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Inhomogeneity and anisotropy effects on the redistribution term in RANS modelling
Remi Manceau, Meng Wang, Dominique Laurence
To cite this version:
Remi Manceau, Meng Wang, Dominique Laurence. Inhomogeneity and anisotropy effects on the redistribution term in RANS modelling. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2001, 438, pp.307-338. �10.1017/S0022112001004451�. �hal-02990731�
Inhomogeneity and anisotropy eets on the
redistribution term in RANS modelling
By R
EMI MANCEAU 1
y, MENG WANG 2
AND DOMINIQUE LAURENCE 1;3
1
LaboratoireNationald'Hydraulique,
EletriitedeFrane,78401Chatou,Frane
2
CenterforTurbuleneResearh,BLDG.500,StanfordUniversity,Stanford,CA94305-3030,
USA
3
DepartmentofMehanialEngineering,UMIST,GeorgeBeggBuilding,SakvilleStreet,
Manhester,M601QD,UK
(Reeived??andinrevisedform??)
A hannel owDNS databaseat Re
=590is used to assessthe validity of modelling
theredistributiontermintheReynoldsstresstransportequationsbyelliptirelaxation.
Themodelassumptionsarefoundtobegloballyonsistentwiththedata. However,the
orrelationfuntion betweentheutuatingveloityand theLaplaianof thepressure
gradient, whih enters the integralequation of the redistribution term, is shown to be
anisotropi.It is elongated in thestreamwisediretion andstrongly asymmetri in the
diretionnormaltothewall,inontrasttotheisotropi,exponentialmodelrepresentation
usedintheoriginalelliptirelaxationmodel.Thisdisrepanyisthemainauseforthe
slightampliationoftheenergyredistributionintheloglayeraspreditedbytheellipti
relaxationequation.Newformulationsofthemodelareproposedinordertoorretthis
spurious behaviour, by aounting for the rapid variations of the length sale and the
asymmetrial shape of theorrelation funtion. These formulationsdo not relyon the
useofwallehoorretiontermstodamptheredistribution.Thebeliefthatthedamping
isduetothesoalled\walleho"eetisalled intoquestionthroughthepresentDNS
analysis.
1. Introdution
Inseondmomentlosures,oneofthemostimportantand diÆulttasksistomodel
the pressure gradient{veloity orrelation in the Reynolds stress transport equations.
Indeed,sinetheprodutiondoesnotneedanymodellingatthislosurelevel,partiular
attentionmust befousedonthisorrelationtermandonthedissipation. Inahannel
ow(e.g.Mansour,Kim&Moin1988)thepressuregradient{veloityorrelation,whose
eetismainlytoredistributetheenergyamongtheReynoldsstresses(henealledthe
\redistribution term"), is the only soure term in the budgets of the wall-normal and
spanwiseReynoldsstresses;itbalanestheprodutionintheshearstressbudget.
SinethepioneeringworksofChou(1945)andRotta(1951),theloalapproah,whih
algebraiallyrelatestheunlosedredistributiontermtothe Reynoldsstress anisotropy,
meanstrain,and meanvortiitytensors, hasbeenpopularinthe turbulenemodelling
ommunity.All standardmodelsarebasedonthisapproah.Theredistributiontermis
y Present address: Laboratoire d'
Etudes Aerodynamiques, UMR CNRS 6609, SP2MI,
writtenin anintegralform andsplitinto threeparts,therapid, slowandsurfaeparts.
Theslow part,whih doesnot involveanymean owquantity, ismodelledin terms of
theReynoldsstressanisotropy.Therapidpartisexpressedintermsofprodutsbetween
mean veloity gradients and a fourth-order tensor, based on the assumption that the
mean veloitygradientis loally onstant. This quasi-homogeneousapproah hasbeen
thestartingpointofalmost allseond momentlosuremodels. Inmostofthem, linear
onesin partiular,(e.g.Rotta1951;Naot,Shavit&Wolfshtein1973;Launder,Reee&
Rodi 1975) and even fully nonlinear ones (e.g. Fu, Launder &Tselepidakis 1987), the
surfae part is negleted or modelled by wall eho terms, as suggested by Gibson &
Launder(1978).Inothers, theinueneofthesolid boundaryis aountedforthrough
variable oeÆients, leadingto quasi-linear models, suh asthat of Speziale, Sarkar&
Gatski(1991).InthereentmodelofCraft&Launder(1996),thenonlinearformulation
diretlyinludes wallinduedeets.
