Stochastic modelling of three-dimensional particle rebound from isotropic rough wall surface

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Stochastic modelling of three-dimensional particle

rebound from isotropic rough wall surface

Darko Radenkovic, Olivier Simonin

To cite this version:

Darko Radenkovic, Olivier Simonin. Stochastic modelling of three-dimensional particle rebound from

isotropic rough wall surface. International Journal of Multiphase Flow, Elsevier, 2018, 109, pp.35-50.

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Radenkovic, Darko and Simonin, Olivier Stochastic modelling of three-dimensional particle

rebound from isotropic rough wall surface. (2018) International Journal of Multiphase Flow,

109. 35-50. ISSN 0301-9322

OATAO

Stochastic

modelling

of

three-dimensional

particle

rebound

from

isotropic

rough

wall

surface

Darko

Radenkovic

a,∗

_,

_Olivier

_Simonin

b a Faculty of Mechanical Engineering, University of Belgrade, Belgrade, Serbia

b Institut de Mécanique des Fluides (IMFT), Université de Toulouse, CNRS, Toulouse, France

Keywords:

Three-dimensional rough wall Stochastic modelling Deterministic simulations Particle-wall interaction Shadow effect model Multiple particle rebounds

a

b

s

t

r

a

c

t

Thispaperdescribesanextensionofthetwo-dimensionalapproachtoparticle-roughwallcollision mod-elling(Sommerfeldand Huber,1999;Konan etal.,2009)tothe caseofthree-dimensionalparticle re-boundfromanisotropicroughwallsurface.Thevirtualthree-dimensionalroughwallisrepresentedas aGaussiancorrelatedsurface.Normalvectoranglestatisticaldistributionsareinvestigatedindetailfor suchvirtualroughwalls,andastatisticalmodellingapproachfortheseanglesisproposedandvalidated intheframeofthelowroughnessapproximation.Next,deterministicsimulationsoffullyelasticparticle collisionswiththethree-dimensionalvirtualwallroughnessstructurearecarriedoutforvarious parti-cleincidentangles.Itisshownthatthereboundangle,inthebouncingplaneoftheparticle,obeysthe distributiongivenbythetwo-dimensionalmodellingapproach.However,thethree-dimensionalstructure inducesatransversedeviationbouncinganglethatobeysaGaussiandistributionwithastandard devi-ationthatincreaseswithincreaseinincidentangle.Astatisticalmodellingapproachforthevirtualwall normalvectorseenbyanyparticleforagivenincidentangleisproposedandvalidatedfrom determin-isticsimulationresults.Theprobabilitythatparticlesmakeonlyonereboundisinagreementwiththe two-dimensionalmultiple-collisionmodelassumption.Anewstochasticprocedureforparticle-isotropic roughwallinteractionsinaLagrangianframeworkisdevelopedandverifiedbycomparisonswith deter-ministicsimulationsandavailableexperimentalresults.

1. Introduction

Particle-laden wall-bounded turbulent flows have wide indus-trialapplicationsandforthatreasontheyhavebeenstudied exten-sively using both numerical and experimental approaches.These flowsarehighlycomplexinspaceandtimeandmanymechanisms are often coupled.Experimental studies (Sommerfeld andKussin, 2004;Bensonetal.,2005)andnumericalstudies(Tsujietal.,1987; Sommerfeld,2003;SquiresandSimonin,2006;Vreman,2007; Ko-nanetal., 2011;Breueretal., 2012;MalloupasandvanWachem, 2013; Vreman, 2015) haveindicated that thewall roughnesscan significantlyinfluencebothparticleandfluidflows.Themaineffect ofwallroughnessinconfinedparticle-ladenflowsisre-dispersion of particles by amplification of the particle wall-normal velocity component. Thisleadstomoreuniformparticle profiles,ahigher wall collisionfrequencyandconsequently agreaterpressuredrop in particle-laden flow than in single-phase flow, increasing with solidmassloading.

∗ _{Corresponding author.}

E-mail address: dradenkovic@mas.bg.ac.rs (D. Radenkovic).

Intheparticle-roughwallmodellingapproach,theactualrough wall is often replaced with a virtual wall (see, for example,

Sommerfeld and Huber, 1992). In such an approach, the compu-tationof the interaction between a spherical particle and a true rough surfaceisreplaced withan equivalentinteraction between theparticlecentreandaneffectivevirtualwall.

Deterministicandstochasticmodellingapproachesmaybe dif-ferentiatedaccordingtothemethodusedforpredictionofthewall inclinationseenbyincidentparticles(seeFig.1).Inthe determin-istic wall modelling approach, a rough wall surface is generated beforehandandthewall inclinationseen byanyincident particle is calculated when the distance betweenthe particle centre and theroughsurfaceisequaltotheparticleradius(Vreman,2015)or lessthantheparticleradius(DeMarchisetal.,2016;Milici,2018). Indeterministic simulations whoseresults are shownin this pa-per wall inclination is calculated when particle centre intersects virtualwall whichisidenticaltothe casewherethedistance be-tweenparticlecentreandtrueroughsurfaceisequaltoparticle ra-dius.Inthestochasticmodellingapproach,whentheparticle cen-trereachesparticleradiusdistancefromacertainlimitboundary,a virtualwallinclinationisgeneratedaccordingtoarandomprocess

Fig. 1. Modelling approaches of particle - wall interaction. (a) In stochastic approach, when particle centre comes at a half of particle diameter D p distance from smooth

macroscopic boundary, virtual wall with inclination angle αis generated. (b) In deterministic approach, particle - wall collision is detected when particle centre crosses virtual wall (sketched with dashed line in figure). The virtual wall properties at the true particle - wall contact is taken identical to the one of the crossing point of the particle centre. The sketched roughness height is exaggerated with respect to the particle diameter.

followingagivenprobabilitydistribution dependentonthe parti-cleincidentangles.Deterministicmodelsaremorecomputationally expensiveowing totheneed tofindthe exactpointofimpact of theparticlesurfaceontheroughwall(orinthecaseofvirtualwall generation,theexactpointofimpactoftheparticlecentreonthe virtual wall) and, in practical Lagrangian simulations of particle-ladenflows,stochasticmodellingapproachesarepreferred.

The virtual wall concept was introduced by

Tsuji et al. (1985) and subsequently redefined (Tsuji et al., 1987). In these models, parameters were empirical, determined by comparisons between experimental and numerical studies in a horizontal pipe. The virtual wall was introduced for incident particle angles less than 7°. For larger incident angles, the vir-tual wall was not introduced, assuming that after collision of the particles with the flat wall, the particles returned into the flow with a significant wall normal component. However, more correctparticlerebound wouldbecalculatedifavirtual wallwas introduced forall particleincident angles. The readeris referred toKonan etal.(2009) fora detailedreview ofother virtual wall concepts.

For most surfaces in engineeringpractice, virtual wall rough-nessinclinationsmayberepresentedusingGaussiandistributions, withzeromeanandstandard deviations



γ

that depend onthe wall structure and the particle diameter (Sommerfeld and Hu-ber, 1992; 1995; Konan etal., 2009). However, owing to the in-cident perspective, particles at low incident angles do not see the lee side of roughness with the same probability as the luv side of the roughness. This is called the shadow effect and was originallyintroduced bySommerfeldandHuber (1999).Following

Sommerfeld and Huber (1999), the inclination angle of a virtual wallcan besampledfroma givenmodifiedGaussiandistribution (herereferredtoastheeffectiveSommerfelddistribution)orusing asimplified proceduretoaccount fortheshadoweffect(here re-ferredtoastheshadoweffectmodel).Intheshadoweffectmodel, thevirtual wall angle is sampledfroma truncatedGaussian dis-tributiontoensurethat theparticleincidencetowardsthevirtual wallisrealizable.Finally,inbothapproaches,ifaparticledoesnot returnto theflow aftercollisionwiththewall,virtual wall angle samplingandparticle-wallcollisioncomputationarerepeated.

