[1,2]Benjamin Bugeat

[1]School of Engineering, University of Leicester, Leicester, UK

2]Department of Engineering, University of Cambridge, Cambridge, UK

3]Department of Aerospace Engineering, Middle East Technical University,
Ankara, Turkey

4]Laboratoire d’Hydrodynamique, CNRS, École polytechnique, Institut
Polytechnique de Paris, Palaiseau, France

5]Département Fluides, Thermique, Combustion, Institut Pprime, CNRS-University
of Poitiers-ENSMA, Poitiers, France

# Acoustic resolvent analysis of turbulent jets

<EMAIL_ADDRESS>Ugur Karban  Anurag Agarwal  Lutz Lesshafft  Peter Jordan  *  [
[  [  [

###### Abstract

We perform a resolvent analysis of a compressible turbulent jet, where the
optimisation domain of the response modes is located in the acoustic field,
excluding the hydrodynamic region, in order to promote acoustically efficient
modes. We examine the properties of the acoustic resolvent and assess its
potential for jet-noise modelling, focusing on the subsonic regime. Resolvent
forcing modes, consistent with previous studies, are found to contain
supersonic waves associated with Mach wave radiation in the response modes.
This differs from the standard resolvent in which hydrodynamic instabilities
dominate. We compare resolvent modes with SPOD modes educed from LES data.
Acoustic resolvent response modes generally have better alignment with
acoustic SPOD modes than standard resolvent response modes. For the optimal
mode, the angle of the acoustic beam is close to that found in SPOD modes for
moderate frequencies. However, there is no significant separation between the
singular values of the leading and sub-optimal modes. Some suboptimal modes
are furthermore shown to contain irrelevant structure for jet noise. Thus,
even though it contains essential acoustic features absent from the standard
resolvent approach, the SVD of the acoustic resolvent alone is insufficient to
educe a low-rank model for jet noise. But because it identifies the prevailing
mechanisms of jet noise, it provides valuable guidelines in the search of a
forcing model (Karban et al. An empirical model of noise sources in subsonic
jets. Journal of Fluid Mechanics (2023): A18).

###### keywords:

Resolvent analysis, Turbulent jet, Coherent structures, Jet noise

##  1 Introduction

Jet-noise modelling, despite several decades of study, is still an active
field of research. Recent progress in LES computations have made possible
accurate comparisons with experiments [1, 2]. But robust low-rank models,
useful for extracting the essential physical mechanisms at play as well as
being an efficient tool for industry, remain an important research objective.
The apparently disorganised turbulent jet flow contains coherent structures
[3, 4]. These structures, termed wavepackets as they experience growth and
decay within the turbulent mixing layer, play a central role in jet noise [5,
6]. Modelling efforts have therefore been directed to wavepackets, using
different variations of local stability analysis – parallel [7], WKB [8], PSE
[9, 10] or OWNS [11]. Such approaches are sufficient to recover a large part
of the acoustic field in supersonic jets [12]. This is because Mach wave
radiation, in which a supersonic disturbance generates an acoustic radiation,
is the dominant mechanism of jet noise in this regime [13]. As stability
analysis provides a good model of the single-point, second-order statistics of
these supersonic coherent structures, it is sufficient to deduce the
associated, coupled acoustic wave. Subsonic jet noise, however, requires
higher-order statistical information, and therefore further modelling. In this
regime, the phase speed of coherent structures is subsonic. Only a small
fraction of the energy of wavepackets – which is acoustically matched [14], is
responsible for acoustic radiation. Moreover, this acoustic radiation is
greatly affected by the space-time modulation of wavepackets, termed jitter
[15, 16], and linked to the stochastic nature of the turbulent flow. By
cancelling destructive interferences, jitter may enhance sound pressure levels
up to 30 dB [16, 17]. As a result, information of the two-point statistics of
these structures is required, which, in the spectral domain, appears in the
cross-spectral density (CSD) matrix.

Resolvent analysis has recently gained popularity in the turbulence modelling
community. In this approach, an input-output view of the flow dynamics is
adopted [18, 19]. A linearised Navier-Stokes operator, through its selective
receptivity to forcing input, produces a flow response that is likely to be
statistically prevalent in the turbulent dynamics. This forcing contains the
non-linear interactions in the flow. In the presence of strong amplification
mechanisms, its modelling may not be required to obtain coherent structures
[20]. In this case, the singular value decomposition (SVD) of the resolvent
operator exhibits a large separation between the first and second singular
values, and the first left singular vector alone can provide a good model for
coherent structures.

Resolvent analysis was used to extract hydrodynamic wavepackets in turbulent
jets [21, 22, 23]. This allowed different mechanisms to be identified in the
frequency range associated with jet noise: the Orr mechanism was found to be
dominant at low frequency while Kelvin-Helmholtz wavepackets prevail at larger
frequencies. [24] and [25] furthermore noticed that resolvent analysis could
be extended to a stochastic framework, opening perspectives for the inclusion
of jitter. In the stochastic framework, a correspondence is identified between
resolvent modes and spectral proper orthogonal decomposition (SPOD) modes [26]
in the case where the forcing field is spatially uncorrelated. While this
“white-noise forcing” hypothesis is not _a priori_ valid in turbulent flow, it
has been found to successfully model the coherent dynamics in a range of
different configurations, including flow over a backward-facing step [20],
certain flames [27] and, importantly, jets (as cited above), at least over a
significant range of frequencies. The cited examples share a strong dominance
of the Kelvin–Helmholtz instability in free shear layers. Many counter-
examples exist as well, where the nonlinear forcing of the linear resolvent
operator strongly excites sub-optimal response patterns, and the white-noise
hypothesis is therefore unsuccessful in reproducing the SPOD spectrum. This is
notably often observed in wall-bounded shear flows [e.g. 28, 29, 30].

[6] pointed out fundamental similarities between resolvent analysis and
acoustic analogies [31, 32]. Indeed, the resolvent operator stems from a
reorganisation of the Navier-Stokes equations, resulting in a linear relation
between a forcing and a response; in aeroacoustics, hydrodynamic sources are
similarly associated with an acoustic field, and linked via a linear operator.
Several authors [33, 34, 35] used a modified version of resolvent analysis
that more closely resembles acoustic analogies. Energy norms associated with
the left and right singular vectors (the response and forcing fields,
respectively) can be defined in the SVD of the resolvent operator [36]. By
restricting the response norm to the acoustic field – rather than the whole
numerical domain – the SVD is expected to yield resolvent modes that are
acoustically efficient. This framework will be termed acoustic resolvent in
this paper, as opposed to standard resolvent which includes both the
hydrodynamic and acoustic regions in the response norm.

