Content uploaded by Yu Xia

Author content

All content in this area was uploaded by Yu Xia on Jun 13, 2019

Content may be subject to copyright.

Full Terms & Conditions of access and use can be found at

https://www.tandfonline.com/action/journalInformation?journalCode=gcst20

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.

Submit your article to this journal

Article views: 170

View Crossmark data

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

References

Abou-Taouk, A., Farcy, B., Domingo, P., Vervisch, L., Sadasivuni, S., and Eriksson, L.E. 2016.

Optimized reduced chemistry and molecular transport for large eddy simulation of partially

premixed combustion in a gas turbine. Combust. Sci. Technol.,188,21–39. doi:10.1080/

00102202.2015.1074574

Bauerheim, M., Staﬀelbach, G., Worth, N.A., Dawson, J., Gicquel, L.Y., and Poinsot, T. 2015.

Sensitivity of LES-based harmonic ﬂame response model for turbulent swirled ﬂames and impact

on the stability of azimuthal modes. Proc. Combust. Instit.,35(3), 3355–3363. doi:10.1016/j.

proci.2014.07.021

Boströmm, E. 2015. Investigation of outﬂow boundary conditions for convection-dominated

incompressible ﬂuid ﬂows in a spectral element framework. Master’s thesis. SCI School of

Engineering Sciences, KTH Royal Institute of Technology, Stockholm, Sweden. https://www.

diva-portal.org/smash/get/diva2:804993/FULLTEXT01.pdf

Bulat, G. 2012. Large eddy simulations of reacting swirling ﬂows in an industrial burner. Doctoral

dissertation. Imperial College London, London. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.

ethos.739539.

Bulat, G., Jones, W.P., and Marquis, A.J. 2014. NO and CO formation in an industrial gas-turbine

combustion chamber using LES with the Eulerian sub-grid PDF method. Combust. Flame,161

(7), 1804–1825. doi:10.1016/j.combustﬂame.2013.12.028

Bulat, G., Jones, W.P., and Navarro-Martinez, S. 2015. Large eddy simulations of isothermal

conﬁned swirling ﬂow in an industrial gas-turbine. Int. J. Heat Fluid Fl.,51,50–64.

doi:10.1016/j.ijheatﬂuidﬂow.2014.10.028

Chomiak, J., and Karlsson, A. 1996. Flame liftoﬀin diesel sprays. Symp. (Int.) Combust.,26(2),

2557–2564. doi:10.1016/S0082-0784(96)80088-9

Dowling, A.P. 1997.Nonlinear self-excited oscillations of a ducted ﬂame. J. Fluid Mech.,346,

271–290. doi:10.1017/S0022112097006484

Febrer, G., Yang, Z., and McGuirk, J. 2011. A hybrid approach for coupling of acoustic wave eﬀects

and incompressible LES of reacting ﬂows. The 47th AIAA/ASME/SAE/ASEE Joint Propulsion

Conference & Exhibit, San Diego, California, U.S.A. Paper No. AIAA 2011-6127.

Fedina, E., Fureby, C., Bulat, G., and Meier, W. 2017. Assessment of ﬁnite rate chemistry large eddy

simulation combustion models. Flow, Turb. Combust.,99(2), 385–409. doi:10.1007/s10494-017-

9823-0

Fedina, E., Fureby, C., Bulat, G., and Meier, W. 2017. Assessment of ﬁnite rate chemistry large eddy

simulation combustion models. ﬂow, turb. Combust.,99(2),385–409. doi: 10.1007/s10494-017-

9823-0

Fureby, C., Nordin-Bates, K., Petterson, K., Bresson, A., and Sabelnikov, V. 2015. A computational

study of supersonic combustion in strut injector and hypermixer ﬂow ﬁelds. Proc. Combust.

Instit.,35(2), 2127–2135. doi:10.1016/j.proci.2014.06.113

1000 Y. XIA ET AL.

Goodwin, D.G., Moﬀat, H.K., and Speth, R.L. 2014. Cantera: an object-oriented software toolkit for

chemical kinetics, thermodynamics, and transport processes. http://www.cantera.org. (Version

2.1.2)

Gregory, P.S., Golden, D.M., Frenklach, M., Moriarty, N.W., Eiteneer, B., Goldenberg, M., …

Qin, Z. 2018. GRI-Mech 3.0 (Tech. Rep.). UC Berkeley. http://combustion.berkeley.edu/gri-

mech/

Han, X., Laera, D., Morgans, A.S., Sung, C.J., Hui, X., and Lin, Y.Z. 2018. Flame macrostructures

and thermoacoustic instabilities in stratiﬁed swirling ﬂames. Proc. Comb. Inst., In Press.

doi:10.1016/j.proci.2018.06.147

Han, X., Li, J., and Morgans, A.S. 2015. Prediction of combustion instability limit cycle oscillations

by combining ﬂame describing function simulations with a thermoacoustic network model.

