Description of the flow in a linear cascade with an upstream cavity part 2: Assessing the loss generated using an exergy formulation (draft)

an author's

https://oatao.univ-toulouse.fr/25971

https://doi.org/10.1016/j.compfluid.2019.104360

Fiore, Maxime and Gourdain, Nicolas and Boussuge, Jean-François and Lippinois, Eric Description of the flow in a

linear cascade with an upstream cavity part 2: Assessing the loss generated using an exergy formulation (draft).

(2020) Computers and Fluids, 199. 104360-104370. ISSN 0045-7930

Description

of

the

flow

in

a

linear

cascade

with

an

upstream

cavity

part

2:

Assessing

the

loss

generated

using

an

exergy

formulation

(draft)

M.

Fiore

a,1,∗

_,

_N.

_Gourdain

b

_,

_J.-F.

_Boussuge

a

_,

_E.

_lippinois

c a CERFACS, Computational Fluid Dynamics team, Toulouse, France

b ISAE-Supaero, Dpt. of Aerodynamics, Energetics and Propulsion, Toulouse, France c Safran Aircraft Engines, Moissy-Cramayel, France

Keywords: Large-Eddy simulation Turbulence injection Transition process Aerodynamic losses Exergy analysis Low-pressure turbine

a

b

s

t

r

a

c

t

Purgeairisinjectedincavitiesathubofaxialturbinestopreventhotmainstreamgasingestioninto in-terstagegaps.Thisprocessinducesadditionallossesfortheturbineduetoaninteractionbetweenpurge andmainstreamflow.Todealwiththisissue,thispaperisdevotedtothestudyofalowspeedlinear cas-cadewithanupstreamcavityataReynoldsnumberrepresentativeofalow-pressureturbineusingRANS andLESwithinletturbulenceinjection.Differentrimsealgeometriesandpurgeflowratesare studied. Detailsaboutnumericalmethodsandcomparisonwithexperimentscanbefoundinacompanionpaper. Theanalysisherefocusesonthelossgenerationbasedonthedescriptionoftheflowand influenceof theturbulenceintroduced inthecompanionpaper. Themeasure oflossisbasedonanexergy analy-sis(i.e.energyinthepurpose togeneratework)thatextendsamorecommonmeasure oflossingas turbines,entropy.Thelossanalysisisledforabaselinecasebysplittingthesimulationdomaininthe contributionsrelatedtotheboundarylayersoverthewettedsurfacesandtheremainingdomain(i.e.the complementaryofboundarylayersdomains)wheresecondaryflowsandrelatedlossarelikelytooccur. Theanalysisshowsthestrongcontributionofthebladesuctionsideboundarylayer,secondaryvortices inthe passageandwakeatthetrailingedgeonthelossgeneration.Thestudy ofdifferentpurgeflow ratesshowsincreased secondaryvorticesenergyand subsequentlossforhigherpurgeflowrates.The rimsealgeometrywithaxialoverlappingpromotesadelayeddevelopmentofsecondaryvorticesinthe passagecomparedtosimpleaxialgappromotinglowerlevelsofloss.

1. Introduction

Rotor-statorwheelspaceinlow-pressureturbinesimpliesgaps often referred to cavities under the main flow passage in which hot turbine gas could be ingested. This hot flow could impinge rotor disks leading to potential overheating anddamages forthe turbine[3]. Some relativecold air isblownfrom thecompressor to feedandseal turbinecavitiesandprevent partiallyhot-gas in-gestion fromthe main annulus[25].Part of thisaircalled purge flow blows intothe mainstreamthrough therimseal. The litera-turegivessomeinsight ontheinfluenceofcavityandpurgeflow onthelossinturbineswithgenerallyadetrimentaleffect[22].The purge flow reinforcesthe horseshoevortex process atthe blade leadingedge asstatedby Kost andNicklas[17] and more

gener-∗ _{Corresponding author.}

E-mail addresses: fiore@cerfacs.fr , maxime.fiore@cerfacs.fr (M. Fiore).

1 orcid = 0 0 0 0-0 0 03-1740-4581

allyof secondary flows in the passage.In a rotor-stator configu-rationwithcavityin-between, Reidetal.[33] reportedthe influ-enceofmixingprocessattherimsealinterfaceandperturbed ro-torsecondary flowsby purgeflow asmechanismsinducing addi-tionallossesintheturbine. However,thedirect linkbetweenthe purgeflowandlossesgeneratedinthemainannulusremains gen-erallydifficulttodrawandquantifyespecially whenusinglossin totalpressure.Theentropy/exergyapproachcanprovideadditional insightin the mechanisms ofloss. The notionof loss ingas tur-bineshasbeenwidelystudiedfromageneralandtheoreticalsight by Horlock [15]comparing a wide rangeof gas turbine architec-tureandintroducing the notion ofavailability function or equiv-alently exergy. This quantity makes possible to account forboth theenergycontainedintheflowwithtotalenthalpyandthelevel ofirreversibilityintheflowwithentropythatwillreducethe en-ergy available to generate work. This analysis is compliant with theuseofentropytoaccount fortheloss generatedinagas tur-bine that is currently widely used to assess the loss generation

