Testing of subgrid scale (SGS) models for large-eddy simulation (LES) of turbulent channel flow

UNIVERSITY OF BOLOGNA

SCHOOLOF ENGINEERING ANDARCHITECTURE

Master of Siene Degree inAerospae Engineering

Testing of subgrid sale (SGS)

models for large-eddy simulation

(LES) of turbulent hannel ow

Master thesis in Applied Aerodynamis

SUPERVISORS:

Prof. Alessandro Talamelli

Prof. Arne V. Johansson

ASSISTANT SUPERVISOR:

Dr. Geert Brethouwer

STUDENT:

Matteo Montehia

Aademi Year 2013-2014

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Sub-grid sale(SGS) modelsare requiredin order to model theinuene of theunresolved small

sales on the resolved sales in large-eddy simulations (LES), the ow at the smallest sales of

turbulene.

In the following work two SGS models are presented and deeply analyzed in terms of auray

throughseveralLESswithdierentspatialresolutions,i.e. gridspaings.

The rstpart ofthis thesisfouses onthe basitheory ofturbulene, the governingequations of

uiddynamisandtheiradaptationtoLES.Furthermore,twoimportantSGSmodelsarepresented:

oneistheDynamieddy-visositymodel(DEVM),developedby[Germanoetal.,1991℄,whilethe otheristheExpliitAlgebraiSGS model(EASSM),by[Marstorpetal.,2009℄.

In addition, somedetails about the implementation of the EASSM in a Pseudo-Spetral

Navier-Stokesode[Chevalieret al.,2007℄arepresented.

Theperformaneof thetwoaforementionedmodelswillbeinvestigatedin thefollowinghapters,

bymeansofLESofahannel ow,withfritionReynoldsnumbers

Re

τ

= 590

upto

Re

τ

= 5200

, with relatively oarse resolutions. Data from eah simulation will be ompared to baseline DNS

data.

Resultshaveshownthat,in ontrasttotheDEVM, theEASSM haspromising potentialsforow

preditionsathighfritionReynoldsnumbers:thehigherthefritionReynoldsnumberisthebetter

theEASSM willbehaveandtheworsetheperformanesoftheDEVMwillbe.

The betterperformaneof theEASSM is ontributedto theabilityto aptureowanisotropyat

thesmallsalesthroughaorretformulationfortheSGSstresses.

Moreover,aonsiderableredutionintherequiredomputationalresouresanbeahievedusing

the EASSM ompared to DEVM. Therefore, the EASSM ombines auray and omputational

VersioneItaliana

Nelle Simulazioni agrandi vortii (Large-eddy simulations, LES), dei modelli per la sala di

sottogriglia(Sub-grid sale, SGS) sono neessariper riprodurrel'inuenza delle piúpiole sale

dellaturbolenza,quindinonrisolte,suquelleheinveevengonorisoltedirettamente.

Nel seguente lavorosono presentati due modelli SGS, laui auratezzaverrá poi analizzata

at-traversovarieLESadiverserisoluzionispaziali,equindidiversiintervallididierenziazione.

Laprima partedellatesisionentrasullateoriadellaturbolenza,partendodalleequazioni

osti-tutivedellauidodinamia,noallaloroversioneperLES.

Due importanti modelli SGS sono stati presentati: il primo é il modello Dynami eddy-visosity

model(DEVM),di[Germanoetal.,1991℄,ilseondoéilmodelloSGSEspliitoAlgebrio,Expliit AlgebraiSGS model (EASSM),di[Marstorpetal.,2009℄.

Sarannoinoltrefornitidettagliaggiuntivisull'implementazionedelmodelloEASSMsuunodiedi

FluidodinamiaComputazionale(ComputationalFluidDynamis,CFD),Pseudo-Spettrale

Navier-Stokessviluppato nel Linné Flow Centre del Dipartimento di Ingegneria Meania del KTH di

Stoolma,da[Chevalieret al.,2007℄.

I seguentiapitoli verterannosull'analisi dellastimadi unussoin unanale,hannel ow, fatta

daimodellidesrittiinpreedenza,perunbassonumerodiReynoldsbasatosull'attrito,

Re

τ

= 590

, no a

Re

τ

= 5200

. I datiottenutida iasunasimulazioneverrannoonfrontationdatidi Simu-lazioni NumeriheDirette (DiretNumerial Simulations, DNS).

Dai risultati ottenuti si puó onludere he, dierentemente dal modello DEVM, l'EASSM ha

promettenti potenzialitá nella stima del usso ad alti numeri di Reynolds

Re

τ

: piú alto é tale numero,piúilmodelloEASSMdarárisultatiaurati,mentreleperformanesdelDEVM

peggior-eranno.

LemiglioriperformanedelmodelloEspliitoAlgebriopossonosenz'altroessereattribuiteallasua

abilitádialolareinmanieraorrettal'anisotropiaallepiolesaletramiteunaformulazione

or-rettadeglistressdisottogriglia,SGS.

Inonlusione,datalaridottaquantitádirisorseomputazionalirihiestapereettuaresimulazioni

rispetto alDEVM, tale modelloombina auratezza eeienza omputazionale,tanto he puo'

1 Turbulene 5

1.1 IntrodutiontoTurbulene . . . 5

1.2 NumerialapproahestoN-S solution . . . 12

1.3 Apartiularase: TurbulentChannelFlow . . . 17

2 Large Eddy Simulation 25 2.1 Thelteringoperation . . . 26

2.2 GoverningequationsofLESandthelosure problem . . . 27

2.3 An exampleofalosure: theEddyVisosityModel(EVM) . . . 27

2.4 Subgrid-saledissipation . . . 28

3 Subgrid-salestress modelsfor LES 31 3.1 TensorialpolynomialformulationoftheSGSstresstensor . . . 31

3.2 ThedynamiproedureforEVM(DEVM) . . . 33

3.3 ExpliitAlgebraiSGS stressmodel . . . 34

4 Implementationin a CFD ode 39 4.1 Theneedforauray: spetralmethods. . . 39

5.1 MeanValues . . . 49

5.2 Root-meansquaredValues,rms . . . 50

5.3 Vortialstrutures . . . 51

6 Results 53

6.1 LESat

Re

τ

= 590

. . . 54

6.2 LESat

Re

τ

= 2000

. . . 64

6.3 LESat

Re

τ

= 5200

. . . 72

Provaad immaginare unpennahio di fumo fuoriusente dauna pipa di un uomopensante,

seduto su diuna sediaadondolo. Anhe in questopioloaspettodellavita quotidiana la

turbu-lenza gioaunruolofondamentale.

Il noto sioRihardFeynman denii laturbulenza ome il piúimportante problema dellasia

lassiaheanoranonéstatorisolto.

Nelorsodegliannidiversisienziatihannoprovatodi omprendereil veroomportamentoditale

fenomeno, la ui omplessitá giae nelle equazioni he lodesrivono. Per unuido Newtoniano,

la turbolenzaé denita dalle equazioni di Navier-Stokes, unsistema di equazioni non-lineari alle

derivateparziali,lauisoluzioneanalitiaanoranonéstataottenuta.

Tuttavia, degli approi alternativi per ottenere delle soluzioni sono stati sviluppati nora: un

metodoonsistenell'eettuaredegliesperimenti: tramitegalleriedelventosiamoingradodi

ripro-durreussiinsvariateondizioni,dallesituazionilassihedistratolimite(motosudiunaparete)

eussoinunanale,noaasipiúompliati, omeilussoattornoadunorpotozzo(omeuna

automobile)eunorpoaerodinamio,omeunaeroplano.

Gliesperimentihannoomeneprinipalel'analisidellequalitádelusso(omepressionee

velo-itá) attraversodelle sondeedelle tenihe di visualizzazione, omela Partile ImageVeloimetry

(PIV).

[Talamellietal.,2009℄nell'Universitádi Bologna.

D'altroanto,unapproiodiversoéquellodisfruttarel'analisinumeriaepassarealla

implemen-tazionedelsistemaN-S suodii diFluidodinamiaComputazionale.

Tuttavia, la soluzione numeria e ompleta fornita dalle Simulazioni Numerihe Dirette (DNS),

in grado di desriverela turbolenza aqualsiasi sala, non ésemprepossibileaausa dellegrandi

risorse omputazionalirihieste,in partiolare adelevati numeridi Reynolds. Perquesto motivo,

l'obiettivoprinipaledelprogettoCICLoPEéquellodiomprendereapienoilfenomenodella

tur-bolenzaqualoranonsiapossibileotteneredatitramiteDNS.

Comunque,latenia DNSnon élasola in gradodi riprodurre numeriamente unpreiso usso.

