Pingpong ball avalanche experiments
J. McElwaine K. Nishimura
jim@orange.lowtem.hokudai.ac.jp
Institute of Low Temperature Science,
University of Hokkaido,
North 19 West 8, KitaKu,
Sapporo 00600819, Japan
Pingpong ball avalanche experiments have been carried out for the
last three years at the Miyanomori ski jump in Sapporo, Japan, to
study threedimensional granular flows. Up to 550,000 balls were
released near the top of the landing slope. The balls then flowed
past video cameras positioned close to the flow, which measured
individual ball velocities in three dimensions, and air pressure tubes
at different heights. The flows developed a complicated
threedimensional structure with a distinct head and tail, lobes and
``eyes''. ``Eyes'' have been observed in laboratory granular flow
experiments and the other features are similar not only to snow
avalanches, but also to other largescale geophysical flows. The
velocities attained showed a remarkable increase with the number of
released balls. A power law for this relation is derived by
similarity arguments. The air pressure data is used to deduce the
structure of the air flow around the avalanche and in conjunction with
the kinetic theory of granular matter to estimate the balance of
forces in the avalanche head.
snow avalanche, twophase flow, granular
flow
Snow avalanches have been measured and observed in the Shiai
valley, Kurobe since 1989. Though there are partial data on the
internal velocity distribution for both dense and powder
parts [Gubler, 1987,Kawada et al., 1989,Nishimura et al., 1989,Nishimura et al., 1993a,Dent et al., 1994,Nishimura & Ito, 1997]
the data are insufficient to constrain and discriminate between
current avalanche models, for a recent survey of current models
see [Harbitz, 1999, Harbitz (1999)], and thus insufficient to allow
a quantitative understanding of the dynamics and internal structure of
snow avalanches. The poor quality of the data is because of the
unpredictability, scarcity and intense destructive power of
avalanches.
Avalanches can be modelled in the laboratory using granular materials
on inclined planes, usually in water for powder
avalanches [TochonDanguy & Hopfinger, 1975,Hopfinger & TochonDanguy, 1977,Beghin & Brugnot, 1983,Hermann et al., 1987,Beghin & Olagne, 1991,Keller, 1995]
or air for dense avalanches [Hutter, 1991,Hutter et al., 1995,Nishimura et al., 1993b,Greve & Hutter, 1993,Greve et al., 1994].
Laboratory experiments are much easier to perform than field
experiments and are usually, easily repeatable. However, the small
size of the granular particles used makes direct observation of
individual particles difficult, and only a few similarity parameters
are typically satisfied [Keller, 1995]. For
example, no laboratory experiments have yet been carried out in which
a dense granular flow becomes a turbulent suspension by entraining the
ambient fluid, though in some
experiments [Rzadkiewicz et al., 1997] a small number
of the grains may enter suspension. Instead experimental models
of powder snow models in water tanks use a denser fluid or a premixed
turbulent suspension. Laboratory granular flows also rarely exhibit
the complex threedimensional structure which is characteristic of
avalanches and other large geophysical flows. For these reasons for
the last five years large scale granular flow experiments have been
carried out using golf balls and pingpong balls. The first
experiments were carried out on long (20
) chutes and
more recently on the Miyanomori ski jump. Pingpong balls are
particularly suitable, since they reach terminal velocity in only a
few metres, so fully developed flows occur even on short slopes.
These experiments have been described in
several papers [Nishimura et al., 1996,Nishimura et al., 1998,Keller et al., 1998].
The aim of these experiments is to elucidate the dynamics of twophase
granular flows rather than to directly extrapolate the results to snow
avalanches. The experiments provide detailed data and provide insights
on the physically significant dynamical processes controlling
avalanches. The hope is that this will lead to a theory of snow
avalanches based on physical processes with no free parameters.
The kinetic theory of granular matter provides only poor agreement
with
experiments [Jenkins & Savage, 1983,Haff, 1983,Lun et al., 1984,Jenkins & Richman, 1988,Johnson et al., 1990,Anderson & Jackson, 1992,Jenkins, 1994],
but does provide a theoretical framework for discussing stresses in
granular flows. Another approach is the direct simulation of granular
flows using the discrete element
method [Campbell & Brennen, 1985,Campbell & Gong, 1986,Cleary & Campbell, 1993,Campbell et al., 1995,Hanes et al., 1997].
These simulations have increased the understanding of granular flows,
including twophase flows, but these simulations have not yet
accurately dealt with particles strongly coupled to fluids or
threedimensional anisotropic flows.
The pingpong ball avalanches can be described by well known
equations. The air flow obeys the NavierStokes equations and
individual pingpong balls follow Newton's laws, whereby the force on
a particle is a function of gravity, particleparticle contacts,
particleground contacts and air drag. The noslip boundary condition
between particles and the air flow determine the drag force. For small
numbers of particles at low Reynolds numbers in closed domains these
equations can be directly
solved [Glowinski et al., 1996,Hu, 1996,Blackmore et al., 1999],
but for this experiment it is currently impossible, because of the
large number of particles and the large range of length and time
scales. Particleparticle collisions occur over time intervals of
order of
whereas the duration of the flow in these
experiments is around
. The length scales in these
experiments are given by the length of the ski jump (
),
the volume of the flow (
),
the diameter of the balls (
) and the compression of
the balls during collisions (
).
This papers discusses two complementary approaches for describing the
experiments. The first is to consider the flow as a single object
moving down the ski jump and to use similarity arguments to deduce
gross features of the flow. The second approach is to use twophase
flow equations that couple the NavierStokes equation for the air
flow to the kinetic theory equations for the ball flow using an
(empirical) drag force.
The experiments were undertaken at the Miyanomori ski jump (the normal
hill for the 1972 Olympics) in Sapporo, Japan. The landing slope was
long and
high
(Fig. 1) and covered with an artificial surface.
Figure 1:
Cross section of the landing slope of the Miyanomori ski
jump. Marked distances are measured from the top of the landing
slope.

