# A Time-Domain Linear Method for Jet Noise Prediction and Control Trend
Analysis

Chitrarth Prasad<EMAIL_ADDRESS>Datta V. Gaitonde  Department of
Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH
43210<EMAIL_ADDRESS>
###### Abstract

Large-scale turbulent structures in the form of coherent wavepackets play a
significant role in the generation of prominent shallow-angle noise radiation
of jets. Economical prediction tools often model these wavepackets in the
frequency-domain using stability modes of the mean flow. The use of
simplifying choices, such as parabolized equations and azimuthal
decomposition, provide efficient methods but can impose constraints on rate of
streamwise variation of the mean state or geometric complexity. The current
investigation develops a time-domain linearized Navier-Stokes-based approach
predicated on the mean basic state for two goals: (i) to obtain the radiated
shallow-angle noise field, including that from imperfectly expanded jets
containing shock trains, and (ii) to estimate noise control trends with
actuator frequency. A previously developed implicit linearization technique
repurposing native non-linear Navier-Stokes code capabilities avoids any
additional constraints on nozzle geometry, while its time-domain nature
facilitates control analysis through transient pulse response. Two other
integral components of the method are the sifting of linearized perturbations
to isolate the acoustic component according to Doak’s momentum potential
theory (MPT), and subsequently Dynamic Mode Decomposition (DMD) to analyze the
response in different spectral ranges. Comparisons with well-validated Large
Eddy Simulation (LES) databases in different spectral ranges show accurate
model predictions for super-radiative shallow angle noise, including for hot
jets from military-style nozzles. For control frequency effectiveness, the
transient perturbation growth characteristics are processed to delineate the
distinct effects of the excitation on vorticity versus the MPT acoustic
variable. For a specific jet with extensive published experimental data using
plasma actuators, it is shown that the method correctly predicts noise
amplification at lower frequencies and reduction at higher values, at the
observed crossover location. Considerations on the costs associated with the
approach, which exploits linearity to extract results at multiple frequencies
with each simulation, are outlined.

###### keywords:

Wavepackets , Time-domain , Linear , Prediction , Control, Momentum Potential
Theory

††journal: Journal of Sound and Vibration

##  1 Introduction

Jet noise forms a major component of the total noise radiated by aircraft,
especially in the aft direction. Extreme noise levels such as those produced
by low bypass-ratio supersonic military engines are responsible for noise-
induced hearing loss among military personnel working in close proximity. Jet-
associated noise pollution is also a major consideration, especially in
communities close to airbases and military training routes. The effects of
aircraft noise on the health of both children and adults are well documented
in literature [1, 2]. Public demand has motivated the exploration of noise
reduction methods for existing and future tactical aircraft [3].

The acoustic field is an unwanted byproduct of the mixing of exhaust gases
with the ambient fluid. Large-scale structures in the jet plume have been
established as primary contributors, especially in the case of supersonic jets
[4, 5, 6]. The spatio-temporal footprint of these large-scale structures takes
the form of coherent wavepackets, which constitute efficient sources of the
peak downstream radiation [7]. Several studies have shown that coherent
wavepacket structures can be modeled in terms of linear stability modes of the
long time-mean of the turbulent flow field. Different modeling approaches have
been developed in the literature; the most popular among these are based on
the parabolized stability equations (PSE) [8, 9]. Although convenient, PSE are
deficient in many regards – these have been discussed in Refs. [10, 11], which
propose more accurate methods based on one-way Euler or Navier-Stokes
equations. Other techniques include global eigenvalues [12] and resolvents
[13, 14, 15, 16].

The application of these frequency-domain methods has generally focused on
circular nozzles and perfectly-expanded operating conditions, where they are
highly efficient and effective in predicting noise characteristics, especially
for supersonic jets. Application of mean-flow basic states to predict noise
from imperfectly expanded jets remain relatively scarce. Some techniques
assume slow streamwise variation of flow properties, which can be violated in
the presence of shock-cells. The exploitation of azimuthal decompositions can
also introduce difficulties in conceptual extension to non-axisymmetric
geometries. Nonetheless, the success of such simple models motivates the
continued vigorous development of mean-flow-based approaches for more complex
situations. In particular, they may reduce the large cost-function associated
with the aeroacoustic analysis and optimization of new nozzle designs and
noise control technologies, for which first-principles or cut-and-try
approaches are expensive and inhibiting.

The present effort develops an approach based on linearized perturbations
about the mean turbulent state to (i) predict the primary features of the
shallow angle noise field applicable even to imperfectly expanded cases and
(ii) estimate the effect of actuator frequency on control efforts. The mean
turbulent state may be obtained in different ways. One method is to use a
Reynolds Averaged Navier-Stokes (RANS) solution validated by comparison with
experimental data, such as in Ref. [17]. However, since our goal is to develop
the method, here we use the mean flow from well-validated LES databases; this
allows different features of the perturbation field to be compared directly
with the LES solution as a truth model. Two databases are considered: the
first, designated Jet-A, is similar to those considered in the modeling
studies of Refs. [13, 9, 15]. Specifically, an unheated ideally-expanded Mach
1.3 jet issuing from a sleeve nozzle is considered as discussed in Refs. [18,
19, 20]. The second case, Jet B, is a more realistic over-expanded heated Mach
1.36 jet from a faceted military-style nozzle, previously examined in Refs.
[21, 22, 23]. A summary of the two LES databases is provided in $\S$2.

The proposed method consists of three components. The first adapts a recently
developed time-domain Mean Flow Perturbation (MFP) approach, originally
proposed by Touber and Sandham [24] to analyze the stability of mean shock-
wave turbulent boundary layer flows. We use the variant proposed by Ranjan _et
al._. [25], which provides a matrix-free method to obtain the principal
stability modes of 2D and 3D mean flows on curvilinear meshes. The MFP
approach, discussed in $\S$3.1, generates solutions to the linearized Navier-
Stokes equations (LNSE) by modifying the native, non-linear Navier-Stokes (NS)
solver used to generate the mean flow. The linearization is implicitly
performed by applying a relatively straightforward constraining body-force,
computed only once and in situ, to maintain the invariance of the base state.
This circumvents tedious explicit linearization by repurposing existing NS
solver capabilities for LNSE analyses. As discussed in Ref. [25], the body-
force term provides a measure of versatility since it facilitates decoupling
of the method used to obtain the mean flow from that for the implicit LNSE;
thus, dissipative shock-capturing schemes employed to generate the mean flow
can be replaced by higher-order methods more suited for LNSE. In fact, the
codes employed to obtain the two LES databases are different, but the MFP
procedure for both is applied using only one code.

The time-domain nature of the present approach offers crucial benefits. First,
the method is applicable to all configurations that can be computed by the
underlying non-linear Navier-Stokes procedures; the solvers employed here can
treat complex configurations with curvilinear meshes. Second, the approach
facilitates the analysis of transients in perturbation growth; this enables
extraction of data on control effectiveness. Since the cost of time-domain
methods is typically a concern, some considerations are put forward on the
expense associated with the present method. Although the time-domain approach
facilitates application to general geometries, the choice of on- and off-
design axisymmetric cases is predicated on their well understood dynamics.
Separate efforts to understand and model the dynamics of non-circular jet
geometries, currently being pursued with LES [26] and PSE [27], will be
addressed in a future effort.

