Experiments and Simulations of a Gravitational Granular Flow Instability

Jan Ludvig Vinningland,1,∗ Øistein Johnsen,1 Eirik G. Flekkøy,1 Renaud Toussaint,2 and Knut Jørgen Måløy1

1Department of Physics, University of Oslo, P. 0. Box 1048, 0316 Oslo, Norway2Institut de Physique du Globe de Strasbourg, CNRS, Université Louis Pasteur, 5 rue Descartes, 67084 Strasbourg Cedex, France

(Dated: November 1, 2006)

A gravitational instability is observed, numerically and experimentally, as a column of dense granular mate-rial positioned above a gap of air falls in a gravitational field. A characteristic pattern of fingers emerges alongthe interface defined by the grains, and a transient coarsening of the structure is caused by a coalescence ofneighboring fingers. The coarsening is limited by the production of new fingers as the separation of the ex-isting fingers reaches a certain distance. The experiments and simulations are shown to be comparable, bothqualitatively and quantitatively. The characteristic inverse length scale of the structures, calculated as the meanof the solid fraction power spectrum, relaxes towards a stable value shared by the numerical and experimentaldata. Investigations are also performed on the sensitivityto changes in various control parameters of the system,such as the density of the grains, the shape of the initial interface, and the dissipation in the system. Increasingthe density makes the system evolve faster towards breakthrough, changing the interface only affects the initialstructures, and increasing the dissipation only has an effect above a certain value of the restitution coefficient.By means of the Fourier transform the growth of the wave numbers that constitute the power spectrum of thebulk solid fraction and the interface are analyzed. The results show that the initial evolution of the instability ischaracterized by an approximate exponential growth which subsequently breaks down as non-linear processestake over and govern the final dominating wave number. This granular instability is therefore an initial, transientreminiscence of the Rayleigh-Taylor instability. However, the later stages of this granular instability displayprocesses that are not observed in its hydrodynamic analog.

PACS numbers: 81.05.Rm, 47.20.Ma, 47.11.-j, 45.70.QjKeywords: granular flow, Rayleigh-Taylor instability, pattern formation, simulation, dissipation

I. INTRODUCTION

Sands and powders are indispensable materials in our mod-ern society but, despite their widespread use, a complete the-oretical description of granular systems and their dynamicalproperties are still not given. An improved understanding ofgranular flows would provide valuable contributions to manyindustrial applications such as pneumatic transport, fluidizedbeds, catalytic cracking [1–3]. Granular materials are also in-volved in a host of intriguing natural, geological phenomena,most of which are still in need of a proper understanding, ase.g. sedimentation [4, 5], erosion and river evolution [6],un-derwater avalanches and turbidites [7], and soil fluidizationduring earthquakes [8].

The relatively complete description of classical fluid dy-namics is also useful in the interpretation of granular flows.Instability is a central problem in fluid dynamics [9, 10], andover the past few years some classical hydrodynamic insta-bilities have also been reported in granular systems. Exam-ples of such are the Saffman-Taylor instability [11] in a gran-ular suspension [12] or in gas-grain mixtures [13], the Kelvin-Helmholtz instability [14] in a vibrated granular mixture [15],the Richtmyer-Meshkov instability in a shock propagating atthe interface between two types of grains [16], as well as somenovel instabilities in submarine avalanches [17] and in a tubeof sand [18, 19].

While we study a gas-grain system, the liquid-grain inter-

∗Electronic address:janlv@fys.uio.no

face in a Hele-Shaw cell was investigated experimentally andtheoretically by Völtz et al. [20]. Sieved glass beads of 80µm in diameter was used in their experiments, and a behav-ior very similar to the classical Rayleigh-Taylor instability[10, 21] was observed. In contrast, the instability discussedin this paper arises in a dry granular material and displaysprocesses of tip splitting and renucleation of fingers. Our ex-perimental and numerical data compares favorably both qual-itatively and quantitatively, despite the two-dimensionality ofthe model and the neglection of interparticle friction and gasinertia. Linear stability analysis is not adequate to predict thefinal dominating wavelength because non-linear effects takeover and govern the final wavelength. Nevertheless, a linearstability analysis of the initial evolution of the instability isprovided, revealing a transient linearity in the development ofthe structures. The linearity breaks down as the non-lineartip-splitting and finger nucleation processes start to dominate.The parameter space of the model is explored by changing thedensity of the grains, the shape of the initial granular interface,and the granular dissipation.

The motivation for the simulations and experiments pre-sented in this paper is to study a granular version of theRayleigh-Taylor instability [21] known from hydrodynamicswhere a dense Newtonian liquid is positioned on top of a lessdense liquid in a gravitational field. In our case, the densefluid is replaced by a granular material which, in contrast tothe liquid, is not influenced by surface tension.

The simulation of two-phase flow is particularly interestingfor many engineering purposes, and historically most simu-lations on two-phase flows have treated the solid phase as acontinuum, which allowed numerical techniques known fromfluid dynamics to be applied. A continuum approach to gran-

ular media (see e.g. Ref. [22]) is however only approximate,and will in cases of discontinuous density variations breakdown completely. Many interesting phenomena in granularmedia are in addition closely related to its particularity.Acomplete description of the interactions between a continuumfluid phase and a discrete solid phase would require the fullNavier-Stokes equation for the fluid, coupled with movingboundary conditions given by the surfaces of the grains andthe geometry of the container, together with differential equa-tions for the grains. Such a detailed scheme consumes pro-hibitive amounts of computational resources, and a numberof techniques (see Ref. [23] for a brief summary) have beendeveloped in order to reduce the computational efforts whilepreserving the physics. In contrast, we use a hybrid techniquethat affords a continuum description of the gas phase and aparticulate description of the granular phase. Our model ne-glects friction, one space coordinate, the inertia of the air,and is able to simulate large systems consuming moderateamounts of computer resources.