Standard pingpong balls with a diameter of
and mass of
per ball, were placed in a large box,
or
along from the top of the landing slope. Flow was
initiated by opening the hinged door on the front of the box. The
balls then flowed down the slope and past the measurement sensors,
which were all placed near the middle of the slope
down
from the top (
from the front of the box). The
experimental procedure is described in more detail
in [Nishimura et al., 1996, Nishimura et al. (1997)].
A video camera was set pointing perpendicularly down at the slope
(Fig. 2) at a height of
. Balls
closer to the lens of the camera appear larger than those which are
further away. Thus the coordinate of a ball can be calculated by
measuring the size of a ball in the video picture, since all the balls
have the same diameter. The and coordinates of a ball are
given directly by its position in the video frame. Comparing
positions for the same ball from adjacent video fields gives the
threedimensional velocity of a ball (timeaveraged over the
between video fields.) The necessary camera
corrections and detailed method are described in [Keller et al., 1998, Keller et
al. (1998)].
Figure 2:
Schematic of the video camera for measuring ball positions

Measuring air velocity in particulate flows is very difficult. In
snow avalanches ordinary meteorological anemometers are inaccurate
because of the snow particles, and are usually
destroyed [Kawada et al., 1989,Nishimura et al., 1989].
[Nishimura et al., 1996, Nishimura et al. (1996)]
and [Nishimura & Ito, 1997, Nishimura and Ito,
(1997)] have developed the use of
pressure measurements (sampling frequency
) for
inferring air speed. A tube connected to a pressure difference sensor
is set so that the open end points downwards, perpendicular to the
main flow direction. Note that this is not a Pitot tube since each
tube has only one opening and the pressure difference is measured with
respect to the air pressure some distance from the flow. Bernoulli's
law then gives

(1) 
where is the pressure difference, is the air
density and is the air speed parallel to the slope. However, this
equation is only valid when the air flow is perpendicular to the end
of the pipe and the local static pressure is known. Also the sensor
itself disturbs the flow. The Reynolds number for the flow around the
tube is approximately (wind speed
, tube
diameter
), so the flow will be partially turbulent
around the sensor. The interaction of the pressure sensor with the
flow coupled with the rapid pressure fluctuations as a result of the
turbulent flow field would lead to inaccurate measurements, if solely
based on Eq. 1. Therefore the pressure tubes were
calibrated by measuring the static pressure depression in a wind
tunnel over a range of velocities. Four of these air pressure sensors
were placed
down the slope at heights of
,
,
and
.
Several video cameras were placed to the side of the slope and at the
bottom of the slope. As can be seen in Fig. 3 the
leading edge of the avalanche is clearly visible. The position of the
front was measured and used to calculate the front velocity.
Figure 3:
Front view of a 550,000 ball avalanche at the Miyanomori
ski jump. The horizontal lines are 5 m apart and the lowest one is
90 m from the top of the landing slope.

