Table Of ContentA Multiphase First Order Model for
Non-Equilibrium Sand Erosion, Transport and Sedimentation
L.Preziosi
DepartmentofMathematicalSciences–PolitecnicodiTorino
CorsodegliAbruzzi24,10129,Torino,Italy.
D.Fransos
OptiflowCompany
5 27BoulevardCharlesMoretti,F-13014Marseille,France
1
L.Bruno
0
2 DepartmentofArchitectureandDesign–PolitecnicodiTorino
n CorsodegliAbruzzi24,10129,Torino,Italy.
a
J
5
1
] Abstract
n
y Threephenomenaareinvolvedinsandmovement: erosion,windtransport,andsedimentation. Thispaperpresentsa
d comprehensiveeasy-to-usemultiphasemodelthatincludeallthreeaspectswithaparticularattentiontosituationsin
-
u whicherosionduetowindshearandsedimentationduetogravityarenotinequilibrium. Theinterestisrelatedtothe
l factthatthesearethesituationsleadingtoachangeofprofileofthesandbed.
f
.
s Keywords: multiphasemodel,sandtransport,erosion,sedimentation
c
PACS:81.05.Rm
i
s 2000MSC:76T15,76T25
y
h
p
[ 1. Introduction
1 When the shear stress exerted by wind on a sandy surface is sufficietly strong, sand grains are lifted from the
v
sandbedandaretransportedbywindtosedimentdownstream. Theraisingsandgrainsfollowaballistictrajectory
7
influencedbydragandgravity,eventuallyimpactingagainonthesurfaceandinducingnewparticlestodetachfrom
7
3 the surface. This phenomenon, knows as saltation, generates a layer close to the sand bed with a typical maximum
5 height of 10-20 cm. Saltation is the main reason of erosion of sandy surfaces and together with the consequent
0 sedimentation of sand particles it is the main reason of dune motion and accumulation of sand in specific regions
.
2 whererecirculationoccurs.
0 The engineering interest in understanding and simulating the dynamics of windblown sand, e.g. dune fields of
5 loosesand,isdictatedbytheirinteractionwithanumberofhumaninfrastructuresinaridenvironments,suchasroads
1
andrailways,pipelines,industrialfacilities,farmlands,townsandbuildingsasshowninFig. 1[1].
:
v Moving intruder sand dunes, soil erosion and/or sand contamination can be comprehensively ascribed, from a
i
X phenomenologicalpointofview,tonon-equilibriumconditions,wherethetwoprocesses,erosionandsedimentation,
donotbalance,leadingtotheerosionordepositionofsandonthesoilandeventuallytotheevolutionofthatinterface.
r
a Inotherterms,suchnon-equilibriumsituationsarethemostinterestingcasesfromtheapplicativepointofview.
Bagnold [8] was the first who studied sand erosion and postulated a relation for the sand flux, determining the
importanceofwindspeedandoftherelatedshearstressonthesandsurface.Laterauthors[5,12,17,21,26,28,30,31]
introducedseveralcorrectionstoBagnold’srule,butallthemodelshaveincommontheobservationthatasandgrain
isejectedfromasandbedifandonlyiftheshearstressatthesurfaceislargerthanathresholdvalue.
Sauermann et al. [30] observed that saltation reaches a steady state after a transitory phase of 2 seconds. After
thisperiodthetrajectoriesarestatisticallyequivalentfortheensembleofgrains. Thisphenomenonhappensbecause
PreprintsubmittedtoAppliedMathematicsLetters February19,2015
Figure1:Windblownsandinteractionwithanthropicactivities:farmlands(a),towns(b),railways(c)
thenewejectedparticlesincreasethesandconcentrationinthesaltationlayerandthisreducesthespeedofsaltating
grains. So, asteadystateisreachedwhenallparticlesareejectedwiththesamevelocity(seealso[2,3,6,31]). A
nice mathematical models of the saltation phenomenon is proposed by Herrmann and Sauermann [13] who studied
the dynamics of the surface of a dry granular bed dividing the sand bed into a non-moving time-dependent region
providingsandmassandanothertime-dependentregionaboveitinwhichsandparticlescanmovetransportedbythe
wind. Theyproposeamodelaveragedovertheverticalcoordinate,presentingafreeboundary.
