Simon M. Heinzmann
1
Ins i u e o Ene gy Sys ems and Fluid-Enginee ing,
ZHAW Zu ich Uni e si y o Applied Sciences,
Win e hu 8401, Swi ze land;
Depa men o Mechanical and P ocess
Enginee ing,
ETH Z€
u ich,
Z€
u ich 8092, Swi ze land
e-mail: [email p o ec ed]
Ha ish S. Gopalak ishnan
Ins i u e o Ene gy Sys ems and Fluid-Enginee ing,
ZHAW Zu ich Uni e si y o Applied Sciences,
Win e hu 8401, Swi ze land
Bi u e Wood
Ansaldo Ene gia Swi ze land,
Baden 5400, Swi ze land
Mi ko R. Bo hien
Ins i u e o Ene gy Sys ems and Fluid-Enginee ing,
ZHAW Zu ich Uni e si y o Applied Sciences,
Win e hu 8401, Swi ze land
Linea S abili y Analysis
o T ans e sal The moacous ic
Modes in Rehea Combus o s
Gas u bines ea u ing sequen ial combus ion a chi ec u es a e ideally sui ed o deli e
CO2- ee, uel- lexible, and on-demand powe o he g id. A sequen ial combus o is
comp ised o wo axially s aged lames. Typically, he i s s age is a swi l-s abilized
p opaga ing lame. The ho p oduc s o he i s s age a e dilu ed wi h ai be o e addi ional
uel is injec ed in he second s age. The esul ing i ia ed mix u e leads o au o-igni ion in he
second combus o . The sequen ial a chi ec u e can be le e aged o a y he uel spli
be ween he combus ion chambe s o accoun o di e en eac i i ies o al e na i e uels.
The moacous ically s able combus o s a e c i ical o ensu e low emissions, high eliabili y,
and mechanical in eg i y. Unde speci ic o -design condi ions, he au o-igni ion-s abilized
second s age can be subjec o ans e sal ins abili ies. This a icle p esen s a ini e-elemen -
coupled me hod o model he he moacous ic beha io o he Ansaldo Ene gia H-class GT36
ehea combus ion s age in a e y cos -e ec i e manne . To do so, he au o-igni ion lame is
di ided in o mul iple pa s o which he lame ans e unc ion (FTF) me hodology
commonly used o plana wa es is applied. The s abili y o ans e se eigenmodes is
assessed and alida ed agains expe imen s pe o med unde engine condi ions. We show
ha he amewo k can co ec ly p edic he s abili y o high- equency combus o modes.
Consequen ly, i can be used o ge eliable s abili y es ima es o high- equency
he moacous ic modes in indus ial ehea combus ion chambe s and o ensu e ha such
ins abili ies a e a oided. [DOI: 10.1115/1.4069453]
Keywo ds: he moacous ics, ans e se modes, high- equency, ehea combus ion, s abil-
i y analysis
1 In oduc ion
To coun e ac global wa ming, a apid educ ion o g een-house
gas emissions is needed. The elec ici y sec o is o eseen o lead he
way in e ms o ne ze o emissions by 2050 [1]. In o de o achie e
his, he u u e ene gy p oduc ion will s ongly ely on enewable
ene gy sys ems such as sola -, wind-, o hyd opowe . Howe e ,
hese sus ainable ene gy solu ions a e subjec o luc ua ing
elec ici y p oduc ion on a b oad ime-scale [1]. To balance he
in e mi en na u e o ene gy p oduc ion and consump ion, excess
powe p oduced om enewable ene gy sys ems can be chemically
s o ed in enewable uels such as hyd ogen, ammonia, o syn he i-
cally p oduced con en ional uels. Subsequen ly, hese uels can
be i ed in gas u bines o p oduce en i onmen ally iendly on-
demand powe , making hem an ideal asse in he u u e ene gy
landscape [2].
To i e a a ie y o enewable uels in gas u bines, he cons an
p essu e sequen ial combus ion a chi ec u e, as implemen ed in he
Ansaldo Ene gia GT36, is a p omising solu ion [2–4]. A sequen ial
combus o consis s o wo combus ion chambe s a anged in se ies
(Fig. 1). The mul ibu ne i s s age consis s o mul iple p opaga ion-
s abilized bu ne s [4]. Downs eam o he i s s age, dilu ion ai is
admixed in he dilu ion ai mixe . Fu he downs eam, uel is
admixed in he sequen ial s age cen e body bu ne (CBB), and due
o he high empe a u e o he esul ing mix u e, an au o-igni ion-
s abilized lame, also called ehea lame, esul s [2,3,5]. To enable
he combus ion o uels wi h di e en eac i i ies, he uel and ai
spli can be a ied be ween he wo combus ion s ages. This allows
o une he combus ion p ocess in each s age sepa a ely o accoun
o he di e en eac i i ies o gas u bine uels, he eby expanding
he ope a ional lexibili y o he engine [5–7].
The moacous ically s able combus ion s ages a e impo an o
ensu e low emissions, high eliabili y, and mechanical in eg i y.
Mul iple me hods exis o pe o m he moacous ic analysis o
combus ion chambe s. High- ideli y me hods like di ec nume ical
simula ions and la ge eddy simula ions p o ide esul s wi h high
accu acy. Howe e , simula ing a ull-scale indus y combus o o
all ele an ope a ing condi ions emains compu a ionally a oo
expensi e. Compu a ionally mo e a o dable modeling echniques
on he o he hand ha e shown o also ep oduce he moacous ic
beha io s accu a ely. These me hods a e ypically based on ini e-
elemen (FE) modeling echniques o ne wo k-modeling app oaches
[8–11]. Nicoud e al. [8] showed ha he ini e-elemen -me hod
(FEM) can be used o classi y he s abili y o he moacous ic
longi udinal eigenmodes in wo-dimensional duc s using a
Tu bo Expo: Tu bomachine y Technical Con e ence & Exposi ion (GT2025), June
16–20, 2025. GT2025.
1
Co esponding au ho .
Manusc ip ecei ed July 4, 2025; inal manusc ip ecei ed July 21, 2025; published
online Sep embe 24, 2025. Edi o : Je zy T. Sawicki.
Jou nal o Enginee ing o Gas Tu bines and Powe DECEMBER 2025, Vol. 147 / 121019-1
Copy igh V
C 2025 by ASME; euse license CC-BY 4.0
Downloaded om h p://asmedigi alcollec ion.asme.o g/gas u binespowe /a icle-pd /147/12/121019/7530914/g p-25-1285.pd by gues on 06 Oc obe 2025
Helmhol z sol e . Lae a e al. [9] accu a ely calcula ed he g ow h
a es o spinning and s anding azimu hal modes o he labo a o y-
scale MICCA combus ion chambe using a 3D Helmhol z sol e .
