Experimental velocity fields and forces for a cylinder penetrating into a granular medium

0.83 MB·English

by  A. Seguin

Experimental velocity fields and forces
for a cylinder penetrating into a granular medium
A. Seguin1, Y. Bertho1, F. Martinez1, J. Crassous2 and P. Gondret1
1Univ Paris-Sud, Univ Paris 6, CNRS, Lab FAST,
Bˆat. 502, Campus Univ, F-91405 Orsay, France and
2Univ Rennes 1, Institut de Physique de Rennes (UMR UR1-CNRS 6251),
Bˆat. 11A, Campus de Beaulieu, F-35042 Rennes, France
Wepresenthereadetailedgranularflowcharacterizationtogetherwithforcemeasurementsforthe
quasi-bidimensional situation of a horizontal cylinder penetrating vertically at a constant velocity
indrygranularmatterbetweentwoparallelglasswalls. Inthevelocityrangestudiedhere,thedrag
3
1 forceon thecylinderdoesnot dependon thevelocity V0 andismainly proportional tothecylinder
diameter d. While the force on the cylinder increases with its penetration depth, the granular
0
velocityprofilearoundthecylinderisfoundtobestationarywithfluctuationsaroundameanvalue
2
leading to the granular temperature profile. Both mean velocity profile and temperature profile
n exhibit strong localization near the cylinder. The mean flow perturbation induced by the cylinder
a
decreasesexponentiallyawayfrom thecylinderonacharacteristiclengthλthatismainlygoverned
J
by the cylinder diameter for a large enough cylinder/grain size ratio d/dg: λ ∼ d/4+2dg. The
8 granular temperature exhibits a constant plateau value T0 in a thin layer close to the cylinder of
extension δT0 ∼ λ/2 and decays exponentially far away with a characteristic length λT of a few
t] grain diameters (λT ∼3dg). Thegranular temperature plateau T0 that scales as V02dg/d is created
f by the flow itself from the balance between the “granular heat” production by the shear rate V0/λ
o
s overδT0 close tothe cylinderand thegranular dissipation far away.
.
t
a PACSnumbers: 45.70.-n,45.50.-j,83.80.Fg
m
-
d I. INTRODUCTION pling between lift and drag forces is important [13]. All
n these physical phenomena require a better knowledge of
o theflowcharacterizationaroundmovingobjectsingrains
c The characterization of forces on moving objects in andtheassociatedforces. Inthiswork,weshallfocuson
[ granular matter is important in fields ranging from fluid
low velocity intrusions.
mechanics to geophysics and biophysics with, for in-
2
stance, the practical situations of meteoritic impacts on Many studies have been performed on the drag force
v
1 planetsorasteroids[1]andofthemotionoflivingorgan- experienced by objects in relative vertical or horizontal
8 ismsinsand[2]. Asamatteroffact,abetterunderstand- motion with grains, but fewer studies have been done
7 ing of meteoritic impacts requires a better knowledge of concerning grain flow characterization. In dense granu-
5 theforcesexperiencedbyimpactors,andthisgaveriseto larmatter,thedragforcemeasuredinvelocity-controlled
.
2 numerous physical studies at the laboratory scale ([3, 4] experiments has been found not to depend on the veloc-
1 andreferencesherein)orby numericalsimulations[5, 6]. ityatlowvelocities,andtobeproportionaltothesurface
2 Most of these studies have identified mainly two terms area of the object and roughly to its depth [14]. As in
1
in the forces experiencedby animpactor: One term pro- classical fluids, the corresponding drag force has been
:
v portional to the square of the velocity and independent shown to depend on the exact shape of the object [15].
i of the depth during the initial stage of the penetration This force may depend on the packing fraction [16], on
X
(at high velocity) and one term proportionalto the pen- the vibrationofthe grains[17], oronpossible dynamical
r
a etration depth and independent of the velocity at the aireffects[18]. Amarouch`eneet al. reportedakinematic
end of the penetration (at low velocity) [3, 4, 6]. The studyoftheflowaroundathindiskplacedinaHele-Shaw
stopping time has been shown to display a non intuitive cell and submitted to a vertical granular chute flow [19].
behavior, with a smaller value for larger impact velocity The perturbed streamlines were observed to be located
[3, 4, 6]. These behaviors for the penetration of the im- in a parabolic shape near the object with a triangular
pactor may depend on the packing fraction of the grains region of non flowing or slowly creeping grains, clearly
[7]; the packing fraction also determines the grain ejec- due to the stabilizing friction force from the two close
tion,whichcangiverisetoaspectacularupwardsvertical walls [20, 21]. The measured velocity profile exhibited a
jet for a fluidized packing [8, 9] or to an opening corona nonlinearbutmonotonicvariationawayfromtheobject.
for a dense packing [10, 11], quite similar to what is ob- Chehataetal. performedthesamekindofexperimentfor
served for splashes in water of droplets or solid spheres a cylinder in a much larger channel, and they measured
[12]. Moreover,the motionofcertainliving organismsin atthe sametime the velocityfieldusinga particleimage
grains, such as sand snakes, is interesting to understand velocimetry (PIV) technique and the drag force experi-
in order to, for example, create artificial robotics able to enced by the cylinder in the grain chute flow [22]. They
move in grains [2]. In these last phenomena, the cou- reported measurements of the flow vorticity and veloc-
2
ity fluctuations that showed shear localization near the
cylinder,andtheynoticedthatthedragforceisindepen- force sensor
dent of the grain velocity and proportional to the cylin-
der diameter. In a horizontal bidimensional experiment
cylinder
of vibrated disks at different packing fraction close to
jamming, the measuredvelocityfield aroundanintruder
high speed
draggedataconstantforcealsoexhibitedshearlocaliza- camera
d
tion[23]. Inahorizontalquasibidimensionalexperiment b
in which a thin but large cylinder is pulled relative to a grains
dense packing of millimetric alumina beads, Takehara et H
al. [24] investigated motion at high velocities for which
L
theforceisshowntoincreasequadraticallywithvelocity, V0
andtheyobservedhighgrainfluctuationmotionsandve-
locity perturbationina narrowzonearoundthe cylinder
vertical
that increases with the cylinder diameter. In the case of
linear stage
a dilute flow, the drag force is also observed to be pro-
portional to the square of the velocity, and proportional
to the cylinder diameter and to the effective density of FIG. 1. (Color online) Sketch of the experimentalsetup.
