of 30
PHYSICAL REVIEW FLUIDS
9
, 104604 (2024)
Superresolution and analysis of three-dimensional velocity fields of
underexpanded jets in different screech modes
Chungil Lee
,
1
,
*
Yuta Ozawa
,
2
Takayuki Nagata
,
3
Tim Colonius,
4
and Taku Nonomura
3
1
Institute of Fluid Science,
Tohoku University
, 980-0811 Sendai, Japan
2
Aviation Technology Directorate,
Japan Aerospace Exploration Agency
, 181-0015 Tokyo, Japan
3
Department of Aerospace Engineering,
Nagoya University
, 464-8603 Nagoya, Japan
4
Division of Engineering and Applied Science,
California Institute of Technology
,
Pasadena, California 91125, USA
(Received 26 December 2023; accepted 20 August 2024; published 18 October 2024)
Time-resolved (TR), three-dimensional (3D) velocity fields of screeching, under-
expanded jets are estimated using non-time-resolved particle image velocimetry and
simultaneous TR microphone measurements. Specifically, we aim to reconstruct TR 3D
velocity fluctuation fields associated with the A2, B, and C modes of a screeching jet
using a linear regression model and to analyze screech dynamics of these modes. The
linear regression model is constructed on the basis of a linear relationship between the
velocity and acoustic fields. Three nozzle pressure ratios (NPRs) of 2.30, 2.97, and 3.40
are employed. The dominant azimuthal modes for three cases are investigated using
azimuthal Fourier coefficients of the acoustic data obtained by the azimuthal array of
eight microphones placed near the nozzle exit. The dominant azimuthal modes at NPRs
of 2.30, 2.97, and 3.40 are
m
=
0, 1, and 1, respectively. The first two proper orthogonal
decomposition (POD) modes in these azimuthal modes are dominant at all NPRs and
are associated with screech. 3D velocity fluctuation fields associated with screech are
reconstructed from these leading POD modes of the acoustic data. The reconstructed
3D velocity fluctuation fields at NPRs of 2.97 and 3.40 exhibit two helical structures
with opposite rotation directions. The present results demonstrate that, in the B mode,
the flapping structure exhibits random clockwise and counterclockwise rotations over an
extended time domain, while maintaining a consistent direction within short time domains.
In addition, in the C mode, two helical structures with opposite rotation directions, as well
as the flapping structure, are observed.
DOI:
10.1103/PhysRevFluids.9.104604
I. INTRODUCTION
A shock-containing jet, e.g., an underexpanded supersonic jet, is generated due to the pressure
difference between the ambient and the choked nozzle. The interaction between shock cells and
turbulent structures results in the generation of two noise components: broadband shock-associated
noise and screech tones [
1
]. The screech tone exhibits high amplitude at a discrete frequency. Powell
[
2
] found discrete stages of the screech frequency and named them A1, A2, B, C, and D modes.
*
Contact author: chungil.lee.e7@tohoku.ac.jp
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0
International
license. Further distribution of this work must maintain attribution to the author(s) and the
published article’s title, journal citation, and DOI.
2469-990X/2024/9(10)/104604(30)
104604-1
Published by the American Physical Society
CHUNGIL LEE
et al.
Each mode exhibits different azimuthal features of the jet. The A1 and A2 modes are axisymmetric
modes, the C mode is a helical mode, and the B and D modes are flapping modes [
3
,
4
]. The screech
frequency in the same stage decreases as the nozzle pressure ratio (NPR) increases.
Powell [
2
] identified that screech tones are generated by resonant processes involving the
Kelvin-Helmholtz (KH) wavepacket traveling downstream and a freestream acoustic wave traveling
upstream. This acoustic wave excites the thin shear layer near the nozzle exit, producing new
downstream-traveling structures. Subsequently, several screech-frequency prediction models have
been introduced based on this theory [
5
7
]. These models can predict the approximate screech
frequency, but they are incapable of distinguishing the multiple screech modes during the staging
process. Shen and Tam [
8
] first suggested that screech may not be closed by a freestream acoustic
wave. They asserted that the resonance is closed by upstream-traveling freestream acoustic waves
for the A1 and B modes and by upstream-traveling guided-jet modes for the A2 and C modes.
The recent works for a screech-frequency prediction model showed that the resonance mechanism
may be closed by not a freestream acoustic wave, but rather the guided-jet modes [
9
15
]. These
guided-jet modes are generated from triadic interactions between the KH wavepacket and stationary
shock structures [
13
]. The prediction model based on the resonance between downstream-traveling
KH waves and upstream-traveling guided-jet modes provided better agreement with experimental
data than the classical prediction model using freestream acoustic waves [
11
].
The screech-frequency prediction model and the measurement of the jet are essential to ana-
lyze the screech mechanism with the aim of developing noise control devices [
16
18
]. Screech
tones have long been investigated experimentally using several measurement systems, such as
particle image velocimetry (PIV) [
19
,
20
], pressure-sensitive paint (PSP) [
21
23
], and schlieren
visualization [
24
29
]. In addition, data-driven analyses such as proper orthogonal decomposition
(POD) [
19
,
30
32
], spectral proper orthogonal decomposition (SPOD) [
13
,
33
], and dynamic mode
decomposition (DMD) [
13
,
34
] have been used to study screech. These techniques are methods
used to extract coherent structures associated with screech. Although the physical mechanisms of
screech tones have been well understood through mode decomposition techniques and the analysis
of time-resolved (TR) data acquired by the schlieren measurement [
13
,
29
,
30
,
34
], the investigation
of unsteady dynamics of screech using TR PIV data has remained a challenge due to limitations in