The study of [33], via the acoustic resolvent, was carried out on a subsonic
turbulent jet at Reynolds number $Re\simeq 10^{3}$. The numerical domain
included a nozzle, where the forcing field was constrained to be localised.
Good agreements were found with direct numerical simulation (DNS). The authors
noted, however, that most of the acoustic radiation was produced at the nozzle
exit, hindering interpretations regarding jet noise. [34] more extensively
analysed the acoustic resolvent framework in turbulent jets at $Re=10^{6}$,
ranging from subsonic to supersonic Mach numbers. The SVD was presented,
including tens of singular values, showing that a clear separation of more
than one order of magnitude existed between the first and the second ones in
supersonic jets. In subsonic jets, however, singular values were all found to
be of the same order of magnitude. The authors concluded on the need to
incorporate more than one mode in jet-noise modelling through acoustic
resolvent analysis – even in supersonic jets, where subsequent modes were
reported to participate in side-line noise. [35] recently used the acoustic
resolvent to propose a novel modelling approach of jet noise. In order to
model the second-order statistics of the acoustic field, the authors proposed
a low-rank model based on the SVD of the acoustic resolvent, enriched by a
model describing the projection of the forcing field onto the right singular
vectors involving two calibration constants. Encouraging results were obtained
for the jet-noise prediction at low angles compared to LES data at $Re\simeq
10^{6}$.

The aim of the present paper is to examine the properties of the acoustic
resolvent and to assess its potential for jet-noise modelling, focusing on the
subsonic regime. In particular, we aim at identifying and providing a physical
interpretation of the fundamental differences between the standard and
acoustic resolvent approaches. Emphasis will be put on the physical nature of
the resolvent forcing modes in order to get new insights into the modelling of
jet-noise sources. We will discuss the structure and phase speed of theses
modes, which have been overlooked in the subsonic regime so far. Unphysical
modes, not reported in the literature, will furthermore be shown to arise, and
hinder the use of the acoustic resolvent. Comparisons with SPOD modes
extracted from LES calculations will provide valuable insights to this end.
The influence of the nozzle, which can greatly affect jet noise [2, 37], will
also be investigated; previous studies on acoustic resolvent [33, 34, 35] did
not discuss it. Overall, the strengths, but also the limitations of the
acoustic resolvent approach, will be highlighted. Mean flows obtained from
high-fidelity LES calculations at $Re\simeq 10^{6}$ will be used to carry out
the resolvent analysis.

The paper is organised as follows. The theoretical and numerical framework are
introduced in section 2. Afterwards, a detailed comparison of the SVD of the
standard and acoustic resolvents is presented 3.1. The effect of the nozzle is
analysed in section 3.2. The acoustic resolvent modes are then more
specifically studied in section 3.3. We examine their physical features, their
alignment with SPOD modes and the effect of the Mach number. A discussion
about the physical interpretation of the SVD of the acoustic resolvent and its
potential for jet-noise modelling follows in section 4, before concluding the
paper in section 5.

##  2 Resolvent analysis

###  2.1 Mean flow

Resolvent analysis involves the linearisation of the governing equations
around a mean flow. We use a database obtained from LES calculations using the
code ’Charles’; details about the numerical procedure and the experimental
validation can be found in [2]. In the present work, a statistically steady
and axisymmetric turbulent jet flow is considered (figure 1). The streamwise
and radial directions are noted $x$ and $r$, respectively. Note that a nozzle
flow is included in the numerical domain. The state vector $\mathbf{q}$ is
expressed with the primitive variables
$\mathbf{q}=(\nu\>\>\mathbf{u}\>\>p)^{t}$. Here, $\nu$ is the specific volume,
$\mathbf{u}$ the velocity vector in cylindrical coordinates and $p$ the
pressure. All physical quantities are made dimensionless using the jet
diameter $D$ and the density and velocity at the nozzle exit along the centre
line $\rho_{\mathrm{j}}$ and $U_{\mathrm{j}}$. In the reference case, the Mach
and Reynolds numbers are $M=U_{\mathrm{j}}/c_{\mathrm{j}}=0.9$ and
$Re=\rho_{\mathrm{j}}U_{\mathrm{j}}D/\mu_{\mathrm{j}}=1.01\times 10^{6}$,
respectively, where $c$ is the speed of sound and $\mu$ is the molecular
viscosity. The Strouhal number will be defined as $St=fD/U_{\mathrm{j}}$.

Figure 1: Mean streamwise velocity at $M=0.9$ and $Re=1.01\times 10^{6}$.

###  2.2 Resolvent matrix

The discretised compressible non-linear Navier-Stokes equations are written as
the following dynamical system:

$\displaystyle\frac{\partial\mathbf{q}}{\partial t}=\mathbf{N}(\mathbf{q}).$
(1)

Introducing the classical Reynolds decomposition
$\mathbf{q}(\mathbf{x},t)=\mathbf{\bar{q}}(\mathbf{x})+\mathbf{q^{\prime}}(\mathbf{x},t)$,
the governing equation of the perturbations reads

$\displaystyle\frac{\partial\mathbf{q^{\prime}}}{\partial
t}=\mathbf{A}\mathbf{q^{\prime}}+\mathbf{f}.$ (2)

The right-hand term has been decomposed such that the Jacobian matrix
calculated around the mean flow $\mathbf{\bar{q}}$,
$\mathbf{A}=\partial\mathcal{N}/\partial\mathbf{q}|_{\mathbf{\bar{q}}}$,
appears. This constitutes the linear part of the equation while $\mathbf{f}$
contains the remaining non-linear terms. The analysis is now carried out in
the frequency domain and restricted to _axisymmetric_ two-dimensional
perturbations; the azimuthal wavenumber is always zero and does not enter the
problem as a parameter. The governing equation becomes

$\displaystyle
i\omega\mathbf{\hat{q}}=\mathbf{A}\mathbf{\hat{q}}+\mathbf{\hat{f}},$ (3)

where $\omega$ is the angular frequency . Following the modelling approach of
turbulent flows proposed by [18] and [19], the term $\mathbf{\hat{f}}$
represents the intrinsic turbulent background forcing originating from the
coupling of different space and time scales. Then, $\mathbf{\hat{q}}$ can be
seen as the linear response to this forcing. Because $\mathbf{\hat{q}}$
contains both energy amplification mechanisms in the hydrodynamic and acoustic
regions, this approach can be used for jet-noise modelling. One can further
note a strong resemblance with acoustic analogies [31, 32]. The resolvent
matrix can finally be introduced as the link, in the frequency domain, between
the forcing and the response fields. Dropping the hat notations for the rest
of the paper, the following relation holds:

$\displaystyle\mathbf{q}=\mathbf{R}\mathbf{f},$ (4)

where $\mathbf{R}=\left(i\omega\mathbf{I}-\mathbf{A}\right)^{-1}$ and
$\mathbf{I}$ is the identity matrix.

