Modal dynamics of self-excited azimuthal instabilities in an annular combustion chamber

Combustion and Flame 160 (2013) 2476–2489

Contents lists available at SciVerse ScienceDirect

Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e

Modal dynamics of self-excited azimuthal instabilities in an annular combustion chamber Nicholas A. Worth, James R. Dawson ⇑ Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK

a r t i c l e

i n f o

Article history: Received 8 October 2012 Received in revised form 18 April 2013 Accepted 30 April 2013 Available online 8 June 2013 Keywords: Azimuthal instabilities Circumferential instabilities Spinning modes Standing modes Flame–flame interactions Annular combustion chamber

a b s t r a c t In this paper we describe the time-varying amplitude and its relation to the global heat release rate of self-excited azimuthal instabilities in a simple annular combustor operating under atmospheric conditions. The combustor was modular in construction consisting of either 12, 15 or 18 equally spaced premixed bluff-body flames around a fixed circumference, enabling the effect of large-scale interactions between adjacent flames to be investigated. High-speed OH⁄ chemiluminescence imaged from above the annulus and pressure measurements obtained at multiple locations around the annulus revealed that the limit cycles of the modes are degenerate in so much as they undergo continuous transitions between standing and spinning modes in both clockwise (CW) and anti-clockwise (ACW) directions but with the same resonant frequency. Similar behaviour has been observed in LES simulations which suggests that degenerate modes may be a characteristic feature of self-excited azimuthal instabilities in annular combustion chambers. By modelling the instabilities as two acoustic waves of time-varying amplitude travelling in opposite directions we demonstrate that there is a statistical prevalence for either standing m = 1 or spinning m = ±1 modes depending on flame spacing, equivalence ratio, and swirl configuration. Phaseaveraged OH⁄ chemiluminescence revealed a possible mechanism that drives the direction of the spinning modes under limit-cycle conditions for configurations with uniform swirl. By dividing the annulus into inner and outer annular regions it was found that the spin direction coincided with changes in the spatial distribution of the peak heat release rate relative to the direction of the bulk swirl induced along the annular walls. For standing wave modes it is shown that the globally integrated fluctuations in heat release rate vary in magnitude along the acoustic mode shape with negligible contributions at the pressure nodes and maximum contributions at the pressure anti-nodes. Crown Copyright Ó 2013 Published by Elsevier Inc. on behalf of The Combustion Institute. All rights reserved.

1. Introduction The occurrence of self-excited thermo-acoustic instabilities is a well known problem in gas turbine combustion systems [1]. In annular combustion chambers the first azimuthal mode is typically the most unstable [2,3] as the circumference is greater than the length of the combustor. Azimuthal modes become self-excited when fluctuations in the heat release rate couple with acoustic waves and propagate azimuthally in both directions around the combustion chamber. This gives rise to large pressure fluctuations which have the potential to cause significant structural damage and therefore it is highly desirable to eliminate instabilities during the design phase. Currently, this is not possible as we do not possess the necessary fundamental understanding of the relevant physical mechanisms that excite and drive azimuthal modes. A large part of this problem has been the lack of data from well controlled laboratory experiments needed to elucidate the behaviour ⇑ Corresponding author. E-mail address: [email protected] (J.R. Dawson).

of self-excited azimuthal instabilities. The aim of this paper is to begin to address this problem. For quite some time theoretical and experimental research has focused on laminar and turbulent axisymmetric flames subjected to longitudinal oscillations [4–15]. The standard approach has been to develop low-order acoustic models that test the linear stability of a given system [5,9] taking advantage of the fact that the acoustics remain linear although the presence of a turbulent flame renders the problem fundamentally non-linear. The non-linearity is generally ascribed to the flame response which is simplistically modelled in the form of a transfer function and then incorporated into linear network models. More recently advanced non-linear approaches such as the flame describing function (FDF) have been adopted which can capture the finite amplitude response of the system, i.e. limit-cycle amplitudes [6,16,17] as well as mode switching [18]. All of these approaches require validation against experiments, ideally from those obtained under realistic operating conditions. Comparatively less research has gone into low-order and nonlinear modelling of azimuthal modes in annular combustion cham-

0010-2180/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier Inc. on behalf of The Combustion Institute. All rights reserved. http://dx.doi.org/10.1016/j.combustflame.2013.04.031

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

bers [19,3,20,21]. In these models the non-linear response of the flame is assumed to behave as if it were a single axisymmetric flame subjected to longitudinal oscillations. A corollary of this is that large-scale interactions with neighbouring flames do not occur. The validity of these assumptions has been the subject of recent research in both industrially relevant and simplified geometries [22–26] and analytically by [27]. Collectively, these studies indicate the that flame response to transverse forcing is inherently different from longitudinal forcing. Simulating the response at the nodal locations of a standing azimuthal mode, O’Connor and Lieuwen [25] found that the flow response of an annular swirling jet changes from an axisymmetric mode when positioned at the pressure anti-node to a helical type mode at the pressure node. From an analytical perspective, Acharya et al. [27] considered the effect of swirl on axisymmetric premixed laminar flame when subjected to transverse acoustic excitation. Their model suggests that the heat release rate produced at pressure node is negligible and maximum at the pressure anti-node. Staffelbach et al. [28] performed a LES of a full annular geometry with realistic burners which resulted in self-excited azimuthal spinning modes and demonstrated that the thermo-acoustic response of each burner was the same, i.e. azimuthal waves drive uniform longitudinal pulsations through each of the burners. A recent LES simulation by the same group [29] showed over a limited time-series that self-excited azimuthal modes alternated between standing and spinning modes. Using a dynamical systems approach Noiray and Schuermans [30] found that modelling azimuthal instabilities with Van der Pol oscillators can capture these modal dynamics however this approach cannot offer any insight into the response of the heat release rate to these different modes. An experimental study of self-excited azimuthal instabilities in a laboratory scale annular combustion chamber with different flame spacings was recently carried out by Worth and Dawson [31]. High-speed OH⁄ chemiluminescence was used to visualise the heat release rate response to spinning modes. The effect of flame spacing resulted in large-scale modifications to the mean and phase-averaged heat release rate around the annulus. The authors also reported that similar modal dynamics to Wolf et al. [29] were observed. Clearly, a better physical understanding of azimuthal modes is needed. An emerging issue is whether or not self-excited azimuthal modes exhibit a preference for standing or spinning modes or whether modal dynamics are a characteristic feature of azimuthal instabilities in annular combustion chambers. Building upon our previous work [31] this paper presents an experimental investigation into the modal dynamics of self-excited azimuthal

2477

instabilities in a model annular gas turbine combustor with 12, 15 and 18 flames. The apparatus, operating conditions and measurement techniques used in this study are discussed in Section 2. The effect of swirl configuration and flame spacing on the mean flame structure obtained from high-speed OH⁄ chemiluminescence viewed from above the annular combustor are presented in Section 3. Section 4 describes the methodology used to identify the spinning and standing wave modes from pressure time-series data enabling a statistical description of the modal dynamics as a function of the different experimental configurations. Finally, the response of the phase-averaged fluctuations in heat release rate for standing and spinning wave modes in Section 5 are analysed and discussed.

