7th US National Technical Meeting
of the Combustion Institute
Hosted by the Georgia Institute of Technology, Atlanta, GA
March 20-23, 2011
Mechanisms for Flame Response in a Transversely Forced
Flame
J. O’Connor, C. Vanatta, J. Mannino, T. Lieuwen
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30308, USA
This paper analyzes the factors controlling the flame response to transverse acoustic forcing.
Significant prior work has focused on longitudinal instabilities, where the flame transfer function
is often defined as the heat release fluctuation divided by the axial velocity fluctuation in the
burner nozzle. This single input transfer function definition does not generalize well for
transverse instabilities, where the flame disturbance mechanism is not solely due to axial velocity
fluctuations. In this work, we discuss the relative importance of axial and transverse acoustic
velocity disturbances over a range of frequencies. Results show that the disturbance field is highly
dependent on system acoustics. A velocity transfer function, describing the coupling between
transverse acoustics in the combustor and longitudinal acoustics at the nozzle, is formulated and
results are shown for both the non-reacting and reacting cases.
1. Introduction
Combustion dynamics, a coupling between resonant combustor acoustics and flame heat
release fluctuations, has been a problem with propulsion and power generation technologies
since the middle of the twentieth century [1]. Initially explained by Rayleigh [2], this coupling
can lead to high-cycle fatigue, reduced operability, and increased emissions. For gas turbines,
these instabilities have become more pronounced as engines have been optimized for low
emissions output [3]. The main emissions abatement strategy, lean combustion, has lead to a rise
in the severity of these instabilities and the more frequent appearance of transverse instabilities in
these engines.
Transverse instabilities are a common instability mode in rockets [4-6], augmenters [7-9],
and annular combustors [10, 11], but have only recently become a significant issue for canannular gas turbine combustors [12]. Traditionally, longitudinal instabilities have been the
dominant mode in can-annular engines and significant work has been done to understand this
instability [13-15]. More recently, work has been initiated to shed light on the flame response
characteristics and coupling mechanisms for transversely forced flames [16-20].
Quantification of global flame response is often in the form of a flame transfer function. In
the case of velocity coupled flame response, the definition of the flame transfer function is given
as the normalized flame heat release fluctuation divided by a normalized reference velocity
fluctuation [21], as is shown in Equation 1.
FF
Q ( fo , A) Q
uref ( fo , A) u
(1)
Flame transfer functions have been measured and calculated for longitudinally excited flames
in several studies [22-27]. Here, the reference velocity has usually been defined as the axial
velocity fluctuation at the edge of the nozzle exit, measured using a two microphone technique or
hot-wire anemometry. An important goal in determining these transfer functions is that they
isolate the flame response, and can be used as a submodel in a larger system dynamics model.
However, because the actual velocity field along the flame front, u’(x), may vary substantially in
amplitude and phase from u’ref, FF should not be interpreted as describing the flame response
alone – it also depends upon certain features of the combustor system as well.
The flame is excited directly by acoustic velocity disturbances, as well as acoustically excited
vortical disturbances. The flame responds quite differently to velocity disturbances arising from
acoustic and vortical disturbances because of their substantially different phase speeds and length
scales. As such, the same value of u’ref may lead to very different characteristics of the vortical
velocity field downstream [28]. To illustrate, the heat release expression for the longitudinally
forced case is broken into two constituent disturbance parts in Equation 2
Q
FLuL,a
F FL, uv
uL,a ( FL
F FL, )
(2)
where u’L,a is the longitudinal acoustic velocity disturbance (and the assumed reference velocity
in this expression), FL is the flame transfer function describing the response of the flame to the
longitudinal acoustic disturbance, FL,ω is the velocity transfer function between the acoustic
velocity disturbance and the vortical velocity disturbance formed at the dump plane, and Fω is
the flame transfer function between the vortical velocity disturbance and the resultant flame
response. Significantly, it shows that by using a single reference velocity, u’L,a in this case, the
flame transfer function FF is not only a function of the actual flame response, FL and Fω, but also
the shear layer response, FL,ω. Thus, the exact same flame could exhibit different transfer
functions, FF, if the shear layer response is different.
This issue becomes more problematic in transversely forced flames, where more sources of
flow disturbances exist [16]. The incident transverse acoustic perturbation may directly disturb
the flame, similar to the work by Ghosh et al. [29] for rocket injectors. The transverse acoustic
pressure field also leads to longitudinal acoustic fluctuations at the flame nozzle region, as
shown by Staffelbach et al. [18] in simulation, and in experimental results from O’Connor et al.