###  2.3 Singular value decomposition

The resolvent matrix contains all the information needed to compute the
response to a given forcing in the frequency domain. A singular value
decomposition (SVD) can be performed in order to find a low-rank approximation
of this matrix. From a physical point of view, this amounts to the definition
of a gain between the energy of the response and that of the forcing and an
identification of the forcing fields that generate the largest values of this
gain. The motivation behind the approach is to explore if, with only a few
modes, a majority of the fluctuation of the system can be recovered; in other
words, building a low-order model. The approach detailed hereafter introduces
a tailored resolvent framework to study jet noise [33, 34, 35]. A measurement
matrix $\mathbf{B}$ is introduced, such that

$\displaystyle\mathbf{f}$ $\displaystyle=\mathbf{B}\mathbf{z}.$ (5)

This allows the forcing field to be restricted to a specified region and/or
specific components of the forcing vector. The modified resolvent matrix
$\mathbf{H}=\mathbf{R}\mathbf{B}$ provides a link between the two new
variables:

$\displaystyle\mathbf{q}=\mathbf{H}\mathbf{z}.$ (6)

The gain between the energy of the response and the forcing, noted
$\sigma^{2}$, is now defined as

$\displaystyle\sigma^{2}=\frac{\langle\mathbf{q},\mathbf{q}\rangle_{\mathrm{R}}}{\langle\mathbf{z},\mathbf{z}\rangle_{\mathrm{F}}}.$
(7)

Two scalar products have been introduced. In a discrete framework, they are
defined from weight matrices such that
$\langle\mathbf{q},\mathbf{q}\rangle_{\mathrm{R}}=\mathbf{q}^{*}\mathbf{M}_{\mathrm{R}}\mathbf{q}$
and
$\langle\mathbf{z},\mathbf{z}\rangle_{\mathrm{F}}=\mathbf{z}^{*}\mathbf{M}_{\mathrm{F}}\mathbf{z}$.
The choice of the norms is detailed in section 2.5. Note that
$\mathbf{M}_{\mathrm{F}}$ has to be a positive-definite matrix. This is not
necessary for $\mathbf{M}_{\mathrm{R}}$, which can be positive semi-definite
[33]. From equations (6) and (7), the gain can then be recast as a generalised
Rayleigh product:

$\displaystyle\sigma^{2}=\frac{\mathbf{z}^{*}\mathbf{H}^{*}\mathbf{M}_{\mathrm{R}}\mathbf{H}\mathbf{z}}{\mathbf{z}^{*}\mathbf{M}_{\mathrm{F}}\mathbf{z}}.$
(8)

We now aim at computing the forcing vectors that generate the largest (or
optimal) gain. Following [36], this problem can be solved as a generalised
SVD, which reads

$\displaystyle\mathbf{H}$
$\displaystyle=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{*}\mathbf{M}_{\mathrm{F}},$
(9) $\displaystyle\mathbf{U}^{*}\mathbf{M}_{\mathrm{R}}\mathbf{U}$
$\displaystyle=\mathbf{V}^{*}\mathbf{M}_{\mathrm{F}}\mathbf{V}=\mathbf{I}.$
(10)

The matrices $\mathbf{V}$ and $\mathbf{U}$ contain the right-singular and
left-singular vectors $\mathbf{v}_{i}$ and $\mathbf{u}_{i}$, respectively. The
vectors $\mathbf{v}_{i}$ correspond to the optimal and sub-optimal forcing
modes generating the optimal and sub-optimal gains $\sigma_{1}>\sigma_{2}>...$
(equation (7)), which are contained in the diagonal matrix $\mathbf{\Sigma}$.
The optimal and sub-optimal response modes, associated with these forcing
modes, are the vectors $\mathbf{u}_{i}$, which are readily computed using
${\mathbf{U}=\mathbf{H}\mathbf{V}\mathbf{\Sigma}^{-1}}$.

###  2.4 Low-rank approximation of the response

Adopting a stochastic approach is relevant to deal with turbulent flows. The
second-order statistics of the system can be modelled within the resolvent
framework by introducing the cross-spectral density (CSD) matrices of the
response and the forcing $\mathbf{P}_{qq}$ and $\mathbf{P}_{zz}$,
respectively. [24] and [25] showed that

$\displaystyle\mathbf{P}_{qq}=\mathbf{H}\mathbf{P}_{zz}\mathbf{H}^{*}.$ (11)

Combined with the SVD of the resolvent in equation (9), $\mathbf{P}_{qq}$ can
be expressed as

$\displaystyle\mathbf{P}_{qq}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{*}\mathbf{M}_{\mathrm{F}}\mathbf{P}_{zz}\mathbf{M}_{\mathrm{F}}^{*}\mathbf{V}\mathbf{\Sigma}\mathbf{U}^{*}.$
(12)

An interesting simplification occurs under the assumption
$(\mathbf{M}_{\mathrm{F}}^{1/2})^{*}\mathbf{P}_{ff}\mathbf{M}_{\mathrm{F}}^{1/2}=\mathbb{I}$.
When $\mathbf{M}_{\mathrm{F}}=\mathbb{I}$, this assumption means that the
forcing field is being modelled as white noise in space [24]. In this case,
because of the orthonormality properties of the SVD, equation (12) can be
rewritten as

$\displaystyle\mathbf{P}_{qq}=\mathbf{U}\mathbf{\Sigma}^{2}\mathbf{U}^{*},$
(13)

which shows that resolvent modes are, in this case, identical to SPOD modes
[26]. A low-rank approximation of the PSD of the state vector - which lies on
the diagonal of $\mathbf{P}_{qq}$, can easily be computed as

$\displaystyle
PSD(\mathbf{q})\simeq\sum_{k=1}^{n_{\mathrm{res}}}\sigma_{k}^{2}|\mathbf{u}_{k}|^{2}.$
(14)

where $n_{\mathrm{res}}$ the number of resolvent modes retained, i.e. the
order of the truncation of the resolvent basis used to build the response.

###  2.5 Energy measure

A measure of the energy of the forcing and response fields has to be chosen
and implemented through matrices $\mathbf{M}_{\mathrm{F}}$ and
$\mathbf{M}_{\mathrm{R}}$. In compressible flows, it is common to use the
energy measure introduced by [38], which provides a measure of the energy of
compressible disturbances that is free of non-physical growth, as shown by
[39]. This energy can be derived for the primitive variables considered in our
study by using the linear relationships between variables, which yields

$E_{\mathrm{Chu}}=\frac{1}{2}\int_{\Omega}\left(\frac{1}{\bar{\nu}}|\mathbf{u}|^{2}+\frac{\gamma\bar{p}}{(\gamma-1)\bar{\nu}^{2}}\nu^{2}+\frac{2}{(\gamma-1)\bar{\nu}}\nu
p+\frac{1}{(\gamma-1)\bar{p}}p^{2}\right)r\mathrm{d}r\mathrm{d}x.$ (15)

The integration domain $\Omega$, in the physical space $(x,r)$, can be chosen
in order to compute the energy of the response in a specific portion of the
numerical domain, thereby setting the definition of the gain in equation (7).
As for the energy of the forcing, $\Omega$ is chosen such that it matches the
size of vector $\mathbf{z}$ in order to ensure that the matrix
$\mathbf{M}_{\mathrm{F}}$ is positive-definite.

###  2.6 Standard and acoustic resolvents: set-up

Two radically different choices of measurement regions of the response energy
will be compared in this paper. These choices are illustrated in figure 2. As
seen in section 2.5, the definition of this region is set through $\Omega$. We
will call standard resolvent the approach in which this domain extends over
$r_{\mathrm{r}}/D\in[0;6]$. This includes the hydrodynamic field (i.e. mixing
layer), which is usually the region of interest in resolvent analysis as it is
where large energy growths takes place. We will refer to acoustic resolvent
the calculations set with $r_{\mathrm{r}}/D\in[4;6]$, which is located in the
near acoustic field, away from the hydrodynamic region. The remaining
parameters are fixed as follows. The optimisation region of the response
axially extends over $x_{\mathrm{r}}/D\in[0;20]$. The forcing region, set
through the control matrix $\mathbf{B}$, extends over
$x_{\mathrm{f}}/D\in[0;20]$ and $r_{\mathrm{f}}/D\in[0;1]$ in the jet region
as well as in the nozzle over $x_{\mathrm{f}}/D\in[-2;0]$ and
$r_{\mathrm{f}}/D\in[0;0.5]$.

