Parallel Simulation of Complex Unsteady Flows with Variational Multiscale LES and Hybrid RANS/LES

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Parallel Simulation of Complex Unsteady Flows with Variational Multiscale LES and Hybrid RANS/LES

Hilde Ouvrard, Bruno Koobus, Maria Vittoria Salvetti, Simone Camarri, Stephen F. Wornom, Alain Dervieux

To cite this version:

Hilde Ouvrard, Bruno Koobus, Maria Vittoria Salvetti, Simone Camarri, Stephen F. Wornom, et

al.. Parallel Simulation of Complex Unsteady Flows with Variational Multiscale LES and Hybrid

RANS/LES. [Research Report] RR-6917, INRIA. 2009, pp.17. �inria-00381570�

9 -6 3 9 9 IS R N IN R IA /R R -- 6 9 1 7 -- F R + E N G

Thème NUM

Parallel Simulation of Complex Unsteady Flows with Variational Multiscale LES and Hybrid RANS/LES

H. Ouvrard, B. Koobus, M.-V. Salvetti, S. Camarri, S. Wornom, A. Dervieux

N° 6917

Avril 2009

H. Ouvrard

∗

, B. Koobus

†

, M.-V.Salvetti

‡

, S.Camarri

§

, S.Wornom

¶

, A. Dervieux

k

ThèmeNUMSystèmesnumériques

ProjetPumas

Rapportdereherhe n°6917Avril200917pages

Abstrat: WestudyanewhybridRANS/VariationalMultisaleLES(VMS-LES) modelforblu

body ows. The simulationshave been arried outusing aparallelized solver, basedon amixed

niteelement/nitevolumeformulationonunstruturedgrids. Parallelperformanesareanalysed.

ThebehaviorofaVMS-LESmodelwithdierentsubgrid salemodelsisinvestigatedfortheow

pastairularylinderatReynoldsnumberRe=3900. Seond,anewstrategyforblendingRANS

andLESmethodsin ahybridmodelisdesribedandappliedto thesimulationoftheowaround

airularylinderat Re=140000.

Key-words: Blu-bodyows,variationalmultisale LES,hybridRANS/LESapproah, parallel

simulation

∗

UniversityofMontpellierII,34095Montpellier,Frane,houvrarddarboux.math.univ-montp2.fr

†

UniversityofMontpellierII,34095Montpellier,Frane,koobusmath.univ-montp2.fr

‡

UniversitàdiPisa,ViaG.Caruso,56122Pisa,Italy,mv.salvettiing.unipi.it

§

UniversitàdiPisa,ViaG.Caruso,56122Pisa,Italy,s.amarriing.unipi.it

¶

Lemma,Sophia-Antipolis,Frane,stephen.wornomsophia.inria.fr

k

INRIA,BP.93,06902Sophia-Antipolis,Frane,alain.dervieuxinria.fr

ave VMS-LES et un modèle hybride RANS/LES

Résumé: OnétudieunnouveaumodélehybridedeturbulenereposantsurRANSetsurdesmod-

èlesVMS-LES.Lesalulssontréalisésàl'aided'unalgorithmeparallèlebasésuruneformulation

mixte-élément-volumenissurmaillagenon-struturé.Lesperformanessurarhitetureparallèle

sont analysées. Le omportement d'un modèle VMS-LES ave diérents modèles sous-grille est

étudié pour leas d'un éoulementautour d'un ylindre ànombre deReynolds Re=3900. Puis,

une nouvelle stratégie de ombinaison d'un modèle RANS ave un modèle LES sous forme d'un

modèlehybrideestdériteet appliquéeàlasimulationd'unéoulementautourd'unylindreàun

nombredeReynoldsRe=140000.

Mots-lés : Éoulement autour d'un obstale arrondi, variational multisale LES, approhe

hybrideRANS/LES,simulationparallèle

1 Introdution

TheapproahinvolvingtheReynolds-AveragedNavier-Stokesequations(RANS)iswidelyusedfor

the simulation of omplex turbulent ows. Howeverthese models are not suient to properly

simulate omplexowswith massiveseparationssuh asthe ow around blu bodies. The LES

approahgivesgenerallymoreauratepreditionsbut requireshigheromputationalost.

The traditional LES approah is based on a ltering operation, the large energy-ontaining

salesareresolvedandthesmallestsalesaremodeledusingasub-gridsale(SGS)model. Usual

LESsubgridstressmodelingstrategies,asforinstanetheSmagorinskymodel,arebasedontheas-

sumptionofanuniversalbehaviorofthesubgridsales. Withinthisassumption,energy-ontaining

eddies shouldnotbeltered. ThenlargeReynolds numbersannot beaddressedwith reasonable

oarse meshes,exept, in partiularregionsof detahed eddies. Evenin theaseof lowReynolds

numberordetahededdies, apartiularattentionmustbepaidto energetieddies. Forexample,

thelassialeddy-visositymodelsarepurelydissipative. Oftenunabletomodelbaksatter,they

apply,instead,dampingtolargeresolvedenergeti eddies.

