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Fluid-acoustic interactions in self-sustained oscillations in turbulent cavity
flows. I. Fluid-dynamic oscillations
Hiroshi Yokoyama1,a兲and Chisachi Kato2
1Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-8656, Japan
2Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
共Received 25 March 2009; accepted 18 September 2009; published online 23 October 2009兲
The fluid-acoustic interactions in a flow over a two-dimensional rectangular cavity are investigated
by directly solving the compressible Navier–Stokes equations. The upstream boundary layer is
turbulent. The depth-to-length ratio of the cavity is 0.5. Phase-averaged flow fields reveal the
mechanism for the acoustic radiation. Large-scale vortices form in the shear layer that separates
from the upstream edge of the cavity. When a large-scale vortex collides with the downstream wall,
the low-pressure fluid in the vortex spreads along the downstream wall. As a result, a local
downward velocity is induced by the local pressure gradient, causing the upstream fluid to expand.
Finally, an expansion wave propagates to the outside of the cavity. The large-scale vortices originate
from the convective disturbances that develop in the shear layer. The disturbances grow due to the
Kelvin–Helmholtz instability, similar to the growth of those in a laminar cavity flow. To clarify the
mechanism for the generation of the initial convective disturbances, computations for
backward-facing step flows with an artificial acoustic source are also performed. As the artificial
acoustic waves become more intense, the initial convective disturbances in the shear layer become
more intense while the spatial growth rate of these disturbances does not change. This means that
the initial convective disturbances in the shear layer are induced by the acoustic waves. © 2009
American Institute of Physics.关doi:10.1063/1.3253326兴
I. INTRODUCTION
Self-sustained oscillations with fluid-acoustic interac-
tions in a flow over a cavity often radiate intense tonal
sound. If unsuppressed, the sound levels in open cavities can
reach 160 dB at a freestream Mach number of 0.8.1These
strong oscillations can cause fatigue in nearby components
such as aircraft wheel wells and landing gear compartments.
The intense tonal sound radiated by the deep cavity of a
train-car gap is a problem to be overcome in the develop-
ment of faster trains.2Large-amplitude pressure pulses can
occur in gas transport systems with closed side branches
even when the freestream Mach number is lower than 0.2.3,4
In short, prediction of oscillations during the design stage
and invention of ways to suppress them are important in the
development of many flow-related industrial products.
Many researchers over the past 50 years have investi-
gated the mechanism of self-sustained oscillations in a cavity
flow. Rossiter5described an oscillation mechanism similar to
that presented for edge tones by Powell.6In this mechanism,
the interactions of vortices with the downstream edge radiate
acoustic waves, which leads to the formation of new vortices
at the upstream edge. Rossiter derived a semiempirical for-
mula to explain this: fL/u1⬁=共n−
␥
兲/共M+1/
兲, where fis
the frequency of the radiated tonal sound, Lis the cavity
length, u1⬁is the freestream velocity, nis a positive integer,
Mis the freestream Mach number,
is the ratio of the con-
vection velocity of the vortices to the freestream velocity,
and
␥
is a constant for the phase correction. The frequencies
predicted by this formula agree with the peak frequencies of
the tonal sound measured in his experiments for a cavity
flow with an upstream turbulent boundary layer 共turbulent
cavity flow兲at high Mach numbers 共M⬎0.4兲.5East7mea-
sured pressure fluctuations in a deep cavity with a large
depth-to-length ratio 共D/L⬎1兲in a turbulent boundary layer
at low Mach numbers 共M⬍0.2兲. He found that intense os-
cillations occur only when the shear layer instability caused
by the fluid-acoustic interactions described by Rossiter5is
coupled with the acoustic resonance of the cavity.
Rockwell and Naudascher8classified the self-sustained
oscillations in a cavity flow into two categories. They called
oscillations due to shear layer instability without the reso-
nance described by Rossiter5“fluid-dynamic oscillations”
and those due to the coupling between the shear layer insta-
bility and the acoustic resonance of the cavity “fluid-resonant
oscillations.” For fluid-dynamic oscillations in a laminar cav-
ity flow, Knisely and Rockwell9found by experiments that
disturbances generated at the upstream edge by acoustic
feedback are amplified in the shear layer by the Kelvin–
Helmholtz 共KH兲instability, resulting in the formation of vor-
tices. On the basis of the feedback loop model described by
Rossiter,5Rowley et al.10 suggested a criterion for the onset
of fluid-dynamic oscillations in a laminar cavity flow, which
takes into account the cavity configuration and the
freestream Mach number. This criterion agrees with their
two-dimensional computational results obtained using the
compressible Navier–Stokes equations.10 They did not dis-
cuss oscillations in a turbulent cavity flow. With regard to
a兲Electronic mail: yokoyama@iis.u-tokyo.ac.jp.
PHYSICS OF FLUIDS 21, 105103 共2009兲
1070-6631/2009/21共10兲/105103/13/$25.00 © 2009 American Institute of Physics21, 105103-1
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
fluid-resonant oscillations in a turbulent cavity flow, by using
particle image velocimetry, Oshkai and Yan3showed that
large-scale vortices form in the shear layer and that they
become acoustic sources. However, the mechanism for the
formation of these vortices and that for the acoustic radiation
have not been described in detail.
Our objective is to clarify in detail the mechanism for
the formation of the large-scale vortices by acoustic feedback
and that for the acoustic radiation by these vortices with
regard to the self-sustained oscillations of a turbulent cavity
flow by directly solving the compressible Navier–Stokes
equations. As mentioned, it is important to suppress the os-
cillations of turbulent cavity flows in various industrial prod-
ucts. In this paper, the fluid-dynamic oscillations are focused
on. In a subsequent paper, the fluid-acoustic interactions of
fluid-resonant oscillations will be focused on.
II. NUMERICAL METHODS
A. Flow configurations
The flow over a two-dimensional cavity, as shown in
Fig. 1, is investigated. The incoming boundary layer is tur-
bulent. The parameters are shown in Table I. The freestream
Mach number Mis 0.3. The ratio of the momentum thickness
to the cavity length
/L, where
is calculated from a sepa-
rate computation for the flow over a flat plate without a cav-
ity, is 0.04 at the position of the upstream edge of the cavity.
