing slope for a–c d = 2.0 mm (glassbeads), d–f d = 3.5 mm (mill stones), g–i d = 5.8 mm (polystyrenebeads), and j–l d = 2.0 mm (sand). Solid curves are the curves whichsatisfy v(z, t)/(gd)0.5 = (vs(t)/(gd)0.5) exp(−z/(λ/d)), as proposedby du Pont et al. [22]

particel size, d [mm]

0 1 2 3 4 5 6 7

λ/d

0

1

2

3

4

5

6

7

d=2.0 mm, glass beadd=3.5 mm, mill stoned=5.8 mm, polystyrene beadd=2.0 mm, sandBonamy et al.(2002)du pont et al.(2005)

Orpe et al.(2007)drum

Fig. 12 Variation in characteristic length λ against particle size d

M =∞∫

0

v2dy ≈ U2 · H (15)

Eqs. 14 and 15 yield the following two relationships for Uand H

U = M

Q(16a)

H = Q2

M(16b)

The exponential velocity profile given by Eq. 11 yieldsU = vs(t)/2 and H = 2λ (where λ is the characteris-tic length scale defined in Eq. 11). While the power-lawvelocity profile suggests, U = vs(t)(η + 1)/(2η + 1), H =(2η + 1)h/(η + 1)2 according to Eq. 12. Experimentally, astime passes, both U and H decrease linearly (t∗ ∼ 0.5). Anincrease in the characteristic velocity is associated with theincrease in the characteristic flow depth for various mate-rials as shown in Fig. 16. All data are consistent with alinear regression curve, except for those of the glass beads(d = 2 mm), which are associated with a greater e (i.e., coef-ficient of restitution, see Table 1). A similar relation betweenthe characteristic velocity and the characteristic flow depthhas been found elsewhere [25]. Furthermore, the character-istic length, λ, and the index η are verified based on Eqs. 11,12, and 16. Both the model and experimental observationsreveal that λ/h = (2η + 1)/2(η + 1)2.

As presented in Fig. 4, the saltating of glass beads mayincrease the local velocities along the inclined slope. How-ever, increasing the bed slope slightly affects U and H .The flow rate at the bottom of the slope is determined byintegrating the velocity profile there. Figure 17a depicts the

123

48 H.-T. Chou, C.-F. Lee

v(y)/vs

0.0 0.2 0.4 0.6 0.8 1.0

y/h

0.0

0.2

0.4

0.6

0.8

1.0

η=2.4, t*=0.1η=2.1, t*=0.3η=1.9, t*=0.5

(a)

v(y)/vs

0.0 0.2 0.4 0.6 0.8 1.0

y/h

0.0

0.2

0.4

0.6

0.8

1.0

η=2.9, t*=0.1η=2.7, t*=0.3η=2.8, t*=0.5

(b)

v(y)/vs

0.0 0.2 0.4 0.6 0.8 1.0

y/h

0.0

0.2

0.4

0.6

0.8

1.0

η=2.2, t*=0.1η=2.1, t*=0.3η=2.0, t*=0.5

(c)

v(y)/vs

0.0 0.2 0.4 0.6 0.8 1.0

y/h

0.0

0.2

0.4

0.6

0.8

1.0

η=3.9, t*=0.1η=3.8, t*=0.3η=3.7, t*=0.5

(d)

Fig. 13 Normalized velocity profiles at various angles of inclination,corresponding to Fig. 10, for a d = 2 mm (glass beads), b d = 3.5 mm(mill stones), c d = 5.8 mm (polystyrene beads), and d d = 2 mm(sand). Best-fit curves are given by Eq. 13 with various coefficients η.Colors and symbols: white (t∗ = 0.1), gray (t∗ = 0.3), black (t∗ =0.5); circle (θ = 0o), downward-pointing triangle (θ = 5o), and square(θ = 10o). Curves Bagnold profile (gray solid lines), linear profile (graydashed lines)

histogram of the scaled flow rate Qout/(gd3)0.5. The effectsof the material property (size, surface friction) and bed slopeis noted mainly when t∗ < 0.6. All experimental runs showthat Qout is close to zero because the flowing layer is shal-low after t∗ > 0.6. Experimental observations reveal thatthe flowing phenomenon has three stages—(1) solid slid-ing (t∗ < 0.06), when both Q and h increase rapidly andapproach their peaks; (2) a transition of the granular material(t∗ = 0.06 ∼ 0.2) from the solid to the fluid phase, and (3)surface flow (t∗ > 0.2), in which both U and H decreasegradually with time. All data are reasonably fitted by thefollowing regression