Startingfromtheseremarks,weinvestigatetheappliationoftheVariationalMultisale(VMS)

onept of Hughes. The VMS approah wasoriginally introdued by Hughes [5, 6℄ for the LES

ofinompressible owsandimplementedin aFourierspetralframework usingafrequenyuto

for the sale separation (small and large sales). In this approah, the Navier-Stokes equations

arenotlteredbut aretreatedby variationalprojetion,andtheeetof theunresolvedsalesis

modeledonlyintheequationsrepresentingthesmallresolvedsales. TheVMS-LESapproah(even

with simplesubgrid salemodels asSmagorinsky'smodel)and dynami LESmodelshaveshown

similar order of auray, but theformer is lessomputationally expensive and does notrequire

anyadhotreatement(smoothingandlipping ofthedynamionstant,asusuallyrequiredwith

dynamiLESmodels)inordertoavoidstabilityproblems. Inthiswork,weonsidertheVMS-LES

implementationpresentedin[10℄forthesimulationofompressibleturbulentowsonunstrutured

gridswithin amixed nite volume/niteelementframework. Weinvestigatetheeet of subgrid

salemodelsin ourVMS-LESmethod forthesimulationofablu-body ow.

Anothermajor diultyfor thesuessof LESforthesimulationofomplexowsisthefat

thattheostofLESinreasesastheowReynoldsnumberisinreased. Indeed,thegridhastobe

neenoughtoresolveasigniantpartoftheturbulentsales,andthisbeomespartiularlyritial

inthenear-wallregions. Anewlassofmodelshasreentlybeenproposedintheliteratureinwhih

RANSandLESapproahesareombinedtogetherinordertoobtainsimulationsasaurateasin

theLESasebutatreasonableomputationalosts. Amongtheseso-alledhybridmodelsdesribed

in theliterature,the Detahed EddySimulation (DES)[24℄ hasreeivedthelargestattention. In

previousworks,weproposedanew strategyforblending RANSandLES approahesin ahybrid

model [20, 18℄. To this purpose, as in [12℄, the ow variables are deomposed in a RANS part

(i.e. the averagedow eld), aorretion partthat takesinto aount the turbulent large-sale

utuations, and athird partmade of the unresolved orSGS utuations. The basiidea is to

solvetheRANSequationsinthewholeomputationaldomainandtoorrettheobtainedaveraged

oweld byadding,wherethegridisadequatelyrened, theremainingresolvedutuations. We

searh herefor a hybridization strategy in whih the RANS and LESmodels are blended in the

omputationaldomain followingagiven riterion. Tothisaim, ablendingfuntionis introdued,

θ

^{,whih smoothlyvaries between0and1. Inpartiular, twodierentdenitions of theblending}

funtion

θ

^{areproposedandexaminedinthispaper. Theyarebasedontheratiosbetween(i)two}

eddy visositiesand (ii) twoharateristilength sales. The RANSmodel usedin theproposed

hybridapproahis alow-Reynoldsnumberversion[4℄of thestandard

k − ε

^{model, whilefor the}

LESparttheVariationalMultisaleapproah(VMS)isadopted [5℄.

In this paper, we present VMS-LES and RANS/VMS-LES parallel simulations of blu-body

ows,by aomputationaluiddynamis (CFD)softwarewhihombinesmeshpartitioning teh-

niques and adomain deomposition method. These simulationsrequire thedisretization of the

uid equations on large three-dimensional meshes with small time-steps. Therefore theyrequire

intensiveomputationalresoures(in terms ofCPU and memory)and parallel omputation is of

partiular interest for suh appliations. We shall desribe in short our solution algorithm and

ompareitsperformanefortwodierentparallearhitetures.

2 Turbulene modeling

2.1 Variational Multisale LES

Inthispaper,weonsidertheKoobus-Farhat VMSimplementation[10℄forthesimulationofom-

pressibleturbulentows. Itusestheowvariabledeomposition[5℄,[3℄:

W = W

|{z}

LRS

+ W ^′

|{z}

SRS

+W ^SGS

⁽¹⁾

where

W

^{is thelargeresolvedsale (LRS)omponentof}

W

^,

W ^′

^{is itssmall resolvedsale (SRS)}

omponent,and

W ^SGS

^thenon-resolvedomponent. Thedeompositionoftheresolvedomponent is obtainedbyprojetiononto twoomplementaryspaes

W

^{(LRS spae)and}

W ^′

^{(SRS spae)of}

theresolvedsalespae:

W ∈ W ; W ^′ ∈ W ^′ .

⁽²⁾

AprojetoroperatorontotheLRSspae

W

^{isdenedbyspatialaveragingonmaroells,obtained}

bynite-volumeagglomerationwhihsplitsthebasis/testfuntions

φ l

^{intolargesalebasisdenoted}

φ _l

^{,andsmallsalebasisdenoted}

φ ^′ _l

^.

