Convergence of Fourier-based time methods for turbomachinery wake passing problems

This is an author-deposited version published in:

http://oatao.univ-toulouse.fr/

Eprints ID: 12004

To link to this article: DOI: 10.1016/j.jcp.2014.08.029

URL:

http://dx.doi.org/10.1016/j.jcp.2014.08.029

To cite this version:

Gomar, Adrien and Bouvy, Quentin and Sicot,

Frédéric and Dufour, Guillaume and Cinnella, Paola and François,

Benjamin

Convergence

of

Fourier-based

time

methods

for

turbomachinery wake passing problems. (2014) Journal of Computational

Physics, vol. 278. pp. 229-256. ISSN 0021-9991

O

pen

A

rchive

T

oulouse

A

rchive

O

uverte (

OATAO

)

OATAO is an open access repository that collects the work of Toulouse researchers and

makes it freely available over the web where possible.

Any correspondence concerning this service should be sent to the repository

Convergence

of

Fourier-based

time

methods

for

turbomachinery

wake

passing

problems

Adrien Gomar

a

,

∗

,

1

,

Quentin Bouvy

a

,

2

,

Frédéric Sicot

a

,

3

,

Guillaume Dufour

b

,

4

,

Paola Cinnella

c

,

5

,

Benjamin François

a

,

d

,

1

a_{CERFACS,CFDTeam,42avenueGaspardCoriolis,31057ToulouseCedex 1,France}

b_{UniversitédeToulouse,InstitutSupérieurdel’Aéronautiqueetdel’Espace(ISAE),10avenueEdouardBelin,31400Toulouse,France} c_{DynFluidLab.,ArtsetMétiersParisTech,151Boulevarddel’Hopital,75013Paris,France}

d_{AirbusOperationsS.A.S.,316routedeBayonne,31060ToulouseCedex9,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Keywords:

Fourier-based time methods Harmonic balance Convergence Spectralaccuracy Contra-rotating open rotor Turbomachinery Wake passing

The convergence of Fourier-based time methods applied to turbomachinery flows is assessed.Thefocusisontheharmonicbalancemethod,whichisatime-domain Fourier-basedapproachstandingasanefficientalternativetoclassicaltimemarchingschemesfor periodicflows.Intheliterature,noconsensusexistsconcerningthenumberofharmonics needed to achieve convergence for turbomachinery stage configurations. In this paper it is shown that the convergence of Fourier-based methods is closely related to the impulsive natureoftheflowsolution,whichinturbomachinesisessentiallygovernedby thecharacteristicsofthepassingwakesbetweenadjacentrows.Asaresultoftheproposed analysis,apriori estimatesareprovidedfortheminimum numberofharmonicsrequired toaccuratelycomputeagiventurbomachineryconfiguration.Theirapplicationtoseveral contra-rotating open-rotorconfigurationsisassessed,demonstratingthepracticalinterest oftheproposedmethodology.

1. Introduction

Theindustrialdesignofturbomachineryisusuallybasedonsteadyflowanalysis,forwhichthereferencesimulationtool are thethree-dimensionalReynolds-AveragedNavier–Stokes (RANS)steadycomputations.However, thisapproachfindsits limitswhenunsteadyphenomenabecomedominant. Insuchacontext,engineersnowneedtoaccount forunsteady-flow effectsasearlyaspossibleinthedesigncycle,whichmakes efficiencyofunsteadycomputationsakeyissue.Aspecificity ofturbomachineryflowsistheirperiodicity,atleastasfarasthemeanfieldpropertiesareconsidered.

*

Corresponding author.

E-mailaddresses:adrien.gomar@gmail.com(A. Gomar), quentin.bouvy@cerfacs.fr(Q. Bouvy), frederic.sicot@cerfacs.fr(F. Sicot), guillaume.dufour@isae.fr (G. Dufour), paola.cinnella@ensam.eu(P. Cinnella), benjamin.francois@cerfacs.fr(B. François).

1 _Ph.D. 2 Study Engineer. 3 Senior Researcher. 4 Associate Professor. 5 _Professor.

Fig. 1. Convergence of harmonic balance computations for a rotor/stator configuration from Sicot et al.[7].

Fourier-basedtime methodsforperiodic flowshaveundergone major developmentsinthelast decade(see He [1]for instance)astheyallowtoreduce thecomputationalcostofunsteadysimulationsascomparedtostandardtime-marching techniques.The basicidea isto decomposethetime-dependentflowvariablesintoFourierseries,whicharetheninjected intotheequationsoftheproblem.Thetime-domainproblemisthusmadeequivalenttoafrequency-domainproblem,where thecomplexFouriercoefficientsarethenewunknowns.Atthispoint,twostrategiescoexisttoobtainthesolution.Thefirst one isto directlysolve theFouriercoefficientsequations, usinga dedicatedfrequency-domainsolver, asproposed by He andNing[2,3].ThesecondstrategyistocasttheproblembacktothetimedomainusingtheinverseFouriertransform,as proposedbyHall[4,5]withtheHarmonicBalance(HB)method.Theunsteadytime-marchingproblemisthustransformed intoasetofsteadyequationscoupledby asourcetermthat representsaspectral approximationofthetime-derivativeof theinitialequations.Themainbenefitofsolvinginthetimedomainisthatsuchamethodologycanbeeasilyimplemented inanexistingRANSsolver,takingadvantageofall classicalconvergence-acceleratingtechniquesforsteadystateproblems. TheHBapproachhasdemonstratedasignificantreductionoftheCPUtime,typicallyonetotwoordersofmagnitude[6–8]. EfficiencyoftheHBmethodresultsfromatrade-offbetweenaccuracyandcostsrequirements.Ononehand,theaccuracy ofFourier-basedtime methodsdependsonthenumberofharmonicsusedtorepresentthefrequencycontentofthetime signal;ontheotherhand,computationalcostsandmemoryconsumptionofthecomputationsalsoscalewiththenumber ofharmonics.

Theoretical resultsabout theconvergence of spectral methods (see e.g. Canuto etal.[9] fora comprehensivereview) predictconvergenceofthenumericalsolutionstarting fromagivennumberofharmonics,providedthattheapproximated functionsatisfiessomeregularityrequirements[10].Nevertheless,thisnumberofharmonicsisconfiguration-dependentand hardlypredictable.Inthiswork,wefocusontheinterfacebetweentwoadjacentbladerowsinaturbomachinerystage con-figuration.Therefore,eventhoughHBmethodsarecapabletohandlemulti-stageconfigurations[6],thispaperonlyfocuses onrotor/statororrotor/rotorstages.StudiesontheconvergenceofFourier-basedtime methodsforturbomachinery simu-lationshavebeenpreviously reportedintheliterature,butwithscatteredresults.Forinstance,usinga frequency-domain approach, Vilmin etal. [11] obtain accurate solutions using 5 harmonics for a compressor stage and3 harmonics for a centripetalturbinestage. Fora transonic compressorstage with forcedbladevibration, Ekici etal.[12] use upto 7 har-monicswithatime-domainharmonicbalanceapproach.Finally,forasubsoniccompressorstage,Sicot etal.[7]reportthat 4 harmonicsistheminimalrequirementtoproperlycapturewakeinteractionsasillustratedinFig. 1.

Theprecedingexamplesshowthatnoconsensusexistsintheliteratureconcerningthenumberofharmonicsneededto achieveconvergence,evenforsimilarconfigurations.Thegoalofthepresentpaperistwofold:toanalyzetheconvergenceof Fourier-basedtimemethod,withfocusonturbomachineryapplications,andtoprovideacriterionfortheminimalnumber ofharmonicsrequiredtoachieveaspecifiedaccuracylevel.

Thepaperisorganized asfollows:first,werecallthedesignprinciplesofthetime-domainharmonicbalanceapproach andtheoretical resultsabout theconvergence ofFourier-based methods.Second, the HB methodis applied tothe linear advectionequationsupplementedwithunsteadyboundaryconditionsofdifferentdegreesofsmoothness,to highlightthe impactofsolutionregularityonHBconvergence.Third,amodelproblemrepresentativeofaturbomachinerywake-passing configuration is set up, and different errormeasures are introduced to compare the numerical and analytical solutions. Theseerrormeasuresallow finallytodefineapredictiontool,whichisappliedtocontra-rotatingopenrotorsimulations.

