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Near- and Far-Field Pressure Skewness and Kurtosis in Heated Supersonic Jets
From Round and Chevron Nozzles
Conference Paper · June 2013
DOI: 10.1115/GT2013-95774
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1 Copyright © 2013 by ASME
Proceedings of ASME Turbo Expo 2013
GT2013
June 3-7, 2013, San Antonio, Texas, USA
GT2013-95774
NEAR- AND FAR-FIELD PRESSURE SKEWNESS AND KURTOSIS IN HEATED
SUPERSONIC JETS FROM ROUND AND CHEVRON NOZZLES
Pablo Mora, Nick Heeb, Jeff Kastner,
Ephraim J. Gutmark School of Aerospace Systems
University of Cincinnati Cincinnati,Ohio,45221
Email: [email protected], [email protected]
K. Kailasanath Naval Research Laboratory
Email: [email protected]
ABSTRACT When the turbulent structures in the shear layer of high-
speed jets travel at supersonic convective speeds relative to the
ambient speed of sound, they radiate Mach waves that become
the dominant component of the overall perceived noise. This is
consistent with the OASPL in the far field reaching a maximum
in same direction as the Mach wave angle. When the speed of
the supersonic jet exceeds a certain level, the steepening of the
wave-front in the near field produces a noise feature called
“crackle.” Both pressure wave steepening and crackle cannot
be recognized in the spectrum of the pressure signal, but in the
temporal waveform of the pressure. The statistics of the
pressure signal and its time derivative, particularly skewness,
have become standard measures of crackle in heated
supersonic jets. Previous studies showed that it is possible to
reduce far-field pressure skewness with the implementation of
notched and chevron nozzles, and to mitigate Mach Wave
radiation with secondary flow techniques. In this paper, we
investigate the effect of chevrons on the pressure and dP/dt
high-order statistics of a Md = 1.5 converging-diverging round
conical nozzle, both in the near and far fields. Cold and heated
jets, To = 300 K and 600 K, are tested at over, design, and
under-expanded conditions. Far-field results of the heated jet
showed that chevrons effectively reduce elevated levels of
skewness and kurtosis of the pressure and dP/dt. These
reductions are remarkable especially around the Mach Wave
angle, the region in which high-order statistics tend to
propagate. Near-field results corroborated the effectiveness of
chevrons in the skewness reduction.
INTRODUCTION With the development of new and improved supersonic
aircraft technologies, propulsion systems are challenged to
become faster and more efficient. But achieving higher speeds
causes a significant increase in noise levels near the vicinity of
the aircraft and in the far field. Intense jet noise is the main
contributor to overall sound pressure levels, and affects those
who work on aircraft carriers or live near airports and military
bases. As a result, mitigating jet noise has become a priority in
order to reduce the annoyance factor and noise-induced hearing
loss.
In supersonic jet engines, the dominant noise sources are
associated with shock and expansion wave diamonds in the
potential core, the shear layer, and the mixing region. The main
factors that determine the jet characteristics, consequently the
noise generation, are the shape of the nozzle, the
thermodynamic conditions of the air, and whether the jet engine
is stationary or in forward flight. For a supersonic C-D nozzle,
three main types of noise are present in the frequency spectrum:
screech, broadband shock-associated noise, and mixing noise
from fine-scale and large-scale turbulent structures. For the
particular case of a perfectly expanded bell shape C-D nozzle,
only mixing noise is identified. Both screech and broadband
shock-associated noise are not generated because shock cells
are not present in this type of nozzle when perfectly expanded.
Conical nozzles are more representative of the variable exit
geometries used on current military fighter aircraft [1].
Chevrons and fluidic injection are both techniques that have
been investigated for supersonic jet noise reduction. Chevrons
generate streamwise vortices that enhance the mixing of the
high-speed jet plume with the low-speed ambient fluid,
2 Copyright © 2013 by ASME
reducing low-frequency noise [2]. Chevrons decreased the
shock cell spacing and as a consequence a reduction in the
broadband shock-associated noise was observed. Screech tones
were also diminished by breaking down the feedback loop
process. On the other hand, chevrons present a penalty in thrust
and are known to increase high-frequency noise [3]. The higher
the chevrons penetrate the flow, the higher the mixing is
enhanced and therefore a greater reduction in peak noise is
obtained. Chevrons tend to perform best in under-expanded
conditions due to the positive pressure at the nozzle exit, and do
not perform as well in over-expanded conditions due to the
negative pressure at the nozzle exit.
