# Plug-&-play generation of non-Gaussian states of light at a telecom
wavelength

Mohamed F. Melalkia  1,$\dagger$ Tecla Gabbrielli  2,3,$\dagger$ Antoine
Petitjean  1 Léandre Brunel  1 Alessandro Zavatta  2,3 Sébastien Tanzilli  1
Jean Etesse  1 and Virginia D’Auria1,4  1Université Côte d’Azur, CNRS,
Institut de Physique de Nice, Parc Valrose, 06108 Nice Cedex 2, France
2European Laboratory for Non-linear Spectroscopy (LENS), University of
Florence, Via nello Carrara 1, 50019 Sesto Fiorentino (FI), Italy
3Istituto Nazionale di Ottica (CNR-INO), CNR, Largo Enrico Fermi 6, 50125
Firenze, Italy
4Institut Universitaire de France, France<EMAIL_ADDRESS>
††journal: optica††articletype: Research Article

This work marks an important progress towards practical quantum optical
technologies in the continuous variable regime, as it shows the feasibility of
experiments where non-Gaussian state generation entirely relies on plug-&-play
components from guided-wave optics technologies. This strategy is demonstrated
experimentally with the heralded preparation of low amplitude Schrödinger cat
states based on single-photon subtraction from a squeezed vacuum. All stages
of the experiment are based on off-the-shelf fiber components. This leads to a
stable, compact, and easily re-configurable realization, fully compatible with
existing fibre networks and, more in general, with future out-of-the-
laboratory applications.

##  1 Introduction

Optical quantum technologies in continuous variable (CV) regime find
applications in quantum metrology [1, 2], quantum communication [3, 4, 5], and
quantum computing [6, 7]. At the same time, many CV quantum information
protocols requires the ability of generating non-Gaussian quantum states, such
as Fock [8] and Schrödinger cat-like states [9], that are essential to fault-
tolerant optical quantum information processing [10] or long distance quantum
communication [11, 12]. A common strategy to benefit from these resources is
to implement conditional state preparation, often starting from a (Gaussian)
squeezed state [13]. This idea has been demonstrated in multiple CV
experimental realizations [9, 8, 14], including the recent trend of employing
a nonlinear waveguide as squeezing source [15]. Reported realizations,
however, rely entirely, or largely, on bulk-optics setups that are generally
hardly scalable and require careful spatial alignments [13].

This work addresses the need for compact and stable realizations that can be
easily assembled out-of-the-laboratory, as demanded to push current CV quantum
optical experiments towards practical demonstrations. More specifically, it
shows the feasibility of non-Gaussian state generation setups fully employing
plug-&-play components from guided-wave technologies. Strategically, we
applied this concept to single-photon subtraction from a squeezed vacuum
state. Such a scheme, commonly used for the preparation of small amplitude
Schrödinger cat preparation [13], can be considered as an archetypal
configuration, gathering Gaussian and non-Gaussian resources and lying at the
heart of multiple complex architectures[13]. Squeezed states at a telecom
wavelength are generated by single-pass spontaneous parametric down conversion
(SPDC) in a commercial periodically poled lithium niobate ridge waveguide
(PPLN/RW), whose input and output are fiber-pigtailed. Both photon subtraction
operation and heralded state characterization via homodyne detection (HD) are
then implemented using standard fiber components from classical telecom
technology. The clear non-Gaussian behaviour of the measured heralded state
proves the pertinence of our approach. This result opens the way to robust CV
optical realizations compatible with existing telecommunication fiber networks
and with considerably reduced resource overhead, spatial alignment, and mode
matching issues.

##  2 Experimental setup

Figure 1: Experimental setup. A fiber-coupled continuous wave (CW) laser
delivers two outputs at $\lambda=$1560.44\text{\,}\mathrm{n}\mathrm{m}$$ and
$\lambda=$780.22\text{\,}\mathrm{n}\mathrm{m}$$. The
$\lambda=$780.22\text{\,}\mathrm{n}\mathrm{m}$$ output passes through a
polarization controller (PC) and a variable fiber optical attenuator (VOA) to
be used as pump source for the PPLN/RW generating the single-mode squeezed
vacuum. Downstream of the PPLN/RW, squeezed light is sent to a 95:5 fiber
beam-splitter. The $5\%$ output undergoes a spectral filtering stage (SFS) and
is sent to a superconducting nanowire single-photon detector (SNSPD), whose
signal heralds the non-Gaussian state generation. The $95\%$ output is sent to
a homodyne detection (HD) made of a 50:50 fiber beam-splitter and a commercial
balanced detector (BD). The HD local oscillator is given by the laser output
at $\lambda=$1560.44\text{\,}\mathrm{n}\mathrm{m}$$, passing through a fiber
phase shifter (PS) and a polarization controller (PC).

Fig. 1 shows the detailed experimental setup. The master source is a
continuous wave (CW) fiber-coupled laser (Muquans) delivering outputs at
$\lambda=$1560.44\text{\,}\mathrm{n}\mathrm{m}$$ and its second harmonic (SH)
at half of the wavelength ($\lambda=$780.22\text{\,}\mathrm{n}\mathrm{m}$$).
The SH output is sent to fiber polarization controller (PC) and an optical
variable attenuator (VOA) that allow adjusting its polarization and intensity.
This set of plug-&-play telecom components is then directly connected to the
PPLN/RW (NTT Electronics corp). Single-mode squeezed vacuum state at
$\lambda=$1560.44\text{\,}\mathrm{n}\mathrm{m}$$ is generated via type-0
frequency degenerate SPDC and collected by the optical fiber at the PPLN/RW
output with an efficiency $\eta_{c}$. Propagation losses inside the nonlinear
waveguide lead to a reduced PPLN/RW transmission efficiency $\eta_{pl}$. As
largely documented in the literature [13], heralded single-photon subtraction
can be implemented by sending squeezed light to a low reflectivity beam-
splitter followed by a single-photon detection stage. In our experiment, a
95:5 fiber beam-splitter (f-BS) splits squeezed light into two paths: the
$5\%$ output is sent to the heralding path, whereas the $95\%$ output,
carrying the heralded non-Gaussian state, goes to the HD for the measurement
of its CV properties. Following an approach that we already validated for
squeezing experiments [16], the HD is realized by means of a 50:50 f-BS whose
outputs are connected to a commercial balanced detection (Insight) showing a
bandwidth of $300\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. The HD local
oscillator (LO) is provided by the fundamental laser output
($\lambda=$1560.44\text{\,}\mathrm{n}\mathrm{m}$$), whose phase, $\theta$, is
controlled via a fiber phase shifter. The overall detection efficiency is
$\eta_{HD}=\eta_{t}\cdot\eta_{pd}\cdot\eta_{el}=0.72$, where $\eta_{t}=0.94$
includes measured losses in the f-BSs, $\eta_{pd}=0.80$ is the photodiode
efficiency as given by the manufacturer, and $\eta_{el}=0.96$ accounts to the
residual HD electronic noise. The HD output is monitored using an oscilloscope
(Teledyne LeCroy) for data post-processing. The CV data acquisition on the
oscilloscope is triggered by photon counting events from the heralding path.
Light on the heralding path undergoes a cascade of optical spectral filters
allowing to comply with the strong mismatch between the HD electronical
bandwidth ($300\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$) and the broad
squeezing optical emission bandwidth (a few THz). Indeed, photon-counting
events of wavelengths far away from the CW LO could trigger the CV detection
of signals whose non-Gaussian features are out-of-the-reach of HD electronics.
This would weave data corresponding to the desired non-Gaussian states with
those relative to squeezed vacuum, eventually resulting in the measurement of
a mixed state. Such an effect further combines with the fact that single-pass
squeezing can be not pure over its whole bandwidth [17]. Spectral filtering in
the heralding path allows circumventing these imperfections [17, 18]. In our
realization, the $5\%$ of tapped light from the squeezed state is filtered
using a commercial $500\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ FWHM fiber
Bragg grating filter (FBG from AOS GmbH) and an in-house fiber-coupled Fabry-
Pérot cavity with a target bandwidth of
$10\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. Note that, as an alternative,
plug-&-play optical filters with bandwidths of a few tens of MHz are
commercially available. Filtered light is then sent to a polarization
controller and to a superconducting nanowire single-photon detector (SNSPD
from IDQuantique). The heralding rate is of
$3\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ and the dark count rate is about
$80\text{\,}\mathrm{H}\mathrm{z}$, giving a signal to noise ratio, SNR, of
$15.7\text{\,}\mathrm{d}\mathrm{B}$.

