Proceedings of ASME Turbo Expo 2012
GT2012
June 11-15, 2012, Copenhagen, Denmark
GT2012-70053
INFLUENCE OF TRANSVERSE ACOUSTIC MODAL STRUCTURE ON THE FORCED
RESPONSE OF A SWIRLING NOZZLE FLOW
Jacqueline O’Connor, Tim Lieuwen
Georgia Institute of Technology
Atlanta, Georgia, USA
ABSTRACT
This study describes continued investigations of the
response of a swirling flow to transverse acoustic excitation.
This work is motivated by transverse combustion instabilities in
annular gas turbine engine architectures. This instability
provides a spatially varying acoustic velocity disturbance field
around the annulus, so that different nozzles encounter different
acoustic disturbance fields. In this study, we simulate this effect
by looking at a standing wave acoustic field where the nozzle is
located at either a velocity anti-node, referred to as out-of-phase
forcing, and a velocity node, referred to as in-phase forcing.
The out-of-phase forcing condition provides an asymmetric
forcing field about the center plane of the flow and excites an
asymmetric response in the flow field; the in-phase forcing
provides a symmetric forcing condition and results in symmetric
flow response near the nozzzle.
The symmetric versus asymmetric flow response was
measured in two ways. First, in the r-x plane where axial and
radial components of velocity are measured using high-speed
particle image velocimetry (PIV), a helical and ring vortex
rollup of the shear layers is evident in the asymmetric and
symmetric forcing condition, respectively. Additionally, the
swirling motion of the jet is measured in the r-θ plane at two
downstream locations and a spatial decomposition is used to
calculate the strength of azimuthal modes in radial velocity
fluctuations. At the forcing frequency, the m=0 mode is
strongly excited at the nozzle exit with symmetric forcing, while
asymmetric forcing results in a strong peak in the m=1 mode, or
the first helical mode. The results in this plane of view are
congruent with those in the r-x plane. Further downstream,
however, the mode strengths change as the vorticity is religned
and natural asymmetries of the swirling jet set in.
Finally, the low frequency self-excited motion of the vortex
core was measured and characterized in the unforced flow. It is
composed of an m=-1 and m=-2 mode, and the physical
interpretation of these mode numbers is highlighted. High
amplitude acoustic forcing decreases the amplitude of
oscillation of this structure in both the in-phase and out-ofphase forcing, but to varying degrees.
NOMENCLATURE
Am
Counter-rotating mode amplitude
Bm
Co-rotating mode amplitude
D, Dout
Outer diameter of nozzle
Din
Inner diameter of nozzle
N
Number of azimuthal modes
R
Radius of centerbody
S
Swirl number
SL
Laminar flame speed
m
Azimuthal mode number
r
Radial direction
t
Time
Velocity vector
u
Fourier transform of velocity
û
u
Axial velocity
Axial velocity fluctuation
u
uo
Bulk velocity
ur,int
Radial velocity integrated in radial direction
v
Transverse velocity
Transverse velocity fluctuation
v
x
Axial direction
z
In-plane direction
Ω
Vorticity
β
Flame aspect ratio
θ
Azimuthal direction
ω
Angular frequency
1
Copyright © 2012 by ASME
INTRODUCTION
Combustion instabilities have plagued the development and
operation of high performance combustion technologies for
almost a century. These instabilities, the result of a coupling
between flame heat release fluctuations and resonant acoustics
inside the combustion chamber [1], have proven to be
destructive to engine hardware, burdensome on engine
operating costs, and detrimental to engine performance and
emissions. In particular, instabilities have caused major
challenges in the development of lean, premixed combustors
used in low NOx gas turbines [2].
A variety of mechanisms exist through which coupling
between the resonant acoustics and flame heat release
fluctuations can occur [3]. In premixed systems, two types of
coupling mechanisms – “equivalence ratio coupling,” where the
fuel flow is pulsed as a result of the acoustic fluctuations [4],
and “velocity coupling,” by which the flame area is modulated
through the action of velocity fluctuations in the flow field [5-8]
– are the dominant sources of flame heat release fluctuations.
In this study, we consider the action of velocity coupling for
flames undergoing transverse flow oscillations.
Much research has considered the response of flames to
longitudinal acoustic forcing, a significant issue in can-annular
combustion systems [9-12].
