Content uploaded by Cristobal Gonzalez Diaz
Author content
All content in this area was uploaded by Cristobal Gonzalez Diaz on Sep 23, 2015
Content may be subject to copyright.
ICSV22, Florence (Italy) 12-16 July 2015 1
!"#$%&'"()*%#+!+"*(,)*-$*+"'*%(#,(!+(#.*+(
"!/'&0(1'&2(+#+3%044*&)'"(!+5(+#+3.!)!66* 6(
1!66%(
Cristobal Gonzalez Diaz and Pedro Cobo Parra
Instituto de Tecnologias Fisicas y de la Informacion (ITEFI),
Consejo Superior de Investigaciones Cientificas (CSIC),
Serrano, 144, 28006, Madrid (Spain)
e-mail: cristobal.g.diaz@csic.es
When airflow passes over an open cavity, due to vortex shedding at the upstream edge of the
cavity and the geometry or shape of the cavity, high-level aero-acoustic noise may be gener-
ated. When the incident acoustic waves produced by the airflow couples with the acoustic
cavity resonance frequencies intensive tones are generated in and around the cavity at reso-
nant discrete frequencies. Cavity noise is one of the most important airframe noises. Sound
radiation due to the acoustic cavity resonances in an open cavity could be achieved by
changing its geometry or shape. This paper presents simulated results, using Finite Element
Methods, of the influence in the resonance frequencies of an open cavity with non-
symmetric and non-parallel walls. Different parameters of the open cavity, such as the angle
between the walls of the open cavity will be considered to optimise its geometry.
78('9:;<=>?:@<9(
The interaction of flow with an open cavity is of certain interest both in aeroacoustics and noise
control [1, 2]. Airframe noises are generated by the interaction between the vortex streets in the
turbulent wake or between the vortices and the solid body edge(s). The phenomena are further com-
plicated by a possible aero-acoustic feedback loop or a possible Helmholtz fluid resonance. Thus,
the main consequence of the flow over the open cavity is the generation of cavity noise [3]. Cavity
noise is one of the most important airframe noises [2, 3]. When flow passes over a cavity or open-
ing, due to vortex shedding at the upstream edge of the cavity, intensive tone noises may be gener-
ated. Strong tonal oscillations occur in a feedback loop between the two edges of the cavity open-
ing. First, the vortices generated and shed from the upstream edge of the cavity convect down-
stream, impinge on the other edge, and produce acoustic waves. Then, as the acoustic waves propa-
gate either inside or outside of the cavity to the upstream edge, where receptivity of the wall jet
shear layer is high, a new instability wave (vortex) is stimulated and shed, a feedback cycle is thus
completed. The tone noise generated in this way may be categorized as fluid-dynamic oscillation
noise. Most high-speed flow (supersonic, transonic, or high subsonic flows) noises are generated by
the fluid dynamic feedback oscillation mode. However, at low flow speed [2], depending on the
geometry of the cavity, another type of tone noise due to fluid resonant oscillation within the cavity
may occur. The sound waves inside the cavity may be longitudinal or transverse depending on the
The 22nd International Congress on Sound and Vibration
ICSV22, Florence, Italy, 12-16 July 2015 2
aspect ratio of the cavity. For a deep cavity at low flow speed, both major tone noise-generating
mechanisms may coexist. If one or more walls of a cavity undergo displacement that is large
enough to exert feedback control on the shear layer perturbations during the cavity oscillation, the
excitation is defined as fluid elastic. The vibrating structure has the same function as the resonating
wave in a fluid resonant oscillation. A resonant type process is used to amplify the perturbed shear
layer flow. Wheel wells and Auxiliary Power Unit compartments during the airplane landing and
take off operations, the bogie section and the shallow cavity accommodating the pantograph of a
high speed train, or the open sunroofs and open windows of a car driven at high speed are sources
of high level cavity tones. Several researches have reported numerical methods to provide the ei-
genvalues (resonances frequencies) and eigenfunctions (normal modes) of open cavities without
flow. Koch [4] proposed to find the acoustic resonances of 2D and 3D rectangular open cavities by
solving Helmholtz wave equation by finite-element methods (FEM). Ortiz et al. [5] suggested the
use of the Image Source Model (ISM) to obtain the impulse response of an open cavity. This meth-
od gives the acoustic resonances of a 3D open cavity by a fast and efficient method, which models
the time response at any point in the medium as the convolution of the source waveform with the
impulse response of the cavity.