Crackle is another component of high-speed jet noise,
identified first by Ffowcs Williams et al. [4]. In this report,
crackle was defined as “spasmodic bursts of a rasping fricative
sound not dissimilar to that made by the irregular tearing of
paper.” When listening to the jet noise of a Rolls-
Royce/SNECMA Olympus 593, this crackling noise was
identified to be the highest contributing factor to overall
subjective annoyance. The main problem arises when trying to
quantify this noise component. Crackle is not identifiable in the
spectrum of the pressure signal, but in the waveform of the
pressure signal. The crackling jet pressure time trace contains a
characteristic pattern of quick strong compressions followed by
slow and weak expansions. From here, it was determined that
the normalized skewness of the temporal waveform of the
pressure or the third moment of the Probability Density
Function of the signal, normalized by the standard deviation, is
a metric for quantifying crackle. While the skewness of a
Gaussian distribution equals zero, positive values of skewness
represent a “greater probability of positive values and
asymmetric probability density function” [5]. After test runs
with the Olympus 593, baseline and afterburner conditions, it
was confirmed that signals from jets with skewness higher than
0.4 relate to crackling jets, while signals with skewness less
than 0.3 do not [4]. The highest crackle was identified at 60o
from jet axis at the downstream, the same direction as the Mach
wave radiation angle.
Krothapalli et al. [6] and Petitjean et al. [7] showed that jet
temperature does influence crackle intensity and therefore
skewness. Non-linear propagation was discarded as a cause of
skewness generated in the near field. Although investigations
have shown that non-linear propagation through turbulent air
may induce positive skewness and crackle-like impression in a
signal, sound pressure levels of scaled experiments were not
sufficient for non-linear propagation to influence crackle and
skewness within the near field [4]. Petitjean et al. [8] estimated
that non-linear effects would not be accentuated until an
approximate distance of 88 De in the radiation direction of 145o.
Krothapalli [6] observed through Schlieren visualization that
Mach waves and crackle were generated in the near field. For
high temperature jets, strong waves with sharp density gradients
(crackle) propagate in the same direction as the Mach waves.
The frequency of occurrence of these crackling waves was
proportional to the jet temperature. Ffowcs Williams et al. [4]
stated that crackle and near-field skewness were generated by
the steepening of the waveform near the source at the jet. A
recent computational study by Nichols et al. [9] verified that
crackling waves seemed to be generated in the shear layer of the
jet, specifically in the region where the core flow transitions to
subsonic.
McInerny [5] proposed that the time derivative of the
pressure (dP/dt) would be a better indicator of crackle. In a
study done on the far-field acoustic signals of the Delta, Scout
and Titan IV rockets, it was proven that the rms, skewness and
kurtosis of the dP/dt are more sensitive indicators of crackle.
The dP/dt statistics allow us to recognize whether shock-like
structures are expected in a signal. While the statistics of the
pressure signal allow the one-sidedness of the waveform to be
identified, they do not contain information regarding the
steepening of the wave. [5, 10]. Gee et al. [10] also showed that
the higher order statistics or moments of the Probability Density
Function (PDF) of the dP/dt need to be considered when
analyzing cracklings jets.
Ffocws Williams et al. [4] briefly analyzed how adding
notches to a round nozzle could mitigate crackle. Skewness was
reduced to values below 0.3, and crackle was not perceived
during experimentation. Papamoschou et al. [11] showed that
Mach Wave radiation could be mitigated with a secondary flow.
This indicates that, under specific operating conditions, the
convective Mach number of large turbulent structures in the jet
core becomes subsonic relative to the secondary flow.
Similarly, the large turbulent structures of the secondary flow
travel subsonic relative to the surrounding ambient. With
subsonic convective Mach numbers in both mixing layers, no
Mach waves are generated. In a more recent study, Martens et
al. [12] showed that, similar to notches, chevrons decreased
OASPL and pressure skewness levels in the far field of the full
scale F404 static engine and a 1/6th
scaled model of the same
nozzle. In our study, we analyze the effect of implementing
chevrons at the exit of a round nozzle, focusing on
understanding the generation of pressure and dP/dt high-order
statistics at the near- and far-fields of the heated supersonic jet.
EXPERIMENTAL SETUP The tests described in this paper were performed in the
Supersonic Heated Jet Rig, which is part of the Gas Dynamics
and Propulsion Laboratory at the University of Cincinnati. A C-
D round nozzle shown in Fig. 1-a), baseline nozzle with exit
diameter of De = 0.813 inches and design Mach of 1.5, was
utilized to simulate a military aircraft engine nozzle. It was
tested at over-expanded (NPR=2.5), design (NPR=3.67), and
under-expanded (NPR=4.5) conditions, and stagnation
temperatures at the plenum of To = 300 K and 600 K, cold and
heated jets. In comparison to smooth contoured nozzles, the C-
D nozzle for the current report is conical and has a sharp throat.
