Content uploaded by Hiroshi Yokoyama
Author content
All content in this area was uploaded by Hiroshi Yokoyama on Feb 25, 2016
Content may be subject to copyright.
Self-sustained oscillations with acoustic feedback in flows
over a backward-facing step with a small upstream step
Hiroshi Yokoyama
a兲
Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656 Japan
Yuichi Tsukamoto
Hitachi Plant Technologies, Ltd., 4-5-2 Higashi-Ikebukuro, Toshima-ku, Tokyo, 170-8466 Japan
Chisachi Kato
Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505 Japan
Akiyoshi Iida
Faculty of Engineering, Kogakuin University, 2665 Nakano-machi, Hachioji-shi, Tokyo, 192-0015 Japan
共Received 16 January 2007; accepted 6 September 2007; published online 23 October 2007兲
Self-sustained oscillations with acoustic feedback take place in a flow over a two-dimensional
two-step configuration: a small forward-backward facing step, which we hereafter call a bump, and
a relatively large backward-facing step 共backstep兲. These oscillations can radiate intense tonal sound
and fatigue nearby components of industrial products. We clarify the mechanism of these
oscillations by directly solving the compressible Navier-Stokes equations. The results show that
vortices are shed from the leading edge of the bump and acoustic waves are radiated when these
vortices pass the trailing edge of the backstep. The radiated compression waves shed new vortices
by stretching the vortex formed by the flow separation at the leading edge of the bump, thereby
forming a feedback loop. We propose a formula based on a detailed investigation of the phase
relationship between the vortices and the acoustic waves for predicting the frequencies of the tonal
sound. The frequencies predicted by this formula are in good agreement with those measured in the
experiments we performed. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2793170兴
I. INTRODUCTION
Self-sustained oscillations with tonal sound, such as cav-
ity tones or edge tones, frequently take place when solid
bodies are placed in shear layers. These oscillations are de-
scribed as fluid-acoustic interactions and are interesting phe-
nomena both for aerodynamics and for aeroacoustics. More-
over, the prediction and the suppression of the oscillations
are very important for many practical applications. When the
oscillations are unsuppressed, the noise levels in open cavi-
ties in aircraft can exceed 150 dB.
1
These intense oscillations
can fatigue nearby components such as aircraft wheel-wells
and landing-gear configurations.
2
For decades, many researchers have used various ap-
proaches to investigate the relationship between the flow and
the sound in these phenomena. Brown
3
experimentally found
that the peak frequencies of the edge tones change discon-
tinuously against the position of the edge and the mean flow
velocity. Powell
4
proposed that the interactions of vortices in
a jet with a sharp edge generate acoustic waves that lead to
the formation of new vortices, and the frequencies predicted
by his proposal agree well with Brown’s experimental
results.
3
Meanwhile, Rossiter
5
proposed a similar feedback
loop over open cavities and derived a semiempirical formula,
fL/u
0
=共n −
␥
兲/共M +1/
兲, where f is the frequency of the
radiated tonal sound, L is the cavity length, u
0
is the free
stream velocity, n is a positive integer, M is the Mach num-
ber based on the free stream velocity,
is the ratio of the
convection velocity of vortices to the free stream velocity,
and
␥
is a constant for the phase correction. The frequencies
predicted by this formula agree with those measured in
experiments.
6
However, this model does not provide the val-
ues of
and
␥
theoretically, treating them instead as empiri-
cal constants to be determined by the best fit with measured
data.
In theoretical studies, the shear layer was idealized as a
vortex sheet, and the phase distributions of the vortices and
the sound were estimated for edge tones, cavity tones, and
aperture tones. The results enabled researchers to predict the
frequencies of the tonal sound without using empirical or
adjustable parameters 共Bilanin and Covert,
7
Tam and Block,
8
Crighton,
9
and Howe
10
兲. However, the phase distributions
estimated by these studies have not been validated by experi-
ments or computations. In recent years, the direct numerical
simulation 共DNS兲, where both vortices and acoustic waves
are directly simulated, has provided a means of studying the
self-sustained oscillation mechanism in detail
11,12
共see the re-
view of Colonius and Lele
13
兲. However, few studies have
been done about the detailed mechanism of vortex shedding
and acoustic radiation, including the phase relationship be-
tween the vortices and the sound in the self-sustained oscil-
lations. Therefore, the physical meaning of constant
␥
in
Rossiter’s formula
5
remains unknown.
