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Noise reduction using a quarter wave tube with different
orifice geometries
Carl Q. Howarda,∗, Richard A. Craiga
aSchool of Mechanical Engineering, The University of Adelaide, South Australia, 5005
Abstract
It is well known that the acoustic performance of silencing elements decreases
with an increase in exhaust gas flow. Tests were conducted on three orifice
geometries of side-branches on an adaptive quarter-wave tube to determine
which was the least compromised by the high-speed exhaust gas passing over
the side-branch. The side-branch geometries that were tested were a sharp
edge, a backward inclined branch, and a bell mouth. The experimental
results show that the side-branch with a bell-mouth geometry resulted in the
greatest noise reduction by an adaptive quarter-wave tube.
Keywords: acoustic, adaptive passive, semi-active, silencer, muffler,
quarter wave tube
1. Introduction
Quarter-wave tubes are a reactive acoustic device used to attenuate tonal
noise at a fundamental and odd-harmonic frequencies. The quarter-wave
tube is attached to the main duct with a side-branch. When gas flows past
the side-branch at low speeds (less than Mach 0.1), it has been found that the
∗Corresponding authorEmail address: [email protected] (Carl Q. Howard)
Preprint submitted to Applied Acoustics August 3, 2013
acoustic attenuation performance is not degraded significantly [1]. However
when flow speeds increase above Mach > 0.1, then the acoustic attenuation
caused by the quarter-wave tube can be degraded, because the resistive part
of the side branch resonator impedance will usually increase [2].
In the work presented here, the noise reduction of an Adaptive Quarter
Wave Tube (AQWT) with three configurations of side-branch geometries is
described, namely a 90◦ branch with square edges, a side branch that is
backwards inclined at 45◦ to the flow in the main exhaust duct, and a side
branch with a bell-mouth.
2. Previous Work
Lambert [3] showed theoretically that the insertion loss of a side-branch
resonator attached to a main duct is very sensitive to the Mach flow num-
ber, especially at frequencies near resonance. In a companion paper [4], he
experimentally showed that for flows greater than Mach > 0.1, the insertion
loss “for all practical purposes had been destroyed by the flow”.
Knotts and Selamet [5] conducted experimental investigations of several
geometries of the neck of a side-branch to determine which configuration
was least likely to generate unwanted pure tone whistle noise. The shapes
included those with square edges, ramps, bevelled edges, and curved (radius)
edges. Their experimental tests indicated that beveled and radius edges was
effective at reducing unwanted tonal noise generation. Their work focused on
the minimization of flow induced noise, whereas the focus of the work in this
paper is providing the greatest noise reduction of tonal noise by an adaptive
quarter wave tube.
2
Most of the previous work on suppression of flow noise from cavities has
examined shallow cavities where the cavity length to width ratio (L/d) is less
than one. Rockwell and Naudascher [6] provide an extensive review on the
subject of noise generated by flow over cavities.
Singhal [7] examined the influence of air-flow past a quarter wave tube,
that was backwards and forward inclined at 45◦, on the noise generation.
He found that when the device was installed so that the quarter-wave tube
was forwards inclined to the flow, it whistled. He also examined the non-
linear coupling between cavity modes for a quarter-wave tube attached at
90◦ to the main duct. The noise spectrum comprised combinations of sum
and differences of the cavity modes.
Anderson [8] examined the effect of air flow past a Helmholtz resonator
and found that the effect of increasing air flow was to increase the resonance
frequency of the resonator, which implies that the reactive impedance of the
neck decreased. Ingard and Ising [9] found that the effect of high inten-
sity sound levels (above 110dB) impinging on an orifice caused the resistive
impedance to increase. The work reported here is similar to that by Knotts
and Selamet [5], however the emphasis is on examination of the noise reduc-
tion caused by a quarter wave tube with various side-branch geometries in a
realistic application.
