Granular Matter 1, 33–41 c© Springer-Verlag 1998

Interaction of a granular stream with an obstacleVolkhard Buchholtz, Thorsten Poschel

Abstract We investigate numerically the interaction ofa stream of granular particles with a resting obstacle intwo dimensions. For the case of high stream velocity wefind that the force acting on the obstacle is proportionalto the square of the stream velocity, the density and theobstacle size. This behaviour is equivalent to that of non-interacting hard spheres. For low stream velocity a gapbetween the obstacle and the incoming stream particlesappears which is filled with granular gas of high tempera-ture and low density. As soon as the gap appears the forcedoes not depend on the square of velocity of the streambut the dependency obeys another law.

1IntroductionWhen a stream of solid frictionless particles i with radiiRi → 0 which do not interact with each other is scat-tered at an obstacle one expects from momentum conser-vation [1] that the net force F acting on the obstacle obeysthe law

F ∼ ρ · A · v2st. (1)

A is the projection of the obstacle to the stream direction,vst is the velocity of the stream particles, and ρ is the den-sity of the stream. When the stream consists of granularparticles which do interact with each other, i.e. betweenwhich forces and torques are transferred, and which un-dergo damping during collisions one can expect deviationsfrom the behaviour in Eq. 1. The aim of this paper is toinvestigate the interaction between the stream and theobstacle using soft-particle molecular dynamics in two di-mensions. As far as we know the described phenomenahave not been investigated experimentally so far.

The paper is organized in three parts. In section 2 weexplain the numerical experiment in detail. Section 3.1

Received: 10 November 1997

V. Buchholtz, T. PoschelHumboldt-Universitat zu Berlin, Institut fur Physik,Invalidenstraße 110, D-10115 Berlin, Germanyvolkhard@summa.physik.hu-berlin.de,thorsten@itp02.physik.hu-berlin.de

Correspondence to: V. Buchholtz

We thank Joshua Colwell, Sergei E. Esipov, Hans J. Herrmann,Stefan Luding, Lutz Schimansky-Geier, Stefan Soko lowski,and Frank Spahn for fruitful discussion.

contains our results concerning the angular distributionsof the scattered stream, i.e. the distributions of mass, mo-mentum and angular momentum as functions of the scat-ter angle Θ. In 3.2 we investigate the force experienced bythe obstacle from the impacting granular stream, whereasin section 4 we present some continuum properties of theexperiment, i.e. the fields of granular temperature anddensity. In the final section we discuss the results.

2The numerical experimentThe setup of the numerical experiment is sketched in Fig. 1,b is the width of the stream, vst the velocity of the streamparticles. When the obstacle is a sphere we can describe itby its radius Rob. At distance RD from the center of theobstacle with Rob RD there is a screen where we mea-sure the angle dependent distributions of the scatteredstream. In the simulation we used RD ≥ 12 · Rob. Theradii of the stream particles Ri are equally distributed inthe interval [0.05; 0.11] cm. Each of them has translationaland rotational degrees of freedom.

Before interacting with the obstacle the particles movein an uniform stream, all with the same initial velocity vst.When released the particles move towards the obstacle,collide with the obstacle and once cross the screen plane(Fig. 1). After crossing the screen plane the particles areremoved from the calculation.

Initially the stream particles are arranged in columns.All particles which belong to the same column have the

y

b2

b2

x Rob

RD

vst

ρ

Fig. 1. The setup of the numerical experiment

34

same x-position. One particle of each column has beenset to a random y-position, then the column was filled inpositive and negative y-direction up to ymin = −b/2 andymax = b/2 respectively. At the initialization time thedistances between the centers of two neighboring sphereswith radii Ri and Rj in horizontal and vertical direction∆x and ∆y are

∆x = 2 · Rmax√

1/ρ (2a)

∆y = (2 · Rmax + 0.2 ∗ GRND)√

1/ρ, (2b)

where ρ is the density of the stream as used in Eq. 1,Rmax is the radius of the largest stream particle and GRNDis a Gaussian random number with standard deviation ofone. All particles move with the same initial stream ve-locity vst in positive x-direction. The density value ρ = 1in Eq. 2 means that the distance between neighbouringcolumns is 2 Rmax. According to the described methodthe number of particles is not conserved during the sim-ulation. Particles are removed from the calculation whenthey cross the screen plane, and particles are added to thesystem according to the initialization scheme. The totalnumber of particles which are involved in the simulationvaries between N = 4000 and N = 6000.

