Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification

an author's

https://oatao.univ-toulouse.fr/20706

http://doi.org/10.1016/j.jcp.2016.08.043

Magri, Luca and Bauerheim, Michaël and Nicoud, Franck and Juniper, Matthew P. Stability analysis of

thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification. (2016) Journal of

Computational Physics, 325. 411-421. ISSN 0021-9991

Stability

analysis

of

thermo-acoustic

nonlinear

eigenproblems

in

annular

combustors.

Part II.

Uncertainty

quantification

Luca Magri

a

,

b

,

∗

,

Michael Bauerheim

c

,

Franck Nicoud

d

,

Matthew P. Juniper

b

a_{CenterforTurbulenceResearch,StanfordUniversity,CA,UnitedStates} b_{CambridgeUniversityEngineeringDepartment,Cambridge,UnitedKingdom} c_{CAPSLab,ETHZürich,Switzerland}

d_{IMAGUMR-CNRS 5149,UniversityofMontpellier,France}

a

b

s

t

r

a

c

t

Keywords: Thermo-acousticstability Uncertaintyquantification Annularcombustors Adjointmethods

Monte Carloand Active Subspace Identificationmethods are combined with first- and second-orderadjointsensitivitiestoperform(forward)uncertaintyquantificationanalysis of the thermo-acoustic stability of two annular combustor configurations. This method is appliedtoevaluate the risk factor,i.e.,the probability forthe systemtobe unstable. It is shown that the adjoint approach reduces the number of nonlinear-eigenproblem calculationsbyasmuchastheMonteCarlosamples.

1. Introduction

Thermo-acousticoscillationsinvolvetheinteractionofheatrelease(e.g.,fromaflame)andsound.Inrocketandaircraft engines,aswellaspower-generationturbines,heatreleasefluctuationscansynchronizewiththenaturalacousticmodesin thecombustionchamber.Thiscancauselargeoscillationsofthefluidquantities,suchasthestaticpressure,thatsometimes lead to catastrophic failure. It is one of the biggest and mostpersistent problems facing rocket [1] and aircraft engine manufacturers[2].

The output ofany frequency-based stability tool is usually a map of the thermo-acousticeigenvalues in thecomplex plane(blacksquaresinFig. 1).Eachthermo-acousticmodemusthavenegativegrowthrateforthecombustortobelinearly stable. The designprocess is even morecomplex becauseof theuncertainty inthe thermo-acousticparameters p ofthe low-order thermo-acousticmodel. Forexample,the speed of sound, the boundary impedancesand the flamemodel are sensitivetopartlyunknownphysicalparameterssuchastheflowregime,manufacturingtolerances,fuelchanges,oracoustic andheatlosses. Asa consequence,eachmode actuallybelongstoan uncertainregionofthe complexplane(Fig. 1). This uncertainregionismeasuredbytheriskfactor[3],whichcorrespondstotheprobabilitythatthemodeisunstable.Although theprobabilisticestimation(uncertaintyquantification)ofthethermo-acousticstabilityisparamountforpractitioners,there areonlyafewstudiesintheliterature[4–6,3].

UncertaintyQuantification(UQ)ofthermo-acousticstabilitywasperformedinlongitudinalacademicconfigurations con-tainingoneturbulent flameinNdiayeetal.[6].Assumingthatonlytheflamemodelwas uncertain,i.e.,thegain andthe

*

Correspondingauthor.

E-mailaddress:lmagri@stanford.edu(L. Magri). http://dx.doi.org/10.1016/j.jcp.2016.08.043

ω

1

=

−



q

ˆ

+₀

, δ

p

N

{

ω

0

,

p1

} ˆ

q0





ˆ

q+₀

,

∂N {ω,p0} ∂ω





_ω 0

ˆ

q0

 ,

(3)

where q

ˆ

+₀ isthe adjoint eigenfunction,which issolution ofthe adjointeigenproblem

N {

ω

0

,

p0

}

Hq

ˆ

+0

=

0,in which

·,

·

represents aninner product andH isthecomplex transpose.Ifthe unperturbedeigenvalue

ω

0 is N-folddegenerate,and

ˆ

e0,iaretheN independenteigenfunctionsassociatedwithit,weobtainaneigenproblemforthefirst-ordereigenvaluedrift,