Figure 2: Locations of the forcing (green dashed line) and response (yellow
solid line) optimisation regions in the standard and acoustic resolvents.
Colour map is the same as in figure 1.

###  2.7 Numerical method

The resolvent code used to perform our calculation is based on that presented
in [40]. This code is formulated from the non-linear compressible Navier-
Stokes written in conservative form. The resolvent matrix is constructed from
the Jacobian matrix, which is obtained via a finite-difference approximation
of the numerical residual of the non-linear equations, as proposed by [41].
Therefore, in this approach, the equations are first discretised and then
linearised, bypassing the tedious task of linearising the continuous
compressible equations before discretising them. At this point, the Jacobian
matrix is associated with conservative variables. As we are interested, in
particular, in pressure fluctuations in the acoustic field, singular vectors
are expected to be expressed in terms of primitive variables. When turbulent
mean flows are considered, [42] noted that a particular care is required in
order to deduce the Jacobian matrix associated with a set of variables from
that computed for another set. The authors derived a transformation, which is
used in the present paper in order to switch from conservative to primitive
variables; details can be found in [42].

Following [36], the SVD introduced in equation (9) is solved as the following
eigenvalue problem:

$\underbrace{({\bf{M}}_{F}^{-1}{\bf{B}}^{*}{\bf{R}}^{*}{\bf{M_{R}}}{\bf{R}}{\bf{B}})}_{\normalsize{\bf{A}}}{\bf{v}}=\sigma^{2}{\bf{v}}.$
(16)

The eigenvalue $\sigma^{2}$ and the eigenvector ${\bf{v}}$ (which correspond
to the singular value and right-singular vector of the aforementioned SVD) are
solved with the the Krylov-Schur algorithm, using the SLEPC library [43]. This
requires the eduction of a Krylov subspace of the matrix ${\bf{A}}$. Each
vector of this subspace is computed by sequentially solving two linear systems
on the resolvent matrix, one involving ${\bf{R}}$, the other involving
${\bf{R}}^{*}$. To this end, a direct LU method is employed using the PETSc
library [44]. This step constitutes the bottle neck of the algorithm in terms
of both CPU time and RAM. The complete procedure is detailed in [40].

The numerical domain extends over $x/D\in[-2.7;30]$ and $r/D\in[0;30]$. This
includes a nozzle flow in the region $x/D\in[-2.7;0]$ and $r/D\in[0;0.5]$. We
use Dirichlet boundary conditions at the inflow and at the top of the domain,
and Neumann conditions at the downstream border. Symmetry boundary conditions
are used along the axis of the jet. Along the nozzle wall, adiabatic, no slip
conditions are set.

Sponge zones are implemented following [45] and are located at $x/D>25$,
$r/D>25$ and above the nozzle at $x/D<-0.3$. Finally, a Cartesian mesh of size
$595\times 285$ is used throughout this paper. Grid independence of the
results is shown in appendix A.

##  3 Results

###  3.1 Comparison between standard and acoustic resolvents

The first singular values of the standard resolvent are three to four orders
of magnitude larger than those of the acoustic resolvent (figure 3a). This is
because the acoustic approach does not measure the large energy amplification
in the mixing layer, which necessarily decreases the gain defined in equation
(7). In the standard approach, higher frequencies ($St\gtrsim 0.4$) exhibit
larger values, as also noted by several authors [33, 46, 21]. These authors
showed that this corresponds to a switch of leading mode: Kelvin-Helmholtz
(K-H) wavepackets here dominate the dynamics of the system whereas the Orr
mechanism takes over at lower frequency. In the acoustic approach, slightly
larger singular values are observed at low frequencies. They remain constant
above $St=0.5$.

Now focusing on one frequency, $St=0.6$, the sub-optimal modes of the SVD are
shown in figure 3(b). In the standard approach, a well known gain separation
of about two orders of magnitude is observed between $\sigma_{1}^{2}$ and
$\sigma_{2}^{2}$. This indicates that the single-point statistics of the
system can be reasonably well modelled using only the first resolvent mode
[20], which is associated with the K-H instability. No such gain separation
occurs for the acoustic resolvent. Instead, a slow decrease is observed for
the first 18 singular values, after which a quick exponential decay takes
place, as also noted by [34]. The implication of this result regarding the
construction of a low-rank response from resolvent modes is that several modes
are likely to be required to build a response that recovers the energy in the
selected acoustic domain.

Figure 3: Comparison between the SVD of the standard and acoustic resolvents.
(a) First singular value at different frequency. (b) First twenty-six singular
values at $St=0.6$.

The hydrodynamic field of the first three response modes are shown for both
approaches in figure 4. In the standard approach, mode 1 is solely driven by
the K-H wavepacket (figure 4a). Sub-optimal modes differs from it, featuring
structures developing further downstream with noticeably shorter wavelength,
i.e. smaller phase velocity (figure 4c,e). They are associated with the Orr
mechanism [22]. As for the acoustic resolvent, a K-H wavepacket now appears in
every mode (figure 4b,d,f). To further examine these wavepackets, the kinetic
energy density profiles along the axial direction,
$\mathrm{d}E_{k}(x)=\int\frac{1}{2}\bar{\rho}|\mathbf{u}|^{2}r\mathrm{d}r$,
are calculated. The K-H wavepacket in mode 1 grows exponentially immediately
at the nozzle exit and reaches its maximum around $x/D\simeq 6$ (figure 5a).
Sub-optimal modes of the standard resolvent exhibit different behaviours as
the maximum of each mode is reached at a different position from one another.
In the acoustic approach, however, energy profiles are the extremely similar
for each mode outside of the nozzle (figure 5b). Furthermore, they are nearly
identical to that of mode 1 of the standard resolvent (K-H wavepacket). The
following interpretation can be proposed. Sub-optimal response modes do not
feature a K-H wavepacket in the standard approach because of the orthogonality
constraint, which does not allow it. Indeed, resolvent modes are constructed
to be orthogonal to one another with respect to a scalar product defined from
a choice of norm and a selected portion of the domain (section 2.5). The
standard approach includes the mixing layer zone in this domain, preventing
the modes from sharing identical components. In the acoustic approach,
response modes do not have to be orthogonal in this region, and can therefore
share the same behaviour. The flow being linearly unstable, K-H wavepacket
naturally appears if no orthogonality constraints aim at cancelling it. As a
result, it is rather expected to observe K-H wavepackets in all acoustic
resolvent modes.

Figure 4: Real part of the pressure of the three first response modes at
$St=0.6$. Left: standard resolvent. Right: acoustic resolvent. The horizontal
black line represents the wall of the nozzle. Figure 5: Kinetic energy density
profiles of the three first resolvent response modes at $St=0.6$ for the
standard (a) and acoustic (b) resolvents. The black dotted line in (b) is the
profile of mode 1 in (a). Each profile is normalised by its maximum value.

The acoustic field of the optimal mode is made visible in figure 6. In the
standard resolvent framework, a lot of the energy in the acoustic field
results from effects associated with the nozzle, such as scattering or
resonance. In the acoustic approach, the acoustic field is more organised,
featuring a clear directivity. It is very similar to the optimal mode obtained
by [34]. It is striking to observe such different acoustic radiations while
the wavepacket profiles are nearly identical (figure 5b). This raises
questions about the nature of the acoustic sources at play, which will be
discussed in section 4 after having analysed more thoroughly the acoustic
resolvent approach.