the fluid as in classical hydrodynamics with a drag coef-
ficient that increases from about 1 at low Kn to about
2-2.5athighKn[25,26],wheretheKnudsennumberKn
cylinderoflengthbandofdiameterdrangingfrom10to
comparesthemeanfreepathinthegastotheobjectsize.
40mmisfirstmaintainedabovethegrainsinbetweentwo
In a previous work [27], we reported key results for
vertical glass walls. The length b of the cylinder is man-
both the experimental measurements and a hydrody-
ufactured about 0.2mm smaller than the width b of the
namic modeling of the canonical problem of the flow
cell, andthe cellis free to movealongthe axialdirection
around a circular cylinder and the associated drag force
of the cylinder. In this way, the cylinder has only mini-
in the granular case. In the present paper, we report
mal mechanical contact and thus solid friction with the
the detailed experimentalstudy ofthe meangrainveloc-
walls during its vertical displacement, and no grain can
ity profile and fluctuations around a horizontal cylinder
fit between the cylinder and the walls and jam the sys-
penetrating vertically in granular matter together with
tem. As the cell moves upwards, the cylinder penetrates
drag force measurements. In Sec. II, we present the ex-
graduallyata constantvelocityinthe granularmedium.
perimental set-up. Experimental results on the profiles
Theentiredynamicsofthegrainsduringthepenetration
of both the mean velocity and the velocity fluctuations
is extracted by recording their motion at the front glass
are presented in Sec. III, whereas Sec. IV concerns force
wallusingahigh-speedvideocamerathatcantakeupto
measurements. TheresultsarediscussedinSec.Vbefore
1000imagesper secondinthe fullresolution1024×1024
the conclusion.
pixels. The grains are lighted fromthe frontand a black
curtain is put behind the cell so that the grains appear
in a white on black background with a good contrast.
II. EXPERIMENTAL SETUP
The images are taken at a sampling rate f adjusted on
the velocity V0 to have f ≥ V0/dg (e.g. f = 50Hz for
Theexperimentsconsistintheverticalpenetrationofa V0 =10mms−1anddg =1mm)sothatthelargestgrain
horizontalcylinderatagivenvelocityV0intoapackingof displacement between two successive images is smaller
grains. The grains are rather monodisperse sieved glass thanonegraindiameter. Thesuccessiveimagesarethen
beadsofdiameterd rangingfrom0.5to4mmwitharel- analyzed by Particle Image Velocimetry (PIV) software
g
ative dispersion ∆d /d of about 10% around the mean (Davis, LaVision) to get the velocity fields of the grains
g g
value, and density ρ = 2.5103kg m−3, contained in a as shown in Fig. 2. The size of the final interrogation
g
rectangular box of length L = 0.2m, height H = 0.1m windowsusedinthecorrelationtechniqueissettypically
and width b = 40mm (Fig. 1). The granular medium is to one grain diameter to have the best resolution con-
prepared by gently stirring the grains with a thin rod, sidering the discrete nature of the medium. The spatial
and the surface is then flattened using a straightedge. resolutionof the obtained velocity field is thus one grain
We have checked that this preparation leads to repro- diameter. As the cylinder and video camera are fixed in
ducible results with only small variations. The solid vol- thelaboratoryframeofreferenceandthegrainsmoveup,
ume fraction is Φ ≃ 0.62 characteristic of a dense gran- the velocity field of the grains is measured in the frame
ular packing, and the density of the granular medium is of reference of the cylinder. The grain velocity far from
3 −3
thus ρ =ρ Φ ≃1.5 10 kg m . The cell containing the thecylinderthuscorrespondstotheundisturbedupward
g
granularpackingcanbemovedupalongaverticaltrans- imposed velocity V0, whereas the velocity is disturbed
lation guide at a given velocity V0 ranging up to 50mm close to the cylinder. Considering the discrete nature
s−1 by a step by step motor (Mavilor BLS-55). A steel of the granular medium and the finite size of the PIV
3
20 0 0.5
30
zb r
m)40 eθ 0.0
m θ
z (50 er 0
V
60 v / r -0.5
70
-40 -30 -20 -10 0 10 20 30 40
x (mm)
-1.0
FIG. 2. (Color online) Left: Typical instantaneous velocity
fieldobtainedbyPIVmeasurementsforacylinderofdiameter (a)
d=20mmpenetratinginapackingofglassbeadsofdiameter
dg =1mm,atthevelocityV0 =50mms−1. Right: Sketchof
thecylindrical coordinates and used notations. 1.5
1.0
correlationwindowcorrespondingto onegraindiameter, 0.5
the PIV calculation is distorted at a distance from the V0
cylinder surface smaller than half a grain diameter so / θ 0.0
v
thatvelocitymeasurementswillbereportedonlyoutside
-0.5
this zone for radialdistances r from the cylinder such as
r > (d+d )/2. With a force sensor (FGP Instrumenta- -1.0
g
tionFN3030)ofrange50N,wealsomeasureinthemean
timethedragforceexperiencedbythecylinderduringits -1.5 (b)
penetrationin the granularpacking atthe sampling rate
-90 -45 0 45 90
2kHz. Thecylinderisrelatedtotheforcesensorbyaver-
θ (deg)
ticalthinrodofdiameter3mmandlength10cm. With-
outgrainsinthecell,wecheckedthatthefrictionforceof
FIG.3. (Coloronline)All230successiveinstantaneous(a)ra-
the cylinder onthe glass wallsis totally negligible. With dialand(b)azimuthalvelocityprofilesnormalizedbythepen-
grainsin the cell, we made sure that no grainwas fitting etrationvelocityV0 takenattheradiallocationr=15mmat
intothesmallspacingbetweenthecylindersidesandthe regularintervals(∆t=20ms)duringthepenetrationinterval
walls and was distorting the force measurement during 20≤zb ≤70mm (d=20mm, dg =1mm, V0 =10mm s−1).