The acoustic resonances in the open cavity can be attenuated by noise control techniques. One
passive control technique consists of lining the inner walls of the cavity with absorbing material [6,
7]. Whilst porous materials provide wideband absorption, they are discouraged in presence of air-
flow. Since open cavity tones are usually driven by air flow (wheel wells in airplanes, shallow cavi-
ty above bogies in trains, sunroof in cars), alternative absorbers supporting air flowing might be
used. Micro-Perforated Panels (MPP) are recognized as the next generation absorbing materials.
Furthermore, they can be used in presence of air flow [8].
Other passive control technique to attenuate the acoustic resonances of an open cavity is to
change its geometry. In general, the design modification of an open cavity to decrease the sound it
produces enforces design requirements limits to the design freedom. Overall open cavity dimen-
sions are restricted due to operation of the plane (the limitation of runway loads defines number of
wheels and spacing, gear locations defined by lateral stability and rotation before lift-off, brake
cooling system), safety (tyre burst) and cost (weight, system complexity, maintenance). Therefore
few constrains should be taking into account such as the volume, length, width, depth... of an open
cavity where the landing gear should fit.
The major aim of this work is to report the influence in the resonance frequencies of an open
cavity with non-symmetric and non-parallel walls. The geometry of the open cavity used in this
study will be described in Section 2. The numerical procedure – Finite Elements Analysis (FEM)
used and the numerical results will be presented in Section 3 and Section 4 respectively. Finally, the
main conclusions will be disclosed in Section 5.
A8(&BC(<DC9("EF@:G(H(IC<JC:;G((
A cavity with non-parallel lateral walls and depth D, open at the topside is considered in this pa-
per. The floor and opening of the cavity are rectangular with dimensions (Lb, Wb) and (Lt, Wt) re-
spectively, see Figure 1. The inner dimensions of the open cavity with parallels and symmetric wall
are (W, L, D) = (53, 32, 38) cm, in such case Lb = Lt = L and Wb = Wt = W.
A three-dimensional Cartesian coordinate system (X, Y, Z) with the origin at the center of the
cavity is also shown in Figure 1. The front (w1) and rear (w3) walls subtend an angle α1 and α3 re-
spectively with the floor of the cavity, and the lateral walls (w1 and w4) form an angle α2 and α4
respectively with the floor of the cavity. A point source S is positioned at the front wall (w1) of the
cavity, with coordinates (xS, yS, zS) = (-0.195, -0.16+slope_w1, 0.1) m, where slope_w1 depends on
the inclination of the w1, in such a way the point source is always located on the front wall (w1).
The problem consists of calculating the sound signal at the receivers Rn, with coordinates
The 22nd International Congress on Sound and Vibration
ICSV22, Florence, Italy, 12-16 July 2015 3
(xRn, yRn, zRn). A baffle is situated around the open wall of the cavity in order to avoid back radiation
between the source and the measurement points. The sound signal have been simulated in four
points; R1; a lower point close to the floor and opposite wall containing the point source, R2; a mid-
dle point set close to the lateral wall (w4) and finally two points located outside the cavity, R3: a
point above the upper baffle opposite to the wall containing point source and R4: a point also above
the upper baffle close to the lateral wall (w4) farther away from the point source. Table 1 and Fig-
ure 2 summarises the coordinates of the points.
Figure 1: Open cavity with non-parallel walls.
Table 1. Coordinates of the source and receiver points.
Points
X(m)
Y(m)
Z(m)
Source
-0.195
-0.16+slope_w1
0.1
Receiver R1
0.135
0.11
-0.17
Receiver R2
0.215
0.11
0.13
Receiver R3
0.135
0.21
0.21
Receiver R4
0.335
0.16
0.21
Figure 2: Source and Receiver (R1, R2, R3 and R4) positions.
Lt
Wt
Lb
Wb
S
z
y
x
D
w1
w2
w3
w4
Baffle
α1
α2α4
α3
0.2
0.1
0
X
-0.1
-0.2
-0.1
0
0.1
Y
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Z
R1
Source
R2
R3
R4
The 22nd International Congress on Sound and Vibration
ICSV22, Florence, Italy, 12-16 July 2015 4
K8(+>JC;@?EL(D;<?C=>;C(H(,@9@:C(*LCJC9:M(!9ELGM@M(N,*4O(
The Finite Element Method (FEM) software used in this work is Comsol Multiphysics 5.0. The
numerical model is experimentally validated by comparing the measured natural frequencies [5] to
the simulated natural frequencies extracted from the peaks of the Frequency Response Functions
(FRF) when the open cavity has symmetric and parallel walls (Lb = Lt = L and Wb = Wt = W). Ta-
ble 2 shows the modal frequencies of the open cavity extracted from the peaks of FRF; experimen-
tally and simulated. The selected mesh size of the model results in an acceptable balance between
the frequency range of interest and the calculation time. The computational cost increases rapidly
with frequency. In the performed simulation a Monopole Point Source located in the “Source Point”
excites the cavity. A frequency resolution of 1 Hz and frequency range of 1-1000 Hz are chosen for
the frequency domain simulations. The significant peaks of this FRF were identified as the acoustic
resonance frequencies of the cavity.