Figure 4:
Side view of the head of a 550,0000 ball avalanche at the
Miyanomori ski jump.

When the door of the box was opened, the balls at the front of the box
rapidly accelerated down the slope (Figs. 3
and 4). The front velocity was much larger than the
tail velocity (the last balls took several seconds to leave the box).
For a
flow the front of the flow accelerated
approximately linearly with distance until it reached a speed of
after
, whereas the balls in the tail had
a speed of only a few metres per second  similar to the speed of a
single ball. The front velocity was roughly constant for the next
until the slope angle started to decrease. This large
disparity in speed between head and tail caused the flow to elongate
so much that at times it covered more than half the slope. The flows
can be separated into three distinct regions: a short, high, fast
moving head; a longer, lower body moving at the same speed; and a very
long tail moving much slower, consisting of separated balls.
Other macroscopic features of the flow are interesting but hard to
quantify. At the beginning of the flow there are often several waves
within the flow which move faster than the body and coalesce in the
head [Nishimura et al., 1998]. Another obvious feature are two
roughly circular regions of reduced flow height, symmetrically located
about the flow centreline, a little behind the head, called
``eyes'' after [Nohguchi et al., 1997, Nohguchi et al. (1997)].
They can be seen on the third line up from the bottom of
Fig. 3 as the darker regions. Similar patterns
have been reported in laboratory granular flow experiments with
styrene foam particles [Nohguchi, 1996] and with ice
particles (Nohguchi, personal communication). In these experiments
the particles are around
in diameter and the flows
contain 1,000
. For such a feature to
exist in experiments of such different scales suggests that the mean
velocity fields and flow structure are similar in all these
experiments. The ``eyes'' may represent a pair of vortices shed by the
head, but only a detailed quantitative analysis of ball velocities can
confirm this. In the tail the balls are not distributed evenly but
tend to cluster, because of inelastic collapse. (As density in a
granular flow increases the collision rate increases thus increasing
dissipation and reducing granular pressure. The density thus continues
to increase and the collision rate diverges, so that a group of
particles can come to rest in continuous contact in finite time.)
In [Nohguchi, 1996, Nohguchi (1996)] granular flows
experiments with styrene particles were performed and the front
velocity was observed to increase with the number of balls. Similar
increases were observed in these experiments. [Nohguchi, 1996, Nohguchi
(1996)] deduced that the maximum front velocities, ,
for flows which vary only in the number of balls, , should scale
according to

(2) 
In order to derive Eq. 2 the drag force was
assumed to be a linear function of the flow velocity. However, the
result can be obtained without this assumption as follows.
The critical assumption is that there is only one significant length
scale given by

(3) 
where is the volume of all the balls and the ball diameter.
The implied constant of proportionality in Eq. 3 is
constant between experiments with different numbers of balls, but is
not constant along the slope, i.e. the height and width of the flow at
any given position scales with the number of balls.
Equation 3 will not be true initially when the input
box size is important nor will it be true where the flow is only a few
balls thick. However, all the flows with more than
are observed rapidly to reach a selfsimilar shape in
around
and the flows are many particle diameters thick
except in the tail.
The effective gravitational acceleration on the flow is
, where is
the angle of the slope, the friction with the slope, the
acceleration due to gravity and and are the air and
ball densities respectively. After the initial surge from the box the
flow is close to its equilibrium velocity, i.e. it is
accelerating/decelerating slowly, so inertia can be ignored. The
Reynolds number for the air flow is of the order so air
viscosity can be ignored. Under the length scale assumption
(Eq. 3) the nondimensional density ratio
, where is the mass of a single ball, is constant for
different sized flows since . Therefore air density
need not be further considered as a dimensional variable
since it can be substituted by . The dependence on the box size
and ball diameter has already been discussed which leaves only three
variables , and . Thus the only dimensionless combination
that can be formed containing the front velocity is the densiometric
Froude number