The second component comprises sifting of the perturbation fields to separate
noise-related components from primitive variables. The LNSE perturbation
evolution is initiated with random pressure fluctuations on a subset of the
domain. Mean flow gradients subsequently induce perturbations in all other
primitive quantities. Although pressure and other primitive variables are
convenient for analysis since they are directly obtained from experiments or
simulations, they are often not optimal for extraction of acoustic-related
information. For example, pressure includes both hydrodynamic and acoustic
components; the former display vortical and convective characteristics, while
the latter are irrotational and radiating. Techniques to separate the
convecting from radiating pressure fluctuations based on phase speeds of
pressure fluctuations, using methods such as reduced-order models or spectral
techniques, may be found Refs. [28, 29, 30].

In the present work, we adopt Doak’s Momentum Potential Theory (MPT) [31],
presented in $\S$3.2, to extract a pure acoustic component. This robust,
physics-based method exactly decomposes momentum density ($\rho\textbf{u}$)
fluctuations, regardless of magnitude, into its constituent hydrodynamic,
acoustic and thermal components, also designated Fluid Thermodynamic (FT)
modes. Indeed, compared to pressure fluctuations, the acoustic FT component
exhibits wave-packets with superior spatio-temporal coherence and radiative
efficiency because Doak’s MPT effectively filters out the turbulent and
convecting high-energy fluctuations by imposing the physical constraints of
irrotationality and isentropy [32, 33]. In the far-field, where acoustic
sources are negligible, the acoustic component recovers the desired scaled
pressure fluctuations. The advantages of such sifting has been previously
discussed to analyze LES fluctuations [20, 22] and to derive constants in
frequency-based wavepacket models [32]. Thus, the use of the acoustic FT
component for comparison of MFP with LES provides a more effective examination
of the noise radiating mechanisms. This component is also crucial in
establishing control characteristics with actuator frequency as discussed
further below.

The final component of the approach, employed to post-process the sifted time-
domain LNSE perturbations, is the traditional Dynamic Mode Decomposition (DMD)
[34]. The method extracts frequency-specific modal information. When applied
to the MPT acoustic component of MFP-obtained perturbations and LES
fluctuations, it provides a clear strategy for comparison of aspects related
specifically to noise prediction analyses. On the other hand, when applied to
the transient impulse response, DMD yields data on growth rates of different
quantities, which can then be interpreted for control purposes as shown in
this work.

The noise prediction results are presented in $\S$4. The impulse random
pressure fluctuation response is obtained with LNSE using the MFP approach.
The resulting perturbations, which are not restricted to pressure alone, are
then sifted with MPT to extract the acoustic component. The robustness and
insensitivity of the solution to different parameters inherent to the method,
including forcing region, initial perturbation amplitude, and mesh resolution
are demonstrated. For verification, comparisons between the LNSE and the
original LES are presented at different frequencies. Finally, the noise
spectra from the LNSE are compared with those from the LES to demonstrate the
proper variations away from the jet shear layer, focusing as mentioned
earlier, on the shallow angle noise.

The second thrust, analyzing control trends with frequency of excitation, is
based on predicting the relative growth of acoustic versus vortical responses
to perturbation frequency in the peak radiation direction. Since the noise
field is directly related to the spatio-temporal signatures of large-scale
structures in the jet plume, noise reduction techniques seek to enhance fine-
scale mixing between the jet plume and the ambient air to break up
acoustically efficient large-scale structures in the flow. The present
approach, $\S$5 provides a relatively direct method to extract information
about the nature of the response to excitation. For this, the sifted and
primitive variables are subjected to DMD to extract the Ritz values [34] of
linearized perturbation growth. Specifically, the relative growth rates of
vorticity and the MPT acoustic component are compared. The conditions chosen
are again motivated by the need to compare with experimental observations. For
this, the extensive data available on the response of Jet-A to plasma-based
actuation [35] are leveraged. Concluding remarks are put forth in $\S$6.

##  2 LES databases

###  2.1 Jet-A: Unheated perfectly-expanded Mach 1.3 jet

The first database is of an unheated perfectly-expanded Mach 1.3 jet, referred
to as Jet-A. The database has been extensively validated with experimental
data and the flow and acoustic dynamics have been discussed in several prior
publications. Examples include extraction of noise generation components [20]
and the effect of plasma actuators for noise reduction [18, 19].

The database was generated by solving the full 3D compressible Navier-Stokes
equations in generalized curvilinear coordinates. The jet exit conditions are
$U_{j}=391\mathrm{m/s},T_{j}=224\mathrm{K}\text{ and
}\rho_{j}=1.567\mathrm{kg/m^{3}}$. Flow quantities are non-dimensionalized
using these as reference values for velocity, temperature and density
respectively. The nozzle exit diameter, $D=0.0254$m is used as the reference
length and the pressure is normalized by $\rho_{j}{U_{j}}^{2}$. The ambient
temperature is $T_{\infty}=300\mathrm{K}$. The inflow is specified through a
short constant area sleeve in the manner specified by Bogey and Bailly [36].
The nozzle exit is located at $x/D\sim 0.17$.

A third-order upwind biased Roe scheme [37] in conjunction with the van Leer
harmonic limiter [38] is employed for the spatial discretization of the
inviscid fluxes. The viscous fluxes are computed using a second-order central
differencing scheme, and time integration is achieved with an implicit second-
order diagonalized [39] Beam Warming scheme [40]. A detailed description of
mesh resolution studies and boundary conditions can be found in Ref. [18]. The
present database uses a cylindrical structured mesh containing $523$, $526$
and $125$ points in the axial, radial and azimuthal directions respectively.
The mesh is clustered in the radial and axial directions near the centerline
and the nozzle walls with gradual stretching towards outer boundaries.

(a)

(b)

Figure 1: Instantaneous LES flow-field in a 2D streamwise plane. (a) Jet-A:
Vorticity contours superimposed on velocity dilatation contours and (b) Jet-B:
Density gradient contours superimposed over velocity dilatation contours.

Figure 1(a) shows vorticity contours (color) superimposed on velocity
dilatation contours (gray-scale) at a representative time-step on an azimuthal
plane. The vorticity contours track the growth and breakdown of the jet shear
layer as it mixes with the ambient fluid, whereas the background dilatation
contours provide a qualitative picture of the generated acoustic waves. As
expected for a supersonic jet, the acoustic waves display a strong downstream
radiation pattern. This is representative of the superdirective nature of the
jet. Although the database has been extensively validated previously [18, 19,
20], a comparison of far-field pressure spectrum in the peak noise radiation
direction ($\theta=30^{\circ}$ from the jet downstream axis) with experimental
measurements is provided in Fig. 2(a) for completeness. The LES predictions
are obtained using the Ffowcs-Williams Hawkings (FW-H) analogy [41], with the
far-field observer located at $94$ jet diameters from the jet exit. Overall,
the LES predictions are in reasonable agreement with the experimental
measurements. Other results, including comparisons of the mean centerline
velocity obtained from long time averaging of the LES with the PIV and
velocity fluctuation measurements of Ref. [42] may be found in Ref. [18].

(a)

(b)

Figure 2:  Far-field sound pressure level for (a) Jet-A at $94$ jet diameters
from the jet exit at $\theta=30^{\circ}$ from the jet downstream axis, (b)
Jet-B at $100$ jet diameters at three different polar angles. The peak noise
radiation for Jet-A occurs at $\theta_{peak}=30^{\circ}$ and for Jet-B occurs
at $\theta_{peak}=47.5^{\circ}$.