Bo hien e al. [10] and Schue mans e al. [11] demons a ed ha
ne wo k models can p edic he s abili y o eal engine combus o s
o which he compac lame assump ion is sa is ied. Hummel e al.
[12] coupled a FEM simula ion o a ne wo k model o p edic he
s abili y o ans e se modes in a combus o wi h a p opaga ion-
s abilized lame. To he bes o ou knowledge, p edic ion o g ow h
a es o ans e se modes in au o-igni ion-s abilized combus o s has
no been epo ed in he li e a u e un il his poin .
The p incipal goal o his s udy is o de elop a amewo k ha is
capable o p edic ing he linea s abili y o ans e se he moacous-
ic modes in a ealis ic gas u bine combus o . To achie e his
objec i e, we employ a FEM-based app oach o model he
combus o acous ics along wi h a ecen ly ex ended Lag angian
pa icle- acking-based amewo k [13,14] o model he au o-
igni ion-s abilized lame dynamics. The amewo k is alida ed
agains expe imen al da a o he Ansaldo Ene gia GT36 ehea
combus o a wo ope a ing poin s (OP).
2 Valida ion Da a F om GT36 Combus o
In his a icle, he GT36 sequen ial combus ion chambe is
conside ed (see Fig. 1), and he de eloped s abili y p edic ion
amewo k is aimed o ep oduce he s abili y beha io o his
combus o . The sequen ial line connec ing he CBB wi h he
u bine inle is simpli ied in o de o be o a ionally symme ic. I s
adius is adap ed acco dingly o ep oduce he same axial change in
c oss-sec ional a ea as he eal engine geome y (Fig. 1). I is e i ied
ha he in luence on he in es iga ed acous ic modes is negligible in
e ms o modeshape and equency. The he moacous ic simula ions
a e pe o med o wo dis inc ope a ing poin s a engine condi ion.
OP1 co esponds o a he moacous ically uns able combus o
oscilla ing in e mi en ly. A OP2, he combus o is s able. The
di e ence in empe a u e be ween he wo OPs was chosen o be
la ge han 100 K, which also indica es a la ge s able ope a ing
window. Figu e 2shows he p obabili y densi y unc ion (PDF) o
he p essu e PðgÞin conjunc ion wi h he ime signal as well as he
equency spec um o OP1 (Fig. 2(a)) and OP2 (Fig. 2(b)). The
in e mi en na u e o he pulsa ions is clea ly isible in he ime
signal o he p essu e o OP1. The equency spec um shows ha
he main pulsa ions occu a a no malized equency o 1 (all
equency alues in his pape a e no malized using he alue o he
main equency peak o OP1).
3 F amewo k De elopmen
The amewo k p esen ed in his wo k was ini ially de eloped in
wo dimensions (2D) and alida ed agains measu emen s o
he moacous ic oscilla ions obse ed in a labo a o y-scale ehea
combus o [14]. I was shown o be able o p edic he g ow h a e o
he dominan uns able mode. Compa ison o chemiluminescence
imaging showed a quali a i ely accu a e ep esen a ion o he spa ial
hea elease a e (HRR) pe u ba ions [14]. He e, we expand he
amewo k o h ee dimensions (3D) o model ans e se eigenm-
odes o he GT36 CBB. The amewo k componen s a e: a eac i e
Reynolds-a e aged Na ie –S okes (RANS) simula ion o ob ain he
ime-a e aged base s a e o s abili y analysis, a 1D Lag angian
pa icle acking amewo k o compu e he lame esponse [13,14],
and a 3D lame segmen a ion me hod o accoun o he acous ic
noncompac ness o he lame in he ans e sal di ec ion. Fu he , an
analy ical HRR pe u ba ion o mula ion in 3D and a 3D Helmhol z
sol e o compu e he eigen alues a e used.
3.1 Nume ical Se up and Resul s o he Th ee-Dimensional
Reynolds-A e aged Na ie –S okes Compu a ional Fluid
Dynamics Simula ions. In his sec ion, he se up o he compu a-
ional luid dynamics (CFD) simula ion is discussed, and insigh s
in o he esul s he eo a e gi en. The mean empe a u e and HRR
ields a e compu ed wi h eac i e RANS simula ions using he
comme cial so wa e FLUENT. The combus ion is modeled using an
in-house ehea combus ion model based on a p og ess a iable
app oach and a abula ed chemical sou ce e m. The p esumed PDF
me hod is used o he u bulence–chemis y in e ac ion. The model
is implemen ed ia he FLUENT use -de ined unc ion.
The chemical sou ce e m is abula ed as a unc ion o uel and ai
mix u e ac ions and no malized p og ess a iable dimensions. The
slow chemis y is ep esen ed ia a composi e p og ess a iable,
which is de ined as he sum o in e media e and p oduc species.
Fuel, cooling ai mix u e ac ions and unno malized p og ess
a iable anspo equa ions a e sol ed as s anda d FLUENT use -
de ined scala equa ions. The i s momen (mean mix u e ac ions)
o Z ( uel), ZCA (cooling ai ), and Yc(unno malized p og ess
a iable) a e sol ed using Eq. (1) [5], whe e uiis he eloci y
componen , xiis he spa ial coo dina e, and Ce is he e ec i e
di usion coe icien
Fig. 1 Ansaldo Ene gia GT36 cons an p essu e sequen ial combus ion chambe schema ic om
Re . [4]
121019-2 / Vol. 147, DECEMBER 2025 T ansac ions o he ASME
Downloaded om h p://asmedigi alcollec ion.asme.o g/gas u binespowe /a icle-pd /147/12/121019/7530914/g p-25-1285.pd by gues on 06 Oc obe 2025
@�
q~
Z
@ þ@�
q~ui~
Z
@xi
−
@
@xi
Ce
@~
Z
@xi
!¼0
@�
q~
ZCA
@ þ@�
q~ui~
ZCA
@xi
−
@
@xi
Ce
@~
ZCA
@xi
!¼0
@�
q~
YC
@ þ@�
q~ui~
YC
@xi
−
@
@xi
Ce
@~
YC
@xi
!¼�qe_
xc
(1)
The second momen s Z00 (mix u e ac ion a iances) a e sol ed
using he ollowing equa ion [5]:
@�q~
Z00
@ þ@�q~ui~
Z00
@xi
−
@
@xi
Ce
@~
Z00
@xi
!¼2:86l u b
@~
Z
@xi
!2
−
2�q~
Z00
s u b
@�
q~
Z00
CA
@ þ@�
q~ui~
Z00
CA
@xi
−
@
@xi
Ce
@~
Z00
CA
@xi
!¼2:86l u b
@~
ZCA
@xi
!2
−
2�
q~
Z00
CA
s u b
@�q~
Y00
c
@ þ@�q~ui~
Y00
c
@xi
−
@
@xi
Ce
@~
Y00
c
@xi
!¼2:86l u b
@~
Yc
@xi
!2
−
2�q~
Y00
c
s u b
(2)
The u bulence ime scale s u b is modeled as he scaled speci ic a e
o dissipa ion ð0:09xGEKOÞ. Equa ions o Z and ZCA a e conse ed
scala equa ions wi hou a chemical sou ce e m. The anspo
equa ion o Yccon ains a chemical sou ce e m, which is closed
wi h a 2D p esumed be a PDF o he uel and cooling ai mix u e
ac ions and 1D p esumed be a PDF o he no malized p og ess
a iable (C). The no malized p og ess a iable is compu ed as he
a io o he local p og ess a iable Ycand i s co esponding
equilib ium alue Yðc,eqÞ. The chemical sou ce e m is gi en by he
ollowing equa ion [5]:
e_
xC¼ððð1
0
_
xZ ,ZCA,C
ð ÞPDF Z ,ZCA
ð ÞPDF C
ð ÞdZ dZCA dC (3)
In he 3D nume ical analysis, one- wel h o he bu ne sec o
(spanning one ull single injec ion inge ) is conside ed (Fig. 3(a)).
