Ball measurements

A video camera was set pointing perpendicularly down at the slope (Fig. 2) at a height of 0.82 m. Balls closer to the lens of the camera appear larger than those which are further away. Thus the z 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 x and y 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 three-dimensional velocity of a ball (time-averaged over the 1/60 s between video fields.) The necessary camera corrections and detailed method are described in Reference Keller, Ito and NishimuraKeller and others (1998).

Fig. 2.

Schematic of the video camera for measuring ball positions.

Air measurements

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 (Reference Kawada, Nishimura and MaenoKawada and others, 1989; Reference Nishimura, Narita, Maeno and KawadaNishimura and others, 1989). Reference Nishimura and ItoNishimura and Ito (1997) and Reference Nishimura, Nohguchi, Ito and KosugiNishimura and others (1997) have developed the use of pressure measurements (sampling frequency 1 KHz) 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 (Fig. 3). 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

where Δp is the pressure difference, ρa is the air density and v is the air speed parallel to the slope. However, this equation is only valid when the airflow 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 10 000 (wind speed 10 m s–¹, tube diameter 0.01 m), 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 Equation (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 100 m down the slope at heights of 0.01, 0.15, 0.3 and 0.45 m.

Fig. 3.

Schematic of the air-pressure sensors.

Front position

Several video cameras were placed to the side and at the bottom of the slope. As can be seen in Figure 4 the leading edge of the avalanche is clearly visible. The position of the front was measured and used to calculate the front velocity.

Fig. 4.

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.

Front velocity

In Reference Nohguchi, Ozawa and NishimuraNohguchi (1997), granular-flow 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. Reference Nohguchi, Ozawa and NishimuraNohguchi (1997) deduced that the maximum front velocities, u, for flows which vary only in the number of balls, N, should scale according to

In order to derive Equation (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

where V is the volume of all the balls and d is the ball diameter. The implied constant of proportionality in Equation (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 (a;) 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 ≈ 10 000 balls are observed rapidly to reach a self-similar shape in about 10 m, and the flows are many particle diameters thick except in the tail.

The effective gravitational acceleration in the downslope direction of the flow is g* = g(1 − pa/pb) where 9 is the angle of the slope, μ is the coefficient of friction with the slope, g is the acceleration due to gravity, and ρa and _ρb 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 airflow is of the order 106, so air viscosity can be ignored. Under the length-scale assumption in Equation (3) the non-dimensional density ratio _ρaV/(Nm), where m is the mass of a single ball, is constant for different-sized flows since V oc N. Therefore air density ρa need not be further considered as a dimensional variable since it can be substituted by m/L³. The dependence on the box size and ball diameter has already been discussed, which leaves only three variables g*, L and u. Thus the only dimensionless combination that can be formed containing the front velocity is the densiometric Froude number

This must be contrast for different flows, and thus

since L α N¹/³.

In Reference Nishimura, Keller, McElwaine and NohguchiNishimura and others (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 Figure 6 and provides additional justification for Equation (3). As expected, the error is worse for flows with small particle numbers, since they rapidly spread into single ball thickness layers with two significant length scales dN1/2 (width and length) and d (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.

Fig. 6.

Front velocities at the k point for different-sized avalanches.

Flow structure and ball velocities

By analyzing the video film (Reference Keller, Ito and NishimuraKeller and others, 1998), individual particle positions can be calculated, and by identifying balls between adjacent video frames, particle velocities are obtained. Figure 7 shows the perpendicular positions, viewed from the side, 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 (1/60 s). This technique, however, cannot see through ping-pong balls, and in the dense head (volume fraction ≈ 0.2) only the balls from the top 0.2 m can be identified, so there is a blank region in Figure 7, marked “passage of head”, where there is no data.

Fig. 7.

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 I is changing slowly on the time-scale l/u, where u is the mean flow velocity, then we can regard these data as providing a cross-section through the flow in the direction of mean velocity, in this case down the slope. For Figure 7 the mean flow velocity is 15 m s–¹, so 0.1 s corresponds to 1.5 m. The head of 1 m length, 0.4 m height, followed by a body of 0.2 m height is visible. The full flow (not shown in Fig. 7) 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.

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 8 shows that the mean downslope (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 decreases slowly in the dense part of the flow by about 1 m s–¹, and then very rapidly in the less dense top layer by a further 1–2 m s–1. This diffuse top layer of satiating balls moving along approximately parabolic trajectories is visible in Figures 5 and 7 and has been discussed in the literature (Reference Johnson, Nott and JacksonJohnson and others, 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 velocity of these satiating balls is easily explained by the extra air drag 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.

Fig. 8.

Vertical profile of ball downslo_pe (x) velocity. The height and velocity of each data point are an average over 50 balls calculated from video camera measurements of the front and body.

