an author's
https://oatao.univ-toulouse.fr/20704
http://doi.org/10.1016/j.jcp.2016.07.032
Magri, Luca and Bauerheim, Michael and Juniper, Matthew P. Stability analysis of thermo-acoustic nonlinear
eigenproblems in annular combustors. Part I. Sensitivity. (2016) Journal of Computational Physics, 325. 395-410.
ISSN 0021-9991
Stability
analysis
of
thermo-acoustic
nonlinear
eigenproblems
in
annular
combustors.
Part
I.
Sensitivity
Luca Magri
a,
b,
∗
,
Michael Bauerheim
c,
Matthew
P. Juniper
b aCenterforTurbulenceResearch,StanfordUniversity,CA,UnitedStatesbCambridgeUniversityEngineeringDepartment,Cambridge,UnitedKingdom cCAPSLab/,ETHZürich,Switzerland
a
b
s
t
r
a
c
t
Keywords: Thermo-acousticstability Sensitivityanalysis Annularcombustors AdjointmethodsWe present an adjoint-based method for the calculation ofeigenvalue perturbations in nonlinear, degenerate and non-self-adjoint eigenproblems. This method is applied to a thermo-acoustic annular combustor network, the stability of which is governed by a nonlinear eigenproblem. We calculate the first- and second-order sensitivities of the growth rate and frequency to geometric, flow and flame parameters. Three different configurationsareanalysed.Thebenchmarksensitivitiesare obtainedbyfinitedifference, whichinvolvessolvingthenonlineareigenproblematleastasmanytimesasthenumber of parameters.By solving onlyone adjoint eigenproblem,we obtainthe sensitivities to anythermo-acousticparameter,whichmatchthefinite-differencesolutionsatmuchlower computationalcost.
1. Introduction
Thermo-acousticoscillationsinvolvetheinteractionofheatreleaseandsound.Inrocketandaircraftengines,heatrelease fluctuationscan synchronizewiththenaturalacousticmodesinthe combustionchamber. Thiscan causeloudvibrations thatsometimesleadtocatastrophic failure.Itisoneofthebiggestandmostpersistentproblemsfacingrocketandaircraft enginemanufacturers[1].
ManystudieshavedemonstratedtheabilityofLarge-EddySimulation(LES)torepresenttheflamedynamics[2]. How-ever,evenwhenLESsimulations confirmthatacombustorisunstable,theydonot suggesthowtocontrol theinstability. Moreover, LES is computationally expensive. Simpler frequency-based models are therefore often used in academia and industryforpre-design,optimization,controlanduncertaintyquantification.
Thereexisttwomaindifferentclassesoffrequency-basedlow-ordermethodsinthermo-acoustics.
1. Network-based methodsmodelthegeometryofthecombustorasanetwork ofacousticelementswheretheacoustic problem can be solved analytically [3–6]. Jump relations connect these elements, enforcing pressure continuity and massorvolumeconservation[7,8]whileaccountingforthedilatationcausedbyflames.Theacousticquantitiesineach segmentare relatedtothe amplitudesoftheforwardandbackward acousticwaves,whicharedetermined such that
*
Correspondingauthor.E-mailaddress:lmagri@stanford.edu(L. Magri).
thecaseoftheHelmholtzapproach,typicallyoforderten/hundredthousandforindustrialgeometries.If
N
representsthe Helmholtzproblem,thentheeigenfunctionconsistsonlyofthediscretizedacousticpressure.An important source of nonlinearity lies in the flame model, which introduces a time delay appearing as an expo-nentialfunction inthefrequencyspace [12].Other nonlinearities intheeigenvalue mayappearbecause oftheboundary impedances[11].Thesolutionofthesenonlineareigenproblemsandthecalculationofthethermo-acousticgrowthratesand frequencyistheobjectiveofstabilityanalysis.Fordesignpurposes,itisalsoimportanttopredicthowthethermo-acoustic stabilitychangesduetovariationsofthesystem.Thisistheobjectiveofsensitivityanalysis.
1.1. Sensitivityanalysisofeigenproblems
Insituationsthataresusceptibletothermo-acousticoscillations,oftenonlyahandfulofoscillationmodesareunstable. Existingtechniquesexaminehowachangeinoneparameteraffectsalloscillationmodes,whetherunstableornot.Adjoint techniquesturnthisaround.Inasingle calculation,theyexaminehow eachoscillationmodeisaffectedby changesinall parameters.Inotherwords,they providegradientinformationaboutthevariation ofan eigenvaluewithrespecttoallthe parametersinthemodel.Forexample,inasystemwithathousandparameters,theycalculategradientsathousandtimes fasterthanfinite-differencemethods.
Fig. 1bisanillustrationoftheeigenvaluesofathermo-acousticsystem.Twoeigenmodesareunstable(theyhavepositive growthrateandlieinthegreyregion).Therearetwoapproachestodeterminehowthesetwo eigenvaluesareaffectedby eachsystemparameter.Ontheonehand,wecouldchangeeachparameterindependentlyandrecalculatealltheeigenvalues, retainingonly theinformationaboutthe eigenvaluesof interest.Thisis calledthefinite-difference approachin thispaper andrequiresasmanycalculationsasthereare parameters.Ontheotherhand,we coulduseadjointmethodstocalculate howeacheigenvalueisaffectedbyeveryparameter,inasinglecalculation.Thisrequiresasmanycalculationsasthereare eigenvaluesofinterest,whichismanytimessmallerthanthenumberofparameters.
Eigenvaluesensitivitymethodsoriginate fromspectralperturbationtheory [13]andquantummechanics[14].In struc-tural mechanics, the calculation of first- and second-order derivatives of non-degenerate eigensolutions of self-adjoint nonlinear eigenproblems was proposed in aeroelasticityby Mantegazza and Bindolino [15] andonly theoretically by Liu andChen[16].Later,[17,18]foundtheanalyticalexpressionsforthesensitivitiesupton-thorderofageneralself-adjoint non-degenerate eigenproblem withapplication to vibrational mechanics. More recently, Li et al.[19] derived eigensolu-tion sensitivities for self-adjoint problems with relevance both to degenerate and non-degenerate simplified structural mechanical problems.Eigenvalue sensitivity is alsocommonly used inhydrodynamic stability [20–26], where the eigen-valueproblemsaretypicallylinear,orwithquadraticnonlinearities,andnon-degenerate.ThereviewbyLuchiniandBottaro
[27]providesathoroughoverviewofthestate-of-the-artofadjointmethodsinhydrodynamicstability.
