G-equation modelling ofthermo-acoustic oscillations ofpartially-premixed flames

Thermoacoustic Instabilities in GasTurbines and Rocket Engines: Industrymeets AcademiaMay 30 – June 02, 2016Munich, GermanyPaper No.: GTRE-009c©The Author(s) 2016

Bernhard Semlitsch1, Alessandro Orchini1, Ann P. Dowling1 and Matthew P. Juniper1

AbstractSelf-excited oscillations of unsteady heat release and acoustic waves cannot be entirely avoided for all operatingconditions of gas turbines equipped with lean-burn technology. Nevertheless, safe operability can be guaranteed as longas thermo-acoustic limit cycles are diminished to tolerable amplitudes. Numerical simulations aid combustor design toavoid and reduce thermo-acoustic oscillations. Non-linear heat release rate estimation and its modelling are essentialfor the prediction of saturation amplitudes of limit cycles. The heat release dynamics of arbitrarily-complex flames canbe approximated by a Flame Describing Function (FDF). To calculate an FDF, a wide range of forcing amplitudes andfrequencies needs to be considered. For this reason, we present a computationally inexpensive level-set approach,which accounts for equivalence ratio perturbations while modelling the velocity fluctuations analytically. The influenceof flame parameters and modelling approaches on flame describing functions and time delay coefficient distributionsare discussed in detail. The numerically-obtained flame describing functions are compared with experimental data andused in an acoustic network model for limit cycle prediction. A reasonable agreement of the heat release gain andlimit cycle frequency is achieved. However, the phase decay is over-predicted, which can be attributed to the fact thatturbulence is neglected. The lack of turbulent dispersion causes a highly correlated heat release response, which isartificial.

KeywordsThermo-acoustic instability, Non-linear combustion modelling, Flame describing function,G-equation, Equivalence ratio

Introduction

A gas turbine combustor can behave as a self-sustainedacoustic resonator. Acoustic waves generated by unsteadycombustion are reflected back towards their origin,provoking unsteady heat release. If high heat release occurssufficiently in phase with high pressure then periodicor quasi-periodic oscillations can be self-sustained. Theseoscillations can lead to undesirable consequences, such asflame blow off, flashback, or even fatigue of the combustor.Lean-burn flames are particularly receptive to such thermo-acoustic oscillations. Nevertheless, lean burn flames areincreasingly popular in combustion applications becausethey have low NOx emissions, which helps to comply withincreasingly stringent legislation.

Ideally, potentially damaging thermo-acoustic oscillationswill be identified at an early stage in the design process.In order to be practical, this requires numerically-efficienttechniques so that a wide range of designs can be screened.

In gas turbines, thermo-acoustic oscillations usuallycannot be avoided for all operating conditions. However, safeoperability can be assured as long as self-excited oscillationsare restricted to tolerable amplitudes. In such scenarios,flame describing functions F(FA, f), being dependent onforcing amplitudes FA and frequencies f , can be used tocharacterise the non-linear flame response (in terms of heatrelease rate q̇) when forced with a harmonic perturbation,

q̇′

q̇= F(FA, f)

u′BuB

= g · eiϕu′B

uB, (1)

where g is the gain, ϕ is the phase delay, and uB is theimposed velocity disturbance just upstream of the burnerorifice. Overbars and primes denote mean averaged andFourier transformed quantities, respectively. The FDF can bedetermined using experimental measurements or numericalsimulations and expressed as the sum of time delays, ∆τ ,with coefficients kn:

F(FA, f) =q̇′/q̇

u′B/uB≈

N∑n=1

kn(FA)e−i2πfn∆τ . (2)

These parameters, kn and ∆τ , relate the amplification ofthe heat release response to a disturbance generated byacoustic waves at the fuel injection location or flame base andbeing retarded by the disturbance convection time. The timedelay constants can be found by kn(FA) = g(n∆τ, FA)∆τ ,where g(t, FA) is the impulse response of the flame, whichis calculated by applying the one-sided inverse Fouriertransform on the flame describing function. In order toreconstruct the flame describing function by the sum of timedelays, the calculated time delay constants are substituted

1University of Cambridge, Department of Engineering, UK

Corresponding author:Bernhard Semlitsch, University of Cambridge, Department of Engineer-ing, Trumpington Street, Cambridge, Cambridgeshire, CB2 1PZ, UK.Email: bs564@cam.ac.uk

2 Thermoacoustic Instabilities in Gas Turbines and Rocket Engines: Industry meets Academia

into Eq. 2,

F(FA, f) =∫ ∞

0

g(t, FA)e−iωtdt

≈N∑n=1

g(n∆τ, FA)e−iωn∆τ∆τ . (3)

The number of time delays, N , determines the resolution ofthe FDF over a given frequency range. The predicted limitcycle amplitude become more accurate as N increases. Thisis particularly evident for the complex flame shapes foundin industrial applications (see e.g. Macquisten et al. 2014).Highly accurate but numerically expensive methods restrictthe assessment to a few forcing amplitudes and frequencies,which might lead to an inappropriate overall descriptionof the flame dynamics. Therefore inexpensive numericalmethods that can provide sufficiently accurate results topredict flame describing function at many forcing amplitudesand frequencies are desired.

The impulse response or sum of time delay coefficientsreveal the time scales of the processes leading to a heatrelease modulation. Huber and Polifke (2009a,b), Komarekand Polifke (2010) and Blumenthal et al. (2013) were ableto distinguish the contributions of different mechanismsinfluencing the heat release by analysing their contributionto the impulse response. It has been suggested by Huber andPolifke (2009a,b) that gains larger than unity can only beachieved with a combination of positive and negative timedelay coefficients. Blumenthal et al. (2013) indicated thata negative contribution, corresponding to a phase shift, canbe caused by restoration effects for premixed flames. Thesefindings highlight that physical insight can be gained byanalysis of the time delay coefficients.

Dowling (1999) demonstrated that flame describingfunctions obtained from a kinematic flame model can beused to predict thermo-acoustic instabilities appropriately.Such kinematic flame models based on level-set methodsare computationally inexpensive, because only essentialfeatures are incorporated. Methods presented by Graham andDowling (2011) can be used even for relatively complexflame shapes. The drawback of Graham and Dowling’simplementation, which was to track two flame surfaces in aLagrangian sense, is the handling of flame-pinching eventsand the distinction of burned and unburned regions. Dueto these shortcomings, the numerical effort becomes highfor strongly wrinkled flames, which limits the investigationrange.