###  2.2 Jet-B: Heated Over-expanded Mach 1.36 jet

The second LES database used in this study represents a model-scale GE-F404
nozzle operating at a Nozzle Pressure Ratio (NPR) of $3.0$ and a Total
Temperature Ratio (TTR) of $2.5$. This corresponds to an over-expanded
operating condition at $M_{j}=1.36$. The database has also been validated and
used extensively for the development and understanding of fluid injection-
based noise reduction devices in Refs. [21, 22, 23]. The database was
generated with the STAR-CCM+ code using a dynamic Smagorinsky subgrid scale
model on trimmed hexahedral meshes. The use of a commercial code combined with
the off-design nature of the jet makes this an important test case to assess
the robustness and feasibility of the proposed method for a practical jet
noise problem relevant to industry. The LES data is non-dimensionalized using
ambient values of the speed of sound $(c_{\infty}=340.17\mathrm{m/s})$,
temperature $(T_{\infty}=288\mathrm{K})$ and density
($\rho_{\infty}=1.226\mathrm{kg/m^{3}}$). The nozzle exit diameter,
$D=0.0254\mathrm{m}$ is used as the reference length, whereas the pressure is
normalized by $\rho_{\infty}{c_{\infty}}^{2}$. Unlike Jet-A, the physical
nozzle is a part of the LES domain and the nozzle exit is located at $x/D=0$.
The nozzle design consists of a sharp throat, with a faceted diverging section
comprising of $12$ seals and $12$ flaps to facilitate for area adjustment in
operational nozzles. A detailed description of the nozzle geometry is provided
in Ref. [43]. In the LES domain, the nozzle exit area ratio is fixed at
$1.295$ to facilitate a comparison with available experimental data [44]. The
flow through the nozzle is simulated by specifying the values of stagnation
pressure and temperature at the nozzle inlet.

A hybrid 3rd-order MUSCL/central differencing scheme [45] in conjunction with
the min-mod limiter is used for spatial discretization of convective fluxes,
whereas a second-order implicit scheme with sub-iterations is employed for
time integration. The mesh consists of approximately $18.1$ million volumes
with prismatic cells near the solid boundaries where $y^{+}<1$. Other details
of the discretization procedure, mesh distribution and grid resolution studies
may be found in Refs. [46, 22].

Figure 1(b) shows the instantaneous density gradient contours superimposed
over velocity dilatation contours for this jet in a 2D streamwise plane. The
shock-cell structures due to the over-expanded nature of the jet are clearly
visible from the density gradient distribution. The dilatation contours, on
the other hand, exhibit a strong downstream Mach wave radiation signature
characteristic of heated jets. As expected, the downstream radiation is much
more intense when compared with Jet-A (Fig. 1(a)) because of the supersonic
convection of large-scale structures in the jet plume with respect to ambient.
Additionally, the sideline and upstream radiation due to the interaction of
the jet shear layer with the shock-cells is also captured by the dilatation
contours.

The near-field LES results are extrapolated to the far-field at $100$ jet
diameters from the nozzle exit using the FW-H analogy, closely following the
experimental procedure to obtain the far-field microphone signals as described
in Ref. [47]. Figure 2(b) compares the far-field predictions of the LES with
the experimental measurements at three different polar angles. At downstream
locations, where Mach waves are dominant and are the focus of the present
study, the LES predictions accurately capture the shape of the spectra up to
the grid cut-off _St_ values ($\approx 1$) [23]. At sideline locations, the
LES accurately resolves the peak associated with the broadband shock-
associated noise (BBSAN).

##  3 Numerical components of the approach

In this section, we summarize the Mean Flow Perturbation (MFP) and data
sifting with Momentum Potential Theory (MPT), emphasizing aspects of interest
to the present work. The third component, Dynamic Mode Decomposition (DMD), is
used in its standard form as in Ref. [34] and is thus omitted for brevity.

###  3.1 Mean Flow Perturbation (MFP) Formulation

As noted in the introduction, the perturbations are propagated with an
implicit approach which repurposes the baseline NS solver to yield LNSE
solutions about the mean flow basic state. When used to explicitly extract the
stability modes, the method represents a matrix free approach [25]; in the
present work however, the interest is in the time-accurate perturbation
evolution for noise-prediction. For completeness, the implicit linearization
may be summarized as follows. The standard NS equations may be written as:

$\frac{\partial\textbf{Q}}{\partial
t}={\textbf{F}\left(\textbf{Q}\right)}+\textbf{B}_{\text{f}},$ (1)

where $\textbf{B}_{\text{f}}$ is a suitable body-force as defined below and Q
is the state vector. Rewriting Eq. (1) as

$\frac{\partial\left({\overline{\textbf{Q}}+\textbf{Q}^{\prime}}\right)}{\partial
t}={\textbf{F}\left(\overline{\textbf{Q}}+\textbf{Q}^{\prime}\right)}+\textbf{B}_{\text{f}},$
(2)

and using Taylor series about the base flow, $\overline{\textbf{Q}}$, Eq. (1)
can be further rearranged as:

$\frac{\partial\textbf{Q}^{\prime}}{\partial
t}+\left(\frac{\partial\overline{\textbf{Q}}}{\partial
t}-\textbf{F}\left(\overline{\textbf{Q}}\right)-\textbf{B}_{\text{f}}\right)=\frac{\partial\textbf{F}}{\partial\textbf{Q}}\textbf{Q}^{\prime}+\frac{\partial^{2}\textbf{F}}{\partial\textbf{Q}^{2}}\textbf{Q}^{\prime
2}+...\text{(higher order terms)}.$ (3)

Thus, if the standard NS equations are solved with a body-force given by

$\textbf{B}_{\text{f}}=\frac{\partial\overline{\textbf{Q}}}{\partial
t}-\textbf{F}\left(\overline{\textbf{Q}}\right)$ (4)

and the perturbations $(\textbf{Q}^{\prime})$ are small enough so that the
contribution from the non-linear terms is negligible, the procedure
successfully provides the evolution of perturbation according to the LNSE on
the base flow $(\overline{\textbf{Q}})$. The approach effectively linearizes
about the conserved variables (cf. Ref. [48]).

The basic state for MFP is obtained for each LES by time-averaging over a
period of $300D_{j}/U_{j}$ seconds. The evolution of the perturbation flow-
field is obtained after specification of an initial distribution. Different
initial condition possibilities have been discussed in Ref. [25], depending on
the objective. The choices for the present cases, and the underlying rationale
and sensitivity, are detailed in the context of each problem for noise
prediction ($\S$4) as well as the control analysis ($\S$5). The linearized
solution in the form of snapshots suitable for post-processing is then
obtained by marching forward in time, using the NS equations with the modified
body-force term in Eq. (4).

As noted earlier, even though the LES databases are generated using two
separate NS solvers, the same structured code employed for Jet-A LES is also
used for Jet-B to perform the MFP. For this, the mean flow-field for Jet-B is
interpolated onto a nominal structured cylindrical mesh containing $651\times
251\times 125$ mesh points. Regardless of the numerical procedure used to
obtain the basic states, the perturbation field is evolved for both jets with
a third-order Roe scheme for spatial discretization and a third-order Runge-
Kutta scheme for time integration; these are selected due to their widespread
availability in most commercial and in-house solvers. The implicit nature of
the MFP procedure offers significant flexibility in the specification of
boundary conditions as they can be applied to either the perturbation field or
the total flow quantities [25]. In the present work, standard no-slip boundary
conditions are applied to the perturbations at the nozzle walls, whereas at
outer boundaries, characteristic boundary conditions are enforced on the total
flow variables.