The self-sustained oscillations with tonal sound also take
place in a flow over a two-step configuration: a small
forward-backward facing step, which we hereafter call a
a兲
Electronic mail: yokoyama@iis.u-tokyo.ac.jp
PHYSICS OF FLUIDS 19, 106104 共2007兲
1070-6631/2007/19共10兲/106104/8/$23.00 © 2007 American Institute of Physics19, 106104-1
Downloaded 20 Nov 2007 to 133.11.199.19. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
bump, and a relatively large backward-facing step or back-
step 共Fig. 1兲. This configuration appears in various industrial
devices, for example, the small bump on the surface of an
automobile side mirror. The tonal sound caused by the oscil-
lations is uncomfortable for the passengers, so the mecha-
nism of the oscillations needs to be clarified so that the
sounds can be predicted and reduced. However, few studies
have been done on the self-sustained oscillations in this con-
figuration, and there are many open questions about the for-
mation of vortices around the bump and about the acoustic
radiation. The objective of the present work is to answer
these questions, focusing on the phase relationship between
the vortices and the sound.
This work mainly consists of computational studies, but
we also conducted wind-tunnel experiments to validate the
computational results. In Sec. II, we explain the flow con-
figurations, the numerical methods, and the experimental
setup. In Sec. III A, we discuss the pressure fluctuation at a
point far from the backstep and confirm that tonal sound is
radiated. In Sec. III B, we present instantaneous vortices and
acoustic waves in detail. In Sec. III C, acoustic sources are
clarified. In Sec. III D, the phase relationship between the
vortices and the acoustic waves are clarified. Based on this
relationship, we propose a formula for predicting the fre-
quencies of the radiated tonal sound in Sec. III E. The fre-
quencies predicted by this formula are compared with those
measured in the experiments that we performed.
II. METHODOLOGY
A. Flow configurations
The baseline flow configurations and parameters for the
present work are shown in Fig. 1 and Table I, respectively.
The bump is placed in a laminar boundary layer. The
Reynolds number Re
x
based on the value of x measured from
the virtual origin is 4.6⫻10
5
at the leading edge of the
bump. Mach number M based on free-stream velocity u
0
is
0.088 in the experiments, but in the computations we inves-
tigate the flow with higher Mach numbers M =0.3, 0.4, and
0.5 to reduce the computational costs.
Boundary layer thickness is 2.4 h at the leading edge of
the bump, and bump length L
b
is 14.3 h, where h is bump
height. Distance L
1
between the trailing edge of the bump
and that of the backstep is 14.3 h. We have clarified in pre-
liminary experiments that intense tonal sound is radiated
from the flow with these configurations. Backstep height H is
21.6 h in the computations and 71.4 h in the experiments. In
the computations, the flow fields are developed more quickly
by shortening backstep height, which also determines the
necessary streamwise length of the computational domain.
To clarify the effect of backstep height on the mechanism of
the self-sustained oscillations, we compare the computational
results of the baseline configurations with those where back-
step height H is 43.1 h in Sec. III E.
B. Numerical methods
1. Governing equations
We simulate both flow and acoustic fields by directly
solving the three-dimensional compressible Navier-Stokes
equations in a conservative form, which are written as
U
t
+ 共E − E
v
兲
x
+ 共F − F
v
兲
y
+ 共G − G
v
兲
z
=0, 共1兲
where U is the vector of the conservative variables, E, F, and
G are the inviscid fluxes, and E
v
, F
v
, and G
v
are the viscous
fluxes. The spatial derivatives are evaluated by the sixth-
order-accurate compact finite difference scheme 共fourth-
order accurate on the boundaries兲.