With an effective Sommerfeld distribution or by using the shadoweffectmodel,alargenumberofgrazingparticleswere pre-dicted corresponding to particles with smallincident angle. This wasexplainedby the fact that thesemodels did not account for themultiple rebounds(Konan etal., 2007) thatmay occurif the particleisbouncingwithasmallpositiveanglesothattheparticle collideswithanotherasperity inthewall regionbeforereturning totheflow.

Konanetal.(2009)computedinadeterministicsimulationthe probability that a particlewitha small bouncing anglewill have

asecondcollisionwiththewallbeforeleavingthewallregion.An analyticalfunctiondependingonlyonthestandarddeviationofthe wall roughnessangle was proposed forthis probability distribu-tion.Finally,multiplereboundsweremodelledusingthestochastic approach, allowing an effective methodin the Lagrangian frame-work called the rough wall multiple-collision model to be pro-posed.

Vreman(2007)carriedoutaDNS-DPSstudyofgas-particleflow throughaverticalpipewithmodelledparticle-roughwall interac-tion.Thevirtualwallnormalvectorwascalculatedasn+

χ

s/

||

n+

χ

s

||

, where projections of vector s were uniform random val-uesfrom−1/√3to1/√3(||s||<1) andthespecularitycoefficient

χ

=0_.2wasroughlyestimatedforthealuminumpipeusedinthe experiment,sinceroughnesswasnotmeasuredintheoriginal ex-periment.Itwasshownthattheroughwallmodelresultedin bet-teragreementwithexperimentalresultsthanwasthecasewithout wallroughness.

Squires and Simonin (2006) studied the influence of wall roughness on the disperse phase in a vertical channel gas-solid flow. LES-DPS was applied with one-way coupling and without inter-particle collisions. The virtual wall was defined from the normal surface vector n=[sin

(

φ

)

cos

(

θ

)

,cos

(

φ

)

,sin

(

φ

)

sin

(

θ

)

], where

φ

and

θ

hada Gaussian distribution with standard devi-ation



γ

. After each particle-wall impact calculation, if the re-bound wall-normalvelocity componentwasnot directed intothe flow,anglesamplingwasrepeated.Particlewallbouncingwas cal-culatedasfrictionless.

Konanetal.(2011)comparedtheinfluencesoftheshadow ef-fectmodelandroughwallmultiple-collisionmodelontheparticle phasepropertiesfromdetachededdysimulationsofparticle-laden flowinahorizontalroughwallchannel.Theeffectsofwall rough-nesswerelesspronouncedinthecaseoftheshadoweffectmodel owingtothelargenumberofgrazingparticlesgeneratedwhichled toweakerparticleverticaldispersion.Thisproblemwasefficiently solvedwiththeroughwallmultiple-collisionmodel.

Breueret al.(2012) proposed a model forparticle-rough wall interaction inwhich the actual wall surfacewas replacedwith a surfacecoveredbymono-sizedspheresthatrepresentedsimplified sand-grain roughness. The standard deviation of the wall rough-ness angle was determined from the mean roughness height Rz (or RMS roughness Rq) and particle diameter. This choice of pa-rameters permitseasierapplication ofthe modelin practical en-gineering problems. Shadow effects were modelled, but multiple reboundswere omitted.The model wasapplied intheir LES-DPS ofhorizontalparticle-ladenchannelflow.

Malloupas and van Wachem (2013) created a model for

particle-wall interactions ina soft-sphereframework andverified it in an LES-DPS of horizontal particle-laden channel flow. Mul-tiple particle rebounds were treated with successive addition of

virtual walls.The firstvirtualwall wasgenerated,withrespectto theshadoweffectmodelwhenaparticlereachedthewall rough-ness amplitude added to the smooth wall. If after impact with thatvirtualwalltheparticlemovedclosertothewall,another vir-tualwallwasgeneratedandtheimpactwasagaincalculated.This mechanismwasrepeateduntiltheparticleleftthewallregion.

Cheng andZhu(2015) investigatedparticle-wallcollisionsina connectedsystemof virtualwall cellscalledavirtual-wall-group. ItwasshownthataneffectiveSommerfelddistributionwasa sim-plified case of their new probability density function (PDF) for wall roughnessangles. A distinctionwasmade betweenparticles that had positive and negative rebound angles. The rough wall multiple-collision procedure ofKonan etal. (2009)was modified andthe resultingprocedure wasappliedin RANS-DPSsimulation ofparticle-gasflowinaconfinedplanarjet.

Vreman(2015)performedDNS-DPSinaverticaldownward gas-solid channel flow. Particle-rough wall collisionswere calculated in a deterministic manner with the wall roughness modelled as densely packedhalf-sphereswiththeflat sidesfixed toa smooth wall. It was reported that the wall roughness enhanced turbu-lenceattenuation,althoughthediametersofthehalf-sphereswere smallerthantheviscouswallunit.Thenon-uniformpartofmean force that free particles exerted on the gas phase, the so-called two-waycouplingeffect,wasfoundtoinfluenceturbulence atten-uationsignificantly.

Theaimofthisstudyistoinvestigateparticle-wallinteractions inathree-dimensional(3D)framework,foranisotropicrough sur-face, as natural extensions of the work of Sommerfeld and Hu-ber (1999) andKonanet al.(2009) inorder to providea greater level of detail for particle-rough wall interactions, especially re-gardingtransversereboundcharacteristics.

The paperisorganizedasfollows.In Section2,thegeneration oftheisotropicvirtualroughwallsurfaceisdescribedandthe an-glesofthenormalvectortothevirtual wallsurfaceareexamined ina 3D frame andstatisticallymodelled.InSection 3,the proce-dure for the deterministic simulation of particle 3D rough fully elastic rebound is outlined andthe resultsof thissimulation are described. The statistical model for3D particle collision with an isotropicrough virtual wall isproposedandverified inSection4. In Section 5, this3D stochastic model isapplied to the particles undergoing rough wall inelastic frictional collisions. Obtained re-sultsarecomparedwithdeterministicsimulationresults,the two-dimensional (2D) stochastic model predictions, and available ex-perimentaldata.

2. Generationandpropertiesofthevirtualroughwallsurface

2.1. DescriptionofGaussianrandomroughsurfacegeneration A virtual 3D rough wall (see Fig. 2) is created according to the procedure of Garcia and Stoll (1984), as implemented by

Bergström(2012).Inthisprocedure,aGaussiancorrelatedsurface, whichhererepresentsavirtualroughwall,isobtainedintermsof RMSroughnessheighth(intheydirection)andcorrelationlength scalesinthexandzdirections,cLxandcLz,respectively.Inthecase of isotropic surfaces,studied in this work, the correlation length scalesinthexandzdirectionsareidenticalcLx = cLz = cL.

Thefirststepincreatingtheroughsurfaceisthegenerationof uncorrelated Gaussian random numbers yu(x,z) with zeromean andstandarddeviationhonthemeshinthex− z plane.Correlated rough wall surfacecoordinates are then obtainedby convolution withaspatialfilteras:

y

(

x,z

)

= +∞  −∞ +∞  −∞ f

(

x− x,z− z

)

yu

(

x,z

)

dxdz (1)

Fig. 2. Virtual rough wall with RMS roughness height h = 0 . 63 μm and correlation length scale c L = 10 μm. The mesh resolution is δx = δz = 1 μm.

