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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 flame responses in a pressurized gas turbine combustor are
predicted using numerical reacting flow 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 flow
fieldislittleaffected 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 func-
tions (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 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; flame
describing function
Introduction
Numerical prediction of thermoacoustic instability in gas turbine combustors is an
ongoing challenge. Many approaches rely on a model for the flame heat release rate
response (denoted _
q) to velocity perturbation just upstream of the flame (denoted u1),
e.g., Li and Morgans (2015); Han et al. (2015), such as the weakly nonlinear flame
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
ðÞ denotes amplitude fluctuations in the frequency domain and ðÞ 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 figures 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 flow, i.e., not able to be compressed.
COMBUSTION SCIENCE AND TECHNOLOGY
2019, VOL. 191, NOS. 5–6, 979–1002
https://doi.org/10.1080/00102202.2019.1583221
© 2019 Taylor & Francis Group, LLC
Many numerical simulations of premixed flame 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 flame 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 flow speed rather
than the speed of sound then determines the time step and thus allows much larger values,
making incompressible simulations significantly faster than fully compressible ones. With
this approach, a complete FDF for a premixed bluff-body stabilized flame was accurately
predicted (Han et al., 2015), as were the nonlinear FDFs of a swirling premixed flame (Xia
et al., 2017b) and a very long non-swirling flame (Li et al., 2017; Xia et al., 2017a) and,
recently, the linear flame response of a stratified flame (Han et al., 2018), etc.
In most flame LES studies, the chemical reaction is modeled by a simplified 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 flame was recently
investigated (Fedina et al., 2017), to the authors’knowledge there is no similar study for
a forced flame, where an accurate reproduction of the flame 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 flame (Bulat et al., 2014) or linear flame response
(Hermeth et al., 2014) at elevated pressures, representative of real combustor operations,
but none thus far have realistically accounted for the nonlinear forced flame response at
such high pressures.
The present work thus aims to numerically simulate the nonlinear flame response in
a pressurized realistic combustor, using incompressible LES. The effects of sub-grid
combustion models, reaction schemes, and operating pressures on the flame 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 filtered 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
ðÞ denotes density-weighted filtering, 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 diffusion. 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 fluxes 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 filtered
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 offer sufficient accuracy. For the
filtered 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 filtered 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 filtered
e_
ωαnow appears as closed in Eq. (3), which is solved by the Eulerian stochastic field
method (Jones and Navarro-Martinez, 2007). The joint PDF is represented as an ensemble
of stochastic fields. These fields have no direct physical significance, but instead represent
an equivalent stochastic system that has the same one-point PDF as that given by Eq. (3).
Any filtered scalar can now be obtained by averaging over all the corresponding stochastic
fields.
For the PaSR model, a different modeling concept is used. The flow 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”,filling with non-reacting
sub-grid scale flow structures. The burnt products from part (i) are mixed with the
“surrounding flow”due to turbulence, giving the final 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 different
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 flame behavior.
The mixing time is defined 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 fluctuation. 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,isfixed as
0.8. The filtered chemical reaction rate, e_
ωα, is modeled by κas (Han et al., 2015)
e_
ωα’κ_
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 field) 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 field 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 inflow. 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
18,400–120,000 and Mach number in the range 0.02–0.29. 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 fluctuation, 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
jj’5000 Pa.
In order to simulate this combustor, we consider a simplified 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 flame is restricted to the combustion chamber. A velocity inlet
condition consistent with the measured inflow 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 “nonreflecting”outflow
Figure 2. (a) Time-signal of normalized pressure fluctuation, 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 “nonreflecting”outflow
condition is a specificartificial boundary condition, designed to neglect the diffusion effects
near the outlet and assumes that the outflow is purely advective (i.e., “nonreflective”)
(Boströmm, 2015); it inhibits the occurrence of negative velocities at an outflow boundary.
All solid boundaries are defined 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 refinement is applied to the swirler,
premixing chamber and front part of the combustion chamber, to better resolve the flame
behavior.
The numerical schemes used by the present LES are as follows. In BOFFIN, a second-
order central difference 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 difference method), coupled with a Sweby limiter (Sweby, 1984)
improving the stability in regions with rapidly changing gradients but adding numerical
diffusivity. Both BOFFIN and OpenFOAM apply an implicit second-order Crank–
Nicolson scheme for temporal discretisation. A small fixed time step of 5 107sis
used to ensure the Courant–Friedrichs–Lewy number is always below 0.3.