Figure 6: Real part of the pressure of the optimal response mode at $St=0.6$
of the standard (a) and acoustic (b) resolvents. In order to make the acoustic
field visible, the colour bar is capped, in each case, by the maximum absolute
value reached over $r/D>4$. Note that these modes are the same as those
already shown in figure 4.

###  3.2 Nozzle influence

The influence of the nozzle flow is investigated as it was not included in
previous studies [34, 21, 35]. Both standard and acoustic resolvent approaches
are tested using different forcing regions selected through the control matrix
$\mathbf{B}$ (equation (5)). In the previous section, this region extended
over $x_{\mathrm{f}}/D\in[-2;20]$, recalling that the nozzle is located in the
part of the domain where $x<0$. We will now restrict the forcing domain to
$x_{\mathrm{f}}/D\in[-1;20]$ and $x_{\mathrm{f}}/D\in[0;20]$, the latter
meaning that strictly no forcing component is located inside the nozzle. The
influence of this parameter on the singular values is studied at $St=0.6$,
which lies in the typical range of frequency of interest for jet noise. In the
standard approach, the first singular value increases as $x_{\mathrm{f}}$ is
decreased (figure 7a). The following singular values, however, remain
unchanged. This results in significantly larger gain separations as the
forcing region is set more upstream in the nozzle. This supports recent works
that stressed the importance of the nozzle in the amplification of
perturbations [2, 22, 37]. In contrast, the singular values of the acoustic
resolvent remain unchanged by the variation of $x_{\mathrm{f}}$ (figure 7b).

Figure 7: Influence of the upstream location of the forcing region on the SVD
at $St=0.6$. (a) Standard resolvent. (b) Acoustic resolvent.

Examining the structure of the streamwise velocity fields of the standard
resolvent modes can shed light on the behaviour of their associated singular
value (figure 8 and 9). It is not surprising that both the forcing and
response of the sub-optimal modes for $x_{\mathrm{f}}/D>-2$ and
$x_{\mathrm{f}}/D>0$ are identical given that their singular values are the
same. Interestingly, the optimal response is the same whether the forcing
field is allowed in the nozzle or not (figure 8b and 9b). Thus, the difference
found for the first singular value must be explained by a change in the
forcing mode. In the $x_{\mathrm{f}}/D>0$ case, the forcing is concentrated in
the mixing layer at the immediate exit of the nozzle (figure 9a). In the
$x_{\mathrm{f}}/D>-2$ case, it is spread in the near-wall region of the nozzle
(figure 8a). Both forcing fields tend to be located as upstream as possible.
This is indeed the most efficient way to promote energy amplification as the
growth of convective instabilities is then experienced over longer distances.
Besides, the tilted forcing structure spreading in the nozzle shows the action
of the Orr mechanism in the boundary layer. This non-modal growth occurs all
along the nozzle length, further supporting that a longer nozzle induces
larger energy growth. The sub-optimal forcing modes also feature tilted
structures but are mostly located in the jet region rather than in the nozzle,
which explains why the sub-optimal singular values are insensitive to
$x_{\mathrm{f}}$. These modes are purely driven by the Orr mechanism, which is
at play in the mixing layer, as analysed by [22].

Figure 8: Real part of the streamwise velocity of the three first resolvent
modes at $St=0.6$. Left: Forcing. Right: Response. Results are for the
standard resolvent only, using $x_{\mathrm{f}}/D\in[-2;20]$. Figure 9: Same as
figure 8 but using $x_{\mathrm{f}}/D\in[0;20]$.

###  3.3 Focus on the acoustic resolvent

The work presented so far has dealt with highlighting some fundamental
differences between the standard and acoustic resolvents. From now on, we will
focus mostly on the acoustic resolvent. The influence of the forcing region
will first be studied. Afterwards, a detailed analysis of the optimal and sub-
optimal modes will be carried out before assessing the potential of the
acoustic resolvent to build a low-rank model of jet noise. The effect of the
Mach number will eventually be examined.

####  3.3.1 Sensitivity to the radial extent of the forcing region

Three forcing regions of different radial extent are considered (see table 1).
The optimal forcing and response modes for each case are presented in figure
10. Each forcing mode features a streamwise wave structure. In the F3 case,
most of the forcing is located above the mixing layer and has a noticeably
shorter wavelength than in the other cases (figure 10a). The associated
response is made of a low-angle acoustic beam (figure 10d). Without being
strictly identical, the reference and F04 cases are similar to each other, but
greatly differ from the F3 set-up. Their optimal response features an acoustic
radiation of larger angle (figure 10e,f). The reference and F04 cases
constitute a more realistic constraint on the forcing (figure 10b,c) compared
to the F3 case, given that the forcing is actually limited within the shear
layer as shown by [47]. The region $r\in[1,3]$ hardly contains any turbulent
activity, and hence, forcing. From a hydrodynamic point of view, the response
mode includes a K-H wavepacket in each case – even in the case
$r_{\mathrm{f}}/D\in[0;0.4],$ which has a zero-forcing close to the wall of
the nozzle and at the start of the mixing layer. These regions were found to
be the most efficient locations to excite this wavepacket (section 3.2).
However, it is not a necessary condition to force the system in these regions
in order to generate this wavepacket; as mentioned in section 3.1, in the
absence of orthogonality constraints in the mixing layer, these wavepacket
will grow regardless of the shape and the precise location of the upstream
forcing as they result from a linear instability of the flow.

Cases | Forcing pipe | Forcing jet | Response
---|---|---|---
Acoustic resolvent F3 | $[-2;0]\times[0;0.5]$ | $[0;20]\times[0;3]$ | $[0;20]\times[4;6]$
Acoustic resolvent Reference | $[-2;0]\times[0;0.5]$ | $[0;20]\times[0;1]$ | $[0;20]\times[4;6]$
Acoustic resolvent F04 | $[-2;0]\times[0;0.4]$ | $[0;20]\times[0;0.4]$ | $[0;20]\times[4;6]$
|  |  |
Table 1: Nomenclature of the different cases studied in section 3.3.1.  Figure
10: Real part of the pressure of the first resolvent mode at $St=0.6$. Top:
forcing. Bottom: response. Left: F3 case. Centre: reference case. Right: F04
case. The nomenclature is given in table 1. Colour bars of the response modes
are capped as in figure 6.

####  3.3.2 Physical features of the acoustic resolvent modes

In this section, we focus on the reference case of the acoustic resolvent and
discuss the first three resolvent modes. Note that hydrodynamic fields have
already been shown in figure 4, where K-H wavepackets were observed in each
response mode. Unlike mode 1 (figure 11d), the acoustic radiation in mode 2 is
now made of two distinct beams (figure 11e). This behaviour was also reported
by [34]. This results from the orthogonality constraint in the acoustic field.
The forcing field of mode 2 follows the same trend (figure 11b): the
streamwise wave that is found in mode 1 (figure 11a) is now divided into two
portions of space. Mode 3 is very different as the spatial support of the
forcing is essentially located downstream (figure 11c). Moreover, the
wavelength appears shorter. In fact, we will show that the phase velocity is
negative: the forcing field is essentially an upstream-travelling wave. As a
result, the associated acoustic radiation is made of an upstream propagating
wave with a clear directivity and emitted from $x/D\simeq 20$ (figure 11f). In
addition, each forcing mode contains a tilted structure near the nozzle,
similar to the structure that was found to generate the optimal hydrodynamic
response in the standard resolvent (figure 8a). Even though it is not dominant
– most of the energy of the forcing is located downstream of the nozzle, this
indicates that K-H wavepackets play a role in optimal sound-generation.

