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Numerical Prediction of the Flame Describing Function and Thermoacoustic Limit Cycle for a Pressurized Gas Turbine Combustor

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The forced flame responses in a pressurised gas turbine combustor are predicted using numerical reacting flow simulations. Two incompressible LES solvers are used, applying two combustion models and two reaction schemes (4-step and 15-step) at two operating pressures (3 bar and 6 bar). Although the combustor flow field is little affected by these factors, the flame length and heat release rate are found to depend on combustion model, reaction scheme and combustor pressure. The flame responses to an upstream velocity perturbation are used to construct the flame describing functions (FDFs). The FDFs exhibit smaller dependence on the combustion model and reaction chemistry than the flame shape and mean heat release rate. The FDFs are validated by predicting combustor thermoacoustic stability at 3 bar and 6 bar and, for the unstable 6 bar case, also by predicting the frequency and oscillation amplitude of the resulting limit cycle oscillation. All of these numerical predictions are in very good agreement with experimental measurements.
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Combustion Science and Technology
ISSN: 0010-2202 (Print) 1563-521X (Online) Journal homepage: https://www.tandfonline.com/loi/gcst20
Numerical prediction of the Flame Describing
Function and thermoacoustic limit cycle for a
pressurised gas turbine combustor
Yu Xia, Davide Laera, William P. Jones & Aimee S. Morgans
To cite this article: Yu Xia, Davide Laera, William P. Jones & Aimee S. Morgans (2019)
Numerical prediction of the Flame Describing Function and thermoacoustic limit cycle for a
pressurised gas turbine combustor, Combustion Science and Technology, 191:5-6, 979-1002, DOI:
10.1080/00102202.2019.1583221
To link to this article: https://doi.org/10.1080/00102202.2019.1583221
Published online: 08 Mar 2019.
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Numerical Prediction of the Flame Describing Function and
Thermoacoustic Limit Cycle for a Pressurized Gas Turbine
Combustor
Yu Xia
a,b
, Davide Laera
a
, William P. Jones
a
, and Aimee S. Morgans
a
a
Department of Mechanical Engineering, Imperial College London, London, U.K. ;
b
Applications Team, Fluids
Business Unit, ANSYS UK Ltd, Oxfordshire, Milton Park, Abingdon, U.K.
ABSTRACT
The forced ame responses in a pressurized gas turbine combustor are
predicted using numerical reacting ow simulations. Two
incompressible
1
large eddy simulation solvers are used, applying two
combustion models and two reaction schemes (4-step and 15-step) at
two operating pressures (3 and 6 bar). Although the combustor ow
eldislittleaected by these factors, the ame length and heat
release rate are found to depend on combustion model, reaction
scheme, and combustor pressure. The ame responses to an upstream
velocity perturbation are used to construct the ame describing func-
tions (FDFs). The FDFs exhibit smaller dependence on the combustion
model and reaction chemistry than the ame shape and mean heat
release rate. The FDFs are validated by predicting combustor thermo-
acoustic stability at 3 and 6 bar and, for the unstable 6 bar case, also by
predicting the frequency and oscillation amplitude of the resulting
limit cycle oscillation. All of these numerical predictions are in very
good agreement with experimental measurements.
ARTICLE HISTORY
Received 20 September 2018
Revised 14 January 2019
Accepted 15 January 2019
KEY WORDS
Thermoacoustic limit cycle;
pressurized combustor;
reaction chemistry;
incompressible LES; ame
describing function
Introduction
Numerical prediction of thermoacoustic instability in gas turbine combustors is an
ongoing challenge. Many approaches rely on a model for the ame heat release rate
response (denoted _
q) to velocity perturbation just upstream of the ame (denoted u1),
e.g., Li and Morgans (2015); Han et al. (2015), such as the weakly nonlinear ame
describing function (FDF, denoted F), whose gain, G, and phase, φ, depend on the
amplitude, ^
u1=
u1
jj
, and frequency, ω, of the perturbation as (Noiray et al., 2008)
Fω;^
u1=
u1
jj
ðÞ¼
b_
q=_
q
^
u1=
u1¼Gω;^
u1=
u1
jj
ðÞexp iφω;^
u1=
u1
jj
ðÞðÞ;(1)
where c
ðÞ denotes amplitude uctuations in the frequency domain and ðÞ the time-
averaged quantities.
CONTACT Yu Xia yu.xia13@hotmail.com Department of Mechanical Engineering, Imperial College London,
London SW7 2AZ, U.K.
Color versions of one or more of the gures in the article can be found online at www.tandfonline.com/gcst.
1
Incompressible does not imply constant density but rather that the density is independent of pressure
variations throughout the ow, i.e., not able to be compressed.
COMBUSTION SCIENCE AND TECHNOLOGY
2019, VOL. 191, NOS. 56, 9791002
https://doi.org/10.1080/00102202.2019.1583221
© 2019 Taylor & Francis Group, LLC
Many numerical simulations of premixed ame FDFs use fully compressible large eddy
simulations (LES), e.g., Krediet et al. (2013); Lee and Cant (2017). However, the computa-
tional costs are extremely large due to the very small time step limited by the inverse of
speed of sound. Recently, the fact that the ame primarily responds to hydrodynamic
disturbances (originally excited by the acoustics) has been exploited by using incompres-
sible LES (Febrer et al., 2011; Han and Morgans, 2015): the convective ow speed rather
than the speed of sound then determines the time step and thus allows much larger values,
making incompressible simulations signicantly faster than fully compressible ones. With
this approach, a complete FDF for a premixed blu-body stabilized ame was accurately
predicted (Han et al., 2015), as were the nonlinear FDFs of a swirling premixed ame (Xia
et al., 2017b) and a very long non-swirling ame (Li et al., 2017; Xia et al., 2017a) and,
recently, the linear ame response of a stratied ame (Han et al., 2018), etc.
In most ame LES studies, the chemical reaction is modeled by a simplied scheme
involving only a few reaction steps, e.g., Han et al. (2015); Bauerheim et al. (2015).
Although the impact of reaction chemistry on an unforced methane ame was recently
investigated (Fedina et al., 2017), to the authorsknowledge there is no similar study for
a forced ame, where an accurate reproduction of the ame response is vital for FDF
prediction. This is even more important when the operating pressure is high. Very few
LES cases have considered an unforced ame (Bulat et al., 2014) or linear ame response
(Hermeth et al., 2014) at elevated pressures, representative of real combustor operations,
but none thus far have realistically accounted for the nonlinear forced ame response at
such high pressures.
The present work thus aims to numerically simulate the nonlinear ame response in
a pressurized realistic combustor, using incompressible LES. The eects of sub-grid
combustion models, reaction schemes, and operating pressures on the ame shape and
the FDFs are analyzed. In order to validate the computed FDFs, thermoacoustic predic-
tions are performed by combining with a low order network approach for the acoustic
waves.
Mathematical formulation of incompressible LES
The ltered conservation equations of mass, momentum, species mass fractions, and
enthalpy solved by the incompressible LES are
@
ρ
@tþ@
ρ~
ui
@xi¼0;(2a)
@
ρ~
ui
@tþ@
ρ~
ui~
uj
@xj¼@
ρ
@xiþ@
@xj
μ@
ui
@xjþ@
uj
@xi

@
@xj
τij;(2b)
@
ρ~
Yα
@tþ@
ρ~
ui~
Yα
@xi¼@
@xi
μ
σm
@~
Yα
@xi

@Jα;i
@xiþ
ρ~
_
ωα;(2c)
980 Y. XIA ET AL.
@
ρ~
h
@tþ@
ρ~
ui~
h
@xi¼@
@xi
μ
σm
@~
h
@xi
!
@Jh;i
@xi
;(2d)
where f
ðÞ denotes density-weighted ltering, ρdensity, uvelocity, ppressure, henthalpy
(including the enthalpy of formation), μdynamic viscosity, and σmthe Prandtl or Schmidt
number as appropriate, with the latter assumed the same for all the species. The Lewis
number, Le, is assumed unity, so that the Prandtl and Schmidt numbers are equal,
implying Fourier heat conduction and Fickian diusion. The equation of state used is
the ideal gas equation, ρ¼P0Wg
RT , with Rthe universal gas constant, Wgthe molar mass of
the mixture, Tthe temperature, and P0the constant operating pressure. The temperature
and heat release rates are computed as a function of enthalpy and composition using
JANAF data (National Institute of Standards and Technology (NIST), 1998). Jα;iand Jh;i
are the sub-grid scalar uxes for the αth species (with α¼1;;Nsp

and Nsp the total
number of species) and the enthalpy, h, respectively, both modeled by the sub-grid eddy
viscosity, μsgs.τij ¼ρg
uiuje
uie
uj

is the sub-grid stress tensor, and e_
ωα=e_
ωðYαÞthe ltered
chemical reaction rate, with Yαthe species mass fraction.
In order to model the unknown stress, τij, the dynamic Smagorinsky model (Piomelli
and Liu, 1995) is applied, which has been found to oer sucient accuracy. For the
ltered term, e_
ωα, two models are applied: (i) the probability density function (PDF) model
(Jones and Navarro-Martinez, 2007; Jones and Prasad, 2010), and (ii) the partially stirred
reactor (PaSR) model (Sabelnikov and Fureby, 2013). In the former case, Eq. 2(c,d) are not
solved directly, but rather the spatially ltered joint PDF, e
Psgs, for all the relevant scalars
(e.g., the species mass fractions and enthalpy) is used:
ρ@e
Psgs ψ

@tþρe
uj
@e
Psgs ψ

@xjþP
Ns
α¼1
@
@ψαρ_
ωαψ

e
Psgs ψ
hi
¼
@
@xi
μ
σmþμsgs
σsgs

@e
Psgs ψ

@xi

ρCd
2τsgs P
Ns
α¼1
@
@ψαψαφαðx;tÞ

e
Psgs ψ
hi
;
(3)
where ψαis the sample (composition) space of an arbitrary scalar, φαðx;tÞ, appearing at
location xand time t, and ψthe global composition of all the involved scalars. The ltered
e_
ωαnow appears as closed in Eq. (3), which is solved by the Eulerian stochastic eld
method (Jones and Navarro-Martinez, 2007). The joint PDF is represented as an ensemble
of stochastic elds. These elds have no direct physical signicance, but instead represent
an equivalent stochastic system that has the same one-point PDF as that given by Eq. (3).
