Analyzing a turbulent pipe flow via the one-point structure tensors

Delft University of Technology

Analyzing a turbulent pipe flow via the one-point structure tensors

Vorticity crawlers and streak shadows

Stylianou, F. S.; Pecnik, R.; Kassinos, S. C.

DOI

10.1016/j.compfluid.2016.10.010

Publication date

2016

Document Version

Final published version

Published in

Computers & Fluids

Citation (APA)

Stylianou, F. S., Pecnik, R., & Kassinos, S. C. (2016). Analyzing a turbulent pipe flow via the one-point

structure tensors: Vorticity crawlers and streak shadows. Computers & Fluids, 140, 450-477.

https://doi.org/10.1016/j.compfluid.2016.10.010

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ContentslistsavailableatScienceDirect

Computers

and

Fluids

journalhomepage:www.elsevier.com/locate/compfluid

Analyzing

a

turbulent

pipe

flow

via

the

one-point

structure

tensors:

Vorticity

crawlers

and

streak

shadows

F.S. Stylianou

a

_{, R. Pecnik}

b

_{, S.C. Kassinos}

a,∗

a Computational Sciences Laboratory (UCY-CompSci), Department of Mechanical and Manufacturing Engineering, University of Cyprus, Kallipoleos Avenue 75, Nicosia 1678, Cyprus

b Process & Energy Department, Delft University of Technology, Leeghwaterstraat 39, 2628CB, Delft, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 4 June 2016 Revised 3 October 2016 Accepted 13 October 2016 Available online 14 October 2016

Keywords:

Turbulent pipe flow DNS Structure tensors Turbulence structure Stream vector Inactive structure Active structure

a

b

s

t

r

a

c

t

Effortstoidentifyandvisualizenear-wallstructurestypicallyfocusontheregiony+_5,where large-scalestructureswithsignificantturbulentkineticenergycontentreside,suchasthehigh-speedand low-speedstreaksassociatedwithsweepandejectionevents.Whileitistruethattheleveloftheturbulent kineticenergydropstozeroasoneapproachesthewall,theorganizationofnear-wallturbulencedoes notendaty+≈ 5.Large-scalestructureswithsignificantstreamwiseextentandspatialorganizationexist evenintheimmediateproximityofthewally+_{<5.Thesecoherentstructureshavereceivedless} atten-tionsofar,butitwouldbebothusefulandenlighteningtobringthemtofocusinorder,ononehand, to understandthem,butalsoto analyzetheirinteraction withthe energeticstructures thatreside at somewhathigherdistancesfromthewall.

Wehaverecentlydevelopedarigorousmathematicalandcomputationalframeworkthatcanbeusedfor thecalculationoftheturbulencestructuretensorsinarbitraryflowconfigurations.Inthiswork,weuse thisnewframeworktocompute,forthefirsttime,thestructuretensorsinafully-developedturbulent pipeflow.WeperformDirectNumericalSimulation(DNS)atReynoldsnumberReb=5300,basedonthe bulkvelocityandthepipediameter.Wedemonstratethediagnosticpropertiesofthestructuretensors, byanalyzingtheDNSresultswithafocusonthenear-wallstructureoftheturbulence.We developa neweductiontechnique,basedontheinstantaneousvaluesofthestructuretensors,fortheidentification ofinactivestructures(i.e.large-scale structureswithoutsignificantturbulentkineticenergy).Thisleads tothe visualizationof“vorticity crawlers” and“streakshadows”, large-scalestructureswith lowenergy contentintheextremevicinityofthewall.Furthermore,comparisonwithtraditionaleductiontechniques (suchasinstantaneousiso-surfacesofturbulentkineticenergy)showsthatthestructure-basededuction methodseamlesslycapturesthelarge-scaleenergeticstructuresfurtherawayfromthewall.Wethenshow that the one-pointstructure tensors reflect themorphologyof theinactive structuresin theextreme vicinityofthewallandthatoftheenergy-containinglarge-scalestructuresfurtherawayfromthewall. The emergingcompletepicture oflarge-scale structureshelpsexplain thenear-wallprofiles ofallthe one-pointstructuretensorsandislikelytohaveanimpactinthefurtherdevelopmentofStructure-Based Models(SBMs)ofturbulence.

© 2016TheAuthor(s).PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

One-point measures of large-scale, energy-containing turbu-lencestructuresareimportantinturbulencemodelingandforflow diagnostics.KassinosandReynolds [18]were the firstto develop acomprehensiveone-pointmathematicalformulationthat canbe

∗ _{Corresponding author.}

E-mail addresses: stylianou.fotos@ucy.ac.cy (F.S. Stylianou), r.pecnik@tudelft.nl

(R. Pecnik), kassinos@ucy.ac.cy (S.C. Kassinos).

usedtoquantifydifferentaspectsoftheenergy-containing turbu-lencestructures.Inthisregard,theyshowedthatitispossiblefor twoturbulencefieldstosharethesamecomponentalitystate,i.e.to havethesameReynoldsstresstensorvaluesR_ij,butyethave differ-entunderlyingturbulencestructure.Differencesintheturbulence structure, although undetectable through the componentality in-formation, lead to different dynamic behavior of the turbulence, forexample in response to external deformation or system rota-tion.Hence,acompleteone-pointdescriptionoftheturbulence re-quirestheinformationcontainedinthestructuretensors[21].The

http://dx.doi.org/10.1016/j.compfluid.2016.10.010

F.S. Stylianou et al. / Computers and Fluids 140 (2016) 450–477 451 structure dimensionalityDij givesinformationaboutthedirections

ofindependenceintheturbulence,thestructurecirculicityFijgives informationonthelarge-scalecirculationintheflow,andthe inho-mogeneityC_ijgivesthedegreeofinhomogeneityoftheturbulence. Thethird-rankstropholysisQ_{i jk}∗ becomesimportantwhenmean ro-tationbreaksthereflectionalsymmetryoftheturbulence[21]. Ex-actdefinitionsofthesetensorsaregiveninthenextsection.

One-point turbulence models that use only the Reynolds stresses and the turbulence scales to characterize the turbu-lence are fundamentally incomplete as shown by Kassinos and Reynolds [18]. Contrariwise, Structure-Based turbulence Models (SBMs) [17,18,20,21,35,39] are a class of turbulence models that makeuseoftheone-pointturbulencetensors.SBMsholdpromise forresolvinginherentlimitationsofsimpleeddy-viscosityclosures and of Reynolds Stress Transport (RST) models. However, an ob-stacle inthefurtherdevelopmentofSBMshasbeentherelatively scarce availability of accurate data that could be used formodel calibrationandvalidation.

The one-pointstructure tensorscan notbe extractedfrom ex-periments. Hence, one normally turns to Direct Numerical Simu-lations (DNS) orLarge EddySimulations (LES)for obtaining data on the structure tensors. Even inthis case, however, the specifi-cation ofproper boundary conditionsfor thecomputation ofthe structuretensorsisadauntingtask.Theunderlyingambiguityover how one can compute the structure tensors incomplex domains has discouraged themore widespread inclusionof thetensors in turbulencedatabases.Wehaveonlyrecentlydevelopeda rigorous mathematical andcomputationalframework that canbe used for thecalculationofthestructuretensorsinarbitraryflow configura-tions [48].We willrefertothisastheGeneralFramework(GF).In thepast,adifferentframeworkhadbeenconsidered[18,21],which isonlyapplicableinsimple,wall-bounded,streamwiseperiodic ge-ometries, e.g.fully-developedchannelflow, pipeflow,square duct flow. We willrefer to this asthe Limited Framework (LF). In this work, we use both aforementioned frameworks (GF and LF) to compute,forthefirsttime,thestructuretensorsinfully-developed turbulent pipe flow. We perform direct numerical simulation at ReynoldsnumberReb=5300, basedonthebulk velocityandthe pipediameter.

Themainobjectivesofthecurrentstudyare:

(a) To illustrate that LF and GF lead to different results for the structure tensors. The same was shown in a fully-developed turbulentchannelflowbyVartdal[52].