The ratio of the momentum thickness of the shear layer to
the cavity length
m/Lis 0.06 at x1/L=0.5, where the origin
of the coordinate is at the position of the upstream edge of
the cavity. The momentum thickness is defined as
m=
冕
ymin
ymax u1av
u1⬁
冉
1−u1av
u1⬁
冊
dx2,共1兲
where u1av is the time- and spanwise-averaged streamwise
velocity. The boundary ymin/L= −0.1 is adopted to remove
the effect of the inverse flow in the cavity. Also, ymax/Lis set
to 1.0 because the u1av/u1⬁is approximately 1.0 at this po-
sition. The parameters of Mand
/Lhave the same values as
those used by Mizushima et al.2in their investigation of a
cavity flow. In their experiment, they measured sound radi-
ating from the flow over the cavity of a train-car gap, for
which D/Lranged from 0.5 to 2.6 around the train car, and
found that self-sustained oscillations occurred.
Preliminary computations of cavity flow were performed
for different values of D/L共0.5, 0.9, 1.3, 1.7, 2.1, and 2.5兲
while keeping the cavity length and incoming boundary layer
thickness constant. The results showed that fluid-dynamic
oscillations occurred when D/Lwas 0.5 and that fluid-
resonant oscillations occurred when it was ⱖ0.9.
Here the fluid-dynamic oscillations are clarified for a
cavity flow when D/Lis 0.5 共L/D=2.0,
/D=0.08兲.A
subsequent paper will focus on the fluid-acoustic interactions
of the fluid-resonant oscillations for deep cavities with
D/Lⱖ0.9. The Reynolds number ReL共=
⬁u1⬁L/
, where
⬁and
are, respectively, the density and viscosity of the
free stream兲,is3.0⫻104in the present computation, whereas
it was 6.6⫻105in Mizushima’s experiment.2The effects of
this difference in the Reynolds number on the fluid-acoustic
interactions in fundamental oscillations are small, as ex-
plained in Sec. III D.
Computations for a backward-facing step 共backstep兲
flow, as shown in Fig. 2, are also performed. In the backstep
flow, tonal sound is not radiated by itself. To clarify the
mechanism for the formation of large-scale vortices in the
shear layer of the cavity flow due to acoustic feedback, the
effects of artificial acoustic waves on the shear layer of the
backstep flow are investigated. The boundary layer at the
trailing edge is the same as that at the upstream edge of the
cavity in the cavity flow. The backstep height His the same
as the cavity depth of 0.5L.
B. Governing equations and finite difference
formulation
Both flow and acoustic fields are simultaneously simu-
lated by directly solving the three-dimensional compressible
Navier–Stokes equations in the conserved form
Qt+
xk
共Fk−Fvk兲=0, 共2兲
where Qis the vector of the conserved variables, Fkis the
inviscid flux vector, and Fvkis the viscous flux vector.
The spatial derivatives are evaluated using the sixth-
order-accurate compact finite difference scheme 共fourth or-
der accurate on the boundaries兲.11 The time integration is
done using the third-order-accurate Runge–Kutta method.
Tur
b
u
l
ent
b
oun
d
ary
l
ayer
D
L
θ
x1
x2
x
3
FIG. 1. Configuration for flow over two-dimensional cavity.
TABLE I. Parameters.
Label MD/LReL
/L
m/L
Present computation 0.3 0.5 3.0⫻1040.04 0.06
Train-car gap experimenta0.3 0.5–2.6 6.6⫻1050.04 ¯
aReference 2.
Tur
b
u
l
ent
b
oun
d
ary
l
ayer
H
θ
x1
x2
x3
Acoustic sourc
e
FIG. 2. Configuration for backward-facing step flow.
105103-2 H. Yokoyama and C. Kato Phys. Fluids 21, 105103 共2009兲
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
To reduce the computational cost, large-eddy simulations
共LES兲are performed in the present work. Nevertheless, the
present computational grid incorporated with the above-
mentioned numerical methods adequately resolves the small-
est active vortices in the boundary layer and in the shear
layer in the cavity. To derive the governing equations for
LES, any variable
is decomposed into its grid-scale 共GS兲
¯
and subgrid-scale 共SGS兲
⬙components,
=
¯
+
⬙.共3兲
In a compressible flow, the GS quantity is recast in terms of
a Favre-filtered variable
˜
=
¯
.共4兲
The GS components of the conserved variables and the
fluxes are given as
Q=1
J共
¯
,
¯
u
˜
1,
¯
u
˜
2,
¯
u
˜
3,
¯
e
˜
兲t,共5兲
Fk=1
J
冢
¯
u
˜
k
¯
u
˜
1u
˜
k+p
¯
¯
u
˜
2u
˜
k+p
¯
¯
u
˜
3u
˜
k+p
¯
共
¯
e
˜
+p
¯
兲u
˜
k
冣
,共6兲
Fvk=1
J
冢
0
˜
k1
˜
k2
˜
k3
u
˜
j
˜
kj +
T
˜
xk
冣
,共7兲
˜
kl =
共T
˜
兲
冉
u
˜
k
xl
+
u
˜
l
xk
−2
3
␦
kl
u
˜
n
xn
冊
,共8兲
=
共T
˜
兲C/Pr, 共9兲
where
kl is the viscous stress tensor,
is the thermal con-
ductivity, and
␦
kl is the Kronecker’s delta function. The fluid
is assumed to be the standard air. Sutherland’s formula is
used for the viscosity coefficient
. It is assumed that the
specific heat Cis 1004 J kg−1 K−1 and that the Prandtl num-
ber Pr is 0.72. No explicit SGS model is used here: the
turbulent energy in the GS that should be transferred to SGS
eddies is dissipated by the tenth-order spatial filter, as de-
scribed below. It was shown by many researchers12–14 that
the above-mentioned method combining the low-dissipation
discretization schemes and the explicit filtering correctly re-
produces turbulent flows such as a cavity flow and a round
jet. This filter removes numerical instabilities at the same
time,15
␣
f
ˆi−1 +
ˆi+
␣
f
ˆi+1 =兺
n=0
5an
2共
i+n+
i−n兲,共10兲
where
is a conserved quantity and
ˆis the filtered quantity.
The coefficients anhave the same values as those used by
Gaitonde and Visbal,16 and the parameter
␣
fis 0.45. In Sec.
III, these computational methods are validated.
