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Proceedings of the ASME Turbo Expo 2013, Power for Land, Sea, and Air

GT2013

June 3-7, 2013, San Antonio, Texas, USA

GT2013-95900

DEVELOPMENT OF A FLAME TRANSFER FUNCTION FRAMEWORK FOR

TRANSVERSELY FORCED FLAMES

Jacqueline O’Connor Sandia National Laboratories Livermore, California, USA

Vishal Acharya Georgia Institute of Technology

Atlanta, Georgia, USA

ABSTRACT This paper describes a framework for the development of a

flame transfer function for transversely forced flames. While

extensive flame transfer function measurements have been made

for longitudinally forced flames, the disturbance field

characteristics governing the flame response of a transversely

forced flame are different enough to warrant separate

investigation. In this work, we draw upon previous

investigations of the flame disturbance pathways in a

transversely forced flame to describe the underlying

mechanisms that govern the behavior of the flame transfer

function. Previous transverse forcing studies have shown that

acoustic coupling in the nozzle region can result in both

transverse and longitudinal acoustic fluctuations at the flame,

and that the acoustic coupling is a function of combustor

geometry, and hence, frequency. The results presented here

quantify this coupling across a large range of frequencies using

a velocity transfer function, FTL. The shape of the velocity

transfer function gain indicates that there is strong acoustic

coupling between the main combustor section and the nozzle at

certain frequencies. Next, measured flame transfer functions

are compared with results from theory. These theoretical results

are derived from two level-set models of flame response to

velocity disturbance fields, where velocity inputs are derived

from experimental results. Data at several test conditions are

presented and larger implications of this research are described

with respect to gas turbine combustor design.

NOMENCLATURE

English

D Nozzle outer diameter

F Transfer function

FTF Flame Transfer Function

G Level-set surface

K Velocity ratio

Lf Flame Length

S Nozzle area

R Nozzle radius

RG Rayleigh gain

T Period

d Diameter

f Frequency

h Height

m Mode number

p Pressure

q Heat release

r Radial coordinate

sf Flame speed

sL Laminar flame speed

t Time

u Axial velocity

v Radial velocity

x Axial coordinate

Greek

γ Coherence

ε Error

ζ Flame coordinate

θ Azimuthal coordinate

σ Acoustic field symmetry variable

Subscripts

L Longitudinal

T Transverse

a Acoustic

c Convective

m Mode

norm Normalized

ref Reference

ω Vortical

2

INTRODUCTION Combustion dynamics, a coupling between resonant

combustor acoustics and flame heat release rate fluctuations,

has been a challenge in propulsion and power generation

technologies since the middle of the twentieth century [1].

Initially explained by Lord Rayleigh [2], this coupling can lead

to high-cycle fatigue, reduced operability, and increased

emissions in these technologies. For gas turbines, these

instabilities have become more pronounced as engines have

been optimized for low emissions output [3]. One particular

emissions abatement strategy, lean combustion, has led to a rise

in the severity of these instabilities and the more frequent

appearance of transverse instabilities in these engines.

Transverse instabilities are frequent instability modes in

rockets [4, 5], augmenters [6-8], annular combustors [9, 10],

and even can-annular combustor systems [11]. They have been

a particular focus of research in the rocket community, where a

coupling mechanism known as “injector coupling” has been

reported as a dominant flame disturbance pathway [12, 13].

Here, transverse acoustic modes oscillating over an injector

face cause mass flow fluctuations through each of the injectors.

This leads to fluctuating reactant mass flow, atomization rates,

vaporization rates, mixing, and finally heat release rate. This

transverse to longitudinal acoustic coupling is an important

component of the present study.

Traditionally, longitudinal instabilities have been the

dominant instability in gas turbine engines and significant work

has been done to understand this mode [14-16]. More recently,

work has been initiated to shed light on the flame response

characteristics and coupling mechanisms for transversely forced

flames [17-24].

One of the first studies to propose a coupling mechanism

between the azimuthal acoustic field and the flame heat release

rate fluctuations came from Staffelbach et al. [17]. Here, large

eddy simulation of a full annular helicopter engine combustor

showed several harmonic flame motions associated with the

spinning mode instability. Not only did the flames flap side-to-

side, as may be expected during a transverse instability, but they

also pulsed up and down, despite the lack of a longitudinal

acoustic mode. The authors indicated that this longitudinal

pulsing was the major source of heat release rate fluctuations in

the combustor, and the driving mechanism behind the

thermoacoustic instability.

Several experimental efforts followed [18, 19, 22, 25], two

of which used transversely forced combustors that mimic the

acoustic conditions of a full annular combustor. Experimental

evidence from O’Connor and Lieuwen [18, 26] supported the

transverse to longitudinal coupling mechanism that was seen by

Staffelbach et al. in LES simulations. Additionally, this work

investigated the effect that coherent structures, stemming from

acoustic excitation of the hydrodynamically unstable flow field,

had on flame response at different parts of a standing,

transverse field.

Experiments by Hauser and coworkers [19, 27]

investigated the flame response to transverse acoustic excitation

of the swirler plenum upstream of the dump plane. This

excitation method was fundamentally different from that

explored in the present study, as well as others [25, 28], where

acoustic excitation was applied directly to the combustion

chamber. Despite these differences, flame chemiluminescence

data from Hauser and coworkers revealed similar behavior to

flames under direct transverse acoustic excitation. For

example, the transformation of acoustic perturbations to vortical

perturbations in the swirler nozzle showed a mechanism by

which the acoustic field couples with the vorticity field in

swirling flows. These swirling velocity disturbances lead to

swirling disturbances in the flame, and were shown to effect

flame response even during simultaneous excitation of the

nozzle by both axial and transverse acoustics.

Reduced-order modeling of transverse instabilities by

Acharya et al. [20] and Graham and Dowling [29] has

investigated the response of realistic flame geometries to

asymmetric forcing conditions present during transverse

instabilities. In particular, results by Acharya et al. have shown

that both the mean structure of the flame as well as the modal

content of the fluctuating velocity field determines the

magnitude of the response of the flame to asymmetric

perturbations.

