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Non-stationary local thermoacoustic phase

relationships in a gas turbine combustor

S Kheirkhah, BD Geraedts, P Saini, K Venkatesan, AM Steinberg

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S Kheirkhah, BD Geraedts, P Saini, K Venkatesan, AM Steinberg, "Non-stationary local thermoacoustic phase relationships in a gas turbine combustor", Proceedings of the Combustion Institute, 36 (3) 3873-80 (2017)

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Non-stationary local thermoacoustic phase relationships in a

gas turbine combustorS. Kheirkhaha,∗, B. D. Geraedtsa, P. Sainia, K. Venkatesanb, A. M. Steinberga

aUniversity of Toronto Institute for Aerospace Studies, Toronto ON, M3H 5T6 CanadabGE Global Research Center, Niskayuna NY, 12309 USA

∗The corresponding author email is kheirkhah@utias.utoronto.ca. The corresponding author mailing ad-

dress is as indicated above.

Colloquium: Gas turbine combustion (Colloquium #13)

Method #2 was used for the word count. The total length of the paper is equivalent to 6167 words, with

details tabulated in the following table.

Item Word CountMain text 2917Equations 76References 454Figure 1 302Figure 2 423Figure 3 398Figure 4 403Figure 5 275Figure 6 406Figure 7 272Figure 8 274Total 6200

The authors will pay for color print of the figures.

1

List of Figures

1 (a) Schematic of the high pressure vessel and (b) inset of the combustion chamber. . . . . 6

2 (a) The pressure fluctuations. (b), (c), and (d) are insets of (a) pertaining to increasing,

decreasing, and constant amplitude pressure fluctuations, respectively. (e) is a represen-

tative chemiluminescence signal associated with x/L = 0.1 = y/L = 0.1, and (f) is the

absolute value of the phase difference between the signals presented in (a) and (e). . . . . . 7

3 |∆φp,q| during Type I behavior for a time sequence with constant pressure oscillation am-

plitude. A region of out-of-phase oscillations moves upstream through the nozzle shear

layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 |∆φp,q| during Type II behavior for a time sequence with constant pressure oscillation

amplitude. Transition of an out-of-phase region positioned close to the burner centerline

into two out-of-phase regions close to the nozzle shear layer. . . . . . . . . . . . . . . . . 9

5 Schematics of (a) Type I and (b) Type II behaviors for constant amplitude oscillations. I

and O represent regions with in-phase and out-of-phase p′ and q′, respectively. . . . . . . . 10

6 |∆φp,q| during Type III behavior for a time sequence with increasing pressure oscillation

amplitude. A coherent region of in-phase oscillations develops at x/L < 0.4, which also

extends into the downstream portion of the combustor. . . . . . . . . . . . . . . . . . . . 11

7 Development of an out-of-phase region in the upstream portion of the combustor during a

time period with decreasing oscillation amplitudes (reverse of Type III behavior). . . . . . 12

8 Schematic of Type III behavior for increasing and decreasing amplitude oscillations. I

and O represent regions with in-phase and out-of-phase p′ and q′, respectively. . . . . . . . 12

2

Non-stationary local thermoacoustic phase relationships in agas turbine combustor

S. Kheirkhaha,∗, B. D. Geraedtsa, P. Sainia, K. Venkatesanb, A. M. Steinberga

aUniversity of Toronto Institute for Aerospace Studies, Toronto ON, M3H 5T6 CanadabGE Global Research Center, Niskayuna NY, 12309 USA

Abstract

A framework is presented for analyzing the local instantaneous phase difference between non-stationary

heat release rate and pressure oscillations that is appropriate for use in high-pressure liquid-fueled com-

bustors. High-speed OH∗ chemiluminescence images were used as a qualitative marker of the heat release

rate, from which the local phase shift relative to the pressure was calculated using the Hilbert transform

technique. Two types of behavior were observed during time sequences with constant amplitude pressure

oscillations. For the first behavior, a region with out-of-phase pressure and heat release rate oscillations

moved upstream along the nozzle shear layer towards the nozzle. For the second behavior, transitions

occurred between out-of-phase oscillations in a region along the centerline and a toroidal region close to

the nozzle. For time periods with increasing amplitude pressure fluctuations, in-phase heat release rate and

pressure oscillations developed throughout the upstream portion of the combustor. These in-phase oscilla-

tions could extend along the burner centerline and towards the downstream portion of the combustor. This

behavior is reversed during time periods with decreasing amplitude pressure oscillations; the upstream

portion of the combustor transitions to featuring regions with out-of-phase oscillations.