Adjointeigenvaluesensitivityanalysisofthermo-acousticsystemswasproposedbyMagriandJuniper[28].Theanalysis was appliedtosimplifiedmodels ofcombustorstofindoptimalpassivemechanismsandsensitivityto base-statechanges inaRijketube[28–31],aducteddiffusionflame[32]and,morerecently,toaductedpremixed-flame[33].However,these studiesdealtwithlineareigenvalueproblemsinwhichtheeigenvalueappearsunderalinearterm.
Theextension oftheadjointanalysistononlinear thermo-acousticeigenproblems wasproposed byMagri [34] and Ju-niperetal.[35]basedonideasofspectralperturbationtheoryofnonlineareigenproblems[36].Theyproposedtwodifferent adjoint methods for the prediction of eigenvalue sensitivities to perturbations to generic system’s parameters. The first methodwas basedontheDiscreteAdjoint approach,inwhichtheeigenvaluedriftisobtainedbyrecursive applicationof thelinearadjointformulaateachiterationstepofthenonlinearsolver.Thesecondapproachwasbasedonthelinearization ofthenonlinearoperatoraroundtheunperturbedeigenvalue,whichneedsfeweroperationsthanthefirstapproach.Inthis paperwe usethesecond approachof Juniperetal.[35] andapplyit toan elaborate annularcombustor. Suchfirst-order adjointanalysiswasappliedrecentlytopredictsymmetrybreakinginannularcombustors[37].
1.2. Objectiveandstructureofthepaper
The aimof thispaperisto provide amethod forthe calculationof first- andsecond-order eigenvalue sensitivities of non-self-adjointnonlineareigenproblemswithdegeneracy.Thisframeworkisappliedtotheelaborateannularcombustorof
[38]tocalculatedesign-parametersensitivities.
In Section 2 we present the theory for adjoint sensitivity analysis of nonlinear eigenproblems. We derive first- and second-ordereigenvaluesensitivityrelationsbothfornon-degenerateanddegenerateeigenproblems.InSection3the math-ematicalmodeloftheannularcombustorthermo-acousticnetwork isbriefly described.Forfurtherbackgroundinannular combustors, the readermayrefer to the review by O’Connoret al.[39].In Section 4.1 we validate the adjointformulae againstfinitedifferences,thelatterofwhichprovidethebenchmarksolutionbecausethey donot relyonanyassumption ontheperturbationsize.Threeconfigurationsareconsidered:aweaklycoupledrotationallysymmetriccombustor(CaseA), astronglycoupledrotationallysymmetriccombustor(Case B)andastronglycouplednon-rotationallysymmetric combus-tor(Case C).Theeigenvalue sensitivitiestoperturbations tobothgeometric,flow andflames parametersarecalculatedin Section4.2.Aconcludingdiscussionendsthepaper.
All ofthesestudies arebased ondeterministic analysis, whichassumesperfectknowledge ofthethermo-acoustic pa-rameters.Includinguncertaintiesintheflameparametersinthestabilitycalculationsistheobjectiveofthesecondpartof thispaper[40].
2. Eigenvaluesensitivityofnonlineareigenproblems
We show howto computethe eigenvaluesensitivityvia equationsinvolvingtheadjoint eigenfunctions.Thisapproach combines aderivation withtheContinuous Adjoint (CA)formulation, inwhichthe problemsare governedby continuous operators, withoutexplicitly derivingthe CAequations.The finalsensitivityequations canbe appliedby usinga Discrete Adjoint(DA)philosophy,whichismoreaccurateandeasiertoimplement(e.g.,forthermo-acousticproblems,[28,30,32]).
First,wesolveforthenonlineardirecteigenproblem(1),inwhichtheeigenvalueappearsunderexponential,polynomial and rational terms. Starting from an initial guess for the eigenvalue,we assume that the convergedeigenvalue
ω
0 is anumericalrootofthedispersionrelation
|
det(
N
{
ω
0,
p0}) |<
tol,
(2)where‘det’isthedeterminantand‘tol’isadesiredtolerance.Inlargesystemsweensurecondition (2)throughrelaxation methods[11] insteadofsolvingforthecharacteristicequation.Equation(2)definesanimplicitfunctionbetween
ω
andp,i.e.,
ω
=
ω
(
p)
.Thecorrespondingeigenfunctionqˆ
0 iscalculatedfromthelinearsystemN
{
ω
0,
p0} ˆ
q0=
0.
(3)Theoperator
N
dependsonlyonthefinalconvergedeigenvalue,ω
0.Thekernelofequation(3)canbefoundbycomputingthesingularvector(s)associatedwiththetrivialsingularvalue(s).
Second,bydefiningtheadjointeigenfunction,q
ˆ
+0,andoperator,N
+,throughaHermitianinnerproductinan appropri-ateHilbertspaceˆ
q+0,
N
{
ω
0,
p0} ˆ
q0=
N
{
ω
0,
p0}
+qˆ
+0,
qˆ
0,
(4)wesolvefortheadjointeigenfunctionassociatedwiththeconvergedeigenvalue
ω
0N
{
ω
0,
p0}
Hqˆ
+0=
0.