Numerical predictions based on kinetic flame responserepresentations rely on parameters and simplifying assump-tions. For example, the flow-field is commonly idealised andconvective effects in the supply duct are neglected. Theseassumptions demand quantitative validation with experi-ments to prove reliability. Lieuwen (2003) highlights that thecrucial drawback of reduced-order models is still the lack ofquantitative validation with experiments.

The main limitations of level set methods modellingthe non-linear heat release generated by flames have beenthe fragility to flame-pinching events and the lack ofvalidation. In order to overcome these problems, we presenta level-set method to estimate flame describing functions,which can robustly handle flame pinching and sophisticated

flames, such as an M-shaped flame. The computationalefficiency of this approach allows evaluation of the flameresponse for numerous forcing amplitudes and frequencies.Therefore, the complexity of the heat release response canbe integrated and analysed for very many time delays andcoefficients in acoustic network simulations. We describe themethodology to obtain the heat release rate from kinematicflame simulations in detail and emphasise the validation ofthe approach. The obtained results under different modellingassumptions are compared with available experimental data.Consequences of the modelling assumptions are pointed outand the results are examined critically.

Methodology - Flame front trackingAssuming that chemical reactions between fuel and oxidizeroccur instantaneously in a thin sheet, the flame front can beregarded as discontinuity dividing reactants and products.The kinematics and surface alterations of the flame frontdetermine its dynamic heat release response. The thin flameburns normal to its front at the flame speed, which isdetermined by local chemical reaction rates.

A scalar G field indicating the shortest distance to theflame front can be defined, where an arbitrary fixed valueG0, e.g. zero, can be used to identify the flame front location.Values ofG larger or smaller thanG0 represent the unburnedmixture or products, respectively. By defining the scalar G-field to be smooth and continuous in the spatial domain,the normal n pointing into the unburned mixture can becalculated by−∇G/|∇G|. Therefore, the convective motionof the flame front is dictated by the local flame speed sL andthe local flow velocity u at which the reactants arrive. Thematerial derivative of the level-set describes the flame frontmovement by the so-called G-equation,

DG

Dt=∂G

∂t+

(u− sL

∇G|∇G|

)· ∇G = 0 . (4)

With the integration of the implicitG field, a fully non-lineardescription of the flame dynamics is obtained.

The flame speed sL depends on the local equivalence ratioφ and curvature effects,

sL = c0s0L(1− L κ) = c0c1φc2ec3(φ−c4)

2

(1− L κ) , (5)

where s0L is the burning speed of a flat flame, L is theMarkstein length, and κ = ∇ · n is the local flame curvature.The fuel in the present investigation is ethylene and the flamespeed, s0L, as a function of the local equivalence ratio, φ,can be modelled by fitted factors ci for the specific reactantmedia. An additional constant c0 is introduced (accordinglyto the work by Graham and Dowling (2011)) to alter thelocation of the numerically predicted flame front to that ofthe experimentally-observed turbulent flame brush.

Table 1. Coefficients used for Eq. 5 to represent ethylene.

c0 (−) c1 (m/s) c2 (−) c3 (−) c4 (−)1.51314 1.32176 3.11023 1.72307 0.36196

The unsteady heat release rate of an axisymmetric flamecan be calculated by spatial integration along its front

Semlitsch et al. 3

z

r

∅ 70

∅ 35

∅ 25

∅ 8

z inj

45◦

fuel injection

distance (mm)rB,in 12.5rB,out 17.5rfh 4.0zsr 40.0air supply inlet

fuel injection

open outlet

8050

200

5046

584

580

∅ 35

∅ 100

Figure 1. Sketch of the burner developed by Balachandran(2005). All dimensions are in millimetres.

represented by the iso-surface G = 0,

q̇(t) = 2πρc0s0L∆hR

∫∫(1− Lκ)|∇G|δ(G)rdrdz . (6)

The local enthalpy release ∆hR(φ) of the reaction perunit mass is defined as ∆hR = 3.2 · 106(min(1, φ))/(1 +0.067 φ) Jkg−1. A fast Fourier transform of the heatrelease rate fluctuations q̇(t)− q̇ is performed, in whichthe contribution at the fundamental forcing frequency isextracted to calculate the gain and phase decay as definedin Eq. 1.

Geometry and setupThe geometry of the laboratory combustor illustrated inFig. 1 is selected for these simulations. It was experimentallyinvestigated by Balachandran (2005). Therefore, comparisonof the numerical results with experiments is permitted. Theburner consists of a cylindrical pipe with varying cross-section and centred bluff body. At the combustion chamberexit, the flow discharges into quiescent ambient conditions.Self excited thermo-acoustic instabilities were not observedexperimentally with a combustion chamber height of 80mm. Therefore, it was extended to 350 mm to investigatelimit cycles.

The fuel is injected through tiny holes around thecircumference of the centre body, which are located 55mm upstream of the burner orifice. The air is suppliedfar upstream, where the inlet pressure and temperature isassumed to be at ambient conditions in the simulations, i.e.101325 Pa and 299 K, respectively. The imposed inlet airmass-flow rate (ṁair = 0.005417 kg/s) is estimated such

that a velocity of 10 m/s is reached at the burner exit. Theemployed flame properties are chosen according to the fuelethylene. The fuel mass flow rate is specified in terms of theequivalence ratio φ, which is a parameter in the simulations.The fuel is assumed to mix uniformly with the air at theinjection location.

Flow fieldAnalytical flow and disturbance models meet the require-ments for the present purpose and are computationally effi-cient. The axisymmetric potential flow model developed byGraham and Dowling (2011) is used, in which a sphericalpoint source is placed 40 mm upstream of the burner orifice.This source location zsr determines the directivity of theflow and spread angle of an annular jet exhausting the port.The distance to the spherical source can be expressed asξ =

√r2 + (z − zsr)2. The radial velocity component in the

spherical coordinate system can be written in terms of thevelocity potential P as uξ = ∂P/∂ξ. In order to conservecontinuity and to ensure irrotationality, the velocity potentialneeds to satisfy ∇2P = 0. Solutions have the form P =Auξ

n, where particularly n = −1 satisfies these conditions.With P = Au/

√r2 + (z − zsr)2, the velocity components

in the cylindrical coordinate system can be written as,

ur =−Au r√

(r2 + (z − zsr)2)3(7)

and

uz =−Au (z − zsr)√(r2 + (z − zsr)2)3

. (8)

The potential amplitude Au can be determined such that adesired mean velocity uB at the burner orifice is obtained,

Au = −uB

((rB,out + rB,in

2

)2+ z2sr

), (9)

where rB,out and rB,in are the outer and inner radii ofthe burner orifice, respectively. A velocity perturbation withsinusoidal form can be included into the velocity model bywriting,

Au = Au · (1 + FA sin(2π ft)) , (10)

where FA is the forcing amplitude and f is the forcingfrequency.

