Computational Analysis of the Indirect Combustion Noise Generation Mechanism in a Nozzle Guided Vane in Transonic Operating Conditions

Ceci, Alessandro and Gojon, Romain and Mihaescu, Mihai Computational Analysis of the Indirect Combustion Noise

Generation Mechanism in a Nozzle Guided Vane in Transonic Operating Conditions. (2021) Journal of Sound and

Vibration, 496. 115851-115870. ISSN 0022-460X

ContentslistsavailableatScienceDirect

Journal

of

Sound

and

Vibration

journalhomepage:www.elsevier.com/locate/jsv

Computational

analysis

of

the

indirect

combustion

noise

generation

mechanism

in

a

nozzle

guided

vane

in

transonic

operating

conditions

Alessandro

Ceci

a,∗

_,

_Romain

_Gojon

a,b

_,

_Mihai

_Mihaescu

a

a KTH Royal Institute of Technology, Competence Center for Gas Exchange (CCGEx) Sweden b ISAE-SUPAERO, Université de Toulouse, France

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 29 March 2020 Revised 27 October 2020 Accepted 15 November 2020 Available online 16 November 2020

Keywords:

LES

Indirect combustion noise Aeroacoustic

Entropy noise

Transonic nozzle guide vanes

a

b

s

t

r

a

c

t

Thecombustion noiseinmodernenginesismainlyoriginatingfromtwotypesof mech-anisms. First, chemical reactions inthe combustion chamberleadstoan unsteadyheat releasewhichisresponsibleofthedirectcombustionnoise.Second,hotandcoldblobsof air comingfromthecombustion chamberareadvectedandaccelerated throughturbine stages, givingriseto entropynoise (or indirectcombustion noise).Inthe presentwork, numerical characterizationofindirect combustionnoise ofaNozzleGuide Vanepassage wasassessedusingthree-dimensionalLargeEddySimulations.Thepresentworkoffersan overviewtotheanalytical,computationalandexperimentalstudiesofthetopic.Numerical simulationsareconductedtoreproducetheeffectsofincomingplanarentropywavesfrom thecombustion chamberandtocharacterizethegeneratedacousticpower.Thedynamic featuresoftheflowareaddressedbythemeansoffrequencydomainandmodalanalyses techniquessuchasFourierDecompositionandProperOrthogonalDecomposition.Finally, thepredicted entropynoisefromnumericalcalculationsiscomparedwiththeanalytical results ofan actuator diskmodel for astatorstage. Thepresent paperprovesthat the generatedindirectcombustionnoisecanbesignificantfortransonicoperatingconditions. Thebladeacousticresponseischaracterizedbytheexcitationofalatentdynamicsatthe forcingfrequency ofthe planarentropywaves,and it increasesas theamplitudeofthe incomingdisturbancesincreases.

© 2020TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/)

1. Introduction

IndirectcombustionnoisewasearlyinvestigatedintheworksofCandel[1]andMarbleandCandel[2].Theyproposedan analyticalmodelfortheconvectionofnon-uniformtemperatureregionsthroughanozzleinseveralconfigurations:subsonic, supersonicandsupersonicwithshocksinthedivergentsegment.Comparedtotherealthree-dimensionalflowphenomenon, the model assumes quasi-one-dimensional inviscid flow and compactness of the nozzle (i.e. the length of the incoming

∗_{Corresponding author.}

E-mail addresses: aceci@kth.se (A. Ceci), romain.gojon@isae-supaero.fr (R. Gojon),mihaescu@kth.se (M. Mihaescu).

https://doi.org/10.1016/j.jsv.2020.115851

0022-460X/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

From an experimental point ofview,an extensivestudyof entropynoise hasbeencarried out by Bakeetal. [8].The experimentaldeviceconsistsofastraighttubeflowwithaheatingmoduleandanozzlewheretheflowisaccelerated.This apparatusiswellknownastheEntropyWaveGenerator(EWG)andseveralnumericalinvestigationoftheEWGexperiment can befound inliterature.The resultsshowedthatthe pressuresignalsobtainedintheEWGexperimentcomefromtwo mainmechanisms:theentropy-to-acousticconversionoftheperturbationsviastrongmeanvelocitygradientsinthenozzle, includingthepresenceofshocksinthenozzle,andtheacousticreflectionattheexhaustduetothenonperfectanechoic outlet.

Studieshavealsoshownthatincompactpremixedcombustorsoflow-emissionsystems,acousticoscillationsandentropy fluctuationsinteractwitheach other,withentropy wavescouplingconstructivelyaswell asdestructively withcombustor acoustics[9].

In1977 CumpstyandMarble[10]proposed ananalyticalmethod,basedontheactuatordisktheory,fortheevaluation ofindirectcombustionnoisethroughseveralturbinestages.Itisbasedontheaxialcompactnessassumptionfortheblade geometry(analogouslytothenozzle)butitconsidersa2Dconfigurationtakingintoaccounttheflowdeflectionduetothe circumferentialvelocitycomponentoftheblades,whichinducesvorticityfluctuations.

Leykoetal.[11]analysedthewavegenerationandtransmissionmechanismfromanumericalperspective.Intheirwork theyaccessedtherangeofvalidityofthecompactassumptionforastatorbladerow,comparingtheanalyticalresultswith simulations data.Theincomingdisturbancesused inthesimulationswere planar1D temperaturepulsesandthe acoustic responseoftherowwasevaluated.Inthenumericalsimulations,planarentropywaveswereshowntobestronglydistorted inthe interblade passageathighfrequencies. Anextensiveexperimental studywasrecentlyconductedinthehigh pres-sureturbinefacilityatPolitecnico diMilanointheframework oftheEuropeanfundedprojectRECORD(ResearchonCore NoiseReduction)[12].Twoturbineoperatingconditionswereinvestigated,subsonicandtransonicrespectively.Theentropy waveexcitationwasshowntogenerateadditionalacousticpowercorrelatedtothetemperatureamplitudeoftheincoming disturbances.

The objectiveof thispaperis toconduct accurate LargeEddySimulations (LES)through a simplifiedtransonic Nozzle GuideVane(NGV) stageofahighpressureturbineinordertogain insightsandexplanationoftheentropyto noise con-versionmechanism.Pureentropywavesofdifferentwavelengthsareinjectedinthecomputationaldomainandtheacoustic responseofthebladepassageisevaluateddownstream.

