[Phys. Rev. E 78 , 011306 (2008)]

[Phys. Rev. E 78 , 011306 (2008)]

PHYSICAL REVIEW E 78, 011306

共2008兲

Avalanche dynamics on a rough inclined plane

Tamás Börzsönyi,

1,2, *

Thomas C. Halsey,

3 and Robert E. Ecke

1

1

Condensed Matter & Thermal Physics and Center for Nonlinear Studies, Los Alamos National Lab, New Mexico 87545, USA

2

HIHIResearch Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary

3

ExxonMobil Upstream Research Co., 3120 Buffalo Speedway, Houston, Texas 77098, USA

共Received 28 March 2008; published 21 July 2008

The avalanche behavior of gravitationally forced granular layers on a rough inclined plane is investigated experimentally for different materials and for a variety of grain shapes ranging from spherical beads to highly anisotropic particles with dendritic shape. We measure the front velocity, area, and height of many avalanches and correlate the motion with the area and height. We also measure the avalanche profiles for several example cases. As the shape irregularity of the grains is increased, there is a dramatic qualitative change in avalanche properties. For rough nonspherical grains, avalanches are faster, bigger, and overturning in the sense that individual particles have down-slope speeds u

p

that exceed the front speed u

f

as compared with avalanches of spherical glass beads that are quantitatively slower and smaller and where particles always travel slower than the front speed. There is a linear increase of three quantities:

共i兲 dimensionless avalanche height, 共ii兲 ratio of particle to front speed, and

共iii兲 the growth rate of avalanche speed with increasing avalanche size with increasing tan

r

where flow rule, where both

r

r

is the bulk angle of repose, or with increasing and

P

P

, the slope of the depth averaged reflect the grain shape irregularity. These relations provide a tool for predicting important dynamical properties of avalanches as a function of grain shape irregularity. A relatively simple depth-averaged theoretical description captures some important elements of the avalanche motion, notably the existence of two regimes of this motion.

DOI: 10.1103/PhysRevE.78.011306

PACS number

共s兲: 45.70.Ht, 47.57.Gc

I. INTRODUCTION

Granular materials form phases with strong similarities with ordinary phases of matter—i.e., solids, liquids, or gases

1

兴. In many natural processes, the coexistence of two of

these phases is observed, requiring a complex multiphase description. An example is avalanche formation, which occurs under various circumstances in nature

共snow avalanches, sand avalanches on dunes, rock avalanches, land slides, etc.

兲 as well as in industrial processes involving granular materials. Although laboratory realizations of granular avalanches do not have the full complexity of avalanches encountered in nature, laboratory studies of this phenomena are important for understanding the fundamental behavior of avalanches where there is an interesting combination of stick-slip friction, yield criteria for the solid phase, and the fluidlike motion of the avalanche itself. Although the statistics of avalanche occurrences is a fascinating area of research

2

兴, we focus here on the dynamics of indi-

vidual avalanche events.

There are several classes of experiments for avalanches, each with advantages and disadvantages. One is to slowly rotate a closed cylinder that is about 50% full of granular material

3 – 6

兴. As the cylinder rotates, the angle of the

granular surface exceeds the critical angle

c

, and material starts to flow along the surface. At rapid rotation rates the flow is continuous, but for slower rotation rates, avalanches occur because the rate of depletion of the granular material in the avalanche brings the surface back to the angle of repose

r

faster than the rotation can maintain a surface angle

* [email protected]

1539-3755/2008/78

共1兲/011306共15兲 greater than

c

. Significant advantages of this approach are that there is no need to continuously supply grains to the system and the flow rate is easily controlled by the rotation rate. Further, using transparent boundaries enables direct observation of the velocity profile in the flowing layer because avalanches in this system typically extend to the side walls.

From such optical measurements one finds that the vertical velocity profile of an avalanche follows an exponential decay

5

兴 in contrast to steady flows where it has an upper linear

part. On the other hand, the role of friction at the side boundaries on the vertical velocity profile is unclear and may be quite important because the horizontal velocity profile is a plug flow with two exponential boundary layers at the walls.

A second realization of avalanches is to add granular material to a heap near its peak to induce granular motion

7 – 9

兴.

The heap can be three dimensional or can be confined between rigid barriers, typically transparent glass or plastic plates, to form a unidirectional flow. For high input mass flux near the top, the grains flow continuously down the surface formed by other grains. For lower incoming mass flow, avalanches form intermittently. The transition between continuous and avalanching flow as a function of input mass flux is hysteretic

3 , 10

兴. Studies using diffusing-wave spectroscopy

7

兴 and molecular dynamics simulations 关

11

兴 provided de-

tailed properties of this intermittency. The experimental results show that the microscopic grain dynamics are similar in the continuous and intermittent flow regimes

8

兴. In the con-

tinuous flow regime for a bulk heap, the grain velocity u

z兲 decreases linearly as a function of the depth z below the surface. Below some depth, however, a much slower “creep” motion is observed with an exponential velocity profile

6 , 12 , 13

兴. For the case of avalanches, theoretical results sug-

gest that the avalanche amplitude should depend strongly on the velocity profile

14 , 15

兴.

011306-1 ©2008 The American Physical Society

BÖRZSÖNYI, HALSEY, AND ECKE

20

15

10

h c

5

h s

0

1 1.2

1.4

tan θ / tan θ

r

1.6

FIG. 1.

共Color online兲 The minimum layer thickness h

s

共averaged for eight different materials

兲 at which flow stops 共solid line兲 and an illustrative curve for the critical thickness h

c

at which flow starts

共dashed line兲. The operating conditions for avalanches 共for the present work

兲 fall in the region represented by the hatched zone.

A third laboratory system for avalanche studies is a layer of grains on a rough plane inclined at an angle

␪ with respect to horizontal

16 – 26

兴. Because we use this system in the

experiments reported below, we review some aspects of these inclined layer flows in more detail. Grains on a rough inclined plane form a thin stable static layer even for

c

and the onset of the flow is expected only above a critical thickness h

c

, while the flow subsides at h

s

h

c

27 , 28

兴. The

values of h

c

and h

s

are rapidly decreasing by increasing

␪ as is schematically illustrated in Fig.

1 . Here h

s

is an average of the curves measured for eight different materials

共for the data, see Sec. III

兲 and h

c

is an approximate curve for illustration.

According to Bagnold

29

兴, the shear stress in granular

flows is proportional to the square of the strain rate. This hypothesis has been checked in various configurations and has been proven to work relatively well for dense inclined plane flows

30

兴. The Bagnold stress-stain relationship leads

to a velocity profile of the form u where the surface velocity u

0

z兲=u

0

关1−共hz兲/h

3

/2 depends on h as u

0

h

3

/2

. The above convex velocity profile is recovered in molecular dynamics

共MD兲 simulations for relatively thick flows 关

31 , 32

兴.

As the internal velocity profile is difficult to determine experimentally, the most straightforward experimental test of the Bagnold hypothesis is to measure the height dependence of the surface velocity u

0

¯.

This relationship, called the flow rule

共FR兲, was obtained for homogeneous flows with glass beads or sand

27 , 33 , 34

兴 and

is found to be consistent with the Bagnold flow profiles. Two slightly different forms of the FR are the Pouliquen flow rule

u

/

gh =

P h

/h

s

兲 −

共1兲 and a modified form due to Jenkins

35

兴 which we denote the

Pouliquen-Jenkins flow rule:

u

/

gh =

PJ

h/h

s

共 ␪ 兲兴共tan

2

␪ /tan

2

1

兲, 共2兲 where

1 is the vertical asymptote of the h

s

兲 curve.