2. Experimental methods 2.1. The model annular combustor A general schematic and photograph of the annular combustor is shown in Fig. 1 and is comprised of a number of equally spaced bluff-body stabilised turbulent premixed flames with swirl. Premixed ethylene–air was fed into a cylindrical plenum chamber of inner diameter Dp = 212 mm and length Lp = 200 mm which contains a honeycomb flow straightener and a series of grids for both flow conditioning and acoustic damping. A hemispherical body of diameter Dh = 140 mm was positioned inside the plenum to converge the flow at the entrance to each of the inlet tubes. Each inlet tube was 150 mm long with an inner diameter of D = 18.9 mm and was fitted with a centrally located conical bluff-body of diameter Dbb = 13 mm with a half angle of 45° resulting in a blockage ratio of 50% at the inlet to the annular combustion chamber. The bluff bodies were arranged around a circle of radius R = 85 mm, as shown in Fig. 2, and fixed between upper and lower plates such that they were flush with the annular combustion chamber. To investigate the effect of flame spacing three sets of upper and lower plates were manufactured with the same circumference to hold either 12, 15 or 18 evenly spaced bluff bodies. Since the average speed of sound, c, is fixed by the averaged temperature of the burned gases, the circumference was kept constant to fix the azimuthal acoustic length of the annulus enabling the effect of flame spacing on the excited modes to be investigated. Three flame separation distances S = 1.56D, 1.87D and 2.33D, where S is defined as the arc distance between the bluff-body centres and D is the inner diameter of the inlet tube were investigated.

Fig. 1. Photograph and schematic of the annular combustion chamber.

2478

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

Fig. 2. Left: Overhead schematic illustrating the location of the pressure measurements P1, P2, and P3, flame separation distance S, and smoothing kernels used to analyse the high-speed OH⁄ chemiluminescence data discussed in Section 5. Right: Swirl configurations and swirler specifications. Note: swirl configuration A shows the bulk swirl directions that are induced at the inner and outer annular walls.

Details of the swirler design and swirl configurations investigated are shown in Fig. 2. The swirler consisted of six a = 60° aerodynamically profiled vanes positioned 10 mm upstream of the bluff body. The swirl number based on the swirler dimensions was R = 1.22 according to the expression R ¼ 23 tan a½1  ðdi =do Þ3 = 1  ðdi =do Þ2 , and di and do are the inner and outer swirler diameters. It is important to note that the level of swirl at the exit plane will reduce due to as the flow is converged. Two different swirl configurations were investigated as shown Fig. 2. The first configuration was uniform anti-clockwise (ACW) swirl with each bluff body equipped with the same swirler. Importantly, this configuration induces a bulk swirling flow at each of the annular walls but in opposite directions as denoted by the large arrows in the figure. In the second configuration consecutive bluff bodies around the annulus are fitted with swirlers of alternating direction to eliminate any bulk swirl. The annular enclosure comprised of two concentric stainless steel tubes with the inner and outer annuli being 127 mm and 212 mm in diameter respectively. The outer annulus could also be modified to accommodate a quartz tube as shown in the photograph in Fig. 1. A parametric study was initially carried out to find self-excited circumferential modes with the salient results summarised in Section 2.3.

2.2. Operating conditions and instrumentation Reactant flow rates were controlled using four Alicat mass flow controllers accurate to 0.8% of the reading ±0.2% of the full scale. A constant bulk velocity of 18 m/s was maintained at the exit of each bluff-body giving a Reynolds number of 1.5  104 based on bluffbody diameter. Hot wire measurements were made at 4 quadrant locations to ensure that the flow was uniform to within 1%. Varying the number of flames varies the thermal load however the rationale behind this was to maintain the same inlet boundary conditions of each individual flame, the mean burned gas temperature, and hence c in order to ensure that the resulting spatial variations in heat release rate were due to changes in S only. The equivalence ratio was varied between / = 0.64–1.0 in order to promote self-excited oscillations and characterise the system response. Three pairs of high sensitivity Kulite XCS-093 pressure transducers (sensitivity: 4.2857  103 mV/Pa, range: 0.35 atm, accuracy: ±0.15% full scale) were positioned 120° apart as denoted by

positions P1, P2, and P3 in Fig. 2 to characterise the instability modes. At each location a pair of microphones were mounted flush with the inside walls along tube length to measure both the pressure time-series for mode classification and also the amplitude of the instabilities. The pressure signals were amplified and filtered prior to calculating the magnitude of the velocity fluctuation using the two-microphone technique [32]. Pressure time-series were acquired at 30 kHz with sample lengths of 4.3 s and digitised using a National Instruments 16 bit PCI 6251 card. The signals were analysed spectrally using the fast Fourier transform in order to determine the frequency centred complex amplitude of the velocity, A ¼ u0 ðf Þ= U . To improve the signal to noise ratio the signal was divided into 10 equal sections before spectral analysis and the mean used, providing a frequency resolution of 2.2 Hz.

2.3. Parametric considerations The aim was to find a set of experimental conditions which gave rise to strong self-excited circumferential modes. Given the well known difficulty in producing circumferential modes in the laboratory the parameter space was kept deliberately large and included varying the inner (Li) and outer (Lo) combustor lengths, addition of swirl, fuel type, and flame spacing (S). As mentioned earlier different S’s were achieved by retaining the same annular dimensions and changing the number of flames. Initial tests were carried out using premixed CH4–air mixtures but only longitudinal combustor modes were found. Given that the frequency of the m = 1 mode was expected occur around 1700–1800 Hz, based on the circumference and the adiabatic temperature for the range of / investigated, C2H4–air mixtures were adopted due to their higher flame speeds SL [33]. Different combinations of inner (Li) and outer (Lo) tube lengths of 70, 130, and 300 mm were also tested. Their lengths were based on acoustic considerations, the 70 mm tube lengths were selected to provide a short enough enclosure to damp longitudinal combustor modes whereas the 130 mm tubes were selected to provide an approximate match between the longitudinal and circumferential modes based on estimates of the adiabatic flame temperature. The longest length of 300 mm, whose fundamental mode was below the first circumferential mode, was selected based on [34] who found that a long tube excited azimuthal modes in a liquid fuelled rocket motor configuration. Most combinations resulted

