A novel modal expansion method for low-order modeling of thermoacoustic instabilities in complex geometries

an author's

https://oatao.univ-toulouse.fr/23926

https://doi.org/10.1016/j.combustflame.2019.05.010

Laurent, Charlélie and Bauerheim, Michaël and Poinsot, Thierry and Nicoud, Franck A novel modal expansion

method for low-order modeling of thermoacoustic instabilities in complex geometries. (2019) Combustion and

Flame, 206. 334-348. ISSN 0010-2180

A

novel

modal

expansion

method

for

low-order

modeling

of

thermoacoustic

instabilities

in

complex

geometries

C. Laurent

a,∗

_,

_M.

_Bauerheim

a,d

_,

_T.

_Poinsot

c

_,

_F.

_Nicoud

b a CERFACS, 42 Avenue Gaspard Coriolis, Toulouse Cedex 1 31057, France

b CNRS, IMAG, Université de Montpellier, Montpellier, France c IMFT, Allée du Professeur Camille Soula, Toulouse 31400, France d ISAE-Supaero, 10 Avenue Edouard-Belin, Toulouse Cedex 4 31055, France

Keywords:

Thermoacoustic instabilities Low Order Model Modal expansion State-space Acoustic network

a

b

s

t

r

a

c

t

Thisworkproposesanimprovementtoexistingmethodsbasedonmodalexpansionsusedforthe predic-tionofthermoacousticinstabilitiesinzeroMachnumberflowconditions.Whereastheorthogonalbasis madeoftheacousticeigenmodesofthedomainboundedbyrigidwallsisclassicallyused,analternative methodbasedonamodalexpansionontoanover-completesetofacousticeigenmodesisproposed.This allowsavoidingthemisrepresentationoftheacousticvelocityfieldoftenobservednear nonrigid-wall boundaries.ALowOrderModelnetworkutilizingastate-spaceframeworkisthenbuiltuponthisnovel typeofmodalexpansion.Severaltestcases,goingfromnonreactingductstoacomplexgeometrywith combustion,arestudiedtoassessthepotentialoftheapproach.Themethodologynotonlysuccessfully mitigatesthemisrepresentationintheacousticfieldinthepresenceofnon-rigid-wallboundaries,butit alsodrasticallyimprovestheconvergencespeed.Themodularityofthemethodanditsabilitytohandle complexgeometriesareillustratedbyconsideringaconfigurationfeaturinganannularchamber,an annu-larplenum,aswellasmultipleburners.Thisnoveltechniqueisexpectedtobringworthyimprovements toexistingLowOrderModelsusingmodalexpansionsforthepredictionofcombustioninstabilities.

1. Introduction

Since their early study by Rayleigh [1], thermoacoustic insta-bilities havebeen a subject of primary scientific interest aswell as a major concern for a number of industrial projects. Experi-mentswithincreasinglycomplexsetupsandadvanceddiagnostics were carried out over the years to study this intricate interplay betweenflamedynamicsandacousticwaves.Progressesin exper-imental workswere accompanied by considerableefforts inboth numericalandtheoreticalstudyofthermoacousticinstabilities. Re-gardingtheformer,LargeEddySimulation(LES)wasprovedasthe mostaccurate tool for the analysisof instabilities in combustors featuringcomplex geometries [2].Yet, thehigh costof thesefull scale simulations led to a growing popularity of alternative and cheapernumericalmethodsrelyingonaseparationoftheacoustic flowandthe complexflamedynamics.The acousticfield is solu-tionoftheHelmholtzequationinthefrequencydomain,whilethe flameresponsetoacousticperturbationsisoftenembeddedintoa

∗ _{Corresponding author.}

E-mail address: [email protected] (C. Laurent).

FlameTransfer Function(FTF)[3],ora FlameDescribingFunction (FDF)representingnonlineareffects[4].

One of the key aspects for the resolution of thermoacoustic eigenmodes is the ability of the method to accurately account for complex geometries that are encountered in industrial com-bustors. Oneofthemoststraightforward approachesisthedirect discretizationoftheHelmholtzequationthatisthensolvedthanks to a Finite Element Method (FEM) solver. State-of-the-art FEM Helmholtz solvers are able tosolve for thermaoustic eigenmodes in complex geometries comprising active flames and dissipative effects[5,6],andcanalsoincorporatetheFDFformalismtocapture nonlinearlimit-cyclebehaviors[7,8].However,directdiscretization FEMHelmholtz solversoftenresultin alarge numberofDegrees ofFreedom (DoF),synonym ofaconsiderablecomputational cost, and only permit little modularity, asany change in the geomet-rical parameters requires a new geometry and mesh generation. In order to circumvent these shortcomings, numerous research groups have opted for the development of Low Order Models (LOMs) enabling even cheaper resolution of thermoacoustic in-stabilities. Low Order Modeling resides in two basic ideas: (1) the number of DoF should be reduced as much as possible in order to permit fastcomputations, and (2)the model should be flexible andhighlymodular,inthe sense that itshould allowfor https://doi.org/10.1016/j.combustflame.2019.05.010

Fig. 1. Schematic representation of an acoustic network. Hashed lines show rigid walls. Gray dotted lines are opening to the atmosphere. The geometry is split into distinct subsystems, that are connected together. The subsystem ( i ) is defined by its volume i , its rigid wall boundary S wi , its boundary opened to the atmosphere S ai , and its

boundaries S ci that are to be connected to other subsystems in the acoustic network. The interface S ci is split into several surface elements S0j located at the connection

points  x0j . It also contains a volume heat source H i .

the straightforward modification of mostgeometrical or physical parameters. The latterpoint is often achieved thanks to a divide

and conquerstrategy, where complexgeometries aredecomposed

into anacoustic networkof simplersubdomains(see Fig.1). Fast and modular LOMs have been promisingly applied to intensive tasks demanding alarge numberofrepeatedresolutions, such as Monte Carlo Uncertainty Quantification [9,10], or passive control throughadjointgeometricaloptimization[11].

ExistingthermoacousticLOMs canbe classifiedintofourmain categories, according to the method employed to describe the acousticfield.

1. The first class ofLOM isa wave-based 1D networkapproach, wherethe acoustic pressure andvelocity are written in func-tion of the Riemann invariants A+ and A−. This method was first successfullyusedin theLOTANtool [12], designedto re-solveinthefrequencydomainlinearlyunstablethermoacoustic modes in simple configurations. More recently and in a sim-ilar fashion, the open-source LOM solver Oscilos [13] devel-oped atImperial College,London,wasusedtoperform for in-stancetime-domain simulationsofthermoacousticlimitcycles inlongitudinalcombustors[14,15]comparabletotheRijketube

[16].Wave-based low order modeling wasalso generalizedto morecomplex cases,including azimuthal modesin configura-tionscomprisingan annularcombustion chamberlinked toan annular plenumthrough multiple burners. This procedure al-lowedBauerheimetal.toconductaseriesofstudiesbasedon a familyof analytical solutions forazimuthal modes in annu-larcombustors[9,17,18].Eventhoughwave-basedLOMsarethe mostadequatetodealwithnetworksoflongitudinalelements whereacousticwavescanbeassumedasplanar,theyalso suf-fer strict limitations: they are indeed unable to capture non-planarmodesincomplexgeometries.