The paper is organized as follows. The following sectionpresents the setup and execution of the experiment, and anoutline of the numerical model and implementation are givenin Sec. III. The numerical and experimental results are pre-sented in Sec. IV together with qualitative and quantitativecomparisons. The numerical results are further analyzed bymeans of the Fourier power spectra of the bulk solid fractionand the interface. The response of the instability to changes inthe controlling parameters is presented at the end of Sec. IV.The paper is completed with a short conclusion in Sec. V.

II. EXPERIMENT

The experimental setup, illustrated in Fig. 1, consists of aclosed Hele-Shaw cell made of two glass plates held togetherby clamps and separated by a 1 mm thick silicone frame. Theinternal dimension of the cell, defined by the silicone frame,is 56 mm by 86 mm, and it is filled with polystyrene beadsand air at atmospheric pressure. The cell pivots on a hingedbar and rotates 130 degrees from a lower to an upper verticalposition, i.e. from A to B in Fig. 1, where it is stopped by a bar.The rotation is performed manually in about 0.2 seconds andthe off-center pivot cause the rotation to slow down the fallingmotion of the grains due to centrifugal forces. Images witha resolution of 512x512 pixels are recorded at a rate of 500frames per second by a high speed digital camera (PhotronFastcam-APX 120K) connected to a computer.

The experiments are performed using Dynoseeds TS140-51 from Microbeads which are monodisperse polystyrenebeads of 140 µm in diameter. The filling of the cell is per-formed horizontally with the upper plate removed and the sil-icone frame adhered to the bottom plate. Small portions ofbeads are carefully deposited on the plate and leveled with thesilicone frame before the upper plate is attached. The cell isflipped a few times after it is closed to allow the grains to forma random loose packed configuration.

The humidity in the lab is important in order to keep theelectrostatic and cohesive properties of the grains at a suitable

high speedcamera

A

B

mµ

Side view

56 mm

hinge

glass plate

silicone

1mm

rotating bar

stopping bar

86 mm

140polystyrenebeads

Front view

clamp

film

Figure 1: Front and side view of the experimental setup. Two cellpositions are superimposed in the side view to illustrate the manualrotation from position A to B.

level to prevent the grains from clustering or sticking to theglass plates. During the filling of the cell and the executionofthe experiment the humidity was kept constant at about 30%.

The experimental results are presented and discussed inSec. IV.

III. SIMULATION

A. Model

The numerical model is a hybrid model that combines a dis-crete description of the grains with a continuum description ofthe gas. The foundation and derivation of the model are pre-sented in detail in Ref. [24, 25], only an outline of the modelis given here.

The granular phase is considered as an agglomeration ofspheres that make up a deformable porous medium describedby coarse grained solid fractionρ(x, y) and velocity fieldsu(x, y), where(x, y) are the two dimensional space coordi-nates. The continuum gas phase is solely described by itspressure fieldP (x, y), hence the inertia and velocity field ofthe gas are not considered in this model. The focus of themodel is to describe fluid flow structures on a scale of a fewgrain diameters, not the flow field on a sub-granular level. Itis justified to neglect the inertia of the fluid as long as theReynolds number is small, which is the case for particles on amicrometer scale sedimenting in air.

The pressure is governed by the equation [24, 25]

φ(((∂P

∂t+ u · ∇P

)))

= ∇ ·(((

Pκ

µ∇P

)))

− P∇ · u , (1)

whereφ = 1−ρ is the porosity,ρ is the solid fraction,κ is thepermeability, andu is the velocity field of the granular phase.

Figure 2: Example of a force network showing the distribution of theinterparticle forceFI among the grains in contact. This is the lowerright corner of a frictionless packing of 1200 grains relaxed undergravity.

P is the gas pressure, andµ is the gas viscosity. This equa-tion is derived from the conservation of air and grain mass,with Darcy’s law [26] for the pressure drop over a volumewith permeabilityκ, the Carman-Kozeny relation [27] for thepermeability, and the isothermal ideal gas law. The empiricalCarman-Kozeny relation is a function of the local solid frac-tion ρ and the grain diameterd

κ(ρ, d) =d2

180

(1 − ρ)3

ρ2. (2)

The dynamics of the grains are governed by Newtons sec-ond law

mdv

dt= mg + FI −

∇P

ρn

, (3)

wherem is the grain mass,v is the grain velocity,FI is theinterparticle force which keeps the grains from overlapping,ρn = ρ ρg/m is the number density, andρg is the mass densityof the material the grains are made of. Contact dynamics [28]is used to calculate the interparticle forceFI, but moleculardynamics techniques [29, 30] could have been used instead.Contact dynamics is an iterative scheme that calculates theinterparticle force distribution consistent with the kinematicconstraints imposed by the impenetrability of the contactinggrains. The iteration stops when the update of the forces be-tween two iterative steps is less than a given value. The re-sulting force network shows how the weight of the grains, andpossible external forces such as the gas pressure, are trans-mitted and carried by the grains in contact and the walls ofthe container. Detailed descriptions of contact dynamics arefound in Refs. [31–33]. Fig. 2 gives an example a force net-work obtained in a frictionless packing of 1200 grains relaxedunder gravity. The direction and magnitude of the forces aregiven by the direction and thickness of the lines. Typical fea-tures of a force network are the heterogeneous distributionofforces and the existence of force chains that transmit a majorfraction of the load.

The interparticle and wall-particle friction are not takenintoaccount in this model. However, there is still dissipation in the

Figure 3: Illustration of the linear smoothing function used to calcu-late the continuous fields from the discrete positions and velocities ofthe grains. The red squares are nodes on the grid, and the gray-levelbackground illustrates the weight map of the central node; black isone, white is zero.

model, governed by a coefficient of restitution that determinethe loss of momentum associated with each impact.

