12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

SPECTRAL PROPER ORTHOGONAL DECOMPOSITION FOR COUPLEDHYDRODYNAMIC/ACOUSTIC FIELDS

Maxime Fiore, Hélène Parisot-Dupuis, Benjamin Etchebarne and Romain Gojon
Department of aerodynamics and Propulsion (DAEP)ISAE-SUPAERO, University of Toulouse
10 Avenue Édouard Belin, 31400 Toulouse, France∗ email: romain.gojon@isae-supaero.fr

ABSTRACT

pler structures (defined as the ones with largest probability tooccur in the turbulent flow) and sorted them in decreasing magnitude according to a norm defined a priori, the energy contained in each mode for example. More recently, a preceding
step corresponding to a discrete Fourier transform (DFT) has
been added to the POD and named spectral POD (SPOD). Thismethod presents the advantage to add up a frequency information on the different phenomena captured by the POD. Thismethod has been described by Schmidt & Colonius (2020), theauthors providing proper theoretical weighting/normalization
for the field variables (density, velocity and temperature fluctuations for compressible flows) in order to capture hydrodynamic flow phenomena. When applying the SPOD treatment
with such normalization to configurations where both acoustic and hydrodynamic interact, the acoustic phenomena generally do not emerge. The POD step of the algorithm could
be able to separate coherent structures, namely the hydrodynamic and acoustic structures. However, the acoustic part isgenerally of very low energy content, corresponding to several
order of magnitudes under the energy level of the hydrodynamic component in the majority of flow configurations. The
POD, which is based on an energy optimal decomposition,
has been tested in such kind of configuration (hydrodynamic
turbulence-acoustic interaction) by Druault et al. (2013) and
Gaudard et al. (2014). It was observed that the first POD
modes act as a filtering technique in the wavenumber space(that contains the main part of the signal energy) but cannot properly separate both components especially in the low
frequency-low wavenumber domain. Therefore, the SPOD being not able to separate the acoustic and hydrodynamic phenomena in its basic implementation, the idea could be to find asuitable normalization/weighting for the input data given to the
SPOD. Two relevant flow quantities to study aero-acoustic interactions are the velocity fluctuations (carrying hydrodynamic
source field) and pressure fluctuations (carrying the acoustic
propagating field). However, the transfer function between hydrodynamic and acoustic fluctuations is generally case dependent and no general theoretical normalization can be derived
for any aero-acoustic fields (i.e. apply a different weighting for
velocity fluctuations related to hydrodynamic source and pressure fluctuations related to acoustic propgation) as the one proposed by Schmidt & Colonius (2020) for hydrodynamic fields.
This paper proposes to apply general pre-processing steps
to the input data of the SPOD to make the hydrodynamic andacoustic fields of the same order of magnitudes in order toproperly analyze the acoustic field with the SPOD treatment
and its interactions with the source hydrodynamic field. This

In a large number of flow configurations, the complexity of the phenomena involved and their potential interaction
can make the analysis of the flow field difficult. In the purpose to decompose the flow in distinct and simpler phenomena, the proper orthogonal decomposition (POD) has beendeveloped and make possible to isolate different phenomena
and sort them in decreasing order of magnitude based on ametric chosen a priori. Recently, the spectral POD (SPOD)
has been developed by adding a preceding discrete Fouriertransform (DFT) step which allows to have the frequency information on the modes retrieved by the POD. This gives
the possibility to isolate several phenomena occurring at asame frequency. Based on the theoretical background proposed by Schmidt et al., this method has been widely validated in hydrodynamic configurations showing its good behavior compared to other modal decomposition methods, namely
DFT or dynamical mode decomposition (DMD). However, inflow configurations where both the hydrodynamic and acoustic
fields interact, without additional treatment, the hydrodynamic
field is generally dominant and only the hydrodynamic modes
emerge. Furthermore, the transfer function between the hydrodynamic (source) field and the resulting acoustic (propagating)field is generally case dependant, making difficult to propose
a weighting/normalization adapted for all aero-acoustic configurations. This paper proposes a general framework basedon some pre-processing steps on the input aero-acoustic fields
to make the hydrodynamic and acoustic fields of comparable magnitude for the SPOD. The purpose is then to analyze
how the hydrodynamic field leads to the acoustic field and thepossible feedback effect. This framework is illustrated on asupersonic impinging jet flow configuration used as an input
data for the SPOD, and based on wall-resolved compressible
large-eddy simulations (LES) where both the hydrodynamic
and acoustic fields are solved.

