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

UNIVERSITY OF BOLOGNA

SCHOOLOF ENGINEERING ANDARCHITECTURE

Master of S ien e Degree inAerospa e Engineering

Testing of subgrid s ale (SGS)

models for large-eddy simulation

(LES) of turbulent hannel ow

Master thesis in Applied Aerodynami s

SUPERVISORS:

Prof. Alessandro Talamelli

Prof. Arne V. Johansson

ASSISTANT SUPERVISOR:

Dr. Geert Brethouwer

STUDENT:

Matteo Monte hia

A ademi Year 2013-2014

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Sub-grid s ale(SGS) modelsare requiredin order to model theinuen e of theunresolved small

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

turbulen e.

In the following work two SGS models are presented and deeply analyzed in terms of a ura y

throughseveralLESswithdierentspatialresolutions,i.e. gridspa ings.

The rstpart ofthis thesisfo uses onthe basi theory ofturbulen e, the governingequations of

uiddynami sandtheiradaptationtoLES.Furthermore,twoimportantSGSmodelsarepresented:

oneistheDynami eddy-vis ositymodel(DEVM),developedby[Germanoetal.,1991℄,whilethe otheristheExpli itAlgebrai SGS model(EASSM),by[Marstorpetal.,2009℄.

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

Navier-Stokes ode[Chevalieret al.,2007℄arepresented.

Theperforman eof thetwoaforementionedmodelswillbeinvestigatedin thefollowing hapters,

bymeansofLESofa hannel ow,withfri tionReynoldsnumbers

Re

τ

= 590

upto

Re

τ

= 5200

,

with relatively oarse resolutions. Data from ea h simulation will be ompared to baseline DNS

data.

Resultshaveshownthat,in ontrasttotheDEVM, theEASSM haspromising potentialsforow

predi tionsathighfri tionReynoldsnumbers:thehigherthefri tionReynoldsnumberisthebetter

theEASSM willbehaveandtheworsetheperforman esoftheDEVMwillbe.

The betterperforman eof theEASSM is ontributedto theabilityto aptureowanisotropyat

thesmalls alesthrougha orre tformulationfortheSGSstresses.

Moreover,a onsiderableredu tionintherequired omputationalresour es anbea hievedusing

the EASSM ompared to DEVM. Therefore, the EASSM ombines a ura y and omputational

VersioneItaliana

Nelle Simulazioni agrandi vorti i (Large-eddy simulations, LES), dei modelli per la s ala di

sottogriglia(Sub-grid s ale, SGS) sono ne essariper riprodurrel'inuenza delle piúpi ole s ale

dellaturbolenza,quindinonrisolte,suquelle heinve evengonorisoltedirettamente.

Nel seguente lavorosono presentati due modelli SGS, la ui a uratezzaverrá poi analizzata

at-traversovarieLESadiverserisoluzionispaziali,equindidiversiintervallididierenziazione.

Laprima partedellatesisi on entrasullateoriadellaturbolenza,partendodalleequazioni

osti-tutivedellauidodinami a,noallaloroversioneperLES.

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

model(DEVM),di[Germanoetal.,1991℄,ilse ondoéilmodelloSGSEspli itoAlgebri o,Expli it Algebrai SGS model (EASSM),di[Marstorpetal.,2009℄.

Sarannoinoltrefornitidettagliaggiuntivisull'implementazionedelmodelloEASSMsuun odi edi

Fluidodinami aComputazionale(ComputationalFluidDynami s,CFD),Pseudo-Spettrale

Navier-Stokessviluppato nel Linné Flow Centre del Dipartimento di Ingegneria Me ani a del KTH di

Sto olma,da[Chevalieret al.,2007℄.

I seguenti apitoli verterannosull'analisi dellastimadi unussoin un anale, hannel ow, fatta

daimodellides rittiinpre edenza,perunbassonumerodiReynoldsbasatosull'attrito,

Re

τ

= 590

,

no a

Re

τ

= 5200

. I datiottenutida ias unasimulazioneverranno onfrontati ondatidi

Simu-lazioni Numeri heDirette (Dire tNumeri al Simulations, DNS).

Dai risultati ottenuti si puó on ludere 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,piúilmodelloEASSMdarárisultatia urati,mentreleperforman esdelDEVM

peggior-eranno.

Lemiglioriperforman edelmodelloEspli itoAlgebri opossonosenz'altroessereattribuiteallasua

abilitádi al olareinmaniera orrettal'anisotropiaallepi oles aletramiteunaformulazione

or-rettadeglistressdisottogriglia,SGS.

In on lusione,datalaridottaquantitádirisorse omputazionaliri hiestapereettuaresimulazioni

rispetto alDEVM, tale modello ombina a uratezza ee ienza omputazionale,tanto he puo'

1 Turbulen e 5

1.1 Introdu tiontoTurbulen e . . . 5

1.2 Numeri alapproa hestoN-S solution . . . 12

1.3 Aparti ular ase: TurbulentChannelFlow . . . 17

2 Large Eddy Simulation 25 2.1 Thelteringoperation . . . 26

2.2 GoverningequationsofLESandthe losure problem . . . 27

2.3 An exampleofa losure: theEddyVis osityModel(EVM) . . . 27

2.4 Subgrid-s aledissipation . . . 28

3 Subgrid-s alestress modelsfor LES 31 3.1 TensorialpolynomialformulationoftheSGSstresstensor . . . 31

3.2 Thedynami pro edureforEVM(DEVM) . . . 33

3.3 Expli itAlgebrai SGS stressmodel . . . 34

4 Implementationin a CFD ode 39 4.1 Theneedfora ura y: spe tralmethods. . . 39

5.1 MeanValues . . . 49

5.2 Root-meansquaredValues,rms . . . 50

5.3 Vorti alstru tures . . . 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 unpenna hio di fumo fuorius ente dauna pipa di un uomopensante,

seduto su diuna sediaadondolo. An he in questopi oloaspettodellavita quotidiana la

turbu-lenza gio aunruolofondamentale.

Il noto si oRi hardFeynman denii laturbulenza ome il piúimportante problema dellasi a

lassi a hean oranonéstatorisolto.

Nel orsodegliannidiversis ienziatihannoprovatodi omprendereil vero omportamentoditale

fenomeno, la ui omplessitá gia e nelle equazioni he lodes rivono. Per unuido Newtoniano,

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

derivateparziali,la uisoluzioneanaliti aan oranonéstataottenuta.

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

metodo onsistenell'eettuaredegliesperimenti: tramitegalleriedelventosiamoingradodi

ripro-durreussiinsvariate ondizioni,dallesituazioni lassi hedistratolimite(motosudiunaparete)

eussoinun anale,noa asipiú ompli ati, omeilussoattornoadun orpotozzo( omeuna

automobile)eun orpoaerodinami o, omeunaeroplano.

Gliesperimentihanno omeneprin ipalel'analisidellequalitádelusso( omepressionee

velo -itá) attraversodelle sondeedelle te ni he di visualizzazione, omela Parti le ImageVelo imetry

(PIV).