2. The second class of LOM relies on modal (or Galerkin) ex-pansions,to expresstheacoustic pressurefield asa combina-tion ofknown acoustic modes. Modal expansion wasfirst in-troducedandformalizedin an acousticcontext by Morse and Ingard intheirinfluential bookTheoreticalAcoustics[19]dated from1968.Inthefieldofthermoacoustics,Zinnetal.[20] and Culick [21–24] were among the first to use it to study com-bustion instabilities in liquid fuel rocket engines. Similarly to thewave-based approach,multiplestudies utilizingmodal ex-pansions are dealing withthe Rijke tube: for example by

Ju-niper [25], WaughandJuniper[26],andBalasubramanianand Sujith [27]. Simplified annular configurations were also ex-amined thanks to pressure modal expansion: Noiray and co-workers [28–30]and Ghirardo et al. [31] conducted a se-ries of theoretical studies in such geometries. More complex modal expansion-based networks were developed for multi-burners chamber-plenum geometries, by Stow and Dowling

[32],Schuermansandco-workers[33,34],andBellucietal.[35]. Their strategy is to perform modal expansions for the pres-sure inthechamber/plenum andto assume acoustically com-pact burners that can be lumped and represented by simple transfer matrices. Unlikewave-based low-ordermodeling, this method is not limited to planar acoustic waves, and can re-solve both azimuthal and longitudinal chamber modes. Even though their approach does not rely on an acoustic network decomposition,Bethkeetal.[36]showedthat arbitrarily com-plex geometriescan beincorporatedin athermoacousticLOM by expanding thepressure onto a setof basis functions com-puted inapreliminarystepthanks toaFEMHelmholtzsolver. Although they appear more general than wave-based LOMs, modal expansion-basedLOMs are alsosubjected tostrict lim-itations, whichmainly resides inthe choice ofthe modal ba-sisemployedtoexpandtheacousticpressure.Thispointisthe mainobjectofthispaper,andisdiscussedinmoredetailslater. 3. ThethirdclassofthermoacousticLOMisamixedmethod,that combines bothRiemanninvariantsandmodalexpansions.The formerareusedtoaccount forlongitudinalpropagation, while the latter are used to account for multi-dimensional geome-tries.ThismixedmethodwasemployedbyEvesqueandPolifke

[37]tomodelazimuthalmodesinmulti-burnersannular cham-ber/plenum configurations. In more recent works [38,39] the mixedstrategy wasfurtherdeveloped toinvestigate nonlinear spinning/standinglimitcyclesintheMICCAannularcombustor. More precisely, in[38] the planaracoustic field inthe ducted burners isrepresented byRiemann invariants, while itis rep-resentedthroughmodalexpansionsinthechamber andinthe plenum.

4. Finally,thelast classofLOMs consistsinthose basedona di-rectspatialdiscretization.Forexample,Sayadi etal.[40]made use of a finite difference scheme to build a dynamical sys-tem representation of a one-dimensional thermoacoustic sys-tem comprisinga volumetricheat source localized within the domain. A more generic direct discretization LOM is the taX

Low-OrderModel developedatTechnical UniversityofMunich

[41]. Its modularity lies in the ability to combine in a same thermoacousticnetwork one-dimensional elementsdiscretized byfinitedifference,geometricallycomplexelementsdiscretized through FEM, and other types of elements such as FTFs and scattering matrices. Since this LOM is built upon direct dis-cretizationoflinearizedpartialdifferentialequations,itcanalso potentially incorporatericherphysics,includingmean flow ef-fectsoracoustic–vortexinteractions.However,thepricetopay forthishighmodularityisalargenumberofDoF:forinstance, in the taX LOM, about O (105_{) DoF were needed to obtain} acoustic eigenmodes of an annular combustor comprising 12 ductedinjectors(butnoactiveflame).

Interestingly,mostmodalexpansionLOMsusedthesametype ofeigenmodes basis,namelythe basiscomposed oftherigid-wall

cavitymodes,orinotherwordsacousticeigenmodessatisfying

ho-mogeneousNeumannboundaryconditions(i.e.zeronormal veloc-ity)overtheentireboundariesofthedomain(andwithoutinternal volumesources).Thispaperfocusesonthenatureoftheacoustic eigenmodesbasisusedtodecomposethepressure,andthe conver-gencepropertiesresultingfromthisexpansion.Althoughthe rigid-wallmodalbasispresentsthehugeadvantageofbeingorthogonal, manyactual systems obviously comprisefrontiers withfar more complexboundaryconditionsthanjustahomogeneousNeumann condition(forexampleaninletoranoutletwheretheimpedance hasafinitevalue).Theuseofsuchbasisthenappearsparadoxical:

howisitpossiblethatasolutionexpressedasarigid-wallmodes

se-riesconvergestowardsasolutionsatisfyinganonrigid-wallboundary

condition?Thissingularityinthepressuremodalexpansionwas

al-readynoticed by Morse andIngard [19],but they did not study itsimpacton theconvergenceofthewhole method.Later,Culick

[24]providedmoreexplanationsaboutthissingularity:themodal expansiondoesnotconvergeuniformlyoverthedomain,butonly inthe less restrictive sense ofthe Hilbert norm (L2 norm); asa result, even though each individual term of the expansion does notsatisfytheappropriateboundarycondition,theinfinitesumof thesetermsmaysatisfy it. Inother words, “thelimit ofthe sum isnot equalto the sumofthe limit” intheneighborhood of the boundary.Aninteresting examinationregardingthissingularityis providedinarecentworkbyGhirardoetal.[42],wherea projec-tionontoamode satisfyingaspecific nonrigid-wallinlet bound-arycondition wasdiscussed. The convergence issuearising from rigid-wallmodalexpansionisevenmoreproblematicinthecaseof acousticLOMswherethegeometryisdecomposedintoanetwork ofsubsystemsthatneedtobecoupledtogetherattheirboundaries (seeFig.1).Foreachindividualsubdomain,thesecoupling bound-ariesarenot rigid-wallbuteach termofthebasis correspondsto arigid-wall:theconvergencesingularitymayariseateachone of thecouplinginterfaces.AlthoughCulick[24]proposedan explana-tiontothissingularitybasedonalocaljustification,onlyveryfew studiesdeal withglobal effects,such asforexample convergence speedoftheeigenfrequencies.

The main contribution of this work is a reformulation ofthe modalexpansion method under the zero Mach number flow as-sumption, allowing for decompositions of the pressure onto an

over-complete family of modes, gathering rigid-wall (u=0) and

pressurerelease(p₌0)modes, insteadoftheclassical rigid-wall modesbasis. Thispaperisorganizedasfollows:inSection 2,the original mathematical derivation of pressure expansion onto the rigid-wallmodal basis is recalled. The problemis then reformu-latedusing expansionsonto an over-complete set ofmodes, also calleda frame. In this frame, non rigid-wall conditions are used for certain elements. Section 3 briefly describes the state-space formalismproposed in[33,34]tocouplesubsystemstogether and build a modular acoustic LOM network. This state-space

formal-ism isthen extended toaccount forthe frameof non-orthogonal modes.InSection4,theconvergencepropertiesareevaluatedand comparedforbothtypesofmodalexpansionsonacanonicalcase which consistsof along duct witha sharpcross-section change. Finally in Section 5, the modularity of the proposed LOM and its abilityto dealwithcases involvingcombustion instabilitiesin complexconfigurationsisdemonstratedbyconsideringanannular chamber/plenumconfigurationfeaturingmultipleburners. 2. Acousticpressuremodalexpansion

InacousticLOMnetworksforcomplexconfigurations,the sys-temis splitinto smallergeometric subsystems. Letusconsider a subsystem(i)asinFig.1,definedasaboundeddomain



i

delim-itedby

∂



i=Swi∪Sai∪Sci.Theprimaryvariableofinterestisthe

acoustic pressure p

(

x,t

)

expressed in the physical space, but its frequency-domaincounterpart pˆ

(

x_,

ω

)

isoftenintroducedto ease the analysis,especiallywhen dealingwithlinearacoustics. These twoquantitiesarerelatedbythe(inverse)Fouriertransformas fol-lows: ˆ p

(

x,

ω

)

=  _+∞ −∞ e − jωt_p

₍

_x,t

₎

_{dt, p}

₍

_x,t

₎

₌ +∞ −∞ e jωt_pˆ

₍

_x,

_ω

₎

_d

_ω

(1)

