19th AIAA/CEAS Aeroacoustics Conference
May 27-29, 2013, Berlin, Germany
AIAA 2013-2081
Quantifying crackle-inducing acoustic shock-structures
emitted by a fully-expanded Mach 3 jet
Woutijn J. Baars∗ and Charles E. Tinney†
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Aerospace Eng. and Eng. Mechanics, The University of Texas at Austin, Austin, TX 78712, USA
The intense acoustic field radiating from an experimental Mach 3 jet is studied through
a survey of the acoustic pressure waveforms over a large spatial domain in the (x, r)-plane of
the jet. Sawtooth-like structures in the waveforms are not the consequence of cumulative
nonlinear acoustic distortions. This was concluded previously from a scaling model based
on an effective Gol’dberg number and the spatial topographies of metrics that quantify
steepened waveforms.1 This implies that acoustic shock-structures are generated by local
mechanisms in, or in close vicinity to, the jet plume. Furthermore, they give rise to the
crackle noise component often observed in the downstream regions of supersonic jets.
The current work aims to quantify crackle in a temporal and spectral fashion. A detection algorithm isolates the shock-structures in the temporal waveforms. Ensemble-averages
of the identified waveform sections are employed to gain an understanding of the crackling
structures. Moreover, PDF’s of the temporal intermittence of these shocks reveal modal
trends and show evidence that crackling shock-structures are present in groups of multiple
shocks. A spectral metric is considered by using wavelet-based time-frequency analyses.
The increase in sound energy is computed by considering the global pressure spectra of the
waveforms and the ones that represent the spectral behavior during instances of crackle.
This energy-based metric is postulated to be an appropriate metric for the level of crackle.
I.
I.A.
Introduction
Supersonic jet noise
The high-intensity noise generated by a shock-free and unheated Mach 3 jet is investigated experimentally.
Unlike subsonic jets, the noise produced by supersonic jets can be categorized into four distinct mechanisms:2, 3 turbulent mixing noise, broadband shock-associated noise, screech and transonic resonance. The
latter three occur when shock waves and expansion fans are present. Restricting one’s attention to turbulent mixing noise produced by shock-free, unheated supersonic jet flows, the more relevant assemblage of
literature reduces to the laboratory-scale jet studies of McLaughlin et al. (1975),4 Tanna & Dean (1975),5
Papamoschou et al. (2010),6 McLaughlin et al. (2010),7 Baars et al. (2011),8 the full-scale flight tests of
Morfey & Howell (1981),9 the laboratory- and full-scale study by Murray & Jansen (2012),10 or the numerical
studies of Howell & Morfey (1987),11 Morris (1977)12 and Seiner et al. (1994).13 Where the development of
robust acoustic analogies is concerned, one may wish to review the work of Morris & Farassat (2002),14 Tam
et al. (2008),15 Tam (2009)16 and Morris (2009).17 Aside from conventional spectral analyses, Laufer et
al. (1976),18 Gallagher & McLaughlin (1981),19 Petitjean et al. (2006)20 and Veltin et al. (2011)21 studied
temporal characteristics of acoustic pressure waveforms in a laboratory (range-restricted) environment, while
Gee et al. (2008)22 focused on the nonlinear propagation of sound from a full-scale static jet engine.
As for subsonic jets, the dominant sound produced by shock-free supersonic jets (considered in this
study) is caused by turbulent mixing noise.23 It has been postulated that turbulent mixing noise consists
of two components. The first and most distinguishable component is generated by the (often supersonic)
convective motion of large turbulent structures that pass along the potential core region of the flow. This
is the source of Mach wave radiation that is observed in the Mach cone and is extensively discussed in the
literature.24, 25, 26, 4, 18, 13, 27, 15, 16 The second component is associated with the fine-scale turbulence within
the shear layer.28
∗ Graduate
† Assistant
Research Assistant, AIAA Student Member.
Professor, AIAA Senior Member. http://www.ae.utexas.edu/facultysites/tinney/
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When considering the radiation of acoustic waveforms from the non-compact jet noise source, nonlinear
acoustic waveform distortions are considered a prerequisite to understanding the process of propagation.
Steepened, shock containing waveforms encountered in acoustic fields of jets are often ascribed to nonlinear
propagation effects without sufficient quantitative evidence supporting this claim. Analogously, mismatches
in acoustic power spectra between far-field signals and predictions, formed by rescaling a closer spectrum
(using 1/r2 dependence) and applying atmospheric absorption, are frequently attributed to nonlinear phenomena (figure 1).
prediction
p(t)
Gpp (f )
steepened waveform
observation
f
Gpp (f )
t
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nozzle
f
jet axis
Figure 1. Schematic of observations that are associated with nonlinear acoustic waveform distortions (not to scale).
The aforementioned observations, such as steepened waveforms and spectral mismatches, are commonly
made in range-restricted chambers since experimental jet noise studies are mostly performed in laboratoryscale environments. This forms an important constraint on acoustic propagation distances. Consequently,
changes in the waveforms caused by cumulative nonlinear distortions are small. An experimental campaign
on an unheated, perfectly expanded, Mach 3 jet in a laboratory-scale environment has been central to the
current research. Previously, it was concluded that sawtooth-like structures in the waveforms are not the
consequence of cumulative nonlinear acoustic distortions.1 This was based on the findings from a scaling
model based on an effective Gol’dberg number and the spatial topographies of metrics that quantify steepened
waveforms. The implication is that acoustic shock-structures are generated by source mechanisms in, or in
close vicinity to, the jet plume. Furthermore, they give rise to the crackle noise component often observed
in the downstream regions of supersonic jets. For the readers perusal, the subject of jet noise crackle is
reviewed in the next section.
I.B.
Jet noise crackle
A distinct component of noise that can arise in supersonic jet flows is known as crackle and is considered
to be part of the turbulent mixing noise category. Although crackle is perceived as the dominant and most
annoying component of supersonic jet noise, it is surprising that only a limited number of studies have focused
on it. The most extensive work dates from the 1970’s, when Ffowcs Williams et al. (1975)29 investigated
crackle characteristics emitting from a static full-scale engine. In that study, crackle was formulated as
“spasmodic bursts of a rasping fricative sound” or as “the sound of an electric arc welder or of a badly
connected loud speaker”. Earlier efforts to identify the source of crackle were motivated by an interest in
developing state-of-the-art military jet engines capable of complying with the norms for jet noise. Conversely,
there is currently no universal consensus on its presence, exact cause and pattern.
I.B.1.
Accomplishments
Acoustic waveforms corresponding to a crackling jet are known to comprise sawtooth-like structures (sharp
compressions followed by more gradual expansions);29 these are further denoted as shock-structures. Figure 2
presents a sample waveform enclosing shock-type structures. Ffowcs Williams et al. (1975)29 noticed that
these shock-structures appear often in groups (figure 2 shows a group of two). On the contrary, Krothapalli
et al. (2000)30 did not find evidence of this.