Keywords:

Thermoacoustic oscillations, Gas turbine combustion, Hilbert transform

Preprint submitted to Proceedings of the Combustion Institute August 16, 2016

1. Introduction

This paper presents a method of experimentally analyzing non-stationary (temporally evolving) ther-

moacoustic oscillations that can be applied in high-pressure, liquid-fueled gas turbine combustors. Whether

pressure (p) and heat release rate (q) oscillation amplitudes increase, decrease, or remain stationary is

determined by the balance between thermoacoustic driving and damping [1–3]. Given quasi-harmonic

oscillations, this is controlled by the local phase shift between the pressure and heat release rate oscilla-

tions, along with the oscillation amplitudes. Coupled local heat release rate and pressure oscillations are

a source (sink) of acoustic energy if they are less than (greater than) 90◦ out of phase. Hence, the nature

of the coupling can be deduced from the phase shift field. The present investigation focuses on the local

phase relationships between the pressure and heat release rate oscillations in a high-pressure liquid fueled

gas turbine combustor.

An underlying challenge in the study of thermoacoustic oscillations is interpretation of experimentally-

accessible q metrics at the relevant conditions, which are elevated-pressure multi-phase flows with large

spatio-temporal variations in equivalence ratio. Quantification of planar laser induced fluorescence (PLIF)-

based heat release rate through, e.g. OH/CH2O PLIF or HCO PLIF, currently is not possible due to

the need for composition-dependent quenching corrections, signal reduction due to pressure-induced line

broadening, and signal trapping (self-absorption) through the high pressure gas [4–6]. Chemiluminescence-

based q measurements are not quantitative in flows with strong equivalence ratio variations, are line-of-

sight integrated, and also suffer from background CO∗2 chemiluminescence as well as signal trapping at

high pressure [6, 7].

These challenges are accentuated for non-stationary oscillations, which may occur during transition

between operating conditions or during nominally stationary operation due to chaotic triggering of a quasi-

deterministic large-scale dynamic process [8–14]. For example, Dawson and Worth [8] observed time-

varying pressure oscillation amplitudes, as well as time-varying phase between pressure signals recorded

at different azimuthal locations in an annular combustor. Studies in a single-element [11–13] combustor

have reported time varying oscillation amplitudes as the flame chaotically transitioned between attached

(V-shaped) and detached (M-shaped) flames. These transitions were associated with local flame based

extinction, formation/attenuation of a precessing vortex core, and upstream flame propagation.

The majority of studies on self-excited thermoacoustically oscillating flames have been performed

at stationary limit-cycle oscillations, e.g. [15–19]. Spatial heat release rate distribution has been quanti-

fied using phase-conditioned flame surface density (derived from OH PLIF) and/or line-of-sight integrated

4

phase-conditioned chemiluminescence measurements [15, 20–22]. The latter commonly are Abel inverted

to produce pseudo two-dimensional phase-conditioned mean fields from the line-of-sight integrated mea-

surements in axisymmetric flows [20, 23]. Recently, phase-conditioned chemiluminescence tomography

has been used to produce three-dimensional images in swirl flames with coherent asymmetries due to pre-

cessing vortex cores [23, 24]. However, phase-conditioning cannot be applied to non-stationary data and,

as described above, these methods of heat release rate measurement have high uncertainties at practical

gas turbine conditions.

The objective of this work is to extend the experimental analysis of internal thermoacoustic oscilla-

tions to non-stationary behavior in high-pressure liquid-fueled flames that are relevant for aeronautical

applications. Specifically, we aim to identify transient regions in which oscillations are being driven or

damped based on the instantaneous phase shift between a heat release rate marker (OH∗ chemilumines-

cence) and the pressure. This framework does not require knowledge of the absolute heat release rate

magnitude, and hence is robust to uncertainty in its measurement. Thus, in addition to this motive, the

presented framework provides a potential means for comparison with numerical simulations based on the

temporally evolving spatial distribution of in-phase and out-of-phase oscillation regions.