INTRODUCTION
The analysis of the aero-acoustic field in flow configurations is generally made difficult for several reasons: the acoustic field is usually induced by complex hydrodynamic phenomena; several hydrodynamic/acoustic phenomena can coexist
with some difficulty to separate their contributions. The latter issue has been investigated in the context of turbulent flowby Lumley (1981). The authors proposed to apply the proper
orthogonal decomposition (POD) to the data obtained from aturbulent flow in order to decompose the complex field in sim-

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Sponge/relaxation zone
20h

ρ/ρ0
2.1

42h

Flat plate (wall no-slip)

12 h
10 h

0.7

ux /u00.8-0.2

uy /u0
0.4

p/p0
1.35

-0.4

0.95

Extracted plane
8.27 hh

Jet inletyzx

Figure 2. Instantaneous fields of density, axial (ux ), tangential (uy ) velocities and pressure in the (x,y) plane (center of thejet nozzle length (z = l/2)).

y

z

x

h

Mach number of M0 = u0 /c = 1.28 and a Reynolds number of
Re = u0 h/ν = 5 × 104 , where u0 , c and ν are respectively thejet velocity, the speed of sound at the nozzle exit, and the kinematic molecular viscosity (the stagnation temperature being
equal to the ambient temperature). At the nozzle inlet, a Blasius laminar boundary-layer profile of thickness 0.075 h anda Crocco-Busemann profile are imposed for velocity and density. The ejection conditions of the jet and the ratios between
the nozzle-to-plate distances and the exit diameter in the simulations are identical to those in the experiments of Thurow
et al. (2002). In order to generate velocity fluctuations at thenozzle exit, low-amplitude random vortical disturbances, notcorrelated in the spanwise direction, are added in the nozzle
boundary layer at x = − 0.25 h using a procedure detailed
in Bogey et al. (2011). The forcing strength is set to α = 0.02.
The unsteady compressible Navier-Stokes equations are
solved in a cartesian coordinate system (x, y, z) by using an explicit six-stage Runge-Kutta algorithm for time integration andlow-dispersion explicit eleven-point finite differences for spatial derivation (Bogey & Bailly (2004); Berland et al. (2007)).
At the end of each time step, a sixth-order eleven-point filtering (Bogey et al. (2009)) is applied to the flow variables inorder to remove grid-to-grid oscillations and to relax turbulent
energy from scales at wave-numbers close to the grid cutoff
wave-number. Thus, the filtering acts as a subgrid-scale modelin the LES (Bogey & Bailly (2004, 2009); Fauconnier et al.
(2013); Kremer & Bogey (2015)). The radiation conditionsof Tam & Dong (1994) are implemented at the inflow and lateral boundaries of the computational domain. A sponge zone
combining grid stretching and Laplacian filtering is also employed in order to damp the turbulent fluctuations before theyreach the lateral boundaries. Adiabatic conditions are imposedat the nozzle walls and at the flat plate. A shock-capturing filtering is also applied in order to avoid Gibbs oscillations nearshocks. It consists in applying a conservative second-order filter at a magnitude determined each time step using a shock
sensor (Bogey et al. (2009)).
The simulations have been carried out using an OpenMPbased in-house solver, and a total of 200,000 iterations arecomputed after the transient period. The temporal discretization is equal to ∆t∼0.003h/u, yielding to a simulation time ofTsim ∼600 h/u. The mesh is composed of 250 × 106 points.

l=3.25 h

Figure 1. Simulation domain projected in the (x, y) plane(top) and iso-surfaces of density associated with 1.25 kg.m−3
coloured by the local Mach number, and of the pressure fluctuations in the plane (x, y) using a colour scale ranging from-7500 to 7500 Pa (bottom).

process is mainly based on two steps: the splitting of the flow
field into a domain dominated by the hydrodynamic effects
and a remaining domain where the acoustic field propagates;
and a normalization of these two regions to have similar levels of fluctuations for the hydrodynamic and acoustic fields.
The process and the analysis of this approach are illustrated
on a normal impinging jet flow configuration simulated using large-eddy simulation (LES) and validated in a previous
study (Gojon et al. (2016)).
The paper is organized as follows: Sec. 1 introduces
the configuration and the numerical setup; The pre-processing
steps to adapt the input fields, the SPOD algorithm and corresponding parameters used for the study are detailed in Sec. 2;
the analysis of the SPOD treatment for the coupled hydrodynamic/acoustic fields is detailed in Sec. 3.