However,thevalidityofthequasi-homogeneousapproximationusedfortherapidpart
is questionable.It assumesthat themean veloitygradient variessuÆiently slowlyto
allowittobetakenoutsidetheintegral,whihisnottheaseinstronglyinhomogeneous
turbulene.Bradshaw,Mansour&Piomelli(1987)usedthehannelowDNSofMansour
etal.(1988)toshowthatthishypothesisisorretdowntoy +
=40,buttotallyinvalid
belowthisvalue.Anotherweaknessofthequasi-homogeneousapproahisthelossofthe
non-loalharateroftheredistributionterm.Theintegralequationforthelatter,whih
involvestwo-pointorrelationsbetweenveloitiesandLaplaianofthepressuregradient,
showsthatitatuallydependsonthemeanowandtheturbulenestateatallpointsof
thedomain. Kim (1989)showedthat in ahannel, exeptin theverynear-wallregion,
theredistributionterm takesontributionsfrom allthedomain, inludingtheopposite
wall.Furthermore,anumberoftheoretialstudies(e.g.Hunt&Graham1978)aswellas
diretnumerialsimulations(Perot&Moin1993)showedthatthestruturesoftheow,
andtheassoiatedlengthsales,arestronglyaetedbythepreseneofasolidboundary
evenin theabseneofmean shear,beauseoftheblokingeet whih isnon-loal.In
partiular,thetwo-pointorrelationsofthewall-normalveloityare,asshownbyHunt
etal.(1989),inuenednearthewallbytheimageeddies.Thesenon-loaleets make
theredistributiontermdiÆult,ifnotimpossible,tomodelin termsofloal variables.
Furthermore,thequasi-homogeneousmodelsannotingeneralbeintegrateddownto
solidboundarieswithoutintroduing orretions,suhasdampingfuntions (there are
exeptions,suhastheCraft&Launder1996model).Dampingfuntionsarenotuniver-
sal,sinetheyarederivedbyttingexperimental orDNSresultswith littletheoretial
justiation.
In order to avoid suh problems, Durbin (1991;1993) introdued a novel approah.
Heproposedto model diretlythetwo-pointorrelationin theintegralequation ofthe
redistribution term, using an isotropi, exponential funtion.A onvolution produt is
obtained,whih an be inverted to givethe so-alledellipti relaxation approah. The
redistribution term is no longer given by an algebrai relation, but rather by a dif-
ferential equation. The non-loal harater is preserved through the ellipti operator
(1 L 2
r 2
),andthemodelanbeintegrateddowntothewall.Anotablefeatureofthis
approahisthat thesouretermof theelliptirelaxationequationanbegivenbyany
quasi-homogeneousmodel.Hene,itenablesthederivationofmodelsvaliddowntosolid
boundaries, from the quasi-homogeneous models ited above, whih have been tested
overawiderangeof dierentows.Eventhoughsomeintuitiveassumptionshavebeen
made,Durbin's modelisbasedonatheoretialapproah, leadingto thehopethatit is
somewhatuniversal,unlikedampingfuntions.
to thev 2
{f (ork " v 2
)model, whihis aversionofthe full Reynoldsstressmodel
reduedtothreetransportequations.Suessfulpreditionsinlude,butarenotlimited
to, ows with adverse pressure gradient and around blu bodies (Durbin 1995), three
dimensional boundary layers (Parneix, Durbin & Behnia 1998), aerodynamis (Lien,
Durbin&Parneix1997),andheat transfer(Behnia, Parneix&Durbin 1998;Maneau,
Parneix&Laurene 2000).