There is a large body of research literature on the acoustic performance of
perforated plates and single small diameter (less than 5mm) circular orifices,
which indicates there is a linear relationship between flow speed and the re-
sistance (real part of the impedance). However, references that describe the
variation in throat impedance with flow speed for large diameters (> 50mm)
3
are not available. This is consistent with comments in the research litera-
ture. Holmberg and Karlsson [10] commented that, “In side branch orifices
the general trend for a grazing flow configuration is an increase in the re-
sistance and a decrease in the reactance with an increasing Mach number.
Analytical modelling of single orifices have been attempted, but it have been
shown that the correlation with experimental data is generally unsatisfac-
tory.” These comments are also consistent with Karlsson and Abom [11],
“Attempts have been made to describe the acoustic impedance of single ori-
fices with analytical models. However, Jing et al. and Peat et al. have shown
that the correlation to measured data is generally unsatisfactory.”
Dequand et al. [12] investigated flow induced acoustic resonances of side-
branches. In their experimental and theoretical studies they were concerned
with two side-branches that were coaxially aligned. They considered side-
branches that had junctions to the main duct with sharp edges and rounded
edges. From their experimental testing they found that there was a significant
variation in the sound pressure response both in amplitude, and flow velocity
depending on whether the cross-junction geometry had a sharp-edged or bell-
mouth opening, which is consistent with our experimental results presented
in this paper.
3. Theory
A Quarter-Wave Tube (QWT) side-branch resonator has an (uncoupled)
resonance frequency fQWT given by
fQWT =c
4Leff
(1)
4
where c is the speed of sound, and Leff is the effective length of the quarter-
wave tube that includes end effects. The speed of sound in air is given by
[13]
c =
√
√
√
√γRT
M(2)
where γ = 1.4 is the ratio of specific heat that is applicable both for di-
atomic molecules and air, R = 8.314 J mol−1K−1 is the molar Universal gas
constant, M = 0.02857 kg mol−1 is the average molar mass, and T is the
gas temperature in Kelvin. These values will be slightly different for exhaust
gases that typically have high concentrations of gaseous carbon dioxide and
water.
These quarter wave tube resonators provide acoustic attenuation at odd
multiples (i.e. 1×, 3×, 5×, . . .) of the fundamental (1×) resonance frequency,
when they are attached to an acoustic duct.
For an engine rotating at constant speed and under constant load, the ex-
citation frequency will remain constant and thus the required length (Leff) of
a QWT remains constant. However, if the engine conditions were to change,
such as by altering the load on the engine, the temperature of the exhaust gas
(T ) changes, which alters the speed of sound in the exhaust gas as indicated
in Eq. (2). Therefore, a fixed length quarter-wave tube will only provide
attenuation for a fixed engine speed and exhaust gas temperature. Whereas
the adaptive quarter-wave tube presented here can be tuned to attenuate
tonal noise over a range of engine speeds and exhaust gas temperatures.
For a four-stroke reciprocating engine, the cylinder firing frequency fcylinder
occurs at half the crankshaft speed multiplied by the number of cylinders in
5
the engine, hence
fcylinder =RPM
60×
cylinders
2(3)
For example, in the case of a V8 engine, rotating at 1500rpm the cylinder
firing frequency is 100Hz.
4. Numerical Predictions
There are several metrics to describe the acoustic performance of muf-
flers that include Insertion Loss (IL), Transmission Loss (TL), and Noise
Reduction (NR) [14].
The transmission loss is the (10 log10) ratio of the incident to transmitted
acoustic power on the muffler. This is simple to calculate using numerical
simulations, but extremely difficult to measure in practice [15, 16], as it
requires the measurement of the incident sound power on the muffler in a
system with flowing high-temperature (500◦C) exhaust gas that is turbulent.
The noise reduction is the difference in the upstream and downstream
sound pressure levels across the exhaust muffler. This is relatively easy to
measure in practice, but difficult to estimate theoretically as the source and
termination impedances must be known. Several methods have been pro-
posed to measure the source impedance of an engine [15, 16, 17, 18]. Munjal
[18] noted that Prasad and Crocker [19] found that the source impedance
of the multi-cylinder inline engine they investigated could be approximated
as an anechoic termination. The method suggested by Boonen and Sas [17],
which involves the use of two microphones and varying the impedance of the
exhaust system, was attempted in this work but gave unsatisfactory results.