2. Time-domainharmonicbalanceapproach

2.1. Derivationforfluiddynamicsapplications

TheUnsteadyReynolds-AveragedNavier–Stokes(U-RANS)equationsarewritteninfinite-volumesemi-discreteformas:

V

∂

W

∂

t

+

R

(

W

) =

0

,

(1)

withV thevolumeofthecell,R theresidualresultingfromthediscretizationofthefluxesandthesourceterms(including theturbulentequations),andW theaverageoftheunknowns(conservativevariables)overthecontrolvolume.

Ifthe meanflowvariables W areperiodic intime withperiod T

=

₂

π

/

ω

,so aretheresiduals R

(

W

)

andtheFourier seriesofEq.(1)is:

∞



k=_−∞

(

ik

ω

V



Wk

+

Rk

)

eikωt

=

0

,

(2)

where



Wkand

RkaretheFouriercoefficientsofW and R(W

)

,respectively,correspondingtothemode k: W

(

t

) =

∞



k=_−∞



Wkeikωt

,

R

(

t

) =

∞



k=_−∞

Rkeikωt

.

(3)

Thecomplexexponentialfamilyforminganorthogonalbasis,theonlywayforEq.(2)tobetrueisthattheweightofevery modek iszero,whichleadstoaninfinitenumberofsteadyequationsinthefrequencydomain:

ik

ω

V



Wk

+

Rk

=

0

,

∀

k

∈ Z.

(4)

McMullen etal.[13–15]solveasubset oftheseequationsuptomode N,

−

N

≤

k

≤

N, yieldingtheNon-LinearFrequency Domain(NLFD)method.

The principle of the time-domain Harmonic Balance (HB) approach [4], sometimes referred to as the Time Spectral Method(TSM)[5,16],istouseanInverseDiscreteFourierTransform(IDFT)to castback thissubset of2N

+

1 frequency-domain equations into the time domain. The IDFT then induces linear relations between Fourier coefficients



Wk anda

uniformsamplingofW at2N

+

_{1 instantsintheperiod:}

Wn

=

N



k=₋N



Wkexp

(

i

ω

n



t

),

0

≤

n

<

2N

+

1

,

(5)

withWn

≡

W

(nt)

and

t

=

T

/(

2N

+

1

)

.Thisleadstoanewsystemof2N

+

1 mathematicallysteadyequationscoupled

byasourceterm:

R

(

Wn

) =

−

V Dt

(

Wn

),

0

≤

n

<

2N

+

1

.

(6)

Equivalently,Eq.(6)mayberewrittenas:

V Dt

(

Wn

) +

R

(

Wn

) =

0

,

0

≤

n

<

2N

+

1

.

(7)

BycomparisonofEq.(7)andEq.(1),itappearsthat thesourcetermV Dt

(W

n

)

plays theroleofaspectralapproximation

ofthetimederivativeinEq.(1).Thisnewtimeoperatorconnectsallthetimeinstantsandcanbeexpressedanalyticallyas:

Dt

(

Wn

) =

N



m=₋N dmWn+m

,

(8) with: dm

=



π T

(

−

1

)

m+1csc

(

πm 2N+1

),

m

=

0

,

0

,

m

=

0

.

(9)

AsimilarderivationcanbemadeforanevennumberofinstantsbutVanderWeide etal.[17] prove thatitcanleadtoa numericallyunstableodd–evendecoupling.

Apseudo-time(

τn

)derivativeisaddedtoEq.(6)tomarchtheequationsinpseudo-timetothesteadystatesolutionsof alltheinstants:

V

∂

Wn

∂

τ

n

2.2.Convergenceofthespectraloperator

Theconvergenceofthe spectraloperator dependsontheregularity oftheapproximatedfunction. Considera function

u

(

t

)

thatiscontinuous,periodicandboundedin

[

₀

,

T

]

_andlet PN

(

u

(

t

))

denoteitstruncatedFourierseries:

PN



u

(

t

)



=

N



k=₋N

ukeikωt

.

(11)

The

L

_{2-normoftheerrorwrites:}



u



₂

=



T 0





u

(

t

)

₋

PN



u

(

t

)

2dt

1/2

.

(12)

Ifu

(

t

)

ism-timescontinuouslydifferentiablein

[

0

,

T

]

(m

≥

1)andits j-thderivative isperiodicon

[

0

,

T

]

forall j

_≤

m

₋

2 then,there existsk0

∈ [

1

,

N

]

suchthatfork

>

k0:

uk

=

O



k−m



,

(13)

where

uk isthek-thFouriercoefficientofu

(

t

)

.Thisequation meansthat,themoreregular thefunctionis,thefasterthe convergencerate ofthe Fouriercoefficients. The property ofthe errorto decay exponentially assoon asthe function is approximatedbya numberofharmonicsgreaterthank0 iscalledspectral accuracy[9].Notethatk0 isnotknownbutis

ratheressentialfortheanalysis.Fork belowk_{0,approximatingthefunction}u(t)_{withitsFourierseriesyieldsunacceptably} higherrors.

3. Linearadvectionofaperiodicperturbation

ToproperlyassesstheconvergenceofthespectraloperatorandthustheHBcomputations,a modelproblemissetup. Weconsiderthelinearadvectionequation:

∂

u

∂

t

+

c

∂

u

∂

x

=

0

,

(14)

withtheconstantadvectionspeedc assumedtobepositive.Theequationissolvedinthedomain

[

₀

,

1

]

_.Periodic perturba-tionsofdifferentshapes(andtherefore,differentsmoothness)areimposedattheleftboundary:

u

(

0

,

t

) =

ul

(

t

),

(15)

whereul isa periodicfunctionofperiod T

=

1

/c.

Theseperturbations areadvectedacrossthecomputationaldomainand

leave through the rightboundary. After a transient oftime length Ttrans

=

1

/

c, thesolution at anypoint x in thespace

domainachievesaperiodicstate.Theexactsolutionforthisperiodicstateisaperiodicfunctionoftheform:

uex

(

x

,

t

) =

ul

(

t

−

x

/

c

).

(16)

Thespacederivativeisdiscretizedbymeansofacenteredfourth-orderfinite-differenceschemeona uniformfineCartesian mesh(

x

=

₀

.

002).Accordingtothetheoryofcharacteristics,thesolutionatthelastmeshpointontherightofthedomain cannotbeimposedandisthusextrapolatedfromtheinside,forthelasttwopoints.Time-discretizationisachievedthrough theHBmethoddescribedinSection2.

Astandardfour-stepRunge–Kuttamethod[18] isusedtopseudo-timemarchtheHBequationstothesteadystate.The CFLnumberinpseudo-timeissetto1toensurestabilityoftheexplicittime-marchingscheme.

Tocompare numerical andexact solutions, the discrete

L

_{2-norm of theerror in time is computed over all the time} instantsateachgridpointsoverthedomain.Then,theaverageinspaceiscomputed.

Inthefollowing,weconsidersolutionsoftheprecedingproblemfordifferentchoicesoftheleftboundarycondition. 3.1. Sumofsinefunctions

Firstofall,afinitesumofsinefunctionsisappliedattheleftboundary:

ul

(

t

) =

cos

(

ω

t

) +

sin

(

2

ω

t

) +

cos

(

3

ω

t

) +

sin

(

4

ω

t

) +

cos

(

5

ω

t

).

(17)

Harmonicbalance computationsarerunwith1to10 harmonics.Foreach computationrangingfrom N

=

1 to N

=

6,we showspatial distributionsofthesolutionatthreetime instants,namelyt

=

_0,t

=

T

/

3 andt

=

_2T

/

3.Sincetheseinstants arenotnecessarilyusedintheHBdiscretization,atemporalinterpolationisperformed.Todoso,thefrequencycontentof theHBsolutionis usedtogetherwithan inverseFouriertransformon thetime-vector

[

0

,

T

/

3

,

2T

/

3

]

.Fig. 2depictsthe resultsofHBcomputationsusing1to6 harmonics.Theanalyticalsolutionisalsoreportedforcomparison.

Fig. 2. Linear advection of a sum of sine functions: numerical solutions at different time instants for different numbers of harmonics.

The accuracy of the solution improves withthe numberof harmonics, until it reachesthe frequencycontent ofthe injected signal, i.e. 5 harmonics. For highersampling levels, the results of HB computationsare superimposed withthe analyticalsolution.

The

L

_{2-normoftherelativeerrorasafunctionofthenumberofharmonicsisshowninFig. 3.Tworesultsaredisplayed:} one forthereferencemesh(2000 gridpoints)andone fora refinedmesh(4000 grid points).TheconvergenceoftheHB

Fig. 3. Linear advection of a sum of sine functions: convergence of the HB method error.