Despitetheremarkablesuess,roomsforimprovingtheellipti relaxationmodelex-
ist. Many of the underlying model assumptions, introdued intuitively, havenot been
validatedbyeither experimentsorDNS.Theobjetiveofthepresentstudyis toevalu-
atetheseassumptionsthroughtheanalysisofahannelowDNSdatabase,andtond
waystoimprovethetheoretialbasisandperformaneofthemodel.Themainissuesto
be examined inlude thevalidity of the two-pointorrelation approximation employed
byDurbin (1991),thevalidityof thelengthsaleused in theelliptioperator,and the
unsatisfatorybehaviourofthemodelinthelogarithmilayer.Afullexplanationofthese
issuesisgivenin x2and x3.Inx4,theresultsoftheDNSanalysis aredisussed.It is
foundthat theelliptirelaxationmodel isgloballyonsistentwiththesimulationdata,
andthattheorrelationlengthsaleisadequatelymodelledbytheturbulentlengthsale
boundednearthewallbytheKolmogorovlengthsale.However,theorrelationfuntion
betweenthe utuatingveloityand theLaplaianof the pressuregradientis strongly
anisotropiandinhomogeneous.Itsapproximationbyanisotropi,exponentialfuntion
isresponsibleforthespuriousampliationoftheenergyredistributionintheloglayer,
aspreditedbythe model. Itis further disoveredthat the so alled\wall eho" eet
inreases the redistribution of energy, ontrary to the general belief. The physial in-
sightsgainedthroughtheDNSstudyareused,inx5,todevelopnewformulationsofthe
modelthatretifytheerroneouslogarithmi-layerbehaviour.Thisisahievedbytaking
intoaounttheinuene ofstronginhomogeneityandanisotropyontheredistribution
term,usingaspatiallyvariablelengthsaleandanasymmetrimodeloftheorrelation
funtion.Unlike somepreviousad ho formulations,thenew formulationsemphasizea
systemati,sientiapproahtoturbulenemodelling,guidedbytheDNSdata.Finally,
x6summarizesthemajorndingsandaomplishmentsofthiswork.
2. Theoretial bakground
2.1. Integral equationof the redistributionterm
Thepressuregradient{veloityorrelationenteringthe Reynoldsstresstransportequa-
tionsis
ij
= u
i p
x
j u
j p
x
i
; (2.1)
where is the density, pis theutuating pressureand u
i
are theutuatingveloity
omponents.Theoverlineindiatesensembleaverage.Traditionally,thistermissplitinto
pressure{strainorrelation and pressure diusion. However, sinethis splittingis non-
unique(Lumley1975)and inonsistentwiththeNavier{Stokesequationsinthelimitof
two-dimensionalturbulene (Speziale 1985),it appearsmore appropriateto model the
pressuregradient{veloityorrelationasawhole.Sinethepressurediusionisnegligible
in themain part ofthe ow,
ij
anberegardedasthe energyredistribution between
theomponentsoftheReynoldsstress,exeptinthenear-wallregion,whereitbalanes
thedierenebetweendissipationandmoleulardiusion.
fromthedivergeneoftheutuatingpartoftheNavier{Stokesequations,
r 2
p
x
k
=
x
k
2 U
i
x
j u
j
x
i +
u
i
x
j u
j
x
i u
i
x
j u
j
x
i
: (2.2)
FollowingKim(1989),itwillbeassumedthattheontributionfromtheinhomogeneous
boundary ondition, or the \Stokes part", is negligible. Aordingly, p=x
k
approxi-
matelysatisesahomogeneousNeumannboundaryondition.
UsingtheGreenfuntionG
ofthedomain,thesolutionof (2.2)takestheform
p
x
k (x)=
Z
r
2 p
x
k (x
0
)G
(x;x
0
)dV(x 0
); (2.3)
where x and x 0
denote position vetors,and dV the elementary volume. The integral
equationoftheredistributiontermanbederivedfrom (2.1)and (2.3):
ij (x)=
Z
ij
(x;x 0
)G
(x;x
0
)dV(x 0
); (2.4)
where
ij (x;x
0
)denotesthetwo-pointorrelationbetweentheveloityandtheLaplaian
ofthepressuregradient:
ij (x;x
0
)= u
i (x)r
2 p
x
j (x
0
) u
j (x)r
2 p
x
i (x
0
): (2.5)
2.2. Theellipti relaxation equation
In(2.4),thetwo-pointorrelationsbetweentheveloityandtheLaplaianofthepressure
gradientneedto bemodelled. Durbin(1991)dened aorrelationfuntion
ij (x;x
0
)=
ij (x
0
;x 0
)f(x;x 0
); (2.6)
andmodelled itby
f(x;x 0
)=exp
r
L
; (2.7)
where r=kx 0
xk and L is the orrelation length sale. This approximation is the
orner-stoneoftheelliptirelaxationmodelandthevalidityof(2.7)isthemainonern
ofthispaper.