[Talamellietal.,2009℄nell'Universitádi Bologna.

D'altro anto,unappro iodiversoéquellodisfruttarel'analisinumeri aepassarealla

implemen-tazionedelsistemaN-S su odi i diFluidodinami aComputazionale.

Tuttavia, la soluzione numeri a e ompleta fornita dalle Simulazioni Numeri he Dirette (DNS),

in grado di des riverela turbolenza aqualsiasi s ala, non ésemprepossibilea ausa dellegrandi

risorse omputazionaliri hieste,in parti olare adelevati numeridi Reynolds. Perquesto motivo,

l'obiettivoprin ipaledelprogettoCICLoPEéquellodi omprendereapienoilfenomenodella

tur-bolenzaqualoranonsiapossibileotteneredatitramiteDNS.

Comunque,late ni a DNSnon élasola in gradodi riprodurre numeri amente unpre iso usso.

Una se ondapossibilitási hiamala simulazione agrandi vorti i (Large-eddysimulation -LES);

questate ni aéingradodiraggiungere, onopportunimodelliperlepi oles aledellaturbolenza,

unasoddisfa entea uratezza omputazionale onunaragionevolequantitádirisorse

omputazion-ali.

A tale avviso, la tesi verterá sull'impiego e il test di due modelli innovativi per le pi ole s ale

dellaturbolenza;inparti olareverrámessaa onfrontol'a uratezzadi ias unmodello onl'altro,

rispettoan he i risultatiforniti da DNS. Lesimulazioni sonostate ompiuteatre diversi numeri

di Reynolds;in parti olarel'ultimo aso,dimaggioreimportanza, érelativoalmassimonumerodi

ReynoldsraggiuntonoradaDNS.

Laprimapartedellatesi omprenderálades rizionedellaturbolenza,siadaunpuntodivistasi o

hematemati o. Saranno onsiderateleequazionidiN-S perunuidoNewtoniano, onusso

in- omprimibileeturbolento, inun anale.

Nelse ondo apitololate ni aLESsaráespressaneidettagli,mentrenelterzoverrannopresentati

duemodellidis alasotto-griglia(sub-grids ale-SGS).Ilquarto apitolospiegherádeidettagli

te -ni isull'implementazionedellesimulazioni,deiduemodelli,edellaparallelizazioneMPIutilizzata.

Nel quinto apitolo enni di metodi statisti i per l'analisi della turbolenza saranno arontati. I

risultativerrannopresentatinelsesto apitolo,einnesarannofornitedelle on lusioninelsettimo

Imagineasmokingplume omingoutfrom apipeofathinking man,thisisatypi aleveryday

life s enariowhereturbulen eplaysamain role. Thephysi ist Ri hardFeynmandened itasthe

most important unsolvedproblemof lassi al physi s.

S ientistsalongtheyearshavetriedto understand therealbehaviourof thisphenomenon, whi h

gains its omplexitybe auseoftheequationsthat des ribesit. ForaNewtonian uid,turbulen e

istra edbyNavier-Stokesequations,asystemofnon-lineardierentialequationswhoseanalyti al

solutionhasnotbeenprovidedyet.

Anyway,alternativeapproa heshasbeendevelopeda rosstheyearsuptonow: themostintuitive

andoldwaytounderstanduidmotionistomakeexperiments: throughtheemploymentofwind

tunnelswe anreprodu etheowinseveral onditions,fromthebasi al hannelowandboundary

layerto themore omplexones,liketheowa rossablu body (likea ar)oranairplane. This

te hniqueinvolves theanalysis of the owproperties(likepressure and velo ity) throughprobes

andadvan ed visualizationte hniquessu hasParti leImageVelo imetry(PIV).

A re entexamplewhoseaimistodes ribehighReynoldsnumberturbulen ea rossapipeis

nev-erthelesstheCICLoPEexperiment,abigproje tdevelopedby

[Talamellietal.,2009℄inUniversityofBologna.

These ondapproa histhustousenumeri alanalysisinordertoimplement odesableto

mainCICLoPEproje t'saimistounderstandturbulen ephenomenafurthertheReynoldsnumber

limitdi tatedbyDire tNumeri alSimulations.

However, DNS is not the only te hnique able to reprodu e numeri allya given ow. A possible

se ond hoi eis alledLarge-eddysimulation;thiste hniqueisabletorea hasatisfying

omputa-tional a ura ywithareasonableamountof omputationalresour es.

InthisthesisNavier-StokesequationsforaNewtonianuidandanin ompressibleowina

turbu-lent hannelows enariowillbedes ribedintherst hapter,togetherwithaphysi aldes ription

ofturbulen e. Inthese ond haptertheLarge-eddysimulation(LES)te hniquewill beexplained

in detail,while in the third onetwoLES sub-grid s alemodelsare shown. In hapter foursome

te hni aldetails oftheimplementationof theLES,ofthemodels,with aparti ularfo usonMPI

parallelizationare pointedout. Sometheory aboutturbulen e statisti swill befa edin thefth

1

TURBULENCE

Everybodyisagenius. Butif youjudgeash byitsability to limb atree,

itwilllive itswholelifebelieving thatitisstupid. (A.Einstein)

1.1 Introdu tion to Turbulen e

Thephenomenonofturbulen eisfoundinseveralappli ations, forexample, ombustiontumbling

in Internal CombustionEngines(ICEs),thewakeofaFormula1 ar,thejetspreadbythenozzle

ofasupersoni air raftengine.

Inautomotiveengineering,forexample,thestudyofaerodynami sarounda arinvolvesthe

har-a terizationofaturbulentwall-boundedow, alledaBoundary Layer. Itwasintensivelystudied

byL.Prandtlin1904;hereturbulen eisthemain responsibleforfri tionandwakedrag.

In asolid ro ketmotor nozzle,there's ageneration ofaplume, whereturbulentmotionsofmany

s ales anbeobserved;fromeddiesandbulges omparableinsizetothewidthoftheplumetothe

and haoti nature,sin ethemotionofeveryeddy isunpredi table.

Figure1.1Large-eddysimulationofjetfromare tangularnozzle. There tangularnozzleis

showningraywithanisosurfa eoftemperature(gold) utalongthe enterplaneofthenozzle

showingtemperature ontours (red/yellow). Thea ousti eldisvisualizedby(blue/ yan)

ontoursofthepressureeldtakenalongthesameplane,fromP.Moin.