Equivalence ratio perturbations The fuel is injected at alocation zinj upstream of the burner orifice. Because themixing process is not considered upstream of the flame,the reactant mixture distribution is passively convected bythe mean flow from the injection to the reaction zone. Theinjected fuel mass flow rate is not significantly influencedby acoustic disturbances and therefore equivalence ratioperturbations can be assumed to be inversely proportional tolocal air velocity fluctuations at the injector nozzle,

φ′inj =φ(

1 + FAu′injuinj

) . (11)The equivalence ratio oscillation φ′B passing the burnerorifice at time t can be expressed as φ′B(r, z, t) =

4 Thermoacoustic Instabilities in Gas Turbines and Rocket Engines: Industry meets Academia

φ′inj(r, z, t− τ), where τ represents the retarded time, whichis required to convect the disturbance from the injectionlocation to the burner orifice. From this point on, theconvection of the equivalence ratio oscillation is simulated.

The mean retarded time can be calculated as τ =∫ z0u−1dz, where u is the mean velocity in the duct. Figure 1

illustrates the cross-sectional change of the duct due to theflame holder, which results in a flow acceleration. Therefore,the time delays are calculated for the straight annular ductand the section constricted by the bluff body separately.The convection distance past the flame holder is |zfh| =∣∣∣ rB,in−rfhtan 45◦ ∣∣∣, while the convection distance in the straightannular section is |zan| = |zinj − zfh|. (rfh is the innerradius of the annular duct.) Since the flow can be assumedto be incompressible in the ducts, the flow velocities areproportional to cross-sectional variations,

uan = uB(r2B,out − r2B,in)(r2B,out − r2fh)

(12)

and

ufh = uB(r2B,out − r2B,in)

(r2B,out − z2 tan2(45◦)). (13)

The retarded times for the two duct sections are,

τan =r2B,out − r2fh

uB(r2B,out − r2B,in)

(|zinj | −

(rB,in − rfh)tan 45◦

)(14)

and

τfh =rB,in − rfh

uB(r2B,out − r2B,in) tan(45◦)·

·(r2B,out −

(rB,in − rfh)2

3

). (15)

ImplementationThe G-field, φ-field, boundary conditions, and domainextensions and discretisations are specified in an input file.The extent of the computational domain is outlined inFig. 2. A regular two-dimensional grid of 441× 441 nodesis employed for the numerical discretisation exploiting theaxisymmetry of the M-shaped flame, where a grid sensitivityanalysis was performed. The numerical mesh starts below theburner orifice in order to facilitate the implementation of theflame boundary conditions at the lip. A strip at the centrelineis excluded to reduce the computational effort.

The flow-field can be calculated directly by the analyticalexpression and is therefore not provided at initialisation.Nevertheless, a uniform equivalence ratio field (set to themean equivalence ratio φ) and a smooth initial G-fieldsatisfying the boundary conditions need to be provided. Thesteady-state flame evolving without perturbations is used asinitial condition for every operating condition.

Executing an iteration, the flow field and velocityoscillations are calculated. The equivalence ratio fluctuationsare convected throughout the domain, where a fifth orderweighted essentially non-oscillatory WENO scheme is usedfor spatial discretisation and an implicit third-order Runge-Kutta total variation diminishing scheme is employed fortime integration.

z

r

flame frontflame tube

combustorwall

bluff body

open outlet

reactant inlet

Figure 2. Sketch of the tubes surrounding the flame front,where the orange tube represents |G| ≤ β and the yellow tubeβ < |G| ≤ γ. (The grid indicates the extent of thetwo-dimensional axisymmetric computational domain.)

Equation 4 shows that the convection of the G-field isonly dependent on the local flow velocity and flame speed.Therefore, only the near-field surrounding the flame frontneeds to be considered, which is illustrated in Fig. 2 by anorange and a yellow region surrounding the red colouredflame. These represent the two concentric tubes around thezero level-set with constant radii 0 < β < γ from the flamefront, which are calculated initially and re-established aftereach iteration. ∗ The G-field is only convected in the interiorof the narrower tube as a signed distance function, whereasconstant extremal values are assigned beyond the outer tube.A smooth transition is computed in-between the inner andouter tube. This can be expressed as,

∂G

∂t+ c(G)

(u + sL

∇G|∇G|

)· ∇G = 0 , (16)

where

c(G) =

1 if |G| ≤ β

(|G| − γ)2 (2|G|+ γ − 3β)(γ − β)3

if β < |G| ≤ γ

0 if |G| > γ(17)

∗Enough information around the flame front needs to be considered tocompute the spatial gradient of the G-field smoothly. The thickness of theflame tubes have been chosen with reference to the computational cell size,where β = 3 cells and γ = β + 3 cells.

Semlitsch et al. 5

The spatial and time derivatives are computed with the samediscretisation schemes as the equivalence ratio fluctuations,which were mentioned in the preceding paragraph. This so-called narrow band level-set method has been described inmore detail by Sethian (2001) and Peng et al. (1999).

Sharp gradients can evolve during the simulationadvancement, as e.g. with pinching events. In order to keepthe G-field smooth, re-initialisation steps are performedHartmann et al. (2010), which consists essentially of solvingthe Eikonal equation |∇G| = 1. This procedure is performedeach time step before a gradient calculation of the updatedG-field is required.

The computation of the heat release rate requires theintegration at a level-set G = 0. Since the flame tubes areused to track the flame front, the flame tubes are rebuiltwith the updated information of the G-field after the coreprocesses of the simulation have been performed. A high-order algorithm is used for integration of the flame surface,which is accomplished by using a one step adaptive meshrefinement, subdividing each cell into five node points.