Nomenclature

Latinletters

(s,c,r) localstreamcoordinates[m] (x,y,z) Cartesiancoordinates[m]

˙

m massflowrate[kg.s−1] ˙

Q internalheatsource[kg.m2.s−2_]

Cx axialchord-length[m]

h enthalpy[kg.m2.s−2_]

k turbulentkineticenergy[kg.m2.s−2]

P_k turbulentkineticenergyproduction[kg.m2_.s−3]

Tu turbulenceintensity[−] u velocity[m.s−1] E energy[kg.m2.s−2] HNGV bladeheight[m] Ma Machnumber[−] q heattransfer[kg.s−3] Re Reynoldsnumber[_−] s entropy[kg.m2.s−2.K−1] Greekletters

χ

exergy[kg.m2_.s−2]

δ

boundarylayerthickness[m]

γ

sealingflowangle,tan−1

(

uz/ux

)

[deg]

κ

artificialviscositycoefficient[_−]

λ

thermalconductivity[kg.m.s−3.K−1]

μ

dynamicviscosity[kg.m−1.s−1]



anergy[kg.m2_.s−2]

τ

viscousstresstensor[kg.m−1.s−2_]

Subscriptsandsuperscripts

•

. productionterm ¯

. Reynolds/temporallyaveragedquantity 0 referencestate

c cavity

eff effectivecontribution in/out inlet/outletcondition m mainannulus

sgs sub-gridscalecontribution tot totalquantity/contribution turb turbulentcontribution visc viscouscontribution

ingas turbines [20,21,24] aspopularized by Denton [7]. In addi-tion, this approach makes possible to account for potential heat andworktransfersbetweentheflowandthegasturbine.The en-tropy/exergyapproachmadepossibleto giveadditionalinsight in the mechanisms of loss associated to purge flow as studied by Reidet al.[33] andZlatinovet al.[42] showingthat a reduction lossescould be achievedby promoting a swirled purge flow be-fore to interact with the main annulus flow. This approach can becoupled withtheComputationalFluid Dynamics(CFD)tool to study the influence of purge flow on the loss generated in the turbine. The current design of turbines is mainly performed us-ing Reynolds Averaged Navier Stokes (RANS) approach where all scales of turbulence are modelled. However, the interaction pro-cesses betweencavityand main annulusflow are inherently un-steady[32].TherelativelylowReynoldsnumberconsideredin low-pressureturbines(Re ~ 105_{)makesmoreandmorefeasible}

wall-resolvedLargeEddySmulations(LES)[34] thatresolve the turbu-lentunsteadyflowfield. Lowerlevelsofturbulencemodellingare obtainedforLESapproachcomparedtoRANS[38,39]thatmay ex-pecta betterresolutionoftheflowfield attheinterface between cavityandmainannulusflow.Fromalossperspective,recent stud-ieshaveshownthattheassessmentoflossinagasturbinecanbe

Fig. 1. View of the experimental set up. Adapted from Schuler [36] .

obtainedmore accurately by LEScompared to RANS [18,20],this beingduetoa betterresolutionof turbulentfield thatdrives the mixing process of momentum andenthalpy whichin turns con-trolstheleveloflossesgeneratedintheflowfield[35].

Thispaper usesthe exergyformalism to drawthe loss gener-ated in a low-speed linear cascade with an upstream cavity in-cludingaparametric studyofthepurgeflow ratesuppliedtothe cavity andthe rim seal geometry. The companion papershowed thegoodbehavior ofRANSapproachto describetheflow fieldin thelinear cascadewiththefiner agreementobtainedforthe LES withinletturbulenceinjection(LESturb.inj.).Thetwosimulationsare

comparedbasedonthe exergyanalysistoprovidethedifferences betweenthetwoapproachesfromalossperspective.TheLES_turb.inj. isthen usedto drawthe exergyanalysisforabaseline case. Fol-lowingthat,theinfluenceofthepurgeflow rateandrimseal ge-ometry on the loss generation is studied in thispaper before to drawconclusions.

2. Configurationandnumericalmethods

Asummaryofthetestcaseandnumericalmethodsisprovided below. Moreinformations can be found in the companionpaper. The test caseconsidered forthisstudyis a low-Machlinear cas-cadecomposedoffivenozzleguidevanereprensentativeofa mod-ern low-pressure turbinedesign installed atKarlsruhe University, Germany (see Fig. 1). The mean Reynolds number based on ax-ial chord andvane exitvelocity is 500,000 typical ofa medium-sizedlow-pressureturbineattake-off[16].Thefree-stream turbu-lence is produced by a turbulence grid positioned at seven axial chord length upstream of the blade leading edge and was mea-sured atmidspan givinga turbulent intensity of Tu= 6% atthe blade leadingedge. Upstream of the blade leadingedge, therim seal is included ina cavity modulelinked tothe test section al-lowingto easily set differentrim seal designs. The purgeflow is supplied to the cavity as a fraction of the mainstream flow, re-spectivelym˙c/m˙m=0,0.5or1%.Mainrigcharacteristicsare

gath-eredinTab.1.Threedifferentrimsealgeometriesarestudied dur-ing the experiments with a first geometrycomposed ofan axial clearance (A) and two geometry using axial overlapping geome-tries: simple (S) and double (D) (see Fig. 2). The different cases studiedwiththedifferentgeometriesandpurgeflowratesare

de-Table 1

Characteristics of the cascade rig.