Combust. Flame,162(10), 3632–3647. doi:10.1016/j.combustﬂame.2015.06.020

Han, X., and Morgans, A.S. 2015. Simulation of the ﬂame describing function of a turbulent

premixed ﬂame using an open-source LES solver. Combust. Flame,162(5), 1778–1792.

doi:10.1016/j.combustﬂame.2014.11.039

Hermeth, S., Staﬀelbach, G., Gicquel, L.Y., Anisimov, V., Cirigliano, C., and Poinsot, T. 2014.

Bistable swirled ﬂames and inﬂuence on ﬂame transfer functions. Combust. Flame,161(1),

184–196. doi:10.1016/j.combustﬂame.2013.07.022

Jones, W.P., Marquis, A.J., and Prasad, V.N. 2012. LES of a turbulent premixed swirl burner using

the Eulerian stochastic ﬁeld method. Combust. Flame,159(10), 3079–3095. doi:10.1016/j.

combustﬂame.2012.04.008

Jones, W.P., and Navarro-Martinez, S. 2007 August. Large eddy simulation of autoignition with

asubgrid probability density function method. Combust. Flame,150(3), 170–187. doi:10.1016/j.

combustﬂame.2007.04.003

Jones, W.P., and Prasad, V.N. 2010. Large eddy simulation of the Sandia ﬂame series (D, E and F)

using the Eulerian stochastic ﬁeld method. Combust. Flame,157, 1621–1636. doi:10.1016/j.

combustﬂame.2010.05.010

Krediet, H., Beck, C., Krebs, W., and Kok, J. 2013. Saturation mechanism of the heat release

response of a premixed swirl ﬂame using LES. Proc. Combust. Instit.,34(1), 1223–1230.

doi:10.1016/j.proci.2012.06.140

Laera, D., Campa, G., and Camporeale, S.M. 2017.Aﬁnite element method for a weakly nonlinear

dynamic analysis and bifurcation tracking of thermo-acoustic instability in longitudinal and

annular combustors. Appl. Energy,187, 216–227. doi:10.1016/j.apenergy.2016.10.124

Laera, D., and Camporeale, S.M. 2017. A weakly nonlinear approach based on a distributed ﬂame

describing function to study the combustion dynamics of a full-scale lean-premixed swirled

burner. J. Eng. Gas Turb. Power,139(9), 091501. doi:10.1115/1.4036010

Lee, C.Y., and Cant, S. 2017. LES of nonlinear saturation in forced turbulent premixed ﬂames. Flow

Turb. Combust.,99(2), 461–486. doi:10.1007/s10494-017-9811-4

Li, J., and Morgans, A.S. 2015. Time domain simulations of nonlinear thermoacoustic behaviour in

a simple combustor using a wave-based approach. J. Sound Vib.,346, 345–360. doi:10.1016/j.

jsv.2015.01.032

Li, J., Xia, Y., Morgans, A.S., and Han, X. 2017. Numerical prediction of combustion instability limit

cycle oscillations for a combustor with a long ﬂame. Combust. Flame,185,2

8–43. doi:10.1016/j.

combustﬂame.2017.06.018

National Institute of Standards and Technology (NIST). 1998.NIST-JANAF Thermochemical Tables,

4th ed., NIST Standard Reference Database 13, NIST, U. S. Department of Commerce.