Forthesakeofsimplicity,inthefollowingthesoundspeedfieldis assumeduniformandthebaselineflowtobeatrest.Notehowever thatthesehypothesesarenotnecessaryandcouldbeomitted.The frequency-domainacousticpressurepˆ

(

x,

ω

)

inthesubsystem



iis

thensolutionofthefollowingHelmholtzequation:

⎧

⎪

⎨

⎪

⎩

c2 0

∇

2pˆ

(

x,

ω

)

− j

αω

pˆ

(

x,

ω

)

+

ω

2pˆ

(

x,

ω

)

=hˆ

(

x,

ω

)

for x∈



i

∇

spˆ=0 for xs∈Swi, pˆ=0 for xs∈Sai

∇

spˆ= fˆ

(

xs,

ω

)

for xs∈Sci (2)

wherethenotation

∇

spˆ=

∇

pˆ

(

xs

)

.nsisintroduced,xsbeinga

geo-metricalpointbelongingtoaboundaryoftheflowdomainandns

thenormalunityvectorpointingoutward.InEq.(2),

α

isan acous-tic losscoefficient, the term ˆh

(

x,

ω

)

is a volume acousticsource, while fˆ

(

x,

ω

)

isasurfaceforcing termimposedontheconnection boundarySci:itisanexternalinputtotheacousticsubsystem

con-tainedin



iexertedbyadjacentsubsystems.Thepressureverifies

rigid-wallboundaryconditiononSwiandisnullonSai.The

volu-mic source hˆ

(

x,

ω

)

represents fluctuations ofheat release, andit maybewrittenasfollows:

ˆ

h

(

x,

ω

)

=− j

ω

(

γ

− 1

)

ω

ˆT

(

x,

ω

)

=− j

ω

(

γ

− 1

)

Hi

(

x

)

Qˆ

(

ω

)

(3)

where

γ

istheheat capacityratioand

ω

ˆT

(

x,

ω

)

isthe local

fluc-tuatingheatreleaserateresultingfromflamedynamics.InEq.(3), ˆ

ω

T

(

x,

ω

)

isdecomposedintoa globalheatreleaserateQˆ

(

ω

)

and

a spatialvolume densityHi

(

x

)

representingthe flameshape (the

integralofHi

(

x

)

over



iisunity).Inthefollowing,onlyoneflame

inthesubdomain



iisconsideredforconciseness,butthe

reason-ing can be extendedwithout difficulty to anynumberof distinct andindependentflameslocatedinasamesubdomain



_i.

In modal expansion based LOMs, on each subsystem



i the

pressure isdecomposed onto a familyformed ofknown acoustic eigenmodesof



_i.Thepurposeisthentoderiveasetofgoverning equationsforthecorresponding modalamplitudes.Thisstep usu-allymakesuseoftheinnerproductdefinedforanyfunctions f

(

x

)

andg

(

x

)

as:



f,g

= 

f

(

x

)

g

(

x

)

d

3_{x (4)}

TheassociatedL2_normisnoted

_||

_f

_||

2.1. Theclassicalrigid-wallmodalexpansion

The usual derivation consists in writing the decomposition of the acoustic pressure onto an orthogonal basis of known acoustic modes

(

ψ

n

(

x

)

)

n1 of the subdomain



i, as pˆ

(

x,

ω

)

=



n

γ

ˆn

(

ω

)

ψ

n

(

x

)

. The set

(

ψ

n

(

x

)

)

n1 is classically chosen as the rigid-wall eigenmodes of the subsystem



_i (without volume sourcesandacousticdamping).Theseeigenmodesverifyrigid-wall conditions (i.e. zero normal velocity) over Swi, but also over the

connection boundary S_ci. In the presence of boundaries that are known tobe openedtothe atmosphere(Sai in Fig.1),the

eigen-modesbasiscanbechosentosatisfytheappropriateconditionon

Sai (i.e.zeropressure), without furtherdifficulty since the

expan-sionbasisisstillorthogonal.Theset

(

ψ

n

(

x

)

)

n1issolutionofthe followingeigenvalueproblem:

⎧

⎨

⎩

c2 0

∇

2

ψ

n+

ω

2n

ψ

n=0 for x∈



i

∇

s

ψ

n=0 for x∈Swi,

ψ

n=0 for x∈Sai

∇

s

ψ

n=0 for x∈Sci (5)

where

ω

n is the eigen-pulsation of the nth eigenmode. By

mak-inguseofthesecondGreen’sidentity(remindedinSupplemental Material A), it can be shown that the set

(

ψ

n

(

x

)

)

n1 defined by Eq.(5)isindeedanorthogonalbasis,that is



ψ

n,

ψ

m

=0forany n=m.

For conciseness, the successive steps required to obtain the modalamplitudes

γ

ˆn

(

ω

)

are notdetailedhere, butthe interested

readeris referred to Supplemental MaterialA, aswell as[19,43]. Toease the formalism,the connectionsurfaceS_ci issplit intoMS

elements



S0jconnecting



i withtheadjacentsubdomains



jat

theboundarypointsx0j.Notethatthesubdomains



jarenot

nec-essarily distinct, since there may exist several connection points between



i and a same neighbor (i.e. we can have x0j1=x0j2,

but



j1=



j2).An exampleofsuch surfacesplittingisshownin

Fig.1. After projectingthepressure field ontothe orthogonal ba-sis

(

ψ

n

(

x

)

)

n1 thankstotheinnerproduct definedinEq.(4),the modalamplitudes

γ

ˆn

(

ω

)

arefoundtobesolutionsofthefollowing

equation:

(

ω

2_{− j}

ωα

₋

ω

2 n

)



ˆ

γ

n

(

ω

)

j

ω

= MS j=1

ρ

0c2₀



S0j

ψ

n

(

x0j

)

n ˆ uj s

(

x0j

)

−

(

γ

− 1

)

H( n) i

n ˆ Q

(

ω

)

(6) where

n=

||

ψ

n

||

22,H( n)

i =



Hi,

ψ

n

istheprojection ofthe flame

shape Hi

(

x

)

onto

ψ

n

(

x

)

, and uˆsj

(

x0j,

ω

)

=− ˆf

(

x0j,

ω

)

/

(

ρ

0j

ω

)

is thenormalvelocityforcingimposedbyadjacentsubsystems



jat

theboundarypointsx0j.

Sincetimederivativeinthephysicalspacecorrespondsto mul-tiplying by j

ω

in the Fourier domain, it is possible to recast

Eq.(6) intothe time-domain. Indoing so, it proves usefulto in-troduce



ˆ_n=

γ

ˆn/

(

j

ω

)

(i.e.



˙n

(

t

)

=

γ

n

(

t

)

)andthenusetheinverse

Fouriertransformtoobtain:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

p

(

x,t

)

=∞ n=1 ˙



n

(

t

)

ψ

n

(

x

)

¨



n

(

t

)

=−

α



˙n

(

t

)

−

ω

n2



n

(

t

)

− MS j=1

ρ

0c20



S0j

ψ

n

(

x0j

)

n uj s

(

x0j,t

)

+

(

γ

− 1

)

H (n) i

n Q

(

t

)

(7)

This dynamical system governs the temporal evolution of the pressure field in the subdomain



i, under the normal velocity

forcing usj

(

x0j,t

)

imposed by adjacent subsystems



j, and

un-der the volume forcing Q (t) imposed by fluctuating flames con-tained within



i. This set of equations was used for example

in[32,33], where the infiniteseries wastruncated up to a finite orderN.It isalso worth notingthat theacoustic velocity canbe calculatedfromthe knowledgeofthemodalamplitudes



n (t)as 

u

(

x,t

)

=−n



n

(

t

)

∇



ψ

n

(

x

)

/

ρ

0. A state-space approach can then be used to couple together the subsystems defining the whole thermoacousticsystemofinterest.Thisformalismwillbedetailed inSection3.