cascade details nominal conditions

Inlet blade angle 37.9 ◦ _{Re 5.6 x 10} 5

Outlet blade angle 66.3 ◦ _{Ma 0.22}

Axial chord C x 75 mm m˙ m 1.13 kg.s −1

HNGV / C x 1.3 ptot,in / p out 1.035

3

Fig. 2. Simulation domain.

noted by a letterforthe rimseal geometryconsidered (A:axial, S: simpleoverlapping,D:doubleoverlapping)andafigureforthe purge flow rate imposed (0: 0%, 05: 0.5%, 1: 1%). For example, theconfigurationA05standsfortheaxialrimsealgeometrywith 0.5% of the mainstream flow supplied in the cavity. RANS simu-lations are performedusing theCFD codeelsA [2]. Thissoftware usesacellcenteredapproachonstructuredmultiblockmeshes.An upwindRoe schemewiththird-orderlimiter is usedforthe con-vective terms[28].Diffusive fluxesare computed witha second-ordercentredscheme.TheWilcox k-

ω

two-equationsmodelwith Zheng’slimiter [41]isused.TheLES_turb.inj.isperformedusingthe in-houseunstructuredAVBPsolver[10].Theconvectiveoperatoris discretizedbythetwo-stepTaylor–Galerkinscheme[5](3rdorder accurate).TheSub-GridScalemodel(SGS)istheWall-Adapting Lo-cal Eddy-viscosity (WALE) [27].A layerof20 prisms innear-wall regions is applied with an expansion ratio of 1.03 and tetraedra fill the remaining domain. The meshes for RANS and LESturb.inj.

aredesignedtofulfilwall-resolvedrequirements[12,29,30] includ-ing agrid refinementfromtheinlet tothe bladeleadingedgeto transportturbulentstructuresgeneratedattheinlet.Thisleadtoa meshof7 _{× 10}6 _{cellsfortheRANSand80 × 10}6 _cellsforthe

LESturb.inj..

The comparison against experiments and the analysis of the flow fieldisprovided inthecompanionpaper.Asummaryofthe main mechanismsdescribedinthecompanionpaperishere pro-posedbased onthe sketchin Fig.3to give aclearerview ofthe correspondingmechanismsofloss.Duetotherelativelyhigh free-stream turbulence(6%) and forthe considered Reynoldsnumber, the boundary layer over the hub, shroud andblade is turbulent withoutseparationontheaftportionofthebladesuctionside.On the pressureside,an enclosed separationbubblebetween10 and 30% chordoccurs. Closeto therimsealinterface upstreamofthe blade,ashearlayerisinducedduetothehighvelocityofmain an-nulusflowcomparedtothecavityflow.Themainannulusflowis deviated downwardsintothecavitywhenfacingthebladedueto potential effect.Amassconservationbalance forthecavityshows that somecavityflowblows intomainstreamatthecenterofthe passagewherepressureislower(F).Thecavityflowfeedsthe ini-tiatingsecondaryvorticesinthebladepassage.Secondaryvortices areinitiatedfacetothebladebytheseparationofthehub bound-arylayerthatrollsupinvorticesknownasthehorseshoevortices. Thehorseshoevortices arechoppedbythebladeleadingedgein two main vortices:the suction side leg (A) that remain close to bladesuctionsideandapressuresidevortex(B)thattravelsinthe passagedueto thecrosspressuregradient betweentwoadjacent

Fig. 3. Main secondary vortices observed in the linear cascade: suction side (A) and pressure side leg of the horse shoe vortex (B), corner vortex(C) and passage vortex (D) including the ingestion process face to the blade due to potential effect (E) and blowing of cavity flow into the annulus at the center of the passage (F).

bladesandentrainstheflowclosetothehub andthecavityflow. Thesetwovorticesgatherclosetothebladesuctionsideand initi-atetwomainvortices:acornervortexclosetothehubandblade suction side(C) anda migratingvortex knownaspassage vortex (D).Thesameprocess ofsecondaryflow developmentcan be ob-servedattheshroudoftheconfigurationrepresentingastatorrow exceptthatnofeedingprocessofsecondaryvorticesbycavityflow isinduced.

3. Exergyanalysisinthesimulationdomain

Theexergyformulationisappliedtothelinearcascade config-uration.This generalassessment isfirst led usingthe RANS sim-ulation.TheRANS andLESturb.inj. arethen comparedwitha

split-tingbetweenmeanandturbulentcontributionintheemphasisto describethe differencesbetweenthetwo approachesfroma loss generation perspective. Once compared, the LESturb.inj. is used to

describethecontributionstolossesoftheboundarylayersoverthe wettedsurfaces(hub,shroudandblade)andintheremaining do-main(thefulldomainwherethesubdomainsrelatedtoboundary layers havebeen removed) where secondary flows are known to developandcontributetolossesintheturbine.Alltheseelements willbe usedtoobtain theregions andmechanismsinducing loss intheturbine.Thegeneraltransportequationforexergy

χ

in dif-ferentialformproposedinEq.(11)ofAppendixAcanbewrittenin integralformconsideringthewholesimulationdomainandsteady flowconditions.