the sampling rate of high-speed cameras and constraints imposed by the laser system.
As mentioned above, spatiotemporally resolved measurements are crucial for understanding the
screech mechanism and are well suited to describe intermittent events, but they can be technically
challenging to acquire. Consequentially, methods to reliably estimate TR flow fields from incom-
pletely resolved data (in space and/or time) have long been sought. Adrian
et al.
[
35
37
] proposed
linear stochastic estimation (LSE) for this task and demonstrated that the conditional flow structure
in isotropic turbulence can be approximated in terms of unconditioned correlation data using LSE.
Bonnet
et al.
[
38
] combined LSE with POD so that spatially and temporally resolved reconstructions
of unsteady flow fields could be visualized using sparse sensor sets. This was originally referred to
as the complementary technique, and formed the framework for many of the LSE-POD combined
techniques that were published soon after. Of particular interest to the current study is the adaption
to the complementary technique presented by Tinney
et al.
[
39
]. In that, POD and Fourier-azimuthal
mode coefficients from non-time-resolved (NTR) PIV measurements of a jet flow were estimated
using Fourier-azimuthal mode coefficients from an array of TR pressure sensors located in the
hydrodynamic periphery of the jet. This modification to the complementary technique employed
the spectral form of LSE [
40
42
]. Similar approaches have been used to study cavity flows [
43
,
44
]
and wakes [
45
47
]. Recently, the methods developed by Tinney
et al.
[
39
] have been revisited by
Ozawa
et al.
[
48
,
49
] and Lee
et al.
[
50
]. In particular, Ozawa
et al.
[
49
] estimated TR velocity
fields associated with the B mode of the screeching jet using NTR PIV data and TR acoustic data.
Lee
et al.
[
51
] adapted the approach of Ozawa
et al.
[
49
] to estimate TR three-dimensional (3D)
density fluctuation fields associated with the B mode using NTR 3D background-oriented schlieren
(3D-BOS) data and TR acoustic data obtained by the azimuthal array of eight microphones placed
near the nozzle exit. They demonstrated that 3D density fluctuation fields associated with the B
104604-2
SUPERRESOLUTION AND ANALYSIS OF ...
mode can be reconstructed from the leading two POD modes in the
m
=
1 azimuthal mode of the
acoustic data. Based on the reconstruction results, it was elucidated that the azimuthal structure
reconstructed from each POD mode is the helical mode, with an opposing rotational direction. In
addition, the temporal amplitude of these structures exhibits variations over time, indicating that
this amplitude not only impacts the azimuthal structure linked to the B mode but also influences its
overall intensity.
The interpretation of the 3D flow structure associated with screech is essential to develop and
improve noise control devices [
16
18
], as described above. Nevertheless, two-dimensional flow
field measurements (e.g., PIV and schlieren) have been almost used and are not ideal for the analysis
of the 3D flow structure. Particularly, the investigation of the azimuthal mode in antisymmetric
geometries (e.g., B and C screech tones) is not straightforward because complex behaviors such
as switching of the rotation direction of the helical structures in the C mode and the azimuthal
mode transition in the B mode appear [
32
,
51
,
52
]. The proposed method by Lee
et al.
[
51
] can
interpret 3D unsteady dynamics of screeching jets. However, the experimental setup of this approach
is not straightforward, and 3D-BOS measurement requires several expensive high-speed cameras to
capture the projection images at different locations.
The objective of the present work is to estimate TR 3D velocity fluctuation fields associated
with screech. The novelty of the proposed method is that 3D velocity fluctuation fields for each
azimuthal mode can be reconstructed using the acoustic data obtained by the azimuthal array of
eight microphones placed near the nozzle exit and the PIV data. Here, the PIV data are not the
velocity field of the azimuthal direction measured by stereo-PIV [
39
,
53
,
54
] or tomographic PIV
[
55
,
56
], but the velocity field traversing along the jet axis. The azimuthal information of screeching
jets which is required to reconstruct a 3D velocity fluctuation field is obtained by the acoustic
data. Furthermore, TR 3D velocity fluctuation fields can be estimated from TR acoustic data by
modifying the approach proposed by Lee
et al.
[
51
]. Therefore, this technique is a straightforward
and cost-effective approach to estimate TR 3D flow fields associated with screech, as compared to
the technique of Lee
et al.
[
51
]. In the present work, we perform a detailed analysis of the A2, B,
and C modes over a wide range of temporal scales using the proposed method, which has not been
reported for the screeching jet noise problem.
The paper is organized as follows. In Sec.
II
, a description of the experimental setup is provided.
In Sec.
III
, the method for estimating TR 3D velocity fluctuation fields associated with screech is
introduced. In Sec.
IV
, the results of the estimation are provided, and we provide new interpretations
of screech dynamics. Finally, concluding remarks are provided in Sec.
V
.
II. EXPERIMENTAL SETUP
The experiment was conducted at the anechoic supersonic jet facility at Tohoku University
[
57
60
]. The details of the experimental facilities can be found in Ref. [
58
]. A cold supersonic
jet and a 10-mm-diam (
D
) converging circular nozzle were employed. The nozzle was designed
based on Ref. [
61
]. The thickness of the nozzle lip was 0.8 mm. The jet operating conditions are
determined by NPR. Here, NPR is defined as the ratio between the stagnation pressure
P
0
measured
by a pressure transducer placed in the plenum chamber and the ambient pressure
P
a
inside the
anechoic room: NPR
=
P
0
/
P
a
. An underexpanded supersonic jet is operated for NPR higher than a
critical value, given by [(
γ
+
1)
/
2]
γ/
(
γ
1)