B. Implementation

A smoothing function is necessary to translate the massmi

and velocityvi of the grains, indexed byi, into continuoussolid fractionρ(x, y) and velocity fieldsu(x, y) on the grid.The fractional value ofmi or vi assigned to each of the fourneighboring grid nodes is determined by the positional weightof the grain, a linear smoothing functions(r − r0) expressedmathematically as [24]

s(r − r0) =

(

1 −∆x

l

)(

1 −∆y

l

)

if ∆x, ∆y < l

0 otherwise ,(4)

wherer(x, y) is the position of the grain,r0(x0, y0) is the po-sition of the grid node,∆x = |x − x0| and∆y = |y − y0| arethe relative distances,l is the lattice constant, and

∑

k s(r −rk) = 1 with k indexing the four sites. The smoothing ofthe grains is illustrated in Fig. 3 where nine grid nodes aredepicted as red squares on a gradient background which isthe positional weight map associated with the central node.Hence, all the grains in Fig. 3 contributes to the field val-ues of this node. A grain positioned on the central node isassigned a weight of one (black), while a grain positionedalong the dotted outline in Fig. 3 is assigned a weight ofzero (white). The weight map, or smoothing function, maybe considered as a collection of equidistant tent functions,distributed over the area in Fig. 3, with maxima rising lin-early from zero to one and back to zero. More explicitly, thesolid fractionρ and average velocityu associated with a nodeat positionr0 are respectivelyρ(r0) =

∑

i s(ri − r0), andu(r0) =

∑

i vis(ri − r0), wherei runs over the number ofparticles [24].

The positional weight used in the calculation of the contin-uous fields is also used when the pressure force evaluated atthe nodes,FP = −∇P/ρn, is distributed back to the grains.

t = 0.00 s t = 0.09 s t = 0.19 s t = 0.28 s t = 0.38 s t = 0.47 s t = 0.56 s t =0.66 s

Figure 4: Images from the experiment with polystyrene beadsof 140 µm in diameter in a closed Hele-Shaw cell that is 56 mm wide, 86 mmhigh, and 1 mm thick. The time of the snapshots is the time evolved since the cell reached the upright position. The experimental setup isillustrated in Fig. 1.

t = 0.00 s t = 0.09 s t = 0.19 s t = 0.28 s t = 0.38 s t = 0.47 s t = 0.56 s t =0.66 s

Figure 5: A series of snapshots from the simulation using polystyrene beads of 140 µm in diameter interacting with air in acell that is 56 mmwide and 68 mm high. The restitution coefficient is 0.5.

Explicitly for a grain atri, FP = −∑

k(∇P/ρn)k s(ri−rk)with a k-index running over the four nodes. In other words,the share ofFP assigned to the grain by the node is equal tothe share ofmi or vi assigned to the node by the grain.

A few approximations are made in the implementation ofthe model. The 2D solid fraction of disks is translated intoa 3D solid fraction of spheres by multiplying the 2D solidfraction with 2/3, which is the ratio of the 3D to the 2Drandom close packed solid fractions. Further, the Carman-Kozeny relation is not valid as the solid fraction approacheszero, and a lower cutoff on the 2D solid fraction,ρmin = 0.25,is introduced. These matters are elaborated in Sec. II C ofRef. [24, 25].

The packing of grains used in the simulations is made of160000 grains with a flat size distribution and a±10% vari-ation in the diameter, i.e. a packing of grains with a meandiameter of 140 µm contains grains that vary from 126 µmto 154 µm in diameter. This minor polydispersity is neces-sary to avoid a hexagonal stacking of the grains. A packing isgenerated by raining grains from a given height, with randomhorizontal positions, in a cell without air. After all the grainshave settled the packing is allowed to further compactify andrelax before the air is introduced. This is to prevent the sourceterm of Eq. (1),−P∇ · u, to introduce local pockets of over-pressure, due to moving grains, that otherwise could perturbthe numerical results.

IV. RESULTS

The rich behavior of the instability issues a number of in-teresting results presented as follows. A qualitative discussion

and comparison of the experimental and numerical images aregiven first, followed by solid fraction and pressure profilesfrom the numerical data. A series of investigations spawnedby the Fourier power spectrum of the solid fraction are pre-sented next. These results include a quantitative comparisonof the experimental and numerical data, and the temporal evo-lution of the wave numbers obtained from the power spectraof the solid fraction. The last subsection investigates there-sponse of the instability when the input parameters, such asdissipation, initial interface, and grain density, are changed.

A. Experimental and numerical images

Figs. 4 and 5 show respectively experimental and numer-ical images of the structures observed when a column ofpolystyrene beads falls under gravity. The common width ofthe systems is 56 mm, while the height of the experimentalcell is 86 mm, and the height of the numerical cell is 68 mm.The experimental images in Fig. 4 are color inverted and con-trast enhanced versions of the original images; the grains areblack and the air is white. The start time of the experimentalimages is set to the instant when the cell reaches the verticalposition. Some grains start to fall during the rotation of thecell and form the pile of grains at the bottom of the first framein Fig. 4.