In the inclined layer system, one can either prepare the layer in a metastable state and mechanically induce a single avalanche

17 – 20

兴 or an intermittent series of avalanches can

011306-2

PHYSICAL REVIEW E 78, 011306

共2008兲 be produced by slowly adding grains near the top of the inclined layer as is done here. In the former case the whole layer becomes metastable

17 – 19

兴, and the shape and propa-

gation of the avalanches depends critically on

␦␪

共the level of metastability

兲, giving rise to downward and also upward expanding avalanches when triggered with a small perturbation. For both kinds of avalanches on an inclined plane the angle at which flow starts or stops depends on the thickness of the layer contrary to the heap experiments, on the properties of the bottom boundary such as surface roughness

共which controls the zero-velocity boundary condition兲, and on the particular granular particle properties including interparticle friction and particle shape. The distinct advantages of this system for the study of avalanches include the robust flow rule relationship in the continuous flow regime

27 , 33 , 36

兴 that relates depth-averaged velocity and height, a

simpler vertical velocity profile, and the ability to adjust the stability of the layer by changing the plane inclination angle

. Note that both flow rules connect the rheology deep in the flowing state with properties of the boundary between the flowing and the stationary states.

The majority of laboratory experiments use spherical beads—an idealized granular material. However, properties of waves, either on a rough incline

34

兴 or in hopper flows

37

兴, depend on the shape anisotropy of the grains. Also, in

recent numerical simulations the velocity profile in Couette flow depends strongly on the angularity of the particles

38

兴.

Thus, it is natural to expect that avalanches will be similarly affected by the shape of the individual granular particles.

Indeed, glass beads and sand show qualitatively different avalanching behavior

39

兴 despite qualitatively similar flow

rules in the steady flow phase. Avalanches formed by sand particles are larger with more dynamic grain motion than avalanches formed by glass beads. By plugging the known flow properties

34

兴 into the depth-averaged model equa-

tions, these basic differences can be explained

39

兴.

In this paper, we present the results of an extensive study, using a set of different materials and describe the details of our experimental methods used for the results presented in

39

兴. Our aim is to capture how the dynamical properties of

the avalanches depend on the grain shape irregularity, including the effects due to deviations from the idealized spherical shape of the overall grain form and effects of the microscopic surface roughness of the particles. We find relations by which we can quantitatively predict major properties of avalanches as a function of the angle of repose slope of the flow rule

P

r

or the

, both of which provide a measure of grain shape irregularity.

We study the case where the layer is initially stable to small perturbations. As new grains are added to the layer at the top section

共5%兲 of the plane, the granular layer becomes locally unstable in a cyclic manner leading to the intermittent formation of avalanches. These avalanches propagate down the plane on top of the stable static layer in a stationary manner; i.e., the shape and velocity of the avalanche does not change appreciably. The dynamical properties of these avalanches are studied far down the plane from where they are formed. We use eight different materials including rough irregular-shaped particles of different sizes and spherical beads. The materials are characterized in Sec. II where the

AVALANCHE DYNAMICS ON A ROUGH INCLINED PLANE

Hopper

Metal Plate

Laser 1

x z y

a.

PHYSICAL REVIEW E 78, 011306

共2008兲 b.

1 mm

Glass Plate

Camera 1

Laser Sheet

Camera 2

θ

Laser 2

FIG. 2.

共Color online兲 Schematic illustration of the experimental setup.

c.

d.

experimental conditions are also described. In Sec. III we discuss how the properties of the avalanches depend dramatically on the shape

共spherical, or irregular兲 of the grains. We describe a simple theory of avalanche flow for glass beads in

Sec. IV. Conclusions are drawn in Sec. V.

II. EXPERIMENT

In this section we describe the experimental apparatus and the characteristics of the granular materials. The experimental techniques and apparatus have been described in detail elsewhere

39 , 40

兴, so only essential features are presented

here.

A. Experimental setup

A sketch of the experimental setup is shown in Fig.

2 . The

granular material flows out of the hopper at a constant flow rate Q. The grains first hit a small metal plate that disperses the material so that the mass flux per unit width F = Q

/W is relatively homogeneous in the y direction transverse

共of total width W

兲 to the inclination direction x. The grains are deposited on the top 5% of the plane. The metal plate helps reduce possible electric charging of the particles during the hopper flow. The granular layer reaches a critical state locally, and an avalanche is formed and travels down the plane. The layer into which the avalanche moved has approximate thickness

h s

. The stable static layer is prepared by releasing grains at the top of the plane and letting the system relax or by letting the hopper run for a longer period of time. The glass plate has dimensions of 220 cm

⫻40 cm with a rough surface prepared either by gluing one layer of the same particles onto it or by using sandpaper with different roughnesses. The two main control parameters of the system are the plane inclination angle

␪ and the nature of the granular material used. The incoming flux and the roughness of the plane could be also varied. As we show in the following, within the limits presented below, variations in these latter parameters do not influence the results presented in this paper. The dynamics of the avalanches are recorded with a fast video camera

共up to

2000 frames per second

兲 about 150 cm below the incoming flow

共camera 1兲.

011306-3

FIG. 3. Microscopic images of the materials:

共a兲 sand with diameter d = 400

⫾100 ␮m, 共b兲 spherical glass beads with d

= 500

⫾100

␮m, 共c兲 fine sand with d=200⫾100 ␮m, and 共d兲 salt with d = 400

⫾100 ␮m.

The vertical profiles of the avalanches along their symmetry axes are recorded with camera 2 using a vertical laser sheet to detect the height differences

共laser 1兲. With another laser sheet

共laser 2兲 slightly inclined with respect to the glass plate, two-dimensional

共2D兲 height profiles of avalanches are reconstructed.

The incoming mass flow rate

共per unit width along yF

0 is fairly constant for all measurements of one material and is in the range 0.02

F

0

艋0.2 g/共cm s兲 depending on the material. The value of F

0 is chosen by determining the flux F

w

needed to produce a wave state where the entire width of the layer is in motion. The properties of such waves were explored experimentally for glass beads to approximately F

w

18

兴. The flux F

/2 to obtain a reasonable number of distinct avalanches in the measurement area. For F

F

w

0 is set

, adjusting F

0 only affects the total number of avalanches and their size distribution rather than the individual avalanche characteristics that are the focus of this paper.

B. Characterization of the materials

The shape of the granular particles has a large impact on the qualitative and quantitative behavior of avalanches in our system whereas other properties such as the mean particle diameter d have less effect. For example, despite some relatively low polydispersity for the materials investigated

共艋⫾30% except for fine sand which had ⫾50%兲, we do not observe any effects attributable to size-induced segregation.

To show the nature of the shape anisotropy qualitatively, we first show in Fig.

3

images of the glass beads

d

= 500

⫾100

= 400

⫾100

␮ m m

兲, fine 共d=200⫾100 ␮ m

兲, and coarse 共d

兲 sand and salt particles 共d=400⫾100 ␮ m

兲.

The glass beads are quite spherical, although no attempt has been made to eliminate slightly aspherical particles and no quantitative analysis of asphericity has been done. The sand has many irregular shapes ranging from approximately spherical to very angular. Finally, the salt grains take on very angular shapes reflecting the cubic symmetry of salt crystals.

For the purposes of our studies, we assume that the grains do

BÖRZSÖNYI, HALSEY, AND ECKE a.

c.

FIG. 4. Microscopic images of the copper particles of size d

= 160

⫾50

␮m and with packing fractions 共a兲

= 0.33,

共c兲

= 0.5, and

共d兲

= 0.63.

= 0.25,

共b兲

␩ not change appreciably with time because the number of realizations of flows is limited and the particle velocities are small. Thus, grains are not subjected to the repetitive collisions that might smooth their shape thereby changing the shape anisotropy or interparticle friction.

The copper materials are used to explicitly introduce a controlled shape distribution and explore the consequences of very different shapes on avalanche behavior. In Fig.

4 , we

show images of commercial

共ECKA Granules GmbH & Co.,

KG.

兲 copper particles 共mean diameter d⬇160⫾50 ␮ m

兲 with very different average shapes. The particles range from highly irregular, dendriticlike shapes to almost spherical. The overall anisotropy is measured by the static packing fraction

␩ for each shape, ranging from 0.25 for irregular particles to

0.63 for spherical copper particles. For comparison, the packing fraction for sand is about 0.56 and for spherical glass beads is 0.63.

The next thing to consider is how the different shapes and properties of the materials affect the basic granular flow properties, in particular

r

and

c

of the bulk material. We do so by observing avalanches on bulk heaps with a high-speed video camera. As material is added at a very small rate to the top of the pile, avalanches form and propagate downwards intermittently. The basic configuration is shown in Fig.