Una seondapossibilitási hiamala simulazione agrandi vortii (Large-eddysimulation -LES);

questateniaéingradodiraggiungere,onopportunimodelliperlepiolesaledellaturbolenza,

unasoddisfaenteauratezzaomputazionaleonunaragionevolequantitádirisorse

omputazion-ali.

A tale avviso, la tesi verterá sull'impiego e il test di due modelli innovativi per le piole sale

dellaturbolenza;inpartiolareverrámessaaonfrontol'auratezzadiiasunmodelloonl'altro,

rispettoanhe i risultatiforniti da DNS. Lesimulazioni sonostate ompiuteatre diversi numeri

di Reynolds;in partiolarel'ultimoaso,dimaggioreimportanza, érelativoalmassimonumerodi

ReynoldsraggiuntonoradaDNS.

Laprimapartedellatesiomprenderáladesrizionedellaturbolenza,siadaunpuntodivistasio

hematematio. Saranno onsiderateleequazionidiN-S perunuidoNewtoniano,onusso

in-omprimibileeturbolento, inunanale.

NelseondoapitololateniaLESsaráespressaneidettagli,mentrenelterzoverrannopresentati

duemodellidisalasotto-griglia(sub-gridsale-SGS).Ilquartoapitolospiegherádeidettagli

te-niisull'implementazionedellesimulazioni,deiduemodelli,edellaparallelizazioneMPIutilizzata.

Nel quinto apitoloenni di metodi statistii per l'analisi della turbolenza saranno arontati. I

risultativerrannopresentatinelsestoapitolo,einnesarannofornitedelleonlusioninelsettimo

Imagineasmokingplumeomingoutfrom apipeofathinking man,thisisatypialeveryday

life senariowhereturbuleneplaysamain role. Thephysiist RihardFeynmandened itasthe

most important unsolvedproblemof lassial physis.

Sientistsalongtheyearshavetriedto understand therealbehaviourof thisphenomenon, whih

gains itsomplexitybeauseoftheequationsthat desribesit. ForaNewtonian uid,turbulene

istraedbyNavier-Stokesequations,asystemofnon-lineardierentialequationswhoseanalytial

solutionhasnotbeenprovidedyet.

Anyway,alternativeapproaheshasbeendevelopedarosstheyearsuptonow: themostintuitive

andoldwaytounderstanduidmotionistomakeexperiments: throughtheemploymentofwind

tunnelsweanreproduetheowinseveralonditions,fromthebasialhannelowandboundary

layerto themoreomplexones,liketheowarossablu body (likeaar)oranairplane. This

tehniqueinvolves theanalysis of the owproperties(likepressure and veloity) throughprobes

andadvaned visualizationtehniquessuhasPartileImageVeloimetry(PIV).

A reentexamplewhoseaimistodesribehighReynoldsnumberturbulenearossapipeis

nev-erthelesstheCICLoPEexperiment,abigprojetdevelopedby

[Talamellietal.,2009℄inUniversityofBologna.

Theseondapproahisthustousenumerialanalysisinordertoimplementodesableto

mainCICLoPEprojet'saimistounderstandturbulenephenomenafurthertheReynoldsnumber

limitditatedbyDiretNumerialSimulations.

However, DNS is not the only tehnique able to reprodue numeriallya given ow. A possible

seondhoieisalledLarge-eddysimulation;thistehniqueisabletoreahasatisfying

omputa-tional auraywithareasonableamountof omputationalresoures.

InthisthesisNavier-StokesequationsforaNewtonianuidandaninompressibleowina

turbu-lenthannelowsenariowillbedesribedinthersthapter,togetherwithaphysialdesription

ofturbulene. IntheseondhaptertheLarge-eddysimulation(LES)tehniquewill beexplained

in detail,while in the third onetwoLES sub-grid salemodelsare shown. Inhapter foursome

tehnialdetails oftheimplementationof theLES,ofthemodels,with apartiularfousonMPI

parallelizationare pointedout. Sometheory aboutturbulene statistiswill befaedin thefth

1

TURBULENCE

Everybodyisagenius. Butif youjudgeash byitsability tolimb atree,

itwilllive itswholelifebelieving thatitisstupid. (A.Einstein)

1.1 Introdution to Turbulene

Thephenomenonofturbuleneisfoundinseveralappliations, forexample,ombustiontumbling

in Internal CombustionEngines(ICEs),thewakeofaFormula1ar,thejetspreadbythenozzle

ofasupersoniairraftengine.

Inautomotiveengineering,forexample,thestudyofaerodynamisaroundaarinvolvesthe

har-aterizationofaturbulentwall-boundedow,alledaBoundary Layer. Itwasintensivelystudied

byL.Prandtlin1904;hereturbuleneisthemain responsibleforfritionandwakedrag.

In asolid roketmotor nozzle,there's ageneration ofaplume, whereturbulentmotionsofmany

salesanbeobserved;fromeddiesandbulgesomparableinsizetothewidthoftheplumetothe

andhaotinature,sinethemotionofeveryeddy isunpreditable.

Figure1.1Large-eddysimulationofjetfromaretangularnozzle. Theretangularnozzleis

showningraywithanisosurfaeoftemperature(gold)utalongtheenterplaneofthenozzle

showingtemperature ontours (red/yellow). Theaousti eldisvisualizedby(blue/yan)

ontoursofthepressureeldtakenalongthesameplane,fromP.Moin.

Whilelaminarowisa smoothand steadyowmotion,where any induedperturbationsare

dampedoutduetotherelativelystrongvisousfores,inturbulentowsotherforesmaybeating

that ounterattheationofvisosity. Ifsuh foresarelargeenough,theequilibrium oftheow

is upset and theuid annot adapt suddenlyto visosity. Thefores that upset this equilibrium

aninlude buoyany,inertia,oreven rotation. Inahannel, visousandinertialforesatingon

theuidareproportionalto

F

v

∝

νL

(1.1)

F

i

∝

V L

2

(1.2)

where

ν

istheuidvisosity,and

L

and

V

aretheharateristiveloityand lengthsales. If

thevisous foresontheuidarelargeomparedwithothers, anydisturbanesintroduedin the

owwilltend to bedamped out. Ontheother hand,iftheinertialfores beome large,theuid

will tend to break upinto eddies. For greater inertialfores, the eddies will break upinto even

smallereddies. This willontinueuntilwereahasmallenoughlengthsale(eddy size)onwhih

The largest of these eddies will be onstrained by the physial size onstraints on the ow (like

hanneldiameter);thesmallereddieswillbeonstrainedbythevisousforeswhihatstrongest

at the smallest length sales. Therefore, one of the diulties assoiated with the predition of

turbulentowis thattherangeoflengthsalesanbeverylarge.

Thedesriptionofturbuleneinvolvesdierentoneptsliketurbuleneenergyprodution,transfer

anddissipation. Rihardson'sfamouspoemgivesagoodideaaboutturbulene:

Big whorlshave littlewhorls

That feed ontheir veloity

Andlittle whorls havelesserwhorls

Andso ontovisosity (in the moleularsense)

Thisisthedesriptionoftheenergyasade onept. Itstatesthatturbulentowsanbe

on-sideredasanagglomerateofeddies ofdierentsizes. Largeenergyontainingeddiesareunstable

and break down andtransferenergy to smallereddies. The proessgoesontill thesmallestone,

theKolmogorovsale,where energyis dissipatedinto heatbyvisouseets.

This isthegreatonlusion thatKolmogorovmadein 1941,andhis rsthypothesis isompletely

basedonthat: atsuientlyhighReynoldsnumber,thestatistisofthesmallsalesareuniversal

andaredeterminedsolely byvisosity,

ν

,andtheenergydissipationrate,

ε

.

Usingdimensionalanalysis,itispossibleto derivetheKolmogorovlengthsale

η

,timesale

t

η

andveloitysale

v

η

:

η

=

ν

3

ε

!1

/

4

,

t

η

=

ν

ε

!1

/

2

,

v

η

= (

νε

)

1

/

4

(1.3)

Then, in aordaneto Rihardson'spoem, Kolmogorovmade aseond hypothesis, basedon

thefat thatatsuientlyhighReynoldsnumberthestatistisofthesaleswhiharesuiently

larger than

η

and muh smallerthan the largestenergeti sales are solely desribed by

ε

. This

hypothesisreferstotheinertialrangeofsales. Thekinetienergyspetrumofthesesalesanbe

desribedby

Figure1.2Turbuleneenergyvswavenumberspetrum,fromJ.M.MDonough.

where

k

isthewavenumberand

C

k

≈

1

.