1

CONFIGURATION/NUMERICAL METHODS

A detailed description of the physics of a supersonic planar jet impinging on a flat plate normally and the corresponding numerical details for LES calculations are provided in
Gojon et al. (2016), but a brief description is provided here.
The impinging jet flow configuration shown in Fig. 1 originates from a planar straight nozzle of height h = 0.002 m,
width l = 3.25 h in the spanwise direction, and whose
lip is 0.5 h thick, in an ambient medium at temperature
T0 = 293 K and pressure p0 = 105 Pa. Different nozzleto-plate distances L have been investigated (3.94 h, 5.5 h,
8.27 h and 9.1 h) but the present study focuses on the 8.27 hcase. At the nozzle exit, the jet is ideally expanded, with a2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Hydrodynamic domain

Acoustic domain

Pressure fluctuation P’red

Normalized
pressure/velocity
fluctuation
1-1
Normalized vorticity
1-1Norm. Vorticity magn.
10Figure 3. Instantaneous normalized vorticity field (left) andtemporally averaged vorticity magnitude field (right) used tocompute the threshold vorticity value.

2

Velocity fluctuation u’red
Figure 4. Instantaneous mixed normalized pressure/velocity
fluctuation field (snapshot) used as an input to the SPOD.

PRE-PROCESSING/SPOD ALGORITHM

The input data corresponds to 4,000 two dimensional cuts
snapshots sampled at the frequency St = f h/u0 = 6.4 where fis the frequency, and containing the variables [ρ,ux ,uy ,p] extracted from the LES simulation from Gojon et al. (2016) (seeFig. 2). The planes are extracted at the center of the jet nozzle
length in the z direction (z = l/2) and the corresponding (x, y)planes are reduced in the x and y directions compared to thefull simulation domain (see Fig. 1). This size domain reduction is performed in order to avoid regions in the sponge zone
and have mesh sizes around the jet sufficiently fine to transport acoustic waves, the mesh being progressively coarsened
as we move away from the jet. The extracted planes extends
from -12 h to 8.27 h in the axial direction, the origin being
the jet lip inlet. In the y direction, the mesh extends from 10 h to 10 h, the origin being the center of the jet. Basedon the numerical parameters of the LES simulation (around
6 points/wavelength) and the coarser mesh size at the boundaries of the plane (∆x ' 10h, ∆y ' 8h), the plane data used as
an input of the SPOD can capture phenomena until St = 5.0.
The spatial domain is first split into a region dominated by hydrodynamic instabilities and the remaining domain consideredas the acoustic propagating domain. To do so, in each cell ofthe (x,y) reduced planes, the z-component of the vorticity ωz
and its modulus (vorticity magnitude) are computed along the
4,000 temporal snapshots (see Fig. 3):|ωz | = |∂ uy ∂ ux
−
|.
∂x
∂y

jet, the hydrodynamic region corresponds to the jet flow, themixing region with the fluid at rest (expansion of the jet flow)
and the region close to the wall corresponding to the two wall
jets created after the impact on the flat plate (see Fig. 3). It canbe noticed that this ”hydrodynamic” region also contains an
acoustic propagation, but the velocity/pressure fluctuations associated to the acoustic waves are several order of magnitude
lower than the hydrodynamic/pressure fluctuations, and thesefluctuations are mixed so that separate acoustic from hydrodynamic is difficult. This is pointed out by Arndt et al. (1997);
Tinney & Jordan (2008); Grizzi & Camussi (2012); Mancinelli
et al. (2017) whose tried to split the fluctuations between nonradiating hydrodynamic and radiating acoustic. For the cells
in the hydrodynamic region, the velocity fluctuations are considered since this variable holds the source term of hydrodynamic fluctuations. In the acoustic region, the pressure fluctuations are considered since this quantity holds the acoustic
propagation. The resulting snapshots consists of mixed fields
in velocity and pressure fluctuations. In order to provide similar amplitudes for these two variables, the velocity and pressure fluctuations are reduced using the global maximum, along
time and in all the corresponding hydrodynamic/acoustic partof the field, of the root mean square (RMS) values of these twovariables yielding to (see Fig. 4):(1)
u0red =The minimum and maximum values of the vorticity magnitude
in all cells and over time |ωz |min and |ωz |max are stored. These
two quantities are then used to build a threshold for the vorticity defined as:|ωz |thres = |ωz |min + 0.1(|ωz |max − |ωz |min ).