When examining waveforms that contain shock-structures, a major difficulty arises from the fact that
these waveforms are similar in shape to those evolved from cumulative nonlinear wave steepening. An
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0.8
p [kP a]
0.6
shock-structures
0.4
0.2
0
−0.2
−0.4
0
1
2
3
4
5
6
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t [ms]
Figure 2. Temporal pressure waveform of far-field jet noise indicating intermittent shock-structures that are responsible
for crackle.
incorrect classification of the shock-structures, as either being crackle or the result of nonlinear waveform
distortion, can result in erroneous conclusions about source locations and source mechanisms.1 In fact, while
pressure waveforms like the ones in figure 2 can be the result of cumulative nonlinear distortions (in the
case of full-scale jet noise for example31 ), it was shown previously that the presence of cumulative effects in
range-restricted laboratories is practically impossible. And so, the observed shock-structures in laboratoryscale data sets such as the one presented here, are the consequence of local nonlinear phenomena and can
be classified as being crackle.
Several -mostly hypothetical- explanations regarding the source mechanisms of crackle are found in the
literature. Ffowcs Williams et al. (1975)29 states that the signatures of crackle are formed within, or in
the near vicinity to, the turbulent jet flow in an almost instantaneous fashion. And so, crackle is thus not
formed with distance during noise propagation (crackle is not a cumulative nonlinear effect). This statement
of Ffowcs Williams et al. (1975)29 seems to be adopted by most researchers and the current work supports
this hypothesis. Their source model can be summarized as follows. Pockets of supersonic air randomly shed
from the jet’s supersonic shear layer and decelerate in the stagnant air. During deceleration, convective
wave pile-up occurs within these pockets and radiates shock-structures efficiently to the far-field. A second
plausible mechanism was presented by Krothapalli et al. (2000).30 Since the pressure signatures closely
resemble acoustic features obtained far from an explosion source, they suggested that micro explosions of
cold air entrained by a hot jet flow can initiate crackle. Others believe that reconnection of the coherent
vortex structures in the jet can induce crackle.32, 33 Some believe that the coalescence of Mach waves in the
near-field cause excessive steepening of the wave fronts.34
Until now, only a few metrics have been applied to pressure-time series to study crackle quantitatively.
The current lack of consensus on reliable quantification methods is believed to play a role in the limited
number of studies on this topic. The skewness of the pressure (third-order moment of the PDF, denoted
as S(p)35 ) as a criterion for crackle was first used by Ffowcs Williams et al. (1975).29 It was suggested
that the crackling sound becomes distinct if S(p) > 0.4 and that the jet is crackle free when S(p) < 0.3. A
shortcoming in this criterion is that the rise times of the compressive parts of the waveform are not taken
into account. The time rates of change of these compressions are important since that determines how strong
the compressive shocks are, and so, the perception of crackle may be better quantified with statistics of the
temporal pressure derivative ṗ, as was noticed by McInerny (1996).36 This was also adopted by Gee et al.
(2007),31 where they showed how a measure of the skewness is not sufficient for concluding the perception of
crackle. A number of studies have now explored skewness values (S(p) and S(ṗ)) as metrics for quantifying
crackle,29, 36, 30, 37, 20, 31, 38, 39 as well as the kurtosis (fourth-order moment of the PDF, denoted as K(p) and
K(ṗ)).36, 31, 39 More specifically, Mora et al. (2013)39 investigated how different jet conditions (exit diameter,
temperature ratio and NPR) influence the skewness and kurtosis of the pressure and pressure derivative.
The crackle study by Krothapalli et al. (2000)30 comprised several experimental studies. The hydrodynamic and acoustic fields of a heated Mach 2 jet were visualized using the schieren method. Strong waves
in the acoustic field suddenly appeared when raising the jet’s total temperature from 580 K to 860 K and
1,250 K. Also, the frequency of occurrence of these strong waves increased with increasing jet temperature
and velocity. This was an interesting finding, since Ffowcs Williams et al. (1975)29 found no evidence that
crackle depends on jet temperature and operating conditions of the nozzle, e.g. perfectly or imperfectly
expanded. Furthermore, it was concluded in the 1970’s that crackle is not one-to-one related to factors such
as a rough combustion process or an afterburner in the case of full-scale engines.
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Quantitative experiments were conducted by Krothapalli et al. (2000)30 at the NASA Ames 80 ft × 40 ft
wind tunnel (anechoic for f > 500 Hz) on an axisymmetric jet (NPR 3.93, Dj = 0.1254 m, Uj = 796 m/s,
T0 = 972 K). Acoustic data were acquired along one arc centered at the jet exit with a radius of r/Dj = 61.
Measurements spanned angles from 35◦ to 135◦ relative to the jet axis. They computed the skewness of
the pressure time series along the arc-array, which varied between 0.20 and 0.35 at angles beyond the Mach
angle. The skewness peaked at 0.70 near the Mach angle and gradually decreased within the Mach cone to
around 0.4. It was concluded that crackle is highly dependent on the angle of observation: it becomes more
significant at shallow angles to the jet axis, peaks in the vicinity of the Mach wave angle, and low levels
of crackle appear at side-line angles. Likewise, Mora et al. (2013)39 found, using a similar arc-array setup,
that the OASPL, pressure skewness, and pressure kurtosis peak near the Mach wave angle. Furthermore,
Krothapalli et al. (2000)30 analysed acoustic data corresponding to a static full-scale engine (NPR 2.52,
Dj = 0.66 m, Uj = 663 m/s, T0 = 1, 200 K). Microphone measurements at a radial distance of r/Dj = 35
and oriented at 40◦ from the jet axis revealed that the time scales of crackle scale with Dj /Uj and were in the
order of 10 ms for the full-scale case. Further quantification involved the computation of the pressure variance
over a temporal moving window (short time variance). By using an amplitude threshold for the short time
variance, they were able to identify shock-structures in the waveform (instances of crackle). Their average
time between shock-structures was around 180 ms (6 Hz). Finally, they inferred that shock-structures occur
for about 5% of the time, but contribute roughly 30% to the total sound energy. The latter was computed
from short term Fourier transforms taken during the times when crackle occurred. Similar crackle trends,
as found from the pressure skewness along the arc-array, were inferred from their energy analyses.
I.B.2.
Challenges when studying crackle
One of the major difficulties in studying crackle is the issue of perception.31 Various observers may perceive
an acoustic waveform as crackle-free, while others do not. And, when it is perceived, what is the degree
of crackle? This issue is aggravated by the fact that there is no unique measure of crackle to assess its
presence in an acoustic measurement. Moreover, spectral representations based on conventional Fourier
analyses are insensitive for two reasons: (1) information is not time-preserved in the process of ensemble
averaging, and (2) crackle causes a wide (broadband), low amplitude energy footprint. Thus, in order to
move forward on a method of quantification, the temporal information must be preserved. A new approach
for quantifying shock-structures is pursued in this work. The current work aims to quantify the shock-type
waveform structures emitted by an experimental Mach 3 jet in a temporal and spectral fashion. The jet
under investigation comprises a Method of Characteristics (MOC) contour and is operated at fully expanded
conditions (shock-free plume) so that its noise field contains solely turbulent mixing noise and crackle. The
experimental campaign is summarized in the next section.