The Fourierdecompositionofthe pressurefluctuationfield highlightsregionofhighpressurewaves’amplitude,which are notpresentinthecaseofnon injectedentropywaves.ProperOrthogonalDecomposition(POD)analyses,ontheother hand,identifythemostcoherentstructuresintheflow-fieldcontaininghighenergymodes.

Thepresentworkshowsthattheentropytoacousticwaveconversionmechanismisofgreatimportanceanditoffersa parametricstudyrelatedtothewavelengthsoftheincomingdisturbances.

Transonic conditions are characterized by trailing edge expansion andcompression waves and it is shown that their dynamics are affected by injecting inletdisturbances witha particular tone. It is possible to observe that, once entropy wavesareinjectedinthedomain,onthesuctionside ofthebladebothacousticresponsesandPODmodesareexcitedby theparticularfrequencyofdisturbances.

Finally, thecomparison ofresultswith theonesfrom an analyticalmodel highlights thedifferencesbetween thetwo approaches.

2. Methodology

2.1. Geometry,computationaldomain,andmesh

ThegeometryconsideredinthepresentstudyisasimplifiedtopologyofarealNGVpassagebasedonexperimentaland numericalstudiesconductedbyYasaetal.[13,14].TheexperimentswereconductedintheKTHtransonictesttunnelfacility. ThedescriptionofthefacilityisclearlyaddressedintheworkofYasaetal.[13].TheNGVdesignparametersofinterest forthepresentstudyaretheaxialchordCax,mid atmid-span,thetrailingedgediameterd andthebladetobladedistanceat

Table 1

NGV main parameters.

Parameter Value Unit

Cax,mid 0.0657 m

d 0.0022 m

H 0.08345 m

Fig. 1. 3D geometry (z-depth magnified 100 times for clarity) and simplified 2D mid-section of the computational domain, together with mesh details (highlighted red boxes). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

consecutiveblades.TheseparametersaresummarizedinTable1.Theoperatingconditionsofthevanehavebeenassessed bylookingatdifferentmeasurementsatspecificinletandoutletlocations[13].

In thisstudy,the baseline caserefers to previous resultsby Ceciet al.[14], withtheaim ofreproducing asimplified set-upofthestudyconductedintheexperiments,withoutanytemperaturefluctuationsinjectedattheinlet.Thesimplified modelisatwo-dimensionalsectionofthebladegeometryatmid-span,extrudedspanwiseinthethirddirection.The com-putationaldomain,togetherwithleadingandtrailingedgesmeshdetails,are visibleinFig.1.Inordertoapplyazimuthal periodicboundaryconditions,theheightofthecomputationaldomainisexactlyequaltothebladetobladedistanceH.

ThenumericalsimulationsarecarriedoutsolvingthecompressibleNavierStokesequationswithadensitybasedsolver, withasub-gridscaleLESmodelfortheturbulentkineticenergy.Thecomputationalgridcountedabout55.5millionvolume cells.Gridstretchingattheinletandoutletofthedomainwasadoptedinordertoeliminatespuriousreflections.Throughout thepaper,thetermbaselinereferstotheunforcednumericalsimulationattheinlettemperatureT1=303.15K.

Foracompleteoverviewofthemeshparameters,boundaryconditions,numericsandvalidationeffortscarriedoutwith respecttothiscase,thereaderisdirectedtoCecietal.[14].1

2.2. Generalflowfeatures

The flow is subsonic atthe inlet of the domain andthe local Mach number is equal to M1=0.15.Then the flow is

acceleratedup tosonicconditionatthe throatofthenozzle guidedvanepassage(at x/Cax,mid≈ 0.75).Passed thethroat,

the flow interactswithcompression andexpansion wavesemitted fromthe shed vortices atthe trailingedge, asshown inFig.2.Thesewavesarealsoimpingingonthesuctionsideofthebladeandthenare reflectedbacktowardsthenormal

Fig. 2. Isocontours of Q-criterion (coloured with the velocity magnitude) superimposed on the divergence of the velocity field.

directionofthesurface.The flowacceleratesfurtheronthesuctionside dueto aPrandtl-Mayerexpansion.Almostatthe endofthesuctionside,itispossibletoobservecompressibilityeffects:asystemofweakshocksinteractingwiththewake ispresentthroughoutthefinal partofthesuctionside.BylookingattheanimationcorrespondingtoFig.2,itseemsthat thesystemofweakshocksismovingwithacharacteristicdynamics.

2D organisedvortices canbeseen fromthe roundedtrailingedge toanaxial distancecorrespondingto x/Cax,mid=0.2

downstream theblade,then theoverall two-dimensionalorganization startstobreakdown insmallerandlessorganized eddies.

Thesheddingfrequencypredictedbythe3DLESisof fs=24kHz,correspondingtoaStrouhalnumberofSts= fsd/U=

0_.176. The Strouhal number is calculated with the length scale d₌0_.0022 m being the trailing edge diameter and the velocityscaleU beingequaltotheonecalculatedfromtheexitMachnumberof0.95.Theexperimentalsheddingfrequency wasfoundtobe21− 22kHzfortransonic/supersonicoperatingconditions[13].

3. Impactofforcedinlettemperaturefluctuations

Asimilarnumericalsetuptothebaselinecaseisadoptedfortheforcedones.Hotandcoldspotsofairareimposedasa planarsinusoidalwavetrainattheinletplane ofthecomputationaldomain.Theaimistoobservetheentropytoacoustic conversionofinflowdisturbancesthroughtheNGVpassage.Thefrequencycontentandtheacousticresponseoftheblade rowareanalysedinordertoobservetheeffectsoftheforcing.Then,theeffectofincreasingtheinlettemperatureisstudied withthe aimofsimulatingmore realisticscenarios andto addressthe effectsonthe NGVacousticresponse.The forcing wavehascharacteristictemporalfrequency f1=1000HzandamplitudeA=4.8%,accordingto

˜

T1=T1[1+Asin

(

2

π

f1t

)

] (1)

ForamorecomprehensivereadingoftheforcedcaseatthetemperatureT₌303_.15K,thereaderisagainredirected to theresultsfoundin[14].