The costs associated with the time-domain approach relative to those using
frequency-based methods depend on the objectives. Some considerations on the
computational expense of the current approach are now presented. The effort to
obtain the body-force $\textbf{B}_{\text{f}}$ of Eq. (4) is negligible, since
it is only computed once. Thus, the cost of each iteration per mesh point for
MFP is similar to that for the LES. The cost of sifting is also similar to
that required for the LES. However, several factors substantially lower the
cost of MFP versus LES by orders of magnitude. (i) The mesh resolution
required to support the mean flow is far smaller than that for the LES, since
the finer convecting scales are averaged out. A practical demonstration of
this is presented in $\S$4. (ii) The linear nature of MFP evolution
facilitates the use of high-order spatio-temporal methods which further
reduces costs. (iii) The MFP equations are only marched for a small fraction
of the time required to obtain statistically converged LES data. (iv) A single
time-domain solution provides snapshot data for the entire range of
frequencies of interest (assuming the time-step-size is chosen appropriately
to resolve these temporal scales). If superdirectivity is of interest, with
focus on the axisymmetric $m=0$ mode, the simulation becomes essentially two-
dimensional. As an example, the Jet-A LES calculation required approximately
$90{,}000$ CPU hours with 768 Intel® Xeon® 8268s Skylake (2.9GHz) processors
($\sim 120$ hours of wall-clock time). In contrast, the corresponding MFP-
based results below were obtained in $24$ CPU hours ($\sim 30$ minutes wall-
clock) on a Linux desktop workstation with 48 Intel® Xeon® E7-4809 v4 (2.10
GHz) processors.

###  3.2 Data sifting for noise radiation studies

The implicit LNSE provides the evolution of fluctuations of primitive
variables. Although $p^{\prime}$ is the quantity of ultimate interest in the
far-field, its value in the near-field is not purely representative of
propagating acoustic content. Rather, it also includes a hydrodynamic
component [49], whose influence adversely affects noise prediction techniques
[32]. The extraction of a variable representing acoustic radiating components
from the primitive fluctuations is highly advantageous for developing
aeroacoustic theories and inform noise control techniques.

A robust physics-based method for simulation data that contain information on
fluctuations of all variables is Doak’s MPT, which separates radiating and the
non-radiating spatio-temporal components. A summary is provided for
completeness; the theory may be found in Ref. [31], with application to jet
noise LES analysis in Refs. [20, 22]. The key step is a decomposition of the
momentum density $(\rho\textbf{u})$ into FT modes representing hydrodynamic,
acoustic and thermal components:

$\rho\textbf{u}=\bar{\textbf{B}}+\textbf{B}^{\prime}-\nabla\psi^{\prime}$ (5)

where

$\nabla\cdot\bar{\textbf{B}}=0,\hskip
14.45377pt\nabla\cdot\textbf{B}^{\prime}=0$ (6)

Here $\bar{\textbf{B}}$ is the mean solenoidal component,
$\textbf{B}^{\prime}$ is the fluctuating solenoidal component and
$\psi^{\prime}$ is the fluctuating irrotational scalar potential. Eqs. (5) and
(6) when used with the mass conservation equation yield,

$\nabla^{2}\psi^{\prime}=\frac{\partial\rho^{\prime}}{\partial t}$ (7)

The irrotational scalar potential can be further split into acoustic and
thermal FT modes: $\psi^{\prime}=\psi^{\prime}_{a}+\psi^{\prime}_{t}$, which
are described by,

$\nabla^{2}\psi_{a}^{\prime}=\frac{1}{c^{2}}\frac{\partial
p^{\prime}}{\partial t},\hskip
14.45377pt\nabla^{2}\psi_{t}^{\prime}=\frac{\partial\rho^{\prime}}{\partial
S}\frac{\partial S^{\prime}}{\partial t}$ (8)

where $\psi^{\prime}_{a}$ and $\psi^{\prime}_{t}$ are the acoustic and thermal
components of the irrotational momentum density, the entropy $S$ is calculated
relative to the jet exit conditions and is given by,

$S=C_{p}[\log(T/T_{j})-R\log(p/p_{j})]$ (9)

Here $C_{p}$ is the specific heat at constant pressure and $R$ is the ideal
gas constant.

The acoustic FT mode is extracted from the LES and MFP data by solving the
Poisson equation for $\psi^{\prime}_{a}$ at every time instant. The Poisson
equation is discretized to second-order accuracy in generalized curvilinear
coordinates and solved using the BiCGSTAB algorithm [50]. The boundary
conditions are similar to the ones used previously to to analyze LES
fluctuations for both Jet-A [20] and Jet-B [22]. The outer boundaries are
assumed to be purely acoustic; this results in
$(\rho\textbf{u})^{\prime}=-\nabla\psi^{\prime}_{a}$, which is integrated
along the boundaries to provide Dirichlet boundary conditions to the Poisson
solver. A zero-gradient Neumann condition for $\psi^{\prime}_{a}$ is enforced
at the centerline due to the axisymmetric nature of the jets considered.

##  4 Noise Prediction

###  4.1 Growth of Perturbations

The Mach number field of the time-mean flow for Jet-A and Jet-B are shown in
Fig. 3. For the latter, a shock train is initiated inside the nozzle because
of its military-style design. In the present case, since Jet-A and Jet-B are
both axisymmetric, and the focus is on superdirectivity analysis, a 2D slice
suffices because of the dominance of the axisymmetric mode [13].

(a)

(b)

Figure 3: Mean Mach number contours showing the regions of initial
perturbation for (a) Jet-A and (b) Jet-B.

The first step is the specification of the initial perturbation field. We take
guidance from efforts to obtain stability modes such as those for supersonic
jets in Ref. [51] and flows past airfoils and cylinders in Ref. [25]. The
wavepacket dynamics are governed by the Kelvin-Helmholtz (K-H) instability in
the jet shear layer [51]. In the context of input-output systems, the K-H
instability is associated with a white-noise forcing in the thin shear layer
in the vicinity of the nozzle exit [13, 52]. In the present investigation
therefore, for both jets, we consider random pressure forcing of the form

$p^{\prime}(\textbf{x},t)=\epsilon_{0}\text{RAND}(\textbf{x})\delta(t-t_{0})p_{\infty}.$
(10)

Here $\text{RAND}(\textbf{x})$ generates a random normalized distribution in
space with values between $-1$ and $1$. The perturbations are scaled by the
freestream value of pressure ($p_{\infty}$) and an initial perturbation
magnitude $\epsilon_{0}$. The use of the Dirac-delta function ensures that the
forcing is only applied at time $t_{0}$ i.e., the perturbation is an initial
condition and yields an impulse response. This choice aids in establishing the
linear independence of snapshots which can be crucial for stability mode
extraction [25] and growth rate determination ($\S$5). The uncorrelated nature
of the initial forcing ensures that the range of frequencies encompasses
frequencies of known principal modes and is consistent with K-H forcing in
previous studies of jet noise with input-output systems. In order to assess
the sensitivity of the MFP solution to the perturbation magnitude
$\epsilon_{0}$ and the forcing location, four test cases are presented in
Table 1. The first three cases, Pert01 through Pert03, are computed with an
initial perturbation of $\epsilon_{0}=10^{-5}$ for both jets. The sub-domains
where this initial perturbation is applied are marked as D0 for Jet-A, while
for Jet-B, two regions D1 and D2 are considered as shown in Fig. 3. The effect
of perturbation magnitude and mesh resolution is examined using a fourth test-
case, Pert04 which uses an initial perturbation with $\epsilon_{0}=10^{-6}$ in
the sub-domain D2 and is marched forward on a coarser mesh obtained by
extracting every alternate point in the streamwise and radial directions of
the LES mesh. Further ramifications of these choices of forcing location, mesh
resolution and the $\epsilon_{0}$ value are discussed in context below.