2479

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

in various strong self-excited axial modes with and without swirl. For Lo = Li = 70 mm all instabilities were damped. The strongest self-excited circumferential oscillations occurred when using swirl and the annular enclosure was comprised of inner and outer tubes lengths of Li = 130 mm and Lo = 300 mm respectively. In the absence of swirl, no combination of Lo and Li resulted in self-excited circumferential instabilities suggesting that the mean flame shape and swirl-associated instabilities are important [17,24,26]. Typical examples of pressure (p0 ) and fluctuating heat release rate (q0 ) time-series and their spectra measured at P1, P2, and P3 for the S = 1.56D configuration, ACW swirl configuration are shown in Fig. 3. The first thing to notice is the sinusoidal nature of p0 shown in Fig. 3a and that each signal is 120° out-of-phase. The time-series of q0 on the other hand is not sinusoidal but clearly shows that the oscillations are also 120° out-of-phase. Note that the frequency resolution of the OH* chemiluminescence measurements is 14.4 kHz is less than half of the pressure measurements. The frequency peak in Fig. 3b and the relative phase lag between P1, P2, and P3 shows that this is the first azimuthal mode and that it is spinning in the ACW direction. The spectra of both p0 and q0 do not exhibit any higher harmonics which suggests that these types of instabilities may be amenable to non-linear modelling using the flame describing function approach [16,18]. Note that although the spectral peaks appear discrete in these figures the energy is spread over a frequency range of 20 Hz.

information. For the self-excited cases 14,328 images (full buffer) were captured at a frame rate of 14.4 kHz at a reduced resolution of 628  640 pixels. A calibration plate was used for image dewarp and minimise perspective effects. Analysis of the pressure time-series data revealed that the instability mode spontaneously switches between m = 1 and m = ±1 circumferential modes and that there was a small modulation in the resonant frequency. Both of these effects needed to be taken into account when compiling phase-averages of the global heat release rate. The fact that the instability frequency is never an exact multiple of the camera acquisition frequency generates a relative drift between the fixed frame rate of the camera and the instability frequency, and therefore acquiring a large dataset enabled the calculation of phase-averaged information without the need for triggering. However, prior to compiling phase averages a procedure was needed to classify the instability modes which is described in detail in Section 4. Once the modes were classified into standing modes and spinning modes according to their direction, the pressure oscillation peaks from one of the sensor locations at P1, P2, and P3 were located and each pressure cycle was normalised by the local period of the oscillation and divided into 18 equal non-dimensional time bins. The corresponding chemiluminescence images were then binned accordingly. Each normalised time bin was populated with a minimum of 200 images.

2.4. High-speed chemiluminescence imaging

3. Mean flame structure

To measure the global heat release rate the full annulus was visualised from overhead via an air cooled mirror immersed in the exhaust gases and angled at 45° [31]. Chemilunimescence images were captured using a Photron SA1.1 high-speed CMOS camera having a 10242 maximum pixel resolution at 5.4 kHz coupled with a LaVision IRO high-speed two-stage intensifier, fitted with a Cerco 2178 UV lens 100F/2.8 and a UV filter (270–370 nm). A total of 2000 images were obtained at a frame rate of 0.5 kHz for averaged

Figure 4 shows the spatial distribution of the normalised mean heat release rate under stable conditions evaluated from 2000 OH⁄ images for three flame separation distances S = 1.56D, 1.87D and 2.33D with ACW swirl configuration. Figure 4d shows the effect of alternating swirl for the S = 1.56D case. A circular mask was applied to cover reflections from the inner wall of the outer enclosure duct, and also to cover the region obscured by the inner enclosure duct as a result of camera perspective.

2000

p2

p1

200

p3

1000 100 0

−1000 0

1

2

0

0

1000

2000

3000

4000

5000

0

1000

2000

3000

4000

5000

−3

x 10

0.15 q1

q2

0.02

q3

0.015 0

0.01 0.005

−0.15

0

1

2

0

−3

x 10

Fig. 3. Pressure and integrated OH⁄ chemiluminescence time-series of a typical self-excited azimuthal mode measured at locations P1, P2, and P3. (a) Exhibits a phase difference of 120° and (b) the resonant frequency of the m = +1 mode for the 18 flame (S = 1.56D) configuration respectively. (c) and (d) Show the corresponding fluctuating heat release rate and spectra. Operating conditions: LM enclosure, U = 18 m s1 and / = 0.85.

2480

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

Fig. 4. The effect of flame spacing, S, on the mean global OH⁄ chemiluminescence viewed from the top of the annulus under stable conditions. (a)–(c) Are for the ACW swirl configuration whereas (d) shows the effect of alternating swirl. The bluff-bodies are visible as the blue discs. U = 18 m s1, / = 0.80. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In all cases the bluff body positions appear as circular regions of negligible (blue) global heat release rate in their wakes. Clearly the effect of S on the mean flame structure is dramatic. For S = 2.33D the heat release rate structure is consistent with individual swirl flames, albeit not perfectly uniform. Some large-scale interactions with neighbouring flames are observed where there are interconnected regions of heat release rate especially towards the inner annular wall. On the other hand, for S = 1.56D and 1.87D largescale flame merging occurs between adjacent flames around the whole annulus as shown by the location of the peak intensity regions as found by [35,31]. The mean flame structure is now highly asymmetric with the majority of the heat release rate occurring in the merged regions and comparably little along the inner and outer walls. The effect of the swirl configuration on the mean flame structure can be discerned by comparing Fig. 4c and d. With ACW swirl the merged regions are angled in the ACW direction at the outer wall and in the CW direction at the inner wall, i.e. the two bulk swirl directions shown in Fig. 2. Alternating swirl results in a more radially symmetric distribution of the merged flame regions around the annulus. The effect of S on the structure of the mean heat release rate can be further assessed by considering the averaged circumferential distribution of the heat release rate about each of the bluff body flame holders. As shown in Fig. 5e, an inner (light grey) and outer (dark grey) annular region was defined between rbb < r < 2D/3 and 2D/3 < r < D respectively where rbb is the bluff body radius. The average circumferential heat release rate hQ(b)i was calculated by integrating the OH⁄ chemuliminesence over each the local azi-

muthal angle, b, in 10° increments and normalising by the maximum of the mean spatially averaged heat release rate, hQmaxi. Figure 5a shows that for the S = 2.33D configuration hQ(b)i tends to a constant value in the inner region indicating that the flame is axisymmetric in the vicinity of the bluff body. The outer annular region shows some asymmetric variations in hQ(b)i at b/p = 1.1 and b/p = 1.7 however their annular location means they are not caused by large-scale flame–flame interactions and most likely result from the variation in confinement with R. As S is reduced the effect of flame merging results in significant large-scale asymmetry with peak values located in the merging regions. The inner region for S = 1.87D retains a relatively constant values of hQ(b)i but not for the closest flame spacing, S = 1.56D. The peaks in the hQ(b)i broaden as S decreases as the flow in the merging region is more developed. The alternating swirl configuration results in a more uniform distribution throughout.