The combus o sec o ansi ion sec ion olume is ma ched o he
ac ual alue o he engine geome y. The domain is meshed wi h a
e ahed al mesh wi h p ism bounda y laye s a he walls. The mesh
size is app oxima ely 20 �10
6
cells wi h e inemen s applied in he
bu ne mixing sec ion and expec ed lame loca ion. The mesh
esolu ion is su icien o esol e he lame b ush wi h mul iple cells,
see Fig. 3(b). The nea -wall egions a e meshed y
þ
<15.0. The
s eady 3D RANS simula ions a e pe o med using FLUENT’s GEKO
model (gene alized k–xGEKO model) wi h uned CMIX and CSEP
pa ame e s (“mixing” and “sepa a ion” pa ame e s in GEKO
model). Fuel is modeled as na u al gas wi h 7% C
2
þ(93.11%
CH
4
, 6.51% C
2
H
6
, and 0.38% ine s by olume). The ho gas inle is
assumed o be a equilib ium composi ion o uel–ai mix u e a
a ge ho gas inle empe a u e. A ci cum e en ially a e aged
empe a u e p o ile is applied a he combus o inle . The
empe a u e p o ile was ob ained om i s s age combus o and
dilu ion ai mixe simula ions. No adia ion hea ans e is included.
A coupled scheme is applied o p essu e– eloci y coupling wi h he
PRESTO! (PREssu e STagge ing Op ions) scheme o p essu e. All
equa ions a e disc e ized using a second o de upwind scheme
(mass, momen um, ene gy, species, and use scala s).
The empe a u e ield on he cen e plane shows ha OP1 (Fig.
4(a)) has a highe eac an s’ empe a u e and a lowe lame
empe a u e compa ed o OP2 (Fig. 4(b)). These condi ions esul
in a highe empe a u e jump ac oss he lame o OP2. The uel
mix u e ac ion dis ibu ions (no shown) a he bu ne mixing
sec ion exi (dump plane) a e e y simila o bo h OPs. In he OP2
case, mo e uel is injec ed o each he highe lame empe a u e. The
no malized HRR on he cen e plane is gi en in Fig. 4(e) o OP1 and
in Fig. 4( ) o OP2. The HRR is no malized by he maximum alue
on he cen e plane. OP1 shows a lowe and mo e dis ibu ed HRR
and a longe lame. Fu he , he adial empe a u e ield a h ee
di e en axial loca ions is shown in Fig. 4(c) o OP1 and Fig. 4(d)
o OP2. The lowe inle empe a u e and highe exi empe a u e o
OP2 a e clea ly isible. Fo comple eness, he HRR cloud is gi en in
Fig. 4(g) o OP1 and Fig. 4(h) o OP2. As is appa en om he
cen e plane HRR igu es, he lame shapes a e di e en .
3.2 Combus o Eigenmodes. The equencies o he moa-
cous ic ins abili ies a e usually close o he na u al acous ic
eigen equencies o he combus o [15]. The acous ic eigenmodes
can be compu ed by sol ing he homogeneous Helmhol z equa ion
(Eq. (4)) in a 3D FEM sol e while conside ing a ime-a e aged
empe a u e dis ibu ion o he lame wi hin he combus o
� 1
q0 ^p
� �þx2
cp0
^p¼0(4)
The Helmhol z equa ion is ob ained by Fou ie ans o ming he
homogeneous wa e equa ion in o he equency domain (eix
Fig. 2 The p obabili y densi y unc ion o he p essu e mea-
su emen PðgÞ, he ime signal, and he co esponding equency
spec um o (a) OP1 and (b) OP2
Fig. 3 (a) Compu a ional domain (po ion) o he CFD simula ion
and (b) longi udinal plane cu o he mesh showing i s esolu ion
oge he wi h he empe a u e con ou plo
Jou nal o Enginee ing o Gas Tu bines and Powe DECEMBER 2025, Vol. 147 / 121019-3
Downloaded om h p://asmedigi alcollec ion.asme.o g/gas u binespowe /a icle-pd /147/12/121019/7530914/g p-25-1285.pd by gues on 06 Oc obe 2025
con en ion). The wa e equa ion i sel can be de i ed using he
linea ized mass- and momen um balance oge he wi h he equa ion
o s a e. Fo he de i a ion, i is assumed ha he low eloci y is
negligible, i.e., low compa ed o he speed o sound. Fu he , any
iscous losses as well as hea ans e a e neglec ed. Ideal gas has
been assumed, and gas p ope ies (hea capaci ies cpand cV) a e
assumed cons an . Small pe u ba ions a e pos ula ed (linea i y).
Using he Helmhol z equa ion, he eigenmodes can be compu ed
which a e also e e ed o as he passi e lame eigenmodes because
he lame is only accoun ed o by inco po a ing he empe a u e
ield in he calcula ion and no as a HRR sou ce e m. The
empe a u e dis ibu ion wi hin he combus o is ob ained om he
CFD mean ields by a ea-weigh ed a e aging ac oss he combus o
diame e , which esul s in a 1D empe a u e ield along he x-axis o
each OP.
He e, we in es iga e i e speci ic ans e se eigenmodes which
co espond o he i s , second, and hi d azimu hal eigenmodes (1T,
2T, and 3T), as well as he i s adial (1R) and i s adial plus i s
longi udinal (1R1L) modes. They a e shown in Fig. 5 oge he wi h
he eigenmode name abb e ia ions and he no malized equencies.