(b) Toexplain thatbothLFandGF arecorrect,andthat the afore-mentioneddifferencesshouldbeattributedtothelackofgauge invarianceofthestructuretensors.

(c) ToprovideargumentsinfavorofusingtheGF forthe compu-tationof thestructure tensors. Forexample, theGF preserves the essence of the structure tensors (as defined in the ho-mogeneous limit)evenin inhomogeneousregions ofthe flow, whereastheLFintroducesseriousdeviations.

(d) To provide a databasefor the developmentand validation of neworexistingSBMs.

(e) Tomanifest thediagnostic properties ofthestructure tensors, by analyzing the DNS results and comparing with traditional eduction techniques (such asinstantaneous iso-surfaces of Q-criterionandturbulentkineticenergy).

(f) To establish a new flow structure characterization technique that allows the identification of inactive structures (i.e. large-scale structures without significant turbulent kinetic energy) basedontheinstantaneousvaluesofthestructuretensors. We believe thatthiscontribution willencouragethe inclusion of the structure tensors in DNS databases, thus accelerating the developmentofstructure-basedmodelsandpromoting theuseof structuretensorsasaflowdiagnostictool.

2. Structuretensors

2.1. Definitions

The structure tensors are determined through the fluctuating streamvector

ψ

_i,definedbytheequations

u_i=



i jk

ψ

_{k, j}

ψ

_k,k =0

ψ

_i,kk =−

ω

i, (1) where u_i and

ω

_i are the fluctuating velocity and vorticity com-ponents,and



ijk isthe Levi-Civita alternatingtensor. Hereafter,a commafollowed by an index denotes partial differentiationwith respectto theimplied coordinate direction.The Einstein summa-tionconventionisimpliedonrepeatedRomanindices.Werequire

ψ



i to be divergence-free so that the simplified Poisson equation inEq.(1)holds,afeaturethatisimportantforthephysical inter-pretationoftheresultingstructuretensorsasexplainedby Kassi-nos etal. [21].To complete the stream vector definitionsuitable boundaryconditionsmustbesupplied[48].

Expressingthe definitionof the Reynolds stressesin terms of thefluctuatingstreamvector,

Ri j=u_iu_j=



ipq



jrs

ψ

q,p

ψ

s,r , (2)

andusingtheidentity



ipq



jrs=det



_δ

i j

δ

ir

δ

is

δ

p j

δ

pr

δ

ps

δ

q j

δ

qr

δ

qs



, (3)

leadstotheconstitutiverelation

Ri j+Di j+Fi j−

(

Ci j+Cji

)

=

δ

i jq2, (4) whereq2_=R

ii=2kistwicetheturbulentkineticenergy.Basedon thisequation,thesecond-rankstructuretensorsaredefinedas Componentality: Ri j=uiuj ri j=Ri j/Rkk (5a)

Dimensionality: Di j=

ψ

k,i

ψ

k, j dˆi j=Di j/Dkk (5b) Circulicity: Fi j=

ψ

i,k

ψ

j,k fˆi j=Fi j/Dkk (5c) Inhomogeneity: Ci j=

ψ

_i_,k

ψ

_k_{, j} cˆi j=Ci j/Dkk. (5d) Unliketheotherstructuretensors,theinhomogeneityC_ij isnot positive semi-definite and thus the trace C_kk=D_kk− Rkk can be negativeoreven zero.Forthisreason, Cij is normalizedin terms of the traces Dkk or Fkk, which by their definition are the same

D_kk=F_kk. Another possibility would have been to normalize all structuretensors withthe trace Rkk,butthischoice isill-defined onsolidboundaries,whereRkkiszero.Onthecontrary,Dkkis non-zeroatthewallsandprovestobethemostmeaningfulchoicefor normalizingallthestructuretensors.

A detailed discussion on the interpretation of each structure tensor is provided by Kassinos et al. [21], but the key features arerecountedhere.Whilethestructuretensorscarry complemen-tary information, the constitutive equation provides a linear de-pendenceamongthem.ThecomponentalityR_ij(theReynoldsstress tensor) givesinformationabout whichcomponents ofthe fluctu-atingvelocityaremoreenergetic.ThedimensionalityDijcarries in-formationaboutthedirectionsofindependenceoftheturbulence. To understand this, notice that the free indices in the definition of Dij are associated to the gradients of

ψ

i, which tend to van-ishalongdirectionsofsubstantialstructureelongationandtendto bestrongestalongdirectionsinwhichshortstructuresarestacked. ThecirculicityFij identifies thedirectionswithlarge-scale circula-tionconcentratedaroundthem.Toappreciatethis, noticethat the freeindicesinthedefinitionofF_ijareassociatedwith

ψ

_i_,whichin turn,throughthePoissonequation

ψ

_i_,kk=−

ω



i,representsa large-scale,smooth version of

ω

_i. Finally,the inhomogeneityCij detects

the inhomogeneity of the turbulence. In fact, the inhomogeneity tensorvanishesidenticallyinhomogeneousflows,ascanbeshown byrecastingtheinhomogeneitydefinitionintotheform

Ci j=

(

ψ

i

ψ

k, j

)

,k−

ψ

i

ψ

k,k j . (6)

Here,thefirsttermiszeroonlyinhomogeneousflows, whilethe secondtermisalwayszeroduetothespecificchoice

ψ

_k,k =0.The inhomogeneityis significant nearsolid boundariesandrelaxesto zerofarawayfromthem.Atintermediatedistancesfromthewall, themagnitude Ckk becomes smallcompared to that ofthe other structuretensors.SincelittleisknownonhowtomodelCijin gen-eral flows, structure-based turbulence models, such as the Alge-braicStructure-Based Model(ASBM) [3,22,27,36,42],are basedon thehomogenized tensors. Theseare obtained by absorbing Cij in-sideD_ijandF_ij, Dcc ij ≡ Dij− 1 2



Cij+Cji



Fcc ij ≡ Fij− 1 2



Cij+Cji



. (7)

Notethat the homogenizedtensors now satisfy Dcc

kk=Fkkcc=Rkk=

q2_.

To complete theone-point tensorial base, an additional third-rankstructuretensormustbedefinedasonecanshowthatit car-riesinformationthatisnot containedinthesecond-ranktensors,

Qi jk=−uj

ψ

i,k=



jrs

ψ

r,s

ψ

i,k. (8)

Usingthedefinitionsofthesecond-rankstructuretensors,onecan showthat



impQm jp=Ri j Qik j− Qjki=



i jpRpk (9a)



impQpm j=Di j− Ci j Qjik− Qi jk=



i jp

(

Dpk− Cpk

)

(9b)



impQjpm=Fi j− Cji Qk ji− Qki j=



i jp

(

Fpk− Ckp

)

. (9c)

Thehomogenizedtensorscanalsobecalculatedfromthe third-ranktensor, Dcc ij = 1 2





impQpmj+



jmpQpmi



Fcc ij = 1 2





impQjpm+



jmpQipm



. (10)

Athird-rank constitutiveequation connectsall the structure ten-sors, Qijk= 1 6



ijkq 2₊1 3



ikpRpj+ 1 3



jip



Dpk− Cpk



+1₃



kjp



Fpi− Cip



+Q_ijk∗ (11)

wheretheStropholysistensor

Q_{i jk}∗ =1

6

(

Qi jk+Qjik+Qjki+Qk ji+Qik j+Qki j

)

(12) is the fully symmetric part of the third-rank structure tensor. Stropholysisliterallymeans “breakingbyrotation”,amnemonicto thefact thatthistensorremains zerointurbulencethathasbeen deformed only by irrotational mean strain. However, mean and framerotationbreakthe reflectionalsymmetry ofturbulenceand generateQ_{i jk}∗ .Oncegenerated,thestropholysiscanbefurther mod-ifiedby irrotational mean strain [18].It is worth notingthat the bi-tracesofthethird-ranktensorare

Qkik=0

Qkki=Qikk=−



u_k

ψ

_i



,k

Q_kik∗ =Q_kki∗ =Q_ikk∗ =−2 3



u_k

ψ

_i



,k,

(13)

whichallvanishinhomogeneousturbulence.