##  3 Data analysis

The properties of the heralded non-Gaussian state can be reconstructed by
performing a quantum state tomography, starting from the state quadratures
measured by the HD [19]. A series of $43000$ homodyne traces, $x_{i}(t)$, of a
duration of $5\text{\,}\mu\mathrm{s}$ each at a sampling rate of
$5\text{\,}\mathrm{G}\mathrm{s}\mathrm{/}\mathrm{s}$ are recorded using the
oscilloscope. Note that the LO phase is spanned with a speed of approximately
$9\pi/s$ and, therefore, it can be considered as a constant over the time
needed to acquire a homodyne trace. For each acquisition $x_{i}(t)$, the
actual $\theta_{i}$ value is inferred from the phase dependency of the
measured squeezed vacuum variance.
As discussed in [17], in the CW regime, the quadrature $X_{i}$ of the heralded
non-Gaussian state can be obtained from raw homodyne data as [20]:

$X_{i}=\int f(t)x_{i}(t)dt,$ (1)

where $f(t)$ is the temporal mode of the heralded non-Gaussian state, as
determined by the filtering stages on the heralding path. When the squeezing
bandwidth is much broader than the bandwidth of the filtering cavity,
$\gamma/\pi$, the theoretical $f(t)$ is expected to be $\sim e^{\gamma
t}u(-t)$, $u(t)$ being the Heaviside step function. Experimentally, $f(t)$ is
obtained as in [21] by computing the autocorrelation function of the heralded
homodyne traces, $\langle x(t)x(t^{\prime})\rangle$, and by numerically
performing its single value decomposition: the eigenfunction corresponding to
the highest eigenvalue gives the optimal $f(t)$. Fig. 2 shows both the
reconstructed temporal mode $f(t)$ (blue) and theoretical one (red); a perfect
agreement between the two curves can be observed and gives a filtering
bandwidth of $\gamma/\pi=$9.75\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ at
FWHM, in agreement with the filtering stage characteristics.

Figure 2: The temporal mode function corresponding to the highest eigenvalue
(blue curve) and the fit with the theoretical model (red curve) as functions
of time. The cascade of the Fabry-Pérot cavity and the fiber Bragg grating
filter in the heralding path gives a filtering bandwidth of
$9.75\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ at FWHM.

By exploiting Eq. (1), the obtained $f(t)$ is used to extract the quadratures
of the squeezed state and of the heralded non-Gaussian state out of the raw
homodyne data. More in details, with reference to Fig. 2, $x_{i}(t)$ data are
post-processed within the time window
$\approx[$-1.25\text{\,}\mu\mathrm{s}$,$-1\text{\,}\mu\mathrm{s}$]$ to extract
the heralded non-Gaussian state quadratures. Data outside this time window are
used to extract the squeezed state quadrature.

##  4 Experimental results and discussion

The tomographic analysis allows measuring, for the non-heralded state, a
squeezing of $-1.80\pm 0.05$ dB and an antisqueezing of $3.36\pm 0.05$ dB.
This corresponds to an initial squeezing of $-5.39\pm 0.05$ dB when corrected
for the total losses experienced by squeezed light,
$\eta_{tot}=\eta_{wg}\cdot\eta_{s}\cdot\eta_{HD}=0.48$. In the expression of
$\eta_{tot}$, $\eta_{s}=0.96$ is the transmission of the subtracting 95:5 f-BS
and $\eta_{wg}$ accounts for both the propagation losses and the guide-to-
fibre coupling occurring at the SPDC stage
($\eta_{wg}=\eta_{pl}\cdot\eta_{c}$). From the measured $\eta_{s}$ and
$\eta_{HD}$, we deduce $\eta_{wg}=0.69$.

The tomography of the heralded non-Gaussian state is performed with the
maximum-likelihood algorithm with $200$ iterations [22]. Fig. 3-(a) shows the
obtained Wigner function of the generated non-Gaussian state. As can be seen,
the generated state has a highly non-Gaussian Wigner function with a minimum
at phase origin $W_{NG}(0,0)=0.016\pm 0.004$. Its profile perfectly matches
the one predicted by theory ($W_{Th}(0,0)=0.015$) following an approach
similar to the one in [23, 24] and taking into account all the experimental
imperfections. This result represents the very first demonstration of non-
Gaussian states in CV regime obtained using a fully guided-wave approach and
it attests its impact towards compact, simple, and plug-&-play setups
involving both Gaussian and non-Gaussian operations.

(a) | (b)
---|---
|
Figure 3: (a) The Wigner function $W_{NG}(x,p)$ of the heralded non-Gaussian
state and (b) the Wigner function $W_{Corr}(x,p)$ of the state corrected by
the homodyne detection efficiency $\eta_{HD}=0.72$.

In this regard, we observe that without modifying $\eta_{wg}$, which is in
agreement with realistic performances in such fibre-coupled ridge waveguides,
our results could be straightforwardly enhanced by simply replacing our
balanced photodetection with one with higher efficiency. By correcting the
data by the HD efficiency ($\eta_{HD}=0.72$) [19], the non-Gaussian state
obtained just after the 95:5 f-BS exhibits a negative Wigner function as
expected for Schrödinger cat states, the corresponding value at the phase
origin being $W_{Corr}(0,0)=-0.065\pm 0.004$ (see Fig. 3-(b)). We point out
that residual losses in the fiber components and low HD efficiency are not
intrinsic to the followed guided-wave approach but only to our specific
realization. In particular, the lack of efficient photodiodes, that represent
the most critical loss factor in the HD, affects in the same way both guided-
wave and bulk realizations.

As concluding remark, we stress that a further improvement of the Wigner
negativity could be obtained by improving $\eta_{wg}$. The combined effect of
propagation and coupling losses at the nonlinear waveguide leads to a
degradation of the purity of the squeezed state on which the subtraction
protocol is applied. This can be circumvented by adopting a full integrated
setup including squeezed state generation, photon subtraction and homodyne
detection, in a configuration similar to the one shown for Gaussian state
generation, manipulation, and detection on lithium niobate circuits [25, 26,
27]. Such a strategy would push the propagation losses down to
$0.03\text{\,}\mathrm{d}\mathrm{B}\mathrm{/}\mathrm{c}\mathrm{m}$ and the
coupling losses to 0, thus giving for a typical chip length of
$5\text{\,}\mathrm{c}\mathrm{m}$, a total $\eta_{wg}=0.97$. In this condition,
the Wigner function negativity would go to its maximal value,
$W_{Corr}(0,0)=-0.22$.

##  5 Conclusion

We have demonstrated that a plug-&-play, fully guided-wave, approach to CV
experiments is definitely reachable with off-the-shelves standard components
and devices. This concept has been pushed, for the first time, to the
generation of non-Gaussian states of light, showing the possibility of
implementing complex setups free from any bulk optics stage. The obtained
realization enables reconfigurable and scalable systems compatible with
integrated photonics [25, 26, 27], adapted to ambitious CV quantum
communication protocols. These ingredients are essential to scale-up
realizations of increasingly complex optical quantum systems capable of
enabling disruptive but realistic quantum information protocols.

Funding This work has been conducted within the framework of the project
OPTIMAL granted by the European Union by means of the "Fond Européen de
développement regional" (FEDER). The authors also acknowledge financial
support from the Agence Nationale de la Recherche (ANR) through the project
HyLight (ANR-17-CE30-0006-01) and the French government through its program
"Investments for the Future" under the Université Côte d’Azur UCA-JEDI project
managed by the ANR (grant agreement ANR-15-IDEX-01).

Acknowledgments Virginia D’Auria acknowledges financial support from the
Institut Universitaire de France (IUF).

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

$\dagger$These authors contributed equally to this work.

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# Manuscript Title:
Pressure-Informed Velocity Estimation in a Subsonic Jet

Songqi Li<EMAIL_ADDRESS>University of Florida, Gainesville, Florida 32611,
USA  Lawrence Ukeiley  University of Florida, Gainesville, Florida 32611, USA

###### Abstract

This work aims to estimate time-resolved velocity field that is directly
associated with pressure fluctuations in a subsonic round jet. To achieve this
goal, synchronous measurements of the velocity field and in-flow pressure
fluctuations were performed at Mach number 0.3. Two different experiment
campaigns were conducted, the first experimental campaign aims to explore the
time-resolved dynamics of the axisymmetric velocity components, and second
experiment focuses on the time-resolved, 2D velocity estimates on a streamwise
plane. Two different methods were utilized to estimate the input-output
relation between velocity and in-flow pressure measurements. A hybrid approach
based on the spectral linear stochastic estimation and the proper orthogonal
decomposition was applied to setup the model in a linear manner, and a
wavelet-based filter was implemented to attenuate the noise level in the
cross-correlation functions. In addition, the pressure-velocity relationship
was also described by neural network architectures based on the multi-layer
perceptron (MLP) and bidirectional long-short-term-memory (LSTM). In both
experimental sets, pressure fluctuations inside the flow are found to be
connected to the streamwise convection of large-scale coherent structures in
the flow. A unique advantage of the bidirectional LSTM method was found among
all estimation schemes is also reported in this work. The estimation result
represents the space-time dynamics of the acoustic sources in the jet flow
field, and it is of great importance to understand the noise generation
mechanism.