Qout(t∗)√gd3

= σ · t∗eβ·t∗ (16)

ξ0.0 0.2 0.4 0.6 0.8 1.0 1.2

η

1.0

1.5

2.0

2.5

3.0

3.5

4.0

d=2.0 mmd=3.5 mmd=5.8 mmd=2.0 mm, sand

SSH rheologyBagnold rheology

increasing t

intermediate transition

(ξ<<1) (ξ∼1, <1)

Fig. 14 Index of SSH velocity profile, η, versus dimensionless num-ber, ξ , for granular media (W = 5 cm) based on experimental data. Itincludes Bagnold, SSH rheology, and intermediate transition regions,based on Bi’s (2005) definition

where σ and β are material parameters. There are two dis-tinct types of curves as shown in Fig. 17a. The material withhigher e (glass beads, dashed curve; σ = 690, β = −12.5)

has a higher flow rate than the material with a lower e (solidcurve; σ = 643, β = −16). The borderline between thefirst and second stages can generally be identified by usingQmax|t=1/β .

Granular materials with smooth surfaces and larger coef-ficient of restitution, such as glass beads, are expected tohave higher mobility. As shown in Fig. 17b, the flow rate,Q/(gd3)0.5 at the midpoint of the inclined free surface isevaluated by using Eq. 11. A comparison with Qout/(gd3)0.5

(Fig. 17a) demonstrates that the flow rate Q/(gd3)0.5 isslightly lower than the outflow rate at the toe when t∗ < 0.2but slightly higher for 0.2 < t∗ < 0.6. The discrepancybetween the flow rates is negligible when t∗ > 0.6, exceptfor the glass beads (d = 2 mm). The flow patterns associatedwith the collapsing process are thus neither steady nor uni-form. The variations in the flow depth are shown in Fig. 18a–c. Sudden sliding causes the flow depth at the downstreamoutlet to exceed that at the midpoint of the inclined surface inthe initial sliding phase (hout > h, Fig. 18a). The discharge atthe outlet is greater than the discharge at the midpoint in theearly sliding stage (Q/(gd3)0.5 < Qout/(gd3)0.5, t∗ ∼ τc).While the collapsing process undergoes in the surface flowregime, the flowing depth at the toe then decreases sharply(hout < h, Fig. 18b). Accordingly, the discharge at the outletis less than that at the midpoint Q/(gd3)0.5 > Qout/(gd3)0.5

in the early surface flow phase. A thin flow layer, with thedepth of a single particle, is observed in the final stage of

123

Falling process of a rectangular granular step 49

Fig. 15 Dimensionless flowrate versus scaled flowing depthfrom various experimental runs.Inset presents these data in threeflow stages (W = 5 cm)

h/d

0 5 10 15 20 25

Q/(

gd3 )0.

5

0

10

20

30 d=2.0 mm(θ=0o)d=2.0mm(θ=5o)d=2.0mm(θ=10o)d=3.5 mm(θ=0o)d=3.5 mm(θ=5o)d=3.5 mm(θ=10o)d=5.8 mm(θ=0o)d=5.8mm(θ=5o)d=5.8mm(θ=10o)d=2.0 mm-sand(θ=0o)d=2.0 mm-sand(θ=5o)d=2.0 mm-sand(θ=10o)

Q/(gd3)0.5=1.17(h/d)-4.84

hjam

h/d0 5 10 15 20

Q/(

gd3 )

0.5

0

5

10

15

20

25

solid sliding, t*=0.05transition, t*=0.20surface flow, t*=0.50

Fig. 16 Dimensionless velocityversus scaled thickness forW = 5 cm at various chuteslopes (solid line plots best-fitlinear regression from allexperimental data). Inset plotsthe relationship between(2η + 1)/2(η + 1)2 and λ/h