4. Characterisation of instability modes 4.1. Modal dynamics A recent LES simulation of self-excited circumferential instabilities in a full annular combustor by [29] found that the instability mode spontaneously switched between spinning and standing circumferential modes. Similar modal dynamics were also observed in our experiments but observed over a longer times. A typical pressure time-series is shown in Fig. 6 and are characterised by time-varying phase and amplitude. The time-series indicates that

2481

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

Fig. 5. Spatially averaged azimuthal distribution of OH⁄ of a single flame evaluated according to the local co-ordinates in (e). The light and dark shaded circles correspond respectively to inner (rbb < r < 2D/3) and outer (2D/3 < r < D) regions of interest, where rbb = 6.5 mm is the bluff body radius. (a)–(d): Shows the effect of flame separation distance on the degree of azimuthal symmetry of OH⁄ under stable operating conditions. (a)–(c) Are for the ACW swirl configuration whereas (d) shows the effect of alternating swirl.

t1

2000

t2

p1

p2

p3

1000

0

−1000 0

0.5

p1

1

p2

1.5

p3

p1

1000

1000

0

0

−1000 0.75

0.751

0.752

2

−1000 0.04

0.041

p2

p3

0.042

Fig. 6. Pressure time-series measured at P1, P2 and P3 illustrating continuous transitions between spinning and standing wave modes. (a) Shows a full 2s interval. (b) and (c) Show local time-series at t1 and t2 corresponding to m = +1 and m = 1 modes respectively. Operating conditions: S = 1.87D, ACW swirl, U = 18m s1, / = 0.85.

2482

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

the modes are degenerate as they switch back and forth from spinning to standing wave modes at the same resonant frequency. Zooming into a portion of the time-series at t1 in Fig. 6b shows that the pressures measured at P1, P2 and P3 are sequentially out-ofphase by 120° corresponding to a m = +1 ACW spinning mode. At t2 in Fig. 6c the instability has switched to a standing wave, m = 1 mode, as shown by the pressures oscillating in-phase and in anti-phase (P1 being located near a pressure node). Herein, m = 1 refers to the standing wave mode, m = +1 refers to an ACW spinning mode and m = 1 refers to a CW spinning mode. For the m = 1 instability an additional complication arises due to the mean swirl convection speed, vh, causes mode to slowly rotate rather than remain stationary. Compared with the angular frequency, x = c/R, the rotation rate, vh/R  x is very slow. However, through calculation of vh(t) the slow rotation of the m = 1 mode can be completely described. This complex modal behaviour of the time-series data in Fig. 6a requires a robust procedure to identify spinning and standing modes so that any changes to the underlying response of the heat release rate could be investigated. In our previous work [31], a sorting and binning procedure based on the relative phase of the pressure peaks measured at P1, P2, and P3 was employed. Circumferential modes were identified when the phase of the signals were within ±51° of their expected values. For example, m = ±1 modes will produce pressure signals which are 0° and ±120° out-of-phase whilst the direction of spin is determined by the order of the signals. The m = 1 mode produces pressure signals in some combination of in-phase and anti-phase. To differentiate between m = 1, ±1 and axial m = 0 modes a frequency criterion was applied. While this approach works adequately, a more complete description of oscillations can be gained by using a modified form of the indicator proposed by Wolf et al. [29] which is described next. At a given angular location hk, the pressure fluctuations, pk, measured at locations P1, P2 and P3 can be modelled as the superposition of two acoustic waves travelling in opposing directions around an annulus of radius, R, as follows:

  pk ¼ Aþ eiðhk v h t=RÞ þ A eiðhk þv h t=RÞ eixt ;

ð1Þ

where A+ and A are the amplitudes of the two waves travelling in opposite directions, vh is the mean swirl convection speed, and x is the angular frequency. Using this simple description a mode indicator, C(t), can be constructed as the sum of the pressure fluctuations at N angular locations according to the expression:

CðtÞ ¼

N 1X p ðhk ; tÞeihk : N k¼1 k

ð2Þ

The phase of this indicator, /C, is characteristic of the excited mode. The m = 1 modes exhibit constant phase whereas m = ±1 modes exhibit linearly increasing (m = 1) or decreasing (m = +1) phase at the rate ±xt. To gain further insight into the modal dynamics the indicator C(t) was divided into small sections of 6 oscillation periods in length denoted as Clocal(t). For an oscillation frequency of 1800 Hz this yields a local time-scale of slocal = 3.3 ms. A least squares method was then used to estimate the values of A+(t), A(t), vh(t), and x(t) which gave the closest estimate of this local indicator. The time-varying behaviour of A+(t), A(t) is used to define the precise character of the circumferential oscillations, and vh(t) is used to assess the spatial orientation and rotation rate of the m = 1 mode. The mode indicator phase, /C, is plotted in Fig. 7a for a typical run over a time period of 1 s when S = 1.87D. This captures the transient nature of the instability as it alternates between standing and spinning modes completing at least 15 transitions during the 1 s sample period. This behaviour is consistent with LES simulations of [29] but also illustrates that these modal dynamics occur over long time durations. The time-varying amplitude of the opposing waves can also be analysed in terms of their ratio A+/A as shown in Fig. 7b. A perfect m = 1 mode implies A+ = A and vh = 0 and therefore their ratio would be unity. However, the presence of bulk swirl or other asymmetries means that in real systems vh – 0 but is c causing the

1

−1

m = −1 m = 1 m = +1

10

−0.5

A+ / A−

φC (rad × 1000)

0

0

10

−1.5 −1

10 −2

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

time (s)

0.6

0.8

1

time (s)

2 m=1 m=±1

vθ t /R / π

m

+1

1

1.5 1 0.5

−1 0

0.2

0.4

0.6

time (s)

0.8

1

0

0

0.2

0.4

0.6

0.8

1

time (s)

Fig. 7. Mode characterisation using C (t) for the S = 1.87D case with ACW Swirl, / = 0.85. (a) Shows the indicator phase time-series, (b) shows the time-varying amplitude ratio and how it release to mode type, (c) compares characterisation method used in [31], and (d) plots the standing wave orientation over time. Note: standing, ACW spinning, and CW spinning correspond to m = 1, +1, and 1 modes respectively.

