Numerical study of the flow and the near acoustic fields of an underexpanded round free jet generating two screech tones

TL;DR: In this paper, the authors explored the flow and near acoustic fields of a supersonic round free jet using a compressible large eddy simulation and found that the regions of highest amplitude in the jet are located in the fifth and the sixth cells of the shock cell structure.

Abstract: The flow and near acoustic fields of a supersonic round free jet are explored using a compressible large eddy simulation. At the exit of a straight pipe nozzle, the jet is underexpanded, and is characterized by a Nozzle Pressure Ratio of 4.03 and a Temperature Ratio of 1. It has a fully expanded Mach number of 1.56, an exit Mach number of 1, and a Reynolds number of 60000. Flow snapshots, mean flow fields and convection velocity in the jet shear layers are consistent with experimental data and theoretical results. Furthermore, two screech tones are found to emerge in the pressure spectrum calculated close to the nozzle. Using a Fourier decomposition of the pressure fields, the two screech tones are found to be associated with anticlockwise helical oscillation modes. Besides, the frequencies of the screech tones and the associated oscillation modes both agree with theoretical predictions and measurements. Moreover, pressure fields filtered at the screech frequencies reveal the presence of hydrodynamic-acoustic standing waves. In those waves, the regions of highest amplitude in the jet are located in the fifth and the sixth cells of the shock cell structure. The two screech tones therefore seem to be linked to two different loops established between the nozzle and the fifth and sixth shock cells, respectively. In the pressure fields, three other acoustic components, namely the low-frequency mixing noise, the high-frequency mixing noise and the broadband shock-associated noise, are noted. The directivity and frequency of the mixing noise are in line with numerical and experimental studies. A production mechanism of the mixing noise consisting of sudden intrusions of turbulent structures into the potential core is discussed. Then, the broadband shock-associated noise is studied. This noise component is due to the interactions between the turbulent structure in the shear layers and the shocks in the jet. By analyzing the near pressure fields, this noise component is found to be produced mainly in the sixth shock cell. Finally, using the size of this shock cell in the classical theoretical model of this noise component, a good agreement is found with the simulation results.

Summary

Introduction

• In non-ideally expanded supersonic jets, several acoustic components including screech noise, mixing noise and broadband shock-associated noise are observed.
• 7,8,9 For subsonic jets, Bogey et al.10 and Bogey and Bailly7 proposed that this acoustic component is due to the intermittent intrusion of turbulent structures into the potential core.
• Finally, the broadband shockassociated noise is examined.

Jets parameters

• The jet originates from a straight pipe nozzle of radius r0, whose lip is 0:1r0 thick.
• Its Reynolds number is Rej ¼ ujDj= ¼ 6 104, where Dj is the nozzle diameter of the ideally expanded equivalent jet and is the kinematic molecular viscosity.
• At the nozzle inlet, a Blasius boundary-layer profile with a thickness of 0:15r0 and a Crocco-Busemann profile are imposed for velocity and density.
• The exit conditions of the jet and the nozzle lip thickness are similar to those in the experiments of Henderson et al.18.

Numerical parameters

• The LES is performed by solving the unsteady compressible Navier–Stokes equations on a cylindrical mesh ðr, , zÞ.
• An explicit six-stage Runge–Kutta algorithm and low-dispersion and low-dissipation explicit eleven-point finite differences are used for time integration and spatial derivation,20,21 respectively.
• At the end of each time step, a high-order filtering is applied to the flow variables in order to remove grid-to-grid oscillations and to dissipate subgrid-scale turbulent energy.
• 27 The axis singularity is treated with the method of Mohseni and Colonius.

Flow snapshots

• In the top figure, isosurfaces of density are displayed in order to show the shock-cell structure.
• Longitudinal structures appear on the outer boundary of the first shock cell.
• The temporal stability of these structures can be seen in the corresponding movie ‘‘Movie 2’’, available online at http://acoustique.ec-lyon.fr/publi/gojon_ija17_movie2.
• Notably, an acoustic component propagating in the upstream direction is visible in the vicinity of the nozzle.
• An empirical model was proposed by Lau et al. 32 to predict the length of the potential core zp for isothermal jets with Mach number up to 2.5.