The parametric study hasbeen conductedin terms ofchanging the referenceinlet statictemperature. The numerical setupisidenticalforalltheforcedsimulations,withthetemperatureschangingaccordinglytothedesiredcasetosimulate. Inaddition,thespecificheatratio

γ

andthespecificheatatconstantpressurecpisalwayskeptconstantduringthe

simu-lation,butwiththeirvaluesbasedontheselectedinletstatictemperatures.Theothertwoanalyses havebeenperformed, respectively,withinlettemperaturesequaltoT1=600KandT1=900K.

Inallsimulatedcases,theMachnumbersM1 andM2is alwayskepttheirvaluesof0.15and0.95respectively.The

refer-encevelocityoftheStrouhalnumberhencechanged,sincesonicconditionsarebasedondifferenttotaltemperaturesand velocityscalesderivedfromM2is=0.95.

Table 2

Forced simulations parameters.

T1 , K A f1 , Hz λs /C ax,mid u1 , m / s t∗

303.15 4 . 8% 1000 0.76 51.90 7.60 600 4 . 8% 1000 1.11 73.02 11.11 900 4 . 8% 1000 1.36 89.43 13.61

Fig. 3. Instantaneous static temperature ratio T /T 1 ; forced inlet temperature T 1 = 303 . 15 K (a), T 1 = 600 K (b) and T 1 = 900 K (c).

In thismannerit isalso possibleto evaluate theeffects ofdifferentwavelengths ofthe incomingentropywaves.The presentparametric studyassesses thedifferentacoustic productionmechanismsasfunctionsof theratio

λ

s/C_ax,mid, with

λ

s beingthe wavelengthof theinjected entropy wave.The physicalsimulated time forthe forcedsimulationis equalto

0.01(s),correspondingto10cyclesattheforcingfrequency f1.Anon-dimensionaltimescaleisdefinedast∗=u1t/Cax,mid,

whereu1iscomputedfromtherelationM1=u1/c1,beingc1theinletspeedofsound.Theforcedsimulationscharacteristics

are specifiedinTable2.Forthechoice oftheparticularforcing frequencyof1000Hz,thereaderisinvitedtoconsult the Methodologysection of[14].Akeyreasonisthehighcomputationalcostofthe3DLES,i.e.conductingallthesimulations oftheparametricstudyforaforcingfrequencyoneorderofmagnitudesmaller,wouldhavebeentoodemandinginterms ofresources.

Fig.3showsthestatictemperatureratioT/T1forthethreeforcedsimulations.Attheinletofthecomputationaldomain,

theverticalstripeshighlightthedifferentwavelengthsoftheentropywaves,givenbythedispersionrelation

λ

s=M1c1/ f1.

The wavelengths normalized by the axial chord C_ax,mid are then:

λ

s/C_ax,mid

|

303.15[K]=0.76,

λ

s/C_ax,mid

|

600[K]=1.11 and

λ

s/Cax,mid

|

900[K]=1.36.It is also possible toobserve that these wavestend to returnto their planar organization

Fig. 4. Amplitude (left) and phase (right) of the FFT of the pressure fluctuation field at f 1 ; baseline case T 1 = 303 . 15 K (a), forced inlet temperature T1 = 303 . 15 K (b), T 1 = 600 K (c) and T 1 = 900 K (d), reference pressure p ref = 20 μPa.

3.1. Fourierdecompositionoffluctuationfields

The Fouriertransformofthetwo dimensionalsnapshots,attheinletforcingfrequency f1 iscomputedfortemperature

andpressurefields.Thenumberofsnapshotsisequalto1800.Thesamplingfrequencywasequaltofourtimesthevortex sheddingone.

Pleasenotethat,adequatesamplingfrequenciesareused,forhighertemperatures,inordertoavoidaliasingand assum-ingaconstantStrouhal(≈ 0.2)numberassociatedwiththesheddingfrequency.

3.1.1. Pressurefluctuations

Forthepressurefluctuationsfield p,theamplitudeandthephasefieldsofthebaselineunforcedcase,atthefrequency

f1, areshowninFig. 4.Inthe presentwork,the amplitudefieldof thepressurefluctuationsandfuture computationsof

soundpressurelevelsareexpressedindecibels,dB,takingasreferencepressurethevalue pref=20

μ

Pa.

Fortheunforced simulation(Fig.4a), itispossible toobservethepresenceofa systemofcompression andexpansion waves(towardsthetrailingedge,suctionside)movingatafrequencyof1000Hz.Thephasefieldofthepressurefluctuations fieldshowsthespatialperiodicityoftheflow.TheamplitudeofthepressurefluctuationsisnegligibleupstreamoftheNGV passage,uptothethroatsection,astherearenoimposedentropywavesinthiscase.

Itcanbeobservedthattheamplitudeofthepressurefluctuationsisincreasinginthebladetrailingedgeregionfromthe unforced case(Fig.4a)tothe forcedones(Fig.4b–d),suggestinga strongermotionoftheweak shocksatthisfrequency. Thisstrongermotionoftheshocksleadstoanacousticemissioninthedownstreamdirection.

ThecaseatT1=900Kpresentsveryhighpressurefieldamplitudesinsuchazone,asitcanbevisualizedfromFig.4d.

Theinteractionoftheentropywaveswiththeacousticfield createdat f1,becomesmoreandmoreimportantastheinlet

Fig. 5. Amplitude (left) and phase (right) of the FFT of the temperature fluctuation field at f 1 ; baseline case T 1 = 303 . 15 K (a), forced inlet temperature T1 = 303 . 15 K (b), T 1 = 600 K (c) and T 1 = 900 K (d).

well,inaccordancewiththeaforementionedtrend.Thisresultsuggeststhattheentropy-to-acousticconversionmechanism isenhancedwhenincreasingtheinlettemperature.

Thephasefieldsofthepressurefluctuationdownstreamofthebladearesimilar,withsharpchangesofphaseidentifying wavefronts.

3.1.2. Temperaturefluctuations

For the temperature field, the same low frequency dynamics as forpressure is visible in the amplitude field of the unforcedcase,Fig.5a.

