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Nonlinear Instability Analysis for PartiallyPremixed Swirl FlamesBernhard Ćosića, Jonas P. Moecka & Christian Oliver Paschereita

a Hermann-Föttinger-Institut, Technische Universität Berlin, Berlin,GermanyAccepted author version posted online: 23 Dec 2013.Publishedonline: 20 May 2014.

To cite this article: Bernhard Ćosić, Jonas P. Moeck & Christian Oliver Paschereit (2014) NonlinearInstability Analysis for Partially Premixed Swirl Flames, Combustion Science and Technology, 186:6,713-736, DOI: 10.1080/00102202.2013.876420

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Combust. Sci. Technol., 186: 713–736, 2014Copyright © Taylor & Francis Group, LLCISSN: 0010-2202 print / 1563-521X onlineDOI: 10.1080/00102202.2013.876420

NONLINEAR INSTABILITY ANALYSIS FOR PARTIALLYPREMIXED SWIRL FLAMES

Bernhard Cosic, Jonas P. Moeck, and Christian OliverPaschereitHermann-Föttinger-Institut, Technische Universität Berlin, Berlin, Germany

Limit-cycle prediction of thermoacoustic instabilities for unstable practical gas turbine com-bustion systems is still a challenge for the gas turbine industry. The nonlinear stabilityanalysis is especially demanding for highly turbulent swirling flames with significant equiv-alence ratio fluctuations. In the present study, a partially premixed swirl-stabilized flameat a Reynolds number of approximately 35,000 is investigated. An experimentally obtainedflame describing function (FDF) is used for the determination of the thermoacoustic oscil-lation frequency and amplitude. Damping is obtained directly from measurements. Themulti-microphone method is used to determine the amplitude dependent transfer functionof the flame as well as the transfer function of the burner and the acoustic response of theboundary conditions. Solving the thermoacoustic modeling framework, with the measuredtransfer functions incorporated, yields frequency and amplitude of the self-excited limitcycle oscillation. The error between model and experimental results is thoroughly assessed.Measurements were made for various lengths of the combustion chamber exhaust gas tubeto verify the results for different frequencies and amplitudes. Good agreement is found forthe entire range of combustor lengths investigated. A sensitivity analysis of the linear flametransfer function to several operational parameters is provided allowing for an assessment ofthe limits of the nonlinear stability analysis approach. Furthermore, the effect of amplitudedependent damping is addressed.

Keywords: Flame describing function; Limit-cycle prediction; Premixed combustion; Stability analysis;Thermoacoustics

INTRODUCTION

Combustion instabilities often restrict the operational range of gas turbine combus-tion chambers and thereby affect the engine efficiency and the emission of pollutantsadversely (Bothien et al., 2013; Lieuwen and Yang, 2005). These mostly self-excitedthermoacoustic instabilities are caused by a positive feedback cycle between heat releaserate and pressure oscillations (Rayleigh, 1878). It is therefore important to predict the occur-rence and amplitude of combustion instabilities early in the design phase and to assess theeffect of different parameters on the thermoacoustic properties of the system.

Received 4 September 2013; revised 6 December 2013; accepted 13 December 2013.

Address correspondence to Bernhard Cosic, Hermann-Föttinger-Institut, Technische Universität Berlin,Müller-Breslau-Str. 8, 10623 Berlin, Germany. E-mail: Bernhard.Cosic@tu-berlin.de

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/gcst.

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Thermoacoustic excitation mechanisms can be broadly classified in two majorgroups, according to the heat release rate fluctuation driving mechanism: (i) velocity pertur-bations and (ii) equivalence ratio perturbations (Huang and Yang, 2009; Zinn and Lieuwen,2005). For a stiff fuel injection system, as commonly encountered in gas turbine burners,equivalence ratio perturbations can be taken to be a direct result of the velocity fluctua-tions at the injector location. The two excitation mechanisms lead to a frequency dependentresponse of the flame to flow disturbances commonly called flame transfer function (FTF).Due to the importance of the time lag between excitation and feedback for a construc-tive phase relation, fundamental analytical models of the flame response are mostly basedon a simple time-lag representation, which was introduced by Crocco and Cheng (1956).An experiments-based approach, which has been applied successfully to gas turbine burn-ers (Paschereit et al., 2002), is to measure the acoustic transfer matrix between pressure andvelocity at the entrance to the burner and downstream of the combustion zone.

The obtained FTF can be incorporated into a thermoacoustic modeling framework tocalculate linear stability limits of the flame–combustor system. Merk (1957) was one of thefirst to apply a frequency-domain type analysis to determine the stability of thermoacousticsystems. Today there are basically two approaches for the determination of linear stability.The first one uses the open-loop gain to assess system stability via a generalized Nyquiststability criterion (Sattelmayer and Polifke, 2003b). Alternatively, the dispersion relation ofthe system can be solved as an eigenvalue problem, which yields the unstable frequencies(Sattelmayer and Polifke, 2003a). The latter method formally requires all subelements inthe model to be evaluated at complex frequencies for the calculation of linear stability.However, if the data were obtained on the basis of experiments or numerical simulations,the response is known only at real frequencies. Here, the effect of the imaginary part of thefrequency can be either ignored, or it can be taken into account by using a physics-based ora parametric model (Schmid et al., 2013).

The type of network stability analysis described above is limited to the linear domain,i.e., to the temporal evolution of small perturbations from the mean state. Since the effectof a finite oscillation amplitude on the flame response is not accounted for, this approachcannot be used for the prediction of the limit-cycle amplitude or for nonlinear instabili-ties (triggering). Until today this remains a key challenge in combustion dynamics relatedresearch.