Mean fields

• The mean axial and radial velocity fields of the jet are presented in Figure 5, where the experimental PIV results of André et al.37 are also displayed for aMj ¼ 1:5 andMe ¼ 1 jet.
• A good agreement is found between the simulation and the experiments.
• Moreover, the variation of the shock-cell size appears to behave linearly.
• The Mach disk position zM and diameter DM can be estimated from the mean velocity field, yielding zM ¼ 2:3r0 and DM ¼ 0:25r0.
• Experimentally, for jets with a NPR exceeding 3.9, Addy36 proposed the following empirical expressions zM Dj ¼ 0:65 ffiffiffiffiffiffiffiffiffiffiffi NPR p ð5Þ DM Dj ¼ 0:36 ffiffiffiffiffiffiffiffiffiffiffi NPR p 3:9 ð6Þ.

Velocity fluctuations

• The rms values of axial and radial velocity fluctuations obtained for the present jet are represented in Figure 9, where the experimental PIV results of André et al.37 are also shown.
• The results obtained in the LES and in the experiment are in fairly good agreement.
• Moreover, the jet shear layer is thicker in the LES than in the experiment.
• The peak rms values of velocity fluctuations in the jet shear layer are represented in Figure 10 as a function of the axial distance.
• In a given cell, it increases gradually and then decreases rapidly on the cell ending.

Convection velocity

• The local convection velocity of the turbulent structures is estimated at the center of the shear layer, where the velocity fluctuations are maximum, as presented in Figure 11.
• It is not constant but varies according to the shock-cell structure, as observed experimentally by André12 for round underexpanded jets.
• Farther downstream, the convection velocity decreases down to the value of 0:60uj, following the decrease of the velocity inside the jet due to the presence of a Mach disk and of an oblique annular shock.
• Similar variations are found for the other cells of the shock cell structure.
• Furthermore, the convection velocity is close to the value 0:35uj ’ 0:5ue at the nozzle exit, as expected for instabilities initially growing in the mixing layers just downstream of the nozzle.

Acoustic results

• Acoustic spectrum near the nozzle exit Two tones emerge 15 dB above the broadband noise at Strouhal numbers St1 ¼ 0:28 and St2 ¼ 0:305.
• Thus, the difference in nozzle lip thickness between the present jet and the experimental study of André12 is unlikely to explain the discrepancy in screech tone frequencies.
• In order to determine whether the jet products alternatively or simultaneously the two screech tones obtained in the spectrum of Figure 13, a Fast Fourier transform is applied using a sliding window in time, of size 35uj=Dj.
• The result is displayed in Figure 14(a) where the sound pressure level is represented as a function of time tuj=Dj and Strouhal number.
• A switch between these two tones is thus observed.

Fluctuating pressure in the jet

• The pressure fields in the (z, r) plane have been recorded every 50th time step.
• Both the frequencies of the screech tones and the associated oscillation modes are consistent with the predictions of Panda41 and the measurements of Powell et al.47.

Mixing noise

• The high-frequency mixing noise cannot be studied from amplitude and phase fields at specific frequencies given the large-frequency bandwidth of this noise component.
• The amplitude field of the fluctuating pressure obtained at the frequency St¼ 0.25 is represented in Figure 20.
• These factors are represented in Figure 21.
• These results indicate that large density deficits appear intermittently on the jet axis in the sixth cell, near the end of the potential core.
• Therefore, the mixing noise seems due to the sudden intrusion of turbulent structures, of low density compared to the exit density, in the potential core, as suggested by Bogey and Baily7 for subsonic jets and by Cacqueray and Bogey52 for an overexpanded jet with an ideally expanded Mach number ofMj ¼ 3:3.

Broadband shock-associated noise

• In order to investigate the broadband shock-associated noise, the amplitude fields of the pressure fluctuations obtained in the (z, r) plane for frequencies St¼ 0.28, 0.40, 0.50 and 0.60 are represented in Figure 22.
• The first frequency corresponds to the lower screech tone frequency.
• This direction is in good agreement with the value of 125 found for a frequency St¼ 0.40 using equation (9).
• Finally, the amplitude field at St¼ 0.60, in Figure 22(d), does not exhibit a clear directivity, but regions of high amplitude appear in the radial direction.
• Therefore, it seems that the constructive interference which produces the broadband shock-associated noise happens mainly between the turbulent structures in the jet shear layers and the shocks of the sixth shock cell.