u0
uRMS

0
Pred
=

P0
PRMS

.

(3)

u0 being either the axial velocity fluctuation (u0x ) or the verticalvelocity fluctuation (u0y ). The corresponding spatio-temporalmatrix Q’ used as an input for the SPOD can be written as:

(2)


0 )
(Pred
1,1
(u0red )2,1

0 )
(Pred
1,2
(u0red )2,2··· ······ ···

0 )(Pred
1,nsnap(u0red )2,nsnap













.
.
.
.
.
..
..
.. ..
..
Q0 = 







(u0 )
00
red nndof −1,1 (ured )nndof −1,2 · · · · · · (ured )nndof −1,nsnap000(Pred )nndof ,1 (Pred )nndof ,2 · · · · · · (Pred )nndof ,nsnap

For each cell of the domain, if the value of the vorticity magnitude overcomes at least one time the threshold value, then thiscell is tagged as being enclosed in the region dominated by hydrodynamic fluctuations. On the contrary, the un-tagged cells
are considered to be in the acoustic domain. For the impinging

3(4)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
where each column represents one snapshot of the full spatial
plane of mixed pressure and velocity fluctuations, and a line
represents the time evolution of a single spatial point and canbe either pressure or velocity fluctuations time signal.
The mathematical basis of the SPOD method is detailedin Sieber et al. (2016) and Towne et al. (2018), its use isdetailed in Schmidt & Colonius (2020) and applications to
fluid flows is provided in Ghate et al. (2020) and Abreu et al.
(2020). The SPOD method relies on the DFT and POD methods. For the present study, 4,000 snapshots (nsnap ) of the 2D
plane are used. The dataset is partitioned into smaller overlapping blocks, each of which represents an ensemble realizationof the flow. It corresponds to a splitting of the initial matrix
Q0 into 8 blocks with 50% overlapping corresponding to 512
snapshots by block. A temporal DFT of each block is thencalculated. To reduce spectral leakage, a Hanning window isused to limit the discontinuities with a hypothetical next period. The frequency resolution is around ∆St = 0.007 to becompared with a frequency resolution of around 0.0015 forc0 area classic DFT analysis. New data matrices denoted Qkformed at the k-th frequency by collecting all of the Fourier
realizations of the different blocks at frequency k. The autocovariance matrix is built for each frequency obtained with the
DFT yielding to:c0 = W (Qc0 )T Qc0 for k ∈ [1, n ]
R
dft
k
k
k

the families (Φl )l∈[1,nmodes ] , (al )l∈[1,nmodes ] being respectively
orthonormal basis. The SPOD treatment provides for eachvariables u0r , Pr0 a set of frequencies k, eigenvectors, eigenvalues, and temporal functions for each of these frequencies(λk,l , Φk,l , ak,l )(k,l)∈[1,ndft ]×[1,nmodes ] .

3

The analysis of the aero-acoustic field of the impinging jet
flow is split into two parts: the analysis of the hydrodynamic
field in the full domain performed by using the vertical velocity
field as a direct output of the LES simulation, and the mixed
vertical velocity/pressure algorithm proposed in this study isused to obtain the decoupling between hydrodynamic sources
and their propagation in the flow domain.