6
In their work they found that the averaged source impedance at high frequen-
cies could be approximated as anechoic. In the modelling work conducted
here, the source impedance was simulated as an anechoic termination.
The insertion loss is the reduction (in decibels) in sound power transmit-
ted through a duct compared to that transmitted with no muffler in place.
The insertion loss can be measured by using a single microphone located
outside the exhaust system, with and without the muffler installed. This
measurement technique was attempted however it was found that the acous-
tic enclosure around the diesel engine did not provide adequate isolation
and the measurements of exhaust noise were contaminated by noise radiat-
ing from the engine and hence the insertion loss results were inaccurate. In
order to make theoretical predictions of insertion loss, the source and termi-
nation impedance must be known, which as described above, is very difficult
to measure in practice.
Numerical predictions of the expected transmission loss of a quarter-wave
tube were made using a lumped-parameter model and Finite Element Anal-
ysis (FEA). One benefit of using FEA is that the transmission loss can be
predicted for the three side-branch geometries considered here.
4.1. Lumped Parameter Model
A lumped parameter model was created to aid in the verification of the
predictions made with finite element analysis. The insertion loss caused by
the installation of a quarter-wave-tube on an infinite duct is given by [20]
IL = 20 log10
∣
∣
∣
∣
∣
∣
1 +Zd
Zr
∣
∣
∣
∣
∣
∣
(4)
7
where the impedance of the infinite duct Zd is given by
Zd =ρc
At
(5)
and the impedance of the quarter wave tube resonator Zr is given by [21]
Zr = Zt + Zc (6)
which comprises the impedance of the throat Zt and the impedance of the
resonator cavity Zc. The impedance of the resonator cavity is given by
Zc = −jc
A2
cot(kLc) (7)
The impedance of the throat Zt is a function of the grazing flow speed and
has been the subject of considerable research. However an equation for the
impedance of a circular orifice with grazing flow is problematic. There is
considerable research reported for small diameter holes, such as in perforated
plates, however there is no expression available for large diameter holes. It
has been found experimentally that the acoustic response varies considerably
for variations in hole diameter, side-branch geometry, and ratio of cross-
sectional areas between the main duct and side-branch resonator. For the
simplest case, where there is no flow, it is assumed that there is no additional
impedance at the throat (Zt = 0). This enables comparison of the theoretical
predictions with the finite element analysis results.
Note that for infinite ducts (i.e. anechoic terminations) the insertion loss
and transmission loss are identical.
4.2. Finite Element Analyses
Finite element models of a circular quarter-wave tube were constructed
using the software Ansys (version 14). Parametric models were created of a
8
circular duct of infinite length, simulated as a finite length duct with anechoic
terminations, with a circular side-branch and quarter wave tube connected to
the main duct. Three side-branch geometries were analyzed: 1) a 90◦ branch,
2) backward inclined at 45◦, and 3) a bell-mouth, as shown in Figure 1,
Figure 2, and Figure 3, respectively.
Figure 1: Finite element model of a circular duct of finite length with anechoic terminations
with a circular side-branch connected at 90◦ with sharp edges.
The transmission loss was calculated using the lumped-parameter model
and finite element analyses, where the diameter of the main duct was D1 =
0.115m, and the side-branch quarter wave tube diameter was D2 = 0.115m,
the speed of sound was (calculated using Eq. (2) c = 538m/s, and the den-
9
Figure 2: Finite element model of circular duct of finite length with anechoic terminations
with a circular side-branch connected to the main exhaust duct backwards inclined at 45◦
with sharp edges.