14
Time integration is per-
formed by a third-order accurate Runge-Kutta method. To
suppress the numerical instabilities associated with the cen-
tral differencing in the compact scheme, we use the tenth-
order-accurate spatial filter shown below,
15
␣
f
ˆ
i−1
+
ˆ
i
+
␣
f
ˆ
i+1
=
兺
n=0
5
a
n
2
共
i+n
+
i−n
兲, 共2兲
where
is a conservative quantity and
ˆ
is the filtered quan-
tity. Coefficients a
n
are the same as the values used by
Gaitonde and Visbal,
16
and parameter
␣
f
is 0.45. We have
clarified in preliminary tests that we can capture acoustic
waves by 10 points per wave 共ppw兲 using the above compu-
tational methods.
2. Computational grid
To evaluate the effect of the three-dimensional distortion
of relatively large vortices in the wake of the backstep on the
radiated sound, we use a three-dimensional computational
grid for the present simulations 共Fig. 2兲. The spanwise extent
of the computational domain is L
z
=3H, and the cross section
in the x-y plane is shown in Fig. 3. We use 11 grids in the
spanwise direction and investigate grid dependence by
changing the number of the grids in the spanwise direction in
Sec. III A. The grid resolution we used for acoustic propaga-
tion was more than 40 grids per wavelength of the funda-
mental frequency to suppress the dissipation of the acoustic
waves, and 11 grids per bump height h so that small flow
structures generated by the bump can be captured. The
present computational domain consists of approximately
2.5⫻ 10
5
structured grids.
FIG. 1. Flow configurations.
TABLE I. Parameters.
Boundary layer Re
x
L
b
/hL
1
/h
␦
/hH/h
Laminar 4.6⫻10
5
14.3 14.3 2.4 21.6
106104-2 Yokoyama et al. Phys. Fluids 19, 106104 共2007兲
Downloaded 20 Nov 2007 to 133.11.199.19. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
3. Initial conditions, boundary conditions,
and buffer zone
All the simulations were initiated by the uniform flow
with free-stream velocity u
0
over the bump and the backstep.
Figure 3 shows the boundary conditions used in the
present studies. The artificial boundaries must allow vortices
and acoustic waves to pass smoothly with minimal distur-
bances. In this study, the nonreflecting boundary conditions
based on the characteristic wave relations
17–19
are used at the
inflow, upper, and outflow boundaries. Nonslip and adiabatic
boundary conditions are used at the wall. The periodic
boundary conditions are used in the spanwise direction.
To minimize errors associated with the artificial bound-
aries, buffer zones 共where we stretch the grid to reduce the
amplitudes of the acoustical and vortical fluctuations before
they reach the boundaries兲 are used near the nonreflecting
boundaries. Moreover, we must keep boundary layer thick-
ness
␦
constant at the leading edge of the bump because the
radiated sound strongly depends on the boundary layer thick-
ness. To achieve desired thickness
␦
at the leading edge of
the bump, Eq. 共1兲 is replaced in the upstream buffer zone by
U
t
+ 共E − E
v
兲
x
+ 共F − F
v
兲
y
+ 共G − G
v
兲
z
=−
0
共a/L
b
兲
2
共U − U
target
兲, 共3兲
where
0
is 0.05, a is the speed of sound, L
b
is the length of
the buffer zone,
is the nondimensional distance from the
beginning of the buffer zone to the inflow boundary 共0 艋
艋1兲, and U
target
is the target vector of the conservative
variables.
20,21
U
target
is obtained from the calculation for the
flat-plate laminar boundary layer.
C. Experimental setup
We conducted our experiments using the suction-type
low-noise wind tunnel shown in Fig. 4. The test section is
composed of a backstep accompanied by a small bump
placed upstream of the backstep. In the spanwise direction,
the test section is terminated by two end walls. One of these
walls is made of a porous plate to suppress the reflections of
the sound, and the other is made of an acrylic plate, which
allows optical observations. The velocity profile and the
sound pressure level are measured, respectively, with a hot
wire anemometer and a sound level meter with a nondirec-
tional 1/2 in. microphone 共Rion NL-31兲.
At a wind speed of 30 m/s, the turbulence intensity is
less than 0.7% and the nonuniformity of the mean flow ve-
locity is less than 0.1%. The velocity profile at the point of
the leading edge of the bump is measured without the bump.