5

[Sreenivasan,1995℄istheKolmogorovonstant. Fromthelastdiagram,it'slearthatturbulenehasdierentbehavioursaordingtothe

wavenum-ber. Basingontheseonepts,weandistinguishfourdierentregions:

1. thelargesale,determinedbytheproblem domaingeometry;

2. theintegralsale

(Λ)

,whih isan

O

(1)

fration(oftentakentobe

∼

0

.

2

)ofthelargesale;

3. the Taylor mirosale whih is an intermediate sale, found in the Kolmogorov's inertial

subrange

(

η

≪

2

π

κ

≪

Λ)

4. theKolmogorov(ordissipation)sale

(

η

)

whihisthesmallestofturbulenesales,theinner

sale

1.1.1 3D Nature of Turbulene

Turbulene is rotational and a three-dimensional phenomenon. It is haraterized by large

utuationsinvortiity,whihareresponsibleforvortexstrethingandlengthsaleredution. These

harateristisareidentially zeroin twodimensionsand theseare thereasonswhyturbulene is

hardtodesribebothanalytiallyandnumerially.

Thesethree-dimensionaldynamialmehanismsarehighlyomplexandnonlinear,howevertheow

anbeassumedas bi-dimensionalforlargesale2Dstrutures. These struturesplayadominant

inthesmallersale,wheretheyarefundamentalformixing,mostofallatmoleularsales(e.g. in

ombustionproblems).

1.1.2 Order & Randomness

Despiteturbulene is haoti, itonsists of ompletely randommotionsthat anaggregatein

oherentstrutures. Typialexamplesareturbulentboundarylayersand homogeneousturbulent

shearows,whihexhibithorseshoe,orhairpinvorties(seegure1.3)thatappeartobeinherent harateristis. Freeshearowslikethemixinglayerrevealoherentvortexstruturesverylearly,

againevenforveryhighturbuleneintensities.

The onepts of order and randomness have also led to some new analyti approahes and new

interpretations in the study of turbulene. The names of these disiplines are known as Chaos,

Bifuration Theory, and DynamialSystems [MDonough,2007℄. These theorieshavebeenfaed forthestudyofturbulene,in partiularintheareaofhydrodynamistabilityandtransitionfrom

laminarowtoturbulentone. Cometoattentionofmathematiians,physiistsandengineers,these

phenomenaisofaremarkablenon-linearity,whihmakesturbuleneunpreditableandomplexto

desribe.

Asanonlinearproblem,itanbeseenthatthesolutionstotheseproblemswiththesamenonlinear

equationswithonlyslightdierenesininitialonditions,willrapidlydiverge. Therefore,asuitable

denition ofturbulene must neessarilyinvolveaomplexdynamial systemwith many degrees

offreedom.

1.1.3 The Reynolds number

Turbulene an be seen also a play between inertial fores and visous. Therefore the ratio

betweenthemisruialinorderto haraterizeaow. TheReynoldsnumberplaysthat role,and

in turbulentowsholds

Re

def

=

U L

ν

≫

1

.

(1.5)

Formany ows of pratialimportane (e.g. a ow on airplanewings) the Reynoldsnumber

anbeontheorderof

Re

∼

10

6

. Thismeansthat thevisousfores,thataremoleularfores,at

in smallersalesthanin thelargeones. However,in anyturbulentowthemoleularvisosityis

alwaysimportantatsomesale. AstheowReynoldsnumberinreases,theregionwhere visous

eetsareremarkable,dereasesinthiknessandtheveloityoftheowhangesveryrapidlyfrom

zeroatthesurfaeto thefree-streamveloityat theouteredgesoftheboundarylayer. Again,we

see thetendenyofthenonlinearinertialtermstogeneratedisontinuitiesat highwavenumbers.

One more,theReynoldsnumberanbeinterpretedalso intermsoflengthandtimesaleratios.

Let'sonsideradutofwidth

L

,withaowveloity

U

. Thetimeauidpartile,withtransverse

veloity

u

′

takesto rossthedut isalledtheinertial time,

T

i

∼

L/u

′

. Atthesameway,visous

foreshaveatimesale,

T

v

∼

L

2

/ν

.

Inaturbulentow,theinertialtime-salewillbemih lessthanthediusivetime-sale,

T

v

T

in

=

u

′

L

ν

>

1

.

(1.6) 1.1.4 Navier-Stokes equations

Turbulene behaviouris ompletely desribed by the governing equations of uid mehanis,

i.e. the ontinuity and Navier-Stokes(N-S) equations. In ase of inompressible ows, they are

expressedin thefollowingwayonnon-dimensionalform:

∂u

i

∂x

i

= 0;

∂u

i

∂t

+

∂u

i

u

j

∂x

j

=

−

∂x

∂p

i

+

1

Re

∂

2

u

i

∂x

j

∂x

j

(1.7)

where

Re

istheharateristiReynoldsnumberoftheow,

u

i

, i

= 1

,

2

,

3

aretheveloity om-ponents,

p

isthepressure. NotethatEinstein'ssummationonventionisusedhere,where

i

= 1

,

2

,

3

.

Together withthemainoweld,sometimes weneedalsoto understandphenomenasthat

Bydenition,thewordpassiverefersto theonditionthattheresultingdensitydierenes areso

small that theeet from the salaron theowis negligible. Soapassivesalaranbe heat or

temperatureinaoworaonentrationofasubstane.

ThereforeisalsopossibletouseNavier-Stokesequations1.7to desribethedevelopmentofa pas-sivesalar,

θ

:

∂θ

∂t

+

∂u

i

θ

∂x

j

=

1

ReP r

∂

2

θ

∂x

j

∂x

j

(1.8)

where

P r

is the Prandtl number, dened as the ratio of momentum diusivity to thermal

diusivity:

P r

def

=

µc

p

1.2 Numerial approahes to N-S solution

Thereareseveralnumerialapproahestosolvethesystem1.7. Themostintuitiveoneisalled DiretNumerial Simulation, DNS,wherebythe governingeuquationsare solvedwithoutmaking

anyassumption, resolvingallthesalesfrom thesmallesttothelargestone. Thereforeitprovides

alltheinformationofaturbulentow,withoutanyapproximations. Sinetheomputationalost

ofDNSsaleswiththeReynoldsnumberis

∼

Re

37

/

14

[ChoiandMoin,2012℄,thisisnotaordable forpratialengineeringanalysesathigh Reynoldsnumber.

Soamorepratialapproahhasbeendeveloped,basedontheReynolds'deomposition[Reynolds,1894℄:

u

i

=

u

i

+

u

′

i

,

θ

=

θ

+

θ

′

(1.9)

wheretheoverlinerepresentstheensembleaveragedquantity,and

u

′

i

and

θ

′

aretheveloityand

salaromponentsutuations,respetively.

UsingtheReynoldsdeompositionin equations1.7and1.8,andtakinganensembleaverageofall terms,theReynolds-averagedN-S (RANS)equationsarederived:

∂u

i

∂t

+

∂u

i

u

j

∂x

j

=

−

∂p

∂x

i

+

1

Re

∂

2

u

i

∂x

j

∂x

j

−

∂u

′

i

u

′

j

∂x

j

,

∂u

i

∂x

i

= 0

(1.10)

∂θ

∂t

+

∂u

i

θ

∂x

j

=

1

ReP r

∂

2

θ

∂x

j

∂x

j

−

∂u

′

j

θ

′

∂x

j

(1.11)

Note that here turbulene is solely desribed by the Reynolds stress tensor

u

′

i

u

′

j

and salar ux vetor

u

′

j

θ

. These terms have to beproperlymodelled in order to lose the problem. Eah model involvesapproximationswhih limitsits auray. Therefore, this approah providesonly

anapproximatesimulationofthemeanow.

The rst simple model was developed by Boussinesq in 1877. It is based on an eddy visosity

formulation

u

′

i

u

′

j

−

2

3

Kδ

ij

=

−

2

ν

T

S

ij

,

S

ij

=

1

2

∂u

i

∂x

j

+

∂u

j

∂x

i

!

(1.12) where

K

=

u

′

i

u

′

i

/

2

isthemeanturbulentkinetienergy,

S

ij

isthemeanstrain-ratetensor,the symmetripartofthemeanveloitygradienttensor,

ν

T

istheeddyvisosityweusetomodelasthe produtofaertainsalewith length

Λ

of theeddiesandveloity

V

. Usingdimensional analysis

ν

T

∼

Λ

V.

(1.13) Eddyvisosityisthenmodeledaordinglyto someharateristisoftheow.

