a time-domain network model for nonlinear thermoacoustic oscillations

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Simon R. Stow1

e-mail: [email protected]

Ann P. Dowling

Department of Engineering,University of Cambridge,Cambridge CB2 1PZ, UK

A Time-Domain Network Modelfor Nonlinear ThermoacousticOscillationsLean premixed prevaporized (LPP) combustion can reduce NOx emissions from gas tur-bines but often leads to combustion instability. Acoustic waves produce fluctuations inheat release, for instance, by perturbing the fuel-air ratio. These heat fluctuations will inturn generate more acoustic waves and in some situations linear oscillations grow intolarge-amplitude self-sustained oscillations. The resulting limit cycles can cause structuraldamage. Thermoacoustic oscillations will have a low amplitude initially. Thus linearmodels can describe the initial growth and hence give stability predictions. An unstablelinear mode will grow in amplitude until nonlinear effects become sufficiently importantto achieve a limit cycle. While the frequency of the linear mode can often provide a goodapproximation to that of the resulting limit cycle, linear theories give no prediction of itsresulting amplitude. In previous work, we developed a low-order frequency-domainmethod to model thermoacoustic limit cycles in LPP combustors. This was based on a“describing-function” approach and is only applicable when there is a dominant modeand the main nonlinearity is in the combustion response to flow perturbations. In thispaper that method is extended into the time domain. The main advantage of the time-domain approach is that limit-cycle stability, the influence of harmonics, and the inter-action between different modes can be simulated. In LPP combustion, fluctuations in theinlet fuel-air ratio have been shown to be the dominant cause of unsteady combustion:These occur because velocity perturbations in the premix ducts cause a time-varyingfuel-air ratio, which then convects downstream. If the velocity perturbation becomescomparable to the mean flow, there will be an amplitude-dependent effect on the equiva-lence ratio fluctuations entering the combustor and hence on the rate of heat release.Since the Mach number is low, the velocity perturbation can be comparable to the meanflow, with even reverse flow occurring, while the disturbances are still acoustically linearin that the pressure perturbation is still much smaller than the mean. Hence while thecombustion response to flow velocity and equivalence ratio fluctuations must be modelednonlinearly, the flow perturbations generated as a result of the unsteady combustion canbe treated as linear. In developing a time-domain network model for nonlinear thermoa-coustic oscillations an initial frequency-domain calculation is performed. The linearnetwork model, LOTAN, is used to categorize the combustor geometry by finding thetransfer function for the response of flow perturbations (at the fuel injectors, say) toheat-release oscillations. This transfer function is then converted into the time domainthrough an inverse Fourier transform to obtain Green’s function, which thus relatesunsteady flow to heat release at previous times. By combining this with a nonlinear flamemodel (relating heat release to unsteady flow at previous times) a complete time-domainsolution can be found by stepping forward in time. If an unstable mode is present, itsamplitude will initially grow exponentially (in accordance with linear theory) until satu-ration effects in the flame model become significant, and eventually a stable limit cyclewill be attained. The time-domain approach enables determination of the limit cycle. Inaddition, the influence of harmonics and the interaction and exchange of energy betweendifferent modes can be simulated. These effects are investigated for longitudinal andcircumferential instabilities in an example combustor system and the results are com-pared with frequency-domain limit-cycle predictions. �DOI: 10.1115/1.2981178�

Introduction

LPP combustion is a method of reducing NOx emissions fromas turbines by mixing the fuel and air more evenly before com-ustion. However, LPP combustion is highly susceptible to ther-oacoustic oscillations. The interaction between sound waves and

1Present address: Rolls-Royce plc, Derby DE24 8BJ, UK.Contributed by the International Gas Turbine Institute of ASME for publication in

he JOURNAL of ENGINEERING FOR GAS TURBINES AND POWER. Manuscript receivedarch 28, 2008; final manuscript received April 25, 2008; published online February

, 2009. Review conducted by Dilip R. Ballal. Paper presented at the ASME Turbo
xpo 2008: Land, Sea and Air �GT2008�, Berlin, Germany, June 9–13, 2008.
ournal of Engineering for Gas Turbines and PowerCopyright © 20

om: http://gasturbinespower.asmedigitalcollection.asme.org/ on 05/05/201

combustion can lead to large pressure fluctuations, resulting instructural damage. Linear models can give predictions for the fre-quency and linear stability of oscillations, and thus indicate pos-sible modes. However, they do not give predictions of oscillationamplitude. Ideally one would wish to design a combustion systemthat is linearly stable; however, this may not be possible. Thusmodels that can predict limit-cycle amplitude are required to helpdesign combustors that oscillate within acceptable bounds. Fur-thermore, such models can be used to investigate the potentialamplitude reduction from damping devices.

We present a low-order theory to predict limit cycles in LPPcombustors, which is based in the time domain and combines a
linear thermoacoustic model with a nonlinear combustion model.
MAY 2009, Vol. 131 / 031502-109 by ASME

4 Terms of Use: http://asme.org/terms

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reviously, we presented a frequency-domain method �1� using aescribing-function approach, which was only valid when a singlerequency dominated. The time-domain approach can be used tonvestigate limit-cycle stability, the influence of harmonics, andhe interaction between different modes.

The thermoacoustic model, described in Sec. 2, is formulated inerms of a network of modules describing the features of the ge-metry. This model can be applied either to long combustorstypical of industrial gas turbines� where the perturbations can bereated as one dimensional, or to thin-annular combustors �typicalf aeroengines� where the perturbations can be treated as circum-erential and longitudinal waves. There have been several studiesn linear combustion-instability models in literature. For example,aschereit et al. �2� described an approach in which the premixernd combustion zone are modeled by a single transfer matrix. Theoefficients of the matrix were found experimentally by using aoud-speaker forcing in a test rig. Krüger et al. �3,4� have usedetworks of one-dimensional modules linked together in bothxial and circumferential directions to model annular combustors.scillations in annular combustors have also been computed bysiao et al. �5,6� using a hybrid computational fluid dynamics

