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 550,000 ball flow the front of the flow accelerated approximately linearly with distance until it reached a speed of 18 ms-1 after 65 m, whereas the balls in the tail had a speed of only a few meters per second -- similar to the speed of a single ball. The front velocity was roughly constant for the next 30 m 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, which we call ``eyes'' after [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 [Hutter and Nohguchi, 1996]. In these experiments the particles are around 1 mm in diameter and the flows contain 1,000- 100,000 particles. 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 collapse3
In [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] deduced that the maximum velocities, v, for flows which vary only in the number of balls, N, should scale according to
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 give by
» 10,000 balls are observed to rapidly (in around 10 m) reach a self-similar shape and the flows are many particle diameters thick except in the tail.
The effective gravitational acceleration on the flow is m the friction, g the acceleration due to gravity and 3) this ratio is constant for different sized flows since Nishimura et al., 1998] the front velocity was measured between the k point and the p point (where the slope angle, 36°, 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 the assumption Eq. 3. As expected the error is worse for small flows, since they rapidly spread into single thickness layers with two significant length scales
Figure 5: Front velocities at the k-point for different sized avalanches.
Flow structure and ball velocities
By analysing the video film, [Keller et al., 1998] calculated individual particle positions, and by identifying balls between adjacent video frames, particle velocities. 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 17 ms(» 0.2) only the balls from the top 0.2 m 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.
6 the mean flow is 15 ms-1 thus 0.1 s corresponds to 1.5 m. The head of 1 m long, 0.4 m high followed by a body 0.2 m high is visible. The full flow (not shown in this figure) has a body of approximately constant height 0.2 m and length 10 m 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 (x) velocity. The height and velocity of each data-point are an average over 50 balls calculated from video camera measurements.
7 shows that the mean down-slope (x) 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 1 ms-1 and then very rapidly in the less dense top layer by a further 1- 2 ms-1. 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 low density 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 air-drag they experience since they move in regions of higher relative air velocity.
The slope of the ski jump 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.)
f the volume fraction, C the fluctuation velocity < ... >denotes an average over particles, m is particle mass and Q[C] denotes the collisional transport of velocity fluctuation. The notation follows [Jenkins and Richman, 1988] where it is shown that in dilute flows the collisional transport term can be ignored.
The square root of the diagonal elements of <CC> (i.e. the velocity standard deviation along the coordinate axis) are shown in Fig. 8. The standard deviation is taken over each video frame. This shows that the perpendicular Anderson and Jackson, 1992,Lun et al., 1984] of granular flow generally postulate that <CC> is isotropic, i.e. it is a multiple of the identity matrix. This is clearly not the case for these flows. Not only do the diagonal elements vary the off diagonal elements are large so that the principle axes of <CC> are not aligned with the flow axes.
In the case of steady flow the mean velocity must be constant and the momentum equation for the flow is
Jenkins, 1987].
For a free surface to be steady and clearly delineated there is a kinematic constraint that f. The top surface in contrast is diffuse, fslowly decreases with z, and we do not expect a condition such as this to be satisfied. Figure 4 shows that the front is indeed very clearly defined which requires that 4.
There is also a dynamical requirement given by Eq. 7 that the forces on the front should balance.
Pressure Measurements
Figure: 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 t=0 and for t<0 the line of best fit (least mean squares) is drawn assuming the pressure distribution in front of a sphere (two free parameters effective radius R and velocity u).
Table 1: Comparison of implied velocities for 150,000(v1) and 300,000(v2) ball avalanches
Table 2: Comparison of implied radii for 150,000(r1) and 300,000(r2) ball avalanches
Landau and 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 1 m, velocity 10 ms-1, Re » 106). A simple approximation is to assume that the flow field is the that of irrotational flow around a sphere. Far from the flow this will be asymptotically true since this is equivalent to a dipole expansion for the velocity field. The flow field has the required symmetries since it is symmetric about y=0 plane and, if the influence of the ground on the air-flow is assumed to be small, the flow field can be reflected in the z=0 plane.
To apply Bernouilli'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 the rest frame of the avalanche head since the slope angle changes slowly.
The velocity distribution around a stationary sphere of radius R in a flow field moving with constant velocity u at infinity is
5 then 9 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 R, and the effective velocity u. The pressure data was sampled at 1000 Hz and passed through a 4 ms width Gaussian filter. 1. The lower three sensors are all in rough agreement with the velocity increasing slightly with height. The difference between these velocities and the head velocity (of order 5 ms-1) 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. 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 3 it can be seen that 5 m back from the front the flow is 10 m wide. The measure radius of curvature in the x-y plane is thus l is close to 1 this can be simplified to
2 shows the much fit obtained with this analysis Air pressure through the front
Figure: Air pressure through the front at 0.01 mfor 300,000 balls. the front reaches the sensor at t = 0 s.
f is
16 and integrating along a streamline
f changes more slowly because it is defined as an average over a volume containing many balls. Suppose the ball concentration is 0 outside the flow and f increases linearly from 0 to 17 gives
10.)
The rapid decrease predicted by this equation is clearly seen in Fig. 10 (and also Fig. 9). The total pressure drops by 84 Pa from 0 s to 0.035 s. The front velocity is around 15 ms-1 so this corresponds to a distance of 0.5 mcertainly much larger than w thus Eq. 18 is 1 the air velocity for this flow at 0.01 m is 8 ms-1 thus 7) can now be integrated with the same approximations to give
19 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 <CC> 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 meter to the left of the flow centre and the location of the pressure measurements. Despite all these difficulties the data does suggest at the significant processes within the avalanches.
The air pressure distribution in front of the ping-pong ball avalanches is well approximated by irrotational flow around around a sphere. The implied air velocities scale as the sixth power of the number of balls in agreement with dimensional analysis and the scaling for the ping-pong ball velocities. The implied radii 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).
Analysis of the forces in the head of the avalanches shows that there is an approximate balance of forces between gravity, granular stress, air drag and air stress. Thus the internal motion and complex structure of the avalanches is a result of the induced air flow. The air drag on and in the head of the avalanche 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 cross-slope fluctuations through collisions, thus supporting the height of the flow. Large air drag forces act throughout the width of the head in contrast to most gravity currents where there is negligible inter-penetration of the ambient fluid and the interaction occurs through only surface forces not body forces.
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.
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 grant-in-aid 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.
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Footnotes
- ... Nishimura1
- Institute of Low Temperature Science, University of Hokkaido, North 19 West 8, Kita-Ku, Sapporo 0060-0819, Japan
- ... models2
- for a recent survey of current models see [Harbitz, 1999]
- ... collapse3
- 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 collisions rate diverges, so that a group of particles can come to rest in continuous contact in finite time.)
- ... true4
- Since it is calculated from averages over two video fields the value at the front isn't known, but extrapolating the curve make it plausible that 8)
- ... line5
- The data is not of sufficiently high quality to warrant a more complicated approach.
Jim McElwaine
1999-10-27
Converted to TechWriterPro+ by PFJ, 07-03-2001
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