Two particles i and j at positions ~ri and ~rj interactif the distance between their centre points is smaller thanthe sum of their radii. In this case we define the particleoverlap ξij = Ri + Rj − |~ri − ~rj |. Two colliding spheresfeel the force

~Fij = FNij · ~nN + FS

ij · ~nS (3)

with the unit vectors in normal direction ~nN = (~ri − ~rj)/|~ri − ~rj | and in shear direction ~nS ,

∣∣~nS∣∣ = 1, ~nN · ~nS = 0.

In viscoelastic approximation the normal force actingbetween colliding spheres has been derived rigorously byBrilliantov et al. [2]

FNij =

Y√

Reffij

(1 − ν2)

(23ξ3/2 + A

√ξ ξ

)(4)

with

A =13

(3η2 − η1)2

(3η2 + 2η1)

(1 − ν2

)(1 − 2 ν)

Y ν2 . (5)

Y , ν and η1/2 are the Young-Modulus, the Poisson ratioand the viscous constants of the particle material. Reff

ij isthe effective radius of the spheres i and j

Reffij =

Ri · Rj

Ri + Rj. (6)

The functional form of Eq. 4 has been found beforeby Kuwabara and Kono [3] using heuristic arguments andhas been derived recently by Morgado and Oppenheim [4]from first principle calculations.

For the shear force we apply the Ansatz by Haff andWerner [5] which has been proven to give correct resultsin many other molecular dynamics simulations of granularmaterial, e.g. [6–9]

FSij = sign

(vrel

ij

)min

γS · meff

ij · ∣∣vrelij

∣∣ , µ · ∣∣FNij

∣∣ . (7)

Table 1. The parameters used in the simulations

α = Y/(1 − ν2) α = 1010 g · cm−0.5 · s−2

β = A · α β = 1000 g · cm−1 · s−1

Shear friction γS = 1000 s−1

Coulomb friction parameter µ = 0.5Material density ρm = 1 g/cm3

The relative velocity vrelij between the particles i and j

with angular velocities Ωi and Ωj and the effective massmeff

ij of the particles i and j are given by

vrelij =

(~ri − ~rj

)· ~nS + Ri · Ωi + Rj · Ωj , (8)

meffij =

mi · mj

mi + mj. (9)

The resulting torques Mi and Mj acting upon the particlesareMi = FS

ij · Ri, Mj = −FSij · Rj . (10)

Eq. 7 takes into account that the particles slide upon eachother for the case that the inequalityµ · ∣∣FN

ij

∣∣ <∣∣FS

ij

∣∣ (11)holds, otherwise they feel some viscous friction.

The particles interact with the spherical obstacle inthe same manner as the particles with each other, i.e. theforce ~Fij acting on the ith particle when colliding withthe obstacle is given by Eq. 3, with Rj = Rob and ~rj isthe position of the obstacle. The mass of the obstacle isinfinite, hence the effective mass (Eq. 9) is meff = mi.The flat obstacle was built up of equal spheres as shownin Fig. 8 with the same material constants as the streamparticles but of infinite mass. Their radii have been R∗

ob =Ri, where Ri is the mean radius of the stream particles.The force acting on a stream particle when colliding withsuch a particle is calculated according to Eq. 3.

For the integration of Newton’s equation of motionwe used the classical Gear predictor-corrector moleculardynamics algorithm [10, 11] of 6th order for the positionsand of 4th order for the rotation of the particles.

In the numerical simulation we used the parametersgiven in Table 1 which are valid for a typical granular ma-terial. The time step for the numerical integration of theGear predictor-corrector scheme was ∆t = 1 · 10−5 s. Us-ing this time step the system behaves numerically stable,i.e. tests with time step ∆t = 2 · 10−5 s lead to the sameresults.

The simulations presented in this paper have beenperformed on the 16 processor message passing parallelcomputer KATJA (http://summa.physik.hu-berlin.de/KATJA/).