Fig. 3. Angles of the virtual wall normal vector with the coordinate axes.

where f

(

x,z

)

=√2

π

cL exp−2



x2_+z2



c2 L (2)

represents a Gaussian filter for isotropic surface generation. The

Eq. (1) is calculated with the fast Fourier transform algorithm (FFT). Standarddeviationoffilteredroughnessy(x,z) is thesame asthestandarddeviationofGaussianrandomnumbersyu(x,z).

Inthedeterministicsimulation,theparticle-wallcollisionis de-tectedwhen a particlecentretrajectory crossesthevirtual rough wall surface through an elementary triangular cell. Unit vectors canthen be formedalong thetwo sidesofthat cellin they− x andy− z planes,andtheir vector product givesthe normal sur-facevectordirectedtowardstheflow.

2.2.Characterizationofthe3Dvirtualroughwallnormalvector Letusdefine(

ξ

,

η

,

ζ

) ∈[0,

π

]3_{,vectoranglesbetweenthe} vir-tualrough wallnormalvectornandtheunitvectorsalongthex, yandzaxes,respectively,asshowninFig.3.

Asshown inFig.3, projectionsofthe virtual normalvector n

aredefinedwith,

nx=cos

(

ξ

)

, ny=cos

(

η

)

, nz=cos

(

ζ

)

,

(

ξ

,

η

,

ζ

)

∈[0,

π

]3

(3)

sothatthevirtualwallnormalunitvectornmaybewrittenas, n=cos

(

ξ

)

i+cos

(

η

)

j+cos

(

ζ

)

k (4)

wherei,jandkaretheorthogonalunitvectorsinthedirectionof thex,y andzaxes, respectively.Scatter plotsofvirtual wall nor-malvectorangles(

η

,

ξ

),(

η

,

ζ

)and(

ζ

,

ξ

)areshowninFig.4,from

-4 50

x[µm]

o

z

y 50 -50 -50

z[µm

]

X

Fig. 4. Scatter plots of virtual rough wall normal vector angles (in degrees): ξ,ηand ζ. The bold numbers represent the correlation coefficients between the corresponding angles. The virtual rough wall examined is characterized by a ratio of RMS height to correlation length scale: h/c L = 0 . 031 , numbers of mesh nodes in the x and z directions Nx = N z = 500 . The dimension of the sampled domain is much larger than the correlation length scale c L .

Fig. 5. Dependence of the RMS virtual wall slope in the x and z directions, σx and σz , respectively, and virtual wall normal vector standard deviations ξ and ζ,

on the ratio of the RMS roughness height h to the correlation length scale c L of the

virtual rough wall. The virtual rough walls examined have N x = N z = 500 nodes. The

sampled domain is much larger than the correlation length scale c L .

angles

ξ

,

η

and

ζ

calculated in every triangular cell of the vir-tualwall.Thecorrespondingdistributions showthatangles

ξ

and

ζ

obeytwouncorrelatedGaussiandistributions,withmeanvalues

ξ

=

ζ

=

π

/2.

FromTsangetal.(2000),itisknownthatforaone-dimensional Gaussianroughnessprofileitholdsthat,

σ

=√2h cL

(5)

where

σ

istheRMSvalueofthewallroughnessslope.

In Fig. 5, (5) is compared with the RMS of the virtual wall slopesalongthexandzdirections,wheretheseslopessx=−nx/ny andsz=−nz/ny,respectively,arecalculatedineverytriangularcell

andprojectionsofthe virtualnormal vectorn aredefinedin(3). Theagreementisverygoodforallvaluescompared.

Theangle

η

canbedirectlycomputedintermsofangles

ξ

and

ζ

:

η

=arcsin



cos2

(

ξ

)

_+cos2

(

ζ

)



_,

η

_∈



_0,

π

2

(6)

Sinceangles

ξ

,

η

and

ζ

arelinkedby(6),theyarenotthree inde-pendentprocesses.

2.3. Statisticalmodellingofthevirtualwallnormalvectoranglesin thecaseoflowroughness

Letusassumethatthevirtualwallnormalvectorangles

ξ

and

ζ

areindependent random variablesashas beenconfirmedwith the(

ζ

,

ξ

)scatterplotsshowninFig.4.Thejointprobability den-sityfunctionP_ξζ(

θ

,

ϕ

) ofangles

ξ

and

ζ

canthen bewrittenas:

P_ξζ

(

θ

,

ϕ

)

=P_ξ

(

θ

)

P_ζ

(

ϕ

)

(

θ

,

ϕ

)

∈[0,

π

]2 (7)

TheisotropicwallroughnessischaracterizedbyequalPDFsof an-gles

ξ

and

ζ

:

P_ξ

(

θ

)

=P_ζ

(

θ

)

,

θ

∈[0,

π

] (8)

withmeanvalues

ξ

=

ζ

=

π

/2andstandarddeviations



ξ

=



ζ

ofangles

ξ

and

ζ

,respectively.

Angles

ξ

and

ζ

maybewritteninthefollowingform:

ξ

=

π

₂ +

ξ

,

ζ

=

π

₂ +

ζ

 (9)

Further,thecaseoflowroughnessisstudiedwhichmeansthatthe effectivevaluesofthevectoranglessatisfytherelations:

|

ξ



_|

_

π

2 and

|

ζ



|



π

2 (10)

IO

i l l

~

I

O

i l l

~

ij:

LAJ

~

ij:

LAJ

~

IOO

~

~

[L

IO

~

O

~

[I]

Ç90 ( 9 0 , .•

:~ 0.02 :~ -0.00

0 5 10 80

ij

(a) Sc

atter

plot

ij -

ç

100 95 ( 90 85 80 100 95

Ç

90 85 100 0 5 10 80 ij

(b) Scatt

e

r p

l

ot

ij - ( 100 80 100

(

c)

Sc

at

ter

plot (

-

ç

100 0.4 ~ ~ ~ ~ -'-.J>

<]

0.3 <JJ,

<l__

0.2

b'

t;;'

0.1 0 O"x

+

O"z X

~Ç

6 ~ (

- -

,/2,h

/cL

o~-~--~-~--~-___,

0 0.05 0.1 0.15 0.2 0.25

h

/cL

Table 1

Virtual wall normal vector angle statistical characteristics (in degrees). The virtual rough walls examined are characterized by the ratio of RMS height to correlation length scale: h / c L . The mean value of the angle ηis ηand the

standard deviations of angles ξ,ηand ζare ξ,ηand ζrespectively; the subscript NS stands for numerical simulation and SM stands for the statistical model (16) and (17) . h / c L ξNS ζNS √ 2·180 π h/ c L ηNS ηNS ηSM ηSM 0.031 2.42 2.45 2.51 3.05 1.60 3.03 1.58 0.037 2.98 3.02 3.00 3.75 1.99 3.73 1.95 0.044 3.54 3.60 3.56 4.49 2.33 4.44 2.32 0.050 3.97 4.00 4.05 5.03 2.58 4.97 2.60 0.056 4.55 4.53 4.54 5.72 2.96 5.70 2.98 0.063 5.08 5.05 5.10 6.39 3.28 6.36 3.33

sothecorrespondingstandarddeviationsoftheanglePDFssatisfy:



ξ



π

2 and



ζ



π

2 (11)

Forlowroughnessangles(inradians),(5)simplifiesto,



ξ

=



ζ

=√2h

cL

(12)

since sx≈

ξ

 andsz≈

ζ

. It can be seen fromFig. 5 that the low roughnessapproximation(12)isvaliduptoapproximately0.15rad (around8°).