This type of nozzle generates two sets of shock diamonds, one
starting at the throat and one at the exit nozzle [1]. These shock
diamonds (and, as a consequence, the broadband shock-
associated noise) build up even at design condition, in contrast
3 Copyright © 2013 by ASME
with smooth contoured nozzles, designed by the method of
characteristics, which would be shock free at design Mach
Number. The conical nozzle profile was selected because it
more accurately models military tactical aircraft exhausts with
variable area ratios. A chevron nozzle, shown in Fig. 1-b), was
also developed and tested under the same conditions. The
nozzle had the same geometry as the baseline, but with 12
chevrons incorporated at the nozzle exit. The chevrons are
equally distributed in the azimuthal direction. Each one is 0.2
De in length, and penetrates the jet 0.08 De, normal to the
divergent profile of the nozzle.
a) b)
Figure 1. Cross sectional view, C-D conical nozzles with
sharp throat: a) Baseline; b) Chevrons. De = 0.542 in. Md =
1.5.
The layout of the Supersonic Heated Rig is shown in Fig. 2.
Near-field measurements were taken with a grid domain of 64
by 36 microphone positions, in the axial and radial directions,
respectively. All positions where evenly spaced with intervals
of 0.5 De axially and radially. The microphone axial rows where
aligned at an angle of 10° from the nozzle axis in order to clear
the shear layer as the jet spreads.
Figure 2. Plan view of UC-GDPL’s Aeroacoustic Test
Facility. Far-field and near-field acoustic arrangements.
Acoustic measurements were recorded in the far field with
an arc array of 9 microphones. All microphones in the array had
a distance of 55 De from the nozzle exit to the microphone
diaphragm. The array spanned upstream angles of ψ = 90o to
downstream angles of 155o. Microphone angle (ψ) is measured
with its pivot centered at the intersection of the centerline and
the exit plane of the nozzle, and with the 0o
angle starting from
the upstream region relative to the jet. The microphones were
unevenly distributed in such a way that more microphone
positions were near the peak noise location for the hot nozzle
case (Mach wave angle direction). The Mach wave angle was
predicted to be around 145o for To = 600K. The final
microphone distribution is shown in Table 1. All microphones
diaphragms were facing normal toward the nozzle exit.
Table 1. Far-field microphone position angles.
900 116
0 125
0 130
0 135
0 140
0 145
0 150
0 155
0
B&K 4954B quarter-inch microphones were mounted in
the near- and far-field domains. The protective grid caps were
removed from all microphones to eradicate the need for
amplitude correction at the high frequencies. Data was taken at
a sampling frequency of 204.8 kHz. For the near and far fields,
2 seconds of data was recorded. Each set of data was split into
blocks of 4096 data points, corresponding to 100 blocks for the
near-field data and 250 blocks for the far-field data. Then, Fast
Fourier Transform was applied to obtain the narrowband noise
spectrum for each group and ensemble averaged over the
blocks. Finally, frequency was non-dimensionalized to obtain
the results as a function of Strouhal number.
RESULTS AND DISCUSSION
Near-field Schlieren results are shown in Fig. 3 for the cold
and heated jets of the baseline and chevron nozzles, design
condition. The appearance of a different number of chevrons at
the nozzle exit in Fig. 3 is a result of the chevron case having
been rotated to a different angle between the cold and heated
conditions. Consistent with the Munday et al. [1] study on
conical nozzles, the baseline jets contain sets of double shocks:
one that is generated at the lips of the nozzle, and another
starting at the nozzle throat. On the other hand, the chevron
nozzle jet core contains a single set of shocks, and the shock
spacing was measured to be shorter. The asymmetry of the
chevron design also makes the compression and expansion
waves look weaker. All of these factors work to reduce the
shock and mixing noise of the jet.
4 Copyright © 2013 by ASME
Figure 3. Near-field Schlieren images. Baseline and chevron
nozzles. NPR = 3.67.
Far-field OASPL for the cold and heated jets at design
condition, baseline and chevron cases, is displayed in Fig. 4.
The OASPL increases alongside stagnation temperature at all
microphone angles in the far field, which is mostly attributed to
the increase in jet velocity [13]. However, velocity scaling does
not account for the high amplitude OASPL in the heated nozzle,
around the 145o microphone direction. Mach Wave radiation is
a component of supersonic heated jets that propagates to the
downstream angles, and becomes the main contributor to the
elevated OASPL values in the far field [6,7]. This is
demonstrated in the heated jet results in Fig. 4, where the peak
of the OASPL is close to the Mach Wave angle at which the
Mach wave radiation propagates.