2.2. Non-localinformation

Even though the structure tensors are one-point correlations they still carry important non-local information aboutthe struc-ture of turbulence. We provide two arguments to support this statement.

First, the fluctuatingstream vector

ψ

_i (the constituentof the structuretensors)isobtainedfromthesolutionofavectorPoisson equation,namely

ψ

_i_,kk=−

ω



i.Thefluctuatingvorticityvectorfield

ω



i actsas the sourceterm for thisvectorPoissonequation.Abasic propertyofthePoissonequationisthatitssolutionatanypointin thedomainreceivessourcetermcontributionsnotonlyfromthat point, but from distant points aswell. Therefore, the fluctuating streamvector willcontainnon-local informationoftheflowfield thatistransferredtothestructuretensors.

Second,thefluctuatingpressurefield(whichcontainsnon-local informationasitemergesfromasolutionofaPoissonequation)is intimatelyconnectedtothestructuretensors.Todemonstratethis, weconsiderasimpleproblemofhomogeneousturbulencesubject to meanrotation.Inthiscase, the Poissonequation fortherapid pressurefluctuations 1

ρp,kkr =−2Gi juj,ireducesto 1

ρ

p,kkr =

ω

i

ω

i, (14)

where Gi j=ui, j is the mean deformation tensor, and

ω

i is the meanvorticityvector.Inhomogeneousturbulencethemean veloc-itygradientsareuniform,andthereforeifwereplacethe fluctuat-ing vorticity with the Poissonequation ofthe fluctuating stream vectorwearriveattherelation _ρ1pr₌

_ω

i

ψ

i.Basedonthis expres-sion,wecanconnecttheCirculicitywiththerapidpressure gradi-ent

1

ρ

2p

r

,kp,kr=

ω

i

ω

jFi j. (15)

Clearly,inthissimpleexampleFijcarriesthenon-localinformation containedintheintensityoftherapidpressuregradient.

2.3. Uniquelydefiningthestructuretensors

In our previous work [48], we have stated that the structure tensorsaregaugeinvariant,whichisactuallymisleading.The prop-erty of gauge invariance should be attributed only to quantities thatareindependentofthespecificgaugeconditionschosento de-fineuniquelythe

ψ

_i,i.e.theEuclidgaugeconditionandboundary gaugecondition[40,48].Asitisshowninthiswork,thestructure tensors donot havethisproperty. Based ontwo differentsets of boundarygaugeconditionsfor

ψ

_i,we havecalculatedtwo differ-ent

ψ

_i fields along withtheir associated structure tensors. Even though both

ψ

_i fields successfully reproduce the same u_i field, theydonotproducethesamevaluesforthestructuretensors.

Here,wealsoproveanalyticallythelackofgaugeinvarianceof thestructuretensors.Theincompressibilityconditionofu_iimplies therelation u_i=



i jk

ψ

k, j, whichisconsidered asthebackboneof thedefinitionfor

ψ

_i.Thisrelationdoesnotdefineuniquelythe

ψ

_i,

sinceaddingagradientofascalarfunction

θ

to

ψ

_i

ψ



i→

ψ

i+

θ

,i (16)

stillsatisfiestherelationbetweenu_iand

ψ

_i.Thisisaconsequence oftheidentity



i jk

θ

,k j=0. (17)

Ifwe apply the gauge transformation Eq.(16)to the definitionof thestructuretensors(apartfromtheReynoldsstress)wecanshow thatthey arenotgaugeinvariant.Toclarifytheissueofgauge in-variance,wewillfocusontheparticularexampleofthestructure dimensionalitytensorDij.Ifweallowforthegaugetransformation

F.S. Stylianou et al. / Computers and Fluids 140 (2016) 450–477 453

ofthestreamvector

ψ

θ_i =

ψ



i+

θ

,i,thenDij transformsaccording to

Dψ_{i j}θ ≡

ψ

θ_k,i

_ψ

θ_{k, j=}Dψ_{i j} +

ψ



k,i

θ

,k j+

θ

,ki

ψ

k, j+

θ

,ki

θ

,k j (18) which is clearly not gauge invariant since D_{i j}ψθ _=Dψ_{i j}. Therefore, differentgauge conditions(chosen touniquely define

ψ

_i) leadto differentvaluesforthestructure tensors.Since thestructure ten-sors were originally definedwith theaim to describe the coher-ent structures of turbulence,we must identifythe specific gauge conditionsthat preservetheir intended meaning.Inthefollowing paragraphs, we provide strong arguments that point to the pre-ferredgaugeconditions.

In homogeneous flows only one gauge conditionis neededto uniquelydefine

ψ

_i.WeimposetheEuclidgaugecondition

ψ

_i_,i=0,

sincethisparticularchoiceimpartsa numberofdesirable proper-tiestothestructuretensors,namely:

(a) theinhomogeneitytensorbecomesidenticallyzeroin homoge-neousflows(seeEq.(6)),

(b) asimplerelationconnectsthecirculicityspectrumtensortothe vorticityspectrumtensorinhomogeneous flows(see[18] Sec-tion2.6),

(c) the differential equation for the stream vector reduces to a Poissontype(see[48]Section3.3),

(d) the relations between

ψ

_i, u_i, and

ω

_i follow a recursive form: [

ψ

_i,i ₌0_,u_i,i ₌0_,

ω

_i,i=0]_, [u_i₌



_{i jk}

ψ

_{k, j} _,

ω

_i₌



_{i jk}u_{k, j}]_, [

ψ

_i_,kk=−



i jkuk, j,ui,kk=−



i jk

ω

k, j].

Ininhomogeneousflows,theEuclidgaugeconditionalone can-not uniquely define

ψ

_i. An additional boundary gauge condition mustbespecified.Therearetwopossibilities:

(a) eitherrestrictthestreamvectorcomponentsthataretangential tothelocalsurfaceboundary:



i jknj

ψ

k



S=



i jknjak,whichleads totheLimitedFramework(LF),

(b) orrestrict thestream vectorcomponentthat is normaltothe localsurfaceboundary: ni

ψ

i



S=niai, whichleadstothe

Gen-eralFramework(GF).

Inbothframeworksthevectorsurfacefielda_imustsatisfy spe-cific conditions that can be found in Stylianou et al. [48], with a rigorousmathematical proof given by Quartapelle [40]. Forthe caseoffully-developed periodicturbulent pipeflow, we can sim-ply set a_i=0 in both formulations. As explained by Quartapelle

[40],theGF isapplicable indomains withanytypeof connected-ness,whiletheLFisonlyapplicabletosimplyconnecteddomains. InadditiontothismathematicalsuperiorityofGF,weprovide be-lowthephysicalargumentsthatpointtothepreferenceofGFover LFforthecomputationofthestructuretensors:

(a) Through the Poisson equation

ψ

_i,kk =−

ω



i, the stream vector

ψ



i representsalarge-scale,smoothversionof

ω

i.Theboundary gaugeconditionenforcedon

ψ

_ishould preservethisproperty, sothat theinterpretationofthestructuretensorsremains un-affectedasthewallboundaryisapproached.Sinceni

ω

i



S=0at thesolidboundaries,theGFgaugeboundaryconditionn_i

ψ

_i



S= 0satisfiesthisrequirement.ThisisnotthecasefortheLF. (b) Asan inhomogeneouswallisapproached, thegaugeboundary

condition should constrain the normal stream vector compo-nent ratherthan the tangentialcomponents.Constraining just the normal stream vector component is less restrictive than constrainingthe twotangentialcomponents.The GFdoestake thiseffectintoaccount,butLFdoesnot.

(c) In this work, we have calculated the structure tensors using boththe GFandtheLF. Comparingtheresultsshowsthatthe GFproduces simplerprofilesforthestructuretensorsthat de-scribemoreaccuratelythestructures ofturbulence.Inviewof

thissimplicity,themodelingofstructuretensorswillbeeasier undertheGF.