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

m = 1 modes to slowly rotate around the annulus. For a perfect m = +1 mode A ? 0, the ratio A+/A ? 1 and vice versa for the m = 1 mode. However, it is clear from Fig. 7b that the time-series of A+/A exhibits the same overall features as Fig. 7a illustrating that the instability rarely approaches pure standing or spinning modes and is predominantly a composite of both. It is therefore useful to define a suitable amplitude threshold to differentiate between predominantly standing and spinning modes. Here, we define a predominantly spinning m = ±1 when A+/A < 0.5 or A+/ A > 2. These threshold limits are denoted by the dashed lines in Fig. 7b indicating that the m = 1 dominant mode for this configuration (S = 1.87D, ACW swirl) with intermittent excursions into m = ±1 modes. Note that the mode phase indicator in Fig. 7a shows that these transitions are not instantaneous, but rather take place over periods of stransition  100 ms. Another interesting feature of the A+/A time-series plot is that it fluctuates continuously. Within the threshold limits, the m = 1 mode is partially stable between the periods 0.05 > t > 0.10 s, 0.27 > t > 0.33 s and 0.53 > t > 0.65 s yet undergo appreciable fluctuations noting the y-axis is logarithmic. By comparison, the m = ±1 modes exhibit transient stability. One interpretation for this configuration is to consider m = 1 to be the preferred mode undergoing time-varying fluctuations in A+ and A, i.e. the m = ±1 modes are unstable and decay to an m = 1 mode. It is important to realise that such an interpretation may be incomplete as it assumes the underlying dynamics that drive the fluctuations in heat release rate remain unchanged. These fluctuations are most likely to emanate from spatial changes in the fluctuating heat release rate and is discussed in Section 5. Figure 7c compares mode classification methods based on the amplitude ratio thresholds (solid lines) with the phase based sorting method used previously [31] (circles). The agreement between the two methods is excellent, each identifying similar modes over the majority of the sample. The only disagreement is seen between t = 0.5–0.6 where the phase based method results in an m = 1 mode but the amplitude ratio classifies it as m = 1. Closer inspection of Fig. 7b shows that this discrepancy is a result of the choice of threshold for A+/A. To completely describe the behaviour the instability depicted by the time-series in Fig. 7 the mean azimuthal convection velocity vh needs to be taken into account as it is non-zero as the m = 1 mode slowly rotates around the annulus noting that vh  c. The degree of rotation can be obtained by tracking the non-dimensional position of the standing wave, vh t/R/p, over time as shown in Fig. 7d. The orientation of the standing wave is denoted by the black squares whereas the grey circles represent the distribution for other modes. This shows that the orientation of the mode changes with time with a full rotation taking approximately 0.25–4 s. However, the rate of change of the orientation is not constant with time, as the mode is occasionally stationary, e.g., at t  0.8 s, and even switches direction of spin. Furthermore, there appears to be some bias in the m = 1 modes, with more gaps present in the distribution in the range 0 < vht/R/p < 1 in comparison with the range 1 < vht/R/p < 2. The angular rotation frequency and orientation preferences are further assessed later in Section 4.2. The continuous nature of the distribution shows that modes other than those classed as m = 1 also have a defined orientation as A+ and A are always finite. 4.2. Statistical description of preferred modes To characterise the modal dynamics and determine whether or not there is a statistically preferred mode joint probability density functions (p.d.f.’s) of A+ and A were constructed and are shown in Fig. 8. The joint p.d.f’s enable the effect of swirl configuration and S to be compared for all operating conditions ranging between

2483

0.85 > / > 1. The amplitude ratio thresholds are marked with dashed lines. The effect of ACW and alternating swirl configurations for each S can be seen by comparing the joint p.d.f’s on the left and right hand sides of Fig. 8. Recall from Fig. 2 that for the ACW swirl configuration each flame is locally subjected to ACW swirl, however in terms of the annular dimensions two opposing bulk swirl directions are established as denoted by the arrows; ACW swirl along the outer annular wall and bulk CW swirl along the inner annular wall. The alternating swirl configuration eliminates these regions of bulk swirl. The effect of S on the preferred mode for the ACW swirl configuration is shown in the joint p.d.f.’s in Fig. 8. As S decreases the statistically preferred mode changes from m = +1 to m = 1 to m = 1. On the other hand, the alternating swirl configuration results in a preference for m = 1 modes for each S as shown in Fig. 8. One possible explanation for this is that the elimination of bulk swirl improves radial symmetry of the merged regions of heat release rate as shown in Fig. 4d. However, each S still exhibits distributions that contain m = ±1 modes. The distributions are widest for S = 1.56D for both swirl configurations and they also give rise to the largest limit-cycle amplitudes. The larger limit-cycle amplitudes are not only due to the increase in power as the limit-cycle amplitudes for S = 1.87D and S = 2.33D are similar. This indicates that breakdown of the mean axisymmetric flame structure caused by large-scale flame merging is important (see Fig. 4 and Ref. [31]). In the case of the ACW swirl configuration, the statistically preferred mode is also strongly influenced by these large-scale interactions between flames. Unlike the other joint p.d.f.’s, Fig. 8a shows a bimodal distribution with the maximum indicating a preference for m = +1 modes but also a second preference for m = 1 modes with probability of approximately half the maximum. Although m = 1 modes are statistically preferred for both swirl configurations when S = 1.87D, the ACW swirl configuration shows a clear bias towards m = 1 modes whereas the p.d.f. for the alternating swirl configuration is narrower with the maximum lying on the principal diagonal. In order to fully understand the relationship between the modal dynamics, S, and swirl configuration these joint p.d.f.’s need to be considered alongside the corresponding heat release rate dynamics which is presented in Section 5. 4.3. Orientation of the m = 1 modes In order to fully characterise the m = 1 modes the mean swirl convection rate, vh/R, for each swirl configuration and S was evaluated and is plotted in Fig. 9 over all operating conditions (0.85 > / > 1). For the ACW swirl configuration both S = 1.56D and 1.87D show a normally distribution about a small negative mean of vh/ R  1 to 3 rad/s. When S = 2.33D deviates from a normal distribution having a positive mean of vh/R = 6 rad/s. There appears to be some consistency with the trends shown by joint p.d.f.’s in Fig. 8, in that the preference for spinning direction appears to be related to the net angular rotation rate. Likewise, the alternating swirl configuration shows a normal distribution about zero for S = 1.56D and 1.87D with a skewed p.d.f., this time to negative values of vh/R for S = 2.33D. Comparing both swirl configurations shows that the distributions are influenced by large-scale flame merging. As expected the rotational speeds of the m = 1 modes are orders of magnitude slower than the time-scale of the instability frequency. It was also found that the m = 1 modes tended to arrange themselves in preferred orientations. In other words, the node locations statistically preferred certain annular locations even though they all slowly rotate with vh. Figure 7d showed that the non-dimensional node positions vht/R/p vary in time and moreover that the m = 1 modes rarely undergo a full rotation around the annulus. An example of the statistically preferred orientation of the m = 1

2484

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

Fig. 8. Joint probability density functions of A+ and A as a function of S and swirl configuration. For uniform ACW swirl figures (a), (c) and (e) shows that effect of S results in a variety mode preferences whereas the alternating swirl configuration in figures (b), (d) and (f) results in a statistical preference for m = 1 modes for all S.

modes is shown in Fig. 10. The p.d.f. of the node locations vht/R is plotted using logarithmic scales. This is a different representation

of the data in Fig. 7d, which is the same case, but demonstrates that the m = 1 modes exhibit a preferred orientation, at 3p/2 and p/2.

2485

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

0.06

0.06

S=1.56D S=1.87D S=2.33D

S=1.56D S=1.87D S=2.33D

0.04

PDF

PDF

0.04

0.02

0.02

0 −50

−25

0

25

50

vθ / R (rad/s)

π/4

−2 −3 −4 π/2

3π/2

5π/4

0

25

50

vh/R, as a function of swirl configuration and flame separation distance, S.

0 −1

−25

vθ / R (rad/s)

Fig. 9. Probability density functions of angular rotation rate,

7π/4

0 −50

3π/4

π Fig. 10. Preferred orientation of the node positions for the m = 1 modes for S = 1.56D, ACW swirl configuration. Each dotted orbit is a logarithmic decade to the denoted power.

Different preferred orientations were found for each S and swirl configuration but are not included here as they may be artefacts of our boundary conditions.