††preprint: This draft was prepared using the LaTeX style file belonging to
Physical Review Fluids

##  I Introduction

The accurate depiction of spatial-temporal activities associated with acoustic
sources in subsonic free jet flows remains a challenging engineering problem
in the fluid dynamics and aeroacoustics communities. Although beamforming
experiments [1, 2, 3] have shown that the region near the end of the potential
core is of great importance with respect to the noise generation, the real-
time dynamics of the source activities in this region cannot be directly
obtained from this approach. On the other hand, time-resolved (TR)
measurements from PIV [4, 5] and hot-wire rakes [6, 7] are capable to reveal
the temporal dynamics of coherent structures in velocity field of the flow.
However, the distillation of the portion of the velocity that directly
contributes to the noise generation in the far field is still a prerequisite
to the evaluation of the noise source mechanisms and their control.

Stochastic estimation (SE), first proposed by Adrian [8], is one of the most
widely applied techniques to refine ”conditional eddies” in turbulent flows
from some dependent and correlated inputs. As elucidated in [9], the goal of
this technique is to utilize conditional information about the flow in an
attempt to estimate the correlated portion of the flow at other locations.
Pioneering work to use this techniques to estimate the time dependence of the
conditional structures can be found in [10, 11, 12, 13] among many others. On
the basis of the traditional SE, the combination of SE with the Proper
Orthogonal Decomposition (POD) has also aroused great interest among
researchers. In this modified approach the POD is utilized to provide reduced-
order representations of the complex turbulent flows, and the original problem
becomes the estimation of time-varying POD expansion coefficients ([14, 15,
16, 17, 18]). In the practical problem of noise generated from turbulent jets,
various types of inputs have been applied in SE to better highlight the source
activities inside the jet shear layer, including the turbulent velocities
[19], far-field acoustics [20], and near-field pressure fluctuations [21, 15].
Apart from the above-mentioned attempts, direct measurements of in-flow
pressure fluctuations [22, 23, 24, 25] provide a novel opportunity to evaluate
the source mechanism directly related to the pressure fluctuations in the
convective turbulent flow. Pressure fluctuations in a subsonic jet has been
experimentally measured and analyzed in Li and Ukeiley [26], in which time-
localized imprints of the wave-packets ([27, 28, 29]) were effectively
extracted from pressure fluctuations in the jet shear layer.

With the booming development of computational power, modern machine-leaning
(ML) methods have come into the researcher’s field of vision for a wide range
of applications. The potential of ML-based techniques to solve complex fluid
dynamical problems have been expounded in [30] and [31]. For the practical
problem associated with this work, i.e., the estimation of real-time source
activities from sequential inputs at discrete locations, pioneering attempts
have been made and yield encouraging results. In a relevant application to the
work reported here Tenney  _et al._ [32] trained a multi-layer perceptron
(MLP) to estimate pressure fluctuations in the near field of a subsonic jet
and the MLP outperforms the stochastic estimation with an improvement of
accuracy. In addition, architectures based on the recurrent neural networks
(RNNs) [33, 34, 35] have also been proposed to estimate or predict real-time
flow dynamics with sequential inputs in time. The RNNs are specialized to
reveal the temporal dynamics of the sequential inputs, which includes simple
RNN, long-short-time-memory (LSTM), and gated recurrent unit (GRU), etc. These
architectures are highlighted by their time-dependent parameter transmission
mechanism and have shown huge success in relevant tasks involving time-
dependent data such as music genre classification [36] and stock price
prediction [37].

This study focuses on the application of experimental measurements of in-flow
pressure fluctuations to elucidate the relationship between pressure and
velocity in a subsonic axisymmetric jet. Synchronized measurements of in-flow
pressure with low-frame-rate PIV were performed to reveal the connections
between velocity and pressure at the sound-generating region. In addition,
time-resolved estimates of turbulent velocity associated to the in-flow
pressure was realized via SE and ML approaches. A hybrid approach combining SE
and POD was adopted in this work, and a wavelet-based filter was implemented
to denoise the cross-correlation functions between pressure and POD expansion
coefficients that were directly calculated from experiments in an attempt to
better highlight the wave-packet structures [27] in the flow. Moreover, two
neural network architectures were also utilized in this work to provide time-
resolved estimates of the velocity field as an alternative approach. Detailed
experimental procedure is described in Section II, followed by the
introduction of the estimation techniques in Section III which includes the
wavelet-filtered stochastic estimation and two neural network architectures.
Section IV reports the main results and a brief summary in Section V concludes
the manuscript.

##  II Experimental Setup

All experiments were carried out in the Anechoic Jet Test Facility at the
University of Florida. Details of this facility in terms of the anechoic room
and the recently installed jet facility can be found in [38] and [39]. The
installed subsonic jet has a convergent nozzle with an exit diameter of
$D=5.08$ cm and an area contraction ratio of 9:1. For all experiments in this
work, the facility was operated under the blow-down mode at $M=0.3$, which
corresponds to a jet exit Reynolds numbers of $Re_{D}=3.8\times 10^{5}$. The
air supply for the jet was controlled with a Fisher regulating valve coupled
to a LabVIEW PID controller that allowed the jet exit Mach number to be
maintained within $1\%$ of the desired value throughout a test.

The fluctuating static pressure inside the flow was measured using both a B&K
4138 1/8” microphone and a GRAS 46DD 1/8” microphone. Each microphone was
equipped with an aerodynamically shaped nosecone to avoid any contamination
from total pressure fluctuations [40]. Although different vendor supplied
nosecone configurations were adopted for the two microphones, in which the B&K
nosecone has a sharp tip and a shorter length and the GRAS nosecone has a
longer and blunt leading edge, Soderman and Allen [41] has shown that the
installation of both types of nosecone configurations will generate
essentially negligible installation effect at the frequency range of interest
in this work (150 ~20kHz). More detailed discussion on in-flow fluctuating
pressure measurements using miniature microphones is available in Li and
Ukeiley [26]. In the current work, the microphones were mounted on a
3D-printed, airfoil-shaped strut which allowed the microphones to be held at
the jet centerline, upper and lower jet liplines, respectively while
generating a minimal disturbance.

To reveal the relationship between pressure and velocity in the flow, pressure
fluctuations were synchronously recorded with velocity measured from PIV, but
at different sampling rates of 80004 Hz and 12 Hz, respectively. A 135 mJ
dual-cavity Litron Nano Nd:YAG laser with a wavelength $\lambda=532$ nm was
used as the light source to illuminate the seeding particles. The flow was
seeded by an ATI Laskin nozzle aerosol generator which produced particles with
diameters around 0.25 µm, and the ambient air was seeded with a Rosco 1700 fog
machine with particle diameters between 0.25 to 0.5 µm. Processing the raw
images in the current experiments was accomplished using DaVis 8.3 from
LaVision. The raw images were divided into interrogation windows, and the
velocity vectors were calculated through a correlation-based multi-pass
routine with decreasing interrogation window sizes. The interrogating window
size for the final pass was 32 pixel $\times$ 32 pixel with $75\%$ overlap
which leads to a vector resolution of around 500 µm in all experimental sets.
The uncertainty of PIV measurements was implemented in DaVis 8.3 using the
method in [42]. This algorithm leads to an average uncertainty in the jet
shear layer at around 4.8 m/s for cross-stream stereoscopic measurements and
2.5 m/s for streamwise planar measurements.

Figure 1: Pressure measurements synchronized with stereo-PIV on cross-stream
planes.

Figure 2: Pressure measurements synchronized with planar PIV on a streamwise
plane.