(4) 
This must be constant for different flows thus

(5) 
since
.
In [Nishimura et al., 1998, Nishimura et al. (1998)] the
front velocity was measured between the k point and the
p point (where the slope angle, , is roughly constant and
steepest, see Fig. 1). The remarkably good fit
between this equation and experiment is seen in
Fig. 5 and provides additional justification for
Eq. 3. As expected the error is worse for small
flows, since they rapidly spread into single thickness layers with two
significant length scales (width and length) and
(height). The height is likely to be the significant length scale in
this range so for small flows we expect the velocities to be
independent of flow size.
Figure 5:
Front velocities at the kpoint for different sized
avalanches.

Flow structure and ball velocities
By analysing the video film, [Keller et al., 1998]
individual particle positions can be calculated, and by identifying
balls between adjacent video frames, particle velocities are obtained.
Figure 6 shows the perpendicular positions and
velocities for a 200,000 ball flow as the particles are advected
beneath the camera. The time interval of one profile is
(
). This technique, however, cannot see
through pingpong balls, and in the dense head (volume fraction
0.2) only the balls from the top
can be
identified thus there is a blank region, marked passage of the
head, in Fig. 6 where there is no data.
Figure 6:
Ball heights in a 200,000 ball avalanche calculated
as the balls are advected beneath a fixed video camera. The lines
show the ball trajectories from one field to the next.

If the structure of any feature in the flow of size is changing
slowly with respect to
, where is the
mean flow velocity, then we can regard this data as providing a cross
section through the flow in the direction of mean velocity, in this
case down the slope. For Fig. 6 the mean flow
velocity is
thus
corresponds to
. The head of
long,
high
followed by a body
high is visible. The full flow (not
shown in Fig. 6) has a body of approximately constant
height
and length
followed by the tail of
the flow which stretches back to the box and consists of separated
balls.
Figure 7:
Vertical profile of ball downslope ()
velocity. The height and velocity of each datapoint are an
average over 50 balls calculated from video camera measurements of
the front and body.

The shape of the velocity profile in a steady shear flow is governed
by the relative magnitude of the drag forces on the upper and lower
surfaces, the body forces and the vertical transport rate of momentum.
Figure 7 shows that the mean downslope ()
velocity of the balls decreases monotonically with height. There is no
visible velocity reduction at the base indicating that surface
friction is unimportant. The mean velocity slowly decreases in the
dense part of the flow by around
and then very rapidly
in the less dense top layer by a further 
. This
diffuse, top layer of saltating balls moving along
approximately parabolic trajectories is visible in
Fig. 4 and Fig. 6 and has been
discussed in the literature [Johnson et al., 1990].
This behaviour is characteristic of dilute energetic flows. In
high density flows, on the other hand, the top surface is well defined
to within a particle diameter. The lower mean velocities of these
saltating balls is easily explained by the extra airdrag they
experience since they move in regions of higher relative air velocity.
The relative air velocity in the bulk of the flow must be much lower
since the flows are in approximate equilibrium and move up to four
times faster than the terminal velocity of an individual ball.
The central part of the ski jump is composed of downslope pointing
bristles and experiments with individual balls show that it is totally
inelastic  the balls bounce to no observable degree  so that
horizontal momentum cannot be converted to vertical momentum by
collisions with the slope, but only in collisions with other balls.
Since vertical motion will rapidly decay through ground collisions,
a priori, one might have expected a high density flow where the
balls are in continuous contact with very small fluctuation
velocities. This is indeed what happens initially when the balls slump
out of the box. However, this dense flow state is unstable and as the
flow accelerates the velocity fluctuations increase and the density
decreases.
Figure 8:
Ball velocity standard deviation for a 300,000 ball
experiment. (The data have been smoothed.)