Any ltered scalar can now be obtained by averaging over all the corresponding stochastic
elds.
For the PaSR model, a dierent modeling concept is used. The ow in a mesh cell is
divided into two parts: (i) a perfectly stirred reactor, where all species are assumed
homogeneously mixed and reacting, and (ii) surroundings,lling with non-reacting
sub-grid scale ow structures. The burnt products from part (i) are mixed with the
surrounding owdue to turbulence, giving the nal species concentrations in the entire,
partially stirred, computational cell (Han et al., 2015). In any cell, the reaction only occurs
COMBUSTION SCIENCE AND TECHNOLOGY 981
within domain (i). The relative proportions of these two domains are controlled by the
reactive volume fraction, κ=τc=ðτcþτmÞ(Chomiak and Karlsson, 1996), governed by the
chemical reaction time, τc, and the turbulent mixing time, τm. To compute τc, an accurate
chemical reaction scheme is needed. Since the accuracy of a reaction scheme generally
increases with the number of reaction steps, in this work two schemes with dierent
complexity levels are considered a 4-step reaction with 7 intermediate species (Abou-
Taouk et al., 2016), and a 15-step scheme with 19 species (Sung et al., 2001)to evaluate
the impact of reaction accuracy on the ame behavior.
The mixing time is dened as τm=Cmffiffiffiffiffiffiffiffiffiffiffiffiffi
τΔτK
p(Fureby et al., 2015), where τΔ=Δ=u0
sgs
denotes the sub-grid mixing time, with Δthe local mesh size and u0
sgs the sub-grid scale
velocity uctuation. τK=ffiffiffiffiffiffiffiffiffiffiffi
ν=sgs
pis the Kolmogorov time, with sgs the sub-grid dissipa-
tion rate and νthe molecular kinematic viscosity. The mixing time constant, Cm,isxed as
0.8. The ltered chemical reaction rate, e_
ωα, is modeled by κas (Han et al., 2015)
e_
ωακ_
ωαρ;e
T;e
Yβ;Cα
1

;with β¼1;2;...;Nsp (4)
where C1is the concentration of the species that is leaving the mesh cell.
In this work, the PDF model (with one eld) is implemented into an incompressible in-
house LES code, BOFFIN (Jones et al., 2012), and the PaSR model is used by the incompres-
sible ReactingFOAM-LES solver in the open-source CFD toolbox, OpenFOAM (version 2.3.0)
(Weller et al., 1998). Since only one eld is used by the PDF model, the resulting PDF reduces
to a δ-function (one unique realization), which is no longer stochastic. Thus the sub-grid scale
turbulence-combustion interactions are neglected in the present BOFFIN study.
Experimental set-up and numerical framework
This work studies the pressurized industrial SGT-100 gas turbine combustor. The entire
rig (Figure 1a) comprises a swirling combustor, a long exhaust pipe, and a spray water
section (not shown) connected to the atmosphere. A cylindrical air plenum upstream of
the combustor provides uniform preheated air inow. The combustor (Figure 1b) consists
of a 12-slot radial swirler entry and a premixing chamber, followed by a dump expansion
into a ,0.5m-long combustion chamber, which has a straight duct of square cross-section
followed by a contraction duct. The exit of the contraction duct is connected to the
Figure 1. (a) Schematic of the SGT-100 combustor rig; (b) detailed structure of the combustor. All
dimensions in millimetre, with the x-origin located at combustion chamber inlet. Images adapted from
Stopper et al. (2010).
982 Y. XIA ET AL.
exhaust pipe via a straight circular pipe. German Natural Gas (Bulat et al., 2014)is
injected at a temperature of 305 K through the swirler entry and mixed with the preheated
air at 685 K, reaching a global equivalence ratio of φ= 0.60 (Stopper et al., 2013). Two
operating pressures,
p= 3 and 6 bar, are used, with the bulk Reynolds number in the range
18,400120,000 and Mach number in the range 0.020.29. This combustor has been
studied experimentally (Stopper et al., 2010,2013) at German Aerospace Centre (DLR)
and numerically (Bulat et al., 2014; Fedina et al., 2017).
The thermoacoustic stability of the combustor was measured to be dependent on the
pressure; being stable at 3 bar, but experiencing limit cycle oscillations at 6 bar (Stopper
et al., 2013). Figure 2a shows the time-signal of the normalized pressure uctuation, p0=
p,
at 6 bar, measured 231 mm beyond the combustion chamber inlet. The averaged spectrum
in Figure 2b exhibits a spectral peak at 216 Hz with amplitude ^
p
jj5000 Pa.
In order to simulate this combustor, we consider a simplied fuel combining all hydro-
carbons into methane, giving a composition of 98.97% CH4,0.27%CO
2and 0.753% N2(Bulat
et al., 2014). The computational domain, shown in Figure 3a, neglects the plenum and the
exhaust pipe since the ame is restricted to the combustion chamber. A velocity inlet
condition consistent with the measured inow rate is imposed at the swirler entry, including
the radial main air inlet and multiple fuel injection holes. The panel air inlet refers to the front
edge of the combustion chamber where a small amount of air enters the domain. The outlet
corresponds to the combustor exit plane where a zero-gradient and a nonreectingoutow
Figure 2. (a) Time-signal of normalized pressure uctuation, p0=
p, at 6 bar, measured at x¼231 mm
with a sampling rate of 10 kHz (enlarged timescale between 20 and 21 s); (b) Power spectrum of time-
signal in (a).
Figure 3. (a) The computational domain (Xia et al., 2018a) and (b) the optimized 7.0-million-cell multi-
block mesh (Bulat, 2012).
COMBUSTION SCIENCE AND TECHNOLOGY 983
condition are applied for pressure and velocity, respectively. Here the nonreectingoutow
condition is a specicarticial boundary condition, designed to neglect the diusion eects
near the outlet and assumes that the outow is purely advective (i.e., nonreective)
(Boströmm, 2015); it inhibits the occurrence of negative velocities at an outow boundary.
All solid boundaries are dened as non-slip adiabatic walls, consistent with previous LES
works on the same combustor (Fedina et al., 2017). The entire domain is discretised with
a multi-block structured mesh comprising 7.0 million cells (see Figure 3b), found to be an
optimal mesh in a previous study (Bulat, 2012). Mesh renement is applied to the swirler,
premixing chamber and front part of the combustion chamber, to better resolve the ame
behavior.
The numerical schemes used by the present LES are as follows. In BOFFIN, a second-
order central dierence scheme is used for all the spatial discretisations, except for the
convective terms in scalar equations where a total variation diminishing (TVD) scheme is
used. OpenFOAM adopts a linear Gaussian interpolation scheme for spatial discretisation
(which is a central dierence method), coupled with a Sweby limiter (Sweby, 1984)
improving the stability in regions with rapidly changing gradients but adding numerical
diusivity. Both BOFFIN and OpenFOAM apply an implicit second-order Crank
Nicolson scheme for temporal discretisation. A small xed time step of 5 107sis
used to ensure the CourantFriedrichsLewy number is always below 0.3.
Unforced LES results and validation
The eects of sub-grid combustion model, reaction chemistry and the pressure on the
unforced mean ow and ame behavior are rst presented. For systematic comparisons,
ve LES cases with dierent modeling assumptions and operating conditions are dened,
as listed in Table 1.
The eects of combustion model are investigated by comparing Cases I and II. Figure 4
shows the time-averaged contours of axial velocity,
u, temperature,
T, and volumetric heat
release rate, _
q, on a symmetry plane, with the top-half of each sub-gure referring to Case
I and the bottom-half to Case II. The mean streamlines are superimposed on the contours
of
uand
Tto allow a clear representation of the inner and outer recirculation zones.
A central vortex core (CVC) extends along the centreline from the exit to the mid-
chamber, with the exit low pressure zone resulting in very high exit velocities on the
centreline, dropping otoward the two sides. This CVC has been observed in previous
experiments (Stopper et al., 2013) and LES (Bulat et al., 2015; Xia et al., 2016,2018a)on
the same combustor.
Table 1. LES cases used to study the eects of dierent factors.
Case No. I II III IV V
LES solver OpenFOAM BOFFIN BOFFIN BOFFIN BOFFIN
Combustion model PaSR PDF PDF PDF PDF
Reaction scheme 4-Step 4-Step 15-Step 4-Step 15-Step
Operating pressure 3 bar 3 bar 3 bar 6 bar 6 bar
984 Y. XIA ET AL.
The mean velocity contours are very similar between Cases I and II (Figure 4a),
suggesting that the combustion model has negligible eect on the unforced ow eld.
For the mean temperature and heat release rate elds, however, some discrepancies exist.
Case I exhibits a longer low temperature zone and a lower heat release rate. A possible
explanation is that OpenFOAM may be more diusive than BOFFIN, due to its use of
a TVD-type scheme for the velocities, which would result in a thicker reaction zone and
corresponding lower peak heat release rate. It is also possible that it is a result of the
dierences in the two combustion models utilized in the two codes.
Figure 5 compares the vertical y-proles of velocity, temperature and mixture fraction
between Cases I and II at four axial locations. The horizontal axis of each sub-gure is thus
split into four segments, each refers to a prole and has the same data range for the plotted
variable. At x= 18.7 mm, very little dierence is observed between the PaSR and PDF models
for either mean or root-mean-square (rms) velocities, conrming that the combustion model
has almost no eect on the ow eld. Both cases accurately capture the experimental data.