The acous ic bounda y condi ions co espond o a p essu e elease a
he inle ðp0¼0Þand ully choked ðu0¼0Þa he ou le . The
in es iga ed combus o eigenmodes (Fig. 5) exhibi a high deg ee o
o a ional symme y which is shown in Fig. 6. The e o e, i he
p essu e along mul iple c oss sec ions (Fig. 6(a)) o a sampling plane
(Fig. 6(c)) o he combus o (pa allel o he symme y axis) is
no malized, he cu es a e close o iden ical (Fig. 6(b)), especially in
he egion close o he lame. Thus, he 3D p essu e ield can be
exp essed by a single no malized 1D modeshape which is locally
scaled wi h a p essu e alue p0ðx e ,y,zÞa a e e ence plane
pe pendicula o he combus o axis. In his case, he e e ence plane
is close o he dump plane. A single no malized p essu e cu e along
a c oss sec ion can be compu ed o all i e eigenmodes. This esul s
in a o al o i e 1D p essu e modeshapes, shown in Fig. 7wi hin he
combus o egion be ween dump plane and ou le . These no malized
eigenmode shapes can subsequen ly be used o compu e he ehea
lame’s HRR esponse o he acous ic p essu e.
3.3 Lag angian F amewo k o Au o-Igni ion Flame
Response Compu a ion. To cha ac e ize he lame esponse o
acous ic pe u ba ions, lame ans e unc ions (FTFs) a e used.
FTFs ela e he lame’s HRR esponse o an acous ic pe u ba ion a
a e e ence loca ion ups eam o he lame. The FTF is a equency
dependen complex alued quan i y whose absolu e alue ep esen s
he gain in ela ion o he acous ic pe u ba ion [16], i.e., how
s ongly he lame eac s o a gi en acous ic ampli ude. The phase o
he FTF desc ibes he co esponding phase di e ence. Usually, he
FTFs a e gi en wi h espec o he acous ic eloci y pe u ba ion (u0)
[17]. While his is applicable o plana wa e pe u bed p opaga ion-
s abilized lames, which a e sensi i e o eloci y pe u ba ions, i
was ound ha ehea lames a e a he insensi i e o eloci y
pe u ba ions [18]. This can be seen by he low FTF gain o eloci y
pe u ba ions shown by Gan e al. [19]. In compa ison, he HRR
gains in esponse o p essu e and empe a u e pe u ba ions a e
signi ican ly highe [19]. The high sensi i i y o au o-igni ion-
s abilized lames o empe a u e and p essu e luc ua ions is due o
he sensi i i y o he igni ion delay ime o small changes in
empe a u e. Thus, due o he di e en sensi i i y o au o-igni ion
lames, i is con enien o compu e he FTFs wi h espec o he
acous ic p essu e p0ins ead o u0.
Flame ans e unc ions can be deduced om expe imen al
measu emen s whe e he lame is acous ically o ced and he HRR
esponse is measu ed [11]. Al e na i ely, CFD can be used [20].
Fu he , analy ical and nume ical models exis o ce ain lame
con igu a ions, o example, he FTF model om Zellhube e al.
[21], Gan e al. [19], and Gopalak ishnan e al. [13,22,23] ha ela e
he HRR esponse o a 1D ehea lame o plana acous ic and
en opic wa es.
To compu e he FTFs o a ans e sely pe u bed noncompac
au o-igni ion lame, he 1D-Lag angian pa icle acking amewo k
om Gopalak ishnan e al. [13,23] is used in his wo k. This is done
by simula ing he igni ion p ocess using a se o 0D homogeneous
cons an p essu e eac o s in CANTERA. To compu e he lame
esponse o a gi en equency, a se o pa icles a e injec ed a he
inle a di e en ime ins ances co esponding o he di e en phases
Fig. 4 Tempe a u e ield on he cen e plane o (a) OP1 and (b) OP2. Radial empe a u e ield a h ee di e en axial
loca ions: (c) OP1 and (d) OP2. HRR on he cen e plane o (e) OP1 and ( ) OP2. HRR cloud o (g) OP1 and (h) OP2.
Fig. 5 Fi e passi e lame eigenmodes o OP1 and hei
co esponding nomencla u es and equencies
121019-4 / Vol. 147, DECEMBER 2025 T ansac ions o he ASME
Downloaded om h p://asmedigi alcollec ion.asme.o g/gas u binespowe /a icle-pd /147/12/121019/7530914/g p-25-1285.pd by gues on 06 Oc obe 2025
o he pe u bing p essu e oscilla ion. As he pa icles a e anspo ed
downs eam, hey a e pe u bed by an acous ic wa e wi h a locally
a ying p essu e ampli ude as de ined by a 1D modeshape (Fig. 7).
The go e ning equa ions o momen um, ene gy, and species mass
balance o each injec ed pa icle in he p esence o acous ic
luc ua ions a e in eg a ed in ime o ob ain he ime e olu ion o he
pa icle empe a u e. F om his da a, he igni ion delay ime as well as
i s modula ion is deduced. The FTFs a e compu ed by ela ing he
p essu e pe u ba ion a he e e ence loca ion (close o he dump
plane) o he in eg a ed HRR pe u ba ions. The lame esponse is
di ided in o wo sepa a e FTFs. FwXðxÞ ep esen s he gain and phase
o he HRR (b_
qa,wX, Eq. (5)) o an eigenmode wi h an eigen equency
Xand he co esponding modeshape w. GwXðxÞdesc ibes he
equency esponse o he axial mo emen (^xig,wX, Eq. (5)) o he
lame. The FTFs a e mul iplied wi h he ela i e p essu e pe u ba ion
a io ^pðx e Þ=p0 o ob ain he hea elease a e and lame mo emen
esponse. _
q0,ais he in eg a ed HRR, and xig,0 is he mean igni ion
leng h. An analy ical s a e space model o he compu ed FTFs is
ob ained ia he VECTORFIT sys em iden i ica ion ool [24]. This
enables he lame esponse compu a ion o complex equency
inpu s and can subsequen ly be in eg a ed in he ini e-elemen sol e
b_
qa,wX
_
q0,a¼FwXx
ð Þ^p x e
ð Þ
p0
^xig,wX
xig,0 ¼GwXx
ð Þ^p x e
ð Þ
p0
(5)
The esponse o he au o-igni ion lame o acous ic pe u ba ions in a
gi en geome ical con igu a ion s ongly depends on he na u e o
he acous ic ield in he conside ed geome y. Fo a duc , he acous ic
ield can be w i en in e ms o a simple ma hema ical exp ession. In
mo e complica ed geome ies as conside ed in his wo k, he
acous ic p essu e ield expe ienced by he pa icle canno be w i en
in e ms o a simple ma hema ical exp ession. Due o his eason, he
FTFs ha e o be compu ed on a mode-by-mode basis, whe e he
acous ic p essu e ield in oduced by each passi e lame eigenmode
is conside ed. This is hen used in he Lag angian amewo k o
compu e he pa icle e olu ion, and subsequen ly he hea elease
esponse o he lame is deduced. This is why he FTF exp essions in
Eq. (5) a e alid o a speci ic eigenmode wX. The impo ance o
modeling he lame esponse o each eigenmode sepa a ely was
in es iga ed in Re . [14] o a 2D geome y. The e i was shown, ha
i a p essu e ac s on he pa icle ea ly on in he igni ion p ocess, i can
hea ily in luence he igni ion delay ime. This consequen ly esul s
in a la ge modula ion o he igni ion delay ime and he e o e in high
FTF gains. In con as , i he p essu e only ac s on he pa icle la e in
he igni ion p ocess, he igni ion delay ime only changes sligh ly,
which leads o smalle FTF gains [14]. A compa ison be ween wo
FTFs is shown in Fig. 8, whe e he in luence o he 1T mode
(diamonds) and a cons an ampli ude sinusoidal pe u ba ion
(ci cles) is compa ed. Di e ences in he lame esponse a e isible
in bo h he gain and he phase and can a y s onge o di e en
eigenmodes o ope a ing condi ions. This unde lines he impo ance
o compu ing he FTFs o each mode sepa a ely unde consid-
e a ion o he acous ic eigenmode shape.