The kinetic theory of granular matter follows that of gases and
describes a system by a particle distribution function , where
is the number
of particles with velocity and range that are
centred at and range at time . The number
density
is the integral of over all velocities
and the volume fraction
. The mean value of any particle property
is defined as

(6) 
The mean velocity field
is thus
, the fluctuation velocity
and the second moment of the fluctuation
velocity
. The granular
temperature
is
the isotropic component of . The stress tensor (sometimes referred
to as the pressure tensor [Jenkins & Savage, 1983]) for a granular
flow is

(7) 
where is the ball density, is particle mass and
denotes the collisional transport of
velocity fluctuation. The notation follows [Jenkins & Richman, 1988, Jenkins and Richman
(1988),] where it is shown that in dilute
flows the collisional transport term can be ignored. The
dilute approximation consists of retaining only terms that are
constant or linear with respect to volume fraction and is valid
when the strength of mean shear relative to velocity fluctuations is
small. This approximation is assumed valid for the rest of the paper.
The square root of the diagonal elements of are the velocity
standard deviations along the coordinate axis and are shown in
Fig. 8. The standard deviation is taken over each video
field. This shows that the perpendicular (), and crossslope
() velocity deviations are roughly similar in the head and
the body,
and
respectively.
However, the downslope () is low initially, then increases
rapidly in the head to reach a maximum of
before
decaying to a roughly constant
in the body. The
results are similar for other flows.
Kinetic theories [Lun et al., 1984,Anderson & Jackson, 1992] of
granular matter often postulate that is isotropic, i.e. the
diagonal stresses , and are identical and
the offdiagonal stresses are zero. This is clearly not the case for
these flows. Figure 8 shows that the diagonal elements
of are never equal.
These data are consistent with video footage in which horizontal velocity
structure is visible and with Fig. 7 that shows that
there is no appreciable vertical shearing.
In the case of steady and uniform flow the mean velocity must be constant
and the momentum equation for the flow is

(8) 
where is the air pressure, is gravity and
is the drag force from the air on the
balls [Jenkins, 1987]. For a free surface to be steady
and clearly delineated there is a kinematic constraint that
vanishes on the surface, where
is the surface normal. That is to say as well as the
mean velocity vanishing normal to the surface so must the velocity
fluctuation. This term, if nonzero, would result in a diffusion
outward from the surface of the volume fraction . The top
surface in contrast is diffuse, slowly decreases with , and
such a condition is not satisfied. Figure 4 shows
that the front is indeed very clearly defined which requires that
. Since is calculated from averages over two video
fields the value at the front is not known, but extrapolating the
curve make it plausible that is indeed zero
(Fig. 8).
There is also a dynamical requirement given by Eq. 8 that
the forces on the front should balance.

(9) 
Unfortunately only a dozen balls or so in each video frame can be
identified, which does not provide enough data to calculate and
derivatives, unless averaged over the whole length of the flow.
Assuming that the only dependence of stress is that required to
counteract gravity the
terms can be dropped. The
terms should be zero on the centre line ()
of the flow and can also also be dropped. The equation is then

(10) 
and can be integrated if is known to provide a variant of
Bernoulli's law. This equation will be returned to after a discussion
of the air pressure data.
Pressure Measurements
Figure 9:
Streamlines for irrotational flow around a stationary
sphere.

Figure 10:
Static air pressure change as the front of a 300,000 ball
avalanche is advected past the sensor at height 0.3 m. The balls
reached the sensor at and for the line of best fit
(least mean squares) is drawn assuming the pressure distribution
in front of a sphere (two free parameters effective radius and
velocity ).

Table 1:
Comparison of implied velocities for 150,000() and 300,000() ball avalanches
height (m) 
(m/s) 
(m/s) 

0.01 
7.55 
8.16 
1.04 
0.15 
8.21 
9.66 
0.95 
0.30 
8.88 
10.13 
0.98 
0.45 
6.81 
8.96 
0.85 

Table 2:
Comparison of implied radii for 150,000() and 300,000() ball avalanches
height (m) 
(m) 
(m) 


0.01 
0.76 
0.88 
1.09 
1.02 
0.15 
0.68 
0.80 
1.06 
1.00 
0.30 
0.87 
1.02 
1.08 
1.01 
0.45 
0.98 
1.08 
1.14 
1.07 

