Relation between enstrophy production and geometry near the turbulent/non-turbulent interface in free shear flows

RELATIONBETWEENENSTROPHYPRODUCTIONANDGEOMETRYNEARTHE

TURBULENT/NON-TURBULENTINTERFACEINFREESHEARFLOWS

TomoakiWatanabe

1

,CarlosB.daSilva

2

,YasuhikoSakai

1

,&KoujiNagata

1

1

DepartmentofMe hani alS ien eandEngineering,NagoyaUniversity,Nagoya,Japan

2

LAETA,IDMEC,InstitutoSuperiorTé ni o,Universidade deLisboa,Lisboa,Portugal

Abstra t Inmanyfreeshearows,su hasmixinglayers,wakesandjetsexhibitasharpturbulent/non-turbulentinterfa e(TNTI)

separatingregionsofturbulentandnon-turbulentorpotentialow.Inthepresentworkthedependen eofenstrophyprodu tiononthe

interfa egeometryneartheTNTIisinvestigatedbyusingdire tnumeri alsimulations(DNS)ofashearfreeturbulen e(SFT)anda

temporallydevelopingplanarjet(PJET).ItisshownthatthegeometryoftheTNTIhasimpa tsontheme hanismgoverningenstrophy

dynami swithintheinterfa elayeritself.Inparti ularitisshownthatenstrophyprodu tionwithintheturbulentsublayerisprimarily

asso iatedwitha onvexshapeoftheinterfa eboththeSFTandPJET.

INTRODUCTION

Turbulent/nonturbulentinterfa es(TNTI)separateturbulentandnonturbulentregionsinmanyfreeshearowssu has

wakes, jetsandmixinglayers[1℄. TheTNTIisalayerwithnitethi kness,whi h onsistsoftheturbulentsublayer

(TSL)andthevis oussuperlayer(VSL):intheVSLvorti ityistransportedfromtheturbulentintotheirrotationalregion

byvis ousdiffusion,whereasinvis idvortexstret hinggovernstheenstrophygenerationwithintheTSL.TheTNTIhas

a onvolutedsurfa e,andits onvolutionsarerelatedtolargevorti itystru turesfromtheturbulentregionunderneath.

Inthisstudy,dire tnumeri alsimulations(DNS)ofashear-freeturbulen e(SFT)andatemporallydevelopingplanar

jet(PJET) areperformedforinvestigatingthe relationshipbetweenthe interfa egeometryandthe enstrophy

produ -tionwithintheTSL.Weusethemean urvatureofthevorti itymagnitudeiso-surfa esfor hara terizingtheinterfa e

geometry.

DNSOFSHEARFREETURBULENCEANDPLANARTURBULENTJETS

TheTNTI isinvestigatedinshearfree turbulen eSFT[3℄ andPJET [2℄by usingDNS,whi hhavebeendetailedin

previousworks[2,3℄,andthereforeonlyashortdes riptionisgivenhere.IntheSFT,the omputationaldomain,whose

sizeis

2π × 2π × 2π

,uses

(512 × 512 × 512)

ollo ationpointsandtheinitialeld onsistsofhomogeneousisotropi turbulen e(

|y| ≤ 0.7π

)andthenon-turbulentowhaszerovelo ity(

0.7π ≤ |y| ≤ π

). Theturbulentowspreadsinto thenon-turbulentowwithtimeintheabsen eofmeanshear.TheTaylorReynoldsnumberis

Re

λ

= u

rms

λ

x

/ν = 100

atthe enteroftheturbulent oreregion(

u

rms

: rmsvelo ity,

λ

x

: Taylormi ros ale,and

ν

: kinemati vis osity). The PJETis arriedoutina omputationaldomainwithsizes

L

x

= 7.5h

,

L

y

= 10h

and

L

z

= 7.5h

inthestreamwise(

x

), lateral(

y

)andspanwise(

z

)dire tions,respe tively,where

h

istheinletnozzlewidthofthejet,andthenumberofgrid pointsis

(N

x

× N

y

× N

z

) = (768 × 1, 024 × 768)

. TheReynoldsnumberis

Re = U

1

h/ν = 6, 000

(

U

1

: initialmean streamwisevelo ityatthejet enterline

y = 0

).TheTaylorReynoldsnumberis

Re

λ

= 98

onthejet enterline.

RESULTS

Anisosurfa eofxedvorti itymagnitude

|ω| = ω

th

isusedforinvestigatingtheTNTI,wherethedete tionthreshold

ω

th

isdeterminedfromtherelationshipbetween

ω

th

andtheturbulentvolume[1℄sothattheiso-surfa eof

|ω|

isatthe outeredgeoftheTNTIlayer.Wedenethisparti ulariso-surfa eastheirrotationalboundaryand onditionalstatisti s

are al ulatedinrespe ttothedistan e(denoted

y

I

)tothisparti ularboundaryasrepresentedinFig.1(a).