The amplitudefield of the forced simulation (Fig. 5b) shows that entropy wavesreduce their intensityafter passing the bladerow.The interactionwiththesystemofshocks isalsovisible.Ontheother hand,theforcedphase field atthe samefrequencyshowstheplanarcharacteroftheincomingentropywaves,whicharenotorganisedinanyparticular man-ner atunforcedconditions.Afterwards,theplanarwavesgetstronglydistortedthroughtheNGVpassageand“streaks”-like structures arevisibleinthemiddleofthepassage.ThosewavesreturnalmostplanarapproximatelyoneaxialchordC_ax,mid

downstreamofthebladerow.

Asinlettemperatureincreases,Fig.5candd,thefeaturesoftheamplitudefieldaresimilar,withanincreasingwidthof thewakeregion.Themaximumamplitudemagnitudedividedbyitsrespectiveinlettemperatureis4_.8%sincethemaximum temperatureisregisteredattheinlet,wheretheforcingisimposed.

The phase field identifies, instead, theplanar character ofthe incoming entropy waveswhich are then distorted and finallypropagateddownstream.Thewavelengthoftheplanarwavesalsoincreaseswithrespecttotheinlettemperature.

Fig. 6. Instantaneous divergence field div (U) at three instants of time (a), (b), (c) (from left to right); forced inlet temperature T 1 = 303 . 15 .

Inconclusion, thepresenceofacousticproductionatthe forcingfrequency f1,withnoticeableamplitudes,isobserved

bothupstreamanddownstreamofthebladepassage.Thedownstreamgeneratednoiseseemstobeemittedfromthesystem ofweakshocksonthesuctionside,asthetrailingedgeweakshocksarealreadymovingatafrequencyof1000Hzinthe unforcedcase(Fig.4a).Whenentropywavesareimposedartheinlet,thebladeacousticresponseisalsocharacterisedby theexcitationofalatentdynamicoftheunforcedcaseat1000Hz.Itseemsthattheweakshockmovementis“lockedin” atthisparticularfrequency,duetothepresenceoftheforcing.

Fig.6showsthedivergenceofthevelocity fieldfortheforcedcaseatT1=303.15Katthreetime instants.Thesystem

ofweakshocksonthesuctionsideofthebladecanberecognisedbythedarkareasofthediv

(

U

)

field,whiletheacoustic wavesarehighlightedbytheredcurves.

Theupstreamnoiseisgeneratedinthesubsonicconvergentregionofthepassageandthenreflectedbackwards.No per-turbationofthesheddingdynamicsisinducedbytheinletforcing,thisresultmaybeduetotheverydifferentfrequencies ofthetwophenomena.

3.2. Properorthogonaldecompositionanalysis

Inthe presentsubsection,ProperOrthogonalDecomposition (snapshotPODmethod[15])is usedinordertoextract the most relevant modeshapes appearing in the 2D snapshots sampled at mid-spanduring the simulations. The number of snapshotsisequalto1800.Studyisperformedfortheunforcedandforcedflow-fields.Theresultingnon-dimensional eigen-vectors(topos)forms anorthonormalbasis.ThereforethecolorscaleofPODresultspresentedinthispaper(from blackto white) ranges the values from−0.003 to 0.003.These are chosen in order to identifymore clearly thefeatures ofeach modeshape.

Throughoutthepaper,theithchrono-modea_ifrequencycontentisexpressedintermsoftheStrouhalnumberwiththe usualdefinitionSt= f d/U,whileitstimehistoryintermsoftimet expressedinseconds.ThemainSt numberstheauthors refertoaretheonesassociatedtothesheddingfrequency fsandforcingfrequency f1,remindingthatthevelocityusedfor

theSt computationistheonecalculatedfromthevaneexitMachnumberof0.95.

3.2.1. PODanalysis,pressurefieldatunforcedinletconditions

Theanalysisisfocusedonthepressurefluctuationfield p,forwhichthePODalgorithmisapplied.Theenergyassociated toeachmodeisplottedinFig.7.

Theappearanceofmodesasacoupleisanevidencethatthemodesrepresentawave-likeperiodicstructureoftheflow: POD modesarerepresented byrealfunctions, andtwo modesareneededto describea travelling wave,representingthe realandcomplexpartsrespectively.

Thefirstcoupleofmodes,associatedwiththesheddingphenomenon,contains17%ofthetotalenergy,whiletheenergy contentper modedropsbelow5% forthesuccessiveones.Torepresent90%ofthetotalenergy,approximately300modes areneeded. ThemodeshapeassociatedwiththevortexsheddingispicturedinFig.8a;onceagainthedominanttoneisat

St=0.176,i.eapproximately24kHz. Thecouple3− 4isassociatedwiththesystemofcompressionandexpansionwaves arisingfromthetrailingedge,asdepictedinFig.8b.Itispossibletoobservethat,evenwithoutthepresenceoftheforcing, thefrequenciesofthismovementareinarangefrom500− 3000Hz,i.e.intheSt rangeof0.00367− 0.02199.Thisfeature, underlinesthepresenceofflowdynamicsinthelowfrequencyrange.

Othercouplesarealsoassociatedwiththelowfrequencydynamicsofthecompressionandexpansionwavesoriginating fromthetrailingedgeandareanalogoustothetemperaturetopo-modes.

3.2.2. PODanalysis,pressurefieldatforcedinletconditions

Looking atthePODofforcedcases,additionalmodeswhichwerenot presentinthebaselinecaseare found.ForT1=

Fig. 7. Cumulative energy percent retaining the first 600 modes (a), and energy associated to each mode (b). Pressure fluctuation field p  _{; baseline case}

T1 = 303 . 15 K.

Fig. 8. First (a) and third (b) POD mode of the pressure fluctuation field p  _{; baseline case T}

1 = 303 . 15 K.

Fig. 9. Cumulative energy percent retaining the first 600 modes (a), and energy associated to each mode (b). Pressure fluctuation field p  _{; forced inlet}

temperature T 1 = 303 . 15 K.

associatedwiththesheddingphenomenon,butthecouple3_{− 4}exhibitsnowatopo-modeattheinletforcingfrequency f1,

associatedwiththeshockmovement.Thetemporaltrendcanbe clearlyvisualizedfromthetime historyoftheassociated temporal coefficient in Fig. 10.The chrono-modeis clearlycharacterized only by the forcing frequency f1,underlining a

strongcorrelation betweenthetopo-modeandthepresenceoftheforcing. Thesheddingmodeisinsteadidenticaltothe unforcedcase.TheratiobetweenthesheddingStnumbersisSts/St1=0.176/0.00733≈ 24.