Thisprovidesthefollowingbalanceequation[1,37]:







(

ρχ

)

ujnjdAIO=Pshaft+

χ

q+



_∇u+



∇T (1)

whereAIO refers tothe inlet/ outletplane ofthesimulation

do-main.Thisequation forexergy

χ

₌

(

htot− h0

)

− T0

(

s− s0

)

gives a

powerbalanceforboth totalenthalpy htotandentropy quantities

s between the inlet and outlet of the domain (left-handside of the equation).Exergy can vary dueto the work transferred with theshaft (Pshaft), powerassociated tothe heat flux(

χ

q), andthe irreversibilitygeneratedinthedomain inducedby velocity gradi-ents(



_∇u) andtemperaturegradients(



∇T).Sincethe configura-tionis static, nowork is transferred withthefluid similarly to a statorrowleadingtoPshaft =0.Noheatistransferred atthe

bor-Fig. 4. Evolution of the flow exergy from the inlet to outlet of the domain (RANS), configuration A05.

Fig. 5. Example of a simulation domain discretized in axial subvolumes V i (green).

For a simple configuration where only the hub boundary layer is considered, V i can

be split in a subvolume associated to the hub boundary layer (red) and a remaining domain that is simply the subvolume V i minus the subvolume associated to the hub

boundary layer (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

derofthedomain(sinceadiabaticwall)leadingto

χ

q=0. There-fore,theexergyvariation betweeninlet andoutletofthedomain canonlydecreaseduetoviscous



_∇u andthermalirreversibilities



∇Tintheflowfield.Incurrentstaticconfigurationwithoutheat transfer, the entropy equation is therefore sufficient to draw the exergyanalysis sincethetotal enthalpy isconserved.The viscous andthermalcontributions intheRANSformalism areobtainedby performinga volume integration ofthe velocity andtemperature gradientsfromtheinlettotheconsideredpositionx[26]:



∇u

(

x

)

=  V,0→x

τ

i j,eff T0 T

∂

ui

∂

xj dV (2)



∇T

(

x

)

=  V,0→x

(

λ

+

λ

turb

)

T0 T2



∂

T

∂

xj



2 dV (3)

where

τ

i j,eff=

(

μ

+

μ

turb

)(

∂

ui/

∂

xj+

∂

uj/

∂

xi

)

is the effective vis-cous stress tensor and ¯. the Reynolds averaged quantities as a directoutput ofthesimulation.Fig. 4showsthe exergydecrease ofthe flowalong the simulation domainwhere theenvironment (reference state) hasbeen taken at conditions (p0 = 101,325 Pa,

T0=300K).Theinlettorimsealleftcornerextendsfromx/Cx= -1to -0.26, rimseal region fromx/Cx = -0.26 to -0.04, blade do-mainfromx/Cx=0to1andbladetrailingedgetotwoaxialchord downstreamoftheblade (outlet) fromx/Cx = 1to3 (seeFig. 5). Thisfigure is obtained by subtracting the total exergyat the in-let of the domain (here normalized to one) to the contributions

Fig. 6. Evolution of the flow anergy from the inlet to outlet of the domain (RANS). Right-hand legend refers to thermal contribution and left-hand to viscous contribu- tion, configuration A05.

Fig. 7. Evolution of the flow anergy production from the inlet to outlet of the do- main (RANS). Right-hand legend refers to thermal contribution and left-hand to vis- cous contribution, configuration A05.

relatedto velocity



_∇u and temperaturegradients



∇T. The de-creaseof exergydueto thesetwo contributions isgenerally said to produceanergy, i.e.useless energyin thepurpose togenerate work.Theviscous andthermalanergyevolutionsalong the simu-lationdomainaregiveninFig.6wheretheleft-handsideabscissa corresponds to viscous term andright-hand side to the thermal one. In addition, the derivative according to the axial coordinate oftheanergythat isherereferred toanergyproductionata sta-tion x of the domain anddenoted



• is provided in Fig. 7. This quantity can be obtainedby splitting the whole domain in axial subdomainsofcharacteristiclengthdxandintegratingthevelocity andtemperaturegradient termsoverthesesubvolumesforwhich thecharacteristiclengthtendstowardszero.Forthethermal con-tribution,themainregionsofanergyproductioncorrespondtothe rimseal extentwherethecold cavityflowstartsto interactwith hotmain annulusflowanddownstreamalongtheblade.The vis-cousanergyproductionisrelativelylow fromtheinlettotherim seal,increasessharplyalong therimseal andbladeextentbefore toreturntowardsintermediatelevelsofanergyproduction down-streamoftheblade.Theviscousanergyproductionkeepsnon-zero contributiondownstreamofthebladewhilethethermal contribu-tionisalmostzeroassoonasoneaxial chorddownstreamofthe nozzleguidevane(x/Cx = 2). Thethermalcontributionisinthe orderofmagnitudeoftentimeslowerthan theviscous contribu-tion.Attheoutletofthedomain,theremaining exergyisaround 98.2 % of the exergyavailable at inlet ofthe domain (compared