U/(gd)0.5

0 1 2 3

H/d

2

4

6

8

10

12

14

d=2.0 mm(θ=0o)d=2.0 mm(θ=5o)d=2.0 mm(θ=10o)d=3.5 mm(θ=0o)d=3.5 mm(θ=5o)d=3.5 mm(θ=10o)d=5.8 mm(θ=0o)d=5.8mm(θ=5o)d=5.8mm(θ=10o)d=2.0 mm-sand(θ=0o)d=2.0 mm-sand(θ=5o)d=2.0 mm-sand(θ=10o)

H/d=2.07+2.59U/(gd)0.5

λ/h0.15 0.20 0.25 0.30 0.35

(2η+

1)/2

(η+1

)2

0.15

0.20

0.25

0.30

0.35d=2.0 mmd=3.5 mmd=5.8 mmd=2.0 mm,sand

avalanching and the flow rates at both midpoint and outletare then close to each other (hout ∼ h, in Fig. 18c).

4 Conclusions

The study investigates the dynamic characteristics and col-lapsing process of a granular step on horizontal and inclinedslopes. The proposed conceptual model of the retreatingupper surface agrees reasonably with all of the experimental

results with a universal sliding time scale, K = ta/(ta+t f ) ≈0.07. With respect to the final deposition angle and the coef-ficient κ that specifies the side-wall effect is not sensitive tothe particle size d. A regressive equation that describes thesidewall effect is proposed for 3 < W/d < 50. Transientvelocity profiles measured from the side wall are character-ized by temporal self-similarity at various times. Due to wallfriction, the velocity profile in the flowing layer can be fit-ted by either an exponential distribution or the power-lawdistribution instead of being the Bagnold velocity profile.

123

50 H.-T. Chou, C.-F. Lee

Fig. 17 Comparisons betweena time evolutions of scaledoutflow rate and b average-depthflow rate in chute experiments(W = 5 cm). Solid and dashedcurves are fitted by the equation,Qout(t)/(gd3)0.5 =σ t∗e(βt∗); σ = 690, β = −12.5(solid curve);σ = 643, β = −16 (dashedcurve)

t*0.0 0.2 0.4 0.6 0.8 1.0

Qou

t / (gd

3 )0.5

0

5

10

15

20

25

30

d=2.0 mm(θ=0o)d=2.0 mm(θ=5o)d=2.0 mm(θ=10o)d=3.5 mm(θ=0o)d=3.5 mm(θ=5o)d=3.5 mm(θ=10o)d=5.8 mm(θ=0o)d=5.8 mm(θ=5o)d=5.8 mm(θ=10o)d=2.0 mm-sand(θ=0o)d=2.0 mm-sand(θ=5o)d=2.0 mm-sand(θ=10o)

surface flow

transition zone

solid sliding

t*0.0 0.2 0.4 0.6 0.8 1.0

UH

/ (gd

3 )0.5

0

5

10

15

20

25

30d=2.0 mm(θ=0o)d=2.0 mm(θ=5o)d=2.0 mm(θ=10o)d=3.5 mm(θ=0o)d=3.5 mm(θ=5o)d=3.5 mm(θ=10o)d=5.8 mm(θ=0o)d=5.8 mm(θ=5o)d=5.8 mm(θ=10o)d=2.0 mm-sand(θ=0o)d=2.0 mm-sand(θ=5o)d=2.0 mm-sand(θ=10o)

surface flow

transition zone

solid sliding

(a) (b)

Fig. 18 Evolution of granularavalanching during a solidsliding—transition (t∗ < 0.2),b surface flow (0.2 < t∗ < 0.6),and c final shallow flow(0.6 < t∗ < 1). Dark grayparticles are in upper flowinglayer; light particles representstatic zone

hout hout

(b)(a)

hout

(c)

The characteristic velocity U/d and the characteristic flowdepth H/d for the transient flow are linearly correlated. Theevolution of flow rates for the falling process of a rectan-gular granular step has three stages with quantification ofthe elapsed time, t∗—(1) initial solid sliding (t ∼ τc, ort∗ < 0.06), (2) transition (0.06 < t∗ < 0.2), and (3) shal-low surface flow (0.2 < t∗ < 1).

Acknowledgments The authors would like to thank the National Sci-ence Council of the Republic of China, Taiwan, for financially sup-porting this research under Contract No. NSC-98-2221-E-008-074. TedKnoy is appreciated for his editorial assistance.

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