The energycontentofthePODfortheforcedcaseisshowninFig.9.Now,thefirst 6modescontain38% ofthetotal energyascomparedtothe30%ofthebaselinecase,withthecouple3− 4retaining12%ofit.ThepressurefluctuationsPOD

Fig. 10. Third POD mode of the pressure fluctuation field p  _{; forced inlet temperature T}

1 = 303 . 15 K.

Fig. 11. Third POD topo-mode of the pressure fluctuation field associated to the dominant frequency f 1 ; forced inlet temperature T 1 = 600 K.

athigherinletforcingtemperatures,instead,ismorecomplexandlessstraightforwardtointerpret.Thisisduetothefact thatsometopo-modesappearmixedtootherfrequencies.

Fortheintermediatewavelengthratio

λ

s/C_ax,mid=1.11(T1=600K),thefirstfourmodesarealwaysassociatedwiththe

sheddingphenomenonandwiththepresenceofcoherentflowstructuresattheforcingfrequency f1.Theycontain24%of

thetotalenergyandtheindividualmodesarenotalwaysassociatedtoasingletemporalfrequency.

ThisaspectcanbeseeninFig.11,whichclearlydepictsthecharacteristicmodeshapeofthepressurefluctuationsfieldat

f1=1000Hzmixedtoasmallfrequencycontentofthesheddingatfs.ThesheddingSt numberatT1=600Kremainsvery

closetotheoneforT1=303.15K,beingequaltoSts=0.173.TheSt numberattheforcingfrequencyisnowSt1=0.00524,

hencetheratioSts/St1=0.173/0.00524≈ 33.Somespatialfeaturesrelatedtovortexsheddingarevisibleinthetopo-mode.

Inaddition,thereisalsoasmallpeakatthesheddingfrequencyinthespectrumofthechrono-mode.Theenergycontent permodeissimilartotheoneinFig.9atT1=303.15K.

For the largest wavelength

λ

s/Cax,mid=1.36 (T1=900 K), instead, the association of a single topo-mode to multiple

temporal frequencies becomeseven stronger than the previous case. There are,again, two modeswhich are prevalently linkedtothesheddingfrequencySts=0.16,buttheyexhibitanoticeabletoneatthefrequencySt1=0.00431.Thesemodes

can bevisualizedinFig.12.Thesamesituationhappensforthetopo-modesmainlyassociatedwiththeforcing frequency

St1=0.00431.ThemodeinFig.12bshowsaclearpeakatSts=0.16,inadditiontotheoneatSt1=0.00431.TheSt number

ratioforthelargestwavelengthcaseisSts/St1=0.16/0.0431≈ 37.

The energycontentassociatedtothefirstfourmode isaround55%,withthe topo-modesassociatedwith f1 retaining

25%ofit.Alltheremainingmodesexhibitenergycontentspermodebelow2%.

Itcanbenoted,again,thatsingleoddmodesaredepictedduetotheappearanceasacouple.

In conclusion,thePOD analysisatallinlet temperatures,despitesuffering ofmixedfrequencycontentinside a mode-shapeatthelargesttemperature,isabletoidentifyadditionalmodeswithcharacteristicshapeat f1,whichwereabsentin

thebaselinecase.

3.2.3. PODanalysis,temperaturefieldatunforcedinletconditions

Fromthemodeshapes’analysisofthetemperature,itisimmediatelypossibletorecognizethedifferentfeatures charac-terizingtheunforcedandtheforcedcases.Thecumulativeenergydistributionamongthetemperaturefluctuationmodes,is showninFig.13forthebaselinecase.Onecanseethatthefirst8modesroughly capture20%oftheenergy,withmode1

Fig. 12. First (a) and third (b) POD topo-modes of the pressure fluctuation field associated to the vortex shedding at frequency f s ; forced inlet temperature

T1 = 900 K.

Fig. 13. Cumulative energy percent retaining the first 600 modes (a), and energy associated to each mode (b). Temperature fluctuation field T  _{; baseline}

case T 1 = 303 . 15 K.

Fig. 14. First POD mode of the temperature fluctuation field T  _{; baseline case T}

1 = 303 . 15 K.

and2containingthehighestenergycontentpermode≈ 5%.Inordertocapture90%oftheenergy,400modesareneeded. An analysis of the chrono- andtopo-modes showedthat the first two PODmodes havea similar periodicity(frequency content)andasimilarstructureinspace.

The firstPOD mode isrepresentedin Fig.14.The dominantfrequencyisSt=0.176 asshowninthe chrono-modesof Fig.14.

Fromaphysicalpointofview,thesestructuresrepresentthevortexsheddingphenomenon.Itisalsopossibletoobserve theacousticwaveswhicharegeneratedbythevorticesatthetrailingedgeoftheblade.

Fig. 15. Third (a) and fifth (b) POD mode of the temperature fluctuation field T  _{; baseline case T 1 = 303 . 15 K.}

Fig. 16. Cumulative energy percent retaining the first 600 modes (a), and energy associated to each mode (b). Temperature fluctuation field T  _{; forced inlet}

temperature T 1 = 303 . 15 K.

Thefollowingcoupleofmodes3_{− 4}isinsteadassociatedwiththesystemofcompressionandexpansionwavesarising fromthetrailingedge.Onecouldnotice,infact,thedarkandlightareasaroundthetrailingedgeinFig.15a.Thefrequency contentisaroundSts=0.016correspondingtoafrequency fs=2180Hz.

The couple 5− 6 is also associated with the shock movement mechanism but with a lower frequency content at

St=0.004 correspondingto f=545Hz,Fig.15b.The frequencycontentofthesheddingiswell separatedfromtheshock movement[14],sincetherearenotopo-modeswhichareassociatedtochrono-modeswithbothfrequencies.Thefrequency content ofthemodesrepresentingthewavestravelling attheendoftheblade arenot always associatedtoa single fre-quency,andevenifdominanttonesaredistinguishable,severalotherfrequenciesappearinthetemporalsignal.

