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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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