Two experimental campaigns were performed in this work. The first experimental
set synchronizes stereo-PIV with single-point pressure measurements on the jet
centerline. The motivation of this experimental campaign originates from the
work of Tinney  _et al._ [15], in which a volumetric, time-resolved estimation
of the turbulent velocity in a Mach 0.85 jet was obtained via stochastic
estimation. In Tinney’s work time-resolved pressure inputs were acquired in
the upstream hydrodynamic periphery, which are associated with the emergence
of the convecting coherent structures. Given the ability to directly measure
the in-flow fluctuating pressure generated by large-scale structures, the
current experimental set aims to perform real-time estimation of velocity
informed from turbulent pressure measured downstream of the estimation field.
However, due to the limited number of microphones that can be placed in the
jet, it is not possible to resolve all dominant azimuthal modes in the jet. On
the other hand, the dominance of the axisymmetric mode has been justified in
[28, 43, 27, 44] with respect to the sound radiation efficiency at shallow
emission angles. Following the argument in [45, 43], the kinematic boundary
conditions will guarantee finite azimuthal mode-0 pressure on the jet
centerline and zero pressure for all higher order modes. Therefore, as an
initial attempt, the advantage of axisymmetry in round jets was taken by
placing the B&K microphone on the jet centerline to capture the temporal
dynamics of the axisymmetric events in pressure. The spatial feature of the
axisymmetric velocity can be obtained from stereo-PIV measurements on multiple
cross-stream planes, and the energetic structures can be extracted in
complementary with the application of azimuthal-Fourier POD [15, 39] to the
mode-0 velocity components. A schematic of the experimental setup is shown in
Figure 2, where the B$\&$K microphone was placed on the jet centerline at
$x/D=8$ and PIV measurements were taken on multiple cross-stream planes
between $x/D=4$ and $6.5$ with a step increment of $\Delta x/D=0.25$. For each
streamwise location, 1600 PIV snapshots were recorded synchronously with time-
lagged pressure.

The second experimental set aimed to characterize space-time dynamics of two-
dimensional velocity fields on a streamwise plane ($x-r$) across the jet
centerline. Planar PIV was utilized to measure velocity vectors within $3\leq
x/D\leq 6.5$, $\lvert r/D\rvert\leq 1.2$. Both B&k and GRAS microphones were
employed in this measurement campaign and were placed on the upper and lower
jet liplines ($r/D=\pm 0.5$), respectively. As displayed in Figure 2, the
nosecone tips were aligned at $x/D=6.6$ which is just downstream of the
velocity field-of-view. To avoid direct interaction with laser light sheet, an
out-of-plane displacement of 5 mm was implemented to the microphones during
the positioning process, and a total of 8000 PIV images were acquired in this
measurement campaign.

##  III Analysis Techniques

Figure 3: Visualization of PIV recording versus fluctuating pressure
measurements, with the goal of estimating time-resolved velocity fields give
pressure inputs. Green frames: PIV snapshots; Yellow frames: velocity
estimates via appropriate input-output models.

One important goal of this study is to estimate time-resolved velocity fields
associated with pressure fluctuations in the flow informed by non-TR PIV
measurements in combination with TR-pressure signals. A graphical illustration
of this procedure is displayed in Figure 3. As a data-driven approach, $N_{t}$
mutually independent velocity snapshots $u_{i}(\mathbf{x},t_{n})$ and the
corresponding time-lagged pressure signals $p^{k}(t_{n}-\tau\sim t_{n}+\tau)$
were first collected to construct the experimental dataset. Here $k$
represents the $k$th microphone ($k=1,2,...I$) and $\tau$ is the maximum time
lag. In both experimental campaigns a fixed time lag $\tau U_{\infty}/D=38$
was chosen to guarantee that the dominant wave-packet structures are preserved
in the cross-correlation functions, and $U_{\infty}$ represents jet exit
velocity. For each velocity snapshot $u_{i}$ at any time $t$, the snapshot POD
will provide the following reduced-order, energy-ranked representation
following [46]:

${u}_{i}(\mathbf{x},t)\approx\sum_{j=1}^{N}\phi_{i}^{(j)}(\mathbf{x})a^{(j)}(t)$
(1)

where $\phi_{i}^{(j)}(\mathbf{x})$ is the eigenfunction of the $j$th POD mode,
and $a^{(j)}(t)$ represents the corresponding POD expansion coefficient. $N$
is the truncation of the first $N$th POD modes. Since the eigenfunctions are
invariant in time, the original problem is equivalent to the TR-estimation of
the POD expansion coefficients $\hat{a}(t)$. This problem can be addressed by
the proposition of appropriate input-output models between time-lagged
pressure and POD expansion coefficients. Once the models are well established,
they can be deployed to estimate time-varying POD expansion coefficients from
pressure inputs, and the results will serve to reconstruct TR-velocity field
in combination with spatial POD eigenfunctions.

In this work two types of models were utilized to estimate the evolution of
POD expansion coefficients in time. The first model is the spectral linear
stochastic estimation (SLSE), which is termed ”SLSE” in short. In addition,
two neural network architectures were also introduced in this work to model
the input-output relationship. Details of both techniques will be presented in
the following.

###  III.1 Velocity Estimation via SLSE-POD

The SLSE-POD is a linear approach which aims to model the relationship between
$\hat{a}$ and $p^{k}(t)$ from the following convolution equation:

$\hat{a}^{(j)}(t)=\sum_{k}\int_{-\tau}^{\tau}h^{(j)}_{k}(\tau^{\prime})p^{k}(t-\tau^{\prime})d\tau^{\prime}.$
(2)

Here $h^{(j)}_{k}$ is the weighting function which requires to be determined
from the experimental dataset. Zhang  _et al._ [18] has shown that this linear
model is equivalent to the following form after taking the Fourier transform
on both sides of Equation 2:

$A(f)=H(f)P(f).$ (3)

Here $A$, $H$, $P$ are Fourier transforms of $\hat{a}$, [$h_{1}$, …,
$h_{k}$]T, [$p^{1}$, …, $p^{k}$]T, respectively. $A\in\mathbb{C}(N\times 1)$,
$H\in\mathbb{C}(N\times I)$ and $P\in\mathbb{C}(I\times 1)$. Multiplying the
complex conjugate $P^{*}$ on both sides of Equation 3 and taking the ensemble
average $E(\cdot)$ over independent realizations, one will get:

$G_{pa}(f)=H(f)G_{pp}(f).$ (4)

$G_{pp}\in\mathbb{C}(I\times I)$ is a Hermitian matrix in which diagonal terms
are the auto-spectral density functions of pressure and off-diagonal terms are
the cross-spectral density functions among different sensors.
$G_{pa}\in\mathbb{C}(N\times I)$ is the cross spectral density function
between pressure and POD coefficients. A direct solution of $G_{pa}$ requires
time-resolved velocity measurements which is not available in the current
work. As suggested by Tinney  _et al._ [47], an indirect approach was adopted
in this study from the definition of cross-correlation between $a$ and $p$:

$R_{ap}(\tau)=\langle a(t_{n})p(t_{n}+\tau)\rangle.$ (5)

Equation 5 indicates that $R_{ap}$ can be computed by orderly shifting the
input signals with respect to each velocity snapshot and ensemble-averaging
from all realizations. Since for stationary flows
$R_{pa}(\tau)=R_{ap}(-\tau)$, the cross spectral density function $G_{pa}$ can
be calculated by taking Fourier transform of $R_{ap}(-\tau)$. Consequently,
the optimized weighting function $H(f)$ can be determined from the following
expression:

$H(f)={G_{pa}}{G_{pp}}^{-1}.$ (6)

Equation 6 is termed ”spectral linear stochastic estimation” (SLSE). The core
difference between the time-lagged LSE and the SLSE lies in the fact that the
former method minimizes the mean-squared error in a macro sense while the
latter performs a series of least-square regressions at all frequencies of
interest. A thorough discussion of both techniques can be found in Tinney  _et
al._ [47]. Once the linear model $H(f)$ is optimized from the training
dataset, a real-time prediction of the velocity field can be performed using
pressure signals of length $T$ from the following equations:

$P=\text{FFT}(p),$ (7) $\hat{A}(f)=H(f)P(f),$ (8)
$\hat{a}(t)=\text{IFFT}(\hat{A}).$ (9)

The output of Equation 9, which is time-resolved estimation of POD expansion
coefficients, will be used in Equation 1 to get spatial-temporal estimation of
the velocity field.