2. Experimental Methodology

The analysis presented here utilizes data from a liquid-fueled gas turbine combustor configuration. A

schematic of the experimental setup is presented in Fig. 1(a). The combustion chamber was installed inside

a high pressure vessel that was equipped with fused silica windows for optical access. The chamber itself,

shown in Fig. 1(b), was comprised of a single-piece of fused silica, having a square cross-section with a

side length of L = 125 mm. The nozzle studied here had dual co-annular air swirlers and multi-point fuel

injection, similar to that described in Ref. [25]. In the present study, fuel was only supplied to the inner

air feed. As shown in the figure, the flow direction is from left to right. Also shown are the mean OH∗

chemiluminescence field and the extents of the measurement domain (described below).

An operating condition that exhibited non-stationary thermoacoustic oscillations at stationary inflow

conditions is discussed here. Air was provided to the nozzle at pressure and temperature of p/pcrit = 0.42

and T/Tcrit = 0.84, respectively, where the critical pressure and temperature are those of a typical single

hydrocarbon surrogate fuel. The combustor was operated with the Jet A fuel at a total power of 600 kW.

Simultaneous OH∗ chemiluminescence and pressure measurements were performed in the present

study. The chemiluminescence imaging suffers from line-of-sight integration and is only considered to

provide a qualitative metric for the distribution of the heat release rate. This line-of-sight integrated nature

5

Nozzle

Combustion

Chamber

OH∗ Camera

x/L

y/L

Fuel

Air

Dump P lane

Air

Pressurized V essel

Cooling Air

Intensifier & Lens

(a) Experimental Setup (b) Combustion Chamber

W indow

F low

0.1 0.85

To Pressure Transducer

0.45

−0.45

Fig. 1. (a) Schematic of the high pressure vessel and (b) inset of the combustion chamber.

of the measurements must be kept in mind during interpretation of the results. Because we are interested

in instantaneous heat release dynamics, the data cannot be Abel inverted to provide pseudo-planar infor-

mation or 3D spatially resolved data. However, the particular analysis framework presented allows for

meaningful insight despite these limitation.

The chemiluminescence system used a high-speed camera (Photron SA-5) equipped with an image

intensifier (Invisible Vision UVi, gate = 25 µs) and UV lens (Nikon 105 mm), operating at a recording

frequency of 10 kHz. Signal in the emission range of the OH∗ chemiluminescence was isolated using a

bandpass filter with a center wavelength of 310 nm and a full-width at half-maximum of 20 nm.

The pressure inside the combustion chamber was measured using a pressure transducer (PCB 102A05),

which was connected to the combustor dump plane by a 1180 mm long and 4 mm inner diameter steel

waveguide. A phase shift correction was applied to match the recorded pressure signal with the com-

bustion chamber pressure based on the waveguide dimensions and combustor operating conditions. The

wavelength corresponding to the measured thermoacoustic pressure oscillation frequency (350 Hz) was

sufficiently long that the oscillation amplitude and phase in the heat release zone can be considered as

uniform. The pressure signal was acquired simultaneously with the intensifier gate from the chemilumi-

nescence system in order to synchronize the different measurements.

3. Data Reduction

Spectral analysis of the pressure signal demonstrated a dominant frequency at approximately 350 Hz.

However, other frequency components and noise also were present in the signal. Removal of local fre-

6

0 200 400 600 800 1000 1200 1400

−10

10

p′(kPa)

0 200 400 600 800 1000 1200 1400

q′

0 200 400 600 800 1000 1200 14000

t (ms)

|Δφp,q|

65 70 75 80 85

−5

5

p′(kPa)

250 255 260 265 610 615 620 625

π

(e)

(f)

(a)

(b) (c) (d)

Fig. 2. (a) The pressure fluctuations. (b), (c), and (d) are insets of (a) pertaining to increasing, decreasing, and constantamplitude pressure fluctuations, respectively. (e) is a representative chemiluminescence signal associated with x/L = 0.1 =y/L = 0.1, and (f) is the absolute value of the phase difference between the signals presented in (a) and (e).

quency components outside the range of the thermoacoustic oscillation is important for the Hilbert trans-

form analysis described below. Hence, a Chebychev-type bandpass filter was applied to the pressure

signal, with the filter center frequency of 350 Hz and a bandwidth of 40 Hz. Care was taken to ensure the

filter did not impose a local phase shift on the signal by filtering over a moving window with a width of

±20 ms.