Table 1: List of test cases for Jet-A and Jet-B with MFP. Case No. | Basic State | Sub-Domain | $\epsilon_{0}$ | Mesh
---|---|---|---|---
Pert01 | Jet-A | D0 | $10^{-5}$ | 523x526 (Same as LES)
Pert02 | Jet-B | D1 | $10^{-5}$ | 651x251 (Same as LES)
Pert03 | Jet-B | D2 | $10^{-5}$ | 651x251 (Same as LES)
Pert04 | Jet-B | D2 | $10^{-6}$ | 326x126 ($1/4^{\text{th}}$ of LES)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Jet-A: Growth of pressure (left) and vorticity (right) perturbations
with time: $t=3.525$ (top), $t=4.775$ (middle), and $t=6.025$ (bottom). Here
$t$ represents the non-dimensional time.

All test cases are marched forward in time with a non-dimensional time-step
$\Delta t=5\times 10^{-4}$. This $\Delta t$ value is found to be sufficient to
avoid convergence issues while keeping a short run-time. A total of $1{,}200$
snapshots are sampled at an effective $\Delta t=0.025$. This corresponds to a
minimum frequency resolution of _St_ $=0.033$ and a Nyquist frequency cut-off
of _St_ $=20$. At this sampling rate, a typical frequency of interest, _St_
$=0.3$, is resolved with 133 snapshots (nine cycles). Figures 4(a), (c) and
(e) show the evolution of pressure perturbations with time at three
representative time instances for Jet-A. At any instant, the pressure
perturbation field has a wavepacket structure resembling the grayscale
dilatation contours of the full LES solution displayed in Fig. 1(a). The
pressure field in Fig. 4 contains content that represent the radiating
component of the flow, as well as hydrodynamic content, similar to that
observed in LES fluctuations [32]. Moreover, as mentioned earlier, the effect
of gradients in the mean flow is to generate perturbations in all other
quantities. For example, Figs. 4(b), (d) and (f) show the evolution of
vorticity perturbations at corresponding time instants. Since the design aim
of many noise reduction technologies is to enhance mixing to break up the
acoustically efficient large-scale structures in the jet plume, the growth of
vorticity perturbations is helpful as one possible measure of the mixing
introduced by the initial perturbations. This connection is leveraged for
analysis of control as further elaborated in $\S$5.

An examination of the evolution of vorticity perturbations offers further
insights into the dynamics uncovered by MFP. In the initial stages of its
evolution, the vorticity perturbations consist of structures inclined against
the shear as shown in the inset in Fig. 4(b), where the vortical structure
marked by the straight-line V1-V2 shows a forward inclination. Figures 4(d)
and (f) track the movement of points V1 and V2 on this structure as it evolves
with time. Due to its vicinity to the jet core, the perturbations at point V2
advect faster than V1, resulting in a deformation of the vortical structure as
it moves downstream. Figure 4(d) shows a time instant when point V2 approaches
V1, and the vortical structure is nearly perpendicular to the jet centerline.
In contrast, Fig. 4(f) represents an instance when point V2 convects further
downstream of V1. At this instant, the vortical structure, which was initially
inclined against the shear in Fig. 4(b) is now inclined with the shear. This
change in orientation of vortical perturbations as they evolve downstream
indicates the existence of the Orr mechanism discussed in Refs. [53, 54].
Additionally, at this instant, the pressure perturbations show the presence of
a secondary wavepacket (marked with a rectangle in Fig. 4(e)) which evolves
further with time. In the context of jet instabilities, the evolution of
wavepackets due to the Orr mechanism is often referred to as non-modal growth
[54] since it occurs in the convectively stable region of the jet (downstream
of the jet exit). Although the Orr mechanism does not play a significant role
in the noise radiation from supersonic jets, the presence of this mechanism in
the current MFP calculations is encouraging relative to PSE-based models,
which do not capture this non-modal growth [54]. In fact, the failure of PSE-
based models to accurately capture the noise radiation of subsonic jets is
often associated with the absence of this mechanism [52]. Resolvent-based jet
noise models, on the other hand, exhibit this mechanism as a response to
nonlinear correlated forcing [52]. This forcing comprises of inclined vortical
structures that are tilted by shear as they are advected downstream analogous
to the insets in Figs. 4(b), (d) and (f). Since MFP does not neglect the
nonlinear interaction between various perturbation modes, which appear as
forcing terms in resolvent-based models, the existence of the Orr mechanism in
the present data is unsurprising. Future efforts will leverage this capability
to extend model sound radiation from subsonic jets.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Jet-B: Growth of pressure (left) and vorticity (right) perturbations
with time: $t=2.2$ (top), $t=5.2$ (middle), and $t=26.875$ (bottom) . The
first few shock-cells are marked for reference.

The corresponding growth of pressure and vorticity perturbations for Jet-B
(Pert04) are shown in Fig. 5. The first few shock-cells immediately downstream
of the nozzle exit are marked in Figs. 5(a), (c) and (e) for reference. The
main downstream radiation signature of the pressure wavepackets in Jet-B is
seen to occur at a larger angle from the jet downstream axis when compared to
Jet-A. This is expected following the wavy-wall analogy [55], which models the
prominent downstream noise radiation of supersonic jets as supersonically
travelling instability waves. From the wavy wall analogy, the directivity of
the downstream radiation is given by $\cos^{-1}(1/M_{c})$, where $M_{c}$ is
the convection Mach number of the instability wave relative to the ambient
speed of sound. Thus, the shift in downstream directivity away from the jet
downstream axis for Jet-B is indicative of a higher convection speed of
wavepackets in Jet-B when compared to Jet-A. Additionally, the interaction of
the growing pressure wavepacket with the shock-cells results in the generation
of upstream travelling waves resembling BBSAN as indicated by the upward
pointing arrows in Figs. 5(c) and 5(e). Although BBSAN is not the focus of the
present study, this observation indicates the generality of MFP, which will be
leveraged in future efforts.

###  4.2 Comparison of Near-field Wavepackets with LES

The comparison of MFP-based predictions of jet noise with the original LES is
performed through the acoustic component discussed in $\S$3.2. Figure 6 shows
the streamwise component of the acoustic component from the LES fluctuation
data for the two jets at a representative time-step. Similar observations can
be made on the downstream directivity for the two jets as in Figs. 4 and 5,
where the main radiation signature of Jet-B, due to its heated nature,
propagates at a larger downstream angle when compared to Jet-A. For both
cases, the acoustic mode is comprised of a coherent wavepacket structure
representative of classical instability waves [56, 57, 58] used to model the
sound generation from jets.