3Results3.1Scattering of the granular streamFirst we investigate the angle dependent distributions ofthe scattered particles for a numerical experiment with a

35

fixed spherical obstacle of radius Rob = 1.5 cm. The differ-ential mass intensity dm of the scattered stream normal-ized by the incoming stream is defined by

dm(Θ) =dmD(Θ)

mst, (12)

where dmD(Θ) is the mass of the particles which are scat-tered into the interval [Θ, Θ + dΘ] and mst is the totalmass of the incoming particles. Θ = 0 is the directiondirectly behind the obstacle.

Figure 2 shows the differential intensity, i.e. the massdm(Θ) of the particles which are scattered into a certaininterval [Θ, Θ + ∆Θ], as a function of the scatter angleΘ. The problem is symmetric due to the symmetry of theincoming particle stream, hence the intensity is symmetricas well. For the sake of better statistics we plot in Fig. 2dm(Θ)+dm(−Θ) over Θ. We find maxima at Θ ≈ ±0.5 =29. The minimum at Θ ≈ 0 is due to the finite size ofthe obstacle, so only few particles are scattered into theregion directly behind the obstacle.

Due to the different sense of the transferred angularmoments resulting from collisions at the lower or at theupper side (in y-direction) of the particles the angularmomentum L(Θ) of the particles which fall into a certaininterval of scatter angle Θ + dΘ has a negative sign forΘ < 0 and a positive for Θ > 0. Hence, the distribu-tion is antisymmetrically. As in the case of the intensitywe find symmetry of the absolute value of the distribu-tion of the angular velocity. Therefore, we draw in Fig. 3L(Θ) − L(−Θ) over Θ. When the particles collide withthe obstacle they lose kinetic energy due to damping. Thelarger the distance |y| of the particles in vertical direc-tion from the obstacle (see Fig. 1) the smaller is the dis-sipated kinetic energy. Hence, the particles moving withlarge y (outer particles) are faster than others with lower|y| (inner particles). Therefore, the stream particles un-dergo more intensive collisions from the outer side (large|y|) than from the inner (smaller |y|). Particles scatteredinto the region behind the obstacle have a higher angular

0

1

2

3

4

5

1.00.50

dm

(arc)Θ

Fig. 2. The differential intensity of the stream scattered at aspherical obstacle over the angle Θ

velocity since they need to be pushed by faster particlesmultiple times from the same side to reach this region.These effects are the reason for the distribution L(Θ) ofthe angular momentum. In agreement with the above ar-guments Fig. 3 shows that the distribution of the absolutevalue of the averaged angular momentum L(Θ) of parti-cles decreases with increasing Θ. In the range Θ ≈ 0 theopposite rotational sense for particles from the upper andfrom the lower side combine to an angular momentum ofzero. In this region the error bars are large according tothe bad statistics.

The averaged absolute value of the momentum |mi~vi| =mi[(v

(x)i )2 + (v(y)

i )2]0.5 of the scattered particles as func-tion of the scatter angle is drawn in Fig. 4. The velocityof the input stream is vst = 500 cm/s. As described abovethe energy loss of the stream is higher for particles scat-tered by low angles Θ. In this region we find large errorbars according to bad statistics – only few particles arescattered directly behind the obstacle. The maxima of thedistribution p(Θ) are close to the intensity maxima. Thereseems to be a preferred region for the scatter angle wherethe traces of the particles are almost in parallel, i.e. whereenergy dissipation is low. We will come back to this as-sumption when we discuss the granular temperature fieldof the stream.

3.2Force experienced by the obstacleThe impact of the stream particles on the target resultsin an effective force between the stream and the obstacle.With the strong simplifying assumptions that the parti-cles have no rotational degree of freedom and the particlesinteract only with the obstacle but not with each other oneobtains from kinetic gas theory a theoretical approxima-tion for the force F

F =dp

dt∼ Rob ρ v2

st. (13)

The force is the transferred momentum per time whichis the mass of the particles colliding with the obstacle in

1.00.50Θ (arc)

0

0.02

0.04

L (g

cm

s)

2-1

Fig. 3. The distribution of angular momentum L(Θ) over thescatter angle Θ

36

1.00.50Θ (arc)

p (g

cm

s)

-1

0

5

10

15

Fig. 4. The distribution of the absolute value of the aver-aged particle momentum over the scatter angle Θ. Again, fora better statistics we took advantage of the symmetry of theproblem

the time dt times the stream velocity. Eq. 13 implies therelations

F ∼ ρ, (14a)F ∼ Rob, (14b)

and

F ∼ v2st. (14c)

We will proceed to investigate each of the relations inEq. 14 numerically for granular particles for which thesimplifying assumptions are evidently not provided.