CFD�/acoustics-based model on a three-dimensional grid of cells.review of linear modeling of combustion instability is given in

efs. �7,8�. Nonlinear modeling of thermoacoustic oscillationsas first motivated by instability in rocket engines �see, for ex-

mple, Refs. �9,10��. For LPP combustion, nonlinear time-domainodels have been developed to calculate limit cycles �11–14�.There is much evidence that equivalence ratio fluctuations are

he dominant cause of heat-release oscillations in LPP combustorssee, for example, Refs. �15,16��. As described in Sec. 3, a simpleaturation model �similar to that in Ref. �14�� is used to model thensteady combustion. However, a more complex flame model,uch as that developed in Ref. �1�, can be used. It is assumed thathis is the main source of nonlinearity determining the amplitudef the limit cycle. An initial calculation using the linearrequency-domain model categorizes the combustor geometry bynding the transfer function for the response of flow perturbationsat the fuel injectors, say� to heat-release oscillations; see Sec. 4.his transfer function is converted to a Green’s function, which

hus relates unsteady flow to heat release at previous times. Inec. 5, we show that by combining this with a nonlinear flameodel �relating heat release to unsteady flow at previous times� a

olution can be found by stepping in time. Thus, whereas in Ref.1� a time-domain nonlinear flame model was converted to therequency domain then combined with the linear acoustic model,ere the acoustic model is converted to the time domain thenombined with the flame model. If an unstable mode is present, itsmplitude will initially grow exponentially �in accordance withinear theory� until saturation effects in the flame model becomeignificant, and eventually a stable limit cycle will be attained. Inec. 6, the model is applied to an example geometry and theesults are compared with limit-cycle predictions from therequency-domain model. Finally, our conclusions are presentedn Sec. 7.

Linear Thermoacoustic Network ModelBefore investigating the nonlinear effects, we give a brief de-

cription of the linear combustion-instability model. We use cy-indrical polar coordinates x, r, and � and write the pressure, den-ity, temperature, and velocity as p, �, T, and �u ,v ,w�,espectively. We will assume a perfect gas, p=Rgas�T. The flow isaken to be composed of a steady axial mean flow �denoted byars� and a small perturbation �denoted by dashes�. We make onef the two assumptions about the combustor geometry: either wessume it is axially long �typical of industrial gas turbines� or thatt has a narrow annular gap compared with its circumferencetypical of aeroengines�. The mean flow for either cases can bessumed to be one dimensional. For the former case, the pertur-
ations will also be one dimensional �at least at frequencies of
31502-2 / Vol. 131, MAY 2009


practical interest�. In the latter case, circumferential and axialvariations of the perturbation could be important but radial varia-tion can be neglected. We consider perturbations with complexfrequency �, i.e., we have p= p�x�+ p��x ,� , t� with p�=Re�p�x ,��ei�t� �and similarly for the other flow variables�. Wemay consider the perturbations as a sum of circumferential modesby writing p�x ,��=�n=−N

N pnein� and so on �where N is sufficientlylarge that the higher-order modes can be neglected�. If the geom-etry is axisymmetric throughout then the circumferential modescan be considered independently. However, if the symmetry isbroken �for instance, by a Helmholtz resonator being present �17��the modes can become coupled and must be calculated together.Note that for the case of a one-dimensional geometry, only theplane waves are present and we simply have N=0.

The geometry is composed of a network of modules describingits features, such as straight ducts, area changes, and combustionzones. For the straight duct modules, wave propagation is used torelate the perturbations at one end of the duct to those at the other.The rest of the modules are assumed to be acoustically compact.Here quasisteady conservation laws for mass, momentum, andenergy are used to relate the flow at the inlet and exit of themodule, accounting for any force or heat input applied within themodule. At a combustion zone, the unsteady heat release is relatedto the flow disturbances by a �linear� flame transfer function.Acoustic boundary conditions are assumed to be known at theinlet and outlet of the geometry. The procedure to find the reso-nant modes is to guess an initial value for � and calculate theperturbations, starting at the inlet and stepping through the mod-ules to the outlet. In general this solution will not match the outletboundary condition. � is then iterated to satisfy this constraint.�For coupled circumferential modes, the relative phase and mag-nitude of the modes at the inlet must also be calculated.� Thefrequency of the mode is then Re�� / �2��, and we define thegrowth rate as −Im��. If the growth rate is negative the mode islinearly stable and would not be expected to be seen in practice,whereas if the growth rate is positive the mode is unstable andwould grow in amplitude until nonlinear effects become importantand a limit cycle is achieved. More details on this linear model aregiven in Ref. �18�, with the procedure for coupled circumferentialmodes described in Ref. �17�.

The above references only consider geometries with a singleflow-pathway from the inlet to the outlet; however, it is possible toextend the model to have multiple pathways. These additional“secondary paths” may originate by splitting of the flow or fromsecondary inlets to the geometry. The paths can later terminate asdead ends or by joining into other paths. Flow from one path toanother through holes can also be incorporated. This means thatcooling flows and staged combustion can be modeled. Schematicsof a possible application and modeling approach are shown in Fig.1. At the start of each secondary path there is an unknown param-eter for the perturbation �such as the ratio of the mass-flux pertur-bations at a flow split�, and at the end of each path there is aboundary condition �such as u=0 at a dead end�. However, due tothe linearity of the model, calculation of these simply involves thesolution of a �P+1�-dimensional complex matrix equation �or�P+1��2N+1�-dimensional for coupled circumferential modes�,where P is the number of secondary paths.

A general review of linear methods for combustion instabilityin LPP gas turbines is given Refs. �7,8�.

3 Nonlinear Flame ModelWe assume that combustion takes place at a single combustion

zone, i.e., there is no radial or axial staging �however, this con-straint can be relaxed; see Sec. 5.3�. This combustion zone isassumed to be acoustically compact in the axial direction �i.e.,short compared with the wavelength of flow perturbations�. Weassume that a ring of D premix ducts or burners is present. For
simplicity, we take these to be identical and uniformly spaced
Transactions of the ASME


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round the circumference, although this need not be the case. Weenote the heat release from duct d by Qd�t�. A flame modelelates Qd�t� to the unsteady flow at previous times. In the ex-mples in Sec. 6, we will use the following simple saturationodel:

Qd�L�t� = − k

md��t − ��md

Qd �1a�

Qd��t� =� Qd�L�t� for�Qd�

L�t�� Qd

�Qd sgn�Qd�L�t�� for�Qd�

L�t�� Qd

� �1b�

here md denotes the �air� mass flow at the fuel injection point inuct d, � is the convection time from there to the combustionone, k is a constant associated with the low-amplitude behavior,nd � is a constant associated with the saturation �0��1�.This nonlinear flame model is very similar to that used in Ref.14�.� The overbars denote unperturbed steady-state values and the

rimes denote the perturbation; hence, Qd�t�= Qd+Qd��t�. �Notehat for small oscillations the unperturbed value is the same as theime-averaged mean, but for larger amplitudes this is not neces-arily the case.� A more complex nonlinear flame model is de-cribed in Ref. �1�. Like Eqs. �1a� and �1b�, this model relates heatelease to a single flow variable in each duct, but models involv-ng more than one variable could also be used. However, to main-ain causality in the time-domain calculation, the model must give

d�t� in terms of flow variables evaluated at earlier times, not theurrent time.