ω

1,andeigendirection,

α

j,asfollows



ˆ

e+_0,i

,

∂

N

{

ω,

p0

}

∂

ω





ω0

ˆ

e0,j



ω

1

α

j

= −



ˆ

e+_0,i

, δ

p

N

{

ω

0

,



p1

} ˆ

e0,j



α

j

,

(4)

for i

,

j

=

1

,

2

,

...,

N. Einstein summationis used,therefore, theinner products in equation (4) are thecomponents ofan

N

×

N matrix and

α

j are thecomponentsofan N

×

1 vector.Among the N eigenvalues drifts,

ω

1,outputted by (4), we select the one with largest growthrate,

ω

1i,because it causes the greatest change in the stability.In thermo-acoustics, degeneracy occursinrotationallysymmetric annular combustorsin whichazimuthal modeshave2-fold degeneracy(see, e.g.,[19]).

Thesecond-ordereigenvaluedriftreads

ω

2

= −

2



ˆ

q+₀

,

∂N {ω,p0} ∂ω





_ω 0

ω

1q

ˆ

1

+ δ

p

N

{

ω

0

,



p1

} ˆ

q1





ˆ

q+₀

,

∂N {ω,p0} ∂ω





_ω 0

ˆ

q0



−

2



ˆ

q+₀

,

1 2 ∂2N {ω,p0} ∂ω2





ω0

ω

2 1

ˆ

q0





ˆ

q+₀

,

∂N {ω,p0} ∂ω





_ω 0

ˆ

q0



.

(5)

Thecalculationoftheperturbedeigenfunctionq

ˆ

1,whichisnecessaryonlyforthecalculationofthesecond-ordereigenvalue drift,isdescribedin[17].

3. UncertaintyquantificationviastandardMonteCarlomethod

PartIofthispapershowedthatusingtheadjointmethodcandrasticallyreducethecomputationalcostofdeterministic sensitivity analysis. Asimilar approach can,thus, be used whenthe parameters are varied randomly.This section shows howtheadjointmethodcanprovideefficientUncertaintyQuantification(UQ)strategiestopredictthermo-acousticstability from aprobabilistic standpoint.We studytwo ofthe three configurationsof Part I,3 i.e., the weakly-coupledrotationally symmetric CaseA,andthe strongly-coupledrotationallyasymmetricCase C,whichisrelevanttoindustrialconfigurations

[20].AstandardMonteCarlomethod(MC)isintegratedwiththeadjointformulationforthecalculationoftheProbability Density Function(PDF) andRiskFactor(RF),thelatterof whichisdefinedastheprobability thatthe systemisunstable, givenaPDFfortheinputparameters[3]

RF

=

∞



0

PDF

(ω

i

)

d

ω

i

.

(6)

In practical applications,theuncertainties are typically greatestin theflame parametersandacoustic damping.Here,we calculatehowthethermo-acousticgrowthrate,

ω

i,whichgovernsthestability,isaffectedby uncertainflameparameters.

Studyinguncertaintyquantificationfortheacousticdamping,forexamplethroughimpedanceboundaryconditions,isjust as straightforward [6,21]. We assume we knowthe maximum and minimum values of the uncertain flame parameters. Using the Principleof MaximumEntropy, we choosethe uniformdistribution forthe input parameters because itis the leastbiasedpossibledistributiongiventheavailableinformation[22].Notethat [6]haveshownthatthePDF shapehasa minoreffectontheriskfactorinthecasetheyconsidered.

By the standard MonteCarlo methodused by [3], M randomvalues of p, calledthe MonteCarlo samplingpMC_{, are}

selected withrespect totheir PDFsandthenonlineareigenproblem (1)issolved M timestoprovide M eigenvalues.This meansthatwiththismethod,whichisthereferencesolution,wehavetosolveforM nonlineareigenproblems.TheMonte Carlo methodalways convergesto the final PDF but suffersfrom slowconvergence, being

∼

O(

1

/

√

M

)

, whichcould be prohibitiveinlargesystemssuchastheHelmholtzequationincomplexgeometries.Thiscallsfortheadjoint-basedmethod. ToavoidthecomputationsoftheM samples,here

∼

O(

104

)

,theMonteCarloanalysisisviewedasarandom perturba-tionaroundtheunperturbedstate.Consequently,theadjointsensitivitiesofSection2areappliedtoobtaintheeigenvalue drifts providing an efficientUQ strategy. Thus, therandom sequenceof parameters pMC _{is usedasa perturbationin the}

first-ordereigenvaluedriftinequation(4),fordegenerateeigenproblems,orinequation(3)fornon-degeneratecases.Then, the perturbedeigenvectoriscalculatedbySVDandthesecond-order driftiscalculatedbyequation (5)foreach sequence

3 _{CaseBistherotationallysymmetricversionofCaseC.Theyhaveasimilarprobabilisticbehavior (notshown)andCaseBisnotreportedherefor}

Table 1

Riskfactors(RF)calculatedbytheMonteCarlomethodasafunctionoftheMonte Carlo samples.MCisthestandardMonteCarlomethod,whichprovidesthebenchmark solution, andADstandsforadjoint.Thestandarddeviationoftheflameindicesand timedelaysis 5%.