Fig. 4. Linear advection of a sum of sine functions: discrete Fourier transform.

computations is slow for N

_≤

_{4. However, when the number of harmonics composing the injected function is reached} (N

=

5), theerrorisminimumandcomputingmoreharmonicsdoesnot changetheerror.It seemsthat theconvergence rateofFourier-basedtime methodsis inherentlylinked tothe spectrumofthe temporalphenomenon that onewants to capture. Here a finitediscrete spectrum composed ofonly five harmonicsis imposed. The value ofthe plateauobtained afterN

=

_{5 isrepresentativeoftheerrorintroducedbythedifferentdiscretizations.Infact,refiningthemeshchangesthis} valuewithout modifyingtheerrorlevels ofthelowerharmonicspoints. Notethatthe erroristhetrueresidual,meaning thatthecomputationiscomparedtotheanalyticalresult.Thisiswhytheonlywaytohaveazeromachinevaluewouldbe tohaveaninfinitenumberofgridpointsandpseudo-iterations.

The temporal discrete Fourier transformof the computational results is compared to the analytical results in Fig. 4. When the number ofharmonics grows in the spectral computations, the Fouriertransform gets closer to the analytical solution.When theHB solutioncontains the whole frequencycontentof theinjected function, thenumericalresults are superimposed with the analytical ones. For intermediate sampling frequencies, as for instance the three-harmonics HB computation,theresolvedharmonicshavehigheramplitudesthantheexactone,sincetheycompensateforharmonicsthat arenotresolved.

Whenthenumberofharmonicscomposing thespectrumofthecomputedsignal isreached, thecomputationalresults aresuperposedwiththeanalyticaloneswithplottingaccuracy,namelyweobtainspectralaccuracy.

3.2.Towardturbomachinerywakes

Considerforsimplicitya turbomachinerystage composed oftworotors. Awake isshedbehind theupstream andthe downstreamrotor.Itisstationaryintheframe ofreferenceattachedtotheupstream wheel.However,whenitcrossesthe rotor–rotorinterface,thewakebecomesunsteadyintheframeofreferenceofthesecondwheel (Fig.5). Thus,anupstream steadyspatialdistortionbecomesunsteadyinthedownstreamrow.

LakshminarayanaandDavino[19] showedthatthewakebehind turbomachinerybladefollowsasimilaritylawforthe velocity.ItcanbeempiricallyapproximatedbyaGaussianfunction:

u

(θ ) =

um

− 

u

·

e−0.693(2

θ

Fig. 5. Characteristic rotor–rotor configuration of a turbomachinery. Dotted lines depict wakes.

whereum denotesthefree-streamvelocity,



u theaxialwakevelocitydeficit,

θ

theazimuthalcoordinateandL thewake width,definedasthefullwidthathalfmaximum.

Thus,inthedownstreamreferenceframe,wakescomingfromtheupstreamwheelcanberepresented,toafirst approx-imation,astheperiodicadvectionofaGaussianfunctionfromtheinter-wheelinterface.

We consideragainthelinearadvectionproblem, withul now takenequalto aGaussianfunction oftheformEq.(18).

ThewidthL issetto10%ofthedomainsize,umissettoc and

u to

10%ofum.

Fig. 6 depicts the HB computationsfor one to six harmonics. The numerical solution convergences toward the exact Gaussian functionwhenincreasingthenumberofharmonics.Whenthisnumberistoosmall,thewidthandthedepthof thewakearebadlyapproximatedbythemethod,andthesolutionexhibitssomespuriousoscillations.

Fig. 7showsthequantitativeconvergenceofthe

L

_{2-normoftherelativeerror.Theconvergencecurvesfortheprevious} functionisalsoreportedforcomparison.Theerrorfollowsnowanearlyexponentialconvergence.

The discrete Fourier transformof the resultsis depicted against theanalytical result inFig. 8. The N

=

2 and N

=

4 computationsbadlycapturetheamplitudesoftheresolvedharmonics.StartingfromN

=

_{6,some ofthelowerfrequencies} are correctly captured,whereas highfrequencies are always under-estimated. This improveswhen further harmonicsare addedtothecomputation.

For a better understanding of the HB convergence behavior, we consider the spectral content of the Gaussian wake model.Precisely,theFouriertransform

g ofaGaussianfunctiong definedas:

g

(

x

) =

Ae−αx2

,

(19)

where A and

α

areconstants,is:

g

(

f

) =

A′e−α′f2

,

(20) where:















A′

=

A



π

α

,

α

′

=

π

2

α

.

(21)

ForthesimilaritylawofLakshminarayanaandDavino,

α

and

α

′_{canbeidentifiedas:}

α

=

0

.

693



2 L

2

,

α

′

₌

1 0

.

693



π

L 2

2

.

(22)

Theexponentialfactorofthewakelaw

α

isinverselyproportionaltoitsFouriercounter-part

α

′_{,meaningthattheirwidth}

willvaryinoppositeway:thethinnerthewake,thewideritsspectrumandvice-versa.

Theconvergencerateisinherentlylinkedtothespectrumoftheconsideredunsteadysignal.Asforthepresentcasewe knowtheanalyticalwakespectrum,wedefine thetheoreticaltruncationerrorastheratiooftheenergycontainedinthe unresolvedpartofthespectrumtotheoverallenergycontentofthefullspectrum:

ε

th

(

f

) =











_∞ f

|

g

(ζ )|

2d

ζ



_∞ 0

|

g

(ζ )|

2d

ζ

.

(23)

Introducingtheerrorfunctiondefinedas:

erf

(

x

) =

_√

2

π

x

0 e−t2 dt

,

(24)

Fig. 6. Linear

advection of a Gaussian function representing a turbomachinery wake: numerical solutions at different time instants for different numbers of

harmonics.

andthecomplementaryerrorfunctiondefinedas:

erfc

(

x

) =

1

−

erf

(

x

),

(25)

Fig. 7. Linear advection of a Gaussian function representing a turbomachinery wake: convergence of the HB method error.

Fig. 8. Linear advection of a Gaussian function representing a turbomachinery wake: discrete Fourier transform.

∞

0





g

(ζ )





2d

ζ =

1 2 ∞

−∞





g

(ζ )





2d

ζ

(26)

=

A′ 2 2



π

2

α

′

,

(27) and: ∞

f





g

(ζ )





2d

ζ =

A ′2 2



π

2

α

′erfc

√

2

α

′f



.

(28)

Thetheoreticaltruncationerrorcanthenbewrittenas:

ε

th

(

f

,

L

) =



erfc



2

α

′

(

L

)

f



.

(29)

One can notice from Eq.(29) that the truncation errordoes not depend on the wakedeficit

u but

only on the wake width L.

Eq.(29)isdepictedinFig. 9.Itcanbeseenthat thewiderthespectrum,thehigherthenumberofharmonicsneeded toreachacertainleveloferror.Moreover,forathinwakewidth(e.g. 2%ofthepitch)thenumberofharmonicsrequiredto captureitwithatruncationerrorof10%isupto25 harmonics.InthelimitofL

→

0,thewakebecomesaDiracfunction which represents the worst possible case. In the preceding example, the Gaussian function had a width of 10% which, accordingtoEq.(29),iscapturedbyusingN

=

7 harmonicsfora10%errortarget.

Fig. 9. Theoretical truncation error of the Lakshminarayana and Davino wake law.

Fig. 10. Definition of pitches and IBPA used with phase-lag boundary conditions.

4. Applicationtoamodelturbomachineryconfiguration

4.1. Extensionoftheharmonicbalanceapproachtoturbomachinerycomputations

Toefficientlyapply theHBapproachtoturbomachineryconfigurations,phase-lagboundaryconditions[20] areusedto cutdownthemeshsizebyusingagridthatspansonlyonebladepassageperrow.Thephase-lagboundaryconditionsare twofold:i)theazimuthalboundariesofapassageandii)thebladerowinterface whichmusthandledifferentrowpitches oneithersides(Fig. 10).Furthermore,intheHBframework,eachrowcapturesthebladepassingfrequencyoftheopposite rowleadingtodifferenttimesamplessolvedineachrow.