(5)IfwefollowedapurelyContinuousAdjoint(CA)approach[28,30,29],we wouldneedtoderiveexplicitlytheHermitian operator
N
H andthecontinuous adjointequations.However, wedonotderive theseequationsexplicitlyandweproceed ononlywiththeabstractexpressionoftheHermitianoperator,inordertoapplytheDiscreteAdjoint(DA)methoddirectly to the final sensitivityrelations, asexplained subsequently.In equation (5), theadjoint eigenfunction canbe found with thesameprocedureas(3).Third,weperturbasystem’sparameterandcalculatetheperturbationoperator,whichwe can evaluatenumericallybyfinitedifferencep
=
p0+
p1
=⇒ δ
pN
{
ω
0,
p1
} =
N
{
ω
0,
p} −
N
{
ω
0,
p0},
(6)where
1.Thisperturbationoperatoristheinputoftheproblemand, therefore,isconstant,i.e.,itdoesnotdependon
ω
.Hence,δ
pN {
ω
0,
p1
}
representsexactlyalltheordersofitsTaylorseries(providingthatp1 issufficientlysmall)
δ
pN
{
ω
0,
p1
} =
∂
N
∂
pp1
+
1 2∂
2N
∂
p2(
p1
)
2+
o(
2
).
(7)Weassumethattheeigenvaluesandeigenfunctionsareanalyticalinthecomplexplanearound
=
0 andω
=
ω
0+ (
p1
)
ω
1+
1 2(
p1
)
2ω
2 and qˆ
= ˆ
q0+ (
p1
)
qˆ
1+
1 2(
p1
)
2qˆ
2,
(8) whereω
1=
dω
dp,
ω
2=
d2ω
dp2 and qˆ
1=
dqˆ
dp,
qˆ
2=
d2qˆ
dp2.
(9)The perturbedeigenproblemmustsatisfyequation (1)andisTaylor-expandeduptothesecond-ordertotalderivativeofp
aroundtheunperturbedeigenvalue
ω
0,yieldingN
ω
0+ (
p1
)
ω
1+
1 2(
p1
)
2ω
2,
p0+
p1
ˆ
q0+ (
p1
)
qˆ
1+
1 2(
p1
)
2qˆ
2=
0,
=⇒
N
{
ω
0,
p0} ˆ
q0+
dN
{
ω
,
p} ˆ
q dp(
p1
)
+
1 2 d2N
{
ω
,
p} ˆ
q dp2(
p1
)
2+
o(
2
)
=
0.
(10)N
{
ω
0,
p0} ˆ
q0+
+ (
p1
)
N
{
ω
0,
p0} ˆ
q1+
∂
N
{
ω
,
p0}
∂
ω
ω0
ω
1qˆ
0+ δ
pN
{
ω
0,
p1
} ˆ
q0+
+ (
p1
)
2 1 2N
{
ω
0,
p0} ˆ
q2+
∂
N
{
ω
,
p0}
∂
ω
ω0
ω
1qˆ
1+ δ
pN
{
ω
0,
p1
} ˆ
q1+
+ (
p1
)
2 1 2∂
2N
{
ω
,
p0}
∂
ω
2ω0
ω
21+
1 2∂
N
{
ω
,
p0}
∂
ω
ω0
ω
2+
∂δ
pN
{
ω
0,
p1
}
∂
ω
ω0
ω
1ˆ
q0+
o(
2
)
=
0.
(11)Importantly,thecrossderivative
∂δ
pN {
ω
0,
p1
} /∂
ω
iszerobecausetheperturbationoperatorδ
pN {
ω
0,
p1
}
isconstant.The unperturbedterm
∼
O(
1)
in equation (11)is trivially zerobecause ofequation (3). Higher orderterms∼
o(
2
)
areneglected.
2.1. First-ordereigenvaluesensitivity
Theequationforthefirstorder
∼
O(
)
isrecastasN
{
ω
0,
p0} ˆ
q1= −
∂
N
{
ω
,
p0}
∂
ω
ω0
ω
1qˆ
0+ δ
pN
{
ω
0,
p1
} ˆ
q0.
(12)The objective is to find the eigenvalue drift
ω
1 due to the perturbationδ
pN
. The adjoint eigenfunction provides a solvabilitycondition forthenon-homogeneoussystem(12)fulfillingtheFredholmalternative2 [41].Mathematically,thisisachievedbyprojectingequation(12)ontotheadjointeigenfunction,q
ˆ
+0ˆ
q+0,
N
{
ω
0,
p0} ˆ
q1= −
ˆ
q+0,
∂
N
{
ω
,
p0}
∂
ω
ω0
ω
1qˆ
0+ δ
pN
{
ω
0,
p1
} ˆ
q0.
(13)Usingequation(5),thedefinitionoftheinnerproduct(4)anditslinearity,yieldsanequationforthefirst-ordereigenvalue drift
ω
1=
−
ˆ
q+0, δ
pN
{
ω
0,
p1
} ˆ
q0ˆ
q+0,
∂N {ω,p0} ∂ωω 0
ˆ
q0,
(14)assumingthat
∂
N {
ω
,
p0} /∂
ω
=
0.Ifthenumberofcomponentsofp is S,andweareinterestedinthefirst-ordersensitivityforeach,equation(14)enablesustoreduce thenumberofnonlinear-eigenproblemcomputationsbycirca S P ,whereP is
the averageof thenumberof iterations neededto obtain
ω
1 by solving thenonlinear eigenproblemperturbed viafinitedifference.
Iftheunperturbedeigenvalue
ω
0 is N-folddegenerate,3 theeigenfunctionexpansionbecomes qˆ
=
Ni=1
α
ieˆ
0,i+
q
ˆ
1+
12
2q
ˆ
2,whereα
iarecomplexnumbersandeˆ
0,iaretheN independenteigenfunctionsassociatedwithω
0.Byrequiringtheright-hand sideofequation (13)to havenocomponentsalong theindependentdirectionse
ˆ
0,i (Fredholmalternative), we obtainaneigenprobleminα
iandeigenvalueω
1 [36]ˆ
e+0,i,
∂
N
{
ω
,
p0}
∂
ω
ω0
ˆ
e0,jω
1α
j= −
ˆ
e+0,i, δ
pN
{
ω
0,
p1
} ˆ
e0,jα
j,
(15)fori
,
j=
1,
2,
...,
N. Einstein summationisused,therefore,theinner products inequation (15)arethe componentsofanN
×
N matrix,α
j arethe componentsofan N×
1 vectorandω
1 is theeigenvalue.Thisequation is definedonlyintheN-folddegeneratesubspace.Inthermo-acoustics,degeneracyoccursinrotationallysymmetricannularcombustorsinwhich
azimuthal modeshave2-fold degeneracy[42–44,37]. Thegeneralizedeigenproblem (15)outputs N first-ordereigenvalue drifts andN unperturbed eigendirections. Weselectthe first-ordereigenvaluedrift,
ω
1, withgreatest growthrate,whichcausesthegreatestchangeinthestability.