Whilelaminarowisa smoothand steadyowmotion,where any indu edperturbationsare

dampedoutduetotherelativelystrongvis ousfor es,inturbulentowsotherfor esmaybea ting

that ountera tthea tionofvis osity. Ifsu h for esarelargeenough,theequilibrium oftheow

is upset and theuid annot adapt suddenlyto vis osity. Thefor es that upset this equilibrium

anin lude buoyan y,inertia,oreven rotation. Ina hannel, vis ousandinertialfor esa tingon

theuidareproportionalto

F

v

∝ νL

(1.1)

F

i

∝ V L

2

(1.2)

where

ν

istheuidvis osity,and

L

and

V

arethe hara teristi velo ityand lengths ales. If

thevis ous for esontheuidarelarge omparedwithothers, anydisturban esintrodu edin the

owwilltend to bedamped out. Ontheother hand,iftheinertialfor es be ome large,theuid

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

smallereddies. This will ontinueuntilwerea hasmallenoughlengths ale(eddy size)onwhi h

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

hanneldiameter);thesmallereddieswillbe onstrainedbythevis ousfor eswhi ha tstrongest

at the smallest length s ales. Therefore, one of the di ulties asso iated with the predi tion of

turbulentowis thattherangeoflengths ales anbeverylarge.

Thedes riptionofturbulen einvolvesdierent on eptsliketurbulen eenergyprodu tion,transfer

anddissipation. Ri hardson'sfamouspoemgivesagoodideaaboutturbulen e:

Big whorlshave littlewhorls

That feed ontheir velo ity

Andlittle whorls havelesserwhorls

Andso ontovis osity (in the mole ularsense)

Thisisthedes riptionoftheenergy as ade on ept. Itstatesthatturbulentows anbe

on-sideredasanagglomerateofeddies ofdierentsizes. Largeenergy ontainingeddiesareunstable

and break down andtransferenergy to smallereddies. The pro essgoesontill thesmallestone,

theKolmogorovs ale,where energyis dissipatedinto heatbyvis ousee ts.

This isthegreat on lusion thatKolmogorovmadein 1941,andhis rsthypothesis is ompletely

basedonthat: atsu ientlyhighReynoldsnumber,thestatisti softhesmalls alesareuniversal

andaredeterminedsolely byvis osity,

ν

,andtheenergydissipationrate,

ε

.

Usingdimensionalanalysis,itispossibleto derivetheKolmogorovlengths ale

η

,times ale

t

η

andvelo itys ale

v

η

:

η =

ν

3

ε

!

1/4

,

t

η

=

ν

ε

!

1/2

,

v

η

= (νε)

1/4

(1.3)

Then, in a ordan eto Ri hardson'spoem, Kolmogorovmade ase ond hypothesis, basedon

thefa t thatatsu ientlyhighReynoldsnumberthestatisti softhes aleswhi haresu iently

larger than

η

and mu h smallerthan the largestenergeti s ales are solely des ribed by

ε

. This

hypothesisreferstotheinertialrangeofs ales. Thekineti energyspe trumoftheses ales anbe

des ribedby

Figure1.2Turbulen eenergyvswavenumberspe trum,fromJ.M.M Donough.

where

k

isthewavenumberand

C

k

≈ 1.5

[Sreenivasan,1995℄istheKolmogorov onstant. Fromthelastdiagram,it's learthatturbulen ehasdierentbehavioursa ordingtothe

wavenum-ber. Basingonthese on epts,we andistinguishfourdierentregions:

1. thelarges ale,determinedbytheproblem domaingeometry;

2. theintegrals ale

(Λ)

,whi h isan

O(1)

fra tion(oftentakentobe

∼ 0.2

)ofthelarges ale;

3. the Taylor mi ros ale whi h is an intermediate s ale, found in the Kolmogorov's inertial

subrange

(η ≪

2π

κ

≪ Λ)

4. theKolmogorov(ordissipation)s ale

(η)

whi histhesmallestofturbulen es ales,theinner

s ale

1.1.1 3D Nature of Turbulen e

Turbulen e is rotational and a three-dimensional phenomenon. It is hara terized by large

u tuationsinvorti ity,whi hareresponsibleforvortexstret hingandlengths aleredu tion. These

hara teristi sareidenti ally zeroin twodimensionsand theseare thereasonswhyturbulen e is

hardtodes ribebothanalyti allyandnumeri ally.

Thesethree-dimensionaldynami alme hanismsarehighly omplexandnonlinear,howevertheow

anbeassumedas bi-dimensionalforlarges ale2Dstru tures. These stru turesplayadominant

inthesmallers ale,wheretheyarefundamentalformixing,mostofallatmole ulars ales(e.g. in

ombustionproblems).

1.1.2 Order & Randomness

Despiteturbulen e is haoti , it onsists of ompletely randommotionsthat anaggregatein

oherentstru tures. Typi alexamplesareturbulentboundarylayersand homogeneousturbulent

shearows,whi hexhibithorseshoe,orhairpinvorti es(seegure1.3)thatappeartobeinherent hara teristi s. Freeshearowslikethemixinglayerreveal oherentvortexstru turesvery learly,

againevenforveryhighturbulen eintensities.

The on epts of order and randomness have also led to some new analyti approa hes and new

interpretations in the study of turbulen e. The names of these dis iplines are known as Chaos,

Bifur ation Theory, and Dynami alSystems [M Donough,2007℄. These theorieshavebeenfa ed forthestudyofturbulen e,in parti ularintheareaofhydrodynami stabilityandtransitionfrom

laminarowtoturbulentone. Cometoattentionofmathemati ians,physi istsandengineers,these

phenomenaisofaremarkablenon-linearity,whi hmakesturbulen eunpredi tableand omplexto

des ribe.

Asanonlinearproblem,it anbeseenthatthesolutionstotheseproblemswiththesamenonlinear

equationswithonlyslightdieren esininitial onditions,willrapidlydiverge. Therefore,asuitable

denition ofturbulen e must ne essarilyinvolvea omplexdynami al systemwith many degrees

offreedom.

1.1.3 The Reynolds number

Turbulen e an be seen also a play between inertial for es and vis ous. Therefore the ratio

betweenthemis ru ialinorderto hara terizeaow. TheReynoldsnumberplaysthat role,and

in turbulentowsholds

Re

def

=

U L

ν

≫ 1.

(1.5)

Formany ows of pra ti alimportan e (e.g. a ow on airplanewings) the Reynoldsnumber

anbeontheorderof

Re ∼ 10

6

. Thismeansthat thevis ousfor es,thataremole ularfor es,a t

in smallers alesthanin thelargeones. However,in anyturbulentowthemole ularvis osityis

alwaysimportantatsomes ale. AstheowReynoldsnumberin reases,theregionwhere vis ous

ee tsareremarkable,de reasesinthi knessandthevelo ityoftheow hangesveryrapidlyfrom

zeroatthesurfa eto thefree-streamvelo ityat theouteredgesoftheboundarylayer. Again,we

see thetenden yofthenonlinearinertialtermstogeneratedis ontinuitiesat highwavenumbers.

On e more,theReynoldsnumber anbeinterpretedalso intermsoflengthandtimes aleratios.

Let's onsideradu tofwidth

L

,withaowvelo ity

U

. Thetimeauidparti le,withtransverse

velo ity

u

′

takesto rossthedu t is alledtheinertial time,

T

i

∼ L/u

′

. Atthesameway,vis ous

for eshaveatimes ale,

T

v

∼ L

2

_/ν

.