The profile agrees with the Blasius profile, so the boundary
layer is apparently laminar 共Fig. 5兲. At a wind speed of
30 m / s, the 99% boundary layer thickness, the momentum
thickness, the displacement thickness, and the shape factor
are 1.7 mm, 0.23 mm, 0.58 mm, 2.6, respectively. The test
section is placed in an anechoic chamber that is covered with
50 mm thick sound-absorbing material, and the background
noise level is suppressed to less than 64 dB at a wind speed
of 30 m / s, which is much lower than the sound pressure
level of the tonal sound of our interest.
To account for errors and uncertainties, we estimated
that the error of the measurement of the streamwise velocity
is 0.2 m/ s at a wind speed of 30 m / s and that of the bump
height is 0.1 mm. We found that the uncertainty about the
FIG. 2. Computational grid 共every fifth grid line is shown for clarity兲.
FIG. 3. Boundary conditions.
FIG. 4. Experimental setup.
FIG. 5. Velocity profile of boundary layer. Measured data 共䊊兲 is compared
with the Blasius profile 共—兲.
106104-3 Self-sustained oscillations with acoustic feedback Phys. Fluids 19, 106104 共2007兲
Downloaded 20 Nov 2007 to 133.11.199.19. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
peak frequency of the tonal sound is about 40 Hz when the
peak frequency is 2370 Hz. To evaluate the repeatability of
the experiments, we repeated the experiments and measured
the frequency spectra 10 times at a frequency resolution of
10 Hz. We found that the 95% uncertainty range of the peak
frequency of the radiated sound was 24 Hz.
III. RESULTS AND DISCUSSION
A. Self-sustained oscillations with tonal sound
Figure 6 shows the fluctuation of the pressure coefficient
Cp 关=p / 共
0
u
0
2
/2兲兴 for M =0.4, where
0
is the density of the
ambient air at the observation point 共Fig. 3兲 after 20 flow-
through times have elapsed since the start of the computa-
tion. This shows that the flow is fully developed from the
initial condition and that the oscillations are occurring. We
also investigated the oscillations when either the bump or the
backstep had been removed 共Fig. 7兲. Figure 8 shows a com-
parison of the frequency spectra of the pressure fluctuations
at the observation point. We averaged the results of the Fou-
rier transform 20 times and the duration of the transform is
about 1300 共h / u
0
兲. This duration corresponds to about 30
periods of the fundamental mode. The results clearly indicate
that the tonal sound with the fundamental frequency
St= fh/u
0
=0.024 is radiated only when both the bump and
the backstep exist. We also found that the fundamental fre-
quencies of the tonal sound are St= 0.025 and 0.023, respec-
tively, for M =0.3 and 0.5 for the baseline configuration. We
investigated the effect of the grid resolution and the width of
the computational domain by conducting the computations
with 21 grids over Lz=3H 共the original width兲 and Lz=6H
for M =0.4. Figure 9 shows that the frequency spectra do not
change from that of the original spectrum. Therefore, we
concluded that the original resolution and width are suffi-
cient to capture the effect of the distortion of the large vor-
tices in the wake of the backstep on the radiated sound.
B. Instantaneous distributions of vortices
and acoustic waves
We investigated the vortical structures and the acoustic
fields in oscillations when the flow is fully developed from
the initial condition. To elucidate the flow structures, the sec-
ond invariant of the velocity gradient tensor Q= 储⍀储
2
−储S储
2
was computed, where ⍀ and S are, respectively, the antisym-
metric and symmetric parts of the velocity gradient tensor.
Regions with Q⬎0 represent vortex tubes. We present the
acoustic fields using density fluctuations. Using these meth-
ods of visualization, we were able to clearly capture the flow
structures and the acoustic waves in the near field. Figure 10
shows instantaneous flow structures and acoustic waves for
M =0.4 from t =0 to t=0.81T, where T represents the period
of the fundamental frequency. A small vortex 共“Vortex 3” at
t=0兲 is shed from the leading edge of the bump and con-
vected downstream. While being convected from the trailing
edge of the bump to the backstep, the vortex merges with its
FIG. 6. Fluctuation of pressure coefficient Cp at the observation point
shown in Fig. 3.
FIG. 7. Configurations without bump 共top兲 and without backstep 共bottom兲.