To demonstrate the adjoint-based eigenvalue sensitivity (14), we consider the generic nonlinear eigenvalue problem representedbya2
×
2 matrixN
11(
ω
)
N
12(
ω
)
N
21(
ω
)
N
22(
ω
)
ˆ
q1ˆ
q2=
0 0.
(16)Wesolveforthecharacteristicequation
F
(
ω
)
=
N
11(
ω
)
N
22(
ω
)
−
N
21(
ω
)
N
12(
ω
)
=
0,
(17)2 Theleft-handsideoperatorrangeisequaltothekerneloftheorthogonalcomplementofitsadjointoperator.
andfind
ω
0 suchthatF(
ω
0)
=
0.Weassumethatthisrootisnon-degenerate.Thecorrespondingdirectandadjointeigen-vectorsare,respectively
ˆ
q0=
−
N
12(
ω
0)/
N
11(
ω
0)
1ˆ
q2,
(18)ˆ
q+0=
−
N
21(
ω
0)
∗/
N
11(
ω
0)
∗ 1ˆ
q+2,
(19)where
N
11(
ω
0)
isassumed=
0 andqˆ
2,qˆ
+2 arearbitrarynon-trivialcomplexnumbers,whicharesetto1.Thedependencyon
ω
0 isdroppedforbrevityfromnowon.Assumingthatthecharacteristicequationdefinesacontinuouslydifferentiablemanifold, the exactfirst-order eigenvaluesensitivityis calculatedby the implicitfunction theorem(also known asDini’s theorem)
∂
ω
∂
p= −
∂
F/∂
p∂
F/∂
ω
= −
N
11∂
N
22/∂
p+
N
22∂
N
11/∂
p−
N
12∂
N
21/∂
p−
N
21∂
N
12/∂
pN
11∂
N
22/∂
ω
+
N
22∂
N
11/∂
ω
−
N
12∂
N
21/∂
ω
−
N
21∂
N
12/∂
ω
.
(20)UsinganEuclideanHermitianinnerproduct,theadjointeigenvaluesensitivity(14),forthisalgebraicproblem,reads
δ
ω
δ
p= −
ˆ
q+0H(∂
N
/∂
p)ˆ
q0ˆ
q+0H(∂
N
/∂
ω
)ˆ
q0,
= −
−
N
21∗/
N
11∗ 1∗∂
N
11/∂
p∂
N
12/∂
p∂
N
21/∂
p∂
N
22/∂
p−
N
12/
N
11 1−
N
∗ 21/
N
11∗ 1 ∗∂
N
11/∂
ω
∂
N
12/∂
ω
∂
N
21/∂
ω
∂
N
22/∂
ω
−
N
12/
N
11 1.
(21)Whenthevector–matrix–vectormultiplicationsareperformed,theadjoint-basedsensitivity(21)coincideswiththe analyt-icalsensitivity(20).Thisillustratesthatequation(14)isanexactrepresentationofthefirst-ordereigenvaluedrift,
δ
ω
/δ
p. 2.2. Second-ordereigenvaluesensitivityTheequationforthesecond-orderisrecastas
1 2
N
{
ω
0,
p0} ˆ
q2= −
∂
N
{
ω
,
p0}
∂
ω
ω0
ω
1qˆ
1+ δ
pN
{
ω
0,
p1
} ˆ
q1+
−
1 2∂
2N
{
ω
,
p0}
∂
ω
2ω0
ω
21+
1 2∂
N
{
ω
,
p0}
∂
ω
ω0
ω
2ˆ
q0=
0.
(22)Thecalculationofthesecond-ordereigenvaluedriftisobtainedbyprojectingequation(22)ontotheadjointeigenfunction, yielding 1 2
ˆ
q+0,
N
{
ω
0,
p0} ˆ
q2=
ˆ
q+0,−
∂
N
{
ω
,
p0}
∂
ω
ω0
ω
1qˆ
1+ δ
pN
{
ω
0,
p1
} ˆ
q1+
ˆ
q+0,
−
1 2∂
2N
{
ω
,
p0}
∂
ω
2ω 0
ω
2 1+
1 2∂
N
{
ω
,
p0}
∂
ω
ω0
ω
2ˆ
q0.
(23)Usingequations(5)and(4)yieldsanequationforthesecond-ordereigenvaluedrift
ω
2= −
2ˆ
q+0,
∂N {ω,p0} ∂ωω 0
ω
1qˆ
1+ δ
pN
{
ω
0,
p1
} ˆ
q1ˆ
q+0,
∂N {ω,p0} ∂ωω 0
ˆ
q0+
−
2ˆ
q+0,
1 2 ∂2N {ω,p 0} ∂ω2ω0
ω
2 1ˆ
q0ˆ
q+0,
∂N {ω,p0} ∂ωω 0
ˆ
q0.