Conclusion

• The flow and near pressure fields of an underexpanded supersonic jet have been described.
• The results, including the shock-cell structure, are consistent with experimental data, empirical and theoretical models.
• The convection velocity of large-scale structures in the jet shear layers is evaluated, and values similar to experimental data are found.
• Moreover, a temporal switch between the two screech tone frequencies is noted.
• Then in the near pressure fields, the low-frequency mixing noise, the high-frequency mixing noise and the broadband shock-associated noise are identified.

Figures

Figure 2. Isosurfaces of density: in purple and red for values of 0.8 and 2.5 kg.m 3, colored by the local Mach number in the top view and by the radial position in the bottom view for the value 1.25 kg.m 3. The pressure field at ¼ 0 and is also shown with a color scale ranging from 2000 to 2000 Pa for the pressure, from white to red. The nozzle is in grey. For the bottom view, a movie is available online at http://acoustique.ec-lyon.fr/publi/gojon_ija17_movie2.avi

Figure 8. (a) Position and (d) diameter of the Mach disk at the end of the first shock cell obtained in the present jet, in the experiment of Addy,36 and using expressions (5) and (6).

Figure 9. Rms values for the (a) axial and (b) radial velocities in the present jet. The colour scale ranges from 0 to 100 m.s 1, from blue to red; the PIV results of André et al.37 for a Mj ¼ 1:5 and Me ¼ 1 jet are shown in the black rectangles.

Figure 14. (a) Sound pressure level at r ¼ 2r0 and z¼ 0 as a function of time and of Strouhal number and (b) sound pressure levels at r ¼ 2r0 and z¼ 0 as a function of the time for the screech tones —— at St1 ¼ 0:28 and – – – at St2 ¼ 0:305. The color scale ranges from 145 to 165 dB/St.

Figure 15. Amplitude (left) and phase (right) fields of fluctuating pressure obtained in the (z, r) plane at the two screech tone frequencies of the simulated jet at (a,b) St1 ¼ 0:28 and (c,d) St2 ¼ 0:305. The colour scales range from 120 to 160 dB/St for the amplitude fields and from to for the phase fields.

Figure 22. Amplitude fields of the pressure fluctuations obtained in the (z, r) plane at the frequencies (a) St¼ 0.28, (b) St¼ 0.40, (c) St¼ 0.50, and (d) St¼ 0.60.

Figure 3. Snapshot of the vorticity norm j!j obtained in the (z, r) plane. The colour scale ranges up to the level of 10uj=Dj, from white to red. The nozzle is in black.

Figure 7. Normalized lengths of the first 10 shock cells obtained for the present jet, and the

Figure 13. Pressure spectrum at r ¼ 2r0 and z¼ 0 as a function of the Strouhal number.

Figure 19. Sound pressure levels on the circle centered on the jet axis at z ¼ 15r0, radius 15r0, as a function of the Strouhal number and of the angle with respect to the downstream direction.

Figure 11. Rms values of velocity fluctuations for the simulated jet with a color scale ranging from 0 to 100 m.s 1, from blue to red; the black line shows the position of the maximum values.

Figure 12. Convection velocity of the turbulent structures in the jet shear layers as a function of the axial position. The dashed vertical grey lines indicate the end of the cells in the shock-cell structure.

Figure 4. (a) Snapshot of the density in the jet and of the fluctuating pressure. The colour scale ranges from 1 to 3 kg.m 3 for the density, from blue to red, and from –2000 to 2000 Pa for the pressure, from white to black; (b) Schlieren picture of an underexpanded jet obtained by André et al.37 for a Mj ¼ 1:55 and Me ¼ 1 jet; a movie is available online at http://acoustique.ec-lyon.fr/publi/gojon_ija17_movie1.avi

Figure 20. Amplitude field of the pressure fluctuations obtained in the (z, r) plane at the frequency St¼ 0.25.