W = X

W l φ _l ; W ^′ = X

W ^′ l φ ^′ _l

⁽³⁾

By variationalprojetion onto

W

^and

W ^′

^{, we obtain the equations governing the largeresolved}

salesandtheequationsgoverningthesmallresolvedsales. AkeyfeatureoftheVMSmodelisthat

weset tozerothemodeledinuene oftheunresolvedsalesonthelargeresolvedones. TheSGS

model is introdued only in theequationsgoverningthesmall resolvedsales, and, by ombining

thesmall and largeresolved saleequations, theresultingGalerkin variationalformulationof the

VMSmodelwrites:

∂(W + W ^′ )

∂t , φ l

+ ∇ · F (W + W ^′ ), φ l

= − τ ^LES (W ^′ ), φ ^′ _l

l = 1, N

⁽⁴⁾

where

F

^{reprents both onvetive and visous terms and}

τ ^LES (W ^′ )

^{is the subgrid sale tensor}

omputedusingtheSRSomponent,denedbyaSGSeddy-visositymodel.

For the purpose of this study, three SGS eddy-visosity models are onsidered: the lassial

model of Smagorinsky[23℄, andtworeent andpromising models, namelythe WALEmodel [16℄

andtheoneofVreman[27℄. MoredetailsonthisVMS-LESapproahanbefoundin[10℄.

2.2 Hybrid RANS/VMS-LES

Asin LabourasseandSagaut[12℄,thefollowingdeompositionoftheowvariablesisadopted:

W = < W >

| {z }

RAN S

+ W ^c

|{z}

correction

+W ^SGS

where

< W >

^{are the RANS ow variables, obtained by applying an averagingoperator to the}

Navier-Stokesequations,

W ^c

^{aretheremainingresolvedutuations(i.e.}

< W > +W ^c

^aretheow

variablesin LES)and

W ^SGS

^{aretheunresolvedorSGS utuations.}

IfwewritetheNavier-Stokesequationsinthefollowingompatonservativeform:

∂W

∂t + ∇ · F (W ) = 0

inwhih

F

^{representsboththevisousandtheonvetiveuxes,fortheaveragedow}

hW i

^weget:

∂hW i

∂t + ∇ · F (hW i) = −τ ^{RAN S} (hW i)

⁽⁵⁾

where

τ ^{RAN S} (hW i)

^{isthelosuretermgivenbyaRANSturbulenemodel.}

Aswellknown,byapplyingalteringoperatorto theNavier-Stokesequations, theLESequa-

tionsareobtained,whihanbewrittenasfollows:

∂hW i + W ^c

∂t + ∇ · F(hW i + W ^c ) = −τ ^LES (hW i + W ^c )

⁽⁶⁾

where

τ ^LES

^{istheSGS term.}

Anequationfortheresolvedutuations

W ^c

^{anthusbederived(seealso[12℄):}

∂W ^c

∂t + ∇ · F (hW i + W ^c ) − ∇ · F (hW i) = τ ^{RAN S} (hW i) − τ ^LES (hW i + W ^c )

⁽⁷⁾

The basi idea of the proposed hybrid model is to solve Eq.(5) in the whole domain and to

orrettheobtainedaveragedowbyaddingtheremainingresolvedutuations(omputedthrough

Eq.(7)), wherever the grid resolution is adequate for a LES. To identify the regions where the

additionalutuations mustbeomputed, we introdue ablending funtion,

θ

^{, smoothly varying}

between

0

^and

1

^{. When}

θ = 1

^{,noorretionto}

hW i

^{isomputedand,thus,theRANSapproahis}

reovered. Conversely,wherever

θ < 1

^{,additionalresolvedutuations areomputed;in thelimit}

of

θ → 0

^{wewanttoreoverafullLESapproah. Thus,thefollowingequationisusedhereforthe}

orretionterm:

∂W ^c

∂t + ∇ · F(hW i + W ^c ) − ∇ · F (hW i) = (1 − θ)

τ ^{RAN S} (hW i) − τ ^LES (hW i + W ^c )

⁽⁸⁾

Although it ouldseem rather arbitraryfrom a physial point of view,in Eq.(8 ) thedamping of

therighthand sidetermthroughmultipliationby

(1 − θ)

^{isaimedtoobtainasmoothtransition}

betweenRANSandLES.Morespeially,wewishtoobtainaprogressiveadditionofutuations

whenthegridresolutioninreasesandthemodelswithesfromtheRANStotheLESmode.

Summarizing, the ingredients of the proposed approah are: a RANS losure model, a SGS

model forLESandthedenitionoftheblendingfuntion.

2.2.1 RANSand LES losures:

FortheLESmode,wewishtoreoverthevariationalmultisaleapproahdesribedinSetion2.1.

Thus, the Galerkin projetionof the equations for averagedow and for orretion terms in the

proposedhybridmodelbeomerespetively:

∂hW i

∂t , ψ l

+ (∇ · F c (hW i), ψ l ) + (∇ · F v (hW i), φ l ) =

− τ ^{RAN S} (hW i), φ l

l = 1, N

(9)

∂W ^c

∂t , ψ l

+ (∇ · F c (hW i + W ^c ), ψ l ) − (∇ · F c (hW i), ψ l ) + (∇ · F v (W ^c ), φ l ) = (1 − θ)

τ ^{RAN S} (hW i), φ l

− τ ^LES (W ^′ ), φ ^′ _l

l = 1, N

(10)

where

τ ^{RAN S} (hW i

^{isthelosuretermgivenbyaRANSturbulenemodeland}

τ ^LES (W ^′ )

^isgiven

byoneoftheSGSlosuresmentionedinSetion 2.1.