(24)Theeigenvalue-driftequations(14),(15),(24)enablethecalculationofthei-thdriftonlybyusingeigenfunctionsupto
(
i−
1
)
-thorder.Thecalculationoftheperturbedeigenfunctionqˆ
1,necessaryforthecalculationofthesecond-ordereigenvalue2.3. Calculationoftheperturbedeigenfunction
The calculation of the perturbed eigenfunction q
ˆ
1 in equation (12) requires solving for a non-homogeneous singularlinear systembecause theinversion operator,
N
−1{
ω
0,
p0}
, doesnot exist.However, thecompatibility condition ensuresthat this linear systemhas (infinite)solutions. For brevity, we define dim
(N )
=
K anduse matrices. In a non-defective degeneratesystem, a completeeigenbasis is{ˆ
q0,
eˆ
i}
,where i=
1,
2,
. . . ,
K−
N, qˆ
0=
Nj
α
jeˆ
0,j andeˆ
i are theremaining non-degenerateeigenfunctions. (Weare assumingthat onlythe0-theigenfunctionis N-folddegenerate.Theextension to other eigenfunctions’ degeneracyisstraightforward.) In general, thecoefficientsα
j are arbitrary,however,when working withperturbations,thesecoefficientsareuniquelydeterminedbythefirst-ordersensitivity(15).Theperturbed eigenfunc-tionisdecomposedasˆ
q1
= ˆ
z+ β
0qˆ
0,
(25)where
β
0 isingeneralacomplexnumber.Bysubstitutingequation(25)into(12),weobtainN
{
ω
0,
p0}ˆ
z= ,
(26)because
N {
ω
0,
p0}(ˆ
z+ β
0qˆ
0)
=
N {
ω
0,
p0}ˆ
z forequation(3).istheright-handsideofequation(12).
ˆ
q1 isthencalculatedasfollows.
•
DecomposeN =
U˜
VH (SingularValueDecomposition,SVD),where˜ =
0
0
.
(27)The submatrix
is diagonalandcontains the K
−
N non-trivialsingular valuesofN {
ω
0,
p0}
.The submatrixisa
N
×
N nullmatrix.Thecolumnsofthe unitarymatrixU are theleft singularvectorsandthecolumnsoftheunitarymatrixV aretherightsingularvectors.
•
Settoanynon-trivialdiagonalmatrix,forexample,theidentitymatrix.
•
Solvefor Y1 Y2= ˜
−1U−1.
(28)•
SetY2=
0 andfindthesolutionˆ
z=
V Y1 0.
(29)Anothermethodforthecalculationofq
ˆ
1 ispresentedinAppendix A.Inthisstudynonormalizationconstraintisimposedand,therefore,
β
0 isarbitrarilysettozero.Thismeansthatweareremovingthenon-uniquenessofq
ˆ
1 byrequiringitnottohaveacomponentalongtheunperturbedeigenfunctionqˆ
0 [36].3. Mathematicalmodelofanannularcombustor
Annularcombustionchambersarecommonlyusedinaircraftgasturbinesbecauseoftheircompactnessandabilityfor efficientlightaround[45,39].Suchconfigurations,however,sufferfromcombustioninstabilitiesduetoazimuthalmodesthat oftenappearatlow frequencies,wheredampingmechanismsarelesseffective. Westudyanannularcombustor configura-tiontypicalofmodernultraLow-NOxcombustionchambers,detailedin[46].ThenetworkmodeldevelopedbyBauerheim etal.[38],whichwas validatedagainstathree-dimensional Helmholtzsolvertopredictthe stabilityofazimuthal modes, is thereforeused inthe presentstudy.Thislow-order model describesacombustion chamber connectedby longitudinal burnersfedbyacommonannularplenum(Fig. 2).
TheAnnular NetworkReductionmethodology[38] analyticallyderivesthedispersion relationdet
(N {
ω
,
p}) =
0 ofthe annularsystem,wheretheoperatorN
isdefinedasN
{
ω
,
p} =
Nb
iR
i(
ω
)
T
i(
ω
,
p)
−
I
,
(30)where
I
istheidentityoperatorand Nb=
19 isthenumberofburners.R
i∈ R
4×4 isthepropagationoperatorthat maps theacousticwavesintheuniformcomponentsofthenetwork,representedbythematricesR
i=
R(
kpxp
)
0 0 R(
kcxc
)
,
(31) R(
kx
)
=
cos(
kx
)
−
sin(
kx
)
sin(
kx
)
cos(
kx
)
,
(32)Fig. 7. Normalizedfirst-ordereigenvaluesensitivities.CalculationwithFiniteDifference(FD)andAdjointmethods(AD).Theangularfrequencysensitivity isshownintheleftpanels,thegrowth-ratesensitivityisshownintherightpanels.CaseAinthetoprow;CaseBinthemiddlerowandCaseCinthe bottomrow.Theadjointsensitivitymatchesthebenchmarksolutiongivenbyfinitedifferences.InCaseC,thesensitivityofn andτisthemeanvalueof thesingle-burnersensitivitiesofFigs. 9 and10.Thefirst-ordereigenvaluedriftisobtainedbymultiplyingthesesensitivitiesby.
Fig. 8. SameasFig. 7butforsecond-ordersensitivities.Thenormalizedsecond-ordersensitivitiesarehigherthanthefirst-ordersensitivitiesofFig. 7.The second-ordereigenvaluedriftis,however,smallerbecauseitisobtainedbymultiplyingthesesensitivitiesby2.
burnertoburner,asshowninFigs. 9 and10.(TheirmeanvaluesareshowninFig. 7e, fandFig. 8e, f.)Thethermo-acoustic systemhasdrasticallydifferentbehavioursdependingonthe burnerbeingperturbed.The first-ordersensitivitiesoscillate in theazimuthal directionandcanbe negative orpositive.This 2-periodicpatternis physicallyrelatedtothe eigenvalue splittingcausedby symmetrybreaking[49,7],whichisduetothe2ndFouriercoefficientoftheflameparameters’spatial distribution (C2n in[49]). From Figs. 9c, d and10c, d,we note that thesecond-order sensitivity patternsare 4-periodic.
Thisoscillationmightbeduetothe4thFouriercoefficientoftheflamedistribution.Thisanalysis,however,isbeyondthe scopeofthispaperandisleftforafollow-onstudy.Byinspection,wefindanaccuratematchbetweenthefinitedifference calculationsandtheadjointpredictions.Thegrowthrateisoverallmostsensitivetothetimedelays
τ
,butthevalueisabout twenty timessmallerthanthecorrespondingrotationallysymmetricCaseBofFig. 7d.Moreover,althoughconfigurationC is similar tothe corresponding rotationallysymmetric CaseB, theparameters to whichit ismostsensitive are different.Table 3
SummaryofthesensitivitiesofFigs. 7,8,9 and10.xxx=strong,xx=mild,x=weak.