2.4. Modelleddistributionofangle

η

andvalidationfrom deterministicsimulation

Using(6),(9)and(10),angle

η

maybewrittenintheframeof thelowroughnessapproximationas,

η

=



ξ

2₊

ζ

2 ₍₁₃₎

Using(13),theprobabilitydensityfunctionP_η(

θ

)ofangle

η

is ob-tained by integration of the bi-Gaussianjoint probability density function P_ξζ on a circle of radius

η

and, for an isotropic rough wall: P_η

(

θ

)

=

_

θ

ξ

2exp

−

θ

2 2



ξ

2



(14) with,bydefinition:  _π/2 0

θ



ξ

2exp

−

θ

2 2



ξ

2



d

θ

=1 (15)

Themeanvalueandvarianceoftheprobabilitydensityfunction ofangle

η

,definedwith(14)are:

η

=



π

2



ξ

(16)



η

2₌



₂₋

π

2



ξ

2 ₍₁₇₎

In Table 1, the meanvalues and standard deviations ofangle

η

obtainedfromthestatisticalmodel(16)and(17)arecompared withtheangle

η

ofthegeneratedvirtualsurface,fordifferent ra-tiosofRMSheighthtocorrelationlengthscalecL.Theagreement betweenthesecomparedvaluesisverygood.The standard devia-tions



ξ

and



ζ

obtainedfromthenumericalsimulationare al-mostidenticalasexpectedforanisotropicroughwall.

3. Numericalsimulationofparticle3Delasticbouncingona roughwallandstatisticalanalysis

3.1. Descriptionofanumericalsimulation

The numerical simulation is realized as follows. An isotropic virtual wall with RMS height h and correlation length scale cL

is generated according to the procedure outlined in Section 2.1. Thesewallgenerationparametersleadtovirtualnormalvector an-gledeviations



ξ

=



ζ

givenby(12).Inthissectionvirtualwalls withvirtual normal vector angle deviations



ξ

₌



ζ

₌2_.5◦ and



ξ

=



ζ

=5◦areexamined.

Starting particle centrecoordinates x andz are sampled from theuniformdistribution,whilethestartingypositionisthesame for all particles and slightly higher than the highest asperity in thesimulateddomain.The pointofimpact oftheparticleon the virtual surfaceisfound anda unit normalvector nis calculated. Projections of the incident velocity vector U−_p are calculated us-ingtransformationmatricesinalocalcoordinatesystem(x,y,z), wheretheyaxisisalongthevirtual wallnormalvectorn (parti-cleincidentpropertiesaredenotedwithsuperscript-andparticle rebound propertieswithsuperscript +). Fullyelasticimpacts are calculated(up+ = up−,

v

p+ = −

v

p−, wp+ = wp−) andthe velocity components obtainedare written back in the general coordinate system(x,y, z). Afterrebound, theparticleis trackedandfurther impactswiththe wallare calculated,ifthey exist.The particleis trackeduntilitovershootsthehighestasperityinthedomain.

10,000particletrajectoriesaresimulatedpersimulation,which issufficienttoobtainconvergedstatistics.In thenumerical simu-lationconcerningtheincidentparticlevelocity,thevelocity projec-tionu−_p isspecifiedasinputdatainadditiontotheangles

α

_p−and

β

−

p,asshowninFig.6.

3.2.Statisticalanalysisof3Dparticlereboundfromisotropicrough wall

PDFs of the first virtual normal vector angles

ξ

and

ζ

seen byparticleswithincidentangles

α

p−and

β

p−,anddifferentvirtual wallscharacterizedwithvirtualnormalvectorstandarddeviations



ξ

=



ζ

,areshowninFig.7andFig.8,respectively.

It can be seen from Fig. 7 that at large incident angles

α

p− (

|

α

−_p

|



ξ

and



ζ

), the distribution of the first angle

ξ

seen byincident particles isGaussianwitha meanvalue equalto

π

/2 andastandard deviationequal tothevalue calculated from(12). Incontrast,astheparticle incidentangle

|

α

p−

|

decreases,thePDF ofthefirstangle

ξ

seenbyincidentparticlesbecomesasymmetric withameanvalueshiftedtowardshighervalues.Bothofthese ef-fectsareenhancedwithincrease inthevirtualwallnormalvector standarddeviation



ξ

₌



ζ

from2.5° to5°.Theseeffectsmaybe duetheso-calledshadoweffectaspointedoutbySommerfeldand Huber(1999).Forlowparticleincidentangles

|

α

p−

|

,thePDFslook veryslightlysensitivetotheparticleincidentangle

β

_p−.

Fig.8showsthatthePDF ofthefirstangle

ζ

seenby incident particlesfollowsthesametrendastheangle

ξ

withameanvalue equalto

π

/2andastandarddeviationequaltothatof

ξ

.Also,the PDF becomesdependenton theparticleincidence angles

α

_p− and

β

−

p withdecreasingvaluesof

|

α

p−

|

.Thiseffectismorepronounced forhighwallroughnessvalues.Therefore,theeffectofincident an-gle

β

_p−forsmallincidentangleamplitudes

α

−_p islargeronthePDF ofangle

ζ

thanonthePDFofangle

ξ

.

Figs. 9and10 show PDFsof particlebouncing angles

α

p+ and

β

+

p computedfromdeterministic simulations,accordingto defini-tionsshowninFig.6.

Fig. 9 showsthe PDFs of angles

α

p+ obtainedfrom determin-isticsimulationsfordifferentincident particleangles

α

−_p and

β

_p−. Atlargeparticleincidentangles

α

p−(

|

α

−p

|



ξ

and



ζ

),the dis-tributionofthereboundangle

α

p+isnearlyGaussianwithamean valueequaltotheabsolutevalueoftheparticleincidentangle

|

α

_p−

|

andastandarddeviationapproximatelyequaltothatofthevirtual wall normal vector



ξ

=



ζ

. When the particle incident angle

|

α

−

p

|

decreases,astrongshifttowardssmallvaluesisobservedand the PDF is nolonger Gaussian. This behaviour is consistent with the2DshadoweffectanalyzedbySommerfeldandHuber(1999).

Fig. 6. Characteristic angles of particle incident velocity U −

p and characteristic angles of particle velocity after rebound U +p . The virtual normal vector n and vector n γ are

defined in (4) and (21) , respectively, and angle γsatisfies (23) .

Fig. 7. PDFs of the first vector angle ξseen by incident particles computed from the deterministic simulations (DS) for isotropic walls characterized by normal vector angle standard deviations ξ= ζ, for different particle incident angles α−

p and βp− .

These PDFs of rebound angle

α

+_p from deterministic simu-lation are then compared with the results of the 2D multiple particle-wallcollisionmodelofKonanetal.(2009)applied inthe incident particle plane. The agreement between these two dis-tributions is very good, which suggests that the 2D model of

Konanetal.(2009)can be appliedto calculatethefinal rebound angle

α

p+inthecaseof3Dparticlereboundfromawall.

However, the 2D approach ofKonan etal.(2009) cannot pre-dict the transverse deviation bouncing angle

β

p+−

β

p−. The PDFs ofthisangle areshownin Fig.10,forcases ofthefirst andfinal particlerebound,fordifferentincidentangles

α

−_p and

β

_p−andtwo virtual wallroughnessvalues.ThesePDFsagree verywell forthe caseofthefirstandfinalrebounds,whichleadstotheconclusion thatthetransverseangledistributionisnotinfluencedbymultiple

y y X X

z

z

0.25 ... v' ... DS,

~ç

=

~

(

=

2.5° 0.25 0.2 --• ---DS,

~ç

=

~

( =

5° 0.2 ~

~

>'-< 0.15

lt

>'-< 0.15

"'

_'

Cl

+

~

Cl

,/,

~

~ 0.1

f

,

~

'7~ ~ 0.1 0.05

'{1

9>

.