Further downstream (x38.7 mm), good agreement between the LES and experiments is
maintained for all velocity proles, otherthan for a small mismatch between the two LES cases
near the centreline of the vrms proles (Figure 5f). This may be due to the dierence in the
CVCs axial length predicted by the two combustion models.
For the temperature, T, both combustion models correctly capture its mean and rms
proles at the rst location of x= 18.7 mm. Dierences between the LES and measure-
ments become more pronounced further downstream. Both LES cases fail to predict the
temperature prole near the top wall, although the experimental data here are not
complete; and this is consistent with other studies on the same rig (Fedina et al., 2017).
Very good agreement is achieved for the mean mixture fraction,
Z, which reveals
a globally lean mixture uniformly distributed in the reaction zone. Although small errors
exist for zrms prole (Figure 5h), bearing in mind the measurement uncertainties (Bulat
et al., 2014), these errors are generally acceptable. It is also noted that the PDF model
slightly overpredicts the mean and rms temperature, consistent with the shorter low
temperature zone predicted by the same model.
Second, the eect of reaction scheme on the unforced LES is studied. To investigate the
4-step and 15-step methane schemes used, their predictions of the laminar burning velocity,
Figure 4. Time-averaged contours of (a) velocity,
u, (b) temperature,
T, and (c) volumetric heat release
rate, _
q, on a symmetry plane, obtained by (top) Case I with PaSR model and (bottom) Case II with PDF
model.
COMBUSTION SCIENCE AND TECHNOLOGY 985
Su, and the adiabatic ame temperature, Ta, are compared to those obtained by a full 325-step
GRI-Mech 3.0 scheme (Gregory et al., 2018). The eect of the constant Schmidt and unity
Lewis number assumptions is evaluated by comparing the LES predictions of Suand Tawith
Figure 5. Vertical y-proles of (ad) mean and (eh) rms ow variables at
p= 3 bar for four axial
locations: x¼18.7, 38.7, 58.7, and 88.7 mm. Solid line: Case I with PaSR model; dotted line: Case II
with PDF model; circle: experimental data (Bulat et al., 2014).
986 Y. XIA ET AL.
those from the chemical solver, Cantera (Goodwin et al., 2014), in which accurate transport
properties are used. Figure 6a shows calculations performed at
p=3barwithanambient
temperature of 650 K, the same conditions as for LES Cases IIII. The Cantera simplied
schemes give similar Suand Tavalues to those of the detailed GRI-Mech scheme across
a range of equivalence ratios. For the operating point of ϕ= 0.6, the Cantera 4-step scheme
slightly overpredicts the laminar burning velocity, with the 15-step scheme matching better
the detailed GRI scheme. The two tested LES solvers slightly underpredict Sufor both the
4-step and 15-step schemes, especially at higher ϕ. These results are as anticipated given the
constant Schmidt and Prandtl number assumption,
2
as it is well known that this leads to an
underprediction of the burning velocity at higher equivalence ratios (Poinsot and Veynante,
2005). Nevertheless, the adiabatic ame temperature and burning velocity are correctly
reproduced by both LES solvers at the combustor operating point of ϕ¼0.6; discrepancies
occur around stoichiometric equivalence ratios.
Following the above validation, the eect of reaction chemistry on the unforced LES is shown
in Figure 7, which compares Cases II and III using BOFFIN code. The impact of reaction
scheme on the mean ow eld is marginal, while the 4-step scheme gives a slightly shorter low
temperature zone and a higher-magnitude heat release rate. This may be due to the slightly
higher laminar burning velocity of the 4-step reaction with BOFFIN at ϕ¼0:6(Figure 6a),
which reduces the axial extent of the reaction zone and increases the fuel burnt at the ame, thus
leading to a higher heat release rate.
Figure 6. Laminar burning velocity (Su) and adiabatic ame temperature (Ta) against equivalence ratio
(ϕ) at (a)
p= 3 bar and (b)
p= 6 bar, both with an ambient temperature of Tm= 650 K. Solid line:
Cantera with GRI-Mech 3.0; Δ: Cantera with 4-step scheme; Ñ: Cantera with 15-step scheme; :
OpenFOAM with 4-step scheme; }: BOFFIN with 4-step scheme; : BOFFIN with 15-step scheme.
The operating point is ϕ¼0.6.
2
Recent computations of laminar ames with BOFFIN using accurate transport properties reproduce the
GRI results.
COMBUSTION SCIENCE AND TECHNOLOGY 987
Figure 8 compares the vertical y-proles of mean and rms variables between Cases II
and III. The match between LES and experiments is generally good for all the variables at
all locations, with errors within the limits of measurement uncertainties (Bulat et al.,
2014), conrming both chemical schemes are accurate. The 4-step scheme slightly over-
predicts the mean temperature,
T, and mass fraction of H2O, but underpredicts the mass
fraction of CH4compared to the 15-step scheme. These are consistent with the small
dierences in the predicted unforced ame (see Figure. 7b,c).
Finally, the eect of operating pressure on the unforced simulations is analyzed.
The burning velocity and temperature computed by the two simple schemes with
BOFFIN are compared with GRI-Mech 3.0 at 6 bar pressure in Figure 6b. Increasing
pressure globally reduces the laminar burning velocity for all equivalence ratios but
does not aect the adiabatic ame temperature, consistent with Poinsot and
Veynante (2005). Both schemes correctly predict the burning velocity and tempera-
ture, with the errors against GRI-Mech 3.0 increasing with ϕ.Finally,theBOFFIN
4-step scheme gives a slightly higher burning velocity than the BOFFIN 15-step
scheme, consistent with their predictions at 3 bar. Although the accurate transport
properties used in Cantera improve the burning velocity prediction at high ϕ,the
two simple BOFFIN schemes are both accurate enough for operation at ϕ¼0.6.
Based on the above validation, the contours of mean ow variables are compared in
Figure 9 between Cases IV and V. Both schemes yield similar mean velocity elds,
indicating the small eect of reaction chemistry on the ow. Due to the faster burning
velocity, the 4-step scheme gives a shorter low temperature zone and a higher ame heat
release rate than the 15-step chemistry. Compared to 3 bar predictions (Figure. 7b,c), the
low temperature zone is much shorter at 6 bar for both schemes, while the heat release
rate is now much higher. These dierences between 3 and 6 bar simulations are mainly
caused by the increase of mixture density associated with a higher pressure and corre-
spondingly a higher chemical reaction rate.
The vertical proles of 6 bar variables are shown in Figure 10 for Cases IV and V. A larger
deviation between the LES and experiments is now evident, especially for some rms variables
(e.g., CH4rms). This may be due to the fact that the combustor is stable at 3 bar, but becomes
unstable at 6 bar with a large-amplitude limit cycle oscillation. Although the 4-step scheme is
suciently accurate for the ow eld, the more detailed 15-step scheme gives improved
predictions for the temperature and species mass fractions, e.g., CH4and H2O.
Figure 7. Time-averaged contours of (a) axial velocity,
u, (b) temperature,
T, and (c) volumetric heat
release rate, _
q, computed at 3 bar by (top) Case II with 4-step scheme and (bottom) Case III with 15-
step scheme.
988 Y. XIA ET AL.
Flame describing functions
The forced ame heat release responses in the analyzed combustor are now computed.
The above simulated unforced ame is submitted to an upstream velocity perturbation, u1,
located at the main air inlet of the swirler entry, varying harmonically as:
Figure 8. Vertical y-proles of (ad) mean and (eh) rms ow variables at 3 bar. Solid line: Case II with
4-step scheme; dotted line: Case III with 15-step scheme; circle: Experimental data (Bulat et al., 2014).
COMBUSTION SCIENCE AND TECHNOLOGY 989
u1¼
u11þAusin 2πfutðÞ½;(5)
with
u15 m/s the mean inow velocity, futhe perturbation frequency and Au¼
u01=
u1
jj
the normalized perturbation amplitude. To construct the FDFs, two forcing
amplitudes (Au= 0.1 and 0.2) across eight forcing frequencies from 200 to 1500 Hz are
used. For each forcing case, a time period of at least 15 forcing cycles (after the initial
transients vanish) has been simulated. The convergence of the computed FDF properties
(e.g., gain and phase) is usually achieved after 1012 cycles. The gain, G, and phase, φ,of
the resulting FDFs are plotted in Figure 1115 for dierent LES cases. Some common
trends are observed for all the FDFs:
(i) The FDF gain has two local maxima with a local minimum in between. The rst
maximum occurs at f200300 Hz, the gain minimum near f¼600 Hz and
the second maximum at f¼800 Hz. These local gain extrema have previously
been found for an atmospheric swirling combustor (Palies et al., 2010), and are
caused by the constructive and destructive interactions between the imposed
longitudinal perturbation and the azimuthal perturbations generated by the ow
swirl. While at atmospheric pressure (Palies et al., 2010), the gain minimum falls
almost to zero, the present FDFs have a less pronounced gain minimum with G
0.5 at f¼600 Hz.
(ii) The FDF phase decreases linearly with frequency, consistent with recent LES
studies (Palies et al., 2010). This is because it mainly depends on the time delay
between the perturbation and the ame response, which is inversely proportional
to the mean ow velocity,
u, and is little aected by other factors. The dynamics
in φnear the frequency of the local gain minimum are much less pronounced for
this pressurized rig than at atmospheric pressure (Palies et al., 2010).
(iii) An increase of the forcing amplitude, Au, always leads to a decrease in the FDF
gain, especially at higher frequencies. This gain saturation is caused by the
leveling-oof the heat release rate oscillations at higher perturbation levels, and
is consistent with previous studies (Han and Morgans, 2015). The FDF phase,
however, shows very little forcing amplitude dependence.