Forinstane, algebrai models(orzero equationmodels) relatelength

Λ

andveloity

V

to the

meanveloityeldandtheowgeometryharateristislikeveloitygradient,distanetothewall,

thiknessoftheshearlayeret. Thesekindofmodelsworkquitewellforthespeiasethatthey

are designedfor,e.g. attahedboundarylayersanddierenttypesof thin shearlayers. However,

theydon'tgivesatisfatoryresultsforgeneralases.

Better results an be ahieved using one-equation models, they typially solve an additional

transport equation for the turbulent kineti energy,

K

, or the eddy visosity,

ν

T

. One-equation models give good results for attahed boundary layers and other thin shear layer ows, but for

omplexows. A goodexampleistheSpalart-Allmaras[SpalartandAllmaras,1992℄model (SA), that solvesfor theeddy visosity. This isverysuitablefor aeronautialappliationsand atually

isthestandardmodelforexternalaerodynamisCFDanalysesatBoeing.

Two-equations models solve two transport equations for two quantities that an be used for

determiningthelengthandveloitysaleneededtoomputetheeddyvisosity. Themostommon

are

K

−

ε

and

K

−

ω

models,wheretransportequationsfortheturbulentkinetienergy,

K

andfor

thedissipationrate,

ε

orthe turbulenefrequeny

ω

aresolved. Nowadaystheimplementationof

these modelsin ommerialCFDodes(e.g., ANSYSFluent)presentsadditionalorretionsthat

might be dependent on non-loal quantities suh as thewall distane. Oneimportant and most

reent example is the Menter SST

K

−

ω

model [Menter,1994℄, whih is suitable for separated ows;itis thestandardturbulenemodelusedatAirbus.

Despiteeddyvisositytwo-equationsmodelsarestilldominatinginindustrialCFD,there'sabig

de-mandformoreauratepreditionofomplexowsituations,inludingonsetofseparation,highly

urvedows,rapidly rotatingowset. Inthesesituations, eddyvisosityBoussinesq'hypothesis

(1.12)doesnotdesribetherealphysiswell. Aneet ofdeorrelation aused byrotationours at high rotation rates, and generally the alteration of prodution to dissipation ratio is a diret

onsequeneofthat.

This phenomena, for instane, is animportant aspetwhih eddy visositymodels doesnot take

into aount,beausethemodelisinsensitivetosystemrotation.

A better alternative to these models are the Reynolds Stress Models. They solve transport

equationsforeahReynoldsstressomponentsderivedfrom themodelledN-S equations,in order

ompu-tationally moreexpensiveandompliated than theothers. However,those dierentialReynolds

stressmodelsanbesimpliedusingtheweakequilibriumassumptionbyRodi[Rodi,1992℄. Details aboutthiswillbeshownin 2. Thealgebrairelationisimpliitin Reynoldsstresses,butthereare some expliit solutions ([Pope,1975℄; [GatskiandSpeziale,1993℄; [WallinandJohansson,2000℄). These modelsarealledExpliitAlgebrai ReynoldsStress Models(EARSM).

Inpartiular,in theEA model[Wallin andJohansson,2000℄ theowanisotropy(

a

ij

)isdesribed asanexpliitexpressionintermsofthe(normalized)meanstrainandrotationtensorswith

addi-tional salarparameters. This leadsto aomparable omputational eorts, as ompared to eddy

visositytwo-equationmodels.

There'salsoaninterestinganalogyforthesalar

θ

modeling. Taylordevelopedananaloguewayto

formulate theeddydiusivitymodel(EDM)forthemeanturbulentsalarux

u

′

i

θ

′

[Taylor,1915℄

u

′

i

θ

′

=

−

D

T

∂θ

∂x

i

,

D

T

=

ν

T

P r

T

(1.14)

where

D

T

istheeddy diusivityoeientand

P r

T

theturbulentPrandtlnumber.

Inananalogouswayto

K

−

ε

model,Nagano

&

Kim[Kim,1988℄developedthe

K

θ

−

ε

θ

model. Thetimesale

τ

θ

=

K

θ

/ε

θ

,isusedtoompute

D

T

∼

Kτ

θ

.

Still, model 1.14 is not ompletely orret. Aording to Bathelor, eddy diusivity assumes an alignmentbetweenthe

u

′

i

θ

′

vetorandthemeansalargradient,soithasto beonsidereditselfa

tensor. Thefollowingexpressionwillholdthen:

u

′

i

θ

′

=

−

D

ij

∂θ

∂x

j

(1.15)

Wheretheeddydiusivitytensoranberewrittenas[DalyandHarlow,1970℄

D

ij

=

−

C

θ

τ

θ

u

′

i

u

′

j

∂θ

∂x

j

(1.16)

and

C

θ

isamodeloeient.

Inthesameway, salaranbemodelled withanExpliit Algebraimodel,in thisasealled the

Expliit AlgebraiSalarFluxModels(EASFM).

Atrade-obetweenaurayandomputationaleortisLarge-eddysimulations(LES)ofturbulent

ows. InLES there'saseparationofsales, inthesense that onlylarge-saleeddiesareresolved,

while the remaining small sales (whih are alled sub-grid sales, SGS) are modelled, one the

resolvedsales havebeenomputed. Theseparationof salesis generallydoneusing agrid. T

of the Navier-Stokesequations. This is thereforeaphysially-oherentapproah sineturbulene

isunstationary.

DespitemoreomputationallyexpensivethanRANS,LESgivesabetterdesriptionofturbulene,

and unlikeDNS,is abletoprovideagoodresolution oftheowin anaeptableamountof

om-putational time. Reently, that timehas been estimatedby Choi&Moin[ChoiandMoin,2012℄, to saleas

∼

Re

26

/

14

.

AfairandsimplieddistintionbetweenDNSandLESanbenotiedhavingalooktothevortial

strutures, for bothof the ases,in gure 1.4. Ata rstglane, we ansee that in LES vortial strutures are underestimatedand fewer,ompared to the DNS,whih is able,instead, to givea

ompleteanddetailed desriptionofthem.

Therst LESmodelwasdevelopedbySmagorinsky[Smagorinsky,1963℄formeteorologial appli-ationsusinganeddyvisosityassumptionintheSGSmodel. Thismodelhasbeenimprovedlater

on by Germano [Germanoetal.,1991℄ introduing the dynami proedure, whih givesa orret asymptotinear-wallbehaviouroftheeddyvisosity,andimprovedtransitionalowspreditions.

However, eddy-visosity models are not anisotropi, that is, they are not able to apture ow

anisotropywell,afeatureoftheow,whih isnotnegletablenearthewalls.

Forthat reasonseveralnon-linear models (whih are ofourse anisotropi),have been developed

reently,and someof themwill bedesribed in detailin hapter 3. In partiular,the aimof this thesis is to test the auray in ow predition of the Expliit Algebrai SGS model (EASSM),

a)

0

0.2

0.4

0.6

0.8

vel_u

-0.0112

0.864

b)

0

0.2

0.4

0.6

0.8

vel_u

-0.0112

0.864

Figure 1.4 Vortial strutures inturbulent hannel ow at

Re

τ

= 590

, visualized by isosurfaes of

λ2

, olored by the veloity magnitude, froma) DNS by P.Shlatter b) LES simulationwiththeExpliitAlgebraimodel.

1.3 A partiular ase: Turbulent Channel Flow

Mathematialmodelsforuiddynamishavebeenalreadydened. Inordertosolvethemodels,

amathematialproblemneedsto bedened inapropertimeandspaedomains.

Aordingtothespaedomain,inadierentgeometry,physialquantitiesandtheowwillbehave

inadierentway. Forthisreasonweneedtosetapartiularase,sothattheowanbeunivoally

lassied.

InthisthesisweareonsideringtheaseofChannelFlow,withthefollowingproperties:

ˆ turbulent,in thesense that theReynoldsnumberis suientlyhigh suhthat the regime

anbeassumedas turbulent;

ˆ fully developed, sothat veloitystatistis areonstant along

x

-axis. Inother words,the

ow is statistially stationary and statistially one-dimensional [Pope,2000℄, with veloity statistisonlyvariablealongthe

y

-axis. Inotherwords:

h

u

i

=

U

=

U

(

y

)

,

h

v

i

=

V

= 0

,

u

′

v

′

=

u

′

v

′

(

y

)

(1.17)

Channelowbelongsto the wall-boundedshear ows lass: owmotionis ontainedbetween

twosolidsurfaes. Therefore,no-slip onditions areimposedonthewalls,wheretheuidveloity

isassumedto bezero. Thefollowingpitureshowsthequalitativebehaviouroftheveloity.