5

where a bulk pile of sand is shown with a line indicating the

5 cm

b.

d.

0.3 mm

θ

c

FIG. 5. Image of a sandpile—a heap of coarse sand particles.

The scale is indicated in the figure and the solid line indicates the local slope used to determine

c

as shown or more generally

r

as well.

PHYSICAL REVIEW E 78, 011306

共2008兲

60

50

40

30

20

10

0

0

50

40

30

20

10

0

0

60 a.

b.

<

θ

<

θ r

> = 20.9

c

> = 24.7

o o

10 20

<

θ r

> = 29.6

o

<

θ c

> = 35.2

o

10 20

30 glass beads (d sand (d av

= 500

µm) av

= 400

µm)

40

30 angle (deg)

40

<

θ

<

θ r

> = 30.5

c

> = 36.5

o o

50 salt (d av

= 400

µm) sand (d av

= 200

µm)

<

θ

<

θ r c

> = 35.1

o

> = 30.6

o

50

60

60

FIG. 6.

共Color online兲 Distributions of the critical angle

c

共solid columns兲 and the angle of repose ␪

r

共open columns兲 measured for 100 avalanches on a three-dimensional sandpile for

共a兲 sand

关red 共gray兲 columns兴 and glass beads 共black columns兲 and 共b兲 fine sand

关red 共gray兲 columns兴 and salt 共black columns兲.

determination of local

c

共just before an avalanche兲 or

共just after an avalanche, not shown兲.

The distribution of the critical angle angle of repose

r

c

␪ is plotted for 100 avalanches for each of

r

and that of the the different materials investigated. The distributions for the glass, sand, and salt are shown in Fig.

6 , and the distributions

for the copper particles are presented in Fig.

7 . The average

values of

c

and

r

are indicated. As expected,

c

and

r

are significantly higher for the piles formed by particles of irregular shape compared to piles of spherical particles. The widths of the distributions are also larger for irregular particles because shape irregularity gives rise to a larger variety of configurations and a wider range of angles at which avalanches start or stop.

For thin layers on an inclined plane, grains start or stop flowing at angles determined by the layer thickness h as well as by their individual properties such as shape or surface roughness. In our experiments, we determine for each material the height h

s

共 ␪ 兲 at which the material stops flowing, the thin layer equivalent of the bulk angle of repose. In a technique described in detail elsewhere

33

兴, we allow material

to flow by slowly adding grains at the top of the plane resulting in intermittent avalanches. Upon stopping the input, the layer comes to rest and the volume of grains on the whole plane is determined. Knowing the surface area of the plate allows a determination of the mean height h

s

with high precision, typically 2%. Alternatively, a large quantity of grains is placed on the inclined plane and a continuous flow persists over the entire plane for 10– 20 s. After the flow subsides, the thickness of the resulting static layer is the same to within experimental error as that determined by the first method.

Many of the details of the characteristics of flow on our inclined plane, including the detailed comparison with the

Pouliquen flow rule

27

兴 and the effects of surface rough-

011306-4

AVALANCHE DYNAMICS ON A ROUGH INCLINED PLANE PHYSICAL REVIEW E 78, 011306

共2008兲

50

40

30

20

10

0

50

40

30

20

10

0

50

40

30

20

10

0

50

40

30

20

10

0

0 a.

b.

c.

d.

copper

η=0.25

<

θ

<

θ c r

> = 33.8

> = 39.0

o o copper

η=0.33

<

θ

<

θ c r

> = 33.5

> = 38.3

o o copper

η=0.5

<

θ

<

θ c r

> = 27.9

> = 32.8

o o copper

η=0.63

<

θ

<

θ c r

> = 23.9

> = 27.1

o o

10 20 30 angle (deg)

40 50 60

FIG. 7.

共Color online兲 Distributions of the critical angle

c

共solid columns兲 and the angle of repose ␪

r

共open columns兲 measured for 100 avalanches on a three-dimensional sandpile for copper with packing fractions

共a兲

共d兲

= 0.25,

= 0.63. Particle sizes d = 160

⫾50

共b兲

␮m.

= 0.33,

共c兲

= 0.5, and ness, have been presented elsewhere

33

兴. We summarize our

results for the particular materials used here by collapsing the data using a normalized ratio of tan

␪ /tan ␪

r

where

r

is determined from the bulk measurement and normalizing the height by the particle diameter d; see Fig.

8 . The solid lines

are best fits to the formula h

s

= ad

/共tan ing values for the fitting parameters

− tan

␣ and

1

1

兲. The resultare indicated in

Table

I . For all the materials, the stopping height h

s

creases with increasing de-

␪ as shown in Fig.

8 , dropping rap-

idly for angles close to the bulk angle of repose and more gradually for larger

. The curves in Fig.

8

are for sandpaper with a roughness of R = 0.19 mm. The corresponding data curves for sand on a surface prepared by glued sand particles are identical within experimental error. The collapse of the data suggests that differences in plane roughness do not translate into significant changes in h

s

. We have also tested the influence of the incoming flux and kinetic energy of the incoming particles on h

s

. For all the above configurations tested using different initial flow rates, the data points fall on the same curve within

⫾5% for tan

/tan

r

⬎1.1. When approaching

r

the measurements become less accurate with the rapid increase of h the data points for tan

s

, leading to a

⫾12% uncertainty of

/tan ␪

r

⬍1.1.

There are several additional remarks about the system that are important. First, the majority of our measurements do not require detailed information about h

c

兲, the thickness at which grains start to flow for a particular

. Thus, we have not made measurements of h

c

. When data about this quantity are needed as in the modeling section presented below, h

c

011306-5

20

15

10

5 salt fine sand glass beads sand copper,

η = 0.25

copper,

η = 0.33

copper,

η = 0.5

copper,

η = 0.63

0

1 1.2

1.4

tan

θ / tan θ

r

1.6

FIG. 8.

共Color online兲 Static layer thickness normalized by d as a function of tan

␪ normalized by tan ␪ to the formula h = ad

/共tan

␪−tan ␪

1

r

. The solid lines are best fits

兲 with the resulting fitting parameters

␣ and ␪

1

s

indicated in Table

I . A vertical solid line indicates

r

. The horizontal bracket near the top of the figure indicates the range of

␪ over which avalanches are measured.

is estimated from measurements in similar systems

27

兴. Sec-

ond, avalanches are observed in the range of 1.04

⬍tan

/tan

r

⬍1.34 as indicated by the bar on the top of

Fig.

8 . For higher

, even if a homogeneous static layer is prepared beforehand, the kinetic energy of the incoming grains is enough to slowly erode the preexisting layer and avalanches can only be observed for a short time.

C. Experimental procedures and analysis

For different materials and for different angles

, we produce avalanches by adding grains continuously at the top of the plane. The main quantities of interest are the velocity of the front or of the individual grains, the lateral extent of the avalanche measured by the avalanche area A, and the height of the layer h as a function of position. These quantities are measured by analyzing images taken with the high-speed video cameras. Two images taken of transmitted light

共cam-

TABLE I. Fitting parameters a and

1 the data in Fig.

8

using the formula h

s

, resulting as best fits to

= ad

/共tan ␪−tan ␪

1

兲.

Sample

Salt

Fine sand

Glass beads

Sand

Copper,

Copper,

Copper,

Copper,

= 0.25

= 0.33

= 0.5

= 0.63

a

0.63

0.35

0.26

0.4

0.66

0.52

0.43

0.35

1

30.0°

31.0°

20.5°

31.1°

32.6°

32.3°

26.9°

23.8°

BÖRZSÖNYI, HALSEY, AND ECKE

a.

b.

10 cm

t

PHYSICAL REVIEW E 78, 011306

共2008兲

x

FIG. 9. Images

共a兲 at ␪=33.6° and 共b兲 at ␪=38.1° using transmitted light taken with camera 1 for sand with d = 400

␮m.

era 1

兲 for coarse sand at

= 33.6° and

= 38.1° are shown in

Fig.

