Segregation Effects in Granular Collapses

Transcription

1 Segregation Effects in Granular Collapses Joshua S. Caplan, Saran Tunyasuvunakool, Stuart B. Dalziel & Nathalie M. Vriend Department of Applied Mathematics and Theoretical Physics University of Cambridge, Cambridge, United Kingdom ABSTRACT: We present the results of an investigation into the collapse of a cylindrical column comprised of two distinct granular media over a horizontal bed. The columns are constructed from two layers of particles with different properties. Each layer contained either small spherical ballotini or large ellipsoidal barley grains. We find that there are significant differences in the flow dynamics between the two possible column configurations and between different layer height ratios. Despite these differences, the run-out distances still follow power laws in the initial aspect ratio, with exponents consistent with previous work on monodisperse columns. Granular segregation is observed in collapses with an initially unstable stratification, leading to differences in the final deposit structure. In particular, we observe a instability in the flow front due to the different frictional properties of the two media. 1 INTRODUCTION Granular materials are ubiquitous in both natural and industrial processes and exhibit a wide range of complex phenomena. Despite much recent research, they remain poorly understood and general constitutive laws are still unknown. One system that has been of particular interest is the collapse of a column of a granular material under its own weight, as illustrated in Figure 1. This problem was first investigated by Lube et al. (004) and Lajeunesse et al. (004) who independently considered the case of a monodisperse column and found that the key variable was the aspect ratio, a h i /r i, of the initial column rather than the mass (or volume) of material. Both papers identified two main regimes depending on whether the initial aspect ratio was greater or smaller than a critical value. Lube et al. found power laws for the non-dimensional runout distance r r i r i = { 1.4a a < a 1/ a > 0.74, (1) while Lajeunesse et al. found similar semi-empirical laws. In contrast, later work by Warnett et al. (013) found that a 0. was a better fit. Neither Lajeunesse et al. nor Lube et al. found that the choice of particles had any significant effects on the flow, but Warnett et al. found a weak dependence on particle size. These axisymmetric experiments have also been extended to quasi-two-dimensional collapses along h i r i h r Figure 1: Illustration of the column collapse problem. rectangular channels (Lube et al. 005, Balmforth and Kerswell 005, Lajeunesse et al. 005, Lube et al. 007). As with axisymmetric collapses, two regimes were found depending on the aspect ratio, but the large a regime followed a different power law that, due to the effects of friction at the walls, depended on the width of the channel. A semi-circular geometry was also considered by (Lajeunesse et al. 005), which behaved the same as the axisymmetric case but allowed more of the internal structure of the collapses to be observed. Various systems involving sloped beds have also been considered (Hogg 007, Mangeney et al. 010, Lube et al. 011, Maeno et al. 013). Collapses have also been successfully modelled numerically using discrete element methods (DEM) (Staron & Hinch 005, Staron & Hinch 00, Zenit 005), using shallow-water-like systems of equations (Mangeney-Castelnau 005, Kerswell 005, Larrieu et al. 00, Doyle et al. 007, Hungr 008), and using the non-newtonian µ(i) rheology (Lagrée et al. 011). All of this previous work has considered only the case on a monodisperse medium whereas real granular media are typically polydisperse. This allows the potential for granular segregation with larger