[16]. 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. The study of Rogers and Marble [7] shows an example of this
coupling in a high blockage-ratio combustor, where a self-excited transverse instability lead to
vortex shedding from the edges of the triangular bluff-body. These disturbance mechanisms and
their pathways are shown in Figure 1.
Transverse Acoustic Excitation
FTL
Longitudinal
Acoustics
FL
FT
FL
FT
Flow Instabilities
F
Flame Response
Figure 1. Velocity disturbance mechanisms present in a transversely forced flame.
Similar to the decomposition of the longitudinally forced disturbance field in Equation 2, the
processes in Figure 1 can be broken into their constituent parts, as shown in Equation 3:
Q
FT uT ,a
FL FTLuT ,a
F ( FT
FL FTL )uT ,a
(3)
The transverse acoustic disturbance, u’T,a, is the assumed reference velocity in this case. In
addition to the terms described above, FT is the flame transfer function describing the flame
response to the transverse acoustic disturbance, u’T,a, and FT,ω is the velocity transfer function
describing the formation of a vortical velocity disturbance by an acoustic velocity disturbance at
the edge of the dump plane.
The focus of this work is the coupling between transverse acoustic fluctuations and
longitudinal acoustic fluctuations in and around the nozzle, characterized by the transfer function
FTL, defined in Equation 4.
FTL
u L ,a ( f o , A)
uT ,a ( f o , A)
(4)
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 one, 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 disturbance. Conversely, if the amplitude of the transfer
function was 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.
As the frequency of transverse acoustic excitation is modulated, the response of the nozzle
changes due to acoustic response of the nozzle section. Studies by Schuller et al. [30] and
Noiray et al. [31] 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 transfer function will therefore
be highly frequency dependent and its magnitude will give an indication of the dominant
acoustic disturbance direction experienced by the flame at the nozzle.
The importance of this 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, as the longitudinal transfer function was. It is important to note, then, that the results
shown in these experiments cannot necessarily be extrapolated to other systems because of the
geometric dependence built into the definition of this transfer function.
2. Experimental Setup
In this section we overview the experimental facility and diagnostic systems used in this
study. For more experimental facility details, see O’Connor et al. [16]. The combustor mimics
an annular combustor configuration and was designed to exhibit a strong transverse acoustic
mode. A swirler nozzle is situated at the center of the chamber. The nozzle section outer and
inner diameters are 31.75 mm and 21.84 mm, respectively, and the swirl number is 0.5. The fuel
is natural gas and the equivalence ratio is 0.95. Six acoustic drivers, three on each side, provide
the acoustic excitation for the system.
The acoustic drivers on either side of the combustor can be controlled independently. By
changing the phase between the signals driving each side of the combustor, different wave
patterns can be created inside the combustor. When the drivers are forced in-phase, a pressure
anti-node and velocity node are created at the center of the experiment. When the drivers are
forced out-of-phase, a pressure node and velocity anti-node are created at the center.
Particle image velocimetry (PIV) is used to measure the velocity field in this experiment. A
LaVision Flowmaster Planar Time Resolved system allows for two-dimensional velocity
measurements at 10 kHz. The reference velocities in the transfer functions were calculated using
the PIV data calculated with 16x16 pixel interrogation windows with 50% overlap and a
resolution of 0.18 mm per pixel. Spatially averaged instantaneous velocities were calculated in
both the axial and transverse direction at the nozzle. The reference axial 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 spatially averaged transverse velocity is calculated along the centerline of the
flow, and integrated along a length of one nozzle outer diameter. These formulae are shown in
Equation 5.
D
uL,a (t )
S
u (x
S
0, r , t )rdr ,
uT ,a (t )
1
v ( x, r
D0
0, t )dx
(5)
All results are non-dimensionalized. The velocities are normalized by the bulk approach
flow velocity, m A (where ρ and A denote approach flow density and annulus area), the spatial
coordinates by the nozzle diameter, D, and the vorticity by the bulk velocity divided by the
annular gap width, Uo/(r2-r1).
3. Velocity fluctuations near the nozzle exit
The time-average flow field is shown in Figure 2, which shows both the time-averaged axial
velocity and vorticity. The flow is from left to right. Key features of the flow are an annular
reactant jet with a central vortex breakdown region and shear layers on the inner and outer edges
of the annular jet. The flame is stabilized in the inner shear layer.
a)
b)
Figure 2. Time-average a) axial velocity and b) vorticity for reacting flow at a bulk velocity
of Uo=10 m/s and a forcing frequency of fo=400 Hz in-phase.
