PierreA. Gremaud 1
andJohnV.Matthews 1
DepartmentofMathematisandCenterfor ResearhinSientiComputation,
NorthCarolina StateUniversity,Raleigh,NC27695-8205,USA
Abstrat. Theproblemofdeterminingthesteadystateowofgranularmaterials
insilos undertheationof gravityis onsidered.Inthe ase ofaMohr-Coulomb
material, the stress equations orrespond to a system of hyperboli onservation
lawswithsouretermsandnonlinearboundaryonditions.Ahigherorder
Dison-tinuousGalerkinmethodisproposedandimplementedforthenumerialresolution
ofthoseequations.TheeÆienyoftheapproahisillustratedbytheomputation
ofthestresseldsinduedinsilos withsharphangesofthewallangle.
1 Introdution
Inthispaper,thesteadystateowofinompressiblegranularmaterials
un-der the ation of gravity is investigated. Of speial interest is the ase of
owsin silosandbins.Indeed,manufaturingindustriesroutinelystoreand
handle vast quantities of raw materials in granular form. The material is
usually retrieved through outlets at the bottom of the ontainers. Serious
diÆulties during the withdrawal proess are often observed. Those range
from deadzones ofmaterials stikingtheontainer's wallstoviolent
vibra-tionsthat anausethe ompleteollapseofthestruture. Inspiteoftheir
ommonness, those problems arepoorlyunderstood. It isproposed hereto
analyzenumeriallythestrutureandpropertiesoftheorrespondingows.
Thisworkis, totheauthors'knowledge,therstappliation ofahigher
or-dernumerialmethod{athirdorderDisontinuousGalerkinmethod{tothis
typeofappliations.
The two main physial assumptions, whih are disussed in Setion 2,
are that only established, steady state, ows are onsidered, and that the
materialis assumedto beeverywhereat yield.Mostof theexistingworkin
this eld deals with steady state owsin onial hoppers, i.e., when using
spherialoordinates,in domainssuhas
f(r;;');r>0;0<
w
;0'2g;
?
ThisresearhwassupportedbytheArmyResearhOÆe(ARO)throughgrants
DAAH04-95-1-0419andDAAH04-96-1-0097,bytheNationalSieneFoundation
whih orresponds to an innite, onverging hopper of half opening angle
w
. The attentiondevoted to this asestems from tworeasons. First, in a
greatnumberofappliations,theontainersareindeedaxisymmetri,ifnot
downrightpieewiseonial.Seond,asaonsequeneoftheinvarianeofthe
domain underthe salingtransformation(r;;') 7!(r;;') where >0,
similaritysolutions, theso-alled radialsolutions,an be onstruted.This
wasrst observed by Jenike [6℄, and hasplayed afundamental role in the
design ofindustrial hopperseversine[7℄, [11℄.Theradialsolutionsanbe
foundnumeriallybysolvingsystemsofordinarydierentialequations,more
preisely, boundary valueproblems. Theirbehavioris well doumented,see
e.g. [4℄, [8℄. Oneshould note, however, that the domain of appliability of
suhanapproahisquitelimited[5℄.Indeed,theveryimportantaseofthe
juntion between onial hoppers of dierent wall angles for instane, see
Figure1p.7,learlyannot betreatedundertheassumption ofradial
sym-metry.Further,eveniftheonsidereddomainshavetheneessarysymmetry
properties,ifandwhentheradialsolutionseetivelyorrespondto
approxi-mationsofwhatisobservedinpratieisnotlear.Thefullsystemofpartial
dierentialequationsdesribingtheproblemsathandhasthustobesolved.
Thisiswhereourontributionlies.
ThemodelispresentedinSetion2.Setion3isdevotedtothedesription
ofthenumerialmethod. Computationalresultsaredisussed andanalyzed
inSetion4.Finally,someonlusionsandremarksonfutureworkareoered
in Setion5.