Finally, since the acoustic pressure is a linear combination of themodalbasisvectors

ψ

n,itnecessarilyverifiesthesame

bound-aryconditions,inparticular

∇

sp=0,viz.u.ns=0onSci.Sincethe

acoustic velocity should not be zero over the boundaries of the (arbitrarilychosen)sub-domain



i, thismayresultina

singular-ityintherepresentationoftheacousticvelocityfield.The impact ofthissingularityontheconvergencepropertiesofthemethodis discussed in Section 4. The following section proposes a mathe-maticalreformulationofthepressuremodalexpansiontomitigate thisundesirablefeature.

2.2.Modalexpansionontoanover-completeframeofacoustic

eigenmodes

The purpose is now to introduce a modal expansion of the acoustic pressure that would allow for satisfying any boundary conditiononSci(andnot

∇

sp=0only).Inthismatter,itis

neces-sarytoretaininthemodalexpansionanadditionaldegreeof free-dom,suchthatboth acousticpressureandnormalacoustic veloc-ityatthe connectionboundary Sci remaina priori undetermined.

Letusthen introduce two distinct families (

ξ

m)m≥ 1 and(

ζ

k)k≥ 1 ofacousticeigenmodesofthesubsystem



_i,characterizedby the twofollowingeigenproblems:

⎧

⎨

⎩

c2 0

∇

2

ξ

m+

ω

2m

ξ

m=0 for x∈



i

∇

s

ξ

m=0 for x∈Swi,

ξ

m=0 for x∈Sai

∇

s

ξ

m=0 for x∈Sci (8)

⎧

⎨

⎩

c02

∇

2

ζ

k+

ω

k2

ζ

k=0 for x∈



i

∇

s

ζ

k=0 for x∈Swi,

ζ

k=0 for x∈Sai

ζ

k=0 for x∈Sci (9)

Both eigenmodes families (

ξ

m)m≥ 1 and (

ζ

k)k≥ 1 verify the same rigid-wall(resp.open)boundaryconditionsonSwi (resp.Sai). The

eigenmodes family (

ξ

m)m≥ 1 is similar to the orthogonal basis (

ψ

n)n≥ 1 used in Section 2.1. Conversely, the eigenmodes family (

ζ

k)k≥ 1 differssinceitverifiesopenboundaryconditionsonSci.

Consider now the eigenmodes family (

φ

n)n≥ 1 formed as the concatenationof (

ξ

m)m≥ 1 and(

ζ

k)k≥ 1:

(

φ

n

)

n1=

(

ξ

m

)

m1∪

(

ζ

k

)

k1.The eigenpulsationsassociatedto theeigenmodes

φ

n are

noted

(

ω

n

)

n1=

(

ω

m

)

m1∪

(

ω

k

)

k1. As a concatenation of two orthogonal bases, (

φ

n)n≥ 1 is not a basis but is instead an over-completesetofeigenmodes, alsocalledaframe[44].The concept of frame wasintroduced in the context of nonharmonic Fourier analysis[45], andlater used in a number offields ranging from waveletanalysis,todigitalimage processingandtime-series fore-casting.Theuseofan over-completeframe toperformmodal ex-pansionsisthemostcrucialelement oftheproposed method.Let usprecisethattheframe(

φ

n)n≥ 1couldbebuiltfromthe concate-nationofothersetsofeigenmodes,aslongasthosedonotapriori

imposeanyconstraintbetweenpressureandvelocityatthe bound-ary Sci. However, using the concatenation of the rigid-wall basis

(

ξ

m)m≥ 1andtheopenatmospherebasis(

ζ

k)k≥ 1hastwomain ad-vantages:(1)they areusually theeasiesttoobtain analyticallyor

numerically,and(2) theframe (

φ

n)n≥ 1 formed by their concate-nationverifiesageneralizedPerseval’s identitywhichensuresthe well-posednessofmodalexpansions[44].

In the following, a compact vectorial notation is introduced to avoid the use of multiple summation symbols: for any in-dexed quantity (fi)i≥ 1 we note f the column-vector such that:

t_f=

₍

_f

0 f1 f2 ...

)

, where t() designates the vector transpose. Foradoublyindexedquantity(fij)i,j≥ 1,wenotefthematrixwhose coefficientsarethef_ij.Conversely,(f)n isthenthcomponentofthe

vectorf.SimilarlytoSection 2.1,thepressuremodalexpansion is soughtundertheformpˆ

(

x,

ω

)

=n

γ

ˆn

(

ω

)

φ

n

(

x

)

,thatis,usingthe

vectorial notation pˆ

(

x,

ω

)

=t

_γ

_ˆ

₍

_ω

₎

_φ

₍

_x

₎

_{. The analytical derivation}

ofthe modalamplitudes

γ

ˆn

(

ω

)

israther long andits details are

notnecessarytounderstandtheremainingofthepaper;itisthus notincludedherebutmadeavailableinSupplementalMaterialB. Asanoutcomeofthisanalyticalderivation,themodalamplitudes

ˆ

γ

n

(

ω

)

arefoundtobesolutionsofthefollowingequation:

(

ω

2_{− j}

ωα

₋

ω

2 n

)



ˆ

γ

n

(

ω

)

j

ω

= MS j=1

ρ

0c20

φ

n⊥

(

x0j

)

uˆsj

(

x0j

)

− MS j=1

ρ

0c20

∇

s

φ

n⊥

(

x0j

)

ϕ

ˆj

(

x0j

)

−

(

γ

− 1

)

H⊥ i,nQˆ

(

ω

)

(10) where:

φ

⊥ n

(

x0j

)

=



S0j



−1

_φ

₍

_x 0j

)



n,

∇

s

φ

⊥ n

(

x0j

)

=



S0j



−1

_∇

s

φ

(

x0j

)



n, H⊥ i,n=



−1_H i



n (11)

Aspreviously thesurfaceSci hasbeendecomposedintoMS plane

surfaceelements



S0jlocatedatx0j.InEq.(11),

=



t

φ

,

φ

isthe

matrixwhose coefficientsare

mn=



φ

m,

φ

n

.This matrix is the

Grammatrixassociatedtotheover-completeframe(

φ

n)n≥ 1:it re-ducestoadiagonalmatrixinthecaseofanorthogonalbasis.The volumesourceˆh

(

x_,

ω

)

isexpressedaccordingtoEq.(3),and_Hiis thecolumnvectorcontainingalltheprojectionsoftheflameshape

Hi

(

x

)

onto the elements of the over-complete frame

(

φ

n

(

x

))

n1. In Eq. (10), uˆj

s

(

x0j

)

is the velocity forcing imposed onto



i by

the adjacent subsystems



_j at the connection points x_0j, while ˆ

ϕ

j

(

x0j

)

=− ˆpj

(

x0j

)

/

(

ρ

0j

ω

)

is an acousticpotential forcing im-posedonto



i atthese boundary points. One maynote that the

secondtermintheright-handside ofEq.(10)hasnocounterpart inEq.(6),contrarytothefirstandthirdterms.