C. Computational grid
Figure 3shows the computational grids for the cavity
and backstep flows. The origins of the coordinates are at the
position of the upstream edge of the cavity and that of the
trailing edge of the backstep, respectively. The spanwise ex-
tents of these computational domains Lware 0.5Land Hfor
the cavity and backstep flows, respectively. This extent cor-
responds to Lw
+=700, where the wall units are based on the
skin-friction coefficient computed for the boundary layer
over a flat plate at the streamwise position corresponding to
the upstream edge of the cavity and the trailing edge of the
backstep, Rex=4.3⫻105. Therefore, approximately six tur-
bulent streaks lie in the spanwise extent. The cross sections
in the x1-x2plane are shown in Fig. 4, where length is non-
dimensionalized by cavity length Land backstep height H
for the cavity and backstep flows, respectively. The effects of
the extent of the computational domain on the results are
discussed in Sec. III C.
To capture the longitudinal vortices in the turbulent
boundary layer, the grid resolutions are set near the wall as
⌬x1
+=38, ⌬x2
+=1, and ⌬x3
+=14, where the wall units are
based on the above-mentioned skin-friction coefficient. To
capture the acoustic propagation accurately, more than 20
grid points are used per acoustic wavelength of the funda-
mental frequency of the oscillations even in the field far from
the cavity. It is clarified in Sec. III that the present computa-
tional grid adequately resolves both the longitudinal vortices
in the turbulent boundary layer and the acoustic waves cre-
ated by the fundamental oscillations. The present computa-
tional domain consists of approximately 5.0⫻106structured
grid points.
FIG. 3. Computational grids for cavity flow 共left兲and backward-facing step
flow 共right兲. Every tenth grid line is shown for clarity.
105103-3 Fluid-acoustic interactions in self-sustained oscillations Phys. Fluids 21, 105103 共2009兲
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
D. Boundary conditions
Figure 4also shows the boundary conditions for the cav-
ity and backstep flows. The inflow and outflow boundaries
are artificial, so that they must allow vortices and acoustic
waves to pass smoothly with minimal numerical distur-
bances. Nonreflecting boundary conditions based on the
characteristic wave relations17–19 are used at these bound-
aries. Nonslip and adiabatic boundary conditions are used at
the wall. The periodic boundary condition is used in the
spanwise 共x3兲direction. The use of the periodic condition is
appropriate in the present study because the disturbances in
the shear layer two dimensionally grow to large-scale vorti-
ces due to the KH instability and radiate two-dimensional
acoustic waves in the fundamental oscillations, as explained
in Sec. IV.
To achieve the desired momentum thickness of the
boundary layer at the upstream edge of the cavity, Eq. 共2兲is
replaced with
Qt+
xk
共Fk−Fvk兲=−
0共a/Ld兲xd
2共Q−Qtarget兲共11兲
for the upstream damping region shown in Fig. 4, where ais
the speed of sound, Ldis the streamwise length of the damp-
ing region, xdis the nondimensional distance from the down-
stream end of the damping region 共0ⱕxdⱕ1兲, and Qtarget is
the target vector of the conserved variables.20,21 The coeffi-
cient
0is set to 0.05. It was demonstrated in a previous
study that acoustic reflection does not occur with this
setting.22 At the inflow boundary of the computational do-
main, the Reynolds number Rex共=
⬁u1⬁x/
, where xis the
distance measured from the origin of the laminar boundary
layer兲and boundary layer thickness
␦
int /Lare assumed to be
9.0⫻103and 0.016, respectively.
Laminar-turbulent transition is prompted by super-
imposing homogeneous freestream turbulence on the solu-
tions for the compressible laminar boundary layer QBlasius for
1.25⬍x2/
␦
int ⬍21.3 共0.02⬍x2/L⬍0.34兲in the damping re-
gion 共−16.0⬍x1/L⬍−15.7兲, where x2= 0 corresponds to the
position of the wall. That is,
Qtarget =QBlasius共x2/
␦
int ⱕ1.25,21.3 ⱕx2/
␦
int兲,
共12兲
Qtarget共t兲=QBlasius +Qfst共t兲共1.25 ⬍x2/
␦
int ⬍21.3兲.
The freestream turbulence Qfst共t兲is obtained by sliding an
instantaneous freezed flow field of homogeneous turbulence
computed by a separate LES. In the simulation of the homo-
geneous turbulence, the initial random field has the Karman
spectrum.23 The turbulence intensity is 6%, and the longitu-
dinal integral scale is about 0.24L. This integral scale corre-
sponds to about 0.06, where is the acoustic wavelength of
the fundamental frequency of the sound radiating from the
cavity flow, and the effect of the freestream turbulence on the
propagation of the radiated acoustic waves is negligibly
small. Also, it was confirmed that the amplitude of the spu-
rious noise generated by the superimposition of this
freestream turbulence is less than 10% of that of the sound
generated by the oscillations on the bottom of the cavity,
where we observe the pressure fluctuations in Sec. IV B.
Therefore, it is concluded that the effects of the freestream
turbulence on the oscillations of the cavity flow are also
small. It is demonstrated in Sec. III B that the boundary layer
becomes a fully developed turbulent state by the upstream
edge of the cavity.
E. Initial flow fields
The computational result for the turbulent flow over a
flat plate is used as the initial flow field for the present com-
putation of the turbulent cavity flow. In the cavity region
共0ⱕx1ⱕLand x2⬍0兲, the velocity is set to 0, and the
pressure and density are set to the same values as those at
x2=0.
III. VALIDATION OF COMPUTATIONAL METHODS
A. Propagation of acoustic waves
The effects of the grid resolution on the propagation of
acoustic waves were preliminarily investigated by using an
artificial monopole. As a result, it was clarified that five grid
points per one wavelength accurately capture the acoustic
waves. Therefore, the present computational grids for the
main computations, for which more than 20 grid points are
used per one acoustic wavelength, sufficiently resolve the
propagation of the acoustic waves for the fundamental
oscillations.