Jietal. [17]coupledak−(cid:15)modelwithamultiphaseapproachinwhichtheslipofthedispersedphaseismodeled
byanalgebraicmodel. Similarturbulentone-dimensionalmodelsareproposedin[24,27,29,30],howeverwithout
amultiphasecoupling. Kangandcoworkers[18,19,20]insteadcoupleamultiphasemodelforthefluidflowwitha
particlemethodforthesandgrains. Asimilarcouplingwasalsousedin[2,4,23]wherehoweverthewindflowwas
computedindipendentlyfromthepresenceofsandparticlesviaasuitableturbulencemodel,typicallythek-(cid:15) model.
Sedimentation has also been widely studied in the literature starting from several applications mainly in envi-
ronmental and chemical engineering. One of the most important component in this phenomenon is the drag force
experiencedbythesedimentingparticlesthathasdrivenalotofattentionbymanyauthorsaswellreviewedin[9].
Differentlyfrompreviouspapers,herewewillproposeacomprehensivemultiphasemodelfortheentireprocess
includingsanderosion,windtransport,andsedimentation,thatworkingalsoinnon-equilibriumconditionsisableto
dealwiththedevelopmentofthestationarysaltationlayerstartingfromgenericinitialandboundaryconditionsandin
particularfromclearairandoversaturatedsituations. Inordertodothatwedevelopaso-calledfirstordermodel(in
time)ofsanderosion,transportanddeposition,thatcanbeeasilytunedusingexperimentaltestcases. Theresulting
advection-diffusionequationforthesuspendedphasecanthenbecoupledwithak−ωmodeldescribingtheturbulent
fluidflow. Themathematicalmodelcanthenbesolvedwiththeaidofthefundamentalerosion/depositionboundary
conditionatthesandbed,thatdependsontheshearstress.
The plan of the paper is then the following. After this introduction, Section 2 presents the mathematical model
mainly focusing on the advective phenomena, on the microscopic dynamics related to the collision between sand
grains,andontheerosionboundarycondition. Theresultofsomenumericalsimulationsfocusingonhowthestation-
aryconditionisreachedwhenwindblowsoveraheterogeneoussandbedarereportedinSection3.
2. TheErosion/Transport/DepositionModel
We considerthe flow ofsand asa multiphase systemcomposed of sandgrains in air. Singlesand grainshave a
densityρˆ andfloatinairwithavolumeratioφ (typicallywellbelow1%),sothatthepartialdensityofsandinairis
s s
ρ = ρˆ φ . Saturationobviouslyimpliesthatφ = 1−φ whereφ isthevolumeratioofair. Themixtureofairand
s s s f s f
sandgrainsisflowingonasandysurfacehavingaclosepackingvolumeratioφ¯ .
s
BecausewindflowisinaturbulentregimethefluidphaseismodelledbytheReynolds-averagedNavier-Stokes
2
equations(RANS)equations. Moreprecisely,ak−ωturbulencemodelisselectedtoprovidetheclosure[25]
ρ∇f·(cid:32)v∂∂fvt=f +0vf ·∇vf(cid:33)=−∇p+∇·[ρf(νa+νt)∇vf]
(1)
∂∂∂∂ωktt ++∇∇··((kωvvf)f)==∇∇··[([ν(νaa++ννt)t∇)∇kω]+]+PkP−ω−γωCkωω2
withstandardboundaryconditions. In(1)kistheturbulentkineticenergy,ωisthespecificdissipationrate,ν andν
a t
are,respectively,airandturbulenceviscosities,P andP aretheproductiontermsforkandω,andγandC aretwo
k ω ω
empiricalcostants.
Indescribingthetransportofsandwestartobservingthatwhilesandparticlesaretrasportedbythewindtheydrift
downwithacharacteristicsedimentationvelocityduetotheactionofgravity[9]. Inaddition,particlecollidegiving
risetoanextra-fluxtermq . Hence,onewecanwritethefollowingequationforthesandvolumeratio
coll
∂φ
s +∇·(φ v +q )=0, (2)
∂t s s coll
where
v =v −w k. (3)
s f sed
This closure can be actually deduced under suitable modelling assumptions from a more general multiphase model
involvingmassandmomentumbalanceforthesuspendedphase.