Inafreespae,usingthemodel(2.7),theredistributiontermanbewrittenas
ij (x)=
Z
ij
(x 0
;x 0
) exp
r
L
4r
| {z }
E(r)
dV(x 0
): (2.8)
Inthis form,
ij
appearsasaonvolutionprodutbetween
ij
and E(r),whihis the
free-spaeGreen funtion assoiatedwith the operator r 2
+1=L 2
. Due to (2.6), the
one-pointorrelationintheintegrandisexpressedasafuntionofx 0
.Ifitwereexpressed
asafuntionofx,theone-pointorrelationouldhavebeentakenoutsidetheintegralin
(2.8),andthenon-loaleetwouldhavebeenlostorentirelyreastintof(x;x 0
),whih
would then bemore diÆultto model. Theonvolutionintegral(2.8)an be inverted,
yieldingtheelliptirelaxationequation:
ij L
2
r 2
ij
= L
2
(u
i r
2 p
x
j +u
j r
2 p
x
i
): (2.9)
ishes. Therefore, Durbin (1991) proposed to replae the right hand side by any quasi-
homogeneousmodel h
ij
,whihleadsto themodel
ij L
2
r 2
ij
= h
ij
: (2.10)
Thismethodprovidesasimplewayofextendingquasi-homogeneousmodelsdowntosolid
boundaries,whenappropriateboundaryonditionsfor
ij
areapplied(Durbin1993).
3. Presentation of the DNS assessment
3.1. Issuestoexamine
Theelliptirelaxationapproahismainly basedontheassumptionthattheorrelation
funtion f(x;x 0
), dened by (2.6), anbe modelled by an exponentialfuntion. This
approximationwasintroduedbyDurbin(1991)onanintuitivebasis,inordertopreserve
the non-loal eet on the redistribution term. However, its validity has never been
hekedbefore,andtheshapeof
ij (x;x
0
)needstobeinvestigated.TheDNSdatabase
of thehannel ow at Re
=590 (Moser,Kim & Mansour, 1999)will be used for this
purpose.
Another aim of this work is to evaluate the orrelation length sale involved in the
model(2.7)fortheorrelationfuntion f(x;x 0
).Iftheturbulentlengthsalewereused
in the whole ow, sineit goesto zero at solid boundaries, the elliptioperator L 2
r 2
wouldvanishatthewall,introduingasingularityinthedierentialequation.Therefore,
Durbin (1991) proposed using the standardturbulentlength sale in the main part of
theow,andtheKolmogorovlengthsaleintheviinityofthewall,i.e.,
L=C
L
max C
3=4
"
1=4
; k
3=2
"
!
: (3.1)
Itis ofinteresttoevaluatepreisely theorrelationlengthsalefrom theDNSdata,in
ordertoassessthevalidityof(3.1).
Theultimateobjetiveofthis workis tondwaysto improvethemodel. Aspointed
outbyWizman etal. (1996),the elliptioperator doesnotbehaveentirelyorretlyin
thelogarithmilayer.Suppose,forinstane,thattheIsotropisationofProdutionmodel
(Naot,Shavit&Wolfshtein1973;Launder, Reee&Rodi 1975),denoted heneforthas
IPmodel,andtheRotta(1951)modelareusedastherapidandslowpartsofthesoure
term h
ij
in(2.10). Theredistributiontermisthengivenby
ij L
2
r 2
ij
= C
1
"
k
u
i u
j 2
3 kÆ
ij
C
2
P
ij 2
3 PÆ
ij
; (3.2)
whereP
ij
= u
i u
k U
j
=x
k u
j u
k U
i
=x
k
andP= 1
2 P
ii
.Inthelogarithmilayer,the
Reynoldsstresses areonstant, andthe prodution and thedissipation behave asy 1
.