Finally, recent work by Worth and Dawson [21] in a full

annular combustor has shown very similar flame behavior and

coupling mechanisms to those measured in transverse forcing

rigs and seen in LES simulation. In particular, the differences

in flame response at different locations in the transverse

acoustic field matched well with single-flame investigations by

O’Connor and Lieuwen [26]. Flame behavior was measured for

both standing-wave and traveling-wave instabilities.

In this work, we focus on the quantification of the global

flame response in the form of a flame transfer function (FTF).

In the case of velocity-coupled flame response, the definition of

the normalized flame transfer function is given as the

normalized flame heat release rate fluctuation divided by a

normalized reference velocity fluctuation [30], as is shown in

Equation (1), where Fnorm is the normalized flame transfer

function, q is the flame heat release rate, u is velocity, and fo is

frequency. Here, the dimensional fluctuation quantities have

been normalized by their time-average quantities.

( )

( )

onorm

ref o

q f qF

u f u

(1)

Flame transfer functions have been measured and

calculated for longitudinally excited flames in several studies

[31-36]. Here, the reference velocity has usually been defined

as the axial velocity fluctuation at the nozzle exit, measured

using a two microphone technique or hot-wire anemometry. An

important motivation for determining these transfer functions is

that they isolate the flame response and can be used as a

submodel in a larger system dynamics model [37-39]. These

models use the measured flame transfer function as a method of

3

predicting the stability of the overall combustion system by

providing an input/output relation between the acoustic

perturbations (input) and flame heat release rate fluctuations

(output).

However, because the actual velocity field along the flame

front, u x , may vary substantially in amplitude and phase

from refu , the FTF should not be interpreted as describing the

flame response alone – it also depends upon certain features of

the combustor system. This dependence is particularly

important for transversely forced flames, which are the focus of

the present work.

The goal of this paper is to discuss the development of a

flame transfer function framework for transversely forced

flames, which has become one of the foremost issues in

combustor development in annular combustor platforms,

including both power generation and aircraft engine

applications. Flame transfer functions are an important part of

many combustion dynamics models, and can be useful in

understanding flame behavior. However, these transfer

functions are only useful if they are properly defined, and

currently very little work has been done to measure and

understand flame transfer functions for transversely forced

flames. Here, we not only aim to review the current literature

available on transverse instabilities, but also discuss the

development of a flame transfer function framework for

transversely forced flames based on the instability mechanisms

that have been previously studied. Data presented at the end of

the paper is used as an example of a measured flame transfer

function in a transversely forced system and these data are

compared to reduced-order models that capture many of the key

velocity-coupling physics.

Flame transfer function framework

The behavior of self-excited combustion instabilities is

dictated by the Rayleigh gain (RG) [2], which is given in

Equation (2),

0

1( ) ( )

T

RG p t q t dtT

(2)

where ( )p t is the pressure fluctuation in time, ( )q t is the

flame heat release rate fluctuation in time, and T is the period

of the fluctuation cycle.

The Rayleigh criterion states that the gain of the

thermoacoustic instability is determined by the phase between

the pressure and heat release rate fluctuations. If the absolute

value of this phase is less than 2 , the Rayleigh gain will be

greater than zero, indicating that the instability will amplify.

Conversely, if this phase is greater than 2 , the instability is

damped. The heat release rate fluctuation term, ( )q t , is

dependent on complicated flow and flame dynamics.

A transfer function approach has been historically used to

describe these flame-flow dynamics. The form of the transfer

function depends on the dominant coupling mechanism. In a

velocity-coupled instability, the heat release rate fluctuation

takes the form,

refq Fu (3)

where q is the heat release rate fluctuation, refu is the

reference velocity fluctuation, and F is the dimensional flame

transfer function describing the relationship between flame heat

release rate fluctuations and reference velocity perturbations.

As discussed above, the acoustic velocity fluctuation at the

base of the flame has often been used as the reference velocity

in the longitudinally forced case. This reference velocity is not

only experimentally tractable, but captures the driving physics

behind the flame response. The initial longitudinal acoustic

velocity fluctuation causes disturbances on the flame surface

through the action of a “base wave” or “root wave” [40, 41],

and can also excite vortical velocity motions that wrinkle the

flame [42-44]. These vortical disturbances can take many

forms, including that of a coherent structure stemming from the

hydrodynamic instabilities in the flow field [45, 46], a swirl

fluctuation that is generated by acoustic velocity fluctuations

passing through the swirler [44, 47], or simply a bulk flow

oscillation, as in the case of conical flame experiments [48].

These velocity disturbance field pathways are shown in Figure

1.

Figure 1. Velocity disturbance mechanisms present in a

longitudinally forced flame.

F can be expressed in terms of the pathways shown in

Figure 1, and is the sum of the complex flame transfer functions

LF and F . For velocity-coupled instabilities, the disturbance

field can be decomposed into acoustic and vortical disturbances

[49], both of which can lead to flame heat release rate

fluctuations. The acoustic disturbances behave according to a

wave equation and travel at the local sound speed. The vortical

disturbances behave according to convective/diffusive relations

and convect at or near the mean flow velocity, much slower than

the local sound speed in these low-Mach number applications.

The resultant flame response is a function not only of the gain

of the flame response to each of these disturbance sources, but

4

also the relative phase between them. To capture this, each of

these sub-transfer functions is a complex quantity.

To illustrate further, the heat release rate expression for the

longitudinally forced case is broken into two constituent

disturbance parts:

, ,L L L aq F F F u (4)

where F is now the sum of two components, the flame transfer

function due to acoustic velocity fluctuations, LF , and the

product of the velocity transfer function describing the coupling

between acoustic and vortical motions, ,LF , and the flame

transfer function describing the flame response to vortical

motion, F . The reference velocity here is the incident

longitudinal acoustic velocity, ,L au . Significantly, it shows that

by using a single reference velocity, ,L au , the flame transfer

function is not only a function of the actual flame response, LF

and F , but also the shear layer response, ,LF . Thus, the

same flame could exhibit different transfer function

characteristics if the shear layer response was different.