Thesedimentationvelocityw canbeevaluatedbythebalanceofdragandbuoyancyforcesandstronglydepends
sed
onthegrainsize.Forinstance,ifwedefinetheparticleReynoldsnumberastheonefeltbythesandgrainsofdiameter
dduringtheirflowandthereforebasedontherelativevelocitybetweenairandsolidparticles,Re =φ |v −v |d/ν ,
s f f s f
thenintheso-calledNewtonregime,correspondingtoparticleReynoldsnumbersabove500,thedragcoefficientC
D
isapproximatelyconstant(forinstance,C =3isusedin[27,30]),sothatonehastheclassicalrelation
d
(cid:115)
4(ρˆ −ρˆ )g
w = s f d. (4)
sed 3ρˆ C
f d
However, at the other extreme, i.e., for particle Reynolds number below few units, corresponding to the so-called
Stokesregime,C ≈ 24,sothatonehastheclassicalStokessedimentationvelocity
d Res
(ρˆ −ρˆ )g
w = s f d2. (5)
sed 18ρˆ ν
f f
Consideringthatinaeoliansandtrasportthephenomenonislimitedtothefirstfewcentimetersfromthegroundand
thattheretheparticleReynoldsnumberisbelow100(seeFig. 2a),abetterevaluationofthesedimentationvelocity
withrespecttoEqs. (4,5)canbeobtainedfittingtheexperimentaldependenceofthedragcoefficientontheparticle
Reynoldsnumberasreviewedin[9]andshowninFig. 2b.
Comingtothecollisiontermq ,alreadyintroducedbyBatchelor[10],neclectingitwouldimplythatsandgrains
coll
are only transported under the action of drag and gravity. However, collisions among particles have the important
non-negligibleeffectofgeneratingasortofdiffusionofsandparticlesfromhighertolowerdensityareas,thatresults
fundamentalinthismodellingframeworkforthestationaryformationofthesaltationlayer.
Forhighvolumeratiosnearclosepacking,Auzeraisetal. [7]suggestedthefollowingnonlinearlawonthebasis
ofexperimentaldata
φk
q =−D ∇ s , with k∈[2,5]. (6)
coll eff φ¯ −φ
s s
3
Figure2:(a)ParticleReynoldsnumberasafunctionofthesandgraindiameterofasedimentingparticleinair.(b)Sedimentationvelocityofsand
grainsinstillairintheStokeslimitofEq. (5)andaccordingtotheexperimentssummarizein[9]. (c)Qualitativedependenceofthediffusivity
coefficientνeff fromthedistancefromthesandbed.
Suchatermenforcestheneedofavoidingthattheclosepackingvolumeratioφ¯ isreachedforthesedimentingmass.
s
However,asφ (cid:28)φ¯ inwind-blownsandapplications,therelationcanbesimplifiedto
s s
q =−ν φ k−1∇φ . (7)
coll eff s s
Fromtheconstitutiveviewpoint,thecollisionterms(6)or(7)canbeconsideredasderivingfromtreatingtheensemble
ofparticlesasagas,sothatthestresstermforthesolidconstituentisisotropicthroughacoefficientthatdependson
theparticledensity. SubstitutingEqs. (7)and(3)into(2)givesthenonlineardegenerateadvection-diffusionequation
∂φ ∂ (cid:16) (cid:17)
s +∇·(φ v )− (φ w )=∇· ν φk−1∇φ . (8)
∂t s f ∂z s sed eff s s
Actually,ifthelimitvaluek=1isalsoallowed,onehasthelinearcasesometimesusedintheliterature.
Thecoefficientν mightbeconsideredascomposedofthreecontributionsν =ν +ν +ν ,thattakeinto
eff eff s turb mol
accountofthepossibledependencefrom
• theshearrate,orbetterthevelocitygradientν =ν (∇φ );
s s s
• the turbulence of the flow, so that this term results from the integration of the CFD simulation in a turbulent
regime;
• themoleculardiffusion,butasreportedin[18],thistermcanbeneglectedwithrespecttootherquantities.