For the cold and heated jets in Fig. 4, the chevron case
shows a significant reduction in OASPL in the downstream
region of the jet, starting at ψ = 135o
for the cold and at ψ =
125o
for the heated jet. In the direction of peak OASPL
propagation, for both temperature conditions, the chevrons
produce a noise reduction of up to 5dB. At the 90o angle,
minimum change in OASPL can be observed between the
baseline and chevron cases.
90 116 125 130 135 140 145 150 155120
122
124
126
128
130
132
134
136
OA
SP
L
o
T0 = 300K Baseline
T0 = 300K Chevrons
T0 = 600K Baseline
T0 = 600K Chevrons
Figure 4. Far field. OASPL vs. Microphone angle (ψ).
Baseline and chevron nozzles. NPR = 3.67.
The spectrum of the acoustic pressure for the baseline and
chevron cases, To = 300 K and 600 K, design condition, at two
different far-field microphone angles, ψ = 90o and 145
o, is
shown in Fig. 5. Broadband shock-associated noise can be
observed for the cold and heated jets, baseline and chevron
cases, in the high frequencies of the ψ = 90o spectrum plots in
Figs. 5-a) and c). Shocks cells are contained even at design
conditions for conical nozzles [1]. Sound pressure levels at the
ψ = 140o position, shown in Figs. 5-b) and d), are notoriously
higher for the heated jet relative to the cold jet. This region in
the aft quadrant is the propagation path for mixing noise from
large turbulent structures. Also, the cold jet spectra from the
baseline contain strong screeching tones, at ψ = 90o and 145
o
microphone positions, compared to the heated jet.
In agreement with previous studies, Figs. 5 shows that
chevrons significantly reduced low-frequency noise, with a
penalty of increased high-frequency noise propagating normal
to the jet axis, as can be observed in Figures 5-a), and c). In the
cold jet, the chevrons diminished the strong screech tone
observed in the baseline spectra at ψ = 90o and 140
o
microphone positions. Heated jet results at the 140o propagation
angle in Fig. 5-c) show that chevrons effectively decreased
mixing noise across all frequencies. Finally, no intense screech
tone were contained in the heated jet, T0 = 600 K, at design
condition, NPR = 3.67, for both nozzle configurations. This
condition becomes an important case of study for comparing
high-order statistics in the acoustic fields of the baseline and
chevron cases.
5 Copyright © 2013 by ASME
100
101
102
80
90
100
110
(a)
= 90o
To =
300K
SP
L [dB
]
100
101
102
80
90
100
110
(b)
= 145o
100
101
102
80
90
100
110
To =
600K
SP
L [dB
]
fDe/U
j
(c)
100
101
102
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90
100
110
fDe/U
j
(d)
Baseline
Chevrons
Baseline
Chevrons
100
101
102
80
90
100
110
(a)
= 90o
To =
300K
SP
L [dB
]
100
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102
80
90
100
110
(b)
= 145o
100
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102
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To =
600K
SP
L [dB
]
fDe/U
j
(c)
100
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102
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fDe/U
j
(d)
Baseline
Chevrons
Baseline
Chevrons
a) b)
100
101
102
80
90
100
110
(a)
= 90o
To =
30
0K
SP
L [
dB
]
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90
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110
(b)
= 145o
100
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To =
60
0K
SP
L [
dB
]
fDe/U
j
(c)
100
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102
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90
100
110
fDe/U
j
(d)
Baseline
Chevrons
Baseline
Chevrons
100
101
102
80
90
100
110
(a)
= 90o
To =
300K
SP
L [dB
]
100
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102
80
90
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110
(b)
= 145o
100
101
102
80
90
100
110
To =
600K
SP
L [dB
]
fDe/U
j
(c)
100
101
102
80
90
100
110
fDe/U
j
(d)
Baseline
Chevrons
Baseline
Chevrons
c) d)
Figure 5. Far-field acoustic spectra. ψ measured from the
upstream. Baseline and chevron nozzles. NPR = 3.67.
Far-field pressure skewness vs. microphone angle is shown
in Fig. 6 for the baseline and chevron cases. Pressure skewness
values of the cold jet are relatively low at all microphone angles
(below 0.2). For the hot baseline jet, a bump of elevated
skewness levels can be observed in the aft quadrant, reaching an
approximate value of 0.35, at a microphone angle of 145o.
Maximum skewness seems to propagate at a direction similar to
the OASPL peak propagation angle. This is consistent with
results shown by Krothapalli et al. Similar propagation angles
suggest that the same source that generates the high levels of
OASPL may be generating high-pressure skewness in the far
field.