5. Phase-averaged global heat release rate So far we have demonstrated that, like the LES simulations of Wolf et al. [29], the time-series of self-excited azimuthal instabilities are characterised by regular transitions between spinning and standing wave modes and performed some statistical analysis to classify their behaviour as function of flame separation distance and swirl configuration. We now examine the structure of the phase-averaged fluctuations in heat release rate for standing and spinning modes. To carry out the phase averaging the identification procedure in Section 4.1 was also used to assign the corresponding raw OH⁄ images into their mode types and then sorted into 18 time-steps, s, normalised by the period of oscillation T = 1/f. At each time-step, the heat release rate q0s were determined and then spatially inte-

  grated over a 50° wide radial sector, q0s , as shown in the schematic in Fig. 2. The radial sector was then rotated in 1° increments to re-evaluate hq0s i. This process was repeated over the whole annulus and divided by the spatial phase-averaged  mean, hq0s i=hq0s i. This amounts to 50% smoothing with a radial kernel 50° wide to remove spatial information and yield representations of the phase-averaged heat release rate similar to those obtained by acoustic models. The left hand side of Fig. 11 shows the radially smoothed phaseaveraged fluctuating heat release rate of the m = +1 for S = 1.56D and both m = +1 and m = 1 modes for S = 1.87D respectively. In each case, the ACW swirl configuration was used. The right hand side of Fig. 11 shows the corresponding normalised amplitudes of pressure, An,p(h) and heat release rate, An,q(h) as functions of annular position, h. The amplitude of the pressure fluctuations Ap(h) was calculated from Eq. (1) using the mean A+, A, and vh/R values evaluated in intervals specified in the caption. The amplitude of the heat release rate fluctuations Aq(h) corresponds to the peak-to-peak values obtained during the smoothed phase-averaged values shown on the lhs of the figure. Both are normalised by their maxima.  Figure 11a shows the hq0s i=hq0s i for a predominantly m = +1 mode for the S = 1.56D case in a sequence of 9 non-dimensional time-steps.   Diametrically opposed regions of maximum and minimum q0s travel around the full circumference in the ACW direction once per cycle. A characteristic feature of the spinning wave mode is that An,q remains approximately constant around the annulus as shown in Fig. 11b. In the case of a perfect travelling wave Ap(h) would also be constant, however, in this case  Aþ =A  3:8. Corresponding fluctuations in An,q arise for the same reasons. The relatively constant value of An,q over the whole annulus indicates that the underlying flame dynamics which drives hq0s i is repeated across all flames with constant phase difference or time-delay. The m = 1 mode in Fig. 11c and d for the S = 1.87D case shows quite different behaviour. For a given orientation of the mode, the diametrically opposed regions of hq0s;min i and hq0s;max i oscillate between the pressure anti-node positions once per cycle, at s/ T  1/9 and 6/9. In Fig. 11d both time- and spatial varying behaviour in both the pressure and heat release rate fluctuations are observed with An,p and An,q varying harmonically and in phase around the annulus. At the pressure anti-node locations (0 and 180°), An,p reaches a maximum, whereas at the pressure nodes (90°, 270°), An,p is over five times weaker. As before, the amplitude does not

2486

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

Fig. 11. Radially integrated heat release rate obtained from OH⁄ chemiluminescence (shown in (a), (e) and (c)) and 1D representation of normalised annular heat release rate An,q(h), and pressure, An,p(h), fluctuation magnitudes (shown in (b), (f) and (d)), showing an m = +1 mode for the S = 1.56D case, and both m = +1 and m = 1 for the S = 1.87D case. The mode orientation intervals for the three cases are: (a) 3p/16 < vht/R < 5p/16; (e) 31p/16 < vht/R < p/16; and (c) 29p/16 < vht/R < 31p/16. Dimensions as in Fig. 4 and U = 18m s1, / = 0.85, LM enclosure.



reduce to zero at the pressure nodes as for this case Aþ =A  0:77. The annular variation An,p and An,q indicates that the underlying flame dynamics change with the acoustic mode shape. Moreover, the flame dynamics at the nodes produce negligible fluctuations in heat release rate compared with the anti-nodes. It is worth while drawing some comparisons with the effect of transverse forcing on a single flame by O’Connor and Lieuwen [25] who found that the flow field of an annular jet changes from predominantly axisymmetric oscillations when located at a pressure anti-node to a helical (flapping) like oscillations when located at a pressure node. According to Fig. 11c and d the axisymmetric flame dynamics, driven by oscillations in the mass flow at the inlet produces maximum heat release rate whereas negligible heat release rate is produced by the helical/flapping modes. An example of a predominantly spinning mode but with a mean amplitude ratio near the threshold value, Aþ =A  2:66 is shown in   Fig. 11e. The variation of q0s over the cycle shows that the regions of maximum and minimum heat release rate spin around the annulus once per oscillation cycle but also vary in magnitude around the annulus as represented by the peaks and troughs that are associated with the m = 1 mode. In this case, the magnitude of the fluctuations do not go to zero at the pressure nodes. Fig. 11f shows that the m = 1 modes result in oscillations in An,q around the annulus whereas m = ±1 modes result in An,q tending to a constant value, an example of which is shown Fig. 11b. Using the same experimental setup with the ACW swirl configuration, Worth and Dawson [31] compared the effect of S on the structure of the phase-averaged heat release rate for m = +1 modes only. For S = 2.33D it was observed that peak fluctuations in heat release rate occurred in crescent shaped regions oriented towards

the outer annular wall. As S was decreased, interactions with neighbouring flames resulted in large inter-connected regions of unsteady heat release rate spanning across multiple flames which were also located along the outer annular wall. Based on these observations, and given that the bulk swirl directions are well defined for the ACW swirl configuration as depicted in Fig. 2, i.e. each burner is equipped with ACW swirlers resulting in bulk swirl in the ACW direction along the outer annular wall and bulk swirl in the CW direction along the inner annular wall, it was conjectured that a correlation between the direction of the spinning modes, the spatial distribution of the unsteady global heat release rate, and the annular bulk swirl directions may exist for this experimental configuration. In other words, it was postulated that if the outer annulus contains a larger proportion of the heat release rate then a statistical preference for the m = +1 mode would occur as the bulk swirl is in the ACW direction and vice versa, noting that m = +1 and m = 1 refers to ACW and CW directions respectively. To investigate the possibility of a correlation between the spin direction, spatial distribution of the unsteady heat release rate, and bulk swirl directions, the radially integrated phase-averaged OH⁄ was re-analysed by dividing the annulus into inner and outer annuli about the mean annular radius, R. Figure 12 compares the magnitude of hq0s i between the inner and outer annuli for m = +1, m = 1, and m = 1 modes for S =1.87D. The m = +1 mode in Fig. 12a shows that peak fluctuations  of q0s in both inner and outer annuli spin in the ACW direction.   However, the magnitude of q0s is greater in the outer annulus where the bulk swirl direction is also in the ACW direction. In   the case of the m = 1 mode in Fig. 12c, peak fluctuations in q0s in both the inner and outer annuli spin in the CW direction, with

2487

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

Fig. 12. Radially integrated heat release rate obtained from phase averaged OH⁄ chemiluminescence (shown in (a), (c) and (e)) and 1D representation of normalised annular heat release rate An,q(h), and pressure, An,p(h), fluctuation magnitudes (shown in (b), (d) and (f)). Inner and outer regions are integrated separately. The mode orientation interval for m = 1 in (e) and (f) is 29p/16 < vht/R < 31p/16. Dimensions as in Fig. 4. For all modes, S = 1.87D, U = 18m s1, / = 0.85, LM enclosure.