Inaturbulentow,theinertialtime-s alewillbemi h lessthanthediusivetime-s ale,

T

v

T

in

=

u

′

L

ν

> 1.

(1.6) 1.1.4 Navier-Stokes equations

Turbulen e behaviouris ompletely des ribed by the governing equations of uid me hani s,

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

expressedin thefollowingwayonnon-dimensionalform:

∂u

i

∂x

i

= 0;

∂u

i

∂t

+

∂u

i

u

j

∂x

j

= −

∂p

∂x

i

+

1

Re

∂

2

_u

i

∂x

j

∂x

j

(1.7)

where

Re

isthe hara teristi Reynoldsnumberoftheow,

u

i

, i = 1, 2, 3

arethevelo ity

om-ponents,

p

isthepressure. NotethatEinstein'ssummation onventionisusedhere,where

i = 1, 2, 3

.

Together withthemainoweld,sometimes weneedalsoto understandphenomenasthat

Bydenition,thewordpassiverefersto the onditionthattheresultingdensitydieren es areso

small that theee t from the s alaron theowis negligible. Soapassives alar anbe heat or

temperatureinaowora on entrationofasubstan e.

ThereforeisalsopossibletouseNavier-Stokesequations1.7to des ribethedevelopmentofa pas-sives alar,

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

k

.

1.2 Numeri al approa hes to N-S solution

Thereareseveralnumeri alapproa hestosolvethesystem1.7. Themostintuitiveoneis alled Dire tNumeri al Simulation, DNS,wherebythe governingeuquationsare solvedwithoutmaking

anyassumption, resolvingallthes alesfrom thesmallesttothelargestone. Thereforeitprovides

alltheinformationofaturbulentow,withoutanyapproximations. Sin ethe omputational ost

ofDNSs aleswiththeReynoldsnumberis

∼ Re

37/14

[ChoiandMoin,2012℄,thisisnotaordable forpra ti alengineeringanalysesathigh Reynoldsnumber.

Soamorepra ti alapproa hhasbeendeveloped,basedontheReynolds'de omposition[Reynolds,1894℄:

u

i

= u

i

+ u

′

i

,

θ = θ + θ

′

(1.9)

wheretheoverlinerepresentstheensembleaveragedquantity,and

u

′

i

and

θ

′

arethevelo ityand

s alar omponentsu tuations,respe tively.

UsingtheReynoldsde ompositionin 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 turbulen e is solely des ribed by the Reynolds stress tensor

u

′

i

u

′

j

and s alar ux ve tor

u

′

j

θ

. These terms have to beproperlymodelled in order to lose the problem. Ea h model involvesapproximationswhi h limitsits a ura y. Therefore, this approa h providesonly

anapproximatesimulationofthemeanow.

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

formulation

u

′

i

u

′

j

−

2

3

Kδ

ij

= −2ν

T

S

ij

,

S

ij

=

1

2

∂u

i

∂x

j

+

∂u

j

∂x

i

!

(1.12) where

K = u

′

i

u

′

i

/2

isthemeanturbulentkineti energy,

S

ij

isthemeanstrain-ratetensor,the symmetri partofthemeanvelo itygradienttensor,

ν

T

istheeddyvis osityweusetomodelasthe

produ tofa ertains alewith length

Λ

of theeddiesandvelo ity

V

. Usingdimensional analysis

ν

T

∼ ΛV.

(1.13)

Eddyvis osityisthenmodeleda ordinglyto some hara teristi softheow.

Forinstan e, algebrai models(orzero equationmodels) relatelength

Λ

andvelo ity

V

to the

meanvelo ityeldandtheowgeometry hara teristi slikevelo itygradient,distan etothewall,

thi knessoftheshearlayeret . Thesekindofmodelsworkquitewellforthespe i asethatthey

are designedfor,e.g. atta hedboundarylayersanddierenttypesof thin shearlayers. However,

theydon'tgivesatisfa toryresultsforgeneral ases.

Better results an be a hieved using one-equation models, they typi ally solve an additional

transport equation for the turbulent kineti energy,

K

, or the eddy vis osity,

ν

T

. One-equation

models give good results for atta hed boundary layers and other thin shear layer ows, but for

omplexows. A goodexampleistheSpalart-Allmaras[SpalartandAllmaras,1992℄model (SA), that solvesfor theeddy vis osity. This isverysuitablefor aeronauti alappli ationsand a tually

isthestandardmodelforexternalaerodynami sCFDanalysesatBoeing.

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

determiningthelengthandvelo itys aleneededto omputetheeddyvis osity. Themost ommon

are

K − ε

and

K − ω

models,wheretransportequationsfortheturbulentkineti energy,

K

andfor

thedissipationrate,

ε

orthe turbulen efrequen y

ω

aresolved. Nowadaystheimplementationof

these modelsin ommer ialCFD odes(e.g., ANSYSFluent)presentsadditional orre tionsthat

might be dependent on non-lo al quantities su h as thewall distan e. Oneimportant and most

re ent example is the Menter SST

K − ω

model [Menter,1994℄, whi h is suitable for separated ows;itis thestandardturbulen emodelusedatAirbus.

Despiteeddyvis ositytwo-equationsmodelsarestilldominatinginindustrialCFD,there'sabig

de-mandformorea uratepredi tionof omplexowsituations,in ludingonsetofseparation,highly

urvedows,rapidly rotatingowset . Inthesesituations, eddyvis osityBoussinesq'hypothesis

(1.12)doesnotdes ribetherealphysi swell. Anee t ofde orrelation aused byrotationo urs at high rotation rates, and generally the alteration of produ tion to dissipation ratio is a dire t

onsequen eofthat.

This phenomena, for instan e, is animportant aspe twhi h eddy vis ositymodels doesnot take

into a ount,be ausethemodelisinsensitivetosystemrotation.

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

equationsforea hReynoldsstress omponentsderivedfrom themodelledN-S equations,in order

ompu-tationally moreexpensiveand ompli ated than theothers. However,those dierentialReynolds

stressmodels anbesimpliedusingtheweakequilibriumassumptionbyRodi[Rodi,1992℄. Details aboutthiswillbeshownin 2. Thealgebrai relationisimpli itin Reynoldsstresses,butthereare some expli it solutions ([Pope,1975℄; [GatskiandSpeziale,1993℄; [WallinandJohansson,2000℄). These modelsare alledExpli itAlgebrai ReynoldsStress Models(EARSM).

Inparti ular,in theEA model[Wallin andJohansson,2000℄ theowanisotropy(

a

ij

)isdes ribed asanexpli itexpressionintermsofthe(normalized)meanstrainandrotationtensorswith

addi-tional s alarparameters. This leadsto a omparable omputational eorts, as ompared to eddy

vis ositytwo-equationmodels.