Figure 9. Time-averaged contours (a) axial velocity,
u, (b) temperature,
T, and (c) volumetric heat
release rate, _
q, computed at 6 bar by (top) Case IV with 4-step scheme and (bottom) Case V with 15-
step scheme.
990 Y. XIA ET AL.
The simulated FDFs are compared between LES cases to evaluate the eects of combustion
model, reaction chemistry and operating pressure. As shown in Figure 11, the FDFs for Cases
I and II are compared at forcing level Au¼0:1. The trends of the gain, G,andphase,φ, both
Figure 10. Vertical y-proles of (ad) mean and (eh) rms ow variables at 6 bar. Solid line: Case IV
with 4-step scheme; dotted line: Case V with 15-step scheme; circle: experimental data from DLR.
COMBUSTION SCIENCE AND TECHNOLOGY 991
generally match well, other than for small gain mismatches at lower frequencies (e.g., 300-
600 Hz), likely caused by the higher heat release rate predicted by the PDF model (Figure 4c).
The eect of stochastic eld number used by the PDF model is also studied based on Case
III. The FDF gain and phase computed by eight stochastic elds are shown in Figure 12,and
are very close to those obtained with only one eld at both perturbation levels for two
frequencies (fu¼600 and 800 Hz). This implies that one eld is sucient for the PDF
model to correctly capture the FDFs in the present combustor. Overall, the eect of sub-
grid combustion model on the forced ame responses is relatively small.
Figure 11. FDFs at
p= 3 bar for forcing amplitude Au¼0.1, obtained for Case I with PaSR model (solid
line with circles) and Case II with PDF model (dashed line with squares). (Top) gain G; (bottom) phase φ.
Figure 12. FDFs at
p= 3 bar, obtained for Case III with PDF model using one eld (solid line with
circles) and spot checks for 8 stochastic elds (squares). (Top) gain G; (bottom) phase φ. (a) Au= 0.1; (b)
Au= 0.2.
992 Y. XIA ET AL.
Second, the eect of reaction chemistry on the FDFs is discussed. The FDFs obtained
by Cases II and III are compared in Figure 13. The magnitudes and trends of the gain and
phase both match well for both forcing levels, with the gains mismatch below 10%. This
good match is also achieved at 6 bar. Although the mean ame shapes dier between the
two reaction schemes at both pressures, with the more detailed 15-step scheme slightly
more accurate, this dierence barely aects the FDFs. The simpler 4-step scheme appears
to be sucient for FDF computation, this providing sucient accuracy at reduced
computational cost.
To further investigate the eect of reaction scheme, the unsteady ame dynamics of
forced Cases II and III are compared in Figure 14, choosing fu= 300 Hz and Au= 0.1 in
order to achieve strong heat release oscillations. Three time snapshots of the heat release
rate eld, _
qðx;y;zÞ, are compared within a forcing period, chosen to correspond to the
normalized uctuation, h_
qi0=h_
qi, being minimum, zero and maximum (with h_
qððð_
qdV
and Vthe domain volume). Although the unforced ame shapes slightly dier between
the two reaction schemes, the ames oscillatory behavior around its mean position is
generally similar. The ame is shortest when h_
qi0=h_
qiis minimum, becoming longer when
h_
qi0¼0, and being longest for the maximum of h_
qi0=h_
qi. The heat release uctuation is
mainly associated with the variation in the ame surface area. Cases II (Figure 14a) and III
(Figure 14b) show that a change in reaction scheme can modify the response of the ame
structure, even at the same pressure. Although the more detailed 15-step scheme yields
a slightly longer ame throughout the forcing period (Figure 14b), consistent with the
unforced ame in Figure 7c,dierences in the detailed ame structure may still result in
a similar ame surface area for the two reaction schemes, explaining the similarity in the
FDF gains shown in Figure 13.
Finally, the eect of operating pressure on the FDFs is analyzed comparing Cases III
and V. Figure 15a shows that at the lower forcing level, Au¼0.1, increasing the pressure
leads to an increase in FDF gain at lower frequencies (f500 Hz). This is in agreement
with a recent experiment by Sabatino et al. (2018).
Figure 13. FDFs at
p= 3 bar for Case II with 4-step scheme (solid line with circles) and Case III with 15-
step scheme (dashed line with squares). (Top) gain G; (bottom) phase φ. (a) Au= 0.1; (b) Au= 0.2.
COMBUSTION SCIENCE AND TECHNOLOGY 993
Figure 14. Snapshots of heat release rate eld, _
qðx;y;zÞ, on a symmetry plane, forced at Au= 0.1, fu=
300 Hz. (I) h_
qi0=h_
qi=0.1; (II) h_
qi0=h_
qi= 0; (III) h_
qi0=h_
qi= 0.1. (a) Case II (4-step scheme,
p= 3 bar); (b)
Case III (15-step scheme,
p= 3 bar); (c) case V (15-step scheme,
p= 6 bar).
Figure 15. FDFs obtained for case III at
p¼3 bar (solid line with circles) and Case V at
p¼6 bar
(dashed line with squares), with (top) gain Gand (bottom) phase φ. (a) Au= 0.1; (b) Au= 0.2.
994 Y. XIA ET AL.
The present FDF also exhibits a strong pressure-dependence of the frequency of its
maximum gain: at 3 bar, the maximum G1.2 is found at f= 200 Hz, while at 6 bar it
increases to G1.4 and shifts to f= 300 Hz (Figure 15a). This is not in agreement with
Sabatino et al. (2018). In their work, a pressure increase from 1 to 4 bar does not vary the
frequency of the maximum gain, with the change of gain level with pressure depending on
the fuel used. They considered a dierent combustor geometry, ame shape, fuel type, etc.,
and their equivalence ratio was adjusted with pressure to ensure the same mean ame
length for all pressures. In the present work, however, a xed ϕis used at both pressures,
giving slightly dierent mean ame lengths, which may lead to dierent coupling between
the ame and the imposed perturbation at two pressures, shifting the frequency of the
maximum FDF gain.
At the higher forcing level of Au¼0:2, the pressure dependence of the gain maximum
is much weaker (Figure 15b), mainly due to the stronger saturation of the ame surface
oscillation at such high perturbations levels. For the FDF phase, the eect of the pressure
is always negligible, consistent with the experiment (Sabatino et al., 2018).
Further insights into the pressure eect are provided by the ame dynamics for Cases
III and V (see Figure. 14b,Figure. 14c). The predicted ame structures are similar,
probably due to the same reaction scheme used. The ame surface area oscillation can
be compared by examining the axial ame length. Increasing the pressure slightly
increases the ame length, likely to be associated with the lower laminar burning velocity.
The frequency dependence of the FDF gain with pressure in Figure 15a can now be
explained: the pressure increase gives a longer ame length and thus a longer ame
response time. At lower frequencies (e.g., 300 Hz), the timescale of the perturbation signal
is also large, giving stronger coupling between the perturbation time and the response
time. In contrast, at higher frequencies, the forcing period is much shorter, leading to
a weaker coupling with the ame response and thus lower FDF gains.
Thermoacoustic limit cycle prediction
In order to validate the above simulated FDFs, they are coupled with the low order
network solver, OSCILOS (Li and Morgans, 2015), in order to predict the thermo-
acoustic stability of the analyzed combustor. OSCILOS has been validated by experi-
ments (Han et al., 2015). It represents the combustor geometry as a network of
connected simple modules, as shown in Figure 16. The length and cross-sectional
area of each module match the original geometry and ow rates. The water spray
section is neglected due to its large acoustic energy dissipation, and the upstream
plenum is ignored as it is preferable to prescribe a physical acoustic boundary condi-
tion at the swirler inlet. The combustion chamber contraction is represented as
a sequence of 50 constant area modules with successively decreasing areas. The mean
ow is accounted for, with the mean ow variables assumed constant within each
module, changing only between modules. The axial distributions of mean velocity and
temperature in the network are reconstructed from the LES mean ow and the
experimental data, respectively.
The acoustic waves are assumed linear and one-dimensional at the low frequencies of
interest (Noiray et al., 2008). Thus, within each module, the acoustic perturbations satisfy
COMBUSTION SCIENCE AND TECHNOLOGY 995
the convected wave equation and can be represented as the sum of downstream and
upstream traveling waves with dierent strengths. These wave strengths are tracked
between modules using linearized ow conservation equations these account for losses
due to stagnation pressure drop at area expansions (Li et al., 2017). The boundary
conditions for the network are dened by the pressure reection coecients, R, denoting
the strength ratio of the reected to incident acoustic waves at an end. In this work, the
network inlet is assumed as highly damped due to the perforated plate installed between
the plenum and the swirler. The inlet reection coecient, Rin, is little aected by the
operating pressure, and it increases in magnitude from Rin
jj
=00.15 and varies in phase
between Rin =0.7πand 0.55πover frequency f=01000 Hz. In contrast, the network
outlet is dened as a slightly damped open end, which does not vary with pressure and has
its magnitude Rout
jj
dropped from 1 to 0.91 and phase Rout from πto 0.84πacross
01000 Hz (Xia et al., 2018b).
Since the present ame has a much shorter axial extent (,100 mm) than the dominant
acoustic wavelengths (,14 m), the ame zone is represented by an innitely thin ame
sheetat x= 45 mm, where the maximum mean heat release rate occurred in experiments
(Stopper et al., 2013). The jump in acoustic wave strengths across the ame is accounted
for using the linearized ow conservation equations across the ame sheet (Dowling,
1997). To account for the eect of acoustic waves on the ame response, a ame model is
prescribed, in this work in the form of an FDF.