3.1 Flame Transfer Function. We now convert the flameodel in Eqs. �1a� and �1b� into the frequency domain. This is not

equired in the time-domain calculations, but gives insight into theffect of the flame model and is required for frequency-domainalculations of linear modes and limit cycles. We aim to find theesponse from Eqs. �1a� and �1b� to a harmonic input

md��t� = Re�mdei�t� �2�

or small disturbances, the heat release is also harmonic and cane written as

Qd��t� = Re�Qdei�t� �3�

he �linear� flame transfer function used in linear-mode calcula-

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ig. 1 Schematics of possible application of thermoacousticetwork model „top… and modeling approach „bottom…

ions is then defined by

ournal of Engineering for Gas Turbines and Power


TflameL �� =

Qd/Qd

md/md

= − ke−i�� 4�

�For a discussion of flame transfer functions, see, for example,Refs. �19,20�.�

For finite disturbances, Qd��t� may not be harmonic but is stillperiodic �with the same period as md��t�� and hence it can be de-scribed by a Fourier series

Qd��t� = Re�m=0

Q�m�eim�t �5�

In frequency-domain limit-cycle calculations, we ignore the

higher-frequency harmonics generated and take Qd= Q�1�. Fromthis we define a nonlinear flame transfer function, which is afunction of frequency and amplitude, A= �md� / md. Evaluating thefirst Fourier coefficient

Q�1� =�

��

0

2�/�

Qd��t�ei�tdt �6�

with Qd��t� and md��t� as in Eqs. �1a�, �1b�, and �2�, it can be shownthat

Tflame��,A�Tflame

L ��=

Q�1�

QL= � 1 for � � 1

1 −2�

�+

2�1 − 1/�2�1/2

��for � � 1

�7�

where �= �k�A /� and �=cos−1�1 /��, with QL being the valuefrom the linearized flame model. Although they are not used inany of the calculations, it is interesting to consider the values ofthe other terms in the expansion equation �5�. For a sinusoidalinput, the flame model produces a bounded sinusoid �see, for ex-ample, the dashed line in the third plot of Fig. 5�. From the sym-metry of this, it is easily seen that no mean or even harmonics are

produced, i.e., Q�m�=0 for even m 0. Furthermore, it can beshown that the odd harmonics are given by

Q�m�

QL= � 0 for � � 1

4

��n2 − 1�cos�n��1 − 1/�2�1/2 −

sin�n��n�

for � � 1 �8�

for odd m 3. Figure 2 shows the variation of Q�m� / QL with � forthis flame model. For ��1, there are no nonlinear effects and sothe values are the same as for the linearized flame model. As � isincreased, the saturation causes the gain of the nonlinear flametransfer function to decrease, with the harmonic terms becomingmore significant relative to this.

4 Flow-Response Transfer FunctionBefore beginning the time-domain calculation, an initial

frequency-domain calculation is required in order to describe theresponse of flow variables to heat-release fluctuations. We expandthe flow variables in circumferential modes, writing the heat re-lease as

Q��,t� = Re�Q��ei�t� �9a�

Q�� = �n=−N

N

Qn�1n�� 9b�

where

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�1n�� = � 1 for n = 0

cos�n�� for n � 0

sin�n�� for n 0 �9c�

nd similarly for the other flow variables. Qn represent the com-

onents of Q in terms of circumferential standing modes and N ishe maximum mode number of interest. Previously, we have de-omposed into spinning modes by taking �1

n��=ein� �see Ref.17��. Using standing modes instead is only a minor change to therequency-domain analysis, but is essential when moving to aime-domain solution �see Sec. 5�. We assume a thin-annular com-ustor, or else we consider only plane waves by taking N=0. Toroceed, we also need information about the azimuthal distribu-ion of the heat-release perturbations. Here we will assume thatombustion is localized to the outlet of each premix duct, and soake

Q�� = �d=1

D

2�� − �d�Qd �10�

here �d=�1+2��d−1� /D are the azimuthal locations of the pre-

ix ducts. �We could consider other distributions, such as Qdniformly distributed over the sector. However, since the circum-erential mode-number of interest is typically much less than theumber of premix ducts, the particular distribution is unlikely toave a big effect.� Comparing Eqs. �9b� and �10� gives

Qn =1

2��

0

2�

Q��2n��d� = �

d=1

D

Qd�2n��d� �11a�

here

�2n�� = � 1 for n = 0

2 cos�n�� for n � 0

2 sin�n�� for n 0 �11b�

The nonlinear flame model requires certain flow variables asnputs. For definiteness, we suppose that only md� is required �as inhe model in Eqs. �1a� and �1b��. In the frequency domain this isˆ

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ig. 2 Variation of Q„m… /QL with �. The solid, dashed, dashed-otted, and dotted lines denote m=1, 3, 5, and 7, respectivelythe value being zero for even m…. Note that Q„1… /QL is equiva-ent to the nonlinear flame transfer function relative to the lin-arized version, Tflame„� ,A… /Tflame

L„�….

d and may be written as

31502-4 / Vol. 131, MAY 2009


md = m�� = �n=−N

N

mn�1n��d� �12�

�see Ref. �1��. The frequency-domain calculation is then as fol-lows. The flow response to unsteady combustion is identified byforcing the components of the heat release. The response to un-

steady heat release in mode n� is determined by setting Qn / Q=�n,n�

. The frequency-domain model �see Sec. 2� is used to cal-culate the perturbations in the flow for this forced rate of heatrelease in the combustor geometry, for a fixed value of �. Due tothe linearity of the problem, this amounts to solving a matrixequation. We define Tn,n�

��= mn / m for this solution. Linearity in

the flow response then gives that for any set of values of Qn�,

mn

m= �

n�=−N

N

Tn,n�

��Qn�

Q�13�

Tn,n�is thus a matrix of transfer functions describing the linear

response of the flow variable to heat-release fluctuations. It givesno information about the flame but instead categorizes the linearwaves generated in the geometry by unsteady combustion. Notethat if the geometry is axisymmetric, the matrix is diagonal withT−n,−n�Tn,n.