#### Wavelet-Based Filtering (WF) of Cross-Correlation Functions

Equation 5 provides an alternative approach to calculate cross-correlation
between pressure and POD expansion coefficients from non-TR velocity data.
However, in practice extraneous noise will appear in the calculation of
$R_{ap}$, and will adversely influence the outcome from SE. To attenuate the
noise level, a wavelet-based filter was implemented in this work to zero out
the portion of the cross-correlation function which possesses a weak or none
wave-packet shape. The continuous wavelet transform (CWT) of a time sequence
$x$ is defined as a convolution between the original signal and the wavelet
function $\psi$ translated to different scales $s$:

$w(s,t)=\int_{-\infty}^{\infty}x(\tau)\psi^{*}(\frac{t-\tau}{s})d\tau.$ (10)

Here the asterisk (∗) indicates the complex conjugate. Details of the
continuous wavelet transform can be found in [48] and [49]. The validity of
the wavelet-based filter originated from the fact that if the turbulent
velocity (i.e. the dominant POD eigenfunctions) and pressure fluctuations both
possess wave-like behaviors driven by the same coherent events in the flow,
the cross-correlation function between the two quantities should also be
dominated by some wave-packet shape. Hence, a wavelet-based filter can be
implemented to preserve the major wave-like events in the cross-correlation
function and effectively reject the noise. The implementation of the wavelet-
based filter starts from the transformation of $R_{ap}(\tau)$ into the time-
scale domain, which will yield complex-valued wavelet coefficients
$w(\tau,s)$. Next, a real-valued threshold $T$ is imposed to obtain the
filtered wavelet coefficients $w^{\prime}$ such that:

$w^{\prime}(\tau,s)=\left\\{\begin{array}[]{ll}w(\tau,s)\text{, if }\lvert
w(\tau,s)\rvert>T;\\\ 0\text{, otherwise.}\end{array}\right.$ (11)

In this work, the complex-valued Morlet wavelet function with the angular
frequency $\omega_{0}=6$ was selected to better highlight the wave-packet
events in the correlation results. The threshold was empirically selected as
$T=0.3\text{ max}(|w|)$, and the filtered cross-correlation function
$R^{\prime}_{ap}$ was obtained from the inverse wavelet transform of
$w^{\prime}$ [48]. An exemplary comparison between raw and filtered cross-
correlation coefficients (see Equation 19 for definition) is displayed in
Figure 4(a), from which the filtered function is seen to better highlight the
portion of the signal with prominent wave-packet shape and high correlation
level. The comparison is also performed in the Fourier space in Figure 4(b),
where the cross-spectrum is found to be significantly smoothed between
$0.1\leq St\leq 1$ after filtering. Although the spectral shape at higher
frequencies doesn’t exhibit significant improvement, the noise amplitude is
still significantly brought down. Since current discussions primarily focus on
the coherent structures in the jet shear layer which governs the hydrodynamic
hump in the low frequency range, the filtered results remains satisfying and
will be employed in the following discussions.

(a)

(b)

Figure 4: (a) A comparison of original and wavelet-filtered cross-correlation
coefficient between the centerline pressure fluctuations at $x/D=8$ and the
first POD mode of the axisymmetric streamwise velocity at $x/D=4$; (b) same
functions in the Fourier domain.

###  III.2 Velocity Estimation via Neural Networks

For the experimental dataset with synchronized streamwise PIV and two-point
pressure measurements, two neural network architectures were proposed in this
study to establish the connection between pressure and POD expansion
coefficients as an alternative approach. The first NN architecture is a many-
to-one model which originates from the multi-layer perceptron (MLP) and the
schematic is presented in Figure 5. For each POD mode $m$, the input layer
collects time-lagged pressure data from two in-flow microphones
$(p_{1}^{1}\sim p_{N}^{1},p_{1}^{2}\sim p_{N}^{2})$. The subscripts $1\sim N$
denote the time-lagged discrete data points and the superscripts $1,2$ are the
indices of the in-flow microphones. Two fully connected linear hidden layers
are included in the architecture, each contains 1024 hidden units
($N_{1}=N_{2}=1024$). In the hidden layers, linear operators are applied to
upscale the input data to a high dimensional space. The output layer
compresses the feature outputs from the last hidden layer into a $1\times 1$
scalar with a nonlinear tanh activation function, which represents the
prediction of the POD expansion coefficient ($\hat{a}^{(m)}_{t}$).

Figure 5: POD coefficient estimation based on the MLP.

(a)

(b)

Figure 6: (a) An LSTM cell at time $t$; (b) The architecture to estimate POD
coefficient based on bidirectional LSTM.

The second NN architecture is constructed based on the bidirectional long-
short-term-memory (LSTM) [50]. An LSTM cell is composed of three gates: an
input gate $i_{t}$, an output gate $o_{t}$, and a forget gate $f_{t}$. These
gates allow LSTM cells to retain and process information over long periods of
time, hence minimizing the effect of a vanishing gradient [51]. Following the
configuration in Figure 6(a), an LSTM cell at time $t$ computes the following
functions:

$i_{t}=\sigma(W_{i}x_{t}+U_{i}h_{t-1}+b_{i}),$ (12)
$f_{t}=\sigma(W_{f}x_{t}+U_{f}h_{t-1}+b_{f}),$ (13)
$g_{t}=\text{tanh}(W_{g}x_{t}+U_{g}h_{t-1}+b_{g}),$ (14)
$o_{t}=\sigma(W_{o}x_{t}+U_{o}h_{t-1}+b_{o}),$ (15) $c_{t}=f_{t}\odot
c_{t-1}+i_{t}\odot g_{t},$ (16) $h_{t}=o_{t}\odot\text{tanh}(c_{t}).$ (17)

In the equations above, $x_{t}$ is the input at time $t$, $h_{t}$ and $c_{t}$
are the hidden state and the cell state at time $t$. $g_{t}$ is the new memory
cell. $i_{t}$, $f_{t}$, $o_{t}$ are the input, forget, and output gates,
respectively. $\sigma$ represents the sigmoid function, and $\odot$ denotes
the element-wise product. $W$, $U$ and $b$ are unknown parameters which need
to be learned from the training dataset. In an LSTM layer the hidden state
$h_{t}$ only receive information from its past states yet the future states
are not included. The bidirectional LSTM ([52]) overcomes this issue by
concatenating the outputs from two LSTM layers of opposite directions. With
this form, outputs from a bidirectional LSTM layer will encompass information
from both past and future states simultaneously.

Taking the advantage of LSTM on handling time sequence inputs, the second NN
includes two bidirectional LSTM layers and a feed-forward output layer, and
the schematic is presented in Figure 6(b). Pressure signals measured from two
microphones at any time $t_{j}$ are concatenated into a $2\times 1$ input
vector $p^{\prime}_{j}=[p_{j}^{1},p_{j}^{2}]^{T}$. The laser burst time
$t_{i}$ is indicated by the subscript $i$ for all variables. In this network,
sequentially arranged inputs $p^{\prime}_{j}\sim p^{\prime}_{N}$ are fed into
the first bidirectional LSTM layer which generates a series of hidden state
vectors $h_{1}^{(1)}\sim h_{N}^{(1)}$. Then the second bidirectional LSTM
layer takes in these vectors and computes the new hidden state vectors
$h^{(2)}$s in a similar approach. The output sizes of both hidden layers were
set to 64. To better highlight the information around the PIV burst moment
which is highly related to the expected output, $h_{i}^{(2)}$ is connected to
the fully-connected output layer to obtain POD coefficient estimates of the
$m$th mode $\hat{a}^{(m)}_{t}$, in which a linear mapping function as well as
a tanh activation function is included in the output layer.

Both NN architectures were trained based on the mini-batch gradient descent
optimization [53]. The experimental dataset contains the modal expansion
coefficients of the leading POD modes from $N_{s}=8000$ mutually independent
velocity snapshots as well as the time-lagged pressure signals. These samples
were randomly shuffled, 80$\%$ were used to train the NN models and the rest
for validation. For each mode, pressure and POD coefficients were re-scaled
into $[-1,1]$. To evaluate the model performance, estimation results from the
training data $\hat{a}^{(m)}$ were compared to the known POD coefficients from
PIV snapshots during the iterative training process. The mean squared error
(MSE) loss was utilized as the criterion of the above-mentioned regression
problem:

$L^{m}=\frac{1}{N_{s}}\sum_{i=1}^{N_{s}}(a_{i}^{(m)}-\hat{a}_{i}^{(m)})^{2}.$
(18)

In contrast to the ”flow” direction of the input data, gradients of the loss
function were back-propagated from the output layer to the neutral network to
optimize the unknown weights and biases in the architectures. The Adaptive
Moment Estimation (ADAM, [54]) was chosen as the optimizer, and the initial
learning rate was set to 1e-3. Each model was trained with 200 epochs for
every POD mode and the size of every mini-batch was set to 64. Models with the
best performance on the validation set were recorded during iterations and
were used for the velocity estimation from pressure inputs. The training
process in this study was carried out on a cloud-based platform with four
Normalized Graphics Processor Units (NGUs).

##  IV Results and Discussions

In the following section the results of the modal analysis of the
experimentally measured velocity fields and pressure informed velocity
estimation will be presented.