Unforced LES results and validation
The effects of sub-grid combustion model, reaction chemistry and the pressure on the
unforced mean flow and flame behavior are first presented. For systematic comparisons,
five LES cases with different modeling assumptions and operating conditions are defined,
as listed in Table 1.
The effects 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-figure 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 offtoward 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 effects of different 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 effect on the unforced flow field.
For the mean temperature and heat release rate fields, 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 diffusive 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
differences in the two combustion models utilized in the two codes.
Figure 5 compares the vertical y-profiles of velocity, temperature and mixture fraction
between Cases I and II at four axial locations. The horizontal axis of each sub-figure is thus
split into four segments, each refers to a profile and has the same data range for the plotted
variable. At x= 18.7 mm, very little difference is observed between the PaSR and PDF models
for either mean or root-mean-square (rms) velocities, confirming that the combustion model
has almost no effect on the flow field. Both cases accurately capture the experimental data.
Further downstream (x38.7 mm), good agreement between the LES and experiments is
maintained for all velocity profiles, otherthan for a small mismatch between the two LES cases
near the centreline of the vrms profiles (Figure 5f). This may be due to the difference in the
CVC’s axial length predicted by the two combustion models.
For the temperature, T, both combustion models correctly capture its mean and rms
profiles at the first location of x= 18.7 mm. Differences between the LES and measure-
ments become more pronounced further downstream. Both LES cases fail to predict the
temperature profile 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 profile (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 effect 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 flame temperature, Ta, are compared to those obtained by a full 325-step
GRI-Mech 3.0 scheme (Gregory et al., 2018). The effect of the constant Schmidt and unity
Lewis number assumptions is evaluated by comparing the LES predictions of Suand Tawith
Figure 5. Vertical y-profiles of (a–d) mean and (e–h) rms flow 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 I–III. The Cantera simplified
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 flame 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 effect 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 flow field 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 flame, thus
leading to a higher heat release rate.
Figure 6. Laminar burning velocity (Su) and adiabatic flame 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 flames with BOFFIN using accurate transport properties reproduce the
GRI results.
COMBUSTION SCIENCE AND TECHNOLOGY 987
Figure 8 compares the vertical y-profiles 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), confirming 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
differences in the predicted unforced flame (see Figure. 7b,c).
Finally, the effect 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 affect the adiabatic flame 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 flow variables are compared in
Figure 9 between Cases IV and V. Both schemes yield similar mean velocity fields,
indicating the small effect of reaction chemistry on the flow. Due to the faster burning
velocity, the 4-step scheme gives a shorter low temperature zone and a higher flame 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 differences 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 profiles 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
sufficiently accurate for the flow field, 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 flame heat release responses in the analyzed combustor are now computed.
The above simulated unforced flame 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-profiles of (a–d) mean and (e–h) rms flow 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
u1’5 m/s the mean inflow 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 10–12 cycles. The gain, G, and phase, φ,of
the resulting FDFs are plotted in Figure 11–15 for different 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 first
maximum occurs at f’200–300 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 flow
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 flame response, which is inversely proportional
to the mean flow velocity,
u, and is little affected 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-offof 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 effects 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-profiles of (a–d) mean and (e–h) rms flow 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 effect of stochastic field number used by the PDF model is also studied based on Case
III. The FDF gain and phase computed by eight stochastic fields are shown in Figure 12,and
are very close to those obtained with only one field at both perturbation levels for two
frequencies (fu¼600 and 800 Hz). This implies that one field is sufficient for the PDF
model to correctly capture the FDFs in the present combustor. Overall, the effect of sub-
grid combustion model on the forced flame 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 field (solid line with
circles) and spot checks for 8 stochastic fields (squares). (Top) gain G; (bottom) phase φ. (a) Au= 0.1; (b)
Au= 0.2.
992 Y. XIA ET AL.
Second, the effect 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 gain’s mismatch below 10%. This
good match is also achieved at 6 bar. Although the mean flame shapes differ between the
two reaction schemes at both pressures, with the more detailed 15-step scheme slightly
more accurate, this difference barely affects the FDFs. The simpler 4-step scheme appears
to be sufficient for FDF computation, this providing sufficient accuracy at reduced
computational cost.