These pathways have long been established in the case of

longitudinal forcing and efforts towards measuring and

predicting both the LF [41] and ,LF F [42, 44] terms have

been made. Transverse modes of combustion instability open

new velocity disturbance pathways that incorporate both

transverse and longitudinal acoustic motion, as well as the

acoustic to vortical coupling mechanisms. The incident

transverse acoustic perturbation may directly disturb the flame,

as seen in the work by Ghosh et al. [50] for rocket injectors.

The transverse acoustic pressure field also leads to longitudinal

acoustic fluctuations in the flame nozzle region, as shown by

Staffelbach et al. [17] in simulation and in experimental results

from O’Connor and Lieuwen [18]. The longitudinal acoustic

disturbance, a result of the fluctuating pressure from the

transverse mode, leads to excitation of a longitudinal acoustic

field in and around the nozzle area. Additionally, vortical

velocity disturbances are excited through both longitudinal and

transverse acoustic excitation. Rogers and Marble [6] show an

example of this coupling in a high blockage-ratio combustor,

where a self-excited transverse instability lead to asymmetric

vortex shedding from the edges of the triangular bluff-body.

These disturbance mechanisms and their pathways are shown in

Figure 2.

Similar to the decomposition of the longitudinally forced

disturbance field in Equation (4), the processes in Figure 2 can

be expressed as:

,T TL L T TL L T aq F F F F F F F u (5)

where TF is the flame transfer function describing the coupling

between flame response and transverse acoustic fluctuations,

TLF is the velocity transfer function describing the coupling

between the transverse and longitudinal acoustic motion at the

nozzle exit, LF is the flame transfer function describing the

coupling between flame response and longitudinal acoustic

fluctuations, TF and

LF are the velocity transfer functions

describing the acoustic to vortical velocity coupling, F is the

flame transfer function describing the coupling between flame

heat release rate fluctuations and vortical motion, and ,T au is

the transverse acoustic velocity, or the reference velocity for

this form of the transfer function. F represents the flame

response to the overall vortical velocity field, taking into

account both the gain and the phase of the disturbances

generated by transverse and longitudinal velocity motions.

Similarly, the sum of the three sources of excitation, transverse

acoustics, longitudinal acoustics, and vortical disturbances, is a

function of both the gain and phase of each of these sub-transfer

functions and gives the overall flame transfer function, F .

Figure 2. Velocity disturbance mechanisms in a

transversely forced flame.

Although quantifying each of these “sub” transfer functions

( TF , LF , TLF , etc.) is an essential step towards a more

complete understanding of flame response to transverse

acoustic excitation, two important considerations stand out.

First, what is the relative contribution of each of these sub-

functions to the overall flame transfer function, F ? Second,

what is the proper reference velocity for a measured flame

transfer function in light of the relative importance of each of

these sub-functions? In the longitudinally forced example, the

5

longitudinal acoustic velocity fluctuation, ,L au , is the proper

reference velocity because it is not only a source of direct flame

disturbance (via LF ), but also the source of the vortical

velocity fluctuations (via LF ) that cause significant flame heat

release rate fluctuation.

For the transversely forced flame, it would seem intuitive

that the transverse acoustic velocity is the proper reference

velocity since it is the “source” disturbance, but several issues

arise. First, where is the location of this reference velocity?

Second, is the direct flame heat release rate fluctuation from the

transverse acoustic excitation branch (TF ) significant enough

to warrant the transverse velocity as a proper reference? Third,

how important is the transverse to longitudinal coupling branch

and could the gain of the transfer function TLF be large enough

to cause longitudinal motions to dominate the flame response?

The answer to the third question may supersede the first two

questions. Determining the characteristics of TLF and the

relative strength of the transverse and longitudinal acoustic

motion is the first step towards developing a flame transfer

function with physical significance. The transverse to

longitudinal velocity transfer function is given as,

,

,

( , )

( , )

L a oTL

T a o

u fF

u f

(6)

This transfer function describes the resulting axial velocity

fluctuation divided by the incident transverse velocity

fluctuation. If the gain of this transfer function is significantly

greater than unity, the flame response may be largely a result of

the longitudinally driven pathways – in this case, the more

appropriate reference velocity for the transversely excited flame

is the longitudinal acoustic velocity. Conversely, if the

amplitude of the transfer function is significantly less than unity,

the dominant acoustic velocity fluctuation would be in the

transverse direction and would drive both the vorticity

generation, through FT,ω, and the flame response.

Additionally, a separate transverse to longitudinal velocity

transfer function should be calculated or measured at different

transverse acoustic field symmetries, as has been done in the

results here and analytical work by Blimbaum et al. [51]. The

behavior of this velocity transfer function may be different for a

nozzle located at a pressure node versus a pressure anti-node,

and different still if there is a traveling component to the

transverse acoustic field. This symmetry dependence is denoted

by the variable σ.

As the frequency of transverse acoustic excitation is

modulated, the amplitude of longitudinal velocity fluctuations

changes due to acoustic response of the nozzle section. Studies

by Schuller et al. [52] and Noiray et al. [53] have both used

external transverse acoustic disturbances to characterize the

resonant frequencies of unconfined burners. This same concept

can be applied to the case of transverse instabilities. The axial

velocity fluctuations will be greatest at the resonant frequencies

of the nozzle cavity (not of the combustor), and this coupling

will be highly dependent upon system geometry. The transverse

to longitudinal velocity transfer function will therefore be

highly frequency dependent and its magnitude will give an

indication of the dominant acoustic disturbance imposed on the

flame at the nozzle.

The acoustic aspect of this coupling has recently been

investigated by Blimbaum et al. [51]. Their results showed that

the near-field acoustic behavior was strongly dependent on the

general waveform of the disturbance in the absence of the

nozzle. Said another way, the location of the nozzle with

respect to global velocity or pressure nodes determined the

response of the acoustic field in and around the nozzle. The

nozzle impedance had a large effect on the transverse to axial

coupling at a pressure anti-node and in the case of traveling-

wave acoustic excitation. In this regard, they showed that the

spatially-averaged pressure to axial velocity relationship was

quite close to the one-dimensional, translated impedance value

at the end of the side branch (nozzle). The notable exception to

this result was when the nozzle was located at a pressure node,

where the axial velocity characteristics are independent of the

nozzle impedance.

