Influence of Transverse Acoustic Modal Structure on … OF TRANSVERSE ACOUSTIC MODAL STRUCTURE ON THE FORCED ... the low frequency self-excited motion of the ... flames undergoing

1 Copyright © 2012 by ASME

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-of-

phase 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

u Velocity vector

u Fourier transform of velocity

u Axial velocity

u Axial velocity fluctuation

uo Bulk velocity

ur,int Radial velocity integrated in radial direction

v Transverse velocity

v Transverse velocity fluctuation

x Axial direction

z In-plane direction

Ω Vorticity

β Flame aspect ratio

θ Azimuthal direction

ω Angular frequency

Proceedings of ASME Turbo Expo 2012 GT2012

June 11-15, 2012, Copenhagen, Denmark

GT2012-70053


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 [13-

16], 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


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 ( 0.6St ) 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 non-

axisymmetric 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, non-

axisymmetric 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-of-

phase) 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.

a)

1

2

3

4

b)

1

2

3

4

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


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 out-

of-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:

0

ˆ , , , ,N

im im

m m

m

u r A r e B r e

(1)

Here, ˆ , ,u r is the complex Fourier transform of the

velocity fluctuation. ,mA r and ,mB 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 1 2N and 1 2N , 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.


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, outer inner oD D u , (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.

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.

a)

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 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)

b)

c)

d)

e)

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.


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 smaller-

scale 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 bottom-

right 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 2m 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 2m and 1m

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 2m and 1m at

frequencies below 200 Hz.


Figure 9. Spectra of several mode numbers integrated over

radius and frequency in the r plane at 1x D , 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 out-

of-phase (asymmetric) forcing on the flow structures described

here. In each of the cases presented here, the in-phase forcing

amplitude was 0.05ov u and the out-of-phase forcing

amplitude was 0.4ov u .

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-of-

phase, 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

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 0m mode) at the

dump plane to a symmetric forcing condition. In the out-of-

phase 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.

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.


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 1m ,

2m , and 1m , 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 1m 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

0m mode; this finding is congruent with the ring vortex

shedding observed in the r-x view. Additionally, the 1m and

1m 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 out-

of-phase forcing, at the forcing frequency, for several mode

numbers.

a)

b)


frequency plotted as a function radius for several mode

numbers for a non-reacting, flow forced at 400 Hz a) in-

phase 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,

1m , 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 1m and

1m , and in both the inner and outer shear layer for mode

0m .

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.



frequency integrated over radii r/D=0-1 plotted as a



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 0m , 1m , and 1m

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 0m 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 0m 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 1m has grown significantly, while the amplitude of

mode 1m 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 1m , 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 1m 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 1m and 1m .

Additionally, theoretical analysis predict that the co-swirling

mode, 1m , is most unstable in swirling flow profiles of this

type [48]. In this case, this the response at 1m may be due

to the natural instability of the jet, while the 1m 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 2m mode, and a pair

of coherent structures described by the 1m 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 2m , 1m , and

for the out-of-phase forcing case, 1m , 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) out-

of-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-


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.

Figure 15. Distribution of mode amplitude at the low

frequencies integrated over radii r/D=0-1 plotted as a



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 2m and 1m , but adds

energy to the mode 1m . The magnitude to which these

effects take hold is dissimilar, particular in the variation in

strength of mode 2m .

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 2m .

a)

b)

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) in-

phase 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-of-

phase 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 Frequency Motion Mode number

No forcing 0<f<5000 Jet

deformation

m=-2

No forcing 0<f<5000 Dual helix m =-1

In-phase –

x/D=0

f=fo Ring vortex m=0

In-phase –

x/D > 0

f=fo Tilted

vortex

m=1

Out-of-phase f=fo Helical

vortex

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


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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