To further investigate the effect of reaction scheme, the unsteady flame 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 field, _
qðx;y;zÞ, are compared within a forcing period, chosen to correspond to the
normalized fluctuation, h_
qi0=h_
qi, being minimum, zero and maximum (with h_
qi¼ððð_
qdV
and Vthe domain volume). Although the unforced flame shapes slightly differ between
the two reaction schemes, the flame’s oscillatory behavior around its mean position is
generally similar. The flame 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 fluctuation is
mainly associated with the variation in the flame surface area. Cases II (Figure 14a) and III
(Figure 14b) show that a change in reaction scheme can modify the response of the flame
structure, even at the same pressure. Although the more detailed 15-step scheme yields
a slightly longer flame throughout the forcing period (Figure 14b), consistent with the
unforced flame in Figure 7c,differences in the detailed flame structure may still result in
a similar flame surface area for the two reaction schemes, explaining the similarity in the
FDF gains shown in Figure 13.
Finally, the effect 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 field, _
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 G’1.2 is found at f= 200 Hz, while at 6 bar it
increases to G’1.4 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 different combustor geometry, flame shape, fuel type, etc.,
and their equivalence ratio was adjusted with pressure to ensure the same mean flame
length for all pressures. In the present work, however, a fixed ϕis used at both pressures,
giving slightly different mean flame lengths, which may lead to different coupling between
the flame 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 flame surface
oscillation at such high perturbations levels. For the FDF phase, the effect of the pressure
is always negligible, consistent with the experiment (Sabatino et al., 2018).
Further insights into the pressure effect are provided by the flame dynamics for Cases
III and V (see Figure. 14b,Figure. 14c). The predicted flame structures are similar,
probably due to the same reaction scheme used. The flame surface area oscillation can
be compared by examining the axial flame length. Increasing the pressure slightly
increases the flame 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 flame length and thus a longer flame
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 flame 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 flow 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
flow is accounted for, with the mean flow 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 flow 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 different strengths. These wave strengths are tracked
between modules using linearized flow 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 defined by the pressure reflection coefficients, R, denoting
the strength ratio of the reflected 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 reflection coefficient, Rin, is little affected by the
operating pressure, and it increases in magnitude from Rin
jj
=0–0.15 and varies in phase
between ffRin =−0.7πand −0.55πover frequency f=0–1000 Hz. In contrast, the network
outlet is defined 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 ffRout from πto 0.84πacross
0–1000 Hz (Xia et al., 2018b).
Since the present flame has a much shorter axial extent (,100 mm) than the dominant
acoustic wavelengths (,1–4 m), the flame zone is represented by an infinitely thin “flame
sheet”at 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 flame is accounted
for using the linearized flow conservation equations across the flame sheet (Dowling,
1997). To account for the effect of acoustic waves on the flame response, a flame 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 satisfied, are identified within OSCILOS using
a“shooting 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
f’231 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 flame response.
For (i), the acoustic wave strength is determined by the geometry, speed of sound, and
boundary conditions, none of which are affected by the pressure in this work. Thus the
flame response is the main source of the stability change. The FDF gain near the frequency
Figure 16. The simplified 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 flame length. The flame surface area
oscillation and heat release fluctuation are subsequently enhanced for the longer flame at
the higher pressure of 6 bar.
For the unstable mode, the final frequency and amplitude of the resulting limit cycle
oscillations are now predicted. The 6 bar FDF of Case V (Figure 15a) is extended from
Au¼0.1–0.5 (with steps of 0.1), for frequencies 200 and 300 Hz, these falling on either
side of the instability frequency. The flame response against forcing level is shown in
Figure 18. The FDF gain falls offwith Auat both frequencies with different 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 gain’s fall-offwith 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= 0.1–0.5. 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 fluctua-
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 fluctuation 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 reflection coefficient
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 reflection coefficient
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 fluctuation 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 reflection coefficient, although
it is more sensitive to the change of outlet reflection coefficient. 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 flame 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 flow is not affected by these factors;
(ii) the PaSR model and the 15-step scheme both give a longer flame with a lower heat
release rate, although due to different reasons; (iii) an increase in the pressure leads to
a higher mean heat release rate. Both combustion models and the used reaction schemes
offer good accuracy for the unforced flow and flame.
The flame 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 effects on the FDFs, regardless of the differences
in the predicted unforced flame. 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 finally 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 financial 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 UK’s ARCHER are acknowledged. We also thank Dr Jim W. Rogerson
and Dr Ghenadie Bulat from Siemens Industrial Turbomachinery Ltd. for their contributions.
Conflicts 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 (2013–2018).
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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