Actually,startingfromtheobviousobservationthatthebehaviourofsandparticlesisisotropic. Objectivity,i.e.,
independence of the constitutive dependence from the reference frame, implies that ν is a scalar isotropic function
s
ofatensor. Bytherepresentationtheoremofisotropicfunctionν canthenonlydependontheinvariantsoftherate
s
of strain tensor D = 1(∇v+∇vT). However, since the flow can be considered as a perturbation of a shear flow in
2 (cid:104) (cid:105)
theverticalplane,theleadingcontributionisthesecondinvariantII = 1 (trD)2−trD2 . Forthisreasonweassume
D 2
thatν = ν (II ). Thisisanimportantgeneralizationbecauseallpapersreferstoadependenceonthewindvelocity
s s D
u∗, which is a well defined quantity only on horizontal surfaces, while more general surfaces like dunes require a
dependencefromanobjectiveinvariantoftherateofstraintensor.
Inordertounderstandthemeaningofthecollisiontermwecanlookforstationaryconfigurationsforwhichall
quantitiesdependonlyonthequotezandvelocitiesaredirectedalongaflatplane(atz=0inthedirectionx). Inthis
case,thestationaryprofileinthesaltationlayercanbeobtainedintegrating
∂φ
ν φk−1 s +w φ =0,
eff s ∂z sed s
jointly with the boundary condition at the sand bed. In the simplest case in which ν is constant and k = 1, one
eff
immediatelyhasanexponentialprofilewithacharacteristiclengthrelatedtothethicknessofthesaltationlayergiven
byδ =ν /w andanintegrationconstantrelatedtotheerosionboundarycondition.
s eff sed
Ingeneral,referringtoFig. 2cthedependenceofthecoefficientonthedistancefromthesandbedshowsastrong
diffusionclosertothesurfaceandalowdiffusionatadistanceclosetotheheightofthesaltationlayer,whileparticles
thatescapefromthesaltationlayerareagainstronglymixedduetotheincreasingdiffusionduetoturbulence.
4
Thegeneralfeaturesoftheerosionboundaryconditioncanbeinferredfromexperimentsknownforalongtime
thatshowthat,generallyspeaking,erosiononlyoccursiftheshearstressatthesurfaceτexceedsathresholdlevelτ,
(cid:112) (cid:112) t
orequivalentlythatu∗ = τ/ρˆ exceedsathresholdlevelu∗ = τ/ρˆ (see[22]andreferenccestherein).
f t t f
Referring to the last notation, because it is the one classically used in the literature (though from the numerical
pointofviewwhatiscomputedisτwhichisthencomparedtoτ)onehasthefluxboundarycondition
t
(cid:16) (cid:17)
q =−ν φk−1∇φ ·n=β u∗2 −u∗2 ,
s eff s s t +
(cid:113)
where (f)+ = (f)2+|f| stands for the positive part of f. Bagnold’s formula [8] u∗t = ρˆsρˆ−fρˆfgd is quite successful in
determiningthedependenceofu∗ fromthegraindiameter. Ontheotherhand, thequantificationoftheparameterβ
t
isnotstraightforwardbecausemostoftheexperimentsmeasuresthehorizontalsandfluxparalleltothesurfacewhile
fortheboundaryconditiononewouldneedsomeknowledgeoftheverticalfluxperpendiculartothesurfacewhichis
theonerelatedtotheejectionofsandgrainsfromthesandbed. Veryrecently,onthebasisoftheirexperimentsHoet
(cid:112)
al. [14,15,16]proposedβ= A ρˆ d/g,sothatthesandfluxduetoerosiontakestheform
H f
wαρˆ (cid:16) (cid:17)
q = fβ u∗2 −u∗2 , (9)
s gd t +
wherew=0.5m/sisthesandgrainejectionverticalvelocityevaluatedexperimentally[11,14]andαisadimension-
lessfreeparametertobefittedtoexperimentalsandfluxprofiles.