~ 0.05 0 0 70 80 90 100 110 70 80 90 100 110

ç

ç

(a)

a;;-=

-

2.5

°,

f3;;-

=

-

30

°

(b)

a;;-

=

-

2

.5°,

f3;;-

=

15

°

0.25 0.25 0.2 0.2 >'-< 0.15

t\

>'-< 0.15

~

Cl V , Cl

w w

~ _0.1

_"'

,

"'

~

w

?

\ 0.1

~

,~

0~

y

'

0.05

w

·~

0.05 ~ 0 0 70 80 90 100 110 70 80 90 100 110

ç

ç

(c)

a;;-

=

-1

2.5

°

,

f3

;;-

=

-

30

°

(d)

a;;-

=

-1

2.5

°,

f3

;;-

=

15

°

0.25 0.25 0.2 0.2 >'-< 0.15 >'-< 0.15 Cl Cl ~ 0.1 ~ 0.1 0.05 0.05 0 0 70 80 90 100 110 70 80 90 100 110

ç

ç

(

e

)

a;;-

=

-32.5°,

f3

;;-

=

-30°

(f)

a;;-

=

-32.5

°,

f3

;;-

=

15

°

Fig. 8. PDFs of the first vector angle ζseen by incident particles computed from the deterministic simulations (DS) for isotropic walls characterized by normal vector angle standard deviations ξ= ζ, for different particle incident angles α−

p and βp− . Legend is the same as in Fig. 7 .

particle-wall collisions. It can be seen that the transverse devia-tion bouncing angle

β

p+−

β

p− obeys a Gaussian distribution with a standarddeviationthatincreaseswithincreasing incidentangle amplitude

|

α

−_p

|

.Itisalsoobservedthatthisstandarddeviation in-creaseswithincreasingvirtual normalvectoranglestandard devi-ations



ξ

=



ζ

. The PDFsof the transverse deviationbouncing angle

β

_p+₋

β

_p− areindependentoftheincidentangle

β

_p−whichis tobeexpected,sincethevirtualwallsareisotropic.

4. Statisticalmodellingofthe3Droughwall-particlecollisions

4.1. Modellingthefirstvirtualwallnormalvectorseenbyparticles forlargeincidentangles

As shown in Figs. 7 and 8, for large particle incident angles

|

α

−

p

|

, with respect to the angle standard deviations,

|

α

−p

|



ξ

and



ζ

, and low wall roughness (11), the first angles

ξ

and

ζ

seenbyparticleshaveGaussian distributions,withzeromeanand standarddeviationequaltothevirtualwallnormalvectorstandard deviations



ξ

=



ζ

.Compared withthiscase, for particles with lowincidentangles

|

α

−_p

|

,thePDFsofangles

ξ

and

ζ

changesince thereisaconditioningeffectbytheincidentparticleangle

α

−p.

Inthissection,statisticalmodellingisdevelopedforlarge inci-dentangles,sothatwecanneglecttheshadoweffectonthePDF ofthewallnormalvectorangleseenbyparticles.

According to Fig. 6, the incident particle velocity U−_p can be written, U−p =

|

U−p

|

cos

(

α

−p

)

t−p +

|

U−p

|

sin

(

α

p−

)

j,

α

p−∈



−

π

2,0

(18)

where t−p is the unit vector collinear with the projection of the incidentparticlevelocityU−p onthehorizontalplane,

t−_p =cos

(

β

_p−

)

i− sin

(

β

−

p

)

k,

β

p−∈[−

π

,

π

] (19) Itcanbenotedthatanyunitvectornseenbyagivenincident particlevelocityU−_p mustverify,

U−p· n<0 (20)

Thisconditionispartoftheshadoweffectleadingtorealizability conditionsforthevirtualwallnormalunitvectorangles.

Letusdefinethe unit vectorn_γ intheincident plane written intermsoftheangle

γ

withthey-axisas,

n_γ =− sin

(

γ

)

t−_p +cos

(

γ

)

j,

γ

∈



−

π

2,

π

2

(21) 0.2 0.15

~

VI '

J \

"-Ci 0.1 i '? p..

+

_i {q

()

~

' ~ 0.05 .

~

\

w

~~ 0 70 80 90 100 110 (

(a)

aP

=

-

2

.

5

°

,

/3p

=

-

30

°

0.2 , -0.15 "-Ci 0.1 p.. 0.05 80 90 100 (

(c) aP

=

-1

2.5

°

,

/3

P

=

-

30

°

110 0.2 -0.15

~

v

"-

v

\

Ci 0.1 p..

' w

"'

-~~

\

0.05

_,

l

_.

_V'G

~

fl}V . • .lf . 'i'i ~-~ 0 70 80 90 100 110 (

(

e

) aP

=

-

32

.

5

°,

/3

p

=

-

30

°

0.2 0.15

Y';,

'y \ "- i

w

Ci 0.1

~

~

p.. ,~

'

0.05 _{!i>i #)} ~ "fi . oflJ VI 'i'i'<,,~ 0 70 80 90 100 110 (

(b)

aP

=

-

2.5

°

,

/3

p

=

15

°

0.2 -0.15 "-Ci _p.. 0.1 0.05 80 90 100 (

(d)

aP

=

-1

2.5

°,

/3

P

=

1

5

°

110 0.2 -0.15 ~

r

V

"-+

~

Ci 0.1 p..

r

~~-

~

-0.05

_x,·

_,

_w

~-(lj </1 ·&. (Y w ~-0 70 80 90 100 110 (

(f)

aP

=

-

32

.

5

°,

/3

p

=

1

5

°

Fig. 9. PDFs of the final rebound angle α+

p of the particle returning to the flow for different initial incident angles αp− and β−p for isotropic surfaces defined with the normal

vector angle standard deviation ξ= ζ. DS stands for deterministic simulation and 2DRWCM for the 2D rough wall multiple-collision model ( Konan et al., 2009 ) applied in the particle incident plane for 2D wall normal vector angle standard deviation γ= ξ.

byimposingthatthescalarprojectionofn_γ ontotheparticle inci-dentvelocityU−p isequaltothatofthewallnormalvectorn,

U−p· nγ =U−p· n (22)

Using(18)and(21),(22)canbewritten,

|

U−_p

|

sin

(

α

_p−−

γ

)

=U−_p · n (23)

whichisequivalenttothefollowingscalarequation,

sin

(

α

p−−

γ

)

=cos

(

β

p−

)

cos

(

α

p−

)

cos

(

ξ

)

+sin

(

α

−p

)

cos

(

η

)

− sin

(

β

−

p

)

cos

(

α

p−

)

cos

(

ζ

)

(24)

The above equation always has a unique solution for

γ

∈ [−

π

/2,

π

/2]. In the frame of the low roughness approximation,

(24)leadstothefirst-orderapproximationfor

γ

,

γ

=cos

(

β

p−

)

ξ

− sin

(

β

p−

)

ζ

 (25) Foralarge particleincidentangle

α

p− andlow wallroughness, since angles

ξ

 and

ζ

 are random independent processes with zeromeanvalues

ξ

=

ζ

₌0andstandarddeviation



ξ

and



ζ

, angle

γ

isarandomprocesswithzeromeanvalue

γ

=0and vari-ance



γ

2_asfollows:



γ

2_=cos2

₍

_β

−

p

)



ξ

2+sin 2

₍

_β

₋

p

)



ζ

2 (26)

Ifangles

ξ

 and

ζ

 areGaussianprocesses,theprobabilitydensity functionP_γ(

θ

)ofangle

γ

isGaussian:

P_γ

(

θ

)

=



1 2

π



γ

2exp

−

θ

2 2



γ

2



,

θ

∈



−

π

2,

π

2

(27)

In Fig. 11, the first vector angle

γ

distribution seen by an inci-dentparticleinthedeterministicsimulationiscomparedwiththe virtual wallangledistributionobtainedfromthe 2Deffective dis-tribution (44) given by Sommerfeld and Huber (1999) calculated in theincident particleplane, with a wall roughnessangle stan-darddeviation



γ

equaltothenormalvector anglestandard de-viation



ξ

₌



ζ

. The agreement between the compared distri-butions is excellent for any particle incident angle

α

p−. At large incident angles

α

p−, the distribution of angle

γ

is nearly Gaus-sian with



γ

₌



ξ

_, inagreement with thestatistical model as-sumption(27).Atlow incidentamplitudeangles

|

α

−p

|

,duetothe shadoweffect,thePDF ofangle

γ

shifts totheright. Inaddition, thePDF of angle

γ

isfound to be independentofthe transverse incidentangle

β

p−.