The importance of this acoustic coupling implies that the

flame transfer function, in the case of transverse forcing, will be

neither decoupled from the hydrodynamic fluctuations nor the

system acoustics. It is important to note, then, that the

quantitative results shown in the following experiments cannot

necessarily be extrapolated to other systems because of the

geometric dependence built into the measurement of this

transfer function. Our goal, then, is to propose a formulation by

which flame transfer functions for transversely forced flames

can be measured and understood.

The remainder of this paper is organized as follows. First,

the experimental facility, experimental and modeling methods,

and data analysis techniques are described. Second, results of

the transverse to longitudinal velocity transfer function are

reported using data from particle image velocimetry (PIV).

Third, several flame transfer functions will be reported and

compared to results from reduced-order modeling techniques.

Finally, we draw conclusions from these results and discuss

further implementation of these transfer function results.

EXPERIMENTAL SETUP AND ANALYSIS The experimental facility used in this study is a rig that

mimics the shape and acoustic modes of an annular combustor.

The flame is located at the center of the combustor and two sets

of three speakers are located on either side of the combustor.

The acoustic drivers are on the end of adjustable tubes that

allow the transverse resonant mode amplitude to be tuned for a

given frequency. In this study, the tubes were kept at a length of

1 meter. Further details of the experimental facility can be

found in Ref. [18]. Figure 3 shows the combustor facility with

the acoustic drivers, adjustable tubes, and settling chamber

6

upstream of the nozzle. The nozzle location is marked in the

figure with an arrow; a closer view of the nozzle is in Figure 4.

Figure 3. Transverse forcing experiment; nozzle location

indicated with white arrow.

Figure 4. Nozzle cavity with swirler, centerbody, and two

microphone ports.

A mixture of premixed air and natural gas at ambient

temperature flows into a large settling chamber, 0.41 meters in

diameter and 0.66 meters tall, which contains a perforated plate

used to condition the flow before it enters the nozzle. The main

test section is ceramic insulated stainless steel with inner

dimensions of 1.14 x 0.35 x 0.08 m, where transverse forcing is

in the longest direction (1.14 m) and the flow direction is in the

next longest direction (0.35 m). The nozzle, shown in Figure 4,

is h=0.095 meters long and has a diameter of douter=0.032

meters. The nozzle contains two microphones, described in

detail later in this section, a distance lmic=0.025 meters apart.

The swirler is located hS=0.038 meters from the tank and a

centerbody, with a diameter of dinner=0.022 meters and length of

hCB=0.051 meters, is located in the center of the swirler,

creating an annular jet at the dump plane. The exit plane of the

test section has four circular exhaust ports, each with a diameter

of 5.08 cm. This constriction on the exhaust was designed to

limit the variation in the hard-wall acoustic boundary condition;

larger exhaust ports could cause significant deviations in the

transverse acoustic field from the intended planar shape.

The standing wave acoustic mode inside the combustor is

created by driving the two sets of speakers at different phases.

When the speakers on each side of the combustor are driven at

the same phase (referred to as “in-phase” or “IP”), an acoustic

pressure anti-node and velocity node are theoretically created

along the centerline of the combustor and flow field. When the

speakers are driven 180⁰ out of phase (referred to as “out-of-

phase” or “OP”), an acoustic velocity anti-node and pressure

node are theoretically created along the centerline.

Pressure and PIV velocity data show that these velocity

node and anti-node assumptions are realistic. While the

velocity fluctuations along the centerline for the out-of-phase

forcing are certainly a maximum, they are not exactly zero in

the in-phase forcing experiments. This is due to both

imbalances in the acoustic diving as well as random turbulent

motion in the vortex breakdown region. Previous studies by the

authors [26] have shown that both the acoustic and resulting

vortical velocity disturbance field are significantly different for

these two forcing configurations.

The time-average flow field is shown in Figure 5. The

swirl number is 0.5 and the bulk flow velocity, calculated using

the mass flow rate and the annular area of the nozzle, is 10 m/s,

resulting in a Reynolds number of approximately 20,500 based

on the outer diameter of the nozzle; the flow exhibits bubble-

type vortex breakdown [54]. The velocities are normalized by

the bulk approach flow velocity, and the spatial coordinates by

the nozzle diameter, D.

The flow field shows the standard features of a swirling

annular jet with vortex breakdown [55, 56]. An annular jet

flows around a central vortex breakdown region, the result of

the absolutely unstable swirling flow. Two spanwise shear

layers are created between the annular jet and the central

recirculation zone as well as the annular jet and the outer fluid.

In this configuration, the flame is a V-flame and is stabilized in

the inner shear layer. During all tests the equivalence ratio was

0.95 and the flame remained attached to the centerbody.

7

Figure 5. Time-average, normalized velocity contours and

streamlines for reacting flow at a bulk velocity of uo=10 m/s

and equivalence ratio of 0.95. Black outlines below the

figure indicate the location and size of the centerbody and

dump plane below the field of view.

Two-dimensional velocity measurements were made using

particle image velocimetry (PIV) and a LaVision Flowmaster

Planar Time Resolved system. The laser is a Litron Lasers Ltd.

LDY303He Nd:YLF laser with a wavelength of 527 nm and a 5

mJ/pulse pulse energy at a 10 kHz repetition rate. The Photron

HighSpeed Star 6 camera has a 640x448 pixel resolution with

20x20 micron pixels on the sensor at the frame rate used. The

seeder particles used were aluminum oxide with a mean

diameter of 2 μm. In the current study, PIV images were taken

at a frame rate of 10 kHz with a time between laser shots of 20

microseconds at a bulk approach velocity of 10 m/s. 500

velocity fields were calculated at each test condition. All data

was imaged through a quartz window at the front of the

combustor that is a 22.9 cm x 22.9 cm and originates 0.64 cm

downstream of the dump plane.

Velocity field calculations were performed using DaVis 7.2

software from LaVision. The velocity calculation was done

using a three-pass operation: the first pass at an interrogation

window size of 64x64 pixels, and the final two passes at an

interrogation window size of 32x32, all with an overlap of 50%.