The central part of the ski jump is composed of down-slope 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.

The kinetic theory of granular matter follows that of gases and describes a system by a particle distribution function f, where f(c, x, t) dc dx is the number of particles with velocity c and range dc that are centred at x and range dx at time t. The number density n(x, f) is the integral of / over all velocities and the volume fraction φ(x, t) = n(x, t)πd₃/6. The mean value of any particle property (c,x,t) is defined as

The mean velocity field u(x,i) is thus (c), the fluctuation velocity C = c – u and the second moment of the fluctuation velocity K(x, t) = (CC). The granular temperature T(x, t) l/3(K_XX + Kyy ,) is the isotropic component of K. The stress tensor (sometimes referred to as the pressure tensor (Reference Jenkins and SavageJenkins and Savage, 1983)) for a granular flow is

where ρ_b is the ball density, m is particle mass and (-)[C] denotes the collisional transport of velocity fluctuation. The notation follows Reference Jenkins and RichmanJenkins 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 φ 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 K are the velocity standard deviations along the coordinate axis and are shown in Figure 9. The standard deviation is taken over each video field. This shows that the perpendicular (K_zz), and cross-slope (K_yy) velocity deviations are roughly similar in the head and the body, 1.5 and 0.5 m s∼\ respectively. However, the downslope (K_xx) is low initially, then increases rapidly in the head to reach a maximum of 2.5 ms–1 before decaying to a nearly constant 0.5 ms–1 in the body. The results are similar for other flows.

Fig. 9.

Ball velocity standard deviation for a 300 000 ball experiment. (Data have been smoothed.)

Kinetic theories (Reference Lun, Savage, Jeffrey and ChepurniyLunn and others, 1984; Reference Anderson and JacksonAnderson and Jackson, 1992) of granular matter often postulate that K is isotropic, i.e. the diagonal stresses K_xx, K_yy and K_zz are identical and the off-diagonal stresses are zero. This is clearly not the case for these flows. Figure 9 shows that the diagonal elements of K are never equal. Since K is symmetric and positive definite it can be decomposed into its principal stresses (eigenvalues) and principal axes (eigenvectors). The rotation of the principal axes away from the coordinate axes (θ_X, θ_y, θ_Z) is shown in Figure 10. The large rotation of the principal axes in the head is clearly visible, in contrast to the much smaller rotation in the body. On symmetry grounds one would expect 9_y to be identically zero if the measurements had been taken along the centre line of the flow, but since the measurements were taken slightly to the side the results are not surprising. The large rotations 0_X and 0_y and small rotation 9_Z suggest a shearing motion in the plane of the slope but no vertical shearing. These data are consistent with video footage in which horizontal velocity structure is visible, and with Figure 8 which shows that there is no appreciable vertical shearing.

Fig. 10.

Rotation angles of the principal axes of the granular stress tensor for a 300 000 hall experiment. (Data have been smoothed.)

In the case of steady and uniform flow, the mean velocity must be constant and the momentum equation for the flow is

where p is the air pressure, g is gravity and f is the drag force from the air on the balls (Reference Jenkins, Hermann, Hovi and LudingJenkins, 1987). For a free surface to be steady and clearly delineated there is a kinematic constraint that n-Kn vanishes on the surface, where n 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 non-zero, would result in a diffusion outward from the surface of the volume fraction φ. The top surface, in contrast, is diffuse, φ slowly decreases with z, and such a condition is not satisfied. Figure 5 shows that the front is indeed very clearly defined, which requires that K_xx = 0. Since K is calculated from averages over two video fields, the value at the front is not known, but extrapolating the curve makes it plausible that K_xx is indeed zero (Fig. 9).

There is also a dynamical requirement given by Equation (8) that the forces on the front should balance.

Unfortunately only a dozen balls or so in each video frame can be identified, which does not provide enough data to calculate y and z derivatives unless averaged over the whole length of the flow. Assuming that the only z dependence of stress is that required to counteract gravity, the ∂/∂z terms can be dropped. The ∂/∂y terms should be zero on the centre line (y = 0) of the flow and can also also be dropped. The equation is then

and can be integrated if/, is known to provide a variant of Bernoulli’s law. We will return to this equation after a discussion of the air-pressure data.

Pressure measurements

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 (Reference Landau and LifshitzLandau and Lifshitz, 1987). In particular, in front of the avalanche head the flow will be irrotational since the Reynolds number is very high (for length 1 m and velocity 10 m s–1, one has Re * 106). A simple approximation is to assume that the flow field is that of irrotational flow around a sphere where the sphere represents the head of the gravity current (cf Fig. 7) in a stationary frame. The flow field has the required symmetries since it is symmetric about the cross-stream (y = 0) 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 (z=0) plane.