FIG. 8. Power spectrum density of pressure coefficient Cp for baseline
configuration 共—兲 compared with a simple backstep without a bump 共---兲
and a bump on a flat plate without a backstep 共¯·兲.
FIG. 9. Power spectrum density of pressure coefficient Cp for the original
case of 11 grids over Lz =3H 共—兲 compared with the case of 21 grids over
Lz=3H 共---兲 and with that of 21 grids over Lz=6H 共¯·兲.
106104-4 Yokoyama et al. Phys. Fluids 19, 106104 共2007兲
Downloaded 20 Nov 2007 to 133.11.199.19. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
upstream vortex and becomes a larger vortex 共“Vortex 2” at
t=0.54T兲. When it passes the trailing edge of the backstep,
the vortex splits into a fast vortex and a slow vortex 共“Vortex
1-1” and “Vortex 1-2” at t= 0.27T兲, and the convection ve-
locity differs between the two vortices depending on the dis-
tance from the wall. A compression wave 共“Compression
wave 2” at 0.27T兲 is radiated between this slow vortex
and its upstream vortex. Meanwhile, an expansion wave
共“Expansion wave 2” at 0.27T兲 is radiated from the vortices
distorted in the wake of the backstep. The behaviors of the
acoustic wave fronts show that the acoustic waves are propa-
gated in the upstream direction along the wall at a velocity of
about a-u
0
.
Figure 11 shows the trajectories of the vortices 共y =1.6 h兲
and those of compression waves 共y = 3.6 h兲 represented by
the local maxima of the second invariant and density, respec-
tively. This figure shows that the shed of a vortex coincides
with the crossing of the compression wave. Figure 12 shows
the vortices and the acoustic waves around the leading edge
of the bump from t = 0.54T to t = 0.81T, including the vectors
of velocity fluctuations. The velocity fluctuation virtually
represents the sound particle velocity, although it is affected
by the vortices near the wall. Due to acoustic expansion, the
gradient of the sound particle velocity in the streamwise di-
rection changes suddenly from a negative to a positive value
just after the most compressed wave 共local maximum line of
density兲 passes. This is because the gradient of pressure fluc-
tuation is very steep around the maximum point, as shown in
Fig. 6. As a result, the vortex formed by the flow separation
at the leading edge of the bump begins to stretch and finally
a new vortex, “Vortex 4,” is shed at t = 0.81 T when the
compression wave 共“Compression wave 2”兲 passes this vor-
tex in the upstream direction. It also shows that there are
trajectories of modes higher than the fundamental mode of
St= 0.24.
C. Acoustic sources
To identify the source positions of the radiated tonal
sound, we investigated the amplitudes of the pressure fluc-
tuations for the fundamental frequency for M =0.4 共Fig. 13兲.
The local maxima of the fluctuations appear in both the up-
stream and downstream regions of the backstep. The up-
stream local maximum represents the source of the compres-
sion waves radiated between the vortex at the trailing edge of
the backstep and its upstream vortex 共t = 0.27 T in Fig. 10兲.
The distance between this point and the trailing edge of the
backstep is L

⬇3.6h. The same value of L

is also obtained
for M = 0.3 and 0.5. This value is important for predicting the
frequencies of the radiated tonal sound because the compres-
sion waves cause new vortices at the leading edge of the
bump. Meanwhile, the local maximum downstream of the
FIG. 10. Contours of density fluctuations
⬘
/
0
and isosurfaces of
Q/ 共u
0
/
␦
兲
2
=0.03. Contour levels range from −5 ⫻10
−3
共dark兲 to 5 ⫻ 10
−3
共light兲.“T” represents the period of fundamental frequency.
FIG. 11. Trajectories of vortices 共⫹兲 and compression waves 共䊊兲 over the
bump.
FIG. 12. Contours of density fluctuations
⬘
/
0
and isosurfaces of
Q/ 共u
0
/
␦
兲
2
=0.03. Contour levels range from −5 ⫻10
−3
共dark兲 to 5 ⫻ 10
−3
共light兲. Vectors represent fluctuations of velocity. “T” represents the period
of fundamental frequency.