II.
II.A.
Experimental campaign
Facility and hardware
The experiments were conducted in the fully anechoic multidisciplinary fluid dynamics facility of The University of Texas at Austin. Details regarding the laboratory and set-up are described in the literature;38, 1
a summary is provided here for the readers perusal. The outer peripheral of the test environment, the anechoic chamber, provides a normal incidence sound absorption coefficient of 99% for f > 100 Hz; the inner
dimensions (wedge tip to wedge tip) are 19 ft(L) × 15 ft(W) × 12 ft(H). A nozzle test rig is installed near
the upstream wall (figure 3a). Air is allowed to enter the chamber through a 4 ft × 4 ft opening behind the
nozzle test rig, which then exhausts through a 6 ft × 6 ft acoustically treated eductor on the opposing wall.
The driving fluid for the cold-flow tests is compressed air stored at 2,100 psig in several tanks comprising
4.25 m3 of water volume storage. The air is fed into a 6 in diameter plenum that houses flow conditioning
elements and which is located immediately upstream of the nozzle contraction. The test were performed
using an in-house fabricated nozzle (figure 3b) that was designed using the Method of Characteristics to have
an exit gas dynamic Mach number of Me = 3.00 (T0 = 273.15 K, γ = 1.4, R = 287.05 J/kg/K). The exit
diameter was constrained to 1 in (25.4 mm) resulting in a throat-to-exit length of 2.30 in. All measurements
were performed with the nozzle operating under perfectly expanded conditions (a nozzle pressure ratio of
N P R = p0 /p∞ = 36.73); the associated mass flow was computed to be 1.04 kg/s.
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(a)
(b)
(c)
Grid measurements:
(in ϕ = −38.5◦ plane)
x/Dj = [5, 145], ∆x/Dj = 10
r/Dj = [25, 95], ∆r/Dj = 10
z
y
r
r
ϕ
nozzle
θ
x, jet axis
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Figure 3. (a) Arrangement of the chamber during grid measurements of the far-field acoustics, (b) the Mach 3 nozzle
mounted to the ø6 in settling chamber and (c) coordinate system and positioning of the 2D grid.
II.B.
Instrumentation
Acoustic data were acquired using four 1/4 in prepolarized, pressure-field, condenser microphones (PCB
377B10 capsules / 426B03 preamplifiers). The microphones have a frequency response range of 4 Hz to
70 kHz with ±1 dB error up to 20 kHz and a dynamic range up to 170 dB (re 20 µPa). A NI PXI-1042Q
system embedded with an eight channel NI PXI-4472 module provided the necessary IEPE conditioned power
(27 VDC and 4 mA) to operate the microphones all the while conditioning the signal to eliminate aliasing
prior to digitization (filter roll-off occurs at 0.84 of the Nyquist frequency). All channels were acquired
synchronously at a rate of 102.4 kS/s with 24 bit resolution for a minimum of 220 samples.
Microphone diaphragms were oriented at grazing incidence to the acoustic wave fronts at all times (plane
of the diaphragm intersecting the complete jet axis40 ) and with grid caps removed. Considering the instrument orientation, planar grid measurements were acquired on a (x, r)-plane oriented at an angle of
ϕ = −38.5◦ (see figure 3c). This uniform grid spanned from 5 Dj to 145 Dj in the axial direction and from
25 Dj to 95 Dj in the radial direction with a uniform spacing of ∆x = ∆r = 10 Dj . An acoustically transparent array was constructed that supported the four microphones. The traversing array was repositioned
in between runs to capture acoustic data at all grid positions.
In order to facilitate subsequent discussion, several grid points were selected in order to form both lineand arc-arrays. Figure 4 provides a visual mapping of these observer points. The first of these comprised
eight microphones forming an artificial arc-array at ρ = 100 Dj ± 1.5% and centered at x = 20Dj . The
precise location of these eight acoustic observers are listed in table 1. Several lines radiating from x = 20Dj
were then formed, labeled A–G in figure 4 and are angled at φ = [22, 35, 45, 52, 65, 86, 94]◦ , respectively.
mic #
1
2
3
4
5
6
7
8
(x, r) /Dj
φ [degr]
ρ/Dj
(115,25)
14.7
98.2
(115,35)
20.2
101.2
(105,55)
32.9
101.2
(95,65)
40.9
99.3
(85,75)
49.1
99.3
(75,85)
57.1
101.2
(55,95)
69.8
101.2
(45,95)
75.3
98.3
Table 1. Microphone grid positions used to form an artificial arc-array.
II.C.
Test conditions
The acoustic data were acquired over a duration of two days. The atmospheric conditions are summarized
in table 2 using subscripts j, ∞ and 0 to denote jet exit, ambient and stagnation conditions, respectively.
Column ‘grid (day 1)’ corresponds to the measurements performed in the range x = [5, 95] Dj , r = [25, 95] Dj ,
whereas column ‘grid (day 2)’ pertains to the remaining section of the grid: x = [105, 145] Dj , r = [25, 95] Dj .
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(G)
(F)
(E)
(D)
(C)
95
(B)
85
75
r/Dj
65
(A)
55
45
35
25
(H)
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0
0 5
15
25
35
45
55
65
75
85
95
105 115 125 135 145
x/Dj
Figure 4. Microphone grid positions used to construct line-arrays (labeled A-H). Lines A-G () radiate from x/Dj = 20;
line H (△) radiates from the nozzle exit. An artificial arc-array (◦) at ρ = 100 Dj ± 1.5% is centered on x = 20Dj .
Jet exit conditions were calculated from standard isentropic relations using an estimate for the dynamic
viscosity, µ (T ), based on Sutherland’s law. More details about the test conditions and jet parameters can
be found in the study by Baars et al. (2012)38 and Baars (2013).1
measured
Mj
N P R = p0 /p∞
p∞ [kP a]
T0 [K]
T∞ [K]
Rel. hum. [%]
grid (day 1)
grid (day 2)
3.00 ± 1%
36.73 ± 4.5%
100.7
100.8
291.2
286.2
293.3
287.2
75.4
63.2
calculated
grid (day 1)
grid (day 2)
Tj [K]
aj [m/s]
a∞ [m/s]
Uj [m/s]
fc = Uj /Dj [kHz]
Tj /T∞
ρ∞ /ρj
Rej
Uc = 0.80Uj [m/s]
Ma = Uj /a∞
Mc = Uc /a∞
φ [degr]
104.0
204.4
343.3
613.3
24.1
0.35
0.35
7.4 · 106
490.6
1.79
1.43
45.6
102.2
202.7
339.7
608.0
23.9
0.36
0.36
7.6 · 106
486.4
1.79
1.43
45.7
Table 2. Summary of the experimental conditions (Mj was controlled to be the fully expanded Mach number Me = 3.00).