the cylinder penetration. In the following we presentve- (– –) Time average value and (···) corresponding standard
locity and force measurements that have been done for deviation. (—) Cosine and sine fits of data by Eq. (2) with
a cylinder/grain size ratio in the range 10 6 d/d 6 80 Ar ≃0.5 and Aθ ≃1.4.
g
andforacylindervelocity0.56V0 650mms−1remain-
ing smaller than (gd )1/2 > 70mm s−1. In this velocity
g where r is the distance from the cylinder center and θ
range,theflowregimeisexpectedtobequasi-static. The
is the angle from the direction of motion (θ > 0 anti-
box length L was chosen large enough (L/d > 5) to not
clockwise) (Fig. 2). We have checked that the granular
play any significant role in the measurements [28]. We
flow is bidimensional: No transverse flow along the y di-
also restrict ourselvesto penetration depth values z not
b rection perpendicular to the glass side wall can be seen
tooclosetoH (z .H−d)sothatthereisnosignificant
b eitherfromtheside(nograinappearsattheglasswallor
role played by the bottom wall [28, 29], and H will thus
disappears from the glass wall during the cylinder pen-
not be a relevant parameter for the present results.
etration) or from the top (no transverse roll motion can
be seen at the free surface). In addition, the results pre-
sented in the following will be for −π/2 < θ < π/2,
III. MEAN VELOCITY PROFILES AND i.e. for the upstream grain flow, and we have checked
VELOCITY FLUCTUATIONS that the downstream flow is very similar, with thus an
upstream-downstream symmetrical flow field character-
istic of low Reynolds numbers flows. Figure 3 shows the
A. Mean velocity
successive instantaneous (a) radial velocity component
v (r,θ,t)and(b)azimuthalvelocitycomponentv (r,θ,t)
r θ
The instantaneousvelocity field obtainedat eachtime takenata givenradiallocationr (here r−d/2=5mm),
t by the PIV technique is decomposed within the cylin- as a function of θ and normalizedby the penetration ve-
drical coordinates: locity V0. These velocity profiles are quite noisy as they
are not averaged at all but they do not show any sys-
v(r,θ,t)=vr(r,θ,t) er+vθ(r,θ,t) eθ, (1) tematic time evolution during the penetration. This ob-
4
servation holds regardless of the radial position and the
1.5
run. Thevelocityprofilearoundthepenetratingcylinder
can thus be considered as stationary and is now time-
averaged during each penetration run. The correspond-
ing time average v(r,θ) = hv(r,θ,t)it of the instanta- Aθ1.0
nFeigo.us3vaenlodcitthyepcroorfirleesspaopnpdeinagrssatsanadtahrdickdedvaisahteiodnliinseinin- , A r 1
dicated by the surrounding two thick dotted lines. The
radial velocity vr(r,θ) is maximum near the direction of 0.5 1-Ar10-1
motion(θ ≃0)andvanishesaroundthe equatorialplane
(θ ≃ ±π/2), and the opposite is true for the azimuthal 10-2
(a) 0 10 20 30 40 50
velocity vθ(r,θ), which is maximal near the equator and 0 r (mm)
vanishesneartheaxisofmotion. Thissuggeststheusual
50
simpleθ dependenceforthetwocylindricalvelocitycom-
ponents:
40
v(r,θ)/V0 =−Ar(r)cosθ er+Aθ(r)sinθ eθ. (2)
)
m 30
The fits of experimental v and v data by cosine and m
r θ (
sinefunctions areshownasthicksolidlines inFig.3and B
are rather close to the averaged profiles. Doing such an 20
analysis for all the radial locations r allows us to obtain
the radial functions Ar(r) and Aθ(r), which prescribes 10
the radial variation of v and v . Such functions are
r θ
plotted in Fig. 4(a) and show a strong flow localization (b)
0
around the penetrating cylinder. As a matter of fact, 0 10 20 30 40 50
A (r) and A (r) tend towardthe value 1, corresponding r (mm)
r θ
toanundisturbedvelocityfieldassoonasr &40mmand
thus r &2d. The radial function of the azimuthal veloc- FIG. 4. (a) Radial dependence of the radial and azimuthal
ity Aθ(r) exhibits an overshooting maximum (Aθ > 1) velocity components (•) Ar and (◦) Aθ, for V0 =10mm s−1,
as a consequence of the incompressibility for the present d = 20mm and dg = 1mm. (—) Best fits by Eqs. (3a) with
two-dimensional (2D) flow. This maximum is located λ≃6.4mmandλs≃9.7mmforAr(r)data,andλ≃7.6mm
very near the cylinder at r ≃ 17mm . d and is fol- and λs ≃8.1mm for Aθ(r) data. Inset: 1-Ar(r) data in log-
linplot. (b)Radialdependenceofthestream function,B(r),
lowed by an inflection point. The radial function A (r)
r obtained either from (•) Ar or (◦) Aθ data, and (—) best fit
of the radial velocity increases from zero at the cylin-
from Eq. (5) with λ ≃ 7.1mm and λs ≃ 9.2mm. The grey
der surface up to the asymptotic value 1 at large r, with
zone corresponds to the cylinder interior (r ≤ 10mm). The
an exponential law as shown by the straight behavior dashed lines stand for (a) A = 1 and (b) B(r) = r which
r,θ
of 1−Ar(r) in the semi-logarithmic plot in the inset of corresponds to an undisturbedvelocity field.