B. Turbulent boundary layer
To confirm that the boundary layer has become a fully
developed turbulent state by the upstream edge of the cavity,
the flow over a flat plate without a cavity is calculated. Fig-
ure 5shows the variation in the computed time- and
spanwise-averaged skin-friction coefficient cfcompared with
x1/L
x2
/
L
0
10.0
0 1.0 7.5
-0.5
Damping region
-16.0
Outflow
Inflow
Adiabatic wall
-15.7
Outflow
x1/H
x2/H
0
2
0.0
0 14.0
-1.0
Damping region
-32
Outflow
Inflow
Adiabatic wall
-31.4
Outflow
Acoustic
source
2.0
FIG. 4. Computational domains and boundary conditions for cavity flow
共top兲and backward-facing step flow 共bottom兲.
105103-4 H. Yokoyama and C. Kato Phys. Fluids 21, 105103 共2009兲
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
that measured in the past experiment24 with the same degree
of freestream turbulence intensity as that of the present com-
putation. The computed skin-friction coefficients are in good
agreement with those measured in the experiment.24 The
boundary layer becomes a fully developed turbulent state at
Rex=4.3⫻105, where the upstream edge of the cavity is to
be located for the main computations.
For the main computation of the cavity flow, the span-
wise cross-correlation coefficient Rof u2along x2
+=15atthe
upstream position of the cavity Rex=3.4⫻105is shown in
Fig. 6. The spanwise extent of the computational domain Lw
+
corresponds to 720. This coefficient is computed from the
flow fields sampled during T+=300 using Eq. 共13兲. The span-
wise periodicity is used when x3+⌬x3is larger than Lw,
R共⌬x3兲=
兰t=0
T+=300兰x3=0
x3=Lwu2
⬘共x3兲u2
⬘共x3+⌬x3兲dx3dt
兰t=0
T+=300兰x3=0
x3=Lwu2
⬘共x3兲u2
⬘共x3兲dx3dt
,共13兲
where u2
⬘is the difference between the instantaneous and
time-averaged velocity components in the x2direction. In
Fig. 6, the local minimum of the coefficient, which appears
at around ⌬x3
+=60, shows that the mean spanwise streak
spacing is about 120. This value is in good agreement with
that presented in the past research.25 This indicates that the
longitudinal vortices, which generate the streaks in the tur-
bulent boundary layer, are adequately captured in the present
computation.
C. Effects of computational domain
It is of critical importance that the computational domain
does not affect the results. To confirm that it does not, two
additional computations for the cavity flow with computa-
tional domains different from the original domain are also
performed.
First, the effects of the vertical length Lhand the stream-
wise length Ldfrom the downstream edge of the cavity to the
downstream outflow boundary are investigated. The calcula-
tions with a domain 共Lh/L=15.0, Ld/L= 9.7 , Lw/L= 0.5兲
larger than the original domain 共Lh/L=10.0, Ld/L
=6.5, Lw/L= 0.5兲are performed. Then, the effects of the
spanwise length Lware investigated by performing the cal-
culation for a cavity flow with a domain with a twice-
extended spanwise length 共Lh/L=10.0, Ld/L= 6.5 , Lw/L
=1.0兲. The grid resolutions of the two extended computa-
tional domains are the same as that of the original computa-
tional domain for the corresponding regions.
Figure 7compares for these three domains the frequency
spectra of the velocity fluctuations at x1/L=0.5 and x2=0.
Frequency fis normalized as St⬅fL/u1⬁. The duration of
the Fourier transform is about 45共L/u1⬁兲, and the results are
averaged 20 times temporally as well as spanwisely. This
duration corresponds to about 36 periods of the fundamental
mode St=0.8 for the cavity flow. Figure 7shows that both
the frequency and the amplitude of the fundamental oscilla-
tions with the two extended domains are virtually identical to
those with the original domain. Therefore, it is concluded
that the original computational domain is sufficiently large
and wide and does not affect the present results.
D. Comparison to experiments
As shown in Table I, the flow parameters in the present
computation have the same values as those in Mizushima’s
experiment2except for the cavity depth and the Reynolds
number. In the experiment, D/Lvaried from 0.5 to 2.6. The
fundamental frequency St=0.8 in the present computation
for the cavity flow closely agrees with that measured in the
experiment 共St=0.78兲, while oscillations at St= 0.4 also oc-
curred in the experiment. Preliminarily, it was demonstrated
that the oscillations at St=0.4 occur for a cavity flow with
FIG. 5. Variations in time- and spanwise-averaged skin-friction coefficients
computed for flat plate boundary layer without cavity.
FIG. 6. Spanwise cross-correlation coefficient of vertical velocity u2along
x2
+=15atRe
x=3.4⫻105for cavity flow.
FIG. 7. Power spectral densities of vertical velocity u2/u1⬁at x1/L=0.5
and x2=0 for cavity flow with original computational domain 共Lh/L
=10.0, Ld/L= 6.5 , Lw/L= 0.5兲, with larger domain 共Lh/L= 15.0 , Ld/L
=9.7, Lw/L= 0.5兲, and with wider domain 共Lh/L=10.0, Ld/L
=6.5, Lw/L= 1.0兲.
105103-5 Fluid-acoustic interactions in self-sustained oscillations Phys. Fluids 21, 105103 共2009兲
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
D/L=1.3 and that they correspond to the fluid-resonant os-
cillations. The fluid-acoustic interactions of the fluid-
resonant oscillations will be investigated in a subsequent pa-
per. Meanwhile, the mode at St= 0.8 corresponds to the fluid-
dynamic oscillations, which are the primary focus of this
paper.
As mentioned, the effect of the difference in the
Reynolds number ReLbetween the present computation and
Mizushima’s experiment2on the fluid-acoustic interactions
in the fundamental oscillations is presumably small for the
following reason. Namely, in the present computation for the
cavity flow, the two-dimensional disturbances in the shear
layer grow to large-scale vortices due to the KH instability of
the inflection point of the averaged velocity profile, as clari-
fied in Sec. IV. This indicates that the interactions between
the large-scale vortices and the turbulent fine-scale vortices
are small, and therefore the fluid-acoustic interactions in the
fundamental oscillations do not strongly depend on the Rey-
nolds number. Hence it can be mentioned that the present
computation captures the fluid-acoustic interactions in the
fluid-dynamic oscillations observed in the experiment.
IV. RESULTS AND DISCUSSION
A. Analysis methods
1. Spatial variations in power spectral density
and phase
The spatial variations in the power spectral density
共PSD兲and the phases of the variables are computed from the
results of the Fourier transform of spanwise-averaged data.