2 and less dense particles rising to the surface of the flow, and smaller and denser particles sinking to the base (Drahun & Bridgwater 1983). In addition, differences in the particle shape can lead to segregationmobility feedback and dramatic effects on the flow (e.g. Pouliquen et al. 1997). In this paper we investigate the effects of segregation on the column collapse problem by considering the collapse of a variety of bidisperse columns. EXPERIMENTAL METHODOLOGY A photograph of the experimental set-up can be seen in figure. In all experiments we used spherical ballotini of diameters between 0.3 mm and 0.4 mm, and ellipsoidal pearl barley grains of approximate size mm 3 mm mm. The materials were poured into an acrylic cylinder of inner radius 47 mm in either a stably stratified (a layer of barley grains above a layer of ballotini) or unstably stratified (ballotini above barley) configuration, as shown in figure 3. The cylinder was then retracted using a pneumatic lift at a speed of approximately 0.7 m s 1 and the column allowed to collapse over a smooth, horizontal PVC sheet. Figure : Photograph of the experimental apparatus. The collapses were filmed using a 1 megapixel high-speed camera at up to 5400 frames per second and still images were taken of the final deposits. A custom perspective correction and circle fitting program was used to calculate the radii of the deposits. Two series of experiments were performed, each considering both column configurations. Series A varied the initial height from 0 mm to 0 mm (0.43 a 4.7), whilst fixing the layer height ratio at Π hballotini /hi = 0.5. For comparison, this series was repeated with columns purely comprised of ballotini, i.e. Π = 1. Series B instead considered columns of fixed initial height hi = 300 mm (a =.4) and varied the layer height ratio of the two layers with 0 Π 1. (a) Stable (b) Unstable Figure 3: Example columns in the two initial configurations considered. Stability is with respect to segregation effects at the interface. 3 RESULTS 3.1 Series A Variations in aspect ratio For stably stratified columns we find similar results to monodisperse collapses. For small aspect ratios (a. 1.1), the collapse begins at the bottom edge with an outward spreading ballotini front. The barley layer then collapses giving an inward moving front separating the undisturbed central region from the collapsing material. When a this fails to reach the centre, leaving a conical frustum, otherwise the deposit is conical. For larger aspect ratios (a & 1.1), a steep central cone remains, surrounded by a wide, thin region. In all cases the barley remains above the ballotini everywhere in the collapse. In contrast, unstably stratified columns behave significantly differently. For smaller aspect ratios we still see the same conical frusta and cones, but a few barley grains are ejected from the main deposit to give a detached ring. For larger aspect ratios, a continuous ring of barley is pushed outwards as can be seen in figure 4(b). At the end of the collapse this can separate from the main deposit. This barley front is unstable, leading to a loss of axisymmetry as seen in figure 5. Towards the edge of the collapse, a small amount of barley grains can be seen at the surface of the ballotini, showing that a limited amount of segregation occurred. 3. Series B Variations in layer height ratio The series B collapses behave similarly to the tallest aspect ratio collapses in series A for most values of Π. The exception is for the stably stratified collapses with Π. 0. where the upper barley layer was able to completely cover the ballotini layer. As with the series A unstably stratified columns, the front then became unstable, leading to asymmetries and the possibility

3 Figure : Deposit from the collapse of a stably stratified column with Π = 0.1 and a =.4 showing asymmetry and detached ring of barley grains. (a) Stable (a = 4.7) (b) Unstable (a = 4.5) Figure 4: Typical collapses of high aspect ratio columns in either configuration. Images are taken at 0.1 s intervals. due to the differing initial potential energies of the columns. To compensate, we can replace the initial aspect ratio with the initial aspect ratio of a column with the same initial potential energy but composed purely of the bottom layer, i.e. we replace a by the rescaled aspect ratio r ρu a a ΠL + (1 ΠL ), ρl (a) a =.3 (b) a =.8 (c) a = 3. (d) a = 4.0 Figure 5: Example of asymmetric deposits in the collapse of unstably stratified columns with Π = 0.5. of a detached barley region, as seen in figure. With the unstably stratified columns we see the same asymmetries as in the series A experiments. We also find a detached region of barley for 0.7. Π < 1, similar to those found in series A with lower aspect ratios and Π = 0.5. Unlike the stably stratified columns, the top layer never overran the bottom layer. 3.3 Run-out distances Figure 7(a) shows the non-dimensional run-out distances r against initial aspect ratio a for the series A collapse. Due to the presence of detached regions of barley after the collapse of an unstably stratified column, a distinction has been made between the radius of the deposit excluding the ring (inner) and including it (outer). Although the curves follow the same general trends, they have different run-out distances () where ρu and ρl are, respectively, the densities of the upper and lower layers, and where ΠL is the ratio of the height of the lower layer to the total column height. This data is shown in figure 7(b), and makes the run-outs for the monodisperse columns, the stably stratified columns, and inner run-out distance for the unstably stratified columns collapse onto a single curve. The data is then well-fitted by the power laws ( a a < ac, (3) r = 1/3 /3 ac a a > ac with ac = 0.95, which is the solid line on the graph. The outer run-out distances for unstably stratified columns, however, does not fall on this curve. In addition, figure 7(b) also includes the data for the series B collapses with the same rescaling of aspect ratio. This rescaling converts variations in Π into variations in a, but is not sufficient to explain the differences in run-out. These differences in run-out can be more clearly seen in figure 8 which shows the variation in run-out as the height ratio changes at fixed aspect ratio. Stably stratified columns show a significant increase in runout with a small ballotini layer (Π 0.1) compared with either the pure barley case or with the predicted values from the series A data. As the amount of ballotini increases, the run-out drops back down, before slowly increasing to a second maximum at Π = 1. For the larger height ratios (Π & 0.7), the predicted runouts are very accurate. Unstably stratified columns, however, only have a single maximum at Π 0.8. In this case the predicted values are a poor fit for the data.