There'salsoaninterestinganalogyforthes alar

θ

modeling. Taylordevelopedananaloguewayto

formulate theeddydiusivitymodel(EDM)forthemeanturbulents alarux

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 diusivity oe ientand

P r

T

theturbulentPrandtlnumber.

Inananalogouswayto

K − ε

model,Nagano

&

Kim[Kim,1988℄developedthe

K

θ

− ε

θ

model. Thetimes ale

τ

θ

= K

θ

/ε

θ

,isusedto ompute

D

T

∼ Kτ

θ

.

Still, model 1.14 is not ompletely orre t. A ording to Bat helor, eddy diusivity assumes an alignmentbetweenthe

u

′

i

θ

′

ve torandthemeans alargradient,soithasto be onsidereditselfa

tensor. Thefollowingexpressionwillholdthen:

u

′

i

θ

′

= −D

ij

∂θ

∂x

j

(1.15)

Wheretheeddydiusivitytensor anberewrittenas[DalyandHarlow,1970℄

D

ij

= −C

θ

τ

θ

u

′

i

u

′

j

∂θ

∂x

j

(1.16)

and

C

θ

isamodel oe ient.

Inthesameway, s alar anbemodelled withanExpli it Algebrai model,in this ase alled the

Expli it Algebrai S alarFluxModels(EASFM).

Atrade-obetweena ura yand omputationaleortisLarge-eddysimulations(LES)ofturbulent

ows. InLES there'saseparationofs ales, inthesense that onlylarge-s aleeddiesareresolved,

while the remaining small s ales (whi h are alled sub-grid s ales, SGS) are modelled, on e the

resolveds ales havebeen omputed. Theseparationof s alesis generallydoneusing agrid. T

of the Navier-Stokesequations. This is thereforeaphysi ally- oherentapproa h sin eturbulen e

isunstationary.

Despitemore omputationallyexpensivethanRANS,LESgivesabetterdes riptionofturbulen e,

and unlikeDNS,is abletoprovideagoodresolution oftheowin ana eptableamountof

om-putational time. Re ently, that timehas been estimatedby Choi&Moin[ChoiandMoin,2012℄, to s aleas

∼ Re

26/14

.

Afairandsimplieddistin tionbetweenDNSandLES anbenoti edhavingalooktothevorti al

stru tures, for bothof the ases,in gure 1.4. Ata rstglan e, we ansee that in LES vorti al stru tures are underestimatedand fewer, ompared to the DNS,whi h is able,instead, to givea

ompleteanddetailed des riptionofthem.

Therst LESmodelwasdevelopedbySmagorinsky[Smagorinsky,1963℄formeteorologi al appli- ationsusinganeddyvis osityassumptionintheSGSmodel. Thismodelhasbeenimprovedlater

on by Germano [Germanoetal.,1991℄ introdu ing the dynami pro edure, whi h givesa orre t asymptoti near-wallbehaviouroftheeddyvis osity,andimprovedtransitionalowspredi tions.

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

anisotropywell,afeatureoftheow,whi h isnotnegle tablenearthewalls.

Forthat reasonseveralnon-linear models (whi h are of ourse anisotropi ),have been developed

re ently,and someof themwill bedes ribed in detailin hapter 3. In parti ular,the aimof this thesis is to test the a ura y in ow predi tion of the Expli it 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  Vorti al stru tures inturbulent hannel ow at

Reτ

= 590

, visualized by isosurfa es of

λ2

, olored by the velo ity magnitude, froma) DNS by P.S hlatter b) LES simulationwiththeExpli itAlgebrai model.

1.3 A parti ular ase: Turbulent Channel Flow

Mathemati almodelsforuiddynami shavebeenalreadydened. Inordertosolvethemodels,

amathemati alproblemneedsto bedened inapropertimeandspa edomains.

A ordingtothespa edomain,inadierentgeometry,physi alquantitiesandtheowwillbehave

inadierentway. Forthisreasonweneedtosetaparti ular ase,sothattheow anbeunivo ally

lassied.

Inthisthesisweare onsideringthe aseofChannelFlow,withthefollowingproperties:

ˆ turbulent,in thesense that theReynoldsnumberis su ientlyhigh su hthat the regime

anbeassumedas turbulent;

ˆ fully developed, sothat velo itystatisti s are onstant along

x

-axis. Inother words,the

ow is statisti ally stationary and statisti ally one-dimensional [Pope,2000℄, with velo ity statisti sonlyvariablealongthe

y

-axis. Inotherwords:

hui = U = U(y),

hvi = V = 0,

u

′

_v

′

_{= u}

′

_v

′

_(y)

(1.17)

Channelowbelongsto the wall-boundedshear ows lass: owmotionis ontainedbetween

twosolidsurfa es. Therefore,no-slip onditions areimposedonthewalls,wheretheuidvelo ity

isassumedto bezero. Thefollowingpi tureshowsthequalitativebehaviourofthevelo ity.

Figure1.5ChannelFlowmeanvelo ityprole,with

u

′

u tuations ontour plotinthe

Furthermore,astatisti allysymmetri owgeometryw.r.t. themid-plane

y = δ

is onrmedby

experiments;thereforethestatisti sof

(u, v, w)

at

y

arethesameasthoseof

(u, −v, w)

at

2δ − y

.

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

parti ularReynoldsnumbers,

Re

τ

def

=

u

τ

δ

ν

(1.18)

Re

b

def

=

uδ

ν

(1.19)

where1.18is basedonthefri tionvelo ity

u

τ

,denedasfollows:

u

τ

def

=

p

τ

W

/ρ

(1.20)

τ

W

isthemeanwallshearstressand

δ

isthe hannelhalf-width. Thebulkvelo ity

u

in 1.19,isdened as

u =

1

2δ

2δ

Z

0

huidy

(1.21)

Foraturbulent hannelow,thefollowingresultholds:

Re

τ

≈ 0.166Re

0.88

b

(1.22)

Notethat thespe iedformula 1.22willbeused forthederivation of

Re

b

, whi h will be on-sideredasaninputquantityinthe omputations.

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

wall-bounded shear ow the hara teristi lengthfor streamwisedevelopmentis mu hlarger thanthe

ross-streamextentoftheregionwithsigni antvelo ityvariation.

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

shearlayerapproximation [JohanssonandWallin, 2012℄. Thisapproximationstatesthatthe har-a teristi streamwisedevelopmentlength,

L

ismu hlargerthan

δ

,theshearlayerthi kness. Aswe

will onrm later,in hannelowthere willbealayer,whose thi kness issmall omparedto the

hara teristi length,wherebyvis ousee ts fri tiondependson.

Thankstothisapproximation,theNavier-Stokesequations anbesimpliedtothethinshearlayer

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 velo ities in 1.23aremean values,and theonly omponentresponsibleforturbulen eisthelastoneontheright-hand-side,whi his alled

theReynoldsstress. Togetherwiththevis ousstress,theReynoldsstressgeneratesthetotalshear

stress.