AsfarthelosureoftheRANSequationsisonerned,thelowReynolds

k − ε

^{modelproposed}

in [4℄isused.

2.2.2 Denitionofthe blending funtionand simpliedmodel:

Asapossiblehoiefor

θ

^{,thefollowingfuntion isusedin thepresentstudy:}

θ = F(ξ) = tanh(ξ ² )

⁽¹¹⁾

where

ξ

^{istheblendingparameter,whihshouldindiatewhetherthegridresolutionisneenough}

toresolveasigniantpartoftheturbuleneutuations,i.e. toobtainaLES-likesimulation. The

hoieoftheblendingparameterislearlyakeypointforthedenitionofthepresenthybridmodel.

In the present study, dierent optionsare proposed and investigated, namely: theratio between

theeddyvisositiesgivenbytheLESandtheRANSlosuresandtheratiobetweentheLESlter

widthand atypiallengthintheRANSapproah.

ToavoidthesolutionoftwodierentsystemsofPDEs andtheonsequentinreaseofrequired

omputationalresoures,Eqs. (9)and(10)anbereasttogetheras:

∂W

∂t , ψ l

+ (∇ · F c (W ), ψ l ) + (∇ · F v (W ), φ l ) =

−θ τ ^{RAN S} (hW i), φ l

− (1 − θ) τ ^LES (W ^′ ), φ ^′ _l

l = 1, N

(12)

Clearly,ifonlyEq. (12)issolved,

hW i

^{isnotavailableateahtimestep. Twodierentoptionsare}

possible: eithertouseanapproximationof

hW i

^{obtainedbyaveragingandsmoothingof}

W

^,inthe

spirit of VMS, orto simplyuse in Eq. (12)

τ ^{RAN S} (W )

^{. Theseond optionis adopted hereasa}

rstapproximation. Wereferto[20,18℄forfurtherdetails.

3 Numerial method and parallelisation strategy

The uidsolverAERO under onsiderationis basedon amixed nite element/nite volume for-

mulation onunstrutured tetrahedralmeshes. Thesheme is vertex-entered, the diusiveterms

aredisretizedusingP1Galerkinniteelementsandtheonvetivetermswithnitevolumes. The

MonotoneUpwindShemeforConservationLawsreonstrutionmethod(MUSCL)isadoptedhere

Figure 1: Speedupand eienyonSGI ICE 8200EX fora266Kvertiesgeometry,measuredon

thealulationofexpliitandimpliitEulerow,from 16oresto512.

and thesheme isstabilized with sixth-orderspatial derivatives. An upwindparameter

γ

^{, whih}

multiplies thestabilization part of thesheme, allowsa diret ontrol of the numerialvisosity,

leadingto afullupwindsheme for

γ = 1

^{andto aentered shemefor}

γ = 0

^{. This}low-diusion MUSCLreonstrution,whihlimitsasfaraspossibletheinterationbetweennumerialandSGS

dissipation,isdesribedin detailin [2℄and [15℄.

Theowequationsareadvanedintimewithanimpliitsheme,basedonaseond-ordertime-

auratebakwarddierene sheme. Thenon-lineardisretised equationsaresolvedbyadefet-

orretion(Newton-like)methodinwhiharstordersemi-disretisationoftheJaobianisused. At

eahtime-step,theresultingsparselinearsystemissolvedbyaRestritedAdditiveShwarz(RAS)

method [21℄. Morespeially,the linearsolveris basedonGMRESwith aRASpreonditioning

and the subdomain problems are solved with ILU(0). Typially, two defet-orretioniterations

requiring eah of them a maximum of 20 RAS iterations are used per time-step. This impliit

shemeislinearlyunonditionallystableandseond-orderaurate.

Forwhatonernstheparallelisationstrategyusedinthisstudy,itombinesmeshpartitioning

tehniques and amessage-passing programmingmodel [9, 13℄. The mesh isassumed to beparti-

tioned into several submeshes,eah one dening a subdomain. Basially thesame serial ode is

going to be exeutedwithin everysubdomain. Modiationsforparallel implementation oured

in themainstepping-loopin orderto takeinto aountseveralassemblyphasesof thesubdomain

results, depending on the uid equations (visous/invisid ows), the spatial approximation and

on the nature of the time advaning proedure (explit/impliit). Beause mesh partitions with

overlapping inur redundant oating-point operations, non-overlappingmesh partitions are ho-

sen. Ithasbeenshownin [13℄that thelatteroptionis moreeientthoughit induesadditional

Figure2: SpeedupandeienyonSGIICE8200EXfora266Kvertiesgeometry: speedupfrom

16oresto512,measures onthealulationofimpliitNavier-stokesow.

ommuniation steps. For our appliations, in a preproessing step we use an automati mesh

partitioner that reates load balaned submeshes induing aminimum amount of interproessor

ommuniations. Dataommuniationsbetweenneighboringsubdomainsareahievedthroughthe

MPIommuniationlibrary.