The duration of the Fourier transform is 10共L/u1⬁兲, which
corresponds to eight periods of the fundamental frequency
for the cavity flow. The variations are averaged 20 times
temporally.
2. Linear stability analysis
Linear stability analysis 共LSA兲is performed in a stan-
dard way10,26,27 and thus is described only briefly here. A
compressible, inviscid, locally parallel formulation is used.
This means that the time-averaged velocity profile slowly
varies in the streamwise direction and that the disturbances
are parallel at each streamwise location. To confirm that the
disturbances for the fundamental frequency are parallel in the
shear layer of the cavity flow, we investigated the spatial
distributions of the phase of velocity u2for the fundamental
frequency along x2/L=0.025, 0, and ⫺0.025, as shown in
Fig. 8, where the reference of the phases is the phase at
x1/L=0.5 and x2/L= 0. As a result, it was clarified that the
three phase distributions approximately agree with each
other and therefore, the disturbances are parallel. The slight
difference near the upstream edge is due to the effects of the
particle velocity of the acoustic waves on the velocity u2.
Also, the normal modes of the form
g共x1,x2,t兲=g
ˆ共x2兲ei共
␣
x1+2
ft兲+ c.c. 共14兲
are assumed, where fis a frequency,
␣
is a complex eigen-
value, gis any flow variable, g
ˆis a complex eigenfunction,
and c.c. denotes the complex conjugate. The modes are in-
serted into the compressible inviscid equations linearized
about the parallel flow, and the eigenvalues and eigenfunc-
tions are found using a Runge–Kutta shooting method. The
present LES data are used to determine the mean flow for the
LSA. Boundary conditions are set such that the disturbances
are exponentially dampened for large +x2, and zero vertical
velocity is imposed at the bottom wall of the cavity.
3. Spanwise cross correlation
The spanwise cross-correlation coefficient Rfof u2for
the fundamental frequency is computed using Eq. 共15兲. The
spanwise periodicity is used when x3+⌬x3is larger than Lw,
Rf共⌬x3兲=
兰t=0
t=20T兰x3=0
x3=Lwu2f共x3兲u2f共x3+⌬x3兲dx3dt
兰t=0
t=20T兰x3=0
x3=Lwu2f共x3兲u2f共x3兲dx3dt ,共15兲
where u2f is the velocity fluctuation for the fundamental fre-
quency and computed using the Fourier transform. The du-
ration of the transform is 10共L/u1⬁兲. This coefficient Rfrep-
resents the two dimensionality of the flow for the
fundamental frequency.
4. Phase-averaging process
In a phase-averaging process, the variables at the same
phase of the fundamental mode are averaged 45 times
temporally for the cavity flow. It has been preliminarily clari-
fied by using a wavelet analysis that the period of the oscil-
lations varies by about 5% temporally. Considering this
variation in the period, the pressure fluctuation on the bottom
FIG. 8. Spatial distributions of the phase in velocity u2along x2/L= 0.025,
0, and ⫺0.025.
FIG. 9. PSD of vertical velocity u2/u1⬁at x1/L=0.5 and x2/L=0 for cavity
flow and at x1/H=1.0 and x2/H=0 for backward-facing step flow.
105103-6 H. Yokoyama and C. Kato Phys. Fluids 21, 105103 共2009兲
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
共x1/L=0.5, x2/L= −0.5兲is used as the reference signal. The
phase at each time is calculated assuming that the period of
the oscillations is the time interval from a peak to a next
peak of this pressure fluctuation. The phase-averaged data
are further averaged in the spanwise direction.
B. Self-sustained oscillations
Figure 9shows the frequency spectrum of the velocity
u2in the shear layer at x1/L=0.5 and x2= 0 for the cavity
flow and that at the corresponding position for the backstep
flow without an artificial acoustic source. Frequency fis nor-
malized as St⬅2fH/u1⬁for the backstep flow, where 2H
corresponds to cavity length L. Self-sustained oscillations
occur at St=0.8 in the cavity flow while oscillations do not
occur in the backstep flow. Note also that the amplitudes of
the fluctuations at other frequencies for the cavity flow are
approximately the same as those for the backstep flow. This
means that the self-sustained oscillations do not strongly af-
fect the amplitude of the turbulent kinetic energy at other
frequencies in the shear layer.
Figure 10 shows the frequency spectrum of the pres-
sure fluctuations on the bottom wall at x1/L=0.5 and
x2/L=−0.5 for the cavity flow and that at x1/H= 1.0 and
x2/H=−1.0 for the backstep flow. The pressure is normalized
as pressure coefficient Cp⬅p/共
⬁u1⬁
2/2兲. Tonal sound is ra-
diated in the self-sustained oscillations at St=0.8 only for the
cavity flow.
C. Vortices
1. Fine-scale vortices
Instantaneous vortices for the cavity flow are illustrated
in Fig. 11. To elucidate the vortices, the second invariant of
the velocity gradient tensor q=储⍀储2−储S储2is computed,
where ⍀and Sare, respectively, the asymmetric and sym-
metric parts of the velocity gradient tensor. Regions with
q⬎0 represent vortex tubes. Before the separation at the
upstream edge, fine-scale longitudinal vortices are active in
the turbulent boundary layer. Soon after the separation, these
vortices become free from the blocking effect of the wall and
become more active than those in the upstream turbulent
boundary layer.
2. Large-scale vortices
In Fig. 12, the contours and vectors represent the phase-
averaged pressure coefficients and velocity vectors, respec-
tively, with the time-averaged components at each position
subtracted. Large-scale vortices are apparent in the shear
layer and form low-pressure regions. These large-scale vor-
tices are composed of the above-mentioned fine-scale
vortices.
Figure 13 shows the spanwise cross-correlation coeffi-
cients Rfof u2for the fundamental frequency along
x1/L=0.25, 0.50, and 0.75 共x2=0兲for the cavity flow. The
coefficients are high at all positions, which means that the
large-scale vortices are coherent in the spanwise direction.
These large-scale vortices form by the growth of the
convective disturbances in the shear layer. In Sec. IV E, it
will be clarified that the acoustic waves only generate the
initial convective disturbances and it does not contribute to
the growth of these disturbances. To clarify the mechanism
of the growth of these convective disturbances, the rate of
the streamwise amplification of the fluctuations in u2for the
fundamental frequency is compared with that predicted by
the LSA in Fig. 14. Note that velocity u2includes a convec-
tive component and a propagative component 共particle veloc-
ity兲. As will be clarified in Sec. IV E, the propagative distur-
FIG. 10. PSD of pressure coefficient Cpat x1/L=0.5 and x2/L=−0.5 for
cavity flow and at x1/H=1.0 and x2/H=−1.0 for backward-facing step flow.