II.D.
Preliminary results
The spatial topography of the overall sound pressure level (OASPL) obtained from the planar grid measurements is presented in figure 5. A strong intensity gradient present along θ = 45◦ (initiating from the jet exit)
is also observed and supports the notion that the Mach wave radiation intensity decays rapidly beyond the
Mach wave radiation angle. Furthermore, this edge remains distinct with outward distance, up to, and likely
beyond the range of consideration. These observations, in combination with the suggestion that the direction
of peak sound intensity coincides with the radiation angle,41 justifies the assumption for the convective speed
of the instability waves responsible for generating Mach waves: Uc = 0.80 Uj . An extensive discussion on
the statistics of the jet flow and acoustic field are provided in the literature.1
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95
128
129
130
131
132
133
123
85
124
75
134
133
132
131
125
126
65
135
130
r/Dj
127
55
136
45
35
129
130
131
132
133
134
137
140
138
143
25
142
139
135
136
141
138
137
134133132
131
129
128
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ρ
0
0 5
15
25
35
45
55
65
75
85
95 105 115 125 135 145
x/Dj
Figure 5. Contours of Overall Sound Pressure Level (OASPL) in dB, pref = 20 µPa, data of the original grid (10 Dj
spacing) are interpolated by a factor of 10.
III.
Methods for shock quantification
From the discussion in § I it has become clear that acoustic waveforms are significantly steepened in
certain areas of the far-field of the jet and that the structures induce crackle. This section pursues methods
to quantify the shock-structures in the acoustic waveforms; the motivations for this are given in itemized
form below:
Shock containing waveforms are of high interest to the jet noise community due to the crackling sound
they produce. Details on source locations and mechanisms can be inferred by studying the shockstructures in the acoustic near/far-field of the jet.
No unique measure of crackling shock-structures exists. A few statistical metrics (skewness and kurtosis
of the pressure and its time derivative) have been applied to waveforms, but shortcomings persist in
that the total waveform is taken into consideration, rather than solely weighing the waveform’s shock
content. Hence, there is a demand for unique metrics that quantify the parts of the waveform that are
solely responsible for crackle: the shock-structures.
Significant reductions in jet noise will be achieved when crackle can be controlled. Quantification is
a first step to the development of future strategies that are tailored to the suppression of crackle in a
novel way.42
Shock-structures are identified using a Shock Detection Algorithm (SDA) that is presented in § III.A. Joint
time-frequency analyses are then discussed (§ III.B) in order to assess the spectral characteristics of these
signatures associated with crackle (§ IV.C).
III.A.
Shock detection algorithm
Several efforts to detect shocks in time series can be found in the literature. McInerny & Ölçmen (2005)43
studied shock contaminated time series acquired in the far-field of a full-scale rocket. Individual shockstructures were identified in order to compute statistics of the pressure rise time versus the shock strength
(pressure jump). Sadler et al. (1998)44 introduced a technique based on time-frequency analyses (waveletbased) to detect shock waves generated by supersonic projectiles. However, the signatures that were detected
were relatively easy to tag, as they were embedded in a low-level background noise.
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Since the aforementioned techniques are not directly applicable to the current data set, a new algorithm
has been developed here. That is, the broad range of frequencies produced by jets comprise amplitudes
that are comparable to the shocks, thus making the latter illusive and difficult to isolate. Figure 6 shows
7 ms of raw pressure data, p(t), from microphone 5 on the arc-array, alongside its pressure derivative, ṗ(t).
Temporal derivatives are computed using a 1st -order forward difference scheme. Shock-structures are acutely
observed in the pressure signal and appear as large-amplitude spikes in the temporal derivative. The fact
that waveform expansions are more gradual than the sharp rises in pressure is well visualized by the skewed
PDF of the pressure derivative, displayed in figure 6d. It is important to realize that computing the pressure
derivative can be problematic when substantial levels of high-frequency noise contaminate the data; further
implications of this will be addressed in § IV.A.
(a), arc-array mic 5
(b)
data
normal
4
2
2
0
0
−2
−2
p/prms
4
−4
0
1
2
3
t [ms]
4
5
6
7
−4
0
+σ
−σ
0.2
6
4
4
2
2
0
0
−2
−2
15
30
45
60
75
90
0.6
(d)
6
−4
0
0.4
B(p/prms )
(c), arc-array mic 5
ṗ/ṗrms
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result of shock detection algorithm:
105
120
135
150
165
tUj /Dj
−4
0
data
normal
0.2
0.4
0.6
B(ṗ/ṗrms )
Figure 6. (a) Pressure time series and (c) associated pressure derivative with markings at which shock-structures
are detected by the SDA and (b,d) associated PDF’s of the signals. Note: figures a,c employ identical time ranges but
non-dimensional time is indicated in figure c.
The SDA begins by identifying all instances in ṗ(t) where the signal’s amplitude exceeds a user defined
threshold of Tṗ = 3 · 10−6σ/(6dt) kPa/ms. Here, σ is the standard deviation of the pressure signal p(t) in Pa,
and dt is the discrete sampling time-step of 10− 5 s. The detected instances are then used to infer amplitudes
and indices of the corresponding pressure extremes; a minimum at earlier times and a maximum at later
times. Next, the SDA omits identified shocks of which the pressure jump (shock strength) does not exceed an
amplitude of Tp = 2.7σ Pa. As a final step, the median time between the pressure minimum and subsequent
maximum is taken as the time instance of each shock, and is tagged as ts = 0 s in the corresponding
local shock time frame. The shock thickness ∆xs and shock rise time ∆ts are simply related by the sound
speed according to ∆xs = a∞ ∆ts . The thresholds Tṗ and Tp are based on a typical shock rise time of
∆ts = 10−6 Tp /Tṗ = 0.0540 ms and results in an effective shock thickness of ∆xs = 10−6 Tp /Tṗ a∞ = 0.0185 m.
The instances when shocks were identified are marked by vertical lines at the top of sub-figure a and c in
figure 6. By visual inspection, the current choice of thresholds results in a fairly robust shock detection
scheme when signals with wildly varying frequencies are examined. Note that the typical rise time and shock
thickness do not depend on the pressure standard deviation σ. Suggestions for future improvements in the
detection and quantification of temporal shocks are discussed elsewhere.1
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III.B.