###  IV.1 Estimation of Axisymmetric Velocity from Cross-Stream PIV and In-
Flow Pressure Measurements

As introduced in the previous section, single-point pressure measurements were
taken synchronously with stereo-PIV on a series of cross-stream planes to
investigate the space-time dynamics of axisymmetric velocity components in the
jet. To extract the axisymmetric velocity component from the PIV snapshots,
velocity fields measured under the Cartesian coordinate were first mapped onto
a polar grid such that the axisymmetric velocity $(u^{(0)}_{x},u^{(0)}_{r})$
can be extracted accordingly. We note that although small portion of the
azimuthal velocity component $u^{(0)}_{\theta}$ still exists after the grid
transformation, theoretical work from Batchelor and Gill [45] has shown that
the mode-0 azimuthal velocity should be zero in round jets. Hence the existing
azimuthal component are believed to be the residue introduced from the
measurement uncertainty and is excepted from the following analysis.

After obtaining 1600 samples of $(u^{(0)}_{x},u^{(0)}_{r})$ at each streamwise
location, the azimuthal-Fourier POD ([15, 39]) was performed to extract the
most energetic spatial structures in the radial direction. The only difference
between the azimuthal-Fourier POD and the snapshot POD originates from an
additional weighting factor $r$ in the eigenvalue decomposition problem, and
the corresponding strategy to tackle this problem has been thoroughly
explained in [6].

(a) Streamwise POD

(b) Radial POD

Figure 7: Energy distribution of the first 5 POD modes at $x/D=4.5$, $5.5$,
and $6.5$.

As suggested by Tinney  _et al._ [15], a $90^{\circ}$ phase difference exists
between the streamwise velocity and its radial counterpart. To better preserve
the phase difference between both velocity components and maintain the
completeness of the turbulent structures in each direction, the scalar-based
azimuthal-Fourier POD was performed on each velocity component individually.
Figure 7 presents the energy distribution of the first 5 POD modes at three
representative axial locations from the streamwise and the radial POD,
respectively. Strong mode-1 dominance for both velocity components can be
observed at all three streamwise locations, and the superposition of the first
2 POD modes takes up around $60\%$ of the energy in the streamwise direction
and $80\%$ in the radial direction. These results imply that at each cross-
stream station, the dominant spatial features of the axisymmetric turbulent
velocity can be effectively captured from the first few modes of the reduced-
order representation, and this property will be utilized to greatly simplify
the computational cost of the stochastic estimation.

(a) $\phi_{x}^{(0)}$ from streamwise POD

(b) $\phi_{r}^{(0)}$ from radial POD

Figure 8: Streamwise evolution of the first two POD eigenfunctions.

Figure 8 shows the shapes of the first two eigenfunctions obtained from the
azimuthal-Fourier POD. The streamwise POD eigenfunctions are seen to evolve
along the jet axis while the radial modes will slightly tilt toward the
ambient side at downstream locations. The most energetic structures of the
first streamwise POD mode are mostly concentrated within $r/D\leqslant 0.5$,
and the first radial mode highlights energetic events between $r/D=0.2\sim
0.8$. Furthermore, the second streamwise mode represents the convection of
some radially compact turbulent structures around the jet lipline, accompanied
by two opposite-signed side-lobes at relatively low amplitudes. Meanwhile the
second radial eigenfunction depicts the appearance of injection-ejection
events across the jet lipline.

(a)

(b)

(c)

Figure 9: Cross-correlation coefficients between the first POD expansion
coefficients and pressure ($C_{ap}$) at $x/D=4,5,6$. Pressure lowpassed at
2000 Hz ($St\approx 1$).

Since SLSE-POD is a correlation-based technique, the time-lagged cross-
correlation coefficients between the first POD modal expansion coefficients
and the in-flow pressure fluctuations ($C_{ap}$) are first examined, and the
results at $x/D=4,5,6$ are presented in Figure 9. Here $C_{ap}$ is defined as:

$C_{ap}(\tau)=\frac{\langle
a({t_{n}})p(t_{n}+\tau)\rangle_{n}}{{rms}(a){rms}(p)}.$ (19)

In an attempt to focus on the coherent, large-scale motions inside the flow,
pressure in Figure 9 was lowpass filtered at 2000 Hz which corresponds to a
Strouhal number of $St\approx 1$. In all instances, wave-like patterns can be
clearly observed from cross-correlation curves, and the correlation level
gradually increases as the velocity measurement plane moves closer to the
pressure probe. For both the first streamwise POD mode and the first radial
POD mode, the peak values of $|C_{ap}|$ always exhibit similar amplitudes at
each axial location, and an apparent phase difference of $\sim 90^{\circ}$
between two curves can be clearly identified. This is consistent to the
observation in [55] where the axial velocity was found to be in antiphase with
in-flow pressure fluctuations whereas the radial velocity was $90^{\circ}$
out-of phase to pressure.

(a) Streamwise POD

(b) Radial POD

Figure 10: Evolution of cross-correlation coefficient $C_{ap}$ between the
first POD mode and pressure fluctuations. Pressure lowpassed at 2000 Hz
($St\approx 1$) and dashed lines represent an empirical convective speed of
$U_{c}=0.7U_{\infty}$.

The space–time correlations $C_{ap}$ calculated from all 11 cross-stream
measurement planes are presented in Figure 10. Since the correlation with
higher order POD modes will decay rapidly, only results from the first POD
mode is shown in the figure. Similar to what has been observed in Figure 9,
wave-packet structures can be found at all streamwise locations with a nearly
uniform convective speed. To quantify this convective behavior, the
propagative pattern in Figure 9 is compared to an empirical convective speed
of $U_{c}=0.7U_{\infty}$ [56] which is represented by dashed lines. Both
velocity components yield good agreements with the empirical slope, which
indicates that $U_{c}=0.7U_{\infty}$ is a good representation of the
convective speed of the axisymmetric wave-packets.

(a) Streamwise POD

(b) Radial POD

Figure 11: Estimation of the space–time dynamics of the mode-1 POD expansion
coefficients via SLSE-POD. Dashed lines represent $U_{c}=0.7U_{\infty}$.

After obtaining cross-correlation functions between the POD expansion
coefficients and in-flow pressure fluctuations at all cross-stream planes, the
SLSE-POD was employed to obtain time-resolved estimation of the POD expansion
coefficients with the use of the the wavelet-based filter discussed above.
Figure 11 presents the time-resolved estimates of the first POD expansion
coefficients from this hybrid strategy. To make the results comparable, the
estimation was realized by using the same pressure signal segment as the
unconditional input for both streamwise and radial estimates. The most
dominant feature from the reconstruction is the streamwise propagation of wavy
patterns in time, which confirms that the convective nature of the dominant
axisymmetric turbulent structures can be precisely captured from this
correlation-based estimation technique. In addition, the $\sim 90^{\circ}$
phase difference is consistently observed between the streamwise expansion
coefficients and the radial ones. The candidate of the convective speed,
$U_{c}=0.7U_{\infty}$, is also plotted in the figures by the dashed lines,
which are seen to match well with the slope of the travelling waves. However,
since SLSE-POD grounds on the correlation between pressure and POD expansion
coefficients, the estimation can only extract the portion of the input that is
linearly correlated with the output. Therefore, as the PIV plane moves
upstream, the coherency between the downstream pressure and the POD expansion
coefficients is gradually weakened, which will directly lead to an attenuation
of the output amplitude. Hence, the spatial envelope of the axially evolving
wave-packet shape can not be fairly captured from this set of experiments. In
order to better describe the real-time amplitudes, more input information from
upstream locations might be necessary.

Figure 12: Five-mode reconstruction of the streamwise turbulent velocity.
Velocity is scaled by the jet exit velocity $U_{\infty}$.

Figure 13: Five-mode reconstruction of the radial turbulent velocity. Velocity
is scaled by the jet exit velocity $U_{\infty}$.

Figure 12 and Figure 13 display the global reconstruction of time-resolved
axisymmetric velocity using the first five POD modes. The space-time evolution
of coherent structures can be clearly observed in the figures across the
sequential instances. The size of each eddy is on the order of the jet
diameter $D$. In addition, the streamwise velocity is oppositely signed across
the jet lipline as displayed in Figure 12, which is a result of the strong
shear events inside the flow.

(a) $x/D=4$

(b) $x/D=5$

(c) $x/D=6$

Figure 14: A comparison of the measured streamwise velocity spectra versus
spectral estimates from wavelet-filtered (WF) SLSE-POD on the jet centerline.
Velocity spectra are scaled by the jet exit velocity $U_{\infty}$.