Figure1.5ChannelFlowmeanveloityprole,with

u

′

utuationsontour plotinthe

Furthermore,astatistiallysymmetriowgeometryw.r.t. themid-plane

y

=

δ

isonrmedby

experiments;thereforethestatistisof

(

u, v, w

)

at

y

arethesameasthoseof

(

u,

−

v, w

)

at

2

δ

−

y

.

Reynolds number is always used to haraterize the ow, in this ase we will refer to two

partiularReynoldsnumbers,

Re

τ

def

=

u

τ

δ

ν

(1.18)

Re

b

def

=

uδ

ν

(1.19)

where1.18is basedonthefritionveloity

u

τ

,denedasfollows:

u

τ

def

=

p

τ

W

/ρ

(1.20)

τ

W

isthemeanwallshearstressand

δ

isthehannelhalf-width.

Thebulkveloity

u

in 1.19,isdened as

u

=

1

2

δ

2

δ

Z

0

h

u

i

dy

(1.21)

Foraturbulenthannelow,thefollowingresultholds:

Re

τ

≈

0

.

166

Re

0

b

.

88

(1.22)

Notethat thespeiedformula 1.22willbeused forthederivation of

Re

b

, whih will be on-sideredasaninputquantityintheomputations.

As previouslystated, hannel ow is a wall-bounded shear ow. In a boundary layer ora

wall-bounded shear ow theharateristilengthfor streamwisedevelopmentis muhlarger thanthe

ross-streamextentoftheregionwithsigniantveloityvariation.

So its behaviour an be observed studying a two-dimensional steady owsenario with the thin

shearlayerapproximation [JohanssonandWallin, 2012℄. Thisapproximationstatesthatthe har-aterististreamwisedevelopmentlength,

L

ismuhlargerthan

δ

,theshearlayerthikness. Aswe

will onrm later,in hannelowthere willbealayer,whose thikness issmall omparedto the

harateristilength,wherebyvisouseets fritiondependson.

Thankstothisapproximation,theNavier-Stokesequationsanbesimpliedtothethinshearlayer

U

∂U

∂x

+

V

∂U

∂y

=

−

1

ρ

dP

0

dx

+

∂

∂y

ν

∂U

∂y

−

u

′

v

′

!

|

{z

}

total shear stress

(1.23)

where

P

0

if thepressureat thewall. Notethat allthe veloities in 1.23aremean values,and theonlyomponentresponsibleforturbuleneisthelastoneontheright-hand-side,whihisalled

theReynoldsstress. Togetherwiththevisousstress,theReynoldsstressgeneratesthetotalshear

stress.

In partiular, usingthe hannel owassumptions given in 1.17, the thin-shearlayerequation be-omes

0 =

−

1

ρ

dP

dx

0

+

d

dy

ν

dU

dy

−

u

′

v

′

!

(1.24)

Integratedinthewall-normaldiretionitreads

0 =

−

1

ρ

dP

dx

0

y

+

ν

dU

dy

−

ν

dU

dy

y

=0

| {z }

u

2

τ

−

u

′

v

′

+ 0

(1.25)

Attheenterline

(

y

=

δ

)

thetotalshearstressiszero,thereforewehavethefollowingondition:

1

ρ

dP

0

dx

=

−

u

2

τ

h

(1.26)

meaningthatthepressuregradientisrelatedtothefritionveloityandthewidthofthehannel.

Plugging this relation in 1.25, we ansee that thetotal shear stress developslinearly arossthe hannel:

ν

dU

dy

−

u

′

v

′

=

u

2

τ

1

−

y

δ

!

(1.27)

whih inwallunitsbeomes

dU

+

dy

+

−

u

′

v

′

+

=

1

−

y

+

δ

+

!

(1.28)

wherethequantities

y

+

and

δ

+

aresaled bytheinner(visous)lengthsale

y

+

def

=

y

l

∗

=

yu

τ

ν

Considerations. Dependingontheregiononsidered,equation1.28assumesdierentforms. Intheouterlayer,wherevisous eetsarenegligible,theleft-hand-sideof1.28beomes

−

u

′

v

′

+

= 1

−

y

+

δ

+

(1.29)

Ontheotherhand,loseto thewall,

y/δ <<

1

,visous eetswill benotnegligibleanymore,

thereforethefollowingrelationholds

dU

+

dy

+

−

u

′

v

′

+

≈

1

(1.30)

thatisompatiblewiththelawofthewall,sinethere'snotinuene oftheReynoldsnumber.

It'saonstantstress region:

U

+

def

=

U

u

τ

= Φ

1

(

y

+

)

(1.31)

u

′

v

′

+

= Φ

2

(

y

+

)

(1.32)

Moreover,for largeReynoldsnumbers,weanalsoassume that thereis an overlap region for

walldistanes

y

,

ℓ

∗

<< y << δ

where

δ

isboundarylayerthiknessand

ℓ

∗

thevisous lengthsale. This isapartiularregion where outerandinnerlayerdesriptionshold simultaneously.

Derivatingthefollowing1.31w.r.t.

y

+

we'llhaveanexpressionwhihisindependentoflengthsale.

Thereforeholds

y

+

d

Φ

1

(

y

+

)

dy

+

≡

const

(1.33)

sothat,oneintegrated,itgivesalogarithmi law:

Φ(

y

+

) =

1

κ

lny

+

+

B

(1.34)

where

κ

= 0

.

38

istheKármánonstantandB=4.1,aordingtoobservations. Inthefollowing

gure6.22ameanveloitydiagramisshown,togetherwiththelawof the wall andtheloglaw, at

Re

τ

= 5200

.

PSfragreplaements

y

+

h

u

i

+

10

0

−

2

10

−

1

10

0

10

1

10

2

10

3

10

4

5 10 15 20 25 30 Figure1.6:

< u >

+

vs

y

+

,at

Re

τ

= 5200

,from[LeeandMoser,2014℄,.:lawofthe wall,:loglaw

Asweanseefromthepreviouspiture,therstveloityregionalledvisoussublayer,follows

thelawofthewall,anditextendsoutto approximately

y

+

= 5

. Despite theabsolutemagnitude

of the turbulent utuations are small in this region, the relative (wall-parallel) intensities are

large. Asweinrease

y

+

,wewillhaveabuer region, wherethemaximumturbuleneprodution

is at

y

+

= 12

and the maximum turbulene intensity at

y

+

= 15

. Log-layer starts between

50

< y

+

<

200

, andit extendsto

y/δ

≈

0

.

15

, where

δ

isthehannel half-width. Beyondthe log

layer,there'snallytheouterregion.

The maximumturbulene prodution and Reynolds stress.

Theturbuleneprodutioninthenear-wallregionofwall-bounded owsanbeformulatedas

P

+

=

−

u

′

v

′

u

2

τ

dU

+

dy

+

≃

1

−

dU

+

dy

+

!

dU

+

dy

+

(1.35)

Thelatterapproximationis validifvisouseetsare small. Fromthis weanderivethatthe

maximumprodutionisfoundwhere

dU

+

dy

+

=

1

2

(1.36) whih leadsto

P

max

+

=

1

4

(1.37)

ThereforetheturbulentprodutionismaximumwhereasbothvisousandReynoldsstressesare

exatlythesame,thatisinthenear-wallregion;itgenerallyhappenswhen

y

+

≈

12

alsopossibletoestimatewherethemaximumReynoldsshearstressours. Derivingequation1.28

w.r.t.

y

+

, and denotingthe Reynoldsshear stress as

τ

, normalized by inner units, the following

relationholds:

d

2

U

+

dy

+

2

+

dτ

+

dy

+

=

−

1

δ

+

(1.38)

ifwealsoassumethattheReynoldsstressismaximumin thelog-region,taking thederivative

1.34w.r.t

y

+

wehave

d

2

U

+

dy

+

2

=

−

1

κy

+

2

(1.39)

meaningthat themaximumoftheReynoldsstressisat

y

max

+

=

r

h

+

κ

=

r

Re

τ

κ

(1.40)

Then,intermsofoutersaleweansaythatthepositionwheretheReynoldsstressreahesits

maximumisproportionalto

Re

−

1

/

2

τ

:

y

max

h

=

κ

−

1

/

2

Re

−

1

/

2

τ

∝

Re

−

1

/

2

τ

(1.41)

ThisalsoanbeprovedusingDNSresultsatdierentfritionReynoldsnumbers. Anexample

ofReynoldsstressprolesis showningure1.7. PSfragreplaements

y/δ

−

u

′

v

′

u

2

τ

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure1.7

−

u

′

v

′

normalizedby

u

τ

vs

y/δ

,from[LeeandMoser,2014℄,

· · ·

:

Re

τ

= 550

,

Themaximum Reynoldsshear stress anbe derived aswell, using the relation1.38and 1.40, togetherwiththelog-law:

τ

max

+

= 1

−

y

+

max

h

+

−

1

κy

max

+

= 1

−

√

2

κh

+

= 1

−

2

√

κRe

τ

.