ItshouldbeclearthatbothLFandGFarecorrect,andthatany differencesintheresultingprofilesofthestructuretensorsare at-tributabletotheir lackofgaugeinvariance.Inlightofthisdegree offreedom,onehastochoose thegaugeconditionthatpreserves theintended meaningofthe structuretensors. Wecomment fur-theronthisinSection3.5.

3. Detailsofthepresentcomputation

DNSoffully-developed incompressible turbulent flowthrough a smooth pipe have been computed previously by Eggels et al.

[10],Loulouetal.[29],Satakeetal.[44],Wagneretal.[55], Fuka-gataandKasagi[12],Veenman[53],Wu andMoin[56]andmore recentlyby El Khouryetal. [11]. What differentiates the current work from the previous studies is the calculation of the struc-turetensors andtheir useforvisualizingthenear-wallstructures. Computingthestructuretensorsinvolvesstatisticalaveragesofthe fluctuating stream vector gradient components. In the following subsections,weprovidedetailedinformationonthenumerical as-pectsofoursimulation.Forvalidation purposes,we compareour results withthe results of Eggels etal. [10], Wu and Moin [56], andEl Khouryet al. [11]. Then, we proceed to present the pro-filesofthestructuretensorsalongthepiperadius,andextractthe physicalinformationconcerningthelarge-scale,energy-containing structuresofturbulence.Finally,wealsodemonstratehowthe in-stantaneousvalues ofthe structure tensors can be used to iden-tify inactive structures, i.e.large-scale structures without signifi-cantenergycontent.Structuresofthistypearelocatedadjacentto thewall.

3.1. Computationalframework

Foroursimulations,wehaveusedtheCDPsoftwaredeveloped attheCenterforTurbulenceResearch(Stanford,NASAAmes).CDP isanunstructured,collocated,nodal-based,finite-volumecodethat solvestheincompressibleNavier–Stokesequations.The fractional-stepmethod[24] isused tonumericallysolve thecontinuityand momentumequations.Briefly,anintermediatevelocityisobtained fromthemomentumequationbyusingthepressurefromthe pre-vious time step. A Poisson systemfor the pressure is solved us-ing the intermediate velocity. The final values of the nodal and face-normalvelocitiesareobtainedbyutilizingthenodaland face-normal pressure gradients to correct the corresponding interme-diatevalues.The finalvelocity satisfiestheincompressibility con-dition.The Crank–Nicolsontimediscretization schemeisusedfor thenodal velocity, present inthe diffusive andnon-linear terms, while the Adams-Bashforthadvancement scheme is used for the face-normalvelocityappearing inthenon-linearterm. Simple in-terpolationschemesareusedfromnodaltofacequantities.Space discretizationof diffusive andconvective termsis treatedvia the Gauss theorem and the summation-by-parts (SBP) operators as explained by Ham et al.[15]. The face-centered gradient related to the diffusive/Laplacian terms are treated via a second-order accurate centered-difference scheme. A very detailed description of the numerical techniques used by this code is reported in

[1,2,14,15,32,33,57].

3.2.Meshdetails

Our computational resources constrained the mesh size to a maximumofapproximately5milliongridpoints.Takingthis limi-tationintoaccount,wecreatedacomputationalmeshthatis suit-able forcapturing all physical phenomenataking place in a tur-bulentpipeflowatlowReynoldsnumbers.Closetothepipewall,

Table 1

Overview of numerical parameters, mean flow properties, and mesh resolution. Since the exact

Reτ of the simulation is not known a priori, during the construction of the mesh we have used

the approximate value Re τ ≈ 180 to identify the approximate viscous units.

Parameter Eggels Veenman Wu and Moin El Khoury Stylianou

Reb = u b D/ν 5300 5299 5300 5300 5300 Reτ= u τR/ν 180 181 181.37 181.05 181.34 ub / u τ 14.73 14.63 14.611 14.637 14.613 Cf = τw / (12ρub2) 9 . 22 ×10 −3 9 . 35 ×10 −3 9 . 369 ×10 −3 9 . 336 ×10 −3 9 . 366 ×10 −3 Tstats / (uRτ) 8.0 20.0 20.53 ∼60 a _98.54 Lx / R 10 10 15 25 15 Nr × N φ× N x 3,145,728 1,785,856 67,108,864 ∼18,670,0 0 0 5,064,108 Nr 96 109 256 – – Nφ 128 128 512 – – Nx 256 128 512 – 361 r_min+ 0.94 0.11 0.17 0.14 0.333 rmax+ 1.88 4.03 1.65 ∼4.44 ∼3.500 Rφ+ max 8.84 8.89 2.22 4.93 4.189 x+ _{7.00 14.10 5.31 [3.03, 9.91] 7.500} a vailable after private communication with El Khoury et al.

viscousforcesdominateandthusthemeshhastobeveryfine.The viscousscales u_ν=u_τ p_ν=

ρ

u2 τ

δ

ν=_u

ν

τ tν=

δ

ν u_τ (19)

forvelocity, pressure, length, and time respectively, are used for normalizationpurposes. Normalization with the viscous scales is symbolizedbysuperscript“+”.

OurcomputationaldomainhasastreamwiseextentofLx=15R. The length was chosen to be the same with the one used in thestudy ofWu and Moin[56]. As they note in their work, ex-perimental data suggests the existence of very large-scale mo-tions that range in length form 8R to 16R, and therefore the choice of Lx=15R is justified. The simulation was performed at bulk Reynolds number Reb=5300 by setting the following val-ues:pipe radius R=1, fluid density

ρ

=1, bulk velocity ub=1, andkinematicviscosity

ν

₌ 1

5300ubD(whereD=2Risthepipe di-ameter). InAppendix A, we explain how thespecification of the previous parametersled to theaveraged pressure gradient dpw

dx = −0.009366ρu2b

R andfrictionvelocityuτ=0.06843ub.Computational detailsofoursimulation,alongwithdetailsofprevioussimulations by other researchers atthe same Reynolds number, are summa-rizedinTable1.

The total number of computational grid points is 5,064,108. In the streamwise direction the number of points is Nx=361, andtherefore the corresponding grid resolution is



x+=7.5. In the r−

φ

plane, the number of points is N_r−φ=14,028. At the wall, the grid is structured with N_φ=270 and a grid resolution ofR



φ

max+ =4.189,



r+min=0.333.Thestructuredgridextendsfor

N_l=26 layers with increasing radial ratio of

λ

₌1_.08. Therefore, thestructuremeshstartsatthewall(r=R)andendsatadistance (measuredfromtheaxisofthepipe):

r=R−



rmin

λ

Nl− 1

λ

− 1 =0.852. (20)

Theinteriorpartofthepipe(unstructured part)consistsof trian-gularprisms.Inthisregion,themaximumradialextentofthe tri-angles isapproximately



r+_max≈ 3.5. The computational mesh in ther−

φ

plane (orequivalentlyin they− z plane) isshown in

Fig.1.

3.3.Cuspat

(

R− r

)

+_{≈ 27}

Forsomelinefiguresdisplayedinupcomingsections,the vari-ablesplottedwithrespecttotheradialdirectionexhibit acuspat around

(

R− r

)

+_{≈ 27.Thisisnotpartofthephysicsoftheflow. It}

is attributedto thetransition ofthe meshfromthe regular hex-ahedra to the triangularprisms. A comparison of the nodal dis-cretizationandcell-centered discretizationofCDPon asymmetric meshes,showsthatthenodal-basedformulation(theoneadopted in ourcomputations) isless sensitive to mesh asymmetries[15]. Allunstructuredcodesarepronetothistypeofmeshsensitivity.