Figure 17. Normalized azimuthal distributions of the fluctuating pressure at z¼ 0 and r ¼ 2r0 (a) filtered around St1 ¼ 0:28 at three times separated by T1=3 and (b) filtered around St2 ¼ 0:305 at three times separated by T2=3; – – – unit circle.

Figure 18. Screech tone frequencies obtained in the experiments of Powell et al.47 and in the present jet as a function of the fully expanded Mach number Mj.

Figure 10. Peak rms values of (a) axial and (b) radial velocity fluctuations as a function of the axial position; —— simulation results and – – – PIV results of André et al.37 for a Mj ¼ 1:5 and Me ¼ 1 jet. The dashed vertical grey lines indicate the end of the cells in the shock-cell structure of the simulated jet.

Figure 5. Mean fields for (a) the axial and (b) the radial velocities. The colour scale ranges from 0 to 600 m.s 1 for the axial velocity, from blue to red, and from 150 to 150 m.s 1 for the radial velocity, from blue to red; the PIV results of André et al.37 for a Mj ¼ 1:5 and Me ¼ 1 jet are displayed in the black rectangles.

Figure 6. Mean total velocity field with a colour scale ranging from 0 to 600 m.s 1, from blue to red; the PIV results of Henderson et al.18 for a jet with similar exit conditions are displayed in the black rectangle.

Figure 21. (a, c) Skewness and (b, d) kurtosis factors of the density fluctuations obtained (a, b) in the (z, r) plane and (c, d) on the jet axis. The dotted line represent the middle of the sixth shock cell, at z ¼ 14:5r0.

Figure 1. Representation of (a) the radial mesh spacings, and (b) the axial mesh spacings.

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a publisher's
https://oatao.univ-toulouse.fr/18720
http://dx.doi.org/10.1177/1475472X17727606
Gojon, Romain and Bogey, Christophe Numerical study of the flow and the near acoustic fields of an
underexpanded round free jet generating two screech tones. (2017) International Journal of Aeroacoustics, vol. 16
(n° 7-8). pp. 603-625. ISSN 1475-472X

Article
Numerical study of the flow
and the near acoustic fields
of an underexpanded round
free jet generating two
screech tones
Romain Gojon and Christophe Bogey
Abstract
The flow and near acoustic fields of a supersonic round free jet are explored using a compressible
large eddy simulation. At the exit of a straight pipe nozzle, the jet is underexpanded, and is
characterized by a Nozzle Pressure Ratio of 4.03 and a Temperature Ratio of 1. It has a fully
expanded Mach number of 1.56, an exit Mach number of 1, and a Reynolds number of 610
4
.
Flow snapshots, mean flow fields and convection velocity in the jet shear layers are consistent
with experimental data and theoretical results. Furthermore, two screech tones are found to
emerge in the pressure spectrum calculated close to the nozzle. Using a Fourier decomposition of
the pressure fields, the two screech tones are found to be associated with anticlockwise helical
oscillation modes. Besides, the frequencies of the screech tones and the associated oscillation
modes both agree with theoretical predictions and measurements. Moreover, pressure fields
filtered at the screech frequencies reveal the presence of hydrodynamic-acoustic standing
waves. In those waves, the regions of highest amplitude in the jet are located in the fifth and
the sixth cells of the shock cell structure. The two screech tones therefore seem to be linked to
two different loops established between the nozzle and the fifth and sixth shock cells, respectively.
In the pressure fields, three other acoustic components, namely the low-frequency mixing noise,
the high-frequency mixing noise and the broadband shock-associated noise, are noted. The
directivity and frequency of the mixing noise are in line with numerical and experimental studies.
A production mechanism of the mixing noise consisting of sudden intrusions of turbulent struc-
tures into the potential core is discussed. Then, the broadband shock-associated noise is studied.
This noise component is due to the interactions between the turbulent structure in the shear
layers and the shocks in the jet. By analyzing the near pressure fields, this noise component is
found to be produced mainly in the sixth shock cell. Finally, using the size of this shock cell in the
Laboratoire de Me
´
canique des Fluides et d’Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon, Universite
´
de Lyon,
Ecully Cedex, France
Corresponding author:
Romain Gojon, Laboratoire de Me
´
canique des Fluides et d’Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon,
Universite
´
de Lyon, 69134 Ecully Cedex, France.
Email: romain.gojon@ec-lyon.fr
International Journal of Aeroacoustics
2017, Vol. 16(7–8) 603–625
! The Author(s) 2017
Reprints and permissions:
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DOI: 10.1177/1475472X17727606
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classical theoretical model of this noise component, a good agreement is found with the simu-
lation results.
Keywords
Large eddy simulation, supersonic jet, screech
Date received: 9 September 2016; accepted: 1 June 2017
Introduction
In non-ideally expanded supersonic jets, several acoustic components including screech
noise, mixing noise and broadband shock-associated noise are observed. The screech noise
is due to an aeroacoustic feedback mechanism established between the turbulent structures
propagating downstream and the acoustic waves propagating upstream. This mechanism
was described by Powell,
1
then by Raman,
2
who proposed that the turbulent structures
developing in the jet shear layers and propagating in the downstream direction interact
with the quasi-periodic shock cell structure of the jet, creating upstream propagating acous-
tic waves. The resonant loop is closed at the nozzle lips where sound waves are reflected back
and excite the shear layers. Moreover, for round jets, Powell
1
identified four modes, labeled
A, B, C, and D, on the basis of the screech frequency evolution with the ideally expanded
Mach number M
j
. Each mode is dominant for a specific ideally expanded Mach number
range and frequency jumps are noted between the modes. Later, Merle
3
showed that mode A
can be divided into modes A1 and A2. Davies and Oldfield
4
studied the oscillation modes of
the jets associated with the five screech modes. They found that A1 and A2 modes are linked
to axisymmetric oscillation modes of the jet, B to sinuous and sometimes helical modes, C to
helical modes and D to sinuous modes. Mixing noise is observed in both subsonic
5
and
supersonic
6
jets. The dominant Strouhal number of this noise component is around 0.2 and
its directivity is well marked around angles of 20