3.1

(5)



(6)

c0 )
The terms of the matrix (Rk k∈[1,ndft ] ) correspond either to nondimensional hydrodynamic fluctuation energy when the spatial
point is in the hydrodynamic region, or to non-dimensional
acoustic energy when the spatial point is in the acoustic region. The converged estimation of the cross-spectral density
tensor is obtained by averaging the spectra over multiple realizations of the flow using Welch’s method. The POD treatmentc0 )is then applied to the matrices (Rk k∈[1,ndft ] at each of the DFT
frequencies. This corresponds to solve the Fredholm eigenvalue problem for each frequency k (or equivalently performc0 )the singular value decomposition of the matrices (Rk k∈[1,ndft ] ):0
A Rk Φl (r, z) drdz = λl Φl (r, z), k ∈ [1, ndft ]R

The 6 first most energetic peaks of SPOD1 for the vertical velocity fluctuation (denoted 1 to 6 in Fig. 5) are shownin Fig. 6. The first peak of SPOD1, at St = 0.092, has an antisymmetrical footprint in the jet and in the acoustic regions withlarge scale coherent structures that extent in the jet region andpropagating acoustic waves outside of the jet. This frequency
is thus related to a symmetric oscillation mode of the jet, thejet axis being an axis of symmetry for the vertical velocity
in presence of a symmetric mode. The second (St = 0.11),
third (St = 0.165), fourth (St = 0.21), fifth (St = 0.24) and sixth(St = 0.29) peaks present a similar flow pattern with symmetrical structures released at the center of the jet with a size thattypically represent the nozzle height. They are thus associatedto an anti-symmetric oscillation mode of the jet. Finally, thedominant peak for the vertical velocity fluctuation corresponds
to the peak 4 of SPOD1 at St = 0.21.

(7)where (λl , Φl ) are the l-th eigenvalue and eigenfunction fromc0 at frequency
the singular value decomposition of the matrix R
kk. The eigenvalues and eigenvectors are sorted from higher
to lower energy content and a fixed number nmodes is generally retained. The temporal functions of the POD treatment are obtained
 by projection of the original fluctuating field0
q0 ∈ u0red , Pred
at a fixed frequency k onto the eigenvectors(Φl )l∈[1,nmodes ] :Zal (t) =A

q0 (r, z,t)Φl (r, z)drdz,

Vertical velocity field in the full domain

The eigenvalues associated to the first 5 (out of the 8) mostenergetic velocity fluctuation SPOD modes (vertical velocity
fluctuation contribution to the turbulent kinetic energy) obtained considering the reduced-size extracted planes are shownin Fig. 5. Each eigenvalue has been non-dimensionalized by
the sum of all eigenvalues for each SPOD mode and over all
frequencies (total turbulent kinetic energy in the flow) to provide the relative contribution of each SPOD mode at each frequency to the vertical velocity contribution in the turbulent kinetic energy balance. The mode 1 (most energetic POD modeover all the frequency range denoted SPOD1) exhibits few
main peaks. At these frequencies, the magnitude ratio withsubsequent SPOD modes is relatively large (generally one order of magnitude) so that the majority of the energy at thesefrequencies is hold by the mode SPOD1. This observationmeans that, at these frequencies, only one mechanism is taking place in the flow domain. Out of these main peaks, thebackground level can be related to the broadband energy contribution of turbulence where the different SPOD modes have
similar energy magnitudes (in general ratios between 1 and 2).
The integration of all the eigenvalues of the SPOD modes over
all frequencies provide the contribution of the vertical velocity fluctuation to the turbulent kinetic energy contained in theextracted plane. This same analysis led for the axial velocity
fluctuation shows that the turbulent kinetic energy is generated
at 47.3 % by the axial velocity fluctuation and 52.7 % by thestudied vertical velocity fluctuation.

where W is the weighting matrix representing the individualarea (dA) of each cell of the spatially discretized plane:dA1,1 dA1,2 · · · · · · dA1,nsnap


..
..
.. ..
..
W =
.
.
.
. .
.
dAnndof ,1 dAnndof ,2 · · · · · · dAnndof ,nsnap

SPOD ANALYSIS(8)4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

4
5

[-]

10−2

6

Eigenvalue

n

10−2

10−3

10−4

0

0.2

0.4

0.6

0.8

10−1

1

SPOD1
SPOD2
SPOD3
SPOD4
SPOD5

1
2

3

4

n

23

λk,l

1

n

Eigenvalue

n

λk,l

10−1

SPOD1
SPOD2
SPOD3
SPOD4
SPOD5

dft
modes
λk,l
∑i=0
∑l=0

dft
modes
λk,l
∑i=0
∑l=0

[-]

100

5 6

10−3

10−4
10−5
0

0.2

0.4

St [-]

0.8

St [-]

Figure 5. SPOD eigenvalues as a function of frequency (St)based on the vertical velocity fluctuations field in the fulldomain for the 5 most energetic SPOD modes (SPOD1 to
SPOD5), logarithmic scale for the y-axis.