9 . These avalanches are localized objects that can be

isolated by differencing of subsequent images because regions were the grains move become visible on a uniform background. The area of the avalanche is determined by the areal extent over which finite displacements in image differencing are measured. Similarly the location of the front of the image differencing region is used to determine the front velocity u

f

.

To track the motion of individual grains inside the avalanches, a sequence of images are extracted from movies of the granular flow dynamics

41

兴. Space-time plots are cre-

ated by taking the intensity along the symmetry axis of the avalanche from movies taken by camera 1. An example is shown in Fig.

10

for sand. The vertical lines starting on the top of the diagrams are the traces of particles on the top of the static layer in front of the avalanche. The tilted trajectory dividing this region from the dynamic region is the trace of the avalanche front, providing a second method for determining u

f

. The traces of moving individual grains inside the avalanche are tilted lines. The change in the tilt of these traces as we go downwards

共towards the end of the avalanche

兲 indicates the decreasing particle velocity u

p

behind the front. By analyzing the space-time plots the mean particle velocity u

p

has been measured at several locations and the velocity profile u

p

x兲 has been determined. For this example using sand particles, the particles in the avalanche core

共close behind the front兲 move faster than the front velocity of the avalanche.

In order to trace the height profile of avalanches, a vertical laser sheet

xz plane兲 is projected onto the plane. Movies are taken of avalanches that are cut by the laser sheet near their center

共symmetry axis兲. The camera is mounted for these experiments on the side at an angle of about 13° with respect to the xy plane

共camera 2 in Fig.

2

兲. By measuring

the speed of the avalanche and taking the intensity only along one vertical line of these movies, the profile of the avalanches can be traced assuming that the profile does not change as it passes through. In Figs.

11

共a兲

and

11

共b兲

, profiles

h s

FIG. 10. Space-time plot for a sand avalanche for

␪=36.8° and

= 0.12 cm. Image size: 5.96 cm in Fig.

20

共a兲 兴.

⫻1.43 s 关velocity profile shown are shown for a sand and for a glass avalanche, respectively.

The images are contracted by 25 times in the horizontal direction. This technique allows the instantaneous profile along one line to be determined.

Another way to visualize the avalanches is to trace a laser line

共laser 2兲 共see Fig.

2

兲 across the direction of avalanche

propagation—i.e., transverse to the plane inclination direction. Using the deflections of this laser line, the whole 2D surface of the avalanche is obtained. A sample image

共from camera 2

兲 is shown in Fig.

12

共a兲

for a sand avalanche. From the image sequence, the 2D height profile is reconstructed and is shown in Fig.

12

共b兲

for the same avalanche and similarly for an avalanche formed by glass beads in Fig.

12

共c兲

.

a.

b.

0.35 cm

10 cm

z x

FIG. 11. Avalanche profiles taken from the side with the help of a laser sheet for

共a兲 sand 共for

␪=33.6°兲 and 共b兲 glass beads 共for ␪

= 22.6°

兲. The horizontal size is contracted by a factor of 25.

011306-6

AVALANCHE DYNAMICS ON A ROUGH INCLINED PLANE PHYSICAL REVIEW E 78, 011306

共2008兲

15 a.

a.

x y

2 cm

10

5 b.

c.

z y x

sand glass beads

FIG. 12.

共Color online兲 共a兲 Image of the laser profile taken with camera 1. The height of the avalanche is obtained as h = h

s

+

x tan

␾, where ␾ is the angle between the plane and the laser sheet

共laser 2兲 and

x is the displacement of the laser line. Height profiles of

共b兲 sand avalanche for

␪=36.8°. Image size 6.9 cm

⫻34.6 cm 共vertical size rescaled by 25⫻兲, maximum height h

m

= 0.35 cm, static layer thickness h

s

= 0.12 cm;

共c兲 glass bead avalanche for

␪=24.3°. Image size 10.3 cm⫻38.4 cm 共vertical size rescaled by 25

⫻兲, maximum height h

m

= 0.28 cm, static layer thickness h

s

= 0.18 cm.

III. RESULTS

The main objective of our experimental investigation is to understand the similarities and differences between avalanches of granular materials with different properties, particularly grain shape irregularity. We first describe some of the qualitative characteristics of the avalanches and then provide quantitative measures of those properties.

The avalanches we investigated are stationary

共do not change with time

兲 above a certain size with an overall shape that depends on the granular material. Glass bead avalanches have an oval shape, Fig.

12

共c兲

, whereas medium sized or

large sand avalanches have two tails, Figs.

9

共b兲

and

12

共b兲

,

which can break off and form very small avalanches. Small sand avalanches typically do not have these tails. Very small avalanches are not stationary, but decelerate, lose grains, and eventually come to rest. The length of the observation area is about 8–12 times the length of small avalanches. Avalanches that are decelerating or that stop in the field of observation are not included in the data presented here. Interacting and merging avalanches are also eliminated.

To demonstrate the stationarity of avalanche size and speed, the time evolution of the front velocity u

f

and the characteristic size

共the square root of the lateral area A兲 is

011306-7

0

20

15

10 b.

5

0

0 1 2 3

t (s)

4 5 6

FIG. 13.

共Color online兲 Time evolution of the 共a兲 lateral size A and

共b兲 the front velocity u

f

of four avalanches for sand with d

= 400

␮m and ␪=35.2°.

shown for a set of avalanches in Figs.

13

共a兲

and

13

共b兲

for sand taken at

= 35.2°. The smallest, slowest moving avalanches have a discernibly downward slope indicating a shrinking, decelerating avalanche. If the decay is exponential, the linear slope would yield a decay time of about 40 s for both the size and speed. For two avalanches of this initial size, one drops below a threshold, about 7 cm

/s, and then quickly shrinks and decelerates until it vanishes. Another with about the same initial conditions, decreases in size and speed for awhile, but seems to recover and survive.

To understand the difference in avalanche survival for small sand avalanches, one needs to take account of the response of the static layer to the passage of an avalanche.

Medium or large sand avalanches typically take material with them along their center and deposit grains along their edges

共see Figs.

9

and

12

兲. The change in the layer height is

about

⫾10% of h

s

. Thus, the different fates of austensibly similar sand avalanches suggests that small differences in residual layer thickness influence their evolution in that a small avalanche that encounters a slightly thinner region behind a recent large avalanche does not survive whereas a small avalanche propagating along the ridge formed by a large avalanche picks up a bit of mass and manages to survive

共at least over the size of the interrogation window兲. No formation of residual ridges left by the wake of the avalanche is observed for small sand avalanches or for avalanches formed by glass beads. Although we have not studied this effect in detail for all the materials, one might surmise that the residual wake structure is a property of irregular grain avalanches whereas the spherical grain avalanches do not have this property.

We now present a quantitative analysis of the speed and size of avalanches whose properties are stationary over the length of the channel, about 200 cm. The avalanche velocity

u f

increases with avalanche size as shown in Figs.

14

共a兲

and

BÖRZSÖNYI, HALSEY, AND ECKE

25 a.

20

15

10

5

0

sand

5

A

1/2

10

(cm)

10 b.

8

glass beads

15

33.6

o

35.2

o

36.8

o

39.5

o

20

PHYSICAL REVIEW E 78, 011306

共2008兲 salt fine sand sand glass beads

0.2

6

5

4

3

2

1

0

0 a.

10 20 30

A

1/2

/ 10 h

s

c.

0 b.

η = 0.25

η = 0.33

η = 0.5

η = 0.63

copper

10 20 30

A

1/2

/ 10 h

s

6

4

22.2

o

23.3

o

24.3

o

25.2

o

26.1

o

2

0 10

A

1/2

20

(cm)

30 40

FIG. 14.

共Color online兲 Avalanche front velocity u

f

as a function of the square root of the lateral avalanche area A for

共a兲 sand and 共b兲 glass beads.

14

共b兲

for fine sand and glass beads, respectively. Similar sets of data were obtained for all eight materials. For anisotropic grains, avalanches move faster with increasing

14

共a兲

. On the other hand, avalanches formed by spherical

beads are independent of

; see Fig.

in the sense that the curves obtained at various plane inclinations collapse; see Fig.