To predict the linear stability of the combustor, the thermoacoustic modes of the above
network geometry are computed using FDFs at Au¼0.1 for Cases III and V with 15-step
scheme. The complex frequencies, ω¼σþi2πf(with σthe growth rate), for which both the
inlet and outlet boundary conditions are satised, are identied within OSCILOS using
ashooting method(Han et al., 2015). The computed modes are marked by white stars in
Figure 17, showing that all modes are predicted to be stable at3 bar, while at 6 bar one mode at
f231 Hz is predicted to be unstable. The predicted stabilities match well with the
experimental observations (Stopper et al., 2013; Xia et al., 2017c), and are unchanged if the
4-step scheme FDFs (Cases II and IV) are used instead of the 15-step ones.
The reason for the stability change with pressure is now considered: thermoacoustic
stability is governed by the combination of (i) acoustic waves and (ii) the ame response.
For (i), the acoustic wave strength is determined by the geometry, speed of sound, and
boundary conditions, none of which are aected by the pressure in this work. Thus the
ame response is the main source of the stability change. The FDF gain near the frequency
Figure 16. The simplied network model of the analyzed combustor (Xia et al., 2018b).
996 Y. XIA ET AL.
of the unstable mode is higher at 6 bar than at 3 bar (Figure 15a), mainly due to the
reduced laminar burning velocity and increased ame length. The ame surface area
oscillation and heat release uctuation are subsequently enhanced for the longer ame at
the higher pressure of 6 bar.
For the unstable mode, the nal frequency and amplitude of the resulting limit cycle
oscillations are now predicted. The 6 bar FDF of Case V (Figure 15a) is extended from
Au¼0.10.5 (with steps of 0.1), for frequencies 200 and 300 Hz, these falling on either
side of the instability frequency. The ame response against forcing level is shown in
Figure 18. The FDF gain falls owith Auat both frequencies with dierent trends.
A stronger saturation occurs at 300 Hz, with the gain dropping by more than 50% as
Auincreases from 0.1 to 0.5. The gain drop at 200 Hz is ,25%. This frequency depen-
dence of the rate of the gains fall-owith forcing level has been observed in previous
Figure 17. Linear stability maps of the analyzed combustor at (a) 3 bar and (b) 6 bar pressure. The
predicted thermoacoustic modes are marked by white stars on the complex fσplane (Xia et al.,
2018b).
Figure 18. (a) Gain, G, and (b) phase, φ, of the FDF at fu¼200 Hz (solid line) and 300 Hz (dash-dotted
line) for perturbation levels of Au= 0.10.5. All calculations performed at
p= 6 bar based on Case V (Xia
et al., 2018b).
COMBUSTION SCIENCE AND TECHNOLOGY 997
numerical (Han et al., 2015) and experimental (Noiray et al., 2008) studies. The FDF phase
shows an almost linearly decreasing trend with forcing level at both frequencies.
This extended FDF is then coupled with OSCILOS to predict the limit cycle frequency
and amplitude. This nonlinear prediction relies on the assumption that the timescale over
which the oscillation amplitude grows is much longer than that of the oscillation itself
(Laera et al., 2017). The frequencies and growth rates of the thermoacoustic oscillations
are predicted across forcing levels, with the zero-growth-rate state taken to correspond to
that at which the limit cycle establishes (Han et al., 2015; Laera and Camporeale, 2017;
Noiray et al., 2008).
The evolutions of frequency, f, and growth rate, σ, of the linearly unstable mode are
shown in Figure 19a over forcing level. Using linear interpolation, the limit cycle pertur-
bation level is Alc
u¼0.3565, with a frequency of flc ¼209 Hz. The latter is very close to
the measured value of 216 Hz (Figure 2b). The axial distribution of the pressure uctua-
tion amplitude, ^
p
jjð
xÞ, under a limit cycle is shown in Figure 19b. At the location where
the pressure signal was measured, x¼231 mm, a uctuation amplitude of ^
plc
¼
4970 Pa is predicted, close to the measured value of 5000 Pa (Figure 2b). The same
predictions are repeated with 4-step chemistry FDFs (Case IV), giving flc = 211 Hz, Alc
u=
0.3559 and ^
p
jj= 4940 Pa, again very close to the experimental data. In light of this
accurate limit cycle prediction, the above simulated FDFs can be considered validated.
The thermoacoustic stabilities and limit cycle predictions are known to be sensitive to the
acoustic boundary conditions, which for the present combustor are unknown as they were
not measured. We therefore investigate the sensitivity of the above predictions (with 15-step
FDFs) to small changes in the upstream and downstream acoustic boundary conditions. If
Rin is taken to have its gain changed by 10% either way, with the outlet reection coecient
unchanged, the predictions change to flc = 210 Hz, Alc
u=0.3557and ^
p
jj= 4947 Pa (for 10%
decrease) and flc = 209 Hz, Alc
u=0.3574and ^
p
jj
= 4993 Pa (for 10% increase). Similarly, if Rout
is rather to have its gain changed by 10% either way, with the inlet reection coecient
unchanged, the predictions change to flc = 205 Hz, Alc
u=0.3825and ^
p
jj= 5280 Pa (for 10%
decrease) and flc = 214 Hz, Alc
u=0.3358and ^
p
jj= 4715 Pa (for 10% increase). Hence, the
Figure 19. (a) Evolutions of frequency (f, solid line) and growth rate (σ, dash-dotted line) of the
unstable mode with Au. Arrows indicate the frequency, flc, and growth rate, σlc , of the limit cycle and
the corresponding forcing level, Alc
u. (b) Axial distribution of pressure uctuation amplitude, ^
p
jj
, when
the limit cycle occurs. Circle refers to the ^
p
jjvalue at the measurement location, x¼0.231 m (Xia
et al., 2018b).
998 Y. XIA ET AL.
predicted limit cycle has a very small dependence on the inlet reection coecient, although
it is more sensitive to the change of outlet reection coecient. An increase of Rin
jj
or
decrease of Rout
jj
are both found to reduce the value of flc but increase Alc
uand ^
p
jj
.
Conclusions
This work simulates the responses of a turbulent swirling ame to upstream perturbation
in a pressurized gas turbine combustor and used them to construct the weakly nonlinear
FDFs. Two incompressible LES solvers are used, applying two sub-grid combustion
models (PaSR and PDF) and two reaction schemes (4-step and 15-step) at two operating
pressures (3 and 6 bar). It is found that (i) the mean ow is not aected by these factors;
(ii) the PaSR model and the 15-step scheme both give a longer ame with a lower heat
release rate, although due to dierent reasons; (iii) an increase in the pressure leads to
a higher mean heat release rate. Both combustion models and the used reaction schemes
oer good accuracy for the unforced ow and ame.
The ame responses to an upstream harmonic velocity perturbation are then computed
across several perturbation frequencies and amplitudes. The constructed FDFs have some
common trends: (i) the FDF gain has two local maxima with one local minimum in
between; (ii) the FDF phase linearly decreases with frequency; (iii) an increase in the
perturbation level always reduces the gain. For a given pressure, the combustion model
and reaction scheme both have very small eects on the FDFs, regardless of the dierences
in the predicted unforced ame. The faster 4-step scheme is thus recommended for FDF
computation. A pressure increase leads to an increase in FDF gain at low frequencies but
to a drop at higher frequencies.
The simulated FDFs are nally validated by performing thermoacoustic predictions using
the low order network approach. The combustor is predicted linearly stable at 3 bar, but
unstable at 6 bar near ,231 Hz, both in agreement with the experimental data. Based on the
unstable mode, the limit cycle is predicted to occur at frequency 209 Hz with a pressure
amplitude of 4970 Pa, both matching the measured data of 216 Hz and 5000 Pa. The
sensitivity of the predicted limit cycle to the acoustic boundary conditions is also discussed.
Acknowledgments
Experimental data from DLR and nancial support from Siemens Industrial Turbomachinery Ltd.,
ERC Starting Grant ACOULOMODE, EPSRC CDT in Fluid Dynamics across Scales and Department
of Mechanical Engineering at Imperial College are all acknowledged. Access to HPC facilities at
Imperial College and via the UKs ARCHER are acknowledged. We also thank Dr Jim W. Rogerson
and Dr Ghenadie Bulat from Siemens Industrial Turbomachinery Ltd. for their contributions.
Conicts of interests
Authors Yu Xia, William P. Jones and Aimee S. Morgans have received funding from the Siemens
Industrial Turbomachinery Ltd.
COMBUSTION SCIENCE AND TECHNOLOGY 999
Funding
This work was funded by the Siemens Industrial Turbomachinery Ltd., the EPSRC Centre for
Doctoral Training (CDT) in Fluid Dynamics across Scales, the Department of Mechanical
Engineering at Imperial College London, and the European Research Council (ERC) Starting
Grant (grant No: 305410) ACOULOMODE (20132018).
ORCID
Yu Xia http://orcid.org/0000-0003-2822-9424
Davide Laera http://orcid.org/0000-0001-6370-4222
Aimee S. Morgans http://orcid.org/0000-0002-0482-9305
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1002 Y. XIA ET AL.
... Interest in thermoacoustics research dwindled in the following decades, until a new generation of low-NOx gas turbines, prone to thermoacoustic instabilities, was introduced in the early 1990s (Correa, 1993;Keller, 1995;Paschereit and Polifke, 1998;Dowling and Morgans, 2005). Rapid developments in experimental techniques and numerical capacities opened the door to a range of new strategies for studying combustion instabilities, including computational fluid dynamics (CFD) simulations (Veynante and Poinsot, 1997;Poinsot and Veynante, 2001;Polifke et al., 2001;Xia et al., 2019), low-order models (Keller, 1995;Paschereit and Polifke, 1998;Dowling and Stow, 2003;Li and Morgans, 2015;Nair and Sujith, 2015;Xia et al., 2019;Gaudron et al., 2020;Fournier et al., 2021), or hybrid methods (Kaess et al., 2008;Silva et al., 2013;Li et al., 2017;Ni et al., 2017;Merk et al., 2018). Next-generation combustors burning carbon-free fuels, such as hydrogen or ammonia, appear to show increased propensity to combustion instabilities (AEsøy et al., 2021;Beita et al., 2021;Lim et al., 2021). ...