Wepresentafewspeedupperformanesmeasuredonthreedierentomputingongurations:

-Conguration1isaSGIICE8200EX with3GHzXeonproessors(jade,Figs.1-2).

-Conguration2isahomogeneouslusterfrom theSophiaAntipolisGrid5000site(Fig.4).

-Conguration3isaheterogeneouslusterfrom theSophiaAntipolisGrid5000site(Fig.5).

Thesequenestobeomputedare:

-200expliittimesteps,forEulermodelorfortheNavier-Stokesmodelusing theVMS-LEStur-

bulenemodel.

- 200 impliittime steps, involvingfor eah 40 RAS-GMRES sweeps,for Euler model orfor the

ompleteNavier-Stokesmodel.

Atypial timefortheexpliitEuler testisfor 32proessors678seonds onSGI ICE8200EX

and1077seondsonGrid5000(ratiois1.56).

Arstommentisthestrongimpatonommuniationspeedonspeedup. Indeedweverifythat

thenewerarhitetureICE8200EXstillshowsagoodspeedupwith512proessors/oresalthough

arathersmallamountofomputationismadeineahproessorsinethenumberofmeshverties

perproessorisabout5000.

Figure3: SpeedupandeienyonSGIICE 8200EXandG5000homogeneouslusterfora266K

vertiesgeometry:speedupfrom16oresto512,measuresonthealulationofexpliitEulerow.

Figure4: SpeedupandeienyonSGIICE8200EXandG5000heterogeneouslusterfora266K

vertiesgeometry:speedupfrom16oresto512,measuresonthealulationofexpliitEulerow.

AseondommentisthattheomplexityoftheNavier-StokesLESmodelhavesomeonsequene

onthespeedup.

Forthesimulationspresentedinthenextsetion,theRoe-Turkelsolverisusedwithanumerial

visosityparameter

γ

^{belonging to the interval}

[0.2, 0.3]

^{. TheCFL numberwashosenso that a}

vortexsheddingyleissampledin around400timestepsforthelow-Reynoldssimulationsandat

least1500timestepsforthesimulationsat

Re =

^140000.

4 VMS-LES Simulations

In this setion, we apply ourVMS-LES methodology to the simulation of a owpast a irular

ylinderatMahnumber

M ∞ = 0.1

^{andatasubritialReynoldsnumber,basedonbodydiameter}

andfreestreamveloity,equalto3900.

Theomputationaldomainsizeis:

−10 ≤ x/D ≤ 25

^,

−20 ≤ y/D ≤ 20

^and

−π/2 ≤ z/D ≤ π/2

^,

where

x

^,

y

^and

z

^denotethestreamwise,transverseandspanwisediretionrespetively. Theylinder ofunit diameter

D

^isenteredon

(x, y) = (0, 0)

^.

For thepurpose of these simulations, the Steger-Warming onditions[25℄ are imposed at the

inowandoutowaswellasontheupperandlowersurfae

(

y = ±H y

^{). Inthespanwisediretionperiodiboundaryonditionsareappliedandontheylinder}

surfaeno-slipboundaryonditionsareset.

The ow domain is disretized by two unstrutured tetrahedral grids: the rst one (GR1)

onsists of approximately

2.9 × 10 ⁵

^{nodes. The averaged distane of the nearest point to the}

ylinderboundaryis

0.017D

^{,whihorrespondsto}

y ⁺ ≈ 3.31

^{. Theseond grid(GR2)isobtained}

from GR1 by rening in a strutured way, i.e. by dividing eah tetrahedron in 4, resulting in

approximately

1.46 ×10 ⁶

^{nodes. Alargenumberof}simulationswerearriedoutbyvaryingdierent parameters,as,forinstane,theSGSmodel,thevalueof

γ s

^orthegridresolution. Wereporthere only the results obtained in some of these simulations. The main parameters of the onsidered

simulations are summarized in Tab.1, together with some of the obtained ow bulk parameters.

Theexperimentalreferenevalueforthemeandrag oeient,

C d

^,is

0.99 ± 0.05

^{from [17℄,whih}

wellagreeswiththoseomputedinwellresolvedLESintheliterature[11,7℄,whileforthevortex-

shedding Strouhal number,

St

^{, values in therange of}

[0.21, 0.22]

^{are generallyobtained. Finally,}

forthemeanreirulationbubblelength,areentexperimentalandnumerialstudy[19℄seemsto

indiateareferenevalueof

l r = 1.51±10%

^{. Fig.5ashowsthemeanpressureoeient}distribution at the ylinder obtained on GR1 in LES and VMS-LES simulations, together with experimental

datafrom[17℄. Fromthedisrepanybetweennumerialresultsandexperimentaldatainthezone

of thenegativepeak itis evident that in all asestheboundary layerevolutionis notaurately

aptured in the simulations, due to the grid oarseness. Another symptom of a too oarse grid

resolution (see the disussion in [11℄) is the underestimation of the mean reirulation length

l r

in allthe simulationsonGR1 (Tab.1). However,somedierenesexist betweentheLESand the