An interface, initially flat in the simulation but slightly per-turbed in the experiment, is defined by the falling grains.When the cell is flipped a pattern of fine fingers emerges alongthe interface which subsequently develop into coarser bubble-finger structures that propagate through the packing. Lookingat the temporal evolution in greater detail it is aptly separated

t = 0.02 s

t = 0.04 s

t = 0.06 s

Figure 6: A closer look at the initial formation of fingers in the sim-ulation. Before the fingers appear there is a transient phaseof ho-mogeneous decompaction. Notice the cusp-shaped geometry of theinterface.

in three stages. The first stage is thenucleation stagechar-acterized by a transition from a homogeneous decompactionto the appearance of the first finger seeds. This stage is onlyobserved in the numerical images, most likely due to the ex-perimental setup which causes perturbations in the cell as therotation suddenly stops. The second stage is thegrowing stagewhere the nucleated fingers grow in size and start to coalescewith their neighboring fingers. The aggregation of grains atthe bottom marks the start of thepropagation stagewhere thegaps separating the fingers are transformed into bubbles prop-agating through the packing. Two competing mechanisms areat play, one producing finer structures, the other producingcoarser structures. The coarsening proceeds as the smallerbubbles, together with the fingers defining them, are left be-hind by the bigger bubbles at an increasing rate. In this pro-cess the fingers defining the smaller bubbles are replaced by asingle wider finger, and the total number of fingers is thus re-duced. Finer structures are produced by new fingers formingon the inner surface of the bubbles that have expand beyonda certain width; the instability repeats itself inside the bub-bles. The new fingers split the bubbles and prevent them fromgrowing indefinitely.

Fig. 6 shows a more detailed series of numerical snapshotsof the early evolution of the instability. The top frame showsthe initial homogeneous decompaction of the bottom layers ofgrains, and it is possible by close observation to discern tinyfinger seeds in the top frame which develop into fingers with acusp-shaped geometry in the next frame. From the second tothe third frame the already present fingers grown in size, andsome new fingers are nucleated. The first and second framerepresent the nucleation stage, while the third frame repre-sents the growing stage.

Fig. 7 shows a snapshot from a simulation where glassbeads of 210 µm are used. The beads are colored in bandsin their initial configuration to illustrate the dynamics ofthemixing and the patterns left behind by the fingers and bubbles.The emerging structures are reminiscent of geological patternsobserved when e.g. water is forced through sediments of sand

Figure 7: A snapshot from a simulation using glass beads of 210 µmin diameter where the initial configuration is colored in horizontalbands to bring out the dynamics of the structures as the instabilityevolves.

[34].A qualitative comparison of the two series of images in

Figs. 4 and 5 renders the two data-sets consistent in manyrespects. The sizes of the bubbles and fingers are compara-ble, and the dynamical processes of finger splitting and fin-ger nucleation are observed in both cases. Nevertheless, somediscrepancies are observed, particularly in the initial and fi-nal evolution of the instability: The incipient decompaction,followed by the emergence of fine fingers, is only observednumerically, and in the latter images of Fig. 5 a horizontaldestabilization of the interface is visible, i.e. some bubblesreach the surface before the others.

The initial evolution of the experimental instability is dis-turbed by two events related to the execution of the experi-ment, which to a great extent explains the initial discrepan-cies. The first event is the rotation of the cell, and the secondevent is the abrupt stop of the cell as it hits the bar. The ro-tation is not fast enough to completely stop the falling motionof the grains, and the 1 mm thickness of the experimental cellallows it to contain up to 7 layers of 140 µm beads betweenthe plates. This will cause the grains to slide and roll downthe inclined plate during the rotation, allowing air and grainsto pass each other in layers (by the so-called Boycott effect[19]), and no fingers are produced. When the cell is almostvertical the falling grains are able to form an interface that fillsa cross section of the cell and fingers may form. However, themechanical shock created as the cell hits the bar generates ashort transverse oscillation in the cell, causing the grains to betossed back and forth between the plates. These oscillationswill erase any structures formed prior to the shock. The re-sult of these two events is visible in the first frames of Fig. 4where the falling grains form an almost homogeneous sheetof grains.

In the numerical images the fingers appear somewhat buck-led and bended compared to the experimental fingers. We be-lieve that this is an artifact caused by the imposed cutoff onthe solid fraction. The cutoff leads to an empty space poros-ity, i.e. in a region with no or very few grains, that is different

0 10 20 30 40 50 60 70

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

height (mm)

P(y

) −

Pat

m

(kP

a)

t = 0.00 ms

t = 0.80 ms

t = 1.60 ms

t = 2.40 ms

(a) Initial vertical pressure profiles

0 10 20 30 40 50 60 70

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

height (mm)

P(y

) −

Pat

m

(kP

a)

t = 0.06 st = 0.12 st = 0.18 st = 0.24 st = 0.30 s

(b) Evolved vertical pressure profiles

Figure 8: Plots of vertical pressure profiles averaged over the systemwidth. (a) shows the evolution of the pressure immediately after thecell is rotated while (b) shows the pressure profiles at latertimes. Thepressure is given as the deviation from one atmosphere.

0 10 20 30 40 50 60 70

0

0.1

0.2

0.3

0.4

0.5

0.6

height (mm)

ρ(y)

t = 0.00 st = 0.06 st = 0.12 st = 0.18 st = 0.24 st = 0.30 s

Figure 9: Plot of vertical solid fraction profiles averaged over thesystem width. The times of the profiles, except for the first profile,coincide with the times in Fig. 8b.

from one. This again leads to overestimated pressure gradi-ents in the most dilute regions of the system, and forces areexerted on the fingers traversing the gap of air. Nevertheless,the shape of the interface is very well reproduced by the sim-

ulations despite the buckling of the fingers.The experimental instability seems to be more stabilized

horizontally than is the case for the numerical instability. Bycomparing images after about 0.4 seconds it is evident thaton a detailed level the numerical images are quite differentfrom the corresponding experimental images. It seems that themost advanced bubble in the numerical images departs fromthe other bubbles at an increasing rate, causing a horizontaldestabilization of the interface. We believe that the reason forthis is found in the zero particle-particle friction and, moreimportantly, the zero particle-wall friction of the numericalmodel. The friction between the glass plates and the grainsin the experiment is quite considerable due to the Janssen re-lation [35], which most likely has a stabilizing effect on thepropagation of the interface.