14

共b兲

.

This observation is not in contradiction with data presented in

18

兴, where avalanche speed for glass beads increases with

decreasing

. In fact, the tendency is the same here for the case of glass beads, as the average avalanche size increases with decreasing

, Fig.

14

共b兲

, so that larger avalanches

共with larger velocity

兲 are observed for less steep inclines. A similar analysis for the anisotropic particles is less conclusive, owing to the wide range of sizes and velocities measured. Nevertheless, after averaging u sizes, the u dent of

f

兲 curve for anisotropic grains is fairly indepen-

␪ as well.

To compare the u

f

A

1

f

for all avalanches of different

/2

兲 curves for the different materials, it is useful to employ dimensionless parameters. For gravitydriven flows on an incline the relevant length scale is h the appropriate velocity scale is mensionalization of the u

f

A

1

/2

gh s

cos

42

s

and

兴. The nondi-

兲 curves by these quantities collapses the data taken at various plane inclinations for each material. The collapsed curves are shown in Fig.

15

共a兲

for sand, salt, and glass beads and in Fig.

15

共b兲

for the copper particles. The dimensionless avalanche velocity increases linearly with increasing dimensionless avalanche size for all

0.1

0

0 0.1

0.2

0.3

0.4

0.5

0.6

tan

θ

r

or tan

θ

1

FIG. 15.

共Color online兲 Dimensionless avalanche velocity as a function of avalanche size for

共a兲 or sand 共⫻兲, glass beads 共䊊兲, salt

共ⴱ兲, fine sand 共䉭兲, and 共b兲 copper particles with

= 0.33

共ⴱ兲,

= 0.5

共⫻兲, and fits to the data.

共c兲 Slope m

u

= 0.63

= 0.25

共䉭兲, ␩

共䊊兲. The dashed lines are best of the linear fits from materials as a function of tan

r

共ⴱ兲 or tan

1

共䊊兲.

共a兲 and 共b兲 for all materials. The data fall into two classes: irregular-shaped particles form avalanches that reach sizes about 2–4 times larger than those formed by spherical beads. Further, the speed of the avalanches from the irregular grains is about 4 times larger than that of the avalanches formed by spherical beads. For a better characterization of these differences, we compare the slope m

Fig.

15

共c兲

, m

u

angle of repose is plotted as a function of the tangent of the

r u

关or of the best fits to the data

1

关shown with dashed lines in Figs.

15

共a兲

and

15

共b兲 兴 for all materials. In

␪ the vertical asymptote of the h

s

兲 curve

兴, both of which provide a measure of grain shape irregularity. We find that m

u

systematically increases with grain shape irregularity. In other words, the avalanche velocity increases with increasing avalanche size systematically for more irregular grains. Note that m tan

r

u

goes to zero at about

= 0.36, implying that for materials with a small angle of repose

共below about ␪

r

= 19.8°

兲 the dimensionless avalanche velocity is small

共about u

f

/

gh s

cos

␪ ⬇0.35兲 and independent of avalanche size.

Another measure of the avalanche size is its height, in particular the maximum height h

m

along the avalanche profile. We determine the height profile using the laser line method described in Sec. II. In Figs.

11

共a兲

and

11

共b兲

, we see

qualitatively that the height of glass avalanches is considerably smaller than that of sand avalanches. A second difference is that after the rapid increase in height at the front, a fast

共exponential like兲 decrease is observed for sand ava-

011306-8

AVALANCHE DYNAMICS ON A ROUGH INCLINED PLANE

30

25

20

15

10

5

0 glass, slope = 1.4

sand, slope = 2.5

slope for glass from [18] a.

30

25

20

15

10

5

0

0

η = 0.63, slope = 1.7

η = 0.5, slope = 2.2

η = 0.33, slope = 3.0

η = 0.25, slope = 3.2

2 4

h s

6

/ d

copper

8 10 b.

c.

12

4

3.5

3

2.5

2

1.5

1

0 0.1

0.2

0.3

0.4

0.5

0.6

tan

θ

r

or tan

θ

1

FIG. 16.

共Color online兲 Avalanche peak height h

m

of the static layer thickness h

共b兲 copper with

= 0.63

= 0.25

共䊊兲. 共c兲 The ratio h

m s

for

共a兲 sand 共⫻兲 and glass beads 共䊊兲,

共䉭兲,

/h

s

共䊊兲. The solid lines are linear fits.

= 0.33

共ⴱ兲,

= 0.5

as a function of tan

r

as a function

共⫻兲, and

共ⴱ兲 or tan ␪

1 lanches, whereas the height variation is straighter for avalanches formed by glass beads.

Numerous avalanche profiles have been recorded for sand, glass beads, and all copper samples at the same sequence of plane inclinations as before to provide a quantitative analysis of avalanche heights. Plotting the avalanche peak height h

m

/d as a function of the thickness of the underlying layer h

s

/d, we find a systematic linear increase as shown for glass beads and sand in Fig.

16

共a兲

, implying that

at lower plane inclinations where the static layer is thicker the avalanche height is proportionally larger. The slope m

h

of the curves gives a measure of the dimensionless avalanche height h

m

/h

s

for a given material which is basically independent of the avalanche size and plane inclination and is m

h

= 1.45 and m

h

= 2.5 for glass beads and sand, respectively.

Also shown is the slope m = 1.55 for glass bead waves on a velvet cloth

18

兴. The close correspondence despite the con-

siderable differences in systems

共cloth versus hard, rough surface

兲 and phenomena 共waves versus avalanches兲 is striking. For the copper particles

关see Fig.

16

共b兲 兴, m

range of 1.7

m

h

⬍3.2 as ␩

h

falls in the changes between 0.25

⬍ ␩

t

PHYSICAL REVIEW E 78, 011306

共2008兲

x

1 cm

a.

0 s

0.1 s

0.2 s

1 cm

1 cm

1 cm

0.3 s

b.

0 s

0.05 s

0.1 s

0.15 s

c.

0 s

0.1 s

0.2 s

0.3 s

d.

0 s

0.2 s

0.4 s

0.6 s

FIG. 17.

共Color online兲 Space-time plots taken at the symmetry axis of avalanches with camera 1.

共a兲 Sand at ␪=36.8° 共␪/␪

= 1.01

兲, 共b兲 copper with per with

= 23.5°

␪/␪

= 0.63 at

c

= 0.95

兲.

= 0.25 at

␪=39.5° 共␪/␪

␪=26.1° 共␪/␪

c c

= 1.01

兲, 共c兲 cop-

= 0.96

兲, and 共d兲 glass at ␪

c

⬍0.63. The general trend is that irregular particles form higher avalanches compared to avalanches of more spherical particles. To quantify this trend we again plot h function of the tangent of the angle of repose asymptote of the h

s

兲 curve兴. As seen in Fig.

r

16

m

/h

关or

共c兲

s

, h

1 as a

m

the

/h

s

increases systematically with tan

r

. This is an important relation as it provides a tool for predicting typical avalanche heights

共for a given material兲 just by measuring the angle of repose of the material.

Having demonstrated that both the dimensionless avalanche height h

m

/h

s

and the growth rate of avalanche speed with increasing avalanche size

共defined as m

u

兲 depend systematically on increasing grain shape irregularity, we can further explore whether such a tendency can be observed in other properties of avalanches. Using the space-time technique described in Sec. II, we can explore the ratio of the mean particle speed near the front u

p

to the front velocity u

f

.

To compare the behavior at the front, four examples are shown in Figs.

17

共a兲

– 17

共d兲

for sand, copper with copper with

= 0.25,

= 0.63, and glass beads, respectively. In Figs.

011306-9

BÖRZSÖNYI, HALSEY, AND ECKE

1.8

1.6

1.4

1.2

1

0.8

0.6

30

20

10

0

0 10 20 u f

(cm/s) a.

sand glass beads

1.8

1.6

1.4

1.2

1

0.8

0.6

b.

copper

η = 0.25

η = 0.33

η = 0.5

η = 0.63

20 25 30

θ (deg)

35 40

FIG. 18.