The first case under consideration is an in-phase forcing test at 400 Hz. In this case, a
pressure anti-node is created over the centerline of the nozzle. The ratio of the acoustic
wavelength to nozzle diameter is 26.7, which indicates that each radial station of the nozzle
experiences roughly the same acoustic excitation from the fluctuating pressure field. Figure 3
shows the spectral content of the axial velocity fluctuations at two downstream locations near the
nozzle.
a)
b)
Figure 3. Spectra of axial velocity fluctuations at a) x/D=0.05 and b) x/D=0.27 for reacting
flow at a bulk velocity of Uo=10 m/s, equivalence ratio of 0.95, and a forcing frequency of
fo=400 Hz in-phase.
It is evident from the plots in Figure 3 that there are significant axial velocity fluctuations (812% of the mean) at the forcing frequency in the nozzle region. At x/D=0.05, the fluctuations
are close to the centerline of the annular jet, showing the longitudinal acoustic motion that results
from the fluctuating transverse acoustic field over the nozzle. Further downstream at x/D=0.27,
the axial velocity fluctuations at the forcing frequency are mainly in the shear layers as a result
of the coherent structures convecting downstream in these regions. Additionally, there is
significant low frequency axial motion in the vortex breakdown region. Similar spectral content,
with a lower magnitude, is also seen in the vortex breakdown region in the cases without
acoustic forcing.
4. Transverse to longitudinal velocity transfer function
Data like those shown above and those from other frequencies tested show that significant
axial fluctuations are excited by the transverse acoustic excitation at certain frequencies. To
describe this coupling, this section quantifies the transverse to longitudinal velocity transfer
function, as described in Equation 2. Figure 4 shows the calculated velocity transfer function 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 and phase 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 error bars.
a)
b)
c)
d)
Figure 4. Transverse to longitudinal velocity transfer function a) gain (|FTL|), b) gain on a
log scale (without error bars), 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 at fo=400 Hz, a
bulk flow velocity of Uo=10 m/s, and equivalence ratio 0.95.
These transfer functions 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 function gains and to estimate the uncertainties [32].
The amplitude results from these transfer functions have two interesting features. First, the
gain has high values, on the order of 6, at 400-500 Hz but drops to very small values by 800 Hz.
Second, the amplitude peaks again at higher frequencies, particularly 1800 Hz. Although the
transfer function amplitude at both these frequencies is large, signifying non-negligible
transverse to axial velocity coupling, the flow response in these two cases is fundamentally
different. This can also be seen by looking at the spectrum as a function of radial location, as
shown in Figure 5.
a)
b)
Figure 5. Axial velocity spectra at x/D=0.05 for a) 400 Hz in-phase and b) 1800 Hz in-phase
acoustic forcing a bulk flow velocity of Uo=10 m/s and equivalence ratio 0.95.
In the 400 Hz case, the axial velocity fluctuations are concentrated in the annular jet, 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 very low near 400 Hz. This trend is true even downstream of the dump plane, as is
shown in Figure 6. Here, the magnitude of the axial velocity fluctuations at the forcing
frequency is shown for the entire flow field. Like the spectra in Figure 5, these plots show that
the spatial distribution of axial velocity fluctuations is significantly different between the low
frequencies and high frequencies, despite the similar magnitude in FTL of the two.
a)
b)
Figure 6. Magnitude of the axial velocity fluctuations at the forcing frequency throughout
the flow field for a) 400 Hz in-phase and b) 1800 Hz in-phase acoustic forcing a bulk flow
velocity of Uo=10 m/s and equivalence ratio 0.95.
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 4. 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.
Through a series of tests at frequencies between 1700 Hz and 1900 Hz in 10 Hz increments,
it was shown that both the flame and flow response were maximized in the 1790 – 1810 Hz
region, indicating that the maximum axial flow oscillations occur in this frequency range. Future
tests using a two-microphone technique in the nozzle will be able to further quantify this effect.
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 [33]. This change in flame shape can be
seen in Figure 7.
a)
b)
Figure 7. 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.
6. Conclusions
The velocity disturbance field of a transversely forced swirl-stabilized flame has several
different pathways by which the flame response can be generated. In this work we focus on the
transverse to longitudinal velocity coupling mechanism, which describes the coupling between
the pressure fluctuations from the transverse acoustic field and the resultant longitudinal motion
in and around the nozzle. A velocity transfer function, FTL, was measured for several different
frequencies and the connection to transfer function gain and nozzle acoustic response was
discussed. In future, the authors plan to measure the velocity transfer function using a twomicrophone method with pressure sensors in the nozzle cavity. Also, a more nuanced definition
of the flame transfer function for a transversely forced flame will be developed and results
reported.
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