Another effect of the zero friction is observed as the right-most bubble in the numerical snapshots reaches the upper sur-face at about 0.45 seconds and leaves behind an open gap inthe upper part of the packing. As grains rush in to fill thisvoid, convection rolls that distort and perturb the finger struc-ture are created. Such convection rolls would to a large extentbe dissipated away if interparticle friction were present in thesimulations.

B. Pressure and solid fraction in the simulation

Spatially averaged vertical pressure and solid fraction pro-files from the numerical data are plotted for different timesinFigs. 8 and 9, respectively. The profiles are calculated by av-eraging the data over the system width. Fig. 8a shows the tem-poral evolution of the vertical pressure profile immediately af-ter the rotation of the cell. The initially flat profile is, within afew milliseconds, transformed into a linearly decreasing func-tion while an overpressure builds up at the bottom of the cell.This corresponds to a fast diffusion of the pressure field (cfEq. (1)), and shows the transient curvature of the pressure pro-file relaxing towards a linear function. This happens at a timescale,ℓ2/(Pℓκ/µ) ≃ 0.1 ms, smaller than the time scale as-sociated with grain motion due to gravity,

√

d/g ≃ 3 ms,whereℓ is the system size, andd the grain diameter. Fig. 8bshows pressure profiles at later times, and the linearity of thepressure in the upper part of the packing, above the interface,is almost unchanged. Fig. 9 shows solid fraction profiles forthe same times as in Fig. 8b. The initially sharp interface issmeared out as the grains start to fall and aggregate at the bot-tom of the cell.

Fig. 10a shows the temporal evolution of the pressure at thebottom of the cell, and Fig. 10b shows the vertical velocity ofthe packing calculated by averaging over all the grains. Dueto the compressibility of the air the pressure is observed toundergo a transient damped oscillation immediately after thecell is rotated. The rotation of the numerical cell is instan-taneous, in contrast to the rotation of the experimental cell,and the sudden acceleration of the grains generates a pressureshock front that propagates into the gap of air. The oscillatingpressure causes the averaged vertical velocity of the grains todisplay a similar oscillation as shown in Fig. 10b.

0 0.005 0.01 0.015 0.020

0.02

0.04

0.06

0.08

0.1

0.12

time (s)

P(y

= 0

) −

P atm (

kPa)

(a) Pressure

0 0.005 0.01 0.015 0.02

−12

−10

−8

−6

−4

−2

0

time (s)

< v

i(y)

> (

mm

/s)

(b) Vertical velocity

Figure 10: (a) shows the temporal evolution of the pressure at thebottom of the cell, while (b) shows the vertical velocity of the pack-ing calculated by averaging over all the grains. Immediately after thecell is flipped a damped oscillation is observed both in pressure andvelocity, which is a signature of the compressibility of theair.

C. Fourier analysis

1. Solid fraction

In addition to the qualitative comparison of the numeri-cal and experimental data, a quantitative validification isper-formed by means of spatial Fourier power spectra of the solidfraction from which an average wave number〈k〉, measuringthe characteristic inverse size of the observed structures, is ob-tained.

The average wave number of the numerical data is calcu-lated as follows. The power spectrumS(k; j) of the solidfraction is calculated by applying the discrete Fourier trans-form [36] on the horizontal linesj of the solid fraction fieldρ(x, y). Herek denotes the horizontal wave numbers ofρ(x),andj is an index running over the horizontal lines of the solidfraction grid. The power spectraS(k; j) are further averagedoverj to produce a single power spectrum,S̄(k), representingthe state of the system at a given time. The mean〈k〉 of theaveraged power spectrum̄S(k) is then finally obtained by the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

time (s)

< k

> (

1/cm

)

simulationexperimentexperiment

(a) Mean wave number

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

time (s)

σ k (1/

cm)

simulationexperimentexperiment

(b) Standard deviation

Figure 11: Mean wave number〈k〉 and standard deviationσk asfunctions of time are plotted in (a) and (b) respectively fortwo ex-periments and one simulation, all using polystyrene beads of 140 µmin diameter.

common definition of the n-th moment

〈 kn 〉 =

∑

kn S̄(k)∑

S̄(k)(5)

with n = 1. The standard deviation

σk =√

〈k2〉 − 〈k〉2 (6)

is also calculated. The unit of〈k〉 is cm−1, and it is the char-acteristic inverse length scale of the observed structures.

The same analysis is used to extract information about thecharacteristic size of the experimental structures. The solidfraction is however not directly accessible in the experimen-tal data, and the values of the image pixels, spanning from 0(black) to 255 (white), are instead used as an estimate of thereal solid fraction. The solid fraction and the pixel value areinverse quantities, i.e. dilute regions have low solid fractionsbut high pixel values since they appear as white in the images.

The experimental images are 322 pixels in width, whereasthe numerical solid fraction grid is only 160 nodes in width.If the mean values of the two power spectra are to be com-parable, the width of the two distributions must be the same.Hence, the experimental power spectrum has an upper cutoff

0 10 20 30 40 50

25

30

35

40

45

50

55

width (mm)

heig

ht (

mm

)

(a) Experiment

0 10 20 30 40 505

10

15

20

25

30

35

40

width (mm)

heig

ht (

mm

)

(b) Simulation

Figure 12: Temporal evolution of the (a) experimental and (b) nu-merical interface. The interface moves upwards with a temporal sep-aration of 0.024 seconds. The times of the first and last interface are0.002 and 0.29 seconds, respectively

given by the upper wave number of the numerical power spec-trum.