MonteCarlo samples

RF via MC RF via 1st-order AD RF via 2nd-order AD

Case A 10,000 33.3% 40.3% 34.9% 20,000 33.9% 41.3% 34.2% 30,000 33.3% 40.6% 33.7% Case C 10,000 40.9% 34.7% 41.5% 20,000 40.8% 34.5% 41.3% 30,000 40.9% 34.6% 41.1% Table 2

Riskfactors(RF)calculatedbytheMonteCarlomethodasafunctionofthestandard deviationoftheflameindexandtimedelayuniformdistributions.MCisthestandard MonteCarlomethodandADstandsforadjoint.

Standard deviation

RF via MC RF via 1st-order AD RF via 2nd-order AD

Case A 1% 15.4% 14.7% 15.4% 2.5% 31.3% 33.2% 31.2% 5% 33.25% 40.3% 33% 10% 34.5% 43.2% 34.9% Case C 1% 18.3% 17.5% 18.3% 2.5% 35.4% 31.1% 35.3% 5% 40.9% 34.7% 40.4% 10% 42.2% 30.0 % 41.5%

ofrandomparameters. Theseareonlyvector–matrix–vector multiplicationsorlower-rank linearsystems.Importantly,the adjointmethodrequiresthecomputation ofonlyone nonlineareigenproblem(1)anditsadjoint,regardlessofthe Monte CarlosamplingM orthenumberofperturbedparameters.

First,we evaluatehowmanyMonteCarlosamplesare neededfortherisk factorstoconverge. Table 1showsthe con-vergence ofthe risk factorfor threeMonte Carlosamplingsof 10,000,20,000 and30,000 imposing a standarddeviation of5%ontheflameindicesandtimedelays.WechoosetheMonteCarlosamplingofM

=

10

,

000 forUQasacompromise betweenaccuracyandcomputationalcost.(Thediscrepancyof

∼

0.6% inthe M

=

20

,

000 caseisduetothebinsizes and consequentnumericalintegrationerror.Ifwe weretorun multipleMonteCarlosimulations,theensembleaverageofthe discrepancywouldtendtozero.)

Secondly,we impose differentstandard deviations to theuniform distributions ofthe flameparameters andcalculate the growth-ratePDFs andrisk factors. Theresults are shownin Table 2. When thestandard deviations are smallerthan 2

.

5%,thefirst-orderadjointmethodprovidesaccurate predictions,althoughitbecomeslessaccurateforlargerdeviations. However,thesecond-orderadjointmethodprovidesaccuratepredictionsoftheriskfactoruptostandarddeviationsof10%, matchingsatisfactorilythebenchmarksolutionbyMC.

InFig. 3we depicttheeigenvalues viaMonteCarlosimulationsobtainedbyMC (first row),first-orderadjoint method (middlerow) andsecond-orderadjointmethod(bottomrow) forthe weaklycoupledCase A.Theclouds(leftpanels) are obtainedbyimposingauniformprobabilitydistributionbetween

±

0

.

1n0

,

±

0

.

1

τ

0,whichrepresenttheuncertaintiesofthe flameparameters(lastrowofTable 2forCaseA).ThePDFsoftheperturbedgrowthratesaredepictedintherightpanels. ThePDF shapeissatisfactorily predictedby thefirst-orderadjointmethod,however,toobtainaccuracyon theriskfactor thesecond-orderadjointformulationisnecessary.Forthiscase,theriskfactorpredictedbyMCis34.5%,byfirst-orderAD is43.2%andbysecond-orderADis34

.

9%.

ThesamequantitiesforCaseCareshowninFig. 4.Forthiscase,theriskfactorpredictedbyMCis42.2%,byfirst-order ADis30%andbysecond-orderADis41

.

5%.ThestronglycoupledconfigurationCismorepronetobeingunstableinpractice thantheweaklycoupledconfigurationA:Forthesamelevelofuncertaintyoftheflameparameters(10%),thegrowthrate isuncertainuptowithin

∼

150s−1 inCase C,whereasitisuncertainuptowithin

∼

20s−1 inCase A.