5

to thereferencestate),i.e.thelinear cascadedeteriorates 1.8%by viscousandtemperaturedissipationprocesstoacceleratetheflow. Theexergyanalysisaccountsforboththecontributionsinducedby viscousandthermalmixingprocesses.Intheremainingofthis pa-per,thefocuswillbegiventotheviscouscontribution.Inthe com-panionpaper,thecomparisonagainstexperimentsshowedthatthe RANS simulation is well able to recover the flow physics dueto the strong influence of free-stream turbulence. However, several recentpapershaveshownthatadetaileddiscussionoflosses (en-tropy generation)canbebetterperformedbyLES[18,20,21,24].In addition,theSLES_turb.inj.showedafineagreementwiththe exper-iments. Therefore, the splitting between mean (laminar and tur-bulent contributions)willbe led forboth RANS andLESturb.inj. in

order to show the differencesbetweenthe two approaches from the lossgenerationsight whilethedetailedanalysisofthe losses generatedinthelinearcascadewillbeledusingtheLESturb.inj..

4. RANS/LEScomparison:viscouslosses

Section 3 provided the total production of viscous an-ergy/entropy along thesimulation domain.In aturbulent flow, it correspondstoameancontributionsometimescalledlaminar con-tributionandaturbulentone[20].Thisformercontributionisonly dueto the meanflow distortion.The second contributionis pro-vided by turbulence that under transfer of energy from large to smallscalesisdissipatedininternalenergy(heat)thatinducethe non locality betweenmean energyflow lost and equivalentheat generated at small scales. The RANS approach provides a direct splitting betweenthemeancontribution throughthe natural vis-cosity of the fluid (

μ

) and the turbulent contribution with the equivalentturbulentviscosity(

μ

turb)[26].Thetotalcontributionis

obtainedbysummingthetwocontributionsasprovidedinEq.(2):





∇u,mean



∇u,turb



=  V



μ

μ

turb



∂

ui

∂

xj +

∂

uj

∂

xi



T0 T

∂

ui

∂

xj dV. (4)

In LESsimulation,the splittingcan beobtained bytaking advan-tageoftheunsteadynatureofthemethod.Thetotalviscous irre-versibilitiesproduced



_∇u canbewrittenas[19–21,23,40]:



∇u=



∇u,mean+Pk (5) where



_∇u,meanisthemeanviscous dissipationandP_kisthe Tur-bulentKineticEnergy (TKE)productionterm. Thesedifferent con-tributionscanbeexpressedas:



∇u,mean=  V

(

μ

+

μ

SGS

)



∂

ui

∂

xj +

∂

uj

∂

xi



T0 T

∂

ui

∂

xj dV (6) Pk=  V−u  iuj

∂

ui

∂

xjdV (7)

where

μ

SGS is the equivalent sub-grid scale viscosity and¯.

rep-resents here the temporal averaging operator. For the RANS and

LES_turb.inj. simulations, the turbulent contribution is larger

com-pared to the mean one and the two approaches exhibit similar trendsoflossgenerationalongthesimulationdomain(seeFig.8). Somedifferencescanbeexhibitedbetweenthetwoapproaches:a decayofturbulentcontributioncanbeobservedfortheLESturb.inj.

fromthe inletto theblade leadingedge (betweenx/Cx = -1and

x/Cx = 0) that canbe associated tothe decayofturbulent struc-tures injected at the inlet of the domain while not observed in RANS. Attherimsealinterface,theproductionoflossisstronger for the RANS with a steep increase that may be induced by an overestimateofturbulentcontributioncomparedtotheLESturb.inj..

Downstream ofthe blade, the decrease ofturbulent contribution

forLESturb.inj.ismadeoveraloweraxialextentcomparedtoRANS

thatmaybeduetoatransportoftheturbulentwakeoveralonger distancefortheRANS.

Fig. 8. Evolution of the viscous anergy production d ∇u /dx along the simulation

including the mean and turbulent contributions for the RANS (a) and TKE produc- tion P k for LES turb.inj. simulation (b), configuration A05.

5. Boundarylayerandsecondaryflowlosses

Basedon theboundarylayer edgedetectionmethod[4,6],the boundary layer thickness can be obtained for the different wet-tedsurfaces ofthedomain (hub,shroud andblade)and the cor-responding volumes _{Vhub, Vshroud, Vblade}. The full simulation do-main lessthe boundary layer contributions provides the remain-ingdomainwithanassociatedvolumeVrem.term.Theexergy

anal-ysisisappliedtotheserestricteddomainstoobtainthe contribu-tions ofthe differentboundary layers andthe remaining domain ina similar mannerto previous studies ofthe losses ingas tur-bines[7,8,14,21].Thedifferentfigures relatedtothecontributions of the boundary layers and remaining domain are given in con-junctionwiththetotal contributionofthedomain atasame ab-scissadenoted