Phase-lagboundaryconditionsonlyaccountfordeterministicrowinteractions,andthereforecannot modelnatural un-steadyphenomenon, such asvortex shedding for instance.Actually, the phase-lag condition isbased on the space–time periodicityofthe flowvariables. Itstatesthat the flowin onepassage

θ

isthesame asthenext passage

θ + θ

butat anothertime t

+

δ

t:

W

(θ + θ,

t

) =

W

(θ,

t

+

δ

t

),

(30)

where

θ

isthepitchoftheconsidered row.Thetime lag

δt can

beexpressed asthephase ofarotatingwavetraveling atthe same speed asthe relative rotation speed of the opposite row:

δt

=

β/

ω

β. The Inter-Blade PhaseAngle

β

(IBPA)

dependsoneach rowbladecount andrelative rotationspeed.It isanalytically givenbyGerolymos andChapin[21].The FouriertransformofEq.(30)impliesthatthespectrumoftheflowina passageisequaltothespectrum oftheneighbor passagemodulatedbyacomplexexponentialdependingontheIBPA:



Wk

(

x

,

r

, θ + θ

G

) = 

Wk

(

x

,

r

, θ )

eikβ

.

Attheazimuthalboundaries,thismodulationcanbecomputedontheflyintheHBframeworkasasampling ofthetime periodisalways knownanditisstraightforward toderive ananalytic formulationinthetime domain(see Ref. [7]). The bladerowinterfaceismorecomplexasthedifferentpitchesandrelativemotionoftherowsrequiretoduplicatetheflowin theazimuthal directionusingthephase-lagperiodicity.Atimeinterpolationalsooccurstotakethedifferenttimesamples into account and a non-abutting mesh technique is applied as the mesh will unlikely have matching cells. To remove spuriouswaves,anover-samplingfollowedbyafilteringareperformed.

Fig. 11. Model turbomachinery configuration.

The time-domain harmonicbalance method has beenimplemented by CERFACS in the elsA solver [22] developedby ONERA. Thiscodesolves theRANS equationsusingacell-centeredapproachonmulti-blocksstructuredmeshes.Usingthe HB method, significant savings in CPUcost have been observed in applications such as rotor/stator interactions [7]and dynamicderivativescomputation[23].

4.2. Numericalsetup

Weconsiderasimplified configurationmodelinga turbomachinerystage.Theconfigurationconsistsofaspatially peri-odicazimuthalperturbationadvecteddownstreamoftheinletboundaryofthecomputationaldomain.Thedomainismade oftwogridblocksinrelativemotionsothat theperturbation,whichissteadyintheupstreamblock, isseenasunsteady by the downstreamone. It isthus representativeof turbomachinerywakes advected across an inter-wheelinterface. The blocksaregeneratedincylindricalcoordinatessuch thatthepresentedconfigurationcanbeassimilatedtoasliceofa tur-bomachinerystage.Withoutlossofgenerality,wesettherotationvelocityoftheupstreamblocktozero(stator).Thestator iscomposedofBstator

=

10 “virtual”bladesandtherotorby Brotor

=

12 “virtual”blades.Thesearetermedvirtualbladesas

no bladeisactuallymeshed. Thepitchratiois representativeofcontra-rotatingopenrotor applicationsinwhichthefirst rowcontainsmorebladesthanthesecond(seeSection5).Moreover,intheseapplications,thenumberofbladesistypically smallerthanclassicalturbomachineryconfigurations.

A wake is axially injected at the inlet of the stator block following the Lakshminarayana and Davino similarity law definedinEq.(18).ItisschematicallyrepresentedinFig. 11.Thisisthusarepresentativemodelproblemofthewakeshed byanupstreamrowthatcrossestherowsinterface,herethestator–rotorinterface.

TheflowismodeledthroughtheEulerequationsinordertoavoidwakethickeningassociatedwithviscouseffects.The velocityisnotimposedattheinletdirectlybutratherthroughthetotalpressureandtotalenthalpydistributions:

p_i0

(θ ) =

piref



1

− 

pi

·

e−0.693(2 θ L)2



_,

₍₃₁₎ h_i0

(θ ) =

hiref



1

− 

hi

·

e−0.693(2 θ L)2



_,

₍₃₂₎

where p_i0 istheinlettotalpressure,

p

i thetotalpressure deficitinthewake,hi0 theinlettotalenthalpy,

h

i thetotal

enthalpydeficitinthewakeandL thewakewidth.Toimposearealisticdistortion,thetotalpressureandenthalpydeficits areestimatedfromaseparate turbomachinerysimulation.Thisleadsto

p

i

=

0

.

025 and

h

i

=

−

0

.

007.Thenegativesign

isduetooverturninginthewake,whichisduetovelocitycomposition,andthereforespecifictorotors.Thestaticpressure ps₁ isimposedattheoutlet:

ps1

=

pref

(

1

+

γ−₂1M_ref2

)

γ γ−1

.

(33)

ThemeanvelocityisthussetbyimposingthetargetmeanMachnumbervalueMref.Attheazimuthalboundaries,phase-lag

conditions[20]areusedtotakeintoaccountforthespace–timeperiodicity.

Roe’sscheme[24]alongwithasecond-orderMUSCLextrapolationisusedforthespatialdiscretizationoftheconvective fluxes.AnimplicitbackwardEulerschemeisusedtomarchtheHBequationsinpseudo-time.

Aparametric studyiscarriedout overthetwo parametersthat influencethetruncationerrordefinedinEq.(23): the numberofharmonicsandthewakewidth.Thenumberofharmonicsusedforthecomputationsrangesfrom1to25.The wake width L, that drives Eqs. (31)and (32), varies between1% and 30% according to a logarithmic scale to ease the visualizationoftheresults.375 computationsareperformedintotal.

Fig. 12. Convergence of the turbomachinery stage computations.

Eachgridblockhasaradialextentoffivegridpoints(i.e. fourcells).Theazimuthalgriddensityinthestatorandrotor blocksiskeptsimilartoguaranteeaconsistentcaptureofthewakeoneachsideoftheinterface.Todoso,if

θ

cell denotes

theazimuthallengthofacellattheinterface,then:

θ

_cell

=

2

π

Bstator 1 Nstator

=

2

π

Brotor 1 Nrotor

,

(34)

whereNstator andNrotor arethenumberofcellsinthestatorandtherotor,respectively.

Meshconvergenceforthethinnestwake(1%ofthepitch) isobtainedwith500 cells intheazimuthaldirectionofthe statorwhichgives600 cellsfortherotorblock.30 gridpointsareputintheaxialdirection.Moreover,aconstantaspectratio of5withrespecttotheazimuthallengthofthecellsiskept,whichsetstheaxiallengthofthecase.Thisleadstoa total numberofgridpoints ofapproximately170 000.Notethatthememoryrequirementofan HBsimulationis2N

+

_{1 times} thatoftheequivalentsteadycase.AnequivalentsteadycomputationtoN

=

_{25 wouldthusrequire}

(

2

×

25

+

₁

)

×

170000

=

8670000 gridpointmesh.ThegridusedforthecomputationsisshowninFig. 11.

ConvergenceoftheiterativeprocedureusedtosolvetheHBequationsisachievedafter3000 iterationsaspresentedin

Fig. 12.Theresidualisdecreasedby morethanthreeordersofmagnitudeforallthesimulations. Smalldiscrepancies are observedontheresidualconvergencefordifferentwakethicknesseswhilethenumberofharmonicsdoesnotinfluencethe convergence(Fig. 12(a)).

4.3.Spectralconvergencestudy

Theprimaryinterestinthissection isthewakecapturingcapabilitiesoftheFourier-basedtimemethodintherotating part.Toanalyzethis,twoerrormeasuresaredefinedandevaluated.

Those measures address the loss of information introduced by the HB approach at the interface. This loss is firstly evaluatedby analyzingthespectrumfromapurely spatialpoint ofview.Then, ahybrid spatial/temporalpoint ofview is taken.Thisfinally allowstoassessthefilteringintroduced bytheharmonicbalancemethodon boththetime andspatial signals.

Infact,inthestatorpart,thewake issteadyandisthusnot filteredbytheHBoperator. Conversely,intherotor part, thesteadywakebecomesunsteadyduetotherelativespeeddifferencebetweenthestatorandtherotor. However,onlya finitenumberofharmonics N isusedtodescribetheunsteadyfield,hencethefiltering.