The unsteady heat release and flow velocity are theessential quantities to construct the flame describing functionand are therefore saved every time step. The requiredstatistical accuracy governs the simulation duration. Dueto the modelling assumptions in the present simulations(absence of sources generating stochastic fluctuations),the heat release rate cycles evolve identically except fornegligible errors caused by numerical uncertainty. Therefore,the computation duration can be restricted to the heat releaseresponse of the flame on one complete forcing cycle (afterthe transient). Nonetheless, repeatability has been ensuredfor the presented data by simulating at least eight fullforcing cycles. Only data acquired after the initial transientis considered for the post-processing.

radial coordinate r (m)

dista

ncedownstre

am

from

burn

erz(m

)

−0.03 −0.02 −0.01 0 0.01 0.02 0.030

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

L = −0.0050mL = −0.0025mL = 0.0000mL = 0.0025mL = 0.0050mL = 0.0075mL = 0.0100mL = 0.0125mL = 0.0150m

Figure 3. Comparison of the unperturbed flame shape fordifferent Markstein lengths at a mean equivalence ratio φ of0.55 with an instantaneous OH PLIF image (Balachandran2005, pp.72).

Validation, comparison to experimental dataThe simulation results are compared to previously computeddata by Graham and Dowling (2011) and experimentalmeasurements obtained by Balachandran (2005) in order toinvestigate the validity and uncertainty of the approach.

Firstly, the flame shape relies on many parameters, suchas the velocity model, flame speed, and Markstein lengths,and is compared to an experimental image of the flamein Fig. 3. The directivity of the steady flame is estimatedconsistently and the best agreement in terms of flame lengthtakes place when flame curvature effects are neglected.However, unsteady flow leads to a wrinkled flame shape evenwithout forcing in the experimental setup. These features arenot represented in the potential flow model, which accountsonly for forced flow disturbances. A perfect comparison withrespect to prior simulation data (Graham and Dowling 2011)using a different modelling formulation was achieved.

Flame describing functions have been measured experi-mentally in the range of 20 to 400 Hz and are compared tothe present simulation results in Fig. 4. The prediction of the

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6

q̇′ /q̇/

u′/u(-)

FA = 10% GrahamFA = 10%FA = 20%FA = 30%FA = 10% Exper imentFA = 20% Exper imentFA = 30% Exper iment

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1

frequency (Hz)

angle

ϕ(rad/π)

Figure 4. For three forcing aptitudes, the flame describingfunctions F(FA, ω) obtained with numerical and experimentalmethods are compared for a mean equivalence ratio φ of 0.55without considering flame curvature effects.

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

u ′/u (-)

q̇′/q̇(-)

40 Hz

160 Hz

G-equ.G-equ. curvature corr.G-equ. φ profileG-equ. turb. profileG-equ. random fluct.CH∗OH∗FSD

Figure 5. The numerically predicted saturation behaviour of theheat release as function of forcing amplitude is compared withexperiments.

6 Thermoacoustic Instabilities in Gas Turbines and Rocket Engines: Industry meets Academia

0 0.01 0.02 0.03 0.04 0.05−1

0

1

2

3

4

5

6

time (s)

q̇norm(-)

0 0.01 0.02 0.03 0.04 0.05−1

0

1

2

3

4

5

6

time (s)

q̇norm(-)

0 0.01 0.02 0.03 0.04 0.05−1

0

1

2

3

4

5

6

time (s)

q̇norm(-)

(a) FA = 8% (b) FA = 17% (c) FA = 25%

0 0.01 0.02 0.03 0.04 0.05−1

0

1

2

3

4

5

6

time (s)

q̇norm(-)

0 0.01 0.02 0.03 0.04 0.05−1

0

1

2

3

4

5

6

time (s)

q̇norm(-)

0 0.01 0.02 0.03 0.04 0.05−1

0

1

2

3

4

5

6

time (s)

q̇norm(-)

(d) FA = 36% (e) FA = 45% (f) FA = 54%

Figure 6. Comparison of the numerically estimated normalised heat release rate q̇norm (red) for different forcing amplitudes withexperimentally measured data (black) (see Balachandran 2005, pp.176) for a mean equivalence ratio φ of 0.55 and a forcingfrequency of 40 Hz. The grey dashed lines indicate the flame curvature effects. (Negative heat release rate values are possible dueto the normalisation chosen by Balachandran (2005); q̇norm = 2 · q̇(t)/q̇ − 1.)

gain agrees reasonably with the measurements, consideringthe simplicity of the velocity model. Nevertheless, the phasedecay is very rapid with the present approach at a forcingamplitude of 10% prior to flame pinching. For higher forcingamplitudes, the phase lag decays less steeply, but remainsover-predicted compared with the experimentally evaluatedbehaviour.

Results by Graham and Dowling (2011) are highlighted inFig. 4 in order to illustrate the differences between Grahamand Dowling’s linearised G-equation model and the presentapproach. In particular, the over-prediction of the phase lagis reduced at high frequencies (> 150 Hz).

The saturation behaviour as a function of the forcingamplitude has been measured for two forcing frequenciesand is compared to the numerically-predicted saturationamplitudes in Fig. 5. For low forcing frequencies (< 100Hz), good agreement is achieved at low forcing amplitudes.However, the heat release response is overestimated at highforcing amplitudes. An oscillatory behaviour of the heatrelease rate as a function of the forcing amplitude arises forhigher forcing frequencies (> 100 Hz).

General trends of the instantaneous measured heat releaserates (see Balachandran 2005, pp.176) are captured withthe numerical approach, as shown in Fig. 6 for differentforcing amplitudes at a frequency of 40Hz. The simulationspredict, in contrast to the experiments, higher maximaand less heat release inbetween peaks. This discrepancycan be related to the neglected flow recirculation andthe temperature distribution used in the computations.The experimental measurements exhibit, in contrast tothe simulations, significant cycle-to-cycle variations, whichmight be induced due to unsteady flow features. Further,dispersive effects translating the locations of the peaks can beobserved in the experimentally measured heat release, whichmay be caused by large-scale turbulence.

Flame describing functions

Firstly, we clarify the individual impact of the simplistic flowand perturbation models on the estimated flame describingfunction, which is illustrated in Fig. 7. Consideringonly velocity perturbations (i.e. no equivalence ratioperturbations), Fig. 7(a), the highest gain occurs at zerofrequency and decays rapidly with small regular oscillationsat approximately every 80 Hz. The velocity perturbationamplitudes depend linearly on the forcing amplitude (seeEq. 10) so the gain is constant with amplitude.

Considering only equivalence ratio perturbations (i.e. novelocity perturbations in the combustion chamber), a moreelaborate gain pattern can be observed, as shown in Fig. 7(b). Dominant features are the high magnitude peaks atlow forcing frequencies, a linear regime at low forcingamplitudes and a regime exposing different heat releasecharacteristics at high forcing amplitudes when the localequivalence ratio can become close to the stoichiometricratio. At intermediate forcing amplitudes, an inclined gaininterference pattern can be observed.