In this case, the acoustic
fluctuations oscillate in the direction of the flow field, creating a
symmetric forcing condition for the flame. This applies to
flames in both laboratory test stands as well as actual engine
configurations. In the case of transverse instabilities, which are
prevalent in annular gas turbine combustion architectures [1316], each nozzle experiences a different disturbance field than
its neighbors. For example, a nozzle situated near a velocity
node experiences large pressure oscillations (that in turn, excite
axial nozzle velocity oscillations), with negligible transverse
velocity oscillations. A nozzle situated near a pressure node
experiences significant transverse flow oscillations. A nozzle in
a traveling wave field experiences both pressure and velocity
oscillations [17]. In this paper, we specifically focus on the
acoustic velocity node and anti-node configurations,
corresponding to symmetric and asymmetric acoustic velocity
fields, respectively.
To explore the response of the flow to these different
acoustic symmetry conditions, we have used high-speed particle
image velocimetry (PIV) to measure the velocity field under a
variety of acoustic forcing conditions. The work presented here
follows closely from our prior publications [17, 18], with the
key new contributions stemming from expanded insight into the
shear layer and vortex breakdown region structures.
Figure 1 illustrates the basic geometry considered in this
study, showing an annular, swirling jet that is excited by a
transverse acoustic field. This notional picture shows the
annular swirling jet with two spanwise shear layers, emanating
from each edge of the annular nozzle, and the large central
recirculation zone downstream of a centerbody. Although the
results presented in the current study are for a non-reacting
flow, the nominal flame position is shown for reference. Here,
the flame is stabilized in the inner shear layer of the centerbody.
Figure 1. Notional drawing of the time-average flow field of
an annular swirling jet with vortex breakdown. The
spanwise shear layers, vortex breakdown bubble, and flame
are shown here. Dotted lines indicate contours of zero axial
velocity and several streamlines are depicted.
Figure 1 shows a typical swirling annular jet with vortex
breakdown. Vortex breakdown stems from the instability of the
swirling jet and arises from the conversion of radial and axial
vorticity to azimuthal vorticity [19, 20]. Though several states
of vortex breakdown are possible [21], the one of primary
interest here is the bubble-type vortex breakdown state (VBB),
which appears at higher swirl numbers. A large literature on the
mechanisms of breakdown and swirling flow structure exists,
e.g., see Liang and Maxworthy [22], Billant et al. [23], and
others [24, 25].
As is also evident in Figure 1, this geometry introduces two
spanwise shear layers originating from the inner and outer
annulus edges, and two streamwise shear layers associated with
the azimuthal flow. These shear layers are subject to the
Kelvin-Helmholtz instability, which can be convectively or
absolutely unstable, depending upon the reverse flow velocity
and swirl number [26]. In addition, the centerbody introduces a
wake flow. In the current study, the swirl number is high
enough such that the vortex breakdown bubble and centerbody
wake have merged to create a single central recirculation zone
[27].
Harmonic forcing can have a significant effect on the
dynamical behavior of this flow field. In this discussion,
though, it is important to recognize the different types of
instabilities present in the flow, particularly their absolute or
convectively unstable nature. This specific flow field can be
roughly broken into the convectively unstable shear layers and
the globally unstable swirling vortex breakdown bubble (VBB).
This demarcation should not be taken too far, though, as
interactions do occur, particularly between the flow structures in
the VBB and the inner shear layer.
Nonetheless, the
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Copyright © 2012 by ASME
dramatically different manner in which these flow structures
respond to forcing makes this distinction a useful one.
First, we address the convectively unstable shear layers.
The free shear layer is a convectively unstable flow, where the
separating vortex sheet rolls up into concentrated regions of
vorticity that pair with downstream distance [28, 29]. As a
disturbance amplifier, shear layers respond strongly to acoustic
forcing [26, 30]. In this case, the vortex passage frequency
locks onto the forcing frequency, generally through a vortex
pairing or collective interaction phenomenon [31]. Acoustic
forcing has been shown to have similar effects on the shear
layers in a variety of flow geometries, including circular,
annular, and swirling jets [32-36]. Shear layer response in the
current geometry has been considered in previous work by the
authors [17, 37].
Consider next forcing effects on the VBB, a result of a
global instability in the flow field. Here, the global instability
of vortex breakdown appears to be key to understanding the
response characteristics. Globally unstable systems execute
self-excited, limit cycle motions, even in the absence of external
forcing [38]. For example, swirl flows often exhibit intrinsic
narrowband oscillations, manifested by structures such as the
precessing vortex core (PVC) or other helical coherent
structures [39]. Because of the globally unstable nature of the
VBB, low amplitude forcing often has minimal impacts upon it.