The force acting on the obstacle fluctuates accordingto the irregularities of the stream, hence, to get reliablevalues we averaged the force for each data point over acertain time that corresponds to approximately 500 rowsof stream particles colliding with the obstacle. As shown

2

4

6

8

10

12

1.00.80.60.40.2

125000 x + 1400ρ

ρ (cm )-2

F (

)4

-2x1

0g

cm s

Fig. 5. The force over the stream density ρ

in the results this time is sufficient to generate very smallerror bars.

As we see from Fig. 5 the simulation is in good agree-ment with Eq. 14a for the force F as a function of thestream density ρ of the input stream. The function F =F (ρ) can be fitted to

F = (125000 · ρ + 1400)g cms2

. (15)

In agreement with Eq. 14b the acting force F is alsolinear in its dependency on the radius Rob of the obsta-cle as drawn in Fig. 6 (crosses and solid line). It can beexpressed by the fit

F = (111 · Rob + 18) 103 g cms2

. (16)

There exists a resulting force F ≈ 18 · 103 g · cm/sec2

for Rob → 0 since as shown below even a very small ob-stacle hinders the stream particles (of finite radius) andresults in a certain resistance for the stream.

Thus, we find the surprising result that the functionalform of the acting force as a function of the density of thestream ρ (Eq. 15) and of the radius Rob of the obstacle(Eq. 16) which we measured in our simulations equals thatof the crude approximation Eq. 13. This coincidence is nottrivial since the stream particles which collide with the ob-stacle build up a characteristic “corona” around the ob-stacle as it can be seen in Fig. 7. The size and shape of thiscorona vary with the stream density ρ and the radius Rob.

For the case of a flat obstacle one observes qualita-tively the same behaviour as for a spherical one. Figure 8shows that in this case the corona which forms the ef-fective obstacle has approximately the same shape as forspherical obstacles. For the force as a function of the ob-stacle size we measure Fflat ∼ 156 · 103 · Rob instead ofF ∼ 111 · 103 · Rob for the simulation with a spherical ob-stacle of the same size and identical simulation parameters(Fig. 6, dashed line). The shape of the obstacle causes only

F (

)5

-2x1

0g

cm s

Spherical obstacleFlat obstacle~111000 x Rob

~156000 x Rob

0

1

2

3

4

5

3210R (cm)ob

Fig. 6. The force over the obstacle size Rob. In agreement withEq. 14b we find a linear function. The solid line shows the forceexperienced by a spherical obstacle, the dashed line shows theforce for a flat obstacle

37

Fig. 7. Snapshot of the simulation. The dissipation duringthe collisions causes a stationary corona which shape and sizedepend on the parameters of the simulation. The colour codesfor the absolute value of the particle velocities, red means highvelocity, blue means low

a change in the prefactor of the force law but leaves thefunctional form unaffected.

The colour in Figs. 7 and 8 codes for the absolute valueof the particle’s velocity. In both cases there is a wide re-gion in front of the obstacle where the velocities are verysmall. In this region the particles almost agglomerate tothe obstacle. The width of this region depends stronglyon the stream velocity and on the material parameters.In Fig. 8 one observes a small gap of low particle densitydirectly in front of the obstacle. In this gap one finds par-ticles of high kinetic energy. The properties of the gap willbe discussed below.

The dependence of the force F on the velocity vst ofthe stream is more complicate. Figure 9 shows that thequadratic behaviour from the estimation Eq. 14c is onlyvalid for high velocities (vst > 700 cm/sec) in the case ofa small obstacle. For lower velocities the curve deviatessignificantly from this function. The data can be fitted to

F ∼ v1.5, (17)

however, we are not sure whether a power law is the ade-quate representation of these data.

Hence, from Eqs. 15–17 we find for our particularsystem

F ∼ ρ · Rob ·

v2st vst > 700 cm/s

v1.5st vst < 700 cm/s.