•_∇u,tottogivethereaderthemagnitudeofthe

con-tributiontothetotalone.Inaddition,sincetheviscousanergy pro-ductionisthesumofvelocitygradients(seeEq.(6)),themain gra-dients directions contributing to the generation of losses can be obtained.Thesegradientscanbe expandedinanycoordinate sys-temandincurrentstudythelocalstreamlinecoordinate(i.e. the Cartesian main axisis oriented withthe stream directionin any points)isused.Thewall-normalcontributions toboundarylayers oncurvedsurfaces(blade)canbe directlyobtainedandthecross contributions can be used to provide the contribution associated to secondary vortices [42]. Fig. 9 shows the viscous anergy pro-ductionrelatedtotheboundarylayersonthedifferentwetted sur-faces(hub,shroudandblade)andtheremainingterm.Theviscous

Fig. 9. Viscous anergy production for the different subdomains based on the LES turb.inj. , configuration A05.

anergyproduction restricted to particular velocity gradients that includesbothlaminarandturbulentcontributionsisalsoprovided insamefigures.Thearea underthedifferentcurvesofFig.9 pro-videsthedifferentcontributionstothetotallossesgenerated:the hub/shroud boundary layers contribute to 25% of losses, 38% for theblade boundarylayer and37%forthe remaining domain.For thehubandshroud,thecontributiontoloseesisalmostconstant alongthesimulationdomainwithanincreasealongtheblade ex-tent between x/Cx = 0 and1. This can be associated to the de-velopmentofa boundary layerover thehub andshroud that

in-Fig. 10. Anergy production at the rim seal interface in the remaining domain based on the LES turb.inj. , configuration A05.

ducevelocitygradientsclosetothewallandasaconsequence vis-cous anergy. This contribution beinginduced by the hub/shroud wall-normalgradient(

∂

us/

∂

r,seeFig.9a).Alongthebladeextent, theblade boundarylayer isa strongcontributionto losses. Simi-larlytohubandshroudboundarylayers,theselossesarerelatedto thewall-normalvelocitygradients associatedtotheblade bound-arylayer (

∂

us/

∂

c,seeFig. 9b).Wheeler etal.[40] reported simi-larloss profiles along the domain in theDNS of ahigh-pressure turbine vane with a strong contribution of the mean strain rate alongthebladeextentassociatedtothebladeboundarylayer.The increase ofthe blade boundarylayer losseson theaft portionof the blade wasalso observed by Lenganiet al.[21] in the LESof alow-pressure turbinewithpassingwakes wheretheincrease of losses wasattributed to small turbulent scales promoted by up-stream wakes.Hammer etal.[14] alsoreportedsimilar influence of the rear partof the blade andthe strong sensitivity oflosses to the boundarylayer state and/or potential boundarylayer sep-aration on the blade suction side. In the remaining domain (see Fig. 9c), several physical phenomena can be identified and con-tributetothegenerationoflosses.Fromtheinletx/Cx =-1tothe rimsealleftcornerx/Cx =-0.26,theanergyproductioncanbe as-sociated to thedissipation of turbulent structures injected atthe inlet ofthe domain.At the rimseal interface extendingbetween

x/Cx = -0.26 and x/Cx = -0.12, the anergy production is related to the axial and tangential velocity gradient in the radial direc-tion (

∂

us/

∂

r and

∂

uc/

∂

r)betweenhigh-momentum mainannulus flowandlow-momentumcavityflow(seeFig.10).Closetorimseal rightcornerbetweenx/Cx=-0.12and-0.04,theanergyproduction stronglyincreases.Thiscontributionisinducedbythevariationsof radial velocityinthetangentialdirection.Thiscorresponds tothe ingestionprocessofmainannulusflowfacetotheblade(negative radial velocity)whilethe cavityflowemerges closeto thecenter of the passage (positive radial velocity). The losses associated to theremainingdomainintheinter-bladechannelbetweenx/Cx =0 andx/Cx=1canbeattributedtosecondaryvortices:pressureand suctionsideofthehorseshoevortex,passageandcornervortices downstream the merging point ofpressure and suction side leg. Thesevortices are continuously fed by thecross flowcomponent induced bythe blade-to-bladepressuregradient. Theenergy pro-videdby thecross pressure gradient ismainly storedinto trans-verseenergytermstothemaindirectionofthevortices.The vor-tices dissipatethissecondary kineticenergy (energycontainedin the cross contributions to the stream one) under a mixing pro-cess. Inaddition,amajorcontributionto lossesofsecondary vor-tices in the passageis induced by strongvariations ofradial ve-locityaccordingto thestreamandcross component. Thepassage

7

Fig. 11. Axial cuts in the passage colored by viscous anergy production related to ∂ u r / ∂ c gradients, configuration A05.