10

0.6

Figure 7. SPOD eigenvalues as a function of frequency (St)based on the mixed field pressure/vertical velocity fluctuations
for the 5 most energetic SPOD modes (SPOD1 to SPOD5),
logarithmic scale for the y-axis.

1

2

1

2

3

4

3

4

10

0

0

y/h

y/h

-10
-12

x/h 0
5

8.27

-10
-12

6

x/h 0
5

8.27

6

Figure 6. Vertical velocity fluctuations SPOD temporalmodes in the full domain for the 6 most energetic structuresof SPOD1.

Figure 8. Mixed vertical velocity/pressure fluctuations
SPOD temporal modes in the full domain for the 6 most energetic structures of SPOD1.

3.2

modes of the SPOD1 are shown in Fig. 8. For the first peak of
the SPOD1, almost no coherent structure of vertical velocity
fluctuation can be exhibited in the hydrodynamic region thatwould indicate that the source of the symmetrical pattern of thepressure fluctuations in the acoustic region is mainly correlated
to the axial velocity large scale structures in the jet. However,for the peaks two to six of SPOD1, coherent structures of vertical velocity fluctuations can be observed. These structures are
symmetrical with respect to the main axis with correspond-

Mixed vertical velocity/pressure fluctuations

The first 5 most energetic mixed vertical velocity/pressure
fluctuation modes (SPOD1 to SPOD5) obtained by considering the hydrodynamic (axial velocity fluctuation imposed) andacoustic domain (pressure fluctuation imposed) are shown in
Fig. 7. The shape of the SPOD1 mode is strongly correlated to the shape of the mode SPOD1 for the vertical velocity
alone. The corresponding 6 first most energetic peaks temporal

5112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
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ing anti-symmetric pressure fluctuation modes in the acoustic
region, as expected. For those modes, the source of the antisymmetrical pattern of the pressure fluctuations in the acoustic
region is mainly correlated to the vertical velocity large scale
structures in the jet. Finally, the dominant peak for the present
mixed vertical velocity/pressure fluctuation corresponds to thepeak 1 of SPOD1 at St = 0.092. This tone is the main acoustic tone of the jet (Gojon et al. (2016)) and becomes dominantwith the pre-processing steps presented in the present paper.

CONCLUSION AND PERSPECTIVESIn the purpose to isolate hydrodynamic, acoustic phenomena and their interaction when using the spectral proper orthogonal decomposition (SPOD) technique, a framework has
been developed that consists in few pre-processing steps applied to the SPOD input data. This method has been used to
study the aeroacoustics of a supersonic impinging jet flow fieldobtained from a LES simulation.
Overall, the mode 1 (most energetic POD mode over all
the frequency range denoted SPOD1) exhibits few main peaks.
At these frequencies, the magnitude ratio with subsequent
SPOD modes is relatively large (generally one order of magnitude) so that the majority of the energy at these frequencies
is hold by the mode SPOD1. This observation means that, atthese frequencies, only one mechanism is taking place in theflow domain. The peaks obtained for SPOD1 are similar to theone obtained in the original study presenting theses jets andusing a classic DFT algorithm. Moreover, the same oscillation
modes of the jet were retrieved. It is interesting to notice thatthe main peak of the SPOD1 mode is not the same when using
the vertical velocity or the proposed mixed velocity/pressure
fluctuations field for the SPOD analysis, highlighting anotheraeroacoustic phenomena. In the future, more complex aeroacoustic flows will be studied in order to verify if the present
methodology is capable to extract reduced order models of theflow with for each mode, its hydrodynamic and acoustic footprints, and their interactions.

ACKNOWLEDGEMENTS
This work was performed using resources from GENCI[CCRT-CINES-IDRIS] (Grant 2021-[A0102A07178]) and
CALMIP (Project 1425-21A).

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