Thus,therighthandsidein(3.2)behavesasy 1
,andtheredistributiontermisthengiven
by
ij
1:51 h
ij
.Thisresultshowsthattheelliptioperatorleadstoanampliationof
theredistribution.Notethatthesameampliationourswithanymodelfor h
ij .
Theoverestimationoftheenergyredistribution bytheRotta&IPmodel inthe log-
arithmilayerhasled anumberof modellerstointroduewalleho typeterms,follow-
ingGibson &Launder (1978).It would bedesirablefor theelliptirelaxationequation
toompensateforthisshortoming.Somemodels,suhastheSpeziale,Sarkar&Gatski
(1991)model, orCraft&Launder (1996)model, orretlyreproduetheredistribution
in thelogarithmilayer.In thisase,it would bepreferablethat theellipti relaxation
Basedontheaboveonsiderations,Wizman etal. (1996)proposedtwonewformula-
tionsoftheellipti relaxationequation.First,theyintroduedaneutralformulationby
takingL 2
in(2.10)inside theLaplaianoperator:
ij r
2
L 2
ij
= h
ij
: (3.3)
Seondly, formodelsthatoverestimatetheredistribution,theyproposed
ij L
2
r
1
L 2
r L 2
ij
= h
ij
; (3.4)
whihexhibitstheexpeteddamping.Laurene&Durbin(1994)andDurbin&Laurene
(1996)suggestedtwootherneutralformulations,givenby
ij
r L 2
r
ij
= h
ij
; (3.5)
and
ij Lr
2
L
ij
= h
ij
: (3.6)
These newformulations havebeenderivedempirially and suer form a lakof jus-
tiations, as emphasizedby the authors themselves. This work aims, througha DNS
analysis,toprovideamoresolidbasisforderivingsuhmodiationstothemodel.The
entral idea is that the orrelation funtion f(x;x 0
) annot be represented by a sim-
pleexponentialfuntion,ontraryto whatwasassumedby Durbin (1991).Indeed,the
preseneof the wall indues ablokingeet, leadingto notonly anelongationof the
turbulent strutures,but alsoan asymmetryin thediretion normalto thewall. Flu-
tuatingquantitiesareorrelatedoverashorterdistaneinthediretiontowardthewall
thanawayfromit.There isplentyof experimentalevidene (Hanjali&Launder 1972;
Sabot 1976)ofthis featurein two-pointorrelationsbetweenomponentsoftheutu-
ating veloity, and onean reasonablydedue that the two-pointorrelations between
theutuating veloity and theLaplaianof the pressuregradientbehavein a similar
manner. The use of the symmetrial orrelation funtion (2.7) leads to overweighting
the region between the point and the wall, whih may be the reason for the spurious
behaviouroftheellipti relaxationequation in thelogarithmilayer.Thisissue willbe
explored in thepresentDNS analysis,in orderto understand howmodiationsto the
elliptirelaxationmodel, suhasthose proposed byWizman et al.(1996),Laurene &
Durbin(1994)andDurbin&Laurene(1996),anbederived.
3.2. Channelow databaseandpost-proessing
Theorrelationfuntionf(x;x 0
)involvestheLaplaianofthepressuregradient,whih
ontainsthree spatial derivatives. Therefore,a very aurateDNS database is needed.
ThehannelowsimulationatRe
=590performedbyMoseret al.(1999)washosen
beauseofitsnumerialauray,thelargenumberofavailablestatistial samples,and
therelativelyhighReynoldsnumber.Thisowwasomputedonagridof384257384
pointsin streamwise(x ),wall-normal(y)andspanwise(z)diretions,respetively.The
omputationaldomainis2h,2handhinx,yandz,wherehdenotesthehannelhalf-
width.Thesimulationodeisbasedonaspetralmethodforspatialderivatives(Fourier
seriesinxandz,andChebyhevpolynomialsiny),andasemi-impliitshemefortime
integration. For statistial averaging, atotal of75 elds (restart les) are available,in
additiontothespatialaveraginginx- andz-diretions.
Thetwo-pointorrelationsbetweentheutuatingveloitiesandtheLaplaianofthe
0