共Color online兲 Ratio of particle velocity u

p

velocity u

f

as a function of the plane inclination for

共a兲 sand 共⫻兲 and glass beads

共䊊兲, 共b兲 copper particles with

= 0.33

共ⴱ兲,

␪=25.2°.

= 0.5

as a function of u

f

共⫻兲, and

= 0.63

= 0.25

and front

共䉭兲,

共䊊兲. The inset of 共a兲 shows u

p

for sand

共⫻兲 at ␪=36.8° and glass beads 共䊊兲 at

17

共a兲

and

17

共b兲

one sees that for anisotropic particles the velocity of particles is larger than the front velocity; in many cases, particles are flying out of the main body of the avalanche and are stopped by the static layer in front of the avalanche. On the contrary for the avalanches formed by spherical beads, Figs.

17

共c兲

and

17

共d兲

, particles are slower

than the avalanche front and particles in the static layer begin moving just before the actual particles in the avalanche arrive at that position. This is clearly seen in Figs.

17

共c兲

and

17

共d兲

by the curved trajectories of particles near the front which are initially at rest. To quantify these qualitative differences between particle and front motion, the average velocity of individual particles u

p

near the avalanche front and the front velocity u

f

are measured for numerous avalanches.

At a given plane inclination

␪ the particle velocity as a function of the front velocity shows a linear dependence, as indicated in the inset of Fig.

18 .

The ratio u

p

/u

f

is plotted in Figs.

18

共a兲

and

18

共b兲

as a function of the plane inclination for sand/glass-beads and copper particles, respectively. For glass beads and sand particles, the separation is very clean with the spherical particles moving slower than the front with 1

u

p

/u

f

⬇0.6, whereas the irregular sand particles overtake the front with 1

u

p

/u

f

⬇1.3. There may be a slight downward trend with increasing

␪ for irregular particles, but the data do not differentiate that trend from a near constant ratio. The ratios

u p

/u

f

are again less clearly separated for copper particles

011306-10

PHYSICAL REVIEW E 78, 011306

共2008兲

2

1.5

1

0.5

a.

0

0

2

1.5

1

0.1

0.2

0.3

0.4

0.5

0.6

tan

θ

r

or tan

θ

1 flow rule with tan

2

θ

r

flow rule with tan

2

θ

1 copper,

η=0.5

sand copper,

η=0.33

0.5

0

0

2

1.5

glass beads

0.1

0.2

β

PJ

0.3

b.

0.4

1

0.5

c.

0

0 0.1

0.2

0.3

0.4

0.5

0.6

β

P

FIG. 19.

共Color online兲 Ratio of the particle velocity u

p

front velocity u the slope

PJ f

as a function of

共a兲 tan

r

共ⴱ兲 or tan of PJFR

33

兴 or 共c兲 the slope

P

1 and

共䊊兲 or 共b兲 of PFR for sand, glass beads, and copper with symbols correspond to the cases when

␤ or

1

共䊊兲.

= 0.33 and

PJ

= 0.5. In

共b兲 the two data was derived using

r

共ⴱ兲 with ratios close to one for the

= 0.63 and

= 0.5 particles.

For whatever reason—perhaps interparticle friction— spherical copper particles are marginal with respect to the separation of particle velocity and front velocity. The irregular copper particles have u

p

/u

f

⬇1.6, considerably greater than 1, and are well separated from the more spherical particles. For the irregular particles there is a definite decrease in u

p

/u

f

with increasing

.

To quantify how the ratio u

p

/u

f

varies with grain shape irregularity, we again consider correlations between the ratio

u p

/u

f

and the angle of repose increase of u

p

/u presented in Fig.

f

r

共or as a function of tan

18

1

r

兲. There is a linear

共or tan

1

兲 with a slope of about 3.5 as shown in Fig.

19

共a兲

for the six materials

. We also compare how the ratio u

p

/u

f

varies with the equation relating u and h in the continuous flow regime—i.e., the granular flow rule. Earlier, we reported values of the slope

PJ

in Eq.

2

兲 for various materials 关

33

and found a systematic increase of shape irregularity—i.e., with increasing

u p

/u

f

as a function of

PJ

with increasing grain

r

共or

1

兲. Plotting

PJ

, we again find a systematic linear increase with a slope of about 2.8. The copper sample with

AVALANCHE DYNAMICS ON A ROUGH INCLINED PLANE PHYSICAL REVIEW E 78, 011306

共2008兲

0.3

a.

glass beads

θ = 24.3

o

1.6

1.4

0.2

1.2

0.1

0

2

1.5

b.

sand

θ = 36.8

o

1

3

2.5

1

0.5

2

1.5

0 1

-100

x / h s

-50

0

FIG. 20.

共Color online兲 Avalanche height profiles 共solid lines兲 taken with the help of a laser sheet and velocity profiles

共䊊兲 measured from space time plots for

共a兲 glass beads 共for

␪=24.3° and

h s

= 0.178 cm

兲 and 共b兲 sand 共for ␪=36.8° and h

s

= 0.12 cm

兲.

= 0.5 seems to be anomalous, perhaps related to the peculiar dynamics of this set of copper in that it is the only copper sample emitting strong sound during shearing. This emission is similar to but much stronger than the sand from the Kelso dune which is known to be an example of “booming sand dunes”

43

兴. Interestingly this peculiarity is not reflected in

the dependence of u

p

/u

f

on

r

; see Fig.

19

共a兲

.

Although the Pouliquen flow rule

关Eq. 共

1

兲兴 does not per-

fectly describe the flow properties of homogeneous flows, especially for the case of copper particles

33

兴, for the clas-

sification of avalanches we have determined

␤ only data for relatively shallow flows where h

/h ting u

p

/u

f

as a function of

P

P s

by taking

⬍10. Plot-

关see Fig.

19

共c兲 兴 we again find

a systematic linear increase. We will come back to this observation in Sec. IV.

Finally, we consider the detailed profiles of velocity and height along the propagating direction of the avalanche. The particle velocity u

p

x兲 is obtained from space-time plots and the height profile from laser line measurements. The measurements are not made simultaneously, but can be compared for avalanches with similar properties. We show profiles for glass beads and coarse sand in Figs.

20

共a兲

and

20

共b兲

with scaled velocity and height on opposite axes. The differences for sand and glass beads are striking. For glass beads, both velocity and height are approximately linear up to the maximum and then fall quickly over a steep but continuous front with a width of order 10h

s

. The slope of the surface behind the front is only slightly shallower than the unperturbed layer with

␦␪

/ ␪ ⬇−0.007, whereas the fractional angular increase near the front is about 0.1—i.e., a difference of about 3°–4° relative to the plane inclination angle.

For sand avalanches, a quite different picture emerges as shown in Fig.

20

共b兲

. First, the velocity maximum and the

height maximum occur at different values of downstream distance with height peaking before velocity. Second, the

FIG. 21.

共Color online兲 Schematic view of avalanches formed in materials consisting of

共a兲 spherical particles 共e.g., glass beads兲 and

共b兲 irregular shaped particles 共such as sand兲. The horizontal scale

共along the slope兲 is compressed by about a factor of 30 relative to the scale perpendicular to the plane. The tip region in

共b兲 represents low density material that spills over the front with speeds greater than the front speed. Note the difference in the mobilization of the underlying layer, especially at the avalanche front.

height and velocity increase faster than linear from the back of the avalanche towards the front. Although we have adjusted the axes to align the maximal velocity and height of the avalanche, one can still see that the functional dependence for scaled velocity greater than one and h

/h

s

艌2 are different. In other words, velocity and height are proportional for small values near the back of the avalanche, but separate above values that, interestingly, differentiate in the mean between the spherical and irregular avalanche behavior. Despite the quite different width of the sand front, of order 25 h

s

, the fractional angular increase at the front is only slightly shallower than for glass beads—i.e.,

␦␪

/

⬇0.07. The difference in profile for the sand arises from the nature of the particles near the front. Whereas the glass beads form a compact avalanche with a sharp but distinct front, the sand particles overtake the front as defined by the motion of the layer underneath creating a dynamics similar to a breaking wave. In the theory section below, we explore a depth-averaged approach for explaining the different avalanche behavior of spherical and irregular granular particles.