Despite the similarity to pressure fluctuations upstream of the core collapse
region, it has been shown elsewhere for Jet-A [32] and Jet-B [33, 22], that
the acoustic component exhibits superior spatio-temporal coherence and
radiative efficiency for the LES data. These observations also hold when using
perturbations propagating on the mean flow. Figure 7 displays the acoustic
component of the perturbation field for each of the two jets after it has
filled the domain. The acoustic components of the LES and MFP are not expected
to be the same, since the former contains all resolved scales in the flow,
while the latter represents propagation on the mean. Nonetheless, the more
compact nature of the acoustic perturbation is easily evident from a
comparision between Fig. 7 and the pressure fields of Figs. 4 and 5. We thus
use $-\frac{\partial\psi_{a}}{\partial x}$ as a measure of the near acoustic
field for the remainder of this investigation.

(a)

(b)

Figure 6: Instantaneous contours of the streamwise component of the acoustic
FT Mode from LES of Jet-A (left) and Jet-B (right).

(a)

(b)

Figure 7: Instantaneous contours of the streamwise component of the acoustic
FT perturbations from MFP of Jet-A (left) and Jet-B (right).

The ability of the acoustic component ($-\frac{\partial\psi_{a}}{\partial x}$)
from the MFP-propagated perturbations to accurately capture the noise
generation trends is now examined by comparing with the corresponding LES
results. For this, we make use of data-driven DMD [34], which processes a
series of snapshots to identify spatial modes that optimally describe the
evolution of the flow from one time instant to another. Briefly, with $N$
snapshots spaced at constant sampling time-step-sizes and assembled in the
subspace U, DMD provides the following representation

$\textbf{U}(\textbf{x},t)=\sum_{n=1}^{N}a_{n}\exp{(\lambda_{n}t)}\Phi_{n}(\textbf{x})$
(11)

where $a_{n}$, $\Phi_{n}(\textbf{x})$ and $\lambda_{n}$ represent the
amplitude, spatial structure and the complex frequency of the $n$-th mode
respectively. The complex frequency can be further split into
$\lambda_{n}=\sigma+i\omega$, where, $\sigma$ and $\omega$ represent the
growth rate and the frequency of the mode respectively. Although DMD modes are
associated with their specific frequencies, they may also be organized by
their amplitudes as defined in Ref. [59].

When used with MFP, DMD provides a robust and economical option to calculate
the growth rates of the unstable modes for a wide range of both internal as
well as external flows [25]. For the statistically stationary LES data, with
the exception of a small number of outliers, the DMD modes are tightly
clustered along a unit circle in the Ritz space and therefore have nearly zero
growth/decay rates. The perturbation growths obtained from MFP are useful
however, especially in the context of control ($\S$5). The comparison of the
LES and MFP results is thus performed by considering DMD modes with the
highest amplitudes from the LES. Three ranges are selected corresponding to
the low- $(0.2\leq St\leq 0.45)$, mid- $(0.45<St\leq 0.75)$ and high-frequency
$(St>0.75)$ regimes. In these, the highest amplitude LES fluctuation DMD modes
are compared with the corresponding MFP perturbation DMD modes at the same
frequencies.

Figure 8 shows a comparison of the near-field
$-\frac{\partial\psi_{a}}{\partial x}$ DMD modes in the form of wavepackets
obtained from both LES and MFP data with increasing frequency for Jet-A.
Figure 9 on the other hand, shows a qualitative picture of the acoustic
radiation due to each of these wavepackets. Both these figures are obtained by
plotting the $-\frac{\partial\psi_{a}}{\partial x}$ DMD modes in different
sub-domains of the flow-field. Several important observations make evident
that the mechanisms identified in the LES-based
$-\frac{\partial\psi_{a}}{\partial x}$ modes are active in the MFP and provide
the proper basis for the prediction technique.

(a) $St=0.35$

(b) $St=0.35$

(c) $St=0.56$

(d) $St=0.56$

(e) $St=0.90$

(f) $St=0.90$

Figure 8: Jet-A: Comparison of the most dominant acoustic DMD modes from LES
(left) with MFP (right) at different _St_ values corresponding to low, mid and
high frequency regimes (Pert01).

(a) $St=0.35$

(b) $St=0.35$

(c) $St=0.56$

(d) $St=0.56$

(e) $St=0.90$

(f) $St=0.90$

Figure 9: Jet-A: Comparison of noise radiation patterns from the wavepackets
shown in Fig. 8 from LES (left) and MFP (right). The radiation patterns are
generated by plotting the $-\frac{\partial\psi_{a}}{\partial x}$ DMD modes in
the domain $0\leq x/D\leq 16$ and $3\leq r/D\leq 8$.

In the low frequency regime ($St=0.35$), the wavepacket represented by the
$-\frac{\partial\psi_{a}}{\partial x}$ LES DMD mode (Fig. 8(a)) displays a
single coherent streamwise structure, which is linked to the K-H instability
of the jet shear layer [52]. This results in a corresponding prominent
downstream radiation pattern as shown with the arrow in Fig. 9(a). This
general wavepacket structure shape, its streamwise extent and radiation
pattern are properly reproduced by the linear technique (Figs. 8(b) and 9(b)).
This is expected because the white-noise nature of the initial perturbation in
the current procedure is similar to K-H forcing in previous studies of jet
noise with input-output systems as discussed previously.

At the next higher frequency, $St=0.56$, focusing first on the LES (Fig.
8(c)), the overall wavepacket in the corresponding DMD mode is more
complicated with the addition of a secondary streamwise coherent structure
downstream of the core collapse. This secondary wavepacket structure appears
related to the Orr mechanism discussed previously. The primary K-H wavepacket
persists at these frequencies (highlighted with a yellow oval) but becomes
more confined to the jet exit i.e., the axial envelope of the wavepacket is
shortened. These features of the LES-derived wavepacket structures at this
frequency are also evident in the corresponding acoustic perturbations from
MFP as shown in Fig. 8(d), which also shows the primary K-H wavepacket as well
as the secondary downstream streamwise coherent structure. The radiation
patterns at $St=0.56$ are shown in Fig. 9(c) for LES and Fig. 9(d) for the
MFP. The increased complexity of the wavepacket is manifested in the existence
of more than one acoustic beam; at this frequency, two prominent directions
are observed as marked by arrows.

At the highest frequency considered, $St=0.9$, the observations on trends and
comparisons between LES and MFP wavepackets (Figs. 8(e) and (f), respectively)
as well as their radiation patterns (Figs. 9(e) and (f), respectively) remain
valid. Specifically, the primary wavepacket becomes shorter, while the
secondary one lengthens. The most prominent radiation patterns appear in more
directions as marked by arrows.

Finally, the upstream radiation footprint at all these frequencies (shown by
circles in Fig. 9) are also exhibited by the perturbation DMD modes. These
observations demonstrate that the primary noise generation mechanisms existing
in the full LES are captured by the MFP. Furthermore, the existence of
sideline radiation signatures in the perturbation field suggests that MFP can
used to model the contribution of large-scale structures to the sideline noise
which may have feedback control benefits with wing mounted sensors [60].

(a) $St=0.30$

(b) $St=0.30$

(c) $St=0.48$

(d) $St=0.48$

(e) $St=0.93$

(f) $St=0.93$

Figure 10: Jet-B (Pert04): Comparison of the most dominant acoustic DMD modes
from LES (left) with MFP (right) at different _St_ values corresponding to
low, mid and high frequency regimes.