Identical spectral filtering was applied to the OH∗ chemiluminescence images. However, the images

first were binned into 1.2 mm × 1.2 mm windows to increase the signal-to-noise ratio and reduce the

computational expense.

Phase-averaging is commonly utilized in thermoacoustic studies to investigate modal behavior of the

combustion chambers, see, for example, [8–10]. However, utilizing this technique in analysis of the

non-stationary signals dynamics is ill-posed, since the relative phase between oscillations is changing

over time; phase-conditioned averaging would blend all of these individual phase shifts. Analysis of

the thermoacoustic coupling requires a rigorous and accurate definition of the temporally evolving phase

shift between the two non-stationary signals, namely the pressure and heat release rate oscillations. The

phase (φ) of a signal (f(t)) can be estimated [26–28] using the Hilbert transform technique, with details

presented in [29]. Specifically, φ is obtained using the following equations.

7

H[f(t)] =1

π

∫ +∞

−∞

f(t′)

t′ − tdt′, (1a)

φ[f(t)] = tan−1{f(t) + iH[f(t)]}. (1b)

The phase difference between two signals, f and g, can then be computed as |∆φf,g(t)| = |φ[f(t)] −

φ[g(t)]|. Accuracy of this phase difference analysis was assessed by using signals (not presented here)

with relatively complex and known analytical phase differences. It was obtained that the Hilber transform

technique produces results that are identical to the known analytical phase difference for the core of the

investigated time period (few milliseconds away from the beginning and end of the signals). Thus, the

Hilbert transform provides a rigorous means of computing the temporally evolving phase shift between

two non-stationary signals.

In order to demonstrate details of the phase difference calculation, the pressure signal and a repre-

sentative chemiluminescence signal taken from x/L = y/L = 0.1 are shown in Figs. 2(a) and 2(e),

respectively. The OH∗ chemiluminescence signal pertains to a randomly selected point in the field. As

shown in the insets of Fig. 2(a), the filtered pressure signal features, increasing (Fig. 2(b)), decreasing

(Fig. 2(c)), and constant (Fig. 2(d)) amplitude pressure oscillations. It is noted that the amplitude of the

OH∗ chemiluminescence oscillations does not follow that of the pressure oscillations because the OH∗

chemiluminescence signal is taken at a particular point in the combustor, and hence does not represent

the volume-integrated heat release rate oscillations. Such integrated data (not presented here) show that

the amplitude of the total heat release rate clearly follows that of pressure fluctuations. Using the Hilbert

transform, the absolute value of the phase difference between these signals was calculated, which is pre-

sented in Fig. 2(f). The absolute value is utilized as it characterizes the nature of the local thermoacoustic

coupling; 0 < |∆φp,q| < π/2 represents locally in-phase (driving) pressure and heat release rate oscilla-

tions, and π/2 < |∆φp,q| < π represents locally out-of-phase (damping) oscillations. As can be seen, the

signals demonstrate temporally varying phase shifts over the entire time span, even at time periods with

stationary oscillation amplitudes.

4. Results

The objective of this section is to demonstrate capability of the Hilbert transform technique for study-

ing characteristic dynamics of the |∆φp,q| fields during constant, increasing, and decreasing pressure am-

plitude oscillations.

8

x/L

y/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

x/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

x/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

π0(d) t = 625 ms(c) t = 620 ms(b) t = 615 ms(a) t = 610 ms

Fig. 3. |∆φp,q| during Type I behavior for a time sequence with constant pressure oscillation amplitude. A region of out-of-phase oscillations moves upstream through the nozzle shear layer.

x/L

y/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

x/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

x/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

π0(a) t = 746 ms (b) t = 754 ms (c) t = 762 ms (d) t = 770 ms

Fig. 4. |∆φp,q| during Type II behavior for a time sequence with constant pressure oscillation amplitude. Transition of anout-of-phase region positioned close to the burner centerline into two out-of-phase regions close to the nozzle shear layer.