Combining the effects of velocity and equivalence ratioperturbations causes essentially the same gain pattern asthat without velocity perturbations, but with an additionalhorizontal interference pattern (which is notable especially athigh forcing amplitudes). Thus, velocity perturbations affectthe gain magnitude indirectly (with the present velocitymodel) by influencing equivalence ratio perturbations withrespect to the flame front rather than generating heat releaseoscillations directly. The frequency of this interference effectis governed by the phase relation between velocity andequivalence ratio perturbations. Figure 7 (d) shows thatshifting the injection location closer to the burner orifice andaltering thereby the phase relation decreases the frequency ofthis horizontal interference pattern.

Semlitsch et al. 7

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uen

cy(H

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Figure 7. The flame describing function gain and the time delay coefficients for a mean equivalence ratio φ of 0.55 are illustratedfor different modelling procedures; (a) only the velocity perturbations (the gain is upscaled by a factor of four for this case), (b) onlyequivalence ratio perturbations, (c) velocity and equivalence ratio perturbations, and (d) velocity and equivalence ratio perturbationswith fuel injection location at zinj = 27.5 mm (closer to the burner) are considered. The number of time delay coefficients is 800,where ∆τ is 0.05 ms.

The time delay coefficients (as defined in Eq. 2) enhancephysical interpretation of the Flame Describing Function.These show the amplitude of the heat release at a range oftimes after the reference velocity signal. From the convectiontime, the heat release location can be inferred, which can alsobe compared with images of the flame. In figure Fig. 7 (a), thehigher values, which occur around τ = 0.012, correspond tothe propagation time of the velocity disturbances to the flametip. In this case, which has no equivalence ratio perturbations,the heat release modulation is generated mostly due to theflame tip motion caused by velocity perturbations. Due to thedefinition of the flame describing function in Eq. 1, a positivecontribution due to velocity perturbations is expected.

For stiff fuel injection systems and quasi-steadyforced oscillations, equivalence ratio perturbations andair stream fluctuations are proportional to each other,φ′/φ ∼ −u′inj/uinj , where the index inj references thevelocity u to the injection location. Because the velocityperturbations in the supply duct are caused by acousticwaves, the velocity fluctuations propagate instantly to theburner orifice, i.e. u′inj/uinj ≈ u′B/uB . Thus, the heatrelease rate response provoked by velocity and equivalenceperturbations are out of phase, which manifests itself as signchange in the time delay coefficients. Therefore, a horizontalborder separates positive and negative coefficients at a timedelay of approximately 0.01 s in Fig. 7 (b), which representsthe intrinsic time delay accounting for the equivalence ratiofluctuation convection from the fuel injection location to theflame base. This intrinsic time delay is altered by a shift ofthe fuel injection location, as shown in Fig. 7 (d).

The curved border visible in the time delay frames (bottomframes) of Fig. 7(b–d) indicates the propagation time of theequivalence ratio oscillations from the injector to the firstflame-pinching event, which can be seen in Fig. 9. This flamepinching leads to the disappearance of the high equivalenceratio perturbations that cause this event. Therefore, onlylow equivalence ratio perturbations remain on the flamefront beyond this propagation time. These are in phase withthe velocity perturbations and contribute positively to thecoefficients at later time delays.

For very low forcing amplitudes, the heat release ratemodulation is concentrated at the flame tip independently ofthe forcing frequency, because relatively small equivalenceratio perturbations do not lead to flame pinching andare therefore not completely burned before reaching theflame tip. Another way to look at this is that the timedelay coefficients of one forcing amplitude represent anintegration over all forcing frequencies, which results in ahigh time delay coefficient associated with the equivalenceratio perturbations modulating the heat release rate at theflame tip.

When equivalence ratio fluctuations are present, it is worthnoting that the velocity perturbations also contribute to thetime delay coefficients. Elevated positive values are notablein Figs. 7 (c) and (d) (as compared with Fig. 7 (b)) attime delays lower that 0.01 s, which mimic the negativeshape with the curved border caused by the equivalence ratioperturbations at later time delays. This demonstrates thatvelocity perturbations affect indirectly the heat release ratewhen equivalence ratio perturbations are present.

8 Thermoacoustic Instabilities in Gas Turbines and Rocket Engines: Industry meets Academia

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Figure 8. Flame describing function gains and time delay constants for different mean equivalence ratios.

Gain distribution The mean velocity and equivalenceratio define the overall distribution of the heat releaserate amplification as functions of forcing amplitude andfrequency, which is discussed further in the followingparagraphs. The evolution for different equivalence ratiosis shown in Fig. 8, where the fuel injection location isunchanged. The horizontal interference pattern caused bythe velocity perturbations can be observed for all meanequivalence ratios appearing at the same forcing frequencies.

Dominant high-amplitude peaks appear at low forcingfrequencies (50 to 150 Hz), at the half wavelength ofthe equivalence ratio oscillation corresponding to the flamelength. Therefore, maximal oscillations in the unsteadyheat release are induced by the alternating appearingequivalence ratio fluctuations. For higher mean equivalenceratios, the flame length becomes shortened and therefore thefrequency at which this peak occurs is increased. Further, thisdominant peak is widened over a larger forcing frequencyrange for higher mean equivalence ratios. The horizontalinterference pattern caused by the velocity fluctuations splitsthis dominant gain elevation at low frequencies for highermean equivalence ratios (φ > 0.7).

At low forcing amplitudes, the equivalence ratioperturbations are too small to cause flame pinching andthe foremost modulation of heat release is generated byequivalence ratio perturbations at the flame tip. When thewavelength of the equivalence ratio perturbations on theflame front is a multiple of the flame length, a highequivalence ratio perturbation disappears at the flame tipand a high equivalence ratio perturbation arises at the flamebase. These effects balance in terms of heat release rategeneration, which evolves therefore relatively constant overa forcing cycle. In contrast, when the high equivalenceratio perturbations on the flame front form odd multiplesof one half, a maximal modulation of the unsteady heat

release is induced. This leads to high gain responses and aregular spaced pattern can be observed in this linear regimeat low forcing amplitudes. The importance of individualequivalence ratio maxima is reduced with the number ofextrema existing on the flame front and this regime becomesnarrower with increasing forcing frequency.