Figure 6 shows how chevrons effectively reduced pressure
skewness generation in the downstream region of the heated
condition. Skewness values dropped to levels below 0.2 at each
microphone position. On the other hand, the cold jet skewness
seems to have increased at most microphone locations.
However, the levels of skewness at this condition, for the
baseline and chevron cases, remained considerably lower than
the magnitudes of the heated jet.
90 116 125 130 135 140 145 150 1550
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Pre
ssure
Skew
ness
o
T0 = 300K Baseline
T0 = 300K Chevrons
T0 = 600K Baseline
T0 = 600K Chevrons
Figure 6. Far field. Pressure Skewness vs. Microphone angle
(ψ). Baseline and chevron nozzles. NPR = 3.67.
Far-field plots in Fig. 7 show elevated intensity of dP/dt
skewness and high directivity toward the downstream direction.
Similar to the pressure skewness, the heated case contains
higher dP/dt skewness magnitudes relative to the cold jet.
Unlike the OASPL and pressure skewness of the hot baseline
jet, dP/dt skewness reached a maximum value at about 147o,
slightly closer to the jet axis. For the cold jet, the peak for the
baseline case is not reached up to 155o, and the skewness levels
continue increasing past this microphone angle.
Similar to the reductions in OASPL and pressure skewness,
Figure 7 shows how chevrons effectively decreased the dP/dt
skewness in the aft region. More than a 50% reduction in
skewness magnitude is observed for the heated jets after the
implementation of the chevrons. The cold jet likewise
demonstrated a reduction in skewness toward the downstream
angles, starting at about ψ = 145o. It is also important to point
out that the dP/dt skewness peak propagation angle was shifted
by the chevron case to the upstream direction, predominantly
observed in the heated condition.
6 Copyright © 2013 by ASME
90 116 125 130 135 140 145 150 1550
0.5
1
1.5
2
dP
/dt S
kew
ness
o
T0 = 300K Baseline
T0 = 300K Chevrons
T0 = 600K Baseline
T0 = 600K Chevrons
Figure 7. Far field. dP/dt skewness vs. Microphone angle
(ψ). Baseline and chevron nozzles. NPR = 3.67.
The graphs in Fig. 8 show the pressure and the dP/dt time
traces for the heated jets for the baseline and chevron cases,
with a microphone position of ψ = 145o (propagation angle of
maximum skewness for the heated jet). The strong shock-like
pressure bursts that generate elevated levels of skewness in the
pressure and dP/dt time series are visible in the pressure signal
of the baseline, shown in Fig. 8-a). These pressure bursts make
the waveform asymmetric with respect to the zero axes, with a
tendency for high positive pressure values and therefore high
skewness. These pressure bursts in the baseline heated jet
contain rapid growths in pressure (shock-like behavior) relative
to the gradual compressions observed for the chevron case in
Fig. 8-b). These sharp compression stages in the baseline jet can
be easily identified in the dP/dt time traces in Fig. 8-c),
appearing as strong positive spikes, reaching values up to 170
MPa. On the other hand, the chevron case in Fig. 8-d) lacks the
strong burst of pressure and the intense spikes in the waveform
of the dP/dt. As a result, a substantial reduction in pressure and
dP/dt skewness is observed at the 145o far-field location in Fig.
7.
1212.5 1213 1213.5
-500
0
500
Pre
ssu
re [M
Pa
]
Baselinesk = 0.35
(a)
1212.5 1213 1213.5
-100
0
100
dP
/dt [M
Pa
/s]
Time [ms](c)
sk = 1.93
1212.5 1213 1213.5
-500
0
500
Chevronssk = 0.12
(b)
1212.5 1213 1213.5
-100
0
100
Time [ms](d)
sk = 0.7
1212.5 1213 1213.5
-500
0
500
Pre
ssu
re [M
Pa
]
Baselinesk = 0.35
(a)
1212.5 1213 1213.5
-100
0
100
dP
/dt [M
Pa
/s]
Time [ms](c)
sk = 1.93
1212.5 1213 1213.5
-500
0
500
Chevronssk = 0.12
(b)
1212.5 1213 1213.5
-100
0
100
Time [ms](d)
sk = 0.7
a) b)
1212.5 1213 1213.5
-500
0
500
Pre
ssu
re [M
Pa
]
Baselinesk = 0.35
(a)
1212.5 1213 1213.5
-100
0
100
dP
/dt [M
Pa
/s]
Time [ms](c)
sk = 1.93
1212.5 1213 1213.5
-500
0
500
Chevronssk = 0.12
(b)
1212.5 1213 1213.5
-100
0
100
Time [ms](d)
sk = 0.7
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-500
0
500
Pre
ssu
re [M
Pa
]
Baselinesk = 0.35
(a)
1212.5 1213 1213.5
-100
0
100
dP
/dt [M
Pa
/s]
Time [ms](c)
sk = 1.93
1212.5 1213 1213.5
-500
0
500
Chevronssk = 0.12
(b)
1212.5 1213 1213.5
-100
0
100
Time [ms](d)
sk = 0.7
c) d)
Figure 8. Far field. Pressure and dP/dt time traces, ψ = 145o.