3.4. Implementationaldetails

Inorder to reachthe fully-developed state asfastaspossible, wehavesettheinitialvelocityfieldto

ui=uiE+ui, (21)

whereu_iE correspondsto theDNSdataofEggelsetal.[10]. Ran-dom velocity fluctuationssatisfying the constrains of zero diver-genceandzerowallvaluewereadded,i.e.theturbulentflowwas trippedwithasolenoidaldisturbance.Onecanachievethisby tak-ingthecurlofaunitrandomvectorfield

ξ

i

u_i=



i jk

ξ

k, j ui



_r=R=0. (22)

The timestep wassetto



t=0.008R/ub.Thistime stepsatisfies the viscous stabilitylimit (VSL)andthe Courant–Friedrichs–Lewy (CFL)criterion.TheVSLgives

VSL=

ν

₍



_r t

min

)

2

=0.895, (23)

whichsatisfiesthestabilityrequirement.The realtimeCFL calcu-latedduringthesimulationisgivenbytherelation

CFL=



t

dV



_|

_u ini

|

dA

2 , (24)

whereniisthenormaltothesurfaceunit vector,andthesurface integrationdAconcerns thecontrolvolume dVofeachgrid node. ThemaximumCFL fluctuatesintime aroundCFLmax ≈ 0.6,while

thesmallestCFLfluctuatesaroundCFLmin≈ 0.2× 10−3.The

maxi-mum value oftheCFL isalsolessthan oneasthe corresponding criteriondictates.

The initial unrealistic and uncorrelated velocity field was evolvedfor30,000timesteps(equivalentto 16_{× 15}R/u_b,enough to allow a particle to travel 16 timesthrough the pipe axial di-mension atthe bulk velocity), in order to ensure that the fully-developedstateisreached. Wealsonotethat theentrancelength neededfora pipeflowtoreachthefully-developed stateisgiven bytheempiricalrelationLe≈ 1.6Re_b1/4,whichforourRebnumber gives27.3R.Thisismanytimeslessthanthe16× 15R.

The entire simulation lasted for 210,000 time steps and thus thecollectionof statisticstook placefor180,000time steps.This

F.S. Stylianou et al. / Computers and Fluids 140 (2016) 450–477 455

Fig. 1. Computational mesh (a) in the y − z plane, and (b) close up of its one quarter. Close to the wall the mesh has structured Cylindrical form, while the core of the pipe is made up of triangular prisms which form the unstructured part of the mesh.

isequivalenttoTstats=96× 15R/ub,enoughtoallowafluid parti-cletotravel96timesthroughthepipeaxialdimensionatthebulk velocity.ExpressingTstats in terms of theeddy-turnovertimeR/uτ

wegetTstats≈ 98.54R/uτ,which isconsiderably longer thanthe

re-spectivetimesusedinpreviousstudiesasindicatedinTable1.To avoidround-off errorsduringthetimeaveragingprocess,the sim-ulation wassplit into3equal partsof60,000 timesteps each.In each part,statisticswere collectedevery 20steps (i.e.3000 sam-ples).Thisapproachwasadoptedfortworeasons:(a)toavoid col-lecting correlated samples and(b) to avoid solving the time de-mandingstream vectorPoissonequationsatevery time step.The finaltimeaveragedquantitieswereobtainedfromasimpleaverage ofthe3aforementionedparts.Inadditiontoaveragingintime,the statisticalsamplewasenhancedbyaveraginginthetwo homoge-neousdirections xand

φ

.Theaveraging inthestreamwise direc-tion isstraightforward,whiletheaveraging inthecircumferential directioninvolvesinterpolationinapolarmesh.

3.5. Boundaryconditions

Theinstantaneous pressureisgivenbytherelation p= dpw

dxx+ ˜

p_,where dpw

dx isthe part that is explicitly setby the pressure gra-dientcontroller(seeAppendixA),and p˜isthepartthatissolved via thepressurePoissonequation inorderforthe incompressibil-ityconditiontobesatisfied.Duetothesymmetryoftheflowthe functional formofthemeanpressure is p

(

x,r

)

=dpw

dx x+g

(

r

)

and therefore it follows that p˜=g

(

r

)

+p

(

x,r,

φ

,t

)

. While the mean pressure p is linear in the x direction, p˜ is periodic. Thus, peri-odic boundary conditions can be assignedto the pressure atthe pipe inletand outletasdone fortheother flow variables. Atthe surfaceofthepipe wall,no-slipboundary conditions(dueto im-permeabilityandviscous forces) areapplied forthevelocity field (i.e.ui

|

r=R=0),alongwithzerowall-normalgradientforthe pres-sure(i.e. d_drp˜



r=R=0).

Forthecalculationofthestructuretensorsoneneedsthe fluc-tuating streamvector, definedvia the threePoisson equationsof

Eq.(1).Theseequationsinvolvethefluctuatingvorticity,whichis unknownsincethemeanvorticityisnotavailableapriori.Toavoid thisdifficulty,wesolvefortheinstantaneousstreamvector,which involves the instantaneous vorticity. Due to thestreamwise peri-odicity,thedomainisconsideredasmultiplyconnectedandas

ex-plained by Stylianou etal. [48] andQuartapelle [40], the proper boundaryconditionsfortheinstantaneousstreamvectorare

GF:

∂ψ

x

∂

r





r=R= 0

ψ

r

|

r=R=0

∂

(

r

ψ

_φ

)

∂

r





r=R= 0. (25)

Thissetofboundary conditionscomprisestheGeneral Framework (GF) forcomputingthestreamvector.Inthisframework,the wall-normalstreamvectorcomponentisrestricted.

Anotherpossibilityistorestrictthewall-tangentialstream vec-torcomponents LF:

ψ

x

|

r=R=! 0

∂

(

r

ψ

r

)

∂

r

|

r=R ! =0

ψ

_φ

|

r=R=! 0. (26) Thissetofboundary conditionscorrespondto theLimited Frame-work(LF) forcomputingthestreamvector.Theexclamationmark isusedtoindicatethattheseboundaryconditionsshouldbeused withcaution.As illustrated byStylianou etal.[48], these bound-aryconditionscreateastreamvectorthatdoesnotreconstructthe correctinstantaneousvelocityvector.Thereisaconstantoffset be-tweentheoriginalandthereconstructedvelocity.Itwasassumed byVartdal[52] thatthisconstantoffsetisresponsible forthe dif-ferencesbetweenthestructuretensorscalculatedviaGFandLF.As amatteroffact,theconstantshiftdoesnotaffectthevaluesofthe structuretensorcomponentssince they involveonlythe fluctuat-ingpartofthevelocity andstreamvectorfields(i.e.theconstant offsetiscanceled out).Therefore,the LFboundaryconditionscan be used for the calculation of the structure tensors, but not for themeanvelocity fieldinapipegeometry.Thedifferencesinthe structuretensorsgeneratedviatheGFandLFareduetothelackof gaugeinvarianceandshouldinfactbeexpected.Thechoiceofthe GFovertheLFmustbebasedonargumentsrelatedtothephysical contentoftheresultingtensors,asoutlinedinSection2.3.

In cylindrical coordinates, both the differential equations and theboundaryconditionsforthestreamvectorcomponentsare de-coupledfromeachother.Sinceourcomputationalsoftwareisbuilt onCartesiancoordinateswetransformtheboundaryconditionsto thesecoordinates.InCartesian coordinates,the Poissonequations forthestreamvectorcomponentsaredecoupled,whilethe bound-aryconditionsare coupled.Thisis illustrated viathe transforma-tions

ψ

φ=− sin

(

φ

)

ψ

y+cos

(

φ

)

ψ

z, (28) where cos

(

φ

)

=y/r and sin

(

φ

)

=z/r. Using a simple first-order differenceschemefortheradialderivative1_{, theradialand}

circum-ferentialboundaryconditionstakethefollowingform

GF:

ψ

W y

ψ

W z

= −1 1+r R



_−sin2

₍

_φ

₎

_sin

₍

_φ

₎

_cos

₍

_φ

₎

sin

(

φ

)

cos

(

φ

)

−cos2

₍

_φ

₎

·

ψ

I y

ψ

I z

(29) LF:

ψ

W y

ψ

W z

= 1 1+r R



_cos2

₍

_φ

₎

_sin

₍

_φ

₎

_cos

₍

_φ

₎

sin

(

φ

)

cos

(

φ

)

sin2

(

φ

)

·

ψ

I y

ψ

I z

(30) whereWstandsforthegridpointonthewall, whileIstandsfor theinternal grid point in the normaldirection. The simplicity of theabove relationisbasedontheassumptionthatthefirst inter-nalgrid pointawayfromthewallmustliealong thewall-normal direction. This is the main reason that we have not used an O-gridmesh.Eventhough



risverysmallatthewall(



r=

δ

ν/3= 1.837× 10−3_{),it shouldnever beneglected since thiswillreduce}

the

ψ

_φ boundaryconditionofEq.(25)to ∂ψ_∂rφ



_

r=R=

0_,andthe

ψ

r boundaryconditionofEq.(26)to ∂ψr

∂r





_r

=R=

0,whichareincorrect. Initially,a sequentialmethodwasadopted tosolvethestream vectorequations,withinner iterationsto ensurethatthe coupled boundaryconditionsare satisfied ateach time step.Eventhough thisprocess givesthecorrectsolution,theconvergenceofthe in-ner iterationswasfound to be slow. Motivatedby the need ofa fasterconvergencerate,a second coupledmethodwasembraced. Thismethodtreatsthelast two componentsofthestreamvector asfully-coupled,makingtheincorporationoftheboundary condi-tionsaneasytask.Anadditionaladvantageofthisapproachisthe eliminationoftheinneriterationprocess.