Although in gas turbines azimuthal modes are often dominant, recent numerical andexperimental work on combustion instabilities in annular combustors indicates that thedominant heat release rate fluctuations also for azimuthal modes occur as a response tomass flow fluctuations through the burner, induced by the circumferential pressure field(Bourgouin et al., 2013; Staffelbach et al., 2009; Wolf et al., 2012). It is therefore admissibleto measure the flame response to longitudinal perturbations and use this information in anacoustic model taking into account the three-dimensional acoustic field of the annular com-bustion chamber and possibly the plenum. This has been done for example by Schuermanset al. (2010), who measured the flame response to longitudinal perturbations in a single-burner rig at engine pressure and predicted the pressure oscillation spectrum in a gas turbinewith good accuracy. From these studies, it can be concluded that the flame describing func-tion measured as the response to longitudinal perturbations in a single-burner set-up isuseful for annular configurations.

The limit cycle amplitude is determined by an equilibrium of oscillation energy gainand loss in the system. Acoustic dissipation increases in most cases linearly with the oscil-lation amplitude. Since the acoustic pressure amplitude typically does not exceed 5% of

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NONLINEAR INSTABILITY ANALYSIS FOR SWIRL FLAMES 715

the static pressure in the combustion chamber, the acoustic field behaves linearly. Thus,only a nonlinearity in the response of the heat release rate fluctuation can lead to a finiteamplitude oscillation. The nonlinear response of the flame can be modeled by an amplitudedependent transfer function, introduced by Dowling (1997), the so-called flame describingfunction (FDF).

It was demonstrated in recent studies (Boudy et al., 2011; Krebs, 2013; Noiray et al.,2008; Palies et al., 2011) that experimentally obtained flame describing functions can beused to predict the limit cycle amplitude with reasonable accuracy. Noiray et al. (2008)investigated the nonlinear response of an unconfined ensemble of laminar Bunsen flamesin an experimental set-up with a variable plenum length. In addition to linear instability andamplitude saturation on the limit cycle, different dynamic phenomena such as triggering ormode switching could be represented by the flame describing function framework. Theapproach was extended to confined laminar flames (Boudy et al., 2011) and to turbulent,perfectly premixed swirl flames at relatively low Reynolds number (Palies et al., 2011).

In the work of Krebs et al. (2013), limit cycle oscillations of a turbulent, partiallypremixed, and perfectly premixed swirl flame were investigated in a setup very similar tothe one used in the present work. The nonlinear oscillation amplitude was calculated basedon flame describing functions from experiments and numerical simulations. Reasonablyaccurate results were obtained; however, for the partially premixed case, the flame describ-ing function, which was measured on the basis of OH∗-chemiluminescence, had to berescaled using the data from numerical simulations. The deviation of the modeled limitcycle amplitude from the measured one was partly attributed to the damping in the model.Also, only one configuration was considered, which makes it generally difficult to assessthe accuracy of the method.

All previous studies that have used experimentally determined flame describing func-tions to estimate the limit cycle oscillation amplitude made use of chemiluminescencesignals to assess the heat release rate oscillations of the flame. However, it is known that aquantitative link between the chemiluminescence of certain radicals, CH∗ or OH∗ for exam-ple, can be established—at best—for perfectly premixed conditions only (Higgins et al.,2001). Since practical combustion systems in gas turbines never meet this requirement, theapplicability of the flame describing function method remains an issue. This is addressedin the present work through the application of the multi-microphone method (MMM) forthe determination of the heat release rate fluctuations. Since this method rests on a directevaluation of the acoustic source and, thus, the heat release rate fluctuation, the degree ofpartial premixedness is irrelevant.

The present work extends the state of the art in flame describing function based non-linear network modeling in several ways. The nonlinear response of a technically premixedflame is studied up to normalized excitation amplitudes of u’/U = 1.4. To overcome theambiguity of chemiluminescence signals in partially premixed conditions, a purely acousticmethod for the determination of the heat release rate fluctuations is used. The measurementresults are compared to traditional chemiluminescence measurements that have been takensimultaneously. The swirl flame has a fairly high Reynolds number of 35,000. Estimatesfor the limit cycle amplitude are made for a range of combustor lengths, thus allowing fora realistic assessment of the accuracy of the method.

In addition, a sensitivity analysis of the linear flame transfer function to several oper-ational parameters is made. A recent study of Kim and Santavicca (2013) identified trendsof several operation parameters on the flame transfer function for two different swirl stabi-lized turbulent burners. One of the major findings of this study is that the flame response is

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very sensitive to changes in mass-flow and thermal power. The sensitivity of the responseof a laminar Bunsen flame was recently investigated by Duchaine et al. (2011) with respectto various parameters, such as the laminar burning velocity and the air inlet temperature.While from the authors’ results it can be seen that already a laminar flame is considerablysensitive to certain parameters, this aspect becomes even more severe in industrial applica-tions, where operational parameters cannot be controlled precisely as in a lab experiment;uncertain input parameters therefore will be common in real combustor configurations.The sensitivity analysis for a fully turbulent partially premixed swirl flame shown in thisstudy allows for an assessment of the limits of the stability analysis approach for industrialconfigurations.

Another novel aspect is that the acoustic damping of the system, which has been asource of uncertainty in previous studies, is accounted for by making use of experimentallydetermined boundary conditions and a measured burner transfer matrix. Furthermore, theeffect of amplitude dependent damping is addressed.

The remainder of this article is structured as follows. First, the theoretical backgroundis explained. The concept of linear and nonlinear thermoacoustic stability analysis and theapplied modeling approaches are briefly introduced. Subsequently, the experimental setupis described, and an assessment of the sensitivity of the linear flame response is presented.The measurement results are then compared to the linear and nonlinear model predictions.Finally, an assessment of the accuracy of the presented modeling methods and experimentalapproaches is made.