Figure 11: First three modes of the acoustic resolvent (reference case) at
$St=0.6$. Top: forcing, real part of the streamwise velocity. Bottom:
response, real part of the pressure is shown. Colour bars of the response
modes are capped as in figure 6.

The phase velocity of the forcing field can be computed as
$c_{\varphi}=\omega/\alpha$ if we assume it is a plane wave in the streamwise
direction. The wavenumber can then be approximated by
$\alpha=\partial\theta/\partial x$ where $\theta$ is the phase of the complex
pressure field. Phase velocity profiles of the first three forcing modes are
shown in figure 12(a). Let us first note that diverging phase values and
spurious oscillations are observed in some portion of the profiles. This is
either caused by additional waves associated with nozzle effects or because
the plane wave assumption does not hold any longer. However, clean phase
velocities are obtained in regions of large forcing, which allows us to
comment on the prevalent forcing mechanisms. Large forcing locations can
indeed be detected by plotting the forcing energy density $\mathrm{d}F(x)$
(figure 12b), which is defined analogously to the kinetic energy density in
section 3.1. Forcing mode 1 and 2 have a supersonic phase velocity with
$c_{\varphi}/a_{\infty}\simeq 1.6$ at the location of maximum energy density,
where $a_{\infty}$ is the speed of sound in the far field. Supersonic waves
are promoted by the acoustic resolvent calculation because they are efficient
acoustic sources [14], maximising the acoustic energy for a given energy
input. Overall, three waves of different phase velocities coexist: the
supersonic forcing, the K-H wavepacket and the acoustic radiation. Forcing
mode 3 is also slightly supersonic and has a negative phase velocity. Its
energy clearly peaks in the downstream region, where an upstream-travelling
acoustic wave is radiated. Such modes are unphysical in jet noise, as will be
discussed later.

The direct link between the forcing and acoustic fields can be clearly shown
by relating the angle of the radiation $\theta$ to the phase velocity of the
forcing which, for a Mach wave, verifies
$\theta=\cos^{-1}\left(\frac{a_{\infty}}{c_{\varphi}}\right)$. The angle is
defined with respect to the centre axis and a value of $0^{\circ}$ would
correspond to downstream radiation. From the calculation of the phase velocity
at the location of the maximum forcing field, the predicted angle is
$\theta=51.3^{\circ}$. Meanwhile, the acoustic radiation observed in the
response mode is $\theta\simeq 51^{\circ}$. While this confirms that the
forcing field is directly responsible for the acoustic radiation, it is
unclear why this specific angle $\theta$ is selected by the resolvent
analysis. Indeed, the phase velocity of the forcing field could a priori take
any values: introduced as an external body force, it is not constrained by the
hydrodynamics. The comparison with SPOD modes carried out in the next section
will shed more light on this matter.

Figure 12: Phase velocity along $r/D=0.49$ (a) and energy density profile (b)
of the first three forcing modes of the acoustic resolvent (reference case) at
$St=0.6$.

####  3.3.3 Comparison with SPOD modes

Calculated from the times series obtained from LES calculations, SPOD modes
provide, for each frequency, an optimal orthogonal basis to represent the
acoustic field. These modes are obtained through the eigendecomposition of the
cross-spectral density (CSD) matrix. [26] gave a comprehensive presentation of
their calculation and properties. SPOD modes are associated with a norm. The
same set-up as that of the resolvent modes is used: Chu’s energy is calculated
over the acoustic domain of the response. The normalised eigenvalues of the
SPOD modes at $St=0.6$ are shown in figure 13, indicating the proportion of
the total energy that each mode contains. Contrary to the SVD of the acoustic
resolvent (figure 3a), the SPOD is low-rank. A separation of more than one
order of magnitude is observed between the first and fourth SPOD modes while
it took nineteen resolvent modes to achieve such a separation. This means that
the acoustic field can be described by a low-rank basis, but that the current
resolvent approach does not provide this basis – at least without further
information.

Figure 13: Eigenvalues of the acoustic SPOD at $St=0.6$.

However, the acoustic resolvent contains relevant features that were not
present in the standard resolvent approach. This can be seen by examining the
first three SPOD modes (figure 14). They are made of a clear acoustic
radiation which, because of the orthogonality constraint, is divided into
different beams as the rank of the mode increases. This feature was also found
in the two first modes of the acoustic resolvent, but absent in the standard
resolvent decomposition. Moreover, the angle of radiation of the first mode is
about $54^{\circ}$, close to the value of $51^{\circ}$ found for the Mach wave
in the first resolvent mode. This indicates that the phase velocity of the
forcing at the origin of this Mach wave must somehow be rooted in the mean
flow properties. The structure of the third SPOD mode confirms that the third
mode of the acoustic resolvent modes, made of an upstream propagating acoustic
wave (figure 11f), is indeed not relevant to the acoustics – even though this
mode is optimal in the given resolvent framework, in which any external
forcing is allowed.

Figure 14: Three first SPOD modes at $St=0.6$, real part of the pressure.
Colour bars are capped as in figure 6.

The alignment between SPOD and resolvent modes can be assessed through a more
global quantity, $\phi$, defined as a normalised scalar product between these
modes:

$\phi=\frac{\langle q_{\mathrm{res}},q_{\mathrm{SPOD}}\rangle}{\sqrt{\langle
q_{\mathrm{res}},q_{\mathrm{res}}\rangle\langle
q_{\mathrm{SPOD}},q_{\mathrm{SPOD}}\rangle}}.$ (17)

The scalar product is associated with Chu’s energy taken over the near-
acoustic domain $x\in[0;25]$ and $r\in[4;6]$. The first SPOD and resolvent
modes are considered. The results are presented for different frequencies in
figure 15. The acoustic resolvent generally produces better alignment with
SPOD modes than the standard resolvent, except at the highest frequencies
considered ($St\simeq 1$). The differences between the two approaches are not
dramatically large but remain significant. Both exhibit a maximum of alignment
around $St=0.3$ where the acoustic resolvent reaches a value of $\phi$ close
to 1. A steep decrease is observed above $St>0.4$. There, values of $\phi$ do
not drop to zero but rather stay around 0.1 and 0.4, indicating that some
acoustic features may still be captured by the resolvent decomposition. In
order to better apprehend the correspondence between these values of $\phi$
and the near-acoustic field, the pressure is plotted along the line $r/D=5$
for each mode at three different frequencies (figure 16). Resolvent and SPOD
modes are compared, and the amplitude of each mode is normalised by its own
maximum value. As expected, the maximum of acoustic radiation is particularly
well predicted at $St=0.3$ by the acoustic resolvent (figure 16a). The
pressure profile is particularly well recovered between around $x/D=10$ and
$x/D=20$, where the signal is the largest. The standard resolvent also
performs well but is not as accurate, predicting a peak further downstream
than that of the SPOD mode, and a decay with $x$ that is not as fast as in the
SPOD mode. The standard resolvent at $St=0.6$, where $\phi\simeq 0.2$,
predicts none of the behaviour of the SPOD mode (figure 16b). Conversely, with
a better alignment at this frequency ($\phi\simeq 0.35$), the acoustic
resolvent mode is able to detect the peak of pressure of the SPOD mode but
does not correctly capture its shape. This tempers the above discussion about
the angle of the acoustic beams observed in the SPOD and resolvent response
fields. Finally, at $St=1$, both approaches produce poor predictions (figure
16c).