... Interest in thermoacoustics research dwindled in the following decades, until a new generation of low-NOx gas turbines, prone to thermoacoustic instabilities, was introduced in the early 1990s (Correa, 1993;Keller, 1995;Paschereit and Polifke, 1998;Dowling and Morgans, 2005). Rapid developments in experimental techniques and numerical capacities opened the door to a range of new strategies for studying combustion instabilities, including computational fluid dynamics (CFD) simulations (Veynante and Poinsot, 1997;Poinsot and Veynante, 2001;Polifke et al., 2001;Xia et al., 2019), low-order models (Keller, 1995;Paschereit and Polifke, 1998;Dowling and Stow, 2003;Li and Morgans, 2015;Nair and Sujith, 2015;Xia et al., 2019;Gaudron et al., 2020;Fournier et al., 2021), or hybrid methods (Kaess et al., 2008;Silva et al., 2013;Li et al., 2017;Ni et al., 2017;Merk et al., 2018). Next-generation combustors burning carbon-free fuels, such as hydrogen or ammonia, appear to show increased propensity to combustion instabilities (AEsøy et al., 2021;Beita et al., 2021;Lim et al., 2021). ...
... OSCILOS can predict the thermoacoustic stability of a given burner by solving one-dimensional flow conservation equations. It has been shown to accurately predict the frequencies and growth rates of thermoacoustic modes appearing in a variety of combustors at a moderate computational cost (Han et al., 2015;Li et al., 2017;Xia et al., 2019;Gaudron et al., 2020). More information about OSCILOS, including the equations that are used to describe each type of module, can be found in Li et al. (2017) and Gaudron et al. (2021). ...
Article
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Predicting the occurrence of thermoacoustic instabilities is of major interest in a variety of engineering applications such as aircraft propulsion, power generation, and industrial heating. Predictive methodologies based on a physical approach have been developed in the past decades, but have a moderate-to-high computational cost when exploring a large number of designs. In this study, the stability prediction capabilities and computational cost of four well-established classification algorithms—the K -Nearest Neighbors, Decision Tree (DT), Random Forest (RF), and Multilayer Perceptron (MLP) algorithms—are investigated. These algorithms are trained using an in-house physics-based low-order network model tool called OSCILOS. All four algorithms are able to predict which configurations are thermoacoustically unstable with a very high accuracy and a very low runtime. Furthermore, the frequency intervals containing unstable modes for a given configuration are also accurately predicted using multilabel classification. The RF algorithm correctly predicts the overall stability and finds all frequency intervals containing unstable modes for 99.6 and 98.3% of all configurations, respectively, which makes it the most accurate algorithm when a large number of training examples is available. For smaller training sets, the MLP algorithm becomes the most accurate algorithm. The DT algorithm is found to be slightly less accurate, but can be trained extremely quickly and runs about a million times faster than a traditional physics-based low-order network model tool. These findings could be used to devise a new generation of combustor optimization tools that would run much faster than existing codes while retaining a similar accuracy.
... The actual combustor geometry deployed in gas turbines was investigated by Bulat et al. (2009) and Sadasivuni et al. (2012) using SAS methods. Xia et al. (2019Xia et al. ( , 2018bXia et al. ( , 2017 recently applied a coupled approach combining incompressible LES with a low order acoustic network solver to study the thermo-acoustically unstable Case B (6 bar). The combustor's FDF was extracted by periodically forcing the inflow velocity at different frequencies and amplitudes, while measuring the subsequent heat release response. ...
... More crucially, the HRR spectrum in Case B also shows a peak at 400 Hz, revealing the thermo-acoustic coupling between the longitudinal pressure mode and the global flame response. The perpetuation of a limit-cycle oscillation with respect to the longitudinal direction can therefore be concluded, as previously stated by Karlis et al. (2019b) and Xia et al. (2019). However, compared to these two prior works, the oscillation frequency in the current simulation is increased by a factor of two. ...
... However, compared to these two prior works, the oscillation frequency in the current simulation is increased by a factor of two. Note that the 400 Hz mode was also predicted by Xia et al. (2019), albeit with a negative growth rate. An improved boundary condition treatment, i.e. more realistic acoustic impedances, may ultimately lead to a more accurate frequency prediction. ...
Thesis
Full-text available
This thesis describes one of the first computational works to investigate the physical feedback mechanisms associated with self-excited, combustion-driven instabilities in gas turbines. For this purpose, a novel numerical method based on large eddy simulation is devised. The method (called BOFFIN) uses a fully compressible formulation to account for acoustic wave propagation and applies a transported probability density function approach for turbulence-chemistry interactions. The latter is solved by the Eulerian stochastic fields method and is complemented by two different 15-step / 19 species chemical reaction schemes. This approach is shown to be flame burning regime independent and therefore highly applicable in the context of partially premixed gas turbine combustion. Combustion instabilities are a phenomenon often encountered in the late design stages of modern gas turbine combustors. Under certain conditions, these types of instabilities can develop into sustained limit-cycle oscillations with potentially severe consequences on a combustor's operating behaviour. In order to study the various physical feedback mechanisms driving such limit-cycle oscillations, two different test cases are simulated in the present work. Firstly, the combined effects of thermo-acoustic and hydrodynamic instabilities are examined in the lab-scale PRECCINSTA model combustor. Secondly, the superposition of a longitudinal and azimuthally spinning instability mode is investigated in the industrial SGT-100 combustor. Amongst the different feedback mechanisms identified and studied in these cases are: mass flow rate and equivalence ratio oscillations, as well as hydrodynamic phenomena such as flame angle oscillations, periodic vortex shedding and a precessing vortex core. It is further demonstrated that in addition to reproducing longitudinal instability modes, the applied LES approach is capable of accounting for modes acting in the transverse direction. Overall, the findings of this research project strongly suggest that BOFFIN is a reliable and accurate method for the prediction of self-excited combustion instabilities in gas turbines.
... In subsequent studies, they added an azimuthal mean flow in the combustion chamber (Bauerheim et al., 2015) and investigated the case of symmetry-breaking due to uncertain parameters using uncertainty quantification (UQ) (Bauerheim et al., 2016). Another example is the open-source research software OSCILOS, developed at Imperial College London, which was used to study limit cycles of non-pressurised and pressurised longitudinal combustors using flame describing functions (FDF) obtained from large eddy simulations (LES) (Han et al., 2015;Xia et al., 2019). There is a limit to the complexity of geometries and mode shapes that can be modelled using a wave-based approach, making it unsuitable for our purposes. ...
Thesis
Thermoacoustic instabilities, which occur due to the feedback between confined flames and acoustics, constitute a major threat to the safe operation of gas turbines and rocket engines. These oscillations can become large enough to cause noise, vibrations or, in the worst cases, extinction of the flame or structural damage. The fact that thermoacoustic systems are sensitive to small changes to their design and operating parameters means that instabilities can appear in the late stages of the design process, requiring costly re-design. In this thesis we address the problem of finding accurately and at reduced computational cost the design changes that most stabilize a thermoacoustic system in the linear regime. We derive a Helmholtz equation with an unsteady heat release source and acoustic impedance boundary conditions. We apply the finite element method with P2 elements to discretize the weak formulation of the Helmholtz equation. We obtain a nonlinear eigenvalue problem for the complex angular frequency, ω, that we solve with the open-source computing platform FEniCS and the library for eigenvalue problems SLEPc. We then derive the formulae for the shape derivatives of the eigenvalues of the Helmholtz equation in Hadamard form for different boundary conditions and in the cases of simple and semi-simple degenerate eigenvalues. These formulae allow us to efficiently calculate the effect of arbitrary boundary perturbations on the frequency and growth rate of the thermoacoustic oscillations, by combining the direct and adjoint eigensolutions. We apply this model and analysis to an electrically heated Rijke tube and a turbulent swirl combustor in 2D. Both systems exhibit an unstable longitudinal mode. We parametrize the shapes using B-splines, calculate the shape derivatives and apply the most stabilizing changes until we make the systems stable. We then study the case of a 30kW laboratory-scale symmetric annular combustor (MICCA) with an unstable azimuthal mode (two-fold degenerate). We perform a shape sensitivity analysis considering both symmetry-breaking and symmetry-preserving boundary perturbations. The second type of changes do not cause the eigenvalues to split and are the most effective at reducing the instability. We then apply shape changes to the plenum and the combustion chamber to reduce the eigenvalue growth rates. We show how adjoint-based sensitivity analysis can be combined with a Helmholtz solver to calculate the influence of the shape of the combustor on the stability of the thermoacoustic modes. By modifying the shape accordingly, we are able to suppress the instability or at least reduce the growth rate of the oscillations. This computational method shows how to significantly alter thermoacoustic oscillations by making small geometry changes. The framework in this thesis can handle arbitrarily complex three-dimensional geometries, which could be useful for the design of industrial combustion systems.
... In this study, the Open Source Combustion Instability Low Order Simulator (OSCILOS) is used to assess the thermoacoustic stability of a large number of randomly generated configurations. OSCILOS is a low-order network model tool that has been used to successfully predict the thermoacoustic stability of different types of combustors [20,22,26,27]. The stability map corresponding to the randomly generated geometry represented in Fig. 1 as computed by OSCILOS for n = 0.98224 and τ = 3.2165 ms. ...