(18)

For large impact velocity vst the corona is in direct con-tact with the obstacle as shown in the snapshots (Figs. 7

Fig. 8. Snapshot of a simulation using a flat obstacle. Similarto Fig. 7 one finds a corona of stream particles shielding theobstacle

and 8). In the region of the transition where the forcestarts to deviate from the quadratic law we observe thatthe agglomerated region of particles having low velocity

107

105

103

103 104102

R = 3 cmob

R = 0.5 cmob

~ v1.5

~ v2

v (cm s )-1

F (

gcm

s)

-1

Fig. 9. The force over the stream velocity vst. Small obstacle:for large velocity vst the force agrees with the crude approxi-mation Eq. 14c. For smaller impact velocity, however, the forcedeviates from this function. The dashed line shows the functionF ∼ v3/2. Large obstacle: the quadratic function remains valideven for smaller vst. We expect a deviation from this functionfor yet smaller impact velocity

38

Fig. 10a–c. For lower impact velocity vst a gap in between theobstacle and the corona is observed. The gap appears at thesame velocity when the force deviates from the quadratic law.

a vst = 2000 cm/s, b vst = 700 cm/s, c vst = 200 cm/s. Againthe colour codes for the absolute value of the particle velocities.The three figures are drawn with the same colour scaling

in front of the obstacle becomes large. In Figs. 7 and 8the size of this region is determined mainly by the size ofthe obstacle, whereas for small stream velocity the “cold”region may become significantly larger than the obstacleitself. Figure 10a–c shows three snapshots of the systemwith stream velocity well above (left), at the transitionpoint (middle) and well below the transition (right).

Moreover close to the transition one observes a gapforming in between the corona and the obstacle. This gapis filled with granular gas at high temperature. In section 4we will discuss the field of granular temperature whichsupports this statement.

The width of this gap depends on the stream velocityas shown in Fig. 11. For vst & 700 cm/s there is no gap. Itappears at vst ≈ 700 cm/s and there is a sharp transition.The width of the gap increases for decreasing stream ve-locity. A similar effect, i.e. the appearance of a gap whenvarying the driving parameters of the system, was foundrecently in an event driven simulation of granular particleswhich are heated (with granular temperature) from belowwhen gravity acts [12]. Similar as in our case there seemsto be a critical value for the driving parameter (in [12] thetemperature) when the gap appears.

For the region of parameters applied in our simula-tions the stream width b = 16 cm does not influence theresults. For obstacle size Rob = 0.5 cm, stream velocityvst = 200 cm/s and stream density ρ = 1 we made testswith b1 = 2 b. We found for the time averaged force actingon the obstacle F (b1) = 1.009 F (b) which fits well into theregion covered by the error bars.

4Continuum representation of the scatter experimentIt has been shown by several authors [13–20 e.g.,] that un-der certain conditions the dynamics of granular material

can be described using hydrodynamics equations. In thissection we want to present field data for the density ρ (~r)and the temperature T (~r) of the granular system. Thefield data have been generated by coarse graining the par-ticle data to a grid with grid constant a = 0.1 cm ≈ Rmax.The average has been taken over a time interval whichcorresponds to approximately 500 rows of stream parti-cles colliding with the obstacle.

Figure 12 shows the density and velocity fields as colorplots for vst = 500 cm/s and three different radii of thespherical obstacle. Red color stands for high and blue forlow values of the density or the velocity. All figures in

103 104102

v (cm s )-1

0.5

1

2

3

4

Gap

(cm

)

Fig. 11. The size of the gap as a function of the stream ve-locity. The value of vst when the gap appears coincides withthe stream velocity when the force which acts on the obstacledeviates from the F ∼ v2

st-behaviour

39

Fig. 12. The fields of density (top row) and granular temperature (bottom row) for three different obstacle sizes

one row have the same scaling for the colours. For thecase of the large sphere (Rob = 1.5 cm) the density infront of the obstacle is very high. The particles whichjoin this high density region are of low temperature. Thetemperature rises up to a characteristic distance from thesurface of the obstacle and has its maximum in the re-gion where the incoming stream particles collide with thecorona which surrounds the obstacle, i.e. the location ofmaximum temperature coincides with the boundary of thecorona. As discussed in the context of Figs. 7 and 8 forsmaller obstacle size (or lower stream velocity) a regionof low density and high temperature forms in front of theobstacle, i.e. the highest density can be found in a certaindistance from the obstacle. The region of highest temper-ature does not necessarily coincide with the border of thecorona.