Wavelet-based time-frequency analysis
Time-frequency analyses are applied to the stationary acoustic data. The motive for applying these techniques is to detect unique spectral content associated with the shock-structures. Such information is otherwise delocalized, i.e. spread out, when conventional Fourier analysis techniques are applied. Wavelet
transforms provide a mathematical framework for performing time-frequency analyses and are reviewed
extensively.45, 46, 47
The continuous wavelet transform is applied in the current work to acoustic pressure time series from
the planar grid measurements. The current implementation of the technique is described in more detail
by Baars & Tinney (2013).48 In general, the wavelet transform is performed by convolving a wavelet with
the signal. This convolution is performed at various time scales l, i.e. frequencies, and so decomposes
the signal in time-frequency space. The progressive complex-valued Morlet wavelet, Eq. 1, is selected for
this study due to its high frequency resolution when compared to other conventional wavelets. However,
in choosing the Morlet wavelet, less temporal resolution is achieved when compared to, for example, the
Mexican hat wavelet; this is particularly noticeable at the low-frequency (large-scale) end of the spectrum.48
It is understood that most of the energy in the crackling signatures (essentially the shock-structures) resides
in the high-frequency end of the spectrum. Hence, if there is poor temporal resolution at low-frequencies it
should not pose any restrictions on the current analysis. Albeit, this obstacle can be overcome by correctly
interpreting the time-frequency results.47
ψ (t/l) = ejωψ t/l e−|t/l|
2
/2
(1)
The Morlet wavelet comprises a harmonic wave whose amplitude is modulated by a Gaussian function. The
non-dimensional frequency is taken as |ωψ | = 6. The temporal convolution of the Morlet wavelet with the
pressure time series p(t), results in complex-valued wavelet coefficients p̃ (l, t), defined as
′
Z
t −t
p̃ (l, t) = p (t′ ) ψ ∗
dt,
(2)
l
with the convolution being performed in the frequency domain.45 In the current study, the resolved band
of frequencies range between 100 Hz < f < fs /2 using a base-2 logarithmic progression of 81 scales in order
to achieve a uniform grid on a logarithmic frequency scale. The spectral results are generally presented in
terms of Strouhal number for the range 0.01 < StDj < 2. The energy density is given by
E (l, t) =
|p̃ (l, t) |2
,
l
(3)
and is known as the Wavelet Power Spectrum (WPS). Finally, the wavelet scale is transformed to an equivalent Fourier frequency, i.e. E (l, t) → E (f, t). The WPS of the pressure waveform from arc-array microphone
5 is shown in figure 7c. In order to observe temporal variations, only a small part (about 14 ms) of the WPS
are presented. For wavelet analyses, when the time scales of transient events are small (which is the case for
stationary signals), time-averaged wavelet spectra can be created following
Z
Edt,
(4)
E = 1/T
T
where T is an arbitrary domain in time. For stationary signals, the single-sided global wavelet spectra
(GW S = 2 · E, and T = length of signal) compare well with the ensemble-averaged Fourier spectra, as is
shown in figure 7d. The GWS resemble a filtered version of the Fourier spectra due to the natural scale
filtering.
III.B.1.
Verifying the SDA by inspection
A visual inspection of the time-frequency topography, alongside the SDA results, is employed here. Pressure
signals that resemble Dirac delta functions cause a wide-spread (almost uniform) excitation over all frequencies. Likewise, edges of step functions or sudden spikes with exponential tails cause an excitation of energy
over a wide range of frequencies. Addison (2002)47 (chapter 2) shows typical WPS signatures associated
with such features. Due to the varying temporal resolution (low resolution at low frequencies and vice versa),
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these signatures manifest themselves as high-amplitude ‘mountain-shape’ formations in the WPS. That is, a
large width at low frequencies (base) that point to spike/edge instances at high frequencies (summit). Sharp
rises in the pressure signal corresponding to crackling shock-structures in the current signals reveal themselves in a similar manner. Figure 7a presents the result of the SDA with the associated WPS in figure 7c;
instances where shocks are detected are indicated by white lines. By inspection, an increase of high-frequency
(a), arc-array mic 5
(b)
data
normal
result of shock detection algorithm:
4
2
2
0
0
−2
−2
p/prms
4
1
2
3
t [ms]
4
5
6
7
−4
0
−σ
0.2
0.4
0.6
B(p/prms )
(d)
(c), arc-array mic 5
100
0
10
0
10
StDj
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−4
0
+σ
90
80
−1
−1
70
10
10
F−BMF
GWS
Shock
60
50
−2
10
0
15
30
45
60
75
90
105
120
135
150
tUj /Dj
−2
10
60
70
80
90
100
Gpp (StDj ) [dB/Hz]
Figure 7.
(a) Pressure time series with markings
at which shock-structures are detected by the SDA, (b) PDF of
the signal, (c) associated WPS, E StDj , t in dB/Hz, pref = 20 µPa and (d) the GWS, Fourier spectrum and shock
spectrum (5% bandwidth moving filter). Note: figures a,c employ identical time ranges but non-dimensional time is indicated
in figure c.
energy occurs when shock-structures are present (i.e. visible for shocks near tUj /Dj = 105, 138 and 148).
Although shock-structures appear to be visible at high frequencies in the WPS, mountain-shapes do not
reveal themselves at low-frequencies. This is caused by the poor temporal resolution of the Morlet wavelet
at low frequencies which smears out the energy of all waveform signatures. The latter is well visualized by
the shock spectrum, which is denoted as Sh(f ) and is formed by ensemble-averaging the WPS at all instances
(ti , a total number of M shocks) where the SDA has detected a shock. This is done according to
M
1 X
E (f, ti ) .
Sh (f ) =
M i=1
(5)
The GWS, PSD’s (using Fourier), and shock spectra are compared in figure 7d. From previous discussions
it became clear that at low-frequencies, the signature of the shock-structures is no different than that of the
entire waveform, and so, the spectra collapse. At the high-frequencies an increase in the shock spectrum is
observed and is representative of crackle.
IV.
Quantifying crackle
Ffowcs Williams et al. (1975)29 and Laufer et al. (1976)18 were the first to focus on individual features
in the acoustic signal associated with Mach wave radiation and crackle. Their work was insightful for the
fundamental knowledge on crackle as was reviewed in § I.B. Now that the shock portions of the waveforms
have been successfully extracted, and for all points on the planar measurement grid, the statistical properties
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of the crackle portions of the acoustic waveforms can be gleaned. This is quantified in three ways: (1) the
shape of the shock-structures are studied throughout the measurement domain (§ IV.A), (2) their temporal
behavior is examined by studying the instances of occurrence (§ IV.B), and (3) the spectral content is
quantified by studying the shock spectrum (§ IV.C).
IV.A.