4.3.1. Spatial-spectrumbasederrormeasure

Thefirst errorquantification

ε1

issetup to quantifythisfilteringby usingonly spatial informationandisdefinedas the

L

_{2-normappliedonthedifferencebetweentherotorandthestatorspectra.Itisequivalenttotheanalyticaltruncation} errordefinedinEq.(23).Indeed,theerrorisdefinedastheratiooftheunresolvedenergyintherotorblocktotheenergy ofthefullspectrum,i.e. thatofthestatorblock:

ε

1

(

N

) =













fmax f=1

|

s θ N

(

f

)

−

r θ N

(

f

)|

2

f

max f=1

|

s θ N

(

f

)|

2

,

(35) where

sθ

N denotesthespatialFouriertransform(indicatedbythe

.

operator)oftheazimuthalextraction(shownby

super-script

θ

)oftheresultofanHB simulationusingN harmonics,inthestator;

r denotesthespectrumofthesignaltransferred totherotor.Thehighestfrequencypresentinthespectrumisdictatedbythespatialdiscretization.Thus, fmax

=

1

/

2

θ

cells,

Fig. 13. Wake of L=5% width extracted in stator and rotor blocks. Signal and spatial Fourier analysis for different computations.

usingthenotationsofEq.(34).Astheazimuthalcellsizeissimilarinbothblocks,thesamesamplingisusedleadingtothe samefrequenciesinbothstatorandrotorspectra.Detailsofthealgorithmusedtocompute

ε1

aregiveninAppendix A.

Theazimuthalvelocitydistributions(lefthand-side)andthecorrespondingspatialspectra(righthand-side)arepresented inFig. 13forarelativewakethicknessof 5%withrespecttothepitchandforHBcomputationsusingN

=

2,5,10and20,

Fig. 14. Evaluation of the error due to the wake capturing using the first error quantificationε1, and comparison with analytical error (dotted lines).

Fig. 15. Discrepancies between the spectrum at the interface and in the rotor block, for varying wake widths and number of harmonics.

respectively.Forthestator,the azimuthaldistribution followsaGaussian functionasexpected. Onthecontrary,therotor distributionisaliasedby theHBdiscretizationandexhibitsspurious oscillationsthattendtodisappearwhenthenumber ofharmonicsusedinthecomputationincreases. For N

=

10,some oscillationsarestill present,butthewakecapturedin themovingblockbeginstoconvergetothatleavingtheupstreamblock.

ThefilteringintroducedbytheHBapproachactsprimarilyonthetimeresolution.Forunder-resolvedHBcomputations, adissipation erroris observed.Thisdissipation isnot spatially uniformandgivesrise todispersion errors onthe spatial spectrumandtospurious high-frequenciesasshowninFig. 13.TheseeffectsvanishwhentheHBcomputationsconverge, i.e. forN

_≥

10.Thisexplainswhythespectrumoftheunresolvedspuriousfrequenciesisimposedtohaveazeroamplitude valuetocompute

ε1

.

Inspectionofthespectra suggeststhesameconclusions. The amplitudeof



ρ

U improveswhen increasing thenumber ofharmonics.Aspreviouslymentioned,forunder-resolvedHBcomputations,adispersionerrorisintroducedandspurious high-frequenciesappearinthe spatial spectraasshowninFig. 13for N

=

_{2 to} N

=

_10.For N

=

_{20,the spectrumofthe} rotor blockmatches that ofthe statorblock. Thisisconsistent withthe theoreticalanalysis, inwhich morethan N

=

₁₀ harmonicsareneededtocapturethewakewithlessthan20% oferrorforthisparticularwakewidth(seeFig. 9).

In summary,for thiswake thickness, the temporal filtering on a simulationinvolving less than tenharmonics is too harshandleadstoasignificantamountofunresolvedenergy,whichdeterioratesthenumericalrepresentationofthewake. Foramore quantitativeanalysis, we compute theerrormeasure

ε1

foreachcomputation ranging overdifferentwake thicknesses and numbers of harmonics. Fig. 14 summarizes the results. Because it quantifies the unresolved energy in comparisontotheresolvedenergy,

ε1

exhibitsabehaviorsimilartothatofthetheoreticalerror

εth

foraGaussianfunction (Eq.(9)).Theiso-errorcontourshaveasimilarshapeastheanalyticalones.Theconclusionsareequivalent:thetruncation errordecreases withthewake thicknessandwiththe numberofharmonics usedto capturethe wake.Nevertheless,for thickerwakesandhighernumbersofharmonics,theerrormeasure

ε1

isover-estimated.Forinstance,around N

=

15 and for L

=

_25%,

ε1

_≈

₁₀−2 _{whereas the theoretical error}

_εth

_{is less than 10}−4_{. The error measure}

_ε1

_{doesnot represent a}

realistic measurefor such width/numberofharmonics combinations,because ofthe spatial Fouriertransformperformed to compute the error. Indeed, asshown in Fig. 15, the Fouriertransform of the spatial signal in the stator block tends toa plateau.The thickerthe wake,the lower thefrequency forwhichthe plateauappears: approximately 15 harmonics for L

=

10% (seeFig. 15(a)) and6 harmonics for L

=

25% (seeFig. 15(b)). Actually,for an N-harmonic HB computation, the spectrum isexplicitly filtered inthe moving block leading to an amplitudeequal to zero above the N-th harmonic.

Fig. 16. Evolution of the spectrum of the inlet boundary condition for different angular pitch.

Therefore, whenthe HB computationsare converged,the differencebetweenthe spatial spectrain thestator andinthe rotorblockisdrivenbytheplateaupresentinthespatialspectrumofthestatorblock.

In fact,thisbehavior islinked tothe windowing ofthesignal ona boundedinterval, thepitch.Tohighlight that,the influenceofa modificationonthe inletboundaryconditionisanalyzed.Theinlet wakedistortionusedinthe model tur-bomachineryconfigurationis originally basedonthe analyticalLakshminarayana andDavino Gaussianlaw (seeEq. (18)). However,thislawisdiscretizedandimposedonaboundedintervalthatspanstheangularpitch.Astherelativethickness increases, theinletconditiondivergesfromtheanalyticalGaussianlawforwhichtheangularpitchistheoreticallyinfinite. ThisisshowninFig. 16throughthespectraofthreeGaussianlaws.Therelativethicknessofthelawsaremodifiedthrough the size ofthe pitch

θ

. Themultiplication by a factor 100 of the pitch leads to adisappearance of theplateau inthe spectrum,whichaccuratelymatcheswiththeFouriertransformofaGaussianfunction.

Fig. 17. Temporal signal seen at loc 1 and loc 2 for an L=5% wake width.

Tosumup, aplateauappearsinthespatial spectrumofthestatorblock. Thisplateauisexplicitlyfilteredintherotor block above the N-th harmonic, leading to an over-estimationof thefirst errormeasure. Thisover-estimation drives the errorvalueforhighernumberofharmonicsandthickerwakes.

4.3.2. Spatial/timedualityerrormeasure

Togetamorerealisticerrormeasure,wetakeagainintoaccounttheenergylossthroughtheinterface,butbasedona spatial/timeduality.Asthislossofenergyispreciselyrelatedtothefilteringintroducedonthetemporalsignal bytheHB approach,theseconderrorquantification

ε2

addressestheresultonthetemporalinformation.

Neartheinterfaceoftheblocks,considerafixedobserverintherotorframeofreference.Thisobserverseesanunsteady wakepassing asthe blockshavea relativespeed difference.The firsterrorquantificationhasshowntheinfluence ofthe number of harmonics on the spatial signal in the rotor block. The error quantification will now point that thisspatial influenceisduetoatemporalfilteringdonebytheHBapproach.

FollowingthesamenotationasinEq.(35),theseconderrormeasureiswrittenas:

ε

2

(

N

) =













fmax f=1

|

s θ N

(

f

)

−

rNt

(

f

)|

2



fmax f=1

|

s θ N

(

f

)|

2

,

(36)

wheresuperscriptt denotesthetemporalversionoftheFouriertransform.Bydefinition,

ε2

quantifiesthematchingbetween aspatialsignalandatemporalinformation.Again,theerrorisdescribedastheunresolvedenergyintherotorblock,divided bytheenergyofthefullspectrum,e.g. thatofthestatorblock.For

ε1

,theamplitudeoftheharmonicsabovetheN-th one wasimposed tozero.Onthecontrary,for

ε2

,thetemporalspectrumintherotorblock isbyessencenullabovethe N-th harmonic,asthefilteringactsontemporalvalues.Detailsofthealgorithmusedtocompute

ε2

aregiveninAppendix B.

Fig. 17showstimesignalsextractedattwodifferentazimuthalpositionsattheinterfaceoftherotorblock,namedloc 1 andloc 2.Thesmallphase-lagbetweenthetwosignalsisduetothespaceshiftbetweenthetwopoints,andisthesame foranychoice ofthe numberofharmonics used inthe computation. Onthe contrary,differencesin termsofamplitude areonlyduetotheuseofan insufficientnumberofharmonics:asthenumberofmodesusedforthetimeapproximation isincreasedfrom N

=

5 to N

=

15,the amplitudeofthe space-shiftedsignalstends to convergetothe samevalue, and spuriousoscillationstendtodisappear.Therefore,inthefollowing,onlyloc 1 willbeconsidered.