with respect to the downstream direction.
This component is mainly generated at the end of the potential core.
7,8,9
For subsonic jets,
Bogey et al.
10
and Bogey and Bailly
7
proposed that this acoustic component is due to the
intermittent intrusion of turbulent structures into the potential core. The broadband shock-
associated noise is produced by the interactions between the turbulence and the shock cell
structure. Martlew
11
was the first to clearly identify this noise. Its central frequency varies
with the angle in the far field, according to experiments.
12–14
Harper-Bourne and Fisher
15
proposed a model which permits to predict the central frequency of this noise component as
a function of the observation angle.
In the present work, the LES of a round supersonic underexpanded jet is carried out in
order to investigate the acoustic mechanisms in non-ideally expanded jets. The jet corres-
ponds to the reference free jet in a study on impinging jets performed by Gojon et al.
16
The
results from this jet were also used to generate schlieren-like images, in a study of Castelain
et al.,
17
in order to asses the quality of the estimation of the convection velocity in the jet
shear layers using schlieren pictures in experiments. In the present paper, the spectral and
hydrodynamic properties of the jet are described and compared with experimental data and
models. Three acoustic components, namely the screech noise, the mixing noise, and the
broadband shock-associated noise, are investigated. In particular, two screech tones are
found in the spectra calculated in the vicinity of the nozzle. The causes of such a result
604 International Journal of Aeroacoustics 16(7–8)