(1.42)

Wealreadyhavetalkedaboutturbuleneanditsprodutioninaqualitativesense,but,inorder

todesribeitproperlyandtounderstandthephenomenoninaompleteanddetailedsense,afous

on the three omponents of theveloity, theyare

u

′

,

v

′

and

w

′

, is neessary. As we'll see in the

nexthapters,theomputationsoftheseomponentsplayanimportantrolein thevalidation ofa

LESmodel.

Inpartiular,it'spossibletoderivetheReynoldsstresstensormakingthesquareofthe

root-mean-squarevalueofthesethree omponents:

u

′

i

u

′

j

=











u

′

u

′

u

′

v

′

u

′

w

′

u

′

v

′

v

′

v

′

v

′

w

′

u

′

w

′

v

′

w

′

w

′

w

′











.

(1.43)

InthefollowingpiturealltheomponentsoftheReynoldsstressareshown. Importantto

un-derlineistherelationbetweenthemaximumturbuleneprodution,andthepeaksintheReynolds

stressproles. PSfragreplaements

y

+

0 5 10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 7 8 9

Figure1.8Varianesofthe veloityutuations, normalizedinwall units,fromDNS of

[LeeandMoser,2014℄at

Re

τ

= 1000

,:

u

′

u

′

+

,:

v

′

v

′

+

,:

w

′

w

′

+

'

Anisotropy of the ow. A fundamental aspet, whih is stronglyrelevantfor theomplete

understandingoftheaforementionedwork,is theoneptofanisotropyin turbulene.

Fromaphysialpointof view,anisotropyis aproperty of theowwhih is notalignedwith the

veloitydiretion andtheveloitygradient.

Forafully-developed hannelowthequantities

u

′

w

′

and

v

′

w

′

= 0

arezero. TheReynoldsstress

tensorbeomes:

τ

ij

=

−

ρ











τ

11

u

′

v

′

0

u

′

v

′

τ

22

0

0

0

τ

33











.

(1.44)

in ase ofisotropi turbulene, the diagonal termsare equal, i.e.

τ

11

=

τ

22

=

τ

33

andall the deviatoritermsarezero. Therefore,foraspeiLESmodel,theabilitytoapturetheanisotropy

oftheowonsistsin reproduingdierentdiagonalterms,i.e.

u

′

u

′

6

=

v

′

v

′

6

=

w

′

w

′

2

LARGE EDDY SIMULATION

Anxietyisthe handmaiden of reativity. (T.S.Eliot)

Reynolds-averaged Navier Stokes (RANS) equations-based simulations are able to solve only

themeanveloityeld oftheow. RANSsimulationsrelyheavilyonmodellingsineallturbulent

motionsaremodelledandthereforetheyarenotalwaysaurate.

InDiretNumerialSimulations(DNS), theunsteadyNavier-Stokesequationsaresolvedwithout

using models. Therefore, DNS is very aurate. However, the omputational powerdemand for

DNSistoolargeforindustrialappliations. Therefore,anewmethodwhihombinesareasonable

owpreditionauray withalimitedamount ofomputationalost hasbeenstronglyrequired

in reentyears.

Large-eddysimulation(LES)representsanalternativethattsthoserequirements. Thispartiular

methodemploysaseparationofsales[Rasam,2014℄: alteringoperationdeomposestheveloity eld (generallytogether withasalarone) intoaresolvedpart,representedbytheomputational

grid,andanunresolvedpart,whihismodelledthroughphysiallyrealistimodels. Inotherwords,

1. thesolutionofthelargesalesofturbuleneonarelativelyoarsegrid;

2. themodelling ofthe smallerunresolvedsales, the so-alledsubgrid-sales (SGS), basedon

theresolvedveloityeld.

2.1 The ltering operation

Step1. Let's onsidernowageneraltimeand spae-dependentfuntion

φ

(

x, t

)

. Theltering

operationonsistsofaonvolutionofakernel

G

∆

onthatfuntion(Leonard,1975),overthedomain

D

ofthegrid:

e

φ

(

x, t

) =

Z

D

φ

(

x, t

)

G

∆

(

x

−

ξ

)

dξ

(2.1)

There are several optionsfor the lter: the ommonly used ones are spetral uto, box and

Gaussianlters. Inspetralodes,spetralltersarethemostsuitableonesforltering,sinethey

at onaspetralspae(where theodeworkson)and aremorepreise. Figure below showsthe

dierenebetweenthedierentkindsoflteringmethods.

Figure2.1.: Sharp-spetrallter,-: Gaussianlter, : Boxlter,

r

=

ξ

,from [Pope,2000℄

eld

e

u

(

x, t

)

is notadeterministivariable, implyingthat

ee

u

(

x, t

)

6

=

u

e

(

x, t

)

,

u

e

′

(

x, t

)

6

= 0

.

2.2 Governing equations of LES and the losure problem

Thelteringoperationleadsto thelteredNavier-Stokesequations:

∂

e

u

i

∂x

i

= 0;

∂

e

u

i

∂t

+

∂

e

u

i

u

e

j

∂x

j

=

−

∂x

∂

p

e

i

+

1

Re

∂

2

e

u

i

∂x

j

∂x

j

−

∂τ

SGS

ij

∂x

j

(2.2)

here

τ

ij

isthe SGS stress tensor. Leonard[Leonard,1974℄ proposed a possibledeomposition ofnon-linearterms(i.e. theSGSstresstensor),inthefollowingway:

τ

SGS

ij

=

u

g

i

u

j

−

e

u

i

u

e

j

=

L

ij

+

C

ij

+

R

ij

(2.3) where

L

ij

=

u

e

g

i

u

e

j

−

u

e

i

e

u

j

,

(2.4)

C

ij

=

e

u

g

i

u

j

′

−

u

e

g

j

u

′

i

(2.5)

R

ij

=

u

g

′

i

u

′

j

,

(2.6) and

u

′

i

=

u

i

−

u

e

i

.

In the Leonard deomposition

R

ij

is the Reynoldssubgrid tensor and represents the interation betweensubgrid-sales;

C

ij

is theross-stresstensor andaountsforlargevssmall sale intera-tions;nally

L

ij

,theLeonardtensor,givestheinterationsbetweenthelargesales.

τ

SGS

ij

is theunknown additivepartin theltered Navier-Stokesequation, and thereforeneeds to bemodelled.

2.3 An exampleofa losure: theEddy Visosity Model (EVM)

Step2. Thereareseveralwaystomodelthesubgridsalestresstensor. Aspreviouslywritten,

the rstandsimplemodel wasdevelopedby Smagorinsky[Smagorinsky,1963℄, formeteorologial appliations. ItoriginatesfromtheRANSmodel,takingintoaountBoussinesq'shypothesis. The

EVMonsistsofalinearformulationofthedeviatoripartof

τ

SGS

ij

,

τ

ij

SGS

=

τ

kk

3

δ

ij

| {z }

isotropic

−

2

ν

SGS

| {z }

EV M

contribution

e

S

ij

(2.7) where

ν

SGS

= (

C

s

∆)

e

2

|

S

e

|

(2.8)

e

S

ij

and

|

S

e

|

are theresolved strain-ratetensorand its magnitude, respetively.

∆

e

is thelter sale,

ν

SGS

istheSGSeddyvisosity,and

C

s

isthemodeloeient,theSmagorinskyoeient.

Notethat

ν

SGS

isaonstantratherthandependentofdiretion. Aordingtothereentpaper of Spalart [Spalart,2015℄, in a simple shear ow with two of the axes aligned with the veloity diretion and thegradientdiretion,suhthat thestraintensor

S

ij

havezeroandequaldiagonal terms, this model predits onstant and equal diagonal terms of the Reynolds stress tensor, i.e.

τ

11

=

τ

22

=

τ

33

.

ThereforeEVManbeonsideredanisotropi model,sine

ν

SGS

doesn'ttakeintoaountany eet ofanisotropy.

As we'll disuss further, this model an be improved using a dynami proedure, where the

C

s

oeientisomputedduringthesimulation.