Fig.4(a). With this finding, assuminganincompressible
flowwithdivv=(1/r)∂(rv )/∂r+(1/r)∂v /∂θ=0,and
r θ
best-fittingvaluesλ≃6.4mmandλ ≃9.7mmforA (r)
takinginto accountthe slipforthe tangentialvelocityat s r
the cylinder surface vθ(r = d/2) 6= 0, we adopt thus the data, and λ ≃ 7.6mm and λs ≃ 8.1mm for Aθ(r) data.
Thefittingcurvesareclosetothedata,andtheλandλ
following expressions for fitting A (r) and A (r) data: s
r θ
values found independently from A (r) and A (r) data
r θ
r−d/2+λs r−d/2 are close, which means that the present dense granu-
A (r)= 1−exp − , (3a)
r r (cid:20) (cid:18) λ (cid:19)(cid:21) lar flow is nearly incompressible. Note that as λs val-
ues are close to half the cylinder diameter the prefactor
r−d/2+λ −λ r−d/2
s
Aθ(r)=1+ λ exp(cid:18)− λ (cid:19),(3b) (r−d/2+λs)/r in Eq.(3a) is close to one,thus explain-
ing the exponential behavior observed for 1−A [inset
r
whereλisthecharacteristiclengthofvelocityradialvari- of Fig. 4(a)].
ation, and the length λ characterizes the velocity slip As the flow can be considered to be bidimensional in
s
tangential to the cylinder surface. As a matter of fact, the(r,θ)plane,onecanintroducethescalarstreamfunc-
Aθ(r =d/2)=λs/λsothatthereissomeslipforλs 6=0. tion ψ(r,θ) related to the velocity field by vθ =−∂ψ/∂r
The slipping length ls defined as (∂vθ/∂r)(r = d/2) = and vr =(1/r)∂ψ/∂θ. Considering the θ dependence for
v (r = d/2)/l is related to λ by l = λ /(2−λ /λ). the velocity field of Eq. (2), the stream function can be
θ s s s s s
As experimentally A (r = d/2) = λ /λ ≃ 1, we have written as
θ s
l ≃ λ . Thus the length λ corresponds roughly here
s s s ψ(r,θ)=−V0 B(r)sinθ, (4)
to the slipping length l . The curves corresponding to
s
Eqs. (3a) are plotted as solid lines in Fig. 4(a) with the where the radial function B(r) is related to the velocity
5
20 2200
-10
15 1155
m) m)
λ (m 10 λ (ms1100 0
5 55 m)
m
00.1 1 10 (a1)00 000.1 1 10 (d1)00 z (b10
V0 (mm s-1) V0 (mm s-1) z-
20 2200 20
15 1155
m) m) 30
λ (m 10 λ (ms 1100 -40 -30 -20 -10 0 10 20 30 40
5 55 x (mm)
0 (b) 00 (e)
0 10 20 30 40 50 0 10 20 30 40 50 FIG.6. TemperaturefieldT(x,z)aroundacylinderofdiam-
d (mm) d (mm)
20 2200 se−te1rind a=p2a0ckminmgopfengreatrinastinofgdwiaimthettehredvge=loc1itmymV.0 = 10mm
15 1155
m) m)
λ (m 10 λ (ms1100 dependonthecylindervelocityV0 [Figs.5(a,d)]withthe
5 55 nearlyconstantvaluesλ=7±1mmandλ =9±1mm.
s
0 (c) 00 (f) The λs values appear to be proportional to the cylin-
0 1 2 3 4 5 0 1 2 3 4 5 der diameter d with the linear fit λ = 0.4d very close
dg (mm) dg (mm) s
to the data [Fig. 5(e)], and no significative dependence
upon the graindiameter d canbe seen in Fig. 5(f) with
FIG. 5. Variations of the characteristic length λ and λs of g
the nearly constant value λ = 18±2mm correspond-
mean velocity profile with (a, d) the velocity V0 (for d = s
20mm and dg = 1mm), (b, e) the cylinder diameter d (for ing again to about 0.4d. The λ values increase linearly
V0 = 5mm s−1 and dg = 1mm) and (c, f) the grain size withthecylinderdiameterdwithaslopearound0.2and
dg (for V0 = 5mm s−1 and d = 40mm). The symbols are a significative non zero value around 3mm correspond-
experimental data and the solid lines correspond to (a) λ = ing here to 3d that may be extrapolated for vanishing
g
7mm, (b) λ=3mm+0.2d, (c) λ=10mm+1.5dg,(d) λs = d [Fig. 5(b)]. The λ variations with the grain diameter
9mm, (e) λs=0.4d, and (f) λs=18mm. d shown in Fig. 5(c) are small but may increase from
g
about 10 to 15mm with an extrapolated value d/4 for
vanishingd andaslopearound1.5. Inconclusion,these
g
radialfunctions A (r) andA (r) byA =B/r andA =
r θ r θ two characteristic lengths λ and λ do not depend on
s
dB/dr. The B(r) curves obtained either from A (r) or
r the penetration velocity V0 and are mainly governed by
A (r) data of Fig. 4(a) are shown in Fig. 4(b). These
θ thecylinderdiameterdwithapossibleweakeffectofthe
curvesareclose,meaningagainthatthepresentgranular
grainsized onλwhenthecylinderdiameterisnotmuch
g
flow is close to being truly incompressible. The average
larger than the grain size.