VMS-LES simulations. In partiular, in LES the disrepany observed in the negative peak of

mean

C p

^{islargerandthedierenesamongthedierentSGSmodelsaremorepronounedthanin}

VMS-LES.Thisisduetothefatthatthenon-dynamieddy-visositymodelshereused,although

mainly ating in thewake,also provideasigniantSGS visosityin the laminarregions, asthe

boundarylayerandthedetahingshearlayers(see,e.g.,Fig.6a). IntheVMS-LESsimulationsthe

spatial distributionof the SGS visosity is qualitativelysimilar to that obtainedin LES, but the

amountis signiantlyreduedeverywhere(omparethe salesof Fig.6aand Fig.6b), and, thus,

also in thelaminar zones. Moreover,wereall that in theVMS-LES approah theSGS visosity

Table1: Mainsimulationparametersandowbulkoeients.

Turb. model SGSmodel Grid

γ s C d

^St

l r

LES Smagorinsky GR1 0.3 1.16 0.212 0.81

LES Vreman GR1 0.3 1.04 0.221 0.97

LES WALE GR1 0.3 1.14 0.214 0.75

VMS-LES Smagorinsky GR1 0.3 1.00 0.221 1.05

VMS-LES Vreman GR1 0.3 1.00 0.22 1.07

VMS-LES WALE GR1 0.3 1.03 0.219 0.94

nomodel - GR1 0.3 0.96 0.223 1.24

nomodel - GR1 0.2 0.94 0.224 1.25

LES WALE GR2 0.3 1.02 0.221 1.22

VMS-LES WALE GR2 0.3 0.94 0.223 1.56

nomodel GR2 0.3 0.92 0.225 1.85

0 20 40 60 80 100 120 140 160 180

−2

−1.5

−1

−0.5 0 0.5 1 1.5

Angle θ (0 at stagnation point) Cp m

LES Smagorinsky LES Vreman LES WALE VMS−LES Smagorinsky VMS−LES Vreman VMS−LES WALE Experiments

0 20 40 60 80 100 120 140 160 180

−1.5

−1.25

−1

−0.75

−0.5

−0.25 0 0.25 0.5 0.75 1

Angle θ (0 at stagnation point) Cp m

VMS−LES WALE LES WALE No model Experiment

(a) (b)

Figure 5: Mean pressure oeient distribution at the ylinder. (a) Simulations on GR1, (b)

SimulationsonGR2.

onlyatsonthe smallestresolvedsales. Thedierentdistribution ofSGS visosityleadsin LES

to additional inauraies, besides those due to grid oarseness and previously disussed, whih

are not present in VMS-LES. For instane, Fig.5a shows that the base pressure is inaurately

predited in all LES simulationsexept for theVreman model, leadingto an inaurate value of

themeandragoeient(Tab.1)whilefortheVMS-LESonestheagreementwiththeexperiments

isfairly good. Thepressuredistributionobtainedin thesimulationswithoutanySGS model(not

shown)is verysimilarto theoneobtainedin theVMS-LES ones,aswellasforlow-orderveloity

statistisinthewakeandforthebulkowparameters,exeptthanforasigniantlyhigher

l r

^given

by theno-model simulations(Tab.1). This isan a-posteriorionrmation that the usedMUSCL

reonstrution indeedintroduesavisosityatingonly onthehighestresolvedfrequenies[2℄,as

the SGS visosity in the VMS approah and that this limits its negative eets. Moreover, the

results obtainedwith two dierent(low)valuesof the parameter

γ s

^{are alsoverysimilar (Tab.1),}

onsistentlywithourpreviousndings[2℄. AsfortheresultsontherenedgridGR2,asexpeted,

in bothLESandVMS-LES theagreementwith thereferenedatais improved. However,in LES

disrepanies are still observed (see, e.g., Fig.5b) due to the exessive introdued SGS visosity,

while with VMS-LESa generalgood agreement isobtained. Note that in this ase, although for

thepressuredistribution,andthusforthedragoeient,thesimulationwithoutanymodelgives

aurate results,thelength ofthe meanreirulationbubble islargely overestimated, dueindeed

tothelakofSGSvisosityinthewake.

(a) (b)

Figure 6: Instantaneous iso-ontours of

µ s /µ

^. Simulations on GR1 with the Smagorinsky SGS model: (a)LES,(b)VMS-LES.

TheFourierenergyspetrumofthespanwiseveloityatP

(3, 0.5, 0)

^{forVremanSGSmodelwith}

LESandVMS-LESontheoarsegridGR1isdisplayedinFig.7. Thefrequenyisnondimensional-

izedbytheStrouhalsheddingfrequeny. ViatheTaylorhypothesisoffrozenturbulene(whih is

justiedsinethemeanonvetionveloityislargeatthatpoint)whihallowstoassumethathigh

(low)timefrequeniesorrespondtosmall(large)saleinspae,weobservethattheenergyinthe

largeresolvedsalesarehigherwithVMS-LES thanwithLES.These resultsorroboratethefat

that in the VMS-LES approah, themodeling of the energy dissipation eets of the unresolved

salesaetsonlythesmallresolvedsales,unlikelyintheLESapproahinwhihthesedissipative

eets atonalltheresolvedsales.