Case Li Sp Si ρp α n τ 1st-order Re(ω1,n) A xxx xx xxx xx xx xx x B x x x x x x xxx C xx x xx x xx x x Im(ω1,n) A x x x x x x xxx B xx x xx x xx xx xxx C xxx xx xxx xx xxx x x 2nd-order Re(ω2,n) A x x x x x x xxx B x x x x x x xxx C xx x xx x xxx x x Im(ω2,n) A x x x x x x xxx B x x x x x x xxx C xx x x x xxx x x
Fig. 9. First- (toprow)andsecond-order(bottomrow)sensitivitiestotheflameindex,n,inCaseC.Thesensitivitiesvarybecausetheconfigurationis non-rotationallysymmetric(Fig. 5).Angularfrequencysensitivityintheleftpanels,growth-ratesensitivityintherightpanels.Theadjointsensitivity matchesthebenchmarksolutiongivenbyfinitedifferences.Thefirst-ordereigenvaluedriftisobtainedbymultiplyingthesesensitivitiesby.
Fig. 10. Same asFig. 9as for the sensitivity to the time delay,τ. The second-order eigenvalue drift is obtained by multiplying these sensitivities by2.
Thismightindicatethatmodellinganannularcombustorasarotationallysymmetricconfigurationmightoverestimateand poorlypredictthesensitivities.
5. Conclusions
Wepresentfirst- andsecond-ordersensitivitiesofeigenvaluesinnonlinearnon-self-adjointeigenproblemswith/without degeneracyviaanadjointmethod.Thisisthefirstapplicationofadjointsensitivityanalysistononlineareigenproblemsas
applied todesign-parametersensitivitystudies inthermo-acoustics.Theadjointsensitivities arecalculatedinanelaborate annularcombustorthermo-acousticnetwork.Twocasesarestudiedasrepresentativecasesofplenum–combustion-chamber dynamics:theweaklycoupledcase,inwhichthecombustion-chambermodeisunstable,andthestronglycoupledcase,in whichtheplenummodeiscoupledwiththecombustion-chambermodethroughtheburners.
We show howto usethe adjointframework tostudythe sensitivitytothe system’sparameters reducing thenumber of computations by a factor equal to the number of the system’s parameters. This is particularly attractive to annular combustors, where the number of flames, thus parameters, is large. We find the strongly coupled case is overall more sensitiveandthesymmetrybreakingmakesthesystemlesssensitive.Thissuggeststhatperfectrotationalsymmetrymight be an exceedingly sensitive model.Moreover, the adjoint sensitivities are not prone to numericalcancellation errors, in contrasttofinitedifferences,becausetheydonotdependonthesizeoftheperturbation.
The sensitivityanalysisshowedthatthe eigenvaluecanbe mostsensitivetogeometric parameters (seecasesA andC inTable 3).However,themanufacturingtolerancesonthegeometryareusuallysmall,i.e., theuncertaintyonthephysical dimensionsoftheannularcombustorissmall.Ontheotherhand,theuncertaintyontheflameparametersand/ordamping islarger[39].Inordertoevaluatetheprobability thatathermo-acousticmodeisunstable,theadjointmethodproposedis extendedtouncertaintyquantificationoftheeigenvaluecalculationinthesecondpartofthispaper[40].
Theadjointframeworkisapromisingmethodfordesigntoobtainquickestimatesofthethermo-acousticsensitivitiesat verycheapcomputationalcost.
Acknowledgements
The authorsaregratefultothe2014CenterforTurbulenceResearchSummerProgram(StanfordUniversity)wherethe ideas of thiswork were born. L.M.and M.P.J acknowledge the European Research Council – Project ALORS 2590620 for financial support.L.M gratefullyacknowledges thefinancial supportreceived fromtheRoyal Academyof Engineering Re-search Fellowships scheme. The authors thank Prof. Franck Nicoud forfruitful discussions. Fig. 1 was adapted from the articleofS.R.StowandA.P.Dowling,Atime-domainnetworkmodelfornonlinearthermoacousticoscillations,ASMETurboExpo, GT2008-50770[9]withpermissionoftheoriginalpublisherASME.
Appendix A. Restrictedmatrixinversionforthecalculationoftheperturbedeigenfunction
The aim is to find the square submatrix of
N {
ω
0,
p0}
with rank=
K−
N, in the subspace of which the matrix isinvertible.First,wepartition
N ˆ
q0asN
{
ω
0,
p0}ˆ
q0=
⎛
⎝
NN11k1 NN1kkk NN13k3 N31 N3k N33⎞
⎠
⎛
⎝
qqˆ
ˆ
00,,1kˆ
q0,3⎞
⎠ ,
(37)whereN1k,N3k areK
×
N submatrices;Nk1,Nk3are N×
K submatricesandNkk isan N×
N submatrix.qˆ
0,1,
qˆ
0,k,
qˆ
0,3 aresubvectors.Thesubvectorq
ˆ
0,kischosentohavenon-trivialcomponents.Hence,thesystemcanberecastasN
{
ω
0,
p0}ˆ
q0=
⎛
⎝
NN11k1 NN13k3 N31 N33⎞
⎠
qˆ
0,1ˆ
q0,3= −
⎛
⎝
NN1kkk N3k⎞
⎠ ˆ
q0,k.
(38)Thematrixontheleft-handsidehasrank
=
K−
N becauseallitscolumnsareindependentsincethe N valuesofqˆ
0,k are non-trivialandtheright-handsideisalinearcombinationoftheleft-handside.Therefore,thecolumnsofN
corresponding to qˆ
0,k on theleft hand-sidecan be removed fromthe matrix without affectingits rank. To reduce the rowspace to a subspace inwhichthematrixhasfull rank,weusethesameargumentasbeforewiththeadjointeigenvectorqˆ
+0.The Ncomponentsq
ˆ
+0,karechosentobenon-trivial.Hence,theN rowscorrespondingtoqˆ
+0,k canberemovedandthefinallinear systembecomesinvertibleinthissubspace,asfollows N11 N13 N31 N33ˆ
q0,1ˆ
q0,3= −
N1k N3kˆ
q0,k.