Gaithersburg, Maryland, U.S.A. doi:10.18434/T42S31

Noiray, N., Durox, D., Schuller, T., and Candel, S. 2008. A uniﬁed framework for nonlinear

combustion instability analysis based on the ﬂame describing function. J. Fluid Mech.,615,

139–167. doi:10.1017/S0022112008003613

Palies, P., Durox, D., Schuller, T., and Candel, S. 2010. The combined dynamics of swirler and

turbulent premixed swirling ﬂames. Combust. Flame,157(9), 1698–1717. doi:10.1016/j.

combustﬂame.2010.02.011

COMBUSTION SCIENCE AND TECHNOLOGY 1001

Piomelli, U., and Liu, J. 1995. Large-eddy simulation of rotating channel ﬂows using a localized

dynamic model. Phys. Fluids,7(4), 839–848. doi:10.1063/1.868607

Poinsot, T., and Veynante, D. 2005.Theoretical and Numerical Combustion, 2nd ed. RT Edwards,

Inc., Philadelphia, PA. p. 34.

Sabatino, F.D., Guiberti, T.F., Boyette, W.R., Roberts, W.L., Moeck, J.P., and Lacoste, D.A. 2018.

Eﬀect of pressure on the transfer functions of premixed methane and propane swirl ﬂames.

Combust. Flame,193, 272–282. doi:10.1016/j.combustﬂame.2018.03.011

Sabelnikov, V., and Fureby, C. 2013. LES combustion modeling for high Re ﬂames using a

multi-phase analogy. Combust. Flame,160(1), 83–96. doi:10.1016/j.combustﬂame.2012.09.008

Stopper, U., Aigner, M., Ax, H., Meier, W., Sadanandan, R., StöHr, M., and Bonaldo, A. 2010. PIV,

2D-LIF and 1D-Raman measurements of ﬂow ﬁeld, composition and temperature in premixed gas

turbine ﬂames. Exp. Therm Fluid Sci.,34(3), 396–403. doi:10.1016/j.expthermﬂusci.2009.10.012

Stopper, U., Meier, W., Sadanandan, R., StöHr, M., Aigner, M., and Bulat, G. 2013. Experimental

study of industrial gas turbine ﬂames including quantiﬁcation of pressure inﬂuence on ﬂow ﬁeld,

fuel/air premixing and ﬂame shape. Combust. Flame,160(10), 2103–2118. doi:10.1016/j.

combustﬂame.2013.04.005

Sung, C.J., Law, C.K., and Chen, J.-Y. 2001. Augmented reduced mechanisms for NO emission in

methane oxidation. Combust. Flame,125(1), 906–919. doi:10.1016/S0010-2180(00)00248-0

Sweby, P.K. 1984. High resolution schemes using ﬂux limiters for hyperbolic conservation laws.

SIAM J. Numer. Anal.,21(5), 995–1011. doi:10.1137/0721062

Weller, H.G., Tabor, G., Jasak, H., and Fureby, C. 1998. A tensorial approach to computational

continuum mechanics using object-oriented techniques. Comput. Phys.,12(6), 620–631.

doi:10.1063/1.168744

Xia, Y., Duran, I., Morgans, A.S., and Han, X. 2016. Dispersion of entropy waves advecting through

combustor chambers. Proceedings of the 23rd International Congress on Sound & Vibration

(ICSV23), Athens, Greece.

Xia, Y., Duran, I., Morgans, A.S., and Han, X. 2018a. Dispersion of entropy perturbations trans-

porting through an industrial gas turbine combustor. Flow, Turb. Combust.,100(2), 481–502.

doi:10.1007/s10494-017-9854-6

Xia, Y., Laera, D., Morgans, A.S., Jones, W.P., and Rogerson, J.W. 2018b. Thermoacoustic limit

cycle predictions of a pressurised longitudinal industrial gas turbine combustor. ASME Turbo

Expo. Paper No. GT2018-75146.

Xia, Y., Li, J., Morgans, A.S., and Han, X. 2017a. Computation of local ﬂame describing functions

for thermoacoustic oscillations in a combustor with a long ﬂame. Proceedings of the 8th European

Combustion Meeting (ECM8), Dubrovnik, Croatia.

Xia, Y., Morgans, A.S., Jones, W.P., and Han, X. 2017b. Simulating ﬂame response to acoustic

excitation for an industrial gas turbine combustor. Proceedings of the 24th International Congress

on Sound & Vibration (ICSV24), London, UK.

Xia, Y., Morgans, A.S., Jones, W.P., Rogerson, J.W., Bulat, G., and Han, X. 2017c. Predicting

thermoacoustic instability in an industrial gas turbine combustor: combining a low order net-

work model with ﬂame LES. ASME Turbo Expo. Paper No. GT2017-63247.

1002 Y. XIA ET AL.