The mean and standard deviation of the power spectrum,〈k〉 andσk, are plotted as functions of time in Figs. 11a and11b, respectively. The data are the same as in Figs. 4 and 5,with an additional data-set from a similar experiment. As theinstability evolves the initial discrepancy is reduced, and theconsistency of the data is quite good. The decrease ofσk inFig. 11b is caused by the increasing length of the fingers. Asthe fingers grow the amplitude of the wave number represent-ing the fingers increases, thus reducing the width of the powerspectrum distribution.

These results show that our gas-grain system is clearly dif-ferent from the liquid-grain system discussed by Völtz et al.[20]. Their liquid-grain system did not display a wave numbershift with time, nor the cusp-shaped geometry of the fingers.

2. The interface

Instead of analyzing the full solid fraction fieldρ(x, y) forthe whole system as in the previous section, the focus is nowthe dynamics of the interface itself.

1 2 3 40

1

2

3

4

5

wave number (1/cm)

ampl

itude

t = 0.05 st = 0.10 st = 0.15 st = 0.20 st = 0.25 st = 0.30 s

(a) Experiment

1 2 3 40

5

10

15

wave number (1/cm)

ampl

itude

t = 0.05 st = 0.10 st = 0.15 st = 0.20 st = 0.25 st = 0.30 s

(b) Simulation

Figure 13: The temporal evolution of the experimental and numericalpower spectrum of every second interface in Fig. 12 plotted in (a)and (b), respectively. The times of the power spectra are given inthe legend box, and the colored circles indicate the location of themaximum wave number at each time.

The function describing the interface is determined by thefollowing procedure. Each vertical line of the solid fractiongrid is scanned from top to bottom, and the first node on theline with a solid fraction less than a given threshold is a pointon the interface. Together these points define the interface.The value of the threshold is determined by plotting the calcu-lated interfaces on top of the numerical snapshots and visuallyconfirm that they correspond well. The overall shape of the in-terface is not very sensitive to a fine-tuning of the threshold.The experimental interface is obtained by the same procedure,only with the solid fraction replaced by the image pixel val-ues. The experimental images are rescaled to the size of thenumerical solid fraction grid before the experimental interfaceis determined. The resulting single-valued functions describ-ing the interface at different times are shown in Figs. 12a and12b for the experiment and simulation, respectively. The twoseries of interfaces have the same temporal separation of 24milliseconds, and they both start (bottom) and stop (top) at0.002 and 0.29 seconds, respectively. Salient features of thesefront evolution plots are the expansion of the bubbles and theappearance of new, wedge shaped fingers visible at the topof the latter interfaces. By comparing Figs. 12a and 12b thefinal shape of the interfaces is in good agreement. The ini-

0 0.05 0.1 0.15 0.2 0.25

−10

−8

−6

−4

−2

0

2

time (s)

ln A

S

k = 0.18 cm−1

k = 0.36 cm−1

k = 0.54 cm−1

k = 0.71 cm−1

k = 0.89 cm−1

0 0.05 0.1 0.15 0.2 0.25

−10

−8

−6

−4

−2

0

time (s)

ln A

S

k = 1.07 cm−1

k = 1.25 cm−1

k = 1.43 cm−1

k = 1.61 cm−1

k = 1.79 cm−1

0 0.05 0.1 0.15 0.2 0.25

−12

−10

−8

−6

−4

−2

0

time (s)

ln A

S

k = 1.96 cm−1

k = 2.14 cm−1

k = 2.32 cm−1

k = 2.50 cm−1

k = 2.68 cm−1

0 0.05 0.1 0.15 0.2 0.25−12

−10

−8

−6

−4

−2

time (s)

ln A

S

k = 2.86 cm−1

k = 3.04 cm−1

k = 3.21 cm−1

k = 3.39 cm−1

k = 3.57 cm−1

Figure 14: The logarithm of the interfacial power spectrum ampli-tudeAI for the 20 lowest wave numbers as functions of time. Thewave numbers are given in the legend boxes, and the vertical linesindicate the time window used to calculate the dispersion relation inFig. 16a.

0 0.05 0.1 0.15 0.2 0.25−4

−3

−2

−1

0

1

2

time (s)

ln A

S

k = 0.18 cm−1

k = 0.36 cm−1

k = 0.54 cm−1

k = 0.71 cm−1

k = 0.89 cm−1

0 0.05 0.1 0.15 0.2 0.25

−4

−3

−2

−1

0

1

2

time (s)

ln A

S

k = 1.07 cm−1

k = 1.25 cm−1

k = 1.43 cm−1

k = 1.61 cm−1

k = 1.79 cm−1

0 0.05 0.1 0.15 0.2 0.25

−4

−3

−2

−1

0

1

time (s)

ln A

S

k = 1.96 cm−1

k = 2.14 cm−1

k = 2.32 cm−1

k = 2.50 cm−1

k = 2.68 cm−1

0 0.05 0.1 0.15 0.2 0.25

−4

−3

−2

−1

0

time (s)

ln A

S

k = 2.86 cm−1

k = 3.04 cm−1

k = 3.21 cm−1

k = 3.39 cm−1

k = 3.57 cm−1

Figure 15: The logarithm of the bulk solid fraction power spectrumamplitudeAS for the 20 lowest wave numbers as functions of time.The wave numbers are given in the legend boxes, and the verticallines indicate the time window used to calculate the dispersion rela-tion in Fig. 16b.

2 4 6 8 10 12

20

40

60

80

100

120

wave number, k (1/cm)

grow

th r

ate,

α I (1/

s)

(a) Interface

2 4 6 8 10 12

20

30

40

50

60

70

80

wave number, k (1/cm)

grow

th r

ate,

α S (1/

s)

(b) Solid fraction

Figure 16: Averaged (a) interface and (b) solid fraction dispersionrelations obtained by a linear fit over the approximately linear part ofthe growth plots in respectively Figs. 14 and 15. The error bars arethe standard deviation within the boxes.

tial interfaces do not coincide equally well due to the externalexperimental disturbances introduced in the cell.