experimental B(r) data can be fitted by the following
empirical function deduced from Eqs. (3a):
B. Velocity fluctuations
r−d/2
B(r)=(r−d/2+λ ) 1−exp − . (5)
s
(cid:20) (cid:18) λ (cid:19)(cid:21)
During each penetration run, we observe that the ve-
For large radial distance r away from the cylinder (r & locity field is stationarywith a well defined time average
2d), B(r) is close to linear in r, which corresponds to value and some erratic time fluctuations. In the preced-
theundisturbedflow. Theinterestingdomainisthusthe ing section, we focusedon the time averagevelocity field
one close to the cylinder (r . 2d) where B(r) deviates v(r,θ), and we now present the time fluctuations whose
significantly from r. The fit of the mean B(r) data of amplitudecanbequantifiedbythegranulartemperature
Fig. 4(b) by the previous equation leads to λ = 7.1mm defined as T(r,θ) = h v(r,θ,t)−v(r,θ) 2i . A typical
t
and λs =9.2mm, which correspond to intermediate val- temperature field T(r,(cid:0)θ) around the cyl(cid:1)inder is shown
ues between the λ and λ values deduced directly from in Fig. 6 in gray scale with black for the lowest value
s
A (r) and A (r) data. In the following, the presented and white for the highest value. A zone of high temper-
r θ
λ and λ values have been obtained by the fit of B(r) ature can be seen very near the cylinder with a radial
s
experimental data. extension of a few millimeters around the cylinder sur-
The λ and λ variations with the penetration velocity face. The precise radial dependence of the temperature
s
V0, the cylinder diameter d, and the grain diameter dg extracted along the θ = 0 streamwise direction in front
are shown in Fig. 5. One can see that neither λ nor λ of the cylinder is plotted in Fig. 7 for different cylinder
s
6
102 102
8
-2s) 1100T010 d/2 δT0 2-2T (mm s)0111000-011 λ (mm)T 642
2
m 10-1 λT 10-2 (a) 0 (d)
m 0.1 1 10 100 0.1 1 10 100
T ( 10-2 2.5 V0 (mm s-1) V0 (mm s-1)
8
10-3 2-2m s) 21..05 mm) 6
10-40 10 20 30 40 T (m010..05 λ (T 42
r (mm) (b) (e)
0 0
0 10 20 30 40 50 0 10 20 30 40 50
d (mm) d (mm)
FIG. 7. Radial dependence of the temperature profile T(r)
5
forθ=0,withd=20mm,dg =1mmanddifferentvelocities 8
s(−▽1)aVn0d=(N0).4Vm0=m5s0−m1,m(•s)−V1.0G=ui2dmelminess−f1o,r(t(cid:3)he)eVy0es=wi1t0hm(—m 2-2m s) 43 mm) 6
)antdherasldoipaelevxatluenesλioTn=δT30.7=m4mmamndcl(o-se--t)otthheepclyaltienaduervasulurefaTce0 T (m0 21 λ (T 42
at r=d/2.
(c) (f)
0 0
0 1 2 3 4 5 0 1 2 3 4 5
dg (mm) dg (mm)
velocities V0. One can see in this semi logarithmic plot
that allthese different profiles havethe same shape with FIG.8. TemperatureplateauT0 andcharacteristiclengthλT
ofradial temperaturevariation asafunction of(a,d)theve-
about a plateau value denoted T0 in a zone of extension
δ close to the cylinder (0 < r −d/2 < δ ) followed locity V0 (for d=20mm and dg =1mm), (b,e) thecylinder
bTy0an exponential decrease T ∼ exp(−r/λTT0) with the dgriaaminestiezredd(gfo(rfoVr0V=0 =5m5mmms−s1−a1nadnddgd==1m40mm)ma)n.dT(ch,efs)ytmhe-
characteristiclength λ at largerr (r−d/2>δ ). The
T T0 bols are experimental data and the solid lines correspond to
differentcurvesatdifferentvelocitiesV0 showthatT0 in- (a)T0=0.05V02,(b)T0 ∝d−1,(c)T0 ∝dg,(d)λT =3.7mm,
creaseswith V0 whereasneither δT0 norλT variessignifi- (e) λT =3.5mm+0.03d, and (f) λT =3.3mm +1.1dg.
cantly as the decreasingparts ofT(r) areabout parallel:
δ ≃4mm and λ ≃3.7mm.
T0 T
The variations of T0 and λT with the velocity V0, the
cylinderdiameterd,andthegraindiameterdg areshown 15
in Fig. 8. The log-log plot of T0 with V0 in Fig. 8(a) ex-
hibits a clear data increase with a slope 2 leading to the
natural scaling T0 ∼ V02. Figures 8(b) and 8(c) show m) 10
that the typical temperature T0 decreases with increas- m
ing cylinder size d and increases with the grain size dg. (T0
δ 5
The T0 dependence upon dg is linear and the T0 depen-
denceupondmaybehyperbolicsuchthatT0 mayfinally
scaleasV2(d /d),whichwillbe supportedbytheplotof
0 g 0
Fig.13(a),asweshallseeinthediscussion(Sec. V).Con-
0 5 10 15
cerning the characteristic length λ of the exponential λ (mm)
T
temperature decrease away from the cylinder, Fig. 8(d)
showsthatλT doesnotdependonV0withthenearlycon- FIG. 9. Radial extension δT0 of the temperature plateau as
a function of thecharacteristic length λ of themean velocity
stantvalueλ =(3.7±0.5)mm. OnecanseeinFig.8(e)
T profile (same symbols as in Fig. 8).