2 The model
Theequationsgoverningthetimedependentowofgranularmaterialunder
gravityare derivedand analyzedin [12℄.Those are foundto belinearly
ill-posed in most ases of pratial interest. To the authors' knowledge, the
situation is not fully understood, mathematially orotherwise.In pratie,
stronglytime-dependentproblems areusually observedin onjuntionwith
funnel ows, i.e., ows for whih the motion essentially takes plae in the
entral part of the silo. This paperdeals exlusively with mass ows, i.e.,
ows for whih all the material is mobilized. In this ontext, established,
steadystateowsanbeobserved.
Thespatial domain is assumed to be axisymmetri,but notneessarily
rightonial. Thepartiles areassumed to haveno motionin the axial (or
') diretion.Even thoughthis assumptionmaybeounterintuitiveto uid
dynamiists,itisanestablishedexperimentalfatforgranularmaterials.The
stresstensortakestheform
T = 2
4 T
rr T
r 0
T
r T
0
0 0 T 3
5
Negletingtheinertialterms,onservationofmomentumyields
rT =g; (2)
inwhihisthedensity,takentobeonstant,andthevetorg isthe
ael-erationduetogravity.
Plasti deformation is assumed everywhere. Constitutive models based
on plastiity are onveniently expressed in terms of the prinipal stresses
i
;i = 1;2;3, i.e., the eigenvalues of the stress tensor T. If the prinipal
stressesare ordered
1
2
3
, thenthe Mohr-Coulombyield ondition
reads 1 3 =
1+sinÆ
1 sinÆ ;
where Æ is the angle of internal frition. This relation anbe derived from
thelawofslidingfrition[7℄,Chap.3.Thisonditionanbeexpressedinthe
originalstressvariables
(T rr T ) 2
+4T 2 r =sin 2 Æ(T rr +T ) 2 : (3)
The Haar-von Karman assumption an be invoked to evaluate the
ir-umferential stress T
''
. Indeed, the Mohr-Coulomb analysis merely states
1 T '' 3
.Foraxisymmetrionverginghoppers,theHaar-vonKarman
assumption states that T
''
is in fat the major prinipal stress. One an
writetheorrespondingequationsin termsoftwounknownsT
rr andT r T rr T r
=f(;;T
rr ;T r ) T r T
=g(;;T
rr ;T
r )
(4)
wheresomesimpliationsresultfromtheuseofthenewvariable = lnr;
wedonotbotherto renamethestressvariables. Therighthand sideterms
aregivenby
f(;;T
rr ;T r )= 3 s 2 T rr
+otT
r 3+s
2 T
+ge
os
g(;;T
rr ;T r )= 1+s 2
otT
rr +3T
r +
1 s
2
otT
ge
sin (5)
where we haveset s=sinÆ, and, forfuture referene, = p
1 s 2
=osÆ.
The equation of state, whih relates T
to the unknowns T
rr and T
r is
the yield ondition (3). It should be notied that (3) is the equation of a
one in the spae (T
rr ;T
r ;T
), whose entral line bisets the (T
rr ;T
)-plane.TheorrespondingrelationbetweenT
andtheunknownsistherefore
notaproperfuntional relation,but ratherassigns thedependent variables
(T
rr ;T
r
) tolie onamanifold: theyield surfae.Thesituation greatly
bearguedthatT
liesonthe\top"oftheyieldsurfae.Aordingly,by
solv-ing(3)forT
,theequationofstateompleting(4, 5)is
T
=h(T
rr ;T
r )
1+s 2
1 s 2
T
rr +2
s
s 2
(1 s 2
) 2
T 2
rr 1
1 s 2
T 2
r
: (6)
This orresponds to the so-alled passive state [7℄, [9℄, as opposed to the
ativestate whih is theother solution from (3), obtainedbyhangingthe
signinfrontoftheradialin(6);see x5.
Rewriting(4)as
U+
F(U)=G(;;U); (7)
with theobvious notation,oneanthen analyzetheeigenvalues
1;2 of the
JaobianF 0
. Afewalulationsleadto
1;2
=tanÆ 1
r
sT
rr T
r
sT
rr T
r
: (8)
A quikanalysis[3℄revealstheeigenvaluestobereal anddistint,provided
that onestays\inthe one",i.e.,j T
r
T
r r
j <tanÆ.Inother words,the steady
statestressequations(4,5,6)formastritlyhyperbolisystemofnonlinear
onservationlawswithsoureterms.Theradialandangularvariables and
anbethoughtofastime-likeandspae-likevariables,respetively.