2.4 Subgrid-sale dissipation

Animportantaspet ofLESistheimpatofSGS ontheresolvedsales. Inotherwords,LES

preditionsarestronglydependentontheontributionof

τ

SGS

ij

tensorontheresolvedkinetienergy

K

=

u

e

i

u

e

j

/

2

. Considerthekinetienergyequation

∂K

∂t

+

∂

∂x

j

(

e

u

j

K

)

|

{z

}

advection

=

−

ν

∂

e

u

i

∂x

j

∂

u

e

i

∂x

j

|

{z

}

viscous

dissipation

−

∂x

∂

i

u

e

i

p

e

+

ν

∂K

∂x

i

−

u

e

i

τ

ij

!

|

{z

}

dif f usion

+

τ

ij

SGS

S

e

ij

| {z }

−

SGS

dissipation

(2.9)

SineLESbydenition,arearriedoutforlargesales,andthegridsaleismuhlargerthan

Kolmogorovone,the visous dissipationterm is sosmall so that it is negligibleompared to the

others. Fromaphysialpointofview,diusiontermtransfersenergyinspae,butnotina

volume-averagedsense [Rasam,2014℄. Forthatreason,anadditionaltermisthereforerequiredinorderto reproduetheorretenergy transferfrom thelargeto thesmallersales. SGS dissipationovers

Π =

−

τ

SGS

ij

S

e

ij

(2.10)

ThemeanSGS dissipationbehavesasasink term, whiletheinstantaneousonegivesnegative

(baksatter)andpositive(forwardsatter)ontributionsinthetransfer.

The following piture showsthe SGS dissipation (red line), together with the dissipation of the

resolvedsales(blue line);theirtotalontributionisshownbytheredline.

PSfragreplaements

y

+

0 -100 -200 -300 -400 -500 -600 -0.25 -0.2 -0.15 -0.1 -0.05 0 0

Figure2.2Dissipationoftheresolvedsales,

ε

,togetherwiththeSGSdissipation,

Π

,and theirsumatinreasingresolutions,fromLESat

Re

τ

= 590

showninhapter6.

ThepreviousonsiderationleadstothefatthataninvestigationonSGSdissipationbehaviour

isveryimportantinanaurayassessmentofLES.

Basedonthispriniple,severalinvestigationshavebeenarriedoutbyChow&Moin[ChowandMoin,2003℄, Ghosal[Ghosal, 1996℄andKravhenko&Moin[Kravhenkoetal.,1996℄.

Geurts &Fröhlih [Geurtsand Fröhlih, 2002℄introdued theSGS ativityparameter, dened as follows

s

=

<ε

SGS

>

<ε

SGS

>

+

<ε

µ

>

(2.11) where

<ε

µ

>

= 2

µ

S

e

ij

S

e

ij

isthevisousdissipation.

With inreasingresolution, SGS dissipation dereaseswhile visous oneinreases, therefore

s

be-omes smaller. A remarkable aspet isthat, theoarser is theresolution , the biggeris

s

. This

an be proved having alook at 2.11: if we use aoarse grid, the resolution will be smallersuh that visous dissipationwill benegligible, and

s

will reah itsmaximumunity value. This ours

small.

Starting from the previous denition, Geurts & Fröhlih [GeurtsandFröhlih,2002℄ dened an errornormas

δ

E

=

E

LES

−

E

e

DN S

e

E

DN S

(2.12)

where

E

LES

isthemeanresolvedkinetienergyintegratedovertheowdomain,while

E

e

DN S

is the samequantity, omputed from ltered DNS datainstead, with the samelter width asin

theLES.

The auray assessment onsists of omputing the relation between

δ

E

and

s

: in the piture below[Rasametal.,2011℄ it is shown that therelativeerror

δ

E

drops almost exponentiallywith dereasing

s

(i.e. high resolution); results referto a LES with the expliit algebraiSGS stress

model (EASSM) [Marstorpetal.,2009℄ at

Re

τ

= 934

, for six dierent resolutions. This result showsthat with inreasing resolution theSGS ontribution beomessmaller and theauray of

the LES higher. Inother words,the resolution of the LES mustbe suiently high to obtain an

aeptable solution.

Figure2.3Normalizederror

δ

E

oftheresolvedkinetienergy,integratedoverthehannel width[Rasametal.,2011℄w.r.t. the ltered DNSvalue vs

s

. Arrow points toinreasing resolutionases.

3

SUBGRID-SCALE STRESS MODELS

FOR LES

Logi willgetyoufrom AtoB.

Imagination willtake youeverywhere. (A.Einstein)

InthishaptertwodierentLESmodelsaregoingtobeanalyzed;therstoneisanimprovement

ofthelassiSmagorinskymodel, theDynami EVM.

Theseondoneisapartiularnon-linearmodelusinganexpliitalgebraiformulationfortheSGS

stresstensor.

Inthefollowingsetionadeomposition oftheSGS stresstensorisgiven.

3.1 Tensorialpolynomial formulation of the SGS stresstensor

A usefulapproah that leadsto several non-linear models is basedon adeomposition of the

SimilartoRANSapproahforReynoldsstresslosures,SGSstresstensoranbeexpressedinterms

ofstrainandrotationratetensors.

Lund &Novikov[LundandNovikov,1993℄ expressedthedeviatoripartof theSGS stresstensor

τ

ij

SGS,d

asageneraltensorialfuntion of theltered strain-rate

S

e

ij

androtation-rate

Ω

e

ij

tensors, theKronekerdelta

δ

ij

andtheltersize

∆

e

as

τ

ij

SGS,d

=

τ

ij

SGS

−

1

3

τ

kk

δ

ij

=

f

(

S

e

ij

,

Ω

e

ij

, δ

ij

,

∆)

e

(3.1) where

e

S

ij

=

1

2

∂

e

u

i

∂x

j

+

∂

e

u

j

∂x

i

!

,

Ω

e

ij

=

1

2

∂

e

u

i

∂x

j

−

∂

e

u

j

∂x

i

!

.

(3.2)

AsPope[Pope,1975℄showedtheReynoldsstressesinRANS,

τ

ij

anbeformulatedasatensor polynomialwithtenelementsofdierentspowersof

S

e

ij

and

Ω

e

ij

andtheirombination.

Coeientsarefuntions of

S

e

ij

and

Ω

e

ij

invariantsorboth,andtheyarederivedusingthe Cayley-Hamiltontheorem:

τ

d

=

10

X

k

=1

β

k

T

(

k

)

,

(3.3)

sothatthetenpolynomialtensorsare:

T

(1)

=

S

e

T

(2)

=

S

e

2

−

1

3

II

S

I

T

(3)

=

Ω

e

2

−

1

3

II

Ω

I

,

T

(4)

=

S

e

Ω

e

−

Ω

e

S

e

T

(5)

=

S

e

2

Ω

e

−

Ω

e

S

e

2

T

(6)

=

S

e

Ω

e

2

+

Ω

e

2

S

e

−

2

3

IV

I

T

(7)

=

S

e

2

Ω

e

2

+

Ω

e

2

S

e

2

−

2

3

V

I

T

(8)

=

S

e

Ω

e

S

e

2

−

e

S

2

Ω

e

S

e

T

(9)

=

Ω

e

e

S

Ω

e

2

−

Ω

e

2

e

S

Ω

e

T

(10)

=

Ω

e

e

S

2

Ω

e

2

−

Ω

e

2

S

e

2

Ω

e

and

β

k

aresalaroeientsthat arefuntionsofvetensorialinvariants

II

S

=

tr

(

S

e

2

)

,

II

Ω

=

tr

(

Ω

e

2

)

,

V

=

tr

(

e

S

2

Ω

e

2

)

.

Thus, equation3.3isthemostgenerlaformulationfor

τ

d

in termsof

S

˜

ij

and

Ω

˜

ij

.

3.2 The dynami proedure for EVM (DEVM)

Regardingto themodeloeient

C

s

forthe eddy-visosity model, it hasbeendemonstrated that adynami omputation of

C

s

(that is, during the simulation),an signiantly improvethe owpreditionauray.

The omputation of

C

s

is done by taking into aount the resolved sales, aordingto a sale invarianeassumption.

Theruialpointofthisdynamiproedure istheso-alledGermanoidentity. Let'snowdenotea

testlter,

∆ = 2

b

∆

e

,

∆ =

e

3

√

Ω

,where

Ω

isthevolumeofaomputationalell. TheGermanoidentity isdened asfollows:

L

ij

=

T

ij

−

b

τ

ij

,

(3.4)

where

T

ij

isthe SGSstress tensorlteredat thetest lterlevel,and

L

ij

=

e

u

d

i

e

u

j

−

be

u

i

be

u

j

Then

L

ij

isappliedin thisway:

L

ij

−

1

3

L

kk

δ

ij

=

−

2

C

s

M

ij

(3.5)

where

M

ij

=

∆

b

2

|

S

be

|

S

be

ij

−

∆

e

2

|

\

S

e

|

S

e

ij

.