Each successive calculation used the previously calculated

velocity field to better refine the velocity vector calculation;

standard image shifting techniques were employed in the

calculation. The correlation peak was found with two, three-

point Gaussian fits, whose typical values ranged from 0.4 to 1

throughout the velocity field. There were three vector rejection

criteria used both in the multi-pass processing steps and the

final post-processing step. First, we rejected velocity vectors

with magnitudes greater than 25 m/s; these were deemed

unphysical in this specific flow field. Second, median filtering

was used to filter points where surrounding velocity vectors had

an RMS value greater than three times the local point. The

median filter rejects spurious vectors that occurred as a result of

issues with imaging, particularly near boundaries or as the result

of window fouling. Third, groups of spurious vectors were

removed; this operation removes errors caused by local issues

with the original image, including window fouling, and were

aggravated by using overlapping interrogation windows.

Finally, vector interpolation was used to fill the small spaces of

rejected vectors. Overall, an average of 8% of vectors were

rejected and replaced with interpolated values.

The chemiluminescence was measured with a Hamamatsu

H5784-04 photomultiplier tube (PMT). CH*

chemiluminescence was filtered using a Newport Physics

bandpass filter centered at 430 nm with a full-width half-

maximum of 10 ±2 nm. The pressure data in the two-

microphone method was taken using two Kistler 211B5

piezoelectric pressure sensors. The pressure sensors are located

lmic=0.0254 m center to center and 0.015 m from the dump

plane in the nozzle cavity. The data was recorded with a

National Instruments NI9205 data acquisition system using

Labview 9. Each channel on the data acquisition board was in

differential mode. Pressure and chemiluminescence data were

acquired at 30 kHz with sample lengths of 15,000 points. Data

were ensemble-averaged using ten ensembles. Uncertainty

estimates for the flame transfer functions were derived from

standard theory, as in Ref. [57].

Data analysis methods

Reference velocities were calculated from two types of

data, two-dimensional PIV data and two-microphone data. In

this section we describe the process by which reference

velocities were calculated from both sources of data.

For the calculation of the transverse to longitudinal velocity

transfer function, TLF , both transverse and longitudinal

reference velocities were calculated from the PIV data. These

reference velocities were calculated by spatially averaging

velocity data over areas of interest. This process reduces

random error caused by choosing data from a single point in the

PIV calculation. The transverse reference velocity is a spatial

average of the transverse velocity fluctuations along the

centerline of the flow over distance of one outer nozzle

diameters downstream. The axial reference velocity was

calculated by integrating the axial velocity at each point along

the radial direction at a downstream distance of x/D=0.05, the

closest location of good data to the dump plane. The

calculation of the spatially integrated velocity is shown in

Equation (7).

,

2

,

( ) ( 0, , )

1( ) ( , 0, )

L a

S

D

T a

D

u t u x r t rdrS

u t v x r t dxD

(7)

8

Here, S is the nozzle exit area and D is the outer diameter

of the nozzle. These spatially integrated velocities were chosen

as being representative of the acoustic fluctuations in both

directions. The spatially-averaged transverse velocity

fluctuations along the centerline seem representative of a

characteristic acoustic velocity in that region because much of

the random motion from the vortex breakdown region should be

removed by the spatial-average. The axial velocity fluctuations,

though, may contain significant vortical velocity components as

the measurement was taken 0.64 cm downstream of the dump

plane.

For the calculation of the flame transfer function, a two-

microphone method was used to calculate acoustic velocity in

the nozzle. This method only measures the longitudinal

acoustic motion in the nozzle as a result of the orientation of the

microphones. A standard two-microphone method [58] was

used to calculate the acoustic velocity.

Reduced-order Modeling

In this section, we briefly summarize the details of the

linear models used to predict the flame transfer function. Each

of these models applies a level-set approach for modeling the

flame response to input velocity disturbances. The key

assumption behind this approach is that the premixed flame is a

thin reaction zone that moves with the local flow field and

propagates normal to itself at the local burning velocity.

Mathematically, this is handled by identifying the flame as the

zero-contour of an implicit function denoted as ( , )G x t . The

evolution of this contour is tracked using the G-equation [59-

61]. The global heat release rate fluctuation is calculated by

integrating the local heat release rate fluctuations, calculated

using the G-equation, in both space (over the entire flame

surface) and time (over an acoustic cycle). The G-equation is

shown in Equation 8.

*

* * * *

f

Gu G s G

t

(8)

Here, *u is the flow velocity at the flame front,

*

fs

denotes the normal propagation speed of the flame front, and

the super-script “*” denotes a dimensional variable. The inputs

to the model are the time-average flame shape, the time-average

flow field, and the time-varying flow fluctuation at the flame

front. This modeling effort follows a series of recent papers

that have focused on predicting this spatially-integrated, global

heat release rate (e.g., [59, 62]) for velocity-coupled

combustion instability.

In this study, we present results from two different

formulations of the G-equation model with varying levels of

fidelity – one that describes the response of an idealized flame

to longitudinal velocity fluctuations, and one that describes the

response of the flame to the more realistic, asymmetric velocity

fluctuations present during transverse instabilities. The first

model considers an idealized axisymmetric V-flame forced by a

longitudinal (axisymmetric) disturbance field. This model

produces well-understood results for the flame transfer function,

and is used here as a reference in order to compare against the

newly measured flame transfer function from the transversely

forced flame. The results of this more basic formulation are

presented both with and without flame-stretch corrections.

In both the no-stretch [40] and stretch-corrected [63]

cases, the explicit flame response dynamics are governed by the

expression in Equation 9., where is the physical flame

coordinate, derived from the contour ( , )G x t using the explicit

transformation ( , , )G x r t . The driver of the flame

dynamics is the flow field fluctuation, where u and v are the

sum of the mean plus the harmonic velocities in each direction.