The fields of granular temperature and density as wellas the appearance of the gap are reminiscent to the be-haviour of a bullet flying in air with supersonic velocity.It has been shown before that rapid granular flows may

contain spatial regions in which the flow is supersonic [16,21–23]. The properties of the flow and the possible em-bedding into a continuum mechanics theory are subjectsof work in progress [23]

5ConclusionWe investigated numerically the properties of a stream ofgranular particles which is scattered by an obstacle andthe force acting upon the obstacle as a function of thestream density ρ, the stream velocity vst and the obstaclesize Rob using molecular dynamics technique. The fieldsof granular temperature and density have been generatedfrom the molecular dynamics data by coarse graining.

Surprisingly we find the force to be proportionally tothe density ρ, the obstacle size Rob, and for high streamvelocities proportionally to v2

st, i.e. F ∼ ρ Rob v2st as one

would expect from a stream of non-interacting hardspheres. For lower velocities we find that the force deviates

40

from the F ∼ v2st law, but still we find a linear law for the

force as a function of Rob and ρ. This deviation occurs ata critical value for the stream velocity vst. At the samevelocity we observe a gap forming in between the surfaceof the obstacle and the corona of stream particles whichagglomerate and rest around the obstacle. This gap riseswhen the stream velocity decreases. The widths of thisgap which is filled with granular gas at high temperaturegrows with decreasing stream velocity.

This agglomeration of particles in front of the obstacleleads to an effective obstacle size Reff

ob larger than Rob.The shape of the corona does not sensitively depend onthe shape of the obstacle, the functional form of the forceF ∼ ρ Rob v2

st remains invariant if the spherical obstacle isreplaced by a flat one.

For the case of a sphere of radius R in a viscous fluidone finds for very low stream velocity (low Reynolds num-ber) experimentally the Stokes law F = 6πη R vst. Thisresult can be derived analytically from the continuummechanical Navier-Stokes equation for incompressible flow[24]

ρ (~vst · ∇)~vst = −∇p + η∇2~vst (19)

as result of the pressure in front and behind the obstacle.The conditions of low Reynolds number and high viscosityassure that streamlines of continuous media cling to eachpoint of the obstacle, which is a condition for the deriva-tion of the Stokes law. For the case of a granular materialthis condition does not hold, i.e. the pressure which actsfrom behind the obstacle is zero, or at least very close tozero since as visible from the snapshots Figs. 7, 8 and 10a–c and from the density fields in Fig. 12 there are almostno particles which collide with the obstacle from behind.According to these considerations we conclude that onecannot expect Stokes law even for very low stream veloc-ity vst because the precondition of a homogeneous mediumwith high viscosity does not hold.

The field data of the granular temperature and thedensity show that for large obstacle size the location ofmaximum temperature coincides with the boundary ofthe corona. For the case of smaller obstacles the highestdensity is found in a certain distance from the obstaclewhereas the region close to the obstacle is filled by a di-luted granular gas of high temperature. For fixed obstaclesize this gap can be found below a certain stream velocityvst. It grows rapidly for decreasing vst.

Instead of using molecular dynamics technique for thesimulation of the dynamics of the granular material whichrequires large computational effort, one could apply an“event-driven algorithm” which works much faster thanmolecular dynamics. Using such algorithms one does notsolve Newton’s equation of motion for each single collisionbetween the grains but one calculates the relative tangen-tial and normal velocities after each collision as functionsof the relative impact velocities (normal and tangential).Therefore one needs the normal and tangential viscoelasticrestitution coefficients which are functions of the materialconstants and the impact velocities. For the approxima-tion that the dilatation is independent of the impact ve-locities Pao and others analytically calculated the normalrestitution coefficient [25, 26]. A more general approachfor the derivation of the coefficients of restitution based on

physical interaction forces between colliding spheres [2–4]has been given in [27]. Effective algorithms for these typeof event driven molecular dynamics simulations can befound e.g. in [28–33].

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