As inSection 2.1for Eq. (6), Eq. (10) can then be recast into thetime-domainbyintroducing



n(t)suchthat

γ

n

(

t

)

=



n

(

t

)

;this

leadsto:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

p

(

x,t

)

= N n=1 ˙



n

(

t

)

φ

n

(

x

)

¨



n

(

t

)

=−

α



˙n

(

t

)

−

ω

2n



n

(

t

)

− MS j=1

ρ

0c20

φ

n⊥

(

x0j

)

usj

(

x0j,t

)

+ MS j=1

ρ

0c20

∇

s

φ

n⊥

(

x0j

)

ϕ

j

(

x0j,t

)

+

(

γ

− 1

)

H⊥i,n Q

(

t

)

(12)

whichisanextensionofEq.(7)whentheexpansionisperformed ontheover-completeframe(

φ

n)n≥ 1insteadoftheorthogonal ba-sis (

ψ

n)n≥ 1. This dynamical system governs the temporal evolu-tion ofthepressure field inthe domain



_i, undernormal veloc-ityforcinguj

s

(

x0s,t

)

andacousticpotentialforcing

ϕ

j

(

x0s,t

)

im-posedby adjacentsubsystems,aswell asthe volumeforcing Q(t) due to the presence of active flames within



_i. In the case of theorthogonalmodalbasis(

ψ

n)n≥ 1 usedintheprevious section, the terms

∇

s

ψ

n⊥

(

x0j

)

vanish(since the eigenmodes satisfy

rigid-wallboundaryconditionsonS_ci),andwehavethesimplerelations

ψ

⊥

n

(

x0j

)

=



S0j

ψ

n

(

x0j

)

/

n andH⊥_i,n=H_i(n)/

n.

Because of the use of the over-complete frame (

φ

n)n≥ 1, the acousticpressureandvelocityarefreetoevolveindependentlyon theboundarySci,whichisthemajorimprovementofthemethod

comparedtoclassicalformalismwherethenormalacoustic veloc-ityu_.nsisnecessarilyzeroontheconnectionboundarySci.

The governing dynamical system of Eq. (12) is a projection of the wave equation (Eq. (2)) onto the modal frame (

φ

n)n≥ 0. However, as this frame is over-complete such projection is ill-conditioned, which constitutes one of the major pitfalls of the proposed method. The first consequence is a numerical diffi-culty to compute the inverse of the frame Gram matrix

−1. In mostcasespresentedinthispaper,theconditionnumberC

(

)

=

||

||

||

−1

_||

_{increases with the size N of the expansion, up to} values ranging from 1018 _{to 10}19_{. The inversion of this poorly} conditioned matrix is achieved thanks to the use of an ade-quate numerical algorithm, based on extra-precision iterative re-finement performed in floating-point quadruple precision. Errors stemming fromthisinversionare systematicallycomputeda

pos-terioriandverified to remainlow. Notethat since the modal

ba-sis size remains in practice limited (typically a few dozens el-ements), the specific inversion procedure used to calculate

−1 does not noticeably increase the computational cost in compar-ison to the classical rigid-wall modal expansion. Secondly, even though

isaccuratelyinverted,theframeover-completenessmay stillresultinpoorlyconditionedspurious componentsinEq.(12). However,theapproachdescribedaboveproduceswell-behaved ex-pansions for the pressure, that is expansions where the terms withthe highestenergyare physicallymeaningful whilelow en-ergytermsrepresentspuriousfluctuations.Thus,an energetic cri-terion for the robust and automatic identification of these spu-rious components was designed, as discussed in more details in

AppendixC.

3. State-spaceformalismforsubsystemscoupling

ThedynamicalsystemderivedinSection2.2allowsustosolve fortheacousticpressureineachindividualsubdomain



i.Inorder

toresolvetheacousticflowinthewholegeometry,individual sub-domainsneedtobeconnectedtogether.Anelegantformulationto connectsubdomainsistouseastate-spaceapproach.Thismethod, alreadyusedin[32–34,41],isadoptedinthisworkbutrequiresto beadapted.Someimplementationdetailsaregivenbelow;further developmentsrelativetostate-spacerepresentationscanbe found incontroltheorytextbooks[46].

Foranyphysicalsystemdescribedbya setofcoordinates X(t) ina phase-space,wecall linearstate-spacerepresentationofthis systemasetofequationsundertheform:

˙

X

(

t

)

=AX

(

t

)

+BU

(

t

)

whereX(t)isthecoordinatesvectorinthephase-space,alsocalled state vector,Ais thedynamicsmatrix,Bistheinput matrix,U(t) theinputvector,Y(t)theoutputvector,Ctheoutputmatrix,andD istheaction,orfeedthroughmatrix.Thefirstequationofthe state-spacerepresentationgovernsthedynamicalevolutionofthestate vector under the forcing exertedby the input vector. The second equation definesa way to compute any desiredoutputsfrom the knowledgeofthestate vectorandtheforcing term.Notethatthe outputY(t)dependsonthestate X(t),butthereverseisnottrue: X(t) evolvesindependentlyof theselected outputY(t).The state-spaceformalism,throughtheRedhefferstar-product[47] (Supple-mentalMaterialC), providesadirectwaytoconnect twosystems represented bytheir state-space realizations,by relatingtheir re-spectiveinputsandoutputs.Asimpleexampleofthisoperationis giveninSection4.

Byconsideringthedynamical systemofEq.(12),itis straight-forward to build a state-space representation for a subdomain



i. A convenient choice is to build the state-vector X(i)(t) from

the modal amplitudes



˙n

(

t

)

and their temporal primitives



n(t).

The input vector U(i)_{(t) contains the normal acoustic velocities}

uj

s

(

x0j,t

)

andthe acousticpotentials

ϕ

j

(

x0j,t

)

imposed by

ad-jacentsubdomains



j.TheheatreleasevolumesourceQ(t)isalso

included inthe input vector. Beside,the computedoutput vector Y(i)_{(t) consistsof the pressures p}

₍

_x

0j,t

)

and the normalacoustic

velocitiesus

(

x0j,t

)

ateverypointx0j ontheconnectionboundary S_ci.Inaddition,anyquantityofinterestthatcanbecomputedfrom themodalamplitudescontainedinthestate-vectorX(i)_(t)mayalso beaddedintheoutputvectorY(i)_{(t).Forconciseness,thedetailed} expressionsofthestate-spacematricesforasubdomain



_iarenot givenhere,butcanbefoundinAppendixA.

After iterativelyapplyingtheRedheffer star-producttoconnect together state-space representationsof every subsystems,the full state-spaceofthewholegeometryisobtainedas:

˙

Xf

(

t

)

=Af Xf

(

t

)

+BfUf

(

t

)

(14)

whereUf_{(t)isanexternalforcingthatcanbeeitherasurfaceora}

volume sourceterm. Intheformer case,coefficientsinthe exter-nalinputmatrixBf_{containstermssimilartothefirst2M}

Scolumns

B(i) _{inEq.(A.1),whileinthelattercaseitcontainstermssimilarto} thelastcolumnofB(i)_{inEq.(A.1).Twoapproachesarethen} possi-ble:(1)Eq.(14)canbeintegratedovertimetoobtainthetemporal evolution oftheacousticflow undertheexternal forcingUf_{(t), or}

(2) thecomplexeigenvaluesandeigenvectorsofthedynamics ma-trixAf_{canbesolvedfor,yieldingtheglobalacoustic}

eigenfrequen-ciesandeigenmodesofthewholedomain.If

λ

n=2

πσ

n+j2

π

fnis

thenth _{complexeigenvalueofthematrixA}f_,thenf

n isthe

eigen-frequency of the nth _{acoustic mode of the whole geometry. In} theabsenceofacousticlosses,volumesourcesorcomplex bound-ary impedances,

σ

n is zero. Conversely, if the system comprises

acousticsources,then2

πσ

n,isthegrowth-rateofthenthacoustic

modeofthewholegeometry:

σ

n>0(resp.

σ

n<0)impliesthatthe

modeisunstable(resp.stable).Themodeshapecanalsobe recon-structedfromthemodalcomponentscontainedintheeigenvector vnassociatedtotheeigenvalue

λ

n.