Baseline and chevron nozzles. NPR = 3.67. To = 600 K.
The kurtosis of the pressure temporal waveform (forth order
statistic) is plotted against microphone angle in Fig. 9, for the
baseline and chevron cases. Kurtosis levels for the baseline cold
jet did not fluctuate much from the “mesokurtic” value of 3.0,
which corresponds to a PDF with a normal distribution.
However, the kurtosis values of the hot baseline jet, similarly to
those of the pressure skewness, are considerably elevated in the
aft quadrant. This bump in the pressure kurtosis shows a
directivity pattern similar to the pressure skewness, with the
maximum values propagating at about ψ = 145o. In a manner
similar to the OASPL and skewness reductions, chevrons
decreased the kurtosis levels for the heated jet, but not for the
cold jet, where a sudden rise in pressure kurtosis can be
observed at the 130o microphone location. The kurtosis peak
propagation angles also shifted to the upstream direction.
7 Copyright © 2013 by ASME
90 116 125 130 135 140 145 150 1553
3.05
3.1
3.15
3.2
3.25
3.3
3.35
3.4
Pre
ssure
Kurt
osis
o
T0 = 300K Baseline
T0 = 300K Chevrons
T0 = 600K Baseline
T0 = 600K Chevrons
Figure 9. Far field. Pressure Kurtosis vs. Microphone angle
(ψ). Baseline and chevron nozzles. NPR = 3.67.
Third and fourth order statistics behave in a similar way for
the pressure and the dP/dt signals. The levels of pressure
kurtosis are much lower compared to the levels of dP/dt
kurtosis for the baseline and chevron cases, especially for the
heated jet. The elevated positive values of dP/dt kurtosis, shown
in Fig. 10, imply that the standard deviation is a consequence of
occasionally intense variances and not due to recurrent
variations with low magnitudes. In other words, the region of
high kurtosis contains occasional strong dP/dt peaks, which are
the slopes of the compressive stages of the pressure bursts.
Similarly to the dP/dt skewness, in Fig. 10 the dP/dt kurtosis
of the heated jet in the baseline jet attains strong values in the
downstream angles. However, the peak of the dP/dt kurtosis
propagates to the microphone angle slightly above145o, which
is closer to the downstream jet axis compared to the OASPL
peak and pressure skewness peak propagation angles. This
implies that the phenomenon responsible for the increase in
higher-order statistics for the pressure and the dP/dt might be
located in a different region of the jet’s shear layer. Similarly,
the chevrons significantly reduced the dP/dt kurtosis levels for
the heated jet. The dP/dt kurtosis peak propagation angle also
experienced a shift in direction, going from ψ = 145o to 140
o,
baseline and chevron cases, respectively. The cold jet also
experienced a reduction in dP/dt kurtosis at the downstream
angles, but the magnitudes were negligible.
90 116 125 130 135 140 145 150 1553
4
5
6
7
8
9
10
dP
/dt K
urt
osis
o
T0 = 300K Baseline
T0 = 300K Chevrons
T0 = 600K Baseline
T0 = 600K Chevrons
Figure 10. Far field. dP/dt Kurtosis vs. Microphone angle
(ψ). Baseline and chevron nozzles. NPR = 3.67.
Figure 11 shows the far-field OASPL, pressure skewness,
and dP/dt skewness, for the heated jet exhausting from the
baseline and chevron nozzles, at over, design, and under-
expanded conditions. Chevrons significantly reduced the
OASPL in the downstream angles, at the design and under-
expanded conditions. For the over-expanded condition, the
implementation of chevrons increased OASPL across all
propagation angles, especially in the region dominated by fine-
scale noise normal to the jet axis [14].
In the pressure and dP/dt skewness plots, higher NPR
produced stronger skewness magnitudes and moved peak
propagation angles further upstream. This is consistent with
previous studies that linked intense skewness levels with higher
stagnation temperatures and therefore higher convective Mach
numbers [7]. Then, observing the downstream angles for the
chevron cases, pressure skewness magnitudes were reduced at
design and under-expanded conditions, while dP/dt skewness
decreased at all operating conditions. Generally, chevrons seem
to mitigate higher-order statistics at the design and under-
expanded conditions, where they reduce the OASPL the most.