Fig. 12. Viscous anergy production along the blade extent in the remaining domain based on the LES turb.inj. , configuration A05.

vortex migrating close to the blade suction side serves as a re-gion of blockage. In a similar manner to a nozzle, this enforces the flow inthe inter-bladechannel to migrateradially.This phe-nomenoninducesstrongvariationofradialvelocitycomponentsin thestreamandcrossdirectiondirection(

∂

ur/

∂

sand

∂

ur/

∂

c gradi-ents,seeFig.11).Theresultinginteractionbetweenthisradialflow and thepassage vortex is not purely a potential flow effect, and shearing occurs betweenthe two flow features that can be also observedincurrentconfiguration(see Fig.12betweenx/Cx = 0.6 andx/Cx =1).Thisphenomenonwasoriginallyobservedby Zlati-nov[42].Thethirdmechanismoflossassociatedtothesecondary vorticesinthepassageistheadditionalfrictiononthehub,shroud and blade suction surfaces where the secondary vortices travel (see Fig. 13). This phenomenon wasinitially observed by Denton and Pullan[8]studying the endwall sources ofloss based on an entropy formulation. Downstreamof the blade from x/Cx = 1 to

x/Cx = 3,thelossesassociated tothe remainingdomain progres-sively decay.The maincontributions tothe decaydownstreamof the trailing edge is twofold: a first contribution related to cross velocity gradients(

∂

us/

∂

c and

∂

uc/

∂

s).Thecrossvelocitygradient corresponds to theshear layerbetween thesuction andpressure side flow promoting the formation of trailing shed vortices. The contributionisstrongclosetothetrailingedgewithaquickdecay sincebecomingalmostzeroassoonasx/Cx=1.4.Asecondprocess relatedto thepassagevortex decaythatinduces entropy produc-tionduetoavariationofradialvelocity(

∂

ur/

∂

c)thatisperformed

Fig. 13. Viscous anergy production in the boundary layer at a constant distance from the wall ( y + = 30) for the hub (left) and blade (right) based on the LES _{turb.inj. ,}

configuration A05.

Fig. 14. Viscous anergy production downstream of the blade in the remaining do- main based on the LES turb.inj. , configuration A05.

over a longer distance since becoming negligible downstream of

x/Cx=1.8-2(seeFig.14).

6. Influenceofpurgeflowrateonviscousanergyproduction The influence of purgeflow rateon the anergy production is analyzed by applying the exergy analysis developed in previous Section 5to the LESturb.inj. performedat the differentpurgeflow

rates available (0, 0.5 and 1% of the main flow rate). The losses associated to the shroud boundary layer have been checked to be unmodifiedby the amountof purgeflow. Indeed,for conven-tionalpurgeflowrates(typically1%ofthemainannulusflowrate), theinfluenceofpurgeflow rateislimitedtothe firsthalfheight ofthe main annulus [36]. The main influenceof purgeflow rate is observed on the hub and remaining domain contribution (see Fig.15a and b). Thepurge flow rateincreases anergyproduction inthe bladepassagefromx/Cx = 0to x/Cx =1 anddownstream ofthe blade. As stated, the anergygenerated in the passageout of the boundary layers and downstream of the blade in the re-mainingdomainmaybeattributedtosecondary flowsdeveloping inthepassage.Thecurrentobservationsupportsafeedingprocess ofthe secondary vortices inthe passageby purge flowthat as a consequence generates more anergy. Due to more energetic

sec-Fig. 15. Influence of the purge flow rate on viscous anergy production for the hub (a) remaining domain (b) and total contributions (c) based on the LES turb.inj. , axial

rim seal.

ondaryvortices,thefrictionprocessatthehubisalsohigherwith a higherpurge flow amount. As a consequence, the total anergy producedalongthedomainincreaseswithanincreasedpurgeflow rate(see Fig. 15c) which is compliant with studies dealing with the influence of purge flow rate on the losses generated in tur-bines[11,31,33].

Fig. 16. Influence of the rim seal geometry (axial, simple and double overlapping) on viscous anergy production for the hub (a) and remaining domain (b) contribu- tions based on the LES turb.inj. , intermediate purge flow rate.

7. Influenceofrimsealgeometryonanergyproduction At the rimseal interface, the anergy productionassociated to azimuthal andaxial velocity gapbetween themain andrim seal flowislowerfortheoverlappinggeometries.Thisisduetothe lo-calrecirculationzone foroverlapping geometriesthatreducesthe shearcomparedtotheaxial geometry(see Fig.16a).The overlap-ping geometries promote an intense recirculation zone that ho-mogenizetheflow attherimseal interface.A mainconsequence is that the cavity purgeflow blows earlierat the rimseal inter-face, i.e. at a lower axial coordinate for the axial rim seal com-paredtotheoverlappinggeometries(seepositiveradialvelocityat therimsealinterfaceforaxialandsimpleoverlappinggeometryin Fig.17).Thisinducesanearlierdevelopmentandstrengtheningof thehorseshoevortexprocess,pressuresideofthehorseshoe vor-texprocessbypurgeflowforaxialgeometrycomparedto overlap-ping ones(seeFig.18wherethepurgeflow canbe trackedsince injectedatalowertemperature).Theradialmigrationofsecondary vortices along blade suction are initiatedearly for the axial rim sealgeometryandpromotesecondaryvorticeswithalarger detri-mentalimpactonlossescomparedtoaxialoverlappinggeometries. 8. Conclusion

Thispaperhasbeendevoted totheanalysisofthe losses gen-eratedina linearcascadewithanupstream cavityrepresentative ofa low-pressureturbinebased onan exergyanalysis.RANSand

9

Fig. 17. Radial cut close to the hub colored by the normalized radial velocity u r for

axial A (a) and simple overlapping S geometry (b).