FIG. 11. Instantaneous isosurfaces of q/共u1⬁/L兲2= 25 for cavity flow.
FIG. 12. 共Color online兲Phase-averaged pressure coefficients and velocity
vectors for cavity flow, where Tis the time period of fundamental
oscillations.
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bances of the acoustic waves are negligible in comparison to
the convective disturbances in the shear layer of x1⬎0.1L.
Therefore, this region is focused on. The amplification
rate computed from the present LES data approximately
agrees with that predicted by the LSA although at around
x1/L=0.2 the rate becomes gradually saturated as the aver-
age velocity gradient becomes more moderate. This means
that the convective disturbances of the fundamental fre-
quency in the shear layer grow into large-scale vortices due
to the KH instability, similar to those in a laminar cavity
flow.9,10 Meanwhile, the mechanism of the generation of the
convective disturbances near the upstream edge 共x1⬍0.1L兲
is not clear because the propagative disturbances of the
acoustic waves are comparable to the convective distur-
bances near the upstream edge. In Sec. IV E, the mechanism
of the generation of the initial convective disturbances by the
acoustic feedback is discussed by decomposing the flow field
into the convective and propagative components for the
backstep flow with an artificial acoustic source.
D. Acoustic radiation mechanism
The contours in Fig. 15 represent the phase-averaged
pressure coefficients with the time-averaged components
subtracted. Figure 15 shows that an acoustic wave is radiated
from the cavity and propagates to the outside of the cavity.
As already mentioned with relation to Fig. 12, large-
scale vortices are apparent in the shear layer. Low-pressure
regions are formed in these large-scale vortices. When the
large-scale vortices are being convected in the shear layer,
the pressure gradient balances with the centrifugal force act-
ing on the rotating fluid in the large-scale vortices. However,
when a large-scale vortex collides with the downstream wall
of the cavity, it is distorted and the low-pressure fluid in the
vortex spreads along the downstream wall, as shown in Fig.
12 共t=0兲. At this time, the low-pressure fluid is not rigidly
rotating, in contrast to the fluid in the large-scale vortices
being convected in the shear layer. This means that the local
pressure gradient no longer balances with the centrifugal
force. As a result, a local downward velocity is induced due
to the local pressure gradient, as shown in Fig. 16, where the
detailed phase-averaged flow field is presented at t=0 in Fig.
12. As shown in Fig. 16, this local downward velocity is
weak in the far upstream region, while it is intense near the
downstream wall. Therefore, the fluid in the upstream region
of the downstream wall expands, an expansion wave is radi-
ated, and finally the radiated expansion wave is propagated
to the outside of the cavity. To the best of the authors’ knowl-
edge, this is the first time that the mechanism of the acoustic
radiation is clarified for the cavity flow in detail.
To roughly estimate the position of the acoustic source
for the cavity flow, we compute the acoustic propagation
from an artificial source placed on the downstream wall
without a free stream, as shown in Fig. 17. Also, as shown
later in Sec. IV E 3, the effects of the mean flow on the
acoustic propagation are negligibly small at the present
Mach number of 0.3. The acoustic source is assumed to be
two dimensional and coherent in the x3共spanwise兲direction.
FIG. 13. Spanwise cross-correlation coefficient Rfof vertical velocity u2for
fundamental frequency along x1/L=0.25, 0.50, and 0.75 共x2=0兲for cavity
flow.
FIG. 14. Growth of PSD of vertical velocity u2/u1⬁along x2= 0 for cavity
flow compared with that predicted by LSA.
FIG. 15. 共Color online兲Phase-averaged pressure coefficients for cavity flow,
where Tis the time period of fundamental oscillations.
FIG. 16. 共Color online兲Phase-averaged pressure coefficients and velocity
vectors near the bottom wall for cavity flow at t=0 in Fig. 12 共the region in
this figure is indicated in Fig. 12兲.
105103-8 H. Yokoyama and C. Kato Phys. Fluids 21, 105103 共2009兲
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The acoustic source is located at two different positions of
共x1a /L,x2a /L兲,共1.0, ⫺0.5兲and 共1.0, 0兲, and both the results
are compared with the result of the cavity flow for M= 0.3.
For doing so, Eq. 共1兲is replaced with Eqs. 共16兲–共18兲
in the regions 共0.8⬍x1/L⬍1.0,−0.5 ⬍x2/L⬍−0.3兲and
共0.8⬍x1/L⬍1.0,−0.2 ⬍x2/L⬍0.2兲,28
Qt+
xk
共Fk−Fvk兲=S,共16兲
S=
冢
S
0
0
0
冉
5
2a2
冊
S
冣
,共17兲
S=Aexp
冋
共− ln 2.0兲
冉
共x1−x1a兲2+共x2−x2a兲2
共0.1L兲2
冊
册
sin 2
ft,
共18兲
where A/
⬁is 3.0⫻10−5, and the frequency fis the same as
the fundamental frequency for the cavity flow. The amplitude
of the pressure fluctuations on the bottom wall at x1/L=0.5
and x2/L=−0.5 is approximately the same as that for the
cavity flow for M= 0.3. The spatial variations in the phase of
the pressure on the bottom wall 共x2/L=−0.5兲and on the
upstream wall 共x1=0兲are shown in Figs. 18 and 19, respec-
tively. Both the spatial variations in the phase for the artifi-
cial acoustic source at 共x1a /L,x2a /L兲=共1.0,−0.5兲without a
free stream are in good agreement with the present compu-
tational results for M= 0.3. However, Fig. 18 shows that the
slope of the pressure phase for the artificial acoustic source at
共x1a /L,x2a /L兲=共1.0,0兲is different near the downstream wall
共x1/L=1.0兲from that of the cavity flow for M= 0.3. This
means the acoustic waves are radiated from 共x1/L,x2/L兲
=共1.0,−0.5兲rather than 共x1/L,x2/L兲=共1.0 , 0兲in the cavity
flow. However, it should also be mentioned that the acoustic
source in the cavity flow is more complex than a monopole
source. Therefore, we only roughly estimate the position of
the acoustic source here.