This behavior is to be contrasted with that exhibited by the
convectively unstable shear layers and the entire VBB itself,
whose space/time position is influenced by the outer flow
oscillations. If the acoustic excitation amplitude is high,
significant changes in the shape and natural oscillations of the
vortex breakdown bubble can occur. This is due to a
phenomenon known as lock-on or entrainment, where the
system oscillations are "entrained" by the external forcing and
oscillate at the forcing frequency rather than the unforced
frequency [40]. Additionally, high amplitude acoustic forcing
can change the time-average shape of the flow, causing the
vortex breakdown bubble to grow in both size and strength [35,
41, 42].
In particular, several factors influence the response of the
PVC to acoustic forcing, including both flow and geometric
parameters. In some cases where the VBB exhibits intrinsic
narrowband oscillations, external excitation at that natural
frequency of oscillation can further amplify it [40]. For
example, LES studies by Iudiciani and Duwig [42] show that
low frequency forcing ( St  0.6 ) resulted in a decrease in the
strength of the PVC fluctuation amplitude, while higher
frequency fluctuations resulted in increases in PVC fluctuation
amplitude.
The presence of transverse forcing adds an additional
degree of freedom to the forced problem because of the nonaxisymmetric nature of the forcing [36, 43, 44]. High Reynolds
number, unforced swirling flows are not instantaneously
symmetric [21]; e.g., swirl biases the strength of co- and
counter-signed helical instabilities.
Moreover, nonaxisymmetric forcing can preferentially excite certain non-
axisymmetric modes in a different manner than they would
otherwise naturally manifest themselves. This effect of forcing
symmetry on the flame response has been measured
experimentally and discussed in previous work by the authors
[17]. Here, the difference in the vortical response of the flow
field between symmetric (in-phase) and asymmetric (out-ofphase) forcing has been shown to have a significant effect on
the character of the flame response. This can be seen in flame
luminescence imaging, examples of which are shown in Figure
2.
1
2
3
4
1
2
3
4
a)
b)
Figure 2. Examples of flame response, via luminescence
imaging, for a) in-phase (symmetric) and b) out-of-phase
(asymmetric) forcing conditions for a flow velocity of 10
m/s, swirl number of 0.5, and equivalence ratio of 0.9.
Images are 0.6 ms apart in each filmstrip.
In these images, it is evident that the flame wrinkling
changes with acoustic field symmetry. For asymmetric forcing,
an asymmetric, helical pattern exists on the flame near the
nozzle exit, while in the symmetric forcing case, a symmetric
ring vortex creates axisymmetric wrinkles on the flame surface,
at least near the flame base. The differences in flow response
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Copyright © 2012 by ASME
that create this flame response effect are the motivating factors
for this work.
The remainder of this paper is organized as follows. First,
the experimental setup is briefly overviewed and several data
analysis methods detailed. Next, the results section begins by
presenting the time-average flow fields, as well as the
dynamical behavior of the unforced flow field. These serve as a
baseline for the forcing results, which show the effect of both
symmetric and asymmetric forcing on the shear layers as well as
the self-excited motion of the vortex breakdown bubble.
Finally, we discuss the implications for these results on the
understanding of transverse instabilities in gas turbine engines.
EXPERIMENTAL SETUP AND ANALYSIS
In this section we overview the experimental facility and
diagnostic systems used in this study. Further details are
provided in O’Connor et al. [37]. The combustor mimics an
annular combustor configuration and was designed to support a
strong transverse acoustic mode. A swirler nozzle is situated at
the center of the chamber with an outer diameter of 31.75 mm,
inner diameter of 21.84 mm, and geometric swirl number of
0.85 [45]. This swirl number calculation used the vane angle of
45⁰, inner diameter of 22.1 mm and outer diameter of 31.8 mm
are used. Six acoustic drivers, three on each side, provide the
acoustic excitation for the system.
The acoustic drivers on either side of the combustor are
controlled independently. By changing the phase between the
signals driving each side of the combustor, different wave
patterns, both standing and traveling waves, can be created.
When the drivers are forced in-phase, an approximate acoustic
pressure anti-node and acoustic velocity node are created at the
center of the experiment – this will be referred to as the
symmetric forcing condition. When the drivers are forced outof-phase, an approximate acoustic pressure node and acoustic
velocity anti-node are created at the center – this will be
referred to as the asymmetric forcing condition. The symmetry
of the acoustic forcing corresponds to the motion of the acoustic
velocity field, as it is the acoustic velocity that is coupling with
the vortical velocity to excite flow response.