For the highly over-expanded case of NPR=2.5, chevrons
decreased the dP/dt skewness but increased OASPL and
pressure skewness magnitudes at the 90o microphone location.
8 Copyright © 2013 by ASME
Baseline Chevrons
125 135 145 155120
125
130
135
140
OA
SP
L
o
125 135 145 155120
125
130
135
140
OA
SP
L
o125 135 145 155
120
125
130
135
140
OA
SP
L
o
125 135 145 1550
0.1
0.2
0.3
0.4
Pre
ssure
Skew
ness
o 125 135 145 155
0
0.1
0.2
0.3
0.4
Pre
ssure
Skew
ness
o
125 135 145 1550
0.1
0.2
0.3
0.4
Pre
ssure
Skew
ness
o
125 135 145 1550
0.5
1
1.5
2
dP
/dt S
kew
ness
o
125 135 145 1550
0.5
1
1.5
2
dP
/dt S
kew
ness
o
125 135 145 1550
0.5
1
1.5
2
dP
/dt S
kew
ness
o
125 135 145 1550
0.5
1
1.5
2
dP
/dt S
kew
ness
o
NPR = 2.5 NPR = 3.67 NPR = 4.5
Figure 11. Far field. OASPL, Pressure and dP/dt Skewness
vs. Microphone angle (ψ). Baseline and chevron nozzles. T0
= 600 K.
Near-field results for the baseline and chevron cases, design
NPR and heated jet, are shown in Figs. 11-13. The OASPL
contour plot for the baseline case is displayed in Fig. 11-a). The
OASPL maximum is radiating toward the downstream region
where the mixing noise and Mach Wave radiation from the
large turbulent structures are maximum. The chevron case on
the other hand, Fig. 11-b), significantly reduces the mixing
noise, and the shock noise is seen to propagate perpendicular to
the jet axis. It is important to notice that the OASPL in Figs. 11-
a) and b) decreases while it propagates to the far field, as
compared to the dP/dt statistics in Figs. 12-c) and 13-a), which
increase alongside with propagation distance.
OASPL
Baseline Chevrons
0 10 20 300
5
10
15
20
X/De
Y/D
e
135
140
145
150
0 10 20 30
0
5
10
15
20
X/De
Y/D
e
135
140
145
150
a) b)
Figure 11. Near field. OASPL: a) Baseline; b) Chevrons. T0
= 600 K, NPR = 3.67.
The pressure skewness for the baseline case, shown in Fig.
12-a), appears to originate near the shear layer and then
propagate to aft region. On the other hand, the chevron case in
Fig. 12-b) appears to have diminished the skewness levels that
were propagated to the downstream, with slight intensification
of the skewness propagated to the upstream. In the far field, it
was also shown that the chevrons amplified positive skewness
levels, which could be linked to the screech tones shifting to the
upstream direction. For the dP/dt skewness in Figs. 12-c) and
d), the elevated levels observed in the baseline case start to
amplify between the 5-10 De downstream of the nozzle exit and
just outside the jet’s shear layer, and intensify as the pressure
signal propagates toward the aft angles in the far field. In Fig.
12-d), the chevrons also significantly decrease the strong levels
of dP/dt skewness that were being generated in the baseline.
Also an upstream shift of the peak propagation angle can be
identified.
9 Copyright © 2013 by ASME
Pressure Skewness
Baseline Chevrons
0 10 20 300
5
10
15
20
X/De
Y/D
e
0
0.1
0.2
0.3
0.4
0 10 20 300
5
10
15
20
X/De
Y/D
e
0
0.1
0.2
0.3
0.4
a) b)
dP/dt Skewness
0 10 20 300
5
10
15
20
X/De
Y/D
e
0
0.5
1
1.5
0 10 20 30
0
5
10
15
20
X/De
Y/D
e
0
0.5
1
1.5
c) d)
Figure 12. Near field. Pressure skewness: a) Baseline; b)
Chevrons. dP/dt skewness: c) Baseline; d) Chevrons. T0 =
600 K, NPR = 3.67.
Pressure kurtosis levels do not deviate much from a value of
3, which corresponds to a normal distribution. Consequently,
not many conclusions can be drawn when observing the near-
field contour plots of the pressure kurtosis. Similar to the far-
field, the dP/dt kurtosis is a more sensitive measure, thus the
graphs shown in Fig. 13 contain more information regarding its
generation and directivity compared to the pressure kutosis. For
the baseline case in Fig. 13-a), the dP/dt kurtosis starts to
develop right after the shear layer of the jet, 5-10 De
downstream of the jet exit. When observing the results in Fig.