4.1. Constant amplitude pressure oscillations

Figures 3 and 4 show the |∆φp,q| contours that demonstrate two common behaviors observed for time

periods during which the pressure oscillations amplitude remained constant. The pressure oscillations

signal associated with the time sequence in Fig. 3 is shown in Fig. 2(d). The pressure oscillations signal

corresponding to Fig. 4 is similar to that demonstrated in Fig. 2(d), and the corresponding inset is not

shown in Fig. 2 for brevity. Regions in which the heat release rate fluctuations were out-of-phase (in-phase)

relative to the pressure oscillations are shown in red (blue). In general, the majority of the out-of-phase

oscillations were in the downstream portion of the combustor (0.4 < x/L < 0.8).

The |∆φp,q| dynamics observed in Figs. 3 and 4 will be referred to as Type I and II behaviors, respec-

tively. The Type I behavior, presented in Fig. 3, occurs over about five oscillations at the thermoacoustic

frequency. During this time period, the portion of the shear layer at y/L < 0 transitions from in-phase to

out-of-phase oscillations. This transition begins in the downstream region (x/L ≈ 0.4), and then moves

upstream through the shear layer, see the dashed ellipses and arrows in Figs. 3(a)- 3(d). The rest of the

|∆φp,q| field remains relatively stationary, though there are similar (but less pronounced) dynamics in the

9

x/L

y/L

x/L

I I

I

I

I

I

I I

(a) Type I (b) Type II

O

O

O

O

Fig. 5. Schematics of (a) Type I and (b) Type II behaviors for constant amplitude oscillations. I and O represent regions within-phase and out-of-phase p′ and q′, respectively.

opposite shear layer (y/L > 0). Other time sequences showed that the Type I dynamics presented in Fig. 3

occurred repeatedly during times with constant oscillation amplitudes, but could take place in the shear

layer at any circumferential position.

The Type II behavior, shown in Fig. 4, is associated with transition between one out-of-phase region

along the centerline (Fig. 4(a)) and two out-of-phase regions (Fig. 4(b-d)) near the nozzle. These regions

and the direction of movement are shown by the dashed ellipses and arrows in Fig. 4. In contrast with

the Type I behavior, which always features movement of an out-of-phase region in the upstream direction,

Type II behavior also occurred in the reverse manner to that shown in Fig. 4. That is, the two out-of-phase

regions near the nozzle may transition into one region along the centerline. It is noted that the line-

of-sight integrated nature of the chemiluminescence measurements implies that the single out-of-phase

region along the centerline could, in fact, be in the shear layer oriented along the line-of-sight. However,

all Type II behavior, in which the transition could happen in either direction involved a region along the

centerline. Hence, this process is not expected to happen inside the shear layer. Furthermore, the two

symmetric regions near the nozzle likely are a manifestation of an axisymmetric region positioned close

to the nozzle.

In order to demonstrate Types I and II behaviors, the associated dynamics are schematically presented

in Figs. 5(a) and 5(b), respectively. In the figures, the symbols I and O correspond to the regions in

which q′ and p′ were in-phase and out-of-phase, respectively. For Type I behavior, the out-of-phase region

transitions from downstream towards upstream and along the shear layer. For Type II, bi-directional

transition of regions with out-of-phase oscillations can occur between the combustor centerline and a

10

x/L

y/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

x/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

x/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

π0(a) t = 65 ms (b) t = 73 ms (c) t = 81 ms (d) t = 89 ms

Fig. 6. |∆φp,q| during Type III behavior for a time sequence with increasing pressure oscillation amplitude. A coherent regionof in-phase oscillations develops at x/L < 0.4, which also extends into the downstream portion of the combustor.

toroidal region near the nozzle indicated by the symmetric circles in Fig. 5(b).

4.2. Increasing and decreasing amplitude pressure fluctuations

During time sequences with increasing pressure oscillations amplitude, the |∆φp,q| fields transition

from a relatively disorganized structure to one with distinct regions of in-phase and out-of-phase oscil-

lations. A typical transition is shown in Fig. 6, which is referred to as Type III behavior. The pressure

fluctuations signal associated with the results in Fig. 6 is demonstrated in Fig. 2(b). For the Type III be-

havior, the upstream portion (x/L < 0.4) initially features a mixture of in-phase and out-of-phase regions,

which develops into a coherent region of in-phase oscillations during the time sequence. The dynamics

of the out-of-phase regions at x/L < 0.4 are similar to the Type II dynamics. Specifically, there exist

transitions between an out-of-phase region along the centerline and regions near the nozzle. There also is

evidence of dynamics in the shear layer at y/L > 0, which transitions between out-of-phase and in-phase

p′ and q′ oscillations. In addition, a relatively large region of in-phase oscillations occasionally develops

in the downstream region of the combustor and along y/L ≈ 0 (Figs. 6(b)- 6(d)) during time periods with

increasing pressure amplitude oscillations. Hence, this behavior is characterized by the formation of a

large coherent region of in-phase p′ and q′ oscillations in the upstream portion of the combustor, which

may extend into the downstream region.