Particle image velocimetry 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. In this study, two sets of PIV data
were taken. The first set of PIV data examined the flow in the
axial flow direction. In this case, the sheet entered the
experiment from a window at the exit plane of the combustor
and reached a width of approximately 12 cm at the dump plane.
This alignment will be referred to as the r-x alignment.
A second set of PIV data were taken to look at the swirling
component of the flow. Here, a laser sheet with a thickness of
approximately 1 mm and a width of 10 cm entered the front
window at a height of 0.7 cm above the dump plane. An
Edmunds Optics 45-degree first surface mirror was placed
above the exit port window and the camera was aligned and
focused on the image on this mirror. Special care was taken to
ensure that there was no distortion in the particle image in the
mirror; mirror/camera alignment was checked often. This
alignment will be referred to as the r-θ alignment.
Several techniques were used to quantify the behavior of
the response of the flow to transverse acoustic forcing. The use
of high-speed PIV diagnostics allowed for spectral analysis of
the data, which was used extensively in this study. First, spectra
of several quantities were calculated using fast Fourier
transforms. 500 images were taken at 10 kHz, resulting in a
spectral resolution of 20 Hz and a maximum resolvable
frequency of 5 kHz.
The frequency domain velocity was also used to
decompose the flow field into spatial azimuthal mode shapes in
the r-θ flow field. This technique has been used by other
authors [46, 47] to calculate the mode shapes in jets and
axisymmetric wakes. In this analysis, the radial and azimuthal
velocities are extracted from the instantaneous velocity fields at
8 radial locations and 21 points around each radius. The
Fourier transformed velocity field is then decomposed as:
N
uˆ  r , ,     Am  r ,   eim  Bm  r ,   e im
m0
Here,
(1)
uˆ  r , ,   is the complex Fourier transform of the
velocity fluctuation.
Am  r ,   and Bm  r ,   are the
complex amplitudes of the modes of the counter-swirling
(clockwise) and co-swirling (counter-clockwise) disturbances,
respectively. Finally, m is the mode number describing the
spatial fluctuation in the azimuthal directions that can have
integer values between   N  1 2 and  N  1 2 , where N is
the number of points measured in the azimuthal direction.
Several results from this analysis are presented in this
paper. It should be noted that each mode strength, Am and Bm,
has not only a spatial modal dependency, but also a frequency
dependency. This means that the spatial mode distributions
over a variety of mode numbers, m, can be plotted at specific
frequencies, or integrated over frequency ranges. The first
analysis approach eliminates the frequency dependence by
calculating mode strengths, ur,int, that have been integrated over
all frequencies (0-5000 Hz). These values are then normalized
by the spectral bandwidth in order to give an average spectral
density (units of Hz-1). This is done to compare amplitudes to
the narrowband analysis described in the second analysis. The
second analysis looks at the mode strength at the forcing
frequency only. This quantity captures the motion due to and in
response to acoustic excitation; in particular, it captures the
convectively unstable response of the shear layers.
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Copyright © 2012 by ASME
RESULTS
Time Average Flow Features
Figure 3 shows the non-reacting, velocity (left) and
azimuthal vorticity normalized by the bulk velocity divided by
the annular gap width,   Douter  Dinner  uo , (right). The axial
velocity plot shows the jets on either side of the centerline with
the reverse flow region in the center that is merged with the
centerbody wake. The two shear layers on each side of the
annular jets are evident in the vorticity plot.
The time-averaged azimuthal velocity field is shown in
Figure 4, obtained at the measurement plane x/D=0 and x/D = 1;
the x/D=0 plane is at the bottom of the field of view in the x-r
plane, 0.7 cm from the dump plane. These time-average views
show a swirling jet with a relatively uniform profile in the radial
direction across the annular width. However, combustor
confinement clearly affects the shape of the jet at this location,
as the flow is able to spread freely in the r-direction, while it is
confined in the z-direction.
b)
Figure 4. Time-average swirling velocity field at a) x/D=0
downstream and b) x/D=1 showing velocity vectors and
contours of velocity magnitude for the non-reacting annular
swirling jet at uo=10 m/s, S=0.85.
Unforced Flow Dynamics
The unforced flow has several inherent dynamical features.