4. Convergenceproperties

In this section both modal expansions presented in

Section 2 are implemented within the LOM state-space frame-work introduced in Section 3, and used to study a canonical case, namely a long quasi-one-dimensional tube comprising a sharp cross-sectionchange. The goalis toshow the limitsofthe rigid-wallmodesdecompositionandtoprovetheperformancesof the over-complete frameapproach ina casewhereboth typesof modalexpansionscanbeevaluatedandcomparedtoananalytical

solution.Inthefollowing,superscriptsOB_(resp.FR_{)refer toresults} obtainedwiththeuseoftheorthogonalbasis(

ψ

n)n≥ 1introduced inSection 2.1(resp. theover-complete frame (

φ

n)n≥ 1 introduced inSection 2.2). SuperscriptsA _{designateanalytical solutions used} forcomparison.

In this example, the long duct with a sudden cross-section changerepresentedinFig. 2isconsidered. Bothends oftheduct areclosedby rigidwalls.Itis decomposedinto 3subsystems, in-cluding2longducts(



1 and



2)withconstantcross-sectionsS1 andS2, andathird subsystem



sc oflengthLsc enclosingthe

re-gionintheneighborhoodofthecross-sectionvariation.

Since both tubes



1 and



2 are long(D1 L1,D2 L2), only plane longitudinalacoustic wavesare considered here. The rigid-wallorthogonalbasesofbothductsarethen:

⎧

⎪

⎨

⎪

⎩



ψ

(1) n

(

x1

)



nN1=



cos



n

π

x1 L1



nN1



ψ

(2) n

(

x2

)



nN2=



cos



n

π

x2 L2



nN2 (15)

where superscript (1)_(resp. (2)_{) refers to the modal basis in}

_

1 (resp.



2), x1 andx2 arethe longitudinal coordinatesin thetwo ducts(Fig.2),andN1 (resp.N2) isthenumberofmodesusedfor thepressuremodalexpansionin



1(resp.



2).Similarlythe over-completeframesintroducedinSection2.2forbothductsaregiven by:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩



φ

(1) n

(

x1

)



nN1=



cos



n

π

x1 L1



nN1/2 



cos



(

2n+1

)

π

x1 2L1

nN1/2



φ

(2) n

(

x2

)



nN2=



cos



n

π

x2 L2



nN2/2 



sin



(

2n+1

)

π

x2 2L2

nN2/2 (16)

These orthogonal bases (Eq. (15)) and over-complete frames (Eq.(16))are consistentwithrigid-wallconditionsatbothendof thelongduct(x1=0andx2=L2).However, theorthogonalbases alsoimposezerovelocitynearthecrosssectionchange(atx1=L1 andx2=0), while the over-complete framesretain an additional degreeoffreedomsuch thatbothvelocity andpressureare a pri-oriundeterminedandfreetoevolveindependentlynearthe cross-section change. In the following, the same number N of eigen-modesareusedformodalexpansionsinbothducts(N1=N2=N). Additionally,all comparisons betweenorthogonal basis and over-complete frame expansions are carried out with the same total numbers of modes N: in other words results from any orthogo-nal basis containing an even number ofvectors N are compared toresultsfroma framecomposed oftwo subfamiliesofsizeN/2. Notealsothat,sinceonlyplanelongitudinalwavesareconsidered here,theconnectionboundariesSc1andSc2donotneedtobe dis-cretized into severalsurface elements



S0j: thereforeMS=1 for

eachsubdomain



1 and



2,andMS=2for



sc.

Modalexpansion isnot performedforthe subdomain



sc

en-closing the cross-section change as its exact geometry has only very little effect on the global eigenmodes of the long duct: in-stead,volume-averagedconservationequationsare usedtoderive a state-space representation of this subsystem. More details are given in Appendix B. The Redheffer star-product (Supplemental MaterialC)isthenappliedrecursivelytoconnect state-space rep-resentationsofsubsystems



1,



sc,and



2.Thestate-space

real-Fig. 3. Pressure mode shape and velocity mode shape of the first mode ( n = 1 ), for N = 4 , 10 , 100 vectors in the OB expansion. Computed solutions (gray dashed lines) are compared to the analytical solutions of Eqs. (19) and (20) (thick dark lines). Closeup views (corresponding to the rectangles) of the pressure and the velocity mode shapes are given on the second row ((d)–(f)) and the fourth row ((j)–(l)) respectively.

results in low relative errors, ranging between 10−7 and 10−8 for all modes 1, 9 and 11, even with small values of N. For in-stance, only about 10 modes in each frames are necessary to accurately capture the eigenfrequency f11. The conditionnumber of the frame Gram matrix, which is an indicator of the expan-sion over-completeness, isC

(

)

=108 _{forN=6.It then} progres-sively deteriorates and reaches 1019 _{at N=50, and saturates to} thisvalueforlargeN.Thisdeteriorationoftheframeconditioning doesnot resultinto adegradationofthe numericalresults. How-ever, itcan be related toa saturation ofthe error,since increas-ingthesizeoftheframebeyondN₌20doesnotresultinsmaller errors.

Finally, it is worth emphasizing that Gibbs fringes have also beenreportedinearlierstudiesemployingorthogonalbasismodal

expansions, for instance by Sayadi et al. [40]. However, the ori-gin of these Gibbs oscillations wasfundamentally different from those observed in the present example: in previous works, this phenomenonwascausedbythepresence ofaninfinitelythin re-gionof fluctuatingheat releaselocated inthe interiorofthe do-main ofinterest. Thisassumption resultsin an exactsolution for the velocity field that isdiscontinuous atthe flame location, and that cannot be accurately represented by a classical Galerkin de-composition.Theover-completeframeexpansionpresentedinthis workaimsatsuppressingGibbsoscillationsduetoa misrepresen-tationoftheacousticfieldsattheboundariesofthesubdomains, anddoesnothavetheabilitytohandleGibbsfringesproducedby a discontinuityin the interiorof the subdomains.Nonetheless, a simpleworkaroundto handlethislattercaseistoreplacethe

in-Fig. 4. Pressure mode shape (a) and velocity mode shape (b) of the first mode ( n = 1 ) for N = 4 vectors in the frame. Numerical solutions (gray lines with × ) are compared to the analytical solutions of Eqs. (19) and (20) (thick dark lines). The local absolute errors for the mode shapes with N = 4 are also plotted (dashed line), with values indicated on the right axis.

Fig. 5. Comparison between the OB and the FR results in terms of frequency rel- ative error ( E OB/FR

fn ), for global modes 1, 9 and 11, in function of the number N of vectors used in the modal expansions. Dark lines are results obtained with rigid- wall eigenmodes expansions ( Eq. (15) ), while gray lines are results obtained with frames modal expansions ( Eq. (16) ).

finitelythinflamebyaslightlythickerregionofheatrelease:this yieldsasolutionforthevelocity fieldthathasasharpyet contin-uous spatialvariation inthe vicinity ofthe flame.Note alsothat infinitelythinflameareinterestingconceptuallyandfroman ana-lyticalperspective; however,physicalflames arealways finiteand sometimesthick enoughtoproducenon-compact effectson ther-moacousticinstabilities[48].

5. Applicationtothermoacousticinstabilitiesinanannular combustor

Inordertodemonstrateitsabilitytopredictthermoacoustic in-stabilities in complex geometries, the state-space LOM based on generalizedframemodalexpansions(FR)isnowappliedtoamore advanced academic configuration comprising active flames in an annular chamber. Results obtained with the classical orthogonal rigid-wall basis (OB) are also provided and compared to those computed witha 3D Finite Element (FE) solver called AVSP [5]. ThegeometrystudiedisdisplayedinFig.6(a),andthe

correspond-inglow-orderacousticnetworkisrepresentedinFig.6(b).It com-prisesan annularplenum(denotedwith thesubscript P), an an-nular chamber (subscript C), and four identical ducted burners (subscript B) of length LB where the active flames are located.

Rigid-wall boundaryconditionsare assumedattheplenum back-planeandthechamberoutletplane.