Figure 15: Alignment between SPOD and resolvent modes. Figure 16: Pressure
along the line $r/D=5$ in the near acoustic field at $St=0.3$ (a), $St=0.6$
(b) and $St=1$ (c).

####  3.3.4 Effect of Mach number

So far, only the case $M=0.9$ has been studied. Two additional Mach numbers
are now considered, $M=0.4$ and $M=1.5$, whose mean flows have also been
obtained by LES calculations. The comparison of the singular values of the
acoustic resolvent at $St=0.6$ for the three Mach numbers is shown in figure
17(a). The supersonic case has a distinct behaviour as a separation of nearly
three orders of magnitude is found between $\sigma_{1}^{2}$ and
$\sigma_{2}^{2}$, as also noted by [34]. This results from the existence of a
supersonic instability wave which directly generates a Mach wave: the optimal
forcing is located at the most upstream location (figure 18a) in order to
trigger a K-H wavepacket to which is attached the acoustic radiation (figure
18d). In other words, the most efficient way to generate acoustic
perturbations coincides with the most efficient way to produce hydrodynamic
perturbations. This is different from the subsonic case previously
investigated, in which the Mach wave was dominantly produced by the supersonic
forcing field rather than the response field. Note that the available data
from LES calculation in the supersonic case does not contain a nozzle: the
mean flow used for the resolvent calculation starts at $x/D=0$. If it was
included, it is very likely that the optimal forcing would be located upstream
in the nozzle as in the $M=0.9$ case of the standard resolvent (figure 8a),
and that the gain separation would be even larger. The second resolvent mode
of the $M=1.5$ case features two acoustic beams with a similar angle to that
of the first mode (figure 18e). It does not result from an instability wave,
but is rather directly produced by the forcing as in the subsonic case. If the
instability wave was triggered, it would indeed generate the same acoustic
radiation as that of the first mode, hence breaking the orthogonality
constraint. Therefore, the sub-optimal modes cannot contain the supersonic K-H
wavepacket. Interestingly, the third mode does not divide the acoustic field
into a third beam (figure 18f), but rather contains a single radiation of
different angle than the two other mode. However, the structures of sub-
optimal modes do not crucially matter in the supersonic regime since, given
the large gain separation, the K-H wavepacket associated with first resolvent
mode is sufficient to model jet noise in this case [12].

Figure 17: First 26 singular values of the acoustic resolvent, renormalised by
$M$, at $St=0.6$ (a) and $St_{a}=St\times M=0.53$ (b) for three Mach numbers.
Figure 18: Same as figure 11 but at $M=1.5$ ($St=0.6$). Note that the
numerical domain in the supersonic case only starts at $x/D=0$, but a larger
domain is displayed to ease comparison with other cases.

The behaviour of the singular values at $M=0.4$ case is similar to that of the
$M=0.9$ case (figure 17a). The first singular values have a comparable
magnitude before experiencing a quick drop. This drop occurs at a smaller rank
than that observed for $M=0.9$. Once again, the forcing modes contain
supersonic waves generating Mach waves (figure 19a,b,c). The first mode is
seen to be upstream propagating which, again, indicates the inability of the
acoustic resolvent to produce a hierarchy of relevant acoustic modes.

Coming back to the singular values presented in figure 17(a), it can be seen
that a normalisation of $\sigma_{i}$ by $M$ was employed. By doing so, the
reference velocity scale is no longer that at the nozzle exit, but rather the
speed of sound in the quiet flow. This is Helmholtz scaling. This choice of
normalisation produces singular values of similar magnitudes for any Mach
numbers. This indicates that these singular values are associated with the
same acoustic mechanism, seen to be linked to the Mach-wave mechanism. The
exception is the previously discussed first singular value at $M=1.5$,
associated instead with a K-H wavepacket, and whose magnitude is indeed much
larger. To examine this further, it is then interesting to compare the
behaviour of the different Mach numbers at a constant Helmholtz number,
defined as $St_{a}=St\times M$. In this case, at $St_{a}=0.53$, the singular
values collapse for any Mach numbers (figure 17b), an observation first made
by [34]. Therefore, the speed of sound is the relevant velocity scale of the
optimisation problem, consistent with the importance of acoustic matching
central to the jet-noise problem [14]. The optimal response modes at this
constant Helmholtz number $St_{a}$ are presented in figure 20. All modes now
have the same acoustic wave length ($\lambda/D=1/St_{a}$). At this frequency,
all optimal modes represent downstream-travelling waves. In the supersonic
case, the acoustic radiation is clearly linked to the hydrodynamic wavepacket,
both fields being synchronised. In the subsonic cases, the acoustic wave
length is distinct from that of the wavepacket.

Figure 19: Same as figure 11 but at $M=0.4$ ($St=0.6$) Figure 20: Optimal
response mode of the acoustic resolvent at three Mach numbers at constant
Helmholtz $St_{a}=0.53$. Colour bars of the response modes are capped as in
figure 6.

##  4 Discussion

###  4.1 Wavepackets

In the subsonic regime, the SVD of the acoustic resolvent identifies planar
supersonic waves as the optimal forcing mode, i.e. the most efficient way to
generate energy in the near acoustic field. While the forcing field varies for
each mode, the hydrodynamic field observed in the response remains nearly
identical for each of them (figure 5b), resulting from the linear convective
instability of the flow. In this picture, the role of hydrodynamic
wavepackets, known to be central in jet noise [5], becomes unclear as the
hydrodynamic properties of the flow appear decorrelated from the acoustic
field. We here propose an interpretation of these observations.

Subsonic jet noise is primarily understood as a result of acoustic matching
between supersonic wavenumbers associated with the spatial envelop of subsonic
wavepackets and the acoustic field [14]. These supersonic components represent
a small fraction of the total energy of the hydrodynamic field: subsonic jet
noise is inefficient in the sense that the ratio between the acoustic and
hydrodynamic energies is very small (in contrast, supersonic instability waves
found in supersonic jets are much more efficient). The supersonic forcing
waves promoted by the acoustic resolvent thus appear consistent with acoustic
matching, as it pinpoints the essential mechanism of jet noise. The fact that
wavepackets do not appear as an input (forcing) of our analysis, but rather as
an output (response) along with acoustic radiation, is associated with the
reorganisation of the Navier-Stokes equations in the resolvent framework.
Here, all non-linear terms are seen as an input while the associated output
develops according to linear mechanisms. Wavepackets, which result from a
linear amplification, are thus observed in the output modes. Furthermore, in
the resolvent framework, forcing modes are introduced as an external force
acting on the flow; the SVD produces modes that are not constrained by the
intrinsic dynamics of the flow and may thus take any shape, only aiming at
satisfying the optimality constraint that is imposed. These forcing modes are
ultimately interpreted as a basis for the intrinsic non-linear turbulent
interactions in the flow [18, 19]). Therefore, the supersonic forcing
structures can emerge from wavepackets that jitter on account of non-linear
effects and contain associated supersonic wavenumbers allowing for acoustic
matching.