Figure 14 presents the comparison between the estimated streamwise velocity
spectra as well as the velocity spectra measured from the hotwire anemometry
at three different locations on the jet centerline. Since the high frequency
content can not be fairly retrieved from wavelet-filtered cross-correlations,
only the low frequency portion ($St<1$) of the estimated spectra are
presented. In general, the velocity spectra from estimation can faithfully
reflect the trend of the measured spectra in the range of $0.1\leq St\leq
0.5$, which can be described as a broadband hump governed by large-scale
structures. The discrepancy of the overall amplitude between the estimation
results and the measured ones comes from the fact that only the portion of
velocity correlated to pressure inputs will be encompassed in the estimated
spectra. The closer the PIV plane is to the pressure probe, the larger the
coherency between the pressure and dominant POD modes, and it will result in
less amplitude discrepancy between measurement and estimation. At $St>0.5$,
the estimated velocity spectra exhibit a steeper decaying rate than the real
spectra. Since the leading POD modes only contribute to the large-scaled
turbulent structures in the flow, the rolling-off of the spectra, which is
dominated by the fine-scale turbulence, can’t be well captured from the
estimation technique.

###  IV.2 Estimation of 2D Velocity from Streamwise PIV and In-Flow Pressure
Measurements

Figure 15: A comparison between pressure spectrum measured from the B&K
microphone at $x/D=6.6$, $r/D=0.5$ and the one from the GRAS microphone at
$x/D=6.6$, $r/D=-0.5$.

In this subsection experimental results from synchronously measured two-
compnent PIV and in-flow pressure signals on the upper and lower jet liplines
will be utilized to provide time-resolved, pressure-informed velocity
estimation on the streamwise plane. As a priority to detailed analysis,
pressure spectra from both GRAS and B&K microphones were first examined and
results are displayed in Figure 15. Although different aerodynamic designs are
employed to the nosecones of the two microphones, careful examination shows
high level of consistency with respect to the spectral shapes from both
microphones. The slight discrepancy between the two pressure spectra, which is
less than 2 dB throughout the frequency range shown in the figure, might arise
from the positioning uncertainties of microphones ($\pm 0.5$mm in all
directions and $1^{\circ}$ in angle), the different frequency response
characteristics of microphones, and the different aerodynamic design of
nosecones. Furthermore, self-noise induced from one in-flow microphone is not
noticed to distort the pressure spectrum measured from the other microphone,
indicating the mutual interaction between both microphones is negligible
compared to the intensive fluctuating pressure level generated by the jet
turbulence.

Figure 16: Energy distribution and the cumulative sum of the first 50 POD
modes under the current field of view.

Figure 17: Streamwise eigenfunctions of the leading POD modes.

Figure 18: Radial eigenfunctions of the leading POD modes.

To extract dominant spatial features from the PIV measurements, the snapshot
POD was performed to decompose the two-dimensional velocity vectors into a
series of energy-ranked modes. Figure 16 presents the energy distribution as
well as the cumulative sum of the first 50 POD modes out of the total of 8000
modes. The energy is seen to be distributed over a wide range of POD modes in
a descending order, and around 55% of the total energy is contained in the
first 50 modes which will be used to reconstruct the time-resolved velocity
fields from SLSE-POD and machine learning approaches. Shapes of the streamwise
and radial components of the leading POD modes are presented in Figure 18 and
Figure 18, respectively. The positive and negative regions of eigenfunctions
in Figure 18 demonstrates that the most energetic region under the current
field of view concentrates inside the jet shear layer. The first two POD
modes, for example, represents the appearance of large-scale structures
spanning between $x/D=5$ to $6.5$. Each of them highlights a structure on one
side of the jet axis such that the combination of the two modes is capable of
representing axisymmetric and antisymmetric patterns of the structures at the
largest scale. Successively, higher order POD modes are gradually governed by
smaller-scaled features which are still centered around the jet liplines.
Although none of the leading POD modes are strictly symmetric or antisymmetric
about the jet axis, the combination of the dominant POD modes is capable to
represent turbulent structures with any symmetrical properties in a similar
approach.

Figure 19: Cross-correlation coefficient $C_{ap}$ between POD expansion
coefficients and pressure fluctuations. The B&K microphone is on the upper
lipline and the GRAS microphone is on the lower lipline.

The cross-correlation coefficient, $C_{ap}$, reveals the relationship between
the dominant POD modes and the measured pressure fluctuations and are
displayed in Figure 19. Since the pressure measurements were performed
downstream of the PIV window, the maximum magnitudes of cross correlation
always appear at some positive time lag. In general, the amplitude of the
correlation coefficient is relatively low (less than 0.22), and will gradually
decrease for higher order POD modes as the dominant spatial structures become
smaller-scaled vortices that are less influential to the pressure on the jet
liplines. Since symmetric properties can not be established from individual
POD modes, correlations with lipline pressure measurements are not fully
symmetric/antisymmetric as well. One representative example is POD mode 4, in
which a clear wave-like pattern with a peak value around 0.2 can be observed
in the correlation with pressure on the upper lipline, however the correlation
with pressure on the lower lipline is at a low level overwhelmed by the
background noise.

(a) Training loss

(b) Validation loss

Figure 20: Evaluation of training and validation losses from both NN
architectures.

Results of velocity estimation from stochastic estimation and machine learning
approaches will be discussed in the following. The learning outcome of the two
neural networks are first examined by evaluating the training and validation
losses of each POD mode, as presented in Figure 20. For all modes, training
and validation losses for both the MLP and bidirectional LSTM architectures
are always within the range of 0.04 to 0.08. However, since the POD expansion
coefficients are energy-ranked, errors appearing in the leading POD modes will
be more likely to influence the estimation results in velocity than higher
order modes. The evaluation of validation losses, which is not directly
involved in the training process, will constitute the generalization error of
the training progress. Validation losses for the both NN schemes are almost
identical to each other although errors from the bidirectional LSTM scheme are
always slightly lower comparatively. The pronounced similarity of loss
functions between two neural networks indicates that the primary relationship
between POD expansion coefficients and time-lagged pressure have been
effectively captured from both well-trained networks.

(a) $a_{1}$

(b) $a_{5}$

(c) $a_{10}$

(d) $a_{50}$

Figure 21: Time-varying estimation of POD modal expansion coefficients. Figure
22: Cross-correlation coefficient between estimated expansion coefficients
from two machine learning schemes.

Figure 21 displays pressure-informed estimation of POD expansion coefficients
from both NN architectures for modes 1, 5,10, 50, respectively. Despite the
expansion coefficients for each mode were scaled into $[-1,1]$ before
training, Figure 21 manifest that the amplitude of prediction results will
gradually reduce with increased POD modes. This is consistent to the fact that
higher POD modes representing smaller-scaled turbulent structures are less
coherent to the pressure fluctuations in the flow. For POD mode 1 and mode 5,
prediction results from MLP and bidirectional LSTM shows remarkable similarity
with respect to large scale fluctuations, in which locations of crests and
troughs in time are mostly identical to each other although discrepancies on
the amplitudes can be observed. Prediction results from both schemes become
less similar for higher order modes, and the temporal evolution hardly matches
each other at mode 50. As a measure of similarity between estimations from the
two NN architectures, time-lagged cross-correlation coefficients between
outputs from both networks are presented in Figure 22 using the same pressure
input. High correlation levels can be observed for the POD mode 1 and 5, which
suggests strong consistency between estimations of the leading POD modes from
both neural network architectures, and the dominant input-output relation
between pressure and the most energetic coherent structures can be well
captured from both trained networks. A decreased correlation level is observed
for higher order POD modes, in which the correlation level decays from 0.42 to
0.08 from mode 10 to mode 50. Since the correlation between pressure and
smaller-scaled vortices becomes weaker, the influence of difference NN
architectures to the prediction outcomes starts to dominate and greatly
reduces the similarity level. Nevertheless, the wave-packet shapes, as well as
the high symmetricity shown in the cross-correlation coefficients, both
indicate that both NN schemes are able to represent the dynamics of the
leading POD modes (i.e. dominant coherent structures) associated with in-flow
pressure to some extent.

(a) $x/D=4$, $r/D=0$

(b) $x/D=4$, $r/D=0.5$

(c) $x/D=5$, $r/D=0$

(d) $x/D=5$,$r/D=0.5$

(e) $x/D=6$, $r/D=0$

(f) $x/D=6$, $r/D=0.5$

Figure 23: A comparison of velocity spectra from neural networks and
stochastic estimations.