Fig. 18describesthespaceandtime spectraoftheaxialmomentum

ρ

U atloc 1,forcomputationsusingN

=

_2,5,10 and20harmonicsandforawakewidthofL

=

_{5%.Thespatialspectrumcontainsthewholewavelengthcontentassociated} totheincomingwake;onthecontrary,duetothefilteringintroducedbytheHBapproach,thetimespectrumiscomposed ofonlyN harmonics.Forcomputationsusinglessthan10timeharmonics,timespectraaretruncated,andtheamplitudeof

Fig. 18. Spatial/time duality for an L=5% wake width.

Fig. 19. Evaluation of the error due to the wake capturing using the second error quantification (ε2), and comparison with analytical error (dotted lines).

ρ

U differsfromthatofthecorrespondingmodeinthespatialspectrum.Asthenumberoftimeharmonicsisincreased,the amplitudeoflowerharmonicsbecomescloserandclosertothatofthecorrespondingharmonicinthereferencesignal,and errorsmovetowardthehigherresolvedharmonics.ForN

=

20,theamplitudesofthe20 resolvedharmonicsaresimilarfor boththetimeandspacespectra.

Insummary,theprecedinganalysisshowsthat,forunder-resolvedHBcomputations,thetimesignalisaffectedbyboth amplitudeandphaseerrors,sincetheenergycontentisredistributedincorrectlyamongtheresolvedharmonics.

Toquantifythiserror,weapplytheerrormeasuredefinedinEq.(36)toHBcomputationsofthemodelturbomachinery problem,correspondingtodifferentchoicesofthewakethicknessanddifferentnumbersofharmonics.Resultsarepresented inFig. 19.The

ε2

errormapisqualitativelyandquantitativelysimilartothe

ε1

onediscussedintheprevioussection.Again, thetruncationerrormeasuredusing

ε2

forthickwakesandhighnumbersofharmonicsdoesnotfollowthetrendobserved forthetheoreticalerror

εth

,duetotheapplicationoftheFouriertransformonaboundedinterval.

Theprecedinganalysisshowsthat,forHBcomputationsthatarewellconvergedintermsinharmonics,thespatial spec-truminthestatorandthetimespectrumintherotorblocktendtomatch,exceptforadditionalspatialerrorsintroducedby theuseofanazimuthalFouriertransformonaboundedinterval,whichconfirmsthevalidityoftheerrormeasuredefined inEq.(36).

Fig. 20. Truncation, computed and analytical errors for four wake widths.

4.4.Comparisonwiththetheoreticalerrormeasure

Theprecedingresultsshowthatapproximatedtruncationerrormeasurescomputedforthemodelturbomachinery prob-lemusinganonlinearflowmodel(Eulerequations)exhibit trends,withrespecttothewakethicknessandnumberofHB harmonics,in closeagreement withthetheoretical error measurederived in Section 3.2fora Gaussian function. Fig. 20

comparesthedifferenterrormeasuresforHBsimulationsofadvectedwakesofvaryingthicknessversusthenumberof har-monicsusedforthetimediscretization.ThiscorrespondstohorizontalcutsofFigs. 9,14 and 19.Foranumberofharmonics higherthanthecutoffharmonicusedinthephase-lagcondition,thethreeerrormeasuresgiveresultsinavery close agree-ment.Afterthatvalue,boththe

ε1

and

ε2

errormeasuresappliedtothemodelturbomachineryproblemexhibitaplateau. Thesameplateauisalsoobservedon

εmxp

errorwhosedefinitionwillbegiveninthenextsection.Theprecedingremarks suggesttheideathat, sinceallerrormeasures providesimilarresults,atleastup tonumbersofharmonicsofinterestfor practicalapplications.Anapriori estimateofthenumberofharmonicsrequiredtoachieveagivenerrorlevelcouldthenbe obtainedbyusingthetheoreticalerrormeasureequation(29),ifaquickestimateofthewakethicknesscharacteristicofa giventurbomachineryproblemisavailable.Inthenextsection,weshow thatareasonableestimate oftheconvergenceof theerrorcanbeobtainedfromapreliminarysteadycomputationbasedonthemixingplaneinterfacecondition.

4.5.Towardan apriorierrorestimate

Inordertodefineanapriori errormeasurethatcanbeusedtoestimatethenumberofharmonicsrequiredtoachievea reasonableconvergenceoftheHBmethod,wesuggesttoevaluatethefilteredspectrumbyusingapreliminarymixingplane steadycomputation.Indeed,ifpotentialeffectsduetothedownstreamrowcanbeneglected,thespatialinformationatthe interfaceinthestatorblock,essentiallyduetotheincomingwakes,canbecapturedwithouttakingintoaccounttherelative motionbetweenthewheels,i.e. bymeans ofa mixingplane computation.Giventheapproximatedazimuthal distribution atthestatorinterface,weconsiderthecumulativeenergycontentofthesignaluptoagivenfrequency f (or,equivalently, toagivenharmonicN

=

f

/

f₁ where f₁ isthefrequencyvalueoftheconsideredunsteadiness). Thecumulativeenergyis definedas: E

(

f

) =



f 0

|

g

(ζ )|

2d

ζ



_∞ 0

|

g

(ζ )|

2d

ζ

,

(37)

where

g isthespectrumofthequantity ofinterest,heretheaxialmomentum.BycomparisonwithEq.(23),therelation betweentherelativeaccumulatedenergyE andthetruncationerror

εmxp

is:

Fig. 21. MU-LS convergence – non-dimensional entropy at 75% span.

Notethatthislasterrormeasureisbasedonlyontheamountofunresolvedenergythatisleftinacomputation ifthe spatialsignalistruncatedatagivencutofffrequency f ,anddoesnotrequireanyinformationfromtherotorblock.Infact, itdependsonlyonthecharacteristicsoftheincomingwake.

TocheckifthenewerrormeasurerepresentsanaccurateestimateofthetruncationerrorofanHBsimulation,wecarry outagainaparametricstudyoftheerrorversusdifferentwakethicknessesandnumbersofharmonics(equivalently,cutoff frequencies),andcompare theresults tothose ofthe aposteriori errormeasures obtained forthemodel turbomachinery problem(

ε

1,

ε

2)andtothetheoreticalerror

ε

th.Resultscorrespondingto

ε

mxparesuperposedtothecorrespondingcurves in Fig. 20.The apriori errormeasure (

εmxp

) matches the theoreticalestimate (

εth

) andtheaposteriori measures (

ε

1,

ε2

)

overawide rangeofharmonics.Similarlytotheaposteriori errors

ε1

and

ε2

,theapriori error

εmxp

exhibitsaplateaufor highnumberofharmonicsandwakethicknesses,duetotheapplicationoftheFouriertransformonaboundedinterval.We alsostressthecloseagreementbetween

εmxp

and

εth

:specifically,estimatesofthenumberofharmonicsneededtocapture 99% ofthecumulativeenergy(equivalently,togeta truncationerrorequalto10%, avaluethat willbejustifiedlater)are identicalforallerrormeasures.

5. Applicationtoacontra-rotatingopenrotorconfiguration

Originally, this study was conducted to understand the convergence issues observed on Contra-Rotating Open Rotor (CROR)configurations.Incontrasttoturbomachineryapplications,convergenceintermsofharmonicshasbeenobservedto beslowonsomeconfigurations.

5.1. Presentationofthecases

Toinvestigatethisissue,twoCRORconfigurationsarestudiedatdifferentoperatingconditions:

1. a Mock-up CROR (noted MU) designed by Safran to be investigated in a wind tunnel (i.e. ground condition: Pi

=

101300Pa and Ti

=

293K). Tworegimes are considered representative of low (LS) andhigh-speed (HS) conditions

(differentrotationspeedsandbladeangles),

2. the Airbus-designedAI-PX7 [25] CROR (noted AI)atcruise condition:high-speedandflight level(i.e. Pi

=

23842 Pa

andTi

=

219

.

6K).

5.2. ResultsofHBcomputations

Figs. 21,22 and23showthenon-dimensionalentropyat75%spancomputedbytheHBmethodforthethree configu-rations.TheMU-LS configurationhasthefastestconvergence.Thereareindeedsomespurious entropywavesdownstream thebladerowinterfaceforN

=

1 and2butnoneareobservedstartingN

=

4.