(39)Now,thesquarematrixontheleft-handsideisinvertiblebecauseithasrank
=
K−
N andthesubspacedimensionisK−
N.Usingthisobservation,wecansolvefortheperturbedeigenvectorsubstitutingequation(25)into(39)
N11 N13 N31 N33ˆ
z1ˆ
z3= −
N1k N3kˆ
q0,k+
1k
3k
.
(40)Setting theeigenvectorto zerobecauseitis alreadyknown,z can
ˆ
be easily foundby solvingthe linearsystem, thefinal solutionofwhichisˆ
q1=
⎛
⎝
zˆ
01ˆ
z3⎞
⎠ + β
0qˆ
0,
(41)References
[1] T.C.Lieuwen,V.Yang,CombustionInstabilitiesinGasTurbineEngines:OperationalExperience,FundamentalMechanisms,andModeling,American InstituteofAeronauticsandAstronautics,Inc.,2005,http://www.lavoisier.fr/livre/notice.asp?id=RAXW2KALXX6OWV.
[2]T.Poinsot,Simulationmethodologiesandopenquestionsforacousticcombustioninstabilitystudies,in:AnnualResearchBriefs,CenterforTurbulence Research,2013,pp. 179–188.
[3] W.Polifke,A.Poncet,C.O.Paschereit,K.Döbbeling,Reconstructionofacoustictransfermatricesbyinstationarycomputationalfluiddynamics,J.Sound Vib.245 (3)(2001)483–510,http://dx.doi.org/10.1006/jsvi.2001.3594.
[4]S.R.Stow,A.P.Dowling,Thermoacousticoscillationsinanannularcombustor,in:ASMETurboExpo,2001,2001-GT-0037.
[5]A.P.Dowling,S.R.Stow,Acousticanalysisofgasturbinecombustors,J.Propuls.Power19 (5)(2003)751–764.
[6] A.P. Dowling, A.S. Morgans, Feedback control of combustionoscillations, Annu. Rev. FluidMech. 37 (2005) 151–182, http://dx.doi.org/10.1146/ annurev.fluid.36.050802.122038.
[7] M.Bauerheim,F.Nicoud,T.Poinsot,Theoreticalanalysisofthemassbalanceequationthroughaflameatzeroandnon-zeroMach numbers,Combust. Flame162 (1)(2015)60–67,http://dx.doi.org/10.1016/j.combustflame.2014.06.017.
[8] L.StrobioChen,S.Bomberg,W.Polifke,Propagationandgenerationofacousticandentropywavesacrossamovingflamefront,Combust.Flame166 (2016)170–180,http://dx.doi.org/10.1016/j.combustflame.2016.01.015.
[9]S.R.Stow,A.P.Dowling,Atime-domainnetworkmodelfornonlinearthermoacousticoscillations,in:ASMETurboExpo,2008,GT2008-50770.
[10]T.Poinsot,D.Veynante,TheoreticalandNumericalCombustion,2ndedition,R. T. Edwards,2005.
[11] F.Nicoud,L.Benoit,C.Sensiau,T.Poinsot,Acousticmodesincombustorswithcompleximpedancesandmultidimensionalactiveflames,AIAAJ.45 (2) (2007)426–441,http://dx.doi.org/10.2514/1.24933.
[12] L. Crocco,Research on combustioninstability inliquid propellantrockets, Symp.,Int., Combust. 12 (1)(1969) 85–99, http://dx.doi.org/10.1016/ S0082-0784(69)80394-2,http://linkinghub.elsevier.com/retrieve/pii/S0082078469803942.
[13]T.Kato,PerturbationTheoryforLinearOperators,2ndedition,Springer,Berlin,Heidelberg,NewYork,1980.
[14]A.Messiah,QuantumMechanics,DoverPublications,1961.
[15] P.Mantegazza,G.Bindolino,Aeroelasticderivativesasasensitivityanalysisofnonlinearequations,AIAAJ.25 (8)(1987)1145–1146,http://dx.doi.org/ 10.2514/3.9758.
[16] Z.-S.Liu,S.-H.Chen,Eigenvalueandeigenvectorderivativesofnonlineareigenproblems,J.Guid.ControlDyn.16 (4)(1993)788–789,http://dx.doi.org/ 10.2514/3.21083.
[17] M.S.Jankovic,Analyticalsolutionsforthenthderivativesofeigenvaluesandeigenvectorsforanonlineareigenvalueproblem,AIAAJ.26 (2)(1988) 204–205,http://dx.doi.org/10.2514/3.9873.
[18]A.L.Andrew,K.-W.E.Chu,Commenton“Analyticalsolutionsforthenthderivativesofeigenvaluesandeigenvectorsforanonlineareigenvalueproblem”, AIAAJ.29 (7)(1990)1990–1991.
[19] L.Li,Y.Hu,X.Wang,Astudyondesignsensitivityanalysisforgeneralnonlineareigenproblems,Mech.Syst.SignalProcess.34(2012)88–105,http:// dx.doi.org/10.1016/j.ymssp.2012.08.011.
[20] A.M.Tumin,A.V.Fedorov,Instabilitywaveexcitationbyalocalizedvibratorintheboundarylayer,J.Appl.Mech.Tech.Phys.25(1984)867–873,http:// dx.doi.org/10.1007/BF00911661.
[21] D.C. Hill,A theoreticalapproachfor analyzing therestabilization ofwakes,NASA Technicalmemorandum103858, http://adsabs.harvard.edu/abs/ 1992aiaa.meetT....H.
[22] F. Giannetti, P. Luchini, Structural sensitivity of the first instability of the cylinder wake, J. Fluid Mech. 581 (2007) 167–197, http://journals. cambridge.org/abstract_S0022112007005654.
[23] D.Sipp,O.Marquet,P.Meliga,A.Barbagallo,Dynamicsandcontrolofglobalinstabilitiesinopen-flows:alinearizedapproach,Appl.Mech.Rev.63 (3) (2010)030801,http://dx.doi.org/10.1115/1.4001478.