The slope of the phase in Fig. 18 shows that the acoustic
waves radiated around the downstream wall 共x1/L=1.0兲
are propagated in the upstream direction 共−x1兲to about
x1/L=0.4 at the sound speed. However, the slope of the
phase becomes approximately flat near the upstream wall
共x1=0兲. This means that the propagation direction of the
acoustic waves changes from the −x1direction to the +x2
direction near the upstream wall. The slope of the phase in
Fig. 19 shows that the acoustic waves are propagated in the
+x2direction along the upstream wall at the sound speed.
E. Backward-facing step flow with artificial acoustic
source
1. Artificial acoustic source
To clarify the mechanism for the generation of the initial
convective disturbances in the shear layer of the cavity flow
by the acoustic feedback, the effects of artificial acoustic
waves on the shear layer of a backstep flow are also inves-
tigated. As shown in Fig. 4, the artificial acoustic source is
placed at 共x1a /H,x2a /H兲=共2.0,−1.0兲, where the backstep
height Hcorresponds to 0.5Lfor the cavity flow. This posi-
tion corresponds to the position of the acoustic source of the
cavity 共the bottom of the downstream wall兲, which was
roughly estimated in Sec IV D. The acoustic source is added
No flow
x1
x
2
Acoustic source
O
L
x1
x2
Acoustic sourc
e
O
L
No flow
FIG. 17. Flow configurations for acoustic propagation from artificial source
without free stream.
FIG. 18. Phase variations in pressure on bottom 共x2= −0.5L兲for fundamen-
tal frequency for cavity flow of M= 0.3 and for two acoustic fields from
artificial acoustic source around x1/L=1.0 and x2/L=−0.5 and around
x1/L=1.0 and x2/L=0 without a free stream.
FIG. 19. Phase variations in pressure on upstream wall 共x1=0兲for funda-
mental frequency for cavity flow of M= 0.3 and for two acoustic fields from
artificial acoustic source around x1/L=1.0 and x2/L=−0.5 and around
x1/L=1.0 and x2/L=0 without a free stream.
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into the backstep flow by replacing Eq. 共1兲in the regions
1.6⬍x1/H⬍2.4 and −1.0⬍x2/H⬍−0.6 with Eqs. 共16兲,
共17兲, and 共19兲,28
S=Aexp
冋
共− ln 2.0兲
冉
共x1−2H兲2+共x2+H兲2
共0.2H兲2
冊
册
sin 2
ft,
共19兲
where Ais the amplitude of the density fluctuations at the
streamwise center of the acoustic source. In order to clarify
the effects of the intensity of the acoustic waves on the
acoustic feedback, three computations with A/
⬁=0,
1.0⫻10−5, and 3.0⫻10−5 are performed, among which the
amplitude of the pressure fluctuations with A/
⬁=3.0e−5 on
the bottom wall at x1/H=1.0 and x2/H= −1.0 is approxi-
mately the same as that at the corresponding position for the
cavity flow. The excitation frequency fis the same as the
fundamental frequency for the cavity flow.
2. Comparison to cavity flow
Figure 20 shows the frequency spectra of the pressure
fluctuations at x1/H=1.0 and x2/H= −1.0 for the backstep
flows with and without the artificial sound of A/
⬁=3.0
⫻10−5 and the pressure fluctuation at the corresponding po-
sition for the cavity flow. It is confirmed that the amplitude
of the artificial sound of A/
⬁=3.0⫻10−5 is approximately
the same as that of the tonal sound in the cavity flow. Also,
the spatial variation in the phase of pressure on the vertical
wall 共x1=0兲for the excitation frequency for the backstep
flow with the acoustic source of A/
⬁=3.0⫻10−5 is com-
pared with the corresponding variation for the cavity flow in
Fig. 21. Figure 21 shows that the spatial variation in pressure
phase is also in good agreement with that for the cavity flow.
Therefore, it is concluded that the acoustic field in the cavity
flow is reproduced in the backstep flow with the artificial
acoustic source of A/
⬁=3.0⫻10−5.
In Fig. 22, the contours and vectors represent the phase-
averaged pressure coefficients and velocities, respectively,
for the backstep flow with the artificial acoustic source of
A/
⬁=3.0⫻10−5. By adding the artificial acoustic source,
large-scale vortices become apparent in the shear layer and
form low-pressure regions like in the cavity flow 共Fig. 12兲.
In Fig. 23, the spanwise cross-correlation coefficients
Rfof u2for the excitation frequency along x1/H=0.5 and
x2=0 for the backstep flow with and without the artificial
acoustic source of A/
⬁=3.0⫻10−5 are compared with the
corresponding coefficient for the cavity flow. The cross-
correlation coefficient becomes as high as 0.9 for the back-
step flow with the artificial acoustic source, although it is
almost zero if the artificial acoustic source is not added. This
is because the above-mentioned large-scale vortices are co-
herent in the spanwise direction like in the cavity flow. Note
also that the cross-correlation coefficient for the backstep
flow with the artificial acoustic source of A/
⬁=3.0⫻10−5 is
higher than that for the cavity flow. This is because the arti-
ficial acoustic source is perfectly coherent in the spanwise
FIG. 20. PSD of pressure coefficient Cpat x1/H= 1.0 and x2/H= −1.0 for
backward-facing step flows with and without artificial acoustic source and at
x1/L=0.5 and x2/L=−0.5 for cavity flow.
FIG. 21. Phase variations in pressure along x1= 0 for backward-facing step
flow with artificial acoustic source and cavity flow.
FIG. 22. 共Color online兲Phase-averaged pressure coefficients and velocity
vectors for backward-facing step flow with artificial acoustic source.
FIG. 23. Spanwise cross-correlation coefficients Rfof vertical velocity u2
along x1/H=0.5 and x2=0 for backward-facing step flows with and without
acoustic source and that along the corresponding line for cavity flow.
105103-10 H. Yokoyama and C. Kato Phys. Fluids 21, 105103 共2009兲
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direction while the large-scale vortices, which become the
acoustic sources for the cavity flow, are not perfectly coher-
ent, as shown in Fig. 13.