In general, the UQ analysis shows that the uncertainty present in the flame parameters can significantly affect the thermo-acoustic stability. Deterministic calculations of the eigenvalue (big dots in the left panels of Figs. 3, 4) are not sufficient for a robust thermo-acoustic stability analysis, i.e., systems that are deterministically stable can have a great probability ofbecomingunstable.Thisisbecausethermo-acousticsystemsarehighly sensitivetochanges insomedesign parameters,asshownin[17].Notethat othersources ofuncertainties,such aspartlyunknown acousticlosses,can affect thestability,althoughthisisnotconsideredhereforsimplicity.

Fig. 5. Partofthespectrumofthecovariancematrix,C.Thedominanteigenvectorsprovidethe Q directionsintheparameterspacealongwhichthe growthratevariesthemost.WerecognizethefirstQ=5 eigenvaluesasactivevariablesbecausetheyaredominantandlesssensitivetotheMonteCarlo sampling,Mcov_{.CaseAisshownintheleftpanelsandCaseCintherightpanels.}

C

= E[∇

p

ω

i

(

∇

p

ω

i

)

T

],

(7)

wherethecolumnvector

∇

p

ω

i

= [∂

ω

i

/∂

p1

∂

ω

i

/∂

p2

. . . ∂

ω

i

/∂

pNp

]

isthegrowthrate’ssensitivitywithrespecttothe

Np thermo-acousticparameters,and

E

istheexpectationoperator.Notethatthisvectorconsistsofpartialderivatives,

therefore,Np eigenvaluesneedtobecalculated.Tocomputethecovariancematrix,weperformaMonteCarlo

integra-tion[13],yielding C

≈

1 Mcov M



cov j=1

[∇

p

ω

i

(

p(j)

)(

∇

p

ω

i

(

p(j)

))

T

],

(8)

wherethevectorofparameters,p(j),isdrawnfromtherelevantuniformPDFofMcov MonteCarlosamples.Asfaras thenumberofcomputationsisconcerned, Mcov

_×

_N

p growthrates,

ω

i,arecalculatedby eitherfinitedifference(FD),

whichrequiressolving Mcov

×

Np nonlineareigenproblems,ortheadjointapproach(AD),whichrequiressolvingonly

onenonlineareigenproblemanditsadjointregardlessofthenumberofparametersandMonteCarlosamples. 2. Identificationoftheactivevariables. C issymmetricand,therefore,admitstherealeigenvaluedecomposition

C

=

W



WT

.

(9)

Basedontherelativeimportanceoftheeigenvalues

λ

j,weselecttheQ dominanteigenvectors,Wk.Thischoicemight

berathersubjectivedependingonthecase[13].Fig. 5showsthattherearegapsbetweenthefirstandsecond eigenval-uesaswellasbetweenthefifthandsixtheigenvalues.Thissuggeststhatfiveactivevariablesshouldbekept.Physically, thefirstgroup (

λ

1) isassociatedwithamean-flameeffect,whilethesecond group(from

λ

2 to

λ

5)corresponds toa symmetry-breakingsplittingeffect,asexhaustivelyexplainedby[3].

3. Developmentofsurrogatemodels. We develop algebraic surrogate models forthe growth rateas a function ofthe activevariables,WT kp,k

=

1

,

...,

Q ,byaleast-squaremethod

ω

_iA S I

(

p

)

=

α

0

+

Q



j=1

α

jWTjp





Linear

+

Q



j=1 Q



k=j

α

jk

(

WTjp

)(

WkTp

)

+

Q



j=1 Q



k=j Q



l=k

α

jkl

(

WTjp

)(

WkTp

)(

WTlp

)





Quadratic and Cubic

.

(10)

The function

ω

i

(

p

)

A S I is also knownas the response surface andneeds MA S I Monte Carlosamples forleast-square

fitting, where MA S I _{isgreater thanthe numberofregressioncoefficients, N}

reg.Here,we comparealinear regression

Fig. 7. 50,000MonteCarloexperimentsobtainedbyevaluatingthecubicleast-squaresurrogatemodelsbyASIwithQ=5 activevariables.Resultsfrom FD-based(firstrow);1st-orderAD-based(secondrow)and2nd-orderAD-based(thirdrow)surrogatemodels.CaseAisshownontheleftandCaseCon theright.Thestraightlineisthecorrectsolution,thehigherthescattering,thelargertheerror.