To use he ex ended 1D-Lag angian FTF amewo k, ce ain
pa ame e s a e needed. The posi ion o he lame in he 1D-
amewo k is calcula ed om he 3D CFD mean ield. Fu he mo e,
he eloci y is a e aged be ween he dump plane and he mean lame
posi ion. Du ing he 1D-Lag angian amewo k compu a ion, i is
assumed ha he p essu e pe u ba ion esul ing om he eigenmode
s a s ac ing om he dump-plane onwa ds. This assump ion is
belie ed alid since high- equency modes wi h ans e sal
componen s a e ypically loca ed locally in he combus ion
chambe , and he CFD p og ess a iable o eac ion emains e y
low in he mixing sec ion. Figu e 9shows he p essu e pe u ba ion
(induced by he 1T modeshape) ha he pa icle expe iences du ing
anspo a ion wi h he mean low. Di e ences a e isible be ween
he modeshape dependen ampli ude (solid line) and a cons an
ampli ude pe u ba ion (dashed line), which esul s in he di e en
FTFs o Fig. 8. The modeshape en elope is shown by he dash-
do ed line.
Fu he assump ions a e made in o de o build he amewo k.
Fi s , he HRR pe u ba ion due o acous ic eloci y a e neglec ed,
and solely he in luence o p essu e, empe a u e, and densi y is
accoun ed o (Eq. (6))
p0x,
ð Þ ¼AwXx
ð Þ^
peix þi/A
T0x,
ð Þ ¼c−1
c
T0
p0
p0x,
ð Þ
q0x,
ð Þ ¼1
c2
0
p0x,
ð Þ
(6)
Fig. 6 P essu e o he 1T mode along di e en c oss sec ions (a)
o di e en poin s on he sampling c oss sec ion (c) on he dump
plane. The p essu e cu es o (a) a e no malized in (b).
Fig. 7 Fi e no malized combus o eigenmodes o OP1
Fig. 8 FTF esul s om he Lag angian amewo k o he 1T
mode (diamonds) and a cons an ampli ude sinusoidal pe u ba-
ion (ci cles)
Jou nal o Enginee ing o Gas Tu bines and Powe DECEMBER 2025, Vol. 147 / 121019-5
Downloaded om h p://asmedigi alcollec ion.asme.o g/gas u binespowe /a icle-pd /147/12/121019/7530914/g p-25-1285.pd by gues on 06 Oc obe 2025
The quan i ies p0ðx, Þ, T0ðx, Þ, and q0ðx, Þco espond o he inpu
alues o he Lag angian amewo k [13] and we e adap ed in Re .
[14] o he ans e se case, which is also shown he e. AwXðxÞ
de e mines he loca ion-dependen ampli ude A o a ce ain
modeshape wwi h eigen equency X. The empe a u e and densi y
pe u ba ions esul om his co espondingly [13]. Since eloci y
pe u ba ions o pu ely ans e sal eigenmodes a e pe pendicula o
he low di ec ion, i is assumed hey ha e a negligible in luence on
he amewo k’s accu acy. Second, all modes a e assumed o be
s anding and hus do no o a e o p opaga e. Gi en he bounda y
condi ions o a p essu e elease inle and choked ou le , as well as he
in es iga ed modeshapes, his is alid. Pos p ocessing o he
p essu e measu emen s showed ha he e is no o a ing componen
o he in es iga ed azimu hal modes.
3.4 Au o-Igni ion Flame Response Modeling. In many cases,
he lame can be assumed compac in longi udinal di ec ion because
he wa eleng h o plana acous ic wa es (o longi udinal eigenm-
odes) is much la ge han he lame hickness. Plana wa es will
acous ically pe u b he lame uni o mly, meaning ha he acous ic
pe u ba ions a e uni o m ac oss he lame on . In his case, he
lame is acous ically compac . Howe e , when he lame is
ans e sely pe u bed by he acous ic wa es, he lame expe iences
a nonuni o m p essu e dis ibu ion. This means ha di e en lame
egions expe ience di e en acous ic pe u ba ions, making he
lame noncompac . To deal wi h his noncompac ness, he lame is
nume ically di ided in o smalle sub lames (blue do s in Fig. 10) in
ans e se z- and y-di ec ions. Fo simplici y, Fig. 10 only shows he
segmen a ion in he z-di ec ion, and no he en i e yz-plane. This
nume ical segmen a ion allows he lame o be pe u bed di e en ly
locally, depending on he modeshape. I is assumed ha each
sub lame is pe u bed by a speci ic alue o he magni ude and
phase o inciden p essu e wa e (^
p). Consequen ly, o each
sub lame, a single alue o he HRR pe u ba ion (b_
qa,N) and axial
loca ion shi (^xig,N) wi h espec o he x-axis is compu ed. The
segmen a ion is di ec ly linked o he esolu ion o he nume ical
mesh, which, in his case, esul s in a leas 40 elemen s pe
wa eleng h and mul iple mesh elemen s in he lame b ush.
Each sub lame is ep esen ed by a 1D Gaussian HRR dis ibu ion
acco ding o Eq. (7) and is de ined o each y- and z-axis combina ion
wi h espec o he coo dina e axis o he GT36 combus o (de ined
in Fig. 3(a))
_
q x,y,z
ð Þ ¼_
q0,ay,z
ð Þ
_
qy,z
ð Þ i i i i i i
2p
pexp −
1
2
x−xig,0 y,z
ð Þ
_
qy,z
ð Þ
!2
0
@1
A(7)
The mean lame posi ion xig,0ðy,zÞ, lame wid h _
qðy,zÞ, and
in eg a ed HRR _
q0,aðy,zÞa e calcula ed om he 3D CFD mean ield
o each ope a ing poin and a e shown in Fig. 11. This is done by
compu ing he pa ame e s using a Gaussian i (mul iple s udies used
Gaussian dis ibu ions [14,19,21,23,25]) o he CFD olume ic
HRR o each 1D c oss sec ion wi h cons an y- and z-coo dina es in
3D, as was shown by Heinzmann e al. in he 2D case [14]. Thus, Fig.