THEORETICAL BACKGROUND AND MODEL DEVELOPMENT

NETWORK MODELING

A well-known technique to represent thermoacoustic systems is network modeling.In this approach, the actual system is divided into several subsystems, each of which is rep-resented by acoustic response functions in the frequency domain. Figure 1 shows the set-upof a network model that is used in the present work. A subsystem is typically represented bya two-input–two-output system that is a mapping between two acoustic variables, usuallyacoustic pressure and velocity or the amplitudes of the upstream and downstream travelingwaves. The generic transfer matrix mapping the upstream acoustic state to that downstreamhas the form

[pds

uds

]=

[T11 T12

T21 T22

] [pds

uds

], (1)

where the elements of the transfer matrix, T11 etc., are complex functions of frequency,and p and u are Fourier transforms of the acoustic pressure and the axial particle velocity,

Figure 1 Network model used in the present work to represent thermoacoustic oscillations observed in the exper-imental test-rig. BTM represents the burner transfer matrix, F the flame transfer matrix/describing function, Zus

and Zds the acoustic impedance upstream of the burner and downstream of the flame, respectively.

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respectively. It is assumed that only plane waves propagate between two elements. In mostcases encountered, this is a valid assumption because the acoustic wavelengths associatedwith the dominant oscillations are large compared with the cross-sectional length scale.

Boundary conditions are represented as single-input–single-output systems, in formof an impedance/admittance or a reflection coefficient. With reference to Figure 1, forexample, the upstream boundary condition takes the form

p1 = Zusu1, (2)

where the impedance Zus is again a complex-valued function of frequency. The definitionof the boundary condition does not have to coincide with the actual physical boundary ofthe set-up. It may be defined at a suitable reference location, at which only plane wavespropagate.

One major advantage of the network modeling approach is that the frequencyresponse of the elements can be obtained by different methods, analytically, experimen-tally, or numerically, and thus offers great flexibility. An experimental determination ofthe upstream impedance, for instance, automatically comprises all relevant effects, such asdamping, temperature inhomogeneities, etc., that possibly affect the plane-wave acousticresponse.

Matching the acoustic variables at the interfaces of all subelements results in a linearhomogeneous system. For the network configuration in Figure 1, this takes the form

⎡⎢⎢⎢⎢⎢⎣

−1 Zus 0 0 0 0B11 B12 −1 0 0 0B21 B22 0 −1 0 00 0 F11 F12 −1 00 0 F21 F22 0 −10 0 0 0 −1 Zds

⎤⎥⎥⎥⎥⎥⎦

︸ ︷︷ ︸S

⎡⎢⎢⎢⎢⎢⎣

p1

u1

p2

u2

p3

u3

⎤⎥⎥⎥⎥⎥⎦ = 0, (3)

where Bij and Fij are the transfer matrix elements of the burner and the flame, respectively.Requiring the existence of a non-trivial solution for the acoustic variables at the interfacesthen gives rise to the dispersion relation

det S(ω) = 0. (4)

Solution of the dispersion relation yields the frequency (real part of ω) and the growth rateof the eigenmodes (negative imaginary part of ω).

The procedure described above, employing the flame transfer function for small dis-turbance levels, only gives information about mode stability. It is however often interestingto estimate the final oscillation level, the limit cycle amplitude. Since the flame is under-stood to be the dominant nonlinearity in the system (Dowling, 1997), the finite amplitudeeffect on the flame needs to be incorporated. This can be done by means of a flame describ-ing function, which in contrast to the flame transfer function has an additional dependenceon the velocity fluctuation amplitude. Using this flame describing function in the networkmodel yields a dispersion relation that also depends on the fluctuation amplitude (Boudyet al., 2011; Noiray et al., 2008; Palies et al., 2011):

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det S(ω, |u2|) = 0. (5)

Here, u2 is the Fourier transform of the axial velocity fluctuation upstream of the flame.An estimate for the limit cycle amplitude and frequency is then obtained by solving Eq. (5)with the additional constraint Im ω = 0. This corresponds to oscillations with zero growth,thus the limit cycle. Stability of the limit cycle furthermore requires ∂(Im ω)/∂|u2| > 0.

Although the dominant nonlinearity in thermoacoustic systems is the flame resonse,high oscillation levels may also introduce noticeable nonlinear effects in the acoustic com-ponents of the system. These nonlinear acoustic effects do not stem from the gas dynamicnonlinearity, as it is inherent in the Euler equations, but from the interaction with thehydrodynamic field. This effect is typically found at discontinuous variations of the cross-sectional area, for example, at a sudden expansion or at the outlet of a duct. These nonlinearacoustic effects can be incorporated in the same manner as the nonlinear flame response,i.e., by extending the transfer function or transfer matrix for the acoustic component to adescribing function with a dependence on the respective acoustic amplitude (Schuller et al.,2009).

FLAME RESPONSE

The effect of the flame is incorporated into the model by means of a flame transferfunction, quantifying the response in heat release rate oscillation to small disturbances inthe upstream velocity, or a flame transfer matrix, formally treating the reaction zone as anacoustic four-pole. The Rankine–Hugoniot relations are used to relate the flame transferfunction (Schuermans et al., 2010) to the acoustic flame transfer matrix (or vice versa) by

p3 = p2 (6)

u3 = [1 + (T3/T2 − 1)F (ω)]u2, (7)

where p denotes the Fourier transform of the acoustic pressure, T is the mean temperature,and subscripts 2 and 3 correspond to conditions upstream and downstream of the flame,respectively (see Figure 1). It is furthermore assumed that the extent of the domain of heatrelease is compact with respect to the acoustic wavelength. F is the flame transfer functionrelating normalized fluctuations in heat release rate (q/q) to normalized fluctuations invelocity:

F (ω) = q/q

u/u.

In case of finite amplitude levels, the flame transfer function is replaced by the flamedescribing function, which has an additional dependence on the velocity fluctuationamplitude: F (ω, |u|).