thatλ does notdepend significantlyonthe cylinderdi-
T
ameter d with a value around λ ≃4mm corresponding
T
thus to a few grain diameters. This is corroborated by
Fig. 8(f), where λ appears to increase linearly with the
T
graindiameterdg ifone ignoresthe singularpointatthe notdependonV0 andismainlygovernedbythe cylinder
smallest grain size dg = 0.5mm. The dependence of the diameter dwith only a weakeffect ofthe grainsize dg in
radial width of the temperature plateau δT0 on V0, d, contrast with the λT size dependence discussed above.
and d is not shown independently, but Fig. 9 displays a
g
clearcorrelationofδ withthecharacteristiclengthλof
T0
mean velocity variations. This means thus that δ does
T0
7
IV. FORCE ON THE CYLINDER We present here force measurements associated with
the velocity profiles presented above and corresponding
to our geometry of a horizontal penetrating cylinder in
The force on an object in relative motion with a gran-
vertical displacement in the granular packing. A typical
ular material has been extensively studied these last few
force variation as a function of the penetration depth
years by many authors in a lot of geometries. The drag
z is shown in Fig. 10(a), where z is the position of
b b
force is observed to vary linearly with the surface area
the cylinder bottom, with thus F = 0 when z = 0.
of the object and increases with the depth z, F ∼ zα, b
Figure 10(a) shows that F increases with z whereas the
b
withpossiblenonlineareffects(α6=1)depending onthe
grainvelocityprofilearoundthecylinderhasbeenshown
directionofmotionandgeometry[14,15,22,29–31]. For
to be stationary in section IIIA. The F(z ) variation is
b
a vertical rod in a horizontal granular flow, α is signifi-
aboutlinear,andifoneconsidersahydrostaticequivalent
cantlylargerthan1(α≃2[14,32]),whichislinkedtothe
“pressure”p=ρgz thatvarieslinearlywith depth inthe
linear increase of the immersed surface with z. For ob-
granular material, this would lead to the Archimedean
jectswhoseimmersedareadoesnotvarywiththe depth, force F = ρgπbd2/4 equal to about 0.2N for the case
A
theexponentαiscloserto1. Fortheverticalpenetration
of Fig. 10(a), which is thus negligible when compared to
ofspheres,cubes,andhorizontalrods,α≃1.2wasfound
the measured drag force.
[30]. For the vertical penetration of spheres and also of
Wedonotobserveanydrasticchangeintherateofin-
cylinders and cones with aspect ratio one and vertical
creaseofF withz ,whichmaybeduetothefactthatwe
b
axis, Ref. [31] found that α ≥ 1,with a α value depend-
do not explore very large depth values nor depth values
ing on the geometry: α ≃ 1 for these cylinders, α ≃ 1.3
close to the bottom wall. The mean slope of F(z ) can
b
for spheres, and α ≃ 1.8 for these cones. This supralin-
be extracted and is displayed in Fig. 10 for different (b)
ear depth dependence of the force (α ≥ 1) is observed
penetration velocities V0, (c) cylinder diameters d, and
atsmalldepthbut isfollowedby asublineardependence
(d) grain diameters d . One can see that the drag force
g
(α ≤ 1) at large depth attributed to the counter force
is approximately independent of V0 and proportional to
generated by the filled-in grains on the top of the ob-
the cylinder diameter d. The F variation with the grain
ject [31]. For a horizontal plate penetrating a granular
diameterd ismorecomplex: WhenF seemstobeabout
g
medium,Ref.[29]foundalinearincreaseoftheforcewith
constant for large graindiameters (d &1mm), it seems
g
depth(α≃1)atsmalldepth,followedbyasaturationat
to increase with decreasing d for small grain diameters
g
larger depth (α≃0) attributed to the Janssen screening
(d .1mm) as already mentioned by [14, 22].
g
wall effect, and finished by an exponential increase very
close to the bottom wall.
V. DISCUSSION
0.8
15 The velocity field and the drag force around a mov-
-1m)0.6 ing cylinder in granular matter show important differ-
N) 10 N m ences compared to the case of a Newtonian fluid. As
F ( 5 >F / z (b00..42 aindmeaptetnedrenoft foafcvt,elothceitydrinagthfoerceexpilnorgedranraunlagremofavtteelorciis-
0 (a)< 0 (c) ties but increases with depth. This differs from Newto-
0 10 20 30 40 50 0 10 20 30 40 50 nian fluids for which the hydrodynamic drag increases
zb (mm) d (mm)
with velocity but is independent of depth, as long as
0.5
-1mm)0.4 -1mm)10..08 flsuuried ovnislcyoslietaydsistiondaepceonndsetnatntofAprrcehsismuerdeesaontbhuaotyparnecsy-
>F / z (N b000...321 >F / z (N b000...642 term. Concerning the velocity field, the perturbation
< 0 (b)< 0 (d) 40 40
0.1 1 10 100 0 1 2 3 4 5
V0 (mm s-1) dg (mm) 30 30
FIG.10. (a)DragforceF onthecylinderasafunctionofits λ / dg20 λ / dsg20
penetration depth zb for V0 = 10mm s−1, d = 20mm, and 10 10
dg = 1mm. Depth force variation hF/zbi, with h·i standing 0 (a) 0 (b)
for an average of zb in the range d/2 ≤ zb ≤ H −d, as a 0 20 40 60 80 100 0 20 40 60 80 100
function of (b) the velocity V0 (for d = 20mm and dg = d / dg d / dg
1mm), (c) the cylinder diameter d (for V0 = 5mm s−1 and
dg =1mm),and(d)thegrainsizedg (forV0 =5mms−1 and FIG. 11. Dimensionless characteristic lengths of the velocity
d = 40mm). Symbols are experimental data and the solid profiles,(a)λ/dg and(b)λs/dg,asafunctionofthesizeratio
linescorrespondtothebestfitsofthedatawith(b)hF/zbi≃ d/dg. (—) Best fit of the data (same symbols as in Fig. 5)
0.29N mm−1 and (c) hF/zbi/d≃0.012N mm−2. with equations (a) λ=2.3dg+0.24d and (b) λs=0.44d.
