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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 610
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
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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 ¼0:5r
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)