11(a)indica es whe e on he yz-plane he s onge and weake HRR
egions a e. Figu e 11(b)de ines he wid h o he HRR in x-di ec ion
o he 3D combus o olume. Figu e 11(c)speci ies he mean lame
loca ion xig,0ðy,zÞ o he 3D combus o olume. Using hese h ee
pa ame e s, he 3D olume ic HRR zone in he combus o olume
can be ebuil analy ically using Gaussian dis ibu ions in x-
di ec ion o e e y poin on he yz-plane.
Ma hema ically, he equa ion o a ans e sely noncompac au o-
igni ion lame ha is pe u bed by an acous ic combus o eigenmode
is gi en by Eq. (12), whose o mula ion is explained in he
ollowing. The ins an aneous HRR is ob ained by accoun ing o he
pe u ba ions in he spa ially in eg a ed HRR
b_
qa,wXy,z
ð Þ ¼_
q0,ay,z
ð ÞFwXx
ð Þ^
p x e ,y,z
ð Þ
p0
(8)
along wi h he luc ua ions in he lame mo ion
^xig,wXy,z
ð Þ ¼xig,0 y,z
ð ÞGwXx
ð Þ^
p x e ,y,z
ð Þ
p0
(9)
and is spa ially dis ibu ed using a Gaussian ke nel as ollows:
Fig. 9 P essu e pe u ba ion expe ienced by he pa icle while
being anspo ed by he mean low. The pe u ba ion induced by
he 1T modeshape is plo ed by he solid line and he cons an
ampli ude pe u ba ion in by he dashed line. The modeshape
en elope is shown by he dash-do ed line.
Fig. 10 Nume ical segmen a ion me hod o a ans e sally pe u bed lame in 2D
121019-6 / Vol. 147, DECEMBER 2025 T ansac ions o he ASME
Downloaded om h p://asmedigi alcollec ion.asme.o g/gas u binespowe /a icle-pd /147/12/121019/7530914/g p-25-1285.pd by gues on 06 Oc obe 2025
_
qwXx,y,z
ð Þ ¼_
q0,ay,z
ð Þ
_
qy,z
ð Þ i i i i i i
2p
p…
exp −
1
2
x−xig,0 y,z
ð Þ−^xig,wXy,z
ð Þ
_
qy,z
ð Þ
!2
0
@1
Ab_
qa,wXy,z
ð Þ
_
q0,ay,z
ð Þ þ1
!
(10)
The spa ial dis ibu ion o he mean HRR is gi en by
_
q0x,y,z
ð Þ ¼_
q0,ay,z
ð Þ
_
qy,z
ð Þ i i i i i i
2p
pexp −
1
2
x−xig,0 y,z
ð Þ
_
qy,z
ð Þ
!2
0
@1
A(11)
The luc ua ing HRR b_
qwXðx,y,zÞ, which is he sou ce e m o he
Helmhol z equa ion, is consequen ly compu ed by sub ac ing he
mean HRR om he ins an aneous coun e pa (Eq. (12))
b_
qwXðx,y,zÞ ¼ _
qwXðx,y,zÞ−_
q0ðx,y,zÞ(12)
The inal exp ession o he inhomogeneous Helmhol z equa ion o
a 3D ans e sely pe u bed and mo ing au o-igni ion lame
(Eq. (13)) is ob ained by subs i u ing he de i ed HRR sou ce e m
(Eq. (12)) on he igh -hand side o he Helmhol z equa ion (Eq. (4)).
The equa ion can be nume ically sol ed in a FE-sol e in he
equency domain o ex ac he eigen equencies and linea g ow h
a es. In his s udy, COMSOL MULTIPHYSICS 6.2 is used
� 1
q0 ^
p
� �þx2
cp0
^
p¼−ixc−1
cb_
qwXx,y,z
ð Þ (13)
4 Resul s
In his sec ion, we discuss he esul s ob ained when using he
amewo k de eloped in Sec. 3 o p edic he linea s abili y o he
GT36 CBB combus o . Fi s , Sec. 4.1 p esen s he expe imen al
measu emen s which se e as a benchma k o alida e he model.
Sec ion 4.2 subsequen ly p esen s he ob ained esul s om he
amewo k.
4.1 Measu emen s o P essu e Pulsa ions in he GT36
Combus o . A bo h in es iga ed OPs, dis inc esonances can be
obse ed. Fo he s able ope a ing poin , he esonances a e clea ly
isible in he equency spec um o he measu emen da a, as shown
in Fig. 12 measu ed by he wo senso s 1 and 2. Senso 1 is loca ed a
he cen e o he dump plane, while senso 2 is loca ed on he wall o
he sequen ial line close o he dump plane. The senso loca ions a e
also indica ed in Fig. 13(b). Figu e 12 indica es ha due o he ac
ha senso 1 is loca ed in he cen e o he dump plane, i is no
possible o measu e azimu hal modes. This is because he cen e o
he dump plane coincides wi h he nodal lines in he p essu e ield o
he 1T, 2T, and 3T modes (Fig. 5). In he con a y, senso 2 on he
sequen ial line can measu e he azimu hal modes’ p essu e
luc ua ions, obse able by he h ee peaks (dashed black line). I ,
howe e , a adial mode is p esen , senso 1 can measu e i s p essu e
luc ua ions, indica ed by he wo peaks (solid blue line) o he 1R
and 1R1 L modes. Table 1compa es he eigen equencies o he 1R
and 1R1 L modes ob ained om he FEM o he co esponding
measu emen s o senso 1. The p edic ed eigen equencies ma ch
e y well wi h he measu ed ones.
To asce ain he azimu hal and longi udinal na u e o he
p edic ed and measu ed eigenmodes and o ensu e a compa ison
be ween modes o simila quali a i e na u e, i is ins uc i e o
analyze he phase ela ion be ween he p essu e luc ua ions a
di e en loca ions. This is done by compu ing ans e unc ions
be ween di e en senso s, bo h o he measu emen s and he FEM
compu a ion. Figu e 13(a)shows he ans e unc ion be ween
senso 1 in he e y cen e o he dump plane and senso 2 on he
sequen ial line wall a a small axial dis ance (indica ed in Fig.
13(b)). The adius o he pola g aph ep esen s he a io o he
absolu e p essu e alues, and he angle shows he phase di e ence
be ween he luc ua ions a he wo conside ed loca ions. In he case
o he FEM (*), a phase di e ence o 180 deg is p esen . This
ma ches he heo e ical expec a ion pe ec ly, as he an inodes o he
1R mode a e loca ed in he cen e and on he ou e walls. A e y close
alue o 200 deg is obse ed om he expe imen al da a (�). The
good ma ch be ween he phase di e ence and in he eigen equency
con i ms ha he measu ed esonance peak a 0.894 co esponds o
he 1R mode.