Although in the general case of flow past bodies of arbitrary form the
actual flow pattern bears almost no relation to the pattern of
potential flow, for streamlined shapes the flow may differ very
little from potential flow; more precisely, it will be potential flow
except in a thin layer of fluid at the surface of the body and in a
relatively narrow wake behind the body [Landau & Lifschitz, 1987]. In
particular in front of the avalanche head the flow will be
irrotational since the Reynolds number is very high (for length of
, velocity
,
).
A simple approximation is to assume that the flow field is that of
irrotational flow around a sphere (Fig. 9) where the
sphere represents the head of the gravity current (cf.
Fig. 6) in a stationary frame. The flow field has the
required symmetries since it is symmetric about the crossstream
() plane and, if the influence of the ground on the airflow is
assumed to be small, the flow field can be reflected in the
perpendicular () plane.
A similar approach to the ambient flow around gravity currents was
pioneered by [von Kármán, 1940, von Kármán (1940)]. He
considered the local flow around where the head meets the ground and
used this to deduce the head angle (
). This is accurate
over distances small compared to the head height. Similar ideas were
also discussed in [Hampton, 1972, Hampton (1972)], but he
considered the ambient flow around semiinfinite debris flows, thus
his approach is correct over scales large compared to the head height
but small compared to the flow length. In contrast the approach in
this paper is equivalent to retaining the first three terms (up to the
dipole) in a multipole expansion and is therefore asymptotically
correct.
To apply Bernoulli's theorem it is most convenient to work in a frame
in which the flow field can be approximated as steady. This is true in
a frame moving with the same velocity as the avalanche head since the
slope angle changes slowly. The velocity distribution around a
stationary sphere of radius in a flow field moving with constant
velocity at infinity is

(11) 
Using Bernoulli's theorem the corresponding pressure distribution is

(12) 
where
. Regarding the pressure
sensors as fixed on the centre line, the data is not of sufficiently
high quality to warrant a more complicated approach, then
, where is taken as the time
when the front reaches the sensor. The output from a pressure sensor
is then

(13) 
Figure 10 shows the result of fitting this curve to
the data from one of the sensors. The equation has three free
parameters: the impact time, which is taken as the point of highest
pressure, the effective radius , and the effective velocity .
The pressure data was sampled at
and passed through a
width Gaussian filter, was taken as
and an additional correction was applied after
calibrating the sensors in the wind tunnel.
The velocities implied by the pressure data are shown in
table 1. The lower three sensors are all in rough
agreement with the velocity increasing slightly with height. The
difference between these velocities and the video derived head
velocity (of order
) is the penetration velocity of the
air into the head. Unsurprisingly this decreases with height as the
air flows over the avalanche rather than into it. The flow velocities
from the top sensor (height 0.45m) are low because it is largely
out of the flow in a region of reduced air velocity. The third column
of table 1 compares the air velocities with scaling
Eq. 2. The agreement for the lowest three sensors
in the flow is very good and provides further evidence in favour of
the length scaling hypothesis.
Though the calculated velocities match the scaling law reasonably well
the radii do not (Table 2). A possible explanation
is as follows. The flow field far from the body is that of a dipole
imposed on constant flow. The magnitude of the dipole is the surface
area of the implied sphere times the velocity . Close to
the front however the flow field, to second order, will be more like
that around an ellipsoid (this is the result of expanding the surface
to second order in the coordinates). The equation fit is influenced by
the region of high pressure difference close to the flow front and the
length scale measured here is actually the local radius of curvature.
Thus
. Video footage and pictures of the slope
shows that the flow front is reasonably approximated by the parabola
where
and is the distance from the
centre line. Thus in Fig. 3 it can be seen that
back from the front the flow is
wide. The
measure radius of curvature in the  plane is thus
independent of the flow scale. This
does not necessarily contradict the scaling hypothesis, because this
is a local length scale and the width of the flow is still expected to
scale as . Thus if scales and is constant the
ratio between front radii is
where
the length scaling ratio. When
is close to 1 this can be simplified to

(14) 
thus the scaling exponent is altered to
. To the same order of approximation can be
taken from the flows for either or . The fourth column of
table 2 shows the much better fit obtained with
this analysis and . This is a very tentative
solution and an explanation is still required as to why the front
should have a constant parabolic shape.
The definition of the calculated radius is somewhat arbitrary.
The pressure data could be equivalently fitted to

(15) 
where is a time constant. The radius is then deduced from .
The above analysis took as the air velocity, but a more natural
choice would be to use the velocity of the coordinate frame,
that is the front velocity if this is known. Since this
velocity has the same scaling this would only result in the implied
radii being multiplied by some constant factor, thus the previous
discussion is unaffected.
Air pressure through the front
Figure:
Air pressure through the front at 0.01 m for
300,000 balls. The front reaches the sensor at
.