Table 2. Modal frequencies of the symmetric and with parallel walls extracted from the peaks of the FRF
[5].
Mode
Fexp (Hz)
Fnum (Hz)
(0,0,0)
157
159
(1,0,0)
374.4
368
(0,1,0)
575
569
(2,0,0)
680.9
673
(2,1,0)
863.4
863
(0,2,0)
1103
1091
(2,2,0)
1290
1269
A Perfectly Matched Layer (PML) has been chosen to simulate the presence of an open bounda-
ry and the radiation of energy to the infinity, this layer absorbs all outgoing wave energy in fre-
quency-domain problems without any impedance mismatch—causing spurious reflections—at the
boundary [9].
P8()CM>L:M(
Assuming a Cartesian coordinates system with origin at the centre of the cavity (see Figure 1 for
further details), the source is located at (xs, ys, zs) = (-0.195, -0.16, 0.1). The impulse responses were
measured at 4 receiver points inside and outside the cavity (Table 1 and Figure 2) for six different
geometry configurations.
1.!Configuration 1: the cavity has symmetric and parallel walls (Lb = Lt = L and Wb = Wt = W),
and α1-4 = 0, that is, a rectangular cavity.
2.!Configuration 2: in this case the front (w1) and rear (w3) walls are parallel, however the angles
between both walls and the floor is not equal to zero, instead α2 = -α4 ≠ 0. The lateral walls (w2
and w4) are not parallels and Lb < Lt and both α1 and α3 are positive.
3.!Configuration 3: in this case all the angles between the walls and the floor are positives but not
equal (α1-4 > 0 and α1 ≠ α2 ≠ α3 ≠ α4) therefore Lb < Lt and Wb < Wt .
4.!Configuration 4: in this configurations the angle between the walls and the floor are positive but
one (α1-3 > 0 and α4 < 0). Also Lb < Lt and Wb < Wt .
5.!Configuration 5: in this case the angle between the walls and floor are equal and negative (α1 =
α2 = α3 = α4 < 0), hence Lb > Lt and Wb < Wt .
6.!Configuration 6: in this case the angle between the walls and floor are also equal but in this case
positive (α1 = α2 = α3 = α4 > 0), hence Lb < Lt and Wb < Wt .
The 22nd International Congress on Sound and Vibration
ICSV22, Florence, Italy, 12-16 July 2015 5
Figure 3 shows the frequency response function between the source and the receiver R1, which is
located close to the floor of the cavity and opposite wall w1, which contains the point source. In this
paper the absorption coefficient of the cavity walls has been assumed as “Hard Boundary Walls”,
therefore the simulations tend to show sharp peaks. It can be seen that by changing the geometry of
the cavity the resonance frequencies of the cavity are attenuated. It can also be observed that at high
frequencies, where the passive strategies are more efficient, the attenuation is higher. The resonance
frequencies are also shifted as the geometry is changed that is due to the fact that the volume of the
cavity is not constant. The influence is higher inside the cavity (Figure 3 and 4) that outside the cav-
ity (Figure 5 and 6), that is due to the fact that the point are located closer to the source, therefore
the influence is of the geometry configuration is higher.
Figure 3: Simulated FRF at the Receiver R1 for different geometry configurations. Blue and solid line =
configuration 1, green and dotted line = configuration 2, red and dashdot line = configuration 3, cyan and
dashed line = configuration 4, magenta and dotted faint line = configuration 5 and black and solid faint line =
configuration 6.
Figure 4: Simulated FRF at the Receiver R2 for different geometry configurations. Blue and solid line =
configuration 1, green and dotted line = configuration 2, red and dashdot line = configuration 3, cyan and
dashed line = configuration 4, magenta and dotted faint line = configuration 5 and black and solid faint line =
configuration 6.
Frequency (Hz)
100 200 300 400 500 600 700 800 900 1000
Relative Level (dB)
-30
-20
-10
0
10
20
30
40
50
1
2
3
4
5
6
Frequency (Hz)
100 200 300 400 500 600 700 800 900 1000
Relative Level (dB)
-30
-20
-10
0
10
20
30
40
50
1
2
3
4
5
6
The 22nd International Congress on Sound and Vibration
ICSV22, Florence, Italy, 12-16 July 2015 6
Figure 4 shows the frequency response function between the source and the receiver R2, which is
set in the upper middle part of the cavity close to the lateral wall (w4) further away from the source.