The flame speed is low for mean equivalence ratios below0.50. Therefore, the flame is impractically elongated andpenetrates the combustor walls. Hence, the importance ofthe heat release modulation at the flame tip diminishes.Equivalence ratio fluctuations generate a higher heat releaserate response, when the wavelength matches the geometricalconstrains of the burner. At higher forcing amplitudes, theflame speed increases and the flame becomes short enoughthat the fronts interact before penetration.

Interesting to note is the manifestation of the wallinteraction in terms of time delay coefficients, where thehigh peak value due to flame tip modulation is replaced by abifurcated value distribution.

An inclined interference pattern generated by equivalenceratio fluctuations and flame pinching can be observed inFig. 8, which is illustrated quantitatively in Fig. 9 by showingthe heat release rate over one forcing cycle at constantforcing amplitude. Abrupt alterations of the unsteady heatrelease can only be provoked by events in which the flamefront disappears suddenly, e.g. flame/wall interactions orpinching effects. Thus, flame pinching is the mechanismgenerating the cusps visible in Fig. 9. On the other hand, theappearance of equivalence ratio patterns at the flame base andtheir propagation on the flame front changes the heat releaserate continuously.

The inclined interference pattern continues beyond thelinear regime, since these patterns are generated byprincipally the same mechanism. However, flame pinchingoccurs within the inclined interference pattern region, but

Semlitsch et al. 9

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not in the linear regime. Figure 9 (2) shows that a maximalinstantaneous heat release is generated when a maximalnumber of high equivalence ratio perturbations are burningon the flame front. The flame is pinching and a highequivalence ratio perturbation is convecting on the flamebase. By the instant shown in Fig. 9 (3), the high equivalenceratio perturbation causing the pinch-off is burned and a lowequivalence ratio perturbation is propagating at the flamebase. This situation causes a very low heat release rate.Therefore, a maximal heat release modulation is reached.The inverse effect takes place (see the heat release rate at 240Hz) when a flame pinching occurs while a low equivalenceratio perturbation is propagating at the flame base.

The inclined interference pattern is caused by theinteraction of at least two equivalence ratio perturbationmaxima. The number of multiple equivalence ratioperturbation maxima that can propagate on the flamesurface is smaller for higher forcing amplitudes and meanequivalence ratios because flame pinching and therefore thedisappearance of peak equivalence ratio perturbations occurafter shorter convection times. Hence, the forcing frequencyneeds to be higher for such events to occur as compared withlower forcing amplitudes or mean equivalence ratios, whichcause the inclination of this interference pattern.

Local equivalence ratios can become close to andeven larger than unity for high forcing amplitudes.Towards stoichiometric combustion, the flame speed andlocal enthalpy release alterations decrease for increasingequivalence ratios and the flame speed exhibits reducesbeyond this limit. Since these two mechanisms govern thegain amplification caused by equivalence ratio perturbations,the gain pattern reduces beyond this point where theequivalence ratio impact saturates and the gain decaysrapidly thereafter. This is even more visible in the time delaycoefficient distribution. As can be seen in Fig. 8, the minimaltime span until flame pinching occurs is independent of themean equivalence ratio. However, the forcing amplitude atwhich it takes place changes with the mean equivalence ratio.At higher forcing amplitudes than this point, the time delaycoefficient amplitudes decay rapidly.

The impact of mean velocity alterations is similarbut opposite to the impact of mean equivalence ratiomodifications. The mean velocity affects the flame length

and thereby the propagation distance of an equivalence ratioperturbation until the flame pinches.

Further modelling The high amplification of the heat releaserate oscillation at low forcing amplitudes generated dueto correlated motion of the flame tip is unrealistic fora turbulent flame. Several additional modelling strategiescan be employed to overcome this problem. The impactof perturbation convection in the supply duct, stochasticfluctuations at the flame and flame curvature effects areaddressed here.

Applying a radially non-uniform equivalence ratioperturbation can be caused by non-equally distributed fuelinjection, which reduces symmetry of the flame setup.Thereby, the coherence of the flame response at the flametip and flame-pinching events is reduced. This causes asignificant reduction of the gain, as shown in Fig. 10 (a),where a cosine distribution† has been employed. Figure 5illustrates that the impact on the gain is seen especially atlarge forcing amplitudes.

The convention in the duct distorts the equivalenceratio perturbation profile discharging at the burner orifice.Incorporating the convection with a turbulent flow profile(Okiishi 1965) into the modelling approach‡, the boundarylayers delay the arrival of the equivalence ratio perturbationsat the flame front. Therefore, the intrinsic time delay isincreased, as can be seen in Fig. 10 (b). Further, thedistribution of equivalence ratio perturbations leads to aflame elongation, which causes a shift of the inclinedinterference gain pattern towards higher forcing frequencies.The coherence of the heat release response and oscillationsdue to the inclined gain interference pattern are reduced,which is more evidently visible in Fig. 5. It was observed thatthe impact on the gain is highly dependent on the particular

†A cosine distributed profile between 0 and π/2 with higher magnitudes atthe inner radius is initially chosen such that the inner equivalence ratio valuecorresponds to the equivalence ratio perturbation amplitude calculated byEq. 11 and the outer value is the mean equivalence ratio. Further, the profileis scaled in order to conserve the mean of the equivalence ratio perturbationamplitude.‡The numerical domain is extended to incorporate the entire supply duct,where the same discretisation scheme is employed as in the combustordomain to convect the equivalence ratio perturbations as passive scalar withthe imposed profile.

10 Thermoacoustic Instabilities in Gas Turbines and Rocket Engines: Industry meets Academia

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turbulent flow profile imposed. A laminar flow profile caneven increase the gain.

Stochastic fluctuations (of 10% of the mean velocity)imposed on the top-hat flow in the supply duct each timestep disperse the equivalence ratio perturbations. Therefore,the gain is reduced as shown in Fig. 10 (c), which can beespecially noted at high forcing frequencies.

Locally high curvature of the flame front causes theflame speed to increase, which occurs especially at theflame tip and with flame-pinching events. Flame wrinklingis damped when the curvature effects are taken into account,

which occurs foremost at high forcing frequencies. At verylow forcing amplitudes, the heat release rate modulation isprimarily generated at the flame tip (with the present velocitymodel). Hence, it is expected that the heat release gain isdispersed in these regions when curvature corrections areconsidered, which can be observed in Fig. 10 (a).