8
2 40 40
10
30 30
2-2mm s) 110001 δ / dgT01200 λ / dgT1200
T (0 0 (a) 0 (b)
-1 0 20 40 60 80 100 0 20 40 60 80 100
10 d / dg d / dg
-2
10
FIG. 13. Dimensionless characteristic length (a) of tempera-
0.01 0.1 1 10 100
V02 dg /d (mm2 s-2) ture plateau δT0/dg and (b) of temperature exponential de-
crease λT/dg as a function of the size ratio d/dg. (—) Fit of
FIG. 12. Temperature plateau T0 as a function of V02dg/d. ethqruoautgiohnt(hae)dδaTt0a=(sa1m.4edgsy+m0b.o1l2sdasanindF(big)s.λ8T a=nd2d9g).+0.07d
(—) Fit of equation T0 =1.5V02dg/d through the data (same
symbols as in Fig. 8).
“heat” production is due to the flow itself, and the scal-
ing of the temperature plateau T0 appears very close to
created by the cylinder in granular matter appears lo- T0 =(1.5±0.5)V02dg/dinthelog-logplotofFig.12where
calized near the cylinder with an exponential radial de- all the data for different V0, d, and dg collapse along a
crease (B(r) ∼ r[1−exp(−r/λ)]), which differs signifi- slope 1. We have seen in the preceding section (Fig. 9)
cantlyfromtheNewtoniancaseineithertheinviscidcase that the radial extension δ of the temperature plateau
T0
forwhichtheperturbationrelaxeswithapowerlawofthe closetothe cylinder islinkedtothe characteristiclength
distance (B(r) = r(1−d2/4r2)) or the viscous case for λ of the radial variations of the mean velocity. The plot
which the perturbation relaxes slowly at large distance of all data of δT0/dg at different V0, d and dg as a func-
from the cylinder (B(r) ≃rln(d/2r)/ln(Re) for Re ≪1 tion of d/d in Fig. 13(a), exhibits a small but non zero
g
and r ≫ d). The present granular case is closer to yield value 1.4±0.2forvanishing d/d andaslope 0.12±0.02
g
stress fluids where a strong localization is also observed corresponding roughly to δ ∼λ/2. As λ, δ is mainly
T0 T0
[33, 34]. All our λ measurements at different velocities, governed by the cylinder diameter d as long as the size
and cylinder and grain diameters, collapse in Fig. 11(a) ratio d/d is larger than about 10. All these findings
g
when made dimensionless with the grain size dg leading mean that the granularheat is created by the shear flow
toalinearincreaseofλ/dg withthe sizeratiod/dg,with V0/λ∼V0/dclosetothecylinder,inaregionofradialex-
a small but non zero value 2.3±0.5 for vanishing d/dg tension ∼λ (more precisely δT0 ∼λ/2). This shear flow
andslope0.24±0.03. Thismeansthatthecharacteristic leads to velocity fluctuations V0dg/λ ∼ V0dg/d at the
length λ of mean velocity radial variation is mainly gov- grain scale that in turn produce the granular tempera-
ernedby the cylinder diameterdaslong asthe sizeratio ture(V0dg/d)2. Theobservedscalingforthetemperature
d/dg is greater than about 10, which is always the case plateau T0 ∼ V02dg/d should come from an equilibrium
here incontrastto smallintruder size situations studied, betweenthecorresponding“heat”productionbythehigh
e.g., in [23]. In contrast, Fig. 11(b) shows undoubtedly shear close to the cylinder and the granular dissipation
that the “slipping length” λs depends only on the cylin- away from the cylinder. The characteristic length λT of
der diameter d with the scaling λs = (0.44±0.4)d very theexponentialdecreaseofthetemperatureobservedfar
close to d/2. away from the cylinder is mainly governed by the grain
AsthetypicalgranulartemperatureT0 ofthegranular size dg: all the data of λT/dg at different V0, d and dg
flow around the cylinder is connected with the relative plotted in Fig. 13(b) as a function of the size ratio d/dg
velocity V0, there is clearly a strong coupling between onlyshowaslightincreasewithd/dg ofslope0.07±0.07
the meanvelocityprofileandthe temperatureprofile. In from the value 2±1 at vanishing d/dg =0.
thatsense,thepresentsituationresemblesthecaseofthe Theflowlocalization,observedinourcaseinthevicin-
motion of a hot cylinder or sphere in a fluid of tempera- ity of the cylinder, is indeed reported in various physical
ture dependent viscosity. This last situation has already systems. It is a common feature of matter with granu-
been studied as it corresponds to the geophysical situa- larity as recently reviewed by [37]. Such shear bands are
tion of the ascending motion of hot diapirs [35, 36]. In currently observed for flow of granular materials, foams,
this last case, the flow is also strongly localized close to or emulsions. In such systems, the thicknesses of the
thehotspherewheretheviscosityismuchlowerthanfar shearbandsareusuallyoffewgrainsorbubbles. Despite
away, as heat diffuses into a fluid layer near the object, the generality of such observations,there is still no clear
thusloweringthefluidviscosity. Indeed,stresscontinuity unifyingframeworkforsuchshearbands[37]. Inthecase
causesregionsoflowviscositytoberegionsoflargestrain ofyieldstressfluids,theshear-bandswidthsmaybelarge
rates, producing a temperature-induced shear band. In compared to the microscopic sizes [38], and shear local-
this lastcase,heatisproducedoutsidethe flowingmate- ization is observedin Newtonian fluid flow close to walls
rial by the hot sphere. In the present granular case, the at temperatures largerthan the temperature of the fluid
9
far away [35, 36]. 20
Note that in a cylindric Couette device, Refs. [39, 40]
15
havereportedavelocityprofileforthesheareddrygrains b
d
idnedpeepnednednetnotfotfhteh“epcryeslisnudreer”aimngpuolsaerdvbeyloacitpyosasnibdlealesxotienr-- ρgzπb 10
nalupwardordownwardair flow. This has to be related F /
to ourcasein whichthe velocity profilearoundthe mov- 5
ing cylinder does not depend on the depth and thus on
theeffective“pressure”. References[39,40]alsoreported 0
a small zone of constant granular temperature T0 ∼ V02 10 20 d3 0/ dg 40 50 60 70 8090100
and extension δ ≃ 2.8d close to the rotating cylinder
T0 g
followed by an exponential decrease at larger distance FIG. 14. Normalized drag force F/(ρgz πdb) on the cylin-
b
with a characteristic length λT ≃ 4.7dg. The temper- der as a function of the size ratio d/dg. Same symbols as in
ature profile we find in the present study for the flow Fig. 10.