Summarizing,ourresultsonrmthat theideaofonentratingtheSGSvisosityonlyonthe

smallest resolvedsales atually permits to use simple eddy-visosity SGS models and to obtain

10 ⁻¹ 10 ⁰ 10 ¹

10 ⁻⁸ 10 ⁻⁷ 10 ⁻⁶ 10 ⁻⁵ 10 ⁻⁴ 10 ⁻³ 10 ⁻²

Normalized time frequency

Energy

VMS−LES Vreman LES Vreman

Figure7: Fourierenergyspetrum: spanwiseveloityforLESVremanandVMS-LESVreman

aurate results, omparableto those obtainedin theliterature with dynami models. We reall

that onunstrutured gridsthisleadstosigniantlymoreaordablesimulations,sinetheostof

thedynamiproeduremaybeomeprohibitive. Inthesamespirit,ithasbeenonrmedthatthe

fatofalsoonentratingthenumerialdissipationonthesmallestresolvedsalespermitstolimit

itsnegativeeets.

Forthis probleminvolving1.5milliondegreesoffreedomand fortwentysheddingylessimu-

lation,thesimulationtimeisabout7hoursona32-proessorIBMPower4omputer.

5 Hybrid RANS/VMS-LES Simulations

The new proposed hybrid model (Flutuation Corretion Model, FCM) has been applied to the

simulation of theowaround airular ylinderat

Re = 140000

^{(based on thefar-eld veloity}

and the ylinder diameter). Thedomain dimensionsare:

−5 ≤ x/D ≤ 15

^,

−7 ≤ y/D ≤ 7

^and

0 ≤ z/D ≤ 2

^{(the symbols are the sameas in Setion 4). Two grids have been used, the rst}

one (GR1) has 4.6

×10 ⁵

^{nodes, while the seond one has (GR2) 1.4}

×10 ⁶

^{nodes. Both grids are}

omposedofastruturedpartaroundtheylinderboundaryandaunstruturedpartintherestof

thedomain. Theinowonditionsarethesameasin theDESsimulationsofTravinet al. [26℄. In

partiular,theowis assumedtobehighly turbulentbysettingtheinowvalueofeddy-visosity

toabout5timesthemoleularvisosityasintheDESsimulationofTravinetal. [26℄. Thissetting

orrespondstoafree-streamturbulenelevel

u ^′2 /U 0

^(where

u ^′

^{istheinletveloityutuationand}

U 0

^isthefree-streammeanveloity)oftheorderof4%. AsdisussedalsobyTravinetal. [26℄,the eet ofsuhahigh leveloffree-streamturbulene isto maketheboundarylayeralmost entirely

turbulentalsoattherelativelymoderateonsideredReynoldsnumber. Theboundarytreatmentis

thesame asfor theVMS-LES simulationsin Setion 4,exept that wall lawsare nowused. The

RANSmodelisthatbasedonthelow-Reynoldsapproah[4℄. TheLESlosureisbasedontheVMS

approah (seeSetion2.1). The SGS models used in the simulations arethose givenin Setion2.

The main parameters haraterizing the simulations arried out with the FCM are summarized

in Tab.2. The main ow bulkparametersobtainedin the present simulationsaresummarized in

Simulation Blendingparameter Grid LES-SGSmodel

FCM1 VR GR1 Smagorinsky

FCM2 LR GR1 Smagorinsky

FCM3 LR GR2 Smagorinsky

FCM4 LR GR1 Vreman

FCM5 LR GR1 Wale

Table2: Simulationnameandtheirmainharateristis

Tab.3,togetherwith theresultsofDESsimulationsin theliteratureand someexperimental data.

Theyhavebeenomputedbyaveragingintime,overatleast20sheddingylesandinthespanwise

diretion. Letusanalyze,rst,thesensitivitytotheblendingparameter,byomparingtheresults

of the simulation FCM1 and FCM2. The results are pratially insensitive to the denition of

the blendingparameter. Conversely, the grid renementprodues adelayin theboundary layer

separationwhihresultsinadereaseof

C ¯ d

^{(ompareFCM2andFCM3). However,notethat, for}

unstrutured grids,the renementhanges theloal qualityof thegrid (in termsofhomogeneity

andregularityoftheelements)andthismayenhane thesensitivityoftheresults. Thesensitivity

totheVMS-LESlosuremodelisalsoverylow(ompareFCM2,FCM4andFCM5). Thisverylow

Datafrom

Re C d C _l ^′

^St

l r θ sep

FCM1 1.410

5

0.62 0.083 0.30 1.20 108

FCM2 1.410

5

0.62 0.083 0.30 1.19 108

FCM3 1.410

5

0.54 0.065 0.33 1.13 115

FCM4 1.410

5

0.65 0.077 0.28 1.14 109(99)

FCM5 1.410

5

0.66 0.094 0.28 1.24 109(100)

Numerialdata

DES[26 ℄ 1.410

5

0.57-0.65 0.08-0.1 0.28-0.31 1.1-1.4 93-99

DES[14 ℄ 1.410

5

0.6-0.81 0.29-0.3 0.6-0.81 101-105

Experiments

[8℄ 3.810

6

0.58 0.25 110

[1℄ 510

6

0.7 112

[22℄ 810

6

0.52 0.06 0.28

Table3: Mainbulkowquantitiesfortheirularylindertestase. SamenotationsasinTab.??.