Although we cannot directly measure the interior profile of the avalanche, it is useful to provide a schematic illustration of the different types of avalanches to summarize what we have learned about them. In Fig.

21 , we show the salient

features of the two avalanche types. The case of shallower avalanches formed in materials with spherical particles is shown in Fig.

21

共a兲

. Here the granular layer fails—i.e., starts

moving—just ahead of the avalanche

共of order 5h

s

– 10h

s

兲, leading to slower maximal particle velocities than the front velocity. The velocity and height of the avalanche are proportional and the variation of both quantities is linear in the

011306-11

BÖRZSÖNYI, HALSEY, AND ECKE PHYSICAL REVIEW E 78, 011306

共2008兲 down-plane direction. The form of the avalanche is reminiscent of a viscous Burgers shock as discussed in more detail below. In contrast, in Fig.

21

共b兲

a sand avalanche is visualized having a larger maximal particle velocity than the front velocity and a considerably higher avalanche height. The avalanche seems to behave like a “breaking wave” with lowdensity particles spilling over the crest at speeds higher than the front speed. The material directly under the front seems to be solid with the avalanche more slowly entraining the underlying material. Direct measurements of this entrainment and the form of the avalanche profile would be very useful but are beyond the scope of the present work.

IV. THEORY

The theory of avalanche behavior is complicated by the existence of a moving fluid phase in contact with a solidlike immobile granular phase. Thus, a complete theoretical picture of avalanche behavior certainly requires a multiphase approach. On the other hand, since the granular material is moving over most of its extent except for small regions at the front and back, a simpler approach neglects the solid state ahead and behind the avalanche. We can then use a depthaveraged approximation, leading to the Saint-Venant shallow flow equations, adapted for granular media by Savage and

Hutter

44

兴. This approach was used in the analysis of wave

formation in dense granular flows

34

兴 and was applied by us

to sand and glass-bead avalanches described previously

39

兴.

down a plane, with the plane parallel to the x direction, is described by the averaged equations for mass,

h

t

+

⳵ 共hu¯

x

= 0,

共3兲 and momentum balance,

⳵ 共hu¯

t

+

⳵ 共hu¯

2

x

=

冉 tan

␮ 共,h兲 − K

h

x

gh cos

.

共4兲

The value of

␣ is determined by the profile of the flow,

= 1 for plug flow

共as in Ref. 关

44

兴兲,

= 4

/3 for a linear flow profile, or 5

/4 for a convex Bagnold profile 关

45

兴. The param-

eter K is determined by the ratio of the normal stresses in the flow: the stress parallel to the bed, lar to the bed,

zz

xx

, and that perpendicu-

. Numerical results show that K

⬇1 for steady-state flows 关

31

兴.

⬅ ␴

xx

/ ␴

zz

The complicated part of this analysis concerns the friction coefficient

,h兲 关

46

兴. For example, if the layer is too thin

at a particular

, the flow stops and a friction coefficient appropriate for the flowing state is no longer valid. This transition from dynamic to static friction and the resulting yield condition for the solid phase in front of and behind the avalanche are not included in our approach and would be difficult to handle in a depth-averaged fashion. We will consider the consequences of this assumption later. For now we assume that the friction coefficient

␮ 共,h兲 is determined by the requirement that the steady flow obeys the rheology specified in Eq.

5

兲 and will thus vary with the particle type. For

simplicity, we consider only the Pouliquen flow rule; the

011306-12

Pouliquen-Jenkins flow rule adds algebraic complexity without improved understanding of this model of avalanche behavior.

From the flow rule relation for steady-state flows

关Eq. 共

1

兲兴

we obtain an expression for h

s h s

=

P h

3

/2

g

gh

.

共5兲

The avalanches we will describe are observed for angles not very far from

1

共i.e., tan

/tan

1

⬍1.4兲. In that range, the relationship

h s

= tan

ad

− tan

1

共6兲 works very well in fitting the data for all the materials. The values for the two fitting parameters a and

Table

I

for sand, glass, and copper. As from

6

1 are shown in

= tan

␪ we obtain

= tan

= tan

1

+

ad

.

h s

共7兲

The slower avalanches we describe correspond to the case of low Froude numbers Fr= u side

共LHS兲 of Eq. 共

4

/

gh cos

兲 scales like Fr

2

␪ beads avalanches correspond to Fr

⬇0.4. Since the left-hand

⬇0.16, we can set the

LHS to zero. Plugging the expression for

. For example, glass

, Eq.

7

兲, into the

RHS, we get tan

− tan

1

d h s

K

h

x

= 0.

共8兲

Substituting the expression for h

s

, Eq.

5

兲, into Eq. 共

8

g

d

冉 tan

− tan

1

K

h

x

P h

3

/2

gh.

Finally we use this form for u

3

兲:

h

t

+

x

g

d

冉 tan

− tan

1

K

h

x

P h

5

/2

h

gh

= 0,

共10兲

共9兲 which yields

h

t

+ a

h

h

x

=

K

P

g

d

x

h

5

/2

h

x

, where

a

h兲 =

gh

5

2

P h h s

3

2

.

共11兲

This equation has solutions similar to those of Burger’s equation

47

兴. Thus, there is a solution consisting of a single

hump propagating down the slope with velocity a

h兲, with a smooth structure determined by the competition between this nonlinear velocity term on the LHS and the dissipative term

共RHS兲 of Eq. 共

11

兲. Numerical solution of this equation yields

AVALANCHE DYNAMICS ON A ROUGH INCLINED PLANE

3

4

3

2

0

1

6

5

h / h s critical

2

1

0.4

β

P

0

FIG. 22.

共Color online兲 Dimensionless avalanche height h

m

共䊊兲 and the ratio of the particle and front velocities u

p

function of

P

/u

f

/h

共ⴱ兲 as a for sand, glass beads and the four copper samples.

s

The solid lines correspond to the criterion a = c

+ for linear velocity profile

共black兲 and for Bagnold velocity profile 关blue 共gray兲兴, while the dotted line is a linear fit to the h

m

dashed line denotes the value of u

p

/u

f

/h

= 1.

s

P

兲 data. A horizontal a fairly linear ramp behind the front, very similar to Fig.

20

共a兲

for glass beads. Further, one can estimate the width of the front by balancing the dispersive term and the dissipative term, taking

h

/ ⳵

x

h/ᐉ and front width. Inserting values for the parameters into Eq.

11

yields a scaled front width

2

h

ᐉ/h

/

s

x

2

h/ᐉ

2

, where

ᐉ is the

⬇6, consistent with the height profile for glass beads in Fig.

20

共a兲

. We can also com-

pare the predicted group velocity a

h兲 with the measured avalanche front velocity for glass beads. Using the peak height h

m

to evaluate a

h兲, we obtain u

f

/ah兲⬇0.6, again consistent agreement between theory and experiment.

The full equations of mass and momentum conservation are hyperbolic with characteristic velocities

c

− 1

兲 +

K

Fr

2

.

共12兲

If the velocity appearing in Eq.

11

兲 does not obey ac

+ then Eq.

11

兲 predicts a structure that moves faster than the

, maximum rate at which information can be propagated in the full system of equations, which is clearly impossible. For faster avalanches such as sand, the small Froude number limit is not valid since Fr

⬎1. In these circumstances, the

Burgers-type solution transforms itself into a truly discontinuous solution traveling at velocity c

+

47

兴, which is de-

scribed by the full system, Eqs.

3

兲 and 共

4

兲, rather than by

Eq.

11

兲.

We can directly test the conditions for which a

c

+ is valid in terms of avalanche height. The normalized avalanche height h

m

/h

s

as taken from the slope of the fits in

Figs.

16

共a兲

and

16

共b兲

is shown in Fig.

22

for glass beads, sand, and all four copper samples as a function of the flow rule slope

h m

/h

s

P

共using a function of tan

r

␥ as a function of

= 0

P

兲. We find a systematic increase of

, as we did when plotting h

m

/h

s

as

关see Fig.