are sought. The production mechanism of the mixing noise is then investigated by evaluating
skewness and kurtosis factors of the fluctuating pressure. Finally, the broadband shock-
associated noise is examined. Notably, a discussion about the lengthscale to use in the
classical model of this noise component is conducted. The paper is organized as follows.
The jet parameters and the numerical methods used for the LES are given in the Parameters
section. The aerodynamic results are analyzed in the Aerodynamic results section, and the
acoustic mechanisms are investigated in the Acoustic results section. Concluding remarks are
provided in the last section.
Parameters
Jets parameters
The large-eddy simulation of a round supersonic jet is performed. The jet originates from a
straight pipe nozzle of radius r
0
, whose lip is 0:1r
0
thick. The jet is underexpanded, and has
a Nozzle Pressure Ratio of NPR ¼ P
r
=P
amb
¼ 4:03 and a Temperature Ratio
TR ¼ T
r
=T
amb
¼ 1, where P
r
and T
r
are the stagnation pressure and temperature and
P
amb
and T
amb
are the ambient values. As for a jet generated by a convergent nozzle,
the exit Mach number of the present jet is M
e
¼ u
e
=c
e
¼ 1, where u
e
and c
e
are the velocity
and speed of sound in the jet. Moreover, the jet is characterized by a fully expanded Mach
number of M
j
¼ u
j
=c
j
¼ 1:56, where u
j
and c
j
are the velocity and the speed of sound in
the ideally expanded equivalent jet. Its Reynolds number is Re
j
¼ u
j
D
j
= ¼ 6  10
4
, where
D
j
is the nozzle diameter of the ideally expanded equivalent jet and  is the kinematic
molecular viscosity. At the nozzle inlet, a Blasius boundary-layer profile with a thickness
of 0:15r
0
and a Crocco-Busemann profile are imposed for velocity and density. The exit
conditions of the jet and the nozzle lip thickness are similar to those in the experiments of
Henderson et al.
18
Finally, low-amplitude vortical disturbances, not correlated in the azi-
muthal direction,
19
are added in the boundary layer in the nozzle, at z ¼0:5r
0
, in order
to generate velocity fluctuations at the nozzle exit. The strength of the forcing is chosen in
order to obtain turbulent intensities of around 6% of the fully expanded jet velocity at the
nozzle exit.
Numerical parameters
The LES is performed by solving the unsteady compressible Navier–Stokes equations on a
cylindrical mesh ðr, , zÞ. An explicit six-stage Runge–Kutta algorithm and low-dispersion
and low-dissipation explicit eleven-point finite differences are used for time integration and
spatial derivation,
20,21
respectively. At the end of each time step, a high-order filtering is
applied to the flow variables in order to remove grid-to-grid oscillations and to dissipate
subgrid-scale turbulent energy. The filtering thus acts as a subgrid scale model.
22–25
The
radiation conditions of Tam and Dong
26
are implemented at the boundaries of the com-
putational domain. A sponge zone combining grid stretching and Laplacian filtering is also
employed to damp the turbulent fluctuations before they reach the boundaries. Moreover,
non-slip adiabatic conditions are used to simulate the nozzle walls. In order to increase the
time step of the simulation, the effective resolution near the origin of the cylindrical
coordinates is reduced.
27
The axis singularity is treated with the method of Mohseni
and Colonius.
28
Finally, a shock-capturing filtering is used in order to avoid Gibbs oscil-
lations near shocks. It consists in applying a conservative second-order filter at a
Gojon and Bogey 605

magnitude determined each time step using a shock sensor.
29
It was successfully used by
Cacqueray et al.
30
for the LES of an overexpanded jet at an equivalent Mach number of
M
j
¼ 3:3.
The simulation is carried out using an OpenMP-based in-house solver, and a total of 250,
000 iterations are computed during the steady state. The temporal discretization is set to
t ¼ 0:002D
j
=u
j
, permitting a simulation time of 500D
j
=u
j
. The cylindrical mesh contains
ðn
r
, n