sity of the gas was ρ = 0.438kg/m3. For these models it was assumed that
there was no damping, and no gas flow. Figure 4 shows the predicted trans-
mission loss versus the normalized frequency, where the excitation frequency
was normalized by the frequency at which the maximum transmission loss
occurred. Note that reference books (e.g. Ref. [1]) will sometimes plot the-
oretical transmission loss versus normalized frequency where the frequency
axis is kL/π, where k is the wavenumber and L is the length of the quarter-
wave-tube. For these analyses where the quarter-wave-tube was backwards
inclined at 45◦, it is difficult to predict the effective length of the quarter wave
tube. This is because the length of tube at the upstream edge of the QWT
is shorter than the length of tube at the downstream edge. For this reason,
the frequency axis was normalized by the frequency at which the maximum
transmission loss occurs. Although Figure 4 shows several results, it can be
10
Figure 3: Finite element model of a circular duct of finite length with anechoic terminations
with a circular side-branch connected to the main exhaust duct at 90◦ with a bell mouth
geometry.
11
seen that the results for the lumped parameter model overlay the predictions
using FEA, and that there is no appreciable change in the theoretical trans-
mission loss (for no flow) when the side-branch geometry is changed from a
square / sharp opening at 90◦ to the main duct, to a bell-mouth, or when
the quarter-wave tube is attached to the main duct at 45◦. However, it will
be shown in the results from the experimental testing with exhaust gas flow
that there is considerable difference in the measured noise reduction due to
the geometry of the side-branch.
0
10
20
30
40
50
60
70
0.9 0.95 1 1.05 1.1
Tran
smis
sio
n L
oss
[d
B]
f/f0 Normalised Frequency
Theoretical Transmission Loss for a QWTNo Flow, No Damping
Ansys: D2/D1=1
Ansys: (2*D2)/(2*D1)=1 Bell
Ansys: D2/D1=1 Bell
Ansys: D2/D1=1 45 deg
Matlab: D2/D1=1
Figure 4: Theoretical predictions of transmission loss of a quarter-wave tube using lumped-
parameter and Finite Element Analyses.
5. Experimental Apparatus
A test platform was built to examine the noise reduction that could be
achieved using an adaptive quarter-wave tube attached to a large diesel
12
PassiveMuffler
Quarter Wave TubeMic in End of Piston
0.0
0m
0.3
75 M
ic 1
0.6
75m
Mic
2
1.3
75m
Mic
3
1.8
40m
Mic
4
2.1
80m
Mic
5
Exhaustfrom Engine
Figure 5: Location of microphones in exhaust duct.
engine, as described in Ref. [22]. A 16-litre, V8 diesel engine (Mercedes
Benz OM 502 LA Power Drive Unit, maximum power 420kW (571hp) at
1800 RPM, continuous rated power 350kW (476 hp) at 1800rpm), was in-
stalled inside an acoustic enclosure. The engine was loaded using a water-
brake dynamometer (Taylor model TD-3100). An adaptive quarter-wave
tube was attached to the main exhaust duct, as shown in Figure 5. The
adaptive quarter-wave tube was tuned by adjusting the position of the pis-
ton in the bore of the quarter-wave tube until the phase angle between a
microphone in the end of the piston and a microphone at the side-branch
(Mic 4) was 90◦. A sliding-Goertzel algorithm was used to determine the
phase angle [23]. The sound in the exhaust was measured using PCB 106B
microphones housed within custom-made water-cooled jackets and attached
at locations shown in Figure 5. The faces of the microphones were flush
mounted with the internal diameter of the exhaust pipe.
13
Tests were conducted to measure the transmission loss of the system with
4 configurations of side-branch geometries as shown in Figure 6: a straight
pipe (i.e. without the adaptive quarter-wave tube attached to the main
exhaust duct), 90◦ with sharp edges, backwards inclined at 45◦, and 90◦ with
a bell-mouth opening.
The diesel engine was operated at various speeds and loads applied by
the water-brake dynamometer. An adaptive quarter-wave tube was used to
attenuate the noise at the cylinder firing frequency. The adaptive quarter-
wave tube comprised a piston in a tube where the position of the piston was
altered by a linear actuator, and hence the effective acoustic path length could
be altered [22]. When the linear actuator was fully extended the quarter-
wave tube had the minimum acoustic path length, and hence can attenuate
a tone at a high frequency, and conversely when the linear actuator was fully
retracted the quarter-wave tube had the maximum acoustic path length, and
hence can attenuate a tone at a low frequency.