3.2.4. PODanalysis,temperaturefieldatforcedinletconditions

Movingtheattentiontotheforcedcases,thesituationchangessignificantly:almostthetotalityoftheenergyis associ-atedwiththefirsttwo modes,whichcontainmorethan90% oftheenergy.Thecumulativeenergyretainedasfunctionof thenumberofmodesandtheenergypermodearepicturedinFig.16.

Thetopo-modeassociatedwiththefirstPODmodeanditsrespectivechrono-modeareshowninFig.17.Theresultsare similar forthesecond PODmode. Thetime signal showsaperfectlyperiodic trendofthetemporal coefficientassociated withthesinglefrequencyof f1=1000Hz.Thespectrumofthe temporalcoefficient,infact,showsthedominanttoneat

St1=0.00733.Thespatialmodeshape alsoshowstheincomingplanarwavetrainofentropywaveswhichgetstrongly

dis-tortedastheyapproachthevanepassage.Theaforementionedtopo-modehasstrongresemblanceswiththeinstantaneous temperaturefieldofFig.3a,onlymissingthemixingfeaturecausedbythevortexshedding.

The other modescalculated with the POD algorithm are associated to the vortex shedding phenomenon and to the shockmovementmechanism.Theirmodeshapesaresimilartotheonesshowedforthebaselinecaseandarenotrepeated. Forthetemperaturefluctuations, almostallthe energyiscontainedinthefirst coupleofmodes1− 2.Thisistrueforall inlettemperatures.InFig.18,itispossibletovisualizethemodeshapesforallinlettemperatures;thechrono-modeisnot presentedsinceitisonlyapuresinusoidalfunctionat1000Hz.Themodeshapesareverysimilar,withincomingwavesof

Fig. 17. First POD mode of the temperature fluctuation field T  _{; forced inlet temperature T}

1 = 303 . 15 K.

Fig. 18. First POD topo-mode, temperature fluctuation field; forced inlet temperature T 1 = 303 . 15 K (a), T 1 = 600 K (b) and T 1 = 900 K (c).

largerwavelengthandathickerwakeregionastheinlettemperatureincreases.Inconclusion,thePODtemperaturemodes, atallforcedconditions,areentirelydominatedbythepresenceofthesinusoidalforcingat f1.

4. Effectsoftheincomingdisturbances

From theprevious analyses,itisthen possibletodrawimportantconclusionsontheeffectoftheinletreference tem-perature.

It hasan effectonchangingthewavelengthoftheincomingdisturbances

λ

s,whichisalsopropagated downstreamof

theblade,makingitthickerastheinlettemperatureincreases.TheStrouhalnumbersassociatedtothesheddingfrequency arealmostconstantforthetwoinlettemperaturesofT1=303.15KtoT1=600K,withtheirrespectivevaluesof0.176and

0.173.Whileitdecreasesdownto0.16fortheinlettemperatureofT1=900K.

TheamplitudefieldsoftheFouriertransformshowaparticularzoneofinterestonthesuctionsideoftheblade,towards itsend.Thisfeatureispresentforallforcedsimulations,withanoutburstregionofveryhighamplitudes(≥ 160dB)forthe highesttemperatureT1=900K.

From thePOD ofthetemperaturefield,all thetemporal signalsofthechrono-modesassociatedwiththe inletforcing frequency show apurely sinusoidaltrend similarto theone in Fig.16.The associatedtopo-modesshow very similar re-semblances totheir respective instantaneousfields inFig.3.The PODalgorithm forthepressure fluctuationfield, on the other end,isnot identifyingcoherentstructuresatasingletemporalfrequencyastheinlettemperatureincreases.Infact, singletopo-modesarelinked,intheirfrequencycontent,tobothvortexshedding,atfrequency fs,andtothemovementof

compressionandexpansionwavestowards theendoftheblade,atfrequency f1.Theimportanceofthemodesrelatedto

f1,anyway,isenhanced forhighertemperatures:theycanrepresentupto25%ofthetotalenergyofthefield p by

them-selves(asinthecaseforT1=900K).Inallsimulations, similarPODtopo-andchrono- modesappearwhena fluctuating

temperatureisimposedattheinlet.Thepresenceofsuchstructuresinthebaselinecaseisnotevident.

Itcan beconcludedthatadditionalacousticpoweriscreatedinallcaseswheretheinlettemperaturefieldisforcedat 1000 Hz.Theforcing isalso ableto excitethe movementof compressionandexpansion wavesemittedfromthe trailing edge.

theunprimedquantitiesrepresentsinsteadtheir localmeanvalue.Thethermodynamicquantitiescpand

γ

arethespecific

heatatconstantpressureandthespecificheatratio.Theoriginalmodelisextendedtoseriesofstator-rotorbladerows,the presentworkonlydealswithasinglestatorNGVpassage.

Thehypothesesofthemodelaresummarizedasfollows: • Compactassumption,

λ

s Cax,mid.

• Lowblade pitch-chordratio,theblade detailscan beneglected andonlythe inletandoutletflowMach numbersand directionsneedtobeconsidered.

• Subsonicaxialflow.

• Radialvariationsareneglected,theflowistreatedastwo-dimensionalintheaxial-tangentialplane. • Theacousticpoweriscalculatedneglectinganydiscontinuitydownstreamoftheturbine.

• Whenentropywavesinteractwiththeturbinestage,arisingpressurewavesarepropagatedfromtherowupstreamand downstream.

• Vorticitywavesareonlyconvecteddownstream.

Each flow-state,upstream anddownstream,is steady,witha uniformstate characterizedby a flowvelocity w, aflow direction

θ

,a uniformpressure p andadensity

ρ

.Threeofthefourconditionsaregivenby conservationofentropy, con-tinuity of mass-flow andconservation of totalenthalpy. Forthe present studythe fourthmatching condition isthe one corresponding tothepresenceofasonicthroatacrosstheNGVpassage.FollowingtheworkofLeykoetal.[11]the reflec-tionandtransmissionacousticcoefficientsforforcedplanarentropywavesareexpressedintermsoftheflowMachnumber

M,theflowdirection

θ

andtheratioofspecificheats

γ

.Theyaredefinedas:

w−₁ ws 1 =−1 2



γ

− 1 2 + cos

θ

1 M1 + sin

θ

1tan

θ

1

η

1M1

(

1− M12

)



−1 , (3)

forthereflectioncoefficient,where

η

=

(

1+

(γ

− 1

)

M2_/2

₎

−1_,and

w+₂ ws 1 = 1− 1 M1cosθ1 1+ 1 M2cosθ2 w−₁ ws 1 , (4)

forthetransmittedone.Thevariablesw±areusedtorefertothetransmitted(+)andreflected(-)acousticwaves.While thevariablews_{identifiestheentropywave.Theunderscripts1and2finallydenotestheupstreamanddownstreamsections}

ofthebladerow,respectively.