S=2.33D

S=1.87D

S=1.56D

1

m=1

m = +1

10

0

10

m = −1

A+/A−

peak magnitudes now located along the inner annulus where the bulk swirl is in the CW direction. Figure 12b and d compares the amplitudes of the global heat release rate for the inner, An,q,inner, outer, An,q,outer and total, An,q, annuli normalised by their maxima. As before, the variations of An,q,inner, An,q,outer, and An,q between cases arise from the differences in amplitude ratio. Figure 12b shows that An,q,outer is up to 60% higher than An,q,inner whereas Fig. 12d shows that An,q,inner is 30–40% greater than An,q,outer. The m = 1 mode in Fig. 12e and f exhibits quite different behaviour. During peak heat release rate at s/T  1/9 and 6/9 the magni  tudes of q0s in the inner and outer regions are similar, with a slight bias towards the outer annulus. A clearer depiction of this is shown in Fig. 12f by comparing An,q,inner and An,q,outer at the anti-node locations. Closer inspection of Fig. 12e also shows that during the oscillation cycle the global heat release rate in the inner and outer annuli spin in opposite directions. This also follows the bulk annular swirl direction, i.e. peak q0n spins ACW in the outer annular region and CW in the inner annular region, meeting at the anti-node locations. Why this should be the case is not presently understood. Analysing all the data for the ACW swirl configuration, the statistical mode preference for all the experimental configurations are collated into a regime diagram. Figure 13 plots the ratio of inner to outer amplitudes of mean heat release rate, hAn,q,inner i/hAn,q,outeri, conditioned on the amplitude ratio A+/A. The previous dependence of the spinning mode direction on the spatial arrangement of the heat release rate for the S = 1.87D case can be clearly identified, with m = +1 and m = 1 modes occurring when the peak heat release rate is located in the outer and inner annuli respectively. Although only m = +1 modes were imaged for the S = 1.56D case, these were also dominated by peak fluctuating heat release rate

−1

10

0.5

1

2

/ Fig. 13. Dependence of amplitude ratio A+/A with the ratio of inner to outer normalised amplitude of the fluctuating heat release rate hAn,q,inneri/hAn,q,outeri on the statistical mode preference for each S with the ACW swirl configuration.

around the outer annuli. The spatial rearrangement of the heat release rate is also clearly observed in the S = 2.33D case, with ACW modes dominated by fluctuations around the outer annuli, and CW modes showing marginally increased heat release rate around the inner annuli. For all separation distances, as the ratio A+/A is increased past the threshold value of 2, the proportion of heat release rate around the inner and outer remains approximately constant taking a value of 0.4–0.5. This behaviour may arise as the amplitude of the opposing wave, A ? 0, leading to no further spatial rearrange-

2488

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489

ment of the heat release rate. Similarly when the amplitude ratio is reduced, hAn,q,inneri/hAn,q,outer i remains approximately constant, albeit taking different values depending on separation distance. The experimental results clearly support the existence of a correlation between the spin direction, spatial distribution of the heat release rate, and bulk swirl directions in the current experimental configuration, however it is important to emphasise that generality has not been established in this study. Whether similar trends will be observed in annular chambers with different burner geometries requires further investigation. In this experiment the bulk swirl directions are well defined for the ACW swirl configuration, whereas in more realistic geometries the flow is more complex and therefore defining the bulk swirl directions is much more problematic. For example, the alternating swirl configuration has each burner fitted with swirlers of alternating sign which should result in cancellation for any even number of burners. This relatively simple change significantly altered the modal dynamics as demonstrated by the joint p.d.f.’s in Fig. 8. Nevertheless, these are intriguing results which require further investigation.

During standing wave azimuthal instabilities, the magnitude of the heat release rate was found to depend on the annular position of each flame relative the mode shape. The spatial response of the heat release rate was found to vary approximately sinusoidally around the annulus, with peak fluctuations produced at the pressure anti-nodes and negligible fluctuations at the pressure nodes. The different spatial and temporal response of the heat release rate to spinning and standing wave modes has significant implications for low-order models, which require a more complete description of the flame response to both azimuthal and longitudinal instabilities in order to make them predictive tools. Acknowledgments NW was funded by the EPSRC under SAMULET Project 2: Combustion Systems for Low Environmental Impact. JRD was funded under the Advanced Research Fellowship scheme. References

6. Conclusions The current paper describes an experimental investigation into the modal dynamics of self-excited azimuthal instabilities in a simple laboratory scale annular combustor. A modular combustor design permitted investigation of 12, 15 and 18 flame configurations, allowing the effect of separation distance to be investigated for a fixed circumferential diameter. In addition two swirler configurations were investigated to investigate the role of bulk annular swirl on the instability characteristics. Simultaneous high-speed OH⁄ chemiluminescence and pressure measurements around the annulus were used to characterise the nature of the azimuthal instabilities. The time-series of the self-excited azimuthal modes were found to be degenerate, switching between standing and spinning modes in both spin directions but maintaining the same resonant frequency. This was found for all configurations investigated. Combined with the LES simulations of [29] suggests that this may be an inherent feature of self-excited azimuthal instabilities in annular combustion chambers. A statistical analysis of the time-series data demonstrated that the instabilities possessed statistically preferred modes based on both separation distance and swirling configuration. For the alternating swirl configuration, in which consecutive swirlers had opposite sign, a statistical preference for standing wave modes was found for all flame separation distances. For the ACW swirling configuration, in which consecutive swirlers had the same sign, the mode preference was a strong function of flame separation distance. The shortest separation distance, S = 1.56D, preferred m = +1 modes, but as S increased a preference for m = 1 and m = 1 modes were found for S = 1.87D and S = 2.33D respectively. These different mode preferences are due to the spatial rearrangement of mean heat release rate and the break down of the mean axisymmetric flame structure as S is reduced. Analysis of the spatial distribution of the heat release rate in the ACW swirling configuration demonstrated a strong correlation between the bulk swirl direction and the direction of spin. When uniform swirl configurations are used, two opposing bulk swirling motions are induced at the inner and outer annular walls. The ACW swirlers used in this experiment induce bulk swirl in the ACW direction along the outer annular wall and bulk swirl in the CW direction along the inner annular wall. By dividing the annulus into inner and outer annuli it was shown that when the peak heat release rate occurs near the inner annular wall then the instability mode spins in the CW direction, and when the peak heat release rate occur near the outer annular wall then the instability mode spins in the ACW direction.