Within the pingpong ball flow the steadystate mass and momentum
equations for the air flow are
To calculate the pressure inside the flow these equations must be
integrated. A simple approximation for the force valid to lowest
order in is

(18) 
where
an
air drag length scale and is the terminal velocity of an
individual ball (
). An analytic solution is not easy
to find. However, if the streamlines are not diverging (or
converging) too rapidly the continuity equation can be approximated by
over short distances if the
streamlines diverge slowly, where is the penetration
velocity of the air into the front. This is most likely to be true
close to the ground where symmetry suggests there will be a streamline
passing straight through the centre of the flow. Substituting this
into Eq. 17 and integrating along this streamline

(19) 
where is distance along a streamline. Even if the surface of the
flow is sharply defined to within a ball diameter changes more
slowly because it is defined as an average over a volume containing
many balls. Suppose the ball concentration is outside the flow and
constant inside the flow. Then increases linearly from
to over a width of order
. Then
integrating Eq. 19 gives

(20) 
accurate for small . Over longer distances the streamlines will
diverge, as the air flow is deflected up and out of the avalanche, the
velocity will decrease and the pressure will increase (see
Fig. 11.)
The rapid decrease in pressure as the flow front goes past the sensor
that is predicted by this equation is clearly seen in
Fig. 11 (and also Fig. 10).
The total pressure drops by
from
to
. The front velocity is around
so this
corresponds to a distance of
certainly much larger than
thus Eq. 20 is
. From
table 1 the air velocity for this flow at
is
thus
so that
. This value for the volume fraction seems
reasonable and is much lower than the maximum packing fraction thus
justifying the dilute approximation.
The momentum equation for the pingpong balls (Eq. 8) can
now be integrated with the same approximations to give

(21) 
At
(using linearly interpolation)
,
and
and
. These quantities are of the same
order showing that the structure of the head is determined by the
balance between air drag, granular stress and gravity. Further back in
the body of the avalanche is approximately
constant and the pressure varies only slowly. Since the effect of
surface drag appears to be small this implies that the air drag on the
top surface balances gravity.
A more detailed analysis of Eq. 21 is not appropriate for
several reasons. The large fluctuations of the air pressure in the
avalanche imply that the flow is turbulent and make interpretation of
the air pressure data very difficult since the sensor measures a
complicated function of the local velocity and local pressure which
can only be simply understood if the direction of the velocity is
known. The ball velocity data also contains a lot of noise since in a
typical frame only a dozen balls can be identified. Though mean values
of velocity are reasonably accurate derivatives of are much less
so. There is an additional problem that the balls that can be
identified may be very special (perhaps only those with low vertical
velocity have been sampled for example) possibly leading to systematic
errors which have not been estimated. In addition the ball position
measurements were taken one metre to the left of the flow centre and
the location of the pressure measurements. Despite all these
difficulties the data does suggest a number of significant processes
within the avalanches.
Figure 12:
Schematic of the air flow round and through a pingpong
ball avalanche.