Letusintroduceanadditionalangle

γ

∗suchthat,

γ

∗_=sin

₍

_β

− p

)

ξ

+cos

(

β

p−

)

ζ

 (28) 0.1 0.08 ~ 0.06 ~ 0.02 -·•·-DS, 6ç = 6( = 2.5°

- * -

2DRWCM, 6"!

=

2.5° - DS, 6ç = 6( = 5° · ··I>··· 2DRWCM, 6"f

=

5°

oi----

---31111aiililllllil

_ _

..._

_._J

0 20 60 80

(

a

)

Œp

=

-2

.5°,

/3p

=

-30

°

0.1 0.08 ~ 0.06 ~ 0.04 0.02 20 40 a+ p 60 80

(c)

Œp

=

-

12

.5°

, /3

p

=

-3

0

°

0.1 0.08 ~ 0.06 ~ 0.04 0.02 0 0 20 40 60 80 a+ p

(

e

)

Œp

=

-32.5

°,

/3p

=

-30

°

0.1 0.08 0.02

oi----

___:ll"-ili•

- - - -

-._J

0 20 40 a+ _p 60

(b)

Œp

=

-2.5

°,

/3p

=

15

°

0.1 0.08 ~ 0.06 ~ 0.04 0.02 20 40 a+ p 60 80 80

(d)

Œp

=

-

12.5

°,

/3

p

=

15

°

0.1 0.08 ~ 0.06 ~ 0.04 0.02 0 0 20 40 60 80 a+ p

(f)

Œp

=

-32.5

°,

/3p

=

15

°

Fig. 10. PDFs of the transverse deviation bouncing angle β+

p −βp− computed from deterministic simulations for different incident angles αp− and βp− for surfaces with normal

vector angle standard deviations ξ= ζ. 1C denotes the distribution with the first rebound angle β+

p , and MC represents the distribution with final rebound angle βp+ .

Angle

γ

∗ is a random process withzeromean value

γ

∗=0 and variance



γ

∗2expressedas:



γ

∗2_=sin2

₍

_β

₋

p

)



ξ

2+cos2

(

β

p−

)



ζ

2 (29) Ifangles

ξ

and

ζ

areGaussianprocesses,probabilitydensity func-tionP_γ∗

(

θ

)

ofangle

γ

∗isGaussian:

P_γ∗

(

θ

)

=



1 2

π



γ

∗2exp

−

θ

2 2



γ

∗2



,

θ

∈



−

π

₂,

π

₂

(30)

Fig. 12 shows PDFs of angle

γ

∗ computed from deterministic simulations using(28)andthemodelGaussian distributiongiven by (30), forthe differentparticleincidentangles

α

−_p and

β

_p−.The agreement betweenthe twoPDFsisexcellent. Althoughangle

γ

∗ dependsontheangles

ξ

and

ζ

,asseenin(28),wherebothangles

ξ

and

ζ

dependontheincidentangles

α

_p− and

β

_p−_,the

γ

∗ distri-butiondoesnotdependontheparticleincidentangles

α

−p and

β

p− and, inparticular,isnot affectedby theshadoweffect.Hence, by construction, angles

γ

and

γ

∗ are orthogonalindependent Gaus-sian processes.Finally,fortheisotropicrough wall,

γ

and

γ

∗ are randomvariablesobeyingtwoindependentGaussian distributions withzeromeansandstandarddeviations



γ

=



γ

∗_,equaltothe identicalstandard deviations



ξ

=



ζ

ofthevirtual wallnormal vectorangledistributions.

Hence, angles (

ξ

,

η

,

ζ

) of the first virtual rough wall normal vector n seen by any incident particle can be computed from a pairofrandomvariables

γ

and

γ

∗givenbyindependentstochastic processes accordingto the Gaussian PDF givenby (27) and(30). Indeed,using(25)and(28),

ξ

and

ζ

arewritten,

ξ

=cos

(

β

p−

)

γ

+sin

(

β

p−

)

γ

∗+

π

₂ (31)

ζ

=− sin

(

β

−

p

)

γ

+cos

(

β

p−

)

γ

∗+

π

₂ (32) and,using(13),

η

maybewritten,

η

=



γ

2₊

γ

∗2 ₍₃₃₎

Therefore, forlarge incident particle velocity angles, no multiple collisionsare expected andthe first elasticparticle-wall collision effectleadstothefollowingexpressionforthefinalbouncing par-ticlevelocityU+_p:

U+_p =U−_p− 2

U−_p· n



n (34)

Thisequationcanbewritteninthefollowingform,

U+_p =U−_p− 2

U−_p· nγ



n_γ− 2

U−_p· n



n− nγ



(35) 2 . - - - ~ - - ~ - - - , · V · liç = li( = 2.5°, lC -·•·-liç =li(= 2.5°, MC 1.5 - + -liç =li(= 5°, lC - l iç =li(= 5°, MC

l

-.:

0.5 -5 0 5 10

f3t -

/3;;

(a)

a;;-=

-2.5

°,

/3;;-

=

-30

°

0.4 - - - ~ - - - - ~ 0.3

...

5:

0.2 0.1 0

y--

~

~

....__:i

-

--

~

_J

-20 -10 0 10 20

f3t

-

/3;;

(

c

)

a;;-=

-12.5

°,

/3

;;-

=

-30

°

0.15 - - - ~ 0.1

...

Cl i:,... 0.05

(

e

)

a;;-

=

-32.

5°,

/3

;;-

=

-

3

0

°

...

Cl i:,...

...

2 1.5 1

i

'l't

0.5 0 -10 -5 0 5 10

f3t

-

/3;;

(b)

a;;-

=

-2.5

°,

/3;;-

=

15

°

0.4 - - - ~ 0.3

I

"'V 'Y . Cl 0.2 i:,...

.

...

0.1 0 ~ * ~~ .,_....__:i- ~1(1,&,M,l;),.i -20 -10 0 10 20

f3t -

/3;;

(d)

a;;-=

-12

.5°,

/3;;-

=

15

°

0.15 . - - - ~ 0.1 0.05

(

f)

a;;-

=

-

32.5

°,

/3;;-

=

15

°

Fig. 11. Comparison of the PDFs of the first vector angle γseen by incident particles in deterministic simulations (DS) for isotropic walls characterized by normal vector angle standard deviation ξ= ζand effective Sommerfeld distribution (ES) calculated in the particle incident plane with angle standard deviation γ= ξ, for different

particle incident angles α− p and βp− .

wherethesecond termrepresentsthe2D elasticbouncingofthe incidentparticle ona virtual surfacewitha normal vectorn_γ in theincidentplaneandthethirdtermrepresentsatransverseeffect ontheparticlevelocitydueto3Dparticlereboundontheisotropic roughwallsurface.