around a cylinder is similar, with a similar scaling of λ
T
but different scalings for T0 and δT0 that should come
from geometric consideration.
VI. CONCLUSIONS
In [27], a model was briefly presented for the granu-
lar flow around the cylinder based on continuous con-
Inthispaper,wehavestudiedbothvelocityprofileand
servation equations for mass, momentum, and granular
force measurements for the granular flow around a hor-
temperature. The phenomenological equations relating
izontal cylinder in the vertical penetration case. While
normal and shear stresses, density, and temperature are
the force increases with the depth, the velocity profile
given by the kinetic theory of granular gases [41] in the
is shown to be stationary during the penetration with a
dense limit. In this approach,the granularmaterialmay
well-defined average and some erratic fluctuations. The
be viewedasafluid whoseviscositydecreases,atagiven
granular flow being close to bidimensional and incom-
pressure,withthegranulartemperature. Fromathermal
pressible, the stream function has been extracted from
pointofview,the heatisproducedbyshearingthe fluid,
the velocity measurements. The flow perturbation is
then diffuses into the material, and is dissipated during
stronglylocalizedclose to the cylinder, exhibiting an ex-
particlescollisions. Itfollowsfromsuchanapproachthat
ponential radial decrease away from the cylinder with a
ashearedzoneincreasesthegranulartemperature,which
characteristiclength λ that scalesmainly with the cylin-
fluidizes the material, increasing the ability to flow. The
der diameter d for large enough cylinder to grain size
numerical simulation of mechanical and heat equations
ratio d/d : λ ≃ 2d +d/4 ≈ d/4 for d/d & 10. The
g g g
shows indeed the presence of such a fluidified zone near
velocity fluctuations quantified by the so-called granu-
the obstacle [27]. The constitutive equation relating the
lar temperature T show also a strong localization near
viscosity to the temperature and the pressure leads to
the penetrating cylinder, with a plateau value T0 in a
a velocity-independent drag force in the creeping flow
narrow crown of extension δ around the cylinder fol-
T0
regime, as observed experimentally.
lowed by an exponential radial decrease with a small
Considering now our observations for the resulting characteristic length λT. The scaling of T0 appears to
2
dragforce onthe cylinder, onecaninfer aneffective fric- be simply T0 ∼ V0dg/d coming from a balance between
tioncoefficientµ=F/(ρgz πdb)basedonahypothetical the “granular heat” production by the shear flow V0/λ
b
effective hydrostatic pressure ρgz at the depth z inside (∼ V0/d for d/dg & 10) over the distance δT0 ∼ λ/2
the granular material. All the data from different V0, d, (∼ d/8 for d/dg & 10) and the granular dissipation far
and dg collapse when plotted as a function of the size away (r > δT0). This granular dissipation which is at
ratio d/dg in Fig. 14, meaning that no other finite size the grainscaleleadsto acharacteristiclengthscalingλT
effect (such as with the container sizes H, L, or b) is ob- for the temperature decrease of a few grain diameters:
served in the present experiment. The effective friction λT ∼3dg.
coefficient appears to be greater than 5 with a plateau Inconclusion,thegranularflowaroundamovingcylin-
value ofabout7±2atlow enoughd/d (d/d .50)and der is very different from the flow of a Newtonian fluid
g g
a possible increase for larger d/d values (d/d & 50). either in a viscous or inertial regime: For the Newto-
g g
This unusual value of the friction coefficient, which is nian fluid case, whatever the regime is, the drag force
about 20 times higher than the usual values of friction increases with the velocity and does not depend on the
coefficientsindensegranularflows[42],hasalreadybeen depth, whereas the contrary is observed for the granu-
mentionedin[3]forthedepth-dependenttermofthedrag lar case for which the drag force does not depend on the
force extracted from impact experiments. This means velocity but increases with the depth. In an upcoming
that the “pressure” is undoubtedly far from hydrostatic paper, we will detail a possible hydrodynamic modeling
in the present situation even if the force variation with ofthisnonclassicalfluidbehaviorbasedonkinetictheory
depth is almost linear. adapted for dense dissipative granular systems as briefly
10
presentedin[27]. The shearlocalizationcharacterizedin orupinpenetrationorwithdrawalsituations,asalready
the present experiment is certainly important to under- observed by [30] and [16].
standtheinfluenceofaclosewallonthemotionofobjects We are grateful to A. Aubertin and R. Pidoux for the
in some practical situations such as the dynamics of im- development of the experimental setup, and E. J. Hinch
pacting spheres [28, 43]. In addition, other experiments and O. Pouliquen for stimulating discussions. This work
have to be performed to understand the very different is supported by the ANR project STABINGRAM No.
drag force experienced by an object when moving down 2010-BLAN-0927-01.
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