106104-5 Self-sustained oscillations with acoustic feedback Phys. Fluids 19, 106104 共2007兲
Downloaded 20 Nov 2007 to 133.11.199.19. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
backstep represents the source of the expansion waves,
which are radiated from the vortices distorted in the wake of
the backstep 共t=0.27 T in Fig. 10兲.
D. Phase distributions of vortices
and acoustic waves
We investigated phase distributions ⌽ of the second in-
variant of the velocity gradient tensor and fluid density. Fig-
ure 14 shows the phase distributions along the observation
line shown at the top for the fundamental frequency for
M =0.3, 0.4, and 0.5. This observation line approximates the
trajectory of the vortex because the amplitude of the second
invariant fluctuations along this line takes its maximum in
the wall normal direction 共Fig. 15兲. The reference point is
above the trailing edge of the backstep, and the phase of the
second invariant at this point is the reference for both the
phase distributions of the second invariant and those of fluid
density.
The phase distributions of the second invariant represent
the convection of the vortex, while those of the fluctuating
density around the leading edge of the bump virtually repre-
sent the upstream propagation of the radiated sound waves
because the density fluctuations caused by the convection of
the vortices are negligibly small around this edge. Figure 14
shows that the phase of the second invariant coincides with
that of the fluctuating fluid density at a downstream point of
the leading edge of the bump. At this point, the compression
waves cause the vortex formed by the flow separation at the
leading edge of the bump to begin to stretch. The distance
between this point and the leading edge of the bump is
L
␣
⬇5h for all the investigated Mach numbers.
The gradient of the phase distributions of the second
invariant also gives the convection velocity of the vortices.
We obtained the averaged convection velocity u
cb
=0.29u
0
on
the bump and u
c1
=0.19u
0
from the trailing edge of the bump
to the backstep. The convection velocity u
cb
on the bump is
approximately confirmed by the trajectories of the vortices in
Fig. 11. The vortices are convected very close to the wall,
and the boundary layer is laminar. As a result, these vortices
are strongly affected by the wall, and the convective veloci-
ties of these vortices are much slower than those of vortices
in free shear layers such as deep cavities. Moreover, the
phase distributions of the second invariant also show that the
distance between the vortex at the trailing edge of the back-
step and its upstream vortex is very narrow. This is because
one of the splitting vortices over the trailing edge of the
backstep 共“Vortex 1-2” in Fig. 10兲 is convected with a much
slower velocity as already mentioned in Sec. III B. A com-
pression wave is radiated between this slow vortex and its
upstream vortex.
E. Formula for frequencies of the tonal sound
The discussion so far focuses on the self-sustained oscil-
lation mechanism and the phase relationship between the
vortices and the acoustic waves. The cycle of the self-
sustained oscillations consists of the following three parts:
FIG. 13. Amplitudes of pressure coefficient Cp for the fundamental fre-
quency. Contour levels range from 10
−10
共dark兲 to 10
−4
共light兲 with the
logarithmic scale.
FIG. 14. Phase changes of second invariant 共solid line兲 and fluctuating fluid
density 共dashed line兲 along the observation line for fundamental frequency.
106104-6 Yokoyama et al. Phys. Fluids 19, 106104 共2007兲
Downloaded 20 Nov 2007 to 133.11.199.19. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
共1兲 A small vortex is shed and begins to be convected at
position L
␣
from the leading edge of the bump when a com-
pression wave passes this point.
共2兲 The vortex is convected downstream at velocity u
cb
on the bump and u
c1
from the trailing edge of the bump to
the backstep.
共3兲 The above-mentioned vortex splits into a fast vortex
and a slow vortex when it passes the backstep. Between this
slow vortex and its upstream vortex, a compression wave is
radiated at position L

from the trailing edge of the backstep.
This wave is propagated at a velocity of a-u
0
in the upstream
direction.
The phase variation of the above processes must be 2
n,
where n is a positive integer. As a result, the frequencies f
n
of the tonal sound can be predicted by the following formula:
f
n
=1/T
n
,
T
n
= 共共L
b
− L
␣
兲/u
cb
+ L
1
/u
c1
+ 共L
b
+ L
1
− L
␣
− L

兲/共a − u
0
兲兲/n, 共4兲
L
␣
= 5.0h, L

= 3.6h, u
cb
= 0.29u
0
, u
c1
= 0.19u
0
.