The streamwise velocity spectra estimated from machine learning architectures
are compared to the ones from SLSE-POD, and results at three representative
streamwise locations are shown in Figure 23. Direct measurements from a
hotwire measurement are also superimposed on the figures. Given the relatively
high noise level between pressure and POD expansion coefficients, the wavelet-
filtered (WF) SLSE-POD is also utilized to generate the spectral estimation
from the wavelet-filtered cross-correlation. For all cases, intensive noise
can be observed in the SLSE spectra after $St=0.5$, which is caused by the
noise contained in the raw cross-correlation functions. Although the
implementation of a wavelet-filter is able to attenuate the noise level by
several orders of magnitude, the empirical choice of the filter threshold will
impair the rolling-off of the spectra and the curves will stop decaying after
$St=1$. In contrast, the high frequency noise is effectively damped in the
spectra predicted from both NN architectures. The bidirectional LSTM spectra
always present a steeper rolling-off rate than the MLP spectra and decay rates
from both networks are faster than the ones from the hotwire measurements.
Since the rolling-off of velocity spectra is contributed from the less-
organized fine-scale turbulence, the POD-based NN architectures, which only
preserve the linear combination of the large-scale spatial features to
reproduce the flow field, are reasonable to possess a faster decay than the
measured spectra. These facts also confirm the effectiveness for NN
architectures to distill useful information associated with the dominant
spatial features from the pressure inputs.

From $St=0.1$ to $0.6$ where large-scale coherent structures are dominant,
four estimation techniques are observed to generate similar spectral
estimations. The high degree of similarity between spectral estimates from
neural networks and linear stochastic estimations proves that relation between
pressure and the large-scale structures appearing in the low-frequency range
is predominantly linear. Spectral densities from SLSE-POD and WF-SLSE-POD are
seen to be the closest to each other, which verifies the assumption that the
wavelet-filtering process will preserve the prevailing wave-packet features
from the noisy cross-correlation functions. As for NN architectures, velocity
spectra predicted from the bidirectional LSTM scheme are more consistent to
the denoised WF-SLSE-POD results. Since it has been verified that the WF-SLSE-
POD is capable to preserve the most significant wave-packet features in the
input-output models, the similarity between bidirectional LSTM and WF-SLSE-POD
demonstrates great advantage of this neural network architecture given the
fact that this method can produce high-quality spectral estimates and no
filtering threshold need to be determined at the same time in an empirical
manner. Such similarity also validates the general advantage of RNN-based
neural networks to effectively handle sequential input data to extract
dominant features.

(a) Streamwise velocity $u_{x}/U_{\infty}$

(b) Radial velocity $u_{r}/U_{\infty}$

Figure 24: Instantaneous velocity reconstruction from bidirectional LSTM.
White dashed lines depict the jet potential core. Figure 25: 2D vorticity
($\omega\frac{D}{U_{\infty}}$) of the reconstructed flow field from
bidirectional LSTM overlaid by velocity vectors.

When comparing to the hotwire measurements, all estimation methods will
underestimate the overall spectral density in general. This is consistent with
the fact that only the portion of velocity associated with pressure inputs can
be well reflected from the estimations. The difference in amplitude will be
gradually narrowed down when $x$ moves from $4D$ to $6D$, indicating a better
relation between pressure and velocity can be constructed when the streamwise
distance is confined. For large-scale events at $x/D=4$, one may observe that
the hump between $St=0.2$ and $0.6$ on the jet lipline, as well as the
broadband event at $St<0.3$ on the jet centerline, are both faithfully
reflected from spectral estimations. However at $5D$ and $6D$ downstream, a
discrepancy on the peak location is found on the jet centerline although the
broadband events between $0.1<St<0.3$ on the jet lipline can still be fairly
captured. For example, at $r=0$, the measured velocity spectrum peaks around
$St=0.3$ at $x/D=5$ and a flattened peak is seen to form between $0.1<St<0.3$
at $x/D=6$. However, for both stochastic estimations and neural networks,
velocity estimates will peak around $St=0.4$ (see Figures 23(c) and 23(e)).
Since the centerline velocity estimation is informed by the pressure recorded
on the jet liplines, the spectral estimations only represent the part of
centerline velocity that is related to the pressure fluctuations on liplines.
In other words, the peak location of spectral estimates won’t necessarily
match the actual peak location without knowing related information from
pressure inputs. The addition of pressure measurements on the jet axis should
be capable to resolve the shift of the peak location.

Figure 24 shows time-resolved velocity reconstruction from the bidirectional
LSTM using the first 50 POD modes which corresponds to the portion of the
overall velocity field that is associated with pressure measured downstream.
The jet potential core is also depicted in all instances based on the jet core
length of $5.73D$. The core length was calculated from the mean velocity
profile based on the definition in [57]. Figures 24(a) and 24(b) show the
temporal evolution of the streamwise and radial velocity components,
respectively. Following the time sequence, the emergence, growth, and orderly
convection of the large-scale structures along the streamwise direction can be
clearly observed inside the jet shear layer in Figure 24(a), and these
structures corresponds to the streamwise propagation of wave-like patterns in
Figure 24(b) where regularly distributed structures in the radial direction
move oppositely to their neighbors. As these structures move downstream, one
can clearly observe the growth and intensification of the turbulent eddies. A
comprehensive evaluation of the velocity reconstruction is presented in Figure
25 in which the 2D vorticity, $\omega=\nabla_{2D}\times u_{i}$ is displayed
and velocity vectors are overlaid on top. Before the end of the potential
core, the dominant feature is the growth and streamwise convection of the
vortical structures. The appearance of counter-rotating vortex pairs can also
be observed inside the jet mixing layer. At the same time, the vorticity is at
an insignificant level inside the potential core which is typically accepted
to be irrotational although flapping motion can be detected by the end of the
jet core. Around the end of the jet core, vortical structures on both sides of
the potential core are merged into some larger structures spanning across the
jet centerline and keep travelling with the flow. These eddies are highlighted
not only by their large sizes but also by the high overall vorticity
magnitude. Since this pressure-informed estimation filters out the part of the
flow that is associated with the downstream pressure, these turbulent
structures emerged downstream of the jet core are highly associated with the
pressure wave-packets in the flow which serves as noise sources propagating to
the acoustic farfield.

##  V Conclusions

In this study, time-resolved source activities inside the jet mixing layer
were estimated based on PIV and downstream in-flow pressure measurements.
Stochastic estimation methods and neural networks were proposed to model the
input-output relationship between in-flow pressure fluctuations and POD
expansion coefficients. For the implementation of stochastic estimation, a
wavelet-based filter was also utilized to extract the dominant the wave-packet
structures from the noisy cross-correlation functions obtained from the
experiments.

Two sets of experiments were performed in combination with the proposed
estimation architectures. The space-time dynamics of the axisymmetric velocity
components was characterized from cross-stream PIV measurements as well as
downstream pressure measurements on the jet axis. For axisymmetric velocity
components, strong mode-1 dominance in space can be observed at all streamwise
locations from the azimuthal-Fourier POD. Cross-correlation between pressure
and the first POD expansion coefficients shows significant wave-like behavior
in the ﬂow at a convection speed of $0.7U_{\infty}$. Time-resolved estimation
of axisymmetric velocity from wavelet-filtered SLSE-POD reproduce the
convective nature of the axisymmetric velocity components. A comparison to the
measured velocity spectra from hotwire measurements demonstrates the capacity
of this hybrid approach to faithfully reflect the broadband hump in the low
frequency range although a discrepancy in the overall spectral amplitude is
observed.

Pressure measurements on the upper and lower jet liplines were synchronously
performed with streamwise planar PIV to obtain real-time estimates of two-
dimensional velocity vectors on a streamwise plane. The most energetic spatial
structures are seen to distribute within a wide range of POD modes, and the
cross-correlation coefficients between lipline pressure and the dominant POD
modes are at a relatively low level. Stochastic estimations and neural
networks were employed to reconstruct the 2D source behaviors using the first
50 POD modes. All estimation approaches utilized in this work are capable of
highlighting the broadband peak at low Strouhal numbers in spite of the
existence of amplitude discrepancies compared to the measured spectra. In the
comparison of velocity spectra estimates, the unique advantage of
bidirectional LSTM architecture to highlight the broadband hump as well as
attenuating the high frequency noise is observed. Real-time velocity
reconstruction from bidirectional LSTM highlights the streamwise convection of
the coherent structures inside the jet mixing layer as well as the formation
of larger eddies downstream of the jet potential core. These results are
beneficial to enhance the understanding of the space-time dynamics of the
acoustic sources in the jet flow field, and the resulting velocity field
estimates could be used to calculate the pressure field via Poisson’s equation
[58] which allows one to evaluate the pressure wave-packets in the flow.

## Acknowledgments

The authors acknowledge the support from Wenyan Li for the development of the
machine learning code. The authors also acknowledge University of Florida
Research Computing (UFRC) for providing computational resources and support
that have contributed to the research results reported in this publication.
This research was funded by the National Science Foundation (NSF) under award
CBET-1704768.

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