First, concentrated vortical regions stemming from the rollup of
the convectively unstable shear layers are present. In this case,
the dominant shear layer mode is helical, typical of swirling
flows [22]. This can be seen in Figure 5, which shows an
instantaneous snapshot of the velocity and vorticity field in the
unforced case.
Figure 3. Time-average velocity streamlines and vectors
(left) and normalized azimuthal vorticity (right) for the
unforced, non-reacting annular swirling jet at uo=10 m/s,
S=0.85.
Figure 5. Instantaneous velocity and normalized azimuthal
vorticity (in color contour) data showing helical vortex
rollup of both inner and outer shear layer, for the unforced,
non-reacting annular swirling jet at uo=10 m/s, S=0.85.
a)
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a)
d)
b)
e)
c)
f)
Figure 6. Coherent jet core motion shown through the
filtered velocity field in the r-θ plane at x/D=1 and a) t=0.5
ms, b) 4.1 ms, c) 10.1 ms, d) 13.1 ms, e) 17.1 ms, f) 20.1 ms,
for a non-reacting, unforced flow at uo=10 m/s, S=0.85.
Gray areas approximate the location of the inner and outer
shear layers.
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Copyright © 2012 by ASME
Additionally, a self-excited structure in the vortex
breakdown region is evident from the r-θ view. Here, data from
the x/D=1 plane is shown as this downstream location shows the
most intense motion in the central recirculation zone. In this
location, the vortex breakdown bubble begins to widen, as can
be seen in Figure 3, and is sufficiently far from the centerbody
boundary condition.
Returning to the central recirculation zone, the structure in
this zone induces fluctuations at low frequencies, 200 Hz and
below, as it precesses about the flow centerline. The frequency
content of these motions is discussed later. To visualize this
structure, filtered velocity data in the r-θ plane are shown in
Figure 6 as a series of images at progressive instances in time.
Here, the velocity data have been low-pass filtered at 200 Hz,
where the dominant fluctuations in the flow occur as described
above, using a second-order Butterworth filter. Additionally,
the gray regions represent the approximate areas of the inner
and outer shear layers at this downstream location.
Two concurrent motions are evident from this series of
velocity fields in Figure 6. The first is a fluctuation in the
overall shape of the jet, while the second is due to two smallerscale coherent structures in the central recirculation zone. First,
as is seen in the time-average image in Figure 4b, the jet at
x/D=1 spreads preferentially towards the top-left and bottomright quadrants of the image. In the time series of the low
frequency motion, a semi-periodic squeezing and contracting of
the jet along the axis along which the jet is biased is observed.
This bias can be clearly seen in Figure 6b and Figure 6d, as
is indicated with dotted lines in time-instances t=0.5 ms and
t=13.1 ms. Motion of the jet in the opposite direction is evident
at the subsequent time, in Figure 6c and Figure 6f and shown
using dotted lines in time-instances t=10.1 ms and t=20.1, with
a transitional time shown in time-instances t=4.1 ms and t=17.1
ms. This motion is indicative of an m  2 mode in the jet;
here the jet pulses in each direction twice as the smaller
structures rotate once around the center of the jet. This motion
seems to indicate the existence of a precessing vortex core, as
described in Ref. [39].
The second motion is smaller scale, and involves two
coherent structures that precess about the center of the flow
field. These can be seen in several of the images in Figure 6,
circled with a dashed line. While these coherent structures
clearly precess around the center of the flow field for parts of
the “cycle,” they also overlap. For example, the structures are
separate in Figure 6a-c for time-instances t=13.1 – 20.1 ms,
while they overlap at time-instances in Figure 6 at t=0.5 – 10.1
ms. This motion exists in addition to the precessing vortex core
described above, and seems to indicate the existence of a
second structure in the vortex core.
Figure 7 shows calculations from the azimuthal modal
decomposition of the radial velocity fluctuations; in this plot,
the modal coefficients have been integrated over the entire
frequency range (0 – 5000 Hz) and several radii (r/D = 0 – 1) in
order to represent the complete energy contained in that mode.
Here, it is evident that modes m  2 and m  1
dominate over the rest of the mode energies; this is congruent
with the physical observations discussed with reference to
Figure 6. The spatial distribution of these modes is shown in
Figure 8, which plots the mode energy (integrated over the
entire frequency spectrum) versus radius for several important
modes.