Figs. 13a and 13b shows, in similar colors, the power spec-trum of every second interface in Figs. 12a and 12b, respec-tively. The location of the maximum wave number for eachindividual power spectrum is indicated by circles. The nu-merical power spectra in Fig. 13b start out with a maximumwave number to the far right, whereas the experimental powerspectra in Fig. 13a start out with a maximum wave number tothe far left. The reason for this discrepant behavior is the verydifferent shape of the initial experimental and numerical inter-faces: The numerical interface is virtually flat, whereas the ex-perimental interface has noise on all wavelengths. The domi-nating wave numbers of the latter power spectra in Figs. 13aand 13b are only slightly shifted with respect to each other,and converge to approximately the same form.

3. Growth rates

We now move on to analyze the growth rates of the wavenumbers as they appear in the power spectra of the numeri-cal data. Following the lines of a linear stability analysiswe

look for possible early exponential growth of the wave num-bers that constitute the power spectrum of the interface andthe solid fraction.

The temporal evolution of the amplitudes for the 20 low-est wave numbers in the power spectrum of the interface areplotted semi-logarithmically in Fig. 14, and the correspondingplots for the averaged bulk solid fraction are shown in Fig. 15.Exponentially growing wave numbers would appear as linearin these plots. The growth of the bulk solid fraction wavenumbers in Fig. 15 is less noisy compared to the growth of theinterface wave numbers in Fig. 14 due to the averaging of thesolid fraction over the system height.

Some of the wave numbers in Figs. 14 and 15 show an ap-proximate linear time-dependence, whereas others are clearlynon-linear. Nevertheless, a linear least squares fit, over atimewindow of 0.6 seconds as indicated in the plots, is obtainedfor all the wave numbers, including those not displayed inFigs. 14 and 15. The resulting growth ratesαI (interface)and αS (solid fraction) are averaged over eight wave num-bers and the dispersion relations are shown in Figs. 16a and16b, respectively. The standard deviation is indicated by theerror bars. Given the approximate linearity of the wave num-ber growth, vigorous conclusions can not be drawn from theseplots. However, while our measurements do not serve to iden-tify a well defined window of linear growth, the growth ratesnevertheless exhibits the dominating, time dependent wavenumber: The maxima at ~4 cm−1 in Figs. 16a and 16b co-incide with the t = 0.05 s maximum of the power spectrum inFig. 13b. However, it is clear that non-linear mechanisms takeover and govern the final dominating wavelength.

In a well controlled experiment of the classical Rayleigh-Taylor instability it is possible to observe exponential growthover a time sufficiently long to erase the influence of the initialnoise. This is however not the case for the granular Rayleigh-Taylor instability since the exponential domain of the wavenumber growth is too short to cancel the initial noise. Hence,the structures that develop early, i.e. on a time scale of about0.1 seconds, are sensitive to the initial structure of the system.Note that this initial structure is not an experimental imperfec-tion but an intrinsic, unavoidable feature of granular packings.

D. Changing simulation parameters

A number of simulations are performed to probe the re-sponse of the instability as the input parameters are changed,i.e. the density of the grains, the roughness of the initial inter-face, and the dissipation of the granular phase.

1. Grain density

A comparison of simulations performed using grains of dif-ferent densities, but with identical start configurations,is pre-sented. The density of the grains are 1.05 g/cm3 (polystyrene)or 2.5 g/cm3 (glass), and the diameter is 140 µm. Fig. 17ashows a plot of the mean wave number for the two densi-ties, and Fig. 17b shows the standard deviation of the power

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

time (s)

< k

> (

1/cm

)

ρg = 1.05 g/cm3 (polystyrene)

ρg = 2.50 g/cm3 (glass)

(a) Mean wave number

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

time (s)

σ k (1/

cm)

ρg = 1.05 g/cm3 (polystyrene)

ρg = 2.50 g/cm3 (glass)

(b) Standard deviation

Figure 17: The temporal evolution of (a) the mean wave number〈k〉 and (b) the standard deviationσk for simulations using grains ofdifferent densitiesρg given in the legend box.

spectra. The final data points of the plots indicate the timeof breakthrough, i.e. when the bubbles reach the upper sur-face. The temporal separation of these points shows that theinstability develops faster if heavier beads are used. The ini-tial reduction of〈k〉 andσk are roughly the same for the twodensities.

2. Initial interface

The response of the instability to different initial interfacesis investigated by three simulations using differently preparedpackings with a variation in the roughness of the interface.

The three interfaces, shown in Fig. 19 where they are de-noted A, B and C in increasing order of roughness, are pre-pared as follows. Interface A is created by displacing thepacking vertically upwards to the top half of the cell, not byrotating the packing as is the usual preparation. A perfectlyflat interface is thus created by the grains from the bottomplate. Interface B is created by removing all grains with theircenters above a given height of the packing, just a few graindiameters away from the original surface, and then rotate thepacking. Interface C is the rotated original packing with asurface determined by the local equilibria of the grains.

0 0.05 0.1 0.15 0.20

1

2

3

4

5

6

time (s)

< k

> (

1/cm

)

interface Ainterface Binterface C

(a) Mean wave number

0 0.05 0.1 0.15 0.20

1

2

3

4

time (s)

σ k (1/

cm)

interface Ainterface Binterface C

(b) Standard deviation

Figure 18: The temporal evolution of the mean wave number〈k〉 andthe standard deviationσk in (a) and (b), respectively, for simulationsusing different initial interfaces. The interfaces are labeled A, B, andC and are shown in Fig. 19.