3.6.Validationofvelocityandpressurestatistics

Themeanvelocityprofilescaledwithinnerandoutervariables ispresented in Fig. 2. Ourresults (denoted by “S”) are in excel-lentagreementwiththeDNSdataofElKhouryetal.[11](denoted by “K”), Wu and Moin [56] (denoted by “M”) and Eggels et al.

[10](denotedby “E”).Inoursimulation,thefirst gridpointaway fromthe wallislocated at

δ

_ν/3,thereforetheviscous sublayer is well-resolved.Thedatainthisregionfollowthetheoretical linear velocitydistributionu+x =

(

R− r

)

+.

At larger distances from the wall, the “log-law” velocity dis-tributionwith“universal” constants (

κ

₌0_.41_,B₌5_.0) isnot fol-lowed.ThisistrueevenatReynoldsnumbersaboveRe_b=20,000,

which is the starting point of the existence of the overlap re-gion(where the argumentsleadingto “log-law” are valid) inthe channel flow. Even at Re_b=44,000 Wu and Moin [56] showed with their DNS data that the assumptions made by Millikan to derive thelog-law are not valid. A numberof studies referenced in the paper of Wu and Moin [56], rule out the applicability of alogarithmicscaling theory forReynolds numbers atleastup to

Reb=230,000.Onlyatthesevery highReb doesaseparation be-tweeninnerandouterscalesarise.Therefore,thelogarithmictrend ofthe data atlow Re_τ (such as the present one)should not be attributedto thelog-law.As explainedby Wu andMoin[56] the approximatelogarithmicvariationofu+_x on

(

R_{− r}

)

+ atlowRe_τ is 1 A second-order scheme for the radial derivative will not improve the overall accuracy of the method since the neighboring node distances in the streamwise and circumferential directions at the wall are one order of magnitude greater than the radial distances (see Table 1 ). Furthermore, the first grid points from the wall are already very close to the boundary, less than 0.5 δν.

dictatedbythenatureofthecurvatureofthemeanvelocity gradi-entprofile.

Themeanpressure difference p+

(

r

)

− p+

(

R

)

asa functionofr

isreported inFig.3. Theplot comparesour resulttothe dataof ElKhouryetal.[11]andWuandMoin[56].Eggelsetal.[10]did notreportthemeanpressuredifference.Closetothepipewallthe two results matcheach other, butas we move towards the pipe centerline small deviations arise. In the range r/R=[0,0.35] our results arecloser tothe results ofEl Khouryet al.[11], whilein theranger_/R=[0_.35_,0_.75] ourdataareclosertothedataofWu andMoin[56].

The Reynolds stress components are presented in Fig. 4. Our dataareinverygoodagreementwiththedataofElKhouryetal.

[11]andWu andMoin[56].The dataofEggelsetal.[10]exhibit smalldeviationsfromthedataofthe remainingcomputations,at leastthenormalcomponents.Thedatafortheshearstress compo-nentfromallcomputationscollapseatthesametrend.

Thepressurefluctuationstatistics arereportedinFig.5.Inthe near-wall region the data of El Khoury et al. [11] and Wu and Moin[56] matcheach other,while inthe outer regionsmall dif-ferencesexist. Ourdataandthe dataof Eggelsetal.[10]) inthe outerregionexhibitalsosmalldifferenceswithrespecttothedata ofWuandMoin[56],butwithreversesigninregard tothedata of El Khouryet al. [11]. The maximum value in our data is the same withthe one ofWu andMoin [56], butits radial location ismatchesbetter byElKhouryetal.[11].Inthenear-wallregion ourdataexhibitthesametrendswiththeonesofElKhouryetal.

[11] andWu andMoin [56]. The discrepancies betweenthe four computationsareattributedtothedifferencesinthedomainsize, themeshresolution,andtheorderofthenumericalschemes ap-plied.

4. Activeandinactivestructures

4.1. Terminology

Inthissection,wedefine theterms“activestructures” and “in-activestructures” andusethemtodistinguishlarge-scalestructures withhighturbulentkineticenergycontent(i.e.active)from large-scale structures with low turbulent kinetic energy (i.e. inactive). Theseshouldnot beconfusedwiththealreadyexisting terminol-ogyof“activemotions” and“inactivemotions”.Theconcept of “ac-tive motions” and “inactive motions” was advanced by Townsend

[49–51] and Bradshaw [6,7], in order to distinguish the motions that contribute tothe wall-normalvelocity fluctuations, fromthe motionsthatcontributeprimarilytothewall-parallelvelocity fluc-tuations.

4.2. Identificationcriteria

Efforts to identify and visualize near-wall structures typically focusinthe region5y+50wherelarge-scale structureswith significant turbulent kinetic energy content reside, such as the high-speed andlow-speed streaks and the associatedsweep and ejectionevents.While itistruethattheleveloftheturbulent ki-netic energy drops to zero as ones approachesthe wall, the or-ganizationofnear-wallturbulencedoesnot endaty+≈ 5. Large-scale structures with significant streamwise extent exist even in the immediate proximity to the wall and it would be both use-ful and enlightening to bring them to focus in order to under-standthem. Furthermore,it wouldbeinteresting toanalyzetheir

F.S. Stylianou et al. / Computers and Fluids 140 (2016) 450–477 457

Fig. 2. Mean axial velocity profile scaled with (a) wall units, and (b) bulk units.

Fig. 3. Normalized mean pressure difference.

interactionwiththemoreenergeticstructuresthatresideat some-whathigherdistancesfromthewall.Tovisualizetheactive struc-tures onecan useiso-surfacesof highvaluesofturbulent kinetic energy. On the other hand, a clearvisualization criterion forthe inactive structuresdoesnot exist.Here,wedevelop onesuch cri-terion.

Our treatment starts by decomposing the fluctuating stream vectorgradienttoasymmetricandanantisymmetricpart

ψ

 i, j= 1 2

(

ψ

i, j+

ψ

j,i

)



Sψ_{i j} +1 2

(

ψ

i, j−

ψ

j,i

)



ψ i j . (31)

Recalling the stream vector definition u_i=



i jk

ψ

k, j, a direct rela-tion between the fluctuatingvelocity vector u_i and the antisym-metrictensor



ψ_{i j} isevident

u_i=



i jk



ψ  k j ⇐⇒



ψ k j = 1 2



ki jui. (32)

Fig. 4. Diagonal and off-diagonal Reynolds stress components normalized with wall units.