Fig. 22. MU-HS convergence – non-dimensional entropy at 75% span.

FortheMU-HSconfiguration,onecanobserveinFig. 22(g)thattheN

=

_{7 HB computationstillpresentssome spurious} wavesdownstreamtheinterface.Itbecomesnegligibleforafinersampling.ThemaindifferencewiththeMU-LS configura-tionisthebladeangle.BycomparingFig. 21andFig. 22,onecanobservethatthebladeangleislower inthehigh-speed case.Therefore,evenifthewakeisofsimilarthicknessdownstreamthefrontrotor,itimpactstheaxialbladerowinterface withalowerangleandthereforelooksthinner.Assumingthattheflowangledownstreamthetrailingedgeisthesameas thebladeincidenceangle

ξ

,thewakethicknessobservedbythebladerowinterface Litf is

Litf

=

L

cos

(ξ )

.

(39)

When

ξ

rises from low-speed to high-speedconfiguration, L will remain almost constant but Litf will decrease andthe

spectrumwidens.

FortheAIconfiguration,Fig. 23showsthattheconvergenceisnotachievedasthefinestHBcomputation(N

=

10)still doesnotcapturethewakecorrectlythroughtheinterface.Itisthickenedbythelowtimeresolution.Althoughthesolveris abletoaccountforanarbitrarynumberoftimesamples,therequiredmemorywouldleadtorunonmoreprocessorsand thusblocksplittingwouldbecomenecessary.Astheadvantageoverclassicaltime-marchingschemewouldvanish,onlyN

=

1to10HBsimulationswereperformed.

TheobservedconvergencedifferencesbetweentheAIandMU-HSconfigurationscan beattributedtotwo main differ-ences:

1. theAIconfigurationisatscalemeaningthattheradialextentisseveraltimeslargerthantheMUconfiguration.Asthe pitch-wiserelativewakewidthisdefinedas

Lpitch

=

L B

Fig. 23. AI-HS convergence – non-dimensional entropy at 75% span.

therelativewakewidthwilldecreaseforhigherradius R.Italsoexplainsthedifferencewithclassicalturbomachinery: asthenumberofblades B canbe oneorderofmagnitudehigherinthelattercasethaninaCRORconfigurationand thediameterlower,therelativewakethicknessesarehigherandthespectrumnarrower.

2. theviscosityisalsodifferent:applyingtheSutherlandlawforairatgroundandflightlevelleadstodynamicviscosities of1

.

807

·

10−5 _{Pa s and1}

_.

₄₃₄

_·

₁₀−5 _{Pa s,respectively.Withlowerviscosity,thebladeboundarylayeristhinnerandthe}

generatedwakesarethinneraswell.Furthermore,themixingwiththemainflowisweakerandthethickeningofthe wakesisalsoslowerleadingtoathinnerwakereachingthebladerowinterface.

5.3. Predictiontoolbasedonthewakethickness

Toestimatethewakethickness,acurvefittingalgorithmisusedtofittheCFDwakestotheLakshminarayanaandDavino Gaussianwakelaw.Onlytherelativespanbetween10%and70%isconsideredaselsewhere,thewakeinteractswiththehub

Fig. 24. Estimation of the relative wake thickness for the three contra-rotating open rotor configurations.

Fig. 25. Reconstructions of a wake depending on the energy content kept in the signal.

boundarylayerandtipvortex.ThisestimationisplottedinFig. 24forthethreeconfigurations.Thewakethicknessisalmost constantalongthespanforthetwoHSconfigurations.Inopposite,theMU-LSshowsanincreaseat50%oftherelativespan. Thisis dueto a large tangential distortion that is attributedto flow separation.Thus, the wake widthestimation isnot reliableinthisregionfortheMU-LSconfigurationasthetangentialdistortionisnolongerGaussian-shaped.Nevertheless, usingFig. 24,thewakewidthsoftheAI-HS,theMU-HSandtheMU-LSareapproximately4%,9.5%and20%,respectively.

At this point, using the theoretical error defined in Eq. (29), the last thing needed to provide an estimation of the required number ofharmonics, is the level ofaccumulated energy (or alternatively the level of error) that ensures the convergenceof the HB method.The level of accumulatedenergy (definedin Eq. (37)) required for a computation to be rigorouslyconvergedisdifficult toestimate. It seemsreasonable, froman engineeringstandpoint, toconsider that a99% accumulationofenergyshouldbea goodcriterion.Toemphasizethat, thereconstructionofa wakeasafunctionoffour levelsofcumulativeenergy E isdepictedinFig. 25.Onecanseethatareconstructionusingonly50%oftheenergyleads to asignal that has neitherthe rightwake deficitnor the correctwidth. Using 90% and95% ofthe energyimprove the resultingshapebutlargesecondary oscillationsremain,withabadcaptureofthewakedeficit. Inopposite,by using99% oftheenergytoreconstructthesignal,onlyminoroscillationsareseenandthewakewidthanddeficitarerecoveredwith morethan95%accuracy.Thus,the99%energythresholdensuresthatthewakewillbecorrectlytransmittedtotheopposite row,whichisthepriorconcernofthispaper.

Therefore,based on thisvalue andthe estimation ofthe wakewidth forall the three CROR configurationsshown in

Fig. 24, one can evaluate the number ofharmonics neededto compute such applications.In fact, based on the analytic formuladerived inSection 3.2andtheequivalenceoftruncationerrorandaccumulatedenergygivenbyEq.(38),onecan deducethenumberofharmonics N neededtocaptureatargetlevelofaccumulatedenergyE foragivenwakewidth:

N

(

E

) =

erfc

−1

_[

₁

₋

_E

_]

√

2

α

′

,

(41)

where

α

′_{isthewakeparameterasdefinedinSection3.2:}

α

′

₍

_L

_{) =}

1 0

.

693



π

L 2

2

.

(42)

Fig. 26. Energyaccumulationbyharmonicsforallspans.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversion ofthisarticle.)

Here, the theoretical estimation of the number of harmonics neededto recover 99% of the energy is then 17, 7 and 3 for, respectively, the AI-HS, the MU-HS andthe MU-LS. These numbers explain whythe AI-HS configuration is still not convergedafterN

=

_{10 harmonics.Infact,such acomputationleadstoacaptureofonly87%ofthesignalenergy.Fig. 25}

supports theargumentthat withthislevelofenergy,the wakeisnot properlycapturedasa90% energysignal doesnot accuratelyestimatethewakedeficitandthickness.

Withthisapproach,onecandeduceapproximatelythenumberofharmonicsneededtocompute suchCROR configura-tionsusingFourier-basedtimemethodsforatargetlevelofaccumulatedenergy.HoweveritislimitedtoGaussianwakes.If thewakeshapeisveryfarfromaGaussiancurveorifanothertangentialdistortionreachestheinterface,thepresent pre-dictiontoolcannotbeused.However,asdemonstratedinSection4.4,theanalyticerrorandtheerrorbasedonanazimuthal Fouriertransformofthedistortionseenjustupstreamtheinterfaceforamixing-planeconfigurationareequivalent. 5.4. PredictiontoolbasedonanazimuthalFouriertransform

Thus, a more general way to analyze the spectrum in a wake is to perform an azimuthal Fourier transform at the rowsinterface ina mixing-planecomputation.Itencompassesboththewakeanalysisdoneabove andalsoanytangential disturbances,asforinstancetheviscosityeffectsnearthehuborthetipvortex.Detailsofthealgorithmusedtocompute thetangentialaccumulatedenergyfromamixingplanecomputationaregiveninAppendix C.

Tohaveaglobalinsightoftheenergycontainedinthetangentialdistortionacrossthewholespan,theenergy accumu-lationisplottedusingacolormapinFig. 26.Threecontourlinesareaddedtoeasetheinterpretation:90%,95%and99%of accumulatedenergy,correspondingtoatruncationerrorofrespectively30%,20%and10%.Thericherspectrumisobserved inthewakeregionbetween10%and70% ofrelative span.Thisistheregionwherethewakeisinfluencedneitherbythe hub boundarylayer norby thetip vortex.Thereforethe wakedrives theconvergenceofHB computations.Resultsare in goodagreementwiththepredictiontoolbasedonthewakethickness.Toemphasizethat,thenumberofharmonicsneeded tohave99%oftheenergyisgiveninTable 1forarelativespanbetween10%and70%.