The temporal evolution of〈k〉 andσk for packings with theinitial interfaces shown in Fig. 19, are plotted in Figs. 18aand 18b, respectively. These plots lead to the conclusion thatthe shape of the initial interface only has an effect on the veryearly stages of the instability. It appears that the slowestevolv-ing interface is B, and that B, not A, is the interface with thesmallest initial noise. This may correspond to the fact thatthe porosity above interface A has larger fluctuations than inB because of the constraint imposed by the flat wall. Afterabout 0.2 seconds the difference in〈k〉 is negligible. The in-creased standard deviation of interface A indicates a slightlyslower and delayed generation of fingers in this case. How-ever, the effect is only temporary and the initial difference is

A

B

C

Figure 19: The interfaces of the packings used in the simulations toinvestigate the sensitivity of instability to initial configuration of theinterface. The interfaces are labeled A, B, and C in increasing orderof roughness.

t = 0.125 s

t = 0.250 s

r = 1.0 r = 0.95 r = 0.80 r = 0.50 r = 0.20 r = 0.050

Figure 20: Snapshots of simulations with different coefficients of restitution,r, given below the snapshots. The time of the simultaneoussnapshots are given to the left of each row.

canceled out. Note that this observation is consistent withtheconclusion at the end of Sec. IV C 3: The exponential domainis too short to cancel the initial noise during the incipientevo-lution of the instability. However, as non-linear effects takeover the system nevertheless evolves towards structures that isinsensitive to the initial noise.

3. Dissipation

A series of simulations are performed where the coefficientof restitution is changed. The motivation is to investigatethe effect on the fingering process and its dynamics. Fig. 20shows two snapshots for each of the six different restitutioncoefficient used in the simulations. The two leftmost snap-shots in Fig. 20, with a restitution of 1.0 and 0.95, distinguishthemselves by a diffuse outline of the interface, fewer fingers,and a general lack of detail compared to the other snapshots.The discrimination of the remaining pairs of snapshots, ob-tained using coefficients of 0.8, 0.5, 0.2, and 0.05, is not asevident because it is difficult to identify significant differencesin these images: The number of fingers is about the same, aswell as the size and shape of the bubbles. It is possible toidentify minor deviations in e.g. the number and shape of therenucleated fingers, but no significant influence of dissipationis observed on the formation of new fingers. The conclusionmust be that the system is invariant under changes in the dis-sipation as long as the restitution coefficient is below a certainthreshold. However, above this threshold the instability is sen-sitive to variations in the restitution, in particular the size andshape of the fingers are affected. The curvature of the bubblesis however only moderately affected by the variations.

The value of the threshold, above which the instability is in-sensitive to variations in the dissipation, is likely to depend onthe average number of impacts the grains participate in. Thenumber of impactsn required to reduce the momentum by agiven fractionF is inversely proportional to the logarithm of

the restitution coefficientr, i.e. n(r) = ln(1 − F )/ ln(r)which follows from the relationrn = 1 − F . A semi-logarithmic plot ofn(r) for three different values ofF isshown in Fig. 21. The actual number of impacts a grain partic-ipates in will, for a given restitution coefficientrt, be greaterthan the number of impacts needed to significantly reducethe momentum of the grains. If the restitution coefficient isreduced beyondrt it will only make a negligible differencebecause the momentum was already very small atrt. FromFig. 21 it is evident that the number of impacts needed to re-duce the momentum by e.g. 90 percent changes drastically asthe restitution coefficient changes from 0.99 (229 impacts)to0.8 (10 impacts). Below this interval the change in numberof impacts is marginal, e.g. atr = 0.6 it takes 5 impacts toreduce the momentum by 90 percent, compared to 10 impactsat r = 0.8. If we assume that a grain on average is involvedin 10 collisions, over a time scale that is small compared tothe dynamical time scale of the structures, 90 percent of itsmomentum is lost ifr = 0.8, and 99 percent is lost ifr = 0.6.Since most of the momentum is already dissipated atr = 0.8

0 0.2 0.4 0.6 0.8 1

1

10

50

100

500

restitution coefficient, r

num

ber

of im

pact

s, n

F = 99.9 %F = 99.0 %F = 90.0 %

Figure 21: Number of impacts required to reduce the momentumofthe grains by the fractionF given in the legend box.

this might explain why the simulations are virtually invariantto a reduction in the restitution coefficient below 0.8.

V. CONCLUSION

A gravitational instability, observed when a column ofdense granular material is positioned above a gap of air ina closed Hele-Shaw cell, is reported both experimentally andnumerically. Qualitative and quantitative comparisons ofnu-merical and experimental data are presented with the conclu-sion that the simulations reproduce the essential experimentalfeatures. Further investigations of the instability, by means ofFourier analysis, indicate a transient growth of the interfacialand solid fraction wave numbers that is close to exponentialearly in the evolution of the instability. Dispersion relations,only valid for small times, are extracted from the numericaldata, and both the interfacial and solid fraction dispersions in-dicate a peak at ~4 cm−1. The response to changes in thesimulation parameters was probed using packings of differ-ent density, different initial interfaces, and by changingthegranular dissipation. Increasing the density leads to an earlier

breakthrough, without any significant change in the coarsen-ing time of the structures. Changing the roughness of the ini-tial interface only has an effect early in the instability, the dif-ference is canceled after about 0.2 seconds. Dissipation onlyaffects the formation of fingers above a certain threshold ofthe restitution coefficient, a behavior that is supported byasimple model for the loss of granular momentum.

The unique behavior of the gas-grain system as comparedto the liquid-grain system investigated by Völtz et al., mightbe caused by the compressibility and low inertia of the air.

As for the experimental work, some refinements of the ex-periment is necessary if the complete evolution of the insta-bility, including the initial fingers, is to be probed.

Acknowledgments

The work was supported by NFR, the Norwegian ResearchCouncil. A special thank to Sean McNamara for useful com-ments, and for making the original F90 implementation of thegas-grain model available during the C++ reimplementation.

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