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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: morapa@mail.uc.edu, ephraim.gutmark@uc.edu

K. Kailasanath Naval Research Laboratory

Email: kailas@lcp.nrl.navy.mil

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

80

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

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

80

90

100

110

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

]

100

101

102

80

90

100

110

(b)

= 145o

100

101

102

80

90

100

110

To =

60

0K

SP

L [

dB

]

fDe/U

j

(c)

100

101

102

80

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

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

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

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

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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