Weinvestigatethelarge-s alegeometryoftheTNTIbyusingthelarge-s alemean urvaturedenedby

H ≡ −∇ · n

LS

, where

n

LS

≡ −∇ω

2

_/|∇ω

2

_|

and

ω

2

isalow-pass-lteredvalueof

ω

2

usedtoremovesmalls alefeaturefromtheTNTI.

Weusedatop-hatlterwithalterwidthof

0.5λ

,where

λ = (λ

x

+ λ

y

+ λ

z

)/3

at

y = 0

.A on aveshapeisrepresented bypositive

H

whilenegative

H

representsa onvexshape(Fig.1(a)). Figure1(b)showstheirrotationalboundaryin thePJETwiththe olorsrepresenting

H/λ

. Figure1( )showstheprobabilitydensityfun tion(pdf)of

H

inSFTand PJET.Inbothows,apeakofthepdfappearsfor

H < 0

( onvex)whilethepdfispositivelyskewedinagreementwith [4℄. Figure1(b)alsoshowsthatalargepartofsurfa eareahasslightlynegativevaluesof

H

althoughlargepositive

H

appearsonthinlines. Figure2showsthe onditionaljointpdfof

H

andenstrophyprodu tion

P

ω

= ω

i

S

ij

ω

j

,(

S

ij

,is thestrain-ratetensor)withintheTSLat

y

I

/λ = −0.25

,forSFTandPJET.Positiveandnegativeenstrophyprodu tion tendstobemoreimportantfor onvexshapes(

H < 0

),although

P

ω

anbelo allyimportantalsofora on aveshapes (

H > 0

)provided

H

issmall.Moreover,

P

ω

is losetozeroforverylargenegative/positive

H

,whi hareoftenobserved

H

λ

3.0

0

-3.0

-L

x

/4 x

L

x

/4

(c)

(b)

-L

z

/4

z

L

z

/4

Turbulent fluid

Isosurface of

y

I

Interface normal direction

Concave (H > 0)

Convex (H < 0)

(a)

-5

0

5

10

10

-4

10

-3

10

-2

10

-1

10

0

Hλ

p

d

f

SFT

PJET

Figure1. (a)Denitionofthedistan efromtheirrotationalboundary

y

I

exhibitingregionsofboth on ave(

H > 0

)and onvex

(

H < 0

)shape. (b)VisualizationoftheirrotationalboundaryinthePJET.The oloroftheirrotationalboundaryshowslarge-s ale

mean urvature

H ≡ −∇ · n

LS

normalizedbytheTaylormi ros ale

λ = (λ

x

+ λ

y

+ λ

z

)/3

at

y = 0

. ( )Pdfoflarge-s alemean urvature

H

oftheirrotationalboundaryinSFTandPJET.

-4

-2

0

2

4

6

8

10

-40

-20

0

20

40

60

80

-4

-2

0

2

4

6

8

10

-4.0x10

5

-2.0x10

5

0.0

2.0x10

5

4.0x10

5

6.0x10

5

8.0x10

5

P

ω

H

λ

P

ω

H

λ

Joint pdf

Joint pdf

(a)

(b)

Figure2.Conditionaljointpdfoflarge-s alemean urvature

H

andenstrophyprodu tion

P

ω

= ω

i

S

ij

ω

j

withintheTSLat

y

I

/λ =

−

0.25

intheSFT(a)andPJET(b).

betweenthelarge-s alevorti alstru turesasinFig.1(b).Itisnoteworthythattheshapeofthesepdfsisverysimilarfor

theSFTandPJETsuggestingthatthemeanshearandlarge-s aleinhomogeneitiesdonotaffe ttheenstrophyprodu tion

me hanism.Futureworkwillfo usonthedetailanalysisontheeffe tsoftheenstrophygoverningme hanisma rossthe

entireTNTIlayer,andontheeffe tsoffreestreamturbulen eontheseme hanisms.

CONCLUSION

Therelationshipbetweentheenstrophyprodu tionandthe mean urvatureofthe irrotationalboundaryisinvestigated

insidetheTSLwithintheTNTIlayerinshearfreeturbulen eandturbulentplanarjets. Thelarge-s alemean urvature

showsthatalargepartoftheinterfa eareais hara terizedbya onvexshape,whilea on aveshapeappearsonlyon

small(thin)lines. Itisshownthattheenstrophyprodu tionislargerinregionswitha onvexshape,andtheobserved

dependen eofenstrophyprodu tiononthelarge-s alemean urvatureisverysimilarbothinSFTandPJET.

Referen es

[1℄C.B.daSilva,J.C.R.Hunt,I.Eames,andJ.Westerweel. Interfa iallayersbetweenregionsofdifferentturbulen eintensity. Annu.Rev.Fluid

Me h.,46:567–590,2014.

[2℄C.B.daSilvaandJ.C.F.Pereira.Invariantsofthevelo ity-gradient,rate-of-strain,andrate-of-rotationtensorsa rosstheturbulent/nonturbulent

interfa einjets.Phys.Fluids,20(5):055101,2008.

[3℄M.A.C.TeixeiraandC.B.daSilva.Turbulen edynami snearaturbulent/non-turbulentinterfa e.J.FluidMe h.,695:257–287,2012.