1
.
89, where
γ
is the specific heat ratio for an ideal air
flow. The ideally expanded jet Mach number
M
j
was calculated assuming that the flow is isentropic
and expressed in terms of the ideally expanded condition. The temperature ratio
T
a
/
T
0
is defined as
the ratio between the stagnation temperature
T
0
measured in the plenum chamber and the ambient
temperature
T
a
. In this paper, the PIV and near-field acoustic measurements were carried out for
NPRs of 2.30, 2.97, and 3.40, and their characteristics are summarized in Table
I
.
The PIV and near-field acoustic measurements were simultaneously performed using a trigger
signal generated by a function generator (WF1974, NF). The experimental setup is shown in Fig.
1
.
Eight 1/4-in. microphones (TYPE4158N, ACO) were used for the near-field acoustic measurement
104604-3
CHUNGIL LEE
et al.
TABLE I. Temperature ratio and Reynolds number for each NPR. Here,
M
e
is the Mach number at the
nozzle exit.
NPR
M
e
M
j
Temperature ratio (
T
a
/
T
0
)Re
2.30
1.0
1.16
1.27
3
.
60
×
10
5
2.97
1.0
1.35
1.36
4
.
62
×
10
5
3.40
1.0
1.45
1.42
5
.
22
×
10
5
and were distributed around the nozzle lip at the radial location 4
D
from the jet axis, as shown in
Fig.
1(b)
. These microphones have a frequency range of 20 Hz to 100 kHz, maximum sound pressure
level (SPL) of 152 dB, and microphone sensitivity of 3.2 mV/Pa. The acoustic data obtained from
eight microphones can be decomposed up to the third azimuthal Fourier mode (
m
=
0–3). A total
of 100 000 samples were recorded using a computer via a data acquisition (DAQ) analyzer (USB-
6363, National Instruments) and a shielded connector block (BNC-2120, National Instruments) at
a sample rate of 200 kHz. The acoustic signal was amplified with a power supply (TYPE5006/4,
ACO). The microphone calibration was performed using an acoustic calibrator (Type2127, ACO)
to ensure a 94 dB response at 1 kHz.
Two-dimensional velocity fields of streamwise and transverse components on the
xy
plane of
the jet were measured by the PIV measurement. The optical system for the PIV measurement
comprises a high-speed camera (Phantom V2640, Vision Research) with single focal length lens
(Nikkor 80–200 mm f/2.8, Nikon) and a double-pulsed laser of Nd:YLF (LDY-300PIV, Litron) with
bandpass optical filters (527
±
10 nm, Edmund Optics). The double-pulsed laser and camera were
synchronized by a function generator (WF1974, NF). The bandpass optical filter of 527
±
2.5 nm
was mounted on the lens of the high-speed camera. The image resolution, pixel size, and bit depth
of the high-speed camera were 1374
×
723 pixels, 13.5 μm, and 12 bits, respectively. A laser sheet
illuminated the
xy
plane traversing the jet axis, and the field of view of 90
×
50 mm
2
was visualized,
as shown in Fig.
1(b)
. The frame-straddling technique was employed to capture two PIV images
with very short time separation because jets in NPRs of 2.30, 2.97, and 3.40 were supersonic flows
(
M
j
>
1
.
10). The temporal interval between images was 1 μs. A total of 2000 image pairs were
recorded at a sampling rate of 4000 fps. This sampling rate is not enough to resolve the screech
phenomenon because the screech frequency in the A, B, and C modes ranges from 10 to 25 kHz,
as reported by the previous works [
4
,
34
]. Seeding particles for PIV are produced by Laskin nozzles
x
y
(
=0
)
High-Speed
Camera
Nd: YLF Laser
Laser Sheet
Mic. 1
Mic. 4
Mic. 5
Mic. 6
Mic. 7
Mic. 8
Field of View
Mic. 2
Mic. 3
Nozzle
4
D
High-Speed
Camera
Nd: YLF Laser
Microphones
-y
(
=
)
(a)
(b)
FIG. 1. (a) Photograph of the experimental setup for the PIV and near-field acoustic measurements and
(b) schematic of the optical configuration for the PIV measurement and the locations of eight microphones.
104604-4
SUPERRESOLUTION AND ANALYSIS OF ...
placed in both the jet generating system and the anechoic room. A glycerin 50% aqueous solution
was utilized, and the particle size was approximately 0.5–2.0 μm. The Stokes number was calculated
based on the diameter of 1 μm at the jet velocity, and its values in NPRs of 2.30, 2.97, and 3.40 were
0.16, 0.20, and 0.21, respectively. It indicates that seeding particles follow the fluid since the Stokes
numbers for all NPRs are less than 1.
The velocity vectors were acquired through a commercial PIV software (Dynamics Studio 6.7,
Dantec Dynamics). Before conducting the PIV analysis, calibration target images were used for
distortion correction because the camera was not perfectly aligned with the laser sheet. The spatial
resolution of the computed velocity field was determined by the size of the spatial interrogation
window. The maximum and minimum sizes of the interrogation window were set to be 32
×
32
and 16
×
16 pixels, respectively. The overlap between interrogation windows was 50%. The grid
spacing corresponded to 0
.
056
D
. Error vectors were reduced using the moving average method.
The results for the uncertainty of the PIV measurement are summarized in the Appendix, and this
uncertainty is regarded as relatively small.
III. PROPOSED METHOD
The proposed method aims to estimate TR 3D velocity fluctuation fields associated with screech,
which cannot be measured by PIV due to the insufficient sampling rate of the high-speed camera and
the absence of azimuthal information. It is well known that the upstream-traveling wave associated