To be exact, the point of the vortex shed is slightly dif-
ferent from the point of the isophase for
and Q, when Fig.
14 is compared with Fig. 11. Therefore, we need to introduce
the phase delay into this formula to predict the exact peak
frequency. However, it is difficult to estimate the delay theo-
retically and the delay is not large 共less than 10% of the
fundamental period兲. The main objective of this paper is to
clarify the self-sustained oscillations around the backstep
with a bump and this simple formula is sufficient for this
objective.
The frequencies predicted by this formula are compared
with those measured in the experiments. As already men-
tioned, the backstep heights for the computational and the
experimental configurations are different. We therefore esti-
mated the effect of the backstep height on the radiated sound.
Figure 16 compares the sound radiated from the original con-
figurations with backstep height H = 21.6 h with that from the
configurations with H = 43.1 h. Figure 16 shows that the dif-
ference in the frequency spectrum is small. We concluded
that the phenomena of the self-sustained oscillations around
the bump do not depend on the backstep height as long as the
backstep height is sufficiently larger than the bump height.
Figure 17 compares the frequency spectra of the pressure
fluctuations at the observation point in the computations with
the sound pressure level measured by the microphone in the
experiment. Mode n = 6 is observed in both the experiment
and the computations, but the lower mode, n = 3, is only ob-
served in the computations. We think this is because of the
difference in Mach numbers. In fact, the amplitude of n =3
decreases and that of n = 6 increases as the Mach number
decreases in our computations. Also, Fig. 18 shows the
changes in measured peak frequency for bump length L
b
where M =0.088 in the experiments. The predicted frequen-
cies are in good agreement with those measured in the ex-
periments. This agreement validates the self-sustained oscil-
lation mechanism clarified by the present computational
FIG. 15. Amplitudes of second invariant fluctuations Q
⬘
/共u
0
/h兲
2
near the
bump. Contour levels range from 10
−8
共dark兲 to 10
0
共light兲 with the loga-
rithmic scale. Dashed line 共---兲 represents the observation line for phase
investigation.
FIG. 16. Power spectrum density of pressure coefficient Cp for baseline
configuration 共—兲 compared with that for the bump with a higher backstep
共---兲.
FIG. 17. Power spectrum density of pressure coefficient Cp in computations
compared with sound pressure level measured in experiment 共circles are
every tenth point兲.
FIG. 18. Change in fundamental frequency for L
b
. Fundamental frequencies
predicted by Eq. 共3兲 共—兲 are compared with experimental results 共䊊兲.
106104-7 Self-sustained oscillations with acoustic feedback Phys. Fluids 19, 106104 共2007兲
Downloaded 20 Nov 2007 to 133.11.199.19. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
study. In the present paper, we did not discuss the absolute
values of the sound pressure level. For the quantitative pre-
diction of the sound pressure level, it is necessary to exactly
simulate the boundary conditions for both acoustic propaga-
tion and flow convection.
IV. CONCLUSION
We numerically investigated flow structures and radiated
sound in a flow over a backstep with a small upstream bump
by directly solving the three-dimensional compressible
Navier-Stokes equations. The results showed that a small
vortex is shed from the leading edge of the bump and con-
vected downstream to the backstep. When this vortex passes
the backstep, it splits into a fast and a slow vortex. A com-
pression wave is radiated between this slow vortex and its
upstream vortex. Meanwhile, expansion waves are radiated
from the vortices distorted in the wake of the backstep. The
compression waves propagated toward the leading edge of
the bump in turn shed new vortices by stretching the vortex
formed by the flow separation at this edge. We also investi-
gated the acoustic sources and the phase distributions of the
second invariant of the velocity gradient tensor and fluctuat-
ing fluid density. As a result, we were able to identify the
phase relationship between the vortices and the acoustic
waves. Based on this relationship, we proposed a formula for
predicting the frequencies of the radiated tonal sound. The
frequencies predicted by this formula are in good agreement
with those measured in the experiments. This agreement
validated the present computational results.