Ensemble-averaged shock-structures
The average shape of the waveform associated with crackle is obtained by ensemble-averaging all instances
where shock-structures have been identified using the SDA described earlier. Figure 8 presents the signatures
along the arc-array; structures are plotted with the median time between pressure minima and maxima
aligned at ts = 0 s. In general, shocks are close to being point-symmetric around ts = 0 s at shallow
(c), φ = 32.9◦ , mic 3
4
4
4
2
0
−2
0
0.1
0.2
p/σ, σ = 97.8 Pa
6
p/σ, σ = 68.6 Pa
2
0
−2
−0.2 −0.1
0
0.1
0.2
2
0
−2
−0.2 −0.1
0
0.1
ts [ms]
ts [ms]
ts [ms]
(d), φ = 40.9◦ , mic 4
(e), φ = 49.1◦ , mic 5
(f), φ = 57.1◦ , mic 6
6
6
4
4
4
p/σ, σ = 118.4 Pa
6
2
0
−2
−0.2 −0.1
0
0.1
0.2
p/σ, σ = 76.2 Pa
p/σ, σ = 55.2 Pa
p/σ, σ = 119.1 Pa
(b), φ = 20.2◦ , mic 2
6
−0.2 −0.1
2
0
−2
−0.2 −0.1
ts [ms]
(g), φ = 69.8◦ , mic 7
6
6
4
4
2
0
−2
−0.2 −0.1
0
ts [ms]
0.1
0.2
0
0.1
0.2
0.2
2
0
−2
−0.2 −0.1
0
ts [ms]
ts [ms]
(h), φ = 75.3◦ , mic 8
(i)
0.1
0.2
0.1
0.2
2
1
2
p/σ
p/σ, σ = 34.4 Pa
p/σ, σ = 37.6 Pa
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(a), φ = 14.7◦ , mic 1
6
0
0
−1
−2
−0.2 −0.1
0
0.1
0.2
−2
−0.2 −0.1
ts [ms]
0
ts [ms]
Figure 8. (a-h) Ensemble-averages (black lines) of the individual shock-structures (grey lines) along the arc-array and
(i) all ensemble-averaged shock-structures superposed.
angles (φ ≤ 20.2◦ ), while the peak (pressure rise transitioning to gradual expansion) becomes more sharp
at positions near the Mach angle (32.9◦ ≤ φ ≤ 49.1◦ ). At angles beyond the Mach angle (φ ≥ 57.1◦ ),
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mic #
∆ts /dt
p+ /σ
p− /σ
∆p [Pa]
1
2
3
4
5
6
7
8
7
6
5
4
4
4
5
5
1.84
1.87
2.06
2.21
2.15
1.94
1.60
1.57
-1.38
-1.38
-1.36
-1.36
-1.43
-1.43
-1.46
-1.47
177.8
223.0
334.3
424.5
423.4
257.4
115.1
104.7
∆p
∆ts
[kPa/ms]
2.54
3.71
6.69
10.61
10.59
6.43
2.30
2.09
Table 3. Statistical properties of the ensemble-averaged shock-structures along the arc-array, visualised in figure 8.
A contour of the average number of shock-structures per second, denoted as Sa , is shown in figure 9.
The maximum number of shocks per second resides along the peak noise path with roughly 1,650 shocks/s
(F)
(G)
(E)
95
4
6
5
6
7
(D)
8
85
9
(C)
(B)
9
8
7
6
5
9.5
75
4
65
3
r/Dj
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structures return to being symmetric but are indicative of high-frequency noise, since (1) pressure bands of
individual shock-structures (grey lines) are wide and have an amplitude similar to isolated shock-structures,
and (2) the amplitudes of the pressure minimum and maximum are nearly equal. Thus classifying these
as crackling shock-structures is questionable. Repeated experiments using transducers capable of resolving
higher frequencies is warranted. For each mean shock-structure, the rise time ∆ts , the pressure maximum
p+ and minimum p− , pressure jump ∆p and the gradient ∆p/∆ts are summarized in table 3.
55
45
(A)
4
6
5
9.5
35
1
2 34
25
0
0 5
2
9.9
15
25
35
45
55
65
75
85
95 105 115 125 135 145
x/Dj
Figure 9. Contour of the normalized average number of shocks per second Sa /Samax × 10, where Samax = 1, 835 s−1 at
(x, r) /Dj = (55, 45).
appearing along this direction. Smaller values of Sa are observed at shallow angles to the jet axis with the
least amount of shocks per second, ≈ 150 shocks/s, occurring at the shallowest angle to the jet, (x, r)/Dj =
(145, 25). At angles much beyond the Mach wave angle, the results are questionable due to the reasons
suggested earlier concerning the discussion of figure 8. Nonetheless, it is inferred from this mapping that
contours follow spherically spreading lines that emanate from the region of most intense sound generation
(x = 17.5 Dj ). It can be concluded that changes in the average number of shocks per second with outward
distance ρ (origin at x = 17.5 Dj ) are minor within, and slightly beyond, the Mach cone. Several findings
regarding the spatial topography of the shock content are itemized as follows,
Within the Mach cone, Sa only depends on polar angle φ. Furthermore, other shock-related measures
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do not change along spreading lines, as will be discussed next (§ IV.B). Physical mechanisms explaining
this behavior are postulated in the next section § IV.B.
An implication of the previous point is that there is an absence of coalescence of shock-structures and
that they travel efficiently, meaning that no shock fronts disappear within the measurement range due
to relaxation effects.
Not only do most shocks reside along the peak noise path (Sa = 1, 650 s−1 ), but they also appear to
be strongest in this direction in an absolute sense (see the pressure jump ∆p in Pa, listed in table 3);
this is partly due to the highest OASPL being along this direction. More importantly, these shocks
are also strongest in terms of the standard pressure deviation of the signal, according to the columns
of p+ and p− in table 3.
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IV.B.
Temporal characteristics
Arrival times of shock-structures are studied in this section, rather than their shapes. At first, in order to
reveal temporal trends, time instances of all shock-structures -detected by the SDA- along radial lines A-G
are visualized in figure 10 for a maximum duration of 1 s. Since it was shown previously (§ IV.A) that there
is sufficient evidence to presume a model where shock-structures travel along radial lines, the ‘bar-code’ type
graphs in figure 10 are interpreted as the arrival time trends of the shocks anywhere along the radial line of
interest.
tUj /Dj · 10−3
0
2
4
6
8
10
12
14
16
18
20
22
A: h∆ti = 2.61ms
B: h∆ti = 1.15ms
C: h∆ti = 0.71ms
D: h∆ti = 0.58ms
E: h∆ti = 1.01ms
F: h∆ti = 1.34ms
G: h∆ti = 1.10ms
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [s]
Figure 10. Temporal instances of shocks along radial lines A-G. Note: signatures along the lines are non-synchronized.
By inspection, it can be seen that shock-structures do not appear randomly within the waveform, but
rather as bursting events. Furthermore, these bursting events of multiple shocks appear to have a large
range of time scales. For example, when considering the temporal instances along line D in figure 10a, high
shock densities are present around 0.2 s and 0.5 s for a duration of about 0.15 s, while this is different when
considering line A or B. To investigate this, the 1st -order intermittence of the shock-structure time instances
ti , is computed as
(∆t)i = ti+1 − ti .
