Collision process between an incident bead and a three-dimensional granular packing
Djaoued Beladjine, Madani Ammi, Luc Oger, and Alexandre Valance
Groupe Matière Condensée et Matériaux, UMR CNRS 6626, F-35042 Rennes cedex, France
Received 28 November 2006; published 14 June 2007
We report on experimental studies of the collision process between an incident bead and a three-dimensional
granular packing made of particles identical to the impacting one. The understanding of such a process and
the resulting ejection of particles is, in particular, crucial to describe eolian sand transport. We present here an
extensive experimental analysis of the collision and ejection process. The analysis is two dimensional in the
sense that we determined only the vertical component V
z
of the ejection velocity of the splashed particles and
the horizontal component V
x
lying in the incident plane. We extracted in particular the distribution of the
ejection velocities for a wide range of impact angles
i
and incident velocity V
i
. We show that the mean
quadratic horizontal velocity of the splashed particles is almost insensitive to changes in the impact angle and
velocity, while the mean quadratic vertical velocity slightly increases with increasing impact velocity as V
i
1/2
.
Moreover, the mean number of splashed particles per collision is found to be dependent on both the impact
angle and velocity, and to scale with the impact speed as V
i
3/2
. A consequence of these outcomes is that the sum
of the kinetic energy of the splashed particles is directly proportional to the kinetic energy of the incident
particle. Finally, we provide the bivariate probability distribution function PV
x
, V
z
of the ejection velocities
and show that it can be approximated by the product of a log-normal distribution and a circular normal one.
DOI: 10.1103/PhysRevE.75.061305 PACS numbers: 45.70.-n, 62.20.-x, 81.05.Rm
I. INTRODUCTION
Sand movement in deserts can cause damage to villages
and seriously perturb the circulation on roads or railways. It
is therefore crucial to understand the mechanisms of sand
transport in order to stop or at least reduce it. Sand grains are
lifted by the wind and accelerated during their flight. These
highly energetic grains, called saltating grains, can travel
over large distance by successive jumps 1,2. As they im-
pact the sand bed surface, they eject other grains from the
bed. These splashed grains, termed reptating grains, contrib-
ute to the augmentation of the sand flux. Some of them can
be promoted to the saltation motion. We are interested here
in the collision process between a saltating grain and a pack-
ing of identical grains. We focused, in particular, on the en-
ergy redistribution through the rebound grain and the ejected
grains. Note that air flow plays a negligible role in the colli-
sion process, since the latter lasts a very short time in com-
parison with the typical time scale of aerodynamics pro-
cesses. This energy redistribution is described, in the
literature, in terms of the splash function 2. Previous ex-
perimental studies 3–5 provided interesting and valuable
information, but the data remain usually sparse and the range
of variation of the impact parameters like the impact veloc-
ity, approach angle are relatively limited. The difficulty
comes from the fact that saltation is a stochastic process, and
a great number of experiments is required to accumulate
enough data for good statistics.
The splash process in eolian sand transport is connected
in some way with the formation of impact craters. Recently,
several studies focused on the morphology and size of im-
pact craters formed in granular media 6,7. The underlying
physical mechanisms governing such processes are indeed of
the same nature as those responsible for the splash. In addi-
tion, the formation of impact craters in granular media cor-
responds to a distinct collisional regime where the impacting
body is much more massive and larger than the particles of
the impacted medium. In the saltation problem, the projectile
and the target are composed of particles of the same nature
and the regime of crater formation is never reached in the
range of impact energy relevant for eolian transport. It is,
however, interesting to note that craters can form at moderate
impact energy in some particular configurations, as at the tip
of a granular heap 8.
The present study aims at giving an extensive view of the
splash function thanks to new data recently obtained from
collision experiments between an incident bead and a three-
dimensional 3D granular packing made of particles identi-
cal to the impacting one. We varied the approach angle
i
and the impacting speed V
i
in a wide range 10°
i
90°
and 50 V
i
/
gd 200, where d is the bead diameter and g
the gravity acceleration and analyzed the projection of the
trajectories of the rebounding and splashed particles onto the
incident plane 0, x , z. We extracted in particular the distri-
bution of the horizontal and vertical ejection velocities V
x
and V
z
, respectively, and proposed laws for the variation of
the mean horizontal and vertical ejection velocity as a func-
tion of the approach angle and impact velocity. We also de-
termined the mean number of splashed particles and its evo-
lution with the impact parameters. Lastly, we provided the
bivariate probability distribution function PV
x
, V
z
of the
ejection speed of the splashed particles in the incident plane,
and showed that it can be approximated by the product of a
univariate log-normal distribution and a univariate circular
normal one.
The paper is organized as follows. In Sec. II, we recall
briefly the experimental and numerical results of the litera-
ture on the splash process. Section III describes the experi-
mental setup used for the collision experiments. In Sec. IV,
we present the experimental results concerning the rebound
process of the incident bead, while Sec. V describes the fea-
tures of the splashed grains. Finally, conclusions and per-
spectives are presented in Sec. VI.
PHYSICAL REVIEW E 75, 061305 2007
1539-3755/2007/756/06130512 ©2007 The American Physical Society 061305-1