Ontheotherhand,thesymmetrictensorSψ_{i j} doesnotdirectly con-tributetothefluctuatingvelocityvectoru_i.Inhomogeneousflows withmeanrotation

ω

i, itcan be shownthat the symmetric ten-sorSψ_{i j} isdirectly relatedtothe gradientofthe rapidfluctuating pressure, 1_ρpr

,i=Sψ



i j

ω

j+12



i jk

ω

juk.Therefore,the symmetric ten-sorSψ_{i j} canbeassumedtoaffectindirectlythefluctuatingvelocity throughthegradientofthefluctuatingrapidpressurethatappears inthemomentum transportequations.Ourcriterionforthe iden-tificationofinactivestructuresinvolvestheinvariantquantity

Qψ≡ −1 2

ψ



i, j

ψ

j,i (33)

andforthis reasonwe call it the Qψ-criterion. One can rewrite theQψ invariantintheform

Qψ=1 2

(

ψ i j



ψ i j − S ψ i j S ψ i j

)

, (34)

Fig. 5. Normalized pressure fluctuation statistics.

which now involves the symmetric Sψ_{i j} and antisymmetric



ψ_{i j} tensors.Usingthisform,itisclearthatQψ_<0wheneverthe sec-ondpartSψ_{i j}Sψ_{i j} islargerthanthefirstpart



ψ_{i j}



ψ_{i j},andtherefore thisconditionidentifies large-scalestructuresthatcontributeonly indirectlytothefluctuatingvelocityfield(i.e.inactivestructures).

Tolinkalltheabovequantitieswiththedefinitionsofthe struc-turetensors,wenotetheexactrelations

Qψ=−1 2C t kk



ψ ij



ψ  ij = 1 2R t kk Sψ_ijSψ_ij=1 2



Dt kk+Ckkt



. (35)

Sincetheaboverelationsrefertoinstantaneousquantities,wehave dropped the time average from the definitions of the structure tensors(hencethe superscriptt). Basedonthe aboverelations,it shouldbeclearthattheQψ<0criterionidentifiesregionswhere

Ct

kk>0,andhence,regionswithpositive valuesofinhomogeneity areoccupied byinactive structures. According tothis connection,

Fig.14ofSection 5canbeusedto identifyregions wherethe in-activestructuresresideonaverage.Basedonthisfigure,the near-wall andcenterline regions havea higher probability to host in-activestructures thantheregion in-betweenthetwo. Inthenear wallregion,turbulentkinetic energydropstozerowhilethefirst invariant ofdimensionalityand inhomogeneity do not.This indi-cates that in this region we can find large-scale coherent struc-turesthat havelowenergycontent. Thisiscounter tothe notion thatlarge-scalestructuresareassociatedwithhighenergycontent, alinkestablishedfromhomogeneousarguments,wherethe inho-mogeneitytensorisidenticallyzero.

WehavedevelopedourstreamvectorbasedQψ-criterionalong thelines thatthe traditionalvelocity-based Qu_{-criterion was} de-veloped[16]. The velocity-based criteria are built to identify co-herentvortexstructures[9].Suchstructuresareextractedfrom re-gionsofQu_{>0,whichreducesto}

_

u

i j



u  i j >Su  i jSu  i j andthereforeto highvaluesofthefirst invariant ofvorticity tensorWt

kk>2S u i jSu



i j. Ourcriterionidentifiesregionsof



ψ_{i j}



_{i j}ψ<S_{i j}ψSψ_{i j} corresponding toQψ <0.Theseargumentsjustifytheuseofinvertedinequality conditionsinthetwocriteria,apointthatshouldbenoted.Having thisinmind, onecan proceed to constructthe

λ

ψ₂ andthe



ψ criteria, all of which will have inverted condition symbols com-paredto the traditional

λ

u

2 and



u 

velocity criteria (for details

see [9]). Any of the new stream vector based criteria (Qψ <0,

λ

ψ2 >0,



ψ 

<0) could be used to identify inactive structures. However,weproposetheuseofQψ sinceitistheonlyonethatis expressibledirectlyintermsofthestructuretensorsandthuscan bemoreeasilycomprehended.Forthesamereason,Qψ canmore easily belinked to structure-basedturbulencemodels inorderto sensitizethemtothepresenceofnear-wallinactivestructures.This isadirectionweplantoexploreinthenearfuture.

As we will show shortly, the conditionQψ >0 is also useful inthat itcan beused toidentifylarge-scale structures withhigh turbulentkineticcontent.Inthissense,theconditionsQψ <0and

Qψ > 0provide a unified structure-based criterion forcapturing bothactiveandinactivestructures.

4.3. Visualizations

Inthissubsection,weusevisualizationcriteriatoidentify tur-bulence structures. For the identification of small-scale vortical structures we use iso-surfaces of positive values of Qu_{. For} ac-tive structureswe useiso-surfaces ofhighvaluesofturbulent ki-netic energy kt_{. Forinactive structures, we usethe newly} devel-opedcriterionbasedontheiso-surfacesofnegativevaluesofQψ. WeshowtheinactivestructuresasextractedfromboththeLFand theGF,inordertohighlighttheeffectofthegaugechoice.

Fig. 6 represents an overview of the various types of turbu-lencestructuresappearinginafully-developedturbulentpipeflow. ThepipedomainisshowninorthogonalCartesianandorthogonal Cylindricalcoordinates.Fortheconstruction oftheradial and cir-cumferential2 _{axes weuse the transformations:r=}



_y2_+z2_,

φ

₌

atan2

(

z,y

)

.Theradialandcircumferential coordinatestakevalues in the range[0, R] and

(

₋

π

_,+

π

] respectively. An artifact of the coordinate transformations is the stretching of the structures in the circumferential direction as we move fromthe wall towards thecenterofthepipe.Thepipewallisillustratedwithgraycolor. High/Low-speed streaks are shown with red/blue color. Vortical structures with right/left handsense ofrotation around the pos-itivex-axisareshownwithgray/blackcolor.Inactivestructuresare shownwithgreencolor.Forthetop/bottompartofFig.6theLF/GF has been used forthe computation of inactive structures. In the electronicsupplementarymaterialonecanfindanimationsoftime consecutive iso-surfaces of thestructures shownin Fig. 6(b)and (d).

Focusing our attention in the near region of high-speed and low-speed streaks, we see an increased vortical activity around thesestructures. Thegenerationmechanismofstreaks[23,26] de-mandsthe existenceofvorticalstructures. Counter-rotating vorti-calstructures thatdrivefluidtowardsthewall (sweepevent)will generatehigh-speedstreaks.Duetothesplattingprocessthe high-speed streaks have larger extent in the circumferential direction than in the radial direction. On the other hand, counter-rotating vortical structures that drive fluid away from the wall (ejection event) will generatelow-speed streaks. Dueto the bursting pro-cessthelowspeedstreakshavesmallerextentinthe circumferen-tialdirectionthanintheradialdirection.

Nowwe moveourattentionto theinactivestructures.Weuse boththeLFandtheGF tocomputetheinactive structures.Inthe case of LF (Fig. 6a,b), the inactive structures reside beneath the streaks as if they are the shadowof the streaks on the wall. In thecaseofGF(Fig.6c,d),theinactivestructuresareagainadjacent to thewall, butthey are located on the sidesofthe streaks.For theconstructionoftheinactivestructureswechoosenegative val-uesofQψ.OtherQψ valuesclosertozerobutstillnegative,will 2 The arctangent function with two arguments atan2(_,_) is used in order to iden- tify the appropriate quadrant of the computed angle.

F.S. Stylianou et al. / Computers and Fluids 140 (2016) 450–477 459

Fig. 6. Visualization of turbulence structures in (a,c) orthogonal Cartesian, and (b,d) orthogonal Cylindrical coordinate system. In (a,b)/(c,d) the LF/GF has been used for the computation of inactive structures. The pipe wall is illustrated with gray color. High/Low-speed streaks with positive/negative streamwise fluctuating velocity u 

x are shown with red/blue color. The streaks are visualized using iso-surfaces of turbulent kinetic energy with value k t = 3 . 20 k max . Vortical structures with positive/negative streamwise fluctuating vorticity ω

x are shown with gray/black color. The vortical structures are visualized using iso-surfaces of the Q u



-criterion with value Q u = 0 . 18 W max

kk . Inactive structures are shown with green color. The Inactive structures are visualized using iso-surfaces of the Q ψ

-criterion with value Q ψ = −1 . 78 C _max

kk . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

indicate regions around the active structures. Animation of the timeconsecutiveiso-surfacesofthestructuresrevealsthatthe in-active structures followthe high-speed andlow-speed streaks in sucha waythattheyare alwaysbeneaththem.Thisisinteresting since theinactivestructures arelocatedinregions withlower lo-calvelocitiesthanthestreaks.Thisindicatestheinterplaybetween

theactiveandinactivestructures andrevealsthenon-localnature ofthesedisturbances.