Thesystemhastobeompletedwith\initial"andboundaryonditions.
The\initial" onditionused hereorrespondsto presribingthestresshigh
up in the hopper,say, on an\ =onstant" surfae,and solve down from
there.Severalargumentsanbeonsideredtojustifythefatthatthestress
information travels downward, see [3℄, [7℄, or [11℄ for more details. In the
alulationspresentedinthispaper,the\initial"onditionisomputedfrom
theradialstresseld;seex4fordetails.Finally,theboundaryonditionsare
givenbythelawofsliding frition.Atanypointonthewall,themagnitude
ofthetangentialstressjT
T
jisproportionalto themagnitudeofthenormal
stressjT
N j, i.e.,
jT
T
j=jT
N j;
where > 0is the oeÆient of wall frition. Ina purely radialgeometry,
theaboveboundaryonditionbeomes
T
r
=T
onthewalls (9)
witha\+"signononesideofthehopperanda\-"signontheother.Observe
that thoseonditionsarenonlinearintheunknownsT
rr ;T
r .
At this point, it is worth mentioning that the equations an in fat be
approahwasforinstanetakenin[8℄and[9℄.However,theuseof this
non-linear hange of variables has the unfortunate side eet of destroying the
onservationformoftheequations,losinginthiswaytheabilitytoompute
shoksin any reliableway. This prevents,for instane, theomputation of
stresses ourring at the juntion between onialhoppers of dierent wall
angles, oneofthemain goalsof this work.Further, asiswell known, many
purely numerialproblems also appear when solvingsystemsin
nononser-vationform.
3 The algorithm
For the sake of simpliity, we only desribethe algorithm in the ase of a
onial hopper. The method used is a formally third order Disontinuous
Galerkinsheme,see[2℄andtherefereneslistedtherein.
Let
0
= lnr
0
,wherer
0
isthevalueoftheradialvariablewestartfrom.
Let
w
>0bethehalf openingangle,andlet=
w
=N bethemeshsize,
N being the number of ells. In this axisymmetri setting, the problem is
spatiallyessentiallyone-dimensional,andthusnoeortshavebeenmadeto
adaptthemesh. Wedene
V =fv2L 1 (0; w ) 2 :vj Kj 2P k (K j ) 2
;j=1;:::;Ng;
whereK j =[ j 1 ; j
℄isthej-thell,with
j
=j,andP k
(K
j
)standsfor
thespae of thepolynomials ofdegreeat mostk in K
j
.We usek =2or3
below;seex4.AsemidisretizationonsistsoflookingforU
h
(;)2V, >
0 ,
suh that U
h ( 0 ;) = V (U( 0
;)), where U(
0
;) is an \initialondition",
seex4,
V
isaprojetionoperatorintoV, and
d d Z Kj U h
(;)v()d+
+ v( j 1=2 )H j 1=2 5 X l=1 ! l F(U h (; j;l )) d d v( j;l ) = 5 X l=1 ! l G(; j;l ;U h
(;))v(
j;l
); 8v2V;j=1;:::;N:
In the previous expression,
+
stands for the usual dierene operator,
+ U j =U j+1 U j
and the oeÆients!
l
and the nodes
j;l
, l=1;:::;5,
j = 1;:::;N stem from the use the lassial 5-point Gaussian quadrature
formula.Weusetheloal Lax-Friedrihsux
H j+1=2 = 1 2 F(U j+1=2
)+F(U + j+1=2 ) j+1=2 (U + j+1=2 U j+1=2 ) ; where j+1=2
isthemagnitudeofthelargesteigenvalueofaproperlyhosen
Roe average matrix A
j+1=2 ( dF ) U=U j+1=2
bakground. Themass matrix anbemade diagonal by hoosing the basis
funtions asLegendrepolynomialsovereahell[1℄.