Fig. 27showsthenon-dimensionalaxialmomentumextractedattherotor/rotorinterfacefromasingle-passage mixing-planecomputationforthethreeconsideredconfigurations.Onecanobservedifferentwakeshapes:theAIHigh-Speed(HS) wakelooks muchthinnerthan theMU-HS,whichlooksthinnerthan theLow-Speed(LS)one. Indeed,the latterdoesnot show awelldelimitedwakestructureallalongthespanexplainingtheestimationofthenumberofharmonicsneededto capturesuchconfigurations.

ThispredictiontoolbasedonanazimuthalFouriertransform,ismoreaccurateasithandleswaketangentialdistortionas wellasanyothertypeofazimuthaldistortions.Thus,itcanbeusedtopredictthenumberofharmonicsneededtocapture

Configuration AI-HS MU-HS MU-LS

Wake thickness 17 7 3

Azimuthal Fourier transform 16 7 4

Fig. 27. Non-dimensional axial momentum(ρU)/(ρU)_∞at the rotor/rotor interface (mixing-plane computations).

acertainlevelofenergyforanyrelativespan.Thecomputationaltimeneededtogettheaccumulatedenergypicturesasin

Fig. 26isnegligible.Infact,ittakeslessthanaminute.

Weverifyaposteriori thatthenumberofharmonicsprovided inTable 1are sufficienttoyieldconvergedHB computa-tions.FortheMU-LS,thepredictiontoolestimate thatfourharmonicsaresufficient.Infact,Fig. 21supportstheargument that four harmonics givesa convergedsimulation as the difference between N

=

_4, N

=

_{5 and} N

=

_{6 HB computations} are barely visible.For theMU-HS, seven harmonics are estimatedto be sufficientwhile visually, it seems that N

=

_{8 is} converged.Infact,onemustkeepinmindthatthesecriteriajustgivealowerboundoftherequirednumberofharmonics neededtogettheconvergenceoftheHBmethod.Indeed,whenrunningan N-harmonicHBcomputation,thetimeperiodis sampledwith2N

+

1 timeinstantswhichis,accordingtotheNyquist–Shannoncriteria[26,27],theminimumsamplingto gettheN-th ofthefundamentalfrequency.Itdoesnotnecessarilymeanthatthelevelofthe N-th harmonicisaccurately predicted.Experienceshowsthatinordertoreachthislevel,onehassometimestorunan N

+

1 orN

+

2 HBcomputations.

6. Conclusions

The accuracy andefficiency ofFourier-based time methods used to solve periodic unsteady problems depend onthe numberofharmonicschosentorepresentthefrequencycontentofthetimesignal.Inthisworkweinvestigatetheaccuracy andconvergencepropertiesofFourier-basedtimeintegrationmethods.Theconvergencerateofthesemethods,intermsof harmonicsrequiredtodescribethesolutionwithagivenlevelofaccuracy,dependsonthespectralcontentofthesolution itself:Fourier-basedtimemethodsareparticularlyefficientforflowproblemscharacterizedbyanarrowFourierspectrum.

Startingfromthis remark,wetry todefine a relevantindicator ofsolution regularityinthe specificcaseof turboma-chineryflows,whichrepresentoneofthemainapplicationsofFourier-basedtimemethodsinFluidMechanics.Tothisaim, weshow thatthe mainsourceofunsteadinessinturbomachinery flowsisduetotherelative motionofwakesgenerated by a givenblade rowwithrespect to thedownstream row.Statistically speaking, thepassing wakes can be seen by the downstreamrowasanazimuthallyadvectedperiodicGaussianpulse, characterizedbyitsthicknessdefinedrelativetothe pitchbetweentwosubsequentbladesandbythevelocitydeficitassociatedtoit.Weshowthatthenarrowerthewake,the largeritsFourierspectrum,andtheslowertheconvergenceofFourier-basedtimemethods.

Inordertoachieve apriori estimatesofthenumberofharmonicsrequiredtoaccurately solveagiventurbomachinery problem,asteady(mixing-plane)simulationispost-processed toextracttangentialvariations oftheaxialmomentumand thecorrespondingspectrumupstreamofthebladerowinterface.Anerrorcriterioncorrespondingto99%oftheaccumulated energyof thespectrum is finally usedto estimate the numberof harmonics.This preliminarystep hasa negligible cost comparedtotheoverall simulation,sincethesteadycomputationisusedto initializetheunsteadyrun,andextractionof thespectrumofthetangentialvariationstakeslessthanaminuteonasingleprocessor.

Fig. A.28. Sketch of the steps needed to compute the first error quantificationε1.

The proposed methodology representsan efficient andreliable operational tool toguide thechoice of thenumber of harmonicsforagiventurbomachineryproblem, andtoevaluatebeforehandtheinterestofapplyingornotaFourier-based timeintegrationschemeinsteadofaclassicaltime-marchingscheme.

Acknowledgements

ThepresentharmonicbalanceformulationwasdevelopedthankstothesupportoftheDirectiondesProgrammes Aéronau-tiquesCivils (FrenchCivil AviationAgency)andoftheAerospaceValley (Midi-PyrénéesandAquitaineworldcompetitiveness cluster).TheauthorswouldalsoliketothankSNECMAfromtheSafrangroupandAirbusOperationsS.A.S.fromtheAirbus groupfortheirkindpermissiontopublishthisstudy.

Appendix A. Detailedalgorithmtocompute

ε

1

Asketchofthestepsusedto evaluate

ε1

fromacomputation isshowninFig. A.28.Twoazimuthal linesareextracted inthestatorandintherotor respectively(step



1).Theseareduplicatedusingthephase-lag conditiontoretrievethefull 2

π

signalinbothblocks.Theaxialmomentum

ρ

U variableisanalyzed.Themainadvantages ofthisvariablearethatitis a representativevariableforthe wake,it isaconservativevariableoftheconsideredgoverningequationsandfinally,it is invariantunderachangeofreferenceframe,unliketherelativevelocitiesforinstance.Then,anazimuthalFouriertransform, denoted

F

_{θ,iscarriedoutoneachazimuthal2}

_π

_{signalsandgivesthefrequencycontentofthewakeinboththestatorand} therotor(step



2).However,duetothetimeinterpolationbetweenthetworowsachievedattheinterface,spuriouseffects can appearupstream theinterfaceasshowninFig. A.29.Forinstance,theeffectsoftherotor blockaresignificant onthe closest cellstotheinterfacefortheN

=

5 computationandstillappearontheverylastscells beforetheinterfaceforthe N

=

_{10 computation.Theyhavedisappeared whenusing} N

=

_{15 harmonics.Tolessen theinfluenceof thisinterpolation,} andthusthespuriouseffects,theextractionoftheaxialmomentumisnotperformedattheclosestcelltotheinterface.If dref istheaxiallengthofablock,theextractionisachievedatdref

/

5 oftheinterfaceupstreamanddownstreamthestator

androtor block, respectively.It representssixtime thelength ofacell intheaxial direction. Asthegoverning equations are theEulerones, thereisnosignificantvariationofthewakethicknesswithin sixcells,whichsupports thishypothesis. Moreover, preliminarystudieshaveshownthat dref

/

5 issufficientto lowerthe spuriouseffectswhile keepingtheresults

consistent.

Appendix B. Detailedalgorithmtocompute

ε

2

The steps to compute the second error quantification for each of the 375 computations, are schematically shown in Fig. B.30. An azimuthal line is extracted in the stator domain, nearby the interface (step



1), as for the first error

Fig. A.29. Occurrence of spurious effects upstream the interface between stator and rotor blocks for a L=5% wake width.

Fig. B.30. Sketch of the steps needed to compute the second error quantification.

quantification.However, inthe rotorblock, a time probingis doneat onepoint givingan unsteady timesignal of

ρ

U(t) (step



1′_{). The azimuthal signal isduplicated usingthe phase-lag condition to retrievethe full 2}

_π

_{signal. The temporal}

andspatialsignalsarethenFouriertransformedsothattheir spectrumcanbecompared(step



2).Thewakeextractionis performedatthe sameaxialdistanceofthe interfaceasforthefirsterrorquantification.Inthiscase, thelocation ofthe pointin therotor block hasa directimpact on theresultsespecially whenthe wakeis under-resolved.To highlightthis impact,the temporalFouriertransform isevaluated attwo differentlocationscalledloc 1 and loc 2.The two points are separatedbyadistance

θ

loc₁₋loc₂

=

dref

/

10 intheazimuthaldirection,asshowninFig. B.30.