In both the no-stretch and stretch-corrected cases, the input

velocity field is modeled as the sum of a constant axial velocity

and axisymmetric, harmonic perturbations in both the axial and

radial directions. The flame is stabilized on a centerbody,

which provides a no-motion boundary condition for at x=0,

r=0. For the no-stretch flame model, the explicit flame

response dynamics are governed by:

2

1Lu v st r r

(9)

Note that f Ls s here. In the no-stretch case, the flame

speed SL is constant along the flame and the dynamics of the

flame are axisymmetric; flame motion is only a function of time

and radial distance from the centerbody. Here, the unsteady

heat release rate fluctuation is only due to flame area

fluctuations. Using the solution to Equation 9, the FTF for the

unstretched flame case can be expressed as [40], where Lf is the

flame length, R is the greatest radial extent of the flame, uo is

the mean flow velocity, and uc is the phase speed of the velocity

disturbance:

2 2

2 2

2

2

( ) (1 )2

( 1)

iSt i St

u

i i St e i St eFTF

St

(10)

where 2

0

2

2

2 2

0

; ;1

( 1);

f

c

f

L uKK

R u

L StSt St

u

(11)

Here, the reference velocity used is the axial velocity

fluctuation at the base of the flame.

The stretched flame results were calculated to compare

against the measured flame transfer function because of the

large frequency range at which the transverse-forcing FTF was

9

measured. Flame stretch can be an important factor at high

frequencies, where the flame wrinkle length-scale is smaller and

as a result, the local flame curvature is higher. Ref. [63] shows

that stretch effects on heat release rate fluctuations, through the

flame-speed-variation mechanism, are on the order of the area-

variation mechanism where * 2

2 1c St O . Here, *

c is a

function of the Markstein length of the fuel/air mixture and 2St

is a modified Strouhal number that describes the characteristics

frequency of flame disturbances. According to this scaling, the

frequency at which stretch effects become comparable to area

fluctuation effects is approximately 1200 Hz for this flame

configuration. It should be noted that the calculation of this

cutoff frequency is highly sensitive to flame parameters such as

flame aspect ratio and flame thickness; flame aspect ratio was

estimated from time-average OH-PLIF images, and flame

thickness was estimated to be 1 mm. In the measured results,

the FTF gain and phase of the no-stretch and stretch-corrected

results begin to diverge significantly at approximately 700 Hz,

where the maximum frequency in these tests was 1800 Hz. This

frequency is on the same order of magnitude as the estimate

from the model scaling, and it is for this reason that we present

the stretch-corrected model for the reader’s reference.

When flame stretch is taken into account, the unsteady heat

release rate fluctuations are the result of fluctuations in flame

area as well as changes in flame speed by stretch effects [63].

The flame dynamics are still governed by the expression in

Equation 9, but now the local flame speed is dependent on the

unstretched laminar flame speed, the Markstein length of local

fuel/air mixture, and both the local flame curvature and

hydrodynamic strain. The solution for the resultant transfer

function was derived in Ref. [63], but is quite extensive and not

reiterated here. Again, the reference velocity used in this

formulation is the axial velocity fluctuation at the base of the

flame.

While these two variations on the axisymmetric flame

response model provide a useful baseline for comparison

against the measured FTF results, a model that accounts for

asymmetries in the input velocity fluctuation may provide a

better representation of the flame dynamics of a transversely

forced flame. In this model (referred to here as the “transverse

model”), the flame dynamics are still governed by the G-

equation formulation, but a measured velocity field is used as

an input to the flame response model. The axial, radial, and

swirling velocities were measured by using high-speed PIV in

four separate laser planes; the first to measure the axial/radial

components as shown in Figure 5, and three planes

perpendicular to the direction of flow at x/D=0, x/D=1, and

x/D=2 to measure the radial and swirling velocity (see Ref. [64]

for more details). The velocity field was interpolated between

these planes to provide both an input time-average and

fluctuating velocity field from the measured flow data. With

measured velocity inputs, this model can capture the differences

in flame response between in-phase and out-of-phase acoustic

forcing. In this study, these velocity-field measurements were

only done at three frequencies, 400 Hz, 800 Hz, and 1200 Hz,

for both in-phase and out-of-phase forcing. As a result, the

transverse model can only be implemented at this limited range

of conditions.

Again, we use a level-set formulation to capture the

dynamics of the explicit flame position, , as shown in

Equation 12.

2 21

1

r x

f

uu u

t r r

sr r

(12)

Here, the flame stabilization boundary condition does not

allow for motion of the base of the flame at x=0, r=0. The

velocity fluctuations are generalized to be non-axisymmetric,

although the time-average flame shape is still assumed to be

axisymmetric. A transformation of this governing equation into

the frequency domain results in the expression in Equation 13,

where “overhats” denote frequency-domain quantities, primes

denote fluctuating quantities, and overbars denote time-

averaged quantities.

,

ˆ ˆ( ) ( )ˆ ˆ ˆ2 ( )t r x r

U r d ri St U r u u

r r dr

(13)

In this cylindrical coordinate system, it is convenient to

express the flame position fluctuation, , as a sum of

circumferential harmonic modes that vary in the theta-direction

at each radial location, r, and at each frequency, ω. This is

analogous to modal decompositions that are frequently applied

to swirling flows; see Ref. [64, 65]. This is expressed as:

ˆ ˆ( , , ) ( , )im

m

m

r e r

(14)

In this way, it can be shown that the flame’s response at

mode number m is a result of the azimuthal flow disturbance of

the same mode number. As such, the global flame heat release

rate can be written as:

2

2

20

2 ( ) 1

ˆ ( ) ( )

1

f

R r

fim m

mR r

Q drdrs r

h dr

rdrs r d dr rQe d

h rd dr

(15)

10

For non-axisymmetric modes ( 0m ), the integral over

is zero, which implies that only the axisymmetric mode

( 0m ) contributes to the global flame area fluctuation in an

axisymmetric flame. An important implication of this result is

that helical modes, while introducing substantial wrinkling of

the flame front, actually lead to no fluctuations in global flame

surface area in axisymmetric flames, a result that has also been

experimentally demonstrated by Moeck et al. [66]. This has

important implications when it comes to the use of flow field

data from experiments as inputs to the model. The fluctuating

flow field input must first undergo a helical-modal

decomposition to extract only its symmetric mode; this spatially

varying symmetric mode is used as the model input. The

expressions in Equation (15) are used to generate the numerator

in Equation (1). This, along with a reference velocity that is

obtained from the measurements, is used to calculate the

complete FTF as defined in that equation. A comparison

between the model prediction and experiments is presented in a

later section.