(3.6) Thesystemofequations3.5isover-determined. Inordertohaveauniquevalueof

C

s

Germano ontrateditin:

C

s

=

−

1

2

h

L

ij

S

e

ij

i

h

S

e

ij

M

ij

i

(3.7)

Moreover,tomake

C

s

variationssmoother,aspatial averaging

h

.

i

hasbeenapplied.

Togetherwithabetterperformaneof theEVM,theGermanoidentitygivesaorretasymptoti

near-wallbehaviourfor

ν

SGS

.

Ithasalsobeenshownthatispossibleto applyGermanoidentityforthedynamiomputationof

3.3 Expliit Algebrai SGS stress model

Anisotropi eets of turbulene are important in several onditions: examplesare near-wall

owbehaviourandboundarylayesseparationaseswithurvature,swirl,rotation.

Asitwasdisussedbefore,theDynamiSmagorinskymodelisanisotropimodel,inthesensethat

theSGSvisosity

ν

SGS

isdiretion-independent.

Therefore,in ordertoaptureowanisotropy,DEVM isnotsuitable.

InthesamespiritasReynoldsstress-basedmodelsareneessaryforRANS,non-linearSGSmodels

areneededforLES.

A reent example of those is the nonlinear dynami SGS stress model by Wang & Bergstrom

[WangandBergstrom,2005℄, whihonsists of three base tensorsand three dynami oeients.

OneofthetermsinthemodelissimilartotheDEVMmodel. Wang&Bergstrom[Wangand Bergstrom,2005℄ showedthatthedynaminon-linearmodelpreditsamorerealistitensorialalignmentoftheSGS

stressthaneddy-visositymodelsandanprovideforbaksatterwithoutlipping oraveragingof

thedynamimodelparameters.

The model here disussed is alled Expliit Algebrai SGS stress model (EASSM), was

devel-oped by Marstorp [Marstorpetal.,2009℄ and is similar to the EARSM by Wallin & Johansson [WallinandJohansson,2000℄, whih is based on a modelled transport equation of the Reynolds stressesandontheassumption thattheadvetionanddiusionof theReynoldsstressanisotropy

arenegligible.

AnalogoustotheReynoldsstressanisotropytensorwedenetheSGSstressanisotropytensoras

a

ij

=

τ

ij

K

SGS

−

2

3

δ

ij

(3.8)

Forsimpliity,we'llonsider

τ

SGS

ij

=

τ

ij

. Moreover,

K

SGS

= (

u

g

i

u

i

−

u

e

i

e

u

i

)

/

2

istheSGSkineti energy.

Inanon-rotatingframethetransportequationfor

a

ij

reads

K

SGS

Da

ij

Dt

|

{z

}

advection

−

∂D

τ

ij

ijk

∂x

k

−

τ

ij

K

SGS

∂D

K

SGS

k

∂x

k

!

|

{z

}

dif f usion

=

−

K

τ

ij

SGS

(

P −

ε

) +

P

ij

−

ε

ij

+ Π

ij

,

(3.9) where

−

D

K

SGS

k

=

−

D

τ

ij

ijk

/

2

is thesumoftheturbulent andmoleularuxes oftheSGS stress andSGS kinetienergy,respetively.

Although the prodution of the SGS stress

P

ij

and SGS kineti energy

P

=

P

ii

/

2

are given in termsof

τ

ij

andlteredgradients,theSGSpressurestrain

Π

ij

andSGSdissipationratetensor

ε

ij

needto bemodelled. Thosetermsare

P

ij

=

−

τ

ik

∂

u

e

j

∂x

k

−

τ

jk

∂

u

e

i

∂x

k

=

K

SGS

"

−

4

3

S

e

ij

−

(

a

ik

S

e

kj

+

S

e

ik

a

kj

) + (

a

ik

Ω

e

kj

−

Ω

e

ik

a

kj

)

#

,

(3.10)

Π

ij

=

2

ρ

(

S

g

ij

p

−

S

e

ij

p

e

)

,

(3.11)

ε

ij

= 2

ν

^

∂u

i

∂x

k

∂u

j

∂x

k

−

∂

u

e

i

∂x

k

∂

e

u

j

∂x

k

!

(3.12)

andtheirmodellingleadsto [Launderetal.,1975℄

Π

ij

=

−

εc

1

a

ij

+

K

SGS

"

3

5

S

e

ij

+

9

c

2

+ 6

11

a

ik

S

e

kj

+

S

e

ik

a

kj

−

2

3

a

km

S

e

mk

δ

ij

!

+

7

c

2

−

10

11

(

a

ik

Ω

e

kj

−

Ω

e

ik

a

kj

)

#

,

(3.13)

ε

ij

=

ε

2

3

δ

ij

,

(3.14)

where

c

1

isarelaxationoeientand

c

2

= 5

/

9

isaparameterofthemodelfortherapid part of

Π

ij

,whihdependsdiretlyonhangesin theresolvedveloitygradients,and

ε

=

ε

ii

/

2

. Like in Wallin & Johansson's [Wallinand Johansson,2000℄ RANS model, the derivation of the EASSM model involves the weak equilibrium assumption, whih implies that the advetion and

diusion terms of the Reynolds stress anisotropy are negleted. In order to simplify the model,

togetherwiththeweakequilibriumassumption,weassumealso that

P

=

ε

. Thisleadsto

0 =

P

ij

−

ε

ij

+ Π

ij

.

(3.15)

Usingthemodellinggivenby3.13and3.14in3.15wehave

c

1

a

ij

=

τ

∗

"

−

11

15

S

e

ij

+

4

9

(

a

ik

Ω

kj

−

Ω

ik

a

kj

)

#

(3.16) notethat

τ

∗

=

K

SGS

/ε

istheSGStime sale.

Finally,equation3.16hasbeensolvedusinganansatz

a

ij

=

β

1

τ

∗

S

e

ij

+

β

4

τ

∗

2

(

S

e

ik

Ω

e

kj

−

Ω

e

ik

S

e

kj

)

,

(3.17) where

β

1

and

β

4

aremodelparametersandfuntionsofthelteredstressandstrain-rate. Using that ansatz,equation3.16beomes

τ

ij

=

K

SGS

"

2

3

δ

ij

+

β

1

τ

∗

e

S

ij

+

β

4

τ

∗

2

(

S

e

ik

Ω

e

kj

−

Ω

e

ik

S

e

kj

)

#

,

(3.18)

whih isthemain EASSM modelformulation. Usingnormalizedstrainandstress-ratetensors

itanberewrittenas

τ

ij

=

2

3

δ

ij

K

SGS

+

β

1

K

SGS

S

e

∗

ij

|

{z

}

eddy

−

viscosity

+

β

4

K

SGS

(

S

e

∗

ik

Ω

e

∗

kj

−

Ω

e

∗

ik

S

e

∗

kj

)

|

{z

}

anisotropy of

SGS stresses

.

(3.19)

Theseondtermontheright-hand-sideisaneddyvisositytermresponsibleforSGSdissipation,

whereasthe third termreproduesanisotropieets of SGS stressesandgivesadisalignmentof

theSGSstressandresolvedstrain-ratetensors.

β

1

and

β

4

oeientshavetheform:

β

1

=

−

33

20

9

c

1

/

4

[(9

c

1

/

4)

2

+

|

Ω

e

∗

|

2

]

,

β

4

=

−

33

20

1

[(9

c

1

/

4)

2

+

|

Ω

e

∗

|

2

]

(3.20) where

|

Ω

e

∗

|

=

p

2

II

∗

Ω

=

q

2

τ

∗

2

e

Ω

ik

Ω

e

ki

≤

0

istheSGStimesale-normalizedseond invariant. Theunknownquantities

K

SGS

and

τ

∗

anbedynamiallyornon-dynamiallyomputed.

Theequation 3.19analso berelatedto thetensorialformulationoftheSGS stresstensor, given in theprevioussetion: therst termontheright-handside reproduestheisotropipartofthe

SGS stress,whiletheseond andthirdtermsanbeonsidered astwopolynomialtensors,forthe

ase

k

= 1

and

k

= 4

.

ThedynamiversionoftheEASSM involvesGermano's dynamiproedure;heretheSGSkineti

energyismodelled intermsofthesquaredSmagorinskyveloitysale

∆

|

S

e

ij

|

[Yoshizawa,1986℄:

K

SGS

=

c

∆

2

|

S

e

ij

|

,

(3.21)

where

∆

e

istheltersale;

|

S

e

ij

|

= (2

S

e

ij

S

e

ij

)

1

/

2

<