4.1 Green’s Function. To use the transfer functions in thetime-domain calculation, they are converted into Green’s func-tions, Gn,n�

, by taking the inverse Fourier transform

Gn,n�

�t� =1

2��

−

Tn,n�

��ei�td� =1

�Re�

0

Tn,n�

��ei�td�

�14�

In practice, Tn,n�� is calculated at a discrete set of frequencies,

f =� / �2��=0,�f ,2�f , . . . , fmax� . The integral is thus approxi-

mated by the corresponding inverse discrete Fourier transform,with Gn,n�

�t� being evaluated at t=0,�t ,2�t , . . . , tmax, where �t=1 / �2fmax

� � and tmax=1 /�f . �t will also be the time step used inthe time-domain calculation. In order to remove the possibility ofhigh-frequency oscillations, while keeping �t small, we may wishto set Tn,n�

to be zero for f � fmax, say.

5 Time-Domain SolutionThe flame model and the flow-response transfer function devel-

oped in Secs. 3 and 4 model the physics of the flow phenomenathat lead to self-excited combustion oscillations. Green’s function�see Eq. �14�� describes the flow fluctuations generated by un-steady heat release, including the reflection and transmission ofsound throughout the combustor geometry. The flame model de-scribes how the flow perturbations lead to further combustion un-steadiness. By combining the two, the development of an oscilla-tion can be predicted. The time-domain calculation proceeds asfollows. For t�0, md� and Qd� are set to low-amplitude randomnoise. This encourages any instabilities in the system to initiatethemselves, in the same way as background turbulent noise wouldin a real engine. The solution is then stepped forward in time,using a time step of �t �see Sec. 4.1�. For each duct, Qd��t� at thenew time value is found from md� at previous times using thenonlinear flame model �suitably discretized if necessary�. In asimilar way to Eqs. �9a�–�9c� and �10�, we write

Q��t,�� = �n=−N

N

Q�n�t��1n�� = �

d=1

D

2�� − �d�Qd��t� �15�

Note that in the time domain, we cannot decompose into a sum ofcircumferential spinning modes. This is why a standing-wave de-

n
composition was chosen in Eq. �9b�. We can then find Q� �t� from


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d��t� using the equivalent to Eq. �11a� as follows:

Q�n�t� =1

2��

0

2�

Q��t,��2n��d� = �

d=1

D

Qd��t��2n��d� �16�

t is thought that the combustion zone is the main source of non-inearity, and we therefore assume that the flow response in theest of the geometry can be treated as linear. �This is the samessumption previously used when calculating limit cycles in therequency domain in Ref. �1�.� Hence the linear frequency-domainalculation in Sec. 2 still applies at finite amplitude. This meanshat Green’s functions can now be used to find m�n�t� from Q�n�t�,sing the time-domain version of Eq. �13�, which is the convolu-ion

m�n�t�m

= �n�=−N

N �0

Gn,n�

�t��Q�n�

�t − t��

Qdt� �17�

inally, md��t� is found from m�n�t� using the equivalent to Eq.12� as follows:

md��t� = m��t,�� = �n=−N

N

m�n�t��1n��d� �18�

he time stepping can then be repeated in this manner until eitherhe initial disturbances decay away or a stable limit cycle ischieved. The mode shape for the limit cycle can be found byaking the Fourier transform of Q�n�t� to find its frequency com-

onents, Qn��, and then using these in forced frequency-domainalculations of the type described in Sec. 2.

Note that if we are only considering plane waves �i.e., N=0�,e only need to calculate the solution for one duct, since the

olution in the others is identical; md��m�0�m� and Qd��Q�0

Q�. Also, in this case we clearly have only a single transferunction and corresponding Green’s function.

In the above analysis, we have chosen to use Green’s functionselating the circumferential modes. We could instead have foundreen’s functions, Gd,d�, relating the flow variable in duct d toeat release in duct d� at earlier times. If the geometry is axisym-etric, by symmetry we have

Td,d� � T1+�d−d��mod D,1 � T1+�d�−d�mod D,1 �19�

herefore, in this case, a single forced perturbation solution perrequency is required and �floor�D /2�+1� Green’s functions muste calculated, as opposed to �N+1� perturbation solutions andN+1� Green’s functions if using circumferential modes. Hencehis may require more or less computational time than theircumferential-mode approach, depending on the number ofodes of interest and on the relative effort of calculating the

erturbation solutions compared with computing the inverse Fou-ier transforms when finding Green’s functions. If there is asym-etry in the geometry and hence modal coupling, D perturbation

olutions and D2 Green’s functions are needed, instead of �2N1� and �2N+1�2, respectively. For an aeroengine, the number ofucts might be around 20–30 while we may only be interested inhe first or second circumferential mode; hence, calculating theuct Green’s functions requires much more effort.

5.1 Single Circumferential Spinning Mode. The resultsrom the time-domain model suggest that the final limit-cycle so-ution is always dominated by single-frequency circumferentialpinning mode �see Sec. 6�. This is in agreement with numericalimulations by Schuermans �21� �see also Ref. �22��, who alsoave an analytical proof that the spinning-mode limit cycle wastable, while the standing mode was not, for a simplified example.ence we now describe a special case where a solution of this

orm is assumed, thus reducing the computation required.
In Eq. �9b�, only a single value of n is now considered, with


�1n��=�2

n��=ein�. Hence only a single transfer function and cor-responding Green’s function are calculated. In the time domain,we only calculate the solution in a single duct, say duct 1. Thesolutions for the other ducts are assumed to be identical except fora phase difference corresponding to the assumed circumferentialmode, i.e., duct d has a phase shift of 2�n�d−1� /D, and so wetake Qn=Q1. Note that it is implicit in this assumption that asingle frequency must be dominant; otherwise, the spinning-wavepattern cannot be present. Therefore, if harmonics are present, thismethod provides only an approximation.

5.2 Sector Rigs. The approach is easily adapted for a thin-annular sector rig, 0��max, instead of a full annulus. In thedecomposition into circumferential mode in Eq. �9b�, n is replacedwith n� /�max, with the modes for negative n no longer present.This leads to minor changes in the frequency-domain analysis,such as the values of the axial wave numbers.

5.3 Multiple Combustion Zones. In the method describedabove, we assumed a single combustion zone, although themethod can still be used if we assume that any additional com-bustion zones respond linearly so that they can be included in theinitial frequency-domain calculation. For multiple nonlinear com-bustion, the approach can be extended to the following. Supposethat there are Z nonlinear combustion zones. Let Qd,z� �t� denote theheat-release perturbation from duct d at zone z, which is related toa flow perturbation, md,z� �t�, by a nonlinear flame model. �Theflame model and the number of ducts, Dz, need not be the same ateach combustion zones.� In the initial frequency-domain calcula-

tion, we set circumferential components Qzn=�z,z�

n,n�

and find the

forced perturbation solution, defining Tz,z�n,n�

��= mzn for this solu-

tion. The general result is then

mzn = �

z�=1

Z

�n�=−N

N

Tz,z�n,n�

��Qz�n�

�20�

As before, these transfer functions can be converted into Green’sfunctions. In the time domain, at each time step we calculate theheat-release perturbation from each duct in each combustion zoneusing the corresponding flame model, and find the flow perturba-tions using Green’s functions.