4.2.Characterizationoftransversebouncingeffectforlargeincident particleangle

Letusdefinetheunitvector normaltoparticleincidentplane: s_p−=t−_p ∧j=sin

(

β

_p−

)

i+cos

(

β

_p−

)

k (36)

Intheframe ofthelow roughnessapproximation (11),the trans-versevirtualwallnormalvectorcomponentmaybewritten,using

(4),(21)and(36),as:

n− nγ =−

γ

∗s−p (37)

Therefore, according to (35) and (37), the 3D effect on elas-ticbouncingvelocityofparticleswithlargeincidentparticleangle maybewritten:

δ

U+_p =2[U−_p· n]

γ

∗s−_p =w+_ps−_p (38)

Eqs.(23)and(38)leadto:

w+_p =2

|

U−_p

|

sin

(

α

−_p

)

γ

∗ (39)

Finally,foranygivenparticleincidentvelocitynorm

|

U−p

|

,velocity w+_p isarandomvariablewithzeromeanandastandarddeviation

σ

+

p givenby,

σ

+

p =2

|

U−p

|

sin

(

|

α

p−

|

)



γ

∗ (40) Bydefinition,thetransversebouncingangle

β

p+−

β

−p iswritten,

sin

(

β

p+−

β

p−

)

=−

w+

p U+pcos

(

α

p+

)

(41)

From(39)and(41),forelasticparticlebouncing,itfollowsthat,

β

+

p =

β

p−− 2tan

(

α

p−

)

γ

∗ (42) Finally,it isfound that angle

β

_p+₋

β

_p− is arandom variablewith zeromeanandastandarddeviation:



β

+

p =2tan

(

|

α

p−

|

)



γ

∗ (43) with



γ

∗givenby(29).

Fig.13showsthetransversebouncingcharacteristicparameters,

σ

+

p and



β

p+, due to 3D elastic rebound on an isotropic rough

0.25 0.2 ~ 0.15 P, 0.1 0.05 0 5 10 15 20 25 1

(a)

a;;-

=

-2.5°,

/3;;-

=

-30°

0.25 0.2 ~ 0.15 P, 0.1 0.05 -20 -10 0 10 20 1

(c)

a;;-=

-12.5

°,

/3;;-

=

-

30°

0.25 0.2 ~ 0.15 P, 0.1 0.05 -20 -10 0 10 20 1

(e)

a;;-=

-32.5

°,

f3

p

=

-30

°

0.25 0.2 ~ 0.15 P, 0.1 0.05 0 5 10 15 20 25 1

(b)

a;;-

=

-2

.5°,

/3;;-

=

15

°

0.25 0.2 ~ 0.15 P, 0.1 0.05 -20 -10 0 10 20 1

(d)

a;;-

=

-12.5°,

/3p

=

1

5°

0.25 0.2 ~ 0.15 P, 0.1 0.05 -20 -10 0 10 20 1

(f)

a;;-

=

-32.5

°,

f3p

=

15

°

Fig. 12. PDFs of the first angle γ∗_{seen by incident particles in deterministic simulation (DS), calculated using (28) , and comparison with the statistical model (SM), equation}

(30) , for different particle incident angles α− p and βp− .

Fig. 13. Dependence of non-dimensional transversal particle characteristics on the particle incident angle α−

p , for virtual walls with normal vector angle standard deviations

ξ= ζ: (a) ratio of standard deviation β+

p of the wall-induced transversal particle angle dispersion to angle standard deviation ξ(b) ratio of standard deviation σp+

of the wall- induced transversal velocity dispersion to the product of incident velocity norm | U −

p| and angle standard deviation ξ. 0.25 0.2

t:s

0.15 ~ ---•---Ll.ç = Ll.( = 2.5°,

os

- ..t;, - Ll.ç = Ll.( = 2.5°, SM - ii.ç = Ll.( = 5',

os

---+---Ll.ç = Ll.( = 5°, SM 0.1 0.05 -20 -10 10 20

(

a

)

aP

=

-

2.5

°,

/3

p

=

-

30

°

0.25 0.2

t:s

0.15 ~ 0.1 0.05 -20 -10 10 20

(c) aP

=

-

12

.5°,

/3

p

=

-

30

°

0.25 0.2

t:s

0.15 ~ 0.1 0.05 -20 -10 10 20

(

e)

a;;-

=

-32

.5°,

/3p

=

-30

°

3 --0--1'>( = 2.5° 2.5 --<>--1'>(=30 - <:J--L'>( = 3.5° 'JJ' 2 -é.-1'>( =40 <l -

*

-

I'>( = 4.5° ..____ 1.5 -a-I'>( = 50

t{'-

- 2 tan(la~I) <l 1 0.5

o

~ - - - ~ - ~ - - ~ - ~

0 10 20 30 40 50

la~

I

(a)

0.25 0.2

t:s

0.15 ~ 0.1 0.05 -20 -10 10 20

(b)

aP

=

-

2.5

°,

/3p

=

15

°

0.25 0.2

t:s

0.15 ~ 0.1 0.05 -20 -10 10 20

(d)

aP

=

-

12

.5°

,

/3p

=

15

°

0.25 0.2

t:s

0.15 ~ 0.1 0.05 -20 -10 10 20

(f)

aP

=

-32.5

°,

/3p

= 15

°

2 --0--1'>( = 2.5° --<>--1'>( = 30 ~ 1.5 'JJ' <l -<;J--L'>( = 3.5° -é.-1'>( = 40 -+ -L'>( = 4.5° 1 o. 1

::,

- e -- 1'>(2 sin(=l5a0 ~I) ..____ \""o.5 0 0 10 20 30 40

la~I

(b)

50

wall, in terms of the particle incident angle

α

_p−. The agreement betweentheparametersfromdeterministicsimulationsandmodel predictions,using(40) and(43), isexcellent, even for low values oftheparticleincidentangleamplitude

|

α

_p−

|

.

4.3.Modellingthefirstvirtualwallnormalvectorseenbyaparticle foranyincidentangle

Forlowincidentparticlevelocityangle

α

−p,oftheorderofthe virtual wall normal vector angle

γ

, the statistical approach pro-posedinSection4.1torepresentthevirtual wallnormalvectorn

doesnot ensure that the realizability conditionU−p· n<0 is sat-isfied.Thisrealizability conditionispartoftheshadoweffect de-scribedby Sommerfeld andHuber (1999)for2D rebound froma rough wall.They proposed a modifiedvirtual wall normalvector angleprobabilitydistribution,conditionedbytheincidentvelocity, whichsatisfies,by construction,the realizabilitycondition. Inthe proposed 3D approach, according to (23), the realizability condi-tionmaybewrittenas

γ

>

α

−

p andleadstotheuseofthe modi-fiedPDF forthe

γ

anglealone.Hence,

γ

∗isassumedtoobeythe standardGaussiandistributiongivenby(30)whereas

γ

isassumed toobeythe effectiveSommerfeld distribution.Therefore,in prac-tice,angles

γ

and

γ

∗arecomputedfromtwoindependentrandom processes

(

γ

,

γ

∗

₎

_∈[−

_π

_/2,

_π

_/2]x[−

_π

_/2,

_π

_/2]:

• Angle

γ

obeystheeffectiveSommerfelddistributionaccounting fortheshadoweffectgivenby:

if

γ

>

α

− p: Pe f f

(

γ |

α

p−

)

= 1



2

π



γ

2 sin

(

α

−p −

γ

)

sin

(

α

−p

)

exp

−

γ

2 2



γ

2



g

(

α

−_p,

γ

)

(44) if

γ

_≤

α

−_p: Pe f f

(

γ |

α

−p

)

=0 (45) with g



α

p−,



γ



=1/  _π/2 α− p 1



2

π



γ

2 sin



α

p−−

γ



sin



α

p−



· exp

−

γ

2 2



γ

2



d

γ

(46)

andstandarddeviation



γ

thatfollowsfrom(26). • Angle

γ

∗obeysastandardGaussianPDFgivenby(30).