In Fig. 24, the spatial variations in the amplitude of the
fluctuations in u2for the excitation frequency for the back-
step flows with and without the artificial acoustic source of
A/
⬁=3.0⫻10−5 are compared with the corresponding
variation for the cavity flow. It is clarified that the spatial
variation in the region of 0ⱕx1/2Hⱕ0.3 is approximately
the same as that in the corresponding region for the cavity
flow. Therefore, it is concluded that the acoustic feedback in
the cavity flow is also reproduced in the backstep flow with
the artificial acoustic source of A/
⬁=3.0⫻10−5 in addition
to the acoustic field, for which a detailed discussion is given
in Sec. IV E 3.
3. Acoustic feedback
Figure 25 shows the variation in the PSD of u2c and that
of u2p for the excitation frequency along x2=0 for the
backstep flow with A/
⬁=3.0⫻10−5, where u2c and u2p
represent the convective and propagative components of u2,
respectively,
u2=u2c +u2p.共20兲
The propagative component u2p is assumed to be the same as
particle velocity, which was calculated for the acoustic
propagation around the backstep from the same artificial
acoustic source without flow. The convective component u2c
is obtained by subtracting the propagative component from
the computed velocity for the backstep flow of M= 0.3. In
Fig. 26, the spatial variation in the pressure phase on the
vertical wall 共x1=0兲for this backstep flow is compared with
that of the acoustic field around the backstep from the same
acoustic source without flow. The acoustic field of the back-
step flow with M= 0.3 is in good agreement with that without
flow. This confirms that the effects of the mean flow on the
acoustic propagation are negligibly small. Figure 25 shows
that the propagative disturbances caused by the acoustic
waves are negligibly small in comparison to the convective
disturbances caused by the large-scale vortices in the region
x1/H⬎0.2.
Figure 27 shows the variations in the PSD of u2c for the
excitation frequency along x2=0 for backstep flows with and
without an artificial acoustic source compared with that pre-
dicted by the LSA. For the backstep flow without an acoustic
source, u2c is equal to u2. Only the curve predicted by the
LSA for the backstep flow without an acoustic source is
shown because the result of the LSA is virtually independent
of the strength of the acoustic source. As shown in Fig. 27,
the convective disturbances become more intense as the ar-
tificial sound becomes more intense. However, all the three
amplification rates computed from the present LES data ap-
proximately agree with that predicted by the LSA in the re-
gion 0.2ⱕx1/H⬍0.4. This means that the growth of the con-
FIG. 24. Variations in PSD of vertical velocity u2/u1⬁along x2= 0 for
backward-facing step flows with and without artificial acoustic source com-
pared with that for cavity flow.
FIG. 25. Variations in PSD of convective component u2c/u1⬁and propaga-
tive component u2p/u1⬁of vertical velocity along x2= 0 for backward-facing
step flows with acoustic source.
FIG. 26. Phase variations in pressure along x1= 0 for backward-facing flows
with artificial acoustic source for M= 0.3 and no flow.
FIG. 27. Variations in PSD of convective component u2c/u1⬁of vertical
velocity along x2=0 for backward-facing step flows with and without acous-
tic source.
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vective disturbances is due to the KH instability and not
affected by the acoustic waves. Meanwhile, the acoustic
waves induce the initial convective disturbances as men-
tioned below. As already shown in Fig. 21, the acoustic
waves are propagated around the upstream edge in the +x2
direction. As a result, the shear layer is moved up and down
due to the particle velocity of the acoustic waves and the
initial convective disturbances are generated in the shear
layer. Therefore, the initial convective disturbances become
more intense as the acoustic waves become more intense, as
shown in Fig. 27.
Since the acoustic field and the acoustic feedback are
reproduced in the backstep flow with the artificial acoustic
source, the above discussion also holds for the acoustic feed-
back in the cavity flow. In the cavity, the initial convective
disturbances are generated by the acoustic waves radiated
due to the collision of the vortices on the downstream wall.
These initial convective disturbances are coherent in the
spanwise direction since the acoustic waves are coherent.
These convective disturbances grow into the coherent large-
scale vortices due to the KH instability independent of the
acoustic waves and the fine-scale vortices, similar to those in
a laminar cavity flow.9,10
V. CONCLUSION
The fluid-acoustic interactions in a flow over a two-
dimensional rectangular cavity were investigated by directly
solving the compressible Navier–Stokes equations. The up-
stream boundary layer was turbulent. The depth-to-length ra-
tio of the cavity was 0.5. Moreover, to clarify the acoustic
feedback for the cavity flow in detail, the effects of artificial
acoustic waves on the shear layer of a backstep flow with the
same Mach number were also investigated.
Phase-averaged flow fields revealed the mechanism of
the acoustic radiation in detail. Two-dimensional large-scale
vortices composed of turbulent fine-scale vortices form in
the shear layer that separates from the upstream edge of the
cavity. When a large-scale vortex collides with the down-
stream wall, the low-pressure fluid in the vortex spreads
along the downstream wall. As a result, a local downward
velocity is induced by the local pressure gradient, and the
upstream fluid expands. In this way, an expansion wave is
radiated.
The large-scale vortices originate from the convective
disturbances that are developed in the shear layer. The am-
plification rate of the convective disturbances agrees with
that predicted by LSA. This means that the disturbances
grow due to KH instability, similar to the growth of those in
a laminar cavity flow. To clarify the mechanism for the gen-
eration of the initial convective disturbances, computations
for backward-facing step flows with an artificial acoustic
source were also performed. As the artificial acoustic waves
became more intense, the initial convective disturbances in
the shear layer became more intense while the amplification
rate was not changed. This means that the initial convective
disturbances in the shear layer are induced by the acoustic
waves and developed into the large-scale vortices due to the
KH instability independent of the acoustic waves and the
fine-scale vortices. This mechanism of the formation of the
large-scale vortices also holds for the cavity flow.
The present study has provided much deeper understand-
ing of the fluid-acoustic interactions in turbulent cavity
flows, which should lead to improved design of various in-
dustrial products and the suppression of the effects of such
interactions.
ACKNOWLEDGMENTS
This research has been supported by a global COE pro-
gram “Global Center of Excellence for Mechanical Systems
Innovation” of the University of Tokyo and by a research
program “Revolutionary Simulation Software” supported by
the Ministry of Education, Culture, Sports, Science and
Technology of Japan 共MEXT兲.
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