The oeÆients of U
h
(;) an then be grouped in a vetor U(). The
unknownvetorU()satisesthefollowingsystemofODEs
d
d
U =F(U)+G(;U) (10)
where F(U) and G(;U) ome respetivelyfrom thedisretization of F(U)
and G(;;U) in (7). Note that we use below an unsplit approah. This
is justied rst by the fat that the soure term G(;U) is not sti and
seond,bytherealizationthatthedeliateinterplaybetweenG(;U)andthe
nonlinear boundary onditions would render the implementation of asplit
algorithmalotmoreinvolvedthanthepresentapproah.
Thedisretizationwith respet to involvesathird orderTVD
Runge-Kutta [13℄ ombined with a loal slope limitingproess. Let > 0be a
onstantinrementin andlet n
=
0
+n;thealgorithmreadsthenas
follows,see e.g.[2℄
{ SetU (0)
=(U
h (
0 ;));
{ Forn=0;:::;N 1,omputeU n+1
h :
1. setU (0)
=U n
h ;
2. fori=1;:::;3,omputetheintermediatestages
U (i)
=
0
i 1
X
j=0
ij U
(j)
+
ij (F(U
(j)
)+G( n
+d
j
;U
(j)
)) 1
A
;
3. setU n+1
h =U
(3)
.
The numerialparametersf
ij g,f
ij
gand fd
j
g,i =1;2;3,j =0;1;2are
respetivelydenedas
1
3=41=4
1=3 0 2=3 1
01=4
0 0 2=3 0
1
1=2
A thorough desription of the loal slope limiting operator , whih is
basedontheuse ofaorretedminmod funtion,anbe found,e.g.,in [2℄;
it isnotrepeatedhere.Notethat both andthe properimplementation
of the boundary onditions (9) requiretransformation to the harateristi
elds,see[3℄forimplementationdetailspertainingtothepresentappliation.
4 Computational results
θ
w
-Γ
0
Γ
0
θ
w
-Θ
w
-P
Q
Q
Θ
w
-
P
Fig.1.Geometrialsituation;left:transitiontoaatterwallangle,right:transition
toasteeperwallangle.Theradialstressisusedtogenerateaninitialonditionon
theurve
0
.Thedomainsofalulationareshaded.
AnypointP admitstworepresentations,namely,(R ;)and(r;),
orre-spondingto thenaturaloordinatesystemsfortheupperandlowerhopper
respetively. The transition is loated through the point Q, see again
Fig-ure 1,where Q=(R
0
;
w )= (r
0 ;
w
). For givenvalues of the material
parametersÆand,thenumerialapproahonsiststhenin
{ generating the radialstress eld T in the upper hopperf(R ;): R >
0;jj
w
g[4℄,[7℄;byonstrution,atapoint(R ;),theradialstress
eldisgivenbyRT();
{ interpolating the radial stress eld on the urve
0
= f(r;) : r =
r
0
;jj
w
g,leadingtoastresstensorS
0 ;
{ hangingtothenewoordinate systemthroughR T
( )S
0
R( ),
whereR()istherotationmatrixofangle;
{ solvingin thelowerhopperf(r;): 0<r <r
0
;0< <
w
gusing the
algorithmdesribedinx3.
Notiethatby(anti)symmetry,oneansolveinonehalfofthedomainonly,
thelawof slidingfrition (9) validon thewallsbeingreplaed bythe
sym-metryonditionT
r
=0ontheentralline.
Some omments are in order. First, the fat that the initial ondition
is generated from the radial stress eld impliitly assumes that this very
solutionissoughtandrealizedbytheproblemintheupperpartofthehopper.
Seond,italsotakesasgrantedthattheradialsolutionreahestheurve
0
unperturbedbythewallorner.Thislastpointislearlysatised,assuming
againadownwardpropagationoftheinformationforthestresses,inaseofa
transitiontoaatterhopper,i.e.,
w <
w
,seeFigure1,left.Ifthetransition
on theharateristi urvesreveals that the dierenein angles should not
betoolarge,namely
w
w
<artanj 1
max
j;
w ;
w >0;
where
max
isthelargesteigenvalueinmodulusofF 0
,see(8),evaluatedfor
the radial stress eld on the urvef(R
0
;) : jj
w
g.Third, the ase
w =
w
anbe used to hek that the algorithm eetively preservesthe
radial solution. The interation between the boundary onditions and the
foringterms rendersthis numeriallydeliate. Further,the radialsolution
itselfmaybeunstable[4℄.Thepresentapproahpreservestheradialsolution
withagreatdegreeofauray,see[3℄formoredetails.Finally,itshouldbe
noted that the above initial ondition is not onsistent with the boundary
ondition(9),unless
w =
w .