Stochastic fluctuations can be imposed frequently (alteredeach time step) or infrequently (imposed over several timesteps) in the combustion chamber, where the fluctuationamplitude has been assumed to increase with the radialdistance from the burner orifice and reach 10% of the mean

Semlitsch et al. 11

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Figure 11. Logarithmic boundary condition error at the outlet as a function of frequency and nonlinear oscillation amplitude at theflame, where the equivalence ratio φ is 0.55 and the combustor length is 350 mm. Modes with positive growth rate are indicated ascircles. The reference case with time delay modelling is shown in (a). The impact of an equivalence ratio perturbation and turbulentflow profile is shown in (b) and (c), respectively. Flame curvature effects are considered in (d). The effect of stochastic flowfluctuations in the combustion chamber and in the supply duct is plotted in (e) and (f), respectively.

velocity at the flame tip. Frequent stochastic fluctuationsdisperse the gain and cause a similar gain pattern as obtainedconsidering flame curvature effects. It is worth noting thatthe linear regime is extended to higher forcing amplitudes.

Infrequent stochastic fluctuations imposed in the combus-tion chamber additionally amplify the gain, since the velocityfluctuations are not contained in the reference velocity uB .As shown in Fig. 10 (c), these infrequent stochastic fluctua-tions overlay the gain generation at frequencies beyond thefirst gain minima. This manifests as a high and low peak atthe intrinsic time delay indicating that the addition acts ashydrodynamic source.

Limit cycle predictionInformation-rich flame describing functions have beenobtained with the aim of predicting thermo-acoustic limitcycles more accurately. An acoustic network approach isused for limit-cycle prediction, which has been described indetail by Stow and Dowling (2004). Essentially, the thermo-acoustic problem is separated into propagation-like sections,which are linked by jump conditions. The heat releaseresponse of the flame represents one of these jump conditionsand is modelled by the sum of time delays approachdescribed in Eq. 2. The governing equations are decomposedinto mean and perturbation contributions, while the latterare reformulated in terms of characteristic waves. Theseperturbations are propagated through the network, startingfrom known inlet conditions towards the outlet conditions.The state of the wave perturbations will not always satisfythe outlet boundary condition and therefore this perturbation

exhibits a positive or negative growth rate. When the outletboundary condition error vanishes for a wave perturbationwith a non-zero amplitude and zero growth rate, a limitcycle is found. The limit cycle is stable when the growthrate decays with increased forcing amplitude and is unstableotherwise.

Many unstable thermo-acoustic modes with positivegrowth rates can be observed in Fig. 11 (a) at low forcingamplitudes when flame curvature effects are neglected.These arise due to the high gain estimated in the linearregime and the steep phase decay. Nonetheless, a limit cycleat a frequency of approximately 350 Hz is predicted, whichis in agreement with the findings by Balachandran (2005),who quotes a frequency of 348 Hz. However, the growthrate of this mode oscillates due to the inclined interferencepattern. An unstable mode manifests at 60 Hz, which hasa long wavelength. Accounting for the room as part of thepiping network leads to a significant reduction of the growthrate for this mode.

When the equivalence ratio perturbation is non-uniform,Fig. 11 (b) shows that the estimated boundary condition errordistribution is similar to that for a uniform equivalence ratioperturbation. However, the limit cycles establish at lowerforcing amplitudes due to the reduced gain.

Modelling the supply duct convection with a turbulentflow profile leads to a slight frequency shift of the limitcycle to 345 Hz and increases its amplitude (see Fig. 11(c)). The reduced gain oscillations observed in Fig. 5 let theunstable mode appear nearly continuously over the forcingamplitudes.

12 Thermoacoustic Instabilities in Gas Turbines and Rocket Engines: Industry meets Academia

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Considering flame curvature effects does not change theestimated limit cycle frequency (as shown in Fig. 11 (d)),but does change its amplitude. Further, fewer unstable modesare seen with low oscillation aptitudes, although they remainunstable until higher amplitudes. These differences can beexplained by comparing Fig. 7 (c) and Fig. 10 (d) morethoroughly. It can be noted that the gain decays rapidly to lowvalues at low oscillation amplitudes when flame curvatureeffects are not considered. In contrast, the gain is initiallylower for low oscillation amplitudes but remains higheruntil higher amplitudes, due to flame curvature effects.Further, the inclined interference pattern shifts towardshigher oscillation amplitudes accounting for flame curvatureeffect, which leads to the increase of the predicted limit cycleamplitude.

Frequent stochastic fluctuations, which are imposed eachtime step, generally reduce the gain. Figure 11 (e) showsthat this reduction leads to fewer unstable modes whenthe stochastic fluctuations are imposed in the combustionchamber. When fluctuations are imposed in the supply duct(see Fig. 11 (f)), the distribution of modes with positivegrowth rates remains similar to the distribution withoutcurvature corrections. Also, the inclined interference patternis shifted to higher forcing amplitudes with this approach,which translates limit cycles to higher amplitudes.

Unsteady heat release oscillations can only add energy tothe acoustic field when they are sufficiently in-phase withthe pressure oscillations. With a steep phase decay of theflame describing function, this condition is easily met overa range of frequencies. Figure 12 shows the predicted phasebehaviour of the flame describing functions consideringflame curvature effects, an equivalence ratio profile, aturbulent flow profile in the supply duct, and stochasticfluctuations in the combustion chamber. It can be noted thatthe phase decay is less steep when a turbulent flow profileis employed to convect the equivalence ratio perturbations inthe supply duct. At approximately 350 Hz, the differencesin the phase lag are small between the modelling approachesand the dominant limit cycle is predicted at this frequencywith all flame describing functions. However, the predicted

limit cycle amplitudes vary significantly depending on themodelling approaches used.

DiscussionFigure 13 summarises the challenges of modelling imper-fectly premixed flames with complex shapes by comparingthe time delay coefficients of experimental data and two G-equation models. The time delay coefficients of the exper-imentally obtained data by Balachandran (2005) exhibit anarrow high-amplitude peak at small time delays. Orchiniand Juniper (2015) performed G-equation simulations of thesame burner geometry, where only the large-scale flow struc-tures were modelled and the influence of equivalence ratioperturbations neglected. Figure 13 reveals that the initial highpeak is similarly represented as within the experimental data,while the large negative time delay coefficients are absent.This indicates that this first peak corresponds to heat releasemodulation caused by large-scale flow structures, which isnot present in the estimation with the present model.