Schuermans et al. (2004) introduced a model for the transfer function of a partiallypremixed flame. This model was found to fit experimental flame response data from theset-up used in the present work reasonably well. The model reads

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NONLINEAR INSTABILITY ANALYSIS FOR SWIRL FLAMES 719

F = e−iωτ1 e−ω2σ 21 /2 − e−iωτ2 e−ω2σ 2

2 /2, (8)

where τ 1 and τ 2 correspond to two time delays, associated with different response mecha-nisms, and σ 1 and σ 2 are the corresponding standard deviations. Eq. (8) yields a maximumgain of unity, which is not observed in the experiments. Therefore, Eq. (8) is endowed withan additional gain when matching with experimental data.

TRANSFER FUNCTION MEASUREMENT

All elements in the network model for the present set-up can be measured with theMMM. This technique uses a decomposition of the plane acoustic field based on multi-ple microphone measurements in a duct, for frequencies sufficiently low so that only planewaves propagate (Paschereit et al., 2002). Using this method, the upstream reflection coeffi-cient is obtained from the microphone array upstream of the burner with acoustic excitationfrom the downstream end. Analogously, the acoustic boundary condition at the combus-tor outlet is determined from the microphone signals on the downstream side when usingexcitation from the upstream end. The burner transfer matrix is obtained from two excita-tion states, employing the speakers on both sides. Mono-frequency excitation is used withrecords of 16–32 seconds length for each frequency.

A significant advantage of using experimental response functions for these elementsis that a non-negligible amount of acoustic damping is associated with them. This dampingeffect is accurately incorporated into the model when using the measured response func-tions. However, as pointed out by, e.g., Schuller et al. (2009), the damping of the acousticboundary conditions can vary with the amplitude. To assess the influence of this effect onthe nonlinear stability analysis, the T12-element of the burner transfer matrix was measuredfor various amplitudes and frequencies because the acoustic velocity amplitude at the flameis for some instabilities in the order of the mean flow velocity. Additionally, the nonlinear-ity of the open-end reflection coefficient was examined, since, in case of a λ/4-mode, avelocity anti-node is situated at the open end of the combustion chamber. For λ/4-modesvelocity amplitudes in the range of six times the mean velocity amplitude were observed atthe combustion chamber exit.

The most important element of the burner transfer matrix is the element T12 becauseit determines the amount of acoustical damping. The other three elements, which werealso measured, can be theoretically deduced by modeling the burner as a compact acousticelement (Bellucci et al., 2005). Consequently, the element T21 is zero, T11 ≈ 1, and T22 isequal to the area ratio between the two acoustic layers connected by the transfer matrix.The measured burner transfer matrix fits reasonably well to this model.

On the left side of Figure 2 the normalized real part of the burner transfer matrix ele-ment T12, which represents the damping of the burner, is shown for various amplitudes andfrequencies, comprising 605 measurement points for 55 frequencies between 50 Hz and300 Hz . On the right side, the normalized imaginary part of T12 is shown, which is respon-sible for the end correction. Both values were normalized with the corresponding value ofthe respective frequency in the linear regime. The ratio between the values measured athigher amplitudes and in the linear regime is for both cases only weakly dependent on theexcitation amplitude, which is shown on the horizontal axis. For self-excited instabilities,amplitudes of up to u’/U = 1 are expected. In the frequency range investigated, a standarddeviation of +/ − 2% from the linear value has to be expected for the real and imaginary

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Figure 2 Real (left) and imaginary (right) part of the burner transfer matrix element T12 for various frequenciesand amplitudes. The vertical axis shows the ratio between higher amplitude values and the linear regime. Thehorizontal axis depicts the acoustic amplitude at the burner entrance.

part of T12. A deviation of 5% in the real and imaginary part of the T12-element affectsthe limit cycle amplitude in the presented study by approximately 1%. Thus only the linearresponse of the burner transfer matrix is taken into account.

As shown in Figure 3 the cold flow open end termination of the combustion cham-ber responds as it would be expected by assuming a typical Levine and Schwinger (1948)correction of the open end termination. The figure shows the isothermal reflection coeffi-cient in the frequency range of 50–300 Hz for various amplitudes of the acoustic velocity.However, around 90 Hz, 102 Hz, 180 Hz, and 204 Hz, the acoustic response deviatesfrom the typical open-end reflection coefficient. These deviations are related to an inter-action of the room acoustics and the combustion chamber exhaust tube. The reflectioncoefficient decreases with increasing amplitudes. The change in the reflection coefficientabsolute value is relatively small and will be therefore neglected in this study. As shown byRämmal and Lavrentjev (2008) and further investigated and modeled by Jörg et al. (2013),

Figure 3 Magnitude of the open-end reflection coefficient for various frequencies and acoustic velocityamplitudes.

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NONLINEAR INSTABILITY ANALYSIS FOR SWIRL FLAMES 721

the damping is significantly increased at the open end termination for the case of a hotjet expanding into a cold environment. Consequently, the reflection coefficient used forthe stability analysis was measured in the linear regime at hot conditions, the results arequalitatively very similar to the reflection coefficient shown in Figure 3.

The response of the flame is also measured with the MMM, in addition to the morecommon technique relying on a chemiluminescence signal. The acoustic method is con-sidered to be the more accurate one (Schuermans et al., 2010). It directly measures theacoustic expansion across the flame, which is proportional to the heat release rate fluc-tuation. The chemiluminescence method, in contrast, can be considered as reliable only inperfectly premixed conditions. With fluctuations in equivalence ratio, as in the present case,the chemiluminescence intensity responds to unsteadiness in the mixture ratio in additionto fluctuations in the heat release rate (Higgins et al., 2001).

The flame transfer matrix elements T11 and T22, relating acoustic pressure and veloc-ity downstream of the flame to those upstream, were calculated from the microphone andthe photomultiplier signals using Eqs. (6) and (7), while the off-diagonal matrix elementsare assumed to be zero (Bellucci et al., 2005). This assumption is simply a statement ofthe Rankine–Hugoniot relations, that hold for compact flames, i.e., for flames whose axialextent is short compared to the relevant acoustic wavelengths. In addition, for T21 ≈ 0, it isalso required that the fuel injection is stiff.