Classical work on gravity currents is based on perfectfluid theory
and assumes that the effects of viscosity and mixing of the fluids at
the interface can be ignored [Benjamin, 1968]. A
major result of Benjamin (1968) is that, except when a gravity
current exactly fills half a cavity, energy dissipation must occur
through the formation of a head and turbulent flow behind it.
Extensions to the basic theory include lower boundary
effects [Simpson, 1972] and a mixing region behind the
head [Simpson, 1986], but there is still assumed to be a clear
boundary at the front of the current.
A complete description of the flow field for a mixture of
Newtonian fluids requires only one velocity field. This is
because there can be no relative motion (at a point) between two
fluids since a noslip condition holds everywhere, thus the velocity
fields for each fluid (where they are defined) must be identical. Thus
mixing between fluids is a slow diffusion process and there are often
well defined boundaries. The stability of boundaries is also enhanced
by surface tension. However, when one of the fluids is a noncohesive
granular fluid there is no surface tension and the granular fluid will
generally have a distinct velocity field. This is because although on
the surface of each grain a no slip condition holds, very large
velocity gradients can exist across a narrow boundary layer, thus the
difference between the ambient fluid velocity field and the granular
velocity field averaged over volumes containing a few grains can be
very large. For grains falling in a gravitational field for example
the relative velocity will be of the order of the terminal velocity.
The standard gravity current theory [Benjamin, 1968]
(correctly) assumes a stagnation point at the front of a gravity
current because the velocity of the ambient and the current must be
equal. This need not be true for granular gravity currents and the air
pressure data shows that there is a significant relative velocity over
the width of the head (Fig. 12.) The drag force is
related to the relative velocity so a large difference between these
avalanches and standard gravity currents is that the drag is a body
force over the head of the avalanche rather than a surface force over
the head's front surface. Analysis of the forces in the head of the
avalanches shows that there is an approximate balance of forces on the
balls between gravity, granular stress and air drag, and that surface
friction is negligible. The air drag is balanced by a large,
anisotropic increase in the granular stress and gravity. This
increase is a result of an increase in the downslope fluctuation
velocity which then leads to an increase in vertical and crossslope
fluctuations through collisions. Though a quantatitive balance of the
vertical forces in the head has not been accomplished the granular
stress and the vertical component of the drag are probably both
significant and lead to the height of the flow. Air drag may also
directly enhance vertical velocity fluctuations. Further back in the
body of the avalanche the granular stresses are constant (downslope)
and the height is lower. Since surface drag is negligible the gravity
must be balanced by air drag forces through the top surface. A likely
mechanism for this is momentum transfer by the saltating particles.
During their high trajectories they have time to exchange considerable
horizontal momentum with slowly moving air and when they collide with
the main body this momentum will be almost perfectly transfered. In
effect there is a drag interaction between the main body and the air
flow over the whole height of the saltating balls. Though this has not
been quantified, this mechanism of momentum transfer is most likely
more efficient than the drag on the upper surface of a smooth gravity
current and helps explains why steady flows occur on such steeps
slopes even with such a large relative density (
.)
The air pressure distribution in front of the pingpong ball
avalanches is well approximated by irrotational flow around around a
sphere. This approach could be extended to the flow behind the head by
comparing the data with turbulent wake theory, but this is difficult
because of the complicated interaction of the pressure sensors with
the air flow when the velocity direction is unknown. The implied air
velocities scale as the sixth power of the number of balls in
agreement with dimensional analysis and the scaling for the pingpong
ball velocities. The length scales implied by the air flow are of the
same order of magnitude as the front height, but only obey the scaling
law if the shape of the head is assumed to have a constant curvature
(in the plane of the slope). Kinetic theory calculations show a
quantatitive balance of forces in the head between gravity, granular
stress and air drag.
The authors gratefully thank the many people who came to Miyanomori
for the experiment, which could not have been done without their help.
Furthermore we wish to thank the workshop staff of the Institute of
Low Temperature Science who made the equipment. This work was partly
supported by grantinaid for cooperative research and science from
the Japanese Ministry of Education, Science and Culture. One of the
authors was supported by an EU/JSPS Fellowship.

Air drag length scale 

Ball velocity 

Ball fluctuation velocity 

Ball diameter 

Position and velocity distribution function 

Drag force between balls and air 

Froude number 

Gravity 

Gravity adjusted for buoyancy, friction and slope angle 

Scaling exponent 

Second moment of ball fluctuation velocity 

Length of a feature in the flow 

Length scale of a flow 

Length scale ratio 

mass of an individual ball 

Ballslope friction 

Surface normal 

Ball number density 

Number of balls in an experiment 

Air pressure 

Air pressure at the flow front 

Ball volume density 

Constant ball volume density in the flow 

General function of balls 

Distance from centre of flow sphere 

Radius of sphere 
, 
Radii of curvature 

Air density 

Ball density 

Stress tensor 

Distance along a streamline 

Time 

Time constant 

Granular temperature 

Slope angle 
, , 
Rotations of the principal axes
of from the coordinate axes 

Mean ball velocity 

Speed of the flow front 

Mean speed of the flow 

Air velocity and speed 
, 
Relative velocity and speed between air and flow front 

Width of the ball volume density variation 

Local Cartesian coordinates aligned with the slope 

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