Equivalence ratio perturbations, which are the focusof this study, manifest themselves as negative coefficientdistribution retarded by convection in the supply duct tohigher time delays. The estimated amplitudes of the negativetime delay coefficients are similar to those contained inthe experimental data. Nevertheless, sharp transitions frompositive to negative and negative to positive time delaycoefficients are predicted with the present approach, whichcorrespond to the convection time of equivalence ratioperturbations travelling in the supply duct and the durationof equivalence ratio perturbations propagating on the flamefront until the first pinching occurs, respectively. In contrast,the experimental data set exhibits a smooth transition.

Sharp time delay coefficient alterations lead to a steepphase decay, whereas smooth distributions result in aflat phase decay (Blumenthal et al. 2013). All modellingapproaches estimate the phase response to be too steep. Itseems therefore that this remains a significant challenge forlow-order flame models.

Figure 3 shows that even the unforced flame shape isdistorted by unsteady flow structures. Further, Fig. 6 revealsthat the instantaneous heat release of the experimental flameexhibits cycle to cycle variations due to unsteady flow. Thusthe heat release does not take place at the same location everycycle. Instead there are small-scale stochastic fluctuationsand the response is dispersed. Sattelmayer (2000) stressesthe importance of turbulent dispersion and the resultantimpulse response distribution over time delays on limitcycle prediction. Shin and Lieuwen (2013) argue similarlyand describe how stochastic flow fluctuations, on average,smooth wiggles of a premixed flame by modifying thelinearised G-equation formulation. Accounting for small-scale stochastic fluctuations or flame curvature effectssmooths the heat release response at the flame tip, especiallyat low forcing amplitudes. However, this does not improvethe comparison to experimental data at a forcing amplitudeof 10%.

Another modelling challenge is to deal with theartificially abrupt coefficient transition at the intrinsic timedelay corresponding to the equivalence ratio perturbationpropagation in the supply duct, which is between 0.01 and

Semlitsch et al. 13

0 0.01 0.02 0.03 0.04−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

time delay (s)

timedelayco

efficent(-)

Orchini et al.experimentwith curvatureno curvaturewith turb. profilestoc. fluctuations

Figure 13. Comparison of time delay coefficients obtained fromsimulation data, experimental data by Balachandran (2005),and data by Orchini and Juniper (2015). The mean equivalenceratio φ is 0.55 and the forcing amplitude FA is 10%.

0.015 s in Fig. 13. With the present approach (neglectinglarge-scale flow structures), a smooth transition in the timedelay coefficients can only be obtained by modelling thedispersion and convection of mixture fraction perturbationsin the supply duct. Alternatively, the smooth transitionmay be caused by the large-scale flow structures andconsequential flame restoration (Blumenthal et al. 2013), assuggested by the fact that the time delay coefficients foundby Orchini and Juniper (2015), which included large-scaleflow structures, have negative values.

For medium/large forcing amplitudes, only low equiva-lence ratio perturbations, which are in-phase with the veloc-ity perturbations, propagate far enough into the combustionchamber to be affected by large-scale flow structures. In thepresent investigation, the velocity perturbations influence thegain by redistributing the equivalence ratio perturbations,rather than by wrinkling the flame and causing a heatrelease modulation directly. Analogously, the large-scaleincompressible flow structures could shift this contributionof the low equivalence ratio perturbations between 0.02 and0.03 s to earlier time delays, thereby spreading them out. Toincorporate this effect, the impact of equivalence ratio andflow perturbations must be coupled.

ConclusionsFlame describing functions (FDFs) have been calculatedfor an axisymmetric M-flame using a level set approach.The FDFs characterise the unsteady heat release causedby imposed velocity fluctuations over a wide range offorcing frequencies and amplitudes. Velocity fluctuationsof the air cause equivalence ratio perturbations becausethe mass flow-rate of fuel is constant, while that of theair varies. Equivalence ratio perturbations form at thefuel injector, which lies upstream of the dump plane, areconvected down a duct and are then convected to theflame front by a simple analytical flow: a spherical sourceterm with oscillating source strength. On the one hand,the FDFs are used to predict limit cycle amplitudes ofthermo-acoustic oscillations. On the other hand, they arealso analysed by decomposing them into a sum of timedelay coefficients, which reveals the physics behind the heatrelease characteristics of the flame.

The numerical results have been compared with experi-mental data. Reasonable comparison was achieved in termsof the heat release gain at constant forcing amplitudes andinstantaneous heat release rates at constant forcing fre-quency. This indicated that the heat release gain can be esti-mated neglecting large-scale flow structures and turbulence,since these phenomena contributed little to the overall gain.

However, large-scale flow structures and turbulenceinfluence the phase decay and the gain oscillations due tothe interference of equivalence ratio perturbations as theytravel along the flame-front. Broadband flow fluctuationssmooth out these interference patterns and disperse the heatrelease response over wider ranges of time delays. Thesesmoothly-distributed time delay coefficients lead to a lesssteep phase decay of the heat release, which is more in linewith experimental data.

In summary, the unphysically steep phase decay of theFDF in this model leads to over-prediction of the numberof unstable thermo-acoustic modes, particularly at highfrequencies. In addition, the inclined interference pattern(Fig. 8), but which would be smoothed out in a turbulentflame, causes the growth rates around a given frequencyto oscillate around zero as the amplitude varies (e.g. inFig. 11 around 350 Hz). Nevertheless, the predicted limitcycle frequency agrees well with experiments and the limitcycle amplitude, although quite strongly affected by thephysical features of the model, is reasonably close to theexperimentally observed amplitude of 0.6. This level setapproach, combining mixture fraction oscillations and asimple velocity model therefore has proven to be a quick andreasonably accurate tool with which to calculate limit cycleamplitudes and frequencies of thermo-acoustic oscillationsfor an axisymmetric burner.

Acknowledgements

This work was conducted within the EU 7th FrameworkProject Joint Technology Initiatives - Clean Sky (AMEL-Advanced Methods for the Prediction of Lean-burn CombustorUnsteady Phenomena), project number: JTI-CS-2013-3-SAGE-06-009 / 641453. This work was performed using the DarwinSupercomputer of the University of Cambridge High PerformanceComputing Service (http://www.hpc.cam.ac.uk/), provided by DellInc. using Strategic Research Infrastructure Funding from theHigher Education Funding Council for England and funding fromthe Science and Technology Facilities Council.

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IntroductionMethodology - Flame front trackingGeometry and setupFlow fieldEquivalence ratio perturbations

ImplementationValidation, comparison to experimental data

Flame describing functionsGain distributionFurther modelling

Limit cycle predictionDiscussionConclusions