5.2. Comparisons

TheresultsoftheanalyticalmodelsareappliedtothecurrenttopologyoftheNGVpassage.

The supersonic dischargeMach number is chosen to be equal to M2=1.05 (just above sonic conditions) in order to

applytheanalyticalmodelwithsupersonicdischarge.ForfixedMachnumbersM1=0.15,M2=1.05andspecificheatratio

γ

=1.4,thereflection(3)andtransmission(4)coefficientoftheacousticwavearenowplottedasfunctionoftheincoming anddischargedirectionsoftheflow,i.e.

θ

1and

θ

2.

As it can be seen from Fig. 19b,the reflected wave only dependson the inflow angle

θ

1. The red dot highlights the

coefficient forthe geometryadoptedinthesimulations,i.e.

θ

1=0.Forthetransmittedwave,instead,theresultsdepends

onbothincominganddischargedirections.Fig.19ashowtheresultsoftheanalyticalmodel.Againthereddotreferstothe actualgeometryofthestudy,with

θ

2=75.Thevalueofthetransmissioncoefficientisw+₂/ws1=0.088,whilethereflection

oneisw−₁/ws

1=0.073.

TheseresultsarenowcomparedwiththeonesfromtheLES.Thecalculationoftheacousticcoefficientswasperformed on twoplanesorthogonaltothe x_−axis,respectivelyatx_{/Cax,mid=−0.}65upstreamandatx_/Cax,mid=2_.40downstreamof theleadingedgeoftheblade.Thelocationx/Cax,mid=0.0correspondstotheleadingedgeoftheblade.Theupstreamplane

is referred asplane 1, whilethe downstream plane islabeled asplane 2. Flow variables were sampledon theplanes of interestbythemeansof75localprobes.Theflowquantitieswerealsoaveragedinthespanwise-direction.

Fig. 19. Transmitted (a) and reflected (b) acoustic waves generated by an incoming entropy wave as function of the model parameters; ( ) analytical model, ( ) actual geometry.

Fig. 20. Average spectrum of the forced temperature wave at plane 1 (a) and at plane 2 (b); ( ) baseline case T 1 = 303 . 15 K, ( ) forced case T 1 = 303 . 15

K.

5.3. Frequencycontentfeatures

Startingfromthespectrum(overallprobes)ofthetemperaturefluctuationfield onplane 1,fromFig.20aitispossible toidentifyapeakatSt1=0.00733fortheforcedcase.Itsamplitudecorrespondsto4.8%oftheinletstatictemperature,as

expected.Nocomparableamplitudes,instead,werefoundforthebaselinecase.Ontheotherhand,forplane2,itispossible to see fromFig.20b, that theentropy wave isattenuated downstream ofthe blade,while thegreatest amplitudein the entirespectrumisstillregisteredforSt1.

Fig.21showsinsteadthepitchwisecharacterofthesewavesontheirrespectiveupstreamanddownstreamplanes. The pitchwise location is normalized by the blade to blade distance H, and goesfrom the lower periodic boundary

y/H=0totheupperperiodicboundaryy/H=1.Again,forthebaseline case,nocomparableamplitudesarefoundalsoin thepitchwisedistribution.Theentropywaveisperfectlyplanarintheupstreampart.Thiscanbe expectedsincetheflow iscompletelyuniforminthisregion.Instead,thewave losesitsperfectlyplanarcharacterontheplanedownstreamofthe NGVpassage.Oncemore,itispossibletoseethattheamplitudeoftheentropywavesisreducedonplane2.

Moving tothe SPL_,spectra upstream anddownstream of the blade arepresented in Fig.22. It ispossible to see the presenceofatoneatforcing frequencySt1 onbothplanes.Thistime,incontrastwiththetemperaturefield,itispossible

toobservethepresenceoflowfrequencycontentsevenforthebaselinecase.SimilarlytoFig.21,thepitchwisecharacterof thepressurewavesontherespectiveupstreamanddownstreamplanesisshowninFig.23,intermsofSPL.

In accordanceto theresultsofthetemperaturefield, pressurewavesarealsoplanar upstreamofthe vane,dueto the uniformityoftheflow.InFig.23aitispossibletovisualizetheadditionalacousticpowergeneratedintheupstreamplane fortheforcedinletconditions.Infact,thedifferenceinSPL(



SPL)betweenthebaselineandtheforcedcaseis≥ 20.

Fig. 21. Pitchwise distribution at the frequency St 1 of the temperature wave amplitude at plane 1 (a) and at plane 2 (b); ( ) baseline case T 1 = 303 . 15 K,

( ) forced case T 1 = 303 . 15 K.

Fig. 22. Average spectrum of the SPL at plane 1 (a) and at plane 2 (b); ( ) baseline case T 1 = 303 . 15 K, ( ) forced case T 1 = 303 . 15 K, reference pressure pref = 20 μPa.

Fig. 23. Pitchwise distribution at the frequency St 1 of the SPL at plane 1 (a) and at plane 2 (b); ( ) baseline case T 1 = 303 . 15 K, ( ) forced case T1 = 303 . 15 K, reference pressure p ref = 20 μPa.

Fig. 24. Average SPL spectrum at plane 1 (a) and at plane 2 (b), forced case T 1 = 600 K, reference pressure p ref = 20 μPa.

Fig. 25. Average SPL spectrum at plane 1 (a) and at plane 2 (b), forced case T 1 = 900 K, reference pressure p ref = 20 μPa.