###  4.2 Perspectives for resolvent-based modelling of subsonic jet-noise

A central message to be taken from our analysis is that, when performing
resolvent analysis, the SVD alone should be considered with caution. While in
systems exhibiting a clear gain separation leading modes can give insight into
underlying mechanisms, in the absence of such gain separation, the resolvent
operator and bases must be considered with circumspection. The jet-noise
problem considered is one such case. And the non-physical, upstream-travelling
modes we identify provide an eloquent illustration of this point.

For problems in which gain separation is lacking—and we here have in mind jet
noise—alternative strategies are required. Two examples are studies of [35]
and [48]. In the former, the characteristics of the forcing projection into
the input space of an acoustic resolvent operator were inferred from the sound
field, and a low-rank model thus obtained. Including eddy viscosity and
restricting the portion of the acoustic field to lower angles (which may
prevent the existence of upstream travelling modes), a rank-1 model was
proposed.

A conclusion of the work we report here is that, for jet noise, forcing data
is necessary to provide clarification of underlying mechanisms. Such data is,
however, extremely difficult to obtain from experiment or simulation when the
problem considered involves high Reynolds number, which is the case of
interest for jet noise [cf. 49]. And when such data is available, interpreting
and modelling the coupling between non-linear scale interaction and sound
remains an daunting task.

Having done the work necessary to obtain useable forcing data from a
moderately resolved, high-Re LES in [49], a second effort [48] was undertaken
to understand how that data might be used to probe the mechanisms by which
non-linear scale interactions drive jet noise. The difficulty of the problem
faced can be appreciated by considering that less than 0.1$\%$ of the forcing
fluctuation energy is responsible for the radiated sound. In [48], dedicated
data-reduction was necessary to first isolate the forcing subspace correlated
with the farfield sound, using the RESPOD technique developed in [50]. But as
this subspace is dominated by forcing components that are not acoustically
matched, it provides neither clear physical interpretation nor guidance for
simplified modelling.

Figure 21: Diagram showing the modelling steps pursued to obtain the empirical
jet noise model introduced in [48]. Given the constant supersonic wave
structure with constant radial support observed in the optimal acoustic
forcing mode (a), the RESPOD forcing mode (b), which generates $\sim 80$% of
the total downstream acoustic energy, is (1) filtered to retain the components
with supersonic phase speed (c), (2) integrated over the radial direction to
obtain a line source (d), which is finally (3) modelled using an empirical
model (e).

Those goals where achieved thanks to the results of the current paper: both
the acoustic resolvent operator and the input bases were crucial. We provide a
brief explanation in what follows.

Despite the lack of a gain separation in the acoustic resolvent, the shapes of
the modes provide the key to understanding the ‘acoustically matched’
components of the forcing field: they show how the forcing field filters
through the resolvent operator, and how that filtering operation extracts the
acoustically matched components from the predominantly silent forcing field.
The acoustic-resolvent input modes exhibit two salient traits: supersonic
phase speed; almost no radial oscillation (cf. Figure 21-a). These traits
motivated the final processing of the LES forcing data (Figure 21-(b-d)) and
led to the elaboration of a rank-1 empirical source model (Figure 21-e).
Without the acoustic resolvent operator we have developed and analysis of its
SVD, the model of [48] would not have been obtained.

And it is noteworthy that the source model identified in that work is (thanks
to the results of the present paper) characterised by a robustness that
permits the computation of jet-noise over a range of operating conditions,
including cases with jets in forward flight. Indeed it was possible to use the
model, coupled with the acoustic resolvent reported here, in an industry
environment. Airbus computed mean fields using industry-standard RANS. They
used the code presented in the current paper to compute acoustic resolvent
operators and then performed jet-noise calculations using the source model of
the companion study [48]. The accuracy of the downstream acoustic predictions
across the range of operating conditions considered was within 2-3 dB.

##  5 Conclusion

We have carried out an acoustic-oriented resolvent analysis on a compressible
turbulent jet. We have termed acoustic resolvent the approach in which the
energy of the response in the SVD is optimised in the near-acoustic field,
excluding the hydrodynamic region. Conversely, the standard resolvent includes
the hydrodynamic domain. While the SVD of the acoustic resolvent was recently
used by different authors [33, 34, 35], the aim of this paper was to shed
light on physical interpretation of the SVD of the acoustic resolvent, and
assess its potential for jet-noise modelling. Fundamental differences have
been pointed out between the acoustic and standard resolvent analyses. Whereas
the latter exhibits a large gain separation between the first and second
singular values, the first singular values of the former slowly decreases
before eventually experiencing a quick exponential decay. Besides, an
organised acoustic radiation with a clear directivity is observed in the
acoustic resolvent modes. This directivity is found similar to that of SPOD
modes. This contrasts with the disorganised, scattered acoustic field found in
the leading mode of the standard resolvent. The influence of the nozzle flow
has been studied for both configurations. The gain separation has been found
sensitive to the upstream location of the forcing modes in the nozzle. This is
because the first mode is driven by a K-H wavepacket which is unstable in the
nozzle and beyond, while sub-optimal modes are associated with the Orr
mechanism at play in the jet region. In the case of the acoustic resolvent, no
influence of the nozzle has been reported since acoustic resolvent modes do
not promote instability waves. Instead, the forcing modes contain supersonic
waves that radiate Mach waves in the response modes, which is an efficient way
to generate acoustic perturbations. While this complies with the optimality
request underpinning the framework of the acoustic resolvent, not all modes
appear relevant to jet noise. The supersonic waves observed in the forcing
modes were discussed and shown to be in line with acoustic matching, from
which jet noise is usually understood. Finally, while the SVD of the acoustic
resolvent alone is deemed insufficient to provide a model for jet noise,
perspectives were discussed, either by improving this framework [35] or by
focusing on forcing models [48].

## Funding

This work has received funding from the Clean Sky 2 Joint Undertaking under
the European Union’s Horizon 2020 research and innovation programme under
grant agreement No 78530. Results reflect only the authors’ view and the JU is
not responsible for any use that may be made of the information it contains.

## Data Availability Statement

Data available on request.

##  Appendix A Mesh convergence

The nomenclature of the different meshes that are used to test the mesh
convergence is given in table 2. Mesh A is the reference mesh used in the
paper and is associated with a numerical domain of height $L_{r}/D=30$. Finer
meshes C and D are tested on a reduced domain $L_{r}/D=12$. To check that
$L_{r}$ had no influence on the results, mesh B was introduced by simply
truncating mesh A to $L_{r}/D=12$. The first six singular values of the
acoustic resolvent (reference case) are plotted in figure 22. This shows
identical results for all the meshes tested.

Mesh | $N_{x}$ | $N_{r}$ | $L_{r}/D$
---|---|---|---
A | 595 | 285 | 30
B | 595 | 229 | 12
C | 993 | 229 | 12
D | 595 | 374 | 12
|  |  |
Table 2: Nomenclature of the meshes Figure 22: Mesh convergence of the SVD of
the acoustic resolvent at $St=0.6$

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