Figure 7. Distribution of mode amplitude integrated from
0-5000 Hz and over radii r/D=0-1 plotted as a function of
mode number at x/D=1 for a non-reacting, unforced flow at
uo=10 m/s, S=0.85.
Figure 8. Distribution of mode amplitude integrated from
0-5000 Hz plotted as a function radius for several mode
numbers at x/D=1 for a non-reacting, unforced flow at
uo=10 m/s, S=0.85. Gray areas approximate the location of
the inner and outer shear layers.
In order to illustrate how this modal energy is distributed in
frequency, Figure 9, plots the spectrum of fluctuations for
several mode numbers. The plot shows that the majority of the
fluctuation energy is in modes m  2 and m  1 at
frequencies below 200 Hz.
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Copyright © 2012 by ASME
about the center plane of the experiment, the shear layers roll up
into ring vortices on both the inner and outer edge of the
annular jet, a symmetric response (the m  0 mode) at the
dump plane to a symmetric forcing condition. In the out-ofphase forcing case, the asymmetric forcing condition, the shear
layers roll up into helical structures, an obviously asymmetric
response to the asymmetric acoustic forcing. The downstream
evolution of these structures will be discussed later. Examples
of this response are shown in Figure 10.
Figure 9. Spectra of several mode numbers integrated over
radius and frequency in the r   plane at x D  1 , for a
non-reacting, unforced flow at uo=10 m/s, S=0.85.
These results provide a baseline for the next analyses,
which investigate the effect of in-phase (symmetric) versus outof-phase (asymmetric) forcing on the flow structures described
here. In each of the cases presented here, the in-phase forcing
amplitude was v uo  0.05 and the out-of-phase forcing
amplitude was v uo  0.4 .
Response at the Forcing Frequency
Before presenting data, two main points should be
emphasized about the response of the flow at the forcing
frequency. The first, a direct result of the nature of a standing
acoustic field, is that the in-phase and out-of-phase forcing
conditions result in very different acoustic velocity fluctuation
amplitudes. As discussed above, in-phase, or symmetric,
forcing results in an approximate velocity node along the
centerline and hence very low amplitude acoustic velocity
fluctuations in the region of the flow. Conversely, the out-ofphase, or asymmetric, forcing condition results in an
approximate velocity anti-node and high amplitudes of acoustic
velocity fluctuations.
The second effect stems directly from the symmetry of the
acoustic forcing. As discussed above, asymmetric forcing
results in asymmetric flow response; this is a trend that has been
observed in a variety of flows and forcing configurations. This
has shown to be a key idea in our analysis of the response of the
swirling jet shear layers. The structures that most readily
respond to acoustic forcing are the convectively unstable shear
layers. As disturbance amplifiers, these structures respond
strongly to acoustic forcing and are highly influenced by the
symmetry of the acoustic field.
In shear layers, the response of these structures to
symmetric verses asymmetric forcing fields is quite different.
In the in-phase forcing case, a symmetric forcing configuration
a)
b)
c)
d)
Figure 10. Comparison of instantaneous (a,c) and filtered
(b,d) velocity at the forcing frequency for a,b) in-phase and
c,d) out-of-phase forcing at 400 Hz, non-reacting flow at
uo=10 m/s, S=0.85. White lines trace the vortex path.
Here, in Figure 10, both the instantaneous as well as the
filtered velocity field at the forcing frequency is shown. This
filtered velocity field is calculated by reconstructing the
velocity field at the forcing frequency only, as detailed in
previous publications [37]. The filtered velocity is appropriate
to show in this case because the shear layers are responding
directly to the acoustics and further inspection and comparison
of the instantaneous and filtered velocity and vorticity fields
reveals great similarity.
Changes in the modal structure of the swirling flow are also
evident between in-phase and out-of-phase forcing. Figure 11
shows the modal decomposition (integrated over all frequencies
and radii between r/D=0-1) for the unforced, out-of-phase
forcing, and in-phase forcing cases at high amplitude forcing at
a downstream location of x/D=0. This location was chosen to
capture the vortex rollup motion in the shear layers, as depicted
in Figure 10.
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Copyright © 2012 by ASME
a)
Figure 11. Distribution of mode amplitude at the forcing
frequency integrated over radii r/D=0-1 plotted as a
function of mode number at x/D=0 for a non-reacting, flow
forced with 400 Hz forcing at uo=10 m/s, S=0.85.