with screech is generated by the interaction between the KH wavepacket (large turbulent structures
of the flow) and shock cell structures [
1
,
2
,
62
]. As mentioned in Sec.
I
, this wave produces new
downstream-traveling flow structures by exciting the thin shear layer on the upstream side and
closes the resonance loop of screech. The previous studies also found that regions of the high
amplitudes in fluctuations of the flow fields are consistent with the effective source locations of
screech [
19
,
34
,
63
,
64
]. These previous results imply that fluctuations of the flow fields are correlated
with the screech generation. Based on these observations, Lee
et al.
[
51
] assumed that density
fluctuation fields associated with screech have a linear relation with acoustic fields of screech.
They proposed a method to estimate 3D density fluctuation fields associated with screech from
acoustic data using this assumption. They showed that 3D density fluctuation fields associated
with screech can be estimated with high accuracy. In addition, there is already an overwhelming
amount of evidence in the open literature to support the hypothesis that velocity fluctuations in the
jet are correlated with pressure fluctuations, as demonstrated by Tinney
et al.
[
41
], Arndt
et al.
[
65
],
and Picard and Delville [
66
], to name a few. Based on these previous studies, we propose the method
to estimate 3D velocity fluctuation fields associated with screech from the acoustic data using the
linear regression model between the simultaneous PIV and acoustic data. Figure
2
shows the flow
chart of the proposed method. The proposed method can be divided into three steps:
(1) NTR PIV data are obtained synchronously with TR acoustic data. The azimuthal Fourier
decomposition for each datum is then required to reconstruct 3D velocity fluctuation fields for each
azimuthal Fourier mode from azimuthal Fourier coefficients of the acoustic data, as similar to the
framework presented by Tinney
et al.
[
39
] and Lee
et al.
[
51
]. However, the PIV data intersecting the
jet axis cannot be decomposed into the azimuthal Fourier mode. In the present work, we construct
the linear regression model for the symmetric and antisymmetric parts of azimuthal Fourier modes,
which correspond to sums over even (
m
=
0, 2) and odd (
m
=
1, 3) azimuthal Fourier modes,
respectively. Therefore, the PIV data are separated into the symmetric and antisymmetric modes
using velocity fluctuation fields at azimuthal angles of 0 and
π
(i.e.,
θ
=
0
). Here, the velocity
field at
θ
=
0
corresponds to the field of view in the PIV measurement, as shown in Fig.
1(b)
.
Azimuthal Fourier coefficients of the acoustic data are calculated and then separated into the
symmetric and antisymmetric modes. POD is then performed on the PIV and acoustic data for
each mode, and their POD coefficients are determined.
(2) The linear regression model for the symmetric and antisymmetric modes is constructed
using NTR POD coefficients of PIV and downsampling POD coefficients of the acoustic data at
104604-5
CHUNGIL LEE
et al.
PIV measurement
(NTR, velocity field)
Microphone measurement
(TR, acoustic field of azimuthal array )
Symmetric
Antisymmetric
:
0
,
2
:
1
,
3
NTR velocity field
Symmetric
Antisymmetric
:
:
Symmetric
Antisymmetric
:
mic
0
,
mic
2
:
mic
1
,
mic
3
NTR POD coefficient
Symmetric
Antisymmetric
:
mic
Downsampling
NTR POD coefficient
Symmetric
Antisymmetric
:
PIV
:
PIV
POD mode
Symmetric
Antisymmetric
:
PIV
:
PIV
Regression
model
POD
POD
Regression coefficient
Symmetric
Antisymmetric
:
:
TR POD coefficient
Symmetric
Antisymmetric
:
est
:
est
POD
recon.
TR Azi. Fourier coefficient
:
(
)
:
(
)
TR 3D velocity field
:
(
,
)
:
(
,
)
Transform from
m
to space
DFT, Time-delay embed
TR POD coefficient
TR Azi. Fourier coefficient
:
mic
Symmetric
Antisymmetric
Symmetric
Antisymmetric
FIG. 2. Flow chart of the proposed method for reconstructing TR 3D velocity fluctuation fields associated
with screech using the regression model.
the same timing with the PIV data. The regression coefficients obtained by this model are utilized
for estimating TR POD coefficients of the velocity field from TR POD coefficients of the acoustic
data.
(3) TR azimuthal Fourier coefficients of the velocity field for the symmetric and antisymmetric
modes, as well as for each azimuthal Fourier mode, are computed using TR POD coefficients of the
velocity field and their corresponding POD modes. Finally, TR 3D velocity fluctuation fields are
reconstructed by transforming TR azimuthal Fourier coefficients from the azimuthal Fourier mode
m
to space
θ
[
39
].
In the following sections, the details of the proposed method will be interpreted.
104604-6
SUPERRESOLUTION AND ANALYSIS OF ...
t
j
t
j
t
j
Mic.1Mic. 2
Mic.8
Time
Acoustic data
DFT
t
j
Time
Azimuthal Fourier coefficient
Time-delay
embed
Acoustic vector
(microphone data)
t
j
=
t
j
(
)
(
)
(
+
)
=0∼3
(
)
(
)
(
+
)
Microphone
data matrix
t
j
=
t
j
t
j
t
j
Time
FIG. 3. Schematic of the calculation process for the time-delay embedded acoustic vector and the micro-
phone data matrix.
A. Definition of the data based on the symmetric and antisymmetric modes
1. PIV data
Streamwise and transverse components of velocity fluctuation fields in time
t
j
are used to
separate the PIV data into the symmetric and antisymmetric modes, as follows:

u
s
x
,
i
(
t
j
)
u
s
r
,
i
(
t
j
)

=
1
2

u
(
x
i
,
y
i
,
t
j
)
+
u
(
x
i
,
y
i
,
t
j
)
v
(
x
i
,
y
i
,
t
j
)
v
(
x
i
,
y
i
,
t
j
)

,
(1)

u
a
x
,
i
(
t
j
)
u
a
r
,
i
(
t
j
)

=
1
2

u
(
x
i
,
y
i
,
t
j
)
u
(
x
i
,
y
i
,
t
j
)
v
(
x
i
,
y
i
,
t
j
)
+
v
(
x
i
,
y
i
,
t
j
)

,
(2)
where
x
and
y
are streamwise and transverse coordinates, respectively, and
u
and
v
refer to stream-
wise and transverse velocity fluctuation components, respectively. The subscripts
x
and
r
represent
the streamwise and radial coordinates that correspond to
x
,
y

0 on the PIV grid, respectively.
Here,
y
i
and
y
i
correspond to radial components at
θ
=
0 and
π
, respectively, as displayed in
Fig.
1(b)
. The superscripts
s
and
a
represent symmetric and antisymmetric parts, respectively, and
1

i

N
PIV
represents the index of grids in the velocity field. The value
N
PIV
is the number of PIV
grids on
x
,
y

0. The velocity vector
q
s
,
a
R
2
N
PIV
×
1
consists of streamwise and radial components
of velocity fluctuation fields on the PIV grid, and is written as
q
s
,
a
(
t
j
)
=

u
s
,
a
x
(
t
j
)
u
s
,
a
r
(
t
j
)