Thirdly,we discussthe complexity andaccuracyofthe surrogate models.Thesealgebraicmodels are computationally cheapprovidingthatthenumberofregressioncoefficientsissmall.Thisnumberisexactlygivenby

Nreg

=

1

+

Q

+ φ



2Q

+

3 2Q

(

Q

−

1

)

+

1 6Q

(

Q

−

1

)(

Q

−

2

)



,

(12)

where

φ

=

0 in linear regression and

φ

=

1 in cubic regression. The last column of Table 3 reports Nreg for the four

surrogatemodelsdeveloped.Ontheonehand,linearregressionischeapbecauseitrequiresonlyNreg

=

6 butthepredicted

risk factors andcoefficientsof determinationare unsatisfactory.This doesnot appreciablyimprove whenall the Q

=

38 (Nreg

=

39)variables areretained.Onthe otherhand,a cubicregressionisneededtoobtain goodpredictionsofthe risk

factor.Retainingallthevariables, Q

=

38,needs Nreg

=

10

,

660,whichmakes thedevelopmentofsuchasurrogate model

computationallytime-consuming.However,retainingonly Q

=

5 activevariablesrequiresthecalculationofonlyNreg

=

56,

whichsignificantlydecreasesthecomplexityoftheoriginalMonteCarloproblemkeepinghighaccuracyontheriskfactors. Insummary,thecubicregressionmodel4 basedonthefirstfiveactivevariablesprovidesanexcellentcompromisebetween computationalcostandaccuracy,alsowhenthegradientsareobtainedbytheadjointmethod.

Finally,toevaluatehowrobustthecalculationoftheriskfactoris,werun16setsofM

=

50

,

000 MonteCarlosimulations andcalculatethemeanandstandarddeviationsoftheresults,asshowninFig. 8forthecubicsurrogatemodelwithQ

=

5. Repeating thisprocedure severaltimes provides an estimation ofthe confidence interval ofthe risk factor. As shownin

Table 3, the first-adjoint method is unsatisfactorily accurate in the weakly coupled regime (RF

=

44

.

5% compared with RF

=

35

.

1% byMC)butinthestronglycoupledregime theestimationismore accurate(RF

=

46

.

2% comparedwithRF

=

43

.

1% byMC).However, Fig. 8 revealsthat the latterestimation is not robust becausethe interval of confidenceby the first-orderadjoint methodis

±

3.56% (paneld). Nevertheless,usingthe second-orderadjoint method(panelf) provides a reliableriskfactorbecausethemeanvalueRF

=

43

.

3% isinagreementwithFD(panelb)andthestandarddeviationissmall,

±

0.41%.Thisanalysisindicatesthat combininga second-orderadjointmethodwiththeASImethodtocompute surrogate modelsisarobustandaccuratemethodtopredictstabilitymarginsofannularcombustors.

4 _{QuadraticmodelsarenotconsideredherebecauseBauerheimetal.[3]showedthatmodeling thecubictermsappreciablyimprovestheaccuracyof}

eigenproblemssolved isreducedbya factorequaltothenumberofMonteCarlosamplesneededto calculatethe covari-ance matrix,which, inthisstudyis 1,140. The surrogatemodel obtainedby cubicregression isfound to be an excellent compromisebetweencomputationalcostandnumericalaccuracy.

Thefirst-orderadjointframeworkisareliabletoolforuncertaintyquantificationaslongastherateofchangeof eigen-valueswithparameters is approximatelylineararound the operatingpoint inquestion and, when it isnot, thestandard deviationsofthesystem’sparametersarenot toolarge.Inthesescenarios, thesecond-orderadjointmethodprovedmore accurateandversatile.Theadjointframework isapromisingmethodfordesign toobtainquickaccurateestimatesofrisk factorsatverycheapcomputationalcost.

Acknowledgements

Theauthorsare gratefultothe2014CenterforTurbulenceResearchSummerProgram(StanfordUniversity)wherethe ideas ofthis work were born. L.M. andM.P.J acknowledge the European Research Council – Project ALORS 2590620 for financialsupport. L.M.gratefullyacknowledgesthefinancial supportreceived fromthe RoyalAcademyofEngineering Re-search FellowshipsScheme.L.M.thanksProf. WolfgangPolifke, Dr.Camilo Silva,Prof. JonasMoeck andGeorgMensah for fruitfuldiscussionsandinterestshowninthisstudy.

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