[1] T. Lieuwen, V. Yang (Eds.), Combustion instabilities in gas turbine engines – operational experience, fundamental mechanisms, and modeling, in: Progress in Astronautics and Aeronautics, vol. 210, AIAA, 2005. [2] W. Krebs, P. Flohr, B. Prade, S. Hoffmann, Thermoacoustic stability chart for high-intensity gas turbine systems, Combustion Science and Technology 174 (2002) 99–128. [3] B. Schuermans, C. Paschereit, P. Monkewitz, Non-linear combustion instabilities in annular gas turbine combustors, in: 44th AIAA Aerospace Sciences Meeting and Exhibit, 2006. [4] M. Fleifil, A. Annaswamy, Z. Ghoneim, A. Ghoniem, Response of a laminar premixed flame to flow oscillations: a kinematic model and thermoacoustic instability results, Combustion and Flame 106 (1996) 487–510. [5] A.P. Dowling, Nonlinear self-excited oscillations of a ducted flame, Journal of Fluid Mechanics 346 (1997) 271–290. [6] A.P. Dowling, A kinematic model of a ducted flame, Journal of Fluid Mechanics 394 (1999) 51–72. [7] J.G. Lee, K. Kim, D. Santavicca, Measurement of equivalence ratio fluctuation and its effect on heat release during unstable combustion, Proceedings of the Combustion Institute 28 (2000) 415–421. [8] T. Schuller, D. Durox, S. Candel, Self-induced combustion oscillations of laminar premixed flames stabilized on annular burners, Combustion and Flame 135 (2003) 525–537. [9] C.J. Lawn, S. Evesque, W. Polifke, A model for the thermoacoustic response of a premixed swirl burner, part i: Acoustic aspects, Combustion Science and Technology 176 (2004) 1331–1358. [10] D. Durox, T. Schuller, S. Candel, Combustion dynamics of inverted conical flames, Proceedings of the Combustion Institute 30 (2005) 1717–1724. [11] C.J. Lawn, W. Polifke, A model for the thermoacoustic response of a premixed swirl burner, part ii: The flame response, Combustion Science and Technology 176 (2004) 1359–1390. [12] J.H. Cho, T. Lieuwen, Laminar premixed flame response to equivalence ratio oscillations, Combustion and Flame 140 (2005) 116–129. [13] R. Balachandran, B. Ayoola, C. Kaminski, A. Dowling, E. Mastorakos, Experimental investigation of the nonlinear response of turbulent premixed flames to imposed inlet velocity oscillations, Combustion and Flame 143 (2005) 37–55. [14] B.D. Bellows, M.K. Bobba, A. Forte, J.M. Seitzman, T. Lieuwen, Flame transfer function saturation mechanisms in a swirl-stabilized combustor, Proceedings of the Combustion Institute 31 (2007) 3181–3188. [15] S.K. Thumuluru, T. Lieuwen, Characterization of acoustically forced swirl flame dynamics, Proceedings of the Combustion Institute 32 (2009) 2893–2900. [16] N. Noiray, D. Durox, T. Schuller, S. Candel, A unified framework for nonlinear combustion instability analysis based on the flame describing function, Journal of Fluid Mechanics 615 (2008) 139–167. [17] P. Palies, D. Durox, T. Schuller, S. Candel, The combined dynamics of swirler and turbulent premixed swirling flames, Combustion and Flame 157 (2010) 1698–1717. [18] F. Boudy, D. Durox, T. Schuller, S. Candel, Nonlinear mode triggering in a multiple flame combustor, Proceedings of the Combustion Institute 33 (2011) 1121–1128. [19] S. Evesque, W. Polifke, C. Pankiewitz, Spinning and azimuthally standing acoustic modes in annular combustors, in: 9th AIAA/CEAS Aeroacoustics Conference, 2003. [20] S.R. Stow, A.P. Dowling, A time-domain network model for nonlinear thermoacoustic oscillations, Journal of Engineering for Gas Turbines and Power 131 (2009) 031502. [21] N. Noiray, M. Bothien, B. Schuermans, Investigation of azimuthal staging concepts in annular gas turbines, Combustion Theory and Modelling 15 (2011) 585–606.

N.A. Worth, J.R. Dawson / Combustion and Flame 160 (2013) 2476–2489 [22] K. Kunze, C. Hirsch, T. Sattelmayer, Transfer function measurements on a swirl stabilized premix burner in an annular combustion chamber, in: ASME Conference Proceedings 2004, pp. 21–29. [23] D. Fanaca, P.R. Alemela, F. Ettner, C. Hirsch, T. Sattelmayer, B. Schuermans, Determination and comparison of the dynamic characteristics of a perfectly premixed flame in both single and annular combustion chambers, in: ASME Conference Proceedings 2008, 2008, pp. 565–573. [24] D. Fanaca, P.R. Alemela, C. Hirsch, T. Sattelmayer, Comparison of the flow field of a swirl stabilized premixed burner in an annular and a single burner combustion chamber, Journal of Engineering for Gas Turbines and Power 132 (2010) 071502. [25] J. O’Connor, T. Lieuwen, Disturbance field characteristics of a transversely excited burner, Combustion Science and Technology 183 (2011) 427–443. 17. [26] M. Hauser, M. Lorenz, T. Sattelmayer, Influence of transversal acoustic excitation of the burner approach flow on the flame structure, Journal of Engineering for Gas Turbines and Power 133 (2011) 041501. [27] V. Acharya, Shreekrishna, D.-H. Shin, T. Lieuwen, Swirl effects on harmonically excited, premixed flame kinematics, Combustion and Flame 159 (2012) 1139– 1150. [28] G. Staffelbach, L. Gicquel, G. Boudier, T. Poinsot, Large eddy simulation of self excited azimuthal modes in annular combustors, Proceedings of the Combustion Institute 32 (2009) 2909–2916.

2489

[29] P. Wolf, G. Staffelbach, L.Y. Gicquel, J.-D. Müller, T. Poinsot, Acoustic and large eddy simulation studies of azimuthal modes in annular combustion chambers, Combustion and Flame 159 (2012) 3398–3413. [30] N. Noiray, B. Schuermans, Azimuthal acoustic modes in annular gas turbine combustion chambers, in: The 19th International Conference on Sound and Vibration, Vilnius, Lithuania, 2011. [31] N.A. Worth, J.R. Dawson, Self-excited circumferential instabilities in a model annular gas turbine combustor: Global flame dynamics, Proceedings of the Combustion Institute (2012). [32] A.F. Seybert, D.F. Ross, Experimental determination of acoustic properties using a two-microphone random-excitation technique, The Journal of the Acoustical Society of America 61 (1977) 1362–1370. [33] S. Davis, C. Law, Determination of and fuel structure effects on laminar flame speeds of c1 to c8 hydrocarbons, Combustion Science and Technology 140 (1998) 427–449. [34] E. Lubarsky, B. Zinn, Heat release distribution in the LRE combustor simulator operating in stable and unstable modes, in: 46th AIAA Aerospace Sciences Meeting and Exhibit AIAA, Reno, Nevada, 2008. [35] N.A. Worth, J.R. Dawson, Cinematographic oh-plif measurements of two interacting turbulent premixed flames with and without acoustic forcing, Combustion and Flame 159 (2012) 1109–1126.