Figure 5: Simulated FRF at the Receiver R3 for different geometry configurations. Blue and solid line =
configuration 1, green and dotted line = configuration 2, red and dashdot line = configuration 3, cyan and
dashed line = configuration 4, magenta and dotted faint line = configuration 5 and black and solid faint line =
configuration 6.
Figure 6: Simulated FRF at the Receiver R4 for different geometry configurations. Blue and solid line =
configuration 1, green and dotted line = configuration 2, red and dashdot line = configuration 3, cyan and
dashed line = configuration 4, magenta and dotted faint line = configuration 5 and black and solid faint line =
configuration 6.
Figure 5 and 6 show the frequency response function between the source and the receiver R3,4,
which are both set outside the cavity above the baffle: R4 opposite to the wall (w3) containing the
point source and R4 close to the lateral wall (w4). As already pointed out previously, the influence
of changing the geometry of the cavity affects less in these points situated outside (Figure 5 and 6)
the cavity that to those situated inside (Figure 3 and 4) the cavity. These points are farther away
Frequency (Hz)
100 200 300 400 500 600 700 800 900 1000
Relative Level (dB)
-15
-10
-5
0
5
10
15
20
25
1
2
3
4
5
6
Frequency (Hz)
100 200 300 400 500 600 700 800 900 1000
Relative Level (dB)
-15
-10
-5
0
5
10
15
20
25
1
2
3
4
5
6
The 22nd International Congress on Sound and Vibration
ICSV22, Florence, Italy, 12-16 July 2015 7
from the source point. However the peaks are also decreased. As expected, the configuration 5
seems to be the less suitable among all the configurations presented in this paper. This is the con-
figuration with walls inclined toward the interior, like a pyramid with a truncate top. Also the con-
figuration 6 seems the best in term of attenuation of the resonance frequencies, however it might not
be the more practical in term of geometry.
Q8(%>JJE;G(E9=("<9?L>M@<9M(
The influence in the amplitude of the resonance frequencies of an open cavity with non-
symmetric and non-parallel walls has been considered in this paper.
The Frequency Response Function between four receiver points, two located inside and two out-
side the cavity have been presented for six different geometry configurations.
It has been shown than changing the geometry of the cavity modifies the resonance frequencies
of it. It has also been observed that at high frequencies, where the passive strategies are more effi-
cient, the attenuation is larger and the influence is higher inside the cavity that outside the cavity.
!?R9<SLC=TJC9:M(
The research leading to these results has received funding from the EU Seventh Framework Pro-
gramme Marie Curie IEF project "ACOCTIA" (FP7-PEOPLE-2011-IEF, Grant Agreement No.
301287).
)*,*)*+"*%(
1.!H.E. Plumblee, J.S. Gibson, L.W. Lassiter, A theoretical and experimental investigation of the
acoustic response of cavities in aerodynamic flow, in: Technical Report No. WADD-TR-62-
75, Air Force Rep., U.S., 1962.
2.!J.E. Rossiter, Wind-tunnel experiments on the flow over rectangular cavities at subsonic and
transonic speeds, Aeronautical Research Council Reports and Memoranda, Technical report
3438, (1964).
3.!X. Gloerfelt, Cavity Noise, in: VKI Lectures: Aerodynamic noise from wall-bounded flows,
Von Karman Institute, 2009.
4.!W. Koch, Acoustic Resonances in Rectangular Open Cavities, AIAA Journal, 43 (2005)
2342-2349.
5.!S. Ortiz, C.L. Plenier, P. Cobo, Efficient modeling and experimental validation of acoustic
resonances in three-dimensional rectangular open cavities, Applied Acoustics, 74 (2013) 949-
957.
6.!S. Ortiz, C. Gonzalez Diaz, P. Cobo, F. Montero de Espinosa, Attenuating open cavity tones
by lining its walls with microperforated panels, Noise Control Engineering Journal, 62 (2014)
145-151.
7.!C. Gonzales Diaz, S. Ortiz, P. Cobo, Attenuation of acoustic resonances in an inclined open
cavity using Micro Perforated Panels, in: 43rd International Congress on Noise Control
Engineering, The Australian Acoustical Society, PO Box 1843, Toowong DC QLD 4066,
AUSTRALIA, Melbourne, Australia, 2014.
8.!P. Cobo, H. Ruiz, J. Alvarez, Double-Layer Microperforated Panel/Porous Absorber as Liner
for Anechoic Closing of the Test Section in Wind Tunnels, Acta Acustica united with
Acustica, 96 (2010) 914-922.
The 22nd International Congress on Sound and Vibration
ICSV22, Florence, Italy, 12-16 July 2015 8
9.!J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves,
Journal of Computational Physics, 114 (1994) 185-200.