3. Simulations
As domain of integration we focus on the flow over a horizontal heterogeneous lane. This is neither an artifi-
cial situation, nor a case of limited importance. In fact, most of the landforms in arid regions and roads are well
approximated by a horizontal flat plane. This geometrical setup is also retained in most of the wind tunnel experi-
mental studies present in the literature that are however mainly addressed to the characterisation of the windblown
sandconcentrationandfluxinuniform,in-equilibriumsteadystateconditions. Nevertheless,uniformandsteadystate
conditionsareexcessivelyidealones. Inthesesituationstheincomingwindalreadytransportsthemaximumallow-
ablesanddensity,sothaterosionanddepositionarebalanced,andhencethesandbedsurfaceisneitherscoured,nor
accumulated.Conversely,inmanyengineeringapplications,attentionmustbepaidtonon-equilibriumconditionsthat
Figure3: Consideredsetupwithalternationoferodibleandnon-erodiblesurfaces,ofequilibriumandnon-equilibriumregions,andasketchofthe
heightofthesaltationlayerδs.
willcauseerosionorsettlementofthesandbed. So,assketchedinFig. 3,ourhorizontalplaneischaracterizedbythe
alternationoferodiblesandyregionsandnon-erodibleregions,correspondingforinstancetoastreet. Inthesimula-
tiontheinflowwindiscleanandwithafullydevelopedlogarithmicallyshapedvelocityprofile,withz =10−5mand
0
u∗rangingbetween0.3and1m/s. Thethresholdvalueforerosionisu∗ =0.25m/sandthegrainsizeisd =0.25mm.
t
FromthesimulationshowninFigures4and5,assoonaswindovercomestheboundarybetweenthenon-erodible
anderodiblesurface(thatisputatadistanceL =1mfromtheinflowboundary)thesaltationlayerstartstodevelop.
w
Figure 4a plots the profiles of the sand volume ratio φ at several points at the beginning of the erodible zone. The
s
5
2e!1
x = !0.1 m x = 4.00 m
1e!1 x = 0.0 m x = 6.40 m
x = 0.1 m x = 6.45 m
x = 0.2 m x = 6.50 m
m] x = 0.4 m x = 6.60 m
z [ x = 0.8 m x = 6.90 m
x = 4.0 m x = 8.00 m
1e!2
0 1 2 3 4 5 0 1 2 3 4 5
(a) !s x 10!5 (b) !s x 10!5
Figure4: Sanddensityprofilesinthesaturation(a)andinthesedimentation(b)regions
modelcorrectlypredictstheprogressiveuptakeofsandandincreaseinthedepthofthesaltationlayer,tillsaturation
is reached because of the equilibrium between erosion and sedimentation. The thickness of the saltation layer at
equilibriumisabout10cminqualitativeagreementwithexperiments.Viceversa,asshowninFig.4b,atthebeginning
ofthesecondnon-erodiblezone, thereisareductionofthewindblownsanddensityintheupperpartofthestream
becauseofthesedimentationprocess,notbalancedbysaltation.
Actually, due to diffusion, some sand also diffuses upstream, mainly very close to the surface where diffusion
is dominant. Referring to Fig. 5c, in this region one can notice an exponential growth of the scaled total sand flux
Q/Q where Q is the sand flux at equilibrium. After the first soil discontinuity at x = 0 convection dominates
sat sat
andQsaturatesinalengthclosetoL =3m(seeFig. 5b),thatisnearlyindependentfromthescaledwindvelocity
sat
u∗/u∗,inqualitativeagreementwith[6].
t
Whentheerodiblesurfaceendsthetotalsandflux Qreadilydecreasesinacharacteristicdistancethatincreases
withthewindvelocityasshowninFig. 5b. ItcanbenoticedfromFig. 5dthatthedecreaseislessthanexponentialin
qualitativeagreementwith[16].
In conclusion, the model (1,8) jointly with the fundamental erosion boundary condition (9) and other standard
initialandboundaryconditionsisablenotonlytodescribetheerosion/transport/sedimentationprocessinstationary
situation, butalsotocaptureallfeaturescharacterizingthedevelopmentofthesaltationlayeruptoequilibriumand
ofitsreduction,thatoccurinnaturewhenthesandysurfacesisheterogeneous.
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