,
(3)
where
u
s
,
a
x
=

u
s
,
a
x
,
1
u
s
,
a
x
,
2
···
u
s
,
a
x
,
N
PIV

T
,
(4)
u
s
,
a
r
=

u
s
,
a
r
,
1
u
s
,
a
r
,
2
···
u
s
,
a
r
,
N
PIV

T
.
(5)
The notation
T
indicates the transpose of a matrix.
2. Microphone data
A multi-time-delay framework is used for constructing the acoustic vector. This approach can
eliminate the need to know the time lag between the PIV and acoustic data required for a linear
regression and improve the estimation accuracy [
41
]. The acoustic vector is obtained in two steps,
as shown in Fig.
3
. First, azimuthal Fourier coefficients ˆ
p
m
C
1
×
1
are calculated by performing
the discrete Fourier transform (DFT) on the acoustic data acquired by eight microphones of the
azimuthal array. In the present work, azimuthal Fourier coefficients for the first four azimuthal
Fourier modes (
m
=
0–3) can be obtained. Then, the acoustic column vector
ˆ
p
m
C
(2
N
td
+
1)
×
1
is
constructed from azimuthal Fourier coefficients of multi-time-delay data centered on the present
104604-7
CHUNGIL LEE
et al.
time
t
j
, and written as
ˆ
p
m
(
t
j
)
=
[
ˆ
p
m
(
t
j
N
td
p
m
(
t
j
(
N
td
1)
)
···
ˆ
p
m
(
t
j
)
···
ˆ
p
m
(
t
j
+
N
td
)
]
T
,
(6)
where
m
is the azimuthal Fourier mode, and
N
td
is the number of past and future data points.
Hereafter, the acoustic vector
ˆ
p
m
is referred to as the microphone data.
B. Proper orthogonal decomposition
POD is a modal analysis to extract dominant structures from a data set. This technique was first
introduced by Lumley [
67
] and is widely used in various turbulent flow fields [
68
]. As reported
by Taira
et al.
[
69
], when
X
C
M
×
N
is a rectangular matrix, eigenvectors of
XX
H
and
X
H
X
are
identical to the left and right singular vectors of
X
, respectively. Here,
X
is the matrix consisting of
column vectors of mean-subtracted flow fields at given instants in time, the notation
H
denotes the
complex conjugate transpose (Hermitian transpose), and
M
and
N
are the numbers of data points in
space and time, respectively. Based on the relation between singular value decomposition (SVD) and
POD, the POD mode and corresponding temporal coefficients of the data matrix can be efficiently
obtained through SVD, as follows:
X
=
USV
H
(7)
=
UZ
.
(8)
The matrices
U
C
M
×
M
and
V
C
N
×
N
are orthogonal spatial and temporal modes, respectively.
The diagonal entries of
S
R
M
×
N
are singular values
σ
, and
σ
2
=
λ
represents energy levels of
each POD mode. Therefore, Eq. (
7
) can be interpreted by POD modes
U
and POD coefficients
Z
=
SV
H
. For dimensionality reduction, the first
r
modes are truncated to the original data matrix.
The matrix with tilde represents the reduced modes; e.g.,
̃
U
denotes the first
r
columns of
U
.
Most LSE-POD approaches have constructed an estimation model between POD coefficients of
flow fields and probe data (pressure or velocity). The aim of the present study is to estimate 3D
velocity fluctuation fields associated with screech from the microphone data using the estimation
model. Therefore, it is necessary to use POD on microphone data to extract modes associated with
screech and to construct the estimation model between POD coefficients of these modes and those
of the velocity field. In addition, the mode decomposition of the microphone data influences the
estimation accuracy of the model. Ozawa
et al.
[
49
] and Lee
et al.
[
51
] reported that the estimation
accuracy decreases as the number of POD modes of the microphone data in the estimation model