Figure 10 shows a comparison of the near-field
$-\frac{\partial\psi_{a}}{\partial x}$ DMD modes for Jet-B for the test case
Pert04. The radiation signature due to each of these wavepackets is also
visible in this figure, and are thus not plotted separately. As expected, due
to the heated nature of the jet, the dominant DMD modes at each frequency have
a single streamwise structure with a strong downstream radiation signature,
analogous to a supersonically convecting wavy wall [55]. This agrees with
previous observations based on Spectral Proper Orthogonal Decomposition (SPOD)
of the acoustic mode in Ref. [22] and follows from the fact that SPOD modes
can be interpreted as optimally averaged DMD modes from an ensemble DMD
problem [61]. Additionally, similar to Jet-A, with an increase in frequency,
the wavelength of the radiation is shorter and the streamwise extent of the
Mach wave emitting region is confined closer to the nozzle exit. This is also
consistent with the noise source characteristics of jets, where the higher
frequencies are radiated from closer to the nozzle exit. The K-H type
wavepacket, its subsequent radiation and trends observed with increasing
frequency are accurately captured by the current procedure despite the
existence of strong shocks in the jet plume; this demonstrates its ability to
model sound from practical shock-containing military configurations.

###  4.3 Spectral Comparison

Near-field spectra are now computed to provide a more quantitative comparison
for all the test cases documented previously in Table 1. For this, we select a
location at $8D$ along several angles $\theta$ measured from the axis – these
locations are in the near acoustic field where the LES and MFP meshes are
sufficiently fine to capture the unsteadiness. The hydrodynamic content at
these locations is negligible and therefore
$-\partial\psi^{\prime}_{a}/\partial x$ is a scaled surrogate for pressure
fluctuations [20, 22]. Figures 11 and 12 display the normalized power spectral
density (PSD) of $p^{\prime}$, from LES and MFP along three angles where the
prominent noise field is obtained for Jet-A and Jet-B, respectively. The PSD
is normalized by the maximum value at _St_ $\geq 0.2$. The general shape of
the reconstructed PSD from MFP matches that of the LES for most of the _St_
values for Jet-A and for $0.2\leq St\leq 1.0$ for Jet-B. This is consistent
with the favorable qualitative comparisons of the DMD modes at these
frequencies shown previously. At the higher angles, MFP overpredicts the PSD
at low frequencies, with the disagreement extending to larger _St_ with
increasing polar angle from the jet downstream axis. This disagreement is
similar to that observed with other stability-based approaches whose
comparisions break down at very low frequencies in more striking ways than
with the present method. One possible reason is associated with the relative
unimportance of the K-H instability at these low _St_ values in the jet flow-
field as discussed in Ref. [52]. When this range of frequencies is of
interest, an alternative approach is to replace the white-noise impulse with a
continuous forcing that explicitly introduces the desired ranges.

Another pertinent observation is that the general trends for Jet-B are
insensitive to the perturbation magnitude and the initial forcing location as
long as the initial forcing encompasses the thin shear layer near the nozzle
exit. Furthermore, the recovered trends for the test cases Pert03 and Pert04,
which share the same forcing location but are computed on different meshes
with $\epsilon_{0}$ values that are an order of magnitude apart, are almost
identical. This not only demonstrates the linearity of the present approach
for these typical parameters but also provides evidence that the mesh
requirements for MFP are considerably less stringent that those of the LES, as
the same spectra are recovered on one-fourth of the original LES mesh.

(a)

(b)

(c)

Figure 11: Jet-A: Normalized power spectral density calculated from MFP
compared with LES at (a) $\theta=30^{\circ}$, (b) $\theta=45^{\circ}$, and (c)
$\theta=60^{\circ}$ at $r/D=8$.

(a)

(b)

(c)

Figure 12: Jet-B: Normalized power spectral density calculated from MFP
compared with LES at (a) $\theta=40^{\circ}$, (b) $\theta=50^{\circ}$, and (c)
$\theta=60^{\circ}$ at $r/D=8$.

##  5 Effect of Control Frequency

The perturbations obtained from the current time-domain procedure, with a
suitably chosen forcing, contain information on growth rates that can be
leveraged to inform the response to active control through excitation. All
components of the current approach, MFP, sifting and DMD contribute to this
endeavor. Although there is no restriction of the approach on properties of
the basic state, here we perform the analysis on Jet-A, because extensive
experimental results using Localized Arc Filament Plasma Actuators (LAFPAs)
and corresponding LES have successfully examined and characterized the effect
of actuator frequency on noise [42, 18]. This facilitates a clear method to
evaluate the trend predictions.

Although LAFPAs can be employed for diagnostic purposes by phase-locking
instabilities, they have been shown to exhibit control authority over
different high-speed flow configurations. A comprehensive review of these
actuators for high-speed flow control can be found in Ref. [62]. Briefly, in
the context of jet noise, LAFPAs are designed to introduce perturbations in
the jet shear layer by producing a high-temperature arc discharge near the
nozzle exit. Since the K-H instability of the jet shear layer is sensitive to
perturbations close to the nozzle exit, mixing enhancements can be obtained by
pulsing these actuators at specific frequencies and wavenumbers, resulting in
noise mitigation.

Figure 13 shows the experimental measurements in the form of differences in
Overall Sound Pressure Levels ($\Delta$OASPL, shown with blue circles),
relative to the no control case, in the direction of peak noise radiation
$(\theta=30^{\circ})$ as a function of frequency [35]. These results
correspond to an in-phase axisymmetric actuation mode in the original
experiment. Note that the $y-$axis is flipped such that the $\Delta$OASPL
values above the horizontal dashed line represent noise reduction and vice-
versa. It is evident that pulsing the actuators at forcing frequencies near
the jet preferred-mode $(St\approx 0.3)$ results in the maximum amplification
of downstream noise. The coherent structures are comprised of large vortex
rings with braids [18]. In contrast, noise mitigation is achieved at higher
forcing frequencies with a maximum $\Delta$OASPL of the order of $-1.5dB$ at
$St\approx 1.6$. At these frequencies, mixing is enhanced without the
formation of large coherent structures.

Figure 13: Noise reduction using LAFPA actuators for Jet-A [35] marked with
circles, and ratio of growth rates for vorticity and
$\bm{-\frac{\partial\psi_{a}}{\partial x}}$ perturbations from MFP marked with
squares. Triangles denote the ratio of growth rates for vorticity and pressure
perturbations from MFP.

Since MFP can accurately capture the spatio-temporal signature of the K-H
instability, it can be tailored to predict the noise mitigation/enhancement
resulting from actuation. This is achieved by restricting the initial
randomized perturbation given by Eq. (10) with $\epsilon_{0}=10^{-5}$ to only
a small region at the nozzle sleeve at the actuator location as in the
simulations of Ref. [19]. The growth of perturbations from the initial forcing
is analyzed using the same steps as before. A total of $1{,}000$ snapshots are
sampled at $\Delta t=0.025$; this combination of $\epsilon_{0}$, $\Delta t$
and number of snapshots is found to be appropriate to maintain a linear growth
of perturbations based on simple proportionality tests between input and
output (not shown). Due to the implicit linearization procedure of MFP, the
contribution from second-order terms in Eq. (3) can become significant as the
perturbation field grows with time. Non-linear perturbation evolution,
although beneficial in some cases (for example, to explore intermittency
[63]), prohibits an accurate assessment of growth rates from DMD which has
control implications.