The AVSP unstructured mesh consists of 3× 106 _tetrahedral cells, while the acoustic network contains 14 subdomains (1 plenum



P, 1 chamber



C, 4 burners



Bi, and 8 cross-section

changes



sc_i), with the addition of 4 active flames. The flames Hi are located at the coordinate

α

LB within each burner, and

are considered asplanarvolume source ofthickness

δ

.Note that the cross-section area SB of the ducted burners is much smaller

thanthearea oftheplenumexitplane andofthechamber back-plane.Thisallowsforthesimplificationoftheplenum-burnerand burner-chamber junctions,by only considering discrete point-like connections.Inother words,thechamber (ortheplenum)is con-nectedto theburner



B_i throughthe subdomain



sci atasingle

point, implying that MS=4 forthe chamber and the plenum(a

singlediscreteconnectionsurface



S0j=SBisusedforeachoneof

the4burners).Thesubdomain



sci issimilar tothecross-section

change described in the previous section, as it essentially en-forcescontinuityofpressureandacousticfluxbetweentheburner end andthe backplane ofthe chamber (or the exit plane of the plenum). Thissimplification also allows usto consider rigid-wall boundaryconditionsatthechamberbackplaneandattheplenum exit plane when defining the eigenmodes for these subdomains (sincethevelocityshouldactuallybenon-zeroonlyat4point-like locationsofinfinitelysmallspatialextent).TheGibbsphenomenon evidenced intheprevious section isthen not expectedtoappear in the chamber and in the plenum, and it is therefore valid to employ orthogonal rigid-wallbases inthesetwo subdomains.On thecontrary,boundaryconditionsatbothendsoftheburnersare expected to differ from rigid-wall oropen atmosphere, and it is thereforenecessarytoemployover-complete frameexpansionsin thesesubdomainsinordertomitigatetheGibbsphenomenonthat mayappear.Thus,theplenumismodeledasa2Dannular subdo-mainofcoordinates(xP,

θ

P),whoseorthogonalbasisis:

ψ

(P) n,m

(

xP,

θ

P

)

=



cos



n

π

xP LP



cos

(

m

θ

P

)

,cos



_n

_π

_xP LP



sin

(

m

θ

P

)



(22)

Fig. 6. (a) The unstructured mesh used in the AVSP FE solver. (b) The low-order thermoacoustic network representing this system, which consists of an annular plenum P ,

4 burners Bi , an annular chamber C , and a set of 8 subdomains sci containing the cross-section changes. Thick dark lines represent rigid-wall boundary conditions. Gray area ( H i ) are the active flames, and the crosses represent the reference points used in the definition of the flame response. The width of the plenum (resp. chamber) in the

radial direction is W P (resp. W C ). All required numerical values for acoustic and flame parameters are indicated in the table.

Pressure in the chamber could be expanded onto an analogous analytical modalbasis. However,this oneis deliberatelyassumed analytically unknown, and a modal basis computed thanks to a preliminaryAVSPsimulationoftheisolatedchamber(withoutthe burnersandtheplenum) isusedinstead.Notethat itis not nec-essary to perform an AVSP simulation foreach LOM simulation: it is indeedpreferable to generate the chamber modalbasis and all related quantities (including thematrix

−1) in a single pre-liminaryAVSPsimulation,andtoassemblethestate-space realiza-tion ofthissubdomain, whichcan then beemployed in asmany LOMsimulations asdesired.In thepresentexample,theuseofa numericallycomputedmodalbasisdemonstratestheabilityofthe presentframeworktocombineinanacousticnetworksubdomains ofdifferenttypesandthustohandleefficientlyarbitrarilycomplex systems.

In the following, the size of themodal bases in the chamber andintheplenumisfixedtoN=12modes. Forcaseswherethe rigid-wallbasisisusedintheductedburners,thisoneisthesame asintheprevious section (Eq.(15)). Ifanover-complete frameis used,itisgivenby:



φ

(B) n

(

xB

)



nN=



cos



_n

_π

_x B LB



nN/2 



sin



_n

_π

_x B LB



nN/2 (23)

Note the difference with the over-complete frame of Eq.(16), as the present one allows pressure andvelocity to evolve indepen-dentlyfromoneanotheratbothendsoftheduct.

Active flames are located within each burner, and the flame shape Hi

(

xB

)

is the rectangular function ofthickness

δ

centered

aroundxB=

α

LB.Theflameresponseismodeledthanks toa

clas-sicalFlameTransferFunction(FTF),relatingthefluctuationsofheat releasetothefluctuationsofacousticvelocityatareferencepoint locatedatx(_Bre f)=

β

LB.Theflamereferencepointislocatedinthe

burners,neartheplenumexit-plane(

β

₌0_.05).Asimple constant-delayFTFisassumed,suchthatinthefrequencydomainthe fluc-tuatingheatreleaseratereads:

ˆ Q

(

ω

)

=Qe− jωτ



ˆ u

(

xB=

β

LB,

ω

)

u

(24)

where Q is the flame power,

τ

is the flame delay,and u is the meanflowspeedthroughtheinjector.Astate-spacerealizationof the time-delay e− jωτ is generatedthanks to a Multi-Pole expan-sion: e− jωτ ≈ MPBF k=1 −2akj

ω

ω

2_+2c kj

ω

−

ω

20k (25)

whereeachterminthesumiscalledaPole BaseFunction(PBF). Thecoefficientsak,ck,w0karedeterminedthankstoarecursive

fit-tingalgorithmrecentlyproposedby Douasbinetal.[49].By mak-inguseoftheinverseFouriertransform,itisthenstraightforward to convert this frequency domain transfer function into a time-domain state-space realization of size 2MPBF× 2MPBF, whose

ex-pressionis providedinAppendix D. Thisprocedureto generatea state-spacerealizationofaFTFwasalreadyusedbyGhirardoetal.

[50].Fortheflame-delay consideredhere, 12PBFs were observed to be sufficientto accurately fit the terme− jωτ, yielding 24DoF foreachflameinthestate-spacerepresentationofthewhole sys-tem. Finally,the18state-space representationsofthe14 acoustic subdomainsand4activeflamesareconnectedtogether.

Asmentionedearlier,theGibbsphenomenonisexpectedto oc-curat both ends ofthe ducted burners.A particularattention is thereforepaidtothetypeofmodalexpansioncarriedoutinthese subdomains.Atotal of4LOMsimulationsare performed,and re-sultsarecomparedtotheonescomputedwiththeFEsolverAVSP, which are used as reference. Results for 10 of the first modes ofthe combustor aresummarized in Table1. First ofall, FR and OB expansions with N=10 modes are both compared to AVSP. Then these computations are repeated with a number of modes increased to N=30. Mode 1 is the combustor Helmholtz mode, andits frequencyandgrowthratewere observedtobe very sen-sitivetotheadditionofacorrectionlengthtotheductedburners. Asthedeterminationoftheoptimalcorrectionlengthisoutofthe scopeofthispaper,differencesregardingtheHelmholtzmodeare not further discussed. The FR expansion with N=10 appears to successfullyresolvethefrequenciesandgrowthratesofthemodes considered,withrelativeerrorscompared toAVSPbelow10%. On thecontrary,theOBexpansionontoN=10modeslargely failsat resolvingthegrowthratesofallbutoneoftheconsideredmodes, withrelativeerrorsupto348%(Mode5).Mode3isthefirst unsta-bleazimuthal modeofthecombustor,andisthereforeof particu-larinterest:FRexpansionyieldsanerrorofonly6%forthegrowth rateofthismode,whereasOBexpansionproducesanerrorof45%. When the size of the expansion basis/frame is increased, results are globally improved for both FR andOB cases. Yet, even with