AdditionalacousticpowerisalsogenerateddownstreamofthepassageasitcanbeseenfromFig.23b.However,inthis case,theacousticwaveisfarfromplanar,withthepresenceofdarkzones,whicharealsovisibleintheamplitudefieldof theFourierTransforminFig.4b.Differencesin



SPL_{≥ 15}arealsoregisteredonplane2.

The upstreamanddownstream spectraofthepressurefluctuationfield areshowninFigs. 24,25forthetwo tempera-turesof600Kand900Krespectively.Twopeaksarepresentbothupstreamanddownstreamofthepassageatthe respec-tivefrequencyidentifiedbySt1.Notableamplitudesatthelowfrequencyrangearestillpresentforbothtemperatures,even

forthebaselinecase.

5.4. Outcomes

Atthispoint,afterhavingshownthepresenceofanadditionalgeneratedacousticsource,theanalyticalmodelis com-paredwiththeresults ofnumericalsimulations.Forthecoefficientsofthe analyticalmodel,theparameter

γ

asa minor influenceontheresults,thereforeonlyonevalueisreported.Theexpressionusedfortheentropyandtheacousticwaves are ws₌ s cp, (5) w±= p

γ

p, (6)

wherews_andw±_{arerespectivelytheentropyandtheacousticwaves.}

The amplitudeofthiscoefficientisaveraged overtheplane ofinterestin ordertogeta singlecoefficient ataspecific frequency.ThefinalresultsarecomparedinTable3.

The value of the transmitted entropy wave ws

2/ws1 is less then unity for all simulated cases, but it increases as the

1/w1,

suggestthat,movingtowardshigherratiosof

λ

s/Cax,mid,theentropy-to-acousticconversionisenhancedacrossthevane.The

coefficientsarealsoincloseragreementwiththeanalyticalmodel,whichisbasedonthecompactassumption. Finally,forthe transmittedacousticcoefficient w+_2/ws

1, itsvalue increasesagainmonotonically forhigher wavelengths.

Forthesimulatedtransonicconditions,theremightbeflowfeatureswhichinteractsstrongerwiththesystemofincoming entropywaves,astheinlettemperatureT1increases.Agoodexampleistheinteractionwiththesystemofcompressionand

expansionwavesthatareemittedfromthetrailingedge,showninFig.5.Inthiscase,onecanobservethatthetransmitted acoustic wave associated to the largestT1 is close tothe value predicted by the analytical model,despite the isentropic

modelconditionsnotapplyingforthesimulatedcases.AstheT1 wouldbefurtherincreased,itmightbepossiblethatthis

coefficientbecomesevengreater,underliningadditionalgeneratednoiseintransonic/supersonicconditions.

6. Conclusions

Theindirectcombustionnoisegenerationmechanismhasbeenaddressedbythemeansof3DLESforasimplifiedNozzle GuideVanepassageof ahighpressureturbine. The solveradoptedforthesimulations wasfirstvalidatedagainst experi-mentaldataavailableforoff-designtransonicconditionswithouttemperatureforcinginapreviouspaper.

Inthecorepartofthepresentwork,apulsatingentropywave-trainwasimposedattheinletofthedomaintosimulate temperaturenon-uniformitiescomingfromthecombustionchamber.

At theforcing frequencyimposed atthe inlet,severalzonesofthe fluid domainrespondwithhighamplitudesof the pressure fluctuations. The response of the NGV passage is stronger as the wavelength

λ

s of the incoming disturbances

is increased. Moreover, thisaspectis complementedby the analysisperformedvia theProperOrthogonal Decomposition method:thebaselinecasedidnotshowanyparticularcoherentstructureattheinletforcingfrequency f1,St1;while

pres-sureandtemperaturefieldsoftheforcedcasesallshowedthepresenceofhighenergymodesatthatparticularfrequency. ThecombinationofthePODtopo-andchrono-modeswasinaccordancewiththeflow-fieldresultsidentifyingthe move-ment ofcompressionandexpansion wavesemitted fromthetrailingedge. Forthepressure fluctuationfield, the chrono-modesidentifiedthecharacteristicfrequencycontent,whilethetopo-modesidentifiedthezoneswithhighestresponse of thefluctuation.

Reflection and transmissioncoefficients are compared to analytical models available inliterature. The 3D simulations show that theentropywavesare highlydistortedbythepassageandlose theirstrength; thiscanbe addressedfromthe entropywave transmissioncoefficientwhichisreducedsignificantlyforallthe wavelengthsofthe incomingdisturbances. Thehypothesisofentropyconservationassumedbytheanalyticalmodelmightnotholdinthecaseoftransonic/supersonic conditionsandwithshortincomingentropywaves.

Thereflected acousticwave, instead,isweakerthantheonepredictedbythecompacttheoryforallsimulations. How-ever,thereflectedacousticcoefficientincreasestowardstheanalyticaloneasT1increases.Finally,thetransmittedacoustic

wave isinfairlygoodagreementwiththeanalyticalmodel,andthetransmissioncoefficientassumesalsolargervaluesfor thehigherT1,suggestingthepresenceofflowfeatures,intransonicconditions,whichmightgenerateadditionalnoise.

Astrongexampleisthedynamicsofthecompression/expansion wavesemittedfromthetrailingedge,whichinteracts withtheincomingentropywaves;i.e.thebladeacousticresponseisstronglyinfluencedby theexcitement ofalatentlow frequencydynamicswhentheplanarentropywavesat1000Hzareimposedattheinlet.Thepresentworkhighlights the features of thispeculiarmechanism, which, intransonic conditions,might additionallycontribute to the generatednoise ratherthantheonlyonepredictedbyisentropicanalyticalmodels.

DeclarationofCompetingInterest

The authors declare that they have no knowncompeting financial interests or personal relationshipsthat could have appearedtoinfluencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

AlessandroCeci:Software,Validation,Formal analysis,Investigation,Writing-originaldraft,Visualization.Romain Go-jon:Conceptualization,Validation,Investigation,Resources,Writing-review&editing,Supervision.MihaiMihaescu: Con-ceptualization,Supervision,Projectadministration,Writing-review&editing.

Acknowledgements

ThesimulationswereperformedatthePDCCenterforHighPerformanceComputing(PDC-HPC)andattheHigh Perfor-manceComputing Center North(HPC2N), on computationalresources provided by theSwedish NationalInfrastructure of Computing(SNIC).

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