Subsequen ly, he ans e unc ion is also compu ed o he mode
wi h a scaled equency o 1.033 o con i m he 1R1 L na u e. Figu e
13(c)shows he same analysis be ween senso 1 and senso 2.
Iden ical phases a e compu ed as o he equency o he i s adial
mode, wi h he only di e ence in ans e unc ion ampli ude. This
occu s because senso 2 is loca ed close o he nodal line in he FEM.
Fo he in es iga ed mode, a second ans e unc ion (Fig. 13(d)) is
compu ed be ween senso 2 and senso 3, which is also loca ed on he
sequen ial line wall, bu u he downs eam. Again, he FEM
p edic s he phase di e ence o be 180 deg, as is o be expec ed due
o he an inode placemen . Due o he close p oximi y o he senso o
he nodal line, he ampli ude a io is high. The ans e unc ion
phase o he expe imen al da a howe e is 240 deg. This does no
pe ec ly ma ch he expec ed esul o 180 deg, bu clea ly indica es
ha he p essu es a hese wo loca ions a e no in-phase. Hence, i is
con i med ha he mode in ques ion has a adial and longi udinal
componen p esen in i s modeshape, and he excellen ma ch in
scaled equencies con i ms he 1R1 L na u e.
The 1R1 L mode is u he analyzed using ime domain analysis
me hods shown in Re . [26]. Figu e 14 displays he p obabili y
densi y unc ion o he bandpass- il e ed (60.037 a ound equency
peak) p essu e measu emen signal PðgÞ, he ime signal, p obabili y
densi y unc ion o he ampli ude PðAÞ, he equency spec um, and
he join p obabili y densi y unc ion Pðg,_g=x0Þ o (a) OP1 and (b)
OP2. All alues a e no malized wi h espec o he measu ed
p essu e a OP1. Clea di e ences a e isible be ween he esul s o
Fig. 12 Scaled equency spec um o he p essu e measu e-
men s om senso 1 loca ed a he cen e o he dump plane (solid
blue line) and om senso 2 loca ed on he wall o he sequen ial
line (dashed black line) o ope a ing condi ion OP2line
Fig. 11 Th ee lame cha ac e iza ion pa ame e s: (a) in eg a ed
HRR _
q0,a(y,z), (b) lame wid h _
q(y,z), and (c) mean lame
loca ion xig,0(y,z) o OP1
Jou nal o Enginee ing o Gas Tu bines and Powe DECEMBER 2025, Vol. 147 / 121019-7
Downloaded om h p://asmedigi alcollec ion.asme.o g/gas u binespowe /a icle-pd /147/12/121019/7530914/g p-25-1285.pd by gues on 06 Oc obe 2025
OP1 and OP2. Whe eas PðgÞis sp ead ac oss a wide ange o
p essu es o OP1, i is concen a ed on a na owe egion o OP2.
Gene ally, o a he moacous ic ins abili y unde going limi cycle
oscilla ions, a bimodal shape would be expec ed o PðgÞ[27–30],
which would show as a ci cula shaped ing in he g,_
g-plane
[27,29,30]. This exac beha io canno be obse ed. This con i ms
ha he he moacous ic ins abili y is no a limi -cycle oscilla ion, as
can also be seen by he in e mi en na u e in he ime signal.
None heless, he high-ampli udes measu ed oge he wi h he b oad
PðgÞcu e and he non-Gaussian-like Pðg,_g=x0Þplo con i m he
he moacous ic ins abili y. The ime signal analysis o OP2 clea ly
indica es a s able mode, o which he acous ic esonances can be
seen. The Gaussian-like shape in he ðg,_
gÞ-plane oge he wi h he
na ow peak in he PðgÞcon i m his.
4.2 Linea S abili y Analysis o T ans e se The moacous ic
Modes. The linea s abili y o he p esen ed i e eigenmodes is
compu ed using he de eloped amewo k. The ollowing poin s
need o be kep in mind when in e p e ing he esul s:
�The FEM is compu ed wi hou any damping e ec s, neglec ing
iscous and he mal damping induced by he wall bounda y
laye s. These a e accoun ed o a pos e io i by he me hod
p oposed by Rome o e al. [31] when in e p e ing he FEM
esul s.
�In he GT36 combus o con igu a ion conside ed he e, speci ic
dampe s we e ins alled a op imal loca ions o e ec i ely
dampen he 1T, 2T, and 3T eigenmodes. The dampe s a e no
inco po a ed in he FEM compu a ion, and hei e ec is
conside ed a pos e io i acco ding o Re . [32].
�An assessmen o he dampe pe o mance [33] in e ms o
g ow h a e educ ion shows ha he 1R1 L mode is damped
signi ican ly less han he 1T, 2T, and 3T modes.
�We compa e linea g ow h a es om he expe imen and he
model below. We also show pulsa ion spec a o he wo
ope a ing condi ions. I should be no ed ha while he
p esen ed model is no able o p edic limi -cycle ampli udes,
he model can be quali a i ely compa ed o p essu e spec a o
assess he s abili y o he sys em, e en i expe imen al g ow h
a es a e no a ailable. This allows o gi e eliable es ima es o
he he moacous ic beha io helping in ha dwa e modi ica-
ions o de ining he necessi y o dampe s wi h a e y cos -
e ec i e simula ion.
Fig. 13 T ans e unc ion esul s be ween wo p essu e senso s
a di e en loca ions, bo h o he measu emen and he FEM
compu a ion. (a) Senso 1 and senso 2 o he 1R mode, (b)
senso con igu a ion and legend, (c) senso 1 and senso 2 o he
1R1 L mode, and (d) senso 2 and senso 3 o he 1R1 L mode.
Table 1 Compa ison be ween he scaled esonance equencies
o he measu emen s and he FEM compu a ion o OP2
Mode Scaled esonance equencies
Measu emen FEM
Fi s adial 0.894 0.909
Fi s adial þ i s longi udinal 1.033 1.049
Fig. 14 The p essu e measu emen ime signal is pos p ocessed o one dis inc equency peak appea ing in he
un il e ed equency spec um o he ime signal. The p obabili y densi y unc ion o he p essu e measu emen s P(g), he
ime signal, p obabili y densi y unc ion o he ampli ude P(A), he equency spec um, and he join p obabili y densi y
unc ion P(g,_
g=x0) a e shown. All alues a e no malized wi h espec o he measu emen s a OP1. The plo s show he
analysis o he 1R1L mode o (a) OP1 and (b) OP2.