In parti ular, 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-normaldire tionitreads

0 = −

1

_ρ

dP

_dx

0

y + ν

dU

dy

− ν

dU

dy











y=0

| {z }

u

2

τ

−u

′

_v

′

_{+ 0}

(1.25)

Atthe enterline

(y = δ)

thetotalshearstressiszero,thereforewehavethefollowing ondition:

1

ρ

dP

0

dx

= −

u

2

τ

h

(1.26)

meaningthatthepressuregradientisrelatedtothefri tionvelo ityandthewidthofthe hannel.

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

ν

dU

dy

− u

′

_v

′

_{= u}

2

τ

1 −

y

δ

!

(1.27)

whi h inwallunitsbe omes

dU

+

dy

+

− u

′

_v

′ +

₌

1 −

y

+

δ

+

!

(1.28)

wherethequantities

y

+

and

δ

+

ares aled bytheinner(vis ous)lengths ale

y

+

def

=

y

l

∗

=

yu

τ

ν

Considerations. Dependingontheregion onsidered,equation1.28assumesdierentforms. Intheouterlayer,wherevis ous ee tsarenegligible,theleft-hand-sideof1.28be omes

− u

′

_v

′ +

= 1 −

y

+

δ

+

(1.29)

Ontheotherhand, loseto thewall,

y/δ << 1

,vis ous ee tswill benotnegligibleanymore,

thereforethefollowingrelationholds

dU

+

dy

+

− u

′

_v

′ +

≈ 1

(1.30)

thatis ompatiblewiththelawofthewall,sin ethere'snotinuen e oftheReynoldsnumber.

It'sa onstantstress region:

U

+

def

=

U

u

τ

= Φ

1

(y

+

)

(1.31)

u

′

_v

′ +

_{= Φ}

2

(y

+

)

(1.32)

Moreover,for largeReynoldsnumbers,we analsoassume that thereis an overlap region for

walldistan es

y

,

ℓ

∗

<< y << δ

where

δ

isboundarylayerthi knessand

ℓ

∗

thevis ous lengths ale. This isaparti ularregion

where outerandinnerlayerdes riptionshold simultaneously.

Derivatingthefollowing1.31w.r.t.

y

+

we'llhaveanexpressionwhi hisindependentoflengths ale.

Thereforeholds

y

+

dΦ

1

(y

+

₎

dy

+

≡ const

(1.33)

sothat,on eintegrated,itgivesalogarithmi law:

Φ(y

+

) =

1

κ

lny

+

_{+ B}

(1.34)

where

κ = 0.38

istheKármán onstantandB=4.1,a ordingtoobservations. Inthefollowing

gure6.22ameanvelo itydiagramisshown,togetherwiththelawof the wall andtheloglaw, at

PSfragrepla ements

y

+

hui

+

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

Reτ

= 5200

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

Aswe anseefromthepreviouspi ture,therstvelo ityregion alledvis oussublayer,follows

thelawofthewall,anditextendsoutto approximately

y

+

_{= 5}

. Despite theabsolutemagnitude

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

large. Aswein rease

y

+

,wewillhaveabuer region, wherethemaximumturbulen eprodu tion

is at

y

+

_{= 12}

and the maximum turbulen e intensity at

y

+

_{= 15}

. Log-layer starts between

50 < y

+

_{< 200}

, andit extendsto

y/δ ≈ 0.15

, where

δ

isthe hannel half-width. Beyondthe log

layer,there'snallytheouterregion.

The maximumturbulen e produ tion and Reynolds stress.

Theturbulen eprodu tioninthenear-wallregionofwall-bounded ows anbeformulatedas

P

+

= −

u

′

_v

′

u

2

τ

dU

+

dy

+

≃

1 −

dU

+

dy

+

!

dU

+

dy

+

(1.35)

Thelatterapproximationis validifvis ousee tsare small. Fromthis we anderivethatthe

maximumprodu tionisfoundwhere

dU

+

dy

+

=

1

2

(1.36) whi h leadsto

P

max

+

=

1

4

(1.37)

Thereforetheturbulentprodu tionismaximumwhereasbothvis ousandReynoldsstressesare

exa tlythesame,thatisinthenear-wallregion;itgenerallyhappenswhen

y

+

_{≈ 12}

alsopossibletoestimatewherethemaximumReynoldsshearstresso urs. 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,intermsofouters alewe ansaythatthepositionwheretheReynoldsstressrea hesits

maximumisproportionalto

Re

−

1/2

τ

:

y

max

h

= κ

−

1/2

_Re

−

1/2

τ

∝ Re

−

1/2

τ

(1.41)

Thisalso anbeprovedusingDNSresultsatdierentfri tionReynoldsnumbers. Anexample

ofReynoldsstressprolesis showningure1.7. PSfragrepla ements

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

uτ

vs

y/δ

,from[LeeandMoser,2014℄,

· · ·

:

Reτ

= 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)

Wealreadyhavetalkedaboutturbulen eanditsprodu tioninaqualitativesense,but,inorder

todes ribeitproperlyandtounderstandthephenomenonina ompleteanddetailedsense,afo us

on the three omponents of thevelo ity, theyare

u

′

,

v

′

and

w

′

, is ne essary. As we'll see in the

next hapters,the omputationsofthese omponentsplayanimportantrolein thevalidation ofa

LESmodel.

Inparti ular,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)

Inthefollowingpi tureallthe omponentsoftheReynoldsstressareshown. Importantto

un-derlineistherelationbetweenthemaximumturbulen eprodu tion,andthepeaksintheReynolds

stressproles. PSfragrepla ements

y

+

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

Figure1.8Varian esofthe velo ityu tuations, normalizedinwall units,fromDNS of

[LeeandMoser,2014℄at

Reτ

= 1000

,:

u

′

u

′

+

,:

v

′

v

′

+

,:

w

′

w

′

+

'

Anisotropy of the ow. A fundamental aspe t, whi h is stronglyrelevantfor the omplete

understandingoftheaforementionedwork,is the on eptofanisotropyin turbulen e.

Fromaphysi alpointof view,anisotropyis aproperty of theowwhi h is notalignedwith the

velo itydire tion andthevelo itygradient.

Forafully-developed hannelowthequantities

u

′

_w

′

and

v

′

_w

′

_{= 0}

arezero. TheReynoldsstress

tensorbe omes:

τ

ij

= −ρ











τ

11

u

′

v

′

0

u

′

_v

′

_τ

22

0

0

0

τ

33











.

(1.44)

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

τ

11

= τ

22

= τ

33

andall the

deviatori termsarezero. Therefore,foraspe i LESmodel,theabilityto apturetheanisotropy

oftheow onsistsin reprodu ingdierentdiagonalterms,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

themeanvelo ityeld oftheow. RANSsimulationsrelyheavilyonmodellingsin eallturbulent

motionsaremodelledandthereforetheyarenotalwaysa urate.