0 20 40 60 80 100 120 140 160 180

−2.2

−1.8

−1.4

−1

−0.6

−0.2 0.2 0.6 1

Angle θ (0 at stagnation point) Cp m

Jones James Roshko Travin et al.

FCM2 FCM4 FCM5

Figure8:

C ¯ p

^{ontheylindersurfaeompared tonumerialand}experimental results

sensitivity hasbeenobservedalso in VMS-LESsimulations atlowReynoldsnumbersee Setion4

and,thus,itseemsmorepeuliartotheVMS-LESapproahratherthantothehybridmodel. The

agreementwiththeDESresultsisfairlygood. Asfortheomparisonwiththeexperiments,asalso

stated in Travin et al. [26℄, sineour simulationsare haraterized by ahigh level of turbulene

intensityattheinow,itmakessense toomparetheresultswithexperimentsathigherReynolds

number,in whih, although thelevelof turbuleneintensity oftheinoming owis verylow,the

transition to turbulene of theboundary layerours upstream separation. The agreement with

these high

Re

experimentsis indeed fairly good, asshown in Tab.3and in Fig.8. The behavior of theseparationanglerequires abriefdisussion. There is asigniantdisrepany betweenthe

valuesobtainedinDESandtheexperimentalones. Foroursimulations,thevaluesof

θ sep

^shownin

Tab.3areestimatedbyonsideringthepointatwhihthe

C p

distributionovertheylinderbeomes nearlyonstant(seee.g. Fig.8),asusuallydoneinexperimentalstudies. Indeed,thereportedvalues

aregenerallyinbetteragreementwiththeexperimentsthanthoseobtainedbyDES.However,ifwe

estimatetheseparationangle fromthestreamlines oftheaverageorinstantaneousveloityelds,

signiantlylowervaluesarefound(reportedinparenthesesinTab.3forthesimulationsFCM4and

FCM5);thesevaluesarelosertothoseobtainedbyDES.Finally,themodelworksinRANSmode

in theboundarylayerandintheshear-layersdetahingfromtheylinder,whileinthewakeafull

VMS-LESorretionisreovered.

Forthisprobleminvolving3.2milliondegreesoffreedomandfortwentysheddingylessimulation,

thesimulationtimeisabout30hoursona32-proessorIBMPower4omputerandabout16hours

ona32-proessorIBMPower5omputer.

6 Conlusion

Inthispaperwehavepresentedparallel simulationsof three-dimensionalturbulentows. An e-

ientimpliittimeadvaninganbeappliedwitharathersmalltimestepandsmallomputational

eort. In theseonditions, agood speedupfor 16-512oresis obtainedwitha reent parallelar-

hiteture. With this tool, we have rst investigated the appliation of a Variational multisale

LESfor thesimulations ofaowpastairularylinder at asubritialReynoldsnumberequal

to

Re =

^{3900. Although arather oarsegrid hasbeen used,this model givesauratepreditions}

of bulkoeientsand showsthat tworeentlydevelopedSGS models, theVreman's model and

the WALE model ombine well in the VMS formulation. Moreover,it appears in this approah

that theinueneoftheSGS modelis weak,but thisseemstogiveasupportto theVMSideaof

adding somedissipation onlyto thesmallestresolvedsales. Inaseondpart,wehavepresented

ahybridRANS/LESapproahusingdierentdenitions ofblendingparameterand SGSmodels.

Forthe losure of the LES part, the VMS approah has been used. This model is validated on

thepreditionofaowaroundairularylinderathigherReynoldsnumber(

Re =

^{140000). The}

resultsobtainedorrelatewellwiththeexperimentalandnumerialdatafromtheliteratureaswell

asthebehavioroftheblendingfuntion.

7 Aknowledgements

Wethank Erilamballaisforkindlyprovidingexperimental dataonerningtheRe=3900test

ase.

Some experiments presented in this paper were arried out using theGrid'5000 experimental

testbed,beingdevelopedundertheINRIAALADDINdevelopmentationwithsupportfromCNRS,

RENATERandseveralUniversitiesaswellasotherfundingbodies(seehttps://www.grid5000.fr).

The authors would like also to aknowledge the support of Centre Informatique National de

l'EnseignementSupérieur(CINES 1

),Montpellier,FRANCE,andthesupportofPACA 2

regionfor

theooperationbetweenINRIAandtheUniversityofMontpellier. TheCINESresultsweremade

ontheSGIICE8200EX parallelmahine.

CINECA(Bologna,Italy),IDRIS(Orsay,Frane)andINRIA-SophiaClusterarealsogratefully

aknowledgedforhavingprovidedomputationalresouresforthisstudy.

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