EXPERIMENTAL SET-UP

An overview of the set-up used for the combustion experiments is shown in Figure 4.The atmospheric test rig is equipped with a movable-block radial swirl burner that allowsfor variations in the swirl number (Leuckel, 1967). Fuel is injected axially, upstreamof a 0.17 m long mixing tube, thus creating technically premixed conditions at theflame. The spatial and temporal unmixedness is significant, which can be concluded from

Figure 4 Overview of the test rig.

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the comparison of chemiluminescence pictures and flame transfer functions of perfectlypremixed and technically premixed conditions (Bobusch et al., 2013). Flashback is avoidedthrough the help of a center body mounted in the mixing tube. An area jump with a ratio of17 represents the inlet to the combustion chamber. This area discontinuity induces vortexbreakdown with the associated recirculation zones and permits the flame to stabilize in theassociated shear layers. The flame tube in the combustion zone is made of quartz glass withan inner diameter of 0.2 m. This provides optical access for the photomultiplier tube (PMT),which is used to measure chemiluminescent light emission from the flame. In order to selectlight emitted from excited radicals, the PMT was equipped with a multichannel optical fiberwith several bandpass filters centered at 308 nm (OH∗), 407 nm (CO∗

2), 431 nm (CH∗), and515 nm (C∗

2).Upstream as well as downstream of the burner, five water-cooled condenser micro-

phones are used to for the plane wave decomposition of the acoustic field. The experimentalrealization of this technique in thermoacoustic systems, based on multiple, axially dis-tributed measurements of the acoustic pressure, is typically called the MMM. It is usedfor the identification of the acoustic properties of the boundary conditions, the burner, andthe flame. In addition to that, the forcing amplitude in terms of the acoustic velocity isdetermined with microphone measurements upstream of the flame. Originally, the wavedecomposition goes back to Seybert and Ross (1977) and Chung and Blaser (1980), whoused only two microphones for the characterization of absorbing materials and mufflers.Poinsot et al. (1986) extended the decomposition method by using multiple microphonesfor the measurement of the reflection coefficient of a premixed flame. This extension allowsfor a simple assessment of the measurement error by the comparison of the reconstructedacoustic field with the pressure measurements. Experimental characterization of transfermatrices of thermoacoustic elements for swirl stabilized flames using the MMM was firstapplied by Paschereit et al. (2002). To ensure good accuracy at low frequencies and avoidbreakdown of the method at frequencies for which half a wavelength fits between twomicrophones, the axial sensor locations are distributed non-uniformly. The error estimatedfor the MMM, by comparing the pressure measurements with the reconstructed acousticfield, were for both microphone sets well below 2% for the frequency range investigated.The burner transfer matrix was measured by loudspeaker excitation upstream and down-stream of the burner. In contrast to that, the flame transfer matrix was measured withupstream excitation only.

The test rig is equipped with four 18-inch 600-W subwoofers to provide for strongacoustic forcing. However, to obtain sufficiently high excitation amplitudes, it is still neces-sary to adjust the upstream tube length. This is achieved by using a trombone-like plenum(Schimek et al., 2011), which allows an extension of about 2 m in length.

For the describing function measurements, the combustion system must be ther-moacoustically stable. However, since the goal is to predict limit cycle frequency andamplitude of combustion instabilities, these measurements have to be made at operatingconditions where the system is nominally unstable. This was achieved by placing an ori-fice downstream of the 0.7 m long microphone exhaust tube, which significantly decreasesthe magnitude of the reflection coefficient, and thus increases the acoustic losses, when thecontraction ratio of the orifice is chosen in a specific manner depending on the flow velocity(Bechert, 1980). The orifice did not have an impact on the flow field of the flame becausethe reduced swirl number (Sρds/ρus, S being the swirl number and ρus and ρds the meandensities upstream and downstream of the flame, respectively) was well below 0.2, whichis a critical value for the impact of outlet boundary conditions on swirl flames (Terhaar

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NONLINEAR INSTABILITY ANALYSIS FOR SWIRL FLAMES 723

et al., 2012). For the measurement of self-excited instabilities, the downstream tube wasreplaced with an adjustable exhaust tube. The length of this exhaust tube can be varied con-tinuously in the range of 0.18 m to 1.1 m, and it was equipped with two microphones 0.3 mdownstream of the flame.

The natural gas–fired combustor was operated at an equivalence ratio of φ = 0.65 anda non-preheated air mass flow of 165 kg/h, resulting in 85 kW of thermal power. A Coriolismass-flow meter was used to measure the fuel mass flow, and a laminar flow element wasused for the air flow. The Reynolds number with respect to the hydraulic diameter of themixing tube was approximately 35,000. The theoretical swirl number (Leuckel, 1967) wasset to 0.7.

SENSITIVITY ANALYSIS

Since a nonlinear analysis with an industry relevant configuration was performed, itis interesting to assess the sensitivity of the flame transfer function shown later in Figure 12to several operation parameters. Gas turbine combustors operating in the field face differingfuel compositions, production tolerances, and operation influences. Caused by limited mea-surement accuracy and variations in the ambient conditions from day to day, the predictionprecision is limited, too. The susceptibility of the flame response to important operationparameters is especially important for the comparison to numerical and to engine data.Additionally, the sensitivity of the flame transfer function can be used for an assessment ofpassive control strategies.

Figure 5 shows on the left side the ratio of the reference flame transfer function,shown in Figure 12, and a transfer function measurement on a different day. The right sideof Figure 5 shows the corresponding phase difference of the flame transfer function. Bothmeasurements were conducted with identical conditions on different days. The repeatabilityof the flame transfer function gain is in the range of ± 5% in the frequency range between105 Hz and 280 Hz. The larger deviations for frequencies higher than 280 Hz are causedby the very small values of the gain in this frequency range. While the gain of the transferfunction is overestimated for frequencies lower than 150 Hz and underestimated for fre-quencies higher than 150 Hz, by one of the measurements, the phase difference is positivefor all frequencies. The phase differences observed are in the range of 10 degrees. The devi-ations can be caused by a number of changes in the ambient conditions, such as changes

Figure 5 Ratio of absolute transfer function values (left) and difference in phase of transfer function (right)measured on different days with an identical configuration but different ambient conditions.