increases. These results indicate that the microphone data contain modes causing overfitting issues.
Therefore, it is important to extract modes associated with screech from the microphone data using
POD.
POD is applied to both PIV and microphone data matrices. The data matrices are given by
ˆ
P
m
=
[
ˆ
p
m
(
t
1
)
ˆ
p
m
(
t
2
)
···
ˆ
p
m
(
t
N
κ
)]
,
ˆ
P
m
C
(2
N
td
+
1)
×
N
κ
,
(9)
Q
s
,
a
=
[
q
s
,
a
(
t
κ
)
q
s
,
a
(
t
2
κ
)
···
q
s
,
a
(
t
N
κ
)]
,
Q
s
,
a
R
2
N
PIV
×
N
,
(10)
where
N
and
κ
are PIV snapshots and the ratio of the sampling rates of the microphone and PIV
measurements, respectively. In this study,
κ
was a fixed parameter as
κ
=
50. Here,
ˆ
P
m
is the
rectangular data matrix because this matrix consists of azimuthal Fourier coefficients including
time-delay components and their snapshots, as shown in Fig.
3
. Therefore, the microphone data
matrix for each azimuthal Fourier mode can be decomposed to POD modes and corresponding
temporal coefficients, as the PIV data matrix. POD modes of the PIV and microphone data matrices
are computed via SVD, i.e.,
ˆ
P
m
=
̃
U
mic
m
̃
Z
mic
m
and
Q
s
,
a
=
̃
U
PIV
s
,
a
̃
Z
PIV
s
,
a
. Here, the first
r
PIV
modes
containing the energy of 80%, and the first
r
mic
=
100 modes were truncated to the PIV and
microphone data, respectively. The truncated POD modes still have sufficient energy to represent
the flow field. The microphone POD mode is hereafter defined as the POD mode obtained from the
microphone data.
104604-8
SUPERRESOLUTION AND ANALYSIS OF ...
C. Estimation model
As mentioned by Lee
et al.
[
51
], TR 3D flow fields associated with screech are estimated by
calculating model coefficients
M
m
from the linear regression model
q
(
t
j
)
=

m
M
m
ˆ
p
m
(
t
j
)
e
im
θ
+
c
.
c
.,
(11)
where
q
is the estimated flow field, and the notation c.c. represents the complex conjugate of the
further term in the sum. The azimuthal Fourier coefficient of the measured flow fields is required to
calculate model coefficients. However, azimuthal Fourier coefficients of velocity fluctuation fields
cannot be obtained because the velocity field on the azimuthal direction cannot be measured using
the present PIV measurement, as stated above. Instead, the measured velocity fluctuation fields can
be separated into the symmetric and antisymmetric modes. In the present work, TR 3D symmetric
and antisymmetric velocity fluctuation fields are estimated by modifying Eq. (
11
), as follows:
q
s
,
a
(
t
j
)
=

m
[
s
,
a
]
M
m
ˆ
p
m
(
t
j
)
e
im
θ
+
c
.
c
.,
(12)
where the sum over
m
[
s
,
a
] represents even and odd azimuthal Fourier modes, respectively.
Model coefficients
M
m
computed by the training data are given by
M
m
=
̃
U
PIV
s
,
a
C
m
̃
U
H
mic
m
,
(13)
where
C
m
C
r
PIV
×
r
mic
is the complex regression coefficient matrix and computed by the linear
regression between POD coefficients of the PIV and microphone data. In the following section,
the calculation of complex regression coefficients
C
m
will be explained in detail.
D. Offline training
The full microphone and PIV data are first separated into the training and test data. The training
data are used to construct the estimation model, and the test data are used for the analysis of
the model accuracy through the randomized
K
-fold cross validation which will be discussed in
Sec.
IV B
.
The linear regression model is constructed by plugging the training data matrices,
ˆ
P
m
and
Q
s
,
a
,
into Eq. (
12
) and given by
Q
s
,
a
=