RESULTS

Transverse to longitudinal velocity transfer function

During a transverse instability, velocity fluctuations in both

the transverse and axial directions are present. As discussed

previously, the relative strength of these fluctuations at the

forcing frequency can be expressed as FTL, the transverse to

longitudinal velocity transfer function. The induced axial

velocity fluctuations are highly dependent on transverse

acoustic mode shape, frequency, and nozzle acoustics. Figure 6

shows the measured transverse to longitudinal velocity transfer

function, FTL, for the non-reacting flow case at a bulk velocity

of uo=10 m/s and the reacting flow case at the same velocity and

an equivalence ratio of 0.95. In the transfer function gain

(Figure 6a, b) and phase (Figure 6c) plots, the reacting in-phase

data are offset by -20 Hz and the non-reacting out-of-phase data

are offset by +20 Hz for clarity of the uncertainty bars.

a)

b)

c)

d)

Figure 6. Transverse to longitudinal velocity transfer

function a) gain (|FTL|), b) gain on a log scale (without

uncertainty estimates), c) phase (<FTL), and d) coherence

(γ2) for reacting flow with in-phase and out-of-phase, and

non-reacting out-of-phase acoustic forcing, a bulk flow

velocity of uo=10 m/s, and equivalence ratio 0.95.

11

These velocity transfer functions, FTL, were obtained from

data where the transverse velocity oscillation magnitude was

nominally 10% of the mean axial velocity. Five ensemble

averages were used to calculate these transfer functions and to

estimate the uncertainties [57].

The amplitude results from these transfer functions have

two interesting features. First, the gain has high values at 400-

500 Hz but drops below unity between 800 Hz and 1400 Hz.

Second, the amplitude peaks again at higher frequencies,

particularly 1500 and 1800 Hz. Although the transfer function

amplitude at both the low and high frequencies is approximately

equal, signifying non-negligible transverse to axial velocity

coupling, the flow response in these two cases is quite different.

This can also be seen by looking at the spatial distribution of

the amplitude of the axial velocity fluctuations at the forcing

frequency, as shown in Figure 7.

In the 400 Hz and 1000 Hz in-phase case, the axial velocity

fluctuations are concentrated in the shear layers, while in the

1800 Hz case the motion takes place across the entire diameter

of the jet, including the vortex breakdown region. Further

downstream, this region stretches even farther in the radial

direction as the jet spreads. Additionally, the coherence of the

axial and transverse velocity fluctuations is nearly unity at 1800

Hz, as well as several other higher frequencies, while the

coherence is low near 400 Hz and 1000 Hz. These plots show

that the spatial distribution of axial velocity fluctuations is

significantly different between the low to mid frequencies and

high frequencies.

This difference in response may be a manifestation of the

nozzle acoustics. A rough calculation of the natural frequency

of the nozzle (a half-wave) is 1800 Hz, in the range of the high

coherence, high amplitude response frequencies seen in Figure

6. This means that the external pressure fluctuation from the

transverse field in the 1800 Hz forcing case is driving the

fluctuations in the nozzle near the resonant frequency, resulting

in a large-scale axial response in and around the nozzle. This

leads to a bulk axial velocity fluctuation in the region of the

nozzle, as can be seen in Figure 7. Through a series of tests at

frequencies between 1700 Hz and 1900 Hz in 10 Hz

increments, it was shown that the flow response was maximized

in the 1790 – 1810 Hz region, indicating that the maximum

axial flow oscillations occur in this frequency range. Even

though 400 Hz and 1000 Hz have different values of FTL, the

flow response at these two frequencies is similar, and very

different from the flow response at 1800 Hz. 400 Hz and 1000

Hz are both still relatively far from the nozzle resonance

frequency near 1800 Hz, and hence do not display the bulk

axial motion characteristic of a strong transverse to longitudinal

acoustic coupling.

a)

b)

c)

Figure 7. Normalized axial velocity fluctuation amplitudes

at the forcing frequency throughout the flow field for a) 400

Hz in-phase, b) 1000 Hz in-phase, and c) 1800 Hz in-phase

acoustic forcing a bulk flow velocity of uo=10 m/s and

equivalence ratio 0.95.

The flow response in the axial direction at the nozzle

resonant frequency affects the entire flow structure, even the

vortex breakdown region, while the flow during off-resonant

frequency excitation responds only in the annular jet core. This

can lead to significant changes in the flame structure, as shown

in the recent work by the authors [64]. This change in time-

average flame shape can be seen in Figure 8.

12

a)

b)

Figure 8. Time-average flame shape for a) no acoustic

forcing and b) high-amplitude acoustic forcing at 1800 Hz

in-phase, for reacting flow at a bulk velocity of Uo=10 m/s,

equivalence ratio of 0.95 [64].

The results from both measurement of FTL and flow

visualization clearly show that the response of the nozzle and

the resultant longitudinal acoustic velocity fluctuations are key

components to understanding disturbance field physics, and in

turn, flame response. The pertinent flame disturbance

pathways, described in Figure 2, change as a function of

frequency. For example, in the range of 600 to 1000 Hz, the

gain of FTL is less than one, signaling that the transverse

acoustics dominate the acoustic disturbance field. However, at

frequencies such as 1500 Hz and 1800 Hz, the nozzle response

is significant enough to induce macro changes in the flow field

and flame, resulting in flame response that is driven by a

longitudinal acoustic motion.

Flame transfer function results

The flame transfer function describes the response of a

flame to an input velocity disturbance. We have measured

flame response for three different acoustic forcing conditions:

in-phase transverse forcing, out-of-phase transverse forcing, and

longitudinal forcing. The results of these tests, using the

longitudinal acoustic velocity measured by the two-microphone

method as the reference velocity, are shown in Figure 9.

a)

b)

c)

13

d)

Figure 9. Flame transfer function a) gain, b) phase, c)

coherence, and d) uncertainty in gain and phase for in-

phase (IP) transverse forcing, out-of-phase transverse

forcing (OP), and longitudinal forcing at a bulk velocity of

uo=10 m/s, equivalence ratio of 0.95.