A similar approach to the ambient flow around gravity currents was pioneered by Reference Von Karmanvon Karman (1940). He considered the local flow around where the head meets the ground and used this to deduce the head angle (60°). This is accurate over distances small compared to the head height. Similar ideas were also discussed in Reference HamptonHampton (1972), but he considered the ambient flow around semi-infinite debris flows, so 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 multi-pole 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 R in a flow field moving with constant velocity v at infinity is

Using Bernoulli’s theorem the corresponding pressure distribution is

where x cos 0 = x • v. Regarding the pressure sensors as fixed on the centre line, the data are not of sufficiently high quality to warrant a more complicated approach, then x(i) = v(R/v – t), where t = 0 is taken as the time when the front reaches the sensor. The output from a pressure sensor is then

Figure 11 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 v. The pressure data were sampled at 1000 Hz and passed through a 4 ms width Gaussian filter, ρa was taken as 1.2 kg m–³ and an additional correction was applied after calibrating the sensors in the wind tunnel.

Fig. 11.

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 v).

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 5 m s–¹) is the penetration velocity of the air into the head. Not surprisingly, this decreases with height as the air flows over the avalanche rather than into it. The flow velocities from the top sensor (height 0.45 m) 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 the scaling Equation (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.

Table 1.

Comparison of implied velocities for 150 000(v₁) and 300 000(v²) ball avalanches

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 π R2u. 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 1/R = l/R, + 1/R². Video footage and pictures of the slope show that the flow front is reasonably approximated by the parabola y = x²/d where d = 5 m and y is the distance from the centre line. Thus in Figure 4 it can be seen that 5 m back from the front the flow is 10 m wide. The measured radius of curvature in the x-y plane is thus R² = d/2 = 2.5 m 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 N¹/3. Thus if R1 scales and R² is constant the ratio between the front radii is

where A = (N’/N)¹/³ is the length-scaling ratio. When A is close to 1 this can be simplified to

and thus the scaling exponent 7 = 1/3 is altered to γ = γ/(l + R¹/R²). To the same order of approximation, R¹ can be taken from the flows for either N or N’. The fourth column of Table 2 shows the excellent fit obtained with this analysis for R²/R¹ = 2.5 and γ = 5/21.

Table 2.

Comparison of implied radii for 150 000(R¹) and 300 000(R²) ball avalanches

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

where t₀ is a time constant. The radius is then deduced from R = t₀v. The above analysis took v as the air velocity, but a more natural choice would be to use the velocity of the coordinate frame, i.e. the front velocity u 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, so the previous discussion is unaffected.

Air pressure through the front

Within the ping-pong ball flow the steady-state mass and momentum equations for the airflow are

To calculate the pressure inside the flow these equations must be integrated. A simple approximation for the force valid to lowest order in φ is

where is an air-drag length scale and u_T is the terminal velocity of an individual ball (7.5 m s-1). 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 (1 – φ) v = v0 over short distances if the streamlines diverge slowly where v0 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 Equation (17) and integrating along this streamline yields

where s is the distance along a streamline. Even if the surface of the flow is sharply defined to within a ball diameter, φ 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 φ_c = const, inside the flow. Then 0 increases linearly from 0 to φ_c over a width w of order Integrating Equation (19) gives

accurate for small s. Over longer distances the streamlines will diverge as the airflow is deflected up and out of the avalanche, the velocity will decrease and the pressure will increase (see Fig. 12).

Fig. 12.

Air pressure through the front at 0.01m for 300 000 balls. The front reaches the sensor at t=0s.

The rapid decrease in pressure as the flow front goes past the sensor that is predicted by this equation is clearly seen in Figure 12 (and also Fig. 11). The total pressure drops by 84 Pa from 0 s to 0.035 s. The front velocity is around 15 m s^–1, so this corresponds to a distance of 0.5 m, certainly much larger than w, and Equation (20) is . From Table 1 the air velocity for this flow at 0.01 m is 8 ms–¹, so v₀ = 1 m s-¹ and φ_c ≈ 0.2. This value for the volume fraction seems reasonable and is much lower than the maximum packing fraction, thus justifying the dilute approximation.

The x component of the momentum equation for the ping-pong balls Equation (8) can now be integrated with the same approximations to give

At t = 0.035 s (using linear interpolation) K_xx = 1.9 m² s–², and . These quantities are all 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, K_xx 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 Equation (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 be simply understood only if the direction of the velocity is known. The ball velocity data also contain a lot of noise since in a typical frame only a dozen balls can be identified. Though mean values of the velocity are reasonably accurate, derivatives of K are much less so. There is an additional problem that the balls that were identified might 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 lm to the left of the flow centre and the location of the pressure measurements. Despite all these difficulties, the data do suggest a number of significant processes within the avalanches.