The noise reduction was measured for each test by comparison of the
noise level when the adaptive quarter-wave tube was tuned to the high-
est frequency (shortest acoustic path length) and when it was tuned to the
cylinder firing frequency. Previous tests [22] showed that when the AQWT
was fully extended (shortest acoustic path length) the sound pressure levels
in the duct were the same as when a straight section of exhaust pipe without
a side-branch was installed.
14
Figure 6: Four configurations of side-branch geometries (from top to bottom): straight
pipe, 90◦ with sharp edges, backwards inclined at 45◦, and 90◦ with a bell-mouth opening.
15
6. Results
The diesel engine and water-brake dynamometer was operated at the con-
ditions listed in Table 1. Cases 1 and 2 were conducted with no-load applied
to the engine, and Cases 3 to 6 had load applied to the engine such that
the exhaust gas temperature at the side-branch was approximately 450◦C.
The flow speed was measured by using a pitot tube and the exhaust gas
temperature slightly upstream of the side-branch.
Case 1 2 3 4 5 6
Speed [rpm] 1800 1500 1800 1700 1600 1500
Load [kW] 0 0 150 140 140 120
Flow speed [m/s] 35.3 24.8 99.0 92.1 87.1 75.7
Table 1: Engine operating conditions for the experimental testing.
The noise reduction at the cylinder firing frequency was measured at the
location of the side-branch (Mic 4) and at a microphone further downstream
(Mic 5), as shown in Figure 7 and Figure 8, respectively.
These results clearly show that for the four cases (3-6) where the engine
was loaded, that the side-branch with the bell-mouth geometry provided sig-
nificantly better noise reduction than the sharp-edge and 45◦ side-branches.
The results for the tests with the 45◦ and 90◦ side-branch when the engine
was loaded shows a general trend of decreasing noise reduction as the engine
speed increased from 1500rpm to 1800rpm. This trend did not occur for the
tests with the bell-mouth side-branch.
These tests were repeated on three separate occasions to confirm these
16
0 10 20 30 40
1500rpm120kW
1600rpm140kW
1700rpm140kW
1800rpm150kW
1500rpmno load
1800rpmno load
Noise Reduction [dB]
Noise Reduction at Mic 4 (T-Branch)
Bell Mouth 45 Degree Branch 90 Degree Branch
Figure 7: Noise Reduction measured at the side-branch (Mic 4) for various engine speeds
and loads.
17
0 10 20 30 40
1500rpm120kW
1600rpm140kW
1700rpm140kW
1800rpm150kW
1500rpmno load
1800rpmno load
Noise Reduction [dB]
Noise Reduction at Mic 5 (Last Mic)
Bell Mouth 45 Degree Branch 90 Degree Branch
Figure 8: Noise Reduction measured downstream of the side-branch (Mic 5) for various
engine speeds and loads.
18
130
135
140
145
150
155
160
165
170
0 1 2
SP
L at
4th
Ord
er[d
B r
e 20
µµ µµPa]
Distance [m]
SPL vs Distance with 45 Branch at 1.8mEGT=450C, Fully Extended
1500rpm 120kW45deg1600rpm 140kW45deg
1700rpm 140kW45deg1800rpm 150kW45deg
1500rpm no load142C 45deg1800rpm no load171C 45deg
Figure 9: Sound pressure level at the cylinder firing order versus distance for various engine
speeds, when the backwards inclined 45◦ side-branch was installed at the 1.8m position.
19
findings and the same conclusions were obtained. Further testing involved
placing a microphone after the passive muffler shown in Figure 5, and it was
found that measured noise reductions were nearly the same as measured at
Mic 4 and Mic 5, which provided confidence that the AQWT attenuates the
noise at the cylinder firing frequency downstream from the side-branch, as
expected.