(6)
The average of all 1st -order intermittencies detected, according to
h∆ti =
M
1 X
(∆t)i ,
M i=1
(7)
is the inverse of the average number of shocks per second (h∆ti = 1/Sa ) and values of these are indicated
in figure 10a for reference. The PDF’s of the 1st -order intermittence, presented in figure 11, reveal temporal
trends along lines A-G.
PDF’s of (∆t)i were determined for all microphone positions on each of the lines A-G (figure 4). Their
envelopes are presented as grey dashed lines and indicate that trends occur along each entire line. This is
in agreement with the findings of constant Sa along each line and the idea that highly-directional shockstructures propagate spherically outward from a compact source. The PDF’s are presented in normalized
form, so that an integral over the intermittence is unity. The x-axis is presented in terms of (∆t)i /h∆ti
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(a)
(b)
3
3
2.5
line (A): φ = 22◦
h∆ti = 2.61ms
2
B((∆t)i /h∆ti)
B((∆t)i /h∆ti)
2.5
1.5
1
0.5
0
0
line (B): φ = 35◦
h∆ti = 1.15ms
2
1.5
1
0.5
0.2
0.4
0.6
0.8
0
0
1
0.2
(∆t)i /h∆ti
0.4
(c)
2.5
line (C): φ = 45◦
h∆ti = 0.71ms
2
B((∆t)i /h∆ti)
B((∆t)i /h∆ti)
1
3
2.5
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0.8
(d)
3
1.5
1
0.5
0
0
0.6
(∆t)i /h∆ti
line (D): φ = 52◦
h∆ti = 0.58ms
2
1.5
1
0.5
0.2
0.4
0.6
0.8
0
0
1
0.2
(∆t)i /h∆ti
0.4
0.6
0.8
1
(∆t)i /h∆ti
Figure 11. (a-d) PDF’s of the 1st -order intermittence (∆t)i of the shock-structures along lines A-D. Note: Average
PDF’s of all individual microphones on radial lines are indicated by black solid lines; upper/lower bounds of all individual PDF’s are
indicated by grey dashed lines.
and indicates that most of the intermittencies are less than average. This is caused by the fact that the
intermittence has a bounded minimum of the closest appearance of two shocks (6dt), while the upper limit
corresponds to the largest instance in between shock-structures detected in the entire waveform. PDF’s have
therefore long tails beyond (∆t)i /h∆ti = 1.
If shock-structures would have appeared periodically in time, the PDF’s would peak at one particular
intermittence. This is clearly not the case, but, modal behavior is observed in the PDF’s (local maxima)
and depend on the angle of observation. Based on the information available, there are a number of plausible
explanations for the physical mechanisms driving these results. As a conceivable cause for the modal temporal behavior, a highly simplified concept of fringe patterning of shock-structure wave fronts is envisioned
(figure 12a). Two sets of equally spaced shock-structure wave fronts are oriented at an angle with respect to
each other. Different trends of arrival times are encountered, depending on the line of observation. However,
these lines are parallel to each other, which contradicts the observations of constant trends along spherically
spreading lines, i.e. lines that diverge from a focal point residing around x = 17.5 Dj . When the fronts are
curved, which can be caused by the natural deceleration of large-scale structures emitting Mach waves for
example (figure 12b), the lines of constant temporal behavior start to diverge. Certain curvatures in different
families of waves emitted by the jet can thus result in the findings presented here.
At the moment, a well-founded understanding on how the proposed concepts can be used to accurately
explain the quantitative results does not exist. It is believed that synchronized measurements of shockstructures (acoustic pressure) and pressure or velocity measurements in the hydrodynamic region of the jet
(preferably spatially extensive) are necessary to continue exploring the proposed concepts of fringe patterning
in combination with curvature effects. Despite of the lack in understanding of the physical mechanisms, it
can still be concluded that shock-structures appear to be present in groups of multiple shocks. This evidence
supports the findings of Ffowcs Williams et al. (1975).29
As a final step, the characteristic time scales of the PDF’s are compared to the PSD’s along line C. In
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(b)
(a)
fringe patterns
curved wave fronts
nozzle
nozzle
x, jet axis
x, jet axis
figure 13, the PDF’s of the 1st -order intermittence along line C are shown, alongside the PSD’s. The modal
time scales of the PDF’s (marked 1-4) are inverted to frequency and indicated in figure 13b. No specific
conclusions can be made, besides the fact that the modal instances reside near the peak of the PSD, which
is not surprising given the fact that shock-structures contribute major amounts of energy (§ IV.C).
(a)
(b)
2.5
100
1
2
3
Gpp [dB/Hz], pref = 20 µPa
3
B((∆t)i /h∆ti)
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Figure 12. (a) Fringe patterns forming from two different set of equally spaced wave fronts and (b) curved wave fronts
due to the deceleration of large-scale turbulence.
4
line (C)
φ = 45◦
2
1.5
1
0.5
0
0
0.2
0.4
0.6
(∆t)i /h∆ti
0.8
1
95
90
85
4 3 2
1
80
75
0.01
0.1
1
2
StDj
Figure 13. (a) PDF’s of the 1st -order intermittence (∆t)i of the shock-structures along line C (peaks are identified by
number 1-4) and (b) Fourier spectra along line C scaled to ρ = 100 Dj (characteristic frequencies corresponding to peak
events 1-4 are identified).
IV.C.
Spectral characteristics
A spectral measure of the level of crackle based on the shock-spectrum (introduced in § III.B.1), is pursued
in this final section on crackle quantification. The approach is tailored towards the perception of crackle by
the human ear.
IV.C.1.
Scaling for perception
The consequence of performing measurements in a laboratory-scale environment is that high frequency
energy, that may be important to one’s perception of crackle, appears well above the upper frequency
threshold of the humar ear (20 kHz). Therefore, laboratory-scale data must be scaled to full-scale conditions
before a measure of crackle is to be computed. An important conclusion of Ffowcs Williams et al. (1975)29
is that crackle can be scaled, meaning that if a full-scale engine crackles, so will the sub-scale version. By
choosing Dj = 24 in, Mj = 1.5, and Tj = 1, 500 K as typical full-scale engine parameters, the exit velocity
is calculated as Uj = 1, 160 m/s. The full-scale scenario is matched to a laboratory-scale nozzle using the
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Strouhal number based on jet diameter, StDj = f Dj /Uj . Given laboratory-scale parameters (Dj = 1 in;
Uj = 610 m/s), the frequency must be lowered by a factor of 12.6 in order to replicate the perception at
full-scale. In practice, microphone signals are now assumed to be sampled at 1/12th the rate of the original
sampling frequency (fs = 100 kHz). The scaling is visualized by presenting acoustic spectra of arc-array
microphone 5 in figure 14. Due to the unavoidable high-frequency limit of the instrumentation, the Nyquist
frequency becomes 50/12 kHz ≃ 4.2 kHz. The absence of spectral information in the 4.2 - 20 kHz audible
range is expected to have only a minor influence on the results due to the roll-off of the PSD’s at these
frequencies. Additionally to the scaling, a weighting has to be applied that accounts for the relative loudness
100
85
GWSL
GWSF
GWSF−A
Shock−A
80
75
70
65
60
1
12 -scaling
(Shock−GW SF )
(GW SF −A)
ear threshold
Nyquist frequency
Gpp [dB/Hz], pref = 20 µPa
90
60
90
gain [%]
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95
· 100%
143%
gain
30
0 1
10
2
10
3
10
4
10
f [Hz]
Figure 14. Laboratory-scale GWS at arc-array microphone 5 (subscript L) scaled down to full-scale (subscript F),
subsequently, an A-weighting is applied (GWSF -A) and the A-weighted shock spectrum is shown. The energy gain
(from dashed to solid black) is 143%.