N=30theOBapproachstillfailsatachievinganacceptable accu-racyforModes3,5,and7.Theerroronthegrowthrateofthefirst unstableazimuthalmodeisof0.3%intheFRcase,whileitisstill overestimatedby18% inthe OBcase. Modes9and10 aremixed modes, similar to those describedby Evesque and Polifke[37]. In mode 9the plenumfirstlongitudinal mode prevails,whilemode 10isthemixed1st_{-azimuthal–1}st_{-longitudinalplenummode,} cou-pled with the 5th _{azimuthal chamber mode. Previous comments} alsoapplies tothese mixedmodes: the over-complete frame

ex-pansionofsize N=10 yieldsan excellent agreementwiththeFE solver, and outperforms even larger orthogonal basis expansions. NotethatintheFRcase,theconditionnumberoftheframeGram matrix used in the 4 ducted burners isof order 108 _{for N=10,} andincreasesto1018_{forN=30.Itwasverifiedthatincreasingthe} framesizedoesnotfurtherdeterioratesitsconditioning:itinstead saturatesat1018_{evenforlargeN.}

The relatively pooraccuracy ofthe OBexpansion LOM is ex-plainedbyacloserexaminationofthemodesshapesinthe burn-ers. Figure 7 shows the shape of the first unstable azimuthal modeofthecombustor(Mode3),plottedoveralinestartingfrom the bottomof theplenum, passing througha burner,andending atthe chamber outletplane. WithN=30, both FR andOB pres-suremodeshapes(Fig.7(a)and(c))arerelativelyclosetotheAVSP computation,exceptfor3Deffectsintheneighborhoodofthe sub-domains connections that cannot be captured. On the contrary,

Fig. 7(d)showsthat theOB expansionproduces significant Gibbs oscillations of the velocity atboth ends of the burner.Note that the Gibbs phenomenon isonly present at theends of the burn-ers,anddoesnotaffectthevelocityfieldwithin thechamber and theplenum. Conversely,Fig.7(b)showsthatthe frameexpansion successfullymitigatesthisGibbsphenomenon.

The conjugation of these spurious oscillations with the pres-enceof activeflames responding tovelocity fluctuationsexplains the large discrepancies observedin thegrowth rates.Indeed,the closeupviewdisplayedinFig.7(d)revealsthattheflamereference point(representedbyastar)liesinaregionwherethevelocityis strongly affectedby Gibbs oscillations. Incontrast,the FR expan-sion(Fig.7(b)) yieldsareferencevelocityclosetotheAVSP refer-encevelocity.Asheatreleasefluctuationsaredirectlyproportional to the reference velocity (Eq. (24)), anymisprediction of the ve-locityintheburnerresultsinapotentiallyerroneousgrowthrate. Thus,should thepointofreferenceliesina regionwhere numer-ical oscillationsare present, thecomputedthermoacousticmodes maystronglydependon unphysicalanduncontrolleddetails such astherelativepositionofthepointofreferenceandtheGibbs os-cillations.Consequently,the orthogonalrigid-wallbasis expansion resultsinaLOMthatishighlysensitivetothelocationoftheflame referencepoint,whichisahighlyundesirablefeatureofa numeri-calmodel.

This example demonstrates the modularity of the proposed LOM,whichcancombineinasamethermoacousticnetworkactive flames,one-dimensionalsubdomains(theburners),2Dsubdomains (theplenum),andcomplex3Dsubdomainsofarbitraryshape(the chamber)forwhichthemodalbasisisnumericallycomputed. Ob-viously,theapproachisnotlimitedtoazimuthaleigenmodes, but is also able to capture anyother form of thermoacoustic eigen-modes.Itisalsoworth comparingthecostassociatedtothe over-completeframe expansionLOMto existingLow-Order Models.As shownabove, N=10 modesweresufficient toachieve a satisfac-tory resolution (with error below 10%) of the first 20 modesof thecombustors (notallshowninTable1). Thestate-spaceofthe whole systemcomprises248DoF:2× 24forthe plenumandthe chamber, 4× 20 forthe straight ducts,8× 3 forthe cross-section changes, and 4× 24 for the active flames. After the preliminary computationofthechambermodalbasiswithAVSP(160CPU sec-onds for12 modes),the LOMcomputationof allthe eigenmodes wasperformedina few CPU secondsonly. Thisiscomparableto the 300DoF necessaryto treata similarannularconfigurationin the workofSchuermans andco-workers[33,35].However, unlike this lattermethod, the presentexample did not assume acousti-cally compact injectorsrepresented aslumped elements, andthe acousticfieldisfullyresolvedwithintheburners.LOMsrelyingon direct discretization ofthe flow domain, although more straight-forwardtoputintoapplication,appearstoresultinmoreDoFand highercost: Emmertetal.[41]required105 _{DoF and38CPUsto}

compute5eigenmodesofa12-injectorsannulargeometrywithout anyactiveflames.Finally,incontrasttomixed-methodLOMsbased onmodalexpansionsalongsideRiemanninvariantsA+/A−_[14],the proposedLow-OrderModel(Eq.(14))canbedirectlyintegratedin timefortemporalsimulationsofthermoacousticacoustic instabil-ities,whereas thetime-domain translation ofRiemann invariants appearstobesomehowconstraining[51].

6. Conclusions

ThisworkaddressedaknownissueinLOMsfor thermoacous-tics, namely the misrepresentation of the acoustic field arising frommodalexpansionsontorigid-walleigenmodesbases,already reported by numerous earlier studies [24]. Under the assump-tion of zero Mach number flow, a reformulation of the classical modal expansion making use of over-complete frames of eigen-modes was proposed for the first time. Two major observations weredrawnfromtheanalysisofasimpleone-dimensional acous-ticproblem:(1)themodalexpansionsingularitywasclearly iden-tified as a Gibbs-like phenomenon affecting the acoustic veloc-ity, resulting in slowconvergence speeds and erroneousvelocity valuesnearjunctionsbetweensubdomains;(2)theover-complete frame expansion was shown to successfully suppress the Gibbs-likephenomenonandtoyieldsignificantly improvedconvergence speeds. Theonly pitfall stemming fromtheframe expansion, lies in its over-completeness that may entail ill-conditionedfeatures, whichmayproducespurious, non-physicaldynamics ofthe pres-sureevolution.Thus,aspecificinversionprocedureisusedto com-pute

−1_, the inverse of the frame Gram matrix. This approach ensures that these spurious components stay negligible in com-parisonto physicallymeaningful components.Morerigorously, in

Appendix C it is demonstrated that spurious eigenmodes arising fromtheframeover-completenesshavelowenergy,andacriterion isderivedtosystematicallyidentifythem.

In a second example, the generalized modal expansion LOM was used to predict thermoacoustic instabilities in an annular combustor.Itwasshowntoyield resultsclosetoa finiteelement solver, whereas the rigid-wall modal expansions failed at accu-ratelypredictingthelineargrowthrates.Thissecondexamplealso demonstratesthemodularityoftheproposedframework,through itsabilitytocombineinasameacousticnetworkhighly heteroge-neousclassesofelementssuch asactive flames,one-dimensional ducts, or complex 3D cavities. From a practical point of view, the implementation of the proposed method only requires min-imal changes to existing algorithms based on rigid-wall expan-sions. Namely, apart from the inversion of the Gram matrix and the identification of spurious eigenmodes, the input/output rela-tions ofeach subdomain composing the acoustic networkshould be adapted to include both pressure and velocity at the bound-aries.Thus,thisnovelmethodisexpectedtobepotentiallyuseful inthefieldofthermoacoutics.

Acknowledgments

The first author would like to thank the French Ministry of HigherEducation,ResearchandInnovationandtheEcoleNormale SuperieuredeParis-SaclayforfundingthisworkthroughtheCDSN scheme.

AppendixA. State-spacerealizationofanacousticsubdomain Thestate-spacerepresentationforasubdomain



ibelongingto

anacousticnetworkisgiveninthissection.Thedynamicalsystem ofEq.(12)governingtheevolutionoftheacousticpressureinthe