The methods that we used to identify the phase relation-
ship between the vortices and the acoustic waves in this pa-
per are also useful for the clarification of other phenomena
associated with fluid-acoustic interactions. Moreover, we be-
lieve that this clarification leads to an improved design guide
for various industrial products for the suppression of the
trouble caused by fluid-acoustic interactions discussed here.
1
H. H. Heller and W. M. Dobrzynski, “Sound radiation from aircraft wheel-
well/landing-gear configurations,” J. Aircr. 14, 768 共1977兲.
2
O. W. Mcgregor and R. A. White, “Drag of rectangular cavities in super-
sonic and transonic flow including the effects of cavity resonance,” AIAA
J. 8, 1959 共1970兲.
3
G. B. Brown, “The vortex motion causing edge tones,” Proc. Phys. Soc.
London 49, 493 共1937兲.
4
A. Powell, “On the edge tone,” J. Acoust. Soc. Am. 33,395共1961兲.
5
J. E. Rossiter, “Wind-tunnel experiments on the flow over rectangular
cavities at subsonic and transonic speeds,” Aero. Res. Counc. R&M, No.
3438 共1964兲.
6
M. A. Kegerise, E. F. Spina, S. Garg, and L. N. Cattafesta, “Mode-
switching and nonlinear effects in compressible flow over a cavity,” Phys.
Fluids 16, 678 共2004兲.
7
A. J. Bilanin and E. E. Covert, “Estimation of possible excitation frequen-
cies for shallow rectangular cavities,” AIAA J. 11, 347 共1973兲.
8
C. K. W. Tam and P. J. W. Block, “On the tones and pressure oscillations
induced by flow over rectangular cavities,” J. Fluid Mech. 89, 373 共1978兲.
9
D. G. Crighton, “The jet edge-tone feedback cycle; linear theory for the
operating stages,” J. Fluid Mech. 234, 361 共1992兲.
10
M. S. Howe, “Edge, cavity and aperture tones at very low Mach num-
bers,” J. Fluid Mech. 330,61共1997兲.
11
L. Larcheveque, P. Sagaut, I. Mary, O. Labbe, and P. Comte, “Large-eddy
simulation of a compressible flow past a deep cavity,” Phys. Fluids 15,
193 共2003兲.
12
C. W. Rowley, T. Colonius, and A. J. Basu, “On self-sustained oscillations
in two-dimensional compressible flow over rectangular cavities,” J. Fluid
Mech. 455, 315 共2002兲.
13
T. Colonius and S. K. Lele, “Computational aeroacoustics: progress on
nonlinear problems of sound generation,” Prog. Aerosp. Sci. 40, 345
共2004兲.
14
S. K. Lele, “Compact finite difference schemes with spectral-like resolu-
tion,” J. Comput. Phys. 103,16共1992兲.
15
K. Matsuura and C. Kato, “Large-eddy simulation of compressible transi-
tional flows in a low-pressure turbine cascade,” AIAA J. 45,442共2007兲.
16
D. V. Gaitonde and M. R. Visbal, “Pad-type higher-order boundary filters
for the Navier-Stokes equations,” AIAA J. 38, 2103 共2000兲.
17
K. W. Thompson, “Time dependent boundary conditions for hyperbolic
systems,” J. Comput. Phys. 68,1共1987兲.
18
T. J. Poinsot and S. K. Lele, “Boundary conditions for direct simulations
of compressible viscous flows,” J. Comput. Phys. 101,104共1992兲.
19
J. W. Kim and D. J. Lee, “Generalized characteristic boundary conditions
for computational aeroacoustics,” AIAA J. 38, 2040 共2000兲.
20
J. B. Freund, “Proposed inflow/outflow boundary condition for direct com-
putation of aerodynamic sound,” AIAA J. 35,740共1997兲.
21
J. Larsson, L. Davidson, M. Olsson, and L. Eriksson, “Aeroacoustic in-
vestigation of an open cavity at low mach number,” AIAA J. 42,2462
共2004兲.
106104-8 Yokoyama et al. Phys. Fluids 19, 106104 共2007兲
Downloaded 20 Nov 2007 to 133.11.199.19. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