Here, it is evident that different forcing configurations have
different effects on the modal content of the flow. For example,
out-of-phase forcing significantly amplifies modes m  1 ,
m  2 , and m  1 , two counter-swirling and a co-swirling
helical modes, respectively. The amplification of these two
modes asymmetrically is indicative of the asymmetric response
of the flow and the helical vortex in the shear layers. The
strong m  1 mode represents the helical vortex rollup evident
in the inner shear layer, as seen in Figure 10.
The in-phase forcing strongly amplifies the symmetric
m  0 mode; this finding is congruent with the ring vortex
shedding observed in the r-x view. Additionally, the m  1 and
m  1 modes are present at very similar amplitudes. In the
inner shear layer region, between r/D=0.2-0.6, the phase
between the fluctuations at these two modes is zero within the
uncertainty in phase estimation (15 degrees). The excitation of
these two modes is also indicative of a symmetric response of
the flow.
Additional information can be garnered from the radial
distribution of fluctuation amplitude of each of these modes.
Figure 12 shows the distribution of energy for in-phase and outof-phase forcing, at the forcing frequency, for several mode
numbers.
b)
Figure 12. Distribution of mode amplitude at the forcing
frequency plotted as a function radius for several mode
numbers for a non-reacting, flow forced at 400 Hz a) inphase and b) out-of-phase forcing at uo=10 m/s, S=0.85.
Gray areas approximate the location of the inner and outer
shear layers at x/D=0, where these data were taken.
In the out-of-phase forcing case, the dominant mode,
m  1 , is prevalent in the inner shear layer, as would be
expected given their convectively unstable nature. This
response is congruent with the helical response seen in the r-x
view in Figure 10. In the in-phase forcing case, the response
seems to focus in the inner shear layer for modes m  1 and
m  1 , and in both the inner and outer shear layer for mode
m  0.
Further downstream, however, the distribution of energy
amongst the modes changes, as is shown in Figure 13, which
plots the distribution of modal amplitude at x/D=1.
9
Copyright © 2012 by ASME
Figure 13. Distribution of mode amplitude at the forcing
frequency integrated over radii r/D=0-1 plotted as a
function of mode number at x/D=1 for a non-reacting, flow
forced with 400 Hz forcing at uo=10 m/s, S=0.85.
In this case, the effect of in-phase, symmetric forcing has
been almost completely lost, manifesting as only a slight
increase in the amplitude of modes m  0 , m  1 , and m  1
over the unforced flow. This may be due to two factors. First,
the action of swirl causes the ring vortices to tilt [22],
introducing an inherent asymmetry in the flow structure
downstream. Additionally, the m  0 mode is stable in a
swirling jet. Theoretical analysis by several authors [48]
indicates that the symmetric mode is stable for these flow
profiles because of the action of swirl. While unswirling jets
often exhibit instabilities of the m  0 mode, the action of swirl
adds an asymmetric bias to the flow and causes the most
unstable modes to be asymmetric. Apparently, the fact that the
m=0 mode is stable causes this disturbance to be rapidly
suppressed.
In the out-of-phase forcing case, however, the amplitude of
mode m  1 has grown significantly, while the amplitude of
mode m  1 has also grown.
The phase between the
fluctuations at these two modes is close to zero, but because the
amplitudes are different, the resulting manifestation of these
modes is a counter-swirling helix resulting from mode m  1 , as
was seen at the x/D=0 station as well.
The strength of the helical shear layer instability is much
higher at x/D=0, as can be seen in Figure 10, because the
amplitude of mode m  1 is so much greater than that of any
other mode number. At x/D=1, however, the strength of the
shear layer vortex has decayed, and this is reflected in the
similarity of the amplitudes of modes m  1 and m  1 .
Additionally, theoretical analysis predict that the co-swirling
mode, m  1 , is most unstable in swirling flow profiles of this
type [48]. In this case, this the response at m  1 may be due
to the natural instability of the jet, while the m  1 response of
the jet stems from the acoustic forcing.
Response of the Self-Excited Motions
In this section, we examine the response of the flow at the
low frequency only, integrated from 0-200 Hz, in order to
capture the influence that acoustic forcing has on the
intrinsically occurring motions. As described with reference to
Figure 6, the swirling jet has two main motions, a jet column
deformation that is described by the m  2 mode, and a pair
of coherent structures described by the m  1 mode. In this
section, we use modal decomposition to see the effect of the
symmetry of forcing. Again, we consider motions at a
downstream distance of x/D=1, where the most vigorous
recirculation zone motion is located.