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in the fuel quality, the cooling efficiency, ambient pressure, sealing quality, temperatureequilibrium of the test facility, and the limited measurement precision. Since the instabilitymeasurements and the flame describing function measurements were not conducted con-secutively but also, for some part, at different days, deviations observed in the range of atleast ± 5% have to be expected.

In order to assess the sensitivity of the flame transfer function to small perturbationsin the operating parameters without effects of day-to-day variations described above, thefollowing results were obtained from measurements immediately following each other inthe range of the operation point.

It is especially important for partially premixed flames to determine the amount offuel correctly. The amount of fuel has an impact not only on the mean thermal power butalso on the amplitude of the equivalence ratio fluctuations and the mixing between injectionpoint and flame front. Figure 6 illustrates the influence of a steady variation of ±3% of thefuel supply. Reducing the equivalence ratio by decreasing the amount of fuel leads to anincreased gain for both frequency regions of strong amplification in the transfer function.Around 100 Hz, the gain is increased by 10%, and for frequencies around 180 Hz, it isincreased by 5%. Contrary to that, by adding 3% fuel, the gain is decreased in the range of100 Hz for 10%. Both variations have a very strong impact around the frequency of 150 Hz,where the gain of the transfer function becomes small. This is to be expected since the slopeof the transfer function is very high around the local minimum at 150 Hz. Small changesin the position of the minimum lead to large deviations in gain. However, the effect onthe phase response indicates an impact on the time lags associated with the flame responseto acoustic perturbations. The time lags are affected because the mean flame position isvery sensitive to the fuel mass flow. While the phase response is diverging for increasingfrequencies, the gain is converging for the two perturbed operation conditions.

Figure 6 Flame transfer function with 3% increased and decreased fuel mass flow. Upper plot shows ratio oftransfer function gain and lower plot difference in phase of the transfer functions with respect to the baselinecondition.

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Figure 7 Flame transfer function with 3% increased and decreased air mass flow. Upper plot shows ratio oftransfer function gain and lower plot difference in phase of the transfer functions with respect to the baselinecondition.

Relatively similar, however differently emphasized, as shown in Figure 7, is the effectof changes in the air mass flow by 3%. Both, variations in the gas mass flow and the airmass flow, modify the equivalence ratio of the flame and thereby it is expected that thechanges are similar. Nevertheless, some differences can be observed because changes inthe air mass flow affect also the mean velocity, which additionally affects the time lags inthe flame response. The deviations around 150 Hz are more pronounced in terms of theflame transfer function gain and phase. Neither the gain nor the phase is affected by themodified air mass flow in the frequency range around 200 Hz. For higher frequencies, atwhich the gain is very low, the effect is again higher. The variations of the phase responseis, besides frequencies close to 150 Hz, of minor importance. Especially the region around100 Hz is very sensitive to changes in the equivalence ratio.

The precision of the gas mass flow meter of ±0.5% and the air mass flow meterof ±3%, introduces additional 5–24% of uncertainty for the flame transfer function gain.Moreover, the measured transfer matrices of the boundary conditions, the burner, and theflame are subject to measurement uncertainties in the order of 1% caused by the MMM.A cumulative error in the range of 0–33% is thus expected for the flame transfer functiongain. The error in the phase of the flame response is more pronounced for higher fre-quencies and is generally in the range between 10–30 degrees. These estimates are basedon repeated measurements of the linear transfer function, however, the nonlinear flameresponse is responsible for the amplitude of the limit cycle oscillation.

The sensitivity of the flame response to changes in the swirl number is importantdue to limited production precision for industrial combustors and for adjustable academicmodel combustors as well as for the comparison to CFD calculations. The swirl numberhas a pronounced effect on the flame transfer function and thus also on the stability char-acteristics, as recently shown by Durox et al. (2013). In the present study, increasing anddecreasing the swirl number by 5% leads to a change in the order of 10% of the FTF (upper

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Figure 8 Flame transfer function with 5% increased and decreased swirl number. Upper plot shows ratio oftransfer function gain and lower plot difference in phase of the transfer functions with respect to the baselinecondition.

plot in Figure 8). The differences of the phase of the flame response are generally smallerthan those associated with perturbations in fuel and air mass flows, and on average about±5 degrees but never exceeding ±10 degrees.

Another source of uncertainty lies in using the chemiluminescent light emission fromthe flame as a measure for the heat release rate. Capturing the light emission with excitedradicals using photomultipliers is sensitive to equivalence ratio fluctuations and is thereforenot suitable for quantitative measurements of the flame response in partially premixed sys-tems. However, this technique is widely used in industry and academia for the qualitativemeasurements of the flame response and for the quantitative measurement of the trans-fer function of perfectly premixed flames. Figure 9 shows the gain of the nonlinear flameresponse for 158 Hz for partially premixed combustion. Since the equivalence ratio fluctua-tions grow non-linearly with the velocity perturbation amplitude, the difference between thephotomultiplier signals and the acoustic method increases, too. The differences between theindividual chemiluminescence signals is caused by the differing sensitivity to equivalenceratio oscillations.