6 ResultsWe now apply the method described above to find time-domain

solutions for an example geometry. The geometry is intended tobe representative of an aeroengine combustor. It consists of anannular plenum chamber, a ring of 20 identical premix ducts, andan annular combustion chamber, and each of these is modeled asa uniform axial duct. The plenum inlet and combustor outlet areassumed to be choked, to model the turbine outlet and compressorinlet, respectively. The fuel injection point �where md is measured�is halfway along the premix ducts, and the combustion zone isassumed to be concentrated at the start of the combustor. Detailsof the geometry and flow conditions are given in Table 1 �therelevance of the long-combustion version is explained in Sec.6.2�.

6.1 Single Unstable Circumferential Mode. Before calculat-ing a time-domain solution for the geometry, we use thefrequency-domain model �Sec. 2� to calculate the linear modes.Note that the axisymmetry of the geometry means that there is nocircumferential modal coupling for linear calculations, and thatpositive and negative values of n are equivalent. There are severalstable linear modes for the geometry; however, only one unstablemode is present �for frequencies less than 3 kHz�. This is a first-order circumferential mode, i.e., n= �1. Its frequency is 521.4 Hzand its growth rate is 56.3 s−1. There is relatively little axial
variation of the pressure perturbation for this mode, except that it
MAY 2009, Vol. 131 / 031502-5


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Tc

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s at a lower level in the plenum compared with the premix ductsnd combustor. �Note that typically we would take k=1 in theame model; however, k=4 was chosen here in order for an un-table mode to be present.�

In the time-domain calculations, we take N=1, i.e., second- andigher-order circumferential waves are ignored. We take fmax1 kHz with �f =1 Hz. The time step used is �t=0.1 ms. Fig-re 3 shows the transfer functions and Green’s functions for theoriginal� example geometry. Note that the values for n= �1 arequal and no coupling terms are present �i.e., Tn,n�

�0 and Gn,n�

0 for n��n�. The results from a time-domain calculation arehown in Fig. 4. As we would expect from the linear predictions,he initial random noise induces an instability in both m�−1 and�1, while m�0 decays away. To begin with, these instabilitiesrow exponentially. During this initial linear phase �t0.3 s�, themplitude of m�−1 was found to be larger for this simulation. Thiss simply because the random noise excited this mode more. Ashe amplitude of m�−1 becomes larger, nonlinear effects becomemportant and its rate of growth diminishes. At around t=0.35 s,t appears that a limit cycle for m�−1 may have been achieved;owever, meanwhile m�1 has grown and nonlinear coupling be-ween the two modes is now important. Between t=0.35 s and.4 s, m�1 continues to grow, while m�−1 decays slightly, until for�0.4 s a stable coupled limit cycle is achieved, which appearso continue indefinitely �the simulation was continued beyond t

able 1 Geometry and flow conditions for original and long-ombustor examples

hoked inlet, pressure 5 MPahoked inlet, temperature 1000 Khoked inlet, mass flow rate 100 kg s−1

lenum, cross-sectional area 0.3 m2

lenum, mean radial location 0.3 mlenum, length 0.1 mremix ducts, number of ducts 20remix ducts, combined area 0.02 m2

remix ducts, mean radial location 0.3 mremix ducts, length before fuel 0.05 mremix ducts, length after fuel 0.05 mombustor, cross-sectional area 0.2 m2

ombustor, mean radial location 0.3 mlame model, convection time � 1.5 mslame model, linear constant k 4lame model, saturation constant � 0.4ombustor, temperature 2500 Kombustor, length �original� 0.3 mombustor, length �long version� 0.707 mhoked outlet -

0 200 400 600 800 10000

0.25

0.5

frequency (Hz)

|Tn,

n |

0 200 400 600 800 1000−1

0

1

frequency (Hz)

phas

e(T

n,n )/

π

0 0.2 0.4 0.6 0.8 1−200

0

200

t (s)

Gn,

n

ig. 3 Transfer functions, Tn,n„�…, and Green’s functions,

n,n„t…, for original geometry. The solid and dashed lines de-

ote n= ±1 „these being equivalent… and n=0, respectively.

31502-6 / Vol. 131, MAY 2009


=10 s�. From the third plot of Fig. 4, we can see that in the finallimit cycle m�−1 and m�1 have the same amplitude but differ inphase by 90 deg. This corresponds to a spinning mode, spinning inthe positive-� direction �i.e., having the form e−i� in the frequencydomain�. The amplitude of m�−1 and m�1 was 0.152m, with thefrequency found to be 519.8 Hz �close to the frequency of theunstable linear mode, as we might expect�. Note that since thetransfer function was only calculated up to fmax=1 kHz, harmon-ics of 519.8 Hz are not seen and only a single frequency ispresent. The same calculation was conducted several times. Dueto the initial random noise differing, in different calculations theamplitude of m�−1 was initially larger, smaller, or very similar tothat of m�1. In each case, however, the final limit cycle was aspinning wave of the same frequency and amplitude �the directionof spin varying from one calculation to another�. A standing-wavelimit cycle was never found in the calculations. This is becausethe standing-wave form is unstable, as can be explained as fol-lows. Suppose a large-amplitude standing wave were present, sayof the cosine form �n=1�. The instability would be strong near to�=0 deg and 180 deg and the flame response saturated there.Now suppose that a small perturbation of the sine form �n=−1�were added. Near to �=90 deg and 270 deg the instability wouldstill be weak, meaning that the sine form would see little satura-tion in the flame response and hence it would grow in amplitude,thus destroying the standing-wave structure �see also Refs.�21,22��. Decreasing �f and �t had a very little effect on thesolution, indicating that the values used were sufficiently small.The effect of increasing fmax is discussed in Sec. 6.3.

As the limit cycle has the form of a single-frequency spinningwave, the approach described Sec. 5.1 should be applicable. Fig-ure 5 shows the results for this simplified method for n=1. Notethat here m�1�m1� and Q1��Q�1. In the third plot of Fig. 5 we seethat, as dictated by the flame model, the heat-release oscillation induct 1 is out of phase with the mass flow perturbation with a lagof 1.5 ms, and bounded between �0.4. Although the interplaybetween the two circumferential modes is now lost, the growthand saturation of the instability are representative of the coupledsolutions calculated previously, and the final spinning-wave limitcycle has frequency and amplitude identical to those before.