Figure 9 shows the sound pressure level measured axially along the ex-
haust pipe when the AQWT was attached to the side-branch with a back-
wards inclination of 45◦, and the AQWT was fully extended (shortest acous-
tic path length). As shown in Figure 5, the location of the side-branch is
at approximately 1.8m. It can be seen that for the cases 3 to 6, where the
engine was loaded, the AQWT is not located at a minima of the sound pres-
sure level. For Case 1, 1800rpm no load, the AQWT appears to be located
near a sound pressure level minima, and hence it could be expected that the
effectiveness of the AQWT could be compromised.
These findings were (pleasantly) unexpected, as the results from the finite
element analyses indicated that the transmission loss does not alter with
the geometry of the side-branch, when there is no gas flow, whereas the
experimental results indicate that the noise reduction is influenced by the
geometry of the side-branch.
Figure 10 and Figure 11 show the spectrum of the sound pressure level
at the location of the side-branch (Mic 4) and at the last microphone in the
exhaust pipe (Mic 5) when the engine was operating at case 5 (1600rpm,
with 140kW of load), with the AQWT fully extended (shortest acoustic path
length) and then tuned to the cylinder firing frequency (107Hz), when the
20
side-branch with the bell-mouth was installed. The results show that when
the Adaptive Quarter Wave Tube was tuned to attenuate noise at the cylinder
firing frequency (107Hz), the noise in the exhaust pipe was reduced by around
25dB at Mic 4 and around 29dB at Mic 5. The spectral results show that
the sound pressure level when the AQWT was tuned to the cylinder firing
frequency (dashed gray line) does not exceed the levels when the AQWT had
the shortest acoustic path length (solid black line), and hence there was no
occurrence of ‘self-noise’ generation as the gas flowed past the side-branch
opening.
21
107, 158
107, 133
100
110
120
130
140
150
160
170
0 100 200 300 400
So
un
d P
ress
ure
Lev
el
[dB
rm
s re
20 µµ µµ
Pa/
sqrt
(Hz)
]
Frequency [Hz]
SPL at Mic 4 (Side Branch) 1600rpm 140kW Bell Mouth
M4 Extended
M4 Tuned
Figure 10: Spectrum of sound pressure level at the location of the side-branch (Mic 4)
when the engine was operating at 1600rpm with 140kW of load with the bell mouth side-
branch installed. The results show that at the cylinder firing frequency of 107Hz, the
AQWT reduced the SPL from 158dB to 133dB.
22
107, 160
107, 131
100
110
120
130
140
150
160
170
0 100 200 300 400
So
un
d P
ress
ure
Lev
el
[dB
rm
s re
20 µµ µµ
Pa/
sqrt
(Hz)
]
Frequency [Hz]
SPL at Mic 5 (Side Branch) 1600rpm 140kW Bell Mouth
M5 Extended
M5 Tuned
Figure 11: Spectrum of sound pressure level downstream of the side-branch (Mic 5) when
the engine was operating at 1600rpm with 140kW of load with the bell mouth side-branch
installed. The results show that at the cylinder firing frequency of 107Hz, the AQWT
reduced the SPL from 160dB to 131dB.
23
7. Conclusions
Experimental testing was conducted using three configurations of side-
branch geometries with an adaptive quarter-wave tube attached to the ex-
haust of a large diesel engine that was loaded by a water-brake dynamometer.
It was shown that the side-branch with the bell-mouth opening provided the
greatest noise reduction and hence was the least affected by the gas flow
past the side-branch compared to the other two side-branch geometries. It
is known that the resistive part of the acoustic impedance of an orifice in-
creases with increasing gas flow velocity. Therefore, one might conclude that
the bell-mouth geometry has lower resistive acoustic impedance compared to
the other geometries. Intuitively this make sense, as the wide opening of the
bell-mouth would have less interaction of oscillating gas particles with the
walls of the side-branch.
The experimental results of the testing of the adaptive quarter-wave tube
and a side-branch with a bell-mouth geometry showed that noise reductions
of up to 35dB at the cylinder firing frequency could be achieved. There was
no evidence that vortex shedding induced self-noise was generated.
24
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