For the incorporation of the flame transfer function in a thermoacoustic modelingframework, the Rankine–Hugoniot relations must be applied [Eq. (7)] if the transfer func-tion is measured with a photomultiplier. The temperature needed for this relation is subjectto measurement uncertainty. The ratio of the upstream and downstream temperature of theflame directly affects the gain of the acoustic flame transfer function. In the presented exper-iments, the temperature was measured 0.3 m downstream of the flame in the center of theexhaust tube. As depicted in Figure 10, taking the temperature of 1200 K of the temperaturemeasurement point to calculate the acoustic transfer function G (G = uds/uus) underesti-mates the gain. Taking heat radiation and convection into account (Lechner and Bothien,2005), which yields a corrected temperature of 1350 K, leads to significantly more accurateresults. The OH∗ photomultiplier results match the MMM obtained transfer function well,besides the ranges around 150 Hz and 95 Hz. The remaining differences can be attributed

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Figure 9 Normalized fluctuation level at 158 Hz for various PMT signals compared to MMM measurement.

Figure 10 Comparison of linear flame transfer function obtained with MMM and PMT-OH∗ for three differentdownstream temperature estimates.

to the susceptibility to equivalence ratio fluctuations. However, using the adiabatic flametemperature of 1800 K overestimates the flame response significantly.

Especially important for the comparison of experimental and numerical results is thecorrect representation of the degree of partial premixedness at the flame. Figure 11 showsexemplarily the magnitude of the flame response at 180 Hz for various splits of partialto perfect premixedness. The degree of partial premixedness was varied by injecting partof the fuel far upstream (x/D > 20) of the burner. With increasing amplitude, the flameresponse starts to saturate around an amplitude of u′/U = 0.2 in all cases. However, with anincreasing amount of partially premixed fuel, an increase in the flame response magnitudeis observed. This indicates that heat release rate fluctuations associated with perturbationsin the equivalence ratio are in phase with fluctuations resulting from perturbations in the

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Figure 11 Flame response at 180 Hz obtained with the MMM for various degrees of partial premixedness (TPM).

flow field. In case of almost perfectly premixed conditions, the flame response is also qual-itatively affected since the response of the perfectly premixed flame increases in magnitudefor higher excitation levels.

EXPERIMENTAL RESULTS AND MODEL PREDICTION CAPABILITIES

In order to predict the limit cycle amplitude, the flame describing function is of cru-cial importance. It is presented in Figure 12. The upper part of the figure shows the gain

Figure 12 Measured linear flame transfer function (blue asterisks) and flame describing function for variousamplitudes (circles). The color of the circles indicates the forcing amplitude at the flame.

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of the amplitude dependent flame transfer function, and the lower part depicts the phaseresponse. The color of the respective markers indicate the acoustic velocity amplitude levelat which the flame response was obtained. Indicating an amplitude level of 10%, the blueline represents the linear flame transfer function measured at a frequency spacing of 3 Hz.Commonly, the linear flame transfer function is measured at an amplitude level up to 10%of the mean velocity.

For frequencies around 200 Hz, nonlinear effects are observed at smaller amplitudesalready. Except for frequencies around 160 Hz and 340 Hz, no significant effect on thephase response was observed. It is reasonable that the change in the phase response around160 Hz is caused by the increasing equivalence ratio fluctuations, since equivalence ratiofluctuations and velocity perturbations are not in phase in this frequency range. The changesin the phase response around 340 Hz are caused by a detachment of the flame at higherforcing amplitudes.

The two regions of high amplification around 100 Hz and 200 Hz show major changesin the gain of the flame response. For frequencies around 100 Hz, amplitudes up to u′/U= 1.4 were achieved, while the realizable amplitudes for frequencies around 200 Hz weresignificantly lower, at approximately u′/U = 0.75. Both regions have a gain of three forlow amplitudes, which is reduced down to around unity for high amplitudes. The flameresponds linearly to forcing around 100 Hz up to an amplitude of u′/U = 0.2. In contrast tothat, the response becomes nonlinear almost immediately for frequencies around 200 Hz.Already small velocity amplitudes cause a significant reduction of the gain in this frequencyregion. It is worth noting that the flame response is increased at higher forcing amplitudesfor frequencies around 145 Hz, which is likely caused by changes in the flame shape andthe mean flow field.

Figure 13 compares the flame describing function measured with the MMM, the pho-tomultiplier results, and the n−τ model [Eq. (8)] for a velocity amplitude of u′/U = 0.63.The parameters of the n−τ model were determined to match the results of the MMM in

Figure 13 Comparison of FDF obtained by MMM and PMT-OH∗ and modeled FDF at an amplitude level ofu′/U = 0.63.

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the frequency range between 100 Hz and 200 Hz, which is the range where the instabilitiesoccur. Flame describing function modeling is used because there is only a very limitednumber of measurement points for amplitudes higher than u’/U = 0.5 available. The n−τ

model smoothes the data, and it is ensured that the flame describing function preserves aphysically meaningful shape, which is lost when applying interpolation between widelyscattered measurement points.

The inaccuracy of the photomultiplier measurements for partially premixed flamescauses a qualitative change of the describing function shape. While the MMM showsthe presence of a node at approximately 160 Hz, the OH∗-photomultiplier suggests a flatresponse at this amplitude level. Depending on the phase between fluctuations in equiva-lence ratio and velocity, the flame response is either overestimated or underestimated bythe photomultiplier measurements. In the case of a constructive phase relation of the twocontributions to the chemiluminescence intensity, the response is overestimated, and for adestructive phase relation it is underestimated.

LIMIT-CYCLE AMPLITUDE PREDICTION

The comparison of the predicted dominant limit cycle amplitudes at the flame, whichwere obtained by the amplitude dependent n−τ model, and the measured dominant insta-bility amplitudes is shown in Figure 14. For both simulation and experimental results, thecolor of the markers indicate the limit cycle amplitude of the dominant instability frequencyin terms of normalized acoustic velocity at the flame.

For the measurement of the self-excited instabilities, the length of the variable down-stream tube was gradually increased from 0.6 m to 1.41 m. The combustor became unstablefor a length of 0.78 m with a frequency of 176 Hz and remained unstable up to the max-imum length of 1.41 m. Throughout the unstable domain, high amplitudes were observed,with decaying frequencies for a combustion chamber length between 0.78 m and 0.94 m.