First, the spectral content of several of these modes is
shown in Figure 14. Here, peaks at the 400 Hz forcing
frequency are evident in both the out-of-phase and in-phase
high amplitude forcing cases, although to a much lesser extent
in the in-phase forcing condition. Modes m  2 , m  1 , and
for the out-of-phase forcing case, m  1 , have significant low
frequency content.
a)
b)
Figure 14. Spectra of several mode numbers integrated
over radius and frequency in the r   plane at x/D=1, for a
non-reacting, flow forced at 400 Hz a) in-phase and b) outof-phase at uo=10 m/s, S=0.85.
Figure 15 shows the response of the low frequency content
to high amplitude acoustic forcing. Here, the in-phase and out-
10
Copyright © 2012 by ASME
of-phase results are compared against the unforced results; each
mode coefficient is integrated between 0-200 Hz and radially
between r/D=0-1.
b)
Figure 15. Distribution of mode amplitude at the low
frequencies integrated over radii r/D=0-1 plotted as a
function of mode number at x/D=1 for a non-reacting, flow
forced with 400 Hz forcing at uo=10 m/s, S=0.85.
Here again, it is evident that different forcing symmetries
have different effects on the modal content of the velocity
fluctuations. In both the in-phase and out-of-phase forcing
cases, high amplitude forcing decreases the amplitude of the
self-excited behavior at modes m  2 and m  1 , but adds
energy to the mode m  1 . The magnitude to which these
effects take hold is dissimilar, particular in the variation in
strength of mode m  2 .
Changes in the radial distribution of this energy are evident
in Figure 16, which shows the mode energy for several modes,
integrated between 0-200 Hz, as a function of radius. Like
before, the in-phase forcing significantly changes the structure
of radial dependence of the modes, particularly mode m  2 .
Figure 16. Distribution of mode amplitude at the low
frequencies plotted as a function radius for several mode
numbers for a non-reacting, flow forced at 400 Hz a) inphase and b) out-of-phase forcing at x/D=1 for uo=10 m/s,
S=0.85. Gray areas approximate the location of the inner
and outer shear layers.
It is evident from data in both the r-x and r-θ planes, as
well as at the forcing frequency and in the low frequency range,
the symmetry of forcing has a significant effect on the response
of the swirling annular flow to transverse acoustic forcing.
CONCLUSIONS
This study has described the effect of transverse acoustic
forcing symmetry on a swirling annular jet. This study was
motivated by transverse instabilities in annular gas turbine
engines, where different nozzles will be subjected to different
portions of the acoustic field. To mimic this effect, two acoustic
forcing conditions were considered: the asymmetric, out-ofphase forcing case with a velocity anti-node along the
centerline, and the symmetric, in-phase forcing case with a
pressure anti-node along the centerline. A summary of the
results is given in Table 1.
Table 1. Summary of results describing coherent structures
and their corresponding frequency ranges/mode numbers
for different acoustic forcing conditions.
Forcing
No forcing
Frequency
0<f<5000
No forcing
In-phase –
x/D=0
In-phase –
x/D > 0
Out-of-phase
0<f<5000
f=fo
a)
f=fo
f=fo
Motion
Jet
deformation
Dual helix
Ring vortex
Mode number
m=-2
Tilted
vortex
Helical
vortex
m=1
m =-1
m=0
m =1, m=-1
The results show that the symmetry of acoustic forcing has
an effect on both the flow at the forcing frequency, the shear
11
Copyright © 2012 by ASME
layer response in particular, and the low frequency, self-excited
motion in the vortex breakdown bubble. This study has
important implications for both understanding and modeling of
the behavior of full combustors undergoing transverse
instabilities. Results from previous studies from the authors, as
well as the results discussed here, indicate that the flame
response would be different in these two cases. This is because
the flame is responding to the velocity disturbances normal to
its surface and the symmetry of forcing changes the topology of
the flow field.
In future work, a wider variety of forcing frequencies,
amplitudes, and flow conditions will be investigated to broaden
our understanding of this issue. Additionally, previous studies
of vortex breakdown dynamics have shown that substantial
changes in the behavior of these structures can arise with
combustion. To that end, further testing will investigate the
effects of heat release on central recirculation zone dynamics.
ACKNOWLEDGMENTS
The authors would like to recognize Travis Smith,
Charlotte Neville, Michael Malanoski, and Michael Aguilar for
their help in data acquisition for this work. 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.
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