Conference Paper
Full-text available
Thermoacoustic instabilities are an undesirable physical phenomenon that can occur in a wide range of combustors. A well-established formalism to predict thermoacoustic stability is based on network models where the combustor is represented by a sequence of connected acoustic modules. This approach has been successfully used to predict the stability of a variety of combustors but can be relatively computationally expensive when a large number of designs is explored. One option to reduce the computational cost of predicting the thermoacoustic stability of a given configuration is to use a data-driven approach as opposed to a physics-based approach. In the former approach, a Machine Learning algorithm is first trained to discriminate between thermoacoustically stable and unstable combustors using examples generated by a (physics-based) acoustic network model. The ML model is then able to predict the thermoacoustic stability of an unknown configuration much faster than a traditional acoustic network model and with a very high accuracy. This approach has been validated in a previous study for simple combustor geometries using a classifier chain (i.e. multiple classifiers). The objective of this study is to generalise those findings by predicting the thermoacoustic stability of more complex combustor geometries using a single classifier. More advanced classification algorithms are thus required to perform that task. Two different types of Deep Neural Network architectures are tested in this study: Deep Multilayer Perceptrons with 3, 6, and 9 hidden layers, and a Convolutional Neural Network. Overall, the Convolutional Neural Network was found to be much more accurate than any of the Multilayer Perceptrons.
... Computational low-order acoustic network models have gained much popularity, as the equations for the mean flow and the acoustics can be solved sequentially [2,[5][6][7][8][9][10][11][12][13][14]. As any parametric analysis to identify stability boundaries requires consideration of many cases; analytical solutions that can characterize the acoustic behaviour for any given geometry, mean flow and acoustic boundary conditions offer significant advantages [15][16][17][18][19][20]. ...
Preprint
Full-text available
In ducts with varying cross-sectional area and sustaining a subsonic non-isentropic mean flow, the axially varying flow conditions affect the acoustic energy balance of the system. This is significant in understanding and controlling thermo-acoustic phenomena, particularly in combustors. This work aims at quantifying the acoustic energy change in such configurations, using the acoustic absorption coefficient, $\Delta$. The acoustic response of the duct to acoustic forcing is determined using an analytical model, neglecting the effect of entropy fluctuations on the acoustic field, and subsequently, $\Delta$ is estimated. The model predictions of $\Delta$ are validated using a linearised Euler equations (LEEs) solver. The model was found to be accurate for Mach numbers below $0.25$, provided the lower frequency limit set by the analytical solution is satisfied. For conically varying area ducts with linear mean temperature gradient, it was observed that $\Delta$ showed very little dependence on frequency, and that the absolute value of $\Delta$ tended to be maximised when the upstream boundary was anechoic rather than non-anechoic. More importantly, $\Delta$ was also observed to show stronger dependence on the mean temperature gradient than area gradient variation for such configurations. Further parametric and optimisation studies for $\Delta$ revealed a crucial finding that a positive mean temperature gradient, representing a heated duct caused acoustic energy absorption. Similarly, a negative mean temperature gradient, representing a cooled duct caused acoustic energy generation -- a key result of this analysis. This behaviour was shown to be consistent with a simplified analysis of the acoustic energy balance. Based on this finding, a linearly proportional reduction in acoustic energy generation was achieved by changing the mean temperature gradient.
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Future combustion power and propulsion systems may operate in premixed regime enabling reduced fuel burn and reduced pollutant emissions. The turbulent premixed regime in those future combustion systems is likely to be in the corrugated regime where modeling the flame as a thin interface propagating into the fresh gas is made possible. The flame displacement speed is thus a key quantity for turbulent combustion in this regime. This quantity is also important for combustion instabilities. Indeed, the flame displacement speed S d combined to the flow speed v determines the flame surface location by determining the flame surface speed w s. The flame surface location has shown to play a major role on combustion instabilities. Research work have also demonstrated the role of the flame displacement speed on the flame response which is used for subsequent combustion instability prediction. In this context, the derivation of flame speed models and flame transfer function models based on this quantity are required. This paper presents the theoretical derivation of flame transfer function coefficients for swirling premixed flames in this context. The derivation is based on the definition of the flame speed for turbulent flame, its perturbed form for oscillating flow, and the kinematic flame-flow speed budget. The obtained results are compared to previous literature data and discussed. The effect of the flame angle, id est the effect of the swirl number on the flame response is also investigated. The present theoretical results suggest the future requirement of comparison between detailed high-speed local flowfield measurements and numerical simulations to evaluate the flame displacement speed budget terms.
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This work simulates a laboratory-scale 3D methane/air burner, which features a bluff body stabilized, lean partially premixed flame experiencing strong limit cycle oscillations. A thin steel liner is installed around the combustion chamber, which heavily interacts with the flow field and produces large amplitude structural deformation via Fluid-Structure Interactions (FSI). An unsteady RANS approach uses the Shear Stress Transport turbulence model and a Flamelet Generated Manifold combustion model to predict the thermoacoustic oscillations in the turbulent reacting flow. The solver also has a built-in finite element Structure Model, which solves the structural governing equations simultaneously with the CFD-computed, finite volume flow equations. This way, a fully coupled, two-way FSI simulation can be performed to predict the thermoacoustic instabilities and the associated solid deformations in the burner. Overall, the predicted strongest pressure oscillation and wall displacement modes (frequency and amplitude) are all in good agreement with the experimental data across different operating conditions. The established workflow may support realistic gas turbine combustor design and prognosis.
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This article focuses on co-simulation of the coupled nonlinear dynamics of flame heat release rate, acoustics, and feedback control in an active thermoacoustic oscillation control system. This work is motivated by an extensive existing body of literature showing the potential of closed-loop active control to suppress thermoacoustic oscillation. Linear algorithms such as linear quadratic Gaussian (LQG) control are often used for thermoacoustic oscillation control, with the important caveat that flame heat release rate oscillation is often highly nonlinear. This creates a need for a tool that can co-simulate the coupled nonlinear thermoacoustic dynamics with linear control. The main contribution of this article is the development of a tool capable of co-simulating nonlinear heat release dynamics using a level-set formulation, together with linear acoustics plus a linear feedback controller. The article demonstrates this framework on an LQG-controlled wedge flame in a unidimensional Rijke tube. The nonlinear thermoacoustic model is formed in a modified level-set solver by connecting the linear acoustics to the original nonlinear flame dynamics via a velocity convection equation. This framework succeeds in preserving the nonlinearity of the thermoacoustics in the Rijke tube without the LQG control. For the flame-driven Rijke tube in this article, LQG control is successful in suppressing the nonlinear thermoacoustic oscillation. In addition, the simulation based on the framework is useful in elucidating the impact of factors such as flame location, flame temperature rise, and the timing of the onset of LQG control on the instability suppression performance and energy needs.
Article
In ducts with varying cross-sectional area and sustaining a subsonic non-isentropic mean flow, the axially varying flow conditions affect the acoustic energy balance of the system. This is significant in understanding and controlling thermo-acoustic phenomena, particularly in combustors. This work aims at quantifying the acoustic energy change in such configurations, using the acoustic absorption coefficient, Δ. The acoustic response of the duct to acoustic forcing is determined using an analytical model, neglecting the effect of entropy fluctuations on the acoustic field, and subsequently, Δ is estimated. The model predictions of Δ are validated using a linearised Euler equations (LEEs) solver. The model was found to be accurate for Mach numbers below 0.25, provided the lower frequency limit set by the analytical solution is satisfied. For conically varying area ducts with linear mean temperature gradient, it was observed that Δ showed very little dependence on frequency, and that the absolute value of Δ tended to be maximised when the upstream boundary was anechoic rather than non-anechoic. More importantly, Δ was also observed to show stronger dependence on the mean temperature gradient than area gradient variation for such configurations. Further parametric and optimisation studies for Δ revealed a crucial finding that a positive mean temperature gradient, representing a heated duct caused acoustic energy absorption. Similarly, a negative mean temperature gradient, representing a cooled duct caused acoustic energy generation – a key result of this analysis. This behaviour was shown to be consistent with a simplified analysis of the acoustic energy balance. Based on this finding, a linearly proportional reduction in acoustic energy generation was achieved by changing the mean temperature gradient.
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This article presents numerical prediction of a thermoacous-tic limit cycle in an industrial gas turbine combustor. The case corresponds to an experimental high pressure test rig equipped with the full-scale Siemens SGT-100 combustor operated at two mean pressure levels of 3 bar and 6 bar. The Flame Transfer Function (FTF) characterising the global unsteady response of the flame to velocity perturbations is obtained for both operating pressures by means of incompressible Large Eddy Simulations (LES). A linear stability analysis is then performed by coupling the FTFs with a wave-based low order thermoacoustic network solver. All the thermoacoustic modes predicted at 3 bar pressure are stable; whereas one of the modes at 6 bar is found to be unstable at a frequency of 231 Hz, which agrees with the experiments. A weakly nonlinear stability analysis is carried out by combining the Flame Describing Function (FDF) predicted by LES with the low order thermoacoustic network solver. The frequency , mode shape and velocity amplitude corresponding to the predicted limit cycle at 209 Hz are used to compute the absolute pressure fluctuation amplitude in the combustor. The numerically reconstructed amplitude is found to be reasonably close to the measured dynamics.
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In the context of combustion noise and combustion instabilities, the transport of entropy perturbations through highly simplified turbulent flows has received much recent attention. This work performs the first systematic study into the transport of entropy perturbations through a realistic gas turbine combustor flow-field, exhibiting large-scale hydrodynamic flow features in the form of swirl, separation, recirculation zones and vortex cores, these being ubiquitous in real combustor flows. The reacting flow-field is simulated using low Mach number large eddy simulations, with simulations validated by comparison to available experimental data. A generic artificial entropy source, impulsive in time and spatially localized at the flame-front location, is injected. The conservation equation describing entropy transport is simulated, superimposed on the underlying flow-field simulation. It is found that the transport of entropy perturbations is dominated by advection, with both thermal diffusion and viscous production being negligible. It is furthermore found that both the mean flow-field and the large-scale unsteady flow features contribute significantly to advective dispersion — neither can be neglected. The time-variation of entropy perturbation amplitude at combustor exit is well-modelled by a Gaussian profile, whose dispersion exceeds that corresponding to a fully-developed pipe mean flow profile roughly by a factor of three. Finally, despite the attenuation in entropy perturbation amplitude caused by advective dispersion, sufficient entropy perturbation strength is likely to remain at combustor exit for entropy noise to make a meaningful contribution at low frequencies.