Inthealulationsbelow,theopeningangles
w and
w
arerespetively
takenas15 o
and 10 o
in arstexperiment,and as10 o
and12 o
,in aseond.
The material parameters orrespond to the ase of orn in a steel hopper.
The angle of internal frition Æ is 32:1 o
, while the angle of wall frition is
11:7 o
,in otherwords,thewallfritionis=tan11:7 o
.
Figure 2 orresponds to the ase of a transition to a steeper hopper,
whereas in Figure 3, the transition is to a shallower hopper. Some
predi-tionsaboutsuhtransitionsanbefoundintheliterature,seee.g.[7℄,x7.12.
Considerationsbasedon analogywithorresponding FluidDynamis
prob-lems and/or on the use of radial solutions have been advaned, prediting
smooth\rarefation"wavesfortransitionstosteeperhoppers,andshoksin
theaseoftransitionstowiderones.Wenoteherethatneitherofthosetypes
of arguments an be fully justied. The present results shed somelight on
thisproblem.
First,intheaseofatransitiontoasteeperhopper,theresultsreported
inFigure 2learlyshowthatthereisindeedformationofararefationwave.
Onean observe,however,thatafter thewavesgenerated onopposite sides
start to interat, they sharpen onsiderably and shoks appear. In Figure
2, both the P 1
and P 2
ases are reported. Although the P 2
results oera
slightlybetterresolution, theyalsosuerfromsmallosillationsthatanbe
observedaroundtheenterline. Thesmallosillationsstemfromnumerial
diÆultieslinkedtotheformoftheforingtermsthere,seetheottermin
(5),and thedeliateinterplaywiththelimitingproedure.Suhosillations
werenotobservedfortheP 1
ase.
Intheaseofatransitiontoashallowerdomain,theresultsofFigure 3
showthe immediateformationofshoks inthestresseld. Inthis ase,the
osillationsoftheP 2
alulationsaremorepronouned;weonlydisplaythe
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
T
rr
/r, P
1
x
y
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
T
r
θ
/r, P
1
x
y
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
T
rr
/r, P
2
y
x
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
T
r
θ
/r, P
2
y
x
Fig.2.Strutureofthestresseldinduedbyatransitionfromanopeningangle
of 15 o
to oneof 10 o
.Valuesof the parameters:Æ = 32:1 o
, =tan(11:7 o
) (orn,
steelwall).Firstrow,P 1
elements;seondrow,P 2
elements.
0.7
0.75
0.8
0.85
0.9
0.95
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
T
rr
/r, P
1
y
x
0.7
0.75
0.8
0.85
0.9
0.95
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
T
r
θ
/r, P
1
y
x
Fig.3.Strutureofthestresseldinduedbyatransitionfromanopeningangle
of 10 o
to oneof 12 o
.Valuesof the parameters:Æ = 32:1 o
, =tan(11:7 o
) (orn,
steelwall).P 1
5 Conluding remarks
We have presented a numerial study of stress elds indued by the
dis-hargeof granularmaterialsinhoppers.Thestress equationsorrespondto
asystemof hyperbolionservationlawswithseveral nonstandardfeatures.
A higherorder DisontinuousGalerkin method hasbeenimplemented. The
orrespondingnumerialresultspartiallyonrmseveral\eduatedguesses"
thathadbeenmadeaboutthestresseldstrutureinduedbyhangesinthe
wallangle.Amoreompletepitureoftheowshouldinvolvetheresolution
oftheveloityequations,whihwillbeoveredin futurepubliations.
Aknowledgments
TheauthorswouldliketothankDavidShaeerandMihaelShearerfor
manyhelpfuldisussions.
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orretiveinsertsforgranularmaterialsinonialhoppers.SubmittedtoInt.J.
NonlinearMeh.
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