The results were compared with frequency-domain limit-cyclepredictions by using Eq. �7�. These predictions assume that thelimit cycle has a single dominant frequency, with harmonics beingignored �details of the technique are given in Refs. �1,8��. Themethod predicts a first-order circumferential spinning-wave limitcycle with frequency 519.8 Hz and amplitude 0.151m, this beingvirtually the same as the time-domain result. If modal coupling isincluded, the method also admits standing-wave solutions at the

0.25 0.3 0.35 0.4 0.45 0.5−0.2

0

0.2

t (s)

m’−

1 /m

0.5 0.501 0.502 0.503 0.504 0.505−0.2

0

0.2

t (s)

m’−

1 /m,m

’1 /m

0.25 0.3 0.35 0.4 0.45 0.5−0.2

0

0.2

t (s)

m’1 /m

Fig. 4 Time-domain solution for original geometry. m�−1„t… /m

and m�1„t… /m are shown; in the third plot, the solid line denotes

the former while the dashed lines denotes the latter. „m�0„t… is

negligible for this solution.…

same frequency. One of the benefits of the time-domain approach



i�m

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s that it reveals that only the spinning-wave limit cycles are stablealthough it may be possible to adapt the frequency-domainethod to give stability predictions�.

6.2 Interaction Between Two Unstable Modes. In order tonvestigate interaction between different modes, the geometry was

odified by lengthening the combustion chamber �see Table 1�.or this geometry there are now two unstable linear modes �belowkHz�. One is a first-order circumferential mode �n= �1� at

10.7 Hz, which is very similar to the unstable mode for theriginal geometry, while the other is a plane wave mode �n=0� at18 Hz, this being approximately a half-wave resonance of theombustion chamber. In order to maximize the chances of theseodes interacting in the time domain, as opposed to one mode

ominating the other, the combustor length was specifically cho-en to give the same growth rate for the two modes, the valueeing 11.6 s−1. The numerical parameters were taken to be theame as before �i.e., N=1, �f =1 Hz, fmax=1 kHz, and �t0.1 ms�. The results from four time-domain calculations are

hown in Figs. 6–9. The different results are entirely due to dif-erences in the random noise used to start the solution. In eachase, there is an initial linear phase where an exponentially grow-ng instability is seen in all three circumferential components. Inalculation 1 �Fig. 6�, initially m�−1 and m�1 have similar ampli-

udes while m�0 is larger. At around t=1.7 s nonlinear interactionetween the modes causes m�−1 and m�1 to begin to decay. Be-ond t=2 s these components become negligible and a stablelane wave limit cycle is established, the amplitude of m�0,.116m, with frequency 619.5 Hz �close to the frequency of the

0.25 0.3 0.35 0.4 0.45 0.5−0.2

0

0.2

t (s)m

’1 /m

0.25 0.3 0.35 0.4 0.45 0.5−0.5

0

0.5

t (s)

Q’ 1/Q

0.5 0.501 0.502 0.503 0.504 0.505−0.5

0

0.5

t (s)

m’1 /m

,Q’ 1/Q

ig. 5 Time-domain solution for original geometry, assumingsingle-spinning wave, n=1. m�1

„t… /m and Q1�„t… /Q are shown;n the third plot, the solid line denotes the former while theashed lines denotes the latter.

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’−

1 /m

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’0 /m

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’1 /m

ig. 6 m�n„t… /m for time-domain solution for long-combustor

eometry „Calculation 1…



linear plane wave mode�. In Calculation 2 �Fig. 7�, m�0 and m�1

initially have similar amplitudes while m�−1 is larger. Eventuallym�0 decays away and a stable spinning-wave limit cycle is set up.This has a frequency of 509.6 Hz �close to that of the linear modefor n= �1� and amplitude 0.123m. In Calculation 3 �Fig. 8�, m�−1

and m�0 initially have similar amplitudes while m�1 is muchsmaller. In this case, at around t=1.7 s, it appears that a limitcycle in m�−1 and m�0 may have been achieved; however, m�1

continues to grow until modal interactions cause m�0 to decay anda spinning limit cycle is formed, which is identical to that inCalculation 2. In Calculation 4 �Fig. 9�, the amplitude of m�−1 isinitially slightly larger than for m�1 while m�0 is slightly larger

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’−

1 /m

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’0 /m

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’1 /m

Fig. 7 m�n„t… /m for time-domain solution for long-combustor

geometry „Calculation 2…

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’−

1 /m

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’0 /m

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’1 /m



1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’−

1 /m

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’0 /m

1.4 1.6 1.8 2 2.2 2.4−0.2

0

0.2

t (s)

m’1 /m



MAY 2009, Vol. 131 / 031502-7


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till. Between t=1.8 s and 2 s it appears that limit cycle is estab-ished in which a spinning wave and a plane wave coexist. How-ver, this is found to be transient and the solution eventually be-omes the same purely spinning limit cycle as in Calculations 2nd 3. Other calculations were conducted and in each case eitherhe plane wave limit cycle or the spinning limit cycle �spinning inne direction or the other� was eventually formed.

Frequency-domain limit-cycle predictions were also calculatedor the long-combustor geometry. The solutions found were alane wave limit cycle, frequency 619.6 Hz, and amplitude.116m, and a first-order circumferential spinning-wave limitycle, frequency 509.7 Hz, and amplitude 0.124m. Again, thesere virtually the same as the two limit cycles found in the time-omain calculations. As before if modal coupling is included,tanding-wave solutions were also found at the same frequency ashe spinning wave.

6.3 Effect of Including Harmonics. In the previous calcula-ions, fmax was always set to be less than twice the limit-cyclerequencies, and so harmonics could not be present in m�. Theffect of allowing harmonics to be present is now investigated. Werst consider the plane wave limit cycle in the long-combustoreometry, setting N=0 so that only plane waves are present, ando Qd��Q�0. Using the same numerical coefficients as before,ime-domain calculations resulted in a limit cycle in which m�0

as sinusoidal with frequency 619.5 Hz and amplitude 0.116m,dentical to the final limit cycle in Calculation 1 in Sec. 6.2. Ofourse, m�0 is not sinusoidal, but a bounded sinusoid �similar tohe dashed line in the third plot of Fig. 5�. As shown in Sec. 3.1,his contains only odd harmonics. A calculation was then con-ucted for fmax=2 kHz �with �f =1 Hz and �t=0.05 ms�. Asxpected, m�0 in the final limit cycle contained a component at thehird-harmonic frequency �1858.8 Hz�. However, the amplitude ofhis component was very low, 0.0006, and it had a negligibleffect on the fundamental �first harmonic�, the limit-cycle solutioneing virtually identical to that before. There are two reasons forhis: First, the amplitude of the limit cycle is not especially high

eaning that saturation effects are moderate and the third har-onic of Q�0 is not large �see Fig. 2�, and second, the flow-

esponse transfer function has low gain at 1858.8 Hz. It is likelyhat a harmonic of an instability could play a significant role if itas close to a resonant frequency of the geometry, and especially

f it was close to another unstable mode or a weakly stable modef the system.