16

共c兲 兴. On the other hand, the

boundary between the two regimes corresponding to a = c

+

011306-13

PHYSICAL REVIEW E 78, 011306

共2008兲 results in a critical thickness h strongly decreases with increasing

Fig.

22

for linear and Bagnold velocity profiles

兲. The range below h

/h

s

with a

c

+

critical

/h

critical s

P

of the flow that

共see the solid lines in corresponds to smooth solutions of Eq.

, whereas above h

/h

critical s

11

兲,

theory predicts discontinuous solutions of the full system traveling at velocity c

The line corresponding to the normalized avalanche height

h m

/h

s

as a function of

P

+

.

crosses this boundary, and the data points for sand and the two anisotropic

共dendritic兲 copper grains fall above

共or at兲 the curve, whereas data points for glass beads and the spherical copper samples fall below the boundary. This observation matches well with the result that the ratio of the particle velocity and the front velocity of the avalanche u

p

/u

f

also increases with increasing

P

and crosses the value 1 at about the same location corresponding to

P

⬇0.4 共see Fig.

22

兲.

Thus, we conclude that the glass bead avalanches, or more generally avalanches formed in materials with small

P

, reflect smooth solutions of Eq.

11

兲, with ac

+

, whereas avalanches in sand or copper samples with highly anisotropic grains, or more generally in materials with higher

P

, represent discontinuous solutions of the full system traveling at velocity c

+

. The latter avalanches propagate into a quiescent bed because they are traveling at the characteristic velocity for the medium. In the experiments the signature of these

“discontinuous” solutions is that grains move faster than the front speed, spilling or breaking ahead of the front in a lowdensity “foamlike” phase. The crossover between these two regimes happens at

P

⬇0.4. The glass bead avalanches are analogous to “flood waves” in river flows, whereas the sand avalanches are analogous to “roll waves” in these flows

47 , 48

兴. Note that ahead of the flowing avalanche, the mov-

ing material propagates into a material at rest, which is presumably in a state close to the critical Mohr-Coulomb state

49

兴. Unlike the flowing state, for which

critical state

xx

⬎ ␴

zz

xx

⬇ ␴

zz

, in this

. Thus the transition region, in which the flow accelerates from rest into a pseudo-steady-state described by the continuum theory, can be viewed as a region of passive Rankine failure, through which the compressive stress parallel to the bed,

xx

, is decreasing with time. The mechanics of this region are complex and cannot be described by the Saint-Venant equations alone.

We should point out that in the linear theory of the instability of steady flows, developed by Forterre and Pouliquen, the criterion a

c

+ corresponds to the stable regime of these flows with respect to wave disturbances

34

兴. Thus, our ob-

servation of discontinuous avalanches for sand and smooth avalanches for glass beads dovetails nicely with their observation that steady flows of sand are far more unstable to such disturbances than are steady flows of glass beads.

The theory as presented has several nice features that semiquantitatively describe the dynamics of avalanches in a range of materials with varying shapes and interparticle friction. Nevertheless, the theory fails to account for several important aspects of the flow. First, the generalized Burgers equation fails to capture the solitonlike nature of the individual avalanches—namely, that they travel at constant speed and maintain uniform shape. One can include more physics by following Pouliquen and Forterre

46

兴 in providing a pa-

rametrization of the friction coefficient that accounts for the

BÖRZSÖNYI, HALSEY, AND ECKE static friction component and hysteresis between flowing and solid phases. Numerical simulations of either the full 2D equations or a 1D version of the Saint-Venant equation with a more realistic friction fail to resolve this difference

50

兴. In

particular, for glass beads one obtains a solitary localized solution, but one with a flat rather than linear height profile behind the front. In that case, the speed is controlled by a front condition rather than by the mechanism consistent with our theory. We find empirically that the predicted velocity

a

h兲 is greater than the measured u

f

, so perhaps a similar control is at play for the glass beads. For sand avalanches, full numerical simulations yield a form similar to our theory, but again the height decays rather than assuming a constant profile. Thus, although the theory presented here captures some of the ingredients of our experimental avalanches, the two phase nature of the avalanches is probably necessary for a more quantitative description of our results.

In addition to the depth-averaged approach used here, it may be possible to use kinetic theory

51 – 53

兴 to describe the

dynamics of avalanches despite the dense granular flow associated with avalanches. In particular, kinetic theory does appear to provide an improved description of the flow rule for the continuous flow regime

33 , 35

兴 relative to the as-

sumptions of Bagnold theory

29

兴. One objection to this ap-

proach is that the flow has a two-phase nature containing a solid phase at the front or back of the avalanche which is outside the scope of kinetic theory. On the other hand, a similar objection applies to our depth averaged approach. We are unaware of avalanche theories that employ kinetic theory and the development of such a description is beyond the scope of our current efforts.

V. CONCLUSIONS

The main conclusion of our experimental study of avalanches is that the form of the avalanche depends on the granular particles. Differences are found not only at the quantitative level, but in the qualitative features

共front propagation, avalanche shape, grain dynamics

兲 which reflect the dramatic differences between smooth grains such as glass beads and irregular grains like sand. Measuring the basic avalanche properties such as velocity and size, we find a systematic linear increase of the normalized avalanche peak height h

m

/h

s

with increasing tan

r

or

P

, where

r

is the angle of repose of the material and

P

is the slope of the depth-averaged flow rule, both measures of grain shape irregularity. Similarly, the slope of the dimensionless velocity versus size curves increases linearly with increasing tan

r

, suggesting that the avalanche velocity increases with increasing avalanche size systematically faster for more anisotropic grains. These two relations enable us to predict typical avalanche sizes and the growth rate of avalanche velocity as a

PHYSICAL REVIEW E 78, 011306

共2008兲 function of increasing size for various materials, by simply measuring

r

or

P

.

When focusing on the dynamics of individual grains

共and comparing it to the propagation of the avalanche

兲, another characteristic difference is found for spherical and nonspherical particles. Particles with irregular shape move considerably faster than the avalanche front, indicating that inertia is an important ingredient in these faster moving avalanches. For avalanches formed by spherical particles, on the other hand, front propagation is transmitted through the contact points

共force chains兲 near the front, thereby inducing motion through a collective yield

共failure兲 condition at the front. In the latter case, the velocity of individual grains is considerably smaller than the front velocity. When quantifying this important property of avalanches we again find a direct correspondence between u

p

/u

f

and either

r

, the angle of repose of the material, or the slope of the depth averaged flow rule

P

共or ␤

PJ

兲. The value u

p

/u

f

increases linearly with increasing tan

r

or

P

. These relationships again provide a tool for predicting the dynamical properties of avalanches by simply measuring

r

or

P

for a given material. The direct correspondence should be valid for moderately polydisperse materials, but may not be valid for highly polydisperse materials where segregation effects become important. Segregation can lead to a spatial variation of h

s

and thereby considerably influence the dynamics of avalanches.

In Sec. IV we have shown that a simple depth-averaged description can capture a strong change in the dynamical behavior of avalanches if we know the basic

共depthaveraged

兲 rheology of the material. Also the striking difference between the behavior of avalanches formed by spherical and nonspherical particles warns us that when modeling granular avalanches or granular dynamics in general, predictions obtained by the simplest models neglecting shape irregularities might not be valid for systems with nonspherical particles. Our measurements, however, provide numerous additional quantitative features of avalanches for grains with different rheologies that cannot be captured by the simple theory presented here, leaving room for further theoretical or numerical studies that can provide a better understanding of avalanching behavior.

ACKNOWLEDGMENTS

The authors benefited from discussions with Brent Daniel,

Anja Slim, Michael Rivera, and Michael Stepanov. One of us

共R.E.E.兲 acknowledges the support of the Aspen Center for

Physics. This research was funded by the U.S. Department of

Energy under Contracts No. W-7405-ENG and No. DE-

AC52-06NA25396. T.B. acknowledges support by the

Bolyai János research program and the Hungarian Scientific

Research Fund

共Contract No. OTKA-F-060157兲.

011306-14

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011306-15

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