Figure 14 Comparison of measured and calculated dominant limit cycle frequencies and amplitudes for variouscombustion chamber lengths.

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The observed frequencies correspond to a λ/4-mode in the downstream section. For furtherincreased lengths, an upstream mode was unstable at a constant frequency of 162 Hz, and adecaying amplitude up to a length of around 1.1 m. For larger combustor lengths, the insta-bility is characterized again by a downstream mode with decaying frequency, starting at120 Hz. Around 1.1 m, two frequencies were observed; this phenomenon will be discussedlater.

The model results show a very good quantitative agreement in terms of frequency andamplitude for the whole range of investigated combustor lengths. Decaying frequencies aswell as the two mode jumps around 0.95 m and 1.1 m are predicted in good agreement withthe experimental results. In contrast to the measurements, the model predicts a decayingfrequency for combustor lengths between 0.97 m and 1.08 m. At 0.8 m, the measurementof the instability amplitude shows a discontinuity where the amplitude level drops from0.65 u’/U to an amplitude of 0.4; this discontinuity is not captured by the model.

Figure 15 allows for a systematic assessment of the modeling error. It shows theabsolute deviation of frequency and amplitude for the whole range of investigated com-bustor lengths. The absolute deviation in terms of frequency is in the range of −10 Hz to10 Hz with a peak deviation of 12 Hz. The median of the relative error for all measurementpoints, which is defined as f /f 0, is 3.5%. Generally, the absolute deviation in terms ofvelocity oscillation amplitude is in the range of −0.1 < u’/U < 0.1, with a peak devia-tion of u’/U = 0.33. The corresponding median of the relative error is 14.6% for all datapoints. Especially around the mode jump at 1.1 m, the prediction deviates strongly from themeasurement in terms of frequency as well as amplitude.

The amplitudes of the secondary frequencies, which were observed around a com-bustion chamber length of 1.06 m–1.1 m are presented in Figure 16. The amplitudes andthe frequencies were accurately predicted by the describing function analysis. Only verysmall secondary peaks at 1.06 m and 1.08 m were not predicted by the model. However, asecondary peak at 1.07 m, which was predicted by the calculation, could not be observed

Figure 15 Absolute deviation of model results and measurements in terms of frequency and limit cycle amplitudefor various combustion chamber lengths.

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Figure 16 Comparison of measured and calculated limit cycle amplitudes and frequencies for two unstable modesfor various combustor lengths.

in the measurements. The two unstable modes were not present simultaneously but ratherquickly alternating in time.

SUMMARY AND CONCLUSIONS

It is important to predict the frequency and the limit cycle amplitude of combus-tion instabilities early in the design phase of combustors. Experimentally obtained flamedescribing functions can be used for the determination of the thermoacoustic oscillationfrequency and amplitude. Until now, this has not been demonstrated for industry relevantcombustion systems.

In the current study, nonlinear stability analysis was applied to an atmospheric,fully turbulent technically premixed swirl combustor for various combustion chamberlengths. The flame response was obtained using the multi-microphone method as well asOH∗-photomultiplier measurements for various forcing amplitudes. The acoustic burnertransfer function and the impedance upstream and downstream of the burner were alsoobtained with the MMM. The acoustic responses of all subsystems were incorporated intoa thermoacoustic modeling framework to determine frequency and amplitude of the self-excited limit cycle oscillation. In order to verify the simulation results, measurements weremade for various lengths of the exhaust gas tube.

An amplitude dependent n–τ model was used to obtain a physically meaningful flameresponse for higher amplitudes, which is not possible with interpolation between widelyscattered measurement points. The results of the acoustic network model are in good agree-ment for the entire range of combustor lengths investigated. The median of the deviationin terms of frequency was smaller than 4% while the median of the deviation in terms ofinstability amplitude was lower than 15%. The flame describing function was also able toreproduce secondary peaks in amplitude and frequency with good accuracy.

An extensive sensitivity analysis revealed that significant deviations and errors areintroduced in the flame transfer function measurement even by very accurate mass flow

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meter systems. It was also shown that nonlinearities at the acoustic boundary conditionsintroduce additional error in the single digit range. When applied to real engines, inaccura-cies caused by production tolerances limit the prediction capabilities further. Unavoidabledifferences, between analytical model and CFD on the one side and the experiment on theother side, in the degree of partial premixedness and the temperature in the combustionchamber, limit the accuracy additionally. The comparison between the photomultiplier andmicrophone measurements revealed that the former can be even qualitatively wrong in thecase of a partially premixed burner.

The presented methodology is proven to be reliable for industry relevant combustionsystems with a reasonable degree of accuracy. At least for natural gas fired combustors, theresults can be scaled to high pressure conditions. However, the correct assessment of thesystem damping for realistic geometries and engine conditions remains challenging.

ACKNOWLEDGMENTS

The authors appreciate the help of Andy Göhrs, Johann Vinkeloe, and the CONFETmembers for their assistance in the lab and helpful discussions.

FUNDING

Financial support from the Research Association for Combustion Engines(Forschungsvereinigung Verbrennungskraftmaschinen e. V.–FVV) is gratefullyacknowledged.

NOMENCLATURE

p acoustic pressureu acoustic velocityu′ acoustic velocity fluctuation amplitudeT transfer matrix elementZ acoustic impedanceB burner transfer matrix elementF flame transfer matrix elementS system matrixω complex frequency in rad/sF flame transfer functionFTF flame transfer functionFDF flame describing functionq heat release rateq’ heat release rate fluctuation amplitudeQ mean heat release rateτ time delayσ standard deviationS swirl numberρ densityMMM multi-microphone methodG acoustic flame transfer function

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U mean velocityD diameter of mixing tubex axial positionφ equivalence ratioPMT photomultiplier tubeλ acoustic wave length

Subscripts and superscripts

ds downstreamus upstream Fourier transform

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