Fig.7illustratestheradiallocationswheretheactiveand inac-tive structures tend to exist. The active structures are generated in the region

(

R_{− r}

)

+_>5. High-speed streaks dominate in the region 5<

(

R− r

)

+_{<12 and low-speed streaks dominate in the}

Fig. 7. Same as in Fig. 6 but different viewpoint (vortical structures are omitted for clarity of the remaining structures). On the top/bottom figure the LF/GF has been used for the computation of inactive structures. The flow direction is towards the reader. The continuous line represents the pipe wall, while the following dashed lines are placed at (R − r)+ = 5 , 12 , 30 , 50 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Visualization of vortical/streaky structures in Cylindrical coordinate system (top/bottom). Translucency is used in order to observe all structures located at different circumferential locations. Inclination angles are shown on the structures. See also the details of Figs. 6 and 7 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

region12<

(

R− r

)

+_{<30. On the other hand,the inactive}

struc-turesare mainlylocatedclosetothewall.The top/bottompartof

Fig.7showstheinactivestructuresascomputedviatheLF/GF. In-activestructurescomputedviatheLFoccupytheregion

(

R− r

)

+_<

5_,whileinactivestructurescomputedviatheGFarelocatedinthe region

(

R− r

)

+_<12.

Representative inclination angles of streaks and vortices are showninFig.8.Itisevidentthatthestreakshavelowerinclination anglesthanthequasi-streamwisevortices.Theinclinationangleof streaks(morespecificallyofthelow-speedstreaks)arearound10o_, whilefor the quasi-streamwisevortices theinclination angle de-pendshighlyontheradiallocationofthestructures.Lateron,we willshowthat theinclinationanglesofthestructuresareingood agreementwiththerotationanglesthatplacethestructuretensors intheirprincipalaxes(seediscussionforFig.23).

Toquantifythemeanstreamwiseextentofthestreaksandthe meanseparation betweenhigh-speedandlow-speed streaks, Wu andMoin[56]usedtwo-pointcorrelationsofthestreamwise

fluc-tuatingvelocity(Figs.29,and31therein).Sincewehavenot com-puted two-point correlationsin our simulation, we use Fig. 9 to qualitativelycomparewiththedataofWuandMoin[56].The vi-sualizations and correlations of Wu and Moin [56] suggest that at Re_τ=180 turbulent pipe flow possesses large-scale, near-wall structures that arecoherentover significant axialdimensions(8R orlarger).ThisisinagreementwithourdataillustratedinFig.9.

Accordingtotheliteraturethemeanseparationbetween high-speed and low-speed streaks is about

φ

r+=50∼ 60. Close to the wall the dataof Wagner etal. [55] (Fig. 15 therein) indicate high-speedtolow-speedstreakazimuthalseparationof

φ

r+≈ 60. For channel flow at Re_τ=180, the near-wall data of Kim et al.

[25](Fig.23therein)indicatespanwiseseparationofz+≈ 55.This means that close to the wall there must be around 18 streaks (counting both high-speed and low-speed) along the azimuthal direction. Our data in Fig. 9 are in accordance to Wagner et al.

F.S. Stylianou et al. / Computers and Fluids 140 (2016) 450–477 461

Fig. 9. Contour plot of the normalized streamwise fluctuating velocity component at r = 0 . 9338 R or (R − r)+ = 12 . The range of the horizontal axis is x : [0, 15 R ] or x + : [0 , 2720] , while the range of the vertical axis is r φ: (−0 . 9338 π, +0 . 9338 π] or φr+ : _{(−532 , +532] . Red/Blue color indicates regions of high/low-speed streaks. A rough} estimation of the streamwise and circumferential extent of the streaks can be extracted from this figure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. A representative schematic of the various types of structures appearing in a turbulent pipe flow at Re b = 5300 . The wall is marked with a thick line while the rest of the lines are placed at (R − r)+ _{= 5 , 12 , 30 , 50 . The streamwise direction is} towards the reader. Two low-speed streaks and one high-speed streak are shown with blue and red color respectively. Vortical structures with right/left hand rota- tion around the positive x -axis are shown with gray/black color. Inactive structures predicted by the LF/GF are illustrated with light/dark green color. (For interpreta- tion of the references to color in this figure legend, the reader is referred to the web version of this article.)

4.4. Arepresentativeschematicofthestructures

Thefigures ofthissectionhighlightthespatialorganizationof arepresentativecollectionofstructureslocatedintheviscouswall region, andallow one to understand the interaction takingplace amongthevarioustypesofstructures.Toassistthereaderwe pro-vide intheelectronic supplementarymaterialaviewpoint anima-tionoftheschematicsofFigs.10–12.

We begin with the well-known quasi-streamwise vortices, which havebeenobserved bothexperimentally [45] and numeri-cally [25].The vorticalstructures of Fig.10 are colored basedon theirstreamwiserotationalsense;i.e.gray/blackcolorcorresponds toright/lefthandrotationaroundthestreamwisedirection.For fur-therunderstanding,Fig.11ashowsthewrappingoffluctuating ve-locitylines around the vorticalstructures, whichis inagreement withtherotationalsenseofthesestructures.Inthepipe circumfer-entialdirection,thevorticalstructuresareorganizedwith alternat-ingsenseofstreamwiserotation.Anysuchpairofcounter-rotating vortices actsasaredistributionengine,eithersendinghigh-speed fluid towards the wall, thus generating a high-speed streak, or ejecting low-speed fluid further away from the wall, thus

gener-atingalow-speedstreak.Asaresultofthesplattingevents high-speedstreaksarelocatedclosertothewall,havinghigherextend in the circumferential direction than in the radial. On the other hand,the ejection events place the low-speed streaks somewhat furtherawayfromthewall,givingthemahigherextendinthe ra-dialdirectionthaninthecircumferential.

AccordingtoFig.11btheareasdirectlyunderthestreakshave the same sign of fluctuating streamwise velocity as with the streaks. Taking into account the no-slip condition on the wall, the fluctuating vorticity vectors become tangent to the wall and adominantcircumferential fluctuatingvorticitycomponentis an-ticipatedin theareas directly underthe streaks. Furthermore,as showninFig.12b,theareasdirectlyunderahigh/low-speedstreak musthavea positive/negativecircumferentialfluctuatingvorticity. The opposite must hold for the areas directlyabove the streaks. Therefore, directly under and above the streaks, the radial com-ponentofthefluctuatingvorticity remainssmallcomparedtothe circumferential component. However, any two neighboring high-speed andlow-speed streaks tend to organize the fluid between them,generatingsignificantradialfluctuatingvorticity.Thesignof thegenerated radial fluctuatingvorticity dependson the circum-ferential arrangement of the streaks, and thus alternates in sign as one moves in the circumferential direction. Low-speed/high-speedpairsofstreaksgeneratepositiveradialfluctuatingvorticity intheirin-betweenregion,whilehigh-speed/low-speedpairs gen-eratenegative radial fluctuatingvorticity. The combinedeffect of allpreviouscommentsexplainsthewrappingoffluctuating vortic-itylinesaroundthestreaksdemonstratedinFig.12a.

We proceed with the newly defined inactive structures. In

Figs. 10–12 the inactive structures predicted by the LF/GFare il-lustrated with light/dark green color. According to the GF, the adjacent to the wall inactive structures are located in areas be-tween the streaks, where the fluctuating vorticity field is reor-ganized from a wall-tangent mode to a wall-normal orientation. Based on Fig.10 the shape ofthe GF inactive structures is lean-ing towardsthe side oflow-speed streaks.Fig. 12bshowsclearly howthe GF inactive structures act asorganizersofthe near-wall fluctuating vorticity field. In their periphery, which reaches be-lowthestreaks,thefluctuatingvorticityvectorsaretangenttothe wallandprimarilyalignedwiththecircumferentialdirection.Only