perceived by the human ear at different frequencies within the audible range (20 Hz - 20 kHz). The human
ear is less sensitive to both low and high audio frequencies, while it is most sensitive to frequencies around
1 kHz. An A-weightinga is selected and is applied to the scaled data to obtain the spectrum denoted by
GWSF -A.
IV.C.2.
Spectral measure of crackle
The study by Krothapalli et al. (2000)30 was the first attempt to investigate how much of the total sound
energy in an acoustic signal was contributed by crackling shock-structures; short time Fourier transforms
were used to investigate this (§ I.B.1). The energy-based metric that is now used for quantifying crackle
is explained in figure 14. The energy increase at instances of crackle is measured as the energy gain from
the GWSF -A spectrum (scaled and A-weighted) to the associated shock spectrum, denoted as Shock-A; the
energy gain is 143% for the spectra shown. When computing the energy gain for each position in the planar
grid, the contour of the percentage gain due to crackle can be constructed (figure 15).
Three important conclusions can be drawn from the results in this section and are itemized as follows.
The crackle energy gain is independent of the distance along paths of constant polar angle φ emanating
from region of most intense sound production. This agrees with the observations of Ffowcs Williams et
al. (1975),29 who state the following on p. 257 of their manuscript: “It seems therefore that observed
crackle is independent of the distance travelled by the sound provided that observations are confined to
a Defined
in national and international standards: ANSI S1.4-1983, S1.42-2001, ISO 226:2003, IEC 61672:2003
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95
85
70
90 110
130
150
170
190
75
r/Dj
65
210
230
250
270
55
45
290
250
270
290
310
330
350
35
25
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100
0
0 5
15
25
150
35
45
200
55
65
75
250
85
300
230
350
95 105 115 125 135 145
x/Dj
Figure 15. Contour of the crackle energy gain in percentage (see figure 14).
positions outside the immediate near field yet close enough to the source that long-term propagation
effects are negligible.”
The current energy-based measure seems to be an appropriate metric for the perception of crackle.
Crackle is perceived dominantly at shallow angles and becomes gradually less pronounced at larger
angles. Figure 14 resembles this trend, while contours of the skewness of the pressure and pressure
derivative38 show a significant decrease in amplitude at shallow angles to the jet.
The energy gain is highest at shallow angles where shock-structures are not as strong as near the Mach
cone edge (table 3). This is due to the steep roll-off of the jet noise spectra at shallow angles, while
the shock spectra remain roughly constant. Therefore, the relative gain is highest at these locations.
V.
Summary and conclusions
Several methods were pursued to quantify acoustic shock-structures in the sound field of the experimental
Mach 3 jet. The absence of any cumulative nonlinear steepening effects in the measurement range of our
experimental Mach 3 jet implied that acoustic shock-structures are generated by local mechanisms in, or in
close vicinity to, the jet plume. The research on the crackling shock-structures was motivated by the fact that
(1) it is perceived as the most annoying component of jet noise, (2) no unique measures of crackle exist, and
(3) significant reductions in jet noise will be achieved when crackle can be controlled. Considering the second
point, only a few statistical metrics have been applied to pressure waveforms in the past with an attempt
to assess the existence of crackle. Shortcomings of using these ensemble-averaged metrics are that they
do not solely weight the waveform’s shock content, but rather take the total waveform into consideration.
Henceforth, crackle is better quantified by studying the shock content.
A shock detection algorithm (SDA) was developed to detect shock fronts in the acoustic pressure waveforms. The algorithm was based on two user defined thresholds of the temporal pressure derivative (pressure
rise time) and the pressure jump across the shock (shock strength). At first, the average shapes of the waveforms associated with crackle were obtained by ensemble-averaging all instances where shock-structures had
been identified using the SDA. Shocks are close to being point-symmetric around the shock front (defined as
the median time between pressure minima and maxima) at shallow angles to the jet axis and have rounded
pressure minima and maxima. At angles near the Mach cone edge, the pressure maxima become more sharp
and are indicative of less pronounced shock relaxation effects. It was concluded that classifying the shockstructures identified outside of the Mach cone as being crackle is highly questionable. Furthermore, within
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the Mach cone, the average number of shocks per second did not vary along radial lines of constant polar
angle φ with a focal point residing in the region of most intense sound generation (x = 17.5 Dj ). And so,
the crackling structures travel efficiently in the sense that they do not disappear due to relaxation within
the measurement range. Trends of temporal shock instances (arrival times) were subsequently studied. Like
the average number of shocks per second, PDF’s of the 1st -order intermittence are independent of the position on each of the radial lines. The revealed modal behavior of the time instances provides evidence that
shock-structures appear to be present in groups of multiple shocks.
A spectral measure of the level of crackle was considered by using wavelet-based time-frequency analyses.
By employing the wavelet transform of acoustic waveforms, acoustic pressure spectra are generated as function of time. Thereafter, the time-frequency energy distribution is used to detect the unique spectral content
associated with the shock fronts identified by the SDA, this is denoted as the shock spectrum. In order compute a practically significant measure, laboratory-scale spectra are scaled to an equivalent full-scale scenario
at first. Secondly, spectra are A-weighted to account for the relative loudness perceived by the human ear at
different frequencies within the audible range. The absolute increase of spectral sound energy between the
global jet noise spectrum and the shock spectrum is independent of outward distance along radial lines and
complies with the findings of constant temporal characteristics along the radial paths with constant polar
angle φ. The increase in sound energy is quantified as the percentage energy gain from the global pressure
spectrum to the shock spectrum. This energy-based metric is postulated to be an appropriate metric for the
perception of crackle since it provides the correct spatial trends: crackle is perceived dominantly at shallow
angles and gradually becomes less pronounced at larger angles.
Acknowledgements
The authors gratefully acknowledge support from the Air Force Office of Scientific Research under grant
number FA9550-11-1-0203, Dr. John Schmisseur, program manager, and the Office of Naval Research, ONR
award number N00014-11-1-0752, Drs. Joseph Doychak and Brenda Henderson, program managers.
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