The main behavoir of the |∆φp,q| field during time periods with decreasing pressure oscillation am-

plitude is similar to the reverse of the processes during increasing oscillation amplitude, with the corre-

sponding representative time sequence demonstrated in Fig. 7. The pressure fluctuations signal associated

with the results in Fig. 7 is presented in Fig. 2(c). Figure 7 shows that, for decreasing pressure oscilla-

tions amplitude, out-of-phase regions are developed in the upstream portion of the combustor, which is

accompanied by increasing coherence of the downstream out-of-phase region.

11

x/L

y/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x/L

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0(b) t = 275 ms(a) t = 250 ms

π

Fig. 7. Development of an out-of-phase region in the upstream portion of the combustor during a time period with decreasingoscillation amplitudes (reverse of Type III behavior).

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

$x/L$

y/L

p′ 00.51

00.51

y/L

I

O

O I I I

O

PossibleandI

x/L

Transition between amplitudes

Mixture

O

x/L

Fig. 8. Schematic of Type III behavior for increasing and decreasing amplitude oscillations. I and O represent regions within-phase and out-of-phase p′ and q′, respectively.

The characteristics of phase relationships associated with increasing/decreasing pressure oscillations

are summarized in the cartoons shown in Figs. 8(a) and (b). As demonstrated in the figures and discussed

earlier, during increasing pressure amplitude oscillations, the upstream portion of the combustor as well

as the region along the centerline transition from featuring mixture of in-phase and out-of-phase to mainly

in-phase thermoacoustic oscillations. This processes is reversed for decreasing pressure amplitude.

5. Concluding Remarks

Dynamics of the phase difference between heat release rate and pressure fluctuations inside a liquid-

fueled gas turbine combustor at elevated pressure was investigated experimentally using high-repetition-

12

rate OH∗ chemiluminescence images and pressure measurements. Using the Hilbert transform technique,

a framework for analysis of the phase difference between the heat release rate and the pressure fluctua-

tions was developed. This technique allows study of the temporal evolution of the local phase difference

between pressure and heat release rate oscillations.

For constant amplitude pressure oscillations, the downstream portion of the combustor was dominated

by a stationary region in which the heat release rate and pressure oscillations were out-of-phase. However,

the upstream portion featured two non-stationary behaviors, which were referred to as Type I and Type II.

For the former, a region of out-of-phase oscillations transitions from downstream towards upstream and

along the nozzle shear layer. In the latter, an out-of-phase region positioned along the nozzle centerline

transitions into a toroidal region positioned close to the nozzle, and vice versa.

During the sequences with increasing pressure oscillations amplitude, the upstream portion of the

combustor became dominated by in-phase heat release rate and pressure oscillations. Simultaneously,

an in-phase region may be developed in the downstream portion of the combustor and along the burner

centerline. This process was reversed for time sequences with decreasing pressure oscillations amplitude.

Specifically, for this type of behavior, the downstream portion of the combustor was dominated by out-of-

phase heat release rate and pressure fluctuations, while out-of-phase regions form in the upstream portion

of the combustor.

The present work demonstrates a rigorous and robust means of describing non-stationary behavior in

thermoacoustically oscillating combustors at practical operating conditions. The frame work developed

here will be directly utilized in two future studies paths. First, we will use the framework along with

the amplitudes of pressure and heat release rate oscillations in order to investigate the structure of local

thermoacoustic energy transfer for non-stationary oscillations. Then, we aim to apply this framework

for multi-dimensional spatially and temporally resolved velocity, OH∗ chemiluminescence, and pressure

oscillations. This will allow to explore the causality connections in themoacoustic oscillations.

Acknowledgments

This investigation is supported by Natural Sciences and Engineering Research Council (NSERC) of

Canada and General Electric under Grant CRDPJ 47740.

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