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Large Eddy Simulations (LES) of a swirl-stabilized natural gas-air flame in a laboratory gas turbine combustor is performed using six different LES combustion models to provide a head-to-head comparative study. More specifically, six finite rate chemistry models, including the thickened flame model, the partially stirred reactor model, the approximate deconvolution model and the stochastic fields model have been studied. The LES predictions are compared against experimental data including velocity, temperature and major species concentrations measured using Particle Image Velocimetry (PIV), OH Planar Laser-Induced Fluorescence (OH-PLIF), OH chemiluminescence imaging and one-dimensional laser Raman scattering. Based on previous results a skeletal methane-air reaction mechanism based on the well-known Smooke and Giovangigli mechanism was used in this work. Two computational grids of about 7 and 56 million cells, respectively, are used to quantify the influence of grid resolution. The overall flow and flame structures appear similar for all LES combustion models studied and agree well with experimental still and video images. Takeno flame index and chemical explosives mode analysis suggest that the flame is premixed and resides within the thin reaction zone. The LES results show good agreement with the experimental data for the axial velocity, temperature and major species, but differences due to the choice of LES combustion model are observed and discussed. Furthermore, the intrinsic flame structure and the flame dynamics are similarly predicted by all LES combustion models examined. Within this range of models, there is no strong case for deciding which model performs the best.
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The mechanisms for nonlinear saturation of a bluff-body stabilised turbulent premixed flame are investigated using LES with the transported flame surface density (TFSD) approach to combustion modelling. The numerical simulation is based on a previous detailed experimental investigation. Results for both the unforced non-reacting and reacting flows are validated against experiment, demonstrating that the fundamental flow features and predicted flame structure are well captured. Key terms in the FSD transport equation are then used to describe the generation and destruction of flame surface area for the unforced reacting flow. In order to investigate the non-linear response of the unsteady heat release rate to acoustic forcing, four harmonically forced flames are considered having the same forcing frequency (160 Hz) but different amplitudes of 10 %, 25 %, 50 % and 64 % of the mean inlet velocity. The flame response is characterised using the Flame Describing Function (FDF). Accurate prediction of the FDF is obtained using the current approach. The computed forced flame structure matches well with the experiment, where effects of shear layer rollup and growth of the vortices on the flame can be clearly observed. Transition to nonlinearity is also observed in the computed FDF. The mechanisms leading to the saturation of the flame response in the higher amplitude case are characterised by inspecting the terms in the FSD transport equation at conditions when the integrated heat release is at its maximum and minimum, respectively. Pinch-off and flame rollup can be seen in snapshots taken at the phase angle of maximum integrated heat release. Conversely, intense vortex shedding and flame-sheet collapse around the shear-layer, as well as small-scale destruction of flame elements in the wake, can be seen in snapshots taken at the phase angle of minimum integrated heat release. The pivotal role of FSD destruction on nonlinear saturation of the flame response is confirmed through the analysis of phase-averaged terms in the FSD transport equation taken at different locations. The phase-averaged subgrid curvature term is found to concentrate in the cusps and downstream regions where flame annihilation is dominant.
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This work numerically investigates the unsteady heat release rate response of a full-scale industrial gas turbine combustor to acoustic perturbations. The combustor contains a lean technically-premixed methane/air flame. Two large eddy simulation solvers are compared, the first being the in-house code BOFFIN which employs a reduced 15-step chemical reaction mechanism; the second is based on the open-source CFD toolbox OpenFOAM and applies both 2-step and 4-step reaction mechanisms. Both are incompressible codes, exploiting the fact that the flame responds to hydrodynamic perturbations excited by the acoustics. For the unforced flow-field, the simulation results agree well with the experiments. The flame heat release rate responses are calculated by applying a harmonic forcing velocity upstream to the flame across two forcing amplitudes and eight forcing frequencies. The obtained frequency responses of flame are known as flame describing functions, which are different between the solvers used, especially on their gains. This indicates that for combustors with industrial complexity, more detailed chemistry mechanisms may be necessary. The phase lags of the flame describing functions generally decrease linearly with forcing frequency, being almost independent of the forcing amplitude and the solver used.
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This work performs a numerical study of the ORACLES combustor which contains a 1-meter-long turbulent premixed flame and sustains low frequency thermoacoustic instabilities. The flame heat release rate response to acoustic disturbances is characterized, but because the flame is long, the axial variation of the response is accounted for using an axial sequence of local flame describing functions (FDF). Low-Mach number large eddy simulations (LES) are performed using the OpenFOAM toolbox, with the turbulence-combustion interaction modelled by the Partially Stirred Reactor (PaSR) model with a global one-step propane/air reaction mechanism. LES results are validated through comparison with experimental data. The computational zone in the combustion chamber is equally split to Ns = 40 axial segments, and the flame heat release rate perturbations in each flame segment extracted and used to construct a sequence of local FDF. The dependence of this on the numbers of flame segments used is investigated.
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The present article investigates the correlation between flame macrostructures and thermoacoustic combustion instabilities in stratified swirling flames. Experiments are carried out in a laboratory scale longitudinal test rig equipped with the Beihang Axial Swirler Independently-Stratified (BASIS) burner, a novel double-swirled combustion system developed by adapting an industrial lean premixed prevaporized (LPP) combustor. At first, the flame macrostructures are investigated and discussed for various total equivalence ratios (ϕtotal) and stratification ratios (SRs). Depending on operating conditions, three different flame types are stabilized in the combustor: two attached flames comprising a stratified flame and a V-shaped flame (V-flame), as well as a lifted flame. Thermoacoustic instabilities are then investigated. The amplitude of the oscillations is found to be more sensitive to SR than the ϕtotal Large amplitude limit cycles are found for low and high values of SR, for which the V-flame and the lifted flame are observed in the combustor, respectively. The flame dynamics are also investigated using local Rayleigh index maps. It is found that for both the lifted flame and V-flame, the major driving force comes from the flame-to-wall impingement region. Coherent structures associated with flame wrinkling are found along the flame brushes of the V-flame. On the contrary, the stratified flame is found to be more thermo-acoustically stable. Finally, incompressible Large Eddy Simulations is used to obtain the flame responses to forcing at 300 Hz, which is very close to the frequencies at which limit cycle oscillations occur. The results show that the global heat release rate response of the stratified flame exhibits a significant phase shift compared to the responses of the other two flame types, and this is the most likely cause of thermoacoustic stabilization.
Article
This paper reports on the effect of pressure on the response of methane–air and propane–air swirl flames to acoustic excitation of the flow. These effects are analyzed on the basis of the flame transfer function (FTF) formalism, experimentally determined from velocity and global OH* chemiluminescence measurements at pressures up to 5 bar. In parallel, phase-locked images of OH* chemiluminescence are collected and analyzed in order to determine the associated flame dynamics. Flame transfer functions and visual flame dynamics at atmospheric pressure are found to be similar to previous studies with comparable experimental conditions. Regardless of pressure, propane flames exhibit a much larger FTF gain than methane flames. For both fuels, the effect of pressure primarily is to modify the gain response at the local maximum of the FTF, at a Strouhal number around 0.5 (176 Hz). For methane flames, this gain maximum increases monotonically with pressure, while for propane flames it increases from 1 to 3 bar and decreases from 3 to 5 bar. At this frequency and regardless of pressure, the flame motion is driven by flame vortex roll-up, suggesting that pressure affects the FTF by modifying the interaction of the flame with the vortex detached from the injector rim during a forcing period. The complex heat transfer, fluid dynamics, and combustion coupling in this configuration does not allow keeping the vortex properties constant when pressure is increased. However, the different trends of the FTF gain observed for methane and propane fuels with increasing pressure imply that intrinsic flame properties and fuel chemistry, and their variation with pressure, play an important role in controlling the response of these flames to acoustic forcing.
Article
The mechanisms for nonlinear saturation of a bluff-body stabilised turbulent premixed flame are investigated using LES with the transported flame surface density (TFSD) approach to combustion modelling. The numerical simulation is based on a previous detailed experimental investigation. Results for both the unforced non-reacting and reacting flows are validated against experiment, demonstrating that the fundamental flow features and predicted flame structure are well captured. Key terms in the FSD transport equation are then used to describe the generation and destruction of flame surface area for the unforced reacting flow. In order to investigate the non-linear response of the unsteady heat release rate to acoustic forcing, four harmonically forced flames are considered having the same forcing frequency (160 Hz) but different amplitudes of 10 %, 25 %, 50 % and 64 % of the mean inlet velocity. The flame response is characterised using the Flame Describing Function (FDF). Accurate prediction of the FDF is obtained using the current approach. The computed forced flame structure matches well with the experiment, where effects of shear layer rollup and growth of the vortices on the flame can be clearly observed. Transition to nonlinearity is also observed in the computed FDF. The mechanisms leading to the saturation of the flame response in the higher amplitude case are characterised by inspecting the terms in the FSD transport equation at conditions when the integrated heat release is at its maximum and minimum, respectively. Pinch-off and flame rollup can be seen in snapshots taken at the phase angle of maximum integrated heat release. Conversely, intense vortex shedding and flame-sheet collapse around the shear-layer, as well as small-scale destruction of flame elements in the wake, can be seen in snapshots taken at the phase angle of minimum integrated heat release. The pivotal role of FSD destruction on nonlinear saturation of the flame response is confirmed through the analysis of phase-averaged terms in the FSD transport equation taken at different locations. The phase-averaged subgrid curvature term is found to concentrate in the cusps and downstream regions where flame annihilation is dominant.