m
[
s
,
a
]
M
m
ˆ
P
m
+
c
.
c
.
(14)
=
̃
U
PIV
s
,
a

m
[
s
,
a
]
C
m
̃
U
H
mic
m
ˆ
P
m
+
c
.
c
.,
(15)
̃
Z
PIV
s
,
a
=

m
[
s
,
a
]
C
m
̃
Z
mic
m
+
c
.
c
.,
(16)
where the notation
·
represents the matrix of subset data of the same instances as the PIV measure-
ment. Note that Eq. (
14
) represents the relationship between the PIV and microphone data at
θ
=
0.
The sum in Eq. (
16
) is flattened and written as
̃
Z
PIV
s
,
a
=
C
s
,
a
Z
mic
s
,
a
,
(17)
104604-9
CHUNGIL LEE
et al.
where
C
s
=
[
C
0
C
2
C
2
]
,
(18)
C
a
=
[
C
1
C
1
C
3
C
3
]
,
(19)
Z
mic
s
=
[
̃
Z
mic
0
̃
Z
mic
2
̃
Z
mic
2
]
T
,
(20)
Z
mic
a
=
[
̃
Z
mic
1
̃
Z
mic
1
̃
Z
mic
3
̃
Z
mic
3
]
T
.
(21)
Here, the * notation represents the complex conjugate. The complex regression coefficients are used
to estimate TR POD coefficients of velocity fluctuation fields for the symmetric and antisymmetric
modes using TR microphone POD coefficients
̃
Z
mic
m
C
r
mic
×
N
κ
. In addition, one of the major
properties of this approach is that TR POD coefficients of velocity fluctuation fields for each
azimuthal Fourier mode can also be estimated because complex regression coefficients of the
symmetric and antisymmetric modes, as well as each azimuthal Fourier mode, can be obtained from
this model. This property leads to reconstruct TR 3D velocity fluctuation fields for each azimuthal
Fourier mode.
The least absolute shrinkage and selection operator (LASSO) regression [
70
] is adopted for
Eq. (
17
), as employed by Lee
et al.
[
51
]. This approach can select the highly correlated microphone
POD modes with the PIV data. Then, complex regression coefficients are calculated using the
chosen microphone POD modes. This approach can also resolve the problem of overfitting that
occurs in the time-domain LSE technique by amplifying the high-order POD modes in the flow with
nonperiodic or quasiperiodic behaviors [
71
]. A group LASSO [
72
], which is group

1
regularization,
was used to encourage sparsity for
̃
Z
mic
m
. The reason why the group-sparsity paradigm was used is
an assumption that the relation between the POD modes of the PIV and microphone data does not
change with time. Then, the objective function is as follows:
minimize
C
s
,
a
1
2
||
̃
Z
PIV
s
,
a
C
s
,
a
Z
mic
s
,
a
||
2
2
+
λ
LASSO
G

g
=
1
||
(
C
s
,
a
)
g
||
2
,
(22)
where the notation ( )
g
represents the
g
th column (group) of
C
, and the number of groups
G
is
determined by the truncated microphone POD modes
r
mic
. The sparsity of complex regression
coefficients
C
is promoted by increasing the regularization factor
λ
LASSO
. In the present study,
various regularization parameters
λ
LASSO
were employed to find the value that achieves the min-
imum estimation error. The impact of the estimation error depending on these parameters will
be discussed in Sec.
IV B 2
. The objective function in Eq. (
22
) is calculated by the fast iterative
shrinkage thresholding algorithm [
73
].
Finally, TR 3D velocity fluctuation fields for the symmetric and antisymmetric modes, as well
as for each azimuthal Fourier mode, can be reconstructed using TR microphone POD coefficients
̃
z
mic
m
and the complex regression coefficients
C
m
obtained by the linear regression model:
q
s
(
t
j
)
=

m
=
[0
,
2]
ˆ
q
s
(
t
j
)
e
im
θ
+
c
.
c
.,
(23)
q
a
(
t
j
)
=

m
=
[1
,
3]
ˆ
q
a
(
t
j
)
e
im
θ
+
c
.
c
.
(24)
Here,
ˆ
q
s
=
̃
U
PIV
s
C
m
̃
z
mic
m
and
ˆ
q
a
=
̃
U
PIV
a
C
m
̃
z
mic
m
are azimuthal Fourier coefficients of the velocity
fluctuation field for the symmetric and antisymmetric modes, respectively.
104604-10