, n
z
Þ¼ð500, 512, 1565Þ’400 million points. The variations of the radial and the axial
mesh spacings are represented in Figure 1. In Figure 1(a), the minimal axial mesh spacing is
located in the jet shear layer, at r ¼r
0
, and is equal to r ¼ 0:0075r
0
. Farther from the jet
axis, the mesh is stretched to reach the maximum value of r ¼ 0:06r
0
for 5r
0
 r  15r
0
.
For r  15r
0
, a sponge zone is implemented. In Figure 1(b), the minimal axial mesh spacing
is found at the nozzle lips, at z ¼0, and is equal to z ¼ 0:0075r
0
. Farther downstream, the
mesh is stretched, leading to z ¼ 0:03r
0
for 5r
0
 z  30r
0
. For z 4 30r
0
, a sponge zone is
applied. In the physical domain the grid is stretched at rates lower than 1%, in order to
preserve numerical accuracy. The maximum mesh spacing of 0:06r
0
in the physical domain
allows acoustic waves with Strouhal numbers up to St ¼ fD
j
=u
j
¼ 5:3 to be well propagated,
where f is the frequency. Finally, note that a similar mesh is used in a convergence study
made in a previous study for the LES of an initially highly disturbed high-subsonic jet.
19
Aerodynamic results
Flow snapshots
Three-dimensional views of the jet are displayed in Figure 2. In the top figure, isosurfaces of
density are displayed in order to show the shock-cell structure. The boundaries of the mixing
layer are also represented using isosurfaces of density. The bottom figure provides a zoomed
view of the nozzle exit region. Longitudinal structures appear on the outer boundary of the
first shock cell. The temporal stability of these structures can be seen in the corresponding
movie ‘‘Movie 2’’, available online at http://acoustique.ec-lyon.fr/publi/gojon_ija17_movie2.
avi. Such structures have been described in several experiments, including those by Arnette
et al.
31
They are due to the small perturbations at the nozzle exit which are amplified by
0 5 10 15 18
0
0.02
0.04
0.06
0.08
(a)
r/r
0
Δr/r
0
−2 0 2 4 6 8 10
0
0.02
0.04
0.06
0.08
(b)
z/r
0
Δz/r
0
Figure 1. Representation of (a) the radial mesh spacings, and (b) the axial mesh spacings.
606 International Journal of Aeroacoustics 16(7–8)

Frequently asked questions

Q1. What are the contributions in "Numerical study of the flow and the near acoustic fields of an underexpanded round free jet generating two screech tones" ?

Then, the broadband shock-associated noise is studied. Finally, using the size of this shock cell in the Laboratoire de Mécanique des Fluides et d ’ Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon, Université de Lyon, Ecully Cedex, France Corresponding author: Romain Gojon, Laboratoire de Mécanique des Fluides et d ’ Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon, Université de Lyon, 69134 Ecully Cedex, France. The Author ( s ) 2017 Reprints and permissions: sagepub. 

Q2. What are the future works in "Numerical study of the flow and the near acoustic fields of an underexpanded round free jet generating two screech tones" ?

The convection velocity of large-scale structures in the jet shear layers is evaluated, and values similar to experimental data are found. The mixing noise component seems due to the sudden intrusion of turbulent structures into the potential core, near its end. 

Q3. What is the strength of the forcing chosen?

The strength of the forcing is chosen in order to obtain turbulent intensities of around 6% of the fully expanded jet velocity at the nozzle exit. 

Q4. How is the convection velocity at the nozzle exit?

the convection velocity is close to the value 0:35uj ’ 0:5ue at the nozzle exit, as expected for instabilities initially growing in the mixing layers just downstream of the nozzle. 

Q5. What was the first to use schlieren pictures in experiments?

The results from this jet were also used to generate schlieren-like images, in a study of Castelain et al.,17 in order to asses the quality of the estimation of the convection velocity in the jet shear layers using schlieren pictures in experiments. 

Q6. How many times did Raman43 observe two screech tones switching in time?

Fornon-ideally expanded jets exiting from a rectangular nozzle with a single-bevelled exit, Raman43 also observed two screech tones switching in time. 

Q7. What is the central frequency of the broadband shock-associated noise?

the size of the sixth shock cell, Ls6 ¼ 2:35r0, located around z ¼ 15r0, is used in the relation (9) to compute the central frequency of the broadband shock-associated noise as a function of angle . 

Q8. What is the mechanism of the broadband shock-associated noise?

In this mechanism, the broadband shock-associated noise is generated by the interactions between the turbulent structures propagating downstream in the jet shear layers and the shocks of the quasi-periodic shock cell structure. 

Q9. What is the mean convection velocity of the cell structures in the jet?

In order to apply equation (8) to the simulated jet, the mean convection velocity is considered equal to 5 uc 4 ¼ 0:65uj in the region 5r0 5 z5 15r0, as suggested in Figure 12. 

Q10. What is the frequency of the first mode of the broadband shock-associated noise?

For screeching jets, Tam et al.13 suggested that the central frequency of the first mode N¼ 1 of the broadband shock-associated noise tends to the screech frequency at ¼ 180 . 

Q11. Where is the local convection velocity of the turbulent structures?

The local convection velocity of the turbulent structures is estimated at the center of the shear layer, where the velocity fluctuations are maximum, as presented in Figure 11.