4 Monodisperse Stable Unstable (inside) Unstable (outside) a A Monodisperse 1 A Stable 0.8 A Unstable (inside) A Unstable (outside) 0. B Stable B Unstable ã (a) Non-dimensional run-out distance against initial aspect ratio a for the series A collapses. (b) Non-dimensional run-out distance against rescaled initial aspect ratio ã for both series of collapses. Figure 7: Log-log graphs of run-out distances Stable Unstable Stable (predicted) Unstable (predicted) Π Figure 8: Non-dimensional run-out distances against layer height ratio Π for series B experiments. Lines are predicted runouts using the power laws in equation (3) after rescaling the aspect ratio using equation (). Error bars show range of run-outs measured over multiple collapse at each layer height ratio. 4 CONCLUSIONS In this paper we have considered the collapse of a bidisperse granular column over a horizontal base. We have shown that stably stratified columns follow similar behaviour to monodisperse columns as the aspect ratio changes. In contrast, in unstably stratified columns a ring of barley grains slides outwards from the deposit causing a significantly larger deposit radius. We also found the ratio of the two media was significant in determining the run-out distance and the final deposit structure. Our first key finding is that segregation does not appear to be a major factor in the collapse dynamics. While some segregation is, as expected, observed in unstably stratified collapses, it is incomplete and only present at the thin edge of the deposit. This is likely due to the very short duration of the collapses (typically less than 1 s) and due to lack of shearing in the flow. While we do find that the initial column configuration is important, much of the differences in run-out can be compensated for by rescaling the aspect ratio using equation (). This suggests that the collapse is driven by the falling column providing a source of energy and matter which allows the front to spread outwards. This is supported by the numerical work of Larrieu et al. (00) who replaced the collapsing column by a volume flux at the centre and were able to reproduce previous experimental observations even at high aspect ratios. The rescaled aspect ratio does not explain all of the behaviour and so we must also consider the flow front. For all unstably stratified columns and, to a lesser extent, for low Π, stably stratified columns, the flow front is composed primarily of barley grains being pushed outwards by ballotini. As the collapses begin to decelerate, the barley grains slide away from the ballotini and spread out over a wide area. This causes an increase in the run-out distance and is the cause of the detached rings of barley observed for certain values of a and Π. When the front edge is composed of ballotini, some ballotini does escape the collapse edge, but it is not enough to be measured as an increased run-out. This sliding process is also likely the cause of the asymmetries observed in those collapses where it is present. The difference in frictional behaviour of the two materials means that variations in the amount of barley at different points in the flow front can affect the distance by which it spreads. This is reminiscent

5 of the fingering instability discovered by Pouliquen et al. (1997), where the build-up of larger, rougher particles at the front of an avalanche causes the front to break up into fingers. In the axisymmetric collapse problem, however, the short time scale of the collapses and the geometry mean that such an instability is, as observed, unlikely to be able to progress further than a small deformation of the edge. The work described in this paper has the significant limitation in that it involves particles with differences in size, density and shape. Our work suggests that, of these differences, the size difference is least important. Size differences are likely the cause of the little segregation observed, but this does not appear to have affected the flow. The success of the rescaled aspect ratio ã suggests that the primary effect of density variations is to the change the column s initial potential energy. The effects of the shape differences are most exciting with dramatic, qualitative differences in the flow and deposit structure. Future work should look to decouple these three effects by careful choice of the particles used. In addition, the lack of segregation means that the initial positions of the different types of particle is the cause of much of the differences between collapses. By considering mixed columns a better understanding of this could be obtained. Mixed columns would also allow segregation to occur throughout the column, rather than being restricted to the interface between the layers, and hence may cause it to have more significant effects. The importance of sliding in the collapse dynamics suggests that base roughness is likely to be significant in this experimental set-up. By using a rougher base barley grain sliding could be inhibited with effects on the flow and final deposits. ACKNOWLEDGEMENTS We would like to thank J. N. McElwaine, E. J. Hinch and J. A. Neufeld for their insights and discussions. We would also like to give particular thanks to D. Page-Croft and the rest of the GKB Laboratory technicians for constructing and maintaining the apparatus used in these experiments. JSC was supported by an EPSRC PhD studentship and NMV was supported by a NERC Postdoctoral Research Fellowship NE/I01047/1. Hogg, A. J. (007). Two-dimensional granular slumps down slopes. Physics of Fluids 19(9), Hungr, O. (008, August). Simplified models of spreading flow of dry granular material. Canadian Geotechnical Journal 45(8), Kerswell, R. R. (005). Dam break with Coulomb friction: A model for granular slumping? Physics of Fluids 17(5), Lagrée, P.-Y., L. Staron, & S. Popinet (011, September). The granular column collapse as a continuum: validity of a two-dimensional Navier Stokes model with a µ(i)-rheology. Journal of Fluid Mechanics 8, Lajeunesse, E., A. Mangeney-Castelnau, & J. P. Vilotte (004). Spreading of a granular mass on a horizontal plane. 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