13-b), chevrons appear to have efficiently reduced dP/dt
kurtosis levels. In general, it was observed that the dP/dt
kurtosis in the near field behaved similarly to the dP/dt
skewness, consistent with the similarities observed for both
statistics in the far field.
dP/dt Kurtosis
Baseline Chevrons
0 10 20 300
5
10
15
20
X/De
Y/D
e
3
4
5
6
7
8
0 10 20 300
5
10
15
20
X/De
Y/D
e
3
4
5
6
7
8
a) b)
Figure 13. Near field. dP/dt Kurtosis: a) Baseline; b)
Chevrons. T0 = 600 K, NPR = 3.67.
CONCLUSIONS Near- and far-field results were shown for a cold and heated
conical nozzle with De = 0.813, Md = 1.5, operating at over,
design, and under-expanded conditions, with and without
chevrons. For the baseline case, far-field results, the OASPL
was significantly higher at all microphone positions for the
heated case, with the peak noise propagation angle consistent
with the peak Mach wave radiation angle. The magnitudes of
pressure and dP/dt skewness and kurtosis for the cold jet were
negligible and near to values corresponding to a normal
distribution. For the heated jet, strong levels of pressure and
dP/dt high-order statistics were observed.
Regarding noise directivity, while OASPL, skewness, and
kurtosis peaks all propagated near the Mach Wave angle, the
dP/dt statistics propagated at a slightly different direction
compared to the OASPL and pressure higher-order statistics
peak propagation angles. Higher NPR also lead to increased
OASPL, pressure, and dP/dt skewness magnitudes, with the
peak Mach wave radiation angle moving upstream with
increasing NPR (higher convective Mach number). The highly
over-expanded condition of NPR=2.5 showed low skewness
and kurtosis levels.
It has previously been demonstrated that notched nozzles
and chevrons decrease pressure skewness in the far-field of a
full-scale engine and a scaled model, potentially resulting in the
mitigation of crackle noise. This study focused on how
chevrons impact both pressure and dP/dt higher-order statistics,
in the near and far-fields of cold and heated jets. Chevrons
reduced the low-frequency noise at the aft angles. For the
heated jet condition, which showed strong skewness and
kurtosis magnitudes in the far field, the implementation of
chevrons efficiently reduced the intense levels of pressure and
dP/dt statistics that propagated near the Mach Wave radiation
angles. It was also observed that chevrons shifted upstream the
peak propagation angle of the pressure and dP/dt skewness and
kurtosis. For the cold jet, chevrons also decreased the dP/dt
skewness and kurtosis magnitudes in the far field, but they were
not effective at the 125o microphone location, where an increase
in magnitude was observed for the pressure statistics. However,
10 Copyright © 2013 by ASME
skewness and kurtosis values for the cold jets issued from the
baseline and chevron nozzles were initially low relative to the
values for the heated jet.
Near-field contour plots were presented for the heated jet at
design condition, all nozzle configurations. In the baseline
results, elevated pressure and dP/dt skewness levels were
already observed in the near field of the jet, validating the
steepening of the wave-front near the shear layer. The dP/dt
skewness in particular seems to be generated near the shear
layer of the jet between 5-10De downstream of the jet exit. For
the chevron cases, a similar pattern was observed in the near
field when compared with the noise mitigation in the far field.
Chevrons reduced the mixing noise represented in intense
OASPL, pressure, and dP/dt skewness and kurtosis levels that
propagate in downstream direction. This demonstrates that the
chevrons target the source of Mach wave radiation and
potentially crackle. A slight increase in pressure and dP/dt
statistics was observed to propagate in the upstream direction,
both in the near and far fields.
ACKNOWLEDGMENTS This research has been sponsored by the Office of Naval
Research (ONR) through the Jet Noise Reduction (JNR) Project
under the Noise Induced Hearing Loss (NIHL) program, as well
as the NRL 6.1 Computational Physics Task Area.
NOMENCLATURE a∞ ambient speed of sound
De diameter at nozzle exit
dP/dt time derivative of the pressure
f frequency
kt kurtosis
Mc convective Mach number (Uc/ a∞)
Md design Mach
NPR nozzle pressure ratio
OASPL overall pressure level
PDF probability density function
Φ Mach wave angle
ψ microphone angle with the upstream axis
sk skewness
St Strouhal number
T∞ ambient temperature
Tj jet temperature at nozzle exit
To nozzle stagnation temperature
TR temperature ratio, Tj/T∞
Uc convective velocity of large turbulent structures
Uj jet velocity at nozzle exit
X/ De axial distance, normalized by exit diameter
r/ De radial distance, normalized by exit diameter
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11 Copyright © 2013 by ASME
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