InDire tNumeri alSimulations(DNS), theunsteadyNavier-Stokesequationsaresolvedwithout

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

DNSistoolargeforindustrialappli ations. Therefore,anewmethodwhi h ombinesareasonable

owpredi tiona ura y withalimitedamount of omputational ost hasbeenstronglyrequired

in re entyears.

Large-eddysimulation(LES)representsanalternativethattsthoserequirements. Thisparti ular

methodemploysaseparationofs ales[Rasam,2014℄: alteringoperationde omposesthevelo ity eld (generallytogether withas alarone) intoaresolvedpart,representedbythe omputational

grid,andanunresolvedpart,whi hismodelledthroughphysi allyrealisti models. Inotherwords,

1. thesolutionofthelarges alesofturbulen eonarelatively oarsegrid;

2. themodelling ofthe smallerunresolveds ales, the so- alledsubgrid-s ales (SGS), basedon

theresolvedvelo ityeld.

2.1 The ltering operation

Step1. Let's onsidernowageneraltimeand spa e-dependentfun tion

φ(x, t)

. Theltering

operation onsistsofa onvolutionofakernel

G

∆

onthatfun tion(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 spe tral uto, box and

Gaussianlters. Inspe tral odes,spe tralltersarethemostsuitableonesforltering,sin ethey

a t onaspe tralspa e(where the odeworkson)and aremorepre ise. Figure below showsthe

dieren ebetweenthedierentkindsoflteringmethods.

Figure2.1.: Sharp-spe trallter,-: Gaussianlter, : Boxlter,

r = ξ

,from [Pope,2000℄

eld

e

u(x, t)

is notadeterministi variable, implyingthat

eeu(x, t) 6= eu(x, t), eu

′

(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

= −

∂

p

_e

∂x

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 possiblede omposition 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

= g

u

e

i

u

e

j

− e

u

i

e

u

j

,

(2.4)

C

ij

= g

e

u

i

u

j

′

− g

u

e

j

u

′

i

(2.5)

R

ij

= g

u

′

i

u

′

j

,

(2.6) and

u

′

i

= u

i

− e

u

i

.

In the Leonard de omposition

R

ij

is the Reynoldssubgrid tensor and represents the intera tion

betweensubgrid-s ales;

C

ij

is the ross-stresstensor anda ountsforlargevssmall s ale

intera -tions;nally

L

ij

,theLeonardtensor,givestheintera tionsbetweenthelarges ales.

τ

SGS

ij

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

2.3 An exampleofa losure: theEddy Vis osity Model (EVM)

Step2. Thereareseveralwaystomodelthesubgrids alestresstensor. Aspreviouslywritten,

the rstandsimplemodel wasdevelopedby Smagorinsky[Smagorinsky,1963℄, formeteorologi al appli ations. ItoriginatesfromtheRANSmodel,takingintoa ountBoussinesq'shypothesis. The

EVM onsistsofalinearformulationofthedeviatori partof

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

| e

S|

(2.8)

e

S

ij

and

| e

S|

are theresolved strain-ratetensorand its magnitude, respe tively.

∆

e

is thelter s ale,

ν

SGS

istheSGSeddyvis osity,and

C

s

isthemodel oe ient,theSmagorinsky oe ient.

Notethat

ν

SGS

isa onstantratherthandependentofdire tion. A ordingtothere entpaper

of Spalart [Spalart,2015℄, in a simple shear ow with two of the axes aligned with the velo ity dire tion and thegradientdire tion,su hthat thestraintensor

S

ij

havezeroandequaldiagonal

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

τ

11

= τ

22

= τ

33

.

ThereforeEVM anbe onsideredanisotropi model,sin e

ν

SGS

doesn'ttakeintoa ountany

ee t ofanisotropy.

As we'll dis uss further, this model an be improved using a dynami pro edure, where the

C

s

oe ientis omputedduringthesimulation.

2.4 Subgrid-s ale dissipation

Animportantaspe t ofLESistheimpa tofSGS ontheresolveds ales. Inotherwords,LES

predi tionsarestronglydependentonthe ontributionof

τ

SGS

ij

tensorontheresolvedkineti energy

K =

u

e

i

u

e

j

/2

. Considerthekineti energyequation

∂K

∂t

+

∂

∂x

j

(

e

u

j

K)

|

{z

}

advection

= − ν

∂

_∂x

e

u

i

j

∂

u

e

i

∂x

j

|

{z

}

viscous

dissipation

−

_∂x

∂

i

u

e

i

p + ν

e

∂K

∂x

i

− e

u

i

τ

ij

!

|

{z

}

dif f usion

+ τ

ij

SGS

S

e

ij

| {z }

−

SGS

dissipation

(2.9)

Sin eLESbydenition,are arriedoutforlarges ales,andthegrids aleismu hlargerthan

Kolmogorovone,the vis ous dissipationterm is sosmall so that it is negligible ompared to the

others. Fromaphysi alpointofview,diusiontermtransfersenergyinspa e,butnotina

volume-averagedsense [Rasam,2014℄. Forthatreason,anadditionaltermisthereforerequiredinorderto reprodu ethe orre tenergy transferfrom thelargeto thesmallers ales. SGS dissipation overs

Π = −τ

SGS

ij

S

e

ij

(2.10)

ThemeanSGS dissipationbehavesasasink term, whiletheinstantaneousonegivesnegative

(ba ks atter)andpositive(forwards atter) ontributionsinthetransfer.

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

resolveds ales(blue line);theirtotal ontributionisshownbytheredline.

PSfragrepla ements

y

+

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

Figure2.2Dissipationoftheresolveds ales,

ε

,togetherwiththeSGSdissipation,

Π

,and theirsumatin reasingresolutions,fromLESat

Reτ

= 590

shownin hapter6.

Theprevious onsiderationleadstothefa tthataninvestigationonSGSdissipationbehaviour

isveryimportantinana ura yassessmentofLES.

Basedonthisprin iple,severalinvestigationshavebeen arriedoutbyChow&Moin[ChowandMoin,2003℄, Ghosal[Ghosal, 1996℄andKrav henko&Moin[Krav henkoetal.,1996℄.

Geurts &Fröhli h [Geurtsand Fröhli h, 2002℄introdu ed theSGS a tivityparameter, dened as follows

s =

<ε

SGS

_>

<ε

SGS

_{> + <ε}

µ

_>

(2.11) where

<ε

µ

_{> = 2µ e}

_S

ij

S

e

ij

isthevis ousdissipation.

With in reasingresolution, SGS dissipation de reaseswhile vis ous onein reases, therefore

s

be- omes smaller. A remarkable aspe t isthat, the oarser is theresolution , the biggeris

s

. This

an be proved having alook at 2.11: if we use a oarse grid, the resolution will be smallersu h that vis ous dissipationwill benegligible, and

s

will rea h itsmaximumunity value. This o urs

small.

Starting from the previous denition, Geurts & Fröhli h [GeurtsandFröhli h,2002℄ dened an errornormas

δ

E

=











E

LES

− e

E

DN S

e

E

DN S











(2.12)

where

E

LES

isthemeanresolvedkineti energyintegratedovertheowdomain,while

E

e

DN S

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

theLES.