As mentioned previously in $\S$1, most noise control methods seek to break up
the noise-efficient large-scale structures in the jet shear layer by enhancing
mixing. Therefore, the actuation frequencies associated with noise reduction
must be vorticity dominated. In the current framework, this reduces to a
problem of identifying the frequencies of the DMD modes that have a high
growth rate of vorticity. At the same time, these frequencies should have a
lower growth rate of the $-\frac{\partial\psi_{a}}{\partial x}$ perturbations
so that the mixing dominates the generation of new acoustic waves due to the
actuation. The growth rates are obtained as the real part of the complex
frequency $\lambda_{n}$ in Eq. (11) by performing DMD separately on both
vorticity as well as $-\frac{\partial\psi_{a}}{\partial x}$ perturbation
snapshots. The frequencies resolved by DMD (and hence their growth rates) are
directly related to the total number of snapshots and their sampling rate.
Therefore, while subjecting the snapshots to DMD, the subspace size was varied
from $200$ to $1{,}000$ snapshots starting from the initial perturbation at
$t=0$ with different sampling frequencies to test for convergence of growth
rates following the guidelines listed in Ref. [25]. Special care was taken to
confirm that the conclusions drawn from this section are insensitive to these
choices.

A suitable measure to examine the relative growth of the vorticity component
to the MPT acoustic variable is to consider the ratio of the real part of the
Ritz values. A map of the ratio of growth rates of vorticity and
$-\frac{\partial\psi_{a}}{\partial x}$ perturbations as a function of _St_
provided in Fig. 13 (squares) using the right $y-$axis. Note that the
horizontal line representing $\Delta$OASPL $=0$ for the experimental data on
the left $y-$axis corresponds to a growth rate ratio of one on the right
$y-$axis. This horizontal line is used to demarcate the “vorticity-dominated”
and “acoustic-dominated” DMD modes as marked in the Fig. 13. The growth rates
are restricted to $St\geq 0.2$, since the noise predictions from MFP deviate
from the LES results at very low frequencies as seen previously in Fig. 11 for
Jet-A. In the limit $St\rightarrow 0$, the growth rates of both quantities of
interest, vorticity and the MPT acoustic variable, become small, and their
ratio becomes ill-behaved.

At lower frequencies $St\leq 0.5$, $-\frac{\partial\psi_{a}}{\partial x}$ is
associated with very high growth rates when compared to vorticity. This
suggests that actuation at these frequencies results in an increase in
acoustic disturbances in the shear layer with lower mixing. This is confirmed
by the experimental observations indicating that actuation in this frequency
band results in an increase in noise radiated from the jet. In contrast, at
higher frequencies the growth of vorticity in the shear layer increases
relative to that of the acoustic perturbations. Thus, actuation at these
higher frequencies results in an increased rate of mixing in the shear layer.
However, acoustic disturbances generated due to actuation at this frequency do
not grow as significantly when compared with mixing. This implies any
actuation in the frequency range $0.65\leq St\leq 2$ should result in a
decrease in noise radiated by the jet. This is in remarkable agreement with
the experimental measurements and supports the observation that the
identification of the “vorticity dominated” frequency range can inform noise
control technologies for nozzle designs.

For completeness, Fig. 13 also shows the ratio of vorticity to pressure
perturbation growth rates as a function of _St_ value (marked with triangles).
As expected, the use of pressure instead of $-\frac{\partial\psi_{a}}{\partial
x}$ in the denominator does not provide such a clear cut demarcation between
the vorticity and acoustic dominated modes, since pressure perturbations also
contain contributions from the non-radiating component. Hence, the growth
rates of pressure perturbations cannot be interpreted as an accurate measure
of sound generation. The sifting of acoustic perturbations is therefore a key
component if the present technique is to be applied for control trend
analysis. If the method is used for noise prediction only, this sifting,
although highly advantageous, is not entirely necessary as away from the jet
shear layer, the pressure perturbations and acoustic component are related
through a simple scaling.

Finally, it must be noted, that the success of the linear method in predicting
noise generation and control trends with actuator frequency does not indicate
that the noise generation from the jet is a linear process. The underlying
turbulent mean basic state is necessary to calculate the growth of
perturbations and ensures that elements of non-linearity that play an
important role in sound generation are retained.

##  6 Conclusion

This investigation presents an implicit time-domain approach to model the
noise radiation from on- and off-design supersonic jets as well as to estimate
noise control trends with actuator frequencies. This is achieved in three
steps. The first involves adding an easily obtained body-force term to an
existing Navier-Stokes (NS) code; this converts the native NS solver into an
implicit linearized NS solver (LNSE) that can compute the linear growth of
perturbations about the long-time mean of a turbulent flow-field. The implicit
linearization avoids additional constraints on configuration and facilitates a
straightforward extension to off-design operation conditions such as those
found in military-style jets, which are otherwise challenging to model using
existing techniques.

To test the robustness of the present approach, two LES databases are employed
as truth models. These represent an ideally-expanded Mach 1.3 jet from a thin
sleeve nozzle using an in-house code and a heated over-expanded Mach 1.36 jet
from a faceted military-style nozzle using a widely used commercial code. The
robustness of the body-force specification allows the use of the same LNSE
solver to compute the evolution of linear perturbations about time-averaged
basic states for both test cases. The forcing employed consists of white-noise
impulse pressure forcing in the jet shear layer. Mean flow gradients then
induce perturbations in all other primitive quantities. The acoustic component
is obtained by sifting the resulting perturbation field using Doak’s momentum
potential theory. By utilizing data-driven Dynamic Mode Decomposition (DMD) to
identify growing spatial modes, it is demonstrated that the primary noise
radiation mechanisms existing in the LES fluctuations are reproduced by the
acoustic perturbations, irrespective of the operating conditions. Moreover,
away from the jet shear layer, the noise spectra from the LNSE matches the
desired variations obtained from the LES for a range of frequencies of
interest at a fraction of the cost of the original LES, highlighting the cost-
effective nature of the present approach.

For control trends, it is postulated that by localizing the initial forcing to
actuator locations as in the experiment, the identification of the “vorticity-
dominated” frequency range, where the growth rate of vorticity perturbations
dominates the growth rate of acoustic perturbations represents the desired
actuator frequencies for noise control. The results are in excellent agreement
with previously reported experimental data and shows the potential of the
approach to model sound radiation and inform control techniques in a robust
and computationally efficient manner. Athough the present study is focused on
axisymmetric jet configurations due to their well-known dynamics, this
approach can be easily extended to non-axisymmetric jets at off-design
operating conditions by leveraging the capabilities of the native NS solver
which is often a limitation of most existing jet noise models. Current efforts
are focused on exploring the use of the method for rectangular jets of
different aspect ratios.

## Acknowledgements

This work was performed in part under the sponsorship of the Office of Naval
Research with Dr. S. Martens serving as Project Monitor (Technical points of
contact: J. Spyropoulos and R. Powers). The views and conclusions contained
herein are those of the authors and do not represent the opinion of the Office
of Naval Research or the U.S. government. The authors are also grateful to Dr.
Rajesh Ranjan for insightful discussions on the implementation of MFP to the
LES data.

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