Fig. 18. Radial cut close to the hub colored by temperature for axial A (a) and sim- ple overlapping geometry S (b).

LESwithinletturbulenceinjectionhavebeencomparedbasedon thisapproach: the lossesrelatedto thedecay ofturbulent struc-turesattheinletarenotobservedintheRANS.Attherimseal in-terface,thelossesrelatedtothesheararetriggeredwithahigher magnitudeinRANScomparedtotheLESwithinletturbulence in-jection and the trailing edge losses are smoothed over a larger axial extent for the RANS. The hub, shroud and blade boundary layer representaround 2/3ofthelossesgeneratedinthe cascade due to wall-normal velocity gradients. In the remaining domain that represent around 1/3 of the loss generated, several mecha-nisms ofloss havebeen identified: the decayof free-stream tur-bulence fromthe inletto blade leadingedge.At therim seal in-terface, the shear layer betweenthe cavitywith low momentum flowandthemainannulusflowwithhighmomentum(bothaxial and azimuthal) induces additional losses. Thermal anergyis also produced since the cavity flow with slightly lower temperature mixes with higher temperature main annulus flow. In the blade passage,secondaryflowsinducelossesundervariousmechanisms: a friction process on the wetted surfaces where secondary vor-ticestravel,ablockageeffectduetothepassagevortexpromoting strong crossradial velocitygradients, anda mixingprocess dissi-patingsecondarykineticenergy.Downstreamofthebladeleading edge, mainlytwo contributionsare addedto thehub andshroud boundarylayer:theturbulentvortexsheddingprocessthatis sig-nificantuntil halfchorddownstream ofthebladetrailingedge.A secondcontributionrelatedtothedecayofsecondaryvorticesover oneaxialchordlength.Thestudyofthedifferentpurgeflowrates availableshowedthatthecavitypurgeflowemerginginthemain annulusissuppliedtosecondaryvortices andmorelossesare in-curred dueto moreenergetic structures withan increasedpurge flowrate.Regardingtherimsealgeometries,theaxialoverlapping geometriespromoteadelayeddevelopmentandfeedingprocessof secondary vortices that inducelesslossesgeneratedcompared to simpleaxialgaps.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgements

TheauthorsarethankfultoSafranAicraftEnginesfortechnical supportandhavingfunded thisresearch project,toONERA for li-censingCerfacstousecode elsA(ONERA-Airbus-Safranproperty). Numerical post-processing have been performed using the free python-based Library Antares. Part of this work was performed using HPC resources from GENCI-CCRT-CINES-IDRIS (Grant 2018-A0042A06074).Experimentaldatawereobtainedwithinthe Euro-peanresearchprojectMainAnnulusGasPathInteractions(MAGPI), AST5-CT-2006-030874.

AppendixA. Energyavailableinthepurposetogeneratework (exergy)

Ingas turbineapplication, the commonquantity to deal with variationofenergyintheflow isthetotalenthalpy htot thatthat

isthe totalenergy(Etot) oftheflow includingthe energyrelated

topressureworkhtot=Etot+p/

ρ

.Atransportequationcanbe

de-rived for total enthalpy htot and entropy s that can be written

as[13]:

∂

(

ρ

htot

)

∂

t +

∂

(

ρ

htot

)

∂

xj =

∂

∂

xj

(

τ

i j uj− qj

)

+

∂

p

∂

t (8)

∂

(

ρ

s

)

∂

t +

∂

(

ρ

sui

)

∂

xi +

∂

(

qi/T

)

∂

xi =1 T

τ

i j

∂

ui

∂

xj +

λ

T2



∂

T

∂

xi



2 +Q˙ T (9)

whereQ˙ is an internal heat source.Exergy isdefined asa com-posite quantitybetweenthe totalenthalpy ofthe flowthat mea-surethetotalenergyintheflowlesstheentropycontributionthat measurethelevel ofirreversibilityintheflow, i.e.the amountof energytobesubtractedtothetotalenthalpy.Exergyisdefined rel-ativelytoadeadstatenoted0sincetheprocessesofenergy trans-ferbetweentheflowandagasturbineoccursinasurrounding en-vironment(generallytheatmosphere) thatcontainsenergy(since underpressurep0)andwithnon-zerotemperatureT0 butwithno

accesstothisenergyinrealprocesses.Exergy

χ

canbeexpressed as:

χ

=

(

htot− h0

)

− T0

(

s− s0

)

. (10)

Based on the transport equations for total enthalpy Eq. (8) and entropy Eq. (9), a transport equation can be derived for ex-ergy[9]withoutinternalheatsourcethatyieldsto:

∂

(

ρχ

)

∂

t +

∂

(

ρχ

uj

)

∂

xj =

∂

∂

xj

(

τ

i j uj−

(

1− T0 T

)

qj

)

− T0 T

τ

i j

∂

ui

∂

xj −

λ

T0 T2



∂

T

∂

xi



2 +

∂

p

∂

t. (11) References

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