The a ura y assessment onsists of omputing the relation between

δ

E

and

s

: in the pi ture

below[Rasametal.,2011℄ it is shown that therelativeerror

δ

E

drops almost exponentiallywith de reasing

s

(i.e. high resolution); results referto a LES with the expli it algebrai SGS stress

model (EASSM) [Marstorpetal.,2009℄ at

Re

τ

= 934

, for six dierent resolutions. This result showsthat with in reasing resolution theSGS ontribution be omessmaller and thea ura y of

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

a eptable solution.

Figure2.3Normalizederror

δE

oftheresolvedkineti energy,integratedoverthe hannel width[Rasametal.,2011℄w.r.t. the ltered DNSvalue vs

s

. Arrow points toin reasing resolution ases.

3

SUBGRID-SCALE STRESS MODELS

FOR LES

Logi willgetyoufrom AtoB.

Imagination willtake youeverywhere. (A.Einstein)

Inthis haptertwodierentLESmodelsaregoingtobeanalyzed;therstoneisanimprovement

ofthe lassi Smagorinskymodel, theDynami EVM.

These ondoneisaparti ularnon-linearmodelusinganexpli italgebrai formulationfortheSGS

stresstensor.

Inthefollowingse tionade omposition oftheSGS stresstensorisgiven.

3.1 Tensorialpolynomial formulation of the SGS stresstensor

A usefulapproa h that leadsto several non-linear models is basedon ade omposition of the

SimilartoRANSapproa hforReynoldsstress losures,SGSstresstensor anbeexpressedinterms

ofstrainandrotationratetensors.

Lund &Novikov[LundandNovikov,1993℄ expressedthedeviatori partof theSGS stresstensor

τ

_ij

SGS,d

asageneraltensorialfun tion of theltered strain-rate

S

e

ij

androtation-rate

Ω

e

ij

tensors, theKrone kerdelta

δ

ij

andtheltersize

∆

e

as

τ

ij

SGS,d

= τ

ij

SGS

−

1

3

τ

kk

δ

ij

= f ( e

S

ij

, e

Ω

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

andtheir ombination.

Coe ientsarefun tions 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)

_{= e}

S

T

(2)

_{= e}

S

2

₋

1

3

II

S

I

T

(3)

_{= e}

_Ω

2

₋

1

3

II

Ω

I

,

T

(4)

_{= e}

S

Ω

e

_{− e}

Ω

S

e

T

(5)

_{= e}

S

2

Ω

e

_{− e}

Ω

S

e

2

T

(6)

_{= e}

S

Ω

e

2

_{+ e}

Ω

2

S

e

₋

2

3

IV I

T

(7)

_{= e}

S

2

Ω

e

2

_{+ e}

Ω

2

S

e

2

₋

2

3

V I

T

(8)

_{= e}

S

Ω

e

S

e

2

_{− e}

S

2

Ω

e

S

e

T

(9)

_{= e}

Ω

e

S

Ω

e

2

_{− e}

Ω

2

e

S

Ω

e

T

(10)

_{= e}

Ω

e

S

2

Ω

e

2

_{− e}

Ω

2

S

e

2

Ω

e

and

β

k

ares alar oe ientsthat arefun tionsofvetensorialinvariants

II

S

= tr(e

S

2

),

II

Ω

= tr( e

Ω

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 pro edure for EVM (DEVM)

Regardingto themodel oe ient

C

s

forthe eddy-vis osity model, it hasbeendemonstrated

that adynami omputation of

C

s

(that is, during the simulation), an signi antly improvethe

owpredi tiona ura y.

The omputation of

C

s

is done by taking into a ount the resolved s ales, a ordingto a s ale

invarian eassumption.

The ru ialpointofthisdynami pro edure istheso- alledGermanoidentity. Let'snowdenotea

testlter,

∆ = 2 e

b

∆

,

∆ =

e

3

√

Ω

,where

Ω

isthevolumeofa omputational ell. TheGermanoidentity isdened asfollows:

L

_ij

_{= T}

_ij

_{− b}

_τ

_ij

_,

(3.4)

where

T

ij

isthe SGSstress tensorlteredat thetest lterlevel,and

L

ij

= d

e

u

i

e

u

j

− be

u

i

beu

j

Then

L

_ij

isappliedin thisway:

L

_ij

₋

1

3

L

kk

δ

ij

= −2C

s

M

ij

(3.5)

where

M

_ij

_{= b}

_∆

2

_|be

_S|be

_S

_ij

_{− e}

_∆

2

_{| e}

\

_{S| e}

_S

_ij

_.

(3.6)

Thesystemofequations3.5isover-determined. Inordertohaveauniquevalueof

C

s

Germano ontra teditin:

C

s

= −

1

2

hL

ij

S

e

ij

i

h e

S

ij

M

ij

i

(3.7)

Moreover,tomake

C

s

variationssmoother,aspatial averaging

h.i

hasbeenapplied.

Togetherwithabetterperforman eof theEVM,theGermanoidentitygivesa orre tasymptoti

near-wallbehaviourfor

ν

SGS

.

Ithasalsobeenshownthatispossibleto applyGermanoidentityforthedynami omputationof

3.3 Expli it Algebrai SGS stress model

Anisotropi ee ts of turbulen e are important in several onditions: examplesare near-wall

owbehaviourandboundarylayesseparation aseswith urvature,swirl,rotation.

Asitwasdis ussedbefore,theDynami Smagorinskymodelisanisotropi model,inthesensethat

theSGSvis osity

ν

SGS

isdire tion-independent.

Therefore,in orderto aptureowanisotropy,DEVM isnotsuitable.

InthesamespiritasReynoldsstress-basedmodelsarene essaryforRANS,non-linearSGSmodels

areneededforLES.

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

[WangandBergstrom,2005℄, whi h onsists of three base tensorsand three dynami oe ients.

OneofthetermsinthemodelissimilartotheDEVMmodel. Wang&Bergstrom[Wangand Bergstrom,2005℄ showedthatthedynami non-linearmodelpredi tsamorerealisti tensorialalignmentoftheSGS

stressthaneddy-vis ositymodelsand anprovideforba ks atterwithout lipping oraveragingof

thedynami modelparameters.

The model here dis ussed is alled Expli it Algebrai SGS stress model (EASSM), was

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

arenegligible.

AnalogoustotheReynoldsstressanisotropytensorwedenetheSGSstressanisotropytensoras

a

ij

=

τ

ij

K

SGS

−

2

3

δ

ij

(3.8)

Forsimpli ity,we'll onsider

τ

SGS

ij

= τ

ij

. Moreover,

K

SGS

= (

u

g

i

u

i

− e

u

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 andmole ularuxes oftheSGS stress andSGS kineti energy,respe tively.

Although the produ tion of the SGS stress

P

ij

and SGS kineti energy

P = P

ii

/2

are given in