We now consider harmonics for the circumferential mode in theriginal geometry. Increasing fmax �with N=1�, we might expecto find harmonics present in m��1; however, this was not seen, theimit cycle being sinusoidal and virtually identical to that before.his is because, although harmonics are present in Qd�, when theontributions from the ducts are combined �through Eq. �16��, forspinning wave, Q��1 become very close to sinusoidal. �This

articularly so since n is small compared with D and saturation isoderate.� The effect of Eq. �16� is to instead transfer distur-

ances at the harmonic frequencies to the higher circumferentialodes. The results for a time-domain calculation for N=3 and

fmax=2 kHz �with �f =1 Hz and �t=0.05 ms� are shown in Fig.0. We see that m��3 are negligible during the linear phase andtart to grow in amplitude once nonlinear effects appear in m��1.he final limit cycle consists of a sinusoidal spinning wave in��1 with frequency 520.2 Hz and amplitude 0.152m �very close

o the result before�, with additionally a sinusoidal spinning waven m��3 with frequency 1560.4 Hz �the third harmonic� and verymall amplitude 0.004m.

The effect of increasing fmax for the single-spinning-mode ap-roximation �Sec. 5.1� was also investigated. In a calculation for

fmax=2 kHz �with �f =1 Hz and �t=0.05 ms�, m�1 had a com-onent at the third-harmonic frequency. This is incorrect because,s we have seen above, the third harmonic should only be present

�3
n m� . However, its amplitude was very low �0.001� and the
31502-8 / Vol. 131, MAY 2009


limit-cycle solution was virtually the same as before. This dem-onstrates that the single-spinning-mode approximation can givegood results, but must be used with caution since the harmonicsare treated incorrectly and potentially this could lead to an inac-curate solution.

6.4 Helmholtz Resonators. We now investigate the effect ofadding Helmholtz resonators to the geometry. Helmholtz resona-tors are damping devices consisting of a large volume connectedto the combustor via a short neck. The modeling of these is de-scribed in Ref. �17�. Attaching a Helmholtz resonator destroys theaxisymmetry of the geometry, leading to modal coupling, andhence a matrix of Green’s functions is required. We first considerattaching a single resonator to the �original� geometry. The reso-nator is placed just before the choked outlet at 0 deg azimuthally.The resonator neck is taken to radius 7 mm and length 30 mm,with a through flow of 10 m s−1 into the combustor. The resona-tor volume is set to 1.3�10−4 m3, and the temperature inside isassumed to be at 1000 K. �This corresponds to a resonator fre-quency of 522.7 Hz.� As discussed in Ref. �17�, for linear theory�i.e., small oscillations� a single resonator does not provide damp-ing for a circumferential mode. The effect instead is to couple thetwo spinning waves to produce a standing wave with a pressurenode at the resonator neck. A time-domain solution was calcu-lated, taking N=1 �with fmax=1 kHz, �f =1 Hz, and �t=0.1 ms�. m�0 and m�1 were found to be negligible, while m�−1

grew in amplitude until a stable limit cycle was achieved, in asimilar way to before. Hence the limit cycle was a standing wavewith a node at 0 deg �and therefore also at 180 deg�. The fre-quency was 519.8 Hz, identical to the case without the resonator,and the amplitude was 0.179m. �This limit cycle is, in fact, thesame as the standing-wave solution found in the frequency-domain calculations in Sec. 6.1.� As in linear theory, the resonatordoes not provide any damping �since the pressure oscillation at theneck is zero� but adjusts the circumferential-mode shape. Notethat the peak amplitude is, in fact, higher than without the reso-nator, although the circumference average is �2 /��0.179m=0.114m, which is lower than before. In practical applications, thepeak amplitude is likely to be the more critical in terms of thestructural damage caused; hence, the results indicate that using asingle Helmholtz resonator for a circumferential mode can poten-tially make the situation worse. The calculation was repeated withthe resonator at 45 deg azimuthally. In this case, m�0 was negli-

−1 1

0.3 0.32 0.34 0.36 0.38 0.4−0.01

0

0.01

t (s)

m’−

3 /m

0.3 0.32 0.34 0.36 0.38 0.4−0.2

0

0.2

t (s)

m’−

1 /m

0.3 0.32 0.34 0.36 0.38 0.4−0.2

0

0.2

t (s)

m’1 /m

0.3 0.32 0.34 0.36 0.38 0.4−0.01

0

0.01

t (s)

m’3 /m

Fig. 10 Time-domain solution for original geometry, with N=3 and fmax=2 kHz. m�n

„t… /m is shown for n=−3, −1, 1, and 3.„m�n

„t… is negligible for even n.…

gible, while both m� and m� grew to form a limit cycle of



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requency 519.8 Hz. The amplitudes of m�−1 and m�1 were both.126m, and they were in phase. This indicates a standing wave ofmplitude �2�0.126m=0.178m, with a node at 45 deg. Hence, ase would expect, the solution is virtually identical to that beforeut rotated by 45 deg. This provides a good check that coupledreen’s functions are working correctly in the numerical scheme.Calculations were then conducted using two Helmholtz resona-

ors for different azimuthal separations �both resonators havinghe same specifications and axial location as before�. With bothesonators at the same azimuthal location, as we would expect, theesults were the same as for a single resonator. For a separation of80 deg, the results were again the same �the symmetry of thetanding wave giving nodes at both 0 deg and 180 deg�. For a 45eg separation, the Helmholtz resonators can give damping. Forinear calculations there is still a single unstable mode. Its fre-uency is slightly shifted �to 522.3 Hz�, while its growth rate isignificantly reduced �to 31.3 s−1� indicating a reduction in thetrength of the instability as a result of damping by the resonators.The presence of the resonators also introduces additional linearodes; however, these modes are stable and therefore of little

nterest.� The resulting limit cycle still has the standing-waveorm, with a node halfway between the resonators �to within 0.5eg�. The frequency is now 521.6 Hz and damping from the reso-ators reduces the amplitude to 0.139m. The results for a separa-ion of 135 deg were very similar �the unstable mode having fre-uency 522.0 Hz and growth rate 30.1 s−1, and the limit-cycleaving frequency 520.9 Hz and amplitude 0.137m, with now anntinode halfway between the resonators�. For a 90 deg separa-ion, the damping is sufficient for there to be no linearly unstable

odes �the largest growth rate being −6.1 s−1� and so no insta-ility is seen in the time-domain calculations, the initial randomoise simply decaying away. �Note that the results in Ref. �17�ndicate that the best resonator separation for an n=1 mode ispproximately 90 deg.� Increasing N �i.e., including more circum-erential modes in the calculation� had a small but noticeable ef-ect. For instance, taking N=4 in the 45 deg separation case, therequency became 522.9 Hz and the amplitude �of m�−1� became.144m �with the amplitude of m�−2 being 0.002m, and the otheromponents being smaller still�. In all cases, the time-domain cal-ulations were in very good agreement with frequency-domainimit-cycle predictions.