[24]P.J.Schmid,L.Brandt,Analysisoffluidsystems:stability,receptivity,sensitivity,Appl.Mech.Rev.66 (2)(2014)024803.
[25] O.Tammisola,F.Giannetti,V.Citro,M.P.Juniper,Second-orderperturbationofglobalmodesandimplicationsforspanwisewavyactuation,J.Fluid Mech.755(2014)314–335,http://dx.doi.org/10.1017/jfm.2014.415,http://www.journals.cambridge.org/abstract_S0022112014004157.
[26] S.Camarri,Flowcontroldesigninspiredbylinearstabilityanalysis,ActaMech.226 (4)(2015)979–1010,http://dx.doi.org/10.1007/s00707-015-1319-1. [27]P.Luchini,A.Bottaro,Adjointequationsinstabilityanalysis,Annu.Rev.FluidMech.46(2014)1–30.
[28] L.Magri,M.P.Juniper,Sensitivityanalysisofatime-delayedthermo-acousticsystemviaanadjoint-basedapproach,J.FluidMech.719(2013)183–202,
http://dx.doi.org/10.1017/jfm.2012.639.
[29] L.Magri,M.P.Juniper,Atheoreticalapproachforpassivecontrolofthermoacousticoscillations:applicationtoductedflames,J.Eng.GasTurbines Power 135 (9)(2013) 091604, http://dx.doi.org/10.1115/1.4024957, http://gasturbinespower.asmedigitalcollection.asme.org/article.aspx?doi=10.1115/ 1.4024957.
[30]L.Magri,M.P.Juniper,Adjoint-basedlinearanalysisinreduced-orderthermo-acousticmodels,Int.J.SprayCombust.Dyn.6 (3)(2014)225–246.
[31]G.Rigas,N.P.Jamieson,L.K.B.Li,M.P.Juniper,Experimentalsensitivityanalysisandcontrolofthermoacousticsystems,J.FluidMech.787(2016)R1.
[32] L.Magri,M.P.Juniper,Globalmodes,receptivity,andsensitivityanalysisofdiffusionflamescoupledwithductacoustics,J.FluidMech.752(2014) 237–265,http://dx.doi.org/10.1017/jfm.2014.328.
[33] A.Orchini,M.P.Juniper,Linearstabilityandadjointsensitivityanalysisofthermoacousticnetworkswithpremixedflames,Combust.Flame165(2015) 97–108,http://dx.doi.org/10.1016/j.combustflame.2015.10.011.
[34]L.Magri,Adjointmethodsinthermo-acousticandcombustioninstability,Ph.D.thesis,UniversityofCambridge,2015.
[35]M.P.Juniper,L.Magri,M.Bauerheim,F.Nicoud, Sensitivityanalysisofthermo-acousticeigenproblemswithadjointmethods,in:SummerProgram, CenterforTurbulenceResearch,2014,pp. 189–198.
[36]E.J.Hinch,PerturbationMethods,CambridgeUniversityPress,1991.
[37]G.A.Mensah,J.P.Moeck,EfficientcomputationofthermoacousticmodesinannularcombustionchambersbasedonBloch-wavetheory,in:ASMETurbo Expo,2015,GT2015-43476.
[38] M.Bauerheim,J.-F.Parmentier,P.Salas,F.Nicoud,T.Poinsot,Ananalyticalmodelforazimuthalthermoacousticmodesinanannularchamberfedby anannularplenum,Combust.Flame161 (5)(2014)1374–1389,http://dx.doi.org/10.1016/j.combustflame.2013.11.014,http://linkinghub.elsevier.com/ retrieve/pii/S0010218013004276.
[39] J.O’Connor,V.Acharya,T.Lieuwen,Transversecombustioninstabilities:acoustic,fluidmechanic,andflameprocesses,Prog.EnergyCombust.Sci.49 (2015)1–39,http://dx.doi.org/10.1016/j.pecs.2015.01.001.
[40]L.Magri,M.Bauerheim,F.Nicoud,M.P.Juniper,Stabilityanalysisofthermo-acousticnonlineareigenproblemsinannularcombustors.PartII. Uncer-taintyquantification,J.Comput.Phys.325(2016)411–421.
[41]J.T.Oden,AppliedFunctionalAnalysis,Prentice-Hall,Inc.,1979.
[42]N.Noiray,B.Schuermans,Onthedynamicnatureofazimuthalthermoacousticmodesinannulargasturbinecombustionchambers,Proc.R.Soc.A469 (2013)20120535.
[43]G.Ghirardo,M.P.Juniper,Azimuthalinstabilitiesinannularcombustors:standingandspinningmodes,Proc.R.Soc.A469(2013)20130232.
[44] M.Bauerheim,P.Salas,F.Nicoud,T.Poinsot,Symmetrybreakingofazimuthalthermo-acousticmodesinannularcavities:atheoreticalstudy,J.Fluid Mech.760(2014)431–465,http://dx.doi.org/10.1017/jfm.2014.578.
[45] M.Boileau,G.Staffelbach,B.Cuenot,T.Poinsot,C.Berat,LESofanignitionsequenceinagasturbineengine,Combust.Flame154 (1–2)(2008)2–22,
http://dx.doi.org/10.1016/j.combustflame.2008.02.006.
[46] M.Bauerheim,A.Ndiaye,P.Constantine,S.Moreau,F.Nicoud,Symmetrybreakingofazimuthalthermoacousticmodes:theUQperspective,J.Fluid Mech.789(2016)534–566,http://dx.doi.org/10.1017/jfm.2015.730.
[47]L.Crocco,S.-I.Cheng,Theoryofcombustioninstabilityinliquidpropellantrocketmotors,Tech.rep.,January1956.
[48]T.Lieuwen,UnsteadyCombustorPhysics,CambridgeUniversityPress,2012.
[49] N.Noiray,M.Bothien,B.Schuermans,Investigationofazimuthalstagingconceptsinannulargasturbines,Combust.TheoryModel.15 (5)(2011) 585–606,http://dx.doi.org/10.1080/13647830.2011.552636.