The measured flame transfer function results are plotted

with the results from a longitudinally forced G-equation model

with and without the flame stretch correction, as well as the

transverse-forcing model. The measured flame transfer function

results follow the trends from these models. The results from

the transverse-forcing model, which directly accounts for the

non-axisymmetry of the disturbance field, match very well with

the experimental gain and phase results, and capture the

differences in flame response between the in-phase and out-of-

phase forcing conditions. Note that the phase results do not

account for the convective delay between the two-microphone

measurement location (1.5 cm upstream of the dump plane) and

the flame. The phase in both of the transverse forcing cases is

relatively constant, particularly at frequencies greater than

approximately 900 Hz. This is a result of the low amplitudes in

the gain curves over this frequency range.

Two issues need to be discussed with respect to these

transfer function results. First, the magnitude of the flame

transfer function gain is low compared to previously reported

longitudinal data, even with the reported uncertainty, which

increases at the low frequencies due to error in the two

microphone method. This is most likely due to two effects.

First, in the lower frequency range, the nozzle acoustics are not

excited and the response of the flame to purely transverse

excitation is very low [20]. This is because the net flux of

reactants through the flame over the course of the acoustic

cycle, in the transverse direction, is negligible and so the flame

heat release rate fluctuation sums to zero. This result also

confirms the findings discussed with respect to the transverse

model: axisymmetric flames only respond to the m=0

component of asymmetric disturbance fields. In the case of

pure transverse forcing and low nozzle response (low

longitudinal fluctuation amplitude), the disturbance field has a

very small m=0 component, and so the global response of this

axisymmetric flame is very low. However, when there is

significant axisymmetric perturbation, where the longitudinal

nozzle acoustics couple strongly with the transverse acoustic

field in the combustor, the flame response is higher due to the

stronger m=0 component of the disturbance field.

Secondly, flame response is inherently low at high

frequencies, as can be seen from the model results in Figure 9.

This stems from the fact that the flame acts like a low pass filter

as a result of the action of kinematic restoration along the flame

front. Short wavelength wrinkles are quickly destroyed as the

flame propagates into itself and contribute very little to the

overall heat release rate fluctuation; this effect is even more

drastic when stretch effects are accounted for in the model.

These points being made, the flame response does increase

at the frequencies at which the nozzle resonates, particularly

from 1540-1800 Hz in both the in-phase and out-of-phase cases.

This intuitively makes sense; the flame is excited by a stronger

longitudinal field that has been amplified by the action of the

nozzle resonance. It should be noted that measurements of the

flame transfer function at system resonances are normally not

recommended as the forcing amplitude may exceed the linear

range. In all these tests, the transverse velocity amplitude along

the centerline was held constant within the linear range, and the

axial velocity amplitude varied as described in the FTL results.

At 1800 Hz in-phase, the axial velocity fluctuation was

approximately 35% of the mean flow velocity, possibly outside

the bounds of what would be considered linear acoustic forcing.

Despite that, the FTF results are shown at this frequency to

emphasize the importance of transverse to longitudinal acoustic

coupling on flame response in transversely forced systems.

This coupling may help identify the conditions under which

transverse instabilities are truly detrimental in gas turbine

combustor geometries. The coupling between the azimuthal

combustor mode and the longitudinal nozzle mode is a function

of combustor geometric and acoustic parameters, such as

annular circumference, nozzle depth, and nozzle impedance

(itself a function of reactant temperature and pressure drop

across the nozzle). The frequencies at which these two modes,

the azimuthal and longitudinal, align will be the most powerful

drivers of self-excited transverse combustion instabilities,

resulting in high amplitude flame response. This type of

coupling can be seen in the LES simulations of Staffelbach et

al. [17].

Understanding of this coupling and the resultant flame

response can have great impact on the way that future gas

turbine combustors are designed. While combustion

instabilities have always been a serious consideration during the

design process [3], these results can provide guidelines for

relative sizing of annular combustor sections and nozzle

geometries over the range of operating conditions (nozzle

impedances) predicted for field operation. By “mistuning” the

nozzle to the annular section, high amplitude transverse

instabilities can be avoided as the transverse to longitudinal

acoustic coupling mechanism will be weak at off-resonance

14

frequencies. Acoustic modeling of proposed designs can be

used to screen these designs for possible coupling and eliminate

harmful transverse dynamics events during operation.

CONCLUSIONS Response of a swirl-stabilized flame to transverse acoustic

excitation is a process that involves not only the coupling of the

flame dynamics with the flow, but the coupling of several

different types of velocity fluctuations that constitute the overall

velocity disturbance field. In particular, the coupling between

transverse acoustics in the main combustor section and

longitudinal acoustics in the nozzle cavity determine the

dominant source of acoustic excitation. Used widely in

predictive thermoacoustic models, flame transfer functions

describe the response of the flame to an incoming velocity

perturbation. In this work, we have proposed a formula for

measuring and understanding the flame transfer function for a

transversely forced flame. First, the transverse to longitudinal

velocity transfer function, FTL, is measured. This velocity

transfer function describes the response of the nozzle to

transverse excitation and is a measure of the relative importance

of the transverse and longitudinal acoustic fields in the region of

the flame. The flame transfer function, with a longitudinal

reference velocity, is used to describe the flame response to the

overall system input as it has been shown that longitudinal

fluctuations are the dominant source of heat release rate

fluctuation. Results from these measurements indicate that

there is significant response of both the nozzle and the flame at

high frequencies, which align with predicted mode shapes in the

nozzle cavity.

The impact of this understanding extends to the very

beginnings of the gas turbine combustor design process, where

decisions on combustor geometry must be made. Assuring that

the mode shapes in the main combustor section and the swirler

nozzle are “mistuned” at nozzle impedances that correspond to

typical engine operating conditions can help eliminate the

occurrence of high amplitude transverse combustion

instabilities in the field.

ACKNOWLEDGMENTS The experimental portion of this work has been partially

supported by the US Department of Energy under contracts

DEFG26-07NT43069 and DE-NT0005054, contract monitors

Mark Freeman and Richard Wenglarz, respectively. The

authors wish to acknowledge Shreekrishna and Preetham for

their help with modeling, and Michael Kolb, Jared Mannino,

and Colin Vanetta for their help in acquiring experimental data.

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