ConclusionsA time-domain network model for nonlinear thermoacoustic os-

illations has been developed. In previous work, a time-domainonlinear flame model was converted to the frequency domainhen combined with the linear acoustic model. Here, the acoustic

odel is converted to the time domain then combined with aame model. Unlike the frequency-domain method, which as-umes a single dominant frequency, in the time domain the effectf harmonics and the interaction between different unstable modesan be investigated.

Time-domain calculations have been conducted on an exampleeometry using a simple saturation flame model. These were com-ared with frequency-domain results, and in all cases good agree-ent was found. Harmonics were not significant for this geometry

nd flame model, although if a harmonic happened to be close tonother unstable mode it could have an important effect. Whereore than one unstable mode was present �and provided oneode was not a harmonic of another� the interaction was such that

he final limit cycle corresponded to one or other of the modeslone, never a combination of modes. Additionally, for a circum-erential instability in an axisymmetric geometry, the spinning-ave form was always preferred to the standing mode.

cknowledgmentThis work was funded by the European Commission, whose

upport is gratefully acknowledged. It is part of the GROWTH



program, research projects ICLEAC: Instability Control of LowEmission Aero Engine Combustors �G4RD-CT-2000-0215� andLOPOCOTEP: Low Pollutant Combustor Technology Programme�G4RD-CT-2001-00447�.

NomenclatureGn,n�

� flow-response Green’s function, relating m�n

to Q�n�

m � mass flowp � pressureQ � heat releaset � time

Tflame � nonlinear flame transfer function, relating Qdto md

Tn,n�

� flow-response transfer function, relating mn to

Qn�

u � axial velocityw � circumferential velocityx � axial coordinate� � circumferential coordinate� � density� � complex frequency

Subscripts� �d � value for premix duct d

Superscripts

� �¯ � unperturbed value� �� perturbation

� �ˆ � complex perturbation, assuming factor of ei�t

� ��m� � frequency harmonic m� �n � circumferential mode n

References�1� Stow, S. R., and Dowling, A. P., 2004, “Low-Order Modelling of Thermoa-

coustic Limit Cycles,” ASME Paper No. GT2004-54245.�2� Paschereit, C. O., Schuermans, B., Polifke, W., and Mattson, O., 2002, “Mea-

surement of Transfer Matrices and Source Terms of Premixed Flames,” ASMEJ. Eng. Gas Turbines Power, 124, pp. 239–247.

�3� Krüger, U., Hürens, J., Hoffmann, S., Krebs, W., and Bohn, D., 1999, “Pre-diction of Thermoacoustic Instabilities With Focus on the Dynamic FlameBehavior for the 3A-Series Gas Turbine of Siemens KWU,” ASME Paper No.99-GT-111.

�4� Krüger, U., Hürens, J., Hoffmann, S., Krebs, W., Flohr, P., and Bohn, D., 2000,“Prediction and Measurement of Thermoacoustic Improvements in Gas Tur-bines With Annular Combustion Systems,” ASME Paper No. 2000-GT-0095.

�5� Hsiao, G. C., Pandalai, R. P., Hura, H. S., and Mongia, H. C., 1998, “Com-bustion Dynamic Modeling for Gas Turbine Engines,” AIAA Paper No. 98-3380.

�6� Hsiao, G. C., Pandalai, R. P., Hura, H. S., and Mongia, H. C., 1998, “Investi-gation of Combustion Dynamics in Dry-Low-Emission �DLE� Gas TurbineEngines,” AIAA Paper No. 98-3381.

�7� Dowling, A. P., and Stow, S. R., 2003, “Acoustic Analysis of Gas TurbineCombustors,” J. Propul. Power, 19�5�, pp. 751–764.

�8� Dowling, A. P., and Stow, S. R., 2005, “Acoustic Analysis of Gas TurbineCombustors,” Combustion Instabilities in Gas Turbine Engines: OperationalExperience, Fundamental Mechanisms, and Modeling �Progress in Astronau-tics and Aeronautics Vol. 210�, T. C. Lieuwen and V. Yang, eds., AIAA, NewYork, Chap. 13, pp. 369–414.

�9� Culick, F. E. C., 1971, “Non-Linear Growth and Limiting Amplitude of Acous-tic Oscillations in Combustion Chambers,” Combust. Sci. Technol., 3�1�, pp.1–16.

�10� Culick, F. E. C., 1989, “Combustion Instabilities in Liquid-Fueled PropulsionSystems: An Overview,” AGARD Conference Proceedings No. 450, Bath, UK,Oct. 6–7, Paper No. 1, AGARD, Neuilly-sur-Seine, France.

�11� Peracchio, A. A., and Proscia, W. M., 1999, “Nonlinear Heat-Release/AcousticModel for Thermoacoustic Instability in Lean Premixed Combustors,” ASMEJ. Eng. Gas Turbines Power, 121�3�, pp. 415–421.

�12� Pankiewitz, C., and Sattelmayer, T., 2003, “Time Domain Simulation of Com-bustion Instabilities in Annular Combustors,” ASME J. Eng. Gas TurbinesPower, 125�3�, pp. 677–685.

�13� Schuermans, B., Bellucci, V., and Paschereit, C. O., 2003, “ThermoacousticModeling and Control of Multi Burner Combustion Systems,” ASME Paper
No. GT2003-38688.
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�14� Bellucci, V., Schuermans, B., Nowak, D., Flohr, P., and Paschereit, C. O.,2004, “Thermoacoustic Modeling of a Gas Turbine Combustor Equipped WithAcoustic Dampers,” ASME Paper No. GT2004-53977.

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a time-domain network model for nonlinear thermoacoustic oscillations

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