121019-8 / Vol. 147, DECEMBER 2025 T ansac ions o he ASME
Downloaded om h p://asmedigi alcollec ion.asme.o g/gas u binespowe /a icle-pd /147/12/121019/7530914/g p-25-1285.pd by gues on 06 Oc obe 2025
Figu e 15 shows he s abili y p edic ion o (a) OP1 and (b) OP2.
The small symbols on he ze o g ow h a e line ep esen he passi e
lame eigenmodes. Analyzing he uns able ope a ing condi ion OP1
i s , i is isible ha he 1T, 2T, and 3T eigenmodes a e p edic ed as
s ongly uns able, which o he a o emen ioned easons, is o be
expec ed. This is in line wi h expe ience om p e ious measu e-
men campaigns indica ing ha he 1T and 2T modes a e ypically
mo e uns able han he in es iga ed 1R1 L mode. Fu he mo e, he
1R mode is p edic ed o be s able and also does no show an
ins abili y in he measu emen s. Finally, he 1R1 L mode is
p edic ed he moacous ically uns able a a equency o 1.005 and
a linea g ow h a e o 1.31% (g ow h a e di ided by i s equency
o oscilla ion). The expe imen al g ow h a e is deduced by i ing an
exponen ial cu e o he bandpass- il e ed p essu e bu s s [34],
esul ing in a mean alue o 0.49%. I he g ow h a e educ ion o
he dampe s is accoun ed o in he model as is desc ibed in Re . [32]
and he bounda y laye losses a e calcula ed by he me hod p oposed
in Re . [31], he esul ing ne g ow h a e is educed o 0.99%. The
di e ence in g ow h a e o 0.50% is likely due o unde es ima ed
damping alues and he simpli ying assump ion o ou me hodol-
ogy. In essence, he amewo k is able o p edic he he moacous ic
ins abili y a a equency o 1.005 as well as o iden i y i s mode
shape, con i med by he expe imen al da a in Sec. 4.1.
Acco ding o he p edic ed g ow h a es o he s able OP2 in
Fig. 15(b), he 1R1 L mode is s able. This is in line wi h he
measu emen s (Fig. 14(b)), o which no ins abili y is obse ed a
ha equency o OP2. The 1R mode is p edic ed o be uns able.
This mode is belie ed o be damped in he eal combus o , as he
dampe s o he 3T mode a e a ideal loca ions and a a simila
damping equency as he 1R mode. In summa y, he amewo k is
able o p edic ha he main p essu e pulsa ion a a equency o 1 is
he 1R1 L mode. Fu he mo e, i shows ha his mode is uns able a
OP1 and s able a OP2.
An addi ional analysis is pe o med, o in es iga e whe he
simila s abili y p edic ions conside ing only he sho e cylind ical
sec ion o he sequen ial line jus downs eam o he a ea expansion
can be made. This in es iga ion is in e es ing as many o he
in es iga ed eigenmodes a e pa icula ly p esen in his cylind ical
sec ion and he smalle domain would esul in a educ ion o
compu a ional ime. The compu a ions o a sho e cylinde a e
done o OP1 and compa ed o he esul s o he eal geome y. Fo
his con igu a ion, he inle is cha ac e ized as a wall. The ou le is
se o anechoic o ma ch i as closely as possible o he ull-leng h
geome y. Using a p essu e elease o wall ou le bounda y
condi ion would place a p essu e node o an inode a he ou le ,
which would no co espond o he condi ion o he ull geome y.
Figu e 16 compa es he s abili y p edic ions om (a) he symme ic
GT36 can combus o and (b) he simpli ied geome y. While he 1T
and 2T modes a e cap u ed well by he simple geome y, he
he moacous ic beha io o he o he modes canno be ep oduced.
Thus, he downs eam bounda y condi ion o he ull combus ion
chambe canno be eplaced by an a i icial ou le u he ups eam
wi h anechoic bounda y condi ion. This is because he e lec i eness
ha he combus ion chambe ou le imposes on he acous ic
eigenmodes is no ep oduced by he anechoic bounda y condi ion.
The e o e, he en i e combus ion chambe needs o be simula ed, as
was ini ially done in Fig. 15.
5 Conclusions
In his pape , we p esen a 3D amewo k based on lame ans e
unc ions and FEM o compu e he eigen equencies and linea
g ow h a es o ans e se he moacous ic eigenmodes in an
indus ial ehea combus ion chambe a engine ope a ing con-
di ions. The he moacous ic modeling app oach cap u es he
essen ial unde lying physics o ehea lame dynamics and consis s
o mul iple componen s: (1) a eac i e CFD calcula ion pe o med
in 3D o ob ain in o ma ion on he lame loca ion, shape, he local
HRR magni udes, and he empe a u e dis ibu ion in he combus-
ion chambe , (2) a Lag angian pa icle- acking algo i hm o ob ain
he ehea lame esponse o acous ic pe u ba ions, and (3) a
nume ical segmen a ion o he lame o cap u e he ans e sal
noncompac ness o he lame. The ma hema ical exp ession o a
ans e sely noncompac ehea lame is hen sol ed in a 3D
Helmhol z sol e . The amewo k is able o iden i y he uns able
mode(s) and p edic he linea g ow h a e, ex ac ed om
measu emen s, o an uns able mode wi h good accu acy. The
alida ion is pe o med agains expe imen al measu emen s o an
uns able and a s able ope a ing poin . The p esen ed me hod can be
used o suppo he indus ial de elopmen p ocess o no el gas
u bine ehea combus ion chambe s. In he ea ly design phase,
especially, i can be used o selec om mul iple ha dwa e a ian s in
a compu a ionally e y e icien way o o o esee dampe s a
ele an equencies.
Acknowledgmen
The esea ch wo k p esen ed in his pape was ca ied ou wi hin
he FLEX4H2 p ojec . The FLEX4H2 p ojec was suppo ed by he
Clean Hyd ogen Pa ne ship and i s membe s Eu opean Union,
Hyd ogen Eu ope, and Hyd ogen Eu ope Resea ch (GA
Fig. 15 S abili y p edic ions: (a) OP1 and (b) OP2. Small symbols
ep esen he passi e lame eigenmodes, and la ge symbols
ep esen he ac i e lame eigenmodes. The g ow h a e is gi en
as a pe cen age o he espec i e ac i e lame eigen equency.
Fig. 16 S abili y p edic ions: (a) GT36 can combus o and (b)
sho cylinde . Small symbols ep esen he passi e lame
eigenmodes, and la ge symbols ep esen he ac i e lame
eigenmodes. The g ow h a e is gi en as a pe cen age o he
espec i e ac i e lame eigen equency.
Jou nal o Enginee ing o Gas Tu bines and Powe DECEMBER 2025, Vol. 147 / 121019-9
Downloaded om h p://asmedigi alcollec ion.asme.o g/gas u binespowe /a icle-pd /147/12/121019/7530914/g p-25-1285.pd by gues on 06 Oc obe 2025