Finally, the first virtual rough wall normal vector angles are computedfromEqs.(31)–(33).

4.4.Modellingofmultipleparticlecollisionswiththevirtualwall

Sommerfeld and Huber (1999) pointed out that the

post-reboundcondition

α

p+>0isneededtoallowtheparticletoreturn tothemainflow.Intheirapplications,theyovercamethisproblem simplybyrepeatingtheparticle-wallrebound procedurewiththe generationofanewfirstvirtualwallnormalvectorangle

γ

. There-fore,Konanetal.(2009)showedthat“grazing” particles2D bounc-ingwithasmall

α

+_{p >}0maysufferseveralparticle-wallcollisions before going back to the main flow. Their approach is based on ananalyticalmodeloftheprobability ofhavingonlyone particle-wall collision and on the repetition of the particle-wall rebound procedure in the caseof a multiple particle-wall collision effect. Thesame methodology maybe extended directly for3D particle reboundfromroughwall.

Thestandarddeviation



β

_p+remainssmall,oftheorderof



ξ

, asshowninFig.13,anddecreasesforparticleswithsmallincident

anglesthat havethehighestprobability ofenduringmultiple col-lisions. Hence, we propose to neglect the 3D deviationeffect on the multiple particle-wall collision probability modelling. There-fore,accordingtoKonanetal.(2009),theprobabilitythatparticles foragivenincidentangle

α

p+makeonlyoneimpactiswrittenas:

P∗

(

n=1

|

α

p+

)

=



tanh

(

ψ

α

+ p



γ

)

if

α

p+≥ 0 0 if

α

p+<0 (47) with

ψ

=1.5.

Fig.14comparestheprobabilitiesthatparticlesmakeonlyone impact at different particle incident angles

α

p− and

β

p− in deter-ministicsimulation withthe theoretical probability that particles makeonlyoneimpact(47),withvariedcoefficient

ψ

.Better agree-mentwithdeterministicsimulationsisobtainedfor

ψ

=2thanfor

ψ

=1_.5_, probablybecause the procedure for generating the sur-face is not identical with that used by Konan et al. (2009). This difference in coefficient

ψ

doesnot have a significant effect on theparticlereboundstatisticsandtheoriginalvalue

ψ

₌1_.5is re-tainedforthispaper.

Itfollowsthattheprobabilitythatparticlesmakeonlyone im-pact does not depend on the transverse incident angle

β

_p− and transversereboundangle

β

_p+:

P∗



n=1

|

α

p+,

β

p+,

β

p−



=P∗



n=1

|

α

+p



(48)

4.5. Stochasticprocedureforcalculationof3Dreboundofaparticle fromanisotropicroughwallwithlowroughnessinaLagrangian framework

InaLagrangianframework,particlesaretrackedintheflowand whenthecentreofanyparticlereacheshalfoftheparticle diam-eterdistancefromtheboundarysurface, themodellingofthe ef-fectivevirtualroughwall iscarriedoutaccordingtothefollowing newstochasticprocedure,derivedinagreementwiththestatistical modelproposedinSection4.3torepresent

γ

and

γ

∗PDFs.

Thisprocedurecanbesummarizedintofivesteps:

(1) Angle

γ

∗issampledaccordingtotheGaussiandistribution(30)

(2) Angle

γ

issampledaccordingtotheeffectiveSommerfeld dis-tribution(44)

(3) Angles

ξ

,

η

and

ζ

are calculated with (31)–(33). The virtual wallnormalvectornisdefinedwith(4).

(4) Aslidingornon-sliding impactiscalculatedforaparticle col-lidingwiththevirtualsmoothwalldeterminedwiththenormal vectornfoundinstep(3)

(5) MultiplecollisionsaretreatedasinKonanetal.(2009). (5.1)ifthereboundangle

α

+_p ≤ 0,theprocedureisrepeatedfrom

step(2).

(5.2) if the rebound angle

α

+p >0, another number s∈[0, 1] is sampledaccordingtotheuniformdistribution.

(5.2.1) ifs∈[0,P∗

(

n=1

|

α

+p

)

],theparticleleaveswallregion (5.2.2) ifs∈[P∗

(

n=1

|

α

p+

)

,1], the procedure isrepeated from

step(2).

4.6. Validationofthemodelledvirtualwallnormalanglesatthefirst particlerebound

Fig.15comparesthePDFsofvirtual wallnormalvectorangles

ξ

,

η

and

ζ

atthe firstwall impactfromdeterministic simulation withthecorrespondingmodelledvirtualwallnormalvectorangles obtainedfromtheprocedureoutlinedinSection4.5,stepsfrom(1) to(4),sinceonlythefirstwall impactismodelled.Different inci-dentangles

α

−p and

β

p−andtwo isotropicvirtualwallsare exam-ined.Theagreementbetweenthe3Ddeterministicsimulationand 3Dstochasticmodelisverygood.

Fig. 14. Comparison of the theoretical probability (47) for different coefficients ψ with probabilities from deterministic simulations that particles make only one rebound before leaving the wall region, for an isotropic wall with virtual wall normal vector angle standard deviations ξ= ζ, for different particle incident angles α−

p and βp− .

Fig. 15. Dependence of the first wall normal vector angle distributions on the particle incident angles α−

p and βp− . Comparison between stochastic model predictions, SM,

and deterministic simulations results, DM.

1.2 ~ -~ 1 +o. CQ

+

;;.

0.8 èi ~ 0.6 Il -E, 0.4

P,

0.2

--e---

a

;;-=

-2.5',/3;;- = -30° __ ,._ a;;-=-12.5',/3;;-=15' - -(47), ,/J

=

1.5 ... (47), ,p

=

2

o~- ~ - ~ - ~ - ~ - ~

0.3 w. 0.2 0 ~ 0.1 0 4

(a)

6ç

=

6(

=

2.

5°

-·••-t.ç = t.( = 2.5', DS - A -t.ç = t.( = 2.5', SM - t.ç = t.( = 5', DS -+-t.ç = t.( = 5', SM

o

1___.,.,...~ ~ --3M l"lil...,.i. 70 80 90 100

ç

(a)

Œp

=

-

2.5

°,

/3p

=

-3

0

°

0.3 w. 0.2 0 ~ 110 5 10 15 20 25 30 T/

(c)

Œp

=

-

2.5°,

/Jp

=

-

30°

0.3 w. 0.2 0 ~ 0.1 0

'--H,---

-

::.._,_-

~

~

-..i

70 80 90 100 110 (

(e)

Œp

=

-2.5

°,

/3p

=

-30

°

~ 1 +o. CQ

+

;:,_

0.8 èi ~ 0.6 Il -E, 0.4

P,

0.2

o~- ~ - ~ - ~ - ~ - ~

0.3 w. 0.2 0 ~ 0.1 0 2 3 4

a;/~ç

(b)

6ç

=

6(

=

5°

0 1__ . . . - ~ ~ __.:!IIIIIIID,,._.._i 70 80 90 100 110

ç

(b)

Œp

=

-

12.5

°,

/3p

=

15

°

0.3 w.0.2

A

0 1 . ~

!

~ 0.1 i 5 10 15 20 25 30 T/

(d)

Œp

=

-

12.5

°,

/3

p =

15

°

0.3 w. 0.2 0 ~ 0.1

0

~.-

